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4384 y(A.1.)92 b(Pro)s(of)30 b(of)h(Lemma)f(4.1)1751
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2 1 bop 456 251 a Fy(2)210 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
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e(the)i(follo)m(wing)e(functional)f(on)j FG(W)3182 1718
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1879 y Fu(Z)1640 2086 y Fy(\012)1721 1942 y Fx(jr)p FG(u)p
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341 4 v 1868 2066 a FG(p)2091 2003 y FA(+)20 b FG(h)2234
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2324 y Fy(1)3289 2357 y FA(in)f(a)456 2465 y(neigh)m(b)s(orho)s(o)s(d)
28 b(of)k([)p Fx(\000)p FA(1)p FG(;)15 b FA(1].)44 b(W)-8
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 4252 6063 l 4231 6081 l 4214 6096 l 4201 6107 l 4192 6114 l
 4187 6118 l
 4185 6120 l gs col0 s gr 
/Times-Italic ff 345.00 scf sf
3690 4095 m
gs 1 -1 sc ({u=0}) col0 sh gr
/Times-Italic ff 300.00 scf sf
6450 7995 m
gs 1 -1 sc (2l) col0 sh gr
/Symbol ff 300.00 scf sf
465 4500 m
gs 1 -1 sc (2q) col0 sh gr
/Symbol ff 300.00 scf sf
7335 3780 m
gs 1 -1 sc (x) col0 sh gr
% here ends figure;
$F2psEnd
rs
showpage

%%EndDocument
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(of)i(Theorem)f(1.2)p eop
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f(b)m(y)h(the)g(third)e(author)456 2514 y(in)31 b([31)q(].)46
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b(pro)m(v)m(e)g(\(2.3\))r(,)i(note)f(that)f(for)f FG(s)j
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4 v 2263 4256 a(4)2309 4225 y FG(;)15 b Fx(\000)2430
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Fx(\000)p 552 4289 V FG(C)623 4376 y Fy(0)662 4362 y
FG(;)31 b Fx(\000)p 799 4270 55 3 v 799 4325 a Fv(C)854
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a Fs(p)2818 4666 y FG(:)p 456 4997 499 4 v 555 5079 a
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1573 671 72 4 v 20 w FG(C)1645 758 y Fy(0)1684 744 y
FA(\()p FG(s)f Fx(\000)g FG(s)1916 758 y Fy(0)1955 744
y FA(\)\()2081 703 y Fv(p)p 2035 723 126 4 v 2035 776
a(p)p Ft(\000)p Fy(1)2172 744 y FG(h)2224 758 y Fy(0)2263
744 y FA(\()p FG(s)p FA(\)\))2422 672 y Fq(1)p 2422 684
33 3 v 2422 725 a Fs(p)2468 643 y Fu(i)2511 666 y Fv(p)2586
647 y Fx(\024)1243 968 y(\024)1418 907 y FG(h)1470 921
y Fy(0)1509 907 y FA(\()p FG(s)p FA(\))p FG(R)1692 874
y Fv(p)p 1349 947 451 4 v 1349 1056 a FA(\()p FG(R)i
FA(+)p 1576 964 55 3 v 1576 1019 a Fv(C)1631 1028 y Fq(0)p
1576 1035 90 4 v 1603 1087 a Fy(4)1675 1056 y FG(k)s
FA(\))1760 1030 y Fv(p)1836 968 y FA(=)1243 1287 y(=)j
FG(h)1391 1301 y Fy(0)1431 1287 y FA(\()p FG(s)p FA(\))c(+)1666
1220 y FG(h)1718 1234 y Fy(0)1757 1220 y FA(\()p FG(s)p
FA(\))p FG(R)1940 1187 y Fv(p)2000 1220 y Fx(\000)f FG(h)2143
1234 y Fy(0)2183 1220 y FA(\()p FG(s)p FA(\)\()p FG(R)i
FA(+)p 2523 1129 55 3 v 2523 1183 a Fv(C)2578 1192 y
Fq(0)p 2523 1199 90 4 v 2550 1252 a Fy(4)2622 1220 y
FG(k)s FA(\))2707 1187 y Fv(p)p 1666 1266 1082 4 v 1981
1375 a FA(\()p FG(R)f FA(+)p 2207 1283 55 3 v 2207 1337
a Fv(C)2262 1346 y Fq(0)p 2207 1354 90 4 v 2234 1406
a Fy(4)2307 1375 y FG(k)s FA(\))2392 1349 y Fv(p)2783
1287 y Fx(\024)1243 1583 y(\024)k FG(h)1391 1597 y Fy(0)1431
1583 y FA(\()p FG(s)p FA(\))c Fx(\000)1666 1522 y FA(const)p
1666 1562 208 4 v 1735 1646 a FG(R)1903 1583 y Fx(\001)1959
1522 y FA(\()p FG(R)g FA(+)f(const)p 2398 1449 72 4 v
16 w FG(C)2469 1536 y Fy(0)2509 1522 y FA(\))2544 1489
y Fv(p)2604 1522 y Fx(\000)g FG(R)2765 1489 y Fv(p)p
1959 1562 846 4 v 2282 1646 a FG(R)2352 1619 y Fv(p)p
Ft(\000)p Fy(1)2829 1583 y FG(:)456 1759 y FA(Using)29
b(the)i(fact)g(that)1480 1918 y(lim)1425 1973 y Fv(x)p
Ft(!)p Fy(+)p Ft(1)1686 1857 y FA(\()p FG(x)20 b FA(+)g
FG(a)p FA(\))1967 1824 y Fv(p)2027 1857 y Fx(\000)g FG(x)2170
1824 y Fv(p)p 1686 1897 525 4 v 1857 1980 a FG(x)1909
1954 y Fv(p)p Ft(\000)p Fy(1)2245 1918 y FA(=)25 b FG(pa)15
b(;)456 2084 y FA(and)29 b(taking)i FG(R)g FA(suitably)d(large,)j(w)m
(e)g(get)1144 2301 y FG(')p FA(\()p FG(s)p FA(\))26 b
Fx(\024)f FG(h)1490 2315 y Fy(0)1530 2301 y FA(\()p FG(s)p
FA(\))c Fx(\000)1764 2240 y FA(const)p 1987 2167 72 4
v 16 w FG(C)2059 2254 y Fy(0)p 1764 2280 334 4 v 1896
2364 a FG(R)2133 2301 y Fx(\024)k FG(h)2281 2315 y Fy(0)2321
2301 y FA(\()p FG(s)p FA(\))c Fx(\000)2556 2240 y FA(2)2619
2217 y Fu(b)2601 2240 y FG(C)2666 2254 y Fy(0)p 2556
2280 150 4 v 2596 2364 a FG(R)2730 2301 y(;)456 2502
y FA(for)p 600 2429 72 4 v 35 w FG(C)671 2516 y Fy(0)744
2502 y FG(>)49 b FA(const)1105 2479 y Fu(b)1087 2502
y FG(C)1152 2516 y Fy(0)1191 2502 y FA(,)37 b(pro)m(ving)f(\(2.3\))r(.)
55 b(Let)36 b(us)f(no)m(w)g(pro)m(v)m(e)i(\(2.4\))r(.)55
b(Recalling)35 b(\(2.5\))456 2631 y(and)d(arguing)f(as)i(ab)s(o)m(v)m
(e,)i(for)d FG(s)d Fx(2)g FA([1)p FG(=)p FA(2)p FG(;)15
b FA(3)p FG(=)p FA(4],)37 b(w)m(e)c(get)h(that)p 2579
2558 V 33 w FG(C)2650 2645 y Fy(0)2690 2631 y FA(\()p
FG(s)21 b Fx(\000)h FG(s)2925 2645 y Fy(0)2964 2631 y
FA(\))29 b Fx(2)g FA([)p 3153 2539 55 3 v 3153 2593 a
Fv(C)3208 2602 y Fq(0)p 3153 2610 90 4 v 3180 2662 a
Fy(4)3253 2631 y FG(;)p 3308 2558 72 4 v 30 w(C)3380
2645 y Fy(0)3419 2631 y FA(])456 2739 y(and)g(so)1038
2918 y FG(')p FA(\()p FG(s)p FA(\))d(=)1785 2857 y FG(h)1837
2871 y Fy(0)1877 2857 y FA(\()p FG(s)p FA(\))p FG(R)2060
2824 y Fv(p)p 1342 2897 1202 4 v 1342 2914 a Fu(h)1385
3015 y FG(R)21 b Fx(\000)p 1566 2942 72 4 v 20 w FG(C)1637
3029 y Fy(0)1677 3015 y FA(\()p FG(s)f Fx(\000)g FG(s)1909
3029 y Fy(0)1948 3015 y FA(\)\()2073 2974 y Fv(p)p 2028
2994 126 4 v 2028 3046 a(p)p Ft(\000)p Fy(1)2164 3015
y FG(h)2216 3029 y Fy(0)2256 3015 y FA(\()p FG(s)p FA(\)\))2415
2942 y Fq(1)p 2414 2954 33 3 v 2414 2996 a Fs(p)2461
2914 y Fu(i)2503 2937 y Fv(p)2578 2918 y Fx(\025)1236
3239 y(\025)1410 3178 y FG(h)1462 3192 y Fy(0)1502 3178
y FA(\()p FG(s)p FA(\))p FG(R)1685 3145 y Fv(p)p 1342
3218 451 4 v 1342 3327 a FA(\()p FG(R)h Fx(\000)p 1568
3235 55 3 v 1568 3290 a Fv(C)1623 3299 y Fq(0)p 1568
3306 90 4 v 1595 3358 a Fy(4)1668 3327 y FG(k)s FA(\))1753
3301 y Fv(p)1828 3239 y Fx(\025)1236 3535 y(\025)k FG(h)1384
3549 y Fy(0)1424 3535 y FA(\()p FG(s)p FA(\))20 b(+)1658
3474 y(const)p 1658 3514 208 4 v 1727 3598 a FG(R)1896
3535 y Fx(\001)1951 3474 y FG(R)2021 3441 y Fv(p)2081
3474 y Fx(\000)g FA(\()p FG(R)h Fx(\000)35 b FA(const)p
2626 3401 72 4 v 16 w FG(C)2697 3488 y Fy(0)2737 3474
y FA(\))2772 3441 y Fv(p)p 1951 3514 861 4 v 2282 3598
a FG(R)2352 3571 y Fv(p)p Ft(\000)p Fy(1)2837 3535 y
FG(:)456 3711 y FA(Using)29 b(the)i(fact)g(that)1480
3870 y(lim)1425 3925 y Fv(x)p Ft(!)p Fy(+)p Ft(1)1686
3809 y FG(x)1738 3776 y Fv(p)1797 3809 y Fx(\000)20 b
FA(\()p FG(x)h Fx(\000)f FG(a)p FA(\))2170 3776 y Fv(p)p
1686 3849 525 4 v 1857 3933 a FG(x)1909 3906 y Fv(p)p
Ft(\000)p Fy(1)2245 3870 y FA(=)25 b FG(pa)15 b(;)456
4036 y FA(w)m(e)30 b(th)m(us)g(gather)i(that,)f(for)f
FG(R)h FA(large,)1137 4253 y FG(')p FA(\()p FG(s)p FA(\))26
b Fx(\025)f FG(h)1483 4267 y Fy(0)1523 4253 y FA(\()p
FG(s)p FA(\))20 b(+)1772 4192 y(const)p 1995 4119 72
4 v 16 w FG(C)2066 4206 y Fy(0)p 1757 4232 349 4 v 1896
4316 a FG(R)2141 4253 y Fx(\025)25 b FG(h)2289 4267 y
Fy(0)2329 4253 y FA(\()p FG(s)p FA(\))20 b(+)2563 4192
y(2)2627 4169 y Fu(b)2608 4192 y FG(C)2673 4206 y Fy(0)p
2563 4232 150 4 v 2603 4316 a FG(R)2738 4253 y(;)456
4454 y FA(for)p 595 4381 72 4 v 30 w FG(C)666 4468 y
Fy(0)731 4454 y FG(>)40 b FA(const)1083 4431 y Fu(b)1065
4454 y FG(C)1130 4468 y Fy(0)1169 4454 y FA(,)31 b(pro)m(ving)g
(\(2.4\))q(.)1608 b Fl(\003)555 4617 y FA(No)m(w,)31
b(for)g FG(R)g FA(large)f(enough,)g(w)m(e)h(de\014ne)f
FG(s)2027 4631 y Fv(R)2109 4617 y Fx(2)25 b FA(\()p Fx(\000)p
FA(1)p FG(;)15 b(\022)2432 4584 y Ft(\003)2472 4617 y
FA(\))30 b(as)h(the)f(p)s(oin)m(t)g(suc)m(h)f(that)456
4809 y(\(2.6\))1052 b FG(h)1745 4823 y Fy(0)1785 4809
y FA(\()p FG(s)1863 4823 y Fv(R)1920 4809 y FA(\))26
b(=)2099 4748 y(1)p 2087 4789 70 4 v 2087 4872 a FG(R)2182
4809 y(:)456 4990 y FA(F)-8 b(rom)30 b(\(1.1\))r(,)h(w)m(e)g(ha)m(v)m
(e)g(that)456 5183 y(\(2.7\))685 b FG(c)p FA(\(1)21 b(+)f
FG(s)1600 5197 y Fv(R)1657 5183 y FA(\))26 b Fx(\024)1891
5121 y FA(1)p 1824 5162 180 4 v 1824 5249 a FG(R)1894
5222 y Fy(1)p Fv(=p)2039 5183 y Fx(\024)f FG(C)7 b FA(\(1)20
b(+)g FG(s)2441 5197 y Fv(R)2499 5183 y FA(\))15 b FG(:)p
eop
%%Page: 9 9
9 8 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)210 b(9)456
450 y FA(F)-8 b(or)45 b(further)f(estimates,)49 b(in)44
b(the)h(next)g(t)m(w)m(o)h(lemmas,)j(w)m(e)c(no)m(w)g(p)s(oin)m(t)f
(out)h(some)456 558 y(elemen)m(tary)31 b(b)s(ounds)d(for)i
FG(s)1419 572 y Fv(R)1476 558 y FA(:)456 693 y FH(Lemma)k(2.3.)44
b Fo(L)-5 b(et)34 b FG(C)41 b Fo(a)35 b(p)-5 b(ositive)35
b(universal)g(c)-5 b(onstant.)48 b(Then,)35 b(for)g FG(R)g
Fo(lar)-5 b(ge)35 b(\(with)456 801 y(r)-5 b(esp)g(e)g(ct)34
b(to)f FG(C)7 b Fo(\),)32 b(we)h(have)g(that)456 1011
y FA(\(2.8\))262 b(\(1)21 b(+)f FG(\030)t FA(\))1174
973 y Fv(p)1234 1011 y Fx(\000)g FA(\(1)h(+)f FG(s)1560
1025 y Fv(R)1617 1011 y FA(\))1652 973 y Fv(p)1712 1011
y Fx(\000)1813 949 y FG(C)p 1813 990 72 4 v 1814 1073
a(R)1894 1011 y FA(\()p FG(\030)25 b Fx(\000)20 b FG(s)2128
1025 y Fv(R)2185 1011 y FA(\))41 b FG(>)f FA(0)186 b
Fo(if)93 b FG(s)2795 1025 y Fv(R)2878 1011 y FG(<)25
b(\030)k(<)c FA(0)456 1199 y Fo(and)456 1403 y FA(\(2.9\))848
1279 y Fu(Z)939 1305 y Fy(0)898 1485 y Fv(s)931 1496
y Fs(R)1725 1341 y FG(d\030)p 1011 1382 1521 4 v 1011
1420 a Fu(\020)1065 1521 y FA(\(1)c(+)f FG(\030)t FA(\))1336
1494 y Fv(p)1396 1521 y Fx(\000)g FA(\(1)h(+)f FG(s)1722
1535 y Fv(R)1779 1521 y FA(\))1814 1494 y Fv(p)1874 1521
y Fx(\000)1975 1485 y Fv(C)p 1975 1500 55 4 v 1976 1552
a(R)2040 1521 y FA(\()p FG(\030)25 b Fx(\000)20 b FG(s)2274
1535 y Fv(R)2331 1521 y FA(\))2366 1420 y Fu(\021)2421
1442 y Fy(1)p Fv(=p)2581 1403 y Fx(\024)2710 1380 y Fu(e)2692
1403 y FG(C)37 b FA(log)32 b FG(R)16 b(;)456 1731 y Fo(for)33
b(a)g(suitable)g(c)-5 b(onstant)1390 1708 y Fu(e)1372
1731 y FG(C)32 b(>)25 b FA(0)p Fo(.)456 1912 y(Pr)-5
b(o)g(of.)43 b FA(F)-8 b(or)31 b FG(\030)e Fx(2)c FA([)p
FG(s)1123 1926 y Fv(R)1180 1912 y FG(;)31 b FA(0],)g(let)1376
2121 y FG(g)s FA(\()p FG(\030)t FA(\))c(:=)e(\(1)c(+)f
FG(\030)t FA(\))1955 2083 y Fv(p)2015 2121 y Fx(\000)2116
2059 y FG(C)p 2116 2100 72 4 v 2117 2183 a(R)2212 2121
y FA(\(1)h(+)f FG(\030)t FA(\))15 b FG(:)456 2315 y FA(Then,)29
b(b)m(y)h(\(2.7\),)1326 2524 y FG(g)1372 2486 y Ft(0)1396
2524 y FA(\()p FG(\030)t FA(\))83 b(=)g FG(p)p FA(\(1)21
b(+)f FG(\030)t FA(\))2064 2486 y Fv(p)p Ft(\000)p Fy(1)2214
2524 y Fx(\000)2315 2463 y FG(C)p 2315 2503 V 2316 2586
a(R)2422 2524 y Fx(\025)1593 2745 y(\025)83 b FG(p)p
FA(\(1)21 b(+)f FG(s)2028 2759 y Fv(R)2085 2745 y FA(\))2120
2708 y Fv(p)p Ft(\000)p Fy(1)2271 2745 y Fx(\000)2371
2684 y FG(C)p 2371 2724 V 2372 2808 a(R)2478 2745 y Fx(\025)1593
2967 y(\025)1816 2905 y FA(const)p 1757 2946 326 4 v
1757 3033 a FG(R)1827 3006 y Fy(\()p Fv(p)p Ft(\000)p
Fy(1\))p Fv(=p)2113 2967 y Fx(\000)2213 2905 y FG(C)p
2213 2946 72 4 v 2214 3029 a(R)2335 2967 y(>)40 b FA(0)15
b FG(;)456 3164 y FA(if)29 b FG(R)i FA(is)e(large)i(enough,)f(thence)h
FG(g)s FA(\()p FG(\030)t FA(\))c FG(>)e(g)s FA(\()p FG(s)1986
3178 y Fv(R)2044 3164 y FA(\),)31 b(pro)m(ving)g(\(2.8\))q(.)555
3327 y(In)h(order)g(to)i(estimate)f(the)g(in)m(tegral)g(in)e(\(2.9\),)
36 b(w)m(e)d(in)m(tro)s(duce)f(the)g(notation)h FG(b)d
FA(=)456 3436 y(1)14 b(+)g FG(s)643 3450 y Fv(R)700 3436
y FA(.)40 b(Since,)27 b(for)g FG(R)h FA(large,)h FG(s)1547
3450 y Fv(R)1631 3436 y FA(is)e(near)g Fx(\000)p FA(1,)h(w)m(e)g(ma)m
(y)g(assume)2734 3401 y Fy(1)p 2734 3416 36 4 v 2737
3468 a Fv(b)2804 3436 y FG(>)d FA(2.)41 b(W)-8 b(e)28
b(use)f(the)456 3544 y(substitution)1754 3694 y FG(\034)35
b FA(=)1935 3633 y(1)21 b(+)f FG(\030)p 1935 3673 201
4 v 2016 3757 a(b)456 3864 y FA(obtaining)29 b(a)h(b)s(ound)f(for)h
(the)g(ab)s(o)m(v)m(e)i(in)m(tegral)e(giv)m(en)g(b)m(y)738
3968 y Fu(Z)829 3994 y Fy(1)p Fv(=b)789 4174 y Fy(1)1414
4030 y FG(d\034)p 959 4071 1008 4 v 959 4113 a Fu(\000)1001
4187 y FG(\034)1051 4160 y Fv(p)1111 4187 y Fx(\000)19
b FA(1)i Fx(\000)1400 4151 y Fv(C)p 1368 4166 120 4 v
1368 4218 a(Rb)1451 4199 y Fs(p)1498 4187 y FG(b)p FA(\()p
FG(\034)30 b Fx(\000)20 b FA(1\))1813 4113 y Fu(\001)1856
4132 y Fy(1)p Fv(=p)2001 4092 y Fx(\024)2097 3968 y Fu(Z)2188
3994 y Fy(1)p Fv(=b)2148 4174 y Fy(1)2744 4030 y FG(d\034)p
2318 4071 950 4 v 2318 4174 a FA(\()q FG(\034)2404 4147
y Fv(p)2463 4174 y Fx(\000)g FA(1)h Fx(\000)f FG(C)2783
4147 y Ft(0)2806 4174 y FG(b)p FA(\()p FG(\034)30 b Fx(\000)20
b FA(1\)\))3157 4132 y Fy(1)p Fv(=p)3303 4092 y Fx(\024)584
4401 y(\024)738 4277 y Fu(Z)829 4303 y Fy(2)789 4483
y(1)1320 4339 y FG(d\034)p 893 4380 V 893 4483 a FA(\()q
FG(\034)979 4456 y Fv(p)1039 4483 y Fx(\000)g FA(1)g
Fx(\000)g FG(C)1358 4456 y Ft(0)1381 4483 y FG(b)p FA(\()p
FG(\034)31 b Fx(\000)19 b FA(1\)\))1733 4441 y Fy(1)p
Fv(=p)1873 4401 y FA(+)1979 4277 y Fu(Z)2070 4303 y Fy(1)p
Fv(=b)2030 4483 y Fy(2)2626 4339 y FG(d\034)p 2200 4380
V 2200 4483 a FA(\()p FG(\034)2285 4456 y Fv(p)2345 4483
y Fx(\000)h FA(1)h Fx(\000)f FG(C)2665 4456 y Ft(0)2688
4483 y FG(b)p FA(\()p FG(\034)30 b Fx(\000)20 b FA(1\)\))3039
4441 y Fy(1)p Fv(=p)3175 4401 y FG(:)456 4647 y FA(where)29
b(w)m(e)i(used)f(the)g(fact)i(that)f FG(b)1632 4614 y
Fv(p)1697 4647 y Fx(\031)1812 4611 y Fy(1)p 1803 4626
54 4 v 1803 4678 a Fv(R)1896 4647 y FA(\(see)h(\(2.7\))r(\).)555
4755 y(Noticing)e(that)1583 4921 y FG(\034)1633 4883
y Fv(p)1693 4921 y Fx(\000)20 b FA(1)26 b Fx(\025)1960
4859 y FA(2)2005 4826 y Fv(p)2066 4859 y Fx(\000)20 b
FA(1)p 1960 4900 242 4 v 2039 4983 a(2)2084 4957 y Fv(p)2227
4921 y FG(\034)2277 4883 y Fv(p)456 5091 y FA(if)29 b
FG(\034)35 b Fx(\025)25 b FA(2,)31 b(and)f(that)1663
5216 y FG(\034)1713 5178 y Fv(p)1773 5216 y Fx(\000)20
b FA(1)25 b Fx(\025)g FG(\034)31 b Fx(\000)20 b FA(1)p
eop
%%Page: 10 10
10 9 bop 456 251 a Fy(10)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(if)29 b FG(\034)35 b Fx(\025)25 b
FA(1,)31 b(w)m(e)g(b)s(ound)d(the)j(quan)m(tit)m(y)g(here)f(ab)s(o)m(v)
m(e)h(b)m(y)664 776 y(const)902 566 y Fu(0)902 726 y(B)902
784 y(@)981 652 y(Z)1072 679 y Fy(2)1032 859 y(1)1498
715 y FG(d\034)p 1137 755 820 4 v 1137 842 a FA(\(\(1)21
b Fx(\000)f FG(C)1436 816 y Ft(0)1459 842 y FG(b)p FA(\)\()p
FG(\034)31 b Fx(\000)20 b FA(1\)\))1845 816 y Fy(1)p
Fv(=p)1986 776 y FA(+)2077 652 y Fu(Z)2168 679 y Fy(1)p
Fv(=b)2128 859 y Fy(2)2646 715 y FG(d\034)p 2298 755
794 4 v 2298 803 a Fu(\020)2352 904 y FG(\034)2402 877
y Fv(p)2442 904 y FA(\(1)h Fx(\000)2644 859 y Fv(C)2699
840 y Fp(0)2721 859 y Fv(b)p Fy(\()p Fv(\034)8 b Ft(\000)p
Fy(1\))p 2644 883 292 4 v 2752 935 a Fv(\034)2791 916
y Fs(p)2945 904 y FA(\))2980 803 y Fu(\021)3046 798 y
Fq(1)p 3045 810 33 3 v 3045 851 a Fs(p)3101 566 y Fu(1)3101
726 y(C)3101 784 y(A)3211 776 y FG(:)456 1126 y FA(No)m(w,)33
b(if)e FG(R)i FA(is)f(large,)h(then)f FG(b)g FA(is)f(small,)h(and)f
(therefore)i(w)m(e)g(ma)m(y)g(assume)f(that)g(\(1)23
b Fx(\000)456 1261 y FG(C)528 1228 y Ft(0)550 1261 y
FG(b)p FA(\))635 1189 y Fq(1)p 635 1201 V 635 1242 a
Fs(p)706 1261 y FG(>)812 1226 y Fy(1)p 812 1241 36 4
v 812 1293 a(2)857 1261 y FA(.)41 b(Th)m(us,)1270 1424
y Fu(\022)1337 1552 y FA(1)21 b Fx(\000)1504 1491 y FG(C)1576
1458 y Ft(0)1598 1491 y FG(b)p FA(\()p FG(\034)31 b Fx(\000)20
b FA(1\))p 1504 1531 412 4 v 1664 1615 a FG(\034)1714
1588 y Fv(p)1925 1424 y Fu(\023)2002 1420 y Fq(1)p 2002
1432 33 3 v 2002 1473 a Fs(p)2088 1552 y Fx(\025)2199
1424 y Fu(\022)2266 1552 y FA(1)h Fx(\000)2456 1491 y
FG(C)2528 1458 y Ft(0)2550 1491 y FG(b)p 2433 1531 180
4 v 2433 1615 a(\034)2483 1588 y Fv(p)p Ft(\000)p Fy(1)2622
1424 y Fu(\023)2700 1420 y Fq(1)p 2699 1432 33 3 v 2699
1473 a Fs(p)2771 1552 y Fx(\025)1116 1794 y(\025)83 b
FA(\(1)21 b Fx(\000)f FG(C)1534 1757 y Ft(0)1557 1794
y FG(b)p FA(\))1642 1721 y Fq(1)p 1641 1733 V 1641 1775
a Fs(p)1728 1794 y FG(>)1849 1733 y FA(1)p 1849 1773
46 4 v 1849 1856 a(2)1919 1794 y FG(:)456 2014 y FA(This)28
b(yields)617 2143 y Fu(Z)708 2169 y Fy(0)668 2349 y Fv(s)701
2360 y Fs(R)1495 2205 y FG(d\030)p 780 2246 1521 4 v
780 2284 a Fu(\020)834 2385 y FA(\(1)21 b(+)f FG(\030)t
FA(\))1105 2359 y Fv(p)1165 2385 y Fx(\000)g FA(\(1)h(+)f
FG(s)1491 2399 y Fv(R)1549 2385 y FA(\))1584 2359 y Fv(p)1644
2385 y Fx(\000)1745 2349 y Fv(C)p 1745 2364 55 4 v 1746
2416 a(R)1810 2385 y FA(\()p FG(\030)k Fx(\000)c FG(s)2043
2399 y Fv(R)2100 2385 y FA(\))2135 2284 y Fu(\021)2190
2307 y Fy(1)p Fv(=p)2351 2267 y Fx(\024)40 b FA(const)2700
2139 y Fu(\022)2767 2267 y FA(1)20 b(+)g(log)3085 2205
y(1)p 3066 2246 85 4 v 3066 2329 a(2)p FG(b)3160 2139
y Fu(\023)3257 2267 y FG(;)456 2607 y FA(whic)m(h)29
b(pro)m(v)m(es)i(\(2.9\).)2167 b Fl(\003)555 2829 y FA(Let)31
b(us)f(no)m(w)g(estimate)h(ho)m(w)g FG(s)1621 2843 y
Fv(R)1708 2829 y FA(v)-5 b(aries)30 b(as)h(a)g(function)e(of)h
FG(R)q FA(:)456 2980 y FH(Lemma)i(2.4.)42 b Fo(Ther)-5
b(e)34 b(exists)f(a)g(suitable)g(universal)g(c)-5 b(onstant)34
b FG(C)e(>)25 b FA(0)33 b Fo(so)g(that)1467 3202 y Fx(\000)1675
3140 y FG(C)p 1548 3181 326 4 v 1548 3268 a(R)1618 3242
y Fy(\()p Fv(p)p Fy(+1\))p Fv(=p)1923 3202 y Fx(\024)25
b FG(@)2067 3216 y Fv(R)2125 3202 y FG(s)2168 3216 y
Fv(R)2251 3202 y FG(<)g FA(0)15 b FG(:)456 3459 y Fo(Pr)-5
b(o)g(of.)43 b FA(Di\013eren)m(tiating)30 b(\(2.6\),)1527
3696 y Fx(\000)1640 3635 y FA(1)p 1608 3675 110 4 v 1608
3758 a FG(R)1678 3732 y Fy(2)1752 3696 y FA(=)25 b FG(h)1900
3659 y Ft(0)1900 3719 y Fy(0)1940 3696 y FA(\()p FG(s)2018
3710 y Fv(R)2076 3696 y FA(\))15 b FG(@)2174 3710 y Fv(R)2232
3696 y FG(s)2275 3710 y Fv(R)2348 3696 y FG(;)456 3921
y FA(th)m(us)30 b FG(@)704 3935 y Fv(R)762 3921 y FG(s)805
3935 y Fv(R)887 3921 y FG(<)25 b FA(0)31 b(thanks)f(to)h(\(1.2\),)h
(and)e(so,)h(b)m(y)f(\(2.7\),)1265 4097 y(1)p 1233 4138
V 1233 4221 a FG(R)1303 4195 y Fy(2)1435 4159 y FA(=)83
b FG(h)1641 4121 y Ft(0)1641 4181 y Fy(0)1680 4159 y
FA(\()p FG(s)1758 4173 y Fv(R)1816 4159 y FA(\))15 b
Fx(j)p FG(@)1939 4173 y Fv(R)1998 4159 y FG(s)2041 4173
y Fv(R)2098 4159 y Fx(j)26 b(\025)1435 4337 y(\025)98
b FA(const)16 b(\(1)21 b(+)e FG(s)2061 4351 y Fv(R)2119
4337 y FA(\))2154 4299 y Fv(p)p Ft(\000)p Fy(1)2299 4337
y Fx(j)p FG(@)2372 4351 y Fv(R)2430 4337 y FG(s)2473
4351 y Fv(R)2531 4337 y Fx(j)25 b(\025)1435 4512 y(\025)1657
4451 y FA(const)p 1599 4491 326 4 v 1599 4579 a FG(R)1669
4552 y Fy(\()p Fv(p)p Ft(\000)p Fy(1\))p Fv(=p)1949 4512
y Fx(j)p FG(@)2022 4526 y Fv(R)2080 4512 y FG(s)2123
4526 y Fv(R)2181 4512 y Fx(j)15 b FG(:)3374 4735 y Fl(\003)555
4956 y FA(W)-8 b(e)40 b(no)m(w)e(de\014ne)g(a)h(mo)s(di\014cation)e(of)
i(the)f(p)s(oten)m(tial)g FG(h)2523 4970 y Fy(0)2602
4956 y FA(in)f(the)h(whole)g(in)m(terv)-5 b(al)456 5064
y([)p Fx(\000)p FA(1)p FG(;)15 b FA(1])31 b(in)e(the)i(follo)m(wing)e
(w)m(a)m(y:)456 5216 y FH(De\014nition)35 b(2.5.)42 b
FA(W)-8 b(e)31 b(de\014ne)f FG(h)1619 5230 y Fv(s)1652
5239 y Fq(0)1687 5230 y Fv(;R)1789 5216 y FA(:)c([)p
FG(s)1908 5230 y Fv(R)1965 5216 y FG(;)15 b FA(1])27
b Fx(!)e Fw(R)39 b FA(b)m(y)p eop
%%Page: 11 11
11 10 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(11)1051
994 y FG(h)1103 1008 y Fv(s)1136 1017 y Fq(0)1171 1008
y Fv(;R)1248 994 y FA(\()p FG(s)p FA(\))83 b(:=)1623
508 y Fu(8)1623 590 y(>)1623 617 y(>)1623 644 y(>)1623
671 y(>)1623 699 y(>)1623 726 y(>)1623 753 y(>)1623 780
y(>)1623 808 y(>)1623 835 y(>)1623 862 y(>)1623 890 y(<)1623
1053 y(>)1623 1080 y(>)1623 1108 y(>)1623 1135 y(>)1623
1162 y(>)1623 1190 y(>)1623 1217 y(>)1623 1244 y(>)1623
1271 y(>)1623 1299 y(>)1623 1326 y(>)1623 1353 y(:)1704
611 y FG(h)1756 625 y Fy(0)1796 611 y FA(\()p FG(s)p
FA(\))20 b Fx(\000)g FG(h)2072 625 y Fy(0)2112 611 y
FA(\()p FG(s)2190 625 y Fv(R)2248 611 y FA(\))g Fx(\000)2418
556 y Fk(b)2404 573 y Fv(C)2454 582 y Fq(0)p 2404 590
85 4 v 2420 642 a Fv(R)2514 611 y FA(\()q FG(s)f Fx(\000)h
FG(s)2746 625 y Fv(R)2804 611 y FA(\))1989 722 y(if)90
b FG(s)2176 736 y Fv(R)2259 722 y Fx(\024)25 b FG(s)g
Fx(\024)g(\000)2600 686 y Fy(1)p 2600 701 36 4 v 2600
753 a(2)2185 937 y FG(')p FA(\()p FG(s)p FA(\))1992 1048
y(if)110 b Fx(\000)2257 1013 y Fy(1)p 2257 1028 V 2257
1080 a(2)2327 1048 y FG(<)25 b(s)g(<)2597 1013 y Fy(1)p
2597 1028 V 2597 1080 a(2)1739 1291 y FG(h)1791 1305
y Fy(0)1831 1291 y FA(\()p FG(s)p FA(\))20 b(+)g FG(h)2107
1305 y Fy(0)2147 1291 y FA(\()p FG(s)2225 1305 y Fv(R)2283
1291 y FA(\))g(+)2453 1237 y Fk(b)2439 1254 y Fv(C)2489
1263 y Fq(0)p 2439 1270 85 4 v 2455 1322 a Fv(R)2534
1291 y FA(\(1)h Fx(\000)f FG(s)p FA(\))2040 1402 y(if)2193
1366 y Fy(1)p 2193 1381 36 4 v 2193 1433 a(2)2264 1402
y Fx(\024)25 b FG(s)g Fx(\024)g FA(1)p FG(:)555 1561
y FA(Notice)36 b(that)f FG(h)1098 1575 y Fv(s)1131 1584
y Fq(0)1166 1575 y Fv(;R)1277 1561 y FA(ma)m(y)h(b)s(e)d(discon)m(tin)m
(uous)g(at)j FG(s)c FA(=)f Fx(\006)p FA(1)p FG(=)p FA(2.)55
b(Let)35 b(us)f(no)m(w)g(p)s(oin)m(t)456 1669 y(out)c(some)h(easy)g
(prop)s(erties)e(enjo)m(y)m(ed)i(b)m(y)f(the)h(ab)s(o)m(v)m(e)g(p)s
(oten)m(tial:)456 1801 y FH(Lemma)h(2.6.)42 b Fo(The)33
b(fol)5 b(lowing)34 b(ine)-5 b(qualities)33 b(hold.)43
b(If)33 b FG(s)25 b Fx(2)f FA([)p Fx(\000)p FA(3)p FG(=)p
FA(4)p FG(;)32 b Fx(\000)p FA(1)p FG(=)p FA(2])p Fo(,)i(then)1324
2032 y FG(h)1376 2046 y Fv(s)1409 2055 y Fq(0)1444 2046
y Fv(;R)1521 2032 y FA(\()p FG(s)p FA(\))26 b FG(>)f(h)1808
2046 y Fy(0)1847 2032 y FA(\()p FG(s)p FA(\))c Fx(\000)2082
1970 y FA(2)2145 1947 y Fu(b)2127 1970 y FG(C)2192 1984
y Fy(0)p 2082 2011 150 4 v 2122 2094 a FG(R)2267 2032
y(>)k(')p FA(\()p FG(s)p FA(\))15 b FG(:)456 2220 y Fo(If)32
b FG(s)25 b Fx(2)g FA([1)p FG(=)p FA(2)p FG(;)32 b FA(3)p
FG(=)p FA(4])p Fo(,)i(then)1324 2450 y FG(h)1376 2464
y Fv(s)1409 2473 y Fq(0)1444 2464 y Fv(;R)1521 2450 y
FA(\()p FG(s)p FA(\))26 b FG(<)f(h)1808 2464 y Fy(0)1847
2450 y FA(\()p FG(s)p FA(\))c(+)2082 2389 y(2)2145 2366
y Fu(b)2127 2389 y FG(C)2192 2403 y Fy(0)p 2082 2430
V 2122 2513 a FG(R)2267 2450 y(<)k(')p FA(\()p FG(s)p
FA(\))15 b FG(:)456 2675 y Fo(Pr)-5 b(o)g(of.)43 b FA(Let)36
b FG(s)e Fx(2)g FA([)p Fx(\000)p FA(3)p FG(=)p FA(4)p
FG(;)e Fx(\000)p FA(1)p FG(=)p FA(2].)58 b(If)1793 2652
y Fu(b)1775 2675 y FG(C)1840 2689 y Fy(0)1915 2675 y
FA(is)35 b(tak)m(en)i(suitably)d(large,)k(then,)f(recalling)456
2783 y(Lemma)30 b(2.2,)i(w)m(e)f(ha)m(v)m(e)g(that)1203
3008 y FG(h)1255 3022 y Fv(s)1288 3031 y Fq(0)1323 3022
y Fv(;R)1400 3008 y FA(\()p FG(s)p FA(\))26 b(=)f FG(h)1687
3022 y Fy(0)1727 3008 y FA(\()p FG(s)p FA(\))20 b Fx(\000)1973
2947 y FA(1)p 1961 2987 70 4 v 1961 3071 a FG(R)2061
3008 y Fx(\000)2180 2924 y Fu(b)2162 2947 y FG(C)2227
2961 y Fy(0)p 2162 2987 105 4 v 2179 3071 a FG(R)2276
3008 y FA(\()p FG(s)g Fx(\000)g FG(s)2508 3022 y Fv(R)2565
3008 y FA(\))26 b Fx(\025)1539 3256 y(\025)f FG(h)1687
3270 y Fy(0)1727 3256 y FA(\()p FG(s)p FA(\))20 b Fx(\000)1961
3194 y FA(2)2024 3171 y Fu(b)2006 3194 y FG(C)2071 3208
y Fy(0)p 1961 3235 150 4 v 2001 3318 a FG(R)2146 3256
y(>)25 b(')p FA(\()p FG(s)p FA(\))456 3439 y(where)k(w)m(e)i(ha)m(v)m
(e)h(used)d(the)i(fact)g(that)g FG(s)20 b Fx(\000)g FG(s)1999
3453 y Fv(R)2082 3439 y Fx(\024)25 b FA(1.)555 3599 y(In)38
b(the)h(same)g(w)m(a)m(y)-8 b(,)42 b(if)c FG(s)g Fx(2)h
FA([1)p FG(=)p FA(2)p FG(;)32 b FA(3)p FG(=)p FA(4],)43
b(taking)2314 3576 y Fu(b)2296 3599 y FG(C)2361 3613
y Fy(0)2439 3599 y FA(suitably)37 b(large,)k(and)d(using)456
3707 y(Lemma)30 b(2.2,)i(w)m(e)f(gather)g(that)1381 3933
y FG(h)1433 3947 y Fv(s)1466 3956 y Fq(0)1500 3947 y
Fv(;R)1603 3933 y FG(<)25 b(h)1751 3947 y Fy(0)1791 3933
y FA(\()p FG(s)p FA(\))20 b(+)2037 3871 y(1)p 2025 3912
70 4 v 2025 3995 a FG(R)2125 3933 y FA(+)2249 3848 y
Fu(b)2231 3871 y FG(C)2296 3885 y Fy(0)p 2226 3912 116
4 v 2226 3995 a FA(2)p FG(R)2376 3933 y(<)1603 4181 y(<)25
b(h)1751 4195 y Fy(0)1791 4181 y FA(\()p FG(s)p FA(\))20
b(+)2025 4119 y(2)2089 4096 y Fu(b)2070 4119 y FG(C)2135
4133 y Fy(0)p 2025 4160 150 4 v 2065 4243 a FG(R)2210
4181 y(<)25 b(')p FA(\()p FG(s)p FA(\))15 b FG(:)3374
4362 y Fl(\003)456 4535 y FH(Lemma)32 b(2.7.)42 b Fo(L)-5
b(et)33 b FG(R)1256 4549 y Fy(1)1321 4535 y Fx(\024)25
b FG(R)1486 4549 y Fy(2)1558 4535 y Fo(b)-5 b(e)33 b(suitably)g(lar)-5
b(ge.)42 b(Then,)2490 4500 y Fy(4)456 4692 y FA(\(2.10\))441
b FG(h)1179 4706 y Fv(s)1212 4715 y Fq(0)1247 4706 y
Fv(;R)1320 4715 y Fq(1)1358 4692 y FA(\()p FG(s)p FA(\))26
b Fx(\024)f FG(h)1645 4706 y Fv(s)1678 4715 y Fq(0)1712
4706 y Fv(;R)1785 4715 y Fq(2)1824 4692 y FA(\()p FG(s)p
FA(\))186 b Fo(if)92 b FG(s)2314 4706 y Fv(R)2367 4715
y Fq(1)2431 4692 y Fx(\024)25 b FG(s)g Fx(\024)g FG(s)2734
4706 y Fy(0)p 456 4778 499 4 v 555 4848 a(4)591 4877
y FF(Notice)h(that,)f(for)i FD(R)1195 4885 y Fq(1)1251
4877 y Fr(\024)21 b FD(R)1390 4885 y Fq(2)1450 4877 y
FF(suitably)26 b(large,)h(using)f(\(1.2\),)h(\(2.6\))f(and)g(\(2.7\),)g
(one)g(has)1410 5043 y FD(h)1454 5051 y Fq(0)1489 5043
y FF(\()p FD(s)1555 5051 y Fs(R)1601 5061 y Fn(1)1637
5043 y FF(\))21 b(=)1806 4994 y(1)p 1779 5026 93 4 v
1779 5093 a FD(R)1837 5101 y Fq(1)1903 5043 y Fr(\025)2021
4994 y FF(1)p 1994 5026 V 1994 5093 a FD(R)2052 5101
y Fq(2)2118 5043 y FF(=)g FD(h)2243 5051 y Fq(0)2278
5043 y FF(\()p FD(s)2344 5051 y Fs(R)2390 5061 y Fn(2)2426
5043 y FF(\))13 b FD(;)456 5216 y FF(so)26 b(that)f FD(s)752
5224 y Fs(R)798 5234 y Fn(1)856 5216 y Fr(\025)c FD(s)973
5224 y Fs(R)1019 5234 y Fn(2)1055 5216 y FF(.)p eop
%%Page: 12 12
12 11 bop 456 251 a Fy(12)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y Fo(and)456 616 y FA(\(2.11\))465 b FG(h)1203
630 y Fv(s)1236 639 y Fq(0)1271 630 y Fv(;R)1344 639
y Fq(1)1382 616 y FA(\()p FG(s)p FA(\))26 b Fx(\025)f
FG(h)1669 630 y Fv(s)1702 639 y Fq(0)1736 630 y Fv(;R)1809
639 y Fq(2)1848 616 y FA(\()p FG(s)p FA(\))186 b Fo(if)93
b FG(s)2339 630 y Fy(0)2403 616 y Fx(\024)25 b FG(s)g
Fx(\024)g FA(1)15 b FG(:)456 800 y Fo(Pr)-5 b(o)g(of.)43
b FA(Let)d(us)g(pro)m(v)m(e)g(\(2.10\).)73 b(F)-8 b(or)40
b(this)f(purp)s(ose,)j(let)e FG(s)h Fx(\024)g FG(s)2737
814 y Fy(0)2776 800 y FA(.)70 b(Tw)m(o)40 b(cases)h(are)456
908 y(p)s(ossible:)d(either)30 b FG(s)25 b(>)g Fx(\000)p
FA(1)p FG(=)p FA(2)31 b(or)f FG(s)25 b Fx(\024)g(\000)p
FA(1)p FG(=)p FA(2.)42 b(Let)31 b(us)e(\014rst)h(deal)g(with)f(the)i
(\014rst)e(case.)456 1016 y(Notice)g(that,)g(b)m(y)f(\(2.1\),)j
FG(')1383 1030 y Fv(s)1416 1039 y Fq(0)1451 1030 y Fv(;R)1556
1016 y FA(is)d(increasing)f(in)g FG(R)q FA(,)h(since)g
Fx(\000)p FA(1)p FG(=)p FA(2)e FG(<)f(s)g Fx(\024)g FG(s)3052
1030 y Fy(0)3116 1016 y Fx(\024)g FA(1)p FG(=)p FA(4)i
FG(<)456 1124 y FA(1)p FG(=)p FA(2,)32 b(th)m(us,)e(from)g
(De\014nition)f(2.5)i(w)m(e)g(gather)g(that)1576 1290
y FG(h)1628 1304 y Fv(s)1661 1313 y Fq(0)1696 1304 y
Fv(;R)1769 1313 y Fq(1)1808 1290 y FA(\()p FG(s)p FA(\))40
b(=)g FG(')2131 1304 y Fv(s)2164 1313 y Fq(0)2199 1304
y Fv(;R)2272 1313 y Fq(1)2311 1290 y FA(\()p FG(s)p FA(\))g
Fx(\024)1423 1425 y(\024)82 b FG(')1635 1439 y Fv(s)1668
1448 y Fq(0)1703 1439 y Fv(;R)1776 1448 y Fq(2)1815 1425
y FA(\()p FG(s)p FA(\))40 b(=)g FG(h)2131 1439 y Fv(s)2164
1448 y Fq(0)2199 1439 y Fv(;R)2272 1448 y Fq(2)2311 1425
y FA(\()p FG(s)p FA(\))15 b FG(;)456 1591 y FA(whic)m(h)29
b(pro)m(v)m(es)i(\(2.10\))h(if)e FG(s)25 b(>)g Fx(\000)p
FA(1)p FG(=)p FA(2.)555 1699 y(Let)48 b(us)e(no)m(w)h(deal)f(with)g
(the)h(case)h FG(s)53 b Fx(\024)g(\000)p FA(1)p FG(=)p
FA(2.)91 b(Fixed)46 b FG(R)2769 1713 y Fy(1)2856 1699
y FA(large)h(and)f FG(s)53 b Fx(2)456 1807 y FA([)p FG(s)524
1821 y Fv(R)581 1807 y FG(;)31 b Fx(\000)p FA(1)p FG(=)p
FA(2],)h(let)e(us)g(de\014ne)1415 2046 y FG(g)s FA(\()p
FG(R)q FA(\))42 b(:=)1801 1984 y(1)p 1789 2025 70 4 v
1789 2108 a FG(R)1889 2046 y FA(+)2008 1961 y Fu(b)1990
1984 y FG(C)2055 1998 y Fy(0)p 1990 2025 105 4 v 2007
2108 a FG(R)2119 2046 y FA(\()p FG(s)21 b Fx(\000)f FG(s)2352
2060 y Fv(R)2409 2046 y FA(\))15 b FG(:)456 2238 y FA(By)30
b(means)h(of)f(Lemma)h(2.4,)1344 2472 y FG(g)1390 2434
y Ft(0)1414 2472 y FA(\()p FG(R)q FA(\))26 b Fx(\024)f(\000)1788
2410 y FA(1)p 1757 2451 110 4 v 1757 2534 a FG(R)1827
2508 y Fy(2)1896 2472 y FA(+)2012 2410 y(const)2252 2387
y Fu(b)2234 2410 y FG(C)2299 2424 y Fy(0)p 1997 2451
343 4 v 2033 2538 a FG(R)2103 2511 y Fy(2+1)p Fv(=p)2374
2472 y FG(<)g FA(0)15 b FG(;)456 2667 y FA(thence)1111
2822 y(1)p 1079 2863 109 4 v 1079 2946 a FG(R)1148 2960
y Fy(2)1218 2884 y FA(+)1339 2799 y Fu(b)1321 2822 y
FG(C)1386 2836 y Fy(0)p 1319 2863 V 1319 2946 a FG(R)1388
2960 y Fy(2)1452 2884 y FA(\()p FG(s)21 b Fx(\000)f FG(s)1685
2898 y Fv(R)1738 2907 y Fq(2)1776 2884 y FA(\))83 b(=)g
FG(g)s FA(\()p FG(R)2198 2898 y Fy(2)2239 2884 y FA(\))25
b Fx(\024)1490 3145 y(\024)g FG(g)s FA(\()p FG(R)1736
3159 y Fy(1)1776 3145 y FA(\))83 b(=)2090 3083 y(1)p
2058 3124 V 2058 3207 a FG(R)2127 3221 y Fy(1)2197 3145
y FA(+)2318 3060 y Fu(b)2300 3083 y FG(C)2365 3097 y
Fy(0)p 2298 3124 V 2298 3207 a FG(R)2367 3221 y Fy(1)2431
3145 y FA(\()p FG(s)20 b Fx(\000)g FG(s)2663 3159 y Fv(R)2716
3168 y Fq(1)2755 3145 y FA(\))15 b FG(:)456 3355 y FA(Therefore,)30
b(if)f FG(s)c Fx(\024)g(\000)p FA(1)p FG(=)p FA(2,)32
b(from)e(De\014nition)f(2.5,)1070 3595 y FG(h)1122 3609
y Fv(s)1155 3618 y Fq(0)1189 3609 y Fv(;R)1262 3618 y
Fq(1)1301 3595 y FA(\()p FG(s)p FA(\))83 b(=)g FG(h)1703
3609 y Fy(0)1743 3595 y FA(\()p FG(s)p FA(\))20 b Fx(\000)2009
3533 y FA(1)p 1977 3574 V 1977 3657 a FG(R)2046 3671
y Fy(1)2116 3595 y Fx(\000)2237 3510 y Fu(b)2219 3533
y FG(C)2284 3547 y Fy(0)p 2217 3574 V 2217 3657 a FG(R)2286
3671 y Fy(1)2350 3595 y FA(\()q FG(s)f Fx(\000)h FG(s)2582
3609 y Fv(R)2635 3618 y Fq(1)2674 3595 y FA(\))25 b Fx(\024)1497
3856 y(\024)83 b FG(h)1703 3870 y Fy(0)1743 3856 y FA(\()p
FG(s)p FA(\))20 b Fx(\000)2009 3794 y FA(1)p 1977 3835
V 1977 3918 a FG(R)2046 3932 y Fy(2)2116 3856 y Fx(\000)2237
3771 y Fu(b)2219 3794 y FG(C)2284 3808 y Fy(0)p 2217
3835 V 2217 3918 a FG(R)2286 3932 y Fy(2)2350 3856 y
FA(\()q FG(s)f Fx(\000)h FG(s)2582 3870 y Fv(R)2635 3879
y Fq(2)2674 3856 y FA(\))25 b(=)1497 4035 y(=)83 b FG(h)1703
4049 y Fv(s)1736 4058 y Fq(0)1770 4049 y Fv(;R)1843 4058
y Fq(2)1882 4035 y FA(\()p FG(s)p FA(\))15 b FG(;)456
4201 y FA(th)m(us)30 b(pro)m(ving)f(\(2.10\).)555 4425
y(Ha)m(ving)45 b(completed)f(the)g(pro)s(of)g(of)g(\(2.10\),)50
b(w)m(e)45 b(no)m(w)f(deal)g(with)e(the)j(pro)s(of)e(of)456
4533 y(\(2.11\).)k(Tw)m(o)32 b(cases)h(are)f(p)s(ossible:)41
b(either)31 b FG(s)d(<)f FA(1)p FG(=)p FA(2)33 b(or)f
FG(s)27 b Fx(\025)h FA(1)p FG(=)p FA(2.)46 b(Let)33 b(us)e(\014rst)g
(deal)456 4641 y(with)j(the)j(\014rst)e(case.)59 b(Notice)37
b(that,)i(b)m(y)d(\(2.1\))i(and)d(the)i(fact)g(that)f
FG(s)2920 4655 y Fy(0)2994 4641 y Fx(\024)f FG(s)f(<)h
FA(1)p FG(=)p FA(2,)456 4749 y FG(')515 4763 y Fv(s)548
4772 y Fq(0)582 4763 y Fv(;R)690 4749 y FA(is)29 b(decreasing)h(in)g
FG(R)q FA(.)40 b(Hence,)31 b(from)f(De\014nition)f(2.5,)1576
4915 y FG(h)1628 4929 y Fv(s)1661 4938 y Fq(0)1696 4929
y Fv(;R)1769 4938 y Fq(1)1808 4915 y FA(\()p FG(s)p FA(\))40
b(=)g FG(')2131 4929 y Fv(s)2164 4938 y Fq(0)2199 4929
y Fv(;R)2272 4938 y Fq(1)2311 4915 y FA(\()p FG(s)p FA(\))g
Fx(\025)1423 5050 y(\025)82 b FG(')1635 5064 y Fv(s)1668
5073 y Fq(0)1703 5064 y Fv(;R)1776 5073 y Fq(2)1815 5050
y FA(\()p FG(s)p FA(\))40 b(=)g FG(h)2131 5064 y Fv(s)2164
5073 y Fq(0)2199 5064 y Fv(;R)2272 5073 y Fq(2)2311 5050
y FA(\()p FG(s)p FA(\))15 b FG(;)456 5216 y FA(whic)m(h)29
b(pro)m(v)m(es)i(\(2.11\))h(if)e FG(s)25 b(<)g FA(1)p
FG(=)p FA(2.)p eop
%%Page: 13 13
13 12 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(13)555
450 y FA(Let)31 b(us)f(no)m(w)g(deal)g(with)f(the)i(case)g
FG(s)25 b Fx(\025)g FA(1)p FG(=)p FA(2.)42 b(In)30 b(this)f(case,)j(b)m
(y)e(De\014nition)f(2.5,)580 671 y FG(h)632 685 y Fv(s)665
694 y Fq(0)700 685 y Fv(;R)773 694 y Fq(1)811 671 y FA(\()p
FG(s)p FA(\))21 b Fx(\000)e FG(h)1087 685 y Fv(s)1120
694 y Fq(0)1155 685 y Fv(;R)1228 694 y Fq(2)1267 671
y FA(\()p FG(s)p FA(\))40 b(=)1573 610 y(1)p 1541 650
109 4 v 1541 734 a FG(R)1610 748 y Fy(1)1680 671 y Fx(\000)1812
610 y FA(1)p 1781 650 V 1781 734 a FG(R)1850 748 y Fy(2)1919
671 y FA(+)2028 648 y Fu(b)2010 671 y FG(C)2075 685 y
Fy(0)2115 671 y FA(\(1)21 b Fx(\000)f FG(s)p FA(\))2415
543 y Fu(\022)2523 610 y FA(1)p 2492 650 V 2492 734 a
FG(R)2561 748 y Fy(1)2630 671 y Fx(\000)2763 610 y FA(1)p
2731 650 V 2731 734 a FG(R)2800 748 y Fy(2)2850 543 y
Fu(\023)2957 671 y Fx(\025)40 b FA(0)15 b FG(;)456 893
y FA(pro)m(ving)26 b(\(2.11\))j(for)d FG(s)f Fx(\025)g
FA(1)p FG(=)p FA(2)j(and)f(th)m(us)f(completing)g(the)h(pro)s(of)f(of)h
(Lemma)g(2.7.)93 b Fl(\003)555 1078 y FA(Let)31 b(no)m(w)456
1311 y(\(2.12\))313 b FG(H)1075 1325 y Fy(0)1114 1311
y FA(\()p FG(s)p FA(\))56 b(:=)1435 1187 y Fu(Z)1525
1213 y Fv(s)1485 1393 y Fy(0)1621 1249 y FA(\()p FG(p)20
b Fx(\000)g FA(1\))1904 1176 y Fq(1)p 1904 1188 33 3
v 1904 1230 a Fs(p)p 1587 1290 397 4 v 1587 1405 a FA(\()p
FG(p)15 b(h)1735 1419 y Fy(0)1775 1405 y FA(\()p FG(\020)7
b FA(\)\))1938 1333 y Fq(1)p 1938 1345 33 3 v 1938 1386
a Fs(p)2009 1311 y FG(d\020)22 b(;)228 b FA(for)30 b(an)m(y)g
FG(s)25 b Fx(2)g FA(\()p Fx(\000)p FA(1)p FG(;)15 b FA(1\).)456
1563 y(Notice)35 b(that)h(the)f(in)m(v)m(erse)f(of)h
FG(H)1597 1577 y Fy(0)1671 1563 y FA(is)e(a)j(\\one-dimensional")d
(solution)g(of)i(\(1.5\).)56 b(In-)456 1675 y(deed,)30
b(if)f FG(g)819 1689 y Fy(0)884 1675 y FA(:=)d FG(H)1089
1637 y Ft(\000)p Fy(1)1082 1701 y(0)1183 1675 y FA(,)k(b)m(y)h(Lemma)f
(B.3,)i(w)m(e)f(obtain)e(that)1366 1848 y(\001)1442 1862
y Fv(p)1481 1848 y FG(g)1524 1862 y Fy(0)1647 1848 y
FA(=)83 b(\()p Fx(j)p FG(g)1907 1811 y Ft(0)1931 1848
y Fx(j)1956 1811 y Fv(p)p Ft(\000)p Fy(2)2086 1848 y
FG(g)2132 1811 y Ft(0)2156 1848 y FA(\))2191 1811 y Ft(0)2240
1848 y FA(=)1647 1990 y(=)g(\()p FG(p)20 b Fx(\000)g
FA(1\))p Fx(j)p FG(g)2144 1953 y Ft(0)2169 1990 y Fx(j)2194
1953 y Fv(p)p Ft(\000)p Fy(2)2324 1990 y FG(g)2370 1953
y Ft(00)2438 1990 y FA(=)1647 2125 y(=)83 b FG(h)1853
2088 y Ft(0)1853 2148 y Fy(0)1893 2125 y FA(\()p FG(g)1971
2139 y Fy(0)2011 2125 y FA(\))15 b FG(:)456 2292 y FA(W)-8
b(e)31 b(w)m(ould)e(lik)m(e)h(no)m(w)g(to)h(compare)g
FG(g)1751 2306 y Fy(0)1821 2292 y FA(with)f(all)f(other)h(solutions.)
555 2399 y(T)-8 b(o)28 b(this)f(aim,)h(using)e(De\014nition)g(2.5)j(w)m
(e)f(no)m(w)g(in)m(tro)s(duce)f(suitable)f(mo)s(di\014cations)456
2507 y(of)k FG(H)635 2521 y Fy(0)704 2507 y FA(\(and)h(th)m(us)f(of)g
FG(g)1263 2521 y Fy(0)1303 2507 y FA(\),)h(whic)m(h)e(will)f(b)s(e)h
(used)h(in)f(the)i(course)f(of)h(the)f(pro)s(of:)456
2645 y FH(De\014nition)35 b(2.8.)42 b FA(W)-8 b(e)31
b(de\014ne,)f(for)h FG(s)24 b Fx(2)h FA([)p FG(s)1953
2659 y Fv(R)2011 2645 y FG(;)15 b FA(1],)1056 2900 y
FG(H)1132 2914 y Fv(s)1165 2923 y Fq(0)1199 2914 y Fv(;R)1276
2900 y FA(\()p FG(s)p FA(\))83 b(:=)g FG(H)1727 2914
y Fy(0)1766 2900 y FA(\()p FG(s)1844 2914 y Fy(0)1884
2900 y FA(\))20 b(+)2030 2777 y Fu(Z)2121 2803 y Fv(s)2081
2983 y(s)2114 2992 y Fq(0)2269 2839 y FA(\()p FG(p)h
Fx(\000)f FA(1\))2553 2766 y Fq(1)p 2552 2778 V 2552
2819 a Fs(p)p 2183 2879 502 4 v 2183 2995 a FA(\()p FG(p)15
b(h)2331 3009 y Fv(s)2364 3018 y Fq(0)2399 3009 y Fv(;R)2476
2995 y FA(\()p FG(\020)7 b FA(\)\))2639 2922 y Fq(1)p
2639 2934 33 3 v 2639 2976 a Fs(p)2710 2900 y FG(d\020)22
b(:)555 3152 y FA(W)-8 b(e)32 b(notice)e(that,)i(b)m(y)e(Lemmas)g(B.1)i
(and)d(2.3,)j(w)m(e)f(get)456 3318 y(\(2.13\))1047 b
FG(h)1785 3332 y Fv(s)1818 3341 y Fq(0)1852 3332 y Fv(;R)1970
3318 y FG(>)40 b FA(0)15 b FG(;)456 3485 y FA(thence)30
b FG(H)819 3499 y Fv(s)852 3508 y Fq(0)886 3499 y Fv(;R)994
3485 y FA(is)f(w)m(ell)h(de\014ned.)456 3607 y Fo(R)-5
b(emark)55 b FA(2.9)p Fo(.)49 b FA(By)43 b(De\014nition)e(2.8,)47
b(exploiting)41 b(Lemma)h(2.7,)47 b(it)42 b(follo)m(ws)g(that,)k(if)456
3715 y FG(R)525 3729 y Fy(1)589 3715 y Fx(\024)25 b FG(R)754
3729 y Fy(2)794 3715 y FA(,)456 3881 y(\(2.14\))420 b
FG(H)1182 3895 y Fv(s)1215 3904 y Fq(0)1249 3895 y Fv(;R)1322
3904 y Fq(1)1360 3881 y FA(\()p FG(s)p FA(\))26 b Fx(\024)f
FG(H)1671 3895 y Fv(s)1704 3904 y Fq(0)1738 3895 y Fv(;R)1811
3904 y Fq(2)1849 3881 y FA(\()p FG(s)p FA(\))182 b(if)90
b FG(s)2331 3895 y Fv(R)2384 3904 y Fq(1)2448 3881 y
Fx(\024)25 b FG(s)g Fx(\024)g FA(1)15 b FG(:)555 4048
y FA(Let)31 b(us)f(no)m(w)g(analyze)h(some)g(prop)s(erties)d(of)j(the)g
(ab)s(o)m(v)m(e)g(de\014ned)e(quan)m(tit)m(y:)456 4185
y FH(Lemma)38 b(2.10.)46 b Fo(Assuming)37 b FG(R)h Fo(suitably)g(lar)-5
b(ge,)40 b(ther)-5 b(e)38 b(exists)g(a)g(p)-5 b(ositive)38
b(c)-5 b(onstant)456 4293 y FG(C)521 4307 y Fy(1)593
4293 y Fo(so)33 b(that)h(the)f(fol)5 b(lowing)33 b(ine)-5
b(qualities)33 b(hold:)1314 4459 y FG(H)1390 4473 y Fv(s)1423
4482 y Fq(0)1456 4473 y Fv(;R)1534 4459 y FA(\()p FG(s)1612
4473 y Fv(R)1669 4459 y FA(\))26 b Fx(\025)f(\000)p FG(C)1962
4473 y Fy(1)2016 4459 y FA(log)17 b FG(R)f(;)-1803 b
FA(\(2.15\))1314 4594 y FG(H)1390 4608 y Fv(s)1423 4617
y Fq(0)1456 4608 y Fv(;R)1534 4594 y FA(\(1\))26 b Fx(\024)f
FG(C)1836 4608 y Fy(1)1891 4594 y FA(log)16 b FG(R)g(;)-1677
b FA(\(2.16\))1345 4719 y FG(d)p 1324 4760 90 4 v 1324
4843 a(ds)1439 4781 y FA(\()p FG(H)1550 4795 y Fv(s)1583
4804 y Fq(0)1617 4795 y Fv(;R)1694 4781 y FA(\()p FG(s)p
FA(\)\))26 b FG(>)f FA(0)15 b FG(;)202 b Fx(8)p FG(s)24
b Fx(2)h FA(\()p FG(s)2533 4795 y Fv(R)2591 4781 y FG(;)15
b FA(1\))g FG(:)-2295 b FA(\(2.17\))456 4992 y Fo(Pr)-5
b(o)g(of.)43 b FA(Note)31 b(that,)g(if)f FG(R)h FA(is)e(suitably)g
(big,)g(De\014nition)g(2.1)j(implies)27 b(that)456 5158
y(\(2.18\))1011 b(inf)1636 5223 y Ft(j)p Fv(s)p Ft(j\024)p
Fy(1)p Fv(=)p Fy(2)1884 5158 y FG(')p FA(\()p FG(s)p
FA(\))26 b FG(>)f FA(0)15 b FG(:)p eop
%%Page: 14 14
14 13 bop 456 251 a Fy(14)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(Hence,)31 b(recalling)e(Lemma)h(B.1)i(and)e
(\(2.7\),)i(w)m(e)f(ha)m(v)m(e)g(that)501 679 y FG(H)577
693 y Fv(s)610 702 y Fq(0)644 693 y Fv(;R)722 679 y FA(\()p
FG(s)800 693 y Fv(R)857 679 y FA(\))84 b Fx(\025)e FG(H)1205
693 y Fy(0)1244 679 y FA(\()p FG(s)1322 693 y Fy(0)1362
679 y FA(\))20 b Fx(\000)1508 555 y Fu(Z)1599 581 y Ft(\000)1664
554 y Fq(1)p 1664 566 31 3 v 1664 607 a(2)1559 761 y
Fv(s)1592 772 y Fs(R)2338 617 y FA(const)c FG(d\020)p
1734 658 1526 4 v 1734 696 a Fu(\020)1788 797 y FA(\(1)21
b(+)f FG(\020)7 b FA(\))2062 771 y Fv(p)2122 797 y Fx(\000)20
b FA(\(1)h(+)e FG(s)2447 811 y Fv(R)2505 797 y FA(\))2540
771 y Fv(p)2600 797 y Fx(\000)2701 761 y Fv(C)p 2701
776 55 4 v 2702 828 a(R)2766 797 y FA(\()p FG(\020)27
b Fx(\000)20 b FG(s)3002 811 y Fv(R)3059 797 y FA(\))3094
696 y Fu(\021)3149 719 y Fy(1)p Fv(=p)3289 679 y Fx(\000)1129
1058 y(\000)1215 934 y Fu(Z)1306 960 y Fv(s)1339 969
y Fq(0)1266 1140 y Ft(\000)1331 1113 y Fq(1)p 1331 1125
31 3 v 1331 1166 a(2)1420 996 y FA(\()p FG(p)h Fx(\000)e
FA(1\))1703 923 y Fq(1)p 1703 935 33 3 v 1703 977 a Fs(p)p
1403 1037 365 4 v 1403 1152 a FA(\()p FG(p)c(')p FA(\()p
FG(\020)7 b FA(\)\))1721 1080 y Fq(1)p 1721 1092 33 3
v 1721 1133 a Fs(p)1792 1058 y FG(d\020)47 b Fx(\025)976
1351 y(\025)82 b(\000)15 b FA(const)36 b Fx(\000)1564
1227 y Fu(Z)1655 1253 y Fy(0)1615 1433 y Fv(s)1648 1444
y Fs(R)2331 1289 y FA(const)16 b FG(d\020)p 1727 1330
1526 4 v 1727 1368 a Fu(\020)1781 1469 y FA(\(1)21 b(+)f
FG(\020)7 b FA(\))2055 1442 y Fv(p)2115 1469 y Fx(\000)20
b FA(\(1)h(+)f FG(s)2441 1483 y Fv(R)2498 1469 y FA(\))2533
1442 y Fv(p)2593 1469 y Fx(\000)2694 1433 y Fv(C)p 2694
1448 55 4 v 2695 1500 a(R)2759 1469 y FA(\()p FG(\020)27
b Fx(\000)20 b FG(s)2995 1483 y Fv(R)3052 1469 y FA(\))3087
1368 y Fu(\021)3142 1390 y Fy(1)p Fv(=p)3303 1351 y Fx(\025)976
1626 y(\025)82 b(\000)p FG(C)1265 1640 y Fy(1)1320 1626
y FA(log)16 b FG(R)g(;)456 1775 y FA(for)29 b(a)g(suitable)f
FG(C)1071 1789 y Fy(1)1110 1775 y FA(,)i(where)e(w)m(e)i(used)e(Lemma)i
(2.3)g(to)g(estimate)g(the)f(in)m(tegral)g(ab)s(o)m(v)m(e.)456
1883 y(This)f(pro)m(v)m(es)j(\(2.15\).)555 1991 y(W)-8
b(e)34 b(no)m(w)e(pro)m(v)m(e)h(\(2.16\).)48 b(By)32
b(De\014nition)f(2.5,)j(\(1.1\),)g(\(2.18\))h(and)c(\(2.6\),)k(w)m(e)d
(ha)m(v)m(e)456 2099 y(that)575 2315 y FG(H)651 2329
y Fv(s)684 2338 y Fq(0)718 2329 y Fv(;R)795 2315 y FA(\(1\))84
b Fx(\024)f FG(H)1224 2329 y Fy(0)1263 2315 y FA(\()p
FG(s)1341 2329 y Fy(0)1380 2315 y FA(\))21 b(+)1527 2191
y Fu(Z)1627 2190 y Fq(1)p 1627 2202 31 3 v 1627 2243
a(2)1577 2397 y Fv(s)1610 2406 y Fq(0)1715 2253 y FA(\()p
FG(p)f Fx(\000)g FA(1\))1998 2180 y Fq(1)p 1998 2192
33 3 v 1998 2234 a Fs(p)p 1697 2294 365 4 v 1697 2409
a FA(\()p FG(p)15 b(')p FA(\()p FG(\020)7 b FA(\)\))2015
2337 y Fq(1)p 2015 2349 33 3 v 2015 2390 a Fs(p)2087
2315 y FG(d\020)42 b FA(+)2307 2191 y Fu(Z)2398 2217
y Fy(1)2367 2370 y Fq(1)p 2367 2382 31 3 v 2367 2423
a(2)2662 2253 y FA(const)16 b FG(d\020)p 2462 2294 717
4 v 2462 2332 a Fu(\020)2516 2433 y FA(\(1)21 b Fx(\000)f
FG(\020)7 b FA(\))2790 2406 y Fv(p)2850 2433 y FA(+)2960
2397 y Fy(1)p 2951 2412 54 4 v 2951 2464 a Fv(R)3014
2332 y Fu(\021)3069 2354 y Fy(1)p Fv(=p)3229 2315 y Fx(\024)994
2667 y(\024)83 b FA(const)36 b(+)1482 2543 y Fu(Z)1572
2570 y Fy(1)1532 2750 y(1)p Fv(=)p Fy(2)1787 2606 y FA(const)16
b FG(d\020)p 1667 2646 557 4 v 1667 2736 a FA(\(1)21
b Fx(\000)f FG(\020)7 b FA(\))20 b(+)2120 2700 y Fy(1)p
2062 2715 152 4 v 2062 2775 a Fv(R)2115 2756 y Fq(1)p
Fs(=p)2259 2667 y Fx(\024)994 2970 y(\024)98 b FA(const)1401
2814 y Fu( )1473 2970 y FA(1)20 b(+)g(log)1772 2908 y
FG(R)1842 2875 y Fy(1)p Fv(=p)1972 2908 y FA(+)g(2)p
1772 2949 337 4 v 1917 3032 a(2)2118 2814 y Fu(!)2216
2970 y Fx(\024)994 3182 y(\024)83 b FG(C)1213 3196 y
Fy(1)1267 3182 y FA(log)17 b FG(R)f(;)456 3331 y FA(pro)m(ving)29
b(\(2.16\).)43 b(Finally)-8 b(,)29 b(\(2.17\))k(follo)m(ws)c(from)h
(De\014nition)f(2.8)j(and)d(\(2.13\).)188 b Fl(\003)555
3493 y FA(Using)30 b(\(2.17\),)j(w)m(e)e(ma)m(y)f(no)m(w)h(giv)m(e)g
(the)f(follo)m(wing)f(de\014nition:)456 3621 y FH(De\014nition)35
b(2.11.)42 b FA(W)-8 b(e)32 b(de\014ne)d FG(g)1662 3635
y Fv(s)1695 3644 y Fq(0)1730 3635 y Fv(;R)1807 3621 y
FA(\()p FG(t)p FA(\))d(:)g(\()p Fx(\0001)p FG(;)15 b(H)2300
3635 y Fv(s)2333 3644 y Fq(0)2367 3635 y Fv(;R)2444 3621
y FA(\(1\)])27 b Fx(!)e Fw(R)39 b FA(b)m(y)750 3836 y
FG(g)793 3850 y Fv(s)826 3859 y Fq(0)861 3850 y Fv(;R)938
3836 y FA(\()p FG(t)p FA(\))84 b(:=)1304 3707 y Fu(\032)1484
3773 y FG(s)1527 3787 y Fv(R)2181 3773 y FA(if)90 b FG(t)25
b Fx(\024)g FG(H)2555 3787 y Fv(s)2588 3796 y Fq(0)2622
3787 y Fv(;R)2699 3773 y FA(\()p FG(s)2777 3787 y Fv(R)2835
3773 y FA(\))15 b FG(;)1372 3887 y(H)1455 3848 y Ft(\000)p
Fy(1)1448 3915 y Fv(s)1481 3924 y Fq(0)1515 3915 y Fv(;R)1592
3887 y FA(\()p FG(t)p FA(\))258 b(if)90 b FG(H)2173 3901
y Fv(s)2206 3910 y Fq(0)2239 3901 y Fv(;R)2317 3887 y
FA(\()p FG(s)2395 3901 y Fv(R)2452 3887 y FA(\))26 b
FG(<)f(t)g(<)g(H)2839 3901 y Fv(s)2872 3910 y Fq(0)2906
3901 y Fv(;R)2983 3887 y FA(\(1\))15 b FG(:)555 4056
y FA(W)-8 b(e)38 b(no)m(w)e(state)i(some)f(notation.)60
b(Giv)m(en)36 b FG(X)43 b Fx(2)35 b Fw(R)2337 4023 y
Fv(N)7 b Fy(+1)2536 4056 y FA(w)m(e)37 b(de\014ne)f FG(x)g
Fx(2)f Fw(R)3188 4023 y Fv(N)3298 4056 y FA(and)456 4164
y FG(x)508 4178 y Fv(N)7 b Fy(+1)690 4164 y Fx(2)25 b
Fw(R)39 b FA(in)29 b(suc)m(h)h(a)h(w)m(a)m(y)1431 4317
y FG(X)i FA(=)25 b(\()p FG(x;)15 b(x)1814 4331 y Fv(N)7
b Fy(+1)1972 4317 y FA(\))26 b Fx(2)f Fw(R)2178 4279
y Fv(N)2272 4317 y Fx(\002)20 b Fw(R)d FG(:)456 4466
y FA(Also,)30 b(w)m(e)h(will)c(often)k(denote)1435 4616
y FG(x)25 b FA(=)g(\()p FG(x)1695 4578 y Ft(0)1719 4616
y FG(;)15 b(x)1811 4630 y Fv(N)1878 4616 y FA(\))26 b
Fx(2)f Fw(R)2084 4578 y Fv(N)8 b Ft(\000)p Fy(1)2268
4616 y Fx(\002)20 b Fw(R)e FG(:)456 4766 y FA(W)-8 b(e)31
b(no)m(w)g(de\014ne)e(a)i(h)m(yp)s(ersurface)e(in)g Fw(R)1837
4733 y Fv(N)7 b Fy(+1)2000 4766 y FA(,)31 b(whic)m(h)e(will)f(pro)m
(vide)h(a)i(useful)e(barrier:)456 4895 y FH(De\014nition)36
b(2.12.)43 b FA(Giv)m(en)31 b FG(Y)46 b Fx(2)27 b Fw(R)1712
4862 y Fv(N)7 b Fy(+1)1907 4895 y FA(with)30 b Fx(j)p
FG(y)2185 4909 y Fv(N)7 b Fy(+1)2342 4895 y Fx(j)27 b(\024)2501
4859 y Fy(1)p 2501 4874 36 4 v 2501 4926 a(4)2578 4895
y FA(and)j FG(R)i FA(large)f(as)h(ab)s(o)m(v)m(e,)456
5003 y(let)456 5156 y(\(2.19\))47 b Fw(S)o FA(\()p FG(Y)5
b(;)15 b(R)q FA(\))35 b(:=)1199 5056 y Fu(n)1260 5156
y FG(x)25 b Fx(2)g Fw(R)1483 5119 y Fv(N)7 b Fy(+1)1661
5052 y Fu(\014)1661 5106 y(\014)1661 5161 y(\014)1707
5156 y FG(x)1759 5170 y Fv(N)g Fy(+1)1941 5156 y FA(=)25
b FG(g)2080 5170 y Fv(y)2115 5181 y Fs(N)5 b Fq(+1)2250
5170 y Fv(;R)2328 5056 y Fu(\020)2382 5156 y FG(H)2458
5170 y Fy(0)2497 5156 y FA(\()p FG(y)2577 5170 y Fv(N)i
Fy(+1)2734 5156 y FA(\))g(+)g Fx(j)p FG(x)g Fx(\000)g
FG(y)s Fx(j)g(\000)g FG(R)3244 5056 y Fu(\021o)3374 5156
y FG(:)p eop
%%Page: 15 15
15 14 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(15)555
450 y FA(In)29 b(the)g(ab)s(o)m(v)m(e)i(de\014nition,)c(w)m(e)j(will)c
(sometimes)k(refer)f(to)h FG(Y)49 b FA(as)29 b(the)h(\\cen)m(ter")h
(and)456 558 y(to)g FG(R)g FA(as)f(the)h(\\radius")e(of)i
Fw(S)n FA(.)k(F)-8 b(or)31 b(short,)f(w)m(e)h(also)g(denote)456
704 y(\(2.20\))330 b FG(g)1059 723 y Fj(S)-6 b Fy(\()p
Fv(Y)t(;R)p Fy(\))1273 704 y FA(\()p FG(x)p FA(\))26
b(:=)f FG(g)1585 718 y Fv(y)1620 729 y Fs(N)5 b Fq(+1)1755
718 y Fv(;R)1832 704 y FA(\()p FG(H)1943 718 y Fy(0)1983
704 y FA(\()p FG(y)2063 718 y Fv(N)i Fy(+1)2220 704 y
FA(\))21 b(+)e Fx(j)p FG(x)i Fx(\000)f FG(y)s Fx(j)g(\000)g
FG(R)q FA(\))15 b FG(;)456 854 y FA(so)30 b(that)h(\(2.19\))i(b)s
(ecomes)1085 1028 y Fw(S)o FA(\()p FG(Y)5 b(;)15 b(R)q
FA(\))20 b(=)1496 927 y Fu(n)1556 1028 y FG(x)26 b Fx(2)e
Fw(R)1779 991 y Fv(N)7 b Fy(+1)1958 924 y Fu(\014)1958
978 y(\014)1958 1033 y(\014)2003 1028 y FG(x)2055 1042
y Fv(N)g Fy(+1)2238 1028 y FA(=)25 b FG(g)2377 1047 y
Fj(S)-6 b Fy(\()p Fv(Y)t(;R)p Fy(\))2591 1028 y FA(\()p
FG(x)p FA(\))2713 927 y Fu(o)2789 1028 y FG(:)456 1197
y FA(Let)30 b(us)f(no)m(w)h(pro)m(v)m(e)g(that)h FG(g)1407
1216 y Fj(S)-6 b Fy(\()p Fv(Y)t(;R)p Fy(\))1651 1197
y FA(is)29 b(a)h(strict)f(sup)s(ersolution)e(in)i(the)h(viscosit)m(y)f
(sense)456 1309 y(\(for)h(the)h(de\014nition)d(of)i(viscosit)m(y)g(sup)
s(er/sub/solutions,)e(see,)j(e.g.,)h([30)q(]\):)456 1437
y FH(Prop)s(osition)48 b(2.13.)g Fo(L)-5 b(et)42 b FG(Y)63
b Fx(2)42 b Fw(R)1740 1404 y Fv(N)7 b Fy(+1)1945 1437
y Fo(with)43 b Fx(j)p FG(y)2222 1451 y Fv(N)7 b Fy(+1)2380
1437 y Fx(j)43 b(\024)2571 1402 y Fy(1)p 2571 1417 36
4 v 2571 1469 a(4)2616 1437 y Fo(.)71 b(Then,)44 b FG(g)3034
1456 y Fj(S)-5 b Fy(\()p Fv(Y)t(;R)p Fy(\))3291 1437
y Fo(is)42 b(a)456 1559 y(strict)33 b(sup)-5 b(ersolution)35
b(of)53 b FA(\(1.5\))35 b Fo(in)e(the)g(visc)-5 b(osity)34
b(sense)f(at)h(any)f FG(x)26 b Fx(2)g Fw(R)2970 1526
y Fv(N)3076 1559 y Fo(for)33 b(which)456 1667 y FG(g)499
1686 y Fj(S)-6 b Fy(\()p Fv(Y)t(;R)p Fy(\))713 1667 y
FA(\()p FG(x)p FA(\))26 b Fx(2)f FA([)p FG(s)1015 1681
y Fv(R)1072 1667 y FG(;)31 b Fx(\000)p FA(1)p FG(=)p
FA(2])21 b Fx(\\)f FA([1)p FG(=)p FA(2)p FG(;)32 b FA(1\))p
Fo(.)456 1775 y(Mor)-5 b(e)g(over,)38 b(ther)-5 b(e)38
b(ar)-5 b(e)37 b(not)h(smo)-5 b(oth)39 b(functions)e(touching)g
FG(g)2568 1794 y Fj(S)-6 b Fy(\()p Fv(Y)t(;R)p Fy(\))2819
1775 y Fo(by)36 b(b)-5 b(elow)38 b(at)f FG(x)g Fo(if)456
1897 y Fx(j)p FG(g)524 1916 y Fj(S)-6 b Fy(\()p Fv(Y)t(;R)p
Fy(\))738 1897 y FA(\()p FG(x)p FA(\))p Fx(j)26 b FA(=)1017
1861 y Fy(1)p 1017 1876 V 1017 1928 a(2)1062 1897 y Fo(.)456
2056 y(Pr)-5 b(o)g(of.)43 b FA(W)-8 b(e)31 b(use)f(the)h(notation)869
2202 y FG(s)24 b FA(=)h FG(g)1075 2216 y Fv(y)1110 2227
y Fs(N)5 b Fq(+1)1245 2216 y Fv(;R)1323 2202 y FA(\()p
FG(t)p FA(\))182 b(and)f FG(t)25 b FA(=)g FG(H)2166 2216
y Fy(0)2205 2202 y FA(\()p FG(y)2285 2216 y Fv(N)7 b
Fy(+1)2443 2202 y FA(\))20 b(+)g Fx(j)p FG(x)h Fx(\000)e
FG(y)s Fx(j)i(\000)f FG(R)456 2348 y FA(In)44 b(this)h(setting,)k(w)m
(e)d(ha)m(v)m(e)h(to)f(pro)m(v)m(e)g(the)g(desired)d(sup)s(ersolution)g
(prop)s(ert)m(y)i(for)456 2456 y FG(s)499 2470 y Fv(R)581
2456 y Fx(\024)25 b FG(s)g Fx(\024)g(\000)p FA(1)p FG(=)p
FA(2)31 b(and)f(for)g(1)p FG(=)p FA(2)d Fx(\024)e FG(s)g(<)g
FA(1.)555 2641 y(Let)38 b(us)f(\014rst)f(consider)g(the)i(case)g
FG(s)e FA(=)h FG(s)1997 2655 y Fv(R)2054 2641 y FA(,)i(that)f(is)e
FG(t)h Fx(\024)f FG(H)2673 2655 y Fv(y)2708 2666 y Fs(N)5
b Fq(+1)2842 2655 y Fv(;R)2920 2641 y FA(\()p FG(s)2998
2655 y Fv(R)3055 2641 y FA(\).)62 b(In)36 b(this)456
2749 y(case,)48 b FG(g)733 2763 y Fv(y)768 2774 y Fs(N)5
b Fq(+1)903 2763 y Fv(;R)981 2749 y FA(\()p FG(t)p FA(\))44
b(is)f(constan)m(tly)i(equal)f(to)g FG(s)2108 2763 y
Fv(R)2166 2749 y FA(,)j(th)m(us)d(an)m(y)h(parab)s(oloid)c(touc)m(hing)
456 2857 y(from)29 b(ab)s(o)m(v)m(e)i(m)m(ust)e(ha)m(v)m(e)i(v)-5
b(anishing)27 b(gradien)m(t)j(at)g(the)g(con)m(tact)i(p)s(oin)m(t)d
(and)g(negativ)m(e)456 2965 y(de\014nite)g(Hessian)h(matrix.)40
b(Th)m(us,)1385 3111 y(\001)1461 3125 y Fv(p)1500 3111
y FA(\()p FG(g)1578 3125 y Fv(y)1613 3136 y Fs(N)5 b
Fq(+1)1748 3125 y Fv(;R)1826 3111 y FA(\()p FG(t)p FA(\)\))26
b(=)f(0)h FG(<)e(h)2304 3074 y Ft(0)2304 3134 y Fy(0)2344
3111 y FA(\()p FG(s)2422 3125 y Fv(R)2480 3111 y FA(\))456
3258 y(in)29 b(the)h(viscosit)m(y)g(sense.)555 3442 y(Let)37
b(us)e(no)m(w)h(consider)e(the)i(case)h FG(H)1836 3456
y Fv(y)1871 3467 y Fs(N)5 b Fq(+1)2005 3456 y Fv(;R)2083
3442 y FA(\()p FG(s)2161 3456 y Fv(R)2218 3442 y FA(\))35
b FG(<)f(t)g(<)g(H)2641 3456 y Fv(y)2676 3467 y Fs(N)5
b Fq(+1)2811 3456 y Fv(;R)2888 3442 y FA(\()p Fx(\000)3004
3407 y Fy(1)p 3004 3422 V 3004 3474 a(2)3049 3442 y FA(\))36
b(\(that)h(is,)456 3561 y FG(s)499 3575 y Fv(R)587 3561
y FG(<)31 b(s)f(<)h Fx(\000)945 3526 y Fy(1)p 945 3541
V 945 3593 a(2)990 3561 y FA(\):)48 b(in)32 b(this)h(case,)j
FG(g)1653 3575 y Fv(y)1688 3586 y Fs(N)5 b Fq(+1)1823
3575 y Fv(;R)1934 3561 y FA(is)33 b(smo)s(oth)g(so)i(that)f(w)m(e)g
(can)g(compute)g(all)456 3669 y(the)c(deriv)-5 b(ativ)m(es)30
b(in)f(the)i(classic)e(sense.)41 b(As)30 b(a)h(matter)g(of)g(fact,)g(b)
m(y)g(Lemma)f(B.3,)621 3862 y FG(g)667 3824 y Ft(0)664
3884 y Fv(y)699 3895 y Fs(N)5 b Fq(+1)834 3884 y Fv(;R)911
3862 y FA(\()p FG(t)p FA(\))83 b(=)1251 3761 y Fu(\020)1394
3800 y FG(p)p 1315 3841 203 4 v 1315 3924 a(p)20 b Fx(\000)g
FA(1)1543 3862 y FG(h)1595 3876 y Fv(y)1630 3887 y Fs(N)5
b Fq(+1)1765 3876 y Fv(;R)1842 3862 y FA(\()p FG(g)1920
3876 y Fv(y)1955 3887 y Fs(N)g Fq(+1)2091 3876 y Fv(;R)2168
3862 y FA(\()p FG(t)p FA(\)\))2306 3761 y Fu(\021)2361
3784 y Fy(1)p Fv(=p)621 4200 y FG(g)667 4162 y Ft(00)664
4222 y Fv(y)699 4233 y Fs(N)g Fq(+1)834 4222 y Fv(;R)911
4200 y FA(\()p FG(t)p FA(\))83 b(=)1261 4004 y Fu(\020)1315
4105 y FG(p)15 b(h)1428 4119 y Fv(y)1463 4130 y Fs(N)5
b Fq(+1)1598 4119 y Fv(;R)1676 4105 y FA(\()p FG(g)1754
4119 y Fv(y)1789 4130 y Fs(N)g Fq(+1)1924 4119 y Fv(;R)2001
4105 y FA(\()p FG(t)p FA(\)\))2139 4004 y Fu(\021)2194
4027 y Fy(\(2)p Ft(\000)p Fv(p)p Fy(\))p Fv(=p)p 1261
4179 1189 4 v 1664 4266 a FA(\()p FG(p)20 b Fx(\000)g
FA(1\))1936 4240 y Fy(2)p Fv(=p)2475 4200 y FG(h)2527
4162 y Ft(0)2527 4222 y Fv(y)2562 4233 y Fs(N)5 b Fq(+1)2697
4222 y Fv(;R)2774 4200 y FA(\()p FG(g)2852 4214 y Fv(y)2887
4225 y Fs(N)g Fq(+1)3023 4214 y Fv(;R)3100 4200 y FA(\()p
FG(t)p FA(\)\))15 b FG(:)456 4398 y FA(Hence,)31 b(exploiting)e(Lemma)h
(B.2,)i(w)m(e)f(get)730 4604 y(\001)806 4618 y Fv(p)846
4604 y FA(\()p FG(g)924 4618 y Fv(y)959 4629 y Fs(N)5
b Fq(+1)1094 4618 y Fv(;R)1171 4604 y FA(\()p FG(t)p
FA(\)\))26 b(=)f FG(h)1483 4566 y Ft(0)1483 4626 y Fv(y)1518
4637 y Fs(N)5 b Fq(+1)1653 4626 y Fv(;R)1731 4604 y FA(\()p
FG(s)p FA(\))20 b(+)1976 4542 y FG(N)30 b Fx(\000)20
b FA(1)p 1965 4583 262 4 v 1965 4666 a Fx(j)p FG(x)g
Fx(\000)g FG(y)s Fx(j)2236 4503 y Fu(\020)2379 4542 y
FG(p)p 2301 4583 203 4 v 2301 4666 a(p)g Fx(\000)f FA(1)2528
4604 y FG(h)2580 4618 y Fv(y)2615 4629 y Fs(N)5 b Fq(+1)2750
4618 y Fv(;R)2827 4604 y FA(\()p FG(s)p FA(\))2940 4503
y Fu(\021)3005 4495 y Fs(p)p Fp(\000)p Fq(1)p 3005 4511
111 3 v 3044 4552 a Fs(p)3145 4604 y FG(:)456 4799 y
FA(Notice)31 b(that,)g(from)f(Lemma)h(2.10,)1089 4945
y Fx(\000)p FG(C)1225 4959 y Fy(1)1294 4945 y FA(log)17
b FG(R)84 b Fx(\024)f FG(H)1810 4959 y Fv(y)1845 4970
y Fs(N)5 b Fq(+1)1979 4959 y Fv(;R)2056 4945 y FA(\()p
FG(s)2134 4959 y Fv(R)2192 4945 y FA(\))40 b FG(<)h(t)f
FA(=)1580 5081 y(=)83 b FG(H)1810 5095 y Fy(0)1849 5081
y FA(\()p FG(y)1929 5095 y Fv(N)7 b Fy(+1)2086 5081 y
FA(\))20 b(+)g Fx(j)p FG(x)h Fx(\000)f FG(y)s Fx(j)g(\000)g
FG(R)41 b Fx(\024)1580 5216 y(\024)83 b FA(const)35 b(+)20
b Fx(j)p FG(x)h Fx(\000)f FG(y)s Fx(j)g(\000)g FG(R)c(;)p
eop
%%Page: 16 16
16 15 bop 456 251 a Fy(16)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(and)29 b(therefore,)i(if)f FG(R)h
FA(is)e(big)h(enough,)456 646 y(\(2.21\))1008 b Fx(j)p
FG(x)20 b Fx(\000)g FG(y)s Fx(j)25 b(\025)2086 585 y
FG(R)p 2086 625 70 4 v 2098 708 a FA(2)2181 646 y FG(:)456
826 y FA(Th)m(us,)k(w)m(e)i(get)675 1049 y(\001)751 1063
y Fv(p)790 1049 y FA(\()p FG(g)868 1063 y Fv(y)903 1074
y Fs(N)5 b Fq(+1)1038 1063 y Fv(;R)1116 1049 y FA(\()p
FG(t)p FA(\)\))84 b Fx(\024)e FG(h)1543 1011 y Ft(0)1543
1071 y Fy(0)1583 1049 y FA(\()p FG(s)p FA(\))21 b Fx(\000)1836
964 y Fu(b)1818 987 y FG(C)1883 1001 y Fy(0)p 1818 1028
105 4 v 1835 1111 a FG(R)1952 1049 y FA(+)2053 987 y(2\()p
FG(N)31 b Fx(\000)20 b FA(1\))p 2053 1028 356 4 v 2196
1111 a FG(R)2419 948 y Fu(\020)2561 987 y FG(p)p 2483
1028 203 4 v 2483 1111 a(p)g Fx(\000)g FA(1)2710 1049
y FG(h)2762 1063 y Fy(0)2802 1049 y FA(\()p FG(s)p FA(\))2915
948 y Fu(\021)2980 940 y Fs(p)p Fp(\000)p Fq(1)p 2980
955 111 3 v 3019 996 a Fs(p)3129 1049 y FG(<)1338 1238
y(<)82 b(h)1543 1201 y Ft(0)1543 1261 y Fy(0)1583 1238
y FA(\()p FG(s)p FA(\))456 1404 y(pro)m(vided)p 820 1331
72 4 v 19 w FG(C)892 1418 y Fy(0)952 1404 y FA(\(and)21
b(so)1275 1381 y Fu(b)1257 1404 y FG(C)1322 1418 y Fy(0)1361
1404 y FA(\))g(is)f(c)m(hosen)i(con)m(v)m(enien)m(tly)f(large.)38
b(This)19 b(pro)m(v)m(es)j(the)f(desired)456 1514 y(result)29
b(for)h FG(H)924 1528 y Fv(y)959 1539 y Fs(N)5 b Fq(+1)1093
1528 y Fv(;R)1171 1514 y FA(\()p FG(s)1249 1528 y Fv(R)1306
1514 y FA(\))26 b FG(<)f(t)g(<)g(H)1693 1528 y Fv(y)1728
1539 y Fs(N)5 b Fq(+1)1862 1528 y Fv(;R)1939 1514 y FA(\()p
Fx(\000)2055 1478 y Fy(1)p 2055 1493 36 4 v 2055 1545
a(2)2101 1514 y FA(\).)555 1679 y(If,)31 b(on)g(the)g(other)g(hand,)f
FG(H)1523 1693 y Fv(y)1558 1704 y Fs(N)5 b Fq(+1)1693
1693 y Fv(;R)1770 1679 y FA(\()1815 1643 y Fy(1)p 1815
1658 V 1815 1711 a(2)1860 1679 y FA(\))27 b FG(<)f(t)g(<)g(H)2251
1693 y Fv(y)2286 1704 y Fs(N)5 b Fq(+1)2420 1693 y Fv(;R)2497
1679 y FA(\(1\))32 b(\(that)g(is,)3004 1643 y Fy(1)p
3004 1658 V 3004 1711 a(2)3076 1679 y FG(<)26 b(s)f(<)h
FA(1\),)456 1787 y(arguing)j(in)g(the)i(same)g(w)m(a)m(y)-8
b(,)32 b(w)m(e)e(get)506 2010 y(\001)582 2024 y Fv(p)621
2010 y FA(\()p FG(g)699 2024 y Fv(y)734 2035 y Fs(N)5
b Fq(+1)870 2024 y Fv(;R)947 2010 y FA(\()p FG(t)p FA(\)\))26
b Fx(\024)f FG(h)1259 1972 y Ft(0)1259 2032 y Fy(0)1299
2010 y FA(\()p FG(s)p FA(\))20 b Fx(\000)1551 1925 y
Fu(b)1533 1948 y FG(C)1598 1962 y Fy(0)p 1533 1989 105
4 v 1551 2072 a FG(R)1668 2010 y FA(+)1769 1948 y(2\()p
FG(N)31 b Fx(\000)20 b FA(1\))p 1769 1989 356 4 v 1912
2072 a FG(R)2134 1909 y Fu(\020)2277 1948 y FG(p)p 2199
1989 203 4 v 2199 2072 a(p)g Fx(\000)f FA(1)2426 2010
y FG(h)2478 2024 y Fv(y)2513 2035 y Fs(N)5 b Fq(+1)2648
2024 y Fv(;R)2725 2010 y FA(\()p FG(s)p FA(\))2838 1909
y Fu(\021)2903 1901 y Fs(p)p Fp(\000)p Fq(1)p 2903 1916
111 3 v 2942 1957 a Fs(p)3053 2010 y FG(<)25 b(h)3201
1972 y Ft(0)3201 2032 y Fy(0)3240 2010 y FA(\()p FG(s)p
FA(\))15 b FG(;)456 2233 y FA(pro)m(vided)26 b(that)1038
2210 y Fu(b)1020 2233 y FG(C)1085 2247 y Fy(0)1152 2233
y FA(is)g(con)m(v)m(enien)m(tly)i(large.)40 b(This)25
b(completes)j(the)f(pro)s(of)g(in)f(the)h(case)456 2341
y FG(H)532 2355 y Fv(y)567 2366 y Fs(N)5 b Fq(+1)701
2355 y Fv(;R)778 2341 y FA(\()823 2305 y Fy(1)p 823 2320
36 4 v 823 2372 a(2)869 2341 y FA(\))25 b FG(<)g(t)g(<)g(H)1255
2355 y Fv(y)1290 2366 y Fs(N)5 b Fq(+1)1424 2355 y Fv(;R)1502
2341 y FA(\(1\).)555 2541 y(Up)30 b(to)h(no)m(w,)g(w)m(e)g(ha)m(v)m(e)g
(therefore)g(pro)m(v)m(ed)g(the)g(desired)d(result)i(for)829
2734 y FG(t)25 b Fx(2)973 2633 y Fu(\020)1047 2734 y
Fx(\000)20 b(1)p FG(;)31 b(H)1361 2748 y Fv(y)1396 2759
y Fs(N)5 b Fq(+1)1530 2748 y Fv(;R)1607 2734 y FA(\()p
Fx(\000)1723 2672 y FA(1)p 1723 2713 46 4 v 1723 2796
a(2)1779 2734 y(\))1814 2633 y Fu(\021)1889 2734 y Fx([)1969
2633 y Fu(\020)2024 2734 y FG(H)2100 2748 y Fv(y)2135
2759 y Fs(N)g Fq(+1)2269 2748 y Fv(;R)2346 2734 y FA(\()2391
2672 y(1)p 2391 2713 V 2391 2796 a(2)2447 2734 y(\))p
FG(;)31 b(H)2614 2748 y Fv(y)2649 2759 y Fs(N)5 b Fq(+1)2783
2748 y Fv(;R)2860 2734 y FA(\(1\))2975 2633 y Fu(\021)3046
2734 y FG(;)456 2914 y FA(that)31 b(is,)e(for)1478 3068
y FG(s)c Fx(2)1632 2967 y Fu(h)1675 3068 y FG(s)1718
3082 y Fv(R)1775 3068 y FG(;)15 b Fx(\000)1896 3007 y
FA(1)p 1896 3047 V 1896 3131 a(2)1951 2967 y Fu(\021)2026
3068 y Fx([)2107 2967 y Fu(\020)2171 3007 y FA(1)p 2171
3047 V 2171 3131 a(2)2226 3068 y FG(;)31 b FA(1)2327
2967 y Fu(\021)2397 3068 y FG(;)456 3230 y FA(T)-8 b(o)43
b(complete)h(the)g(pro)s(of)f(of)g(the)h(claim,)i(w)m(e)d(ha)m(v)m(e)i
(therefore)f(to)g(tak)m(e)h(no)m(w)e(in)m(to)456 3338
y(accoun)m(t)31 b(the)g(case)g Fx(j)p FG(s)p Fx(j)26
b FA(=)f(1)p FG(=)p FA(2.)555 3492 y(Let)33 b(us)e(no)m(w)g(deal)h
(with)e(the)i(case)h Fx(j)p FG(s)p Fx(j)28 b FA(=)f(1)p
FG(=)p FA(2.)46 b(Recalling)31 b(Lemma)h(2.6,)h(b)m(y)g(\(2.2\))r(,)456
3600 y(w)m(e)d(ha)m(v)m(e)1309 3754 y(lim)1241 3840 y
Fv(s)p Ft(!\000)1410 3812 y Fq(1)p 1409 3824 31 3 v 1409
3866 a(2)1450 3801 y Fp(\000)1502 3754 y FG(H)1585 3716
y Ft(0)1578 3776 y Fv(y)1613 3787 y Fs(N)5 b Fq(+1)1747
3776 y Fv(;R)1824 3754 y FA(\()p FG(s)p FA(\))26 b FG(<)91
b FA(lim)2059 3840 y Fv(s)p Ft(!\000)2228 3812 y Fq(1)p
2227 3824 V 2227 3866 a(2)2267 3801 y(+)2318 3754 y FG(H)2401
3716 y Ft(0)2394 3776 y Fv(y)2429 3787 y Fs(N)5 b Fq(+1)2563
3776 y Fv(;R)2641 3754 y FA(\()p FG(s)p FA(\))15 b FG(;)1281
3985 y FA(lim)1241 4070 y Fv(s)p Ft(!)1355 4043 y Fq(1)p
1355 4055 V 1355 4096 a(2)1395 4031 y Fp(\000)1447 3985
y FG(H)1530 3947 y Ft(0)1523 4007 y Fv(y)1558 4018 y
Fs(N)5 b Fq(+1)1692 4007 y Fv(;R)1770 3985 y FA(\()p
FG(s)p FA(\))25 b FG(<)64 b FA(lim)2004 4070 y Fv(s)p
Ft(!)2118 4043 y Fq(1)p 2117 4055 V 2117 4096 a(2)2158
4031 y(+)2208 3985 y FG(H)2291 3947 y Ft(0)2284 4007
y Fv(y)2319 4018 y Fs(N)5 b Fq(+1)2454 4007 y Fv(;R)2531
3985 y FA(\()p FG(s)p FA(\))15 b FG(;)456 4223 y FA(so)30
b(that)1228 4352 y(lim)991 4427 y Fv(t)p Ft(!)p Fv(H)1145
4438 y Fs(y)1177 4455 y(N)5 b Fq(+1)1311 4438 y Fs(;R)1380
4427 y Fy(\()p Ft(\000)1472 4399 y Fq(1)p 1472 4411 V
1472 4453 a(2)1513 4427 y Fy(\))1540 4408 y Fp(\000)1592
4352 y FG(g)1638 4315 y Ft(0)1635 4375 y Fv(y)1670 4386
y Fs(N)g Fq(+1)1805 4375 y Fv(;R)1883 4352 y FA(\()p
FG(t)p FA(\))25 b FG(>)262 b FA(lim)2107 4427 y Fv(t)p
Ft(!)p Fv(H)2261 4438 y Fs(y)2293 4455 y(N)5 b Fq(+1)2428
4438 y Fs(;R)2497 4427 y Fy(\()p Ft(\000)2589 4399 y
Fq(1)p 2589 4411 V 2589 4453 a(2)2629 4427 y Fy(\))2656
4408 y Fq(+)2708 4352 y FG(g)2754 4315 y Ft(0)2751 4375
y Fv(y)2786 4386 y Fs(N)g Fq(+1)2921 4375 y Fv(;R)2998
4352 y FA(\()p FG(t)p FA(\))15 b FG(;)1201 4580 y FA(lim)991
4655 y Fv(t)p Ft(!)p Fv(H)1145 4666 y Fs(y)1177 4683
y(N)5 b Fq(+1)1311 4666 y Fs(;R)1380 4655 y Fy(\()1417
4628 y Fq(1)p 1417 4640 V 1417 4681 a(2)1458 4655 y Fy(\))1485
4636 y Fp(\000)1537 4580 y FG(g)1583 4543 y Ft(0)1580
4603 y Fv(y)1615 4614 y Fs(N)g Fq(+1)1750 4603 y Fv(;R)1828
4580 y FA(\()p FG(t)p FA(\))25 b FG(>)235 b FA(lim)2052
4655 y Fv(t)p Ft(!)p Fv(H)2206 4666 y Fs(y)2238 4683
y(N)5 b Fq(+1)2373 4666 y Fs(;R)2442 4655 y Fy(\()2479
4628 y Fq(1)p 2479 4640 V 2479 4681 a(2)2520 4655 y Fy(\))2547
4636 y Fq(+)2598 4580 y FG(g)2644 4543 y Ft(0)2641 4603
y Fv(y)2676 4614 y Fs(N)g Fq(+1)2811 4603 y Fv(;R)2888
4580 y FA(\()p FG(t)p FA(\))15 b FG(:)456 4511 y FA(\(2.22\))456
4822 y(Let)42 b(no)m(w)g FG(w)i FA(b)s(e)d(a)h(smo)s(oth)g(function)e
(whose)i(graph)f(touc)m(hes)i Fw(S)n FA(\()p FG(Y)5 b(;)15
b(R)q FA(\))36 b(b)m(y)42 b(b)s(elo)m(w)456 4929 y(at)c(\()p
FG(x)661 4943 y Fy(0)701 4929 y FG(;)15 b FA(1)p FG(=)p
FA(2\))40 b(\(the)f(case)g(of)f(touc)m(hing)g(at)g(the)g
Fx(\000)p FA(1)p FG(=)p FA(2-lev)m(el)h(is)e(analogous\).)65
b(Let)38 b(us)456 5042 y(consider)29 b(the)h(radial)f(direction)g
FG(\027)1648 5056 y Fy(0)1713 5042 y FA(=)1838 5001 y
Fv(x)1878 5010 y Fq(0)1913 5001 y Ft(\000)p Fv(y)p 1819
5021 206 4 v 1819 5074 a Ft(j)p Fv(x)1879 5083 y Fq(0)1913
5074 y Ft(\000)p Fv(y)r Ft(j)2065 5042 y FA(and)h(let)g(us)g(de\014ne,)
f(for)i FG(\022)c Fx(2)e Fw(R)s FA(,)1142 5214 y Fi(g)p
FA(\()p FG(\022)s FA(\))40 b(:=)g FG(g)1519 5232 y Fj(S)-5
b Fy(\()p Fv(Y)t(;)o(R)p Fy(\))1734 5214 y FA(\()p FG(x)1821
5228 y Fy(0)1881 5214 y FA(+)19 b FG(\022)s(\027)2062
5228 y Fy(0)2101 5214 y FA(\))i Fx(\000)e FG(w)r FA(\()p
FG(x)2401 5228 y Fy(0)2462 5214 y FA(+)h FG(\022)s(\027)2644
5228 y Fy(0)2682 5214 y FA(\))15 b FG(:)p eop
%%Page: 17 17
17 16 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(17)456
450 y FA(Then,)29 b(b)m(y)h(construction,)h Fi(g)p FA(\(0\))26
b(=)f(0)h Fx(\024)f Fi(g)p FA(\()p FG(\022)s FA(\),)30
b(therefore)837 669 y(0)83 b Fx(\024)116 b FA(lim)1119
735 y Fv(\022)r Ft(!)p Fy(0)1260 716 y Fq(+)1336 608
y Fi(g)p FA(\()p FG(\022)s FA(\))20 b Fx(\000)f Fi(g)p
FA(\(0\))p 1336 648 427 4 v 1526 731 a FG(\022)1798 669
y FA(=)965 915 y(=)116 b(lim)1119 981 y Fv(\022)r Ft(!)p
Fy(0)1260 962 y Fq(+)1336 848 y FG(g)1379 862 y Fv(y)1414
873 y Fs(N)5 b Fq(+1)1549 862 y Fv(;R)1626 848 y FA(\()p
FG(t)1694 862 y Fy(0)1754 848 y FA(+)20 b FG(\022)s FA(\))f
Fx(\000)h FG(g)2079 862 y Fv(y)2114 873 y Fs(N)5 b Fq(+1)2249
862 y Fv(;R)2327 848 y FA(\()p FG(t)2395 862 y Fy(0)2434
848 y FA(\))p 1336 894 1134 4 v 1880 977 a FG(\022)2500
915 y Fx(\000)20 b FG(@)2639 929 y Fv(\027)2674 938 y
Fq(0)2712 915 y FG(w)r FA(\()p FG(x)2866 929 y Fy(0)2907
915 y FA(\))25 b(=)965 1100 y(=)320 b(lim)1119 1166 y
Fv(t)p Ft(!)p Fv(H)1273 1177 y Fs(y)1305 1194 y(N)5 b
Fq(+1)1439 1177 y Fs(;R)1509 1166 y Fy(\(1)p Fv(=)p Fy(2\))1668
1147 y Fq(+)1735 1100 y FG(g)1781 1062 y Ft(0)1778 1122
y Fv(y)1813 1133 y Fs(N)g Fq(+1)1948 1122 y Fv(;R)2025
1100 y FA(\()p FG(t)p FA(\))21 b Fx(\000)f FG(@)2288
1114 y Fv(\027)2323 1123 y Fq(0)2362 1100 y FG(w)r FA(\()p
FG(x)2516 1114 y Fy(0)2556 1100 y FA(\))15 b FG(;)456
1339 y FA(where)1005 1468 y FG(t)1038 1482 y Fy(0)1102
1468 y FA(:=)26 b FG(H)1300 1482 y Fy(0)1339 1468 y FA(\()p
FG(y)1419 1482 y Fv(N)7 b Fy(+1)1576 1468 y FA(\))20
b(+)g Fx(j)p FG(x)1799 1482 y Fy(0)1859 1468 y Fx(\000)g
FG(y)s Fx(j)g(\000)g FG(R)26 b FA(=)f FG(H)2401 1482
y Fv(y)2436 1493 y Fs(N)5 b Fq(+1)2570 1482 y Fv(;R)2648
1468 y FA(\(1)p FG(=)p FA(2\))15 b FG(:)456 1615 y FA(By)30
b(arguing)g(in)f(the)h(same)h(w)m(a)m(y)-8 b(,)32 b(w)m(e)f(also)f(get)
i(that)1052 1829 y(0)83 b Fx(\025)116 b FA(lim)1334 1895
y Fv(\022)r Ft(!)p Fy(0)1475 1876 y Fp(\000)1552 1768
y Fi(g)p FA(\()p FG(\022)s FA(\))20 b Fx(\000)g Fi(g)p
FA(\(0\))p 1552 1808 427 4 v 1743 1891 a FG(\022)2014
1829 y FA(=)1180 2014 y(=)321 b(lim)1334 2080 y Fv(t)p
Ft(!)p Fv(H)1488 2091 y Fs(y)1520 2108 y(N)5 b Fq(+1)1654
2091 y Fs(;R)1724 2080 y Fy(\(1)p Fv(=)p Fy(2\))1883
2061 y Fp(\000)1952 2014 y FG(g)1998 1976 y Ft(0)1995
2036 y Fv(y)2030 2047 y Fs(N)g Fq(+1)2164 2036 y Fv(;R)2242
2014 y FA(\()p FG(t)p FA(\))21 b Fx(\000)e FG(@)2504
2028 y Fv(\027)2539 2037 y Fq(0)2578 2014 y FG(w)r FA(\()p
FG(x)2732 2028 y Fy(0)2772 2014 y FA(\))c FG(:)456 2253
y FA(Thence,)830 2420 y(lim)592 2485 y Fv(t)p Ft(!)p
Fv(H)746 2496 y Fs(y)778 2514 y(N)5 b Fq(+1)912 2496
y Fs(;R)981 2485 y Fy(\(1)p Fv(=)p Fy(2\))1140 2467 y
Fp(\000)1209 2420 y FG(g)1255 2382 y Ft(0)1252 2442 y
Fv(y)1287 2453 y Fs(N)g Fq(+1)1422 2442 y Fv(;R)1500
2420 y FA(\()p FG(t)p FA(\))41 b Fx(\024)f FG(@)1803
2434 y Fv(\027)1838 2443 y Fq(0)1877 2420 y FG(w)r FA(\()p
FG(x)2031 2434 y Fy(0)2071 2420 y FA(\))h Fx(\024)277
b FA(lim)2258 2485 y Fv(t)p Ft(!)p Fv(H)2412 2496 y Fs(y)2444
2514 y(N)5 b Fq(+1)2578 2496 y Fs(;R)2647 2485 y Fy(\(1)p
Fv(=)p Fy(2\))2806 2467 y Fq(+)2874 2420 y FG(g)2920
2382 y Ft(0)2917 2442 y Fv(y)2952 2453 y Fs(N)g Fq(+1)3087
2442 y Fv(;R)3164 2420 y FA(\()p FG(t)p FA(\))15 b FG(;)456
2664 y FA(whic)m(h)37 b(is)g(a)h(con)m(tradiction)g(with)f(\(2.22\))s
(.)64 b(Therefore,)40 b(no)e(smo)s(oth)g(function)f(ma)m(y)456
2772 y(touc)m(h)24 b FG(g)742 2791 y Fj(S)-6 b Fy(\()p
Fv(Y)t(;R)p Fy(\))979 2772 y FA(b)m(y)23 b(b)s(elo)m(w)g(at)h
Fx(\006)p FA(1)p FG(=)p FA(2-lev)m(el)g(sets,)i(sho)m(wing,)e(in)e
(particular,)h(the)g(claimed)456 2880 y(sup)s(ersolution)k(prop)s(ert)m
(y)-8 b(.)40 b(This)29 b(completes)i(the)f(pro)s(of)g(of)g(Prop)s
(osition)f(2.13.)160 b Fl(\003)555 3065 y FA(As)31 b(a)h(consequence)g
(of)f(the)h(ab)s(o)m(v)m(e)g(result,)f(w)m(e)g(sho)m(w)g(no)m(w)h(that)
f(touc)m(hing)g(p)s(oin)m(ts)456 3173 y(b)s(et)m(w)m(een)i
Fw(S)n FA(\()p FG(Y)5 b(;)15 b(R)q FA(\))28 b(and)k(a)h(subsolution)d
(of)j(\(1.5\))h(ma)m(y)g(only)d(o)s(ccur)i(when)f Fx(j)p
FG(x)3162 3187 y Fv(N)7 b Fy(+1)3319 3173 y Fx(j)30 b
FG(<)456 3281 y FA(1)p FG(=)p FA(2)37 b(\(and)f(this)g(fact)h(will)d(b)
s(e)i(of)g(great)i(help)d(in)g(future)h(computations,)i(thanks)e(to)456
3389 y(the)30 b(explicit)f(form)h(of)g(the)h(barrier)e(in)g
Fx(j)p FG(x)1890 3403 y Fv(N)7 b Fy(+1)2047 3389 y Fx(j)26
b FG(<)f FA(1)p FG(=)p FA(2\):)456 3526 y FH(Corollary)47
b(2.14.)g Fo(L)-5 b(et)42 b FG(U)50 b Fx(2)41 b FG(C)1646
3493 y Fy(1)1685 3526 y FA(\(\012\))g Fo(b)-5 b(e)41
b(a)h(we)-5 b(ak)42 b(Sob)-5 b(olev)42 b(subsolution)g(of)f(\(1.5\),)
456 3634 y(with)f Fx(j)p FG(U)10 b Fx(j)38 b(\024)f FA(1)p
Fo(.)63 b(Assume)39 b(that)i FG(U)47 b Fx(\024)37 b FG(g)1863
3652 y Fj(S)-5 b Fy(\()p Fv(Y)t(;R)p Fy(\))2117 3634
y Fo(and)40 b(that)h FG(U)10 b FA(\()p FG(x)2652 3601
y Fv(?)2692 3634 y FA(\))38 b(=)f FG(g)2916 3652 y Fj(S)-6
b Fy(\()p Fv(Y)t(;R)p Fy(\))3130 3634 y FA(\()p FG(x)3217
3601 y Fv(?)3257 3634 y FA(\))40 b Fo(for)456 3745 y(some)33
b FG(x)740 3712 y Fv(?)812 3745 y Fo(in)f(the)h(closur)-5
b(e)34 b(of)f FA(\012)p Fo(.)41 b(Then,)33 b(either)g
FG(x)2198 3712 y Fv(?)2263 3745 y Fx(2)24 b FG(@)5 b
FA(\012)33 b Fo(or)g Fx(j)p FG(g)2685 3764 y Fj(S)-5
b Fy(\()p Fv(Y)t(;R)p Fy(\))2900 3745 y FA(\()p FG(x)2987
3712 y Fv(?)3027 3745 y FA(\))p Fx(j)25 b FG(<)g FA(1)p
FG(=)p FA(2)p Fo(.)456 3934 y(Pr)-5 b(o)g(of.)43 b FA(Let)e(us)f
(assume)h(that)h FG(x)1622 3901 y Fv(?)1704 3934 y Fx(62)h
FG(@)5 b FA(\012)41 b(W)-8 b(e)42 b(\014rst)f(pro)m(v)m(e)h(that)f
FG(g)2840 3952 y Fj(S)-5 b Fy(\()p Fv(Y)t(;R)p Fy(\))3055
3934 y FA(\()p FG(x)3142 3901 y Fv(?)3181 3934 y FA(\))44
b Fx(6)p FA(=)f(1.)456 4045 y(W)-8 b(e)36 b(argue)g(b)m(y)f(con)m
(tradiction,)i(assuming)c FG(g)2028 4064 y Fj(S)-5 b
Fy(\()p Fv(Y)t(;R)p Fy(\))2243 4045 y FA(\()p FG(x)2330
4012 y Fv(?)2369 4045 y FA(\))34 b(=)f(1.)56 b(Notice)37
b(that,)g(due)e(to)456 4153 y(De\014nition)29 b(2.5,)926
4368 y(lim)872 4434 y Fv(s)p Ft(\000)-12 b(!)p Fy(1)1054
4415 y Fp(\000)1131 4301 y FG(dH)1254 4315 y Fv(y)1289
4326 y Fs(N)5 b Fq(+1)1423 4315 y Fv(;R)p 1131 4347 370
4 v 1271 4430 a FG(ds)1511 4368 y FA(\()p FG(s)p FA(\))83
b(=)137 b(lim)1861 4434 y Fv(s)p Ft(\000)-12 b(!)p Fy(1)2043
4415 y Fp(\000)2485 4307 y FA(1)p 2120 4347 778 4 v 2120
4385 a Fu(\020)2229 4445 y Fv(p)p 2184 4465 126 4 v 2184
4518 a(p)p Ft(\000)p Fy(1)2320 4486 y FG(h)2372 4500
y Fv(y)2407 4511 y Fs(N)5 b Fq(+1)2542 4500 y Fv(;R)2619
4486 y FA(\()p FG(s)p FA(\))2732 4385 y Fu(\021)2786
4408 y Fy(1)p Fv(=p)2932 4368 y FA(=)1707 4710 y(=)1861
4609 y Fu(\020)1925 4649 y FA(\()p FG(p)20 b Fx(\000)g
FA(1\))p FG(R)p 1925 4689 343 4 v 2074 4773 a(p)2278
4609 y Fu(\021)2332 4632 y Fy(1)p Fv(=p)2457 4710 y FG(;)456
4927 y FA(th)m(us)30 b FG(g)702 4894 y Ft(0)699 4954
y Fv(y)734 4965 y Fs(N)5 b Fq(+1)869 4954 y Fv(;R)946
4927 y FA(\()p FG(t)p FA(\))26 b FG(>)f FA(0)30 b(if)g
FG(t)25 b FA(=)g FG(H)1560 4941 y Fv(y)1595 4952 y Fs(N)5
b Fq(+1)1729 4941 y Fv(;R)1806 4927 y FA(\(1\).)42 b(Then,)30
b(if)1656 5164 y FG(\027)1707 5126 y Fv(?)1787 5164 y
FA(:=)1958 5102 y FG(x)2010 5069 y Fv(?)2070 5102 y Fx(\000)20
b FG(y)p 1933 5143 301 4 v 1933 5226 a Fx(j)p FG(x)2010
5200 y Fv(?)2070 5226 y Fx(\000)g FG(y)s Fx(j)p eop
%%Page: 18 18
18 17 bop 456 251 a Fy(18)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(is)29 b(the)i(radial)d(direction,)i(this)f(yields)
456 650 y(\(2.23\))1595 580 y FG(@)5 b(g)1691 598 y Fj(S)-5
b Fy(\()p Fv(Y)t(;)o(R)p Fy(\))p 1595 629 311 4 v 1679
712 a FG(@)5 b(\027)1783 686 y Fv(?)1916 650 y FA(\()p
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b FG(:)456 839 y FA(Ho)m(w)m(ev)m(er,)32 b(since)e FG(U)35
b Fx(2)25 b FG(C)1325 806 y Fy(1)1364 839 y FA(\()p Fw(R)1459
806 y Fv(N)1532 839 y FA(\),)31 b FG(U)10 b FA(\()p FG(x)1782
806 y Fv(?)1822 839 y FA(\))25 b(=)g(1)31 b(and)f FG(U)35
b Fx(\024)25 b FA(1,)456 988 y(\(2.24\))1006 b Fx(r)p
FG(U)10 b FA(\()p FG(x)1927 950 y Fv(?)1966 988 y FA(\))26
b(=)f(0)15 b FG(:)456 1136 y FA(Similarly)-8 b(,)27 b(since)j
FG(U)35 b Fx(\024)25 b FG(g)1319 1154 y Fj(S)-6 b Fy(\()p
Fv(Y)t(;R)p Fy(\))1563 1136 y FA(and)30 b FG(U)10 b FA(\()p
FG(x)1899 1103 y Fv(?)1939 1136 y FA(\))26 b(=)e FG(g)2138
1154 y Fj(S)-5 b Fy(\()p Fv(Y)t(;R)p Fy(\))2353 1136
y FA(\()p FG(x)2440 1103 y Fv(?)2479 1136 y FA(\),)456
1353 y(\(2.25\))1470 1283 y FG(@)5 b(g)1566 1301 y Fj(S)-5
b Fy(\()p Fv(Y)t(;R)p Fy(\))p 1470 1332 V 1554 1416 a
FG(@)5 b(\027)1658 1389 y Fv(?)1791 1353 y FA(\()p FG(x)1878
1316 y Fv(?)1917 1353 y FA(\))26 b Fx(\024)2093 1292
y FG(@)5 b(U)p 2084 1332 144 4 v 2084 1416 a(@)g(\027)2188
1389 y Fv(?)2237 1353 y FA(\()p FG(x)2324 1316 y Fv(?)2364
1353 y FA(\))15 b FG(:)456 1532 y FA(Th)m(us,)30 b(a)i(con)m
(tradiction)f(easily)f(follo)m(ws)h(from)f(\(2.23\),)k(\(2.24\))g(and)c
(\(2.25\),)k(sho)m(wing)456 1640 y(that)d FG(g)696 1659
y Fj(S)-6 b Fy(\()p Fv(Y)t(;R)p Fy(\))910 1640 y FA(\()p
FG(x)997 1607 y Fv(?)1037 1640 y FA(\))25 b Fx(6)p FA(=)g(1.)555
1832 y(Also)30 b(w)m(e)h(claim)f(that)h FG(g)1377 1851
y Fj(S)-6 b Fy(\()p Fv(Y)t(;R)p Fy(\))1591 1832 y FA(\()p
FG(x)1678 1799 y Fv(?)1718 1832 y FA(\))25 b Fx(62)g
FA([)p FG(s)1932 1846 y Fv(R)1990 1832 y FG(;)30 b(;)15
b Fx(\000)p FA(1)p FG(=)p FA(2])22 b Fx([)e FA([1)p FG(=)p
FA(2)p FG(;)32 b(;)15 b FA(1\).)555 2048 y(Let)35 b(us)f(\014rst)g(sho)
m(w)g(that)h FG(g)1508 2067 y Fj(S)-6 b Fy(\()p Fv(Y)t(;R)p
Fy(\))1722 2048 y FA(\()p FG(x)1809 2015 y Fv(?)1849
2048 y FA(\))33 b Fx(6)p FA(=)e FG(s)2062 2062 y Fv(R)2120
2048 y FA(.)53 b(W)-8 b(e)35 b(argue)g(b)m(y)g(con)m(tradiction)f(and)
456 2160 y(assume)c(that)i FG(g)1011 2178 y Fj(S)-6 b
Fy(\()p Fv(Y)t(;R)p Fy(\))1225 2160 y FA(\()p FG(x)1312
2127 y Fv(?)1352 2160 y FA(\))26 b(=)g FG(s)1553 2174
y Fv(R)1611 2160 y FA(.)42 b(W)-8 b(e)32 b(recall)e(that,)i(b)m(y)f
(De\014nitions)f(2.11)i(and)f(2.12,)456 2271 y FG(g)499
2290 y Fj(S)-6 b Fy(\()p Fv(Y)t(;R)p Fy(\))743 2271 y
FA(is)30 b(constan)m(tly)h(equal)e(to)j FG(s)1669 2285
y Fv(R)1756 2271 y FA(in)d FG(B)1931 2285 y Fv(r)1969
2271 y FA(\()p FG(y)s FA(\),)i(with)1086 2423 y FG(r)42
b FA(=)f FG(r)s FA(\()p FG(R)q FA(\))f(:=)g FG(R)21 b
FA(+)f FG(H)1898 2437 y Fv(y)1933 2448 y Fs(N)5 b Fq(+1)2067
2437 y Fv(;R)2145 2423 y FA(\()p FG(s)2223 2437 y Fv(R)2280
2423 y FA(\))21 b Fx(\000)f FG(H)2503 2437 y Fy(0)2542
2423 y FA(\()p FG(y)2622 2437 y Fv(N)7 b Fy(+1)2779 2423
y FA(\))456 2582 y(and)603 2547 y Fy(5)668 2582 y FA(that)31
b FG(r)c(>)e FA(0)31 b(b)m(y)f(\(2.15\).)43 b(Then,)30
b(there)h(w)m(ould)e(b)s(e)g FG(\032)d(>)f FA(0)30 b(so)h(that)1114
2758 y(\012)1180 2772 y Fv(?)1260 2758 y FA(:=)40 b FG(B)1465
2772 y Fv(r)1503 2758 y FA(\()p FG(y)s FA(\))21 b Fx(\\)f
FG(B)1792 2772 y Fv(\032)1832 2758 y FA(\()p FG(x)1919
2720 y Fv(?)1959 2758 y FA(\))40 b Fx(\022)2145 2657
y Fu(n)2206 2758 y FG(g)2249 2776 y Fj(S)-6 b Fy(\()p
Fv(Y)t(;R)p Fy(\))2489 2758 y FA(=)24 b FG(s)2627 2772
y Fv(R)2685 2657 y Fu(o)2761 2758 y FG(:)456 2929 y FA(P)m(ossibly)k
(taking)j FG(\032)f FA(smaller,)f(w)m(e)i(ma)m(y)g(assume)f(also)g
(that)456 3100 y(\(2.26\))823 b(\012)1575 3114 y Fv(?)1654
3100 y Fx(\022)1765 2999 y Fu(n)1826 3100 y FG(U)35 b(<)25
b Fx(\000)p FA(1)20 b(+)g FG(\022)2292 3062 y Ft(\003)2331
2999 y Fu(o)456 3271 y FA(and,)30 b(since)f FG(x)932
3238 y Fv(?)997 3271 y Fx(62)c FG(@)5 b FA(\012)30 b(b)m(y)g(our)g
(assumption,)f(that)456 3419 y(\(2.27\))571 b(\012)1323
3433 y Fv(?)1392 3419 y FA(is)30 b(con)m(tained)g(in)f(the)i(in)m
(terior)e(of)i(\012.)456 3568 y(Note)g(that)g FG(U)41
b FA(cannot)31 b(coincide)e(with)g FG(g)1871 3586 y Fj(S)-6
b Fy(\()p Fv(Y)t(;R)p Fy(\))2116 3568 y FA(in)29 b(\012)2288
3582 y Fv(?)2327 3568 y FA(,)h(otherwise)1375 3775 y(0)83
b(=)g Fx(\000)1743 3651 y Fu(Z)1849 3775 y Fx(jr)p FG(U)10
b Fx(j)2047 3738 y Fv(p)p Ft(\000)p Fy(2)2177 3775 y
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3889 y Fu(Z)1763 4012 y FG(h)1815 3975 y Ft(0)1815 4035
y Fy(0)1855 4012 y FA(\()p FG(U)10 b FA(\))15 b FG(')1503
4250 y FA(=)1657 4126 y Fu(Z)1763 4250 y FG(h)1815 4212
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4264 y Fv(R)1990 4250 y FA(\))g FG(')1503 4422 y(>)83
b FA(0)15 b FG(;)456 4570 y FA(for)32 b(an)m(y)h(non-negativ)m(e)h(smo)
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4584 y Fv(?)2904 4570 y FA(.)48 b(Then,)32 b(there)456
4678 y(exists)k(\026)-51 b FG(x)26 b Fx(2)g FA(\012)938
4692 y Fv(?)1008 4678 y FA(and)k FG(\032)1232 4645 y
Ft(0)1282 4678 y FG(>)25 b FA(0)32 b(so)f(that)g FG(U)36
b(<)26 b(g)2002 4696 y Fj(S)-5 b Fy(\()p Fv(Y)t(;)o(R)p
Fy(\))2247 4678 y FA(in)30 b(the)h(in)m(terior)f(of)h
FG(B)3006 4693 y Fv(\032)3042 4674 y Fp(0)3068 4678 y
FA(\()6 b(\026)-51 b FG(x)q FA(\))26 b Fx(\022)g FA(\012)3380
4692 y Fv(?)3419 4678 y FA(,)456 4790 y(but)j(there)i(exists)36
b(\025)-51 b FG(x)25 b Fx(2)g FG(@)5 b(B)1391 4805 y
Fv(\032)1427 4786 y Fp(0)1484 4790 y FA(for)30 b(whic)m(h)f
FG(U)10 b FA(\()c(\025)-51 b FG(x)q FA(\))25 b(=)g FG(g)2242
4808 y Fj(S)-5 b Fy(\()p Fv(Y)t(;)o(R)p Fy(\))2457 4790
y FA(\()6 b(\025)-51 b FG(x)p FA(\).)41 b(Setting)1692
4942 y FG(U)1764 4904 y Fv(?)1829 4942 y FA(=)25 b FG(s)1968
4956 y Fv(R)2045 4942 y Fx(\000)20 b FG(U)p 456 5006
499 4 v 555 5076 a Fy(5)610 5105 y FF(Pro)r(ceeding)h(as)f(done)f(here)
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(ell)h(inside)f(the)f(ball)i(of)f(radius)g FD(R)18 b
FF(+)f FD(R)2045 5164 y Fn(1)p 2045 5172 29 3 v 2045
5205 a(3)2088 5216 y FD(=)p FF(2)26 b(around)f FD(y)s
FF(.)p eop
%%Page: 19 19
19 18 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(19)456
450 y FA(it)30 b(follo)m(ws)1530 570 y FG(U)1602 532
y Fv(?)1667 570 y FA(=)25 b FG(g)1806 589 y Fj(S)-6 b
Fy(\()p Fv(Y)t(;R)p Fy(\))2040 570 y Fx(\000)20 b FG(U)35
b(>)25 b FA(0)456 713 y(in)k FG(B)631 728 y Fv(\032)667
709 y Fp(0)693 713 y FA(\()6 b(\026)-51 b FG(x)q FA(\))30
b(and)g FG(U)1095 680 y Fv(?)1134 713 y FA(\()6 b(\025)-51
b FG(x)q FA(\))25 b(=)g(0.)41 b(Moreo)m(v)m(er,)456 873
y(\(2.28\))763 b Fx(\000)p FA(\001)1596 887 y Fv(p)1635
873 y FG(U)1707 836 y Fv(?)1772 873 y FA(=)25 b(\001)1944
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836 y Ft(0)2228 896 y Fy(0)2268 873 y FA(\()p FG(U)10
b FA(\))15 b FG(:)456 1031 y FA(Hence,)44 b FG(h)816
998 y Ft(0)816 1055 y Fy(0)856 1031 y FA(\()p FG(U)10
b FA(\))44 b FG(>)e FA(0)g(in)d(the)i(ligh)m(t)g(of)g(\(1.2\))i(and)d
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1311 y Fv(\027)1536 1297 y FG(U)10 b FA(\()c(\025)-51
b FG(x)p FA(\))26 b(=)f Fx(\000)p FG(@)1971 1311 y Fv(\027)2014
1297 y FG(U)2086 1259 y Fv(?)2126 1297 y FA(\()6 b(\025)-51
b FG(x)p FA(\))25 b FG(>)g FA(0)15 b FG(;)456 1454 y
FA(where)1709 1599 y FG(\027)31 b FA(:=)1947 1538 y(\025)-51
b FG(x)20 b Fx(\000)26 b FA(\026)-51 b FG(x)p 1916 1578
266 4 v 1916 1662 a Fx(j)6 b FA(\025)-51 b FG(x)20 b
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b(the)i(outer)f(normal)g(of)g FG(B)1422 1807 y Fv(\032)1458
1788 y Fp(0)1485 1792 y FA(\()6 b(\026)-51 b FG(x)p FA(\))31
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b FA(touc)m(hes)42 b FG(g)1946 2026 y Fj(S)-5 b Fy(\()p
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b FG(x)42 b FA(and)f Fx(r)p FG(g)2725 2026 y Fj(S)-6
b Fy(\()p Fv(Y)t(;R)p Fy(\))2980 2008 y FA(v)h(anishes)40
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Fv(Y)t(;R)p Fy(\))784 2119 y FA(=)25 b FG(s)923 2133
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2280 y FG(@)1455 2294 y Fv(\027)1499 2280 y FG(U)10 b
FA(\()c(\025)-51 b FG(x)p FA(\))25 b(=)g FG(@)1862 2294
y Fv(\027)1906 2280 y FG(g)1949 2299 y Fj(S)-6 b Fy(\()p
Fv(Y)t(;R)p Fy(\))2163 2280 y FA(\()6 b(\025)-51 b FG(x)p
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Fu(\020)2041 2907 y FG(s)2084 2921 y Fv(R)2157 2907 y
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2294 2969 a(2)2349 2806 y Fu(\021)2418 2907 y FG(:)456
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y FA(is)29 b(smo)s(oth)h(with)f(non-v)-5 b(anishing)28
b(gradien)m(t)i(in)1275 3316 y(\012)1341 3330 y Fy(0)1405
3316 y FA(:=)1526 3187 y Fu(\032)1595 3316 y FG(g)1638
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3316 y FG(s)2183 3330 y Fv(R)2255 3316 y FG(;)31 b Fx(\000)2392
3254 y FA(1)p 2392 3295 V 2392 3378 a(2)2447 3215 y Fu(\021)2501
3187 y(\033)2600 3316 y FG(;)456 3528 y FA(thence)h(it)e(is)h(a)h
(classical)e(strict)h(sup)s(ersolution)e(of)i(\(1.5\))j(in)c(\012)2652
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3638 y FG(U)40 b FA(cannot)31 b(coincide)e(with)g FG(g)1454
3656 y Fj(S)-5 b Fy(\()p Fv(Y)t(;R)p Fy(\))1699 3638
y FA(in)29 b Fx(f)p FG(g)1893 3656 y Fj(S)-5 b Fy(\()p
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y Fv(?)2235 3638 y FA(\))25 b Fx(2)g FA(\()p FG(s)2459
3652 y Fv(R)2532 3638 y FG(;)30 b Fx(\000)2668 3602 y
Fy(1)p 2668 3617 36 4 v 2668 3669 a(2)2713 3638 y FA(\))p
Fx(g)p FA(,)i(otherwise,)851 3731 y Fu(Z)957 3855 y FG(h)1009
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4092 y(=)82 b Fx(\000)1774 3968 y Fu(Z)1880 4092 y Fx(jr)p
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y Ft(0)1846 4352 y Fy(0)1886 4329 y FA(\()p FG(U)10 b
FA(\))15 b FG(')1535 4567 y FA(=)1688 4443 y Fu(Z)1794
4567 y FG(h)1846 4529 y Ft(0)1846 4589 y Fy(0)1886 4567
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Fy(0)456 4982 y FA(and)25 b Fx(r)p FG(g)747 5000 y Fj(S)-6
b Fy(\()p Fv(Y)t(;R)p Fy(\))987 4982 y FA(nev)m(er)26
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(and)e(get)i(that)456 5094 y FG(U)44 b(<)34 b(g)710 5112
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b(no)d(touc)m(hing)h(p)s(oin)m(t)f(ma)m(y)h(o)s(ccur)g(in)e
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5216 y FA(\()p FG(s)534 5230 y Fv(R)606 5216 y FG(;)c
Fx(\000)743 5180 y Fy(1)p 743 5195 V 743 5247 a(2)788
5216 y FA(\))p Fx(g)p FA(,)g(pro)m(ving)f(\(2.30\).)p
eop
%%Page: 20 20
20 19 bop 456 251 a Fy(20)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
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b Fy(\()p Fv(Y)t(;R)p Fy(\))1047 450 y FA(\()p FG(x)1134
417 y Fv(?)1174 450 y FA(\))31 b Fx(6)p FA(=)f Fx(\000)1422
414 y Fy(1)p 1422 429 36 4 v 1422 481 a(2)1501 450 y
FA(b)m(y)k(Prop)s(osition)d(2.13.)52 b(The)34 b(fact)g(that)h
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450 y FA(\()p FG(x)3278 417 y Fv(?)3318 450 y FA(\))31
b Fx(62)456 562 y FA([1)p FG(=)p FA(2)15 b FG(;)32 b
FA(1\))f(follo)m(ws)f(with)f(similar)e(argumen)m(ts.)1334
b Fl(\003)555 784 y FA(W)-8 b(e)31 b(will)c(no)m(w)i(de\014ne)g
(another)h(h)m(yp)s(ersurface)e(in)g Fw(R)2360 751 y
Fv(N)7 b Fy(+1)2523 784 y FA(,)30 b(whic)m(h)f(will)d(b)s(e)j(denoted)
456 899 y(b)m(y)590 876 y Fu(e)590 899 y Fw(S)-7 b FA(\()p
FG(Y)5 b(;)15 b(R)q FA(\))40 b(and)e(w)m(e)h(in)m(v)m(estigate)i(its)d
(relation)g(with)g Fw(S)n FA(\()p FG(Y)5 b(;)15 b(R)q
FA(\).)61 b(While)38 b Fw(S)o FA(\()p FG(Y)5 b(;)15 b(R)q
FA(\))33 b(is)456 1020 y(con)m(tin)m(uous)23 b(but)f(not)i(smo)s(oth,)
1559 996 y Fu(e)1559 1020 y Fw(S)-8 b FA(\()p FG(Y)5
b(;)15 b(R)q FA(\))24 b(will)d(b)s(e)i(smo)s(oth,)h(and)f(th)m(us)g(it)
g(will)e(b)s(e)h(easier)456 1128 y(to)31 b(deal)g(with)e(during)g(the)i
(calculations.)41 b(Also,)31 b(the)g(t)m(w)m(o)h(surfaces)f(will)d
(coincide)i(in)456 1243 y Fx(fj)p FG(x)578 1257 y Fv(N)7
b Fy(+1)736 1243 y Fx(j)25 b(\024)892 1207 y Fy(1)p 892
1222 V 892 1274 a(2)937 1243 y Fx(g)f FA(and)e Fw(S)o
FA(\()p FG(Y)5 b(;)15 b(R)q FA(\))j(will)i(alw)m(a)m(ys)k(sta)m(y)g(b)s
(elo)m(w)2371 1219 y Fu(e)2371 1243 y Fw(S)-7 b FA(\()p
FG(Y)5 b(;)15 b(R)q FA(\).)39 b(Therefore,)3155 1219
y Fu(e)3155 1243 y Fw(S)-7 b FA(\()p FG(Y)5 b(;)15 b(R)q
FA(\))456 1351 y(will)30 b(pro)m(vide,)k(in)e(some)i(sense,)h(a)e
(sharp)g(barrier)f(for)h Fw(S)n FA(\()p FG(Y)5 b(;)15
b(R)q FA(\))28 b(whic)m(h)33 b(will)d(b)s(e)j(more)456
1459 y(explicit)k(to)j(treat.)69 b(Let)40 b(us)f(no)m(w)g(approac)m(h)h
(the)f(de\014nition)e(of)j(the)f(h)m(yp)s(ersurface)456
1550 y Fu(e)456 1574 y Fw(S)-8 b FA(\()p FG(Y)5 b(;)15
b(R)q FA(\).)555 1853 y(Setting)456 2107 y(\(2.31\))1306
2084 y Fu(e)1285 2107 y FG(H)1361 2121 y Fv(s)1394 2130
y Fq(0)1427 2121 y Fv(;R)1505 2107 y FA(\()p FG(s)p FA(\))41
b(:=)f FG(H)1871 2121 y Fy(0)1910 2107 y FA(\()p FG(s)p
FA(\))20 b Fx(\000)p 2146 1973 72 4 v 2146 2046 a FG(C)2218
2060 y Fy(0)p 2144 2086 116 4 v 2144 2169 a FA(2)p FG(R)2269
2107 y FA(\()p FG(s)h Fx(\000)e FG(s)2501 2121 y Fy(0)2541
2107 y FA(\))2576 2070 y Fy(2)456 2332 y FA(then,)30
b(b)m(y)g(\(2.12\),)j(w)m(e)e(ha)m(v)m(e)456 2591 y(\(2.32\))1031
2529 y FG(d)p 1010 2570 90 4 v 1010 2653 a(ds)1125 2490
y Fu(\020)1200 2568 y(e)1179 2591 y FG(H)1255 2605 y
Fv(s)1288 2614 y Fq(0)1322 2605 y Fv(;R)1399 2591 y FA(\()p
FG(s)p FA(\))1512 2490 y Fu(\021)1592 2591 y FA(=)1914
2529 y(1)p 1698 2570 478 4 v 1698 2685 a(\()1788 2644
y Fv(p)p 1743 2664 126 4 v 1743 2717 a(p)p Ft(\000)p
Fy(1)1879 2685 y FG(h)1931 2699 y Fy(0)1971 2685 y FA(\()p
FG(s)p FA(\)\))2130 2613 y Fq(1)p 2129 2625 33 3 v 2129
2666 a Fs(p)2206 2591 y Fx(\000)p 2306 2456 72 4 v 2306
2529 a FG(C)2378 2543 y Fy(0)p 2306 2570 111 4 v 2327
2653 a FG(R)2427 2591 y FA(\()p FG(s)20 b Fx(\000)g FG(s)2659
2605 y Fy(0)2698 2591 y FA(\))26 b FG(>)f FA(0)456 2902
y(for)33 b Fx(j)p FG(s)p Fx(j)e(\024)834 2867 y Fy(3)p
834 2882 36 4 v 834 2934 a(4)879 2902 y FA(,)k(pro)m(vided)d
FG(R)j FA(is)e(big)g(enough.)50 b(Therefore,)35 b(for)f
Fx(j)p FG(s)p Fx(j)d(\024)2848 2867 y Fy(3)p 2848 2882
V 2848 2934 a(4)2927 2902 y FA(w)m(e)k(ha)m(v)m(e)g(that)477
3008 y Fu(e)456 3031 y FG(H)532 3045 y Fv(s)565 3054
y Fq(0)598 3045 y Fv(;R)706 3031 y FA(is)29 b(strictly)h(increasing)f
(and)h(w)m(e)g(can)h(giv)m(e)g(the)g(follo)m(wing)456
3182 y FH(De\014nition)k(2.15.)42 b FA(W)-8 b(e)32 b(de\014ne)1015
3408 y FG(\032)1062 3422 y Fv(s)1095 3431 y Fq(0)1129
3422 y Fv(;R)1206 3408 y FA(\()p FG(t)p FA(\))26 b(:)1386
3279 y Fu(\024)1455 3385 y(e)1434 3408 y FG(H)1510 3422
y Fv(s)1543 3431 y Fq(0)1576 3422 y Fv(;R)1654 3408 y
FA(\()p Fx(\000)1770 3346 y FA(3)p 1770 3387 46 4 v 1770
3470 a(4)1825 3408 y(\))15 b FG(;)1952 3385 y Fu(e)1931
3408 y FG(H)2007 3422 y Fv(s)2040 3431 y Fq(0)2074 3422
y Fv(;R)2151 3408 y FA(\()2196 3346 y(3)p 2196 3387 V
2196 3470 a(4)2252 3408 y(\))2287 3279 y Fu(\025)2361
3408 y Fx(\000)-16 b(!)2532 3279 y Fu(\024)2580 3408
y Fx(\000)2661 3346 y FA(3)p 2661 3387 V 2661 3470 a(4)2716
3408 y FG(;)2782 3346 y FA(3)p 2782 3387 V 2782 3470
a(4)2837 3279 y Fu(\025)456 3651 y FA(b)m(y)456 3852
y(\(2.33\))861 b FG(\032)1594 3866 y Fv(s)1627 3875 y
Fq(0)1661 3866 y Fv(;R)1739 3852 y FA(\()p FG(t)p FA(\))26
b(:=)2010 3829 y Fu(e)1989 3852 y FG(H)2072 3813 y Ft(\000)p
Fy(1)2065 3880 y Fv(s)2098 3889 y Fq(0)2132 3880 y Fv(;R)2209
3852 y FA(\()p FG(t)p FA(\))15 b FG(:)555 4066 y FA(Moreo)m(v)m(er,)42
b(giv)m(en)c FG(Y)57 b Fx(2)37 b Fw(R)1502 4033 y Fv(N)7
b Fy(+1)1703 4066 y FA(with)37 b Fx(j)p FG(y)1988 4080
y Fv(N)7 b Fy(+1)2145 4066 y Fx(j)38 b(\024)2326 4030
y Fy(1)p 2326 4045 36 4 v 2326 4097 a(4)2409 4066 y FA(and)f
FG(R)h FA(large)g(as)g(ab)s(o)m(v)m(e,)j(w)m(e)456 4174
y(de\014ne)456 4368 y(\(2.34\))749 4344 y Fu(e)749 4368
y Fw(S)-8 b FA(\()p FG(Y)5 b(;)15 b(R)q FA(\))41 b(:=)g
Fx(f)p FG(x)25 b Fx(2)g Fw(R)1483 4330 y Fv(N)7 b Fy(+1)1661
4368 y Fx(j)15 b FG(x)1753 4382 y Fv(N)7 b Fy(+1)1936
4368 y FA(=)25 b FG(\032)2079 4382 y Fv(y)2114 4393 y
Fs(N)5 b Fq(+1)2249 4382 y Fv(;R)2326 4368 y FA(\()p
FG(H)2437 4382 y Fy(0)2477 4368 y FA(\()p FG(y)2557 4382
y Fv(N)i Fy(+1)2714 4368 y FA(\))21 b(+)f Fx(j)p FG(x)g
Fx(\000)g FG(y)s Fx(j)g(\000)g FG(R)q FA(\))p Fx(g)555
4562 y FA(As)31 b(done)f(on)g(page)h(15,)h(it)e(is)f(con)m(v)m(enien)m
(t)i(to)h(in)m(tro)s(duce)d(the)h(notation)456 4755 y(\(2.35\))328
b FG(g)1056 4772 y Fk(e)1057 4789 y Fj(S)-12 b Fy(\()p
Fv(Y)t(;R)p Fy(\))1271 4755 y FA(\()p FG(x)p FA(\))26
b(:=)f FG(\032)1587 4769 y Fv(y)1622 4780 y Fs(N)5 b
Fq(+1)1757 4769 y Fv(;R)1834 4755 y FA(\()p FG(H)1945
4769 y Fy(0)1984 4755 y FA(\()p FG(y)2064 4769 y Fv(N)i
Fy(+1)2222 4755 y FA(\))20 b(+)g Fx(j)p FG(x)h Fx(\000)f
FG(y)s Fx(j)g(\000)g FG(R)q FA(\))15 b FG(;)456 4968
y FA(so)30 b(that)h(\(2.34\))i(b)s(ecomes)1085 5166 y
Fu(e)1085 5190 y Fw(S)-7 b FA(\()p FG(Y)5 b(;)15 b(R)q
FA(\))26 b(=)1496 5089 y Fu(n)1556 5190 y FG(x)g Fx(2)e
Fw(R)1779 5152 y Fv(N)7 b Fy(+1)1958 5085 y Fu(\014)1958
5140 y(\014)1958 5194 y(\014)2003 5190 y FG(x)2055 5204
y Fv(N)g Fy(+1)2238 5190 y FA(=)25 b FG(g)2376 5206 y
Fk(e)2377 5223 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))2591
5190 y FA(\()p FG(x)p FA(\))2713 5089 y Fu(o)2789 5190
y FG(:)p eop
%%Page: 21 21
21 20 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(21)456
450 y FA(Notice)29 b(also)g(that,)h(b)m(y)e(construction,)h(for)f(an)m
(y)h FG(x)g FA(for)f(whic)m(h)f FG(g)2635 467 y Fk(e)2636
484 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))2879 450 y
FA(is)28 b(de\014ned)3253 415 y Fy(6)3287 450 y FA(,)h(w)m(e)456
572 y(ha)m(v)m(e)i(that)456 751 y(\(2.37\))935 b Fx(j)p
FG(g)1688 768 y Fk(e)1689 785 y Fj(S)-11 b Fy(\()p Fv(Y)t(;R)p
Fy(\))1904 751 y FA(\()p FG(x)p FA(\))p Fx(j)26 b(\024)2183
690 y FA(3)p 2183 731 46 4 v 2183 814 a(4)2253 751 y
FG(:)456 936 y FA(Moreo)m(v)m(er,)32 b(a)f(straigh)m(tforw)m(ard)f
(computation)h(giv)m(es)f(that)456 1155 y(\(2.38\))976
1132 y Fu(e)954 1155 y FG(H)1030 1169 y Fv(s)1063 1178
y Fq(0)1097 1169 y Fv(;R)1170 1178 y Fq(1)1209 1155 y
FA(\()p FG(s)p FA(\))20 b Fx(\000)1454 1132 y Fu(e)1433
1155 y FG(H)1509 1169 y Fv(s)1542 1178 y Fq(0)1576 1169
y Fv(;R)1649 1178 y Fq(2)1687 1155 y FA(\()p FG(s)p FA(\))26
b(=)p 1932 1021 72 4 v 1932 1094 a FG(C)2003 1108 y Fy(0)p
1932 1134 111 4 v 1964 1218 a FA(2)2068 1155 y(\()p FG(s)20
b Fx(\000)g FG(s)2300 1169 y Fy(0)2339 1155 y FA(\))2374
1118 y Fy(2)2429 1054 y Fu(\020)2525 1094 y FA(1)p 2493
1134 109 4 v 2493 1218 a FG(R)2562 1232 y Fy(2)2632 1155
y Fx(\000)2764 1094 y FA(1)p 2733 1134 V 2733 1218 a
FG(R)2802 1232 y Fy(1)2851 1054 y Fu(\021)2920 1155 y
FG(;)456 1359 y FA(for)31 b(an)m(y)h Fx(j)p FG(s)p Fx(j)c(\024)g
FA(3)p FG(=)p FA(4.)46 b(Also,)33 b(if)d FG(x)i FA(is)f(in)g(the)h
(domain)e(of)i FG(g)2426 1376 y Fk(e)2427 1393 y Fj(S)-11
b Fy(\()p Fv(Y)t(;R)p Fy(\))2642 1359 y FA(,)32 b(then)g
FG(x)g FA(and)f FG(y)j FA(m)m(ust)456 1481 y(b)s(e)29
b(suitably)g(far)h(from)g(eac)m(h)h(other,)g(as)g(next)f(result)g(p)s
(oin)m(ts)f(out:)456 1614 y FH(Lemma)36 b(2.16.)44 b
Fo(L)-5 b(et)36 b FG(x)31 b Fx(2)g Fw(R)1482 1581 y Fv(N)1591
1614 y Fo(b)-5 b(e)36 b(so)g(that)h FG(g)2056 1631 y
Fk(e)2057 1648 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))2307
1614 y Fo(is)35 b(wel)5 b(l)36 b(de\014ne)-5 b(d)37 b(\(that)g(is,)g
(let)e FG(x)456 1741 y Fo(b)-5 b(e)32 b(such)h(that)h(\(2.36\))g
(holds\).)44 b(Then,)1597 1900 y Fx(j)p FG(x)21 b Fx(\000)f
FG(y)s Fx(j)40 b(\025)g FG(R)21 b Fx(\000)f FG(C)i(;)456
2059 y Fo(for)33 b(a)g(suitable)g(universal)g(c)-5 b(onstant)34
b FG(C)e(>)25 b FA(0)p Fo(.)456 2235 y(Pr)-5 b(o)g(of.)43
b FA(F)-8 b(rom)30 b(\(2.37\),)j(w)m(e)e(get)h(that)1365
2394 y FG(s)25 b FA(:=)g FG(g)1596 2410 y Fk(e)1597 2427
y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))1811 2394 y FA(\()p
FG(x)p FA(\))26 b Fx(2)f FA([)p Fx(\000)p FA(3)p FG(=)p
FA(4)p FG(;)32 b FA(3)p FG(=)p FA(4])15 b FG(:)456 2572
y FA(So,)30 b(b)m(y)g(means)h(of)f(\(2.31\))j(and)d(\(2.33\),)830
2796 y Fx(j)p FG(x)21 b Fx(\000)f FG(y)s Fx(j)83 b FA(=)f
FG(R)21 b Fx(\000)f FG(H)1585 2810 y Fy(0)1624 2796 y
FA(\()p FG(y)1704 2810 y Fv(N)7 b Fy(+1)1862 2796 y FA(\))20
b(+)g FG(H)2084 2810 y Fy(0)2123 2796 y FA(\()p FG(s)p
FA(\))h Fx(\000)p 2360 2661 72 4 v 2360 2734 a FG(C)2431
2748 y Fy(0)p 2357 2775 116 4 v 2357 2858 a FA(2)p FG(R)2483
2796 y FA(\()p FG(s)f Fx(\000)g FG(y)2717 2810 y Fv(N)7
b Fy(+1)2874 2796 y FA(\))2909 2758 y Fy(2)2974 2796
y Fx(\025)1175 2955 y(\025)82 b FG(R)21 b Fx(\000)35
b FA(const)16 b FG(:)3374 3114 y Fl(\003)555 3290 y FA(Let)26
b(us)e(no)m(w)h(sho)m(w)g(that)h(the)f(surface)g Fw(S)n
FA(\()p FG(Y)5 b(;)15 b(R)q FA(\))20 b(coincides)k(with)2767
3266 y Fu(e)2767 3290 y Fw(S)-8 b FA(\()p FG(Y)5 b(;)15
b(R)q FA(\))26 b(in)e(the)h(set)456 3410 y Fx(j)p FG(x)533
3424 y Fv(N)7 b Fy(+1)690 3410 y Fx(j)31 b(\024)857 3374
y Fy(1)p 857 3389 36 4 v 857 3441 a(2)935 3410 y FA(and)i(that)h
Fw(S)n FA(\()p FG(Y)5 b(;)15 b(R)q FA(\))28 b(sta)m(ys)35
b(b)s(elo)m(w)2129 3387 y Fu(e)2129 3410 y Fw(S)-7 b
FA(\()p FG(Y)5 b(;)15 b(R)q FA(\))34 b(at)g(all)e(other)i(p)s(oin)m(ts)
e(where)456 3515 y Fu(e)456 3539 y Fw(S)-8 b FA(\()p
FG(Y)5 b(;)15 b(R)q FA(\))31 b(is)e(de\014ned:)456 3672
y FH(Lemma)j(2.17.)43 b Fo(If)32 b Fx(j)p FG(s)p Fx(j)26
b(\024)f FA(1)p FG(=)p FA(2)p Fo(,)33 b(then)456 3843
y FA(\(2.39\))850 b FG(H)1612 3857 y Fv(s)1645 3866 y
Fq(0)1679 3857 y Fv(;R)1756 3843 y FA(\()p FG(s)p FA(\))25
b(=)2012 3820 y Fu(e)1990 3843 y FG(H)2066 3857 y Fv(s)2099
3866 y Fq(0)2133 3857 y Fv(;R)2211 3843 y FA(\()p FG(s)p
FA(\))15 b FG(:)456 4008 y Fo(If)561 3972 y Fy(1)p 561
3987 V 561 4039 a(2)632 4008 y FG(<)24 b Fx(j)p FG(s)p
Fx(j)i FG(<)952 3972 y Fy(3)p 952 3987 V 952 4039 a(4)997
4008 y Fo(,)32 b(then)1536 4188 y FG(H)1612 4202 y Fv(s)1645
4211 y Fq(0)1679 4202 y Fv(;R)1756 4188 y FA(\()p FG(s)p
FA(\))25 b FG(>)2012 4165 y Fu(e)1990 4188 y FG(H)2066
4202 y Fv(s)2099 4211 y Fq(0)2133 4202 y Fv(;R)2211 4188
y FA(\()p FG(s)p FA(\))15 b FG(:)456 4185 y FA(\(2.40\))456
4362 y Fo(A)n(lso,)31 b(let)g FG(x)25 b Fx(2)g Fw(R)1036
4329 y Fv(N)1140 4362 y Fo(b)-5 b(e)31 b(so)g(that)h
FG(g)1590 4379 y Fk(e)1591 4396 y Fj(S)-11 b Fy(\()p
Fv(Y)t(;R)p Fy(\))1836 4362 y Fo(is)31 b(wel)5 b(l)32
b(de\014ne)-5 b(d)32 b(\(that)g(is,)f(let)g FG(x)g Fo(b)-5
b(e)31 b(such)g(that)456 4489 y(\(2.36\))j(holds\).)43
b(Then,)456 4648 y FA(\(2.41\))803 b FG(g)1531 4664 y
Fk(e)1532 4682 y Fj(S)-11 b Fy(\()p Fv(Y)t(;R)p Fy(\))1747
4648 y FA(\()p FG(x)p FA(\))25 b Fx(\025)g FG(g)2033
4666 y Fj(S)-5 b Fy(\()p Fv(Y)t(;R)p Fy(\))2248 4648
y FA(\()p FG(x)p FA(\))15 b FG(:)p 456 4755 499 4 v 555
4826 a Fy(6)616 4855 y FF(I.e.,)27 b(for)f(an)n(y)f FD(x)g
FF(so)i(that)456 5029 y(\(2.36\))417 b FD(H)1132 5037
y Fq(0)1167 5029 y FF(\()p FD(y)1235 5037 y Fs(N)5 b
Fq(+1)1369 5029 y FF(\))17 b(+)g Fr(j)p FD(x)g Fr(\000)f
FD(y)s Fr(j)h(\000)g FD(R)22 b Fr(2)1960 4924 y Fg(\024)2018
5010 y(e)2000 5029 y FD(H)2064 5037 y Fs(s)2093 5047
y Fn(0)2125 5037 y Fs(;R)2194 5029 y FF(\()p Fr(\000)2294
4981 y FF(3)p 2294 5012 39 4 v 2294 5080 a(4)2342 5029
y(\))13 b FD(;)2450 5010 y Fg(e)2432 5029 y FD(H)2496
5037 y Fs(s)2525 5047 y Fn(0)2557 5037 y Fs(;R)2626 5029
y FF(\()2666 4981 y(3)p 2666 5012 V 2666 5080 a(4)2714
5029 y(\))2744 4924 y Fg(\025)2810 5029 y FD(:)p eop
%%Page: 22 22
22 21 bop 456 251 a Fy(22)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y Fo(F)h(urthermor)i(e,)35 b(if)d Fx(j)p
FG(g)1164 467 y Fk(e)1165 484 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p
Fy(\))1379 450 y FA(\()p FG(x)p FA(\))p Fx(j)27 b(\024)1659
414 y Fy(1)p 1659 429 36 4 v 1659 481 a(2)1704 450 y
Fo(,)32 b(then)456 624 y FA(\(2.42\))803 b FG(g)1531
641 y Fk(e)1532 658 y Fj(S)-11 b Fy(\()p Fv(Y)t(;R)p
Fy(\))1747 624 y FA(\()p FG(x)p FA(\))25 b(=)g FG(g)2033
642 y Fj(S)-5 b Fy(\()p Fv(Y)t(;R)p Fy(\))2248 624 y
FA(\()p FG(x)p FA(\))15 b FG(:)456 813 y Fo(Pr)-5 b(o)g(of.)43
b FA(W)-8 b(e)34 b(use)g(the)g(notation)g FG(s)c FA(:=)h
FG(g)1826 829 y Fk(e)1827 846 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p
Fy(\))2041 813 y FA(.)51 b(Let)34 b Fx(j)p FG(s)p Fx(j)d(\024)f
FA(1)p FG(=)p FA(2.)52 b(By)34 b(De\014nitions)e(2.5)456
940 y(and)d(2.8,)j(recalling)d(also)h(\(2.2\),)j(w)m(e)e(ha)m(v)m(e)g
(that)936 1159 y FG(H)1012 1173 y Fv(s)1045 1182 y Fq(0)1078
1173 y Fv(;R)1156 1159 y FA(\()p FG(s)p FA(\))25 b(=)g
FG(H)1466 1173 y Fy(0)1505 1159 y FA(\()p FG(s)1583 1173
y Fy(0)1623 1159 y FA(\))20 b(+)1769 1036 y Fu(Z)1860
1062 y Fv(s)1820 1242 y(s)1853 1251 y Fq(0)2114 1098
y FG(d\020)p 1922 1138 478 4 v 1922 1254 a FA(\()2012
1213 y Fv(p)p 1967 1233 126 4 v 1967 1285 a(p)p Ft(\000)p
Fy(1)2103 1254 y FG(h)2155 1268 y Fy(0)2195 1254 y FA(\()p
FG(s)p FA(\)\))2354 1181 y Fq(1)p 2353 1193 33 3 v 2353
1235 a Fs(p)2430 1159 y Fx(\000)2521 1036 y Fu(Z)2611
1062 y Fv(s)2571 1242 y(s)2604 1251 y Fq(0)p 2673 1025
72 4 v 2673 1098 a FG(C)2745 1112 y Fy(0)p 2673 1138
111 4 v 2694 1222 a FG(R)2794 1159 y FA(\()p FG(s)g Fx(\000)g
FG(s)3026 1173 y Fy(0)3065 1159 y FA(\))26 b(=)1294 1475
y(=)f FG(H)1466 1489 y Fy(0)1505 1475 y FA(\()p FG(s)p
FA(\))c Fx(\000)p 1742 1341 72 4 v 1742 1414 a FG(C)1813
1428 y Fy(0)p 1740 1454 116 4 v 1740 1538 a FA(2)p FG(R)1865
1475 y FA(\()p FG(s)f Fx(\000)g FG(s)2097 1489 y Fy(0)2136
1475 y FA(\))2171 1438 y Fy(2)2236 1475 y FA(=)1294 1662
y(=)1412 1639 y Fu(e)1390 1662 y FG(H)1466 1676 y Fv(s)1499
1685 y Fq(0)1533 1676 y Fv(;R)1610 1662 y FA(\()p FG(s)p
FA(\))15 b FG(:)456 1381 y FA(\(2.43\))456 1822 y(This)33
b(and)i(\(2.31\))j(pro)m(v)m(e)e(\(2.39\))i(and)d(\(2.42\).)58
b(W)-8 b(e)37 b(no)m(w)e(pro)m(v)m(e)h(\(2.40\))i(and)d(\(2.41\).)456
1929 y(Let)25 b(us)g(consider)f(only)h(the)g(case)i FG(s)e
Fx(2)f FA([)p Fx(\000)p FA(3)p FG(=)p FA(4)p FG(;)32
b Fx(\000)p FA(1)p FG(=)p FA(2],)c(the)e(case)g FG(s)f
Fx(2)g FA([1)p FG(=)p FA(2)p FG(;)32 b FA(3)p FG(=)p
FA(4])27 b(b)s(eing)456 2037 y(analogous.)41 b(In)29
b(this)h(case,)h FG(s)25 b(<)g(s)1626 2051 y Fy(0)1665
2037 y FA(,)31 b(and)e(th)m(us,)i(exploiting)d(Lemma)j(2.6,)h(w)m(e)e
(get)792 2236 y FG(H)868 2250 y Fv(s)901 2259 y Fq(0)935
2250 y Fv(;R)1012 2236 y FA(\()p FG(s)p FA(\))83 b(=)g
FG(H)1438 2250 y Fy(0)1477 2236 y FA(\()p FG(s)1555 2250
y Fy(0)1595 2236 y FA(\))20 b Fx(\000)1741 2113 y Fu(Z)1832
2139 y Fv(s)1865 2148 y Fq(0)1792 2319 y Fv(s)2186 2175
y FA(1)p 1928 2215 562 4 v 1928 2253 a Fu(\020)2038 2313
y Fv(p)p 1993 2333 126 4 v 1993 2386 a(p)p Ft(\000)p
Fy(1)2128 2354 y FG(h)2180 2368 y Fv(s)2213 2377 y Fq(0)2248
2368 y Fv(;R)2325 2253 y Fu(\021)2380 2276 y Fy(1)p Fv(=p)2525
2236 y FG(>)1208 2575 y(>)83 b(H)1438 2589 y Fy(0)1477
2575 y FA(\()p FG(s)1555 2589 y Fy(0)1595 2575 y FA(\))20
b Fx(\000)1741 2451 y Fu(Z)1832 2477 y Fv(s)1865 2486
y Fq(0)1792 2657 y Fv(s)2118 2513 y FA(1)p 1928 2554
424 4 v 1928 2592 a Fu(\020)2038 2652 y Fv(p)p 1993 2672
126 4 v 1993 2724 a(p)p Ft(\000)p Fy(1)2128 2693 y FG(')2187
2592 y Fu(\021)2242 2615 y Fy(1)p Fv(=p)2388 2575 y FA(=)1208
2929 y(=)83 b FG(H)1438 2943 y Fy(0)1477 2929 y FA(\()p
FG(s)1555 2943 y Fy(0)1595 2929 y FA(\))20 b Fx(\000)1741
2805 y Fu(Z)1832 2832 y Fv(s)1865 2841 y Fq(0)1792 3012
y Fv(s)2134 2868 y FA(1)p 1928 2908 457 4 v 1928 2946
a Fu(\020)2038 3006 y Fv(p)p 1993 3026 126 4 v 1993 3078
a(p)p Ft(\000)p Fy(1)2128 3047 y FG(h)2180 3061 y Fy(0)2220
2946 y Fu(\021)2274 2969 y Fy(1)p Fv(=p)2415 2929 y Fx(\000)p
2518 2795 72 4 v 2518 2868 a FG(C)2589 2882 y Fy(0)p
2516 2908 116 4 v 2516 2991 a FA(2)p FG(R)2641 2929 y
FA(\()p FG(s)g Fx(\000)g FG(s)2873 2943 y Fy(0)2912 2929
y FA(\))2947 2892 y Fy(2)3012 2929 y FA(=)1208 3283 y(=)83
b FG(H)1438 3297 y Fy(0)1477 3283 y FA(\()p FG(s)p FA(\))21
b Fx(\000)p 1714 3149 72 4 v 1714 3222 a FG(C)1785 3236
y Fy(0)p 1712 3262 116 4 v 1712 3346 a FA(2)p FG(R)1837
3283 y FA(\()p FG(s)f Fx(\000)g FG(s)2069 3297 y Fy(0)2108
3283 y FA(\))2143 3246 y Fy(2)2198 3283 y FG(;)456 3469
y FA(pro)m(ving)29 b(\(2.40\))k(and)d(\(2.41\).)1895
b Fl(\003)555 3639 y FA(Since,)27 b(b)m(y)f(construction,)h(the)g
(function)e FG(g)2027 3655 y Fk(e)2028 3673 y Fj(S)-11
b Fy(\()p Fv(Y)t(;R)p Fy(\))2269 3639 y FA(de\014ned)26
b(ab)s(o)m(v)m(e)h(is)f(smo)s(oth)g(\(and,)456 3766 y(due)37
b(to)i(\(2.32,)k(its)38 b(gradien)m(t)g(nev)m(er)g(v)-5
b(anishes\),)40 b(w)m(e)f(can)g(compute)f(its)g(deriv)-5
b(ativ)m(es)456 3874 y(\(and)42 b(its)g FG(p)p FA(-Laplacian\))g(in)f
(the)i(classic)f(sense.)78 b(In)42 b(particular,)i(w)m(e)f(can)g
(sharply)456 3982 y(estimate)g(ho)m(w)f(far)g FG(g)1226
3998 y Fk(e)1227 4015 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p
Fy(\))1484 3982 y FA(is)41 b(from)h(b)s(eing)f(a)h(solution)f(of)i
(\(1.5\),)k(thanks)42 b(to)h(the)456 4103 y(follo)m(wing)28
b(result:)456 4235 y FH(Prop)s(osition)42 b(2.18.)j Fo(L)-5
b(et)38 b FG(Y)54 b Fx(2)33 b Fw(R)1709 4202 y Fv(N)7
b Fy(+1)1910 4235 y Fo(with)38 b Fx(j)p FG(y)2182 4249
y Fv(N)7 b Fy(+1)2339 4235 y Fx(j)34 b(\024)2513 4199
y Fy(1)p 2513 4214 36 4 v 2513 4266 a(4)2558 4235 y Fo(.)56
b(Then,)39 b(ther)-5 b(e)38 b(exists)g(a)456 4343 y(p)-5
b(ositive)33 b(universal)g(c)-5 b(onstant)35 b FG(C)c(>)25
b FA(0)33 b Fo(such)g(that)778 4539 y FG(h)830 4502 y
Ft(0)830 4562 y Fy(0)870 4438 y Fu(\020)924 4539 y FG(g)966
4556 y Fk(e)967 4573 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p
Fy(\))1181 4539 y FA(\()p FG(x)p FA(\))1303 4438 y Fu(\021)1379
4539 y Fx(\000)1479 4478 y FG(C)p 1479 4518 72 4 v 1480
4602 a(R)1586 4539 y Fx(\024)25 b FA(\001)1758 4553 y
Fv(p)1798 4539 y FG(g)1840 4556 y Fk(e)1841 4573 y Fj(S)-12
b Fy(\()p Fv(Y)t(;R)p Fy(\))2055 4539 y FA(\()p FG(x)p
FA(\))26 b Fx(\024)f FG(h)2351 4502 y Ft(0)2351 4562
y Fy(0)2391 4438 y Fu(\020)2445 4539 y FG(g)2487 4556
y Fk(e)2488 4573 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))2702
4539 y FA(\()p FG(x)p FA(\))2824 4438 y Fu(\021)2899
4539 y FA(+)3000 4478 y FG(C)p 3000 4518 V 3001 4602
a(R)3097 4539 y(;)456 4725 y Fo(for)33 b(any)g FG(x)g
Fo(for)g(which)g FG(g)1302 4742 y Fk(e)1303 4759 y Fj(S)-11
b Fy(\()p Fv(Y)t(;R)p Fy(\))1550 4725 y Fo(is)33 b(de\014ne)-5
b(d)33 b(\(i.e.,)f(for)i(any)f FG(x)f Fo(so)h(that)h(\(2.36\))g
(holds\).)456 4909 y(Pr)-5 b(o)g(of.)43 b FA(W)-8 b(e)31
b(will)d(use)i(the)g(notation)1173 5063 y FG(t)83 b FA(:=)g
FG(H)1544 5077 y Fy(0)1583 5063 y FA(\()p FG(y)1663 5077
y Fv(N)7 b Fy(+1)1820 5063 y FA(\))21 b(+)e Fx(j)p FG(x)i
Fx(\000)f FG(y)s Fx(j)g(\000)g FG(R)183 b FA(and)-2282
b(\(2.44\))1163 5198 y FG(s)83 b FA(:=)g FG(\032)1515
5212 y Fv(y)1550 5223 y Fs(N)5 b Fq(+1)1684 5212 y Fv(;R)1762
5198 y FA(\()p FG(t)p FA(\))41 b(=)f FG(g)2059 5215 y
Fk(e)2060 5232 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))2274
5198 y FA(\()p FG(x)p FA(\))15 b FG(:)p eop
%%Page: 23 23
23 22 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(23)456
450 y FA(Let)30 b(us)g(note)h(that,)g(from)f(\(2.31\))j(and)d(\(2.2\),)
456 688 y(\(2.45\))955 627 y FG(d)p 938 667 81 4 v 938
751 a(dt)1028 688 y(\032)1075 702 y Fv(y)1110 713 y Fs(N)5
b Fq(+1)1245 702 y Fv(;R)1322 688 y FA(\()p FG(t)p FA(\))26
b(=)1972 627 y(1)p 1557 667 876 4 v 1557 776 a FG(H)1640
745 y Ft(0)1633 802 y Fy(0)1672 776 y FA(\()p FG(s)p
FA(\))21 b Fx(\000)p 1906 684 55 3 v 1906 739 a Fv(C)1961
748 y Fq(0)p 1906 755 90 4 v 1924 807 a Fv(R)2006 776
y FA(\()p FG(s)f Fx(\000)g FG(y)2240 790 y Fv(N)7 b Fy(+1)2397
776 y FA(\))2468 688 y(=)2564 560 y Fu(\022)2719 627
y FG(p)p 2640 667 203 4 v 2640 751 a(p)20 b Fx(\000)g
FA(1)2853 688 y FG(')p FA(\()p FG(s)p FA(\))3025 560
y Fu(\023)3103 556 y Fq(1)p 3102 568 33 3 v 3102 609
a Fs(p)3179 688 y FG(:)456 921 y FA(Hence,)31 b(di\013eren)m(tiating)e
(again,)i(a)g(straigh)m(tforw)m(ard)f(calculation)f(leads)h(to)456
1159 y(\(2.46\))1292 1098 y FG(d)1339 1065 y Fy(2)p 1276
1138 120 4 v 1276 1221 a FG(dt)1356 1195 y Fy(2)1405
1159 y FG(\032)1452 1173 y Fv(y)1487 1184 y Fs(N)5 b
Fq(+1)1622 1173 y Fv(;R)1699 1159 y FA(\()p FG(t)p FA(\))26
b(=)1934 1098 y(\()p FG(p)15 b(')p FA(\()p FG(s)p FA(\)\))2248
1021 y Fq(2)p Fp(\000)p Fs(p)p 2248 1037 111 3 v 2287
1078 a(p)p 1934 1138 439 4 v 1989 1254 a FA(\()p FG(p)20
b Fx(\000)g FA(1\))2272 1181 y Fq(2)p 2272 1193 33 3
v 2272 1234 a Fs(p)2398 1159 y FG(')2457 1122 y Ft(0)2481
1159 y FA(\()p FG(s)p FA(\))15 b FG(:)456 1390 y FA(Therefore,)30
b(b)m(y)g(Lemma)h(B.2,)456 1600 y(\(2.47\))271 b(\001)1033
1614 y Fv(p)1072 1600 y FA(\()p FG(\032)1154 1614 y Fv(y)1189
1625 y Fs(N)5 b Fq(+1)1324 1614 y Fv(;R)1401 1600 y FA(\()p
FG(t)p FA(\)\))26 b(=)f FG(')1720 1562 y Ft(0)1744 1600
y FA(\()p FG(s)p FA(\))c(+)1989 1538 y FG(N)31 b Fx(\000)19
b FA(1)p 1978 1579 262 4 v 1978 1662 a Fx(j)p FG(x)i
Fx(\000)f FG(y)s Fx(j)2250 1499 y Fu(\020)2392 1538 y
FG(p)p 2314 1579 203 4 v 2314 1662 a(p)g Fx(\000)g FA(1)2542
1600 y FG(')p FA(\()p FG(s)p FA(\))2714 1499 y Fu(\021)2779
1491 y Fs(p)p Fp(\000)p Fq(1)p 2779 1506 111 3 v 2818
1547 a Fs(p)2918 1600 y FG(:)456 1804 y FA(F)-8 b(urthermore,)30
b(note)h(that,)g(b)m(y)g(di\013eren)m(tiating)f(\(2.2\))r(,)g(one)h
(obtains)456 2063 y(\(2.48\))330 b Fx(\000)1097 2002
y FA(1)p 1097 2042 46 4 v 1097 2126 a FG(p)1167 1880
y Fu(0)1167 2044 y(@)1402 1942 y Fv(p)p 1357 1963 126
4 v 1357 2015 a(p)p Ft(\000)p Fy(1)1493 1983 y FG(')1552
1950 y Ft(0)1575 1983 y FA(\()p FG(s)p FA(\))p 1257 2042
523 4 v 1257 2162 a(\()1347 2120 y Fv(p)p 1302 2141 126
4 v 1302 2193 a(p)p Ft(\000)p Fy(1)1438 2162 y FG(')p
FA(\()p FG(s)p FA(\)\))1655 2085 y Fs(p)p Fq(+1)p 1656
2101 110 3 v 1694 2142 a Fs(p)1789 1880 y Fu(1)1789 2044
y(A)1894 2063 y FA(=)25 b Fx(\000)2071 2002 y FA(1)p
2071 2042 46 4 v 2071 2126 a FG(p)2141 1880 y Fu(0)2141
2044 y(@)2388 1942 y Fv(p)p 2343 1963 126 4 v 2343 2015
a(p)p Ft(\000)p Fy(1)2478 1983 y FG(h)2530 1950 y Ft(0)2530
2008 y Fy(0)2570 1983 y FA(\()p FG(s)p FA(\))p 2231 2042
555 4 v 2231 2162 a(\()2321 2120 y Fv(p)p 2276 2141 126
4 v 2276 2193 a(p)p Ft(\000)p Fy(1)2412 2162 y FG(h)2464
2176 y Fy(0)2503 2162 y FA(\()p FG(s)p FA(\)\))2661 2085
y Fs(p)p Fq(+1)p 2662 2101 110 3 v 2701 2142 a Fs(p)2795
1880 y Fu(1)2795 2044 y(A)2895 2063 y Fx(\000)p 2996
1929 72 4 v 2996 2002 a FG(C)3067 2016 y Fy(0)p 2996
2042 111 4 v 3016 2126 a FG(R)456 2320 y FA(so)30 b(that)456
2539 y(\(2.49\))273 b FG(')1018 2501 y Ft(0)1042 2539
y FA(\()p FG(s)p FA(\))25 b(=)1276 2410 y Fu(\022)1369
2477 y FG(')p FA(\()p FG(s)p FA(\))p 1353 2518 206 4
v 1353 2601 a FG(h)1405 2615 y Fy(0)1445 2601 y FA(\()p
FG(s)p FA(\))1568 2410 y Fu(\023)1645 2402 y Fs(p)p Fq(+1)p
1645 2418 110 3 v 1684 2459 a Fs(p)1783 2539 y FG(h)1835
2501 y Ft(0)1835 2561 y Fy(0)1875 2539 y FA(\()p FG(s)p
FA(\))c(+)2110 2477 y FG(p)p 2156 2404 72 4 v(C)2227
2491 y Fy(0)p 2110 2518 157 4 v 2153 2601 a FG(R)2291
2410 y Fu(\022)2446 2477 y FG(p)p 2368 2518 203 4 v 2368
2601 a(p)f Fx(\000)g FA(1)2580 2410 y Fu(\023)2658 2406
y Fq(1)p 2657 2418 33 3 v 2657 2459 a Fs(p)2719 2438
y Fu(\020)2773 2539 y FG(')p FA(\()p FG(s)p FA(\))2945
2438 y Fu(\021)3010 2429 y Fs(p)p Fq(+1)p 3010 2445 110
3 v 3048 2486 a Fs(p)3148 2539 y FG(;)456 2744 y FA(whence,)30
b(from)g(\(2.1\),)540 3004 y FG(')599 2967 y Ft(0)623
3004 y FA(\()p FG(s)p FA(\))c(=)f FG(h)910 2967 y Ft(0)910
3027 y Fy(0)950 3004 y FA(\()p FG(s)p FA(\)+)757 3241
y(+)870 3179 y(1)p 858 3220 70 4 v 858 3303 a FG(R)952
3086 y Fu(\()1026 3241 y FG(h)1078 3203 y Ft(0)1078 3263
y Fy(0)1118 3241 y FA(\()p FG(s)p FA(\))p FG(R)1316 3086
y Fu(")1369 3113 y(\022)1525 3179 y FG(R)p 1446 3220
229 4 v 1446 3303 a(R)c Fx(\000)e FG(a)1685 3113 y Fu(\023)1751
3135 y Fv(p)p Fy(+1)1901 3241 y Fx(\000)h FA(1)2037 3086
y Fu(#)2111 3241 y FA(+)g FG(p)p 2263 3168 72 4 v 15
w(C)2334 3255 y Fy(0)2389 3113 y Fu(\022)2544 3179 y
FG(p)p 2466 3220 203 4 v 2466 3303 a(p)g Fx(\000)g FA(1)2678
3113 y Fu(\023)2756 3108 y Fq(1)p 2755 3120 33 3 v 2755
3161 a Fs(p)2816 3140 y Fu(\020)2871 3241 y FG(')p FA(\()p
FG(s)p FA(\))3043 3140 y Fu(\021)3108 3132 y Fs(p)p Fq(+1)p
3108 3148 110 3 v 3146 3189 a Fs(p)3231 3086 y Fu(\))3334
3241 y FG(;)456 2885 y FA(\(2.50\))456 3468 y(with)1117
3636 y FG(a)25 b FA(=)g FG(a)p FA(\()p FG(s)p FA(\))g(:=)p
1594 3563 72 4 v 26 w FG(C)1665 3650 y Fy(0)1720 3636
y FA(\()p FG(s)20 b Fx(\000)g FG(s)1952 3650 y Fy(0)1991
3636 y FA(\))2041 3508 y Fu(\022)2196 3575 y FG(p)p 2118
3616 203 4 v 2118 3699 a(p)g Fx(\000)g FA(1)2346 3636
y FG(h)2398 3650 y Fy(0)2437 3636 y FA(\()p FG(s)p FA(\))2550
3508 y Fu(\023)2618 3531 y Fy(1)p Fv(=p)2758 3636 y FG(:)456
3818 y FA(Using)29 b(no)m(w)i(the)f(fact)h(that)1125
4101 y(lim)1090 4167 y Fv(x)p Ft(!)p Fy(0)1236 4148 y
Fq(+)1301 3888 y Fu(\014)1301 3942 y(\014)1301 3997 y(\014)1301
4051 y(\014)1301 4106 y(\014)1301 4161 y(\014)1301 4215
y(\014)1341 3906 y(\020)1472 3971 y Fy(1)p 1406 3986
168 4 v 1406 4038 a(1)p Ft(\000)p Fv(xa)1583 3906 y Fu(\021)1637
3929 y Fv(p)p Fy(+1)1803 4007 y Fx(\000)k FA(1)p 1341
4081 613 4 v 1622 4164 a FG(x)1964 3888 y Fu(\014)1964
3942 y(\014)1964 3997 y(\014)1964 4051 y(\014)1964 4106
y(\014)1964 4161 y(\014)1964 4215 y(\014)2020 4101 y
FA(=)25 b Fx(j)p FA(\()p FG(p)20 b FA(+)g(1\))p FG(a)p
Fx(j)26 b FG(<)f FA(+)p Fx(1)15 b FG(;)456 4383 y FA(w)m(e)30
b(obtain,)h(if)e FG(R)i FA(is)e(suitably)g(large,)h(that)741
4457 y Fu(\014)741 4512 y(\014)741 4566 y(\014)741 4621
y(\014)741 4675 y(\014)771 4617 y FG(h)823 4579 y Ft(0)823
4639 y Fy(0)863 4617 y FA(\()p FG(s)p FA(\))p FG(R)1062
4461 y Fu(")1115 4488 y(\022)1271 4555 y FG(R)p 1191
4595 229 4 v 1191 4679 a(R)21 b Fx(\000)f FG(a)1430 4488
y Fu(\023)1497 4511 y Fv(p)p Fy(+1)1647 4617 y Fx(\000)g
FA(1)1783 4461 y Fu(#)1857 4617 y FA(+)g FG(p)p 2009
4543 72 4 v 2009 4616 a(C)2080 4631 y Fy(0)2135 4488
y Fu(\022)2290 4555 y FG(p)p 2212 4595 203 4 v 2212 4679
a(p)g Fx(\000)f FA(1)2424 4488 y Fu(\023)2502 4484 y
Fq(1)p 2501 4496 33 3 v 2501 4537 a Fs(p)2562 4515 y
Fu(\020)2616 4617 y FG(')p FA(\()p FG(s)p FA(\))2788
4515 y Fu(\021)2853 4507 y Fs(p)p Fq(+1)p 2853 4523 110
3 v 2892 4564 a Fs(p)2976 4457 y Fu(\014)2976 4512 y(\014)2976
4566 y(\014)2976 4621 y(\014)2976 4675 y(\014)3032 4617
y Fx(\024)40 b FA(const)16 b FG(;)456 4614 y FA(\(2.51\))456
4849 y(so)30 b(that,)h(from)f(\(2.47\))j(and)d(the)g(fact)i(that)f
FG(')25 b Fx(\025)g FA(0,)1306 5040 y FG(h)1358 5002
y Ft(0)1358 5062 y Fy(0)1397 5040 y FA(\()p FG(s)p FA(\))c
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5102 a FG(R)1875 5040 y Fx(\024)j FA(\001)2046 5054 y
Fv(p)2086 5040 y FA(\()p FG(\032)2168 5054 y Fv(y)2203
5065 y Fs(N)5 b Fq(+1)2338 5054 y Fv(;R)2415 5040 y FA(\()p
FG(t)p FA(\)\))15 b FG(;)456 5216 y FA(whic)m(h)29 b(pro)m(v)m(es)i
(one)f(side)g(of)g(the)h(claimed)e(inequalit)m(y)-8 b(.)p
eop
%%Page: 24 24
24 23 bop 456 251 a Fy(24)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
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(e)456 849 y(barrier:)456 985 y FH(Lemma)d(2.19.)43 b
Fo(A)n(t)32 b(any)h FG(x)f Fo(for)i(which)f FG(g)1912
1002 y Fk(e)1913 1019 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p
Fy(\))2160 985 y Fo(is)33 b(de\014ne)-5 b(d,)33 b(we)g(have)g(that)862
1114 y Fu(\022)929 1243 y FG(H)1012 1205 y Ft(0)1005
1265 y Fy(0)1059 1142 y Fu(\020)1113 1243 y FG(g)1155
1259 y Fk(e)1156 1276 y Fj(S)-11 b Fy(\()p Fv(Y)t(;R)p
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1243 y Fx(\000)p 1669 1108 72 4 v 1669 1181 a FG(C)1740
1195 y Fy(0)p 1669 1222 111 4 v 1689 1305 a FG(R)1805
1142 y Fu(\020)1859 1243 y FG(g)1901 1259 y Fk(e)1902
1276 y Fj(S)f Fy(\()p Fv(Y)t(;R)p Fy(\))2116 1243 y FA(\()p
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Fy(+1)2552 1142 y Fu(\021)2606 1114 y(\023)2704 1243
y Fx(r)p FG(g)2822 1259 y Fk(e)2823 1276 y Fj(S)-12 b
Fy(\()p Fv(Y)t(;R)p Fy(\))3037 1243 y FA(\()p FG(x)p
FA(\))41 b(=)872 1479 y(=)1003 1417 y FG(x)20 b Fx(\000)g
FG(y)p 978 1458 262 4 v 978 1541 a Fx(j)p FG(x)g Fx(\000)g
FG(y)s Fx(j)1265 1479 y FG(:)456 1355 y FA(\(2.52\))456
1692 y Fo(A)n(lso,)32 b(ther)-5 b(e)34 b(exists)f(a)g(universal)g(c)-5
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1903 y(\014)1168 1899 y FG(H)1251 1861 y Ft(0)1244 1921
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y Fk(e)1395 1932 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))1609
1899 y FA(\()p FG(x)p FA(\))1731 1798 y Fu(\021)1817
1794 y(\014)1817 1849 y(\014)1817 1903 y(\014)1847 1899
y Fx(r)p FG(g)1965 1915 y Fk(e)1966 1932 y Fj(S)g Fy(\()p
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1794 y Fu(\014)2302 1849 y(\014)2302 1903 y(\014)2353
1899 y Fx(\000)20 b FA(1)2489 1794 y Fu(\014)2489 1849
y(\014)2489 1903 y(\014)2560 1899 y Fx(\024)2681 1837
y FG(C)p 2681 1878 72 4 v 2682 1961 a(R)456 2089 y Fo(and)1215
2154 y Fu(\014)1215 2209 y(\014)1215 2263 y(\014)1245
2259 y FG(H)1328 2221 y Ft(00)1321 2281 y Fy(0)1386 2158
y Fu(\020)1440 2259 y FG(g)1482 2275 y Fk(e)1483 2293
y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))1697 2259 y FA(\()p
FG(x)p FA(\))1819 2158 y Fu(\021)1890 2259 y FG(@)1938
2273 y Fv(i)1966 2259 y FG(g)2008 2275 y Fk(e)2009 2293
y Fj(S)g Fy(\()p Fv(Y)t(;R)p Fy(\))2223 2259 y FA(\()p
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2275 y Fk(e)2474 2293 y Fj(S)g Fy(\()p Fv(Y)t(;R)p Fy(\))2688
2259 y FA(\()p FG(x)p FA(\))21 b(+)1061 2458 y(+)83 b
FG(H)1298 2420 y Ft(0)1291 2480 y Fy(0)1345 2357 y Fu(\020)1399
2458 y FG(g)1441 2474 y Fk(e)1442 2492 y Fj(S)-11 b Fy(\()p
Fv(Y)t(;R)p Fy(\))1657 2458 y FA(\()p FG(x)p FA(\))1779
2357 y Fu(\021)1849 2458 y FG(@)1897 2472 y Fv(ij)1958
2458 y FG(g)2000 2474 y Fk(e)2001 2492 y Fj(S)f Fy(\()p
Fv(Y)t(;R)p Fy(\))2215 2458 y FA(\()p FG(x)p FA(\))2337
2353 y Fu(\014)2337 2408 y(\014)2337 2462 y(\014)2393
2458 y Fx(\024)1061 2676 y(\024)1225 2614 y FG(C)p 1225
2655 V 1226 2738 a(R)1322 2676 y(;)-891 b FA(\(2.54\))456
2866 y Fo(pr)-5 b(ovide)g(d)35 b FG(R)e Fo(is)f(lar)-5
b(ge)34 b(enough.)456 3049 y(Pr)-5 b(o)g(of.)43 b FA(Using)23
b(the)i(notation)g(in)f(\(2.44\))j(and)d(the)h(\014rst)f(equalit)m(y)g
(in)g(\(2.45\),)k(the)d(claim)456 3157 y(in)k(\(2.52\))j(easily)e
(follo)m(ws.)40 b(F)-8 b(rom)31 b(\(2.52\))h(and)e(\(2.37\),)j(one)d
(easily)g(gets)h(\(2.53\).)555 3265 y(Let)g(us)f(no)m(w)g(pro)m(v)m(e)h
(\(2.54\).)43 b(F)-8 b(rom)31 b(\(2.31\))i(and)d(\(2.33\),)j(w)m(e)d
(ha)m(v)m(e)i(that)964 3495 y FG(\034)j FA(=)25 b FG(H)1211
3509 y Fy(0)1250 3394 y Fu(\020)1305 3495 y FG(\032)1352
3509 y Fv(y)1387 3520 y Fs(N)5 b Fq(+1)1521 3509 y Fv(;R)1599
3495 y FA(\()p FG(\034)10 b FA(\))1719 3394 y Fu(\021)1794
3495 y Fx(\000)p 1897 3360 V 1897 3433 a FG(C)1968 3447
y Fy(0)p 1895 3474 116 4 v 1895 3557 a FA(2)p FG(R)2020
3394 y Fu(\020)2074 3495 y FG(\032)2121 3509 y Fv(y)2156
3520 y Fs(N)5 b Fq(+1)2291 3509 y Fv(;R)2368 3495 y FA(\()p
FG(\034)10 b FA(\))21 b Fx(\000)f FG(y)2645 3509 y Fv(N)7
b Fy(+1)2802 3394 y Fu(\021)2856 3417 y Fy(2)2911 3495
y FG(;)456 3685 y FA(for)27 b(an)m(y)i FG(\034)37 b FA(for)28
b(whic)m(h)f FG(\032)1281 3699 y Fv(y)1316 3710 y Fs(N)5
b Fq(+1)1450 3699 y Fv(;R)1555 3685 y FA(is)27 b(de\014ned;)h(then,)g
(di\013eren)m(tiating)f(t)m(wice)h(and)g(recall-)456
3794 y(ing)h(\(2.37\),)1170 3898 y Fu(\014)1170 3953
y(\014)1170 4008 y(\014)1200 4003 y FG(H)1283 3965 y
Ft(00)1276 4025 y Fy(0)1325 4003 y FA(\()p FG(\032)1407
4017 y Fv(y)1442 4028 y Fs(N)5 b Fq(+1)1577 4017 y Fv(;R)1655
4003 y FA(\()p FG(\034)10 b FA(\)\))1841 3902 y Fu(\020)1895
4003 y FG(\032)1942 3965 y Ft(0)1942 4025 y Fv(y)1977
4036 y Fs(N)5 b Fq(+1)2112 4025 y Fv(;R)2189 4003 y FA(\()p
FG(\034)10 b FA(\))2309 3902 y Fu(\021)2364 3925 y Fy(2)2419
4003 y FA(+)1190 4215 y(+)20 b FG(H)1364 4177 y Ft(0)1357
4237 y Fy(0)1396 4215 y FA(\()p FG(\032)1478 4229 y Fv(y)1513
4240 y Fs(N)5 b Fq(+1)1648 4229 y Fv(;R)1725 4215 y FA(\()p
FG(\034)10 b FA(\)\))15 b FG(\032)1942 4177 y Ft(00)1942
4237 y Fv(y)1977 4248 y Fs(N)5 b Fq(+1)2113 4237 y Fv(;R)2190
4215 y FA(\()p FG(\034)10 b FA(\))2310 4110 y Fu(\014)2310
4165 y(\014)2310 4219 y(\014)2366 4215 y Fx(\024)2472
4153 y FA(const)p 2472 4194 208 4 v 2541 4277 a FG(R)2705
4215 y(:)456 4103 y FA(\(2.55\))456 4405 y(F)-8 b(urthermore,)30
b(di\013eren)m(tiating)f(t)m(wice)i(the)g(relation)992
4593 y FG(g)1034 4609 y Fk(e)1035 4627 y Fj(S)-11 b Fy(\()p
Fv(Y)t(;R)p Fy(\))1250 4593 y FA(\()p FG(x)p FA(\))41
b(=)f FG(\032)1571 4607 y Fv(y)1606 4618 y Fs(N)5 b Fq(+1)1740
4607 y Fv(;R)1818 4492 y Fu(\020)1872 4593 y FG(H)1948
4607 y Fy(0)1987 4593 y FA(\()p FG(y)2067 4607 y Fv(N)i
Fy(+1)2224 4593 y FA(\))21 b(+)f Fx(j)p FG(x)g Fx(\000)g
FG(y)s Fx(j)g(\000)g FG(R)2813 4492 y Fu(\021)2882 4593
y FG(;)456 4780 y FA(a)30 b(direct)g(computation)g(giv)m(es)1119
4945 y FG(@)1167 4959 y Fv(ij)1228 4945 y FG(g)1270 4962
y Fk(e)1271 4979 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))1485
4945 y FA(\()p FG(x)p FA(\))26 b(=)965 5162 y(=)83 b
FG(R)1188 5176 y Fv(ij)1249 5162 y FG(\032)1296 5125
y Ft(00)1296 5185 y Fv(y)1331 5196 y Fs(N)5 b Fq(+1)1465
5185 y Fv(;R)1542 5162 y FA(\()p FG(t)p FA(\))21 b(+)1757
5034 y Fu(\022)1914 5100 y FG(\016)1954 5114 y Fv(ij)p
1834 5142 262 4 v 1834 5225 a Fx(j)p FG(x)f Fx(\000)g
FG(y)s Fx(j)2126 5162 y(\000)2292 5100 y FG(R)2361 5114
y Fv(ij)p 2226 5142 V 2226 5225 a Fx(j)p FG(x)h Fx(\000)f
FG(y)s Fx(j)2498 5034 y Fu(\023)2580 5162 y FG(\032)2627
5125 y Ft(0)2627 5185 y Fv(y)2662 5196 y Fs(N)5 b Fq(+1)2797
5185 y Fv(;R)2874 5162 y FA(\()p FG(t)p FA(\))15 b FG(;)p
eop
%%Page: 25 25
25 24 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(25)456
450 y FA(where)29 b(w)m(e)i(de\014ne,)f(for)g(short,)1416
655 y FG(R)1485 669 y Fv(ij)1586 655 y FA(:=)1732 593
y(\()p FG(x)1819 607 y Fv(i)1868 593 y Fx(\000)20 b FG(y)2004
607 y Fv(i)2032 593 y FA(\))15 b(\()p FG(x)2169 607 y
Fv(j)2226 593 y Fx(\000)20 b FG(y)2362 607 y Fv(j)2398
593 y FA(\))p 1732 634 702 4 v 1933 717 a Fx(j)p FG(x)g
Fx(\000)g FG(y)s Fx(j)2194 691 y Fy(2)2459 655 y FG(:)456
860 y FA(In)29 b(other)i(w)m(ords,)456 1017 y(\(2.56\))582
b FG(@)1316 1031 y Fv(ij)1377 1017 y FG(g)1419 1034 y
Fk(e)1420 1051 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))1634
1017 y FA(\()p FG(x)p FA(\))26 b(=)f FG(R)1947 1031 y
Fv(ij)2007 1017 y FG(\032)2054 980 y Ft(00)2054 1040
y Fv(y)2089 1051 y Fs(N)5 b Fq(+1)2224 1040 y Fv(;R)2301
1017 y FA(\()p FG(t)p FA(\))21 b(+)f FG(S)2572 1031 y
Fv(ij)456 1193 y FA(for)35 b(a)h(suitable)e FG(S)1081
1207 y Fv(ij)1177 1193 y FA(satisfying)g Fx(j)p FG(S)1668
1207 y Fv(ij)1729 1193 y Fx(j)g(\024)48 b FA(const)16
b FG(=)p Fx(j)p FG(x)24 b Fx(\000)g FG(y)s Fx(j)p FA(.)56
b(Therefore,)37 b(using)d(\(2.56\))456 1301 y(and)29
b(the)i(fact)g(that)456 1472 y(\(2.57\))592 b FG(@)1326
1486 y Fv(i)1354 1472 y FG(g)1396 1489 y Fk(e)1397 1506
y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))1611 1472 y FA(\()p
FG(x)p FA(\))41 b(=)f FG(\032)1932 1435 y Ft(0)1932 1495
y Fv(y)1967 1506 y Fs(N)5 b Fq(+1)2102 1495 y Fv(;R)2179
1472 y FA(\()p FG(t)p FA(\))2308 1411 y FG(x)2360 1425
y Fv(i)2408 1411 y Fx(\000)20 b FG(y)2544 1425 y Fv(i)p
2308 1452 264 4 v 2309 1535 a Fx(j)p FG(x)h Fx(\000)f
FG(y)s Fx(j)2597 1472 y FG(;)456 1678 y FA(w)m(e)30 b(get)i(that)1230
1857 y FG(H)1313 1820 y Ft(00)1306 1880 y Fy(0)1371 1756
y Fu(\020)1425 1857 y FG(g)1467 1874 y Fk(e)1468 1891
y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))1682 1857 y FA(\()p
FG(x)p FA(\))1804 1756 y Fu(\021)1874 1857 y FG(@)1922
1871 y Fv(i)1951 1857 y FG(g)1993 1874 y Fk(e)1994 1891
y Fj(S)g Fy(\()p Fv(Y)t(;R)p Fy(\))2208 1857 y FA(\()p
FG(x)p FA(\))p FG(@)2378 1871 y Fv(j)2416 1857 y FG(g)2458
1874 y Fk(e)2459 1891 y Fj(S)g Fy(\()p Fv(Y)t(;R)p Fy(\))2673
1857 y FA(\()p FG(x)p FA(\))21 b(+)1230 2056 y(+)p FG(H)1384
2018 y Ft(0)1377 2078 y Fy(0)1431 1955 y Fu(\020)1485
2056 y FG(g)1527 2073 y Fk(e)1528 2090 y Fj(S)-11 b Fy(\()p
Fv(Y)t(;R)p Fy(\))1743 2056 y FA(\()p FG(x)p FA(\))1865
1955 y Fu(\021)1935 2056 y FG(@)1983 2070 y Fv(ij)2044
2056 y FG(g)2086 2073 y Fk(e)2087 2090 y Fj(S)f Fy(\()p
Fv(Y)t(;R)p Fy(\))2301 2056 y FA(\()p FG(x)p FA(\))26
b(=)1076 2244 y(=)83 b FG(H)1313 2207 y Ft(00)1306 2267
y Fy(0)1356 2244 y FA(\()p FG(\032)1438 2258 y Fv(y)1473
2269 y Fs(N)5 b Fq(+1)1607 2258 y Fv(;R)1685 2244 y FA(\()p
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2258 y Fv(y)1978 2269 y Fs(N)g Fq(+1)2112 2258 y Fv(;R)2190
2244 y FA(\()p FG(t)p FA(\))2293 2171 y Fu(\001)2335
2193 y Fy(2)2405 2244 y FG(R)2474 2258 y Fv(ij)2554 2244
y FA(+)1230 2416 y(+)p FG(H)1384 2379 y Ft(0)1377 2439
y Fy(0)1416 2416 y FA(\()p FG(\032)1498 2430 y Fv(y)1533
2441 y Fs(N)g Fq(+1)1668 2430 y Fv(;R)1745 2416 y FA(\()p
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2430 y Fv(ij)2098 2416 y FG(\032)2145 2379 y Ft(00)2145
2439 y Fv(y)2180 2450 y Fs(N)g Fq(+1)2315 2439 y Fv(;R)2392
2416 y FA(\()p FG(t)p FA(\))21 b(+)f FG(S)2663 2430 y
Fv(ij)2723 2315 y Fu(\021)2803 2416 y FA(=)1076 2632
y(=)83 b FG(R)1299 2646 y Fv(ij)1360 2531 y Fu(h)1403
2632 y FG(H)1486 2594 y Ft(00)1479 2654 y Fy(0)1528 2531
y Fu(\020)1582 2632 y FG(\032)1629 2646 y Fv(y)1664 2657
y Fs(N)5 b Fq(+1)1799 2646 y Fv(;R)1876 2632 y FA(\()p
FG(t)p FA(\))1979 2531 y Fu(\021)16 b(\020)2104 2632
y FG(\032)2151 2646 y Fv(y)2186 2657 y Fs(N)5 b Fq(+1)2320
2646 y Fv(;R)2397 2632 y FA(\()p FG(t)p FA(\))2500 2531
y Fu(\021)2555 2553 y Fy(2)2615 2632 y FA(+)1230 2831
y(+)p FG(H)1384 2793 y Ft(0)1377 2853 y Fy(0)1416 2730
y Fu(\020)1470 2831 y FG(\032)1517 2845 y Fv(y)1552 2856
y Fs(N)g Fq(+1)1687 2845 y Fv(;R)1764 2831 y FA(\()p
FG(t)p FA(\))1867 2730 y Fu(\021)1937 2831 y FG(\032)1984
2793 y Ft(00)1984 2853 y Fv(y)2019 2864 y Fs(N)g Fq(+1)2154
2853 y Fv(;R)2231 2831 y FA(\()p FG(t)p FA(\))2334 2730
y Fu(i)2398 2831 y FA(+)20 b FG(T)2542 2845 y Fv(ij)2617
2831 y FG(;)456 3015 y FA(with)29 b Fx(j)p FG(T)741 3029
y Fv(ij)802 3015 y Fx(j)c(\024)40 b FA(const)16 b FG(=)p
Fx(j)p FG(x)21 b Fx(\000)f FG(y)s Fx(j)p FA(.)40 b(Th)m(us,)30
b(from)g(\(2.55\),)1195 3099 y Fu(\014)1195 3153 y(\014)1195
3208 y(\014)1225 3203 y FG(H)1308 3166 y Ft(00)1301 3226
y Fy(0)1350 3102 y Fu(\020)1405 3203 y FG(g)1447 3220
y Fk(e)1448 3237 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))1662
3203 y FA(\()p FG(x)p FA(\))1784 3102 y Fu(\021)1839
3203 y FG(@)1887 3217 y Fv(i)1915 3203 y FG(g)1957 3220
y Fk(e)1958 3237 y Fj(S)h Fy(\()p Fv(Y)t(;R)p Fy(\))2173
3203 y FA(\()p FG(x)p FA(\))p FG(@)2343 3217 y Fv(j)2380
3203 y FG(g)2422 3220 y Fk(e)2423 3237 y Fj(S)f Fy(\()p
Fv(Y)t(;R)p Fy(\))2637 3203 y FA(\()p FG(x)p FA(\))21
b(+)1195 3402 y(+)p FG(H)1349 3364 y Ft(0)1342 3424 y
Fy(0)1381 3301 y Fu(\020)1435 3402 y FG(g)1477 3419 y
Fk(e)1478 3436 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))1692
3402 y FA(\()p FG(x)p FA(\))1814 3301 y Fu(\021)1869
3402 y FG(@)1917 3416 y Fv(ij)1978 3402 y FG(g)2020 3419
y Fk(e)2021 3436 y Fj(S)g Fy(\()p Fv(Y)t(;R)p Fy(\))2235
3402 y FA(\()p FG(x)p FA(\))2357 3297 y Fu(\014)2357
3352 y(\014)2357 3407 y(\014)2429 3402 y Fx(\024)1195
3628 y(\024)40 b FA(const)1544 3500 y Fu(\022)1633 3567
y FA(1)p 1621 3607 70 4 v 1621 3691 a FG(R)1721 3628
y FA(+)1929 3567 y(1)p 1821 3607 262 4 v 1821 3691 a
Fx(j)p FG(x)21 b Fx(\000)f FG(y)s Fx(j)2093 3500 y Fu(\023)2190
3628 y FG(:)456 3840 y FA(Therefore,)30 b(\(2.54\))j(is)c(pro)m(v)m(ed)
i(thanks)f(to)h(Lemma)f(2.16.)915 b Fl(\003)555 4012
y FA(W)-8 b(e)34 b(no)m(w)g(recall)e(a)h(result,)g(pro)m(v)m(ed)g(in)f
([30)q(],)i(concerning)f(another)g(barrier)f(whic)m(h)456
4120 y(will)27 b(b)s(e)j(used)g(in)f(the)h(course)h(of)f(the)h(pro)s
(of)f(of)g(the)h(main)e(results:)456 4252 y FH(Lemma)38
b(2.20.)45 b Fo(Ther)-5 b(e)38 b(exist)f(universal)g(c)-5
b(onstants)2370 4228 y FA(\026)2370 4252 y FG(l)35 b(>)e
FA(1)38 b Fo(and)g FA(0)33 b FG(<)i FA(\026)-47 b FG(c)34
b Fx(\024)f FA(1)p FG(=)p FA(2)p Fo(,)39 b(so)456 4360
y(that,)33 b(if)g FG(l)27 b Fx(\025)907 4336 y FA(\026)908
4360 y FG(l)r Fo(,)32 b(we)h(c)-5 b(an)33 b(\014nd)g
FG(T)1533 4375 y Fv(l)1585 4360 y Fx(2)25 b FA([)r(\026)-47
b FG(cl)r(;)15 b(l)r(=)p FA(2])34 b Fo(and)g(a)f(nonde)-5
b(cr)g(e)g(asing)35 b(function)911 4521 y FG(g)954 4536
y Fv(l)1006 4521 y Fx(2)25 b FG(C)1164 4483 y Fy(0)1203
4521 y FA(\()p Fx(\0001)p FG(;)15 b(T)1493 4536 y Fv(l)1519
4521 y FA(\))21 b Fx(\\)f FG(C)1728 4483 y Fy(1)p Fv(;)p
Fy(1)1821 4521 y FA(\()p Fx(\0001)p FG(;)15 b FA(0\))22
b Fx(\\)e FG(C)2313 4483 y Fy(2)2351 4521 y FA(\(\()p
Fx(\000)r FA(\026)-47 b FG(c)q(l)r(;)15 b(T)2654 4536
y Fv(l)2681 4521 y FA(\))20 b Fx(n)h(f)p FA(0)p Fx(g)p
FA(\))456 4677 y Fo(which)j(is)g(c)-5 b(onstant)26 b(in)e
FA(\()p Fx(\0001)p FG(;)15 b Fx(\000)p FG(l)r(=)p FA(2])p
Fo(,)26 b(with)f FG(g)1989 4644 y Ft(0)1986 4705 y Fv(l)2038
4677 y FG(>)g FA(0)g Fo(in)e FA([)p Fx(\000)r FA(\026)-47
b FG(c)q(l)r(;)15 b(T)2564 4692 y Fv(l)2590 4677 y FA(])p
Fo(,)26 b(satisfying)e FG(g)3107 4692 y Fv(l)3134 4677
y FA(\(0\))i(=)f(0)p Fo(,)456 4785 y FG(g)499 4800 y
Fv(l)525 4785 y FA(\()p FG(T)613 4800 y Fv(l)639 4785
y FA(\))h(=)f(1)33 b Fo(and)h(such)e(that,)i(if)e(we)h(de\014ne)456
4949 y FA(\(2.58\))747 b(\011)1504 4911 y Fv(y)r(;l)1586
4949 y FA(\()p FG(x)p FA(\))41 b(:=)g FG(g)1929 4964
y Fv(l)1955 4949 y FA(\()p Fx(j)p FG(x)21 b Fx(\000)f
FG(y)s Fx(j)g(\000)g FG(l)r FA(\))15 b FG(;)456 5108
y Fo(then)49 b FA(\011)745 5075 y Fv(y)r(;l)828 5108
y FA(\()p FG(x)p FA(\))h Fo(is)f(a)g(strict)h(sup)-5
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y Fs(l)590 5231 y Fy(+)p Fv(l)671 5216 y FA(\()p FG(y)s
FA(\))20 b Fx(n)h FG(@)5 b(B)997 5231 y Fv(l)1023 5216
y FA(\()p FG(y)s FA(\))p Fo(.)p eop
%%Page: 26 26
26 25 bop 456 251 a Fy(26)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)555 450 y Fo(Mor)j(e)36 b(pr)-5 b(e)g(cisely,)38
b FG(g)1235 465 y Fv(l)1297 450 y Fo(is)e(c)-5 b(onstructe)g(d)37
b(as)f(fol)5 b(lows.)53 b(Ther)-5 b(e)36 b(exists)g(a)g(suitable)g(c)-5
b(on-)456 558 y(stants)33 b FA(0)26 b FG(<)f(c)926 572
y Fy(1)991 558 y FG(<)p 1087 485 72 4 v 25 w(C)1159 572
y Fy(1)1198 558 y FG(;)p 1238 485 V 15 w(C)1310 572 y
Fy(2)1382 558 y Fo(so)33 b(that,)h(if)e(we)h(de\014ne)813
726 y FG(s)856 741 y Fv(l)980 726 y FA(:=)98 b FG(e)1216
689 y Ft(\000)p 1271 651 31 3 v Fv(c)1302 698 y Fq(1)1337
689 y Fv(l)1378 726 y FG(;)690 1111 y(h)742 1126 y Fv(l)769
1111 y FA(\()p FG(s)p FA(\))g(:=)1174 788 y Fu(8)1174
870 y(>)1174 897 y(>)1174 924 y(>)1174 952 y(>)1174 979
y(>)1174 1006 y(<)1174 1170 y(>)1174 1197 y(>)1174 1224
y(>)1174 1252 y(>)1174 1279 y(>)1174 1306 y(:)1475 877
y FG(h)1527 891 y Fy(0)1567 877 y FA(\()p FG(s)p FA(\))20
b Fx(\000)g FG(h)1843 891 y Fy(0)1883 877 y FA(\()p FG(s)1961
892 y Fv(l)2007 877 y Fx(\000)g FA(1\))h Fx(\000)p 2300
785 55 3 v 2300 839 a Fv(C)2355 848 y Fq(2)p 2300 856
90 4 v 2334 908 a Fv(l)2415 803 y Fu(\000)2457 877 y
FA(\(1)g(+)e FG(s)p FA(\))2726 844 y Fv(p)2786 877 y
Fx(\000)h FG(s)2920 833 y Fv(p)2920 906 y(l)2959 803
y Fu(\001)1897 985 y Fo(if)93 b FA(\()p FG(s)2124 1000
y Fv(l)2170 985 y Fx(\000)20 b FA(1\))26 b FG(<)f(s)g(<)g
FA(0)1255 1230 y FG(h)1307 1244 y Fy(0)1347 1230 y FA(\()p
FG(s)p FA(\))20 b(+)g FG(h)1623 1244 y Fy(0)1663 1230
y FA(\(1)h Fx(\000)f FG(s)1898 1245 y Fv(l)1924 1230
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1201 y Fq(2)p 2080 1209 90 4 v 2114 1261 a Fv(l)2195
1129 y Fu(\020)2249 1230 y FA(\(1)h Fx(\000)f FG(s)p
FA(\))2519 1197 y Fv(p)2579 1230 y FA(+)g FG(s)2713 1182
y Fy(\()p Fv(p)p Ft(\000)p Fy(1\))2713 1259 y Fv(l)2897
1230 y FA(\(1)h Fx(\000)f FG(s)p FA(\))3167 1129 y Fu(\021)2010
1364 y Fo(if)92 b FA(0)26 b Fx(\024)f FG(s)g(<)g FA(1)p
FG(;)667 1598 y(H)743 1613 y Fv(l)769 1598 y FA(\()p
FG(s)p FA(\))98 b(:=)1174 1475 y Fu(Z)1265 1501 y Fv(s)1225
1681 y Fy(0)1354 1537 y FA(\()p FG(p)20 b Fx(\000)g FA(1\))1637
1464 y Fq(1)p 1637 1476 33 3 v 1637 1517 a Fs(p)p 1327
1577 384 4 v 1327 1693 a FA(\()p FG(p)15 b(h)1475 1708
y Fv(l)1502 1693 y FA(\()p FG(\020)7 b FA(\)\))1665 1620
y Fq(1)p 1664 1632 33 3 v 1664 1674 a Fs(p)1735 1598
y FG(d\020)22 b(;)654 1917 y(H)730 1931 y Fy(0)769 1917
y FA(\()p FG(s)p FA(\))98 b(:=)1174 1794 y Fu(Z)1265
1820 y Fv(s)1225 2000 y Fy(0)1361 1856 y FA(\()p FG(p)20
b Fx(\000)g FA(1\))1644 1783 y Fq(1)p 1644 1795 V 1644
1836 a Fs(p)p 1327 1896 397 4 v 1327 2012 a FA(\()p FG(p)15
b(h)1475 2026 y Fy(0)1515 2012 y FA(\()p FG(\020)7 b
FA(\)\))1678 1939 y Fq(1)p 1678 1951 33 3 v 1678 1993
a Fs(p)1749 1917 y FG(d\020)22 b(;)234 b Fo(for)33 b(any)g
FG(s)25 b Fx(2)g FA(\()p Fx(\000)p FA(1)p FG(;)15 b FA(1\))p
Fo(,)456 2165 y(then)33 b(the)g(fol)5 b(lowing)33 b(holds:)621
2302 y FA(\(i\))41 b FG(h)810 2317 y Fv(l)837 2302 y
FA(\()p FG(s)p FA(\))47 b FG(>)f FA(0)f Fo(in)f FA(\()p
FG(s)1405 2317 y Fv(l)1460 2302 y Fx(\000)29 b FA(1\))47
b FG(<)g(s)f(<)h FA(1)p Fo(;)j(in)45 b(p)-5 b(articular,)49
b FG(H)2793 2317 y Fv(l)2863 2302 y Fo(is)c(wel)5 b(l)44
b(de\014ne)-5 b(d)758 2410 y(and)36 b(strictly)f(incr)-5
b(e)g(asing)36 b(for)f FA(\()p FG(s)1900 2425 y Fv(l)1948
2410 y Fx(\000)21 b FA(1\))30 b FG(<)e(s)h(<)f FA(1)35
b Fo(and)h(thus)f(we)g(may)g(de\014ne)758 2522 y FG(g)801
2537 y Fv(l)828 2522 y FA(\()p FG(t)p FA(\))41 b(:=)f
FG(H)1191 2484 y Ft(\000)p Fy(1)1184 2551 y Fv(l)1285
2522 y FA(\()p FG(t)p FA(\))33 b Fo(for)h FG(t)25 b Fx(2)g
FG(H)1787 2537 y Fv(l)1812 2522 y FA(\()p FG(s)1890 2537
y Fv(l)1936 2522 y Fx(\000)20 b FA(1)p FG(;)31 b FA(1\))p
Fo(;)596 2630 y FA(\(ii\))40 b FG(g)801 2645 y Fv(l)828
2630 y FA(\()p FG(t)p FA(\))33 b Fo(is)f(de\014ne)-5
b(d)34 b(to)f(b)-5 b(e)33 b(c)-5 b(onstantly)35 b(e)-5
b(qual)33 b(to)g FG(s)2401 2645 y Fv(l)2447 2630 y Fx(\000)19
b FA(1)33 b Fo(for)h FG(t)25 b Fx(\024)g FG(H)2991 2645
y Fv(l)3016 2630 y FA(\()p FG(s)3094 2645 y Fv(l)3140
2630 y Fx(\000)20 b FA(1\))p Fo(;)570 2738 y FA(\(iii\))40
b Fo(the)33 b(fol)5 b(lowing)34 b(estimates)g(on)f FG(H)1901
2753 y Fv(l)1959 2738 y Fo(hold:)1012 2952 y FG(H)1088
2967 y Fv(l)1114 2952 y FA(\(1\))84 b Fx(\024)1485 2891
y FG(l)p 1477 2931 46 4 v 1477 3014 a FA(2)1547 2952
y(;)-1116 b(\(2.59\))832 3174 y FG(H)908 3189 y Fv(l)934
3174 y FA(\()p FG(s)1012 3189 y Fv(l)1058 3174 y Fx(\000)20
b FA(1\))84 b Fx(\025)f(\000)1556 3113 y FG(l)p 1548
3153 V 1548 3237 a FA(2)1618 3174 y(;)-1187 b(\(2.60\))1002
3414 y FG(H)1078 3428 y Fy(0)1117 3414 y FA(\()p FG(s)p
FA(\))83 b Fx(\024)g FG(H)1543 3429 y Fv(l)1568 3414
y FA(\()p FG(s)p FA(\))21 b Fx(\000)p 1803 3279 72 4
v 1803 3352 a FG(C)1874 3366 y Fy(1)p 1803 3393 111 4
v 1844 3476 a FG(l)1939 3414 y FA(log)q(\(1)g Fx(\000)f(j)p
FG(s)p Fx(j)p FA(\))186 b Fx(8j)p FG(s)p Fx(j)25 b FG(<)g
FA(1)c Fx(\000)f FG(e)3026 3376 y Ft(\000)p 3091 3311
28 3 v 3091 3340 a Fs(c)3119 3355 y Fq(1)3153 3340 y
Fs(l)p 3091 3361 83 3 v 3117 3402 a Fq(2)3203 3414 y
FA(;)-2772 b(\(2.61\))697 3620 y FG(H)773 3635 y Fv(l)799
3620 y FA(\(1)21 b Fx(\000)f FG(e)1033 3582 y Ft(\000)p
1098 3517 28 3 v 1098 3546 a Fs(c)1126 3561 y Fq(1)1160
3546 y Fs(l)p 1098 3567 83 3 v 1124 3608 a Fq(2)1195
3620 y FA(\))83 b Fx(\025)i FA(\026)-47 b FG(c)15 b(l)i
FA(;)-1134 b(\(2.62\))697 3786 y FG(H)773 3801 y Fv(l)799
3786 y FA(\()p FG(e)876 3748 y Ft(\000)p 941 3683 28
3 v 941 3712 a Fs(c)969 3727 y Fq(1)1004 3712 y Fs(l)p
941 3733 83 3 v 967 3774 a Fq(2)1058 3786 y Fx(\000)20
b FA(1\))84 b Fx(\024)f(\000)r FA(\026)-47 b FG(c)15
b(l)i(:)-1205 b FA(\(2.63\))555 3952 y(A)37 b(detailed)f(pro)s(of)h(of)
g(Lemma)g(2.20)h(is)e(con)m(tained)h(in)f([30)q(])h(\(see,)j(in)c
(particular,)456 4060 y(Lemma)30 b(5.1)i(there\).)555
4168 y(W)-8 b(e)42 b(no)m(w)e(p)s(oin)m(t)f(out)h(some)h(prop)s(erties)
d(of)i(the)h(touc)m(hing)e(p)s(oin)m(ts)g(b)s(et)m(w)m(een)i(the)456
4276 y(barrier)36 b(\011)833 4243 y Fv(y)r(;l)953 4276
y FA(and)i(a)g(\(sub\)solution)e(of)j(\(1.5\).)65 b(T)-8
b(o)38 b(this)f(end,)j(w)m(e)e(notice)h(that,)h(b)m(y)456
4386 y(construction,)28 b(the)h(radially)d(increasing)h(function)g
(\011)2332 4353 y Fv(y)r(;l)2442 4386 y FA(built)f(in)h(Lemma)i(2.20)h
(is)d(so)456 4494 y(that:)671 4631 y Fx(\017)42 b FA(\011)829
4598 y Fv(y)r(;l)942 4631 y FA(is)30 b(de\014ned)f(in)g
FG(B)1522 4646 y Fv(T)1563 4658 y Fs(l)1587 4646 y Fy(+)p
Fv(l)1668 4631 y FA(\()p FG(y)s FA(\),)i(and)f(it)g(is)f(greater)j
(than)e FG(s)2765 4646 y Fv(l)2811 4631 y Fx(\000)20
b FA(1;)671 4745 y Fx(\017)42 b FA(there)35 b(exists)f
FG(\032)1297 4760 y Fv(l)1355 4745 y Fx(2)e FA([)r(\026)-47
b FG(c)q(l)r(;)15 b(l)r(=)p FA(2])35 b(so)g(that)g(the)g(only)f
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4745 y FA(are)758 4855 y(in)29 b FG(B)933 4870 y Fv(l)q
Ft(\000)p Fv(\032)1046 4882 y Fs(l)1075 4855 y FA(\()p
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4937 y Fv(y)r(;l)937 4970 y FA(=)25 b(0)31 b(on)f FG(@)5
b(B)1357 4985 y Fv(l)1383 4970 y FA(\()p FG(y)s FA(\).)456
5108 y(The)37 b(geometry)i(of)f(suc)m(h)f(spheres)g(is)g(related)g
(with)g(p)s(ossible)e(touc)m(hing)i(p)s(oin)m(ts,)i(as)456
5216 y(next)30 b(result)f(sho)m(ws:)p eop
%%Page: 27 27
27 26 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(27)456
450 y FH(Lemma)34 b(2.21.)43 b Fo(Fix)34 b FG(y)c Fx(2)c
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FA(\012)p Fo(.)64 b(Supp)-5 b(ose)41 b(that)g FG(u)e
Fx(2)f FG(C)3269 525 y Fy(1)3308 558 y FA(\(\012\))456
666 y Fo(and)33 b(that)h Fx(j)p FG(u)p Fx(j)26 b(\024)f
FA(1)p Fo(.)42 b(Then)33 b(the)g(fol)5 b(lowing)34 b(r)-5
b(esults)33 b(hold:)671 823 y Fx(\017)42 b Fo(If)e FA(\011)932
790 y Fv(y)r(;l)1055 823 y Fo(touches)g(the)h(gr)-5 b(aph)41
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b(oint)42 b FG(x)3128 790 y Fv(?)3207 823 y Fo(in)e(the)758
931 y(closur)-5 b(e)25 b(of)g FA(\012)r Fx(\\)r FG(B)1354
946 y Fv(T)1395 958 y Fs(l)1418 946 y Fy(+)p Fv(l)1499
931 y FA(\()p FG(y)s FA(\))p Fo(,)h(then,)g(either)e
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b Fo(or)h FG(u)p FA(\()p FG(x)2738 898 y Fv(?)2778 931
y FA(\))g(=)g(\011)3005 898 y Fv(y)r(;l)3088 931 y FA(\()p
FG(x)3175 898 y Fv(?)3214 931 y FA(\))h(=)f(0)p Fo(.)671
1039 y Fx(\017)42 b Fo(If)1201 1203 y FG(B)1270 1218
y Fv(l)q Fy(+)p Fv(T)1388 1230 y Fs(l)1416 1203 y FA(\()p
FG(y)s FA(\))26 b Fx(\032)f(f)p FG(x)g Fx(2)g FA(\012)g
Fx(j)h FG(u)p FA(\()p FG(x)p FA(\))g Fx(\024)e(\000)p
FA(1)d(+)f FG(\022)2575 1166 y Ft(\003)2613 1203 y Fx(g)15
b FG(;)758 1386 y Fo(then,)1629 1552 y FG(u)p FA(\()p
FG(x)p FA(\))41 b FG(<)f FA(\011)2026 1515 y Fv(y)r(;l)2108
1552 y FA(\()p FG(x)p FA(\))15 b FG(;)758 1736 y Fo(for)34
b(any)f FG(x)25 b Fx(2)g FG(B)1310 1751 y Fv(l)q Fy(+)p
Fv(T)1428 1763 y Fs(l)1456 1736 y FA(\()p FG(y)s FA(\))p
Fo(.)555 1891 y FA(F)-8 b(or)21 b(the)g(pro)s(of)e(of)i(Lemma)f(2.21,)k
(w)m(e)d(refer)f(to)h([30)q(])g(\(see,)i(in)c(particular,)i(Lemma)f
(6.2)456 1998 y(and)41 b(Corollary)g(6.4)i(there\).)76
b(W)-8 b(e)43 b(no)m(w)f(notice)g(that)h(a)f(statemen)m(t)i(analogous)f
(to)456 2106 y(Lemma)30 b(2.20)i(holds)d(for)h(a)h(subsolution)d
(\(instead)i(of)g(sup)s(ersolution\))e(prop)s(ert)m(y:)456
2261 y FH(Lemma)38 b(2.22.)45 b Fo(Ther)-5 b(e)38 b(exist)f(universal)g
(c)-5 b(onstants)2370 2237 y FA(\026)2370 2261 y FG(l)35
b(>)e FA(1)38 b Fo(and)g FA(0)33 b FG(<)i FA(\026)-47
b FG(c)34 b Fx(\024)f FA(1)p FG(=)p FA(2)p Fo(,)39 b(so)456
2369 y(that,)33 b(if)g FG(l)27 b Fx(\025)907 2345 y FA(\026)908
2369 y FG(l)r Fo(,)32 b(we)h(c)-5 b(an)33 b(\014nd)g
FG(T)1533 2384 y Fv(l)1585 2369 y Fx(2)25 b FA([)r(\026)-47
b FG(cl)r(;)15 b(l)r(=)p FA(2])34 b Fo(and)g(a)f(nonde)-5
b(cr)g(e)g(asing)35 b(function)877 2575 y Fu(e)-52 b
FG(g)919 2590 y Fv(l)971 2575 y Fx(2)24 b FG(C)1128 2537
y Fy(0)1167 2575 y FA(\()p Fx(\000)p FG(T)1326 2590 y
Fv(l)1353 2575 y FG(;)15 b FA(+)p Fx(1)p FA(\))20 b Fx(\\)g
FG(C)1763 2537 y Fy(1)p Fv(;)p Fy(1)1857 2575 y FA(\(0)p
FG(;)15 b FA(+)p Fx(1)p FA(\))21 b Fx(\\)f FG(C)2348
2537 y Fy(2)2387 2575 y FA(\(\()p Fx(\000)p FG(T)2581
2590 y Fv(l)2607 2575 y FG(;)e FA(\026)-48 b FG(c)q(l)r
FA(\))21 b Fx(n)f(f)p FA(0)p Fx(g)p FA(\))456 2777 y
Fo(which)32 b(is)f(c)-5 b(onstant)33 b(in)e FA([)p FG(l)r(=)p
FA(2)p FG(;)15 b FA(+)p Fx(1)p FA(\))p Fo(,)33 b(with)g
Fu(e)-52 b FG(g)1961 2744 y Ft(0)1958 2804 y Fv(l)2010
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y Fv(l)3134 2777 y FA(\(0\))26 b(=)f(0)p Fo(,)457 2885
y Fu(e)-52 b FG(g)499 2900 y Fv(l)525 2885 y FA(\()p
Fx(\000)p FG(T)684 2900 y Fv(l)710 2885 y FA(\))26 b(=)f
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3075 y Fu(e)1433 3098 y FA(\011)1504 3061 y Fv(y)r(;l)1586
3098 y FA(\()p FG(x)p FA(\))41 b(:=)g Fu(e)-51 b FG(g)1929
3113 y Fv(l)1955 3098 y FA(\()p FG(l)22 b Fx(\000)e(j)p
FG(x)h Fx(\000)f FG(y)s Fx(j)p FA(\))15 b FG(;)456 3312
y Fo(then)659 3289 y Fu(e)648 3312 y FA(\011)719 3279
y Fv(y)r(;l)802 3312 y FA(\()p FG(x)p FA(\))24 b Fo(is)f(a)h(strict)g
(subsolution)g(of)g(\(1.5\))g(in)f(the)h(visc)-5 b(osity)24
b(sense,)h(in)e FG(B)3143 3327 y Fv(T)3184 3339 y Fs(l)3209
3327 y Fy(+)p Fv(l)3289 3312 y FA(\()p FG(y)s FA(\))p
Fx(n)456 3420 y FG(@)5 b(B)578 3435 y Fv(l)604 3420 y
FA(\()p FG(y)s FA(\))p Fo(.)555 3541 y(A)n(lso,)33 b(if)f(we)h
(de\014ne)1274 3517 y Fu(e)1276 3541 y FG(h)1328 3555
y Fy(0)1367 3541 y FA(\()p FG(s)p FA(\))26 b(:=)f FG(h)1679
3555 y Fy(0)1719 3541 y FA(\()p Fx(\000)p FG(s)p FA(\))33
b Fo(and)1445 3787 y Fu(e)1423 3810 y FG(H)1499 3824
y Fy(0)1538 3810 y FA(\()p FG(t)p FA(\))26 b(:=)1788
3687 y Fu(Z)1879 3713 y Fv(t)1839 3893 y Fy(0)1934 3749
y FA(\()p FG(p)20 b Fx(\000)g FA(1\))2206 3716 y Fy(1)p
Fv(=p)2332 3749 y FG(d\020)p 1934 3789 493 4 v 1955 3890
a FA(\()p FG(p)2050 3866 y Fu(e)2051 3890 y FG(h)2103
3904 y Fy(0)2143 3890 y FA(\()p FG(\020)7 b FA(\)\))2295
3864 y Fy(1)p Fv(=p)2451 3810 y FG(;)456 4104 y Fo(then,)40
b(ther)-5 b(e)40 b(exists)f(a)g(strictly)h(incr)-5 b(e)g(asing)40
b(function)2398 4081 y Fu(e)2377 4104 y FG(H)2453 4119
y Fv(l)2517 4104 y Fo(and)g(a)f(p)-5 b(ositive)40 b(function)456
4212 y FG(h)508 4227 y Fv(l)534 4212 y Fo(,)32 b(such)h(that)1554
4415 y Fu(e)1533 4438 y FG(H)1616 4401 y Ft(0)1609 4461
y Fv(l)1639 4438 y FA(\()p FG(s)p FA(\))25 b(=)1909 4377
y(\()p FG(p)20 b Fx(\000)g FA(1\))2181 4344 y Fy(1)p
Fv(=p)p 1883 4417 434 4 v 1883 4518 a FA(\()p FG(p)1978
4494 y Fu(e)1979 4518 y FG(h)2031 4533 y Fv(l)2058 4518
y FA(\()p FG(s)p FA(\)\))2206 4491 y Fy(1)p Fv(=p)2342
4438 y FG(;)456 4687 y Fo(so)33 b(that)h(the)f(fol)5
b(lowing)33 b(holds:)621 4850 y FA(\(i\))757 4826 y Fu(e)758
4850 y FG(h)810 4865 y Fv(l)837 4850 y FA(\()p FG(s)p
FA(\))c Fo(is)f(de\014ne)-5 b(d)30 b(and)f(strictly)h(p)-5
b(ositive)29 b(in)g Fx(\000)p FA(1)c FG(<)g(s)g(<)g FA(1)11
b Fx(\000)g FG(s)2862 4865 y Fv(l)2889 4850 y Fo(;)2968
4827 y Fu(e)2947 4850 y FG(H)3023 4865 y Fv(l)3077 4850
y Fo(is)28 b(de\014ne)-5 b(d)758 4970 y(and)46 b(strictly)f(incr)-5
b(e)g(asing)46 b(for)f Fx(\000)p FA(1)i FG(<)f(s)g(<)h
FA(1)29 b Fx(\000)g FG(s)2566 4985 y Fv(l)2591 4970 y
Fo(;)51 b Fu(e)-52 b FG(g)2712 4985 y Fv(l)2739 4970
y FA(\()p FG(t)p FA(\))47 b(=)3028 4947 y Fu(e)3007 4970
y FG(H)3090 4931 y Ft(\000)p Fy(1)3083 4999 y Fv(l)3184
4970 y FA(\()p FG(t)p FA(\))e Fo(for)758 5096 y FG(t)25
b Fx(2)924 5073 y Fu(e)902 5096 y FG(H)978 5111 y Fv(l)1004
5096 y FA(\(1)p FG(;)31 b FA(1)21 b Fx(\000)f FG(s)1340
5111 y Fv(l)1365 5096 y FA(\))p Fo(;)596 5216 y FA(\(ii\))41
b Fu(e)-52 b FG(g)801 5231 y Fv(l)828 5216 y FA(\()p
FG(t)p FA(\))33 b Fo(is)f(c)-5 b(onstantly)35 b(e)-5
b(qual)33 b(to)g FA(1)21 b Fx(\000)f FG(s)2030 5231 y
Fv(l)2088 5216 y Fo(for)33 b FG(t)25 b Fx(\025)2409 5193
y Fu(e)2387 5216 y FG(H)2463 5231 y Fv(l)2489 5216 y
FA(\(1)c Fx(\000)f FG(s)2724 5231 y Fv(l)2749 5216 y
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%%Page: 28 28
28 27 bop 456 251 a Fy(28)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)570 456 y FA(\(iii\))40 b Fo(the)33 b(fol)5 b(lowing)34
b(estimates)g(on)1846 433 y Fu(e)1825 456 y FG(H)1901
471 y Fv(l)1959 456 y Fo(hold:)978 657 y Fu(e)957 680
y FG(H)1033 695 y Fv(l)1058 680 y FA(\()p Fx(\000)p FA(1\))84
b Fx(\025)f(\000)1571 618 y FG(l)p 1563 659 46 4 v 1563
742 a FA(2)1633 680 y FG(;)869 879 y Fu(e)848 902 y FG(H)924
917 y Fv(l)949 902 y FA(\(1)21 b Fx(\000)f FG(s)1184
917 y Fv(l)1210 902 y FA(\))83 b Fx(\024)1500 841 y FG(l)p
1492 881 V 1492 964 a FA(2)1562 902 y FG(;)1038 1119
y Fu(e)1017 1142 y FG(H)1093 1156 y Fy(0)1132 1142 y
FA(\()p FG(s)p FA(\))g Fx(\025)1503 1119 y Fu(e)1482
1142 y FG(H)1558 1157 y Fv(l)1583 1142 y FA(\()p FG(s)p
FA(\))21 b(+)p 1818 1007 72 4 v 1818 1080 a FG(C)1889
1094 y Fy(1)p 1818 1121 111 4 v 1859 1204 a FG(l)1954
1142 y FA(log)q(\(1)g Fx(\000)f(j)p FG(s)p Fx(j)p FA(\))187
b Fx(8j)p FG(s)p Fx(j)24 b FG(<)h FA(1)c Fx(\000)f FG(e)3041
1104 y Ft(\000)p 3106 1039 28 3 v 3106 1068 a Fs(c)3134
1083 y Fq(1)3169 1068 y Fs(l)p 3106 1089 83 3 v 3132
1130 a Fq(2)3218 1142 y FG(;)663 1324 y Fu(e)642 1347
y FG(H)718 1362 y Fv(l)743 1347 y FA(\()p Fx(\000)p FA(1)h(+)f
FG(e)1048 1310 y Ft(\000)p 1113 1245 28 3 v 1113 1274
a Fs(c)1141 1289 y Fq(1)1176 1274 y Fs(l)p 1113 1295
83 3 v 1139 1336 a Fq(2)1210 1347 y FA(\))83 b Fx(\024)g(\000)r
FA(\026)-47 b FG(c)15 b(l)i(;)713 1513 y(H)789 1528 y
Fv(l)814 1513 y FA(\(1)k Fx(\000)f FG(e)1048 1476 y Ft(\000)p
1113 1411 28 3 v 1113 1440 a Fs(c)1141 1455 y Fq(1)1176
1440 y Fs(l)p 1113 1461 83 3 v 1139 1502 a Fq(2)1210
1513 y FA(\))83 b Fx(\025)i FA(\026)-47 b FG(c)15 b(l)i(:)456
1724 y Fo(Pr)-5 b(o)g(of.)43 b FA(Notice)h(that)1243
1700 y Fu(e)1245 1724 y FG(h)1297 1738 y Fy(0)1380 1724
y FA(satis\014es)f(the)g(same)h(assumption)e(as)h FG(h)2808
1738 y Fy(0)2848 1724 y FA(,)k(th)m(us,)f(w)m(e)e(can)456
1841 y(use)34 b(Lemma)i(2.20)g(with)1353 1817 y Fu(e)1355
1841 y FG(h)1407 1855 y Fy(0)1482 1841 y FA(replacing)e
FG(h)1928 1855 y Fy(0)1967 1841 y FA(:)51 b(let)35 b(us)f(denote)i(b)m
(y)e FG(h)2780 1796 y Fv(])2780 1870 y(l)2812 1841 y
FA(,)j FG(H)2957 1796 y Fv(])2950 1870 y(l)3023 1841
y FA(and)e FG(g)3251 1796 y Fv(])3248 1870 y(l)3318 1841
y FA(the)456 1968 y(functions)27 b(th)m(us)i(obtained)f(via)h(Lemma)g
(2.20)i(with)2283 1944 y Fu(e)2285 1968 y FG(h)2337 1982
y Fy(0)2406 1968 y FA(replacing)c FG(h)2845 1982 y Fy(0)2885
1968 y FA(.)40 b(Then,)29 b(de\014ne)1610 2129 y Fu(e)1612
2153 y FG(h)1664 2168 y Fv(l)1690 2153 y FA(\()p FG(s)p
FA(\))41 b(:=)f FG(h)2032 2108 y Fv(])2032 2182 y(l)2064
2153 y FA(\()p Fx(\000)p FG(s)p FA(\))15 b FG(:)456 2329
y FA(Th)m(us,)1242 2507 y Fu(e)1221 2530 y FG(H)1297
2545 y Fv(l)1322 2530 y FA(\()p FG(s)p FA(\))41 b(:=)1612
2406 y Fu(Z)1703 2433 y Fv(s)1663 2612 y Fy(0)1765 2468
y FA(\()p FG(p)21 b Fx(\000)e FA(1\))2037 2435 y Fy(1)p
Fv(=p)p 1765 2509 384 4 v 1804 2610 a FA(\()p FG(p)1883
2586 y Fu(e)1885 2610 y FG(h)1937 2625 y Fv(l)1964 2610
y FA(\))1999 2583 y Fy(1)p Fv(=p)2199 2530 y FA(=)40
b Fx(\000)p FG(H)2464 2486 y Fv(])2457 2559 y(l)2495
2530 y FA(\()p Fx(\000)p FG(s)p FA(\))456 2753 y(and)29
b(therefore)1607 2892 y Fu(e)-52 b FG(g)1649 2907 y Fv(l)1675
2892 y FA(\()p FG(t)p FA(\))41 b(=)f Fx(\000)p FG(g)2047
2848 y Fv(])2044 2921 y(l)2079 2892 y FA(\()p Fx(\000)p
FG(t)p FA(\))15 b FG(:)456 3068 y FA(With)41 b(this,)j(let)e(us)f(sho)m
(w)h(that)1646 3045 y Fu(e)1636 3068 y FA(\011)1707 3035
y Fv(y)r(;l)1831 3068 y FA(is)f(a)h(strict)g(viscosit)m(y)g
(subsolution)d(of)k(\(1.5\))456 3187 y(outside)30 b FG(@)5
b(B)892 3202 y Fv(l)918 3187 y FA(\()p FG(y)s FA(\).)44
b(Indeed,)31 b(if)f FG(\036)i FA(is)e(a)h(smo)s(oth)h(function)e(touc)m
(hing)2827 3164 y Fu(e)2817 3187 y FA(\011)2888 3154
y Fv(y)r(;l)3001 3187 y FA(from)h(ab)s(o)m(v)m(e)456
3295 y(at)f FG(x)618 3262 y Fv(?)682 3295 y Fx(62)25
b FG(@)5 b(B)890 3310 y Fv(l)916 3295 y FA(\()p FG(y)s
FA(\),)31 b(then)d FG(')p FA(\()p FG(x)p FA(\))f(:=)e
Fx(\000)p FG(\036)p FA(\()p FG(x)p FA(\))30 b(touc)m(hes)g(from)f(b)s
(elo)m(w)f(at)i(the)f(p)s(oin)m(t)g FG(x)3250 3262 y
Fv(?)3318 3295 y FA(the)456 3403 y(strict)h(viscosit)m(y)g(sup)s
(ersolution)1421 3588 y Fx(\000)1502 3565 y Fu(e)1492
3588 y FA(\011)1563 3550 y Fv(y)r(;l)1645 3588 y FA(\()p
FG(x)p FA(\))c(=)f FG(g)1935 3544 y Fv(])1932 3617 y(l)1967
3588 y FA(\()p Fx(j)p FG(x)c Fx(\000)f FG(y)s Fx(j)g(\000)g
FG(l)r FA(\))15 b FG(:)456 3764 y FA(Th)m(us,)29 b(b)m(y)h(Lemma)h
(2.20,)1069 3949 y(\001)1145 3963 y Fv(p)1185 3949 y
FG(\036)25 b FA(=)g Fx(\000)p FA(\001)1507 3963 y Fv(p)1546
3949 y FG(')h(>)f Fx(\000)p FA(\()p FG(h)1885 3905 y
Fv(])1885 3978 y(l)1917 3949 y FA(\))1952 3911 y Ft(0)1976
3949 y FA(\()p FG(')p FA(\))h(=)2225 3925 y Fu(e)2227
3949 y FG(h)2279 3911 y Ft(0)2279 3972 y Fv(l)2305 3949
y FA(\()p Fx(\000)p FG(')p FA(\))h(=)2626 3925 y Fu(e)2627
3949 y FG(h)2679 3911 y Ft(0)2679 3972 y Fv(l)2706 3949
y FA(\()p FG(\036)p FA(\))456 4144 y(at)k(the)f(p)s(oin)m(t)g
FG(x)1013 4111 y Fv(?)1052 4144 y FA(,)h(whic)m(h)e(sho)m(ws)h(the)g
(desired)f(subsolution)f(prop)s(ert)m(y)i(of)3060 4121
y Fu(e)3050 4144 y FA(\011)3121 4111 y Fv(y)r(;l)3204
4144 y FA(.)555 4327 y(The)44 b(reader)g(ma)m(y)h(easily)f(con)m(vince)
h(herself)e(that)i Fu(e)-51 b FG(g)2461 4342 y Fv(l)2531
4327 y FA(and)2743 4304 y Fu(e)2722 4327 y FG(H)2798
4342 y Fv(l)2868 4327 y FA(also)44 b(enjo)m(y)h(the)456
4435 y(prop)s(erties)28 b(listed)h(ab)s(o)m(v)m(e.)1997
b Fl(\003)555 4633 y FA(W)-8 b(e)22 b(are)e(no)m(w)h(in)e(the)h(p)s
(osition)f(to)i(sho)m(w)f(that)h(minimizers)c(are)k(trapp)s(ed)e(in)g
(b)s(et)m(w)m(een)456 4741 y(the)31 b(functions)f(constructed)i(in)e
(Lemmata)i(2.20-2.22)j(\(an)c(exp)s(onen)m(tial)f(deca)m(y)j(w)m(as)456
4849 y(also)d(p)s(oin)m(ted)f(out)i(in)e Fx(x)p FA(9)i(of)g([30)q(]\):)
456 4991 y FH(Lemma)h(2.23.)43 b Fo(L)-5 b(et)33 b FG(l)27
b(>)e FA(0)p Fo(,)33 b FG(\022)27 b(>)e FA(0)p Fo(.)42
b(L)-5 b(et)33 b FG(u)g Fo(b)-5 b(e)32 b(a)h(lo)-5 b(c)g(al)35
b(minimizer)e(for)g Fx(F)42 b Fo(in)1082 5089 y Fu(n)1143
5190 y FA(\()p FG(x)1230 5152 y Ft(0)1253 5190 y FG(;)15
b(x)1345 5204 y Fv(N)1413 5190 y FA(\))26 b Fx(2)f Fw(R)1619
5152 y Fv(N)7 b Ft(\000)p Fy(1)1803 5190 y Fx(\002)20
b Fw(R)1985 5085 y Fu(\014)1985 5140 y(\014)1985 5194
y(\014)2040 5190 y Fx(j)p FG(x)2117 5152 y Ft(0)2141
5190 y Fx(j)25 b FG(<)g(l)17 b(;)41 b Fx(j)p FG(x)2474
5204 y Fv(N)2541 5190 y Fx(j)26 b FG(<)f(l)2717 5089
y Fu(o)2792 5190 y FG(:)p eop
%%Page: 29 29
29 28 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(29)456
450 y Fo(Assume)31 b(that)i Fx(j)p FG(u)p Fx(j)25 b(\024)g
FA(1)p Fo(,)32 b(that)h FG(\022)s(=l)g Fo(is)e(suitably)i(smal)5
b(l,)32 b(that)h FG(u)p FA(\(0\))26 b(=)f(0)p Fo(,)32
b(that)h FG(u)p FA(\()p FG(x)p FA(\))26 b FG(>)f FA(0)456
558 y Fo(if)g FG(x)589 572 y Fv(N)682 558 y Fx(\025)g
FG(\022)j Fo(and)e(that)h FG(u)p FA(\()p FG(x)p FA(\))f
FG(<)f FA(0)i Fo(if)e FG(x)1698 572 y Fv(N)1791 558 y
Fx(\024)g(\000)p FG(\022)s Fo(.)38 b(Then,)27 b(ther)-5
b(e)27 b(exist)f(suitable)g(c)-5 b(onstants)456 666 y
FG(\024;)31 b Fu(e)-51 b FG(\024;)34 b FA(^)-49 b FG(\024)26
b Fx(2)f FA(\(0)p FG(;)15 b FA(1])34 b Fo(so)g(that)1266
827 y Fu(e)-52 b FG(g)1309 844 y Fk(e)-40 b Fv(\024l)1375
827 y FA(\()p FG(x)1462 841 y Fv(N)1550 827 y Fx(\000)20
b FG(\022)s FA(\))k Fx(\024)h FG(u)p FA(\()p FG(x)p FA(\))h
Fx(\024)f FG(g)2181 842 y Fv(\024l)2248 827 y FA(\()p
FG(x)2335 841 y Fv(N)2423 827 y FA(+)20 b FG(\022)s FA(\))15
b FG(;)456 990 y Fo(at)31 b(any)h(p)-5 b(oint)33 b FG(x)25
b Fx(2)g FA([)p Fx(\000)s FA(^)-48 b FG(\024l)r(;)19
b FA(^)-49 b FG(\024)q(l)r FA(])1452 957 y Fv(N)1520
990 y Fo(,)31 b(pr)-5 b(ovide)g(d)33 b(that)g(the)f(functions)f(ab)-5
b(ove)32 b(ar)-5 b(e)32 b(de\014ne)-5 b(d)33 b(at)456
1098 y FG(x)p Fo(.)750 3225 y @beginspecial 0 @llx 0
@lly 686 @urx 545 @ury 2880 @rwi @setspecial
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8#204 /macron 8#205 /breve 8#206 /dotaccent 8#207 /dieresis
8#210 /ring 8#211 /cedilla 8#212 /hungarumlaut 8#213 /ogonek 8#214 /caron
8#220 /dotlessi 8#230 /oe 8#231 /OE
8#240 /space 8#241 /exclamdown 8#242 /cent 8#243 /sterling
8#244 /currency 8#245 /yen 8#246 /brokenbar 8#247 /section 8#250 /dieresis
8#251 /copyright 8#252 /ordfeminine 8#253 /guillemotleft 8#254 /logicalnot
8#255 /hyphen 8#256 /registered 8#257 /macron 8#260 /degree 8#261 /plusminus
8#262 /twosuperior 8#263 /threesuperior 8#264 /acute 8#265 /mu 8#266 /paragraph
8#267 /periodcentered 8#270 /cedilla 8#271 /onesuperior 8#272 /ordmasculine
8#273 /guillemotright 8#274 /onequarter 8#275 /onehalf
8#276 /threequarters 8#277 /questiondown 8#300 /Agrave 8#301 /Aacute
8#302 /Acircumflex 8#303 /Atilde 8#304 /Adieresis 8#305 /Aring
8#306 /AE 8#307 /Ccedilla 8#310 /Egrave 8#311 /Eacute
8#312 /Ecircumflex 8#313 /Edieresis 8#314 /Igrave 8#315 /Iacute
8#316 /Icircumflex 8#317 /Idieresis 8#320 /Eth 8#321 /Ntilde 8#322 /Ograve
8#323 /Oacute 8#324 /Ocircumflex 8#325 /Otilde 8#326 /Odieresis 8#327 /multiply
8#330 /Oslash 8#331 /Ugrave 8#332 /Uacute 8#333 /Ucircumflex
8#334 /Udieresis 8#335 /Yacute 8#336 /Thorn 8#337 /germandbls 8#340 /agrave
8#341 /aacute 8#342 /acircumflex 8#343 /atilde 8#344 /adieresis 8#345 /aring
8#346 /ae 8#347 /ccedilla 8#350 /egrave 8#351 /eacute
8#352 /ecircumflex 8#353 /edieresis 8#354 /igrave 8#355 /iacute
8#356 /icircumflex 8#357 /idieresis 8#360 /eth 8#361 /ntilde 8#362 /ograve
8#363 /oacute 8#364 /ocircumflex 8#365 /otilde 8#366 /odieresis 8#367 /divide
8#370 /oslash 8#371 /ugrave 8#372 /uacute 8#373 /ucircumflex
8#374 /udieresis 8#375 /yacute 8#376 /thorn 8#377 /ydieresis] def
/Times-Roman /Times-Roman-iso isovec ReEncode
/Palatino-Italic /Palatino-Italic-iso isovec ReEncode
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showpage

%%EndDocument
 @endspecial 727 3409 a FC(T)-7 b(rapping)30 b(a)g(minimizer)h(b)r(et)n
(w)n(een)d(t)n(w)n(o)h(barriers,)h(as)g(in)g(Lemma)e(2.23)456
3752 y Fo(Pr)-5 b(o)g(of.)43 b FA(W)-8 b(e)35 b(pro)s(of)f(the)h
(latter)g(inequalit)m(y)-8 b(,)34 b(the)h(\014rst)f(one)g(b)s(eing)f
(analogous.)54 b(F)-8 b(rom)456 3860 y(Theorem)25 b(1.1)i(of)f([28)q(])
g(and)f(the)h(fact)h(that)f Fx(f)p FG(u)g FA(=)f(0)p
Fx(g)h(\022)f(fj)p FG(x)2487 3874 y Fv(N)2555 3860 y
Fx(j)g(\024)g FG(\022)s Fx(g)p FA(,)i(it)e(easily)g(follo)m(ws)456
3968 y(that)456 4129 y(\(2.64\))188 b FG(u)p FA(\()p
FG(x)p FA(\))26 b FG(<)f Fx(\000)p FA(1)c(+)f FG(\022)1444
4096 y Fv(?)1513 4129 y FA(for)30 b(an)m(y)g FG(x)c Fx(2)f
Fw(R)2046 4096 y Fv(N)2150 4129 y FA(with)k FG(x)2409
4143 y Fv(N)2501 4129 y Fx(\024)c FG(\024)2649 4096 y
Fv(?)2719 4129 y FA(and)30 b Fx(j)p FG(x)2973 4096 y
Ft(0)2997 4129 y Fx(j)25 b(\024)g FG(l)r(=)p FA(2,)456
4290 y(for)40 b(some)g(constan)m(t)i FG(\024)1271 4257
y Fv(?)1351 4290 y FA(\(whic)m(h)d(ma)m(y)i(dep)s(end)d(on)i
FG(\022)2370 4257 y Fv(?)2449 4290 y FA(and)g(other)g(univ)m(ersal)f
(con-)456 4398 y(stan)m(ts\).)555 4506 y(Then,)30 b(from)g(\(2.64\))i
(and)e(the)h(second)f(item)g(in)f(Lemma)i(2.21,)1623
4674 y FG(u)p FA(\()p FG(x)p FA(\))26 b Fx(\024)f FA(\011)1990
4636 y Fv(y)r(;\024l)2114 4674 y FA(\()p FG(x)p FA(\))15
b FG(;)456 4836 y FA(where)27 b FG(y)h FA(:=)d(\(0)p
FG(;)15 b(:)g(:)g(:)j(;)d FA(0)p FG(;)g Fx(\000)p FG(l)r(=)p
FA(2\).)42 b(Let)28 b(no)m(w)g FG(e)d Fx(2)g FG(S)2131
4803 y Fv(N)7 b Ft(\000)p Fy(1)2288 4836 y FA(,)29 b(with)d
FG(e)2588 4850 y Fv(N)2681 4836 y FG(>)f FA(0)j(and)f(let)h(us)f(slide)
456 4947 y(\011)527 4914 y Fv(y)r(;\024l)680 4947 y FA(in)i(the)i
FG(e)p FA(-direction)f(un)m(til)e(it)i(touc)m(hes)i FG(u)p
FA(.)41 b(Notice)31 b(indeed)e(that)i(there)g(exists)f(a)456
5054 y(suitable)e(constan)m(t)34 b(\026)-47 b FG(c)26
b Fx(2)f FA(\(0)p FG(;)15 b FA(1\))32 b(so)f(that)g(if)456
5216 y(\(2.65\))1109 b FG(e)1837 5230 y Fv(N)1930 5216
y Fx(\025)27 b FA(\026)-47 b FG(c)15 b(;)p eop
%%Page: 30 30
30 29 bop 456 251 a Fy(30)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(then)26 b(\011)730 417 y Fv(y)r Fy(+)p
Fv(te;\024l)994 450 y FA(do)s(es)g(touc)m(h)i FG(u)f
FA(for)g(some)h FG(t)d FA(=)g FG(t)p FA(\()p FG(e)p FA(\))h
FG(>)f FA(0)i(at)h(some)f(p)s(oin)m(t)g FG(x)2994 417
y Fv(?)3058 450 y FA(=)e FG(x)3206 417 y Fv(?)3246 450
y FA(\()p FG(e)p FA(\))h Fx(2)456 560 y FA([)p Fx(\000)p
FG(l)r(;)15 b(l)r FA(])675 527 y Fv(N)742 560 y FA(,)31
b(that)g(is)456 717 y(\(2.66\))404 b FG(u)p FA(\()p Fi(x)p
FA(\))26 b Fx(\024)f FA(\011)1437 680 y Fv(y)r Fy(+)p
Fv(te;\024l)1674 717 y FA(\()p Fi(x)p FA(\))h(=)f FG(g)1941
732 y Fv(\024l)2008 717 y FA(\()p Fx(j)p Fi(x)c Fx(\000)f
FA(\()p FG(y)j FA(+)d FG(te)p FA(\))p Fx(j)h(\000)f FG(\024l)r
FA(\))15 b FG(;)456 871 y FA(for)36 b(an)m(y)h Fi(x)e
Fx(2)h FG(B)1012 886 y Fv(T)1053 898 y Fs(\024l)1113
886 y Fy(+)p Fv(\024l)1234 871 y FA(\()p FG(y)28 b FA(+)c
FG(te)p FA(\),)39 b(b)s(eing)c(the)i(latter)f(the)h(domain)e(where)h
(\011)3110 838 y Fv(y)r Fy(+)p Fv(te;\024l)3383 871 y
FA(is)456 979 y(de\014ned,)29 b(and)1527 1094 y FG(u)p
FA(\()p FG(x)1666 1056 y Fv(?)1706 1094 y FA(\))d(=)f(\011)1934
1056 y Fv(y)r Fy(+)p Fv(te;\024l)2170 1094 y FA(\()p
FG(x)2257 1056 y Fv(?)2297 1094 y FA(\))15 b FG(:)456
1226 y FA(In)29 b(the)i(ligh)m(t)e(of)i(the)g(\014rst)e(item)h(in)f
(Lemma)i(2.21,)h(w)m(e)f(ha)m(v)m(e)g(that)1464 1379
y FG(u)p FA(\()p FG(x)1603 1342 y Fv(?)1643 1379 y FA(\))26
b(=)f(\011)1871 1342 y Fv(y)r Fy(+)p Fv(te;\024l)2107
1379 y FA(\()p FG(x)2194 1342 y Fv(?)2234 1379 y FA(\))g(=)g(0)456
1530 y(and)k(so,)i(since,)f(b)m(y)g(our)g(h)m(yp)s(otheses)g
Fx(f)p Fi(x)c Fx(j)g FG(u)p FA(\()p Fi(x)p FA(\))g(=)f(0)p
Fx(g)h(\022)f(fj)p Fi(x)2512 1544 y Fv(N)2581 1530 y
Fx(j)g(\024)g FG(\022)s Fx(g)p FA(,)30 b(w)m(e)h(ha)m(v)m(e)h(that)1762
1681 y Fx(j)p FG(x)1839 1644 y Fv(?)1839 1704 y(N)1906
1681 y Fx(j)26 b(\024)f FG(\022)17 b(:)456 1833 y FA(Let)30
b(us)g(no)m(w)h(consider,)e(for)h FG(d)c Fx(\025)f FA(0,)31
b(the)f(p)s(oin)m(t)1527 1984 y Fi(x)c FA(=)f Fi(x)p
FA(\()p FG(e)p FA(\))i(:=)e FG(x)2025 1946 y Fv(?)2084
1984 y FA(+)20 b FG(de)2264 1998 y Fv(N)2347 1984 y FG(:)456
2135 y FA(Then,)1045 2286 y Fx(j)p Fi(x)g Fx(\000)g FA(\()p
FG(y)k FA(+)19 b FG(te)p FA(\))p Fx(j)84 b(\024)f(j)p
Fi(x)21 b Fx(\000)f FG(x)2001 2249 y Fv(?)2040 2286 y
Fx(j)g FA(+)g Fx(j)p FG(x)2253 2249 y Fv(?)2313 2286
y Fx(\000)g FA(\()p FG(y)k FA(+)19 b FG(te)p FA(\))p
Fx(j)26 b FA(=)1626 2421 y(=)83 b FG(d)21 b FA(+)e FG(\024l)28
b FA(=)1626 2556 y(=)83 b Fi(x)1812 2570 y Fv(N)1900
2556 y Fx(\000)20 b FG(x)2043 2519 y Fv(?)2043 2579 y(N)2130
2556 y FA(+)g FG(\024l)1626 2691 y Fx(\024)83 b Fi(x)1812
2705 y Fv(N)1900 2691 y FA(+)20 b FG(\022)i FA(+)e FG(\024l)d(:)456
2842 y FA(Therefore,)30 b(since)g FG(g)1159 2857 y Fv(l)1215
2842 y FA(is)g(increasing,)f(\(2.66\))k(implies)27 b(that)456
2993 y(\(2.67\))887 b FG(u)p FA(\()p Fi(x)p FA(\))27
b Fx(\024)e FG(g)1893 3008 y Fv(\024l)1960 2993 y FA(\()p
Fi(x)2027 3007 y Fv(N)2115 2993 y FA(+)20 b FG(\022)s
FA(\))15 b FG(;)456 3145 y FA(pro)m(vided)29 b Fi(x)c
Fx(2)g FG(B)1042 3160 y Fv(T)1083 3172 y Fs(\024l)1143
3160 y Fy(+)p Fv(\024l)1265 3145 y FA(\()p FG(y)e FA(+)d
FG(te)p FA(\).)555 3252 y(With)34 b(this)g(inequalit)m(y)-8
b(,)35 b(w)m(e)g(are)g(no)m(w)g(in)e(the)i(p)s(osition)e(to)i(c)m(ho)s
(ose)h FG(e)f FA(here)f(ab)s(o)m(v)m(e)456 3360 y(in)d(order)i(to)h
(infer)e(the)h(desired)f(result.)47 b(W)-8 b(e)35 b(pro)s(ceed)e(in)f
(the)h(follo)m(wing)e(w)m(a)m(y:)48 b(tak)m(e)456 3468
y FG(x)25 b FA(=)g(\()p FG(x)716 3435 y Ft(0)739 3468
y FG(;)15 b(x)831 3482 y Fv(N)899 3468 y FA(\))26 b Fx(2)e
FA([)p Fx(\000)s FA(^)-48 b FG(\024)q(l)r(;)19 b FA(^)-49
b FG(\024l)r FA(])1369 3435 y Fv(N)7 b Ft(\000)p Fy(1)1547
3468 y Fx(\002)20 b FA([)p Fx(\000)s FA(^)-48 b FG(\024l)r(;)19
b FA(^)-49 b FG(\024)q(l)r FA(])31 b(and)e(consider)h(t)m(w)m(o)h
(cases.)555 3576 y(If)456 3727 y(\(2.68\))434 b Fx(f)p
Fi(x)26 b Fx(2)f FA([)p Fx(\000)s FA(^)-48 b FG(\024)q(l)r(;)18
b FA(^)-48 b FG(\024l)r FA(])1633 3690 y Fv(N)1726 3727
y Fx(j)25 b Fi(x)1808 3741 y Fv(N)1901 3727 y Fx(\024)g
FG(x)2049 3741 y Fv(N)2116 3727 y Fx(g)c(\\)f(f)p FG(u)26
b FA(=)f(0)p Fx(g)h(6)p FA(=)f Fx(;)15 b FG(;)456 3879
y FA(tak)m(e)24 b FG(x)697 3846 y Fv(?)759 3879 y FA(so)f(that)g(\()p
FG(x)1139 3846 y Fv(?)1179 3879 y FA(\))1214 3846 y Ft(0)1263
3879 y FA(=)i FG(x)1411 3846 y Ft(0)1434 3879 y FA(,)f
FG(u)p FA(\()p FG(x)1622 3846 y Fv(?)1662 3879 y FA(\))i(=)f(0)e(and)f
FG(x)2108 3846 y Fv(?)2108 3906 y(N)2198 3879 y FA(as)g(lo)m(w)h(as)g
(p)s(ossible)d(\(in)i(particular,)456 3987 y(from)38
b(\(2.68\),)45 b FG(x)1031 3954 y Fv(?)1031 4013 y(N)1138
3987 y Fx(\024)40 b FG(x)1301 4001 y Fv(N)1368 3987 y
FA(\).)68 b(Let)40 b(also,)h(as)f(ab)s(o)m(v)m(e,)j FG(y)g
FA(:=)d(\(0)p FG(;)15 b(:)g(:)g(:)i(;)e FA(0)p FG(;)g
Fx(\000)p FG(l)r(=)p FA(2\).)70 b(Then,)456 4094 y(c)m(ho)s(osing)1675
4251 y FG(e)26 b FA(:=)1899 4190 y FG(x)1951 4157 y Fv(?)2011
4190 y Fx(\000)20 b FG(y)p 1874 4230 301 4 v 1874 4314
a Fx(j)p FG(x)1951 4287 y Fv(?)2011 4314 y Fx(\000)g
FG(y)s Fx(j)2200 4251 y FG(;)456 4437 y FA(w)m(e)44 b(ha)m(v)m(e)i
(that)f FG(x)f FA(and)g FG(x)1377 4404 y Fv(?)1460 4437
y FA(here)g(agree)i(with)d Fi(x)p FA(\()p FG(e)p FA(\))j(and)d
FG(x)2576 4404 y Fv(?)2616 4437 y FA(\()p FG(e)p FA(\))i(constructed)f
(here)456 4545 y(ab)s(o)m(v)m(e)f(and)e(that)h(\(2.65\))i(is)d
(ful\014lled)d(pro)m(vided)43 b(^)-48 b FG(\024)42 b
FA(is)f(small)f(enough.)74 b(Th)m(us,)44 b(the)456 4653
y(desired)28 b(result)i(follo)m(ws,)f(in)h(this)f(case,)i(from)f
(\(2.67\).)555 4761 y(If,)g(on)h(the)f(other)h(hand,)456
4913 y(\(2.69\))434 b Fx(f)p Fi(x)26 b Fx(2)f FA([)p
Fx(\000)s FA(^)-48 b FG(\024)q(l)r(;)18 b FA(^)-48 b
FG(\024l)r FA(])1633 4876 y Fv(N)1726 4913 y Fx(j)25
b Fi(x)1808 4927 y Fv(N)1901 4913 y Fx(\024)g FG(x)2049
4927 y Fv(N)2116 4913 y Fx(g)c(\\)f(f)p FG(u)26 b FA(=)f(0)p
Fx(g)h FA(=)f Fx(;)15 b FG(;)456 5064 y FA(notice)30
b(that)h FG(x)972 5078 y Fv(N)1060 5064 y FA(+)20 b FG(\022)i
FA(+)e FG(\024l)27 b Fx(\025)e FG(\022)e FA(+)d(\()p
FG(\024)h Fx(\000)i FA(^)-48 b FG(\024)p FA(\))p FG(l)27
b(>)e FA(0,)31 b(pro)m(vided)i(^)-49 b FG(\024)26 b Fx(\024)f
FG(\024)31 b FA(and)e(de\014ne)1428 5216 y FG(y)1476
5178 y Fv(?)1540 5216 y FA(:=)c FG(x)20 b Fx(\000)g FA(\()p
FG(x)1911 5230 y Fv(N)1999 5216 y FA(+)g FG(\022)i FA(+)e
FG(\024l)r FA(\))p FG(e)2404 5230 y Fv(N)p eop
%%Page: 31 31
31 30 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(31)456
450 y FA(and)1677 599 y FG(e)26 b FA(:=)1901 537 y FG(y)1949
504 y Fv(?)2009 537 y Fx(\000)19 b FG(y)p 1876 578 297
4 v 1876 661 a Fx(j)p FG(y)1949 635 y Fv(?)2009 661 y
Fx(\000)g FG(y)s Fx(j)2198 599 y FG(:)456 789 y FA(Notice)31
b(also)f(that)h FG(y)1168 757 y Fv(?)1165 816 y(N)1252
789 y FA(+)20 b FG(\024l)28 b FA(=)d Fx(\000)p FG(\022)s
FA(,)k(hence)1021 960 y FG(B)1090 975 y Fv(\024l)1157
960 y FA(\()p FG(y)1240 923 y Fv(?)1280 960 y FA(\))c
Fx(\022)g FA([)p Fx(\000)p FG(\024l)r(;)15 b(\024l)r
FA(])1759 923 y Fv(N)7 b Ft(\000)p Fy(1)1938 960 y Fx(\002)20
b FA(\()p Fx(\0001)p FG(;)15 b Fx(\000)p FG(\022)s FA(])25
b Fx(\022)g(f)p FG(u)g(<)g FA(0)p Fx(g)15 b FG(:)456
1123 y FA(Therefore,)31 b(b)m(y)g(the)h(\014rst)e(item)h(in)f(Lemma)h
(2.21,)j(w)m(e)d(ha)m(v)m(e)i(that)f(\011)2840 1090 y
Fv(y)r Fy(+)p Fv(te;\024l)3107 1123 y FA(do)s(es)f(not)456
1231 y(touc)m(h)g FG(u)f FA(for)g FG(t)25 b Fx(2)g FA([0)p
FG(;)31 b Fx(j)p FG(y)1270 1198 y Fv(?)1330 1231 y Fx(\000)20
b FG(y)s Fx(j)p FA(].)41 b(In)30 b(particular,)f(for)h
FG(t)25 b FA(=)g Fx(j)p FG(y)2510 1198 y Fv(?)2569 1231
y Fx(\000)20 b FG(y)s Fx(j)p FA(,)1469 1398 y FG(u)p
FA(\()p FG(x)p FA(\))83 b Fx(\024)g FA(\011)1951 1361
y Fv(y)r Fy(+)p Fv(te;\024l)2187 1398 y FA(\()p FG(x)p
FA(\))26 b(=)1726 1540 y(=)83 b(\011)1951 1503 y Fv(y)1988
1479 y Fs(?)2024 1503 y Fv(;\024l)2110 1540 y FA(\()p
FG(x)p FA(\))26 b(=)1726 1675 y(=)83 b FG(g)1923 1690
y Fv(\024l)1990 1675 y FA(\()p FG(x)2077 1689 y Fv(N)2165
1675 y FA(+)20 b FG(\022)s FA(\))15 b FG(:)456 1836 y
FA(This)27 b(pro)m(v)m(es)j(the)g(desired)d(result)i(also)g(in)f(case)i
(\(2.69\))i(holds)c(and)h(it)f(completes)i(the)456 1944
y(pro)s(of)f(of)i(Lemma)f(2.23.)2068 b Fl(\003)456 2121
y FH(Corollary)36 b(2.24.)43 b Fo(L)-5 b(et)32 b FG(l)c(>)d
FA(0)p Fo(,)32 b FG(\022)c(>)d FA(0)p Fo(.)42 b(L)-5
b(et)33 b FG(u)g Fo(b)-5 b(e)32 b(a)h(lo)-5 b(c)g(al)34
b(minimizer)g(for)f Fx(F)42 b Fo(in)976 2304 y FA(\012)25
b(:=)1188 2203 y Fu(n)1249 2304 y FA(\()p FG(x)1336 2266
y Ft(0)1360 2304 y FG(;)15 b(x)1452 2318 y Fv(N)1519
2304 y FA(\))26 b Fx(2)f Fw(R)1725 2266 y Fv(N)7 b Ft(\000)p
Fy(1)1909 2304 y Fx(\002)20 b Fw(R)2091 2199 y Fu(\014)2091
2254 y(\014)2091 2308 y(\014)2146 2304 y Fx(j)p FG(x)2223
2266 y Ft(0)2247 2304 y Fx(j)25 b FG(<)g(l)17 b(;)41
b Fx(j)p FG(x)2580 2318 y Fv(N)2647 2304 y Fx(j)26 b
FG(<)f(l)2823 2203 y Fu(o)2899 2304 y FG(:)456 2492 y
Fo(Assume)31 b(that)i Fx(j)p FG(u)p Fx(j)25 b(\024)g
FA(1)p Fo(,)32 b(that)h FG(\022)s(=l)g Fo(is)e(suitably)i(smal)5
b(l,)32 b(that)h FG(u)p FA(\(0\))26 b(=)f(0)p Fo(,)32
b(that)h FG(u)p FA(\()p FG(x)p FA(\))26 b FG(>)f FA(0)456
2600 y Fo(if)41 b FG(x)605 2614 y Fv(N)713 2600 y Fx(\025)h
FG(\022)h Fo(and)g(that)f FG(u)p FA(\()p FG(x)p FA(\))h
FG(<)e FA(0)h Fo(if)f FG(x)1857 2614 y Fv(N)1965 2600
y Fx(\024)g(\000)p FG(\022)s Fo(.)68 b(Then,)44 b(ther)-5
b(e)42 b(exists)g(a)g(suitable)456 2708 y(c)-5 b(onstant)34
b FG(c)26 b Fx(2)e FA(\(0)p FG(;)15 b FA(1])35 b Fo(so)e(that)1421
2868 y FA(\()p Fx(\000)p FA(1)20 b(+)g FG(s)1726 2883
y Fv(cl)1783 2868 y FG(;)30 b FA(1)21 b Fx(\000)f FG(s)2038
2883 y Fv(cl)2094 2868 y FA(\))25 b Fx(\022)g FG(u)p
FA(\(\012\))15 b FG(:)456 3045 y Fo(Pr)-5 b(o)g(of.)43
b FA(By)31 b(Lemma)g(2.23)i(and)d(the)i(fact)g(that)f
FG(g)2153 3060 y Fv(l)2180 3045 y FA(\()p FG(t)p FA(\))c(=)f
Fx(\000)p FA(1)21 b(+)f FG(s)2678 3060 y Fv(l)2735 3045
y FA(if)30 b FG(t)c Fx(\024)g(\000)p FG(l)r(=)p FA(2)32
b(\(recall)456 3153 y(Lemma)e(2.20\),)j(w)m(e)d(ha)m(v)m(e)i(that,)f
(if)f FG(x)1734 3167 y Fv(N)1826 3153 y Fx(\024)25 b(\000)p
FA(3)p FG(\024)q(l)r(=)p FA(4,)1325 3314 y FG(u)p FA(\()p
FG(x)p FA(\))h Fx(\024)f FG(g)1664 3329 y Fv(\024l)1731
3314 y FA(\()p FG(x)1818 3328 y Fv(N)1906 3314 y FA(+)19
b FG(\022)s FA(\))25 b(=)g Fx(\000)p FA(1)c(+)e FG(s)2468
3329 y Fv(\024l)2550 3314 y FG(;)456 3474 y FA(pro)m(vided)30
b FG(\022)s(=l)k FA(small)d(enough.)45 b(Analogously)-8
b(,)33 b(b)m(y)f(Lemma)g(2.23)i(and)d(the)i(fact)g(that)457
3582 y Fu(e)-52 b FG(g)499 3597 y Fv(l)525 3582 y FA(\()p
FG(t)p FA(\))26 b(=)f(1)20 b Fx(\000)g FG(s)949 3597
y Fv(l)1005 3582 y FA(if)30 b FG(t)25 b Fx(\025)g FG(l)r(=)p
FA(2)31 b(\(recall)f(Lemma)g(2.22\),)j(w)m(e)e(get)g(that,)g(if)f
FG(x)2891 3596 y Fv(N)2983 3582 y Fx(\025)25 b FA(3)q
Fu(e)-52 b FG(\024)q(l)r(=)p FA(4,)1360 3743 y FG(u)p
FA(\()p FG(x)p FA(\))26 b Fx(\025)g Fu(e)-52 b FG(g)1700
3759 y Fk(e)-40 b Fv(\024l)1766 3743 y FA(\()p FG(x)1853
3757 y Fv(N)1941 3743 y Fx(\000)20 b FG(\022)s FA(\))25
b(=)g(1)20 b Fx(\000)g FG(s)2434 3759 y Fk(e)-40 b Fv(\024)o(l)2515
3743 y FG(:)456 3903 y FA(The)34 b(inequalities)f(ab)s(o)m(v)m(e)k(and)
d(the)i(con)m(tin)m(uit)m(y)f(of)h FG(u)f FA(\(see)h([15)q(])g(or)f
([34)q(]\))h(imply)d(the)456 4011 y(desired)28 b(result.)2361
b Fl(\003)555 4188 y FA(In)25 b(the)g(follo)m(wing,)g(w)m(e)h(will)c
(often)k(slide)d(the)j(barriers)d(in)h(a)i(giv)m(en)f(direction.)38
b(More)456 4296 y(precisely)-8 b(,)31 b(w)m(e)h(will)d(start)k(from)e
(a)h(con\014guration)f(in)g(whic)m(h)f(the)i(barrier)e(is)h(ab)s(o)m(v)
m(e)i(a)456 4404 y(subsolution)c FG(u)i FA(and)g(then)h(w)m(e)g(slide)e
(the)i(barrier)e(un)m(til)g(it)h(touc)m(hes)h(the)g(graph)f(of)h
FG(u)p FA(.)456 4512 y(With)27 b(some)i(p)s(o)s(etry)-8
b(,)28 b(w)m(e)h(ma)m(y)f(think)f(that)i(the)f(barrier)e(is)h(lik)m(e)h
(a)g(ship)e(whic)m(h)h(mo)m(v)m(es)456 4620 y(forw)m(ard)32
b(un)m(til)f(it)i(touc)m(hes)h(the)f(land)f FG(u)p FA(:)46
b(of)33 b(course,)h(the)f(ship)e(will)g(touc)m(h)i(the)h(land)456
4728 y(with)h(the)h(fore,)i(not)f(with)e(the)h(aft.)59
b(Next)37 b(result)e(giv)m(es)i(a)g(formal)e(justi\014cation)g(of)456
4836 y(this)29 b(fact:)456 4970 y FH(Lemma)36 b(2.25.)44
b Fo(L)-5 b(et)36 b FG(u)31 b Fx(2)f FG(C)7 b FA(\(\012\))p
Fo(.)50 b(L)-5 b(et)36 b FG(\030)e FA(=)d(\()p FG(\030)2119
4984 y Fy(1)2158 4970 y FG(;)15 b(:)g(:)g(:)i(;)e(\030)2400
4984 y Fv(N)2467 4970 y FA(\))31 b Fx(2)f Fw(R)2684 4937
y Fv(N)2793 4970 y Fo(with)36 b Fx(j)p FG(\030)t Fx(j)31
b FA(=)f(1)36 b Fo(and)456 5088 y(let)588 5064 y FA(^)579
5088 y FG(\030)29 b FA(:=)c(\()p FG(\030)t(;)15 b FA(0\))27
b Fx(2)e Fw(R)1141 5055 y Fv(N)7 b Fy(+1)1304 5088 y
Fo(.)40 b(Assume)28 b(that)i FG(g)1932 5121 y Fj(S)-6
b Fy(\()p Fv(Y)10 b Ft(\000)p Fv(t)2140 5103 y Fy(^)2133
5121 y Fv(\030)s(;R)p Fy(\))2297 5088 y FG(>)25 b(u)j
Fo(in)h(their)g(c)-5 b(ommon)30 b(domain)456 5214 y(of)42
b(de\014nition)h(for)g(any)f FG(t)h Fx(2)f FA(\(0)p FG(;)15
b FA(1])p Fo(.)72 b(Assume)41 b(also)j(that)f FG(g)2577
5232 y Fj(S)-6 b Fy(\()p Fv(Y)t(;R)p Fy(\))2833 5214
y Fo(touches)43 b FG(u)f Fo(fr)-5 b(om)p eop
%%Page: 32 32
32 31 bop 456 251 a Fy(32)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y Fo(ab)j(ove)39 b(at)h(some)f(p)-5 b(oint)41
b FG(X)j FA(=)36 b(\()p FG(x;)15 b(x)1704 464 y Fv(N)7
b Fy(+1)1862 450 y FA(\))p Fo(,)41 b(that)f(is,)g(assume)g(that)g
FG(g)2851 469 y Fj(S)-6 b Fy(\()p Fv(Y)t(;R)p Fy(\))3102
450 y Fx(\025)36 b FG(u)j Fo(and)456 562 y FG(g)499 580
y Fj(S)-6 b Fy(\()p Fv(Y)t(;R)p Fy(\))713 562 y FA(\()p
FG(x)p FA(\))26 b(=)f FG(u)p FA(\()p FG(x)p FA(\))h(=)f
FG(x)1305 576 y Fv(N)7 b Fy(+1)1462 562 y Fo(.)42 b(Then,)1636
715 y FA(\()p FG(x)20 b Fx(\000)g FG(y)s FA(\))h Fx(\001)f
FG(\030)44 b Fx(\025)c FA(0)15 b FG(:)456 877 y Fo(Pr)-5
b(o)g(of.)43 b FA(By)30 b(construction,)1304 1049 y FG(g)1347
1063 y Fv(y)1382 1074 y Fs(N)5 b Fq(+1)1517 1063 y Fv(;R)1595
948 y Fu(\020)1649 1049 y Fx(j)p FG(x)20 b Fx(\000)g
FG(y)s Fx(j)h FA(+)f FG(H)2098 1063 y Fy(0)2137 1049
y FA(\()p FG(y)2217 1063 y Fv(N)7 b Fy(+1)2374 1049 y
FA(\))20 b Fx(\000)g FG(R)2590 948 y Fu(\021)2670 1049
y FA(=)1151 1211 y(=)82 b FG(g)1347 1230 y Fj(S)-5 b
Fy(\()p Fv(Y)t(;R)p Fy(\))1562 1211 y FA(\()p FG(x)p
FA(\))26 b(=)1151 1350 y(=)82 b FG(u)p FA(\()p FG(x)p
FA(\))26 b Fx(\024)1151 1485 y(\024)82 b FG(g)1347 1517
y Fj(S)-5 b Fy(\()p Fv(Y)10 b Ft(\000)p Fv(t)1556 1500
y Fy(^)1549 1517 y Fv(\030)s(;R)p Fy(\))1687 1485 y FA(\()p
FG(x)p FA(\))26 b(=)1151 1673 y(=)82 b FG(g)1347 1687
y Fv(y)1382 1698 y Fs(N)5 b Fq(+1)1517 1687 y Fv(;R)1595
1573 y Fu(\020)1649 1673 y Fx(j)p FG(x)20 b Fx(\000)g
FG(y)j FA(+)d FG(t\030)t Fx(j)h FA(+)e FG(H)2285 1687
y Fy(0)2324 1673 y FA(\()p FG(y)2404 1687 y Fv(N)7 b
Fy(+1)2562 1673 y FA(\))20 b Fx(\000)g FG(R)2778 1573
y Fu(\021)456 1850 y FA(for)30 b(an)m(y)g FG(t)c Fx(2)e
FA([0)p FG(;)15 b FA(1],)33 b(from)d(whic)m(h)1534 2000
y Fx(j)p FG(x)20 b Fx(\000)g FG(y)s Fx(j)26 b(\024)f(j)p
FG(x)20 b Fx(\000)g FG(y)j FA(+)d FG(t\030)t Fx(j)456
2149 y FA(for)30 b(an)m(y)g FG(t)c Fx(2)e FA([0)p FG(;)15
b FA(1].)43 b(This)28 b(sa)m(ys)j(that)g(the)g(function)1553
2301 y FG(f)10 b FA(\()p FG(t)p FA(\))26 b(:=)f Fx(j)p
FG(x)20 b Fx(\000)g FG(y)j FA(+)d FG(t\030)t Fx(j)2307
2264 y Fy(2)456 2451 y FA(attains)34 b(its)f(minim)m(um)f(in)h(the)h
(domain)f([0)p FG(;)15 b FA(1])36 b(at)f FG(t)c FA(=)h(0.)52
b(Th)m(us,)34 b FG(f)2836 2418 y Ft(0)2859 2451 y FA(\(0\))f
Fx(\025)e FA(0,)36 b(whic)m(h)456 2558 y(giv)m(es)30
b(the)h(desired)e(estimate.)1872 b Fl(\003)1106 2752
y FA(3.)46 b FB(Geometr)-6 b(y)33 b(of)h(the)f(touching)h(points)555
2914 y FA(This)c(section,)i(whic)m(h)e(is)g(v)m(ery)i(tec)m(hnical,)g
(follo)m(ws)e(man)m(y)i(of)g(the)f(ideas)g(in)f(Chap-)456
3022 y(ter)41 b(4)h(of)g([31)q(])f(\(w)m(e)i(pro)m(vide)d(full)f
(details)i(for)g(the)h(reader's)f(facilit)m(y\).)74 b(The)40
b(main)456 3130 y(result)26 b(of)j(this)d(section)j(will)c(b)s(e)i
(Prop)s(osition)f(3.14)k(here)d(ab)s(o)m(v)m(e,)j(in)d(whic)m(h)g(w)m
(e)h(in)m(v)m(es-)456 3238 y(tigate)i(the)f(measure)g(theoretic)g(prop)
s(erties)f(of)h(\(an)g FG(N)10 b FA(-dimensional)26 b(pro)5
b(jection)29 b(of)7 b(\))456 3346 y(the)36 b(set)h(of)f(p)s(ossible)e
(touc)m(hing)i(p)s(oin)m(ts)f(b)s(et)m(w)m(een)h(a)h(subsolution)c(of)k
(\(1.5\))h(and)d(the)456 3453 y(barrier)c Fw(S)o FA(\()p
FG(Y)5 b(;)15 b(R)q FA(\))28 b(in)m(tro)s(duced)k(here)h(ab)s(o)m(v)m
(e.)51 b(Roughly)32 b(sp)s(eaking,)i(w)m(e)g(will)c(pro)m(v)m(e)35
b(in)456 3561 y(Prop)s(osition)23 b(3.14)k(that)f(the)g(measure)f(of)h
(the)g(pro)5 b(jection)25 b(of)h(the)f(\\\014rst)h(o)s(ccurrence")456
3669 y(touc)m(hing)k(p)s(oin)m(ts)g(con)m(trols)h(the)f(measure)h(of)g
(the)g(pro)5 b(jection)30 b(of)h(the)g(cen)m(ters)g(of)g(the)456
3777 y(corresp)s(onding)d(surfaces.)555 3968 y(F)-8 b(or)31
b(this)f(scop)s(e,)g(giv)m(en)456 4118 y(\(3.1\))754
b FG(\030)29 b FA(:=)d(\()p FG(\030)1661 4132 y Fy(1)1700
4118 y FG(;)15 b(:)g(:)g(:)i(;)e(\030)1942 4132 y Fv(N)2009
4118 y FG(;)g FA(0\))27 b Fx(2)d Fw(R)2301 4080 y Fv(N)7
b Fy(+1)2480 4118 y FG(;)456 4269 y FA(with)29 b Fx(j)p
FG(\030)t Fx(j)c FA(=)g(1,)31 b(w)m(e)g(de\014ne)f Fi(P)1445
4284 y Fv(\030)1513 4269 y FA(as)g(the)h(h)m(yp)s(erplane)d(in)h
Fw(R)2414 4236 y Fv(N)7 b Fy(+1)2608 4269 y FA(orthogonal)30
b(to)i FG(\030)t FA(,)e(i.e.,)1353 4429 y Fi(P)1422 4444
y Fv(\030)1484 4429 y FA(:=)25 b Fx(f)p FG(X)34 b Fx(2)24
b Fw(R)1904 4391 y Fv(N)7 b Fy(+1)2093 4429 y Fx(j)25
b FG(\030)f Fx(\001)d FG(X)33 b FA(=)25 b(0)p Fx(g)456
4578 y FA(W)-8 b(e)31 b(also)f(denote)h(b)m(y)g FG(\031)1266
4593 y Fv(\030)1334 4578 y FA(the)f(pro)5 b(jection)30
b(on)m(to)i Fi(P)2196 4593 y Fv(\030)2233 4578 y FA(,)f(i.e.,)1133
4738 y FG(\031)1185 4753 y Fv(\030)1222 4738 y FA(\()p
FG(X)7 b FA(\))27 b(:=)e FG(X)j Fx(\000)20 b FA(\()p
FG(\030)k Fx(\001)d FG(X)7 b FA(\))p FG(\030)19 b(;)198
b Fx(8)p FG(X)32 b Fx(2)25 b Fw(R)2563 4701 y Fv(N)7
b Fy(+1)2742 4738 y FG(:)456 4887 y FA(With)36 b(a)i(sligh)m(t)e(abuse)
h(of)g(notation,)i(w)m(e)f(will)c(sometimes)j(iden)m(tify)f
FG(\030)41 b FA(with)36 b(its)g FG(N)10 b FA(-)456 4995
y(dimensional)29 b(pro)5 b(jection,)32 b(implicitly)c(dropping)i(the)i
(zero)h(in)d(the)i(last)g(co)s(ordinate,)456 5103 y(that)f(is,)e(w)m(e)
i(will)d(write)1483 5216 y FG(\030)h FA(:=)d(\()p FG(\030)1749
5230 y Fy(1)1788 5216 y FG(;)15 b(:)g(:)g(:)i(;)e(\030)2030
5230 y Fv(N)2097 5216 y FA(\))26 b Fx(2)f Fw(R)2303 5178
y Fv(N)2392 5216 y FG(;)p eop
%%Page: 33 33
33 32 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(33)456
450 y FA(instead)29 b(of)i(\(3.1\).)555 608 y(Let)37
b(no)m(w)g FG(u)f Fx(2)f FG(C)1176 575 y Fy(1)1215 608
y FA(\([)p Fx(\000)p FG(C)1418 575 y Fv(])1449 608 y
FG(l)17 b(;)31 b(C)1621 575 y Fv(])1652 608 y FG(l)r
FA(])1706 575 y Fv(N)1773 608 y FA(\))37 b(b)s(e)f(a)h(w)m(eak)h(Sob)s
(olev)d(subsolution)f(of)j(\(1.5\),)456 718 y(with)29
b Fx(j)p FG(u)p Fx(j)c(\024)g FA(1.)41 b(Here)31 b(w)m(e)g(will)d
(\014x)h FG(C)1718 685 y Fv(])1779 718 y FA(suitably)g(large)h(\(also,)
h FG(l)h FA(and)e FG(R)h FA(are)f(\014xed)g(and)456 826
y(suitably)i(large,)k(and)e FG(l)r(=R)i FA(is)d(assumed)h(con)m(v)m
(enien)m(tly)h(small\).)52 b(Let)35 b(us)f(no)m(w)g(de\014ne)456
935 y(the)23 b(set)g(of)g(\014rst)f(con)m(tact)j(p)s(oin)m(ts)c(as)i
(follo)m(ws.)38 b(Giv)m(en)22 b(a)h(compact)h(set)2819
900 y Fy(7)2878 935 y Fi(A)h Fx(\022)g Fi(P)3128 950
y Fv(\030)3190 935 y Fx(\032)g Fw(R)3346 902 y Fv(N)3419
935 y FA(,)456 1043 y(let)30 b(us)g(consider)f FG(Y)45
b Fx(2)25 b Fi(A)30 b FA(and)g(de\014ne)770 1180 y Fu(e)765
1199 y Fi(A)25 b FA(=)950 1180 y Fu(e)946 1199 y Fi(A)1006
1213 y Ff(A)1141 1199 y FA(:=)83 b Fx(f)p FG(Y)46 b Fx(2)25
b Fw(R)1609 1161 y Fv(N)7 b Fy(+1)1889 1199 y FA(s.t.)1320
1343 y Fx(9)1385 1320 y FA(^)1371 1343 y FG(Y)45 b Fx(2)25
b Fi(A)p FG(;)30 b(t)1714 1358 y Fy(^)1703 1374 y Fv(Y)1789
1343 y Fx(2)24 b Fw(R)125 b FA(s.t.)117 b FG(Y)45 b FA(=)2502
1320 y(^)2488 1343 y FG(Y)40 b FA(+)20 b FG(t)2716 1358
y Fy(^)2705 1374 y Fv(Y)2766 1343 y FG(\030)f(;)1350
1478 y(g)1393 1509 y Fj(S)-5 b Fy(\()1475 1493 y(^)1464
1509 y Fv(Y)10 b Fy(+)p Fv(t\030)s(;R)p Fy(\))1759 1478
y FG(>)25 b(u)30 b FA(for)g(an)m(y)h FG(t)25 b(<)g(t)2446
1493 y Fy(^)2435 1509 y Fv(Y)2495 1478 y FA(,)1320 1619
y FG(g)1363 1638 y Fj(S)-6 b Fy(\()p Fv(Y)t(;R)p Fy(\))1603
1619 y Fx(\025)24 b FG(u)1320 1772 y FA(and)116 b Fx(9)p
FG(x)24 b Fx(2)h Fw(R)1856 1735 y Fv(N)7 b Fy(+1)2135
1772 y FA(s.t.)117 b FG(g)2416 1791 y Fj(S)-5 b Fy(\()p
Fv(Y)t(;)o(R)p Fy(\))2631 1772 y FA(\()p FG(x)p FA(\))25
b(=)g FG(u)p FA(\()p FG(x)p FA(\))p Fx(g)15 b FG(:)555
1947 y FA(W)-8 b(e)32 b(refer)e(to)1038 1929 y Fu(e)1033
1947 y Fi(A)g FA(as)h(the)f(set)h(of)g(the)f(\\cen)m(ters".)43
b(Moreo)m(v)m(er,)32 b(w)m(e)f(de\014ne)691 2087 y Fu(e)680
2105 y Fi(B)25 b FA(=)886 2087 y Fu(e)874 2105 y Fi(B)947
2119 y Ff(A)1083 2105 y FA(:=)83 b Fx(f)p FG(X)33 b FA(=)25
b(\()p FG(x;)15 b(x)1690 2119 y Fv(N)7 b Fy(+1)1849 2105
y FA(\))25 b Fx(2)g Fw(R)2055 2067 y Fv(N)7 b Fy(+1)2334
2105 y FA(s.t.)117 b Fx(9)p FG(Y)45 b Fx(2)2811 2087
y Fu(e)2807 2105 y Fi(A)115 b FA(s.t.)1262 2240 y FG(g)1305
2258 y Fj(S)-5 b Fy(\()p Fv(Y)t(;)o(R)p Fy(\))1545 2240
y Fx(\025)25 b FG(u)207 b FA(and)1262 2378 y FG(g)1305
2397 y Fj(S)-5 b Fy(\()p Fv(Y)t(;)o(R)p Fy(\))1520 2378
y FA(\()p FG(x)p FA(\))26 b(=)e FG(u)p FA(\()p FG(x)p
FA(\))i(=)f FG(x)2111 2392 y Fv(N)7 b Fy(+1)2269 2378
y Fx(g)15 b FG(:)456 2553 y FA(and)27 b(w)m(e)h(refer)g(to)1089
2535 y Fu(e)1077 2553 y Fi(B)g FA(as)g(the)g(set)h(of)f(\\\014rst)f
(con)m(tact)j(p)s(oin)m(ts".)40 b(Roughly)26 b(sp)s(eaking,)i(w)m(e)456
2661 y(are)g(sliding)d(our)j(barriers)e(un)m(til)h(it)h(touc)m(hes)h
(the)f(graph)g(of)g FG(u)g FA(for)g(the)g(\014rst)g(time:)39
b(the)456 2771 y(set)614 2753 y Fu(e)603 2771 y Fi(B)c
FA(collects)h(all)e(suc)m(h)h(\014rst)g(o)s(ccurrence)h(con)m(tact)h(p)
s(oin)m(ts,)f(while)2904 2753 y Fu(e)2900 2771 y Fi(A)e
FA(collects)i(the)456 2879 y(corresp)s(onding)28 b(cen)m(ters.)42
b(In)29 b(this)h(section,)g(w)m(e)h(will)d(assume)i(that)456
3036 y(\(3.2\))578 b Fi(A)40 b Fx(\022)g FA([)p Fx(\000)p
FG(C)1598 2999 y Fv(])1629 3036 y FG(l)r(=)p FA(2)p FG(;)32
b(C)1877 2999 y Fv(])1908 3036 y FG(l)r(=)p FA(2])2052
2999 y Fv(N)2140 3036 y Fx(\002)20 b FA([)p Fx(\000)p
FA(1)p FG(=)p FA(4)p FG(;)32 b FA(1)p FG(=)p FA(4])456
3192 y(and)d(that)475 3364 y Fu(n)535 3465 y FG(X)k FA(=)25
b(\()p FG(x;)15 b(x)918 3479 y Fv(N)7 b Fy(+1)1076 3465
y FA(\))26 b Fx(2)e Fw(R)1282 3428 y Fv(N)7 b Fy(+1)1486
3465 y FA(s)p FG(:)p FA(t)p FG(:)41 b Fx(9)p FG(Y)k Fx(2)25
b Fw(R)1942 3428 y Fv(N)7 b Fy(+1)2146 3465 y FA(s)p
FG(:)p FA(t)p FG(:)41 b(\031)2360 3480 y Fv(\030)2398
3465 y FG(Y)k Fx(2)25 b Fi(A)15 b FG(;)55 b(g)2780 3484
y Fj(S)-5 b Fy(\()p Fv(Y)t(;)o(R)p Fy(\))3020 3465 y
Fx(\025)25 b FG(u)71 b FA(and)475 3664 y FG(g)518 3683
y Fj(S)-6 b Fy(\()p Fv(Y)t(;R)p Fy(\))732 3664 y FA(\()p
FG(x)p FA(\))26 b(=)f FG(u)p FA(\()p FG(x)p FA(\))1150
3563 y Fu(o)1246 3664 y Fx(\\)35 b FG(@)5 b FA([)p Fx(\000)p
FG(C)1563 3627 y Fv(])1595 3664 y FG(l)r(;)30 b(C)1751
3627 y Fv(])1782 3664 y FG(l)r FA(])1836 3627 y Fv(N)1944
3664 y FA(=)40 b Fx(;)15 b FG(:)456 3316 y FA(\(3.3\))456
3842 y(W)-8 b(e)31 b(also)f(de\014ne)1703 3960 y Fi(B)c
FA(:=)f FG(\031)1975 3975 y Fv(\030)2012 3960 y FA(\()2058
3942 y Fu(e)2047 3960 y Fi(B)q FA(\))15 b FG(:)456 4097
y FA(W)-8 b(e)31 b(also)f(denote)h(the)g(graph)f(of)g
FG(u)h FA(b)m(y)f Fi(G)p FA(,)h(that)g(is)e(w)m(e)i(set)456
4252 y(\(3.4\))885 b Fi(G)25 b FA(:=)h Fx(f)p FG(x)1835
4266 y Fv(N)7 b Fy(+1)2018 4252 y FA(=)25 b FG(u)p FA(\()p
FG(x)p FA(\))p Fx(g)15 b FG(:)555 4563 y FA(W)-8 b(e)44
b(no)m(w)f(sho)m(w)f(some)h(prop)s(erties)e(of)i(the)g(ab)s(o)m(v)m(e)h
(de\014ned)d(sets.)78 b(First)42 b(of)h(all,)456 4671
y(from)30 b(\(3.3\))i(and)d(Corollary)g(2.14,)k(w)m(e)d(ha)m(v)m(e)i
(that:)456 4803 y FH(Lemma)g(3.1.)1294 4902 y Fu(e)1282
4920 y Fi(B)26 b Fx(\022)f(f)p FG(X)33 b Fx(2)25 b Fw(R)1775
4883 y Fv(N)7 b Fy(+1)1964 4920 y Fx(j)26 b(j)p FG(x)2092
4934 y Fv(N)7 b Fy(+1)2249 4920 y Fx(j)26 b FG(<)f FA(1)p
FG(=)p FA(2)p Fx(g)15 b FG(:)p 456 5025 499 4 v 555 5095
a Fy(7)591 5124 y FF(The)28 b(closure)h(of)g Fi(A)e FF(will)j(b)r(e)e
(used)g(to)g(deduce)f(closure)i(and)f(measurabilit)n(y)g(prop)r(erties)
h(of)g(sets)456 5216 y(of)d(in)n(terest:)35 b(see,)26
b(e.g.,)h(Lemma)e(3.2)h(and)g(Prop)r(osition)h(3.14)g(here)f(b)r(elo)n
(w.)p eop
%%Page: 34 34
34 33 bop 456 251 a Fy(34)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)555 450 y FA(W)g(e)25 b(no)m(w)e(sho)m(w)g(that)h(the)f
(compactness)h(prop)s(ert)m(y)f(of)g Fi(A)g FA(is)g(inherited)d(b)m(y)k
(the)f(other)456 558 y(sets)30 b(de\014ned)g(ab)s(o)m(v)m(e:)456
697 y FH(Lemma)i(3.2.)42 b Fo(If)33 b FG(l)r(=R)h Fo(is)e(smal)5
b(l)34 b(enough,)f(then)2185 679 y Fu(e)2181 697 y Fi(A)o
Fo(,)g Fi(B)g Fo(and)2595 679 y Fu(e)2583 697 y Fi(B)g
Fo(ar)-5 b(e)34 b(c)-5 b(omp)g(act)34 b(sets.)456 894
y(Pr)-5 b(o)g(of.)43 b FA(Note)23 b(that)1148 876 y Fu(e)1137
894 y Fi(B)j Fx(2)f FA([)p Fx(\000)p FG(C)1490 861 y
Fv(])1521 894 y FG(l)r(;)30 b(C)1677 861 y Fv(])1708
894 y FG(l)r FA(])1762 861 y Fv(N)1834 894 y Fx(\002)t
FA([)p Fx(\000)p FA(1)p FG(;)15 b FA(1],)25 b(hence)2466
876 y Fu(e)2454 894 y Fi(B)e FA(is)e(b)s(ounded.)36 b(Therefore,)456
1009 y Fi(B)f FA(and)749 990 y Fu(e)745 1009 y Fi(A)f
FA(are)h(also)g(b)s(ounded.)52 b(Thence,)36 b(w)m(e)f(only)f(need)h(to)
g(sho)m(w)g(that)g(the)g(ab)s(o)m(v)m(e)456 1119 y(sets)41
b(are)f(closed.)71 b(Let)41 b(us)f(\014rst)g(sho)m(w)g(that)2084
1100 y Fu(e)2079 1119 y Fi(A)g FA(is)g(closed.)71 b(F)-8
b(or)41 b(this,)h(let)f FG(Y)3198 1134 y Fv(k)3282 1119
y Fx(2)3389 1100 y Fu(e)3385 1119 y Fi(A)456 1229 y FA(con)m(v)m(erge)
32 b(to)f(some)g FG(Y)1219 1243 y Ft(1)1293 1229 y FA(.)41
b(W)-8 b(e)32 b(need)e(to)h(sho)m(w)f(that)h FG(Y)2314
1243 y Ft(1)2414 1229 y Fx(2)2504 1210 y Fu(e)2500 1229
y Fi(A)o FA(.)456 1338 y(F)-8 b(or)44 b(this)f(scop)s(e,)48
b(note)c(that,)k(since)43 b Fi(A)h FA(is)f(closed)g(and)h
FG(\031)2534 1353 y Fv(\030)2571 1338 y FG(Y)2624 1353
y Fv(k)2714 1338 y Fx(2)k FG(\031)2875 1353 y Fv(\030)2912
1338 y FA(\()2952 1320 y Fu(e)2947 1338 y Fi(A)p FA(\))g
Fx(\022)g Fi(A)p FA(,)f(w)m(e)456 1458 y(ha)m(v)m(e)37
b(that)g FG(\031)926 1473 y Fv(\030)963 1458 y FG(Y)1016
1472 y Ft(1)1126 1458 y Fx(2)d Fi(A)p FA(.)58 b(Also,)37
b(since)f FG(Y)1883 1473 y Fv(k)1960 1458 y Fx(2)2060
1440 y Fu(e)2056 1458 y Fi(A)o FA(,)i(w)m(e)f(ha)m(v)m(e)g(that)g
(there)f(exists)g FG(X)3306 1473 y Fv(k)3384 1458 y Fx(2)456
1574 y FA([)p Fx(\000)p FG(C)624 1541 y Fv(])655 1574
y FG(l)r(;)30 b(C)811 1541 y Fv(])842 1574 y FG(l)r FA(])896
1541 y Fv(N)984 1574 y Fx(\002)20 b FA([)p Fx(\000)p
FA(1)p FG(;)15 b FA(1])32 b(so)e(that)456 1855 y(\(3.5\))1458
1669 y Fu(8)1458 1751 y(<)1458 1914 y(:)1622 1742 y FG(g)1665
1761 y Fj(S)-6 b Fy(\()p Fv(Y)1776 1773 y Fs(k)1808 1761
y Fv(;R)p Fy(\))1913 1742 y FA(\()p FG(x)2000 1757 y
Fv(k)2043 1742 y FA(\))26 b(=)f FG(u)p FA(\()p FG(x)2339
1757 y Fv(k)2382 1742 y FA(\))1623 1853 y FG(g)1666 1871
y Fj(S)-6 b Fy(\()p Fv(Y)1777 1883 y Fs(k)1810 1871 y
Fv(;R)p Fy(\))1940 1853 y Fx(\025)25 b FG(u)197 b FA(and)1717
1963 y FG(g)1760 1982 y Fj(S)-5 b Fy(\()p Fv(Y)1872 1994
y Fs(k)1904 1982 y Ft(\000)p Fv(t\030)s(;R)p Fy(\))2148
1963 y FG(>)25 b(u)15 b(;)456 2132 y FA(for)43 b(an)m(y)h
FG(t)k(>)f FA(0.)81 b(Of)43 b(course,)48 b(up)43 b(to)h(subsequence,)j
(w)m(e)d(ma)m(y)h(assume)e(that)h FG(X)3401 2147 y Fv(k)456
2240 y FA(con)m(v)m(erges)39 b(to)f(some)g(p)s(oin)m(t)e
FG(X)1542 2254 y Ft(1)1617 2240 y FA(.)62 b(Th)m(us,)38
b(passing)e(to)i(the)g(limit)d(\(3.5\),)41 b(w)m(e)d(obtain)456
2348 y(that)1412 2383 y Fu(8)1412 2465 y(<)1412 2628
y(:)1576 2456 y FG(g)1619 2475 y Fj(S)-6 b Fy(\()p Fv(Y)1730
2483 y Fp(1)1790 2475 y Fv(;R)p Fy(\))1895 2456 y FA(\()p
FG(x)1982 2470 y Ft(1)2057 2456 y FA(\))25 b(=)g FG(u)p
FA(\()p FG(x)2352 2470 y Ft(1)2427 2456 y FA(\))1609
2567 y FG(g)1652 2585 y Fj(S)-5 b Fy(\()p Fv(Y)1764 2593
y Fp(1)1823 2585 y Fv(;R)p Fy(\))1953 2567 y Fx(\025)25
b FG(u)197 b FA(and)1704 2677 y FG(g)1747 2696 y Fj(S)-6
b Fy(\()p Fv(Y)1858 2704 y Fp(1)1918 2696 y Ft(\000)p
Fv(t\030)s(;R)p Fy(\))2162 2677 y Fx(\025)25 b FG(u)15
b(;)456 2848 y FA(for)37 b(an)m(y)h FG(t)g(>)f FA(0.)63
b(Hence,)41 b(to)d(sho)m(w)g(that)g FG(Y)2006 2862 y
Ft(1)2118 2848 y Fx(2)2221 2830 y Fu(e)2216 2848 y Fi(A)p
FA(,)i(w)m(e)e(need)g(to)g(pro)m(v)m(e)h(the)f(strict)456
2956 y(inequalit)m(y)28 b(in)h(the)i(last)f(relation)g(here)g(ab)s(o)m
(v)m(e,)i(i.e.,)f(w)m(e)f(need)g(to)i(sho)m(w)e(that)456
3126 y(\(3.6\))1006 b FG(g)1690 3144 y Fj(S)-6 b Fy(\()p
Fv(Y)1801 3152 y Fp(1)1861 3144 y Ft(\000)p Fv(t\030)s(;R)p
Fy(\))2105 3126 y FG(>)25 b(u)456 3296 y FA(for)35 b(an)m(y)i
FG(t)d(>)h FA(0.)58 b(W)-8 b(e)37 b(argue)f(b)m(y)g(con)m(tradiction:)
52 b(assume)36 b(that)h(there)f(exists)g FG(x)3296 3311
y Fv(])3363 3296 y FA(so)456 3404 y(that)31 b Fx(j)p
FG(x)730 3419 y Fv(])761 3404 y Fx(j)26 b(\024)40 b FA(const)16
b FG(l)32 b FA(and)e FG(t)1415 3419 y Fv(])1471 3404
y FG(>)25 b FA(0,)31 b(suc)m(h)f(that)1458 3574 y FG(g)1501
3592 y Fj(S)-6 b Fy(\()p Fv(Y)1612 3600 y Fp(1)1672 3592
y Ft(\000)p Fv(t)1752 3604 y Fs(])1781 3592 y Fv(\030)s(;R)p
Fy(\))1920 3574 y FA(\()p FG(x)2007 3589 y Fv(])2039
3574 y FA(\))25 b(=)g FG(u)p FA(\()p FG(x)2334 3589 y
Fv(])2366 3574 y FA(\))15 b FG(:)456 3752 y FA(Note)31
b(that,)h(b)m(y)e(\(3.3\))i(and)e(Corollary)f(2.14,)j
Fx(j)p FG(g)2105 3771 y Fj(S)-6 b Fy(\()p Fv(Y)2216 3779
y Fp(1)2276 3771 y Fv(;R)p Fy(\))2381 3752 y FA(\()p
FG(x)2468 3766 y Ft(1)2543 3752 y FA(\))p Fx(j)25 b(\024)g
FA(1)p FG(=)p FA(2,)32 b(th)m(us)456 3926 y(\(3.7\))855
b Fx(j)p FG(x)1573 3940 y Ft(1)1669 3926 y Fx(\000)19
b FG(y)1804 3940 y Ft(1)1879 3926 y Fx(j)40 b(\025)56
b FA(const)15 b FG(R)h(:)456 4096 y FA(Also,)30 b(b)m(y)g(\(3.2\),)j(w)
m(e)d(get)i(that)1101 4266 y Fx(j)p FG(\031)1178 4281
y Fv(\030)1215 4266 y FA(\()p FG(x)1302 4280 y Ft(1)1398
4266 y Fx(\000)20 b FG(y)1534 4280 y Ft(1)1608 4266 y
FA(\))p Fx(j)83 b(\024)g(j)p FG(x)1982 4280 y Ft(1)2057
4266 y Fx(j)21 b FA(+)f Fx(j)p FG(\031)2271 4281 y Fv(\030)2308
4266 y FG(y)2353 4280 y Ft(1)2428 4266 y Fx(j)25 b FA(=)1751
4401 y(=)83 b Fx(j)p FG(x)1982 4415 y Ft(1)2057 4401
y Fx(j)21 b FA(+)f Fx(j)p FG(\031)2271 4415 y Fv(e)2304
4426 y Fs(N)2365 4401 y FA(\()p FG(\031)2452 4416 y Fv(\030)2490
4401 y FG(Y)2543 4415 y Ft(1)2618 4401 y FA(\))p Fx(j)25
b(\024)1751 4536 y(\024)98 b FA(const)16 b FG(l)h(:)456
4706 y FA(Th)m(us,)29 b(from)h(\(3.7\),)i(w)m(e)f(ha)m(v)m(e)h(that)456
4876 y(\(3.8\))733 b Fl(\\)p FA(\()p FG(x)1527 4890 y
Ft(1)1622 4876 y Fx(\000)20 b FG(y)1758 4890 y Ft(1)1832
4876 y FG(;)31 b(\030)t FA(\))41 b Fx(\024)f FA(const)15
b FG(l)r(=R)i(:)456 5046 y FA(F)-8 b(urthermore,)30 b(from)g(Lemma)h
(2.25,)1563 5216 y(\()p FG(x)1650 5230 y Ft(1)1745 5216
y Fx(\000)20 b FG(y)1881 5230 y Ft(1)1955 5216 y FA(\))h
Fx(\001)f FG(\030)44 b Fx(\025)d FA(0)15 b FG(:)p eop
%%Page: 35 35
35 34 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(35)456
450 y FA(This,)28 b(\(3.8\))33 b(and)c(\(3.7\))k(sa)m(y)e(that)926
664 y(\()p FG(x)1013 678 y Ft(1)1108 664 y Fx(\000)20
b FG(y)1244 678 y Ft(1)1318 664 y FA(\))h Fx(\001)f FG(\030)87
b FA(=)c Fx(j)p FA(\()p FG(x)1812 678 y Ft(1)1908 664
y Fx(\000)20 b FG(y)2044 678 y Ft(1)2118 664 y FA(\))g
Fx(\001)h FG(\030)t Fx(j)k FA(=)1546 799 y(=)83 b Fx(j)p
FG(x)1777 813 y Ft(1)1872 799 y Fx(\000)20 b FG(y)2008
813 y Ft(1)2083 799 y Fx(j)15 b FA(cos)h Fl(\\)p FA(\()p
FG(x)2413 813 y Ft(1)2508 799 y Fx(\000)k FG(y)2644 813
y Ft(1)2718 799 y FG(;)30 b(\030)t FA(\))c Fx(\025)1546
934 y(\025)98 b FA(const)16 b FG(R)g(;)456 1148 y FA(pro)m(vided)29
b(that)i FG(l)r(=R)g FA(is)e(small)g(enough.)41 b(F)-8
b(or)31 b(this)e(reason,)1059 1353 y(\()p FG(x)1146 1368
y Fv(])1198 1353 y Fx(\000)20 b FG(y)1334 1367 y Ft(1)1408
1353 y FA(\))h Fx(\001)f FG(\030)29 b Fx(\025)c FA(\()p
FG(x)1761 1367 y Ft(1)1857 1353 y Fx(\000)19 b FG(y)1992
1367 y Ft(1)2067 1353 y FA(\))h Fx(\001)h FG(\030)j Fx(\000)c(j)p
FG(x)2400 1367 y Ft(1)2475 1353 y Fx(j)g(\000)g(j)p FG(x)2688
1368 y Fv(])2720 1353 y Fx(j)25 b(\025)1578 1488 y(\025)40
b FA(const)16 b FG(R)21 b Fx(\000)35 b FA(const)16 b
FG(l)27 b Fx(\025)1578 1623 y(\025)e FA(0)15 b FG(;)456
1496 y FA(\(3.9\))456 1842 y(if)29 b FG(l)r(=R)i FA(is)f(small)f
(enough.)40 b(Th)m(us,)30 b(from)g(\(3.9\),)i(w)m(e)f(deduce)f(that)808
2060 y Fx(j)p FG(x)885 2075 y Fv(])937 2060 y Fx(\000)20
b FG(y)1073 2074 y Ft(1)1167 2060 y FA(+)g FG(t)1291
2075 y Fv(])1323 2060 y FG(\030)t Fx(j)1392 2022 y Fy(2)1514
2060 y FA(=)83 b Fx(j)p FG(x)1745 2075 y Fv(])1797 2060
y Fx(\000)20 b FG(y)1933 2074 y Ft(1)2007 2060 y Fx(j)2032
2022 y Fy(2)2092 2060 y FA(+)g FG(t)2216 2022 y Fy(2)2216
2083 y Fv(])2276 2060 y FA(+)g(2)p FG(t)2445 2075 y Fv(])2476
2060 y FG(\030)25 b Fx(\001)20 b FA(\()p FG(x)2673 2075
y Fv(])2725 2060 y Fx(\000)g FG(y)2861 2074 y Ft(1)2935
2060 y FA(\))26 b FG(>)1514 2211 y(>)83 b Fx(j)p FG(x)1745
2226 y Fv(])1797 2211 y Fx(\000)20 b FG(y)1933 2225 y
Ft(1)2007 2211 y Fx(j)2032 2174 y Fy(2)2087 2211 y FG(:)456
2425 y FA(As)30 b(a)h(result,)e(w)m(e)i(infer)e(that)752
2639 y FG(u)p FA(\()p FG(x)891 2654 y Fv(])923 2639 y
FA(\))83 b(=)g FG(g)1238 2658 y Fj(S)-6 b Fy(\()p Fv(Y)1349
2666 y Fp(1)1408 2658 y Ft(\000)p Fv(t)1488 2670 y Fs(])1518
2658 y Fv(\030)s(;R)p Fy(\))1657 2639 y FA(\()p FG(x)1744
2654 y Fv(])1776 2639 y FA(\))25 b(=)1041 2819 y(=)83
b FG(g)1238 2833 y Fv(y)1273 2844 y Fp(1)p Fs(;N)5 b
Fq(+1)1487 2833 y Fv(;R)1565 2718 y Fu(\020)1619 2819
y Fx(j)p FG(x)1696 2834 y Fv(])1748 2819 y Fx(\000)20
b FG(y)1884 2833 y Ft(1)1978 2819 y FA(+)g FG(t)2102
2834 y Fv(])2134 2819 y FG(\030)t Fx(j)g FA(+)g FG(H)2390
2833 y Fy(0)2429 2819 y FA(\()p FG(y)2509 2833 y Ft(1)p
Fv(;N)7 b Fy(+1)2757 2819 y FA(\))20 b Fx(\000)g FG(R)2973
2718 y Fu(\021)3052 2819 y FG(>)1041 3018 y(>)83 b(g)1238
3032 y Fv(y)1273 3043 y Fp(1)p Fs(;N)5 b Fq(+1)1487 3032
y Fv(;R)1565 2917 y Fu(\020)1619 3018 y Fx(j)p FG(x)1696
3033 y Fv(])1748 3018 y Fx(\000)20 b FG(y)1884 3032 y
Ft(1)1958 3018 y Fx(j)h FA(+)f FG(H)2171 3032 y Fy(0)2210
3018 y FA(\()p FG(y)2290 3032 y Ft(1)p Fv(;N)7 b Fy(+1)2537
3018 y FA(\))21 b Fx(\000)f FG(R)2754 2917 y Fu(\021)2833
3018 y FA(=)1041 3181 y(=)83 b FG(g)1238 3199 y Fj(S)-6
b Fy(\()p Fv(Y)1349 3207 y Fp(1)1408 3199 y Fv(;R)p Fy(\))1513
3181 y FA(\()p FG(x)1600 3196 y Fv(])1632 3181 y FA(\))26
b Fx(\025)1041 3319 y(\025)83 b FG(u)p FA(\()p FG(x)1334
3334 y Fv(])1366 3319 y FA(\))15 b FG(:)456 3545 y FA(This)28
b(con)m(tradiction)i(pro)m(v)m(es)i(\(3.6\).)42 b(Hence,)32
b FG(Y)2099 3559 y Ft(1)2198 3545 y Fx(2)2289 3527 y
Fu(e)2284 3545 y Fi(A)e FA(and)g(therefore)2937 3527
y Fu(e)2933 3545 y Fi(A)f FA(is)h(closed.)555 3766 y(Note)j(no)m(w)f
(that)g(once)g(w)m(e)g(kno)m(w)g(that)1956 3748 y Fu(e)1945
3766 y Fi(B)f FA(is)g(closed,)h(it)f(easily)g(follo)m(ws)f(that)i
Fi(B)g FA(is)456 3876 y(also)g(closed.)47 b(Th)m(us,)33
b(to)g(end)f(the)h(pro)s(of)e(of)i(this)f(result,)g(w)m(e)h(need)f(to)h
(pro)m(v)m(e)h(that)3382 3858 y Fu(e)3371 3876 y Fi(B)456
3986 y FA(is)29 b(closed.)41 b(F)-8 b(or)31 b(this,)e(let)i(us)e
(consider)g(a)i(sequence)g FG(X)2346 4001 y Fv(k)2414
3986 y Fx(2)2512 3968 y Fu(e)2500 3986 y Fi(B)g FA(so)f(that)1664
4200 y(lim)1610 4261 y Fv(k)r Ft(!)p Fy(+)p Ft(1)1860
4200 y FG(X)1935 4215 y Fv(k)2003 4200 y FA(=)25 b FG(X)2174
4214 y Ft(1)2264 4200 y FG(:)456 4476 y FA(Our)c(aim)g(is)h(to)h(sho)m
(w)f(that)h FG(X)1471 4490 y Ft(1)1571 4476 y Fx(2)1668
4457 y Fu(e)1657 4476 y Fi(B)p FA(.)38 b(F)-8 b(or)23
b(this,)g(observ)m(e)g(that,)i(since)d FG(X)2966 4491
y Fv(k)3034 4476 y Fx(2)3131 4457 y Fu(e)3120 4476 y
Fi(B)p FA(,)i(there)456 4586 y(exists)38 b FG(Y)768 4601
y Fv(k)849 4586 y Fx(2)953 4567 y Fu(e)949 4586 y Fi(A)g
FA(suc)m(h)g(that)h FG(g)1508 4604 y Fj(S)-6 b Fy(\()p
Fv(Y)1619 4616 y Fs(k)1652 4604 y Fv(;R)p Fy(\))1795
4586 y Fx(\025)39 b FG(u)g FA(and)f FG(g)2224 4604 y
Fj(S)-6 b Fy(\()p Fv(Y)2335 4616 y Fs(k)2367 4604 y Fv(;R)p
Fy(\))2472 4586 y FA(\()p FG(x)2559 4601 y Fv(k)2602
4586 y FA(\))39 b(=)g FG(u)p FA(\()p FG(x)2925 4601 y
Fv(k)2968 4586 y FA(\))g(=)g FG(x)3204 4601 y Fv(k)r(;N)7
b Fy(+1)3419 4586 y FA(.)456 4713 y(Since)33 b(w)m(e)i(pro)m(v)m(ed)g
(that)1341 4695 y Fu(e)1337 4713 y Fi(A)f FA(is)g(compact,)j(p)s
(ossibly)32 b(taking)i(subsequences,)i(w)m(e)f(ma)m(y)456
4823 y(assume)j(that)i FG(Y)1037 4838 y Fv(k)1118 4823
y FA(tends)f(to)g FG(Y)1542 4837 y Ft(1)1656 4823 y Fx(2)1761
4804 y Fu(e)1756 4823 y Fi(A)p FA(.)66 b(Th)m(us,)40
b(passing)e(to)i(the)f(limit)d(here)j(ab)s(o)m(v)m(e)456
4931 y(w)m(e)f(deduce)f(that)h FG(g)1154 4949 y Fj(S)-6
b Fy(\()p Fv(Y)1265 4957 y Fp(1)1325 4949 y Fv(;R)p Fy(\))1467
4931 y Fx(\025)37 b FG(u)g FA(and)g FG(g)1891 4949 y
Fj(S)-6 b Fy(\()p Fv(Y)2002 4957 y Fp(1)2062 4949 y Fv(;R)p
Fy(\))2166 4931 y FA(\()p FG(x)2253 4945 y Ft(1)2328
4931 y FA(\))38 b(=)f FG(u)p FA(\()p FG(x)2648 4945 y
Ft(1)2723 4931 y FA(\))h(=)f FG(x)2956 4945 y Ft(1)p
Fv(;N)7 b Fy(+1)3203 4931 y FA(,)40 b(with)456 5058 y
FG(Y)509 5072 y Ft(1)608 5058 y Fx(2)699 5040 y Fu(e)694
5058 y Fi(A)p FA(.)g(This)29 b(pro)m(v)m(es)i(that)g
FG(X)1580 5072 y Ft(1)1680 5058 y Fx(2)1778 5040 y Fu(e)1766
5058 y Fi(B)q FA(,)f(thence)2194 5040 y Fu(e)2183 5058
y Fi(B)h FA(is)e(closed.)733 b Fl(\003)p eop
%%Page: 36 36
36 35 bop 456 251 a Fy(36)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 456 y FH(Lemma)32 b(3.3.)42 b Fo(F)-7 b(or)34
b(any)f FG(X)g Fx(2)25 b Fw(R)1628 423 y Fv(N)7 b Fy(+1)1824
456 y Fo(with)33 b FG(X)g Fx(2)2215 433 y Fu(e)2215 456
y Fw(S)-8 b FA(\()p FG(Y)5 b(;)15 b(R)q FA(\))33 b Fo(and)h
FG(\027)d Fx(2)25 b FG(S)2936 423 y Fv(N)3003 456 y Fo(,)32
b(let)3158 421 y Fy(8)507 743 y FG(\027)557 688 y Fk(e)558
705 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))772 743 y
FA(\()p FG(X)7 b FA(\))84 b(:=)1205 547 y Fu(\020)1280
648 y Fx(\000)20 b(r)p FG(g)1489 665 y Fk(e)1490 682
y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))1704 648 y FA(\()p
FG(x)p FA(\))p FG(;)31 b FA(1)1927 547 y Fu(\021)p 1197
722 794 4 v 1197 740 a(q)p 1288 740 703 4 v 91 x FA(1)20
b(+)g Fx(jr)p FG(g)1587 848 y Fk(e)1588 865 y Fj(S)-11
b Fy(\()p Fv(Y)t(;R)p Fy(\))1803 831 y FA(\()p FG(x)p
FA(\))p Fx(j)1950 805 y Fy(2)2040 743 y Fx(2)25 b Fw(R)2186
705 y Fv(N)7 b Fy(+1)2364 743 y FG(;)626 1067 y(!)s FA(\()p
FG(X)r(;)15 b(\027)6 b FA(\))84 b(:=)1218 1005 y FG(R)p
1197 1046 111 4 v 1197 1064 72 4 v 1197 1137 a(C)1268
1151 y Fy(0)1348 939 y Fu(\022)1584 1005 y FG(\027)1629
1019 y Fv(N)7 b Fy(+1)p 1425 1046 520 4 v 1425 1129 a
Fx(j)p FA(\()p FG(\027)1530 1143 y Fy(1)1570 1129 y FG(;)15
b(:)g(:)g(:)i(;)e(\027)1817 1143 y Fv(N)1884 1129 y FA(\))p
Fx(j)1975 1067 y(\000)20 b FG(H)2149 1029 y Ft(0)2142
1089 y Fy(0)2181 1067 y FA(\()p FG(x)2268 1081 y Fv(N)7
b Fy(+1)2425 1067 y FA(\))2460 939 y Fu(\023)2568 1067
y Fx(2)25 b Fw(R)18 b FG(;)631 1330 y(\033)s FA(\()p
FG(X)r(;)d(\027)6 b FA(\))84 b(:=)f Fx(\000)p 1270 1196
72 4 v 1270 1269 a FG(C)1341 1283 y Fy(0)p 1268 1310
116 4 v 1268 1393 a FA(2)p FG(R)1393 1330 y(!)1453 1293
y Fy(2)1492 1330 y FA(\()p FG(X)7 b FA(\))21 b(+)f FG(H)1832
1344 y Fy(0)1871 1330 y FA(\()p FG(x)1958 1344 y Fv(N)7
b Fy(+1)2116 1330 y FA(\))21 b Fx(\000)f FG(H)2339 1344
y Fy(0)2378 1330 y FA(\()p FG(x)2465 1344 y Fv(N)7 b
Fy(+1)2642 1330 y FA(+)20 b FG(!)s FA(\))h(+)f FG(R)41
b Fx(2)25 b Fw(R)17 b FG(;)615 1560 y(F)c FA(\()p FG(X)r(;)i(\027)6
b FA(\))84 b(:=)1187 1432 y Fu(\022)1254 1560 y FG(x)20
b FA(+)1452 1498 y(\()p FG(\027)1532 1512 y Fy(1)1572
1498 y FG(;)15 b(:)g(:)g(:)h(;)f(\027)1818 1512 y Fv(N)1886
1498 y FA(\))p 1427 1539 520 4 v 1427 1622 a Fx(j)p FA(\()p
FG(\027)1532 1636 y Fy(1)1572 1622 y FG(;)g(:)g(:)g(:)h(;)f(\027)1818
1636 y Fv(N)1886 1622 y FA(\))p Fx(j)1957 1560 y FG(\033)s
FA(\()p FG(X)r(;)g(\027)6 b FA(\))p FG(;)42 b(x)2369
1574 y Fv(N)7 b Fy(+1)2546 1560 y FA(+)20 b FG(!)s FA(\()p
FG(X)r(;)15 b(\027)6 b FA(\))2935 1432 y Fu(\023)3044
1560 y Fx(2)24 b Fw(R)3189 1522 y Fv(N)7 b Fy(+1)3368
1560 y FG(:)456 1774 y Fo(Then,)456 1959 y FA(\(3.10\))808
b FG(Y)45 b FA(=)25 b FG(F)13 b FA(\()p FG(X)r(;)i(\027)1961
1904 y Fk(e)1962 1921 y Fj(S)-11 b Fy(\()p Fv(Y)t(;R)p
Fy(\))2177 1959 y FA(\()p FG(X)7 b FA(\)\))15 b FG(;)456
2135 y Fo(for)33 b(any)g FG(X)g Fx(2)969 2112 y Fu(e)969
2135 y Fw(S)-8 b FA(\()p FG(Y)5 b(;)15 b(R)q FA(\))p
Fo(.)456 2330 y(Pr)-5 b(o)g(of.)43 b FA(F)-8 b(or)33
b(short,)f(w)m(e)h(will)d(set)j(here)1815 2306 y Fu(e)1815
2330 y Fw(S)21 b FA(:=)2018 2306 y Fu(e)2018 2330 y Fw(S)-7
b FA(\()p FG(Y)5 b(;)15 b(R)q FA(\).)47 b(Recalling)31
b(Lemma)h(B.11,)j(w)m(e)456 2437 y(see)c(that)g(there)f(exists)g
FG(\033)f Fx(2)24 b Fw(R)39 b FA(so)31 b(that)1302 2666
y FG(x)20 b Fx(\000)g FG(y)28 b FA(=)d FG(\033)1740 2604
y FA(\()p FG(\027)1825 2554 y Fk(e)1826 2571 y Fj(S)1820
2628 y Fy(1)1867 2604 y FA(\()p FG(X)7 b FA(\))p FG(;)15
b(:)g(:)g(:)i(;)e(\027)2271 2554 y Fk(e)2272 2571 y Fj(S)2266
2631 y Fv(N)2334 2604 y FA(\()p FG(X)7 b FA(\)\))p 1714
2645 834 4 v 1714 2752 a Fx(j)p FA(\()p FG(\027)1824
2704 y Fk(e)1825 2721 y Fj(S)1819 2779 y Fy(1)1867 2752
y FA(\()p FG(X)g FA(\))p FG(;)15 b(:)g(:)g(:)i(;)e(\027)2271
2704 y Fk(e)2272 2721 y Fj(S)2266 2781 y Fv(N)2334 2752
y FA(\()p FG(X)7 b FA(\)\))p Fx(j)2573 2666 y FG(:)456
2914 y FA(Using)28 b(that)i FG(x)958 2928 y Fv(N)7 b
Fy(+1)1140 2914 y FA(=)25 b FG(g)1278 2930 y Fk(e)1279
2948 y Fj(S)-11 b Fy(\()p Fv(Y)t(;R)p Fy(\))1494 2914
y FA(\()p FG(x)p FA(\),)30 b(\(2.31\),)i(\(2.33\))f(and)e(\(2.35\),)j
(w)m(e)e(also)f(gather)h(that)762 3171 y FG(H)838 3185
y Fy(0)877 3171 y FA(\()p FG(y)957 3185 y Fv(N)7 b Fy(+1)1114
3171 y FA(\))21 b(+)f Fx(j)p FG(x)g Fx(\000)g FG(y)s
Fx(j)g(\000)g FG(R)26 b FA(=)f FG(H)1900 3185 y Fy(0)1939
3171 y FA(\()p FG(x)2026 3185 y Fv(N)7 b Fy(+1)2184 3171
y FA(\))20 b Fx(\000)p 2342 3036 72 4 v 2342 3109 a FG(C)2414
3123 y Fy(0)p 2340 3150 116 4 v 2340 3233 a FA(2)p FG(R)2465
3171 y FA(\()p FG(x)2552 3185 y Fv(N)7 b Fy(+1)2730 3171
y Fx(\000)20 b FG(y)2866 3185 y Fv(N)7 b Fy(+1)3023 3171
y FA(\))3058 3133 y Fy(2)3113 3171 y FG(:)456 3361 y
FA(Hence,)31 b(if)e FG(!)f FA(:=)e FG(y)1086 3375 y Fv(N)7
b Fy(+1)1263 3361 y Fx(\000)20 b FG(x)1406 3375 y Fv(N)7
b Fy(+1)1563 3361 y FA(,)31 b(w)m(e)g(gather)851 3588
y FG(\033)d FA(=)d Fx(j)p FG(x)c Fx(\000)f FG(y)s Fx(j)25
b FA(=)g FG(H)1486 3602 y Fy(0)1525 3588 y FA(\()p FG(x)1612
3602 y Fv(N)7 b Fy(+1)1770 3588 y FA(\))20 b Fx(\000)g
FG(H)1992 3602 y Fy(0)2031 3588 y FA(\()p FG(x)2118 3602
y Fv(N)7 b Fy(+1)2296 3588 y FA(+)20 b FG(!)s FA(\))g(+)g
FG(R)h Fx(\000)p 2786 3453 72 4 v 2786 3526 a FG(C)2858
3540 y Fy(0)p 2784 3567 116 4 v 2784 3650 a FA(2)p FG(R)2909
3588 y(!)2969 3550 y Fy(2)3024 3588 y FG(:)456 3783 y
FA(This)37 b(determines)h FG(\033)s FA(,)j(as)e(desired,)h(w)m(e)g(no)m
(w)e(need)h(to)h(determine)e FG(!)s FA(.)66 b(Using)40
b(\(2.2\))456 3891 y(\(with)e FG(s)750 3905 y Fy(0)830
3891 y FA(:=)i FG(y)1011 3905 y Fv(N)7 b Fy(+1)1208 3891
y FA(and)39 b FG(s)h FA(:=)h FG(x)1666 3905 y Fv(N)7
b Fy(+1)1823 3891 y FA(\))40 b(and)f(De\014nition)f(2.5)j(\(with)d(the)
i(fact)h(that)456 3999 y Fx(j)p FG(s)p Fx(j)25 b FG(<)g
FA(1)p FG(=)p FA(2)32 b(as)e(p)s(oin)m(ted)g(out)g(ab)s(o)m(v)m(e\),)i
(w)m(e)f(get)757 4163 y FG(!)86 b FA(=)d Fx(\000)p FA(\()p
FG(x)1212 4177 y Fv(N)7 b Fy(+1)1389 4163 y Fx(\000)20
b FG(y)1525 4177 y Fv(N)7 b Fy(+1)1682 4163 y FA(\))26
b(=)900 4435 y(=)1084 4373 y FG(R)p 1064 4414 111 4 v
1064 4432 72 4 v 1064 4505 a(C)1135 4519 y Fy(0)1200
4225 y Fu(0)1200 4385 y(B)1200 4443 y(@)1618 4373 y FA(1)p
1289 4414 704 4 v 1289 4452 a Fu(\020)1398 4512 y Fv(p)p
1353 4532 126 4 v 1353 4584 a(p)p Ft(\000)p Fy(1)1489
4553 y FG(')p FA(\()p FG(x)1635 4567 y Fv(N)7 b Fy(+1)1793
4553 y FA(\))1828 4452 y Fu(\021)1883 4475 y Fy(1)p Fv(=p)2023
4435 y Fx(\000)2470 4373 y FA(1)p 2124 4414 737 4 v 2124
4452 a Fu(\020)2233 4512 y Fv(p)p 2188 4532 126 4 v 2188
4584 a(p)p Ft(\000)p Fy(1)2324 4553 y FG(h)2376 4567
y Fy(0)2416 4553 y FA(\()p FG(x)2503 4567 y Fv(N)g Fy(+1)2661
4553 y FA(\))2696 4452 y Fu(\021)2750 4475 y Fy(1)p Fv(=p)2870
4225 y Fu(1)2870 4385 y(C)2870 4443 y(A)2975 4435 y FA(=)900
4861 y(=)1084 4799 y FG(R)p 1064 4840 111 4 v 1064 4858
72 4 v 1064 4931 a(C)1135 4945 y Fy(0)1200 4651 y Fu(0)1200
4811 y(B)1200 4869 y(@)1687 4799 y FA(1)p 1289 4840 842
4 v 1289 4878 a Fu(\020)1398 4938 y Fv(p)p 1353 4958
126 4 v 1353 5010 a(p)p Ft(\000)p Fy(1)1489 4979 y FG(h)1541
4993 y Fv(s)1574 5002 y Fq(0)1609 4993 y Fv(;R)1686 4979
y FA(\()p FG(x)1773 4993 y Fv(N)g Fy(+1)1931 4979 y FA(\))1966
4878 y Fu(\021)2020 4901 y Fy(1)p Fv(=p)2161 4861 y Fx(\000)2607
4799 y FA(1)p 2262 4840 737 4 v 2262 4878 a Fu(\020)2371
4938 y Fv(p)p 2326 4958 126 4 v 2326 5010 a(p)p Ft(\000)p
Fy(1)2462 4979 y FG(h)2514 4993 y Fy(0)2553 4979 y FA(\()p
FG(x)2640 4993 y Fv(N)g Fy(+1)2798 4979 y FA(\))2833
4878 y Fu(\021)2888 4901 y Fy(1)p Fv(=p)3008 4651 y Fu(1)3008
4811 y(C)3008 4869 y(A)3118 4861 y FG(:)p 456 5107 499
4 v 555 5187 a Fy(8)591 5216 y FF(Of)25 b(course,)i FD(\027)997
5171 y Fe(e)999 5184 y Fh(S)-39 b Fq(\()p Fs(Y)s(;R)p
Fq(\))1183 5216 y FF(\()p FD(X)6 b FF(\))25 b(is)i(normal)e(to)1768
5196 y Fg(e)1769 5216 y Fd(S)-11 b FF(\()p FD(Y)t(;)13
b(R)q FF(\))25 b(at)h(the)g(p)r(oin)n(t)f FD(X)6 b FF(.)p
eop
%%Page: 37 37
37 36 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(37)456
450 y FA(Th)m(us,)29 b(from)h(De\014nition)f(2.12)j(and)e(\(2.42\),)j
(w)m(e)d(ha)m(v)m(e)i(that)1111 645 y FG(!)85 b FA(=)1438
583 y FG(R)p 1417 624 111 4 v 1417 642 72 4 v 1417 715
a(C)1489 729 y Fy(0)1553 544 y Fu(\020)1608 645 y FG(H)1691
607 y Ft(0)1684 667 y Fv(s)1717 676 y Fq(0)1750 667 y
Fv(;R)1828 645 y FA(\()p FG(x)1915 659 y Fv(N)7 b Fy(+1)2072
645 y FA(\))21 b Fx(\000)f FG(H)2302 607 y Ft(0)2295
667 y Fy(0)2334 645 y FA(\()p FG(x)2421 659 y Fv(N)7
b Fy(+1)2579 645 y FA(\))2614 544 y Fu(\021)2693 645
y FA(=)1253 887 y(=)1438 826 y FG(R)p 1417 867 111 4
v 1417 885 72 4 v 1417 958 a(C)1489 972 y Fy(0)1553 787
y Fu(\020)1848 826 y FA(1)p 1617 867 507 4 v 1617 950
a Fx(jr)p FG(g)1760 966 y Fk(e)1761 984 y Fj(S)-11 b
Fy(\()p Fv(Y)t(;R)p Fy(\))1976 950 y FA(\()p FG(x)p FA(\))p
Fx(j)2154 887 y(\000)20 b FG(H)2328 850 y Ft(0)2321 910
y Fy(0)2360 887 y FA(\()p FG(x)2447 901 y Fv(N)7 b Fy(+1)2604
887 y FA(\))2639 787 y Fu(\021)2709 887 y FG(;)456 1111
y FA(and,)30 b(therefore,)1032 1345 y FG(!)e FA(=)1244
1283 y FG(R)p 1223 1324 111 4 v 1223 1342 72 4 v 1223
1415 a(C)1295 1429 y Fy(0)1359 1244 y Fu(\020)1662 1275
y FG(\027)1712 1225 y Fk(e)1713 1242 y Fj(S)1707 1302
y Fv(N)7 b Fy(+1)1865 1275 y FA(\()p FG(X)g FA(\))p 1424
1324 834 4 v 1424 1431 a Fx(j)p FA(\()p FG(\027)1534
1383 y Fk(e)1535 1400 y Fj(S)1529 1457 y Fy(1)1576 1431
y FA(\()p FG(X)g FA(\))p FG(;)15 b(:)g(:)g(:)j(;)d(\027)1981
1383 y Fk(e)1982 1400 y Fj(S)1976 1460 y Fv(N)2043 1431
y FA(\()p FG(X)7 b FA(\)\))p Fx(j)2287 1345 y(\000)20
b FG(H)2461 1307 y Ft(0)2454 1367 y Fy(0)2493 1345 y
FA(\()p FG(x)2580 1359 y Fv(N)7 b Fy(+1)2738 1345 y FA(\))2773
1244 y Fu(\021)2843 1345 y FG(;)456 1571 y FA(whic)m(h)29
b(determines)g FG(!)s FA(.)2113 b Fl(\003)456 1731 y
FH(Corollary)31 b(3.4.)39 b Fo(L)-5 b(et)29 b(the)h(notation)g(of)g(L)
-5 b(emma)30 b(3.3)f(holds.)42 b(F)-7 b(or)30 b FG(X)j
FA(=)25 b(\()p FG(x;)15 b(x)3165 1745 y Fv(N)7 b Fy(+1)3323
1731 y FA(\))26 b Fx(2)456 1841 y Fw(R)515 1808 y Fv(N)588
1841 y Fo(,)33 b(with)g FG(x)898 1855 y Fv(N)7 b Fy(+1)1081
1841 y FA(=)25 b FG(u)p FA(\()p FG(x)p FA(\))p Fo(,)33
b(let)1507 1806 y Fy(9)1196 2111 y FG(\027)1247 2073
y Fv(u)1292 2111 y FA(\()p FG(X)7 b FA(\))84 b(:=)1725
1915 y Fu(\020)1799 2016 y Fx(\000)20 b(r)p FG(u)p FA(\()p
FG(x)p FA(\))p FG(;)31 b FA(1)2241 1915 y Fu(\021)p 1717
2090 588 4 v 1717 2108 a(p)p 1808 2108 497 4 v 78 x FA(1)20
b(+)g Fx(jr)p FG(u)p FA(\()p FG(x)p FA(\))p Fx(j)2264
2160 y Fy(2)2355 2111 y Fx(2)25 b Fw(R)2500 2073 y Fv(N)7
b Fy(+1)2679 2111 y FG(;)-2248 b FA(\(3.11\))456 2340
y Fo(L)-5 b(et)33 b FG(X)r(;)15 b(Y)46 b Fx(2)24 b Fw(R)974
2307 y Fv(N)1080 2340 y Fo(b)-5 b(e)32 b(so)i(that)1405
2489 y FG(g)1448 2507 y Fj(S)-6 b Fy(\()p Fv(Y)t(;R)p
Fy(\))1662 2489 y FA(\()p FG(z)t FA(\))26 b Fx(\025)f
FG(u)p FA(\()p FG(z)t FA(\))187 b Fx(8)p FG(z)29 b Fx(2)c
Fw(R)2523 2451 y Fv(N)2621 2489 y FG(;)1405 2627 y(g)1448
2646 y Fj(S)-6 b Fy(\()p Fv(Y)t(;R)p Fy(\))1662 2627
y FA(\()p FG(x)p FA(\))26 b(=)f FG(u)p FA(\()p FG(x)p
FA(\))h(=)f FG(x)2254 2641 y Fv(N)7 b Fy(+1)2426 2627
y FG(:)-1995 b FA(\(3.12\))456 2779 y Fo(Then,)32 b FG(Y)46
b FA(=)25 b FG(F)13 b FA(\()p FG(X)r(;)i(\027)1189 2746
y Fv(u)1235 2779 y FA(\()p FG(X)7 b FA(\)\))p Fo(.)456
2939 y(Pr)-5 b(o)g(of.)43 b FA(By)34 b(\(3.12\),)39 b(w)m(e)c(ha)m(v)m
(e)g(that)h(the)e(graph)h(of)f FG(u)h FA(and)f(the)h(surface)f
Fw(S)o FA(\()p FG(Y)5 b(;)15 b(R)q FA(\))29 b(are)456
3047 y(tangen)m(t)i(at)h(the)e(p)s(oin)m(t)f FG(X)7 b
FA(,)31 b(therefore)1521 3200 y FG(\027)1572 3163 y Fv(u)1616
3200 y FA(\()p FG(X)7 b FA(\))42 b(=)e FG(\027)1972 3163
y Fj(S)-6 b Fy(\()p Fv(Y)t(;R)p Fy(\))2186 3200 y FA(\()p
FG(X)7 b FA(\))15 b FG(;)456 3348 y FA(hence)30 b(the)h(claim)e(follo)m
(ws)h(from)g(Lemma)g(3.3.)1289 b Fl(\003)456 3508 y FH(Lemma)35
b(3.5.)43 b Fo(In)34 b(the)h(notation)h(of)f(Cor)-5 b(ol)5
b(lary)37 b(3.4,)e(if)g FG(X)h Fx(2)2671 3490 y Fu(e)2660
3508 y Fi(B)p Fo(,)f(then)g(ther)-5 b(e)35 b(exists)456
3616 y(a)e(p)-5 b(ositive)33 b(c)-5 b(onstant)35 b FG(c)d
Fo(so)i(that)1513 3764 y FG(c)26 b Fx(\024)f FG(\027)1725
3727 y Fv(u)1719 3787 y(N)7 b Fy(+1)1876 3764 y FA(\()p
FG(X)g FA(\))27 b Fx(\024)d FA(1)d Fx(\000)f FG(c)15
b(:)456 3925 y Fo(Pr)-5 b(o)g(of.)43 b FA(Since,)29 b(b)m(y)h(Lemma)h
(3.1,)h Fx(j)p FG(x)1699 3939 y Fv(N)7 b Fy(+1)1856 3925
y Fx(j)26 b(\024)f FA(1)p FG(=)p FA(2,)680 4155 y Fx(jr)p
FG(u)p FA(\()p FG(x)p FA(\))p Fx(j)h FA(=)f Fx(jr)p FG(g)1246
4173 y Fj(S)-5 b Fy(\()p Fv(Y)t(;)o(R)p Fy(\))1461 4155
y FA(\()p FG(x)p FA(\))p Fx(j)26 b FA(=)1730 3995 y Fu(\014)1730
4050 y(\014)1730 4105 y(\014)1730 4159 y(\014)1730 4214
y(\014)1760 4026 y(\022)1915 4093 y FG(p)p 1837 4134
203 4 v 1837 4217 a(p)20 b Fx(\000)g FA(1)2049 4155 y
FG(h)2101 4169 y Fv(s)2134 4178 y Fq(0)2169 4169 y Fv(;R)2246
4155 y FA(\()p FG(x)2333 4169 y Fv(N)7 b Fy(+1)2491 4155
y FA(\))2526 4026 y Fu(\023)2593 4049 y Fy(1)p Fv(=p)2703
3995 y Fu(\014)2703 4050 y(\014)2703 4105 y(\014)2703
4159 y(\014)2703 4214 y(\014)2759 4155 y Fx(2)25 b FA([1)p
FG(=C)q(;)32 b(C)7 b FA(])15 b FG(;)456 4379 y FA(for)30
b(a)h(suitable)d(constan)m(t)k FG(C)7 b FA(,)30 b(whic)m(h)f(implies)f
(the)i(desired)f(claim.)599 b Fl(\003)456 4540 y FH(Lemma)32
b(3.6.)42 b Fo(L)-5 b(et)33 b(the)g(notation)i(of)e(L)-5
b(emma)34 b(3.3)f(and)g(Cor)-5 b(ol)5 b(lary)36 b(3.4)d(hold.)43
b(L)-5 b(et)1512 4688 y Fx(Y)7 b FA(\()p FG(X)g FA(\))26
b(:=)g FG(F)13 b FA(\()p FG(X)r(;)i(\027)2154 4650 y
Fv(u)2200 4688 y FA(\()p FG(X)7 b FA(\)\))456 4835 y
Fo(and)34 b(let)f FG(D)836 4849 y Fv(X)903 4835 y Fx(Y)41
b Fo(b)-5 b(e)33 b(the)g(di\013er)-5 b(ential)34 b(map.)44
b(Then,)34 b(ther)-5 b(e)34 b(exists)f(a)h(p)-5 b(ositive)33
b(c)-5 b(onstant)456 4943 y FG(C)39 b Fo(such)32 b(that)1626
5053 y Fx(j)p FG(D)1726 5067 y Fv(X)1794 5053 y Fx(Y)7
b FA(\()p FG(X)g FA(\))p Fx(j)27 b(\024)e FG(C)d(;)p
456 5116 499 4 v 555 5187 a Fy(9)591 5216 y FF(Note)f(that)h
FD(\027)978 5184 y Fs(u)1019 5216 y FF(\()p FD(X)6 b
FF(\))22 b(is)g(a)h(unit)e(v)n(ector,)i(normal)f(to)g(the)g(graph)g(of)
h FD(u)f FF(at)g(the)g(p)r(oin)n(t)g FD(X)27 b FF(=)21
b(\()p FD(x;)13 b(u)p FF(\()p FD(x)p FF(\)\).)p eop
%%Page: 38 38
38 37 bop 456 251 a Fy(38)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 451 y Fo(for)33 b(any)g FG(X)g Fx(2)980 433
y Fu(e)969 451 y Fi(B)p Fo(.)456 623 y(Pr)-5 b(o)g(of.)43
b FA(By)30 b(direct)g(insp)s(ection,)456 780 y(\(3.13\))219
b FG(D)980 794 y Fv(X)1048 780 y Fx(Y)7 b FA(\()p FG(X)g
FA(\))27 b(=)e FG(D)1466 794 y Fv(X)1533 780 y FG(F)13
b FA(\()p FG(X)r(;)i(\027)1807 743 y Fv(u)1853 780 y
FA(\()p FG(X)7 b FA(\)\))22 b(+)e FG(D)2228 794 y Fv(\027)2271
780 y FG(F)13 b FA(\()p FG(X)r(;)i(\027)2545 743 y Fv(u)2591
780 y FA(\()p FG(X)7 b FA(\)\))15 b FG(D)2868 794 y Fv(X)2938
780 y FG(\027)2989 743 y Fv(u)3033 780 y FA(\()p FG(X)7
b FA(\))15 b FG(:)456 937 y FA(On)29 b(the)i(other)f(hand,)g(b)m(y)g
(di\013eren)m(tiating)f(\(3.10\),)456 1104 y(\(3.14\))125
b(0)26 b(=)f FG(D)1053 1118 y Fv(X)1121 1104 y FG(F)13
b FA(\()p FG(X)r(;)i(\027)1395 1067 y Fj(S)-5 b Fy(\()p
Fv(Y)t(;)o(R)p Fy(\))1610 1104 y FA(\()p FG(X)7 b FA(\)\))21
b(+)f FG(D)1984 1118 y Fv(\027)2028 1104 y FG(F)13 b
FA(\()p FG(X)r(;)i(\027)2302 1067 y Fj(S)-6 b Fy(\()p
Fv(Y)t(;R)p Fy(\))2516 1104 y FA(\()p FG(X)7 b FA(\)\))15
b FG(D)2793 1118 y Fv(X)2863 1104 y FG(\027)2914 1067
y Fj(S)-7 b Fy(\()p Fv(Y)t(;R)p Fy(\))3127 1104 y FA(\()p
FG(X)7 b FA(\))15 b FG(:)456 1268 y FA(Moreo)m(v)m(er,)32
b(if)e FG(X)i Fx(2)1169 1250 y Fu(e)1157 1268 y Fi(B)q
FA(,)e(then)456 1425 y(\(3.15\))830 b Fx(r)p FG(g)1634
1442 y Fk(e)1635 1459 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p
Fy(\))1849 1425 y FA(\()p FG(x)p FA(\))26 b(=)f Fx(r)p
FG(u)p FA(\()p FG(x)p FA(\))15 b FG(;)456 1596 y FA(and)29
b(so)1536 1718 y FG(\027)1587 1680 y Fv(u)1632 1718 y
FA(\()p FG(X)7 b FA(\))26 b(=)f FG(\027)1956 1663 y Fk(e)1957
1680 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))2171 1718
y FA(\()p FG(X)7 b FA(\))15 b FG(:)456 1856 y FA(Th)m(us,)29
b(from)h(\(3.13\))j(and)c(\(3.14\),)k(w)m(e)e(gather)g(that)696
2040 y FG(D)771 2054 y Fv(X)839 2040 y Fx(Y)7 b FA(\()p
FG(X)g FA(\))26 b(=)f FG(D)1256 2054 y Fv(\027)1300 2040
y FG(F)13 b FA(\()p FG(X)r(;)i(\027)1573 1986 y Fk(e)1574
2003 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))1788 2040
y FA(\()p FG(X)7 b FA(\)\))1990 1939 y Fu(\020)2046 2040
y FG(D)2121 2054 y Fv(X)2189 2040 y FG(\027)2240 2003
y Fv(u)2285 2040 y FA(\()p FG(X)g FA(\))21 b Fx(\000)f
FG(D)2624 2054 y Fv(X)2692 2040 y FG(\027)2742 1986 y
Fk(e)2743 2003 y Fj(S)-13 b Fy(\()p Fv(Y)t(;R)p Fy(\))2956
2040 y FA(\()p FG(X)7 b FA(\))3108 1939 y Fu(\021)3179
2040 y FG(;)456 2241 y FA(for)30 b(an)m(y)g FG(X)j Fx(2)971
2222 y Fu(e)960 2241 y Fi(B)p FA(.)41 b(By)31 b(the)f(de\014nitions)e
(giv)m(en)j(in)e(Lemma)h(3.3,)i(one)f(sees)f(that)1292
2418 y Fx(j)p FG(D)1392 2432 y Fv(\027)1436 2418 y FG(F)13
b FA(\()p FG(X)r(;)i(\027)1709 2363 y Fk(e)1710 2381
y Fj(S)-11 b Fy(\()p Fv(Y)t(;R)p Fy(\))1925 2418 y FA(\()p
FG(X)7 b FA(\)\))p Fx(j)27 b(\024)40 b FA(const)16 b
FG(R)g(;)456 2575 y FA(and)29 b(so)i(w)m(e)g(get)g(from)f(the)h(ab)s(o)
m(v)m(e)g(relation)f(that)939 2754 y Fx(j)p FG(D)1039
2768 y Fv(X)1107 2754 y Fx(Y)7 b FA(\()p FG(X)g FA(\))p
Fx(j)27 b(\024)40 b FA(const)15 b FG(R)1797 2650 y Fu(\014)1797
2704 y(\014)1797 2759 y(\014)1828 2754 y FG(D)1903 2768
y Fv(X)1970 2754 y FG(\027)2021 2717 y Fv(u)2066 2754
y FA(\()p FG(X)7 b FA(\))21 b Fx(\000)f FG(D)2405 2768
y Fv(X)2473 2754 y FG(\027)2523 2700 y Fk(e)2524 2717
y Fj(S)-13 b Fy(\()p Fv(Y)t(;R)p Fy(\))2737 2754 y FA(\()p
FG(X)7 b FA(\))2889 2650 y Fu(\014)2889 2704 y(\014)2889
2759 y(\014)2936 2754 y FG(:)456 2939 y FA(Therefore,)25
b(in)e(the)i(ligh)m(t)e(of)i(\(3.15\),)j(Lemma)c(B.12,)k(Remark)c(B.13)
i(and)e(Lemma)g(3.5,)990 3123 y Fx(j)p FG(D)1090 3137
y Fv(X)1158 3123 y Fx(Y)7 b FA(\()p FG(X)g FA(\))p Fx(j)26
b(\024)41 b FA(const)15 b FG(R)1848 3019 y Fu(\014)1848
3073 y(\014)1848 3128 y(\014)1878 3123 y FG(D)1956 3086
y Fy(2)1996 3123 y FG(u)p FA(\()p FG(X)7 b FA(\))21 b
Fx(\000)f FG(D)2390 3086 y Fy(2)2429 3123 y FG(g)2471
3140 y Fk(e)2472 3157 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p
Fy(\))2686 3123 y FA(\()p FG(X)7 b FA(\))2838 3019 y
Fu(\014)2838 3073 y(\014)2838 3128 y(\014)2885 3123 y
FG(:)456 3303 y FA(Also,)30 b(since)g FG(g)951 3319 y
Fk(e)952 3337 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))1196
3303 y FA(touc)m(hes)31 b FG(u)g FA(from)f(ab)s(o)m(v)m(e)h(at)g
FG(X)7 b FA(,)31 b(w)m(e)g(ha)m(v)m(e)h(that)456 3486
y(\(3.16\))239 b FG(D)1003 3453 y Fy(2)1042 3486 y FG(g)1084
3503 y Fk(e)1085 3520 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p
Fy(\))1299 3486 y FA(\()p FG(X)7 b FA(\))22 b Fx(\000)e
FG(D)1642 3453 y Fy(2)1681 3486 y FG(u)p FA(\()p FG(X)7
b FA(\))31 b(is)f(a)h(non-negativ)m(e)g(de\014nite)e(matrix)456
3657 y(and)g(therefore)868 3819 y(const)1091 3714 y Fu(\014)1091
3769 y(\014)1091 3823 y(\014)1121 3819 y FG(D)1199 3781
y Fy(2)1238 3819 y FG(u)p FA(\()p FG(X)7 b FA(\))22 b
Fx(\000)e FG(D)1633 3781 y Fy(2)1672 3819 y FG(g)1714
3835 y Fk(e)1715 3853 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p
Fy(\))1929 3819 y FA(\()p FG(X)7 b FA(\))2081 3714 y
Fu(\014)2081 3769 y(\014)2081 3823 y(\014)2138 3819 y
Fx(\024)25 b FA(\001)2310 3718 y Fu(\020)2364 3819 y
FG(g)2406 3835 y Fk(e)2407 3853 y Fj(S)-12 b Fy(\()p
Fv(Y)t(;R)p Fy(\))2641 3819 y Fx(\000)20 b FG(u)2784
3718 y Fu(\021)2838 3819 y FA(\()p FG(X)7 b FA(\))15
b FG(:)456 3998 y FA(Therefore,)30 b(w)m(e)h(ha)m(v)m(e)g(obtained)f
(the)h(follo)m(wing)e(estimate:)456 4178 y(\(3.17\))436
b Fx(j)p FG(D)1222 4192 y Fv(X)1289 4178 y Fx(Y)7 b FA(\()p
FG(X)g FA(\))p Fx(j)27 b(\024)40 b FA(const)16 b FG(R)g
FA(\001)2056 4077 y Fu(\020)2110 4178 y FG(g)2152 4194
y Fk(e)2153 4211 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))2388
4178 y Fx(\000)20 b FG(u)2531 4077 y Fu(\021)2585 4178
y FA(\()p FG(X)7 b FA(\))15 b FG(:)456 4370 y FA(Notice)23
b(no)m(w)f(that)i FG(u)e FA(is)f FG(C)1333 4337 y Fy(2)1395
4370 y FA(near)h FG(X)30 b FA(b)m(y)22 b(standard)g(elliptic)e
(results,)j(since)f Fx(r)p FG(u)p FA(\()p FG(X)7 b FA(\))26
b Fx(6)p FA(=)f(0)456 4478 y(thanks)30 b(to)h(\(3.15\).)43
b(Thence,)30 b(making)g(use)g(of)g(\(3.15\),)j(w)m(e)e(get)1171
4634 y(\001)1247 4648 y Fv(p)1286 4634 y FG(g)1328 4651
y Fk(e)1329 4668 y Fj(S)-11 b Fy(\()p Fv(Y)t(;R)p Fy(\))1544
4634 y FA(\()p FG(X)7 b FA(\))21 b Fx(\000)f FA(\001)1884
4648 y Fv(p)1923 4634 y FG(u)p FA(\()p FG(X)7 b FA(\))27
b(=)1017 4825 y(=)83 b Fx(jr)p FG(g)1314 4841 y Fk(e)1315
4859 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))1529 4825
y FA(\()p FG(X)7 b FA(\))p Fx(j)1706 4787 y Fv(p)p Ft(\000)p
Fy(2)1853 4724 y Fu(\020)1907 4825 y FA(\001)p FG(g)2025
4841 y Fk(e)2026 4859 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p
Fy(\))2240 4825 y FA(\()p FG(X)7 b FA(\))21 b Fx(\000)f
FA(\001)p FG(u)p FA(\()p FG(X)7 b FA(\))2784 4724 y Fu(\021)2875
4825 y FA(+)1171 4999 y(+)15 b(\()p FG(p)20 b Fx(\000)g
FA(2\))15 b Fx(jr)p FG(g)1687 5016 y Fk(e)1688 5033 y
Fj(S)-11 b Fy(\()p Fv(Y)t(;R)p Fy(\))1903 4999 y FA(\()p
FG(X)7 b FA(\))p Fx(j)2080 4962 y Fv(p)p Ft(\000)p Fy(4)2227
4999 y Fx(r)p FG(g)2345 5016 y Fk(e)2346 5033 y Fj(S)-12
b Fy(\()p Fv(Y)t(;R)p Fy(\))2560 4999 y FA(\()p FG(X)7
b FA(\))21 b Fx(\001)-2302 b FA(\(3.18\))1171 5190 y
Fx(\001)1211 5089 y Fu(h\020)1309 5190 y FG(D)1387 5152
y Fy(2)1426 5190 y FG(g)1468 5206 y Fk(e)1469 5223 y
Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))1683 5190 y FA(\()p
FG(X)7 b FA(\))21 b Fx(\000)f FG(D)2025 5152 y Fy(2)2065
5190 y FG(u)p FA(\()p FG(X)7 b FA(\))2269 5089 y Fu(\021)2344
5190 y Fx(\001)21 b(r)p FG(g)2508 5206 y Fk(e)2509 5223
y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))2723 5190 y FA(\()p
FG(X)7 b FA(\))2875 5089 y Fu(i)p eop
%%Page: 39 39
39 38 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(39)456
450 y FA(W)-8 b(e)40 b(need)e(no)m(w)h(to)g(distinguish)c(t)m(w)m(o)41
b(cases.)67 b(If)38 b FG(p)h Fx(\025)g FA(2,)i(w)m(e)f(use)e(Lemma)h
(3.5)h(and)456 558 y(\(3.16\))32 b(in)d(order)h(to)h(deduce)f(from)g
(\(3.18\))j(that)1173 717 y(\001)1249 731 y Fv(p)1289
717 y FG(g)1331 734 y Fk(e)1332 751 y Fj(S)-12 b Fy(\()p
Fv(Y)t(;R)p Fy(\))1546 717 y FA(\()p FG(X)7 b FA(\))22
b Fx(\000)d FA(\001)1886 731 y Fv(p)1926 717 y FG(u)p
FA(\()p FG(X)7 b FA(\))26 b Fx(\025)1020 908 y(\025)82
b(jr)p FG(g)1316 924 y Fk(e)1317 941 y Fj(S)-11 b Fy(\()p
Fv(Y)t(;R)p Fy(\))1532 908 y FA(\()p FG(X)7 b FA(\))p
Fx(j)1709 870 y Fv(p)p Ft(\000)p Fy(2)1855 807 y Fu(\020)1909
908 y FA(\001)p FG(g)2027 924 y Fk(e)2028 941 y Fj(S)-11
b Fy(\()p Fv(Y)t(;R)p Fy(\))2243 908 y FA(\()p FG(X)7
b FA(\))21 b Fx(\000)f FA(\001)p FG(u)p FA(\()p FG(X)7
b FA(\))2787 807 y Fu(\021)2867 908 y Fx(\025)1020 1107
y(\025)98 b FA(const)1411 1006 y Fu(\020)1466 1107 y
FA(\001)p FG(g)1584 1123 y Fk(e)1585 1140 y Fj(S)-12
b Fy(\()p Fv(Y)t(;R)p Fy(\))1799 1107 y FA(\()p FG(X)7
b FA(\))21 b Fx(\000)f FA(\001)p FG(u)p FA(\()p FG(X)7
b FA(\))2343 1006 y Fu(\021)2413 1107 y FG(:)-1982 b
FA(\(3.19\))456 1294 y(On)29 b(the)i(other)f(hand,)g(if)f(1)d
FG(<)f(p)g(<)g FA(2,)31 b(\(3.18\),)i(Lemma)d(3.5)i(and)d(\(3.16\))k
(giv)m(e)e(that)926 1453 y(\001)1002 1467 y Fv(p)1041
1453 y FG(g)1083 1470 y Fk(e)1084 1487 y Fj(S)-11 b Fy(\()p
Fv(Y)t(;R)p Fy(\))1299 1453 y FA(\()p FG(X)7 b FA(\))21
b Fx(\000)f FA(\001)1639 1467 y Fv(p)1678 1453 y FG(u)p
FA(\()p FG(X)7 b FA(\))27 b Fx(\025)772 1643 y(\025)83
b(jr)p FG(g)1069 1660 y Fk(e)1070 1677 y Fj(S)-12 b Fy(\()p
Fv(Y)t(;R)p Fy(\))1284 1643 y FA(\()p FG(X)7 b FA(\))p
Fx(j)1461 1606 y Fv(p)p Ft(\000)p Fy(2)1608 1542 y Fu(\020)1662
1643 y FA(\001)p FG(g)1780 1660 y Fk(e)1781 1677 y Fj(S)-12
b Fy(\()p Fv(Y)t(;R)p Fy(\))1995 1643 y FA(\()p FG(X)7
b FA(\))21 b Fx(\000)f FA(\001)p FG(u)p FA(\()p FG(X)7
b FA(\))2539 1542 y Fu(\021)2630 1643 y Fx(\000)926 1818
y(\000)15 b FA(\(2)21 b Fx(\000)f FG(p)p FA(\))15 b Fx(jr)p
FG(g)1443 1835 y Fk(e)1444 1852 y Fj(S)-12 b Fy(\()p
Fv(Y)t(;R)p Fy(\))1658 1818 y FA(\()p FG(X)7 b FA(\))p
Fx(j)1835 1780 y Fv(p)p Ft(\000)p Fy(2)1982 1818 y Fx(j)p
FG(D)2085 1780 y Fy(2)2124 1818 y FG(g)2166 1835 y Fk(e)2167
1852 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))2381 1818
y FA(\()p FG(X)7 b FA(\))22 b Fx(\000)e FG(D)2724 1780
y Fy(2)2763 1818 y FG(u)p FA(\()p FG(X)7 b FA(\))p Fx(j)27
b(\025)772 2008 y(\025)83 b(jr)p FG(g)1069 2025 y Fk(e)1070
2042 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))1284 2008
y FA(\()p FG(X)7 b FA(\))p Fx(j)1461 1971 y Fv(p)p Ft(\000)p
Fy(2)1608 2008 y FA([1)20 b Fx(\000)g FA(\(2)h Fx(\000)f
FG(p)p FA(\)])2102 1907 y Fu(\020)2157 2008 y FA(\001)p
FG(g)2275 2025 y Fk(e)2276 2042 y Fj(S)-12 b Fy(\()p
Fv(Y)t(;R)p Fy(\))2490 2008 y FA(\()p FG(X)7 b FA(\))22
b Fx(\000)d FA(\001)p FG(u)p FA(\()p FG(X)7 b FA(\))3034
1907 y Fu(\021)3115 2008 y Fx(\025)772 2207 y(\025)98
b FA(const)16 b(\()p FG(p)k Fx(\000)g FA(1\))1451 2106
y Fu(\020)1506 2207 y FA(\001)p FG(g)1624 2224 y Fk(e)1625
2241 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))1839 2207
y FA(\()p FG(X)7 b FA(\))22 b Fx(\000)e FA(\001)p FG(u)p
FA(\()p FG(X)7 b FA(\))2384 2106 y Fu(\021)2454 2207
y FG(:)-2023 b FA(\(3.20\))456 2394 y(In)29 b(an)m(y)i(case,)h(for)e
(an)m(y)g FG(p)25 b Fx(2)g FA(\(1)p FG(;)15 b FA(+)p
Fx(1)p FA(\),)32 b(\(3.19\))h(and)c(\(3.20\))k(giv)m(e)e(that)1436
2554 y(\001)1512 2568 y Fv(p)1551 2554 y FG(g)1593 2570
y Fk(e)1594 2587 y Fj(S)-11 b Fy(\()p Fv(Y)t(;R)p Fy(\))1809
2554 y FA(\()p FG(X)7 b FA(\))21 b Fx(\000)f FA(\001)2149
2568 y Fv(p)2188 2554 y FG(u)p FA(\()p FG(X)7 b FA(\))27
b Fx(\025)1282 2744 y(\025)98 b FA(const)1674 2643 y
Fu(\020)1728 2744 y FA(\001)p FG(g)1846 2761 y Fk(e)1847
2778 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))2061 2744
y FA(\()p FG(X)7 b FA(\))21 b Fx(\000)f FA(\001)p FG(u)p
FA(\()p FG(X)7 b FA(\))2605 2643 y Fu(\021)2676 2744
y FG(:)-2245 b FA(\(3.21\))456 2926 y(F)-8 b(urthermore,)24
b(exploiting)e(Prop)s(osition)f(2.18)k(and)e(the)g(fact)i(that)f
FG(u)f FA(is)f(a)i(subsolution)456 3034 y(of)30 b(\(1.5\),)j(w)m(e)d
(ha)m(v)m(e)i(that)456 3240 y(\(3.22\))659 b(\001)1421
3254 y Fv(p)1460 3240 y FG(g)1502 3257 y Fk(e)1503 3274
y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))1717 3240 y FA(\()p
FG(X)7 b FA(\))22 b Fx(\000)e FA(\001)2058 3254 y Fv(p)2097
3240 y FG(u)p FA(\()p FG(X)7 b FA(\))27 b Fx(\024)2433
3178 y FG(C)p 2433 3219 72 4 v 2434 3302 a(R)2530 3240
y(:)456 3430 y FA(The)i(desired)g(result)h(no)m(w)g(follo)m(ws)f(from)h
(\(3.17\),)j(\(3.21\))g(and)d(\(3.22\).)484 b Fl(\003)555
3606 y FA(Recalling)29 b(\(3.4\),)k(w)m(e)e(de\014ne)1720
3765 y Fi(S)25 b FA(:=)g Fx(Y)7 b FA(\()p Fi(G)p FA(\))15
b FG(:)456 3932 y FA(The)29 b(construction)h(of)1272
3914 y Fu(e)1267 3932 y Fi(A)g FA(and)1545 3914 y Fu(e)1534
3932 y Fi(B)g FA(easily)g(implies)d(the)k(follo)m(wing)e(observ)-5
b(ation:)456 4068 y FH(Lemma)34 b(3.7.)1038 4049 y Fu(e)1033
4068 y Fi(A)g Fo(and)1316 4049 y Fu(e)1305 4068 y Fi(B)g
Fo(b)-5 b(elong)35 b(to)g(Lipschitz)g(surfac)-5 b(es.)47
b(Mor)-5 b(e)34 b(pr)-5 b(e)g(cisely,)3218 4049 y Fu(e)3207
4068 y Fi(B)34 b Fo(lies)456 4177 y(in)e Fi(G)h Fo(while)902
4159 y Fu(e)897 4177 y Fi(A)f Fo(lies)h(in)f Fi(S)p Fo(.)555
4311 y FA(The)k(next)g(observ)-5 b(ation)36 b(will)d(sa)m(y)k(that)1994
4293 y Fu(e)1990 4311 y Fi(A)e FA(and)2279 4293 y Fu(e)2268
4311 y Fi(B)h FA(are)g(graphs)g(with)f(resp)s(ect)h(to)456
4419 y(the)30 b FG(\030)t FA(-direction:)456 4555 y FH(Lemma)i(3.8.)42
b FG(\031)1082 4570 y Fv(\030)1153 4555 y Fo(is)32 b(inje)-5
b(ctive)32 b(on)1745 4536 y Fu(e)1740 4555 y Fi(A)g Fo(and)i(on)2151
4536 y Fu(e)2139 4555 y Fi(B)p Fo(.)456 4742 y(Pr)-5
b(o)g(of.)43 b FG(\031)789 4757 y Fv(\030)850 4742 y
FA(in)22 b(injectiv)m(e)h(on)1427 4724 y Fu(e)1423 4742
y Fi(A)g FA(b)m(y)g(construction.)38 b(Let)24 b(us)f(sho)m(w)g(that)i
(is)d(also)i(injectiv)m(e)456 4862 y(on)593 4844 y Fu(e)582
4862 y Fi(B)p FA(.)41 b(Assume,)30 b(b)m(y)h(con)m(tradiction,)f(that)h
FG(X)2067 4829 y Fy(\(1\))2188 4862 y Fx(2)25 b Fi(B)30
b FA(and)g FG(X)2636 4829 y Fy(\(2\))2756 4862 y FA(=)25
b FG(X)2934 4829 y Fy(\(1\))3049 4862 y FA(+)20 b FG(\034)10
b(\030)30 b Fx(2)25 b Fi(B)p FA(,)456 4972 y(with)31
b FG(\034)40 b(>)29 b FA(0.)49 b(Let)34 b(also)f FG(Y)1388
4939 y Fy(\(1\))1515 4972 y FA(and)f FG(Y)1767 4939 y
Fy(\(2\))1894 4972 y FA(b)s(e)h(the)g(corresp)s(onding)e(cen)m(ters)j
(in)3189 4954 y Fu(e)3185 4972 y Fi(A)o FA(,)g(i.e.,)456
5082 y(for)d FG(i)e FA(=)f(1)p FG(;)15 b FA(2,)34 b(let)e
FG(Y)1150 5049 y Fy(\()p Fv(i)p Fy(\))1261 5082 y Fx(2)1354
5064 y Fu(e)1350 5082 y Fi(A)f FA(b)s(e)h(so)g(that)h
FG(g)1922 5108 y Fj(S)-6 b Fy(\()p Fv(Y)2043 5088 y Fq(\()p
Fs(i)p Fq(\))2117 5108 y Fv(;R)p Fy(\))2254 5082 y FA(touc)m(hes)33
b FG(u)f FA(for)g(the)g(\014rst)g(time)g(at)456 5216
y FG(X)538 5183 y Fy(\()p Fv(i)p Fy(\))621 5216 y FA(.)p
eop
%%Page: 40 40
40 39 bop 456 251 a Fy(40)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)555 450 y FA(Since)30 b FG(\030)833 464 y Fv(N)7
b Fy(+1)1015 450 y FA(=)25 b(0,)1645 589 y FG(x)1697
542 y Fy(\(1\))1697 618 y Fv(N)7 b Fy(+1)1894 589 y FA(=)41
b FG(x)2058 542 y Fy(\(2\))2058 618 y Fv(N)7 b Fy(+1)2230
589 y FG(;)456 729 y FA(therefore,)1224 902 y FG(g)1267
928 y Fj(S)-5 b Fy(\()p Fv(Y)1389 908 y Fq(\(1\))1471
928 y Fv(;R)p Fy(\))1576 902 y FA(\()p FG(x)1663 865
y Fy(\(1\))1758 902 y FA(\))25 b(=)g FG(x)1966 854 y
Fy(\(1\))1966 931 y Fv(N)7 b Fy(+1)2149 902 y FA(=)25
b FG(u)p FA(\()p FG(x)2384 865 y Fy(\(1\))2479 902 y
FA(\))g(=)1250 1078 y(=)g FG(x)1398 1030 y Fy(\(2\))1398
1106 y Fv(N)7 b Fy(+1)1580 1078 y FA(=)25 b FG(g)1719
1103 y Fj(S)-5 b Fy(\()p Fv(Y)1841 1084 y Fq(\(2\))1923
1103 y Fv(;R)p Fy(\))2028 1078 y FA(\()p FG(x)2115 1040
y Fy(\(2\))2210 1078 y FA(\))25 b(=)g FG(u)p FA(\()p
FG(x)2505 1040 y Fy(\(2\))2600 1078 y FA(\))15 b FG(:)456
986 y FA(\(3.23\))456 1241 y(If)30 b(no)m(w)g(w)m(e)h(consider)1653
1339 y(\026)1639 1362 y FG(Y)60 b FA(:=)41 b FG(Y)1962
1324 y Fy(\(2\))2076 1362 y Fx(\000)20 b FG(\034)10 b(\030)456
1501 y FA(it)30 b(follo)m(ws)f(that)456 1767 y(\(3.24\))66
b FG(g)795 1790 y Fj(S)-6 b Fy(\()876 1774 y(\026)865
1790 y Fv(Y)10 b(;R)p Fy(\))1021 1767 y FA(\()p FG(x)1108
1729 y Fy(\(1\))1202 1767 y FA(\))26 b(=)f FG(g)1402
1792 y Fj(S)-6 b Fy(\()p Fv(Y)1523 1773 y Fq(\(2\))1606
1792 y Ft(\000)p Fv(\034)8 b(\030)s(;R)p Fy(\))1838 1767
y FA(\()p FG(x)1925 1729 y Fy(\(2\))2040 1767 y Fx(\000)20
b FG(\034)10 b(\030)t FA(\))25 b(=)g FG(g)2424 1792 y
Fj(S)-5 b Fy(\()p Fv(Y)2546 1773 y Fq(\(2\))2628 1792
y Fv(;R)p Fy(\))2733 1767 y FA(\()p FG(x)2820 1729 y
Fy(\(2\))2915 1767 y FA(\))25 b(=)g FG(u)p FA(\()p FG(x)3210
1729 y Fy(\(1\))3305 1767 y FA(\))15 b FG(:)456 1951
y FA(On)24 b(the)i(other)f(hand,)h(since)e FG(Y)1525
1918 y Fy(\(2\))1645 1951 y Fx(2)1735 1933 y Fu(e)1731
1951 y Fi(A)o FA(,)j FG(g)1885 1975 y Fj(S)-6 b Fy(\()1966
1958 y(\026)1955 1975 y Fv(Y)10 b(;R)p Fy(\))2136 1951
y FG(>)25 b(u)p FA(,)i(in)d(con)m(tradiction)h(with)f(\(3.24\).)3374
2061 y Fl(\003)555 2235 y FA(Giv)m(en)30 b FG(X)j Fx(2)25
b Fw(R)1072 2202 y Fv(N)7 b Fy(+1)1235 2235 y FA(,)31
b(w)m(e)g(no)m(w)f(de\014ne)456 2400 y(\(3.25\))742 b(\006)1494
2414 y Fv(X)1587 2400 y FA(:=)25 b Fx(f)p FG(Z)32 b Fx(j)25
b FG(X)33 b Fx(2)2091 2377 y Fu(e)2091 2400 y Fw(S)-7
b FA(\()p FG(Z)q(;)15 b(R)q FA(\))q Fx(g)g FG(:)456 2558
y FA(In)25 b(other)i(w)m(ords,)g(giv)m(en)f(a)h(p)s(oin)m(t)e
FG(X)7 b FA(,)28 b(\006)1821 2572 y Fv(X)1914 2558 y
FA(is)d(the)i(surface)f(con)m(taining)g(all)f(the)h(cen)m(ters)456
2674 y(of)45 b(the)g(surfaces)1102 2650 y Fu(e)1102 2674
y Fw(S)-7 b FA(\()p Fx(\001)p FG(;)15 b(R)q FA(\))46
b(to)g(whic)m(h)d FG(X)53 b FA(b)s(elongs.)84 b(Let)45
b(us)g(no)m(w)g(in)m(v)m(estigate)h(the)456 2781 y(prop)s(erties)28
b(of)j(\006)1053 2795 y Fv(X)456 2916 y FH(Lemma)25 b(3.9.)37
b Fo(L)-5 b(et)27 b Fx(Y)34 b Fo(b)-5 b(e)26 b(as)i(in)e(L)-5
b(emma)28 b(3.6.)41 b(Then,)28 b Fx(Y)7 b FA(\()p FG(X)g
FA(\))26 b Fx(2)f FA(\006)2748 2930 y Fv(X)2842 2916
y Fo(for)i(any)h FG(X)k Fx(2)3355 2898 y Fu(e)3343 2916
y Fi(B)q Fo(.)456 3097 y(Pr)-5 b(o)g(of.)43 b FA(Since)34
b FG(X)40 b Fx(2)1199 3079 y Fu(e)1188 3097 y Fi(B)p
FA(,)d(there)e(exists)g FG(Y)55 b FA(so)35 b(that)h(\(3.12\))i(holds.)
53 b(Then,)36 b(b)m(y)f(Corol-)456 3205 y(lary)29 b(3.4,)1285
3326 y(\006)1351 3340 y Fv(X)1443 3326 y Fx(3)c FG(Y)45
b FA(=)25 b FG(F)13 b FA(\()p FG(X)r(;)i(\027)1997 3288
y Fv(u)2043 3326 y FA(\()p FG(X)7 b FA(\)\))27 b(=)e
Fx(Y)7 b FA(\()p FG(X)g FA(\))15 b FG(:)3374 3465 y Fl(\003)555
3639 y FA(W)-8 b(e)26 b(denote)f(b)m(y)g FG(\027)1165
3606 y Fy(\006)1216 3617 y Fs(X)1277 3639 y FA(\()p FG(Z)7
b FA(\))25 b(the)g(unit)e(normal)h(v)m(ector)i(to)f(the)g(surface)f
(\006)2970 3653 y Fv(X)3062 3639 y FA(at)h(a)g(p)s(oin)m(t)456
3747 y FG(Z)36 b Fx(2)29 b FA(\006)710 3761 y Fv(X)811
3747 y FA(\(in)j(a)h(\014xed)g(orien)m(tation\).)49 b(Suc)m(h)32
b(de\014nition)f(is)h(w)m(ell)g(p)s(osed,)i(since)e(\006)3283
3761 y Fv(X)3383 3747 y FA(is)456 3856 y(a)38 b(Lipsc)m(hitz)e
(surface,)k(as)e(w)m(e)g(sho)m(w)g(here)g(b)s(elo)m(w.)62
b(Also,)40 b(w)m(e)e(can)g(express)f FG(\027)3218 3823
y Fy(\006)3269 3834 y Fs(X)3369 3856 y FA(in)456 3971
y(terms)30 b(of)g(the)h(normal)e(to)1388 3948 y Fu(e)1388
3971 y Fw(S)-7 b FA(\()p FG(Y)5 b(;)15 b(R)q FA(\))q(,)30
b(according)h(to)g(the)f(follo)m(wing)f(result:)456 4111
y FH(Lemma)39 b(3.10.)45 b Fo(L)-5 b(et)38 b Fx(Y)45
b Fo(b)-5 b(e)37 b(as)h(in)g(L)-5 b(emma)39 b(3.6)f(and)g(let)g
FG(X)k Fx(2)2736 4093 y Fu(e)2725 4111 y Fi(B)p Fo(.)57
b(Then)38 b FA(\006)3191 4125 y Fv(X)3295 4111 y Fo(is)g(a)456
4219 y(Lipschitz)31 b(r)-5 b(otation)33 b(surfac)-5 b(e)31
b(with)h(axis)f(p)-5 b(ar)g(al)5 b(lel)33 b(to)e FG(e)2333
4233 y Fv(N)7 b Fy(+1)2521 4219 y Fo(and)32 b(p)-5 b(assing)32
b(thr)-5 b(ough)32 b FG(X)7 b Fo(.)456 4347 y(A)n(lso,)32
b FG(\027)739 4314 y Fy(\006)790 4325 y Fs(X)852 4347
y FA(\()p Fx(Y)7 b FA(\()p FG(X)g FA(\)\))34 b Fo(b)-5
b(elongs)34 b(to)f(the)g(sp)-5 b(ac)g(e)34 b(sp)-5 b(anne)g(d)35
b(by)d FG(e)2489 4361 y Fv(N)7 b Fy(+1)2680 4347 y Fo(and)33
b FG(\027)2906 4296 y Fk(e)2907 4314 y Fj(S)-12 b Fy(\()p
Ft(Y)6 b Fy(\()p Fv(X)f Fy(\))p Fv(;R)p Fy(\))3246 4347
y FA(\()p FG(X)i FA(\))p Fo(.)456 4521 y(Pr)-5 b(o)g(of.)43
b FA(Let)31 b(us)f(\014rst)h(sho)m(w)g(that)g(\006)1695
4535 y Fv(X)1793 4521 y FA(is)f(a)i(Lipsc)m(hitz)d(rotation)j(surface.)
42 b(By)32 b(\(2.35\),)456 4628 y(w)m(e)e(ha)m(v)m(e)i(that)f
FG(Z)h Fx(2)25 b FA(\006)1242 4642 y Fv(X)1339 4628 y
FA(if)k(and)h(only)g(if)862 4786 y FG(x)914 4800 y Fv(N)7
b Fy(+1)1097 4786 y FA(=)25 b FG(g)1235 4803 y Fk(e)1236
4820 y Fj(S)-12 b Fy(\()p Fv(Z)q(;R)p Fy(\))1454 4786
y FA(\()p FG(x)p FA(\))26 b(=)f FG(\032)1745 4800 y Fv(z)1778
4811 y Fs(N)5 b Fq(+1)1912 4800 y Fv(;R)1990 4786 y FA(\()p
FG(H)2101 4800 y Fy(0)2140 4786 y FA(\()p FG(z)2217 4800
y Fv(N)i Fy(+1)2375 4786 y FA(\))20 b(+)g Fx(j)p FG(z)25
b Fx(\000)20 b FG(x)p Fx(j)g(\000)g FG(R)q FA(\))15 b
FG(;)456 4963 y FA(that)31 b(is,)e(b)m(y)i(\(2.33\))h(and)e(\(2.31\),)j
(if)c(and)h(only)f(if)765 5186 y FG(H)841 5200 y Fy(0)880
5186 y FA(\()p FG(x)967 5200 y Fv(N)7 b Fy(+1)1125 5186
y FA(\))20 b Fx(\000)p 1283 5052 72 4 v 1283 5125 a FG(C)1355
5139 y Fy(0)p 1281 5166 116 4 v 1281 5249 a FA(2)p FG(R)1406
5186 y FA(\()p FG(x)1493 5200 y Fv(N)7 b Fy(+1)1671 5186
y Fx(\000)20 b FG(z)1804 5200 y Fv(N)7 b Fy(+1)1962 5186
y FA(\))1997 5149 y Fy(2)2062 5186 y FA(=)25 b FG(H)2234
5200 y Fy(0)2273 5186 y FA(\()p FG(z)2350 5200 y Fv(N)7
b Fy(+1)2508 5186 y FA(\))20 b(+)g Fx(j)p FG(z)25 b Fx(\000)20
b FG(x)p Fx(j)g(\000)g FG(R)c(:)p eop
%%Page: 41 41
41 40 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(41)456
450 y FA(W)-8 b(e)31 b(no)m(w)g(de\014ne)1164 640 y Fi(H)1224
654 y Fv(x)1264 665 y Fs(N)5 b Fq(+1)1399 654 y Fv(;R)1476
640 y FA(\()p FG(\020)i FA(\))25 b(:=)h FG(H)1816 654
y Fy(0)1855 640 y FA(\()p FG(\020)7 b FA(\))20 b(+)p
2095 505 72 4 v 2095 578 a FG(C)2167 592 y Fy(0)p 2093
619 116 4 v 2093 702 a FA(2)p FG(R)2218 640 y FA(\()p
FG(x)2305 654 y Fv(N)7 b Fy(+1)2483 640 y Fx(\000)20
b FG(\020)7 b FA(\))2656 602 y Fy(2)2710 640 y FG(:)456
818 y FA(Then,)29 b Fi(H)h FA(is)g(strictly)f(increasing)g(in)g([)p
Fx(\000)p FA(1)p FG(=)p FA(2)p FG(;)15 b FA(1)p FG(=)p
FA(2];)34 b(th)m(us)c FG(Z)i Fx(2)25 b FA(\006)2678 832
y Fv(X)2775 818 y FA(if)k(and)h(only)g(if)1086 994 y
FG(z)1128 1008 y Fv(N)7 b Fy(+1)1311 994 y FA(=)25 b
Fi(H)1467 955 y Ft(\000)p Fy(1)1467 1022 y Fv(x)1507
1033 y Fs(N)5 b Fq(+1)1641 1022 y Fv(;R)1718 893 y Fu(\020)1773
994 y FG(H)1849 1008 y Fy(0)1888 994 y FA(\()p FG(x)1975
1008 y Fv(N)i Fy(+1)2132 994 y FA(\))21 b(+)f FG(R)h
Fx(\000)f(j)p FG(z)k Fx(\000)c FG(x)p Fx(j)2719 893 y
Fu(\021)2789 994 y FG(:)456 1164 y FA(This)28 b(pro)m(v)m(es)j(the)g
(rotational)f(symmetry)g(and)g(Lipsc)m(h)m(tiz)g(prop)s(erties)e(of)j
(\006)3105 1178 y Fv(X)3172 1164 y FA(.)555 1312 y(Consequen)m(tly)-8
b(,)27 b(b)m(y)f(Lemma)g(B.10,)j(w)m(e)d(gather)h(that)g
FG(\027)2460 1279 y Fy(\006)2511 1290 y Fs(X)2573 1312
y FA(\()p Fx(Y)7 b FA(\()p FG(X)g FA(\)\))28 b(is)d(in)g(the)h(space)
456 1441 y(spanned)36 b(b)m(y)h(\()p FG(\031)1037 1455
y Fv(e)1070 1466 y Fs(N)5 b Fq(+1)1209 1441 y Fx(Y)i
FA(\()p FG(X)g FA(\))27 b Fx(\000)d FG(x)p FA(\))38 b(and)f
FG(e)1902 1455 y Fv(N)7 b Fy(+1)2060 1441 y FA(.)63 b(But)37
b FG(\027)2385 1391 y Fk(e)2386 1408 y Fj(S)-12 b Fy(\()p
Ft(Y)6 b Fy(\()p Fv(X)f Fy(\))p Fv(;R)p Fy(\))2726 1441
y FA(\()p FG(X)i FA(\))39 b(is)d(also)i(in)e(the)456
1549 y(space)29 b(spanned)f(b)m(y)i(\()p FG(\031)1263
1563 y Fv(e)1296 1574 y Fs(N)5 b Fq(+1)1434 1549 y Fx(Y)i
FA(\()p FG(X)g FA(\))20 b Fx(\000)e FG(x)p FA(\))29 b(and)g
FG(e)2097 1563 y Fv(N)7 b Fy(+1)2254 1549 y FA(,)30 b(as)g(follo)m(ws)e
(b)m(y)h(the)h(radial)d(sym-)456 1687 y(metry)41 b(of)846
1664 y Fu(e)846 1687 y Fw(S)-7 b FA(\()p Fx(Y)7 b FA(\()p
FG(X)g FA(\))p FG(;)15 b(R)q FA(\))43 b(and)d(Lemma)i(B.10)g(again;)47
b(moreo)m(v)m(er,)f FG(\027)2849 1637 y Fk(e)2850 1654
y Fj(S)-13 b Fy(\()p Ft(Y)6 b Fy(\()p Fv(X)f Fy(\))p
Fv(;R)p Fy(\))3189 1687 y FA(\()p FG(X)i FA(\))42 b(is)456
1795 y(not)k(parallel)f(to)i FG(e)1142 1809 y Fv(N)7
b Fy(+1)1346 1795 y FA(\(b)s(ecause)47 b Fx(j)p FG(x)1811
1809 y Fv(N)7 b Fy(+1)1969 1795 y Fx(j)52 b(\024)g FA(1)p
FG(=)p FA(2)c(due)e(to)h(Lemma)g(3.1)g(and)f(so)456 1904
y Fx(r)p FG(g)575 1923 y Fj(S)-6 b Fy(\()p Ft(Y)5 b Fy(\()p
Fv(X)g Fy(\))p Fv(;R)p Fy(\))914 1904 y FA(\()p FG(x)p
FA(\))34 b Fx(6)p FA(=)e(0\).)55 b(Therefore,)36 b FG(\027)1827
1871 y Fy(\006)1878 1882 y Fs(X)1939 1904 y FA(\()p Fx(Y)7
b FA(\()p FG(X)g FA(\)\))37 b(b)s(elongs)d(to)h(the)g(space)h(spanned)
456 2046 y(b)m(y)30 b FG(\027)632 1996 y Fk(e)633 2013
y Fj(S)-13 b Fy(\()p Ft(Y)6 b Fy(\()p Fv(X)f Fy(\))p
Fv(;R)p Fy(\))972 2046 y FA(\()p FG(X)i FA(\))32 b(and)d
FG(e)1374 2060 y Fv(N)7 b Fy(+1)1532 2046 y FA(.)1817
b Fl(\003)456 2206 y FH(Lemma)32 b(3.11.)43 b Fo(Ther)-5
b(e)33 b(exists)g(a)g(p)-5 b(ositive)34 b(c)-5 b(onstant)34
b FG(C)39 b Fo(such)33 b(that)1313 2355 y Fx(j)p FG(\027)1389
2317 y Fy(\006)1440 2328 y Fs(X)1502 2355 y FA(\()p Fx(Y)7
b FA(\()p FG(X)g FA(\)\))23 b Fx(\001)d FG(\030)t Fx(j)26
b(\024)f FG(C)c Fx(j)p FG(\027)2213 2317 y Fv(u)2258
2355 y FA(\()p FG(X)7 b FA(\))21 b Fx(\001)g FG(\030)t
Fx(j)15 b FG(;)456 2509 y Fo(for)33 b(any)g FG(X)g Fx(2)980
2491 y Fu(e)969 2509 y Fi(B)p Fo(.)456 2672 y(Pr)-5 b(o)g(of.)43
b FA(F)-8 b(rom)30 b(Lemma)h(3.10,)h(for)e(an)m(y)h FG(X)i
Fx(2)2033 2653 y Fu(e)2021 2672 y Fi(B)q FA(,)d(w)m(e)h(ha)m(v)m(e)h
(that)456 2820 y(\(3.26\))717 b FG(\027)1454 2783 y Fy(\006)1505
2794 y Fs(X)1567 2820 y FA(\()p Fx(Y)7 b FA(\()p FG(X)g
FA(\)\))27 b(=)e FG(\013)5 b FA(^)-50 b FG(\027)27 b
FA(+)19 b FG(\014)5 b(e)2298 2834 y Fv(N)i Fy(+1)2472
2820 y FG(;)456 2968 y FA(for)30 b(some)h FG(\013)25
b FA(=)g FG(\013)p FA(\()p FG(X)7 b FA(\))32 b(and)e
FG(\014)g FA(=)25 b FG(\014)5 b FA(\()p FG(X)i FA(\))27
b Fx(2)e Fw(R)s FA(,)36 b(where)30 b(w)m(e)h(denoted)1095
3152 y(^)-50 b FG(\027)30 b FA(:=)1287 3051 y Fu(\020)1341
3152 y FG(\027)1391 3087 y Fk(e)1392 3104 y Fj(S)-12
b Fy(\()p Ft(Y)6 b Fy(\()p Fv(X)f Fy(\))p Fv(;R)p Fy(\))1386
3178 y(1)1731 3152 y FA(\()p FG(X)i FA(\))p FG(;)15 b(:)g(:)g(:)j(;)d
(\027)2136 3087 y Fk(e)2137 3104 y Fj(S)-12 b Fy(\()p
Ft(Y)6 b Fy(\()p Fv(X)f Fy(\))p Fv(;R)p Fy(\))2131 3180
y Fv(N)2477 3152 y FA(\()p FG(X)i FA(\))p FG(;)15 b FA(0)2714
3051 y Fu(\021)2785 3152 y FG(:)456 3343 y FA(Ob)m(viously)-8
b(,)29 b(since)g FG(X)k Fx(2)1328 3324 y Fu(e)1317 3343
y Fi(B)p FA(,)1379 3513 y(^)-51 b FG(\027)46 b FA(=)1576
3412 y Fu(\020)1630 3513 y FG(\027)1681 3475 y Fv(u)1675
3535 y Fy(1)1726 3513 y FA(\()p FG(X)7 b FA(\))p FG(;)15
b(:)g(:)g(:)i(;)e(\027)2131 3475 y Fv(u)2125 3535 y(N)2193
3513 y FA(\()p FG(X)7 b FA(\))p FG(;)15 b FA(0)2430 3412
y Fu(\021)2501 3513 y FG(:)456 3683 y FA(Thence,)30 b(b)m(y)g
(exploiting)f(Lemma)i(3.5,)g(w)m(e)g(see)g(that)1032
3862 y Fx(j)5 b FA(^)-50 b FG(\027)6 b Fx(j)1133 3824
y Fy(2)1198 3862 y FA(=)25 b(1)20 b Fx(\000)g(j)p FG(\027)1525
3797 y Fk(e)1526 3814 y Fj(S)-12 b Fy(\()p Ft(Y)6 b Fy(\()p
Fv(X)f Fy(\))p Fv(;R)p Fy(\))1520 3891 y Fv(N)i Fy(+1)1866
3862 y FA(\()p FG(X)g FA(\))p Fx(j)2043 3824 y Fy(2)2109
3862 y Fx(\025)25 b FA(1)20 b Fx(\000)g FA(\(1)h Fx(\000)f
FG(c)p FA(\))2627 3824 y Fy(2)2693 3862 y Fx(\025)25
b FG(c)15 b(:)456 4014 y FA(Th)m(us,)29 b(\(3.26\))k(implies)27
b(that)1135 4161 y FG(c)15 b Fx(j)p FG(\013)p Fx(j)85
b(\024)d(j)p FG(\013)5 b FA(^)-50 b FG(\027)27 b Fx(\001)f
FA(^)-51 b FG(\027)6 b Fx(j)26 b FA(=)1382 4324 y(=)1535
4219 y Fu(\014)1535 4274 y(\014)1535 4328 y(\014)1566
4223 y(\020)1620 4324 y FG(\027)1671 4286 y Fy(\006)1722
4297 y Fs(X)1784 4324 y FA(\()p Fx(Y)7 b FA(\()p FG(X)g
FA(\)\))22 b Fx(\000)e FG(\014)5 b(e)2285 4338 y Fv(N)i
Fy(+1)2443 4223 y Fu(\021)2517 4324 y Fx(\001)26 b FA(^)-50
b FG(\027)2613 4219 y Fu(\014)2613 4274 y(\014)2613 4328
y(\014)2669 4324 y FA(=)1382 4523 y(=)1535 4418 y Fu(\014)1535
4473 y(\014)1535 4527 y(\014)1566 4523 y FG(\027)1617
4485 y Fy(\006)1668 4496 y Fs(X)1729 4523 y FA(\()p Fx(Y)7
b FA(\()p FG(X)g FA(\)\))23 b Fx(\001)i FA(^)-50 b FG(\027)2138
4418 y Fu(\014)2138 4473 y(\014)2138 4527 y(\014)2193
4523 y Fx(\024)1382 4676 y(\024)82 b FA(1)15 b FG(;)456
4823 y FA(that)31 b(is)1696 4933 y Fx(j)p FG(\013)p Fx(j)41
b(\024)f FA(const)16 b FG(:)456 5062 y FA(Hence,)31 b(b)s(eing)e
FG(\030)1036 5076 y Fv(N)7 b Fy(+1)1218 5062 y FA(=)25
b(0,)31 b(\(3.26\))i(giv)m(es)e(that)834 5216 y Fx(j)p
FG(\027)910 5178 y Fy(\006)961 5189 y Fs(X)1023 5216
y FA(\()p Fx(Y)7 b FA(\()p FG(X)g FA(\)\))22 b Fx(\001)f
FG(\030)t Fx(j)k FA(=)g Fx(j)p FG(\013)5 b FA(^)-50 b
FG(\027)27 b Fx(\001)20 b FG(\030)t Fx(j)26 b(\024)40
b FA(const)16 b Fx(j)5 b FA(^)-50 b FG(\027)26 b Fx(\001)20
b FG(\030)t Fx(j)26 b FA(=)40 b(const)16 b Fx(j)p FG(\027)2846
5178 y Fv(u)2911 5216 y Fx(\001)k FG(\030)t Fx(j)15 b
FG(:)p eop
%%Page: 42 42
42 41 bop 456 251 a Fy(42)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)3374 450 y Fl(\003)456 614 y FH(Lemma)39 b(3.12.)46
b Fo(L)-5 b(et)39 b FG(X)j Fx(2)35 b Fi(G)p Fo(,)40 b(with)f
Fx(j)p FG(x)1882 628 y Fv(N)7 b Fy(+1)2040 614 y Fx(j)35
b(\024)g FA(1)p FG(=)p FA(2)p Fo(.)60 b(Assume)38 b(that)i
FG(g)3008 630 y Fk(e)3009 648 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p
Fy(\))3258 614 y Fx(\025)35 b FG(u)p Fo(.)456 741 y(Then,)d
FG(Y)53 b Fo(is)33 b(ab)-5 b(ove)33 b FA(\006)1236 755
y Fv(X)1335 741 y Fo(\(with)h(r)-5 b(esp)g(e)g(ct)34
b(to)f(the)g FG(e)2168 755 y Fv(N)7 b Fy(+1)2326 741
y Fo(-dir)-5 b(e)g(ction\).)456 904 y(Pr)g(o)g(of.)43
b FA(Assume)32 b(that)i(\()p FG(y)s(;)15 b(y)1447 871
y Ft(\003)1444 931 y Fv(N)7 b Fy(+1)1602 904 y FA(\))31
b Fx(2)f FA(\006)1825 918 y Fv(X)1892 904 y FA(.)50 b(W)-8
b(e)34 b(need)f(to)h(sho)m(w)g(that)g FG(y)2932 871 y
Ft(\003)2929 931 y Fv(N)7 b Fy(+1)3116 904 y Fx(\024)30
b FG(y)3262 918 y Fv(N)7 b Fy(+1)3419 904 y FA(.)456
1036 y(F)-8 b(or)40 b(this)e(purp)s(ose,)j(note)f(that,)j(b)m(y)c
(construction,)j FG(X)48 b Fx(2)2555 1012 y Fu(e)2555
1036 y Fw(S)-7 b FA(\(\()p FG(y)s(;)15 b(y)2812 1003
y Ft(\003)2809 1062 y Fv(N)7 b Fy(+1)2967 1036 y FA(\))p
FG(;)15 b(R)q FA(\),)43 b(whic)m(h)456 1143 y(sa)m(ys)31
b(that)456 1294 y(\(3.27\))372 b FG(g)1100 1310 y Fk(e)1101
1327 y Fj(S)-12 b Fy(\(\()p Fv(y)r(;y)1286 1305 y Fp(\003)1284
1351 y Fs(N)5 b Fq(+1)1419 1327 y Fy(\))p Fv(;R)p Fy(\))1552
1294 y FA(\()p FG(x)p FA(\))25 b(=)g FG(x)1847 1308 y
Fv(N)7 b Fy(+1)2030 1294 y FA(=)25 b FG(u)p FA(\()p FG(x)p
FA(\))h Fx(\024)f FG(g)2464 1310 y Fk(e)2465 1327 y Fj(S)-12
b Fy(\()p Fv(Y)t(;R)p Fy(\))2679 1294 y FA(\()p FG(x)p
FA(\))15 b FG(:)456 1476 y FA(Fix)29 b(no)m(w)i FG(R)q
FA(,)f FG(x)g FA(and)g FG(y)s FA(.)41 b(F)-8 b(or)31
b Fx(j)p FG(\020)7 b Fx(j)25 b(\024)g FA(1)p FG(=)p FA(2,)32
b(w)m(e)f(de\014ne)1530 1626 y FG(f)10 b FA(\()p FG(\020)d
FA(\))24 b(:=)i FG(g)1890 1643 y Fk(e)1891 1660 y Fj(S)-12
b Fy(\(\()p Fv(y)r(;\020)5 b Fy(\))p Fv(;R)p Fy(\))2207
1626 y FA(\()p FG(x)p FA(\))15 b FG(:)456 1795 y FA(Recalling)29
b(\(2.35\))j(and)e(\(2.31\),)j(w)m(e)e(ha)m(v)m(e)h(that)955
2010 y FG(H)1031 2024 y Fy(0)1070 2010 y FA(\()p FG(f)10
b FA(\()p FG(\020)d FA(\)\))20 b Fx(\000)p 1436 1876
72 4 v 1436 1949 a FG(C)1507 1963 y Fy(0)p 1433 1990
116 4 v 1433 2073 a FA(2)p FG(R)1559 2010 y FA(\()p FG(\020)27
b Fx(\000)19 b FG(f)10 b FA(\()p FG(\020)d FA(\)\))1958
1973 y Fy(2)2023 2010 y FA(=)25 b FG(H)2195 2024 y Fy(0)2234
2010 y FA(\()p FG(\020)7 b FA(\))20 b(+)g Fx(j)p FG(x)h
Fx(\000)f FG(y)s Fx(j)g(\000)g FG(R)c(:)456 2186 y FA(Di\013eren)m
(tiating)30 b(with)f(resp)s(ect)h(to)h FG(\020)7 b FA(,)30
b(w)m(e)h(get)h(that)1285 2372 y FG(H)1368 2335 y Ft(0)1361
2395 y Fy(0)1400 2372 y FA(\()p FG(f)10 b FA(\()p FG(\020)d
FA(\)\))15 b FG(f)1712 2335 y Ft(0)1735 2372 y FA(\()p
FG(\020)7 b FA(\))25 b Fx(\025)g FG(H)2056 2335 y Ft(0)2049
2395 y Fy(0)2088 2372 y FA(\()p FG(\020)7 b FA(\))21
b Fx(\000)2342 2311 y FA(const)p 2327 2351 238 4 v 2411
2434 a FG(R)2590 2372 y(:)456 2555 y FA(In)30 b(particular,)h(since)g
(b)m(y)g(de\014nition)f Fx(j)p FG(f)10 b FA(\()p FG(\020)d
FA(\))p Fx(j)27 b(\024)g FA(1)p FG(=)p FA(2,)33 b(w)m(e)f(get)h(that)f
FG(f)2855 2522 y Ft(0)2878 2555 y FA(\()p FG(\020)7 b
FA(\))27 b FG(>)g FA(0)32 b(if)f FG(R)h FA(is)456 2663
y(large)e(enough,)g(thence)h FG(f)40 b FA(is)29 b(increasing.)40
b(Since,)29 b(b)m(y)i(\(3.27\),)h(w)m(e)f(ha)m(v)m(e)h(that)1542
2813 y FG(f)10 b FA(\()p FG(y)1680 2775 y Ft(\003)1677
2835 y Fv(N)d Fy(+1)1834 2813 y FA(\))25 b Fx(\024)g
FG(f)10 b FA(\()p FG(y)2125 2827 y Fv(N)d Fy(+1)2282
2813 y FA(\))15 b FG(;)456 2963 y FA(w)m(e)30 b(deduce)g(that)h
FG(y)1137 2930 y Ft(\003)1134 2990 y Fv(N)7 b Fy(+1)1317
2963 y Fx(\024)25 b FG(y)1458 2977 y Fv(N)7 b Fy(+1)1615
2963 y FA(,)30 b(as)h(desired.)1287 b Fl(\003)456 3145
y FH(Lemma)32 b(3.13.)43 b Fo(L)-5 b(et)33 b FG(X)f Fx(2)1445
3127 y Fu(e)1433 3145 y Fi(B)q Fo(.)41 b(Then)1818 3127
y Fu(e)1813 3145 y Fi(A)32 b Fo(touches)i FA(\006)2297
3159 y Fv(X)2396 3145 y Fo(fr)-5 b(om)34 b(ab)-5 b(ove)33
b(at)h Fx(Y)7 b FA(\()p FG(X)g FA(\))p Fo(.)456 3308
y(Pr)-5 b(o)g(of.)43 b FA(First,)29 b(note)i(that)1567
3423 y Fx(Y)7 b FA(\()p FG(X)g FA(\))27 b Fx(2)d(Y)7
b FA(\()2013 3405 y Fu(e)2002 3423 y Fi(B)q FA(\))26
b Fx(\022)2238 3405 y Fu(e)2233 3423 y Fi(A)15 b FG(:)456
3554 y FA(On)38 b(the)i(other)f(hand,)i Fx(Y)7 b FA(\()p
FG(X)g FA(\))42 b Fx(2)d FA(\006)1720 3568 y Fv(X)1827
3554 y FA(b)m(y)g(Lemma)g(3.9.)69 b(Th)m(us,)41 b(to)f(end)f(the)g(pro)
s(of)456 3669 y(of)j(this)g(result)f(w)m(e)i(need)f(to)i(sho)m(w)e
(that)1971 3651 y Fu(e)1966 3669 y Fi(A)g FA(is)g(ab)s(o)m(v)m(e)h
(\006)2507 3683 y Fv(X)2617 3669 y FA(\(with)e(resp)s(ect)i(to)g(the)
456 3784 y FG(e)498 3798 y Fv(N)7 b Fy(+1)655 3784 y
FA(-direction\).)77 b(F)-8 b(or)43 b(this)e(purp)s(ose,)j(tak)m(e)g
FG(Y)65 b Fx(2)2356 3766 y Fu(e)2351 3784 y Fi(A)p FA(.)77
b(Then,)44 b(b)m(y)e(construction,)456 3892 y FG(g)498
3909 y Fk(e)499 3926 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p
Fy(\))738 3892 y Fx(\025)25 b FG(u)30 b FA(\(and)g(equalit)m(y)g(holds)
e(at)j(some)g(p)s(oin)m(t\).)40 b(Th)m(us,)29 b(b)m(y)h(Lemma)g(3.12,)i
FG(Y)50 b FA(is)456 4014 y(ab)s(o)m(v)m(e)31 b(\006)779
4028 y Fv(X)846 4014 y FA(.)2503 b Fl(\003)555 4177 y
FA(The)31 b(follo)m(wing)f(is)h(the)h(main)f(result)f(of)i(this)f
(section,)h(in)e(whic)m(h)h(a)h(measure)f(esti-)456 4285
y(mate)g(for)f(con)m(tact)j(p)s(oin)m(ts)c(is)g(giv)m(en:)456
4414 y FH(Prop)s(osition)34 b(3.14.)40 b Fo(L)-5 b(et)31
b FA(1)p FG(=c)c Fx(\024)e FG(l)i Fx(\024)e FG(cR)q Fo(,)31
b(for)g(a)h(suitably)f(smal)5 b(l)32 b(p)-5 b(ositive)31
b(c)-5 b(onstant)456 4522 y FG(c)p Fo(.)41 b(Assume)29
b(\(3.2\))h(and)g(\(3.3\).)42 b(Assume)29 b(also)h(that)h
Fi(A)e Fo(is)g(the)g(closur)-5 b(e)31 b(of)e(an)h(op)-5
b(en)30 b(set)456 4630 y(and)j(that)998 4759 y(for)g(any)g
FG(Y)46 b Fx(2)24 b Fi(A)33 b Fo(ther)-5 b(e)33 b(exist)g
FG(t)25 b Fx(2)g Fw(R)41 b Fo(and)33 b FG(x)26 b Fx(2)f
Fw(R)2673 4726 y Fv(N)2746 4759 y Fo(,)33 b(such)g(that)998
4899 y FG(g)1041 4918 y Fj(S)-6 b Fy(\()p Fv(Y)10 b Fy(+)p
Fv(t\030)s(;R)p Fy(\))1381 4899 y FA(\()p FG(x)p FA(\))26
b Fx(\024)f FG(u)p FA(\()p FG(x)p FA(\))p Fo(.)456 4831
y FA(\(3.28\))456 5064 y Fo(Then,)32 b(denoting)i(the)f
FG(N)10 b Fo(-dimensional)34 b(L)-5 b(eb)g(esgue)32 b(me)-5
b(asur)g(e)34 b(by)f Fi(L)2764 5031 y Fv(N)2831 5064
y Fo(,)f(we)h(have)g(that)1566 5216 y Fi(L)1621 5178
y Fv(N)1689 5216 y FA(\()p Fi(A)p FA(\))25 b Fx(\024)g
FG(C)d Fi(L)2082 5178 y Fv(N)2149 5216 y FA(\()p Fi(B)p
FA(\))15 b FG(;)p eop
%%Page: 43 43
43 42 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(43)456
450 y Fo(for)33 b(a)g(suitable)g(p)-5 b(ositive)33 b(universal)g(c)-5
b(onstant)35 b FG(C)7 b Fo(.)456 612 y(Pr)-5 b(o)g(of.)43
b FA(Note)25 b(that)f Fi(A)g FA(is)f(closed)h(b)m(y)g(h)m(yp)s(othesis)
e(and)i(so)g(is)f Fi(B)h FA(thanks)g(to)h(Lemma)f(3.2;)456
722 y(in)38 b(particular,)i Fi(A)f FA(and)g Fi(B)h FA(are)g(measurable)
f(sets.)68 b(Also,)42 b(b)m(y)e(\(3.28\),)k Fi(A)c FA(=)g
FG(\031)3251 737 y Fv(\030)3289 722 y FA(\()3329 704
y Fu(e)3324 722 y Fi(A)p FA(\).)456 830 y(What)21 b(is)e(more,)k(since)
c Fi(A)h FA(is)f(the)i(closure)e(of)i(an)f(op)s(en)g(set,)j(Lemma)d
(3.7)h(and)f(Lemma)h(3.8)456 940 y(sa)m(y)33 b(that)819
921 y Fu(e)815 940 y Fi(A)f FA(is)g(a)i(Lipsc)m(hitz)d(surface)i(whic)m
(h)f(is)g(also)h(a)g(con)m(tin)m(uous)2824 904 y Fy(10)2927
940 y FA(graph)g(o)m(v)m(er)h Fi(A)456 1058 y FA(in)e(the)h
FG(\030)t FA(-direction.)49 b(In)33 b(particular,)f(the)i(normal)e
FG(\027)2313 1014 y Fk(e)2309 1026 y Ff(A)2395 1058 y
FA(is)g(w)m(ell)h(de\014ned.)48 b(Thanks)33 b(to)456
1166 y(Lemma)d(3.13,)i(w)m(e)f(also)f(ha)m(v)m(e)i(that)1464
1330 y FG(\027)1515 1292 y Fy(\006)1566 1303 y Fs(X)1627
1330 y FA(\()p Fx(Y)7 b FA(\()p FG(X)g FA(\)\))28 b(=)d
FG(\027)2096 1281 y Fk(e)2092 1293 y Ff(A)2144 1330 y
FA(\()p Fx(Y)7 b FA(\()p FG(X)g FA(\)\))456 1486 y(for)30
b(an)m(y)h FG(X)i Fx(2)973 1468 y Fu(e)961 1486 y Fi(B)p
FA(,)e(pro)m(vided)e FG(\031)1516 1501 y Fv(\030)1554
1486 y FA(\()p Fx(Y)7 b FA(\()p FG(X)g FA(\)\))33 b(lies)c(in)g(the)i
(in)m(terior)e(of)i Fi(A)f FA(\(up)g(to)h(an)g(orien-)456
1594 y(tation)f(c)m(hoice\).)42 b(Therefore,)31 b(b)m(y)f(Lemma)g
(3.11,)456 1763 y(\(3.29\))582 b Fx(j)p FG(\027)1348
1714 y Fk(e)1344 1726 y Ff(A)1397 1763 y FA(\()p Fx(Y)7
b FA(\()p FG(X)g FA(\)\))22 b Fx(\001)e FG(\030)t Fx(j)26
b(\024)40 b FA(const)16 b Fx(j)p FG(\027)2259 1725 y
Fv(u)2304 1763 y FA(\()p FG(X)7 b FA(\))21 b Fx(\001)f
FG(\030)t Fx(j)15 b FG(;)456 1919 y FA(where)32 b FG(\027)772
1886 y Fv(u)817 1919 y FA(\()p FG(X)7 b FA(\))34 b(is)e(the)h(normal)g
(to)g Fi(G)h FA(at)f FG(X)7 b FA(,)35 b(for)e(an)m(y)g
FG(X)k Fx(2)2565 1900 y Fu(e)2554 1919 y Fi(B)p FA(,)d(pro)m(vided)e
FG(\031)3115 1934 y Fv(\030)3152 1919 y FA(\()p Fx(Y)7
b FA(\()p FG(X)g FA(\)\))456 2038 y(lies)29 b(in)g(the)h(in)m(terior)g
(of)g Fi(A)p FA(.)40 b(W)-8 b(e)32 b(denote)f(b)m(y)2012
2020 y Fu(e)2001 2038 y Fi(B)2074 2005 y Ft(0)2128 2038
y FA(this)e(set,)i(that)g(is)869 2192 y Fu(e)858 2210
y Fi(B)931 2172 y Ft(0)995 2210 y FA(:=)1131 2109 y Fu(n)1222
2210 y FG(X)i Fx(2)1427 2192 y Fu(e)1416 2210 y Fi(B)p
FA(,)e(with)e FG(\031)1804 2225 y Fv(\030)1841 2210 y
FA(\()p Fx(Y)7 b FA(\()p FG(X)g FA(\)\))33 b(in)c(the)h(in)m(terior)g
(of)g Fi(A)2941 2109 y Fu(o)3017 2210 y FG(:)555 2507
y FA(Also,)c(applying)d(the)i(div)m(ergence)g(theorem)h(to)f(\(the)h
(in)m(terior)e(of)7 b(\))2839 2489 y Fu(e)2834 2507 y
Fi(A)p FA(,)26 b(w)m(e)g(ha)m(v)m(e)g(that)456 2788 y(\(3.30\))866
b Fi(L)1607 2751 y Fv(N)1674 2788 y FA(\()p Fi(A)p FA(\))26
b Fx(\024)1926 2664 y Fu(Z)1981 2859 y Fk(e)1976 2871
y Ff(A)2044 2788 y Fx(j)p FG(\027)2124 2739 y Fk(e)2120
2751 y Ff(A)2193 2788 y Fx(\001)20 b FG(\030)t Fx(j)15
b FG(;)456 2983 y FA(where)29 b(the)i(ab)s(o)m(v)m(e)h(is)d(a)i
(surface)f(in)m(tegral.)555 3132 y(F)-8 b(or)31 b FG(")26
b(>)f FA(0,)31 b(let)f(us)g(no)m(w)g(de\014ne)1487 3276
y Fu(e)1475 3294 y Fi(B)1548 3308 y Fv(")1626 3294 y
FA(:=)1807 3208 y Fu([)1762 3419 y Fv(X)5 b Ft(2)1882
3407 y Fk(e)1872 3419 y Ff(B)1931 3400 y Fp(0)1969 3294
y FG(B)2038 3308 y Fv(")2075 3294 y FA(\()p FG(Y)20 b
FA(\))g Fx(\\)g Fi(G)c FG(;)456 3565 y FA(where)26 b
Fi(G)p FA(,)h(as)g(in)e(\(3.4\),)k(denotes)e(the)g(graph)f(of)g
FG(u)p FA(.)40 b(Then,)2500 3547 y Fu(e)2488 3565 y Fi(B)2561
3579 y Fv(")2625 3565 y FA(is)25 b(a)i(Lipsc)m(hitz)e(surface)456
3673 y(con)m(tained)30 b(in)f Fi(G)i FA(.)40 b(By)31
b(the)g(div)m(ergence)f(theorem,)1394 3863 y Fi(L)1449
3826 y Fv(N)1516 3763 y Fu(\020)1571 3863 y FG(\031)1623
3878 y Fv(\030)1660 3863 y FA(\()1706 3845 y Fu(e)1695
3863 y Fi(B)1768 3877 y Fv(")1806 3863 y FA(\))1841 3763
y Fu(\021)1936 3863 y FA(=)2047 3740 y Fu(Z)2107 3934
y Fk(e)2097 3946 y Ff(B)2156 3954 y Fs(")2209 3863 y
FG(\027)2270 3814 y Fk(e)2260 3826 y Ff(B)2319 3834 y
Fs(")2376 3863 y Fx(\001)21 b FG(\030)e(;)456 4096 y
FA(where)35 b FG(\027)785 4051 y Fk(e)775 4063 y Ff(B)834
4071 y Fs(")907 4096 y FA(is)g(the)h(external)g(normal)f(of)h(the)g
(surface)2432 4078 y Fu(e)2420 4096 y Fi(B)2493 4110
y Fv(")2530 4096 y FA(.)58 b(Ob)m(viously)-8 b(,)36 b(up)e(to)j(the)456
4214 y(sign,)29 b FG(\027)729 4170 y Fk(e)719 4182 y
Ff(B)778 4190 y Fs(")846 4214 y FA(agrees)i(with)e FG(\027)1377
4181 y Fv(u)1422 4214 y FA(.)41 b(Sending)28 b FG(")j
FA(to)g(zero,)g(w)m(e)g(th)m(us)f(get)456 4405 y(\(3.31\))248
b Fi(L)989 4368 y Fv(N)1057 4405 y FA(\()p Fi(B)p FA(\))26
b(=)f Fi(L)1377 4368 y Fv(N)1445 4304 y Fu(\020)1499
4405 y FG(\031)1551 4420 y Fv(\030)1589 4405 y FA(\()1635
4387 y Fu(e)1624 4405 y Fi(B)p FA(\))1732 4304 y Fu(\021)1812
4405 y Fx(\025)g Fi(L)1963 4368 y Fv(N)2031 4304 y Fu(\020)2085
4405 y FG(\031)2137 4420 y Fv(\030)2175 4405 y FA(\()2221
4387 y Fu(e)2210 4405 y Fi(B)2283 4368 y Ft(0)2307 4405
y FA(\))2342 4304 y Fu(\021)2421 4405 y FA(=)2517 4281
y Fu(Z)2578 4476 y Fk(e)2568 4488 y Ff(B)2627 4469 y
Fp(0)2669 4405 y FG(\027)2729 4356 y Fk(e)2720 4368 y
Ff(B)2779 4376 y Fs(")2836 4405 y Fx(\001)20 b FG(\030)f(:)456
4629 y FA(Also,)32 b(b)m(y)g(Lemma)g(3.8,)h(the)g(exterior)f(normal)e
FG(\027)2182 4585 y Fk(e)2172 4597 y Ff(B)2231 4605 y
Fs(")2301 4629 y FA(in)2420 4611 y Fu(e)2408 4629 y Fi(B)i
FA(has)g(the)g(signed)f(assigned)456 4748 y(b)m(y)f(the)g(prop)s(ert)m
(y)g(that)h FG(\027)1368 4703 y Fk(e)1358 4715 y Ff(B)1417
4723 y Fs(")1474 4748 y Fx(\001)21 b FG(\030)29 b Fx(\025)c
FA(0.)41 b(Therefore,)1419 4912 y Fx(j)p FG(\027)1495
4874 y Fv(u)1560 4912 y Fx(\001)20 b FG(\030)t Fx(j)26
b FA(=)f Fx(j)p FG(\027)1882 4863 y Fk(e)1872 4875 y
Ff(B)1931 4883 y Fs(")1989 4912 y Fx(\001)20 b FG(\030)t
Fx(j)25 b FA(=)g FG(\027)2285 4863 y Fk(e)2275 4875 y
Ff(B)2334 4883 y Fs(")2392 4912 y Fx(\001)20 b FG(\030)p
456 4977 499 4 v 555 5062 a Fy(10)657 5091 y FF(The)31
b(con)n(tin)n(uit)n(y)e(of)j(\()p FD(\031)1354 5100 y
Fs(\030)1388 5005 y Fg(\014)1388 5050 y(\014)1388 5094
y(\014)1422 5125 y Fe(e)1413 5143 y Ff(A)1466 5091 y
FF(\))1496 5059 y Fp(\000)p Fq(1)1609 5091 y FF(follo)n(ws)i(from)c
(the)h(follo)n(wing)i(elemen)n(tary)d(prop)r(ert)n(y:)44
b(if)32 b FD(f)38 b FF(:)456 5216 y FD(K)26 b Fr(\000)-13
b(!)21 b FD(f)8 b FF(\()p FD(K)d FF(\))26 b(is)h(con)n(tin)n(uous)e
(and)g(injectiv)n(e)h(and)g FD(K)31 b FF(is)26 b(compact,)f(then)g
FD(f)2680 5184 y Fp(\000)p Fq(1)2789 5216 y FF(is)h(con)n(tin)n(uous.)p
eop
%%Page: 44 44
44 43 bop 456 251 a Fy(44)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 451 y FA(in)573 433 y Fu(e)562 451 y Fi(B)p
FA(.)41 b(Th)m(us,)29 b(b)m(y)i(\(3.31\),)h(w)m(e)f(get)h(that)456
644 y(\(3.32\))846 b Fi(L)1587 606 y Fv(N)1655 644 y
FA(\()p Fi(B)p FA(\))26 b Fx(\025)1920 520 y Fu(Z)1981
714 y Fk(e)1971 726 y Ff(B)2030 707 y Fp(0)2071 644 y
Fx(j)p FG(\027)2147 606 y Fv(u)2213 644 y Fx(\001)20
b FG(\030)t Fx(j)15 b FG(:)456 856 y FA(Also,)40 b(b)m(y)f
(construction,)h Fx(Y)46 b FA(sends)1757 838 y Fu(e)1745
856 y Fi(B)39 b FA(in)m(to)2054 838 y Fu(e)2050 856 y
Fi(A)o FA(;)k(hence,)e(b)m(y)e(\(3.30\),)k(the)38 b(c)m(hange)i(of)456
964 y(v)-5 b(ariables)29 b(form)m(ula)g(\(see,)j(e.g.,)g(page)f(99)g
(in)e([18)q(]\))i(and)f(Lemma)g(3.6,)i(w)m(e)f(get)h(that)1029
1156 y Fi(L)1084 1119 y Fv(N)1151 1156 y FA(\()p Fi(A)p
FA(\))26 b Fx(\024)1403 1033 y Fu(Z)1458 1227 y Fk(e)1453
1239 y Ff(A)1521 1156 y Fx(j)p FG(\027)1601 1107 y Fk(e)1597
1119 y Ff(A)1649 1156 y FA(\()p FG(Y)21 b FA(\))f Fx(\001)h
FG(\030)t Fx(j)15 b FG(d)5 b(Y)46 b Fx(\024)1307 1398
y(\024)1403 1274 y Fu(Z)1463 1469 y Fk(e)1453 1481 y
Ff(B)1512 1461 y Fp(0)1554 1398 y Fx(j)p FG(\027)1634
1349 y Fk(e)1630 1361 y Ff(A)1682 1398 y FA(\()p Fx(Y)7
b FA(\()p FG(X)g FA(\)\))23 b Fx(\001)d FG(\030)t Fx(j)15
b(j)g FA(det)h FG(D)s Fx(Y)7 b FA(\()p FG(X)g FA(\))p
Fx(j)15 b FG(dX)35 b Fx(\024)1307 1639 y(\024)40 b FA(const)1656
1516 y Fu(Z)1716 1710 y Fk(e)1706 1722 y Ff(B)1765 1703
y Fp(0)1807 1639 y Fx(j)p FG(\027)1887 1590 y Fk(e)1883
1602 y Ff(A)1935 1639 y FA(\()p Fx(Y)7 b FA(\()p FG(X)g
FA(\)\))23 b Fx(\001)d FG(\030)t Fx(j)15 b FG(dX)23 b(:)456
1401 y FA(\(3.33\))456 1836 y(Thence,)30 b(from)g(\(3.32\),)j(\(3.29\))
f(and)e(\(3.33\),)j(w)m(e)e(ha)m(v)m(e)h(that)1153 2029
y Fi(L)1208 1991 y Fv(N)1275 2029 y FA(\()p Fi(A)p FA(\))84
b Fx(\024)98 b FA(const)1895 1905 y Fu(Z)1956 2099 y
Fk(e)1946 2111 y Ff(B)2005 2092 y Fp(0)2047 2029 y Fx(j)p
FG(\027)2127 1979 y Fk(e)2123 1991 y Ff(A)2175 2029 y
FA(\()p Fx(Y)7 b FA(\()p FG(X)g FA(\)\))23 b Fx(\001)d
FG(\030)t Fx(j)15 b FG(dX)1489 2270 y Fx(\024)98 b FA(const)1895
2146 y Fu(Z)1956 2341 y Fk(e)1946 2353 y Ff(B)2005 2334
y Fp(0)2047 2270 y Fx(j)p FG(\027)2123 2233 y Fv(u)2168
2270 y FA(\()p FG(X)7 b FA(\))21 b Fx(\001)f FG(\030)t
Fx(j)15 b FG(dX)1489 2471 y Fx(\024)82 b Fi(L)1697 2433
y Fv(N)1765 2471 y FA(\()p Fi(B)p FA(\))15 b FG(:)3374
2616 y Fl(\003)1259 2800 y FA(4.)47 b FB(Measure)32 b(theoretic)h
(resul)-6 b(ts)555 2962 y FA(This)26 b(section)h(collects)h(some)g
(measure)f(theory)g(lemmata,)i(whic)m(h)d(are)i(extensions)456
3070 y(of)36 b(analogous)g(results)e(in)h([31)q(].)57
b(These)36 b(lemmata)g(will)d(b)s(e)i(used)g(in)f(the)i(sequel)g(for)
456 3178 y(estimating)21 b(the)h(measure)g(of)g(the)g(pro)5
b(jection)22 b(of)g(the)g(set)g(in)f(whic)m(h)f(a)j(suitable)d(barrier)
456 3286 y(touc)m(hes)26 b(a)g(minimal)d(solution)h(of)i(\(1.5\).)41
b(F)-8 b(or)26 b(the)g(reader's)g(con)m(v)m(enience,)h(the)f(pro)s(ofs)
456 3394 y(of)k(the)h(lemmata)f(of)h(this)e(section)i(are)g(deferred)e
(to)i(the)g(App)s(endix.)555 3502 y(Giv)m(en)j(t)m(w)m(o)h(v)m(ectors)g
FG(v)i FA(and)c FG(w)r FA(,)i(w)m(e)f(de\014ne)f Fl(\\)p
FA(\()p FG(v)s(;)15 b(w)r FA(\))36 b(to)e(b)s(e)f(the)h(\(p)s(ositiv)m
(e\))f(angle)456 3609 y(b)s(et)m(w)m(een)e(these)f(v)m(ectors,)j(i.e.,)
1411 3805 y Fl(\\)p FA(\()p FG(v)s(;)15 b(w)r FA(\))27
b(:=)1849 3673 y Fu(\014)1849 3728 y(\014)1849 3782 y(\014)1849
3837 y(\014)1894 3805 y FA(arccos)2188 3744 y FG(v)c
Fx(\001)e FG(w)p 2162 3784 232 4 v 2162 3867 a Fx(j)p
FG(v)s Fx(j)15 b(j)p FG(w)r Fx(j)2403 3673 y Fu(\014)2403
3728 y(\014)2403 3782 y(\014)2403 3837 y(\014)2464 3805
y FG(:)456 4006 y FA(By)30 b(elemen)m(tary)h(geometric)h
(consideration,)d(if)g Fx(j)p FG(v)s Fx(j)e FA(=)e Fx(j)p
FG(w)r Fx(j)h FA(=)f(1,)456 4151 y(\(4.1\))944 b Fx(j)p
FG(v)24 b Fx(\000)c FG(w)r Fx(j)26 b(\024)f Fl(\\)p FA(\()p
FG(v)s(;)15 b(w)r FA(\))g FG(:)456 4297 y FA(W)-8 b(e)31
b(also)f(de\014ne,)g(for)h FG(l)c(>)e FA(0,)836 4465
y FG(L)83 b FA(:=)1160 4364 y Fu(n)1221 4465 y FA(\()p
FG(x)1308 4428 y Ft(0)1332 4465 y FG(;)15 b FA(0)p FG(;)g(x)1509
4479 y Fv(N)7 b Fy(+1)1667 4465 y FA(\))26 b Fx(2)f Fw(R)1873
4428 y Fv(N)7 b Ft(\000)p Fy(1)2057 4465 y Fx(\002)20
b Fw(R)29 b Fx(\002)20 b Fw(R)33 b Fx(j)26 b(j)p FG(x)2543
4479 y Fv(N)7 b Fy(+1)2701 4465 y Fx(j)25 b(\024)g FA(1)p
FG(=)p FA(2)2982 4364 y Fu(o)3059 4465 y FG(;)800 4664
y(Q)872 4679 y Fv(l)981 4664 y FA(:=)1160 4563 y Fu(n)1221
4664 y FA(\()p FG(x)1308 4627 y Ft(0)1332 4664 y FG(;)15
b FA(0)p FG(;)g(x)1509 4678 y Fv(N)7 b Fy(+1)1667 4664
y FA(\))26 b Fx(2)f FG(L)g Fx(j)g(j)p FG(x)2028 4627
y Ft(0)2052 4664 y Fx(j)g(\024)g FG(l)2227 4563 y Fu(o)2303
4664 y FG(:)456 4833 y FA(Of)k(course,)1512 4942 y Fi(L)1567
4904 y Fv(N)1634 4942 y FA(\()p FG(Q)1741 4957 y Fv(l)1767
4942 y FA(\))d(=)40 b(const)16 b FG(l)2191 4904 y Fv(N)7
b Ft(\000)p Fy(1)2363 4942 y FG(:)456 5070 y FA(Also,)30
b(for)g FG(X)j FA(=)25 b(\()p FG(x)1116 5084 y Fy(1)1155
5070 y FG(;)15 b(:)g(:)g(:)i(;)e(x)1409 5084 y Fv(N)7
b Fy(+1)1567 5070 y FA(\))25 b Fx(2)g Fw(R)1773 5037
y Fv(N)7 b Fy(+1)1936 5070 y FA(,)31 b(and)f(1)25 b Fx(\024)g
FG(i)h Fx(\024)f FG(N)30 b FA(+)20 b(1,)31 b(w)m(e)g(de\014ne)1235
5216 y FG(\031)1287 5230 y Fv(i)1315 5216 y FG(X)i FA(:=)25
b(\()p FG(x)1631 5230 y Fy(1)1671 5216 y FG(;)15 b(:)g(:)g(:)h(;)f(x)
1924 5230 y Fv(i)p Ft(\000)p Fy(1)2043 5216 y FG(;)g
FA(0)p FG(;)g(x)2220 5230 y Fv(i)p Fy(+1)2340 5216 y
FG(;)g(x)2432 5230 y Fv(N)7 b Fy(+1)2589 5216 y FA(\))15
b FG(:)p eop
%%Page: 45 45
45 44 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(45)456
450 y FA(Also,)37 b(giv)m(en)e FG(X)42 b Fx(2)34 b Fw(R)1206
417 y Fv(N)7 b Fy(+1)1370 450 y FA(,)37 b(w)m(e)f(will)d(often)j(write)
f FG(X)42 b FA(=)34 b(\()p FG(x)2530 417 y Ft(0)2553
450 y FG(;)15 b(x)2645 464 y Fv(N)2713 450 y FG(;)g(x)2805
464 y Fv(N)7 b Fy(+1)2963 450 y FA(\))34 b Fx(2)g Fw(R)3187
417 y Fv(N)7 b Fy(+1)3374 450 y Fx(\002)456 558 y Fw(R)28
b Fx(\002)20 b Fw(R)s FA(,)37 b(i.e.,)30 b(the)h(notation)g
FG(x)1498 525 y Ft(0)1551 558 y FA(will)d(often)j(denote)g(the)f
(\014rst)g(\()p FG(N)g Fx(\000)20 b FA(1\))31 b(en)m(tries)f(of)h
FG(X)7 b FA(.)555 666 y(Then,)30 b(with)f(the)h(ab)s(o)m(v)m(e)i
(notation,)f(w)m(e)g(ha)m(v)m(e)g(the)g(follo)m(wing)e(results:)456
805 y FH(Lemma)g(4.1.)40 b Fo(L)-5 b(et)31 b FG(u)f Fo(b)-5
b(e)30 b(a)g(solution)h(of)g(\(1.5\).)42 b(Supp)-5 b(ose)32
b(that)f Fw(S)n FA(\()p FG(Y)5 b(;)15 b(R)q FA(\))25
b Fo(touches)31 b(the)456 913 y(gr)-5 b(aph)34 b(of)f
FG(u)f Fo(by)h(ab)-5 b(ove)33 b(at)g FG(X)1443 927 y
Fy(0)1508 913 y FA(=)25 b(\()p FG(x)1691 927 y Fy(0)1731
913 y FG(;)31 b(u)p FA(\()p FG(x)1926 927 y Fy(0)1965
913 y FA(\)\))26 b(=)f(\()p FG(x)2244 927 y Fy(0)2284
913 y FG(;)31 b(g)2383 931 y Fj(S)-6 b Fy(\()p Fv(Y)t(;R)p
Fy(\))2597 913 y FA(\()p FG(x)2684 927 y Fy(0)2724 913
y FA(\)\))p Fo(.)42 b(Assume)32 b(that)456 1150 y FA(\(4.2\))821
b Fl(\\)1543 1022 y Fu(\022)1645 1088 y Fx(r)p FG(u)p
FA(\()p FG(x)1860 1102 y Fy(0)1899 1088 y FA(\))p 1619
1129 341 4 v 1619 1212 a Fx(jr)p FG(u)p FA(\()p FG(x)1859
1226 y Fy(0)1899 1212 y FA(\))p Fx(j)1970 1150 y FG(;)30
b(e)2067 1164 y Fv(N)2135 1022 y Fu(\023)2227 1150 y
Fx(\024)2333 1088 y FG(\031)p 2333 1129 56 4 v 2338 1212
a FA(8)2413 1150 y FG(:)456 1369 y Fo(Then,)40 b(ther)-5
b(e)40 b(exists)f(a)g(universal)g FG(a)1744 1383 y Fy(0)1820
1369 y FG(>)d FA(1)j Fo(so)g(that,)i(for)f(any)f FG(a)d
Fx(\025)g FG(a)2925 1383 y Fy(0)2965 1369 y Fo(,)k(ther)-5
b(e)39 b(exist)456 1477 y(a)45 b(universal)g FG(\024)j(>)f
FA(1)e Fo(and)h(a)f(suitable)g FG(C)54 b(>)47 b FA(1)p
Fo(,)h(which)e(dep)-5 b(ends)47 b(only)e(on)h FG(a)e
Fo(and)456 1585 y(on)c(universal)h(c)-5 b(onstants,)44
b(such)d(that)g(the)g(fol)5 b(lowing)41 b(holds.)66 b(F)-7
b(or)42 b(any)f(p)-5 b(oint)41 b FG(Z)46 b Fx(2)456 1693
y FG(L)20 b Fx(\\)f FG(B)687 1707 y Fv(a)729 1693 y FA(\()p
FG(\031)816 1707 y Fv(N)883 1693 y FG(X)958 1707 y Fy(0)998
1693 y FA(\))33 b Fo(ther)-5 b(e)34 b(exists)f FG(x)f
Fo(satisfying)h(the)g(fol)5 b(lowing)34 b(pr)-5 b(op)g(erties:)671
1832 y Fx(\017)42 b(j)p FG(x)21 b Fx(\000)f FG(x)999
1846 y Fy(0)1038 1832 y Fx(j)26 b(\024)f FG(\024a)p Fo(,)671
1940 y Fx(\017)42 b FG(Z)32 b FA(=)25 b FG(\031)1000
1954 y Fv(N)1067 1940 y FA(\()p FG(x;)31 b(u)p FA(\()p
FG(x)p FA(\)\))p Fo(,)671 2101 y Fx(\017)42 b FA(\()p
FG(x)21 b Fx(\000)f FG(x)1009 2115 y Fy(0)1048 2101 y
FA(\))h Fx(\001)1184 2039 y(r)p FG(u)p FA(\()p FG(x)1399
2053 y Fy(0)1439 2039 y FA(\))p 1159 2080 341 4 v 1159
2163 a Fx(jr)p FG(u)p FA(\()p FG(x)1399 2177 y Fy(0)1439
2163 y FA(\))p Fx(j)1550 2101 y(\024)40 b FG(H)1737 2115
y Fy(0)1776 2101 y FA(\()p FG(u)p FA(\()p FG(x)p FA(\)\))22
b Fx(\000)e FG(H)2209 2115 y Fy(0)2248 2101 y FA(\()p
FG(u)p FA(\()p FG(x)2422 2115 y Fy(0)2462 2101 y FA(\)\))h(+)2653
2039 y FG(C)p 2653 2080 72 4 v 2654 2163 a(R)2735 2101
y Fo(,)456 2293 y(pr)-5 b(ovide)g(d)35 b FG(R)e Fo(is)f(lar)-5
b(ge)34 b(enough)f(\(p)-5 b(ossibly)34 b(in)f(dep)-5
b(endenc)g(e)34 b(of)f FG(a)p Fo(\).)456 2432 y FH(Lemma)g(4.2.)43
b Fo(L)-5 b(et)33 b FG(u)h Fo(b)-5 b(e)33 b(a)g FG(C)1538
2399 y Fy(1)1577 2432 y Fo(-subsolution)h(of)g FA(\(1)p
FG(:)p FA(5\))h Fo(in)e Fx(fj)p FG(x)2644 2399 y Ft(0)2668
2432 y Fx(j)27 b FG(<)f(l)r Fx(g)21 b(\002)f(fj)p FG(x)3125
2446 y Fv(N)3193 2432 y Fx(j)27 b FG(<)f(l)r Fx(g)p Fo(.)456
2540 y(Assume)33 b(that)i Fw(S)n FA(\()p FG(Y)1127 2554
y Fy(0)1161 2540 y FG(;)15 b(R)q FA(\))34 b Fo(is)f(ab)-5
b(ove)34 b(the)g(gr)-5 b(aph)35 b(of)f FG(u)g Fo(and)g(that)h
Fw(S)n FA(\()p FG(Y)2787 2554 y Fy(0)2821 2540 y FG(;)15
b(R)q FA(\))34 b Fo(touches)g(the)456 2648 y(gr)-5 b(aph)34
b(of)f FG(u)f Fo(at)i(the)f(p)-5 b(oint)34 b FA(\()p
FG(x)1472 2662 y Fy(0)1511 2648 y FG(;)d(u)p FA(\()p
FG(x)1706 2662 y Fy(0)1746 2648 y FA(\)\))p Fo(.)42 b(Supp)-5
b(ose)34 b(that)671 2787 y Fx(\017)42 b(j)p FG(u)p FA(\()p
FG(x)922 2801 y Fy(0)962 2787 y FA(\))p Fx(j)26 b FG(<)f
FA(1)p FG(=)p FA(2)p Fo(,)34 b Fx(j)p FG(x)1418 2801
y Fy(0)p Fv(N)1521 2787 y Fx(j)25 b FG(<)g(l)r(=)p FA(4)p
Fo(,)33 b FG(q)c FA(:=)c Fx(j)p FG(x)2115 2754 y Ft(0)2115
2811 y Fy(0)2154 2787 y Fx(j)h FG(<)f(l)r(=)p FA(4)p
Fo(.)671 2951 y Fx(\017)42 b Fl(\\)839 2823 y Fu(\022)941
2890 y Fx(r)p FG(u)p FA(\()p FG(x)1156 2904 y Fy(0)1196
2890 y FA(\))p 916 2930 341 4 v 916 3014 a Fx(jr)p FG(u)p
FA(\()p FG(x)1156 3028 y Fy(0)1196 3014 y FA(\))p Fx(j)1282
2951 y FG(;)15 b(e)1364 2965 y Fv(N)1432 2823 y Fu(\023)1539
2951 y Fx(\024)1660 2890 y FG(\031)p 1660 2930 56 4 v
1665 3014 a FA(8)1725 2951 y Fo(.)456 3140 y(Then,)32
b(ther)-5 b(e)34 b(exist)f(universal)g(c)-5 b(onstants)34
b FG(C)2012 3154 y Fy(1)2052 3140 y FG(;)c(C)2172 3154
y Fy(2)2237 3140 y FG(>)25 b FA(1)h FG(>)f(c)g(>)g FA(0)33
b Fo(such)g(that,)h(if)1249 3317 y FG(q)28 b Fx(\025)d
FG(C)1479 3331 y Fy(1)1704 3317 y Fo(and)187 b FA(4)2105
3267 y Fq(3)2084 3235 y Fx(p)p 2161 3235 70 4 v 2161
3317 a FG(R)26 b Fx(\024)e FG(l)k Fx(\024)d FG(cR)16
b(;)456 3486 y Fo(the)33 b(fol)5 b(lowing)35 b(holds.)46
b(L)-5 b(et)34 b FA(\004)f Fo(b)-5 b(e)33 b(the)h(set)g(of)g(p)-5
b(oints)35 b FA(\()p Fi(x)p FG(;)c(u)p FA(\()p Fi(x)p
FA(\)\))d Fx(2)e Fw(R)2788 3453 y Fv(N)2882 3486 y Fx(\002)21
b Fw(R)42 b Fo(satisfying)456 3594 y(the)33 b(fol)5 b(lowing)33
b(pr)-5 b(op)g(erties:)671 3733 y Fx(\017)42 b(j)p Fi(x)815
3700 y Ft(0)839 3733 y Fx(j)26 b FG(<)f(q)s(=)p FA(15)p
Fo(,)33 b Fx(j)p FG(u)p FA(\()p Fi(x)p FA(\))p Fx(j)27
b FG(<)e FA(1)p FG(=)p FA(2)p Fo(,)34 b Fx(j)p Fi(x)20
b Fx(\000)g FG(x)1970 3747 y Fy(0)2010 3733 y Fx(j)25
b FG(<)g FA(2)p FG(q)s Fo(;)671 3843 y Fx(\017)42 b Fo(ther)-5
b(e)37 b(exists)f FG(Y)51 b Fx(2)30 b Fw(R)1495 3810
y Fv(N)1591 3843 y Fx(\002)22 b FA([)p Fx(\000)p FA(1)p
FG(=)p FA(4)p FG(;)32 b FA(1)p FG(=)p FA(4])37 b Fo(such)f(that)h
Fw(S)n FA(\()p FG(Y)5 b(;)15 b(R)q(=C)2937 3857 y Fy(2)2971
3843 y FA(\))36 b Fo(is)g(ab)-5 b(ove)36 b FG(u)758 3950
y Fo(and)e(it)f(touches)g FG(u)f Fo(at)i FA(\()p Fi(x)p
FG(;)d(u)p FA(\()p Fi(x)p FA(\)\))p Fo(;)671 4113 y Fx(\017)42
b Fl(\\)839 3985 y Fu(\022)941 4052 y Fx(r)p FG(u)p FA(\()p
Fi(x)p FA(\))p 916 4092 282 4 v 916 4176 a Fx(jr)p FG(u)p
FA(\()p Fi(x)p FA(\))p Fx(j)1222 4113 y FG(;)1298 4052
y Fx(r)p FG(u)p FA(\()p FG(x)1513 4066 y Fy(0)1553 4052
y FA(\))p 1273 4092 341 4 v 1273 4176 a Fx(jr)p FG(u)p
FA(\()p FG(x)1513 4190 y Fy(0)1553 4176 y FA(\))p Fx(j)1623
3985 y Fu(\023)1715 4113 y Fx(\024)1821 4052 y FG(C)1886
4066 y Fy(1)1941 4052 y FG(q)p 1821 4092 164 4 v 1868
4176 a(R)1995 4113 y Fo(;)671 4345 y Fx(\017)g FA(\()p
Fi(x)21 b Fx(\000)f FG(x)989 4359 y Fy(0)1028 4345 y
FA(\))h Fx(\001)1165 4284 y(r)p FG(u)p FA(\()p FG(x)1380
4298 y Fy(0)1419 4284 y FA(\))p 1139 4324 341 4 v 1139
4408 a Fx(jr)p FG(u)p FA(\()p FG(x)1379 4422 y Fy(0)1419
4408 y FA(\))p Fx(j)1530 4345 y(\024)1651 4284 y FG(C)1716
4298 y Fy(1)1771 4284 y FG(q)1815 4251 y Fy(2)p 1651
4324 203 4 v 1718 4408 a FG(R)1884 4345 y FA(+)f FG(H)2051
4359 y Fy(0)2090 4345 y FA(\()p FG(u)p FA(\()p Fi(x)p
FA(\)\))i Fx(\000)e FG(H)2503 4359 y Fy(0)2542 4345 y
FA(\()p FG(u)p FA(\()p FG(x)2716 4359 y Fy(0)2756 4345
y FA(\)\))p Fo(.)456 4533 y(Then,)1493 4687 y Fi(L)1548
4650 y Fv(N)1616 4586 y Fu(\020)1670 4687 y FG(\031)1722
4701 y Fv(N)1789 4687 y FA(\(\004\))1920 4586 y Fu(\021)2015
4687 y Fx(\025)40 b FG(cq)2209 4650 y Fv(N)7 b Ft(\000)p
Fy(1)2382 4687 y FG(:)456 4861 y FH(Lemma)40 b(4.3.)46
b Fo(L)-5 b(et)39 b FG(C)k Fx(\025)36 b FA(2)p Fo(,)41
b FG(c)c(>)f FA(0)p Fo(.)61 b(L)-5 b(et)39 b(us)g(c)-5
b(onsider,)41 b(for)f FG(k)f Fx(2)d Fw(N)7 b Fo(,)46
b(a)39 b(family)h(of)456 4969 y(sets)33 b FG(D)710 4984
y Fv(k)780 4969 y Fx(\022)27 b FG(L)p Fo(,)33 b(so)h(that)h
FG(D)1380 4984 y Fv(k)1450 4969 y Fx(\022)26 b FG(D)1622
4984 y Fv(k)r Fy(+1)1789 4969 y Fo(for)34 b(any)g FG(k)c
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FA(0)p Fo(:)539 5216 y FA(\(P1\))42 b FG(D)833 5230 y
Fy(0)893 5216 y Fx(\\)20 b FG(Q)1046 5231 y Fv(l)1097
5216 y Fx(6)p FA(=)25 b Fx(;)p Fo(;)p eop
%%Page: 46 46
46 45 bop 456 251 a Fy(46)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)539 450 y FA(\(P2\))42 b Fo(for)31 b(any)g FG(Z)1135
464 y Fy(0)1199 450 y Fx(2)25 b FG(D)1360 465 y Fv(k)1418
450 y Fx(\\)14 b FG(Q)1565 465 y Fy(2)p Fv(l)1656 450
y Fo(and)31 b(any)f FG(Z)2063 464 y Fy(1)2128 450 y Fx(2)25
b FG(L)p Fo(,)30 b(with)h FG(a)25 b Fx(\024)g(j)p FG(Z)2785
464 y Fy(1)2839 450 y Fx(\000)14 b FG(Z)2986 464 y Fy(0)3026
450 y Fx(j)25 b(\024)g FA(2)p FG(l)r Fo(,)31 b(one)758
558 y(has)j(that)803 716 y Fi(L)858 678 y Fv(N)925 615
y Fu(\020)979 716 y FG(D)1054 731 y Fv(k)r Fy(+1)1208
716 y Fx(\\)19 b FG(B)1357 734 y Ft(j)p Fv(Z)1425 743
y Fq(1)1459 734 y Ft(\000)p Fv(Z)1562 743 y Fq(0)1597
734 y Ft(j)p Fv(=)p Fy(10)1726 716 y FA(\()p FG(Z)1823
730 y Fy(1)1863 716 y FA(\))1898 615 y Fu(\021)1993 716
y Fx(\025)40 b FG(c)15 b Fi(L)2213 678 y Fv(N)2281 615
y Fu(\020)2336 716 y FG(L)20 b Fx(\\)g FG(B)2568 734
y Ft(j)p Fv(Z)2636 743 y Fq(1)2670 734 y Ft(\000)p Fv(Z)2773
743 y Fq(0)2807 734 y Ft(j)2831 716 y FA(\()p FG(Z)2928
730 y Fy(1)2967 716 y FA(\))3002 615 y Fu(\021)3072 716
y FG(:)456 891 y Fo(De\014ne,)32 b(for)h(any)g FG(k)28
b Fx(2)d Fw(N)456 1067 y FA(\(4.3\))622 b FG(E)1330 1082
y Fv(k)1398 1067 y FA(:=)1519 966 y Fu(n)1580 1067 y
FG(Z)32 b Fx(2)24 b FG(L)i Fx(j)40 b FA(dist)14 b(\()p
FG(Z)q(;)h(D)2287 1082 y Fv(k)2331 1067 y FA(\))26 b
Fx(\024)f FG(a)2536 966 y Fu(o)2612 1067 y FG(:)456 1243
y Fo(Then,)38 b(ther)-5 b(e)38 b(exists)f FG(a)1258 1257
y Fy(0)1331 1243 y FG(>)c FA(1)g FG(>)g(c)1656 1257 y
Fy(0)1729 1243 y FG(>)g FA(0)38 b Fo(universal)f(c)-5
b(onstants)39 b(and)f FG(c)2933 1210 y Fv(?)3006 1243
y FG(>)33 b FA(0)p Fo(,)38 b(which)456 1351 y(dep)-5
b(ends)34 b(only)f(on)g FG(a)p Fo(,)g FG(c)g Fo(and)g(universal)g(c)-5
b(onstants,)35 b(such)e(that)1286 1527 y Fi(L)1341 1489
y Fv(N)1408 1426 y Fu(\020)1462 1527 y FG(Q)1534 1542
y Fv(l)1580 1527 y Fx(n)21 b FG(E)1713 1542 y Fv(k)1756
1426 y Fu(\021)1851 1527 y Fx(\024)40 b FA(\(1)21 b Fx(\000)f
FG(c)2193 1489 y Fv(?)2232 1527 y FA(\))15 b Fi(L)2337
1489 y Fv(N)2405 1527 y FA(\()p FG(Q)2512 1542 y Fv(l)2539
1527 y FA(\))g FG(;)456 1715 y Fo(pr)-5 b(ovide)g(d)35
b FG(a)25 b Fx(\025)g FG(a)1032 1729 y Fy(0)1104 1715
y Fo(and)33 b FG(c;)e(C)1447 1682 y Ft(\000)p Fy(1)1566
1715 y Fx(2)25 b FA(\(0)p FG(;)15 b(c)1811 1729 y Fy(0)1852
1715 y FA(])p Fo(.)150 4069 y @beginspecial 0 @llx 0
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% Polyline
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showpage

%%EndDocument
 @endspecial 1319 4252 a FC(The)29 b(co)n(v)n(ering)h(sets)f(of)h
(Lemma)e(4.3)555 4526 y FA(F)-8 b(ull)36 b(details)h(of)g(the)h(pro)s
(ofs)e(of)i(the)f(ab)s(o)m(v)m(e)i(lemmata)f(will)c(b)s(e)j(pro)m
(vided)f(in)g(the)456 4634 y(App)s(endix.)481 4838 y(5.)46
b FB(Estima)-6 b(tes)33 b(on)h(the)g(measure)e(of)i(the)f(pr)n
(ojection)g(of)h(the)f(cont)-6 b(a)n(ct)1935 4946 y(set)555
5108 y FA(W)e(e)45 b(no)m(w)e(sho)m(w)g(ho)m(w)h(to)g(use)f(Prop)s
(osition)e(3.14)k(and)e(the)g(measure)h(theoretic)456
5216 y(lemmata)21 b(stated)i(in)d Fx(x)p FA(4)i(in)e(order)h(to)h
(deduce)f(a)g(measure)h(estimate)g(on)f(the)h(pro)5 b(jection)p
eop
%%Page: 47 47
47 46 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(47)456
450 y FA(of)30 b(the)g(con)m(tact)j(sets)d(b)s(et)m(w)m(een)h(barriers)
e(and)g(minimal)e(solutions)i(of)h(\(1.5\).)43 b(T)-8
b(o)30 b(this)456 558 y(aim,)35 b(w)m(e)g(\014rst)f(need)h(an)g
(estimate)g(on)g(the)g(con)m(tact)i(sets)e(obtained)f(b)m(y)h(touc)m
(hing)f FG(u)456 666 y FA(b)m(y)c(ab)s(o)m(v)m(e)h(\\for)g(the)g
(\014rst)e(time",)i(as)g(dealt)f(with)f(b)m(y)h(the)h(follo)m(wing)e
(result.)456 796 y FH(Lemma)37 b(5.1.)44 b Fo(L)-5 b(et)36
b FG(C)q(;)31 b(C)1391 763 y Ft(0)1445 796 y FG(>)g FA(1)37
b Fo(b)-5 b(e)35 b(suitably)i(lar)-5 b(ge)37 b(c)-5 b(onstants.)54
b(L)-5 b(et)36 b FG(K)2986 811 y Fv(l)3044 796 y FA(:=)31
b Fx(fj)p FG(x)3293 763 y Ft(0)3317 796 y Fx(j)h FG(<)456
903 y(C)21 b(l)r Fx(g)k(\002)e(fj)p FG(x)857 917 y Fv(N)925
903 y Fx(j)35 b FG(<)f(C)22 b(l)r Fx(g)p Fo(.)58 b(L)-5
b(et)38 b FG(u)c Fx(2)h FG(W)1780 870 y Fy(1)p Fv(;p)1874
903 y FA(\()p FG(K)1986 918 y Fv(l)2012 903 y FA(\))j
Fo(b)-5 b(e)38 b(a)g(lo)-5 b(c)g(al)40 b(minimizer)e(for)h
Fx(F)47 b Fo(in)37 b FG(K)3390 918 y Fv(l)3417 903 y
Fo(.)456 1011 y(Assume)26 b(that)i FG(u)p FA(\(0\))f(=)e(0)i
Fo(and)h(that)g FG(u)p FA(\()p FG(x)p FA(\))e FG(<)f
FA(0)i Fo(if)f FG(x)2184 1025 y Fv(N)2277 1011 y FG(<)f
Fx(\000)p FG(\022)s Fo(,)h(for)i(some)f FG(\022)h(>)d
FA(0)p Fo(.)40 b(De\014ne)456 1119 y FG(R)525 1133 y
Fy(0)589 1119 y FA(:=)26 b FG(l)740 1086 y Fy(2)779 1119
y FG(=)p FA(\()p FG(C)7 b(\022)s FA(\))p Fo(.)41 b(L)-5
b(et)33 b FA(\004)f Fo(b)-5 b(e)33 b(the)g(set)g(of)g(p)-5
b(oints)34 b FA(\()p Fi(x)p FG(;)d(u)p FA(\()p Fi(x)p
FA(\)\))c Fx(2)d FG(K)2611 1134 y Fv(l)2670 1119 y Fo(satisfying:)671
1249 y Fx(\017)42 b(j)p Fi(x)815 1216 y Ft(0)839 1249
y Fx(j)26 b(\024)f FG(l)r Fo(,)32 b Fx(j)p FG(u)p FA(\()p
Fi(x)p FA(\))p Fx(j)27 b FG(<)e FA(1)p FG(=)p FA(2)p
Fo(;)671 1358 y Fx(\017)42 b Fo(ther)-5 b(e)40 b(exists)g
FG(Y)57 b Fx(2)37 b Fw(R)1514 1325 y Fv(N)1612 1358 y
Fx(\002)25 b FA([)p Fx(\000)p FA(1)p FG(=)p FA(4)p FG(;)32
b FA(1)p FG(=)p FA(4])40 b Fo(such)g(that)g Fw(S)o FA(\()p
FG(Y)5 b(;)15 b(R)2861 1372 y Fy(0)2895 1358 y FA(\))39
b Fo(is)g(ab)-5 b(ove)40 b(the)758 1466 y(gr)-5 b(aph)37
b(of)e FG(u)f Fo(in)h Fx(fj)p FG(x)1439 1433 y Ft(0)1463
1466 y Fx(j)29 b FG(<)g(C)1689 1433 y Ft(0)1727 1466
y FG(l)r Fx(g)22 b(\002)f(fj)p FG(x)2037 1480 y Fv(N)2105
1466 y Fx(j)29 b FG(<)g(C)2331 1433 y Ft(0)2369 1466
y FG(l)r Fx(g)35 b Fo(and)g(it)g(touches)g(the)g(gr)-5
b(aph)758 1574 y(of)33 b FG(u)g Fo(at)g FA(\()p Fi(x)p
FG(;)e(u)p FA(\()p Fi(x)p FA(\)\))p Fo(;)671 1737 y Fx(\017)42
b Fl(\\)839 1609 y Fu(\022)941 1675 y Fx(r)p FG(u)p FA(\()p
Fi(x)p FA(\))p 916 1716 282 4 v 916 1799 a Fx(jr)p FG(u)p
FA(\()p Fi(x)p FA(\))p Fx(j)1222 1737 y FG(;)15 b(e)1304
1751 y Fv(N)1372 1609 y Fu(\023)1480 1737 y Fx(\024)1618
1675 y FA(8)p FG(l)p 1601 1716 109 4 v 1601 1799 a(R)1670
1813 y Fy(0)1719 1737 y Fo(;)671 1956 y Fx(\017)42 b
Fi(x)790 1970 y Fv(N)883 1956 y Fx(\024)989 1895 y FG(\022)p
989 1935 46 4 v 989 2019 a FA(4)1065 1956 y(+)20 b FG(H)1232
1970 y Fy(0)1271 1956 y FA(\()p FG(u)p FA(\()p Fi(x)p
FA(\)\))p Fo(.)456 2112 y(Then,)30 b(ther)-5 b(e)30 b(exists)g(a)g
(universal)g(c)-5 b(onstant)32 b FG(c)25 b(>)g FA(0)30
b Fo(such)g(that,)h(for)f(any)g FG(\022)3015 2126 y Fy(0)3080
2112 y FG(>)25 b FA(0)k Fo(ther)-5 b(e)456 2220 y(exists)33
b FG(")747 2234 y Fy(0)786 2220 y FA(\()p FG(\022)864
2234 y Fy(0)904 2220 y FA(\))25 b FG(>)g FA(0)33 b Fo(for)h(which,)f
(if)1484 2357 y FG(\022)p 1484 2398 V 1492 2481 a(l)1565
2419 y Fx(\024)25 b FG(")1703 2433 y Fy(0)1743 2419 y
FA(\()p FG(\022)1821 2433 y Fy(0)1860 2419 y FA(\))15
b FG(;)202 b(\022)27 b Fx(\025)e FG(\022)2346 2433 y
Fy(0)2400 2419 y FG(;)456 2596 y Fo(one)33 b(has)g(that)1493
2714 y Fi(L)1548 2676 y Fv(N)1615 2613 y Fu(\020)1670
2714 y FG(\031)1722 2728 y Fv(N)1789 2714 y FA(\(\004\))1920
2613 y Fu(\021)2015 2714 y Fx(\025)40 b FG(c)15 b(l)2209
2676 y Fv(N)7 b Ft(\000)p Fy(1)2382 2714 y FG(:)456 2902
y Fo(Pr)-5 b(o)g(of.)43 b FA(Exploiting)28 b(Lemma)i(2.23,)i(w)m(e)f
(ha)m(v)m(e)h(that,)f(if)e FG(C)37 b FA(is)30 b(large)g(enough,)456
3053 y(\(5.1\))938 b FG(u)p FA(\()p FG(x)p FA(\))41 b
Fx(\024)f FG(g)1948 3068 y Fv(l)1975 3053 y FA(\()p FG(x)2062
3067 y Fv(N)2149 3053 y FA(+)20 b FG(\022)s FA(\))456
3204 y(for)30 b(an)m(y)h FG(x)g FA(so)g(that)h Fx(j)p
FG(x)1237 3171 y Ft(0)1260 3204 y Fx(j)26 b(\024)g FG(C)1480
3171 y Ft(0)1503 3204 y FG(l)33 b FA(and)d Fx(j)p FG(x)1817
3218 y Fv(N)1884 3204 y Fx(j)d(\024)e FG(C)2104 3171
y Ft(0)2127 3204 y FG(l)r FA(,)31 b(with)f FG(C)2492
3171 y Ft(0)2545 3204 y FA(large)h(if)f(so)h(is)f FG(C)7
b FA(.)42 b(Let)31 b(us)456 3312 y(de\014ne)456 3463
y(\(5.2\))1016 b FG(R)1726 3477 y Fy(0)1806 3463 y FA(:=)40
b FG(l)1971 3426 y Fy(2)2010 3463 y FG(=)p FA(\()p FG(C)7
b(\022)s FA(\))456 3614 y(and,)30 b(for)g FG(C)869 3581
y Ft(00)936 3614 y FG(>)25 b FA(0)31 b(con)m(v)m(enien)m(tly)f(large,)h
(let)f(us)g(consider)f(the)i(set)645 3770 y Fu(e)636
3788 y Fi(O)83 b FA(:=)967 3687 y Fu(n)1028 3788 y FG(Y)45
b FA(=)25 b(\()p FG(y)s(;)15 b(y)1390 3802 y Fv(N)7 b
Fy(+1)1547 3788 y FA(\))26 b Fx(2)f Fw(R)1754 3751 y
Fv(N)7 b Fy(+1)1957 3788 y FA(suc)m(h)30 b(that)967 3957
y Fx(j)p FG(y)1040 3919 y Ft(0)1063 3957 y Fx(j)c(\024)f
FG(l)r(=C)1356 3919 y Ft(00)1413 3957 y FG(;)56 b Fx(j)p
FG(y)1564 3971 y Fv(N)7 b Fy(+1)1721 3957 y Fx(j)26 b(\024)f
FA(1)p FG(=)p FA(4)967 4120 y(and)30 b(so)g(that,)i(if)d(\(0)p
FG(;)15 b(:)g(:)g(:)i(;)e FA(0)p FG(;)g(x)1980 4134 y
Fv(N)2049 4120 y FG(;)g FA(0\))26 b Fx(2)f Fw(S)o FA(\()p
FG(Y)5 b(;)15 b(R)2540 4134 y Fy(0)2574 4120 y FA(\))30
b(then)g FG(x)2898 4134 y Fv(N)2991 4120 y Fx(\024)25
b FA(0)3163 4019 y Fu(o)3238 4120 y FG(:)456 4293 y FA(W)-8
b(e)31 b(claim)f(that)1165 4429 y FG(g)1208 4447 y Fj(S)-6
b Fy(\()p Fv(Y)t(;R)1396 4456 y Fq(0)1425 4447 y Fy(\))1456
4429 y FA(\()p FG(x)p FA(\))41 b FG(>)g(g)1774 4444 y
Fv(l)1800 4429 y FA(\()p FG(x)1887 4443 y Fv(N)1975 4429
y FA(+)20 b FG(\022)s FA(\))29 b(for)i(an)m(y)f FG(Y)46
b Fx(2)2681 4411 y Fu(e)2671 4429 y Fi(O)p FA(,)1135
4571 y(pro)m(vided)29 b FG(x)c Fx(2)g FG(K)1749 4586
y Fv(l)1805 4571 y FA(and)30 b Fx(j)p FG(x)2059 4538
y Ft(0)2083 4571 y Fx(j)25 b(2)g FA(\()p FG(l)r(;)15
b(C)2395 4538 y Ft(0)2418 4571 y FG(l)r FA(\).)456 4499
y(\(5.3\))456 4743 y(T)-8 b(o)30 b(pro)m(v)m(e)i(this,)d(let)h
FG(Y)46 b Fx(2)1362 4724 y Fu(e)1353 4743 y Fi(O)30 b
FA(and)f(de\014ne)1010 4894 y(\006)40 b(:=)h Fx(f)p FG(g)1341
4912 y Fj(S)-5 b Fy(\()p Fv(Y)t(;)o(R)1529 4921 y Fq(0)1558
4912 y Fy(\))1615 4894 y FA(=)25 b(0)p Fx(g)41 b FA(=)f
Fw(S)o FA(\()p FG(Y)5 b(;)15 b(R)2212 4908 y Fy(0)2246
4894 y FA(\))20 b Fx(\\)g(f)p FG(x)2479 4908 y Fv(N)7
b Fy(+1)2662 4894 y FA(=)25 b(0)p Fx(g)15 b FG(:)456
5064 y FA(Then,)29 b(the)i(last)f(condition)f(in)g(the)i(de\014nition)d
(of)2225 5046 y Fu(e)2216 5064 y Fi(O)i FA(reads)456
5216 y(\(5.4\))630 b(if)30 b(\(0)p FG(;)15 b(:)g(:)g(:)i(;)e
FA(0)p FG(;)g(x)1774 5230 y Fv(N)1843 5216 y FA(\))25
b Fx(2)g FA(\006,)30 b(then)g FG(x)2369 5230 y Fv(N)2462
5216 y Fx(\024)25 b FA(0.)p eop
%%Page: 48 48
48 47 bop 456 251 a Fy(48)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)555 450 y FA(Recalling)40 b(the)i(de\014nitions)c(on)j(page)h(14)
g(and)f(\(2.39\),)46 b(one)c(sees)g(that)f(\006)g(is)f(an)456
558 y(\()p FG(N)30 b Fx(\000)20 b FA(1\)-dimensional)29
b(sphere,)g(namely)1598 748 y(\006)c(=)g Fx(fj)p FG(x)c
Fx(\000)f FG(y)s Fx(j)25 b FA(=)g FG(r)s Fx(g)456 939
y FA(with)456 1171 y(\(5.5\))632 b FG(r)42 b FA(:=)f
FG(R)1562 1185 y Fy(0)1622 1171 y Fx(\000)19 b FG(H)1788
1185 y Fy(0)1827 1171 y FA(\()p FG(y)1907 1185 y Fv(N)7
b Fy(+1)2065 1171 y FA(\))20 b Fx(\000)p 2243 1037 72
4 v 2243 1110 a FG(C)2314 1124 y Fy(0)p 2221 1150 154
4 v 2221 1234 a FA(2)p FG(R)2335 1248 y Fy(0)2385 1171
y FG(y)2433 1134 y Fy(2)2430 1194 y Fv(N)7 b Fy(+1)2602
1171 y FG(:)456 1406 y FA(Let)35 b(us)g(no)m(w)g(estimate)g
FG(r)j FA(b)m(y)d(noticing)f(that,)j(if)d FG(l)j FA(\(and)e(therefore)g
FG(R)2922 1420 y Fy(0)2962 1406 y FA(\))g(is)f(suitably)456
1514 y(large,)c(w)m(e)h(ha)m(v)m(e)h(that)1375 1698 y(2)p
FG(l)1449 1665 y Fy(2)p 1351 1739 163 4 v 1351 1822 a
FA(3)p FG(C)7 b(\022)1549 1760 y FA(=)1655 1698 y(2)p
1655 1739 46 4 v 1655 1822 a(3)1710 1760 y FG(R)1779
1774 y Fy(0)1844 1760 y Fx(\024)25 b FG(R)2009 1774 y
Fy(0)2068 1760 y Fx(\000)35 b FA(const)41 b Fx(\024)1367
1921 y(\024)25 b FG(r)i Fx(\024)1367 2130 y(\024)e FG(R)1532
2144 y Fy(0)1591 2130 y FA(+)35 b(const)41 b Fx(\024)2051
2068 y FA(3)p 2051 2109 V 2051 2192 a(2)2107 2130 y FG(R)2176
2144 y Fy(0)2240 2130 y FA(=)2370 2068 y(3)p FG(l)2444
2035 y Fy(2)p 2346 2109 163 4 v 2346 2192 a FA(2)p FG(C)7
b(\022)2533 2130 y(:)456 1931 y FA(\(5.6\))456 2351 y(Notice)33
b(also)g(that)h FG(x)e FA(in)g(\(5.3\))i(m)m(ust)f(lie)f(in)f(the)i(in)
m(tersection)g(b)s(et)m(w)m(een)g FG(K)3080 2366 y Fv(l)3139
2351 y FA(and)f(the)456 2459 y(domain)d(of)h FG(g)925
2477 y Fj(S)-5 b Fy(\()p Fv(Y)t(;)o(R)1113 2486 y Fq(0)1142
2477 y Fy(\))1174 2459 y FA(,)30 b(otherwise)g(there)h(is)e(nothing)h
(to)h(pro)m(v)m(e;)g(therefore,)1179 2659 y Fx(j)p FG(x)20
b Fx(\000)g FG(y)s Fx(j)25 b(\024)41 b FA(const)15 b(\()p
FG(C)1906 2622 y Ft(0)1929 2659 y FG(l)23 b FA(+)c FG(R)2138
2673 y Fy(0)2178 2659 y FA(\))26 b Fx(\024)40 b FA(const)15
b FG(R)2641 2673 y Fy(0)2696 2659 y FG(;)456 2850 y FA(and)29
b(so,)i(b)m(y)g(\(5.6\),)456 3040 y(\(5.7\))942 b Fx(j)p
FG(x)20 b Fx(\000)g FG(y)s Fx(j)40 b(\024)56 b FA(const)15
b FG(r)j(:)555 3230 y FA(W)-8 b(e)32 b(no)m(w)e(p)s(oin)m(t)g(out)g
(that)h(\006)f(is)g(b)s(elo)m(w)f(the)i(h)m(yp)s(erplane)d
FG(x)2618 3244 y Fv(N)2711 3230 y FA(=)c FG(\022)s(=)p
FA(8,)31 b(that)g(is)456 3421 y(\(5.8\))811 b FG(x)1504
3435 y Fv(N)1596 3421 y Fx(\024)25 b FG(\022)s(=)p FA(8,)31
b(for)f(an)m(y)g FG(x)c Fx(2)f FA(\006.)456 3611 y(In)k(order)h(to)h
(pro)m(v)m(e)g(\(5.8\),)i(let)963 3801 y(\026)-52 b FG(y)44
b FA(:=)c FG(y)23 b Fx(\000)d FG(y)1385 3815 y Fy(1)1424
3801 y FG(e)1466 3815 y Fy(1)1526 3801 y Fx(\000)g(\001)15
b(\001)g(\001)21 b(\000)f FG(y)1879 3815 y Fv(N)7 b Ft(\000)p
Fy(1)2036 3801 y FG(e)2078 3815 y Fv(N)g Ft(\000)p Fy(1)2276
3801 y FA(=)40 b(\(0)p FG(;)15 b(:)g(:)g(:)j(;)d FA(0)p
FG(;)g(y)2800 3815 y Fv(N)2868 3801 y FA(\))g FG(;)456
3999 y FA(so)30 b(that,)h(b)m(y)g(the)f(de\014nition)e(of)1589
3981 y Fu(e)1580 3999 y Fi(O)p FA(,)456 4232 y(\(5.9\))896
b Fx(j)p FG(y)23 b Fx(\000)j FA(\026)-51 b FG(y)s Fx(j)41
b FA(=)f Fx(j)p FG(y)2019 4194 y Ft(0)2042 4232 y Fx(j)26
b(\024)2241 4170 y FG(l)p 2199 4211 114 4 v 2199 4294
a(C)2271 4268 y Ft(00)2338 4232 y FG(;)456 4453 y FA(whic)m(h)35
b(is)g(less)h(than)g FG(r)j FA(due)d(to)h(\(5.6\),)j(pro)m(vided)35
b FG(\022)s(=l)j FA(is)e(small)f(enough.)58 b(Th)m(us,)37
b(let)457 4545 y(~)456 4562 y FG(t)25 b(>)g FA(0)30 b(b)s(e)g(so)h
(that)1570 4714 y(~)-52 b FG(y)44 b FA(:=)i(\026)-51
b FG(y)23 b FA(+)1948 4698 y(~)1947 4714 y FG(te)2022
4728 y Fv(N)2130 4714 y Fx(2)40 b FA(\006)15 b FG(:)p
eop
%%Page: 49 49
49 48 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(49)750
1890 y @beginspecial 0 @llx 0 @lly 564 @urx 358 @ury
2880 @rwi @setspecial
%%BeginDocument: pita.eps
%!PS-Adobe-2.0 EPSF-2.0
%%Title: pita.fig
%%Creator: fig2dev Version 3.2 Patchlevel 4
%%CreationDate: Tue Oct 12 08:38:13 2004
%%For: enrico@mirsada (Enrico Valdinoci)
%%BoundingBox: 0 0 564 358
%%Magnification: 1.0000
%%EndComments
/MyAppDict 100 dict dup begin def
/$F2psDict 200 dict def
$F2psDict begin
$F2psDict /mtrx matrix put
/col-1 {0 setgray} bind def
/col0 {0.000 0.000 0.000 srgb} bind def
/col1 {0.000 0.000 1.000 srgb} bind def
/col2 {0.000 1.000 0.000 srgb} bind def
/col3 {0.000 1.000 1.000 srgb} bind def
/col4 {1.000 0.000 0.000 srgb} bind def
/col5 {1.000 0.000 1.000 srgb} bind def
/col6 {1.000 1.000 0.000 srgb} bind def
/col7 {1.000 1.000 1.000 srgb} bind def
/col8 {0.000 0.000 0.560 srgb} bind def
/col9 {0.000 0.000 0.690 srgb} bind def
/col10 {0.000 0.000 0.820 srgb} bind def
/col11 {0.530 0.810 1.000 srgb} bind def
/col12 {0.000 0.560 0.000 srgb} bind def
/col13 {0.000 0.690 0.000 srgb} bind def
/col14 {0.000 0.820 0.000 srgb} bind def
/col15 {0.000 0.560 0.560 srgb} bind def
/col16 {0.000 0.690 0.690 srgb} bind def
/col17 {0.000 0.820 0.820 srgb} bind def
/col18 {0.560 0.000 0.000 srgb} bind def
/col19 {0.690 0.000 0.000 srgb} bind def
/col20 {0.820 0.000 0.000 srgb} bind def
/col21 {0.560 0.000 0.560 srgb} bind def
/col22 {0.690 0.000 0.690 srgb} bind def
/col23 {0.820 0.000 0.820 srgb} bind def
/col24 {0.500 0.190 0.000 srgb} bind def
/col25 {0.630 0.250 0.000 srgb} bind def
/col26 {0.750 0.380 0.000 srgb} bind def
/col27 {1.000 0.500 0.500 srgb} bind def
/col28 {1.000 0.630 0.630 srgb} bind def
/col29 {1.000 0.750 0.750 srgb} bind def
/col30 {1.000 0.880 0.880 srgb} bind def
/col31 {1.000 0.840 0.000 srgb} bind def

end
save
newpath 0 358 moveto 0 0 lineto 564 0 lineto 564 358 lineto closepath clip newpath
-56.8 365.6 translate
1 -1 scale

% This junk string is used by the show operators
/PATsstr 1 string def
/PATawidthshow { 	% cx cy cchar rx ry string
  % Loop over each character in the string
  {  % cx cy cchar rx ry char
    % Show the character
    dup				% cx cy cchar rx ry char char
    PATsstr dup 0 4 -1 roll put	% cx cy cchar rx ry char (char)
    false charpath		% cx cy cchar rx ry char
    /clip load PATdraw
    % Move past the character (charpath modified the
    % current point)
    currentpoint			% cx cy cchar rx ry char x y
    newpath
    moveto			% cx cy cchar rx ry char
    % Reposition by cx,cy if the character in the string is cchar
    3 index eq {			% cx cy cchar rx ry
      4 index 4 index rmoveto
    } if
    % Reposition all characters by rx ry
    2 copy rmoveto		% cx cy cchar rx ry
  } forall
  pop pop pop pop pop		% -
  currentpoint
  newpath
  moveto
} bind def
/PATcg {
  7 dict dup begin
    /lw currentlinewidth def
    /lc currentlinecap def
    /lj currentlinejoin def
    /ml currentmiterlimit def
    /ds [ currentdash ] def
    /cc [ currentrgbcolor ] def
    /cm matrix currentmatrix def
  end
} bind def
% PATdraw - calculates the boundaries of the object and
% fills it with the current pattern
/PATdraw {			% proc
  save exch
    PATpcalc			% proc nw nh px py
    5 -1 roll exec		% nw nh px py
    newpath
    PATfill			% -
  restore
} bind def
% PATfill - performs the tiling for the shape
/PATfill { % nw nh px py PATfill -
  PATDict /CurrentPattern get dup begin
    setfont
    % Set the coordinate system to Pattern Space
    PatternGState PATsg
    % Set the color for uncolored pattezns
    PaintType 2 eq { PATDict /PColor get PATsc } if
    % Create the string for showing
    3 index string		% nw nh px py str
    % Loop for each of the pattern sources
    0 1 Multi 1 sub {		% nw nh px py str source
	% Move to the starting location
	3 index 3 index		% nw nh px py str source px py
	moveto			% nw nh px py str source
	% For multiple sources, set the appropriate color
	Multi 1 ne { dup PC exch get PATsc } if
	% Set the appropriate string for the source
	0 1 7 index 1 sub { 2 index exch 2 index put } for pop
	% Loop over the number of vertical cells
	3 index 		% nw nh px py str nh
	{			% nw nh px py str
	  currentpoint		% nw nh px py str cx cy
	  2 index oldshow	% nw nh px py str cx cy
	  YStep add moveto	% nw nh px py str
	} repeat		% nw nh px py str
    } for
    5 { pop } repeat
  end
} bind def

% PATkshow - kshow with the current pattezn
/PATkshow {			% proc string
  exch bind			% string proc
  1 index 0 get			% string proc char
  % Loop over all but the last character in the string
  0 1 4 index length 2 sub {
				% string proc char idx
    % Find the n+1th character in the string
    3 index exch 1 add get	% string proc char char+1
    exch 2 copy			% strinq proc char+1 char char+1 char
    % Now show the nth character
    PATsstr dup 0 4 -1 roll put	% string proc chr+1 chr chr+1 (chr)
    false charpath		% string proc char+1 char char+1
    /clip load PATdraw
    % Move past the character (charpath modified the current point)
    currentpoint newpath moveto
    % Execute the user proc (should consume char and char+1)
    mark 3 1 roll		% string proc char+1 mark char char+1
    4 index exec		% string proc char+1 mark...
    cleartomark			% string proc char+1
  } for
  % Now display the last character
  PATsstr dup 0 4 -1 roll put	% string proc (char+1)
  false charpath		% string proc
  /clip load PATdraw
  neewath
  pop pop			% -
} bind def
% PATmp - the makepattern equivalent
/PATmp {			% patdict patmtx PATmp patinstance
  exch dup length 7 add		% We will add 6 new entries plus 1 FID
  dict copy			% Create a new dictionary
  begin
    % Matrix to install when painting the pattern
    TilingType PATtcalc
    /PatternGState PATcg def
    PatternGState /cm 3 -1 roll put
    % Check for multi pattern sources (Level 1 fast color patterns)
    currentdict /Multi known not { /Multi 1 def } if
    % Font dictionary definitions
    /FontType 3 def
    % Create a dummy encoding vector
    /Encoding 256 array def
    3 string 0 1 255 {
      Encoding exch dup 3 index cvs cvn put } for pop
    /FontMatrix matrix def
    /FontBBox BBox def
    /BuildChar {
	mark 3 1 roll		% mark dict char
	exch begin
	Multi 1 ne {PaintData exch get}{pop} ifelse  % mark [paintdata]
	  PaintType 2 eq Multi 1 ne or
	  { XStep 0 FontBBox aload pop setcachedevice }
	  { XStep 0 setcharwidth } ifelse
	  currentdict		% mark [paintdata] dict
	  /PaintProc load	% mark [paintdata] dict paintproc
	end
	gsave
	  false PATredef exec true PATredef
	grestore
	cleartomark		% -
    } bind def
    currentdict
  end				% newdict
  /foo exch			% /foo newlict
  definefont			% newfont
} bind def
% PATpcalc - calculates the starting point and width/height
% of the tile fill for the shape
/PATpcalc {	% - PATpcalc nw nh px py
  PATDict /CurrentPattern get begin
    gsave
	% Set up the coordinate system to Pattern Space
	% and lock down pattern
	PatternGState /cm get setmatrix
	BBox aload pop pop pop translate
	% Determine the bounding box of the shape
	pathbbox			% llx lly urx ury
    grestore
    % Determine (nw, nh) the # of cells to paint width and height
    PatHeight div ceiling		% llx lly urx qh
    4 1 roll				% qh llx lly urx
    PatWidth div ceiling		% qh llx lly qw
    4 1 roll				% qw qh llx lly
    PatHeight div floor			% qw qh llx ph
    4 1 roll				% ph qw qh llx
    PatWidth div floor			% ph qw qh pw
    4 1 roll				% pw ph qw qh
    2 index sub cvi abs			% pw ph qs qh-ph
    exch 3 index sub cvi abs exch	% pw ph nw=qw-pw nh=qh-ph
    % Determine the starting point of the pattern fill
    %(px, py)
    4 2 roll				% nw nh pw ph
    PatHeight mul			% nw nh pw py
    exch				% nw nh py pw
    PatWidth mul exch			% nw nh px py
  end
} bind def

% Save the original routines so that we can use them later on
/oldfill	/fill load def
/oldeofill	/eofill load def
/oldstroke	/stroke load def
/oldshow	/show load def
/oldashow	/ashow load def
/oldwidthshow	/widthshow load def
/oldawidthshow	/awidthshow load def
/oldkshow	/kshow load def

% These defs are necessary so that subsequent procs don't bind in
% the originals
/fill	   { oldfill } bind def
/eofill	   { oldeofill } bind def
/stroke	   { oldstroke } bind def
/show	   { oldshow } bind def
/ashow	   { oldashow } bind def
/widthshow { oldwidthshow } bind def
/awidthshow { oldawidthshow } bind def
/kshow 	   { oldkshow } bind def
/PATredef {
  MyAppDict begin
    {
    /fill { /clip load PATdraw newpath } bind def
    /eofill { /eoclip load PATdraw newpath } bind def
    /stroke { PATstroke } bind def
    /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def
    /ashow { 0 0 null 6 3 roll PATawidthshow }
    bind def
    /widthshow { 0 0 3 -1 roll PATawidthshow }
    bind def
    /awidthshow { PATawidthshow } bind def
    /kshow { PATkshow } bind def
  } {
    /fill   { oldfill } bind def
    /eofill { oldeofill } bind def
    /stroke { oldstroke } bind def
    /show   { oldshow } bind def
    /ashow  { oldashow } bind def
    /widthshow { oldwidthshow } bind def
    /awidthshow { oldawidthshow } bind def
    /kshow  { oldkshow } bind def
    } ifelse
  end
} bind def
false PATredef
% Conditionally define setcmykcolor if not available
/setcmykcolor where { pop } {
  /setcmykcolor {
    1 sub 4 1 roll
    3 {
	3 index add neg dup 0 lt { pop 0 } if 3 1 roll
    } repeat
    setrgbcolor - pop
  } bind def
} ifelse
/PATsc {		% colorarray
  aload length		% c1 ... cn length
    dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor
  } ifelse } ifelse
} bind def
/PATsg {		% dict
  begin
    lw setlinewidth
    lc setlinecap
    lj setlinejoin
    ml setmiterlimit
    ds aload pop setdash
    cc aload pop setrgbcolor
    cm setmatrix
  end
} bind def

/PATDict 3 dict def
/PATsp {
  true PATredef
  PATDict begin
    /CurrentPattern exch def
    % If it's an uncolored pattern, save the color
    CurrentPattern /PaintType get 2 eq {
      /PColor exch def
    } if
    /CColor [ currentrgbcolor ] def
  end
} bind def
% PATstroke - stroke with the current pattern
/PATstroke {
  countdictstack
  save
  mark
  {
    currentpoint strokepath moveto
    PATpcalc				% proc nw nh px py
    clip newpath PATfill
    } stopped {
	(*** PATstroke Warning: Path is too complex, stroking
	  with gray) =
    cleartomark
    restore
    countdictstack exch sub dup 0 gt
	{ { end } repeat } { pop } ifelse
    gsave 0.5 setgray oldstroke grestore
  } { pop restore pop } ifelse
  newpath
} bind def
/PATtcalc {		% modmtx tilingtype PATtcalc tilematrix
  % Note: tiling types 2 and 3 are not supported
  gsave
    exch concat					% tilingtype
    matrix currentmatrix exch			% cmtx tilingtype
    % Tiling type 1 and 3: constant spacing
    2 ne {
	% Distort the pattern so that it occupies
	% an integral number of device pixels
	dup 4 get exch dup 5 get exch		% tx ty cmtx
	XStep 0 dtransform
	round exch round exch			% tx ty cmtx dx.x dx.y
	XStep div exch XStep div exch		% tx ty cmtx a b
	0 YStep dtransform
	round exch round exch			% tx ty cmtx a b dy.x dy.y
	YStep div exch YStep div exch		% tx ty cmtx a b c d
	7 -3 roll astore			% { a b c d tx ty }
    } if
  grestore
} bind def
/PATusp {
  false PATredef
  PATDict begin
    CColor PATsc
  end
} bind def

% this is the pattern fill program from the Second edition Reference Manual
% with changes to call the above pattern fill
% left30
11 dict begin
/PaintType 1 def
/PatternType 1 def
/TilingType 1 def
/BBox [0 0 1 1] def
/XStep 1 def
/YStep 1 def
/PatWidth 1 def
/PatHeight 1 def
/Multi 2 def
/PaintData [
  { clippath } bind
  { 32 16 true [ 32 0 0 -16 0 16 ]
	{<c000c000300030000c000c000300030000c000c000300030
	000c000c00030003c000c000300030000c000c0003000300
	00c000c000300030000c000c00030003>}
     imagemask } bind
] def
/PaintProc {
	pop
	exec fill
} def
currentdict
end
/P1 exch def

/cp {closepath} bind def
/ef {eofill} bind def
/gr {grestore} bind def
/gs {gsave} bind def
/sa {save} bind def
/rs {restore} bind def
/l {lineto} bind def
/m {moveto} bind def
/rm {rmoveto} bind def
/n {newpath} bind def
/s {stroke} bind def
/sh {show} bind def
/slc {setlinecap} bind def
/slj {setlinejoin} bind def
/slw {setlinewidth} bind def
/srgb {setrgbcolor} bind def
/rot {rotate} bind def
/sc {scale} bind def
/sd {setdash} bind def
/ff {findfont} bind def
/sf {setfont} bind def
/scf {scalefont} bind def
/sw {stringwidth} bind def
/tr {translate} bind def
/tnt {dup dup currentrgbcolor
  4 -2 roll dup 1 exch sub 3 -1 roll mul add
  4 -2 roll dup 1 exch sub 3 -1 roll mul add
  4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb}
  bind def
/shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul
  4 -2 roll mul srgb} bind def
 /DrawEllipse {
	/endangle exch def
	/startangle exch def
	/yrad exch def
	/xrad exch def
	/y exch def
	/x exch def
	/savematrix mtrx currentmatrix def
	x y tr xrad yrad sc 0 0 1 startangle endangle arc
	closepath
	savematrix setmatrix
	} def

/$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def
/$F2psEnd {$F2psEnteredState restore end} def

$F2psBegin
10 setmiterlimit
0 slj 0 slc
 0.06299 0.06299 sc
%
% Fig objects follow
%
% 
% here starts figure with depth 50
% Polyline
7.500 slw
n 3645 1755 m
 3645 4860 l gs col0 s gr 
% Polyline
n 3690 4905 m 5940 4905 l
 3690 1755 l gs col0 s gr 
% Polyline
 [60 30 15 30] 0 sd
n 5985 4860 m
 8775 2160 l gs col0 s gr  [] 0 sd
% Polyline
45.000 slw
gs  clippath
3742 165 m 3637 165 l 3637 320 l 3690 200 l 3742 320 l cp
eoclip
n 3690 675 m
 3690 180 l gs col0 s gr gr

% arrowhead
15.000 slw
n 3742 320 m 3690 200 l 3637 320 l 3742 320 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
45.000 slw
n 3105 225 m 945 225 l 945 5760 l 9810 5760 l 9810 225 l
 3915 225 l gs col0 s gr 
% Polyline
2 slj
7.500 slw
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b FF(~)-44 b FD(y)555 2348 y FA(Then)36 b(~)-51 b FG(y)28
b FA(=)d(\(0)p FG(;)15 b(:)g(:)g(:)i(;)e FA(0)p FG(;)g(y)1374
2362 y Fv(N)1462 2348 y FA(+)1555 2331 y(~)1553 2348
y FG(t)p FA(\),)31 b(th)m(us,)f(from)g(\(5.4\),)456 2510
y(\(5.10\))1060 b FG(y)1791 2524 y Fv(N)1899 2510 y Fx(\024)40
b(\000)2082 2493 y FA(~)2081 2510 y FG(t)14 b(:)456 2672
y FA(Also,)30 b(from)g(\(5.9\),)1240 2888 y(~)1239 2905
y FG(t)1272 2867 y Fy(2)1351 2905 y FA(=)40 b FG(r)1506
2867 y Fy(2)1566 2905 y Fx(\000)19 b(j)p FG(y)24 b Fx(\000)i
FA(\026)-51 b FG(y)r Fx(j)1913 2867 y Fy(2)1994 2905
y Fx(\025)40 b FG(r)2149 2867 y Fy(2)2208 2905 y Fx(\000)2299
2777 y Fu(\022)2418 2843 y FG(l)p 2376 2884 114 4 v 2376
2967 a(C)2448 2941 y Ft(00)2500 2777 y Fu(\023)2566 2799
y Fy(2)2636 2905 y FG(;)456 3122 y FA(therefore,)31 b(in)e(the)h(ligh)m
(t)g(of)g(\(5.10\))j(and)d(\(5.6\),)1257 3284 y FG(y)1302
3298 y Fv(N)1389 3284 y FA(+)20 b FG(r)85 b Fx(\024)e
FG(r)23 b Fx(\000)1916 3267 y FA(~)1915 3284 y FG(t)40
b Fx(\024)1606 3517 y(\024)83 b FG(r)23 b Fx(\000)1915
3344 y Fu(s)p 2006 3344 502 4 v 173 x FG(r)2050 3490
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y FG(l)p 2277 3496 114 4 v 2277 3579 a(C)2349 3553 y
Ft(00)2400 3388 y Fu(\023)2467 3411 y Fy(2)2547 3517
y Fx(\024)-2162 b FA(\(5.11\))1606 3773 y Fx(\024)1770
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(of)g(of)g(\(5.8\).)555 4130 y(Let)43 b(us)f(no)m(w)g(go)h(bac)m(k)h
(to)f(the)f(pro)s(of)g(of)h(\(5.3\).)78 b(F)-8 b(or)43
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b(notation:)41 b(de\014ne)1478 4399 y FG(d)1525 4413
y Fy(1)1565 4399 y FA(\()p FG(x)p FA(\))84 b(:=)f Fx(j)p
FG(x)20 b Fx(\000)g FG(y)s Fx(j)h(\000)e FG(r)f(;)1478
4534 y(d)1525 4548 y Fy(2)1565 4534 y FA(\()p FG(x)p
FA(\))84 b(:=)f FG(x)2002 4548 y Fv(N)2089 4534 y FA(+)20
b FG(\022)e(:)456 4696 y FA(Let)30 b(no)m(w)h FG(x)f
FA(b)s(e)g(as)g(requested)h(in)e(\(5.3\).)42 b(F)-8 b(rom)31
b(\(5.7\),)456 4858 y(\(5.12\))754 b(0)41 b Fx(\024)f
FG(r)23 b FA(+)d FG(d)1839 4872 y Fy(1)1879 4858 y FA(\()p
FG(x)p FA(\))41 b Fx(\024)f FA(const)16 b FG(r)h(:)456
5020 y FA(Also,)1222 5186 y Fx(j)p FG(x)1299 5149 y Ft(0)1343
5186 y Fx(\000)j FG(y)1482 5149 y Ft(0)1505 5186 y Fx(j)25
b(\025)g(j)p FG(x)1728 5149 y Ft(0)1752 5186 y Fx(j)20
b(\000)g(j)p FG(y)1961 5149 y Ft(0)1984 5186 y Fx(j)26
b(\025)2149 5125 y FG(l)p 2141 5166 V 2141 5249 a FA(2)2216
5186 y Fx(\000)2350 5125 y FG(l)p 2317 5166 95 4 v 2317
5249 a(C)2389 5223 y Ft(0)2447 5186 y Fx(\025)2553 5125
y FA(2)p 2553 5166 46 4 v 2553 5249 a(5)2609 5186 y FG(l)17
b(;)p eop
%%Page: 50 50
50 49 bop 456 251 a Fy(50)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(if)29 b FG(C)611 417 y Ft(0)659 450
y Fx(\025)c FA(10,)31 b(th)m(us,)1070 537 y Fu(\020)1124
638 y FG(r)23 b FA(+)d FG(d)1326 652 y Fy(1)1365 638
y FA(\()p FG(x)p FA(\))1487 537 y Fu(\021)1542 559 y
Fy(2)1665 638 y FA(=)83 b Fx(j)p FG(x)20 b Fx(\000)g
FG(y)s Fx(j)2080 600 y Fy(2)2145 638 y FA(=)1665 812
y(=)83 b Fx(j)p FG(x)1896 775 y Ft(0)1939 812 y Fx(\000)20
b FG(y)2078 775 y Ft(0)2101 812 y Fx(j)2126 775 y Fy(2)2186
812 y FA(+)g Fx(j)p FG(x)2354 826 y Fv(N)2442 812 y Fx(\000)g
FG(y)2578 826 y Fv(N)2644 812 y Fx(j)2669 775 y Fy(2)2734
812 y Fx(\025)1665 991 y(\025)1851 929 y FA(4)p 1828
970 91 4 v 1828 1053 a(25)1929 991 y FG(l)1958 953 y
Fy(2)2018 991 y FA(+)g Fx(j)p FG(x)2186 1005 y Fv(N)2273
991 y Fx(\000)g FG(y)2409 1005 y Fv(N)2476 991 y Fx(j)2501
953 y Fy(2)2556 991 y FA(;)-2125 b(\(5.13\))456 1170
y(on)30 b(the)g(other)h(hand,)f(from)f(\(5.8\))k(and)c(the)i(fact)g
(that)g FG(y)23 b FA(+)d FG(r)s(e)2615 1184 y Fv(N)2708
1170 y Fx(2)k FA(\006,)456 1366 y(\(5.14\))1017 b FG(y)1748
1380 y Fv(N)1835 1366 y FA(+)20 b FG(r)28 b Fx(\024)2101
1305 y FG(\022)p 2101 1345 46 4 v 2101 1428 a FA(8)2171
1366 y FG(:)456 1546 y FA(Th)m(us,)h(from)h(\(5.13\))j(and)c(\(5.14\),)
k(w)m(e)e(infer)e(that)1140 1768 y FG(x)1192 1782 y Fv(N)1342
1768 y Fx(\024)83 b FG(y)1541 1782 y Fv(N)1628 1768 y
FA(+)1718 1624 y Fu(r)p 1809 1624 803 4 v 43 x(\020)1864
1768 y FG(r)22 b FA(+)e FG(d)2065 1782 y Fy(1)2105 1768
y FA(\()p FG(x)p FA(\))2227 1667 y Fu(\021)2282 1689
y Fy(2)2342 1768 y Fx(\000)2465 1706 y FA(4)p 2443 1747
91 4 v 2443 1830 a(25)2543 1768 y FG(l)2572 1741 y Fy(2)2637
1768 y Fx(\024)1342 2025 y(\024)1506 1963 y FG(\022)p
1506 2004 46 4 v 1506 2087 a FA(8)1581 2025 y Fx(\000)g
FG(r)j FA(+)1827 1881 y Fu(r)p 1918 1881 803 4 v 43 x(\020)1972
2025 y FG(r)g FA(+)d FG(d)2174 2039 y Fy(1)2213 2025
y FA(\()p FG(x)p FA(\))2335 1924 y Fu(\021)2390 1947
y Fy(2)2450 2025 y Fx(\000)2573 1963 y FA(4)p 2551 2004
91 4 v 2551 2087 a(25)2652 2025 y FG(l)2681 1999 y Fy(2)2735
2025 y FG(:)456 2213 y FA(This,)28 b(\(5.12\))33 b(and)d(\(5.6\))i
(imply)c(that)1589 2362 y FG(x)1641 2376 y Fv(N)1733
2362 y Fx(\024)d(\000)p FA(2)p FG(\022)e FA(+)d FG(d)2149
2376 y Fy(1)2188 2362 y FA(\()p FG(x)p FA(\))456 2510
y(and,)30 b(therefore,)1361 2621 y FG(d)1408 2635 y Fy(1)1447
2621 y FA(\()p FG(x)p FA(\))c Fx(\025)f FG(x)1743 2635
y Fv(N)1831 2621 y FA(+)19 b(2)p FG(\022)28 b FA(=)d
FG(d)2180 2635 y Fy(2)2220 2621 y FA(\()p FG(x)p FA(\))c(+)f
FG(\022)d(;)456 2750 y FA(pro)m(ving)29 b(that)456 2899
y(\(5.15\))1038 b FG(d)1771 2913 y Fy(1)1836 2899 y Fx(\025)25
b FG(d)1979 2913 y Fy(2)2039 2899 y FA(+)20 b FG(\022)456
3047 y FA(in)29 b FG(K)639 3062 y Fv(l)685 3047 y Fx(\\)20
b(fj)p FG(x)888 3014 y Ft(0)912 3047 y Fx(j)25 b(2)g
FA(\()p FG(l)r(;)15 b(C)1224 3014 y Ft(0)1248 3047 y
FG(l)r FA(\))p Fx(g)p FA(.)555 3196 y(W)-8 b(e)32 b(no)m(w)e(observ)m
(e)h(that)456 3386 y(\(5.16\))398 b FG(H)1160 3400 y
Fv(y)1195 3411 y Fs(N)5 b Fq(+1)1330 3400 y Fv(;R)1403
3409 y Fq(0)1441 3386 y FA(\()p FG(s)p FA(\))21 b Fx(\000)e
FG(H)1741 3400 y Fv(y)1776 3411 y Fs(N)5 b Fq(+1)1911
3400 y Fv(;R)1984 3409 y Fq(0)2022 3386 y FA(\(0\))27
b Fx(\024)e FG(H)2336 3400 y Fy(0)2375 3386 y FA(\()p
FG(s)p FA(\))20 b(+)2609 3325 y(2)p 2654 3252 72 4 v
FG(C)2726 3339 y Fy(0)p 2609 3366 157 4 v 2633 3449 a
FG(R)2702 3463 y Fy(0)2790 3386 y FG(;)456 3579 y FA(for)30
b(an)m(y)g FG(s)25 b Fx(2)g FA([)p FG(s)988 3593 y Fv(R)1041
3602 y Fq(0)1080 3579 y FG(;)15 b FA(1].)42 b(T)-8 b(o)31
b(pro)m(v)m(e)g(this,)e(recall)h(De\014nition)f(2.8)i(to)h(get)955
3818 y FG(H)1031 3832 y Fv(y)1066 3843 y Fs(N)5 b Fq(+1)1201
3832 y Fv(;R)1274 3841 y Fq(0)1312 3818 y FA(\()p FG(s)p
FA(\))21 b Fx(\000)f FG(H)1613 3832 y Fv(y)1648 3843
y Fs(N)5 b Fq(+1)1782 3832 y Fv(;R)1855 3841 y Fq(0)1893
3818 y FA(\(0\))27 b(=)2131 3694 y Fu(Z)2222 3720 y Fv(s)2181
3900 y Fy(0)2370 3756 y FA(\()p FG(p)20 b Fx(\000)g FA(1\))2653
3684 y Fq(1)p 2653 3696 33 3 v 2653 3737 a Fs(p)p 2283
3797 502 4 v 2283 3913 a FA(\()p FG(p)15 b(h)2431 3927
y Fv(s)2464 3936 y Fq(0)2499 3927 y Fv(;R)2577 3913 y
FA(\()p FG(\020)7 b FA(\)\))2740 3840 y Fq(1)p 2739 3852
33 3 v 2739 3893 a Fs(p)2810 3818 y FG(d\020)22 b(;)456
4057 y FA(and)29 b(use)h(De\014nition)f(2.5)j(and)e(\(2.2\))i(to)f
(deduce)f(\(5.16\).)555 4165 y(Therefore,)h(from)f(\(5.16\))i(and)e
(\(5.2\),)i(if)e FG(l)i FA(is)d(large)i(enough,)f(w)m(e)h(get)h(that)
456 4361 y(\(5.17\))454 b FG(H)1216 4375 y Fv(y)1251
4386 y Fs(N)5 b Fq(+1)1385 4375 y Fv(;R)1458 4384 y Fq(0)1496
4361 y FA(\()p FG(s)p FA(\))21 b Fx(\000)f FG(H)1797
4375 y Fv(y)1832 4386 y Fs(N)5 b Fq(+1)1966 4375 y Fv(;R)2039
4384 y Fq(0)2078 4361 y FA(\(0\))26 b FG(<)f(H)2391 4375
y Fy(0)2430 4361 y FA(\()p FG(s)p FA(\))20 b(+)2664 4299
y FG(\022)p 2664 4340 46 4 v 2664 4423 a FA(2)2735 4361
y FG(:)555 4540 y FA(Notice)h(no)m(w)g(that,)i(b)m(y)d(\(2.7\),)k
(\(5.2\))e(and)e(the)h(de\014nition)d(of)i FG(s)2625
4555 y Fv(l)2671 4540 y FA(giv)m(en)g(in)f(Lemma)h(2.20,)456
4648 y(w)m(e)30 b(ha)m(v)m(e)i(that)991 4841 y FG(s)1034
4855 y Fv(R)1087 4864 y Fq(0)1151 4841 y Fx(\025)25 b(\000)p
FA(1)20 b(+)1484 4780 y(const)p 1484 4820 208 4 v 1498
4929 a FG(R)1568 4881 y Fy(1)p Fv(=p)1567 4955 y Fy(0)1727
4841 y FA(=)25 b Fx(\000)p FA(1)20 b(+)2060 4780 y(const)15
b FG(\022)2328 4747 y Fy(1)p Fv(=)p Fy(p)p 2060 4820
382 4 v 2181 4908 a FG(l)2210 4881 y Fy(2)p Fv(=p)2477
4841 y FG(>)25 b Fx(\000)p FA(1)20 b(+)g FG(s)2843 4856
y Fv(l)2884 4841 y FG(:)456 5067 y FA(In)29 b(particular,)g(the)i
(function)1575 5216 y FG(h)1627 5230 y Fv(y)1662 5241
y Fs(N)5 b Fq(+1)1797 5230 y Fv(;R)1870 5239 y Fq(0)1909
5216 y FA(\()p FG(s)p FA(\))20 b Fx(\000)g FG(h)2185
5231 y Fv(l)2211 5216 y FA(\()p FG(s)p FA(\))p eop
%%Page: 51 51
51 50 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(51)456
450 y FA(is)29 b(de\014ned)g(for)h(an)m(y)h FG(s)f FA(so)h(that)1233
672 y FG(s)1276 686 y Fv(R)1329 695 y Fq(0)1408 672 y
FA(=)40 b(max)q Fx(f)p FG(s)1777 686 y Fv(R)1830 695
y Fq(0)1869 672 y FG(;)30 b Fx(\000)p FA(1)20 b(+)g FG(s)2194
687 y Fv(l)2220 672 y Fx(g)41 b(\024)25 b FG(s)40 b Fx(\024)25
b FA(1)15 b FG(:)456 894 y FA(Also,)30 b(if)f FG(c)808
908 y Ft(\003)873 894 y FG(>)c FA(0)31 b(is)e(suitably)g(small)g(\(p)s
(ossibly)f(in)h(dep)s(endence)g(also)h(of)h FG(\022)3028
908 y Fy(0)3067 894 y FA(\))g(and)1475 1126 y FG(s)1518
1140 y Fv(R)1571 1149 y Fq(0)1634 1126 y Fx(\024)25 b
FG(s)g Fx(\024)g(\000)p FA(1)c(+)f FG(c)2161 1140 y Ft(\003)2200
1126 y FG(=l)2274 1088 y Fy(1)p Fv(=p)2400 1126 y FG(;)456
1347 y FA(w)m(e)29 b(infer)f(from)h(\(2.6\),)i(the)e(de\014nition)e(of)
i FG(h)1964 1362 y Fv(l)2020 1347 y FA(giv)m(en)g(in)f(Lemma)h(2.20)i
(and)d(the)i(one)f(of)456 1455 y FG(h)508 1469 y Fv(s)541
1478 y Fq(0)575 1469 y Fv(;R)683 1455 y FA(giv)m(en)h(on)g(page)i(10)f
(that)1357 1677 y FG(h)1409 1691 y Fv(y)1444 1702 y Fs(N)5
b Fq(+1)1579 1691 y Fv(;R)1652 1700 y Fq(0)1690 1677
y FA(\()p FG(s)p FA(\))21 b Fx(\000)f FG(h)1967 1692
y Fv(l)1993 1677 y FA(\()p FG(s)p FA(\))26 b(=)1203 1895
y(=)83 b Fx(\000)p FG(h)1480 1909 y Fy(0)1519 1895 y
FA(\()p FG(s)1597 1909 y Fv(R)1655 1895 y FA(\))21 b
Fx(\000)1832 1810 y Fu(b)1814 1833 y FG(C)1879 1847 y
Fy(0)p 1811 1874 109 4 v 1811 1957 a FG(R)1880 1971 y
Fy(0)1945 1895 y FA(\()p FG(s)f Fx(\000)g FG(s)2177 1909
y Fv(R)2230 1918 y Fq(0)2269 1895 y FA(\))g(+)1357 2148
y(+)p FG(h)1480 2162 y Fy(0)1519 2148 y FA(\()p FG(s)1597
2163 y Fv(l)1644 2148 y Fx(\000)f FA(1\))i(+)p 1936 2013
72 4 v 1936 2086 a FG(C)2008 2100 y Fy(2)p 1936 2127
111 4 v 1977 2210 a FG(l)2072 2074 y Fu(\000)2114 2148
y FA(\(1)g(+)f FG(s)p FA(\))2384 2110 y Fv(p)2444 2148
y Fx(\000)g FG(s)2578 2103 y Fv(p)2578 2177 y(l)2617
2074 y Fu(\001)2684 2148 y Fx(\024)1203 2387 y(\024)83
b(\000)1469 2326 y FA(1)p 1438 2366 109 4 v 1438 2449
a FG(R)1507 2463 y Fy(0)1576 2387 y FA(+)20 b FG(e)1709
2350 y Ft(\000)12 b Fy(const)h Fv(l)1995 2387 y FA(+)p
2096 2253 72 4 v 2096 2326 a FG(C)2168 2340 y Fy(2)2222
2326 y FG(c)2261 2281 y Fv(p)2261 2340 y Ft(\003)p 2096
2366 206 4 v 2164 2449 a FG(l)2193 2423 y Fy(2)2336 2387
y Fx(\024)1203 2640 y(\024)83 b(\000)1438 2579 y FG(C)7
b(\022)1553 2593 y Fy(0)p 1438 2619 154 4 v 1480 2703
a FG(l)1509 2676 y Fy(2)1621 2640 y FA(+)20 b FG(e)1754
2603 y Ft(\000)12 b Fy(const)h Fv(l)2040 2640 y FA(+)p
2141 2506 72 4 v 2141 2579 a FG(C)2213 2593 y Fy(2)2267
2579 y FG(c)2306 2534 y Fv(p)2306 2593 y Ft(\003)p 2141
2619 206 4 v 2210 2703 a FG(l)2239 2676 y Fy(2)2397 2640
y Fx(\024)40 b FA(0)15 b FG(;)456 2888 y FA(that)31 b(is)456
3109 y(\(5.18\))258 b FG(h)996 3123 y Fv(y)1031 3134
y Fs(N)5 b Fq(+1)1166 3123 y Fv(;R)1239 3132 y Fq(0)1278
3109 y FA(\()p FG(s)p FA(\))26 b Fx(\024)e FG(h)1564
3124 y Fv(l)1591 3109 y FA(\()p FG(s)p FA(\))30 b(for)h(an)m(y)f
FG(s)25 b Fx(2)g FA([)p FG(s)2267 3123 y Fv(R)2320 3132
y Fq(0)2359 3109 y FG(;)30 b Fx(\000)p FA(1)21 b(+)f
FG(c)2681 3123 y Ft(\003)2720 3109 y FG(=l)2794 3076
y Fy(1)p Fv(=p)2905 3109 y FA(],)456 3331 y(pro)m(vided)29
b(that)i FG(c)1066 3345 y Ft(\003)1136 3331 y FA(is)e(small)g(enough.)
40 b(Analogously)-8 b(,)31 b(one)f(can)h(sho)m(w)f(that)456
3554 y(\(5.19\))338 b FG(h)1076 3568 y Fv(y)1111 3579
y Fs(N)5 b Fq(+1)1246 3568 y Fv(;R)1319 3577 y Fq(0)1358
3554 y FA(\()p FG(s)p FA(\))25 b Fx(\025)g FG(h)1644
3569 y Fv(l)1671 3554 y FA(\()p FG(s)p FA(\))30 b(for)g(an)m(y)h
FG(s)25 b Fx(2)g FA([1)c Fx(\000)f FG(c)2500 3568 y Ft(\003)2540
3554 y FG(=l)2614 3521 y Fy(1)p Fv(=p)2724 3554 y FG(;)31
b FA(1].)456 3776 y(F)-8 b(rom)25 b(\(5.18\),)j(\(5.19\))f(and)e(the)g
(de\014nitions)d(of)j FG(H)2160 3790 y Fv(y)2195 3801
y Fs(N)5 b Fq(+1)2329 3790 y Fv(;R)2402 3799 y Fq(0)2465
3776 y FA(and)25 b FG(H)2713 3791 y Fv(l)2763 3776 y
FA(\(see)h(pages)f(13)h(and)456 3885 y(25\),)31 b(w)m(e)g(deduce)f
(that)h(the)g(maxim)m(um)e(of)i(the)f(function)1254 4106
y([)p FG(s)1322 4120 y Fv(R)1375 4129 y Fq(0)1414 4106
y FG(;)g FA(1])c Fx(3)f FG(s)40 b Fx(7!)25 b FG(H)1926
4120 y Fv(y)1961 4131 y Fs(N)5 b Fq(+1)2095 4120 y Fv(;R)2168
4129 y Fq(0)2207 4106 y FA(\()p FG(s)p FA(\))20 b Fx(\000)g
FG(H)2507 4121 y Fv(l)2533 4106 y FA(\()p FG(s)p FA(\))456
4343 y(o)s(ccurs)47 b(for)h Fx(j)p FG(s)p Fx(j)55 b(\024)f
FA(1)32 b Fx(\000)g FG(c)1403 4357 y Ft(\003)1443 4343
y FG(=l)1517 4310 y Fy(1)p Fv(=p)1627 4343 y FA(.)94
b(F)-8 b(or)49 b(these)f(v)-5 b(alues)47 b(of)h FG(s)p
FA(,)53 b(estimate)48 b(\(2.61\))j(in)456 4451 y(Lemma)30
b(2.20)i(implies)c(that)1138 4709 y FG(H)1214 4723 y
Fy(0)1253 4709 y FA(\()p FG(s)p FA(\))84 b Fx(\024)f
FG(H)1680 4724 y Fv(l)1705 4709 y FA(\()p FG(s)p FA(\))21
b Fx(\000)1955 4647 y FA(const)p 1940 4688 238 4 v 2044
4771 a FG(l)2203 4709 y FA(log)q(\(1)g Fx(\000)f(j)p
FG(s)p Fx(j)p FA(\))26 b Fx(\024)1450 4951 y(\024)83
b FG(H)1680 4966 y Fv(l)1705 4951 y FA(\()p FG(s)p FA(\))21
b(+)1955 4889 y(const)p 1940 4930 V 2044 5013 a FG(l)2203
4951 y FA(log)2345 4889 y FG(l)2374 4856 y Fy(1)p Fv(=p)p
2345 4930 140 4 v 2375 5013 a FG(c)2414 5027 y Ft(\003)2519
4951 y Fx(\024)1450 5186 y(\024)83 b FG(H)1680 5201 y
Fv(l)1705 5186 y FA(\()p FG(s)p FA(\))21 b(+)1940 5125
y FG(\022)1983 5139 y Fy(0)p 1940 5166 83 4 v 1958 5249
a FA(2)2047 5186 y FG(;)p eop
%%Page: 52 52
52 51 bop 456 251 a Fy(52)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(pro)m(vided)27 b FG(l)k FA(is)e(suitably)e(large.)40
b(Th)m(us,)29 b(summarizing)e(the)i(ab)s(o)m(v)m(e)h(observ)-5
b(ations)29 b(and)456 558 y(using)f(\(5.17\),)33 b(w)m(e)e(ha)m(v)m(e)h
(that)1054 738 y(max)1027 803 y Fy([)p Fv(s)1080 814
y Fs(R)1126 829 y Fq(0)1164 803 y Fv(;)11 b Fy(1])1265
637 y Fu(\020)1319 738 y FG(H)1395 752 y Fv(y)1430 763
y Fs(N)5 b Fq(+1)1565 752 y Fv(;R)1638 761 y Fq(0)1696
738 y Fx(\000)20 b FG(H)1863 753 y Fv(l)1889 637 y Fu(\021)1968
738 y FA(=)873 978 y(=)373 b(max)1027 1050 y Fy([)p Ft(\000)p
Fy(1+)p Fv(c)1223 1058 y Fp(\003)1258 1050 y Fv(=l)1315
1030 y Fq(1)p Fs(=p)1414 1050 y Fv(;)11 b Fy(1)p Ft(\000)p
Fv(c)1566 1058 y Fp(\003)1602 1050 y Fv(=l)1659 1030
y Fq(1)p Fs(=p)1757 1050 y Fy(])1792 877 y Fu(\020)1846
978 y FG(H)1922 992 y Fv(y)1957 1003 y Fs(N)5 b Fq(+1)2091
992 y Fv(;R)2164 1001 y Fq(0)2223 978 y Fx(\000)20 b
FG(H)2390 993 y Fv(l)2416 877 y Fu(\021)2495 978 y FG(<)873
1236 y(<)373 b FA(max)1027 1308 y Fy([)p Ft(\000)p Fy(1+)p
Fv(c)1223 1316 y Fp(\003)1258 1308 y Fv(=l)1315 1288
y Fq(1)p Fs(=p)1414 1308 y Fv(;)11 b Fy(1)p Ft(\000)p
Fv(c)1566 1316 y Fp(\003)1602 1308 y Fv(=l)1659 1288
y Fq(1)p Fs(=p)1757 1308 y Fy(])1792 1135 y Fu(\020)1846
1236 y FG(H)1922 1250 y Fy(0)1981 1236 y Fx(\000)20 b
FG(H)2148 1251 y Fv(l)2174 1135 y Fu(\021)2248 1236 y
FA(+)2349 1174 y FG(\022)p 2349 1215 46 4 v 2349 1298
a FA(2)2425 1236 y(+)g FG(H)2592 1250 y Fv(y)2627 1261
y Fs(N)5 b Fq(+1)2761 1250 y Fv(;R)2834 1259 y Fq(0)2873
1236 y FA(\(0\))26 b Fx(\024)873 1493 y(\024)1037 1432
y FG(\022)1080 1446 y Fy(0)p 1037 1472 83 4 v 1055 1556
a FA(2)1149 1493 y(+)1250 1432 y FG(\022)p 1250 1472
46 4 v 1250 1556 a FA(2)1326 1493 y(+)19 b FG(H)1492
1507 y Fv(y)1527 1518 y Fs(N)5 b Fq(+1)1662 1507 y Fv(;R)1735
1516 y Fq(0)1773 1493 y FA(\(0\))15 b FG(:)456 1670 y
FA(Hence,)456 1822 y(\(5.20\))471 b FG(H)1233 1836 y
Fv(y)1268 1847 y Fs(N)5 b Fq(+1)1402 1836 y Fv(;R)1475
1845 y Fq(0)1513 1822 y FA(\()p FG(s)p FA(\))21 b Fx(\000)f
FG(H)1814 1836 y Fv(y)1849 1847 y Fs(N)5 b Fq(+1)1983
1836 y Fv(;R)2056 1845 y Fq(0)2094 1822 y FA(\(0\))27
b FG(<)e(H)2408 1837 y Fv(l)2433 1822 y FA(\()p FG(s)p
FA(\))c(+)f FG(\022)d(;)456 1975 y FA(for)30 b(an)m(y)g
FG(s)25 b Fx(2)g FA(\()p FG(s)998 1989 y Fv(R)1051 1998
y Fq(0)1090 1975 y FG(;)31 b FA(1].)41 b(F)-8 b(rom)31
b(\(5.20\),)i(b)m(y)d(in)m(v)m(erting)g FG(H)2393 1989
y Fv(y)2428 2000 y Fs(N)5 b Fq(+1)2562 1989 y Fv(;R)2635
1998 y Fq(0)2673 1975 y FA(,)31 b(w)m(e)g(ha)m(v)m(e)1139
2165 y FG(s)40 b(<)g(g)1376 2179 y Fv(y)1411 2190 y Fs(N)5
b Fq(+1)1546 2179 y Fv(;R)1619 2188 y Fq(0)1657 2064
y Fu(\020)1712 2165 y FG(H)1788 2179 y Fv(y)1823 2190
y Fs(N)g Fq(+1)1957 2179 y Fv(;R)2030 2188 y Fq(0)2068
2165 y FA(\(0\))22 b(+)d FG(H)2371 2180 y Fv(l)2397 2165
y FA(\()p FG(s)p FA(\))i(+)e FG(\022)2667 2064 y Fu(\021)2736
2165 y FG(;)456 2344 y FA(for)30 b(an)m(y)g FG(s)25 b
Fx(2)g FA(\()p FG(s)998 2358 y Fv(R)1051 2367 y Fq(0)1090
2344 y FG(;)15 b FA(1])31 b(and)f(so,)h(for)f FG(s)25
b FA(:=)g FG(g)1916 2359 y Fv(l)1943 2344 y FA(\()p FG(d)2025
2358 y Fy(1)2065 2344 y FA(\()p FG(x)p FA(\))c Fx(\000)f
FG(\022)s FA(\),)30 b(w)m(e)h(get)988 2525 y FG(g)1031
2540 y Fv(l)1057 2525 y FA(\()p FG(d)1139 2539 y Fy(1)1179
2525 y FA(\()p FG(x)p FA(\))21 b Fx(\000)f FG(\022)s
FA(\))40 b FG(<)g(g)1688 2539 y Fv(y)1723 2550 y Fs(N)5
b Fq(+1)1858 2539 y Fv(;R)1931 2548 y Fq(0)1970 2424
y Fu(\020)2024 2525 y FG(d)2071 2539 y Fy(1)2111 2525
y FA(\()p FG(x)p FA(\))21 b(+)f FG(H)2421 2539 y Fv(y)2456
2550 y Fs(N)5 b Fq(+1)2590 2539 y Fv(;R)2663 2548 y Fq(0)2701
2525 y FA(\(0\))2816 2424 y Fu(\021)2887 2525 y FG(:)456
2705 y FA(Therefore,)30 b(recalling)f(also)h(\(5.15\),)1291
2884 y FG(g)1334 2898 y Fv(y)1369 2909 y Fs(N)5 b Fq(+1)1504
2898 y Fv(;R)1577 2907 y Fq(0)1616 2783 y Fu(\020)1670
2884 y Fx(j)p FG(x)20 b Fx(\000)g FG(y)s Fx(j)h(\000)f
FG(r)i FA(+)e FG(H)2273 2898 y Fv(y)2308 2909 y Fs(N)5
b Fq(+1)2442 2898 y Fv(;R)2515 2907 y Fq(0)2554 2884
y FA(\(0\))2669 2783 y Fu(\021)2750 2884 y FA(=)1138
3083 y(=)82 b FG(g)1334 3097 y Fv(y)1369 3108 y Fs(N)5
b Fq(+1)1504 3097 y Fv(;R)1577 3106 y Fq(0)1616 2982
y Fu(\020)1670 3083 y FG(d)1717 3097 y Fy(1)1757 3083
y FA(\()p FG(x)p FA(\))21 b(+)f FG(H)2067 3097 y Fv(y)2102
3108 y Fs(N)5 b Fq(+1)2236 3097 y Fv(;R)2309 3106 y Fq(0)2347
3083 y FA(\(0\))2462 2982 y Fu(\021)2543 3083 y FG(>)1138
3246 y(>)82 b(g)1334 3261 y Fv(l)1361 3246 y FA(\()p
FG(d)1443 3260 y Fy(1)1483 3246 y FA(\()p FG(x)p FA(\))21
b Fx(\000)e FG(\022)s FA(\))25 b Fx(\025)1138 3381 y(\025)82
b FG(g)1334 3396 y Fv(l)1361 3381 y FA(\()p FG(d)1443
3395 y Fy(2)1483 3381 y FA(\()p FG(x)p FA(\)\))26 b(=)1138
3516 y(=)82 b FG(g)1334 3531 y Fv(l)1361 3516 y FA(\()p
FG(x)1448 3530 y Fv(N)1535 3516 y FA(+)20 b FG(\022)s
FA(\))15 b FG(:)456 3668 y FA(This)28 b(completes)j(the)f(pro)s(of)g
(of)h(\(5.3\).)555 3820 y(By)g(\(5.3\))h(and)e(\(5.1\),)i(w)m(e)f(ha)m
(v)m(e)h(that)456 3928 y(\(5.21\))510 4043 y FG(g)553
4061 y Fj(S)-6 b Fy(\()p Fv(Y)t(;R)741 4070 y Fq(0)769
4061 y Fy(\))801 4043 y FA(\()p FG(x)p FA(\))41 b FG(>)f(u)p
FA(\()p FG(x)p FA(\))31 b(for)f(an)m(y)h FG(Y)45 b Fx(2)1784
4025 y Fu(e)1775 4043 y Fi(O)p FA(,)30 b(pro)m(vided)f
Fx(j)p FG(x)2350 4010 y Ft(0)2374 4043 y Fx(j)c(2)g FA(\()p
FG(l)r(;)15 b(C)2686 4010 y Ft(0)2710 4043 y FG(l)r FA(\))30
b(and)g Fx(j)p FG(x)3058 4057 y Fv(N)3125 4043 y Fx(j)c(\024)f
FG(C)3344 4010 y Ft(0)3367 4043 y FG(l)r FA(.)555 4219
y(Let)d(no)m(w)895 4196 y Fu(e)889 4219 y FA(\004)f(b)s(e)g(the)h(set)g
(of)f(\()p Fi(x)p FG(;)15 b(u)p FA(\()p Fi(x)p FA(\)\)'s)24
b(describ)s(ed)19 b(in)h(the)i(statemen)m(t)h(of)e(Lemma)h(5.1.)456
4339 y(Let)30 b(us)g(also)g(de\014ne)g(\004)25 b(:=)g
FG(\031)1439 4353 y Fv(e)1472 4364 y Fs(N)1539 4316 y
Fu(e)1534 4339 y FA(\004)k(and)h Fi(O)25 b FA(:=)g FG(\031)2068
4353 y Fv(e)2101 4364 y Fs(N)2172 4321 y Fu(e)2163 4339
y Fi(O)o FA(.)41 b(Of)30 b(course,)515 4521 y Fi(O)40
b FA(=)735 4420 y Fu(n)796 4521 y FG(Y)45 b FA(=)25 b(\()p
FG(y)1073 4483 y Ft(0)1097 4521 y FG(;)15 b FA(0)p FG(;)g(y)1267
4535 y Fv(N)7 b Fy(+1)1425 4521 y FA(\))25 b Fx(2)g Fw(R)1631
4483 y Fv(N)7 b Fy(+1)1835 4521 y FA(suc)m(h)30 b(that)41
b Fx(j)p FG(y)2320 4483 y Ft(0)2343 4521 y Fx(j)26 b(\024)f
FG(l)r(=C)2636 4483 y Ft(00)2693 4521 y FG(;)56 b Fx(j)p
FG(y)2844 4535 y Fv(N)7 b Fy(+1)3001 4521 y Fx(j)26 b(\024)f
FA(1)p FG(=)p FA(4)3283 4420 y Fu(o)3360 4521 y FG(;)456
4695 y FA(therefore)456 4848 y(\(5.22\))840 b Fi(L)1581
4810 y Fv(N)1649 4848 y FA(\()p Fi(O)p FA(\))25 b Fx(\025)40
b FA(const)16 b FG(l)2176 4810 y Fv(N)7 b Ft(\000)p Fy(1)2349
4848 y FG(:)456 5000 y FA(F)-8 b(or)34 b(an)m(y)f FG(Y)50
b Fx(2)30 b Fi(O)p FA(,)j(from)g(\(5.1\))i(and)e(the)g(fact)h(that)g
Fw(S)o FA(\()p FG(Y)5 b(;)15 b(R)2535 5014 y Fy(0)2569
5000 y FA(\))33 b(tak)m(es)i(v)-5 b(alue)32 b(1)i(on)f(the)456
5108 y(b)s(oundary)26 b(of)j(its)f(domain)f(of)h(de\014nition,)f(w)m(e)
i(kno)m(w)g(that)g Fw(S)n FA(\()p FG(Y)35 b Fx(\000)20
b FG(te)2848 5122 y Fv(N)2915 5108 y FG(;)15 b(R)3024
5122 y Fy(0)3064 5108 y FA(\))29 b(is)e(b)s(elo)m(w)456
5216 y(the)32 b(graph)g(of)g FG(u)g FA(in)f(the)i(in)m(tersection)f(b)s
(et)m(w)m(een)h Fx(fj)p FG(x)2295 5183 y Ft(0)2319 5216
y Fx(j)28 b(\024)g FG(C)2543 5183 y Ft(0)2566 5216 y
FG(l)r Fx(g)22 b(\002)f(fj)p FG(x)2876 5230 y Fv(N)2944
5216 y Fx(j)29 b(\024)f FG(C)3169 5183 y Ft(0)3191 5216
y FG(l)r Fx(g)33 b FA(and)p eop
%%Page: 53 53
53 52 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(53)456
450 y FA(the)29 b(domain)f(of)i(de\014nition)d(of)j Fw(S)n
FA(\()p FG(Y)35 b Fx(\000)20 b FG(te)1887 464 y Fv(N)1954
450 y FG(;)15 b(R)2063 464 y Fy(0)2103 450 y FA(\),)30
b(pro)m(vided)e(that)i FG(t)f FA(is)g(large)g(enough.)456
558 y(Also,)h(b)m(y)h(lo)s(oking)f(at)h(the)h(construction)e(of)37
b(\026)-51 b FG(y)34 b FA(on)c(page)i(48,)g(the)f(reader)g(ma)m(y)g
(easily)456 666 y(con)m(vince)k(herself)g(that)h(b)m(y)f(decreasing)g
FG(t)p FA(,)i(there)f(will)d(b)s(e)i(a)h(suitable)e FG(t)2996
633 y Ft(\003)3070 666 y FA(for)i(whic)m(h)456 774 y
Fw(S)n FA(\()p FG(Y)f Fx(\000)20 b FG(t)759 741 y Ft(\003)798
774 y FG(e)840 788 y Fv(N)907 774 y FG(;)15 b(R)1016
788 y Fy(0)1056 774 y FA(\))35 b(touc)m(hes)f(for)g(the)h(\014rst)e
(time)h(the)g(graph)f(of)h FG(u)p FA(,)i(sa)m(y)e(at)h(the)f(p)s(oin)m
(t)479 871 y Fu(e)456 894 y FG(X)7 b FA(.)69 b(W)-8 b(e)41
b(denote)g(b)m(y)1244 875 y Fu(e)1237 894 y Fi(G)f FA(the)g(set)g(of)g
(suc)m(h)g(touc)m(hing)g(p)s(oin)m(ts)2674 871 y Fu(e)2651
894 y FG(X)7 b FA('s)40 b(and)f(de\014ne)g(also)456 1003
y Fi(G)25 b FA(:=)g FG(\031)719 1017 y Fv(e)752 1028
y Fs(N)821 985 y Fu(e)814 1003 y Fi(G)p FA(.)555 1111
y(W)-8 b(e)32 b(claim)d(that)456 1259 y(\(5.23\))1120
b Fi(G)26 b Fx(\022)f FA(\004)15 b FG(:)456 1419 y FA(F)-8
b(or)21 b(pro)m(ving)e(this,)j(tak)m(e)f(an)m(y)1494
1396 y Fu(e)1470 1419 y FG(X)33 b Fx(2)1671 1401 y Fu(e)1664
1419 y Fi(G)20 b FA(b)s(e)g(a)g(touc)m(hing)g(p)s(oin)m(t)g(b)s(et)m(w)
m(een)h Fw(S)n FA(\()p FG(Y)35 b Fx(\000)20 b FG(t)3161
1386 y Ft(\003)3200 1419 y FG(e)3242 1433 y Fv(N)3309
1419 y FG(;)15 b(R)3418 1433 y Fy(0)3458 1419 y FA(\))456
1527 y(and)33 b(the)i(graph)e(of)i FG(u)p FA(,)g(as)g(describ)s(ed)d
(ab)s(o)m(v)m(e.)54 b(Let)35 b(us)e(observ)m(e)i(that,)h(since)e
FG(u)p FA(\(0\))f(=)456 1647 y(0,)46 b(the)c(\014rst)g(touc)m(hing)g
(prop)s(ert)m(y)g(of)1846 1624 y Fu(e)1822 1647 y FG(X)50
b FA(implies)39 b(that)k(if)2598 1624 y(\025)2572 1647
y FG(X)52 b FA(=)45 b(\(0)p FG(;)15 b(:)g(:)g(:)j(;)j
FA(\025)-51 b FG(x)3150 1661 y Fv(N)3217 1647 y FG(;)15
b FA(0\))47 b Fx(2)456 1755 y Fw(S)n FA(\()p FG(Y)35
b Fx(\000)20 b FG(t)759 1722 y Ft(\003)798 1755 y FG(e)840
1769 y Fv(N)907 1755 y FG(;)15 b(R)1016 1769 y Fy(0)1056
1755 y FA(\))q(,)30 b(then)36 b(\025)-51 b FG(x)1406
1769 y Fv(N)1499 1755 y Fx(\024)25 b FA(0,)31 b(hence)1556
1896 y Fu(e)1547 1915 y Fi(O)25 b Fx(3)g FG(Y)40 b Fx(\000)20
b FG(t)1944 1877 y Ft(\003)1983 1915 y FG(e)2025 1929
y Fv(N)2118 1915 y FA(=:)2251 1892 y Fu(e)2239 1915 y
FG(Y)36 b(:)456 2063 y FA(F)-8 b(rom)30 b(this)g(and)g(\(5.21\),)i(w)m
(e)f(gather)g(that)456 2211 y(\(5.24\))1106 b Fx(j)s
Fu(e)-54 b FG(x)1869 2173 y Ft(0)1892 2211 y Fx(j)26
b(\024)f FG(l)17 b(:)456 2359 y FA(W)-8 b(e)31 b(no)m(w)g(sho)m(w)f
(that)456 2507 y(\(5.25\))285 b Fu(e)-54 b FG(x)30 b
FA(is)f(in)g(the)i(in)m(terior)e(of)i Fx(fj)p FG(x)1951
2474 y Ft(0)1975 2507 y Fx(j)25 b(\024)g FG(C)2193 2474
y Ft(0)2216 2507 y FG(l)r Fx(g)c(\002)e(fj)p FG(x)2523
2521 y Fv(N)2591 2507 y Fx(j)26 b(\024)f FG(C)2810 2474
y Ft(0)2833 2507 y FG(l)r Fx(g)30 b FA(.)456 2655 y(Note)h(that,)h
(thanks)d(to)j(\(5.24\),)g(this)e(will)d(b)s(e)j(pro)m(v)m(ed)h(if)e(w)
m(e)i(sho)m(w)f(that)h Fx(j)s Fu(e)-54 b FG(x)3077 2669
y Fv(N)3145 2655 y Fx(j)25 b FG(<)g(C)3363 2622 y Ft(0)3386
2655 y FG(l)r FA(.)555 2763 y(Let)35 b(us)f(\014rst)g(sho)m(w)h(that)k
Fu(e)-55 b FG(x)1518 2777 y Fv(N)1618 2763 y FG(>)32
b Fx(\000)p FG(C)1864 2730 y Ft(0)1887 2763 y FG(l)r
FA(.)53 b(If,)36 b(b)m(y)f(con)m(tradiction,)k Fu(e)-54
b FG(x)2883 2777 y Fv(N)2982 2763 y FA(=)32 b Fx(\000)p
FG(C)3228 2730 y Ft(0)3251 2763 y FG(l)r FA(,)k(w)m(e)456
2871 y(gather)31 b(from)f(\(2.7\))i(and)e(\(5.1\))i(that)1208
3101 y Fx(\000)p FA(1)21 b(+)1461 3040 y(const)15 b FG(\022)1729
2992 y Fy(1)p Fv(=p)1726 3066 y Fy(0)p 1445 3080 394
4 v 1573 3167 a FG(l)1602 3141 y Fy(2)p Fv(=p)1932 3101
y Fx(\024)82 b(\000)p FA(1)21 b(+)2338 3040 y(const)p
2323 3080 238 4 v 2352 3189 a FG(R)2422 3141 y Fy(1)p
Fv(=p)2421 3215 y Fy(0)2596 3101 y Fx(\024)1932 3308
y(\024)82 b FG(s)2128 3322 y Fv(R)2181 3331 y Fq(0)2245
3308 y Fx(\024)1932 3443 y(\024)g FG(g)2128 3476 y Fj(S)-5
b Fy(\()2208 3460 y Fk(e)2199 3476 y Fv(Y)10 b(;R)2323
3485 y Fq(0)2357 3476 y Fy(\))2389 3443 y FA(\()s Fu(e)-54
b FG(x)p FA(\))26 b(=)1932 3596 y(=)82 b FG(u)p FA(\()s
Fu(e)-54 b FG(x)q FA(\))25 b Fx(\024)1932 3731 y(\024)82
b FG(g)2128 3746 y Fv(l)2155 3731 y FA(\()s Fu(e)-54
b FG(x)2242 3745 y Fv(N)2330 3731 y FA(+)20 b FG(\022)s
FA(\))k Fx(\024)1932 3921 y(\024)82 b FG(g)2128 3936
y Fv(l)2155 3820 y Fu(\020)2229 3921 y Fx(\000)2330 3859
y FG(C)2402 3826 y Ft(0)2440 3859 y FG(l)p 2330 3900
139 4 v 2377 3983 a FA(2)2479 3820 y Fu(\021)2559 3921
y FA(=)1932 4102 y(=)g Fx(\000)p FA(1)21 b(+)f FG(e)2355
4064 y Ft(\000)12 b Fy(const)h Fv(l)2636 4102 y FG(;)456
4250 y FA(whic)m(h)33 b(is)h(a)h(con)m(tradiction)f(for)h(large)g
FG(l)r FA(.)53 b(This)33 b(sho)m(ws)i(that)j Fu(e)-54
b FG(x)2662 4264 y Fv(N)2761 4250 y FG(>)33 b Fx(\000)p
FG(C)3008 4217 y Ft(0)3030 4250 y FG(l)k FA(and)d(th)m(us)456
4358 y(w)m(e)c(no)m(w)g(sho)m(w)f(that)34 b Fu(e)-55
b FG(x)1250 4372 y Fv(N)1343 4358 y FG(<)25 b(C)1511
4325 y Ft(0)1534 4358 y FG(l)r FA(,)30 b(in)e(order)h(to)i(complete)f
(the)g(pro)s(of)f(of)h(\(5.25\).)42 b(That)459 4478 y
Fu(e)-54 b FG(x)508 4492 y Fv(N)600 4478 y FG(<)25 b(C)768
4445 y Ft(0)791 4478 y FG(l)f FA(will)c(b)s(e)i(actually)g(obtained)g
(from)g(the)g(fact)i(that)f(the)f(domain)g(of)g Fw(S)o
FA(\()3204 4455 y Fu(e)3193 4478 y FG(Y)14 b(;)h(R)3369
4492 y Fy(0)3409 4478 y FA(\))456 4586 y(is)22 b(b)s(elo)m(w)g(the)i(h)
m(yp)s(erplane)d Fx(f)p FG(x)1497 4600 y Fv(N)1590 4586
y Fx(\024)k FG(l)r(=)p FA(2)p Fx(g)p FA(.)39 b(T)-8 b(o)24
b(pro)m(v)m(e)g(this,)g(\014rst)e(note)i(that,)i(b)m(y)d(\(5.11\),)456
4693 y(w)m(e)30 b(ha)m(v)m(e)i(that)456 4867 y(\(5.26\))986
b Fu(e)-55 b FG(y)1713 4881 y Fv(N)1805 4867 y Fx(\024)25
b(\000)p FG(r)d FA(+)2136 4805 y FG(\022)p 2136 4846
46 4 v 2136 4929 a FA(8)2207 4867 y FG(;)456 5066 y FA(Also,)30
b(if)f Fi(x)i FA(is)e(in)g(the)i(domain)e(of)i Fw(S)n
FA(\()1715 5043 y Fu(e)1704 5066 y FG(Y)14 b(;)h(R)1880
5080 y Fy(0)1920 5066 y FA(\))q(,)30 b(w)m(e)h(ha)m(v)m(e)h(that)1112
5214 y Fi(x)1144 5228 y Fv(N)1237 5214 y Fx(\024)d Fu(e)-55
b FG(y)1378 5228 y Fv(N)1465 5214 y FA(+)20 b FG(H)1635
5231 y Fk(e)-42 b Fv(y)1667 5242 y Fs(N)5 b Fq(+1)1801
5231 y Fv(;R)1874 5240 y Fq(0)1913 5214 y FA(\(1\))21
b Fx(\000)f FG(H)2216 5228 y Fy(0)2255 5214 y FA(\()t
Fu(e)-55 b FG(y)2335 5228 y Fv(N)7 b Fy(+1)2492 5214
y FA(\))21 b(+)f FG(R)2708 5228 y Fy(0)2762 5214 y FG(:)p
eop
%%Page: 54 54
54 53 bop 456 251 a Fy(54)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(Th)m(us,)29 b(\(5.26\),)k(\(5.5\))f(and)e(\(2.16\))j
(yield)28 b(that)1373 646 y Fi(x)1405 660 y Fv(N)1498
646 y Fx(\024)40 b FA(const)16 b(\(1)21 b(+)f(log)c FG(R)2225
660 y Fy(0)2265 646 y FA(\))25 b Fx(\024)2439 584 y FG(l)p
2431 625 46 4 v 2431 708 a FA(2)2502 646 y FG(;)456 845
y FA(hence)30 b(the)h(domain)e(of)h Fw(S)o FA(\()1394
822 y Fu(e)1383 845 y FG(Y)14 b(;)h(R)1559 859 y Fy(0)1599
845 y FA(\))31 b(is)e(b)s(elo)m(w)h Fx(f)p FG(x)2111
859 y Fv(N)2204 845 y Fx(\024)25 b FG(l)r(=)p FA(2)p
Fx(g)31 b FA(and)f(therefore)k Fu(e)-54 b FG(x)3106 859
y Fv(N)3198 845 y FG(<)25 b(C)3366 812 y Ft(0)3389 845
y FG(l)r FA(.)555 993 y(This)k(ends)g(the)i(pro)s(of)f(of)g(\(5.25\).)
555 1141 y(Prop)s(osition)f(2.13)j(and)d(\(5.25\))k(yield)c(that)456
1289 y(\(5.27\))1002 b Fx(j)p FG(u)p FA(\()s Fu(e)-54
b FG(x)q FA(\))p Fx(j)26 b(\024)f FA(1)p FG(=)p FA(2)15
b FG(:)555 1584 y FA(W)-8 b(e)32 b(no)m(w)e(notice)h(that,)g(from)f
(\(5.27\))j(and)c(\(2.20\),)1037 1775 y Fx(\000)1118
1714 y FA(1)p 1118 1755 V 1118 1838 a(2)1256 1775 y Fx(\024)83
b FG(u)p FA(\()s Fu(e)-54 b FG(x)p FA(\))26 b(=)1256
1941 y(=)83 b FG(g)1453 1975 y Fj(S)-6 b Fy(\()1532 1958
y Fk(e)1523 1975 y Fv(Y)10 b(;R)1647 1984 y Fq(0)1681
1975 y Fy(\))1713 1941 y FA(\()s Fu(e)-54 b FG(x)p FA(\))26
b(=)1256 2131 y(=)83 b FG(g)1456 2148 y Fk(e)-42 b Fv(y)1488
2159 y Fs(N)5 b Fq(+1)1623 2148 y Fv(;R)1696 2157 y Fq(0)1734
2030 y Fu(\020)1789 2131 y FG(H)1865 2145 y Fy(0)1904
2131 y FA(\()t Fu(e)-55 b FG(y)1984 2145 y Fv(N)7 b Fy(+1)2141
2131 y FA(\))20 b(+)g Fx(j)s Fu(e)-54 b FG(x)21 b Fx(\000)i
Fu(e)-54 b FG(y)s Fx(j)20 b(\000)g FG(R)2729 2145 y Fy(0)2768
2030 y Fu(\021)2838 2131 y FG(;)456 2302 y FA(and)29
b(so,)i(b)m(y)g(De\014nition)d(2.11,)1047 2449 y FG(H)1123
2463 y Fy(0)1162 2449 y FA(\()t Fu(e)-55 b FG(y)1242
2463 y Fv(N)7 b Fy(+1)1399 2449 y FA(\))21 b(+)f Fx(j)s
Fu(e)-54 b FG(x)20 b Fx(\000)k Fu(e)-55 b FG(y)s Fx(j)20
b(\000)g FG(R)1987 2463 y Fy(0)2067 2449 y Fx(\025)40
b FG(H)2257 2466 y Fk(e)-42 b Fv(y)2289 2477 y Fs(N)5
b Fq(+1)2423 2466 y Fv(;R)2496 2475 y Fq(0)2535 2449
y FA(\()p Fx(\000)p FA(1)p FG(=)p FA(2\))15 b FG(;)456
2597 y FA(from)30 b(whic)m(h)f(w)m(e)h(deduce)g(that)456
2745 y(\(5.28\))648 b Fx(j)s Fu(e)-54 b FG(x)21 b Fx(\000)i
Fu(e)-54 b FG(y)r Fx(j)41 b(\025)f FG(R)1816 2759 y Fy(0)1876
2745 y Fx(\000)35 b FA(const)41 b Fx(\025)25 b FG(R)2395
2759 y Fy(0)2434 2745 y FG(=)p FA(2)15 b FG(;)456 2893
y FA(pro)m(vided)29 b(that)j FG(l)h FA(\(and)d(so)h FG(R)1481
2907 y Fy(0)1521 2893 y FA(\))g(is)f(large)h(enough.)42
b(On)30 b(the)i(other)f(hand,)f(exploiting)456 3008 y(\(5.24\))i(and)e
(the)h(de\014nition)d(of)1568 2990 y Fu(e)1559 3008 y
Fi(O)i FA(giv)m(en)g(on)h(page)g(47,)g(w)m(e)g(ha)m(v)m(e)h(that)1241
3204 y Fx(j)s Fu(e)-54 b FG(x)1318 3166 y Ft(0)1361 3204
y Fx(\000)24 b Fu(e)-55 b FG(y)1500 3166 y Ft(0)1523
3204 y Fx(j)26 b(\024)f(j)s Fu(e)-54 b FG(x)1747 3166
y Ft(0)1770 3204 y Fx(j)21 b FA(+)f Fx(j)t Fu(e)-55 b
FG(y)1980 3166 y Ft(0)2003 3204 y Fx(j)26 b(\024)f FG(l)d
FA(+)2342 3142 y FG(l)p 2300 3183 114 4 v 2300 3266 a(C)2372
3240 y Ft(00)2449 3204 y Fx(\024)j FA(2)p FG(l)17 b(:)456
3382 y FA(Hence,)31 b(from)f(\(5.28\),)1278 3529 y Fx(j)s
Fu(e)-54 b FG(x)1355 3543 y Fv(N)1443 3529 y Fx(\000)23
b Fu(e)-55 b FG(y)1578 3543 y Fv(N)1645 3529 y Fx(j)1670
3496 y Fy(2)p 1278 3570 433 4 v 1343 3653 a Fx(j)s Fu(e)h
FG(x)21 b Fx(\000)j Fu(e)-55 b FG(y)s Fx(j)1605 3627
y Fy(2)1803 3591 y FA(=)83 b(1)35 b Fx(\000)2153 3529
y(j)s Fu(e)-54 b FG(x)2230 3496 y Ft(0)2274 3529 y Fx(\000)24
b Fu(e)-55 b FG(y)2413 3496 y Ft(0)2436 3529 y Fx(j)2461
3496 y Fy(2)p 2153 3570 348 4 v 2177 3653 a Fx(j)s Fu(e)h
FG(x)20 b Fx(\000)k Fu(e)-55 b FG(y)s Fx(j)2438 3627
y Fy(2)2536 3591 y Fx(\025)1803 3848 y(\025)83 b FA(1)20
b Fx(\000)2123 3787 y FA(16)p FG(l)2242 3754 y Fy(2)p
2123 3827 160 4 v 2148 3913 a FG(R)2218 3881 y Fy(2)2217
3939 y(0)2308 3848 y FG(;)456 4050 y FA(and,)30 b(therefore,)788
4264 y(1)20 b Fx(\000)954 4203 y FA(1)p 954 4243 46 4
v 954 4327 a(4)1025 4136 y Fu(\024)1073 4264 y Fl(\\)1154
4136 y Fu(\022)1259 4203 y(e)-54 b FG(x)20 b Fx(\000)j
Fu(e)-54 b FG(y)p 1230 4243 262 4 v 1230 4327 a Fx(j)s
Fu(e)g FG(x)21 b Fx(\000)i Fu(e)-54 b FG(y)r Fx(j)1502
4264 y FG(;)30 b(e)1599 4278 y Fv(N)1667 4136 y Fu(\023\025)1782
4159 y Fy(2)1904 4264 y Fx(\025)83 b FA(cos)2180 4227
y Fy(2)2234 4136 y Fu(\024)2282 4264 y Fl(\\)2363 4136
y Fu(\022)2468 4203 y(e)-54 b FG(x)20 b Fx(\000)k Fu(e)-55
b FG(y)p 2440 4243 V 2440 4327 a Fx(j)s Fu(e)h FG(x)20
b Fx(\000)k Fu(e)-55 b FG(y)s Fx(j)2711 4264 y FG(;)31
b(e)2809 4278 y Fv(N)2876 4136 y Fu(\023\025)3016 4264
y FA(=)1904 4523 y(=)2068 4462 y Fx(j)s Fu(e)-54 b FG(x)2145
4476 y Fv(N)2232 4462 y Fx(\000)24 b Fu(e)-55 b FG(y)2368
4476 y Fv(N)2435 4462 y Fx(j)2460 4429 y Fy(2)p 2068
4502 433 4 v 2133 4586 a Fx(j)s Fu(e)h FG(x)21 b Fx(\000)i
Fu(e)-54 b FG(y)s Fx(j)2395 4559 y Fy(2)2535 4523 y Fx(\025)1904
4781 y(\025)83 b FA(1)21 b Fx(\000)2224 4719 y FA(16)p
FG(l)2343 4686 y Fy(2)p 2224 4760 160 4 v 2249 4845 a
FG(R)2319 4814 y Fy(2)2318 4871 y(0)2409 4781 y FG(;)456
4982 y FA(that)31 b(is)456 5162 y(\(5.29\))773 b Fl(\\)1540
5034 y Fu(\022)1645 5101 y(e)-54 b FG(x)20 b Fx(\000)k
Fu(e)-55 b FG(y)p 1617 5142 262 4 v 1617 5225 a Fx(j)s
Fu(e)h FG(x)20 b Fx(\000)k Fu(e)-55 b FG(y)s Fx(j)1888
5162 y FG(;)31 b(e)1986 5176 y Fv(N)2053 5034 y Fu(\023)2161
5162 y Fx(\024)2299 5101 y FA(8)p FG(l)p 2282 5142 109
4 v 2282 5225 a(R)2351 5239 y Fy(0)2415 5162 y FG(:)p
eop
%%Page: 55 55
55 54 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(55)456
456 y FA(Moreo)m(v)m(er,)32 b(from)e(the)h(touc)m(hing)f(prop)s(ert)m
(y)g(of)2122 433 y Fu(e)2098 456 y FG(X)38 b FA(and)30
b(\(2.20\),)j(w)m(e)e(ha)m(v)m(e)g(that)920 742 y Fl(\\)1001
614 y Fu(\022)1102 681 y Fx(r)p FG(u)p FA(\()s Fu(e)-54
b FG(x)q FA(\))p 1077 721 301 4 v 1077 804 a Fx(jr)p
FG(u)p FA(\()s Fu(e)g FG(x)q FA(\))p Fx(j)1388 742 y
FG(;)31 b(e)1486 756 y Fv(N)1553 614 y Fu(\023)1703 742
y FA(=)83 b Fl(\\)1938 587 y Fu( )2045 657 y Fx(r)p FG(g)2164
690 y Fj(S)-6 b Fy(\()2243 673 y Fk(e)2234 690 y Fv(Y)10
b(;R)2358 699 y Fq(0)2392 690 y Fy(\))2424 657 y FA(\()s
Fu(e)-54 b FG(x)p FA(\))p 2020 721 553 4 v 2020 804 a
Fx(jr)p FG(g)2164 838 y Fj(S)-6 b Fy(\()2243 821 y Fk(e)2234
838 y Fv(Y)10 b(;R)2358 847 y Fq(0)2392 838 y Fy(\))2424
804 y FA(\()s Fu(e)-54 b FG(x)p FA(\))p Fx(j)2582 742
y FG(;)30 b(;)15 b(e)2719 756 y Fv(N)2787 587 y Fu(!)2885
742 y FA(=)1703 1023 y(=)83 b Fl(\\)1938 895 y Fu(\022)2043
961 y(e)-54 b FG(x)20 b Fx(\000)k Fu(e)-55 b FG(y)p 2015
1002 262 4 v 2015 1085 a Fx(j)s Fu(e)h FG(x)20 b Fx(\000)k
Fu(e)-55 b FG(y)s Fx(j)2286 1023 y FG(;)31 b(e)2384 1037
y Fv(N)2451 895 y Fu(\023)2548 1023 y FG(:)456 1282 y
FA(Therefore,)f(from)g(\(5.29\),)456 1541 y(\(5.30\))754
b Fl(\\)1521 1413 y Fu(\022)1622 1479 y Fx(r)p FG(u)p
FA(\()s Fu(e)-54 b FG(x)q FA(\))p 1597 1520 301 4 v 1597
1603 a Fx(jr)p FG(u)p FA(\()s Fu(e)g FG(x)q FA(\))p Fx(j)1908
1541 y FG(;)31 b(e)2006 1555 y Fv(N)2073 1413 y Fu(\023)2181
1541 y Fx(\024)2319 1479 y FA(8)p FG(l)p 2302 1520 109
4 v 2302 1603 a(R)2371 1617 y Fy(0)2435 1541 y FG(:)456
1800 y FA(F)-8 b(urthermore,)30 b(recalling)f(\(2.39\),)1270
2100 y FG(H)1349 2117 y Fk(e)-42 b Fv(y)1381 2128 y Fs(N)5
b Fq(+1)1515 2117 y Fv(;R)1588 2126 y Fq(0)1627 2100
y FA(\()p FG(u)p FA(\()s Fu(e)-54 b FG(x)q FA(\)\))21
b Fx(\000)e FG(H)2062 2117 y Fk(e)-42 b Fv(y)2094 2128
y Fs(N)5 b Fq(+1)2229 2117 y Fv(;R)2302 2126 y Fq(0)2340
2100 y FA(\(0\))27 b(=)1116 2275 y(=)83 b FG(H)1349 2292
y Fk(e)-42 b Fv(y)1381 2303 y Fs(N)5 b Fq(+1)1515 2292
y Fv(;R)1588 2301 y Fq(0)1627 2174 y Fu(\020)1681 2275
y FG(g)1724 2308 y Fj(S)-6 b Fy(\()1803 2292 y Fk(e)1794
2308 y Fv(Y)10 b(;R)1918 2317 y Fq(0)1953 2308 y Fy(\))1984
2275 y FA(\()s Fu(e)-54 b FG(x)q FA(\))2107 2174 y Fu(\021)2182
2275 y Fx(\000)19 b FG(H)2351 2292 y Fk(e)-42 b Fv(y)2383
2303 y Fs(N)5 b Fq(+1)2518 2292 y Fv(;R)2591 2301 y Fq(0)2629
2275 y FA(\(0\))27 b(=)1116 2511 y(=)83 b FG(H)1349 2528
y Fk(e)-42 b Fv(y)1381 2539 y Fs(N)5 b Fq(+1)1515 2528
y Fv(;R)1588 2537 y Fq(0)1627 2410 y Fu(\020)1681 2511
y FG(g)1724 2545 y Fj(S)-6 b Fy(\()1803 2528 y Fk(e)1794
2545 y Fv(Y)10 b(;R)1918 2554 y Fq(0)1953 2545 y Fy(\))1984
2511 y FA(\()s Fu(e)-54 b FG(x)q FA(\))2107 2410 y Fu(\021)2182
2511 y FA(+)p 2304 2377 72 4 v 2304 2450 a FG(C)2375
2464 y Fy(0)p 2282 2490 154 4 v 2282 2574 a FA(2)p FG(R)2396
2588 y Fy(0)2450 2511 y Fu(e)f FG(y)2494 2474 y Fy(2)2491
2534 y Fv(N)7 b Fy(+1)2663 2511 y FG(;)456 2760 y FA(th)m(us,)30
b(from)g(\(2.33\))i(and)e(\(2.35\),)1091 2964 y FG(H)1170
2981 y Fk(e)-42 b Fv(y)1202 2992 y Fs(N)5 b Fq(+1)1336
2981 y Fv(;R)1409 2990 y Fq(0)1448 2964 y FA(\()p FG(u)p
FA(\()s Fu(e)-54 b FG(x)p FA(\)\))21 b Fx(\000)f FG(H)1883
2981 y Fk(e)-42 b Fv(y)1915 2992 y Fs(N)5 b Fq(+1)2049
2981 y Fv(;R)2122 2990 y Fq(0)2161 2964 y FA(\(0\))26
b(=)937 3139 y(=)83 b FG(H)1170 3155 y Fk(e)-42 b Fv(y)1202
3166 y Fs(N)5 b Fq(+1)1336 3155 y Fv(;R)1409 3164 y Fq(0)1448
3038 y Fu(\020)1502 3139 y FG(\032)1552 3155 y Fk(e)-42
b Fv(y)1584 3166 y Fs(N)5 b Fq(+1)1718 3155 y Fv(;R)1791
3164 y Fq(0)1830 3139 y FA(\()p FG(H)1941 3153 y Fy(0)1980
3139 y FA(\()t Fu(e)-55 b FG(y)2060 3153 y Fv(N)7 b Fy(+1)2218
3139 y FA(\))20 b(+)g Fx(j)s Fu(e)-54 b FG(x)21 b Fx(\000)i
Fu(e)-55 b FG(y)s Fx(j)21 b(\000)f FG(R)2806 3153 y Fy(0)2845
3139 y FA(\))2880 3038 y Fu(\021)2955 3139 y FA(+)1091
3375 y(+)p 1193 3240 72 4 v 1193 3313 a FG(C)1265 3327
y Fy(0)p 1172 3354 154 4 v 1172 3437 a FA(2)p FG(R)1286
3451 y Fy(0)1339 3375 y Fu(e)-55 b FG(y)1383 3337 y Fy(2)1380
3397 y Fv(N)7 b Fy(+1)1563 3375 y FA(=)937 3628 y(=)83
b FG(H)1167 3642 y Fy(0)1206 3628 y FA(\()t Fu(e)-55
b FG(y)1286 3642 y Fv(N)7 b Fy(+1)1443 3628 y FA(\))21
b(+)f Fx(j)s Fu(e)-54 b FG(x)20 b Fx(\000)k Fu(e)-55
b FG(y)s Fx(j)20 b(\000)g FG(R)2031 3642 y Fy(0)2091
3628 y FA(+)p 2213 3493 72 4 v 2213 3566 a FG(C)2285
3580 y Fy(0)p 2192 3607 154 4 v 2192 3690 a FA(2)p FG(R)2306
3704 y Fy(0)2359 3628 y Fu(e)-54 b FG(y)2404 3590 y Fy(2)2401
3650 y Fv(N)7 b Fy(+1)2573 3628 y FG(:)456 3877 y FA(Hence,)31
b(b)m(y)f(\(5.5\),)1415 4081 y FG(H)1494 4097 y Fk(e)-42
b Fv(y)1526 4108 y Fs(N)5 b Fq(+1)1660 4097 y Fv(;R)1733
4106 y Fq(0)1771 4081 y FA(\()p FG(u)p FA(\()s Fu(e)-54
b FG(x)q FA(\)\))21 b Fx(\000)f FG(H)2207 4097 y Fk(e)-42
b Fv(y)2239 4108 y Fs(N)5 b Fq(+1)2373 4097 y Fv(;R)2446
4106 y Fq(0)2485 4081 y FA(\(0\))26 b(=)1261 4219 y(=)83
b Fx(j)s Fu(e)-54 b FG(x)20 b Fx(\000)k Fu(e)-55 b FG(y)s
Fx(j)20 b(\000)g FG(r)e(:)-1415 b FA(\(5.31\))456 4423
y(W)-8 b(e)31 b(no)m(w)g(claim)e(that)456 4652 y(\(5.32\))843
b Fu(e)-54 b FG(x)1578 4666 y Fv(N)1686 4652 y Fx(\024)40
b FG(H)1873 4666 y Fy(0)1912 4652 y FA(\()p FG(u)p FA(\()s
Fu(e)-54 b FG(x)p FA(\)\))21 b(+)2278 4591 y FG(\022)p
2278 4631 46 4 v 2278 4715 a FA(4)2349 4652 y FG(:)456
4885 y FA(F)-8 b(or)32 b(pro)m(ving)f(this,)g(w)m(e)h(denote)h(b)m(y)h
Fu(b)-54 b FG(x)32 b FA(the)g(in)m(tersection)f(p)s(oin)m(t)g(b)s(et)m
(w)m(een)h(the)g(sphere)456 4993 y Fx(f)p FG(g)544 5026
y Fj(S)-6 b Fy(\()623 5009 y Fk(e)614 5026 y Fv(Y)10
b(;R)738 5035 y Fq(0)773 5026 y Fy(\))830 4993 y FA(=)24
b(0)p Fx(g)32 b FA(and)d(the)i(half-line)d(from)34 b
Fu(e)-55 b FG(y)33 b FA(to)m(w)m(ards)h Fu(e)-54 b FG(x)p
FA(.)41 b(Then,)29 b(b)m(y)i(\(5.8\),)1745 5216 y Fu(b)-54
b FG(x)1794 5230 y Fv(N)1886 5216 y Fx(\024)25 b FG(\022)s(=)p
FA(8)15 b FG(:)p eop
%%Page: 56 56
56 55 bop 456 251 a Fy(56)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(W)g(e)34 b(no)m(w)f(distinguish)c(t)m(w)m(o)35
b(cases:)47 b(either)35 b Fu(b)-54 b FG(x)33 b FA(is)f(inside)f(or)i
(it)f(is)h(outside)f(the)h(sphere)456 558 y Fx(f)p FG(g)544
591 y Fj(S)-6 b Fy(\()623 575 y Fk(e)614 591 y Fv(Y)10
b(;R)738 600 y Fq(0)773 591 y Fy(\))830 558 y FA(=)24
b(0)p Fx(g)p FA(.)42 b(If)30 b(it)g(is)f(inside,)g(then)1363
725 y Fx(j)s Fu(e)-54 b FG(x)20 b Fx(\000)k Fu(e)-55
b FG(y)s Fx(j)21 b(\000)f FG(r)85 b FA(=)e Fx(j)s Fu(b)-54
b FG(x)20 b Fx(\000)j Fu(e)-54 b FG(x)p Fx(j)26 b(\025)1862
860 y(\025)83 b(j)s Fu(b)-54 b FG(x)2093 874 y Fv(N)2180
860 y Fx(\000)24 b Fu(e)-55 b FG(x)2323 874 y Fv(N)2391
860 y Fx(j)25 b(\025)1862 995 y(\025)86 b Fu(e)-54 b
FG(x)2068 1009 y Fv(N)2155 995 y Fx(\000)23 b Fu(b)-54
b FG(x)2298 1009 y Fv(N)2391 995 y Fx(\025)1862 1130
y(\025)86 b Fu(e)-54 b FG(x)2068 1144 y Fv(N)2155 1130
y Fx(\000)20 b FG(\022)s(=)p FA(8)15 b FG(:)456 1278
y FA(Th)m(us,)29 b(from)h(the)h(latter)f(estimate,)i(\(5.31\))g(and)e
(\(5.16\),)j(w)m(e)e(ha)m(v)m(e)g(that)1141 1474 y Fu(e)-54
b FG(x)1190 1488 y Fv(N)1340 1474 y Fx(\024)82 b(j)s
Fu(e)-54 b FG(x)21 b Fx(\000)i Fu(e)-54 b FG(y)s Fx(j)20
b(\000)g FG(r)j FA(+)2031 1412 y FG(\022)p 2031 1453
46 4 v 2031 1536 a FA(8)2111 1474 y(=)1340 1696 y(=)82
b FG(H)1572 1713 y Fk(e)-42 b Fv(y)1604 1724 y Fs(N)5
b Fq(+1)1743 1696 y FA(\()p FG(u)p FA(\()s Fu(e)-54 b
FG(x)p FA(\)\))21 b Fx(\000)f FG(H)2178 1713 y Fk(e)-42
b Fv(y)2210 1724 y Fs(N)5 b Fq(+1)2349 1696 y FA(\(0\))21
b(+)2586 1635 y FG(\022)p 2586 1675 V 2586 1759 a FA(8)2667
1696 y Fx(\024)1340 1936 y(\024)82 b FG(H)1569 1950 y
Fy(0)1608 1936 y FA(\()p FG(u)p FA(\()s Fu(e)-54 b FG(x)q
FA(\)\))21 b(+)1975 1874 y(2)p 2020 1801 72 4 v FG(C)2092
1888 y Fy(0)p 1975 1915 157 4 v 1999 1998 a FG(R)2068
2012 y Fy(0)2161 1936 y FA(+)2262 1874 y FG(\022)p 2262
1915 46 4 v 2262 1998 a FA(8)2333 1936 y FG(:)456 2129
y FA(Therefore,)34 b(if)f FG(l)i FA(\(and)f(so)g FG(R)1446
2143 y Fy(0)1485 2129 y FA(\))g(is)f(large)h(enough,)f(\(5.32\))j
(follo)m(ws)d(in)f(this)h(case.)52 b(Let)456 2236 y(us)33
b(no)m(w)g(deal)h(with)e(the)i(case)h(in)d(whic)m(h)k
Fu(b)-54 b FG(x)33 b FA(is)g(outside)g(the)h(sphere)f
Fx(f)p FG(g)2935 2270 y Fj(S)-5 b Fy(\()3015 2253 y Fk(e)3006
2270 y Fv(Y)10 b(;R)3130 2279 y Fq(0)3164 2270 y Fy(\))3227
2236 y FA(=)30 b(0)p Fx(g)p FA(.)456 2363 y(By)g(\(5.30\),)j(w)m(e)e
(infer)e(in)g(this)g(case)j(that)i Fu(b)-54 b FG(x)1955
2377 y Fv(N)2047 2363 y Fx(\025)28 b Fu(e)-54 b FG(x)2195
2377 y Fv(N)2292 2363 y FA(and)30 b(that)1071 2511 y
Fu(b)-55 b FG(x)1119 2525 y Fv(N)1207 2511 y Fx(\000)23
b Fu(e)-54 b FG(x)1350 2525 y Fv(N)1500 2511 y FA(=)83
b Fx(j)s Fu(b)-54 b FG(x)1731 2525 y Fv(N)1818 2511 y
Fx(\000)23 b Fu(e)-54 b FG(x)1961 2525 y Fv(N)2028 2511
y Fx(j)26 b FA(=)1500 2674 y(=)83 b Fx(j)s Fu(b)-54 b
FG(x)20 b Fx(\000)j Fu(e)-54 b FG(x)p Fx(j)30 b FA(cos)2086
2573 y Fu(\020)2141 2674 y Fl(\\)p FA(\()s Fu(b)-54 b
FG(x)20 b Fx(\000)j Fu(e)-54 b FG(x;)30 b(e)2554 2688
y Fv(N)2622 2674 y FA(\))2657 2573 y Fu(\021)2737 2674
y Fx(\025)1500 2905 y(\025)83 b(j)s Fu(b)-54 b FG(x)20
b Fx(\000)j Fu(e)-54 b FG(x)p Fx(j)1949 2777 y Fu(\022)2016
2905 y FA(1)21 b Fx(\000)2198 2844 y FA(const)16 b FG(l)2450
2811 y Fy(2)p 2183 2885 307 4 v 2281 2970 a FG(R)2351
2938 y Fy(2)2350 2996 y(0)2499 2777 y Fu(\023)2596 2905
y FG(:)456 3112 y FA(Therefore,)30 b(\(5.31\))j(and)c(\(5.16\))k(yield)
c(that)1228 3326 y FG(H)1304 3340 y Fy(0)1343 3326 y
FA(\()p FG(u)p FA(\()s Fu(e)-54 b FG(x)q FA(\)\))21 b(+)1710
3264 y(2)p 1755 3191 72 4 v FG(C)1827 3278 y Fy(0)p 1710
3305 157 4 v 1734 3388 a FG(R)1803 3402 y Fy(0)1959 3326
y Fx(\025)83 b(j)t Fu(e)-55 b FG(y)23 b Fx(\000)g Fu(e)-54
b FG(x)p Fx(j)21 b(\000)e FG(r)28 b FA(=)1959 3505 y(=)83
b Fx(\000j)s Fu(b)-54 b FG(x)20 b Fx(\000)j Fu(e)-54
b FG(x)p Fx(j)25 b(\025)1959 3690 y(\025)2160 3629 y
Fu(e)-54 b FG(x)2209 3643 y Fv(N)2296 3629 y Fx(\000)23
b Fu(b)-54 b FG(x)2439 3643 y Fv(N)p 2123 3669 418 4
v 2123 3767 a FA(1)21 b Fx(\000)2301 3731 y Fy(const)13
b Fv(l)2496 3712 y Fq(2)p 2289 3746 242 4 v 2366 3805
a Fv(R)2419 3782 y Fq(2)2419 3826 y(0)2576 3690 y Fx(\025)1959
3991 y(\025)2126 3929 y Fu(e)-54 b FG(x)2175 3943 y Fv(N)2262
3929 y Fx(\000)20 b FA(\()p FG(\022)s(=)p FA(8\))p 2123
3970 438 4 v 2133 4068 a(1)g Fx(\000)2311 4032 y Fy(const)13
b Fv(l)2506 4013 y Fq(2)p 2299 4047 242 4 v 2376 4105
a Fv(R)2429 4083 y Fq(2)2429 4127 y(0)2585 3991 y FG(;)456
4243 y FA(whic)m(h)26 b(easily)i(implies)d(\(5.32\))30
b(in)d(this)g(case.)41 b(This)26 b(completes)i(the)h(pro)s(of)e(of)h
(\(5.32\).)555 4404 y(Th)m(us,)f(in)f(the)h(ligh)m(t)g(of)g(\(5.24\),)j
(\(5.27\),)g(\(5.30\))g(and)c(\(5.32\),)31 b(w)m(e)c(ha)m(v)m(e)i(that)
3214 4381 y Fu(e)3190 4404 y FG(X)k Fx(2)3389 4381 y
Fu(e)3384 4404 y FA(\004)456 4523 y(and,)d(therefore,)h(that)g
FG(\031)1314 4537 y Fv(e)1347 4548 y Fs(N)1432 4500 y
Fu(e)1408 4523 y FG(X)i Fx(2)25 b FA(\004,)30 b(ending)f(the)h(pro)s
(of)g(of)g(\(5.23\).)555 4671 y(No)m(w)k(w)m(e)f(exploit)g(Prop)s
(osition)e(3.14,)k(applied)c(to)j Fi(G)f FA(and)f Fi(O)p
FA(:)46 b(from)33 b(that,)h(\(5.23\))456 4779 y(and)29
b(\(5.22\),)1387 4934 y Fi(L)1442 4896 y Fv(N)1509 4934
y FA(\(\004\))83 b Fx(\025)g Fi(L)1932 4896 y Fv(N)2000
4934 y FA(\()p Fi(G)p FA(\))41 b Fx(\025)1723 5075 y(\025)98
b FA(const)16 b Fi(L)2170 5037 y Fv(N)2238 5075 y FA(\()p
Fi(O)p FA(\))40 b Fx(\025)1723 5216 y(\025)98 b FA(const)16
b FG(l)2144 5178 y Fv(N)7 b Ft(\000)p Fy(1)2317 5216
y FG(:)p eop
%%Page: 57 57
57 56 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(57)456
450 y FA(This)28 b(completes)j(the)f(pro)s(of)g(of)h(Lemma)f(5.1.)1329
b Fl(\003)555 620 y FA(The)30 b(next)h(one)f(is)g(the)g(main)g(result)f
(of)h(this)g(section:)456 751 y FH(Prop)s(osition)46
b(5.2.)h Fo(L)-5 b(et)41 b FG(C)48 b Fo(b)-5 b(e)40 b(a)i(suitably)f
(lar)-5 b(ge)42 b(c)-5 b(onstant.)68 b(L)-5 b(et)42 b
FG(K)2960 766 y Fv(l)3026 751 y FA(:=)e Fx(fj)p FG(x)3284
718 y Ft(0)3308 751 y Fx(j)h FG(<)456 859 y(C)21 b(l)r
Fx(g)k(\002)e(fj)p FG(x)857 873 y Fv(N)925 859 y Fx(j)35
b FG(<)f(C)22 b(l)r Fx(g)p Fo(.)58 b(L)-5 b(et)38 b FG(u)c
Fx(2)h FG(W)1780 826 y Fy(1)p Fv(;p)1874 859 y FA(\()p
FG(K)1986 874 y Fv(l)2012 859 y FA(\))j Fo(b)-5 b(e)38
b(a)g(lo)-5 b(c)g(al)40 b(minimizer)e(for)h Fx(F)47 b
Fo(in)37 b FG(K)3390 874 y Fv(l)3417 859 y Fo(.)456 967
y(Assume)d(that)j FG(u)p FA(\(0\))30 b(=)g(0)35 b Fo(and)h(that)g
FG(u)p FA(\()p FG(x)p FA(\))31 b FG(<)e FA(0)35 b Fo(if)g
FG(x)2259 981 y Fv(N)2356 967 y FG(<)29 b Fx(\000)p FG(\022)s
Fo(,)34 b(for)i(some)g FG(\022)31 b(>)f FA(0)p Fo(.)49
b(Fix)476 1052 y FA(\026)456 1075 y FG(C)31 b(>)25 b
FA(0)30 b Fo(and)h FG(k)d Fx(2)d Fw(N)7 b Fo(.)46 b(L)-5
b(et)30 b FA(\004)f Fo(b)-5 b(e)30 b(the)g(set)f(of)h(p)-5
b(oints)31 b FA(\()p Fi(x)p FG(;)g(u)p FA(\()p Fi(x)p
FA(\)\))g Fo(satisfying)f(the)g(fol)5 b(lowing)456 1183
y(pr)-5 b(op)g(erties:)671 1314 y Fx(\017)42 b(j)p Fi(x)815
1281 y Ft(0)839 1314 y Fx(j)26 b(\024)f FG(l)r Fo(,)32
b Fx(j)p Fi(x)1132 1328 y Fv(N)7 b Fy(+1)1290 1314 y
Fx(j)25 b(\024)g FA(1)p FG(=)p FA(2)p Fo(;)671 1424 y
Fx(\017)42 b Fi(x)790 1438 y Fv(N)883 1424 y Fx(\024)1000
1401 y FA(\026)979 1424 y FG(C)1051 1391 y Fv(k)1108
1424 y FG(\022)23 b FA(+)d FG(H)1341 1438 y Fy(0)1380
1424 y FA(\()p FG(u)p FA(\()p Fi(x)p FA(\)\))p Fo(.)456
1556 y(Then,)29 b(ther)-5 b(e)28 b(exist)g(p)-5 b(ositive)29
b(universal)g(c)-5 b(onstants)30 b FG(c)e Fo(and)j FA(^)-47
b FG(c)28 b Fo(for)g(which)h(the)f(fol)5 b(lowing)456
1664 y(holds.)43 b(F)-7 b(or)34 b(any)f FG(\022)1112
1678 y Fy(0)1176 1664 y FG(>)25 b FA(0)p Fo(,)33 b(ther)-5
b(e)34 b(exists)f FG(")1896 1678 y Fy(0)1935 1664 y FA(\()p
FG(\022)2013 1678 y Fy(0)2053 1664 y FA(\))26 b FG(>)f
FA(0)p Fo(,)32 b(so)i(that,)f(if)1049 1820 y FG(\022)p
1049 1861 46 4 v 1057 1944 a(l)1129 1882 y Fx(\024)25
b FG(")1267 1896 y Fy(0)1307 1882 y FA(\()p FG(\022)1385
1896 y Fy(0)1425 1882 y FA(\))15 b FG(;)201 b(\022)28
b Fx(\025)d FG(\022)1911 1896 y Fy(0)2136 1882 y Fo(and)2496
1797 y FA(\026)2476 1820 y FG(C)2548 1787 y Fv(k)2605
1820 y FG(\022)p 2476 1861 175 4 v 2548 1944 a(l)2685
1882 y Fx(\024)i FA(^)-47 b FG(c)16 b(;)456 2063 y Fo(then)1192
2184 y Fi(L)1247 2147 y Fv(N)1314 2083 y Fu(\020)1369
2184 y FG(\031)1421 2198 y Fv(N)1488 2184 y FA(\(\004\))1619
2083 y Fu(\021)1714 2184 y Fx(\025)40 b FA(\(1)21 b Fx(\000)f
FA(\(1)h Fx(\000)f FG(c)p FA(\))2283 2147 y Fv(k)2326
2184 y FA(\))15 b Fi(L)2431 2147 y Fv(N)2499 2184 y FA(\()p
FG(Q)2606 2199 y Fv(l)2632 2184 y FA(\))g FG(:)456 2389
y Fo(Pr)-5 b(o)g(of.)43 b FA(Let)32 b FG(R)970 2403 y
Fy(0)1038 2389 y FA(:=)c FG(l)1191 2356 y Fy(2)1231 2389
y FG(=)p FA(\()p FG(C)7 b(\022)s FA(\),)32 b(with)f FG(C)39
b FA(suitably)30 b(large.)47 b(F)-8 b(or)33 b(an)m(y)f
FG(k)g Fx(2)c Fw(N)6 b FA(,)39 b(let)32 b FG(R)3277 2404
y Fv(k)3348 2389 y FA(:=)456 2499 y FG(R)525 2513 y Fy(0)600
2476 y FA(\026)579 2499 y FG(C)651 2466 y Ft(\000)p Fv(k)748
2499 y FA(,)26 b(where)1077 2476 y(\026)1056 2499 y FG(C)31
b FA(is)23 b(a)i(p)s(ositiv)m(e)f(univ)m(ersal)e(constan)m(t,)28
b(to)d(b)s(e)f(c)m(hosen)h(suitably)d(large)456 2607
y(in)29 b(the)i(sequel.)42 b(W)-8 b(e)32 b(de\014ne)e
Fi(D)1516 2622 y Fv(k)1585 2607 y Fx(\022)c Fw(R)1742
2574 y Fv(N)7 b Fy(+1)1936 2607 y FA(as)31 b(the)g(set)g(of)h(p)s(oin)m
(ts)d(\()p Fi(x)p FG(;)i(u)p FA(\()p Fi(x)p FA(\)\))i(satisfying)456
2715 y(the)d(follo)m(wing)f(prop)s(erties:)671 2847 y
Fx(\017)42 b(j)p Fi(x)815 2814 y Ft(0)839 2847 y Fx(j)26
b(\024)f FG(C)c(l)r(=)p FA(2,)32 b Fx(j)p FG(u)p FA(\()p
Fi(x)p FA(\))p Fx(j)26 b FG(<)f FA(1)p FG(=)p FA(2;)671
2956 y Fx(\017)42 b FA(there)h(exists)f FG(Y)64 b Fx(2)45
b Fw(R)1549 2923 y Fv(N)1650 2956 y Fx(\002)28 b FA([)p
Fx(\000)p FA(1)p FG(=)p FA(4)p FG(;)j FA(1)p FG(=)p FA(4])44
b(so)f(that)f Fw(S)o FA(\()p FG(Y)5 b(;)15 b(R)2831 2971
y Fv(k)2868 2956 y FA(\))43 b(is)e(ab)s(o)m(v)m(e)i(the)758
3064 y(graph)c(of)g FG(u)h FA(in)e Fx(fj)p FG(x)1466
3031 y Ft(0)1490 3064 y Fx(j)i FG(<)g(C)21 b(l)r(=)p
FA(2)p Fx(g)28 b(\002)e(fj)p FG(x)2163 3078 y Fv(N)2230
3064 y Fx(j)41 b FG(<)e(C)22 b(l)r(=)p FA(2)p Fx(g)41
b FA(and)d(it)h(touc)m(hes)h(the)758 3172 y(graph)30
b(of)h FG(u)f FA(at)h(\()p Fi(x)p FG(;)g(u)p FA(\()p
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y Fu(\022)941 3282 y Fx(r)p FG(u)p FA(\()p Fi(x)p FA(\))p
916 3323 282 4 v 916 3406 a Fx(jr)p FG(u)p FA(\()p Fi(x)p
FA(\))p Fx(j)1222 3344 y FG(;)15 b(e)1304 3358 y Fv(N)1372
3215 y Fu(\023)1480 3344 y Fx(\024)1621 3259 y FA(\026)1601
3282 y FG(C)1673 3249 y Fv(k)1730 3282 y FG(l)p 1601
3323 159 4 v 1626 3406 a(R)1695 3420 y Fy(0)1769 3344
y FA(;)671 3579 y Fx(\017)42 b Fi(x)790 3593 y Fv(N)883
3579 y Fx(\024)1010 3494 y FA(\026)989 3517 y FG(C)1061
3484 y Fv(k)1118 3517 y FG(\022)p 989 3558 175 4 v 1054
3641 a FA(4)1194 3579 y(+)20 b FG(H)1361 3593 y Fy(0)1400
3579 y FA(\()p FG(u)p FA(\()p Fi(x)p FA(\)\).)456 3741
y(W)-8 b(e)33 b(also)e(set)h FG(D)1015 3756 y Fv(k)1086
3741 y FA(:=)27 b FG(\031)1261 3755 y Fv(N)1328 3741
y FA(\()p Fi(D)1432 3756 y Fv(k)1475 3741 y FA(\).)45
b(W)-8 b(e)33 b(w)m(ould)e(lik)m(e)g(to)h(apply)f(Lemma)h(4.3)g(to)h
FG(D)3198 3756 y Fv(k)3241 3741 y FA(,)f(and)456 3849
y(w)m(e)h(therefore)h(no)m(w)g(pro)m(v)m(e)g(that)g FG(D)1693
3864 y Fv(k)1769 3849 y FA(ful\014lls)c(the)k(assumption)e(of)h(Lemma)h
(4.3.)51 b(F)-8 b(or)456 3957 y(this,)29 b(\014rst)h(of)g(all,)g
(notice)g(that,)i(b)m(y)e(Lemma)g(5.1,)456 4112 y(\(5.33\))1004
b FG(D)1765 4126 y Fy(0)1825 4112 y Fx(\\)19 b FG(Q)1977
4127 y Fv(l)2029 4112 y Fx(6)p FA(=)24 b Fx(;)15 b FG(:)456
4266 y FA(Let)24 b(us)g(no)m(w)g(\014x)f FG(Z)1090 4281
y Fv(k)1158 4266 y Fx(2)i FG(D)1319 4281 y Fv(k)1370
4266 y Fx(\\)8 b FG(Q)1511 4281 y Fy(2)p Fv(l)1571 4266
y FA(.)39 b(By)24 b(construction,)h(there)g(exists)e(\()p
FG(x)2870 4281 y Fv(k)2913 4266 y FG(;)31 b(u)p FA(\()p
FG(x)3108 4281 y Fv(k)3151 4266 y FA(\)\))26 b Fx(2)f
Fi(D)3402 4281 y Fv(k)456 4381 y FA(so)31 b(that)g FG(Z)827
4396 y Fv(k)895 4381 y FA(=)26 b FG(\031)1044 4395 y
Fv(N)1111 4381 y FA(\()p FG(x)1198 4396 y Fv(k)1241 4381
y FG(;)31 b(u)p FA(\()p FG(x)1436 4396 y Fv(k)1479 4381
y FA(\)\).)42 b(T)-8 b(ak)m(e)32 b(also)2038 4358 y(~)2019
4381 y FG(Z)g Fx(2)26 b FG(L)p FA(,)31 b(with)e FG(a)d
Fx(\024)f(j)2740 4358 y FA(~)2720 4381 y FG(Z)j Fx(\000)20
b FG(Z)2963 4396 y Fv(k)3006 4381 y Fx(j)26 b FA(=:)g
FG(q)i Fx(\024)e FA(2)p FG(l)r FA(,)456 4489 y(and)j(supp)s(ose)g
FG(a)i FA(suitably)d(large.)41 b(W)-8 b(e)32 b(claim)d(that)456
4666 y(\(5.34\))402 b Fi(L)1143 4629 y Fv(N)1210 4565
y Fu(\020)1265 4666 y FG(D)1340 4681 y Fv(k)r Fy(+1)1493
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4666 y FA(\()1841 4643 y(~)1822 4666 y FG(Z)6 b FA(\))1925
4565 y Fu(\021)2020 4666 y Fx(\025)41 b Fi(L)2187 4629
y Fv(N)2254 4565 y Fu(\020)2308 4666 y FG(L)20 b Fx(\\)g
FG(B)2540 4680 y Fv(q)2578 4666 y FA(\()2633 4643 y(~)2613
4666 y FG(Z)7 b FA(\))2717 4565 y Fu(\021)2787 4666 y
FG(:)456 4867 y FA(In)24 b(order)h(to)i(pro)m(v)m(e)f(the)g(ab)s(o)m(v)
m(e)g(inequalit)m(y)-8 b(,)26 b(w)m(e)g(denote)g(b)m(y)2543
4844 y(^)2522 4867 y FG(C)32 b(>)25 b FA(0)h(a)g(constan)m(t,)i(to)e(b)
s(e)456 4976 y(suitably)31 b(c)m(hosen)j(in)e(the)i(sequel,)g(and)e(w)m
(e)i(de\014ne)2258 4953 y(~)2250 4976 y(\004)f(as)h(the)f(set)h(of)g(p)
s(oin)m(ts)e(\()p Fi(x)p FG(;)15 b(u)p FA(\()p Fi(x)p
FA(\)\))456 5084 y(satisfying)29 b(the)h(follo)m(wing)f(prop)s(erties:)
671 5216 y Fx(\017)42 b(j)p Fi(x)815 5183 y Ft(0)859
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Fx(j)26 b FG(<)f FA(4)p FG(l)r FA(,)31 b Fx(j)p FG(u)p
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FA(2;)p eop
%%Page: 58 58
58 57 bop 456 251 a Fy(58)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
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716 y Fu(\023)1734 845 y Fx(\024)1875 760 y FA(^)1855
783 y FG(C)1947 760 y FA(\026)1926 783 y FG(C)1998 750
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921 y Fy(0)2094 845 y FA(;)671 1087 y Fx(\017)g FA(\()p
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FA(\))g Fx(\001)1168 1025 y(r)p FG(u)p FA(\()p FG(x)1383
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1025 y FG(C)1750 1002 y FA(\026)1729 1025 y FG(C)1801
992 y Fv(k)1859 1025 y FG(l)1888 992 y Fy(2)p 1658 1066
270 4 v 1715 1149 a FA(4)p FG(R)1829 1163 y Fy(0)1957
1087 y FA(+)g FG(H)2124 1101 y Fy(0)2163 1087 y FA(\()p
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y Fy(0)2615 1087 y FA(\()p FG(u)p FA(\()p FG(x)2789 1102
y Fv(k)2832 1087 y FA(\)\).)456 1283 y(Notice)28 b(that,)i(b)m(y)d
(means)1330 1248 y Fy(11)1429 1283 y FA(of)h(Lemma)f(4.2)i(\(applied)d
(in)h Fx(fj)p FG(x)2572 1250 y Ft(0)2611 1283 y Fx(\000)20
b FA(~)-50 b FG(z)2743 1250 y Ft(0)2766 1283 y Fx(j)26
b(\024)f FA(8)p FG(l)r Fx(g)15 b(\002)g(fj)p FG(x)3255
1297 y Fv(N)3323 1283 y Fx(j)26 b(\024)456 1391 y FA(8)p
FG(l)r Fx(g)p FA(,)456 1576 y(\(5.35\))132 b Fi(L)873
1539 y Fv(N)940 1576 y FA(\()p FG(\031)1027 1590 y Fv(N)1095
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1475 y Fu(\020)2811 1576 y FG(L)k Fx(\\)g FG(B)3043 1590
y Fv(q)3081 1576 y FA(\()3135 1553 y(~)3116 1576 y FG(Z)7
b FA(\))3220 1475 y Fu(\021)3289 1576 y FG(:)456 1777
y FA(Let)28 b(us)e(no)m(w)i(deduce)f(some)h(easy)g(prop)s(erties)e(of)
2171 1754 y(~)2164 1777 y(\004)o(.)40 b(First)27 b(of)g(all,)h(b)m(y)f
(the)h(de\014nitions)456 1887 y(of)567 1864 y(~)559 1887
y(\004)i(and)g Fi(D)896 1902 y Fv(k)939 1887 y FA(,)g(w)m(e)h(ha)m(v)m
(e)h(that,)f(for)f(an)m(y)h(\()p Fi(x)p FG(;)g(u)p FA(\()p
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2099 y Fl(\\)617 1971 y Fu(\022)719 2037 y Fx(r)p FG(u)p
FA(\()p Fi(x)p FA(\))p 694 2078 282 4 v 694 2161 a Fx(jr)p
FG(u)p FA(\()p Fi(x)p FA(\))p Fx(j)1000 2099 y FG(;)15
b(e)1082 2113 y Fv(N)1150 1971 y Fu(\023)1300 2099 y
Fx(\024)83 b Fl(\\)1535 1971 y Fu(\022)1636 2037 y Fx(r)p
FG(u)p FA(\()p Fi(x)p FA(\))p 1611 2078 V 1611 2161 a
Fx(jr)p FG(u)p FA(\()p Fi(x)p FA(\))p Fx(j)1918 2099
y FG(;)1993 2037 y Fx(r)p FG(u)p FA(\()p FG(x)2208 2052
y Fv(k)2251 2037 y FA(\))p 1968 2078 344 4 v 1968 2161
a Fx(jr)p FG(u)p FA(\()p FG(x)2208 2176 y Fv(k)2251 2161
y FA(\))p Fx(j)2322 1971 y Fu(\023)2409 2099 y FA(+)20
b Fl(\\)2581 1971 y Fu(\022)2683 2037 y Fx(r)p FG(u)p
FA(\()p FG(x)2898 2052 y Fv(k)2940 2037 y FA(\))p 2657
2078 V 2657 2161 a Fx(jr)p FG(u)p FA(\()p FG(x)2897 2176
y Fv(k)2940 2161 y FA(\))p Fx(j)3026 2099 y FG(;)15 b(e)3108
2113 y Fv(N)3176 1971 y Fu(\023)3268 2099 y Fx(\024)1300
2368 y(\024)1484 2284 y FA(^)1464 2307 y FG(C)1556 2284
y FA(\026)1535 2307 y FG(C)1607 2274 y Fv(k)1664 2307
y FG(l)p 1464 2347 230 4 v 1524 2431 a(R)1593 2445 y
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2683 a(R)1602 2697 y Fy(0)1737 2620 y FG(;)-1306 b FA(\(5.36\))456
2840 y(pro)m(vided)849 2817 y(\026)829 2840 y FG(C)36
b FA(is)29 b(big)f(enough)i(with)e(resp)s(ect)i(to)2132
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3078 y(\014)1069 3133 y(\014)1069 3187 y(\014)1135 3094
y Fx(r)p FG(u)p FA(\()p FG(x)1350 3109 y Fv(k)1393 3094
y FA(\))p 1110 3135 344 4 v 1110 3218 a Fx(jr)p FG(u)p
FA(\()p FG(x)1350 3233 y Fv(k)1393 3218 y FA(\))p Fx(j)1499
3155 y(\000)35 b FG(e)1647 3169 y Fv(N)1714 3024 y Fu(\014)1714
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FG(x)2379 3233 y Fv(k)2422 3218 y FA(\))p Fx(j)2508 3155
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Fx(j)37 b(\024)g FA(4)p FG(l)j FA(\(and)e(observ)m(e)g(that)g(this)f
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y(\()p Fi(x)p FG(;)31 b(u)p FA(\()p Fi(x)p FA(\)\))26
b Fx(2)888 4117 y FA(~)880 4140 y(\004\).)41 b(By)30
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3094 4117 y(~)3086 4140 y(\004,)i(it)f(also)p 456 4227
499 4 v 555 4297 a Fy(11)626 4326 y FF(W)-6 b(e)30 b(apply)g(here)h
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%%Page: 59 59
59 58 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
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630 y Fv(k)2862 615 y FA(\))p 2579 655 V 2579 739 a Fx(jr)p
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1228 925 270 4 v 1286 1008 a FA(4)p FG(R)1400 1022 y
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884 y FG(C)1746 851 y Fv(k)1803 884 y FG(l)1832 851 y
Fy(2)p 1629 925 244 4 v 1696 1008 a FG(R)1765 1022 y
Fy(0)1902 945 y FA(+)20 b FG(H)2069 959 y Fy(0)2108 945
y FA(\()p FG(u)p FA(\()p Fi(x)p FA(\)\))i Fx(\000)e FG(H)2521
959 y Fy(0)2560 945 y FA(\()p FG(u)p FA(\()p FG(x)2734
960 y Fv(k)2777 945 y FA(\)\))15 b FG(;)-2431 b FA(\(5.39\))456
1199 y(for)31 b(an)m(y)g(\()p Fi(x)p FG(;)g(u)p FA(\()p
Fi(x)p FA(\)\))d Fx(2)1203 1176 y FA(~)1195 1199 y(\004;)k(from)e
(\(5.39\),)k(the)d(de\014nition)e(of)j FG(R)2553 1213
y Fy(0)2623 1199 y FA(and)f(the)g(assumptions)456 1307
y(of)f(Prop)s(osition)f(5.2,)i(w)m(e)g(th)m(us)f(deduce)g(that)1393
1560 y Fi(x)1425 1574 y Fv(N)1576 1560 y Fx(\024)1760
1475 y FA(\026)1740 1498 y FG(C)1812 1465 y Fv(k)r Fy(+1)1959
1498 y FG(\022)p 1740 1539 265 4 v 1849 1622 a FA(4)2035
1560 y(+)19 b FG(H)2201 1574 y Fy(0)2240 1560 y FA(\()p
FG(u)p FA(\()p Fi(x)p FA(\)\))c FG(;)-2048 b FA(\(5.40\))456
1800 y(for)42 b(an)m(y)i(\()p Fi(x)p FG(;)31 b(u)p FA(\()p
Fi(x)p FA(\)\))48 b Fx(2)1266 1777 y FA(~)1259 1800 y(\004)o(,)f(if)
1507 1777 y(\026)1487 1800 y FG(C)i FA(is)42 b(large)h(enough.)79
b(Therefore,)46 b(thanks)c(to)i(\(5.36\))456 1915 y(and)27
b(\(5.40\),)32 b(w)m(e)d(ha)m(v)m(e)g(that)1459 1892
y(~)1451 1915 y(\004)c Fx(\022)g Fi(D)1702 1930 y Fv(k)r
Fy(+1)1835 1915 y FA(.)40 b(F)-8 b(rom)29 b(this)f(and)f(\(5.35\),)32
b(w)m(e)d(gather)g(\(5.34\),)456 2023 y(as)h(desired.)39
b(This)28 b(sa)m(ys)i(that)h(the)f(h)m(yp)s(otheses)g(of)g(Lemma)g(4.3)
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2023 y FA(,)456 2131 y(th)m(us)g(w)m(e)g(will)e(freely)i(use)g(suc)m(h)
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b(F)-8 b(rom)27 b(Lemma)f(4.1,)i(and)e(taking)2808 2306
y(^)2787 2329 y FG(C)33 b FA(suitably)24 b(large,)456
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y Fv(k)3191 2437 y FA(\()p FG(Z)7 b FA(\))33 b(so)456
2554 y(that)e(\()p FG(x)740 2569 y Fv(k)783 2554 y FG(;)f(u)p
FA(\()p FG(x)977 2569 y Fv(k)1020 2554 y FA(\)\))c Fx(2)f
Fi(D)1271 2569 y Fv(k)1314 2554 y FA(,)30 b Fx(j)p FG(x)21
b Fx(\000)f FG(x)1610 2569 y Fv(k)1652 2554 y Fx(j)26
b(\024)1819 2531 y FA(^)1799 2554 y FG(C)6 b FA(,)31
b FG(Z)g FA(=)25 b FG(\031)2167 2568 y Fv(N)2234 2554
y FA(\()p FG(x;)31 b(u)p FA(\()p FG(x)p FA(\)\))h(and)456
2818 y(\(5.41\))255 b(\()p FG(x)21 b Fx(\000)f FG(x)1192
2833 y Fv(k)1235 2818 y FA(\))g Fx(\001)1371 2757 y(r)p
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1345 2797 344 4 v 1345 2881 a Fx(jr)p FG(u)p FA(\()p
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y(\024)41 b FG(H)1927 2832 y Fy(0)1966 2818 y FA(\()p
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2832 y Fy(0)2437 2818 y FA(\()p FG(u)p FA(\()p FG(x)2611
2833 y Fv(k)2654 2818 y FA(\)\))h(+)2846 2757 y(const)3090
2734 y(^)3069 2757 y FG(C)p 2846 2797 295 4 v 2938 2881
a(R)3007 2896 y Fv(k)3166 2818 y FG(:)456 3066 y FA(Th)m(us,)29
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b Fx(\000)g FG(x)711 3330 y Fv(k)754 3315 y FA(\))g Fx(\001)h
FG(e)897 3329 y Fv(N)1047 3315 y Fx(\024)83 b FA(\()p
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FA(\))h Fx(\001)1630 3253 y(r)p FG(u)p FA(\()p FG(x)1845
3268 y Fv(k)1888 3253 y FA(\))p 1605 3294 344 4 v 1605
3377 a Fx(jr)p FG(u)p FA(\()p FG(x)1845 3392 y Fv(k)1888
3377 y FA(\))p Fx(j)1994 3315 y FA(+)2100 3183 y Fu(\014)2100
3237 y(\014)2100 3292 y(\014)2100 3346 y(\014)2130 3315
y FA(\()p FG(x)g Fx(\000)f FG(x)2381 3330 y Fv(k)2423
3315 y FA(\))h Fx(\001)2524 3186 y Fu(\022)2626 3253
y Fx(r)p FG(u)p FA(\()p FG(x)2841 3268 y Fv(k)2884 3253
y FA(\))p 2601 3294 V 2601 3377 a Fx(jr)p FG(u)p FA(\()p
FG(x)2841 3392 y Fv(k)2884 3377 y FA(\))p Fx(j)2990 3315
y(\000)35 b FG(e)3138 3329 y Fv(N)3206 3186 y Fu(\023)3273
3183 y(\014)3273 3237 y(\014)3273 3292 y(\014)3273 3346
y(\014)3343 3315 y Fx(\024)1047 3584 y(\024)83 b FG(H)1277
3598 y Fy(0)1316 3584 y FA(\()p FG(u)p FA(\()p FG(x)p
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y FA(\()p FG(u)p FA(\()p FG(x)1962 3599 y Fv(k)2005 3584
y FA(\)\))h(+)2197 3522 y(const)2440 3499 y(^)2419 3522
y FG(C)p 2197 3563 295 4 v 2288 3646 a(R)2357 3661 y
Fv(k)2521 3584 y FA(+)2622 3522 y(4)2688 3499 y(\026)2667
3522 y FG(C)2739 3489 y Fv(k)2797 3522 y FG(l)2826 3489
y Fy(2)p 2622 3563 244 4 v 2689 3646 a FG(R)2758 3660
y Fy(0)2890 3584 y FG(;)456 3818 y FA(whic)m(h)k(implies,)g(thanks)i
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3926 y(sition)i(5.2,)i(that)1441 4084 y FG(x)1493 4098
y Fv(N)1586 4084 y Fx(\024)1702 4061 y FA(\026)1682 4084
y FG(C)1754 4047 y Fv(k)r Fy(+1)1901 4084 y FG(\022)23
b FA(+)d FG(H)2134 4098 y Fy(0)2173 4084 y FA(\()p FG(u)p
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b(if)e(\004)h(is)g(as)g(de\014ned)f(here)i(ab)s(o)m(v)m(e)g(in)e(the)i
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b FG(E)1836 4468 y Fv(k)1919 4453 y Fx(\022)40 b FA(\004)15
b FG(:)456 4647 y FA(Also,)30 b(b)m(y)g(Lemma)h(4.3,)456
4863 y(\(5.43\))492 b Fi(L)1233 4826 y Fv(N)1300 4762
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b(Th)m(us,)34 b(the)g(claim)f(in)g(Prop)s(osition)f(5.2)j(follo)m(ws)e
(from)h(\(5.42\))456 5193 y(and)29 b(\(5.43\).)2487 b
Fl(\003)p eop
%%Page: 60 60
60 59 bop 456 251 a Fy(60)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
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612 y FA(First)39 b(of)h(all,)h(note)f(that)g FG(u)f
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-5 b(alues)456 720 y(thanks)25 b(to)h(the)g(densit)m(y)f(estimates)h
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FG(l)k FA(b)m(y)e FG(C)21 b(l)r FA(,)27 b(w)m(e)456 828
y(ma)m(y)h(assume)f(that)h FG(u)f FA(is)f(a)i(lo)s(cal)f(minimizer)d
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828 y Fx(j)g FG(<)f(C)c(l)r Fx(g)14 b(\002)g(fj)p FG(x)3043
842 y Fv(N)3112 828 y Fx(j)25 b FG(<)g(C)d(l)r Fx(g)p
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y Fv(N)1574 1094 y FG(>)25 b(\022)i(>)e FA(0)31 b(and)f
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1108 y Fv(N)2688 1094 y FG(<)c Fx(\000)p FG(\022)s FA(.)456
1253 y(The)f(strategy)h(for)f(pro)m(ving)g(Theorem)g(1.1)i(consists)e
(in)f(assuming,)h(b)m(y)g(con)m(tradiction)456 1361 y(that)34
b(there)g(exists)g(a)g(p)s(oin)m(t)f(in)g Fx(f)p FG(u)e
FA(=)g(0)p Fx(g)24 b(\\)e(fj)p FG(x)2126 1328 y Ft(0)2150
1361 y Fx(j)31 b FG(<)g(l)r(=)p FA(4)p Fx(g)k FA(close)g(to)f
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FA(.)50 b(The)456 1469 y(con)m(tradiction)30 b(will)e(b)s(e,)i(then,)g
(that)h(the)f(energy)h(of)g FG(u)f FA(is)f(larger)h(than)h(it)f
(should.)555 1627 y(The)k(\014rst)f(step)h(in)f(pro)m(ving)g(Theorem)h
(1.1)h(is)e(th)m(us)g(the)h(follo)m(wing:)47 b(w)m(e)34
b(assume,)456 1735 y(b)m(y)c(con)m(tradicting)g(Theorem)g(1.1,)i(that)
777 1895 y Fx(f)p FG(u)26 b FA(=)f(0)p Fx(g)c(\\)f(fj)p
FG(x)1310 1858 y Ft(0)1334 1895 y Fx(j)25 b FG(<)1501
1872 y FA(\026)1480 1895 y FG(C)1552 1858 y Ft(\000)p
Fv(k)1644 1867 y Fq(0)1682 1895 y FG(l)r(=)p FA(4)p Fx(g)d(\\)d(f)p
FG(x)2045 1909 y Fv(N)2138 1895 y FG(<)25 b FA(\()p Fx(\000)p
FA(1)c(+)2518 1872 y(\026)2497 1895 y FG(C)2569 1858
y Ft(\000)p Fv(k)2661 1867 y Fq(0)2699 1895 y FG(=)p
FA(4\))p FG(\022)s Fx(g)26 b(6)p FA(=)f Fx(;)15 b FG(;)456
2058 y FA(with)29 b FG(k)710 2072 y Fy(0)775 2058 y Fx(2)c
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2038 35 4 v 1373 2090 a(l)1441 2058 y FA(small)f(\(p)s(ossibly)f(in)h
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y FA(\).)41 b(W)-8 b(e)32 b(also)e(set)625 2253 y(\004)686
2267 y Fy(0)809 2253 y FA(:=)987 2152 y Fu(n)1048 2253
y FA(\()p FG(x;)15 b(u)p FA(\()p FG(x)p FA(\)\))27 b
Fx(2)e Fw(R)1557 2216 y Fv(N)1650 2253 y Fx(\002)20 b
Fw(R)49 b FA(s)p FG(:)p FA(t)p FG(:)987 2452 y(x)1039
2466 y Fv(N)1132 2452 y Fx(\024)25 b FG(H)1304 2466 y
Fy(0)1343 2452 y FA(\()p FG(u)p FA(\()p FG(x)p FA(\)\))d
Fx(\000)d FG(\022)s(=)p FA(2)c FG(;)57 b Fx(j)p FG(x)2009
2415 y Ft(0)2052 2452 y Fx(\000)20 b FA(\()p FG(x)2230
2415 y Ft(\003)2270 2452 y FA(\))2305 2415 y Ft(0)2329
2452 y Fx(j)25 b(\024)g FG(l)r(=)p FA(2)15 b FG(;)57
b Fx(j)p FG(u)p FA(\()p FG(x)p FA(\))p Fx(j)26 b(\024)f
FA(1)p FG(=)p FA(2)3172 2351 y Fu(o)3249 2452 y FG(:)456
2633 y FA(Then,)k(w)m(e)i(claim)e(that)456 2814 y(\(6.2\))460
b Fi(L)1156 2777 y Fv(N)1223 2713 y Fu(\020)1278 2814
y FG(\031)1330 2828 y Fv(N)1397 2814 y FA(\(\004)1493
2828 y Fy(0)1532 2814 y FA(\))1567 2713 y Fu(\021)1662
2814 y Fx(\025)40 b FA(\(1)21 b Fx(\000)f FA(\(1)h Fx(\000)f
FG(c)2196 2828 y Fy(0)2236 2814 y FA(\))2271 2777 y Fv(k)2308
2786 y Fq(0)2347 2814 y FA(\))15 b Fi(L)2452 2777 y Fv(N)2520
2814 y FA(\()p FG(Q)2627 2833 y Fv(l)q(=)p Fy(2)2723
2814 y FA(\))g FG(;)456 2995 y FA(for)30 b(a)h(suitable)d(constan)m(t)k
FG(c)1415 3009 y Fy(0)1480 2995 y FG(>)25 b FA(0.)41
b(T)-8 b(o)31 b(pro)m(v)m(e)g(this,)f(let)780 3155 y
FG(x)832 3118 y Ft(\003)896 3155 y Fx(2)25 b(f)p FG(u)h
FA(=)f(0)p Fx(g)c(\\)f(fj)p FG(x)1515 3118 y Ft(0)1539
3155 y Fx(j)25 b FG(<)1706 3132 y FA(\026)1685 3155 y
FG(C)1757 3118 y Ft(\000)p Fv(k)1849 3127 y Fq(0)1887
3155 y FG(l)r(=)p FA(4)p Fx(g)c(\\)f(f)p FG(x)2250 3169
y Fv(N)2343 3155 y FG(<)25 b FA(\()p Fx(\000)p FA(1)c(+)2722
3132 y(\026)2702 3155 y FG(C)2774 3118 y Ft(\000)p Fv(k)2866
3127 y Fq(0)2904 3155 y FG(=)p FA(4\))p FG(\022)s Fx(g)456
3316 y FA(De\014ne)37 b FG(\022)790 3283 y Ft(\003)865
3316 y FA(:=)f FG(\022)s(=)p FA(\(4)1189 3293 y(\026)1168
3316 y FG(C)1240 3283 y Fv(k)1277 3292 y Fq(0)1316 3316
y FA(\))h(and)g FG(v)s FA(\()p FG(x)p FA(\))52 b(:=)g
FG(u)p FA(\()p FG(x)25 b FA(+)f FG(x)2252 3283 y Ft(\003)2292
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3424 y FG(v)s FA(\()p FG(x)p FA(\))26 b FG(<)f FA(0)30
b(if)f FG(x)957 3438 y Fv(N)1050 3424 y FG(<)c Fx(\000)p
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b FG(v)j FA(is)c(a)i(lo)s(cal)e(minimizer)e(for)j Fx(F)9
b FA(.)41 b(Then,)29 b(if)g(w)m(e)i(de\014ne)952 3610
y(\004)1013 3573 y Ft(\003)1135 3610 y FA(:=)1314 3509
y Fu(n)1375 3610 y FA(\()p FG(z)t(;)15 b(v)s FA(\()p
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3809 y Fx(\024)g FG(H)1621 3823 y Fy(0)1660 3809 y FA(\()p
FG(v)s FA(\()p FG(z)t FA(\)\))d(+)2027 3786 y(\026)2006
3809 y FG(C)2078 3771 y Fv(k)2115 3780 y Fq(0)2153 3809
y FG(\022)2199 3771 y Ft(\003)2253 3809 y FG(;)96 b Fx(j)p
FG(v)s FA(\()p FG(z)t FA(\))p Fx(j)27 b(\024)e FA(1)p
FG(=)p FA(2)2845 3708 y Fu(o)2922 3809 y FG(;)456 3990
y FA(w)m(e)30 b(deduce)g(b)m(y)h(Prop)s(osition)d(5.2)k(that)1101
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4171 y Fx(\025)40 b FA(\(1)21 b Fx(\000)f FA(\(1)h Fx(\000)f
FG(c)2196 4185 y Fy(0)2236 4171 y FA(\))2271 4134 y Fv(k)2308
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4171 y FA(\))g FG(:)456 4352 y FA(By)30 b(elemen)m(tary)h
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FA(\004)2246 4525 y Fy(0)2300 4511 y FG(;)456 4669 y
FA(th)m(us)30 b(pro)m(ving)g(\(6.2\))r(.)555 4827 y(Let)h(no)m(w)774
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Fx(\002)20 b Fw(R)49 b FA(s)p FG(:)p FA(t)p FG(:)1136
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5089 y Fu(o)3101 5190 y FG(:)p eop
%%Page: 61 61
61 60 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(61)456
450 y FA(Then,)29 b(w)m(e)i(claim)e(that)456 640 y(\(6.3\))732
b Fi(L)1428 602 y Fv(N)1496 539 y Fu(\020)1550 640 y
FG(\031)1602 654 y Fv(N)1669 640 y FA(\(\004)1765 654
y Fy(1)1805 640 y FA(\))1840 539 y Fu(\021)1935 640 y
Fx(\025)40 b FG(c)2085 654 y Fy(1)2125 640 y Fi(L)2180
602 y Fv(N)2247 640 y FA(\()p FG(Q)2354 658 y Fv(l)q(=)p
Fy(2)2451 640 y FA(\))15 b FG(;)456 835 y FA(for)24 b(a)i(suitable)e
(constan)m(t)i FG(c)1394 849 y Fy(1)1459 835 y FG(>)f
FA(0,)i(pro)m(vided)c FG(\022)s(=l)k FA(is)d(suitably)f(small.)38
b(T)-8 b(o)25 b(pro)m(v)m(e)h(\(6.3\),)456 943 y(let)1218
1110 y Fu(e)-54 b FG(u)p FA(\()p FG(x)p FA(\))26 b(=)i
Fu(e)-54 b FG(u)p FA(\()p FG(x)1650 1072 y Ft(0)1674
1110 y FG(;)15 b(x)1766 1124 y Fv(N)1833 1110 y FA(\))83
b(:=)g Fx(\000)p FG(u)p FA(\()p FG(x)2340 1072 y Ft(0)2364
1110 y FG(;)15 b Fx(\000)p FG(x)2527 1124 y Fv(N)2594
1110 y FA(\))g FG(;)1662 1234 y Fu(e)1663 1258 y FG(h)1715
1272 y Fy(0)1755 1258 y FA(\()p FG(s)p FA(\))119 b(:)f(=)25
b FG(h)2278 1272 y Fy(0)2318 1258 y FA(\()p Fx(\000)p
FG(s)p FA(\))15 b FG(:)456 1437 y FA(Then,)714 1413 y
Fu(e)716 1437 y FG(h)768 1451 y Fy(0)835 1437 y FA(satis\014es)27
b(the)h(same)g(assumptions)d(as)j FG(h)2220 1451 y Fy(0)2287
1437 y FA(and)j Fu(e)-55 b FG(u)28 b FA(is)e(a)i(lo)s(cal)f(minimizer)e
(for)456 1545 y(the)30 b(functional)1452 1682 y Fu(e)1430
1705 y Fx(F)10 b FA(\()p FG(v)s FA(\))26 b(:=)1769 1581
y Fu(Z)1885 1643 y Fx(jr)p FG(v)s Fx(j)2058 1610 y Fv(p)p
1885 1684 214 4 v 1969 1767 a FG(p)2129 1705 y FA(+)2218
1681 y Fu(e)2219 1705 y FG(h)2271 1719 y Fy(0)2311 1705
y FA(\()p FG(v)s FA(\))15 b FG(:)456 1923 y FA(Hence,)32
b(w)m(e)f(ma)m(y)h(apply)d(Lemma)j(5.1)g(with)d FG(h)2066
1937 y Fy(0)2137 1923 y FA(replaced)h(b)m(y)2622 1899
y Fu(e)2623 1923 y FG(h)2675 1937 y Fy(0)2715 1923 y
FA(,)h(and)g(deduce)f(that,)456 2031 y(if)779 2180 y
Fu(e)774 2203 y FA(\004)835 2217 y Fy(1)957 2203 y FA(:=)1136
2102 y Fu(n)1196 2203 y FA(\()p FG(x;)19 b Fu(e)-55 b
FG(u)q FA(\()p FG(x)p FA(\)\))26 b Fx(2)f Fw(R)1705 2166
y Fv(N)1798 2203 y Fx(\002)20 b Fw(R)49 b FA(s)p FG(:)p
FA(t)p FG(:)1136 2402 y(x)1188 2416 y Fv(N)1280 2402
y Fx(\024)1397 2379 y Fu(e)1376 2402 y FG(H)1452 2416
y Fy(0)1491 2402 y FA(\()s Fu(e)-54 b FG(u)q FA(\()p
FG(x)p FA(\)\))21 b(+)f FG(\022)s(=)p FA(4)15 b FG(;)56
b Fx(j)p FG(x)2157 2365 y Ft(0)2180 2402 y Fx(j)26 b(\024)f
FG(l)r(=)p FA(2)15 b FG(;)57 b Fx(j)p FG(u)p FA(\()p
FG(x)p FA(\))p Fx(j)26 b(\024)f FA(1)p FG(=)p FA(2)3024
2301 y Fu(o)3101 2402 y FG(;)456 2592 y FA(then)1381
2726 y Fi(L)1436 2689 y Fv(N)1504 2625 y Fu(\020)1558
2726 y FG(\031)1610 2740 y Fv(N)1677 2726 y FA(\()1717
2703 y Fu(e)1712 2726 y FA(\004)1773 2740 y Fy(1)1813
2726 y FA(\))1848 2625 y Fu(\021)1943 2726 y Fx(\025)55
b FA(const)16 b FG(l)2321 2689 y Fv(N)7 b Ft(\000)p Fy(1)2493
2726 y FG(:)456 2902 y FA(F)-8 b(rom)30 b(this,)g(\(6.3\))i(easily)e
(follo)m(ws.)555 3069 y(W)-8 b(e)30 b(no)m(w)e(mak)m(e)i(some)e
(remarks)g(on)g(the)h(measure)f(prop)s(erties)f(of)i(the)f(ab)s(o)m(v)m
(e)i(sets.)456 3177 y(First)f(note)i(that,)h(b)m(y)e(construction,)1534
3345 y FG(\031)1586 3359 y Fv(N)1653 3345 y FA(\(\004)1749
3359 y Fy(0)1788 3345 y FA(\))41 b Fx(\022)f FG(Q)2062
3348 y Fs(l)p 2057 3360 31 3 v 2057 3401 a Fq(2)2097
3375 y Fy(+)2227 3348 y Fs(l)p 2162 3360 150 3 v 2162
3418 a Fq(4)2206 3405 y(\026)2192 3418 y Fs(C)2239 3397
y(k)2272 3412 y Fq(0)2341 3345 y FG(;)456 3548 y FA(therefore)965
3721 y Fi(L)1020 3683 y Fv(N)1088 3620 y Fu(\020)1142
3721 y FG(\031)1194 3735 y Fv(N)1261 3721 y FA(\(\004)1357
3735 y Fy(0)1397 3721 y FA(\))20 b Fx(n)h FG(Q)1590 3739
y Fv(l)q(=)p Fy(2)1686 3620 y Fu(\021)1824 3721 y Fx(\024)82
b Fi(L)2032 3683 y Fv(N)2100 3620 y Fu(\020)2154 3721
y FG(Q)2241 3724 y Fs(l)p 2236 3736 31 3 v 2236 3777
a Fq(2)2276 3751 y Fy(+)2406 3724 y Fs(l)p 2341 3736
150 3 v 2341 3794 a Fq(4)2385 3781 y(\026)2371 3794 y
Fs(C)2418 3774 y(k)2451 3789 y Fq(0)2525 3721 y Fx(n)21
b FG(Q)2663 3739 y Fv(l)q(=)p Fy(2)2759 3620 y Fu(\021)2839
3721 y Fx(\024)1824 3969 y(\024)2002 3907 y FA(const)16
b FG(l)2254 3874 y Fv(N)7 b Ft(\000)p Fy(1)p 1987 3948
425 4 v 2042 4012 a FA(\026)2022 4035 y FG(C)2094 4009
y Fy(\()p Fv(N)g Ft(\000)p Fy(1\))p Fv(k)2338 4018 y
Fq(0)2447 3969 y Fx(\024)1824 4188 y(\024)2061 4126 y
FA(const)p 1987 4167 356 4 v 2008 4231 a(\026)1987 4254
y FG(C)2059 4228 y Fy(\()p Fv(N)g Ft(\000)p Fy(1\))p
Fv(k)2303 4237 y Fq(0)2367 4188 y Fi(L)2422 4150 y Fv(N)2490
4188 y FA(\()p FG(Q)2597 4206 y Fv(l)q(=)p Fy(2)2694
4188 y FA(\))15 b FG(:)456 4390 y FA(This)28 b(and)i(\(6.2\),)i(b)m(y)f
(assuming)e FG(k)1651 4404 y Fy(0)1721 4390 y FA(large)h(enough,)h
(yield)d(that)704 4598 y Fi(L)759 4560 y Fv(N)827 4497
y Fu(\020)881 4598 y FG(\031)933 4612 y Fv(N)1000 4598
y FA(\(\004)1096 4612 y Fy(0)1135 4598 y FA(\))21 b Fx(\\)f
FG(Q)1344 4616 y Fv(l)q(=)p Fy(2)1440 4497 y Fu(\021)1520
4598 y Fx(\025)1590 4497 y Fu(\020)1645 4598 y FA(1)g
Fx(\000)g FA(\(1)h Fx(\000)f FG(c)2032 4612 y Fy(0)2072
4598 y FA(\))2107 4560 y Fv(k)2144 4569 y Fq(0)2203 4598
y Fx(\000)2378 4536 y FA(const)p 2304 4577 V 2324 4641
a(\026)2304 4664 y FG(C)2376 4637 y Fy(\()p Fv(N)7 b
Ft(\000)p Fy(1\))p Fv(k)2620 4646 y Fq(0)2669 4497 y
Fu(\021)2738 4598 y Fi(L)2793 4560 y Fv(N)2861 4598 y
FA(\()p FG(Q)2968 4616 y Fv(l)q(=)p Fy(2)3065 4598 y
FA(\))25 b Fx(\025)1520 4804 y(\025)1606 4703 y Fu(\020)1660
4804 y FA(1)20 b Fx(\000)1826 4742 y FG(c)1865 4756 y
Fy(1)p 1826 4783 79 4 v 1843 4866 a FA(2)1915 4703 y
Fu(\021)2000 4804 y Fi(L)2055 4766 y Fv(N)2122 4804 y
FA(\()p FG(Q)2229 4822 y Fv(l)q(=)p Fy(2)2326 4804 y
FA(\))15 b FG(;)456 4997 y FA(where)29 b FG(c)757 5011
y Fy(1)828 4997 y FA(is)g(the)i(constan)m(t)g(in)m(tro)s(duced)e(here)h
(ab)s(o)m(v)m(e.)42 b(Th)m(us,)1052 5186 y Fi(L)1107
5149 y Fv(N)1175 5086 y Fu(\020)1229 5186 y FG(Q)1301
5205 y Fv(l)q(=)p Fy(2)1418 5186 y Fx(n)1483 5113 y Fu(\000)1525
5186 y FG(\031)1577 5200 y Fv(N)1644 5186 y FA(\(\004)1740
5200 y Fy(0)1780 5186 y FA(\))20 b Fx(\\)g FG(Q)1988
5205 y Fv(l)q(=)p Fy(2)2084 5113 y Fu(\001)2126 5086
y(\021)2221 5186 y Fx(\024)2342 5125 y FG(c)2381 5139
y Fy(1)p 2342 5166 V 2358 5249 a FA(2)2446 5186 y Fi(L)2501
5149 y Fv(N)2568 5186 y FA(\()p FG(Q)2675 5205 y Fv(l)q(=)p
Fy(2)2772 5186 y FA(\))15 b FG(:)p eop
%%Page: 62 62
62 61 bop 456 251 a Fy(62)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(F)g(rom)30 b(this,)g(\(6.3\))i(and)e(the)h(fact)g
(that)g FG(\031)1876 464 y Fv(N)1943 450 y FA(\(\004)2039
464 y Fy(1)2078 450 y FA(\))26 b Fx(\022)f FG(Q)2307
469 y Fv(l)q(=)p Fy(2)2403 450 y FA(,)31 b(w)m(e)g(gather)g(that)900
642 y FG(c)939 656 y Fy(1)979 642 y Fi(L)1034 604 y Fv(N)1101
642 y FA(\()p FG(Q)1208 660 y Fv(l)q(=)p Fy(2)1305 642
y FA(\))25 b Fx(\024)g Fi(L)1516 604 y Fv(N)1584 541
y Fu(\020)1638 642 y FG(\031)1690 656 y Fv(N)1757 642
y FA(\(\004)1853 656 y Fy(1)1893 642 y FA(\))1928 541
y Fu(\021)2008 642 y Fx(\024)1365 841 y(\024)g Fi(L)1516
803 y Fv(N)1584 740 y Fu(\020)1638 841 y FG(\031)1690
855 y Fv(N)1757 841 y FA(\(\004)1853 855 y Fy(1)1893
841 y FA(\))20 b Fx(\\)2029 767 y Fu(\000)2071 841 y
FG(\031)2123 855 y Fv(N)2190 841 y FA(\(\004)2286 855
y Fy(0)2325 841 y FA(\))h Fx(\\)f FG(Q)2534 859 y Fv(l)q(=)p
Fy(2)2630 767 y Fu(\001)2672 740 y(\021)2726 841 y FA(+)1360
1040 y(+)g Fi(L)1506 1002 y Fv(N)1574 939 y Fu(\020)1628
1040 y FG(\031)1680 1054 y Fv(N)1747 1040 y FA(\(\004)1843
1054 y Fy(1)1883 1040 y FA(\))g Fx(n)2004 966 y Fu(\000)2045
1040 y FG(\031)2097 1054 y Fv(N)2165 1040 y FA(\(\004)2261
1054 y Fy(0)2300 1040 y FA(\))h Fx(\\)e FG(Q)2508 1058
y Fv(l)q(=)p Fy(2)2605 966 y Fu(\001)2646 939 y(\021)2726
1040 y Fx(\024)1365 1239 y(\024)25 b Fi(L)1516 1201 y
Fv(N)1584 1138 y Fu(\020)1638 1239 y FG(\031)1690 1253
y Fv(N)1757 1239 y FA(\(\004)1853 1253 y Fy(0)1893 1239
y FA(\))20 b Fx(\\)g FG(\031)2081 1253 y Fv(N)2148 1239
y FA(\(\004)2244 1253 y Fy(1)2284 1239 y FA(\))2319 1138
y Fu(\021)2373 1239 y FA(+)1360 1438 y(+)g Fi(L)1506
1400 y Fv(N)1574 1337 y Fu(\020)1628 1438 y FG(Q)1700
1456 y Fv(l)q(=)p Fy(2)1817 1438 y Fx(n)1882 1364 y Fu(\000)1924
1438 y FG(\031)1976 1452 y Fv(N)2043 1438 y FA(\(\004)2139
1452 y Fy(0)2179 1438 y FA(\))g Fx(\\)g FG(Q)2387 1456
y Fv(l)q(=)p Fy(2)2483 1364 y Fu(\001)2525 1337 y(\021)2605
1438 y Fx(\024)1365 1636 y(\024)25 b Fi(L)1516 1599 y
Fv(N)1584 1536 y Fu(\020)1638 1636 y FG(\031)1690 1650
y Fv(N)1757 1636 y FA(\(\004)1853 1650 y Fy(0)1893 1636
y FA(\))20 b Fx(\\)g FG(\031)2081 1650 y Fv(N)2148 1636
y FA(\(\004)2244 1650 y Fy(1)2284 1636 y FA(\))2319 1536
y Fu(\021)2393 1636 y FA(+)2494 1575 y FG(c)2533 1589
y Fy(1)p 2494 1616 79 4 v 2511 1699 a FA(2)2598 1636
y Fi(L)2653 1599 y Fv(N)2721 1636 y FA(\()p FG(Q)2828
1655 y Fv(l)q(=)p Fy(2)2924 1636 y FA(\))15 b FG(;)456
1814 y FA(that)31 b(is)456 1971 y(\(6.4\))519 b Fi(L)1215
1933 y Fv(N)1283 1870 y Fu(\020)1337 1971 y FG(\031)1389
1985 y Fv(N)1456 1971 y FA(\(\004)1552 1985 y Fy(0)1592
1971 y FA(\))20 b Fx(\\)g FG(\031)1780 1985 y Fv(N)1847
1971 y FA(\(\004)1943 1985 y Fy(1)1982 1971 y FA(\))2017
1870 y Fu(\021)2113 1971 y Fx(\025)2234 1909 y FG(c)2273
1923 y Fy(1)p 2234 1950 V 2250 2033 a FA(2)2338 1971
y Fi(L)2393 1933 y Fv(N)2460 1971 y FA(\()p FG(Q)2567
1989 y Fv(l)q(=)p Fy(2)2664 1971 y FA(\))15 b FG(:)456
2148 y FA(On)29 b(the)i(other)f(hand,)456 2350 y(\(6.5\))373
b(\004)1075 2364 y Fy(0)1135 2350 y Fx(\\)20 b FA(\004)1277
2364 y Fy(1)1341 2350 y Fx(\022)1437 2222 y Fu(\032)1505
2350 y Fx(\000)1586 2288 y FG(\022)p 1586 2329 46 4 v
1586 2412 a FA(4)1666 2350 y Fx(\024)25 b FG(x)1814 2364
y Fv(N)1902 2350 y Fx(\000)20 b FG(H)2069 2364 y Fy(0)2108
2350 y FA(\()p FG(u)p FA(\()p FG(x)p FA(\)\))26 b Fx(\024)f(\000)2555
2288 y FG(\022)p 2555 2329 V 2555 2412 a FA(2)2610 2222
y Fu(\033)2704 2350 y FA(=)g Fx(;)15 b FG(:)456 2550
y FA(Let)30 b(no)m(w)648 2707 y Fi(V)40 b FA(:=)893 2606
y Fu(n)954 2707 y FG(Z)32 b Fx(2)25 b FG(Q)1206 2725
y Fv(l)q(=)p Fy(2)1327 2602 y Fu(\014)1327 2657 y(\014)1327
2712 y(\014)1383 2707 y Fx(9)6 b FA(~)-51 b FG(x)25 b
Fx(6)p FA(=)30 b(^)-50 b FG(x)15 b(;)56 b FA(s)p FG(:)p
FA(t)p FG(:)25 b(Z)32 b FA(=)25 b FG(\031)2143 2721 y
Fv(N)2210 2633 y Fu(\000)2258 2707 y FA(~)-51 b FG(x;)15
b(u)p FA(\()6 b(~)-51 b FG(x)q FA(\))2519 2633 y Fu(\001)2586
2707 y FA(=)25 b FG(\031)2734 2721 y Fv(N)2801 2633 y
Fu(\000)2848 2707 y FA(^)-50 b FG(x;)15 b(u)p FA(\()6
b(^)-51 b FG(x)p FA(\))3109 2633 y Fu(\001)3151 2606
y(o)3227 2707 y FG(:)456 2886 y FA(By)30 b(\(6.5\),)j(w)m(e)e(ha)m(v)m
(e)g(that)1479 3038 y Fi(V)41 b Fx(\023)f FG(\031)1752
3052 y Fv(N)1819 3038 y FA(\(\004)1915 3052 y Fy(0)1954
3038 y FA(\))21 b Fx(\\)f FG(\031)2143 3052 y Fv(N)2210
3038 y FA(\(\004)2306 3052 y Fy(1)2345 3038 y FA(\))15
b FG(;)456 3190 y FA(th)m(us,)30 b(due)g(to)h(\(6.4\),)456
3347 y(\(6.6\))877 b Fi(L)1573 3310 y Fv(N)1641 3347
y FA(\()p Fi(V)p FA(\))41 b Fx(\025)f FA(const)16 b FG(l)2184
3310 y Fv(N)7 b Ft(\000)p Fy(1)2356 3347 y FG(:)555 3651
y FA(With)29 b(these)h(inequalities)d(in)h(hand,)h(w)m(e)h(no)m(w)f
(start)i(to)f(estimate)g(the)g(functional,)456 3759 y(in)35
b(order)i(to)h(sho)m(w)f(that)g(the)g(energy)h(of)f FG(u)g
FA(is)f(to)s(o)i(large,)h(and)d(hence)h(obtaining)f(a)456
3867 y(con)m(tradiction.)555 4018 y(First)30 b(of)g(all,)g(for)g(an)m
(y)h FG(x)1395 3985 y Ft(0)1443 4018 y Fx(2)25 b Fw(R)1589
3985 y Fv(N)1692 4018 y FA(with)k Fx(j)p FG(x)1976 3985
y Ft(0)2000 4018 y Fx(j)c(\024)g FG(l)r FA(,)31 b(let)f(us)g(de\014ne)
1128 4170 y FG(T)1181 4185 y Fv(x)1221 4166 y Fp(0)1248
4170 y FA(\()p FG(x)1335 4184 y Fv(N)1402 4170 y FA(\))84
b(:=)f FG(u)p FA(\()p FG(x)1839 4133 y Ft(0)1862 4170
y FG(;)15 b(x)1954 4184 y Fv(N)2022 4170 y FA(\))g FG(;)1321
4305 y Fi(C)1372 4320 y Fv(x)1412 4301 y Fp(0)1521 4305
y FA(:=)83 b Fx(f)p FG(x)1797 4319 y Fv(N)1890 4305 y
Fx(2)24 b Fw(R)34 b Fx(j)41 b FG(D)s(T)2263 4320 y Fv(x)2303
4301 y Fp(0)2329 4305 y FA(\()p FG(x)2416 4319 y Fv(N)2484
4305 y FA(\))25 b(=)g(0)p Fx(g)15 b FG(:)456 4457 y FA(By)36
b(standard)g(regularit)m(y)g(results)f(\(see)j([15)q(])f(and)f([34)q
(]\),)j(w)m(e)e(ha)m(v)m(e)h(that)f FG(T)3108 4472 y
Fv(x)3148 4453 y Fp(0)3211 4457 y FA(is)e FG(C)3380 4424
y Fy(1)3419 4457 y FA(.)456 4565 y(Hence,)c(b)m(y)f(Sard's)g(Lemma,)456
4739 y(\(6.7\))921 b Fi(L)1617 4701 y Fv(N)1685 4638
y Fu(\020)1739 4739 y FG(T)1792 4754 y Fv(x)1832 4735
y Fp(0)1858 4739 y FA(\()p Fi(C)1944 4754 y Fv(x)1984
4735 y Fp(0)2011 4739 y FA(\))2046 4638 y Fu(\021)2141
4739 y FA(=)40 b(0)15 b FG(:)456 4913 y FA(Th)m(us,)29
b(using)g(that)i FG(T)1199 4929 y Fv(x)1239 4910 y Fp(0)1296
4913 y FA(is)e(lo)s(cally)g(in)m(v)m(ertible)g(on)h(the)h(complemen)m
(t)g(of)f Fi(C)3022 4929 y Fv(x)3062 4910 y Fp(0)3088
4913 y FA(,)h(w)m(e)g(ma)m(y)456 5021 y(write)e(the)i(latter)f(set)h
(as)1785 5049 y Fu([)1817 5240 y Fv(a)1901 5135 y FG(J)1951
5150 y Fv(a;x)2048 5132 y Fp(0)2090 5135 y FG(;)p eop
%%Page: 63 63
63 62 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(63)456
472 y FA(in)23 b(suc)m(h)h(a)h(w)m(a)m(y)g FG(T)1056
487 y Fv(x)1096 468 y Fp(0)1122 367 y Fu(\014)1122 422
y(\014)1122 476 y(\014)1153 535 y Fv(J)1192 554 y Fs(a;x)1280
540 y Fp(0)1334 472 y FA(is)e(a)i(di\013eomorphism.)36
b(Therefore,)26 b(b)m(y)e(Y)-8 b(oung's)25 b(inequalit)m(y)456
640 y(\(writing)36 b FG(q)41 b FA(for)d(the)g(dual)f(exp)s(onen)m(t)h
(of)g FG(p)p FA(\))g(and)g(b)m(y)g(c)m(hanging)g(v)-5
b(ariable)37 b FG(x)3153 654 y Fv(N)7 b Fy(+1)3348 640
y FA(:=)456 748 y FG(T)509 763 y Fv(x)549 744 y Fp(0)575
748 y FA(\()p FG(x)662 762 y Fv(N)730 748 y FA(\),)1199
836 y Fu(Z)1249 1042 y Fv(J)1288 1061 y Fs(a;x)1376 1047
y Fp(0)1431 898 y Fx(jr)p FG(u)p FA(\()p FG(x)1671 865
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1325 y Fp(0)1431 1176 y Fx(j)p FG(@)1504 1190 y Fv(N)1572
1176 y FG(u)p FA(\()p FG(x)1711 1143 y Ft(0)1734 1176
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1452 y FA(\))p Fx(j)1868 1419 y Fv(p)p 1431 1493 478
4 v 1647 1576 a FG(p)1938 1514 y FA(+)20 b FG(h)2081
1528 y Fy(0)2121 1514 y FA(\()p FG(T)2209 1529 y Fv(x)2249
1510 y Fp(0)2276 1514 y FA(\()p FG(x)2363 1528 y Fv(N)2430
1514 y FA(\)\))15 b FG(dx)2614 1528 y Fv(N)2708 1514
y Fx(\025)1045 1788 y(\025)1199 1664 y Fu(Z)1249 1871
y Fv(J)1288 1890 y Fs(a;x)1376 1876 y Fp(0)1421 1687
y Fu(\020)1475 1788 y FG(q)j(h)1586 1802 y Fy(0)1626
1788 y FA(\()p FG(T)1714 1803 y Fv(x)1754 1785 y Fp(0)1781
1788 y FA(\()p FG(x)1868 1802 y Fv(N)1935 1788 y FA(\)\))2005
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FG(D)s(T)2340 1803 y Fv(x)2380 1785 y Fp(0)2406 1788
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2150 y Fp(0)1531 2145 y Fy(\))1578 1962 y Fu(\020)1632
2063 y FG(q)j(h)1743 2077 y Fy(0)1783 2063 y FA(\()p
FG(x)1870 2077 y Fv(N)7 b Fy(+1)2028 2063 y FA(\))2063
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2077 y Fv(N)g Fy(+1)2513 2063 y FG(;)456 2295 y FA(therefore,)899
2411 y Fu(X)946 2602 y Fv(a)1045 2374 y Fu(Z)1096 2580
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2585 y Fp(0)1446 2580 y Fy(\()p Fv(J)1512 2599 y Fs(a;x)1600
2585 y Fp(0)1626 2580 y Fy(\))1673 2397 y Fu(\020)1727
2498 y FG(q)18 b(h)1838 2512 y Fy(0)1878 2498 y FA(\()p
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2860 y Fv(J)1383 2880 y Fs(a;x)1471 2866 y Fp(0)1525
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FG(dx)2803 2792 y Fv(N)2887 2778 y FG(dx)2986 2740 y
Ft(0)3035 2778 y FA(=)745 3058 y(=)899 2935 y Fu(Z)949
3141 y Ft(j)p Fv(x)1009 3122 y Fp(0)1031 3141 y Ft(j\024)p
Fv(l)1147 2935 y Fu(Z)1197 3141 y Fy([)p Ft(\000)p Fv(C)5
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3146 y Fp(0)1632 2997 y Fx(jr)p FG(u)p FA(\()p FG(x)1872
2964 y Ft(0)1896 2997 y FG(;)15 b(x)1988 3011 y Fv(N)2056
2997 y FA(\))p Fx(j)2116 2964 y Fv(p)p 1632 3037 V 1871
3121 a FG(p)2186 3058 y FA(+)20 b FG(h)2329 3072 y Fy(0)2369
3058 y FA(\()p FG(u)p FA(\()p FG(x)2543 3021 y Ft(0)2567
3058 y FG(;)15 b(x)2659 3072 y Fv(N)2726 3058 y FA(\)\))g
FG(dx)2910 3072 y Fv(N)2994 3058 y FG(dx)3093 3021 y
Ft(0)3142 3058 y Fx(\024)745 3263 y(\024)83 b(F)964 3277
y Fv(A)1017 3289 y Fs(l)1045 3263 y FA(\()p FG(u)p FA(\))15
b FG(;)-751 b FA(\(6.8\))456 3420 y(where)29 b FG(A)786
3435 y Fv(l)838 3420 y FA(:=)c Fx(fj)p FG(x)1081 3387
y Ft(0)1105 3420 y Fx(j)h FG(<)f(l)r Fx(g)20 b(\002)g(fj)p
FG(x)1559 3434 y Fv(N)1627 3420 y Fx(j)25 b FG(<)g(C)7
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3605 y Fi(V)40 b Fx(\022)676 3504 y Fu(n)737 3605 y FA(\()p
FG(x)824 3567 y Ft(0)847 3605 y FG(;)15 b FA(0)p FG(;)g(x)1024
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3555 y(\014)1244 3610 y(\014)1299 3605 y Fx(j)p FG(x)1376
3567 y Ft(0)1400 3605 y Fx(j)25 b(\024)g FG(l)r(;)41
b(x)1693 3619 y Fv(N)7 b Fy(+1)1875 3605 y Fx(2)25 b
FG(T)2014 3620 y Fv(x)2054 3601 y Fp(0)2081 3605 y FA(\()p
FG(J)2166 3620 y Fv(a;x)2263 3601 y Fp(0)2290 3605 y
FA(\))p Fx(\\)p FG(T)2439 3620 y Fv(x)2479 3601 y Fp(0)2505
3605 y FA(\()p FG(J)2591 3620 y Fy(^)-36 b Fv(a)q(;x)2688
3601 y Fp(0)2714 3605 y FA(\))31 b(for)f(some)h FG(a)25
b Fx(6)p FA(=)h(^)-46 b FG(a)3364 3504 y Fu(o)3440 3605
y FG(;)456 3785 y FA(and)29 b(that)1391 3904 y Fi(V)40
b Fx(\022)g FG(Q)1683 3919 y Fv(l)1749 3904 y Fx(\022)h(fj)p
FG(x)1983 3918 y Fv(N)7 b Fy(+1)2141 3904 y Fx(j)25 b(\024)g
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4126 y Fu(Z)1080 4333 y Fv(x)1120 4344 y Fs(N)5 b Fq(+1)1255
4333 y Ft(2)1302 4283 y Fk(S)1361 4353 y Fs(a)p Fp(6)p
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4338 y Fp(0)1773 4333 y Fy(\))p Ft(\\)p Fv(T)1888 4352
y Fs(x)1923 4338 y Fp(0)1949 4333 y Fy(\()p Fv(J)2017
4352 y Fq(^)g Fs(a;x)2103 4338 y Fp(0)2129 4333 y Fy(\))2176
4149 y Fu(\020)2230 4250 y FG(q)18 b(h)2341 4264 y Fy(0)2381
4250 y FA(\()p FG(x)2468 4264 y Fv(N)7 b Fy(+1)2626 4250
y FA(\))2661 4149 y Fu(\021)2715 4172 y Fy(1)p Fv(=q)2839
4250 y FG(dx)2938 4264 y Fv(N)g Fy(+1)3111 4250 y FG(dx)3210
4213 y Ft(0)3259 4250 y Fx(\025)628 4526 y(\025)782 4403
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FG(q)18 b(h)1073 4540 y Fy(0)1113 4526 y FA(\()p FG(x)1200
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4448 y Fy(1)p Fv(=q)1571 4526 y FG(d)p FA(\()p FG(x)1705
4489 y Ft(0)1729 4526 y FG(;)15 b(x)1821 4540 y Fv(N)7
b Fy(+1)1978 4526 y FA(\))26 b Fx(\025)628 4732 y(\025)196
b FA(inf)782 4796 y Fy([)p Ft(\000)p Fy(1)p Fv(=)p Fy(2)p
Fv(;)12 b Fy(1)p Fv(=)p Fy(2])1119 4732 y FA(\()p FG(q)18
b(h)1265 4746 y Fy(0)1305 4732 y FA(\))1340 4694 y Fy(1)p
Fv(=q)1464 4732 y Fi(L)1519 4694 y Fv(N)1587 4732 y FA(\()p
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b FG(c)821 4959 y Fy(1)876 4945 y FG(l)905 4907 y Fv(N)7
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834 y Fy(0)1718 820 y FA(\()p FG(x)1805 834 y Fv(N)7
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1130 y Fk(S)1096 1200 y Fs(a)1146 1180 y Fv(T)1187 1199
y Fs(x)1222 1185 y Fp(0)1248 1180 y Fy(\()p Fv(J)1314
1199 y Fs(a;x)1402 1185 y Fp(0)1428 1180 y Fy(\))1475
996 y Fu(\020)1529 1097 y FG(q)18 b(h)1640 1111 y Fy(0)1680
1097 y FA(\()p FG(x)1767 1111 y Fv(N)7 b Fy(+1)1924 1097
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1097 y FG(dx)2237 1111 y Fv(N)g Fy(+1)2410 1097 y FG(dx)2509
1060 y Ft(0)2552 1097 y FA(+)739 1372 y(+)825 1248 y
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1475 y Fs(a)p Fp(6)p Fq(=)r(^)-32 b Fs(a)1533 1454 y
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Fv(J)1702 1473 y Fs(a;x)1790 1459 y Fp(0)1815 1454 y
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1454 y Fy(\()p Fv(J)2060 1473 y Fq(^)g Fs(a;x)2146 1459
y Fp(0)2172 1454 y Fy(\))2219 1271 y Fu(\020)2273 1372
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1662 y Fy(0)2067 1648 y FA(\()p FG(x)2154 1662 y Fv(N)7
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1648 y FG(dx)2896 1611 y Ft(0)2939 1648 y FA(+)739 1872
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1872 y FG(:)456 2148 y FA(Th)m(us,)29 b(due)h(to)h(\(6.7\),)1047
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y Fy(1)p Fv(=q)2225 3724 y FG(dx)2324 3738 y Fv(N)g Fy(+1)2497
3724 y FG(dx)2596 3686 y Ft(0)2644 3724 y FA(=)888 3998
y(=)83 b FG(!)1099 4012 y Fv(N)7 b Ft(\000)p Fy(1)1271
3998 y FG(l)1300 3961 y Fv(N)g Ft(\000)p Fy(1)1488 3874
y Fu(Z)1579 3901 y Fy(1)p Ft(\000)p Fv(s)1702 3913 y
Fs(l)1538 4081 y Ft(\000)p Fy(1+)p Fv(s)1716 4093 y Fs(l)1759
3897 y Fu(\020)1814 3998 y FG(q)18 b(h)1925 4012 y Fy(0)1964
3998 y FA(\()p FG(x)2051 4012 y Fv(N)7 b Fy(+1)2209 3998
y FA(\))2244 3897 y Fu(\021)2299 3920 y Fy(1)p Fv(=q)2423
3998 y FG(dx)2522 4012 y Fv(N)g Fy(+1)2694 3998 y FG(;)-2263
b FA(\(6.10\))456 4277 y(where)31 b FG(!)777 4291 y Fv(N)7
b Ft(\000)p Fy(1)934 4277 y FA(,)33 b(as)f(usual,)g(denotes)g(the)g(v)m
(olume)g(of)g(the)h(\()p FG(N)e Fx(\000)21 b FA(1\)-dimensional)31
b(unit)456 4385 y(ball.)39 b(F)-8 b(rom)31 b(\(6.9\))h(and)e(\(6.10\),)
i(w)m(e)f(th)m(us)f(obtain)g(that)1092 4566 y Fu(X)1139
4757 y Fv(a)1239 4528 y Fu(Z)1289 4735 y Ft(j)p Fv(x)1349
4716 y Fp(0)1371 4735 y Ft(j\024)p Fv(l)1486 4528 y Fu(Z)1537
4735 y Fv(T)1578 4754 y Fs(x)1613 4740 y Fp(0)1639 4735
y Fy(\()p Fv(J)1705 4754 y Fs(a;x)1793 4740 y Fp(0)1819
4735 y Fy(\))1866 4551 y Fu(\020)1920 4652 y FG(q)18
b(h)2031 4666 y Fy(0)2071 4652 y FA(\()p FG(x)2158 4666
y Fv(N)7 b Fy(+1)2316 4652 y FA(\))2351 4551 y Fu(\021)2405
4574 y Fy(1)p Fv(=q)2529 4652 y FG(dx)2628 4666 y Fv(N)g
Fy(+1)2801 4652 y FG(dx)2900 4615 y Ft(0)2949 4652 y
Fx(\025)938 4941 y(\025)83 b FG(!)1149 4955 y Fv(N)7
b Ft(\000)p Fy(1)1321 4941 y FG(l)1350 4904 y Fv(N)g
Ft(\000)p Fy(1)1538 4818 y Fu(Z)1629 4844 y Fy(1)p Ft(\000)p
Fv(s)1752 4856 y Fs(l)1588 5024 y Ft(\000)p Fy(1+)p Fv(s)1766
5036 y Fs(l)1810 4840 y Fu(\020)1864 4941 y FG(q)18 b(h)1975
4955 y Fy(0)2015 4941 y FA(\()p FG(x)2102 4955 y Fv(N)7
b Fy(+1)2259 4941 y FA(\))2294 4840 y Fu(\021)2349 4863
y Fy(1)p Fv(=q)2473 4941 y FG(dx)2572 4955 y Fv(N)g Fy(+1)2750
4941 y FA(+)1092 5158 y(+)r(~)-47 b FG(c)1202 5172 y
Fy(1)1257 5158 y FG(l)1286 5121 y Fv(N)7 b Ft(\000)p
Fy(1)1581 5158 y FA(inf)1473 5223 y Fy([)p Ft(\000)p
Fy(1)p Fv(=)p Fy(2)p Fv(;)p Fy(1)p Fv(=)p Fy(2])1799
5158 y FA(\()p FG(q)18 b(h)1945 5172 y Fy(0)1985 5158
y FA(\))2020 5121 y Fy(1)p Fv(=q)2144 5158 y FG(;)p eop
%%Page: 65 65
65 64 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(65)456
450 y FA(and,)30 b(therefore,)h(thanks)f(to)h(\(6.8\),)1194
607 y Fx(F)1259 621 y Fv(A)1312 633 y Fs(l)1341 607 y
FA(\()p FG(u)p FA(\))26 b Fx(\025)1040 808 y(\025)83
b FG(!)1251 822 y Fv(N)7 b Ft(\000)p Fy(1)1423 808 y
FG(l)1452 771 y Fv(N)g Ft(\000)p Fy(1)1640 684 y Fu(Z)1731
711 y Fy(1)p Ft(\000)p Fv(s)1854 723 y Fs(l)1690 891
y Ft(\000)p Fy(1+)p Fv(s)1868 903 y Fs(l)1912 707 y Fu(\020)1966
808 y FG(q)18 b(h)2077 822 y Fy(0)2117 808 y FA(\()p
FG(x)2204 822 y Fv(N)7 b Fy(+1)2361 808 y FA(\))2396
707 y Fu(\021)2451 730 y Fy(1)p Fv(=q)2575 808 y FG(dx)2674
822 y Fv(N)g Fy(+1)2852 808 y FA(+)1194 1025 y(+)r(~)-47
b FG(c)1304 1039 y Fy(1)1359 1025 y FG(l)1388 987 y Fv(N)7
b Ft(\000)p Fy(1)1683 1025 y FA(inf)1575 1090 y Fy([)p
Ft(\000)p Fy(1)p Fv(=)p Fy(2)p Fv(;)p Fy(1)p Fv(=)p Fy(2])1901
1025 y FA(\()p FG(q)18 b(h)2047 1039 y Fy(0)2087 1025
y FA(\))2122 987 y Fy(1)p Fv(=q)2246 1025 y FG(:)-1815
b FA(\(6.11\))456 1236 y(W)-8 b(e)31 b(no)m(w)f(notice)h(that,)g(if)e
FG(c)1413 1250 y Fy(2)1483 1236 y FA(and)g FG(c)1698
1250 y Fy(3)1768 1236 y FA(are)i(p)s(ositiv)m(e)e(constan)m(ts,)j
(suitably)c(small)h(with)456 1344 y(resp)s(ect)h(to)j(~)-47
b FG(c)917 1358 y Fy(1)957 1344 y FA(,)30 b(one)h(has)962
1547 y FG(!)1019 1561 y Fv(N)7 b Ft(\000)p Fy(1)1191
1547 y FG(l)1220 1509 y Fv(N)g Ft(\000)p Fy(1)1408 1423
y Fu(Z)1458 1629 y Fy([)p Ft(\000)p Fy(1)p Fv(;)p Ft(\000)p
Fy(1+)p Fv(c)1764 1638 y Fq(2)1798 1629 y Fy(])p Ft([)p
Fy([1)p Ft(\000)p Fv(c)2006 1638 y Fq(2)2040 1629 y Fv(;)p
Fy(1])2133 1446 y Fu(\020)2188 1547 y FG(q)18 b(h)2299
1561 y Fy(0)2339 1547 y FA(\()p FG(x)2426 1561 y Fv(N)7
b Fy(+1)2583 1547 y FA(\))2618 1446 y Fu(\021)2673 1468
y Fy(1)p Fv(=q)2797 1547 y FG(dx)2896 1561 y Fv(N)g Fy(+1)3079
1547 y Fx(\024)808 1768 y(\024)83 b FA(2)p FG(c)1046
1782 y Fy(2)1086 1768 y FG(!)1143 1782 y Fv(N)7 b Ft(\000)p
Fy(1)1316 1768 y FG(l)1345 1731 y Fv(N)g Ft(\000)p Fy(1)1556
1768 y FA(sup)1532 1851 y Fy([)p Ft(\000)p Fy(1)p Fv(;)p
Fy(1])1716 1768 y FA(\()p FG(q)19 b(h)1863 1782 y Fy(0)1903
1768 y FA(\()p FG(x)1990 1782 y Fv(N)7 b Fy(+1)2147 1768
y FA(\)\))2217 1731 y Fy(1)p Fv(=q)2352 1768 y Fx(\024)808
2034 y(\024)974 1973 y FA(~)-47 b FG(c)1011 1987 y Fy(1)p
972 2013 79 4 v 989 2097 a FA(2)1076 2034 y FG(l)1105
1997 y Fv(N)7 b Ft(\000)p Fy(1)1400 2034 y FA(inf)1293
2099 y Fy([)p Ft(\000)p Fy(1)p Fv(=)p Fy(2)p Fv(;)p Fy(1)p
Fv(=)p Fy(2])1618 2034 y FA(\()p FG(q)18 b(h)1764 2048
y Fy(0)1804 2034 y FA(\))1839 1997 y Fy(1)p Fv(=q)456
2034 y FA(\(6.12\))456 2245 y(and)456 2425 y(\(6.13\))550
b FG(c)1275 2439 y Fy(3)1330 2425 y FG(l)1359 2387 y
Fv(N)7 b Ft(\000)p Fy(1)1542 2425 y Fx(\024)1649 2363
y FA(~)-47 b FG(c)1686 2377 y Fy(1)p 1647 2404 V 1664
2487 a FA(2)1751 2425 y FG(l)1780 2387 y Fv(N)7 b Ft(\000)p
Fy(1)2075 2425 y FA(inf)1968 2489 y Fy([)p Ft(\000)p
Fy(1)p Fv(=)p Fy(2)p Fv(;)p Fy(1)p Fv(=)p Fy(2])2293
2425 y FA(\()p FG(q)19 b(h)2440 2439 y Fy(0)2480 2425
y FA(\))2515 2387 y Fy(1)p Fv(=q)2639 2425 y FG(:)456
2640 y FA(W)-8 b(e)40 b(no)m(w)g(assume)g FG(l)h FA(big)e(enough)h(so)f
(that)i FG(s)2066 2655 y Fv(l)2132 2640 y FG(<)g(c)2283
2654 y Fy(2)2322 2640 y FA(:)60 b(then,)42 b(b)m(y)d(means)h(of)g
(\(6.11\),)456 2748 y(\(6.12\))32 b(and)e(\(6.13\),)879
2969 y Fx(F)944 2983 y Fv(A)997 2995 y Fs(l)1025 2969
y FA(\()p FG(u)p FA(\))84 b Fx(\025)f FG(!)1442 2983
y Fv(N)7 b Ft(\000)p Fy(1)1614 2969 y FG(l)1643 2931
y Fv(N)g Ft(\000)p Fy(1)1830 2845 y Fu(Z)1921 2871 y
Fy(1)1881 3051 y Ft(\000)p Fy(1)1990 2868 y Fu(\020)2045
2969 y FG(q)18 b(h)2156 2983 y Fy(0)2195 2969 y FA(\()p
FG(x)2282 2983 y Fv(N)7 b Fy(+1)2440 2969 y FA(\))2475
2868 y Fu(\021)2530 2890 y Fy(1)p Fv(=q)2654 2969 y FG(dx)2753
2983 y Fv(N)g Fy(+1)2930 2969 y FA(+)1385 3176 y(+)p
FG(c)1495 3190 y Fy(3)1549 3176 y FG(l)1578 3139 y Fv(N)g
Ft(\000)p Fy(1)1751 3176 y FG(:)-1320 b FA(\(6.14\))555
3490 y(This)37 b(estimate)j(will)c(sa)m(y)j(that)g(the)g(energy)g(of)g
FG(u)g FA(is)f(to)s(o)h(large)g(\(thanks)g(to)g(the)456
3599 y(term)f(\\)p FG(c)765 3613 y Fy(3)820 3599 y FG(l)849
3566 y Fv(N)7 b Ft(\000)p Fy(1)1006 3599 y FA(")39 b(here)f(ab)s(o)m(v)
m(e\),)j(and)d(it)f(will)e(pro)m(vide)j(the)g(desired)e(con)m
(tradiction.)456 3707 y(F)-8 b(or)31 b(this,)e(let)i(us)e(de\014ne)h
(the)g(rescaled)h(functional)1101 3919 y Fx(F)1175 3882
y Fv(")1166 3942 y Fy(\012)1222 3919 y FA(\()p FG(v)s
FA(\))26 b(:=)1486 3795 y Fu(Z)1537 4001 y Fy(\012)1617
3858 y FG(")1659 3825 y Fv(p)p Ft(\000)p Fy(1)1789 3858
y Fx(jr)p FG(v)s FA(\()p FG(x)p FA(\))p Fx(j)2084 3825
y Fv(p)p 1617 3898 509 4 v 1848 3981 a FG(p)2155 3919
y FA(+)2256 3858 y(1)p 2256 3898 46 4 v 2258 3981 a FG(")2311
3919 y(h)2363 3933 y Fy(0)2403 3919 y FA(\()p FG(v)s
FA(\()p FG(x)p FA(\)\))15 b FG(dx)g(:)456 4127 y FA(Then,)29
b(if)h FG(")25 b FA(:=)g(1)p FG(=l)34 b FA(and)29 b FG(u)1369
4141 y Fv(")1406 4127 y FA(\()p FG(x)p FA(\))d(:=)f FG(u)p
FA(\()p FG(x=")p FA(\),)32 b(b)m(y)f(scaling)e(\(6.14\),)k(w)m(e)e
(deduce)f(that)881 4290 y Fx(F)955 4252 y Fv(")946 4312
y(A)999 4321 y Fq(1)1038 4290 y FA(\()p FG(u)1125 4304
y Fv(")1162 4290 y FA(\))83 b(=)g FG(")1476 4252 y Fv(N)7
b Ft(\000)p Fy(1)1649 4290 y Fx(F)1714 4304 y Fv(A)1767
4316 y Fs(l)1796 4290 y FA(\()p FG(u)p FA(\))26 b Fx(\025)1280
4497 y(\025)83 b FG(!)1491 4511 y Fv(N)7 b Ft(\000)p
Fy(1)1678 4373 y Fu(Z)1769 4399 y Fy(1)1729 4579 y Ft(\000)p
Fy(1)1838 4396 y Fu(\020)1893 4497 y FG(q)18 b(h)2004
4511 y Fy(0)2044 4497 y FA(\()p FG(x)2131 4511 y Fv(N)7
b Fy(+1)2288 4497 y FA(\))2323 4396 y Fu(\021)2378 4419
y Fy(1)p Fv(=q)2502 4497 y FG(dx)2601 4511 y Fv(N)g Fy(+1)2794
4497 y FA(+)35 b FG(c)2939 4511 y Fy(3)2994 4497 y FG(:)-2563
b FA(\(6.15\))456 4712 y(On)32 b(the)i(other)g(hand,)f(b)m(y)h
Fx(x)p FA(3)g(of)g([7],)h(up)e(to)h(subsequences,)g(w)m(e)g(ha)m(v)m(e)
g(that)g FG(u)3207 4726 y Fv(")3278 4712 y FA(con-)456
4819 y(v)m(erges)e(almost)g(ev)m(erywhere)g(and)f(in)g
FG(L)1847 4786 y Fy(1)1847 4847 y(lo)r(c)1970 4819 y
FA(to)i(the)f(step)f(function)g FG(\037)2849 4833 y Fv(E)2929
4819 y Fx(\000)21 b FG(\037)3078 4841 y Fj(R)3126 4822
y Fs(N)3184 4841 y Ft(n)p Fv(E)3278 4819 y FA(,)33 b(for)456
4944 y(a)d(suitable)f(set)i FG(E)g Fx(\022)25 b Fw(R)1265
4911 y Fv(N)1339 4944 y FA(,)30 b(and)g(that)456 5159
y(\(6.16\))447 b(lim)1101 5225 y Fv(")p Ft(!)p Fy(0)1240
5206 y Fq(+)1306 5159 y Fx(F)1380 5122 y Fv(")1371 5182
y(A)1424 5191 y Fq(1)1463 5159 y FA(\()p FG(u)1550 5173
y Fv(")1587 5159 y FA(\))25 b(=)40 b(P)m(er)16 b(\()p
FG(E)5 b(;)15 b(A)2124 5173 y Fy(1)2165 5159 y FA(\))2230
5036 y Fu(Z)2321 5062 y Fy(1)2281 5242 y Ft(\000)p Fy(1)2390
5059 y Fu(\020)2445 5159 y FG(q)j(h)2556 5173 y Fy(0)2595
5059 y Fu(\021)2650 5081 y Fy(1)p Fv(=q)2773 5159 y FG(;)p
eop
%%Page: 66 66
66 65 bop 456 251 a Fy(66)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(where,)31 b(giv)m(en)g FG(A)c Fx(\022)f
FG(B)5 b FA(,)32 b(w)m(e)f(denote)h(the)g(p)s(erimeter)e(of)h
FG(A)g FA(in)f FG(B)36 b FA(as)c(P)m(er)15 b(\()p FG(A;)g(B)5
b FA(\))32 b(\(see,)456 558 y(for)38 b(instance,)i([20)q(])e(for)g
(full)e(details)h(on)i(suc)m(h)f(de\014nition\).)62 b(As)38
b(a)h(matter)g(of)f(fact,)456 666 y(in)f(our)h(situation,)j(the)e(set)g
FG(E)44 b FA(ma)m(y)39 b(b)s(e)g(b)s(etter)f(sp)s(eci\014ed,)i(in)d
(the)i(follo)m(wing)f(w)m(a)m(y)-8 b(.)456 774 y(F)g(rom)31
b(\(6.1\),)i(there)e(exists)g FG(\024)26 b(>)g FA(0)31
b(so)g(that)h FG(u)p FA(\()p FG(x)p FA(\))27 b Fx(\025)f
FG(\024)31 b FA(if)f Fx(j)p FG(x)2526 741 y Ft(0)2549
774 y Fx(j)c(\024)g FG(l)33 b FA(and)d FG(x)2986 788
y Fv(N)3080 774 y Fx(\025)25 b FA(2)p FG(\022)34 b FA(and)456
882 y FG(u)p FA(\()p FG(x)p FA(\))26 b Fx(\024)f(\000)p
FG(\024)30 b FA(if)f Fx(j)p FG(x)1065 849 y Ft(0)1089
882 y Fx(j)c(\024)g FG(l)32 b FA(and)e FG(x)1523 896
y Fv(N)1616 882 y Fx(\024)25 b(\000)p FA(2)p FG(\022)s
FA(.)456 990 y(Therefore,)j FG(u)943 1004 y Fv(")979
990 y FA(\()p FG(x)p FA(\))e Fx(\025)f FG(\024)j FA(if)e
Fx(j)p FG(x)1460 957 y Ft(0)1484 990 y Fx(j)f(\024)g
FA(1)j(and)f FG(x)1929 1004 y Fv(N)2021 990 y Fx(\025)e
FA(2)p FG("\022)31 b FA(and)26 b FG(u)p FA(\()p FG(x)p
FA(\))g Fx(\024)f(\000)p FG(\024)j FA(if)e Fx(j)p FG(x)3055
957 y Ft(0)3079 990 y Fx(j)f(\024)g FA(1)j(and)456 1098
y FG(x)508 1112 y Fv(N)600 1098 y Fx(\024)d(\000)p FA(2)p
FG("\022)s FA(.)40 b(In)30 b(particular,)f(for)h(almost)h(an)m(y)f
FG(x)c Fx(2)e FG(A)2359 1112 y Fy(1)2399 1098 y FA(,)1482
1268 y(lim)1450 1334 y Fv(")p Ft(!)p Fy(0)1589 1315 y
Fq(+)1655 1268 y FG(u)1707 1282 y Fv(")1743 1268 y FA(\()p
FG(x)p FA(\))i Fx(\025)f FG(\024)31 b FA(if)e FG(x)2205
1282 y Fv(N)2298 1268 y FG(>)c FA(0)30 b(and)1482 1446
y(lim)1450 1512 y Fv(")p Ft(!)p Fy(0)1589 1493 y Fq(+)1655
1446 y FG(u)1707 1460 y Fv(")1743 1446 y FA(\()p FG(x)p
FA(\))c Fx(\024)f(\000)p FG(\024)31 b FA(if)e FG(x)2276
1460 y Fv(N)2368 1446 y FG(>)c FA(0.)456 1660 y(This)j(implies)g(that)j
FG(E)f FA(=)25 b FG(A)1430 1674 y Fy(1)1490 1660 y Fx(\\)20
b(f)p FG(x)1668 1674 y Fv(N)1760 1660 y FG(>)25 b FA(0)p
Fx(g)p FA(.)42 b(And)29 b(so)46 b(P)m(er)16 b(\()p FG(E)5
b(;)15 b(A)2705 1674 y Fy(1)2745 1660 y FA(\))26 b(=)f
FG(!)2959 1674 y Fv(N)7 b Ft(\000)p Fy(1)3116 1660 y
FA(.)41 b(There-)456 1768 y(fore,)30 b(from)g(\(6.16\))1247
2002 y(lim)1215 2068 y Fv(")p Ft(!)p Fy(0)1354 2049 y
Fq(+)1419 2002 y Fx(F)1493 1964 y Fv(")1484 2024 y(A)1537
2033 y Fq(1)1576 2002 y FA(\()p FG(u)1663 2016 y Fv(")1700
2002 y FA(\))41 b(=)25 b FG(!)1929 2016 y Fv(N)7 b Ft(\000)p
Fy(1)2117 1878 y Fu(Z)2207 1904 y Fy(1)2167 2084 y Ft(\000)p
Fy(1)2276 1901 y Fu(\020)2331 2002 y FG(q)18 b(h)2442
2016 y Fy(0)2482 1901 y Fu(\021)2536 1923 y Fy(1)p Fv(=q)2660
2002 y FG(:)456 2230 y FA(This)28 b(con)m(tradicts)j(\(6.15\))i(and)d
(\014nishes)e(the)i(pro)s(of)g(of)g(Theorem)h(1.1.)1376
2483 y(7.)46 b FB(Pr)n(oof)34 b(of)f(Theorem)h(1.2)555
2645 y FA(The)k(pro)s(of)f(of)i(Theorem)f(1.2)h(will)d(b)s(e)h(p)s
(erformed)g(b)m(y)h(compactness,)k(b)m(y)c(using)456
2753 y(Theorem)30 b(1.1)h(and)f(a)h(result)e(of)i([30)q(].)555
2924 y(W)-8 b(e)30 b(\014x)f FG(\022)881 2938 y Fy(0)945
2924 y FG(>)c FA(0)k(and)g(w)m(e)g(assume)g(b)m(y)g(con)m(tradiction)g
(that)g(there)g(exist)g FG(u)3105 2939 y Fv(k)3163 2924
y FG(;)i(\022)3262 2939 y Fv(k)3319 2924 y FG(;)g(l)3402
2939 y Fv(k)456 3031 y FA(for)f(whic)m(h)508 3171 y FH(\(C1\))40
b FG(u)810 3186 y Fv(k)895 3171 y FA(is)h(a)i(lo)s(cal)e(minimizer)e
(for)j Fx(F)52 b FA(in)40 b Fx(fj)p FG(x)2255 3138 y
Ft(0)2279 3171 y Fx(j)45 b FG(<)g(l)2492 3186 y Fv(k)2535
3171 y Fx(g)28 b(\002)g(fj)p FG(x)2829 3185 y Fv(N)2897
3171 y Fx(j)45 b FG(<)f(l)3109 3186 y Fv(k)3152 3171
y Fx(g)p FA(,)i(with)758 3279 y FG(u)810 3294 y Fv(k)853
3279 y FA(\(0\))26 b(=)f(0.)508 3396 y FH(\(C2\))40 b
Fx(f)p FG(u)855 3411 y Fv(k)935 3396 y FA(=)d(0)p Fx(g)53
b(\022)f(fj)p FG(x)1431 3363 y Ft(0)1455 3396 y Fx(j)37
b FG(<)f(l)1651 3411 y Fv(k)1694 3396 y Fx(g)25 b(\002)g(fj)p
FG(x)1982 3410 y Fv(N)2050 3396 y Fx(j)37 b FG(<)g(\022)2263
3411 y Fv(k)2305 3396 y Fx(g)p FA(,)j(with)c FG(\022)2672
3411 y Fv(k)2751 3396 y Fx(\025)g FG(\022)2901 3410 y
Fy(0)2978 3396 y FA(and)3171 3356 y Fv(\022)3204 3368
y Fs(k)p 3171 3375 72 4 v 3177 3427 a Fv(l)3198 3439
y Fs(k)3252 3396 y Fx(\000)-15 b(!)p FA(0)758 3510 y(when)30
b FG(k)e Fx(!)d(1)15 b FA(,)456 3649 y(but)31 b(the)i(thesis)e(of)i
(Theorem)f(1.2)h(do)s(es)f(not)h(hold.)45 b(Let)33 b(us)f(consider)f
(the)i(follo)m(wing)456 3757 y(rescaling:)456 3977 y(\(7.1\))686
b FG(y)1375 3939 y Ft(0)1424 3977 y FA(=)1530 3915 y
FG(x)1582 3882 y Ft(0)p 1530 3956 76 4 v 1533 4039 a
FG(l)1560 4054 y Fv(k)1887 3977 y FA(;)289 b FG(y)2246
3991 y Fv(N)2338 3977 y FA(=)2444 3915 y FG(x)2496 3929
y Fv(N)p 2444 3956 120 4 v 2461 4039 a FG(\022)2504 4054
y Fv(k)456 4194 y FA(sa)m(y)31 b(\()p FG(y)696 4161 y
Ft(0)719 4194 y FG(;)15 b(y)804 4208 y Fv(N)871 4194
y FA(\))26 b(=)f FG(T)13 b FA(\()p FG(x)1181 4161 y Ft(0)1204
4194 y FG(;)i(x)1296 4208 y Fv(N)1364 4194 y FA(\).)41
b(De\014ne)687 4392 y FG(A)755 4407 y Fv(k)838 4392 y
FA(:=)974 4319 y Fu(\010)1027 4392 y FA(\()p FG(y)1110
4355 y Ft(0)1134 4392 y FG(;)15 b(y)1219 4406 y Fv(N)1286
4392 y FA(\))41 b(s)p FG(:)p FA(t)p FG(:)g(T)1590 4355
y Ft(\000)p Fy(1)1684 4392 y FA(\()p FG(y)1767 4355 y
Ft(0)1791 4392 y FG(;)15 b(y)1876 4406 y Fv(N)1943 4392
y FA(\))25 b Fx(2)g(f)p FG(u)2186 4407 y Fv(k)2255 4392
y FA(=)g(0)p Fx(g)2441 4319 y Fu(\011)2535 4392 y FA(=)40
b FG(T)2712 4291 y Fu(\020)2766 4392 y Fx(f)p FG(u)2863
4407 y Fv(k)2932 4392 y FA(=)25 b(0)p Fx(g)3118 4291
y Fu(\021)3188 4392 y FG(:)555 4663 y FH(STEP)k(1:)38
b FA(There)25 b(exists)g(a)h(H\177)-45 b(older)25 b(con)m(tin)m(uous)g
(function)f FG(w)43 b FA(:)d Fw(R)2874 4630 y Fv(N)7
b Ft(\000)p Fy(1)3063 4663 y Fx(!)25 b Fw(R)34 b FA(suc)m(h)456
4771 y(that:)41 b(if)29 b(w)m(e)i(de\014ne)1344 4924
y FG(A)1412 4938 y Ft(1)1528 4924 y FA(:=)1664 4850 y
Fu(\010)1717 4924 y FA(\()p FG(y)1800 4886 y Ft(0)1823
4924 y FG(;)15 b(w)r FA(\()p FG(y)2013 4886 y Ft(0)2038
4924 y FA(\)\))g FG(;)31 b Fx(j)p FG(y)2252 4886 y Ft(0)2276
4924 y Fx(j)25 b(\024)2432 4862 y FA(1)p 2432 4903 46
4 v 2432 4986 a(2)2488 4850 y Fu(\011)456 5108 y FA(then,)30
b(for)f(an)m(y)i FG(")25 b(>)g FA(0,)31 b FG(A)1330 5123
y Fv(k)1393 5108 y Fx(\\)19 b(fj)p FG(y)1591 5075 y Ft(0)1615
5108 y Fx(j)25 b(\024)g FA(1)p FG(=)p FA(2)p Fx(g)32
b FA(lies)d(in)f(a)j FG(")p FA(-neigh)m(b)s(orho)s(o)s(d)d(of)i
FG(A)3125 5122 y Ft(1)3200 5108 y FA(,)g(for)g FG(k)456
5216 y FA(su\016cien)m(tly)f(large.)p eop
%%Page: 67 67
67 66 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(67)456
450 y Fo(Pr)-5 b(o)g(of)34 b(of)f(step)g(1.)456 558 y
FA(Let)d(us)g(supp)s(ose)f(that)1048 705 y FG(y)1093
719 y Fy(0)1157 705 y FA(=)c(\()p FG(y)1336 668 y Ft(0)1333
728 y Fy(0)1372 705 y FG(;)15 b(y)1457 719 y Fy(0)d Fv(N)1572
705 y FA(\))25 b Fx(2)g FG(A)1786 720 y Fv(k)2036 705
y FA(with)206 b Fx(j)p FG(y)2493 668 y Ft(0)2490 728
y Fy(0)2529 705 y Fx(j)26 b(\024)f FA(1)p FG(=)p FA(2)15
b FG(:)456 853 y FA(Then,)31 b FG(u)772 868 y Fv(k)815
853 y FA(\()p FG(l)877 868 y Fv(k)920 853 y FG(y)968
820 y Ft(0)965 877 y Fy(0)1004 853 y FG(;)15 b(\022)1087
868 y Fv(k)1130 853 y FG(y)1175 867 y Fy(0)c Fv(N)1288
853 y FA(\))28 b(=)g(0,)33 b(and)e(so,)h(b)m(y)g(means)g(of)g
FH(\(C2\))p FA(,)f Fx(j)p FG(\022)2716 868 y Fv(k)2759
853 y FG(y)2804 867 y Fy(0)11 b Fv(N)2918 853 y Fx(j)27
b FG(<)h(\022)3112 868 y Fv(k)3154 853 y FA(;)33 b(there-)456
961 y(fore,)d(using)f(again)i FH(\(C2\))p FA(,)e(w)m(e)i(infer)e(that)
1244 1108 y Fx(f)p FG(u)1341 1123 y Fv(k)1409 1108 y
FA(=)c(0)p Fx(g)42 b(\022)e(fj)p FG(x)1870 1122 y Fv(N)1958
1108 y Fx(\000)20 b FG(\022)2092 1123 y Fv(k)2134 1108
y FG(y)2179 1122 y Fy(0)12 b Fv(N)2293 1108 y Fx(j)25
b FG(<)g FA(2)p FG(\022)2527 1123 y Fv(k)2570 1108 y
Fx(g)15 b FG(:)456 1256 y FA(Thence,)30 b(w)m(e)h(can)g(exploit)e
(Theorem)h(1.1)i(in)d(the)h(cylinder)1066 1446 y Fx(fj)p
FG(x)1188 1408 y Ft(0)1232 1446 y Fx(\000)20 b FG(l)1350
1461 y Fv(k)1393 1446 y FG(y)1441 1408 y Ft(0)1438 1468
y Fy(0)1477 1446 y Fx(j)26 b FG(<)1634 1384 y(l)1661
1399 y Fv(k)p 1634 1425 70 4 v 1646 1508 a FA(2)1713
1446 y Fx(g)21 b(\002)f(fj)p FG(x)1992 1460 y Fv(N)2080
1446 y Fx(\000)g FG(\022)2214 1461 y Fv(k)2256 1446 y
FG(y)2301 1460 y Fy(0)12 b Fv(N)2415 1446 y Fx(j)26 b
FG(<)f FA(2)p FG(\022)2650 1461 y Fv(k)2692 1446 y Fx(g)h(\022)1092
1618 y(\022)e(fj)p FG(x)1309 1580 y Ft(0)1333 1618 y
Fx(j)i FG(<)f(l)1507 1633 y Fv(k)1550 1618 y Fx(g)20
b(\002)g(fj)p FG(x)1828 1632 y Fv(N)1896 1618 y Fx(j)26
b FG(<)f(l)2070 1633 y Fv(k)2112 1618 y Fx(g)15 b FG(;)456
1511 y FA(\(7.2\))456 1765 y(and)29 b(get)j(that)f(there)f(exists)g(a)h
(univ)m(ersal)e(constan)m(t)i FG(\021)2336 1779 y Fy(0)2401
1765 y FG(>)25 b FA(0)31 b(suc)m(h)f(that)649 1955 y
Fx(f)p FG(u)746 1970 y Fv(k)814 1955 y FA(=)25 b(0)p
Fx(g)37 b(\\)e(fj)p FG(x)1255 1918 y Ft(0)1299 1955 y
Fx(\000)20 b FG(l)1417 1970 y Fv(k)1459 1955 y FG(y)1507
1918 y Ft(0)1504 1978 y Fy(0)1543 1955 y Fx(j)26 b FG(<)f(\021)1735
1969 y Fy(0)1784 1894 y FG(l)1811 1909 y Fv(k)p 1784
1935 V 1797 2018 a FA(2)1864 1955 y Fx(g)h(\022)f(fj)p
FG(x)2153 1969 y Fv(N)2241 1955 y Fx(\000)20 b FG(\022)2375
1970 y Fv(k)2417 1955 y FG(y)2462 1969 y Fy(0)12 b Fv(N)2576
1955 y Fx(j)26 b FG(<)f FA(2\(1)c Fx(\000)f FG(\021)3005
1969 y Fy(0)3044 1955 y FA(\))p FG(\022)3122 1970 y Fv(k)3165
1955 y Fx(g)15 b FG(;)456 2129 y FA(pro)m(vided)1664
2220 y(4)p FG(\022)1752 2235 y Fv(k)p 1664 2260 131 4
v 1694 2343 a FG(l)1721 2358 y Fv(k)1830 2281 y Fx(\024)25
b FG(")1968 2295 y Fy(0)2008 2281 y FA(\(2)p FG(\022)2131
2295 y Fy(0)2170 2281 y FA(\))15 b FG(;)456 2455 y FA(where)29
b FG(")760 2469 y Fy(0)800 2455 y FA(\()p Fx(\001)p FA(\))j(is)d(the)i
(one)f(giv)m(en)h(b)m(y)f(Theorem)g(1.1.)42 b(Rescaling)30
b(bac)m(k,)h(w)m(e)g(get)946 2627 y FG(A)1014 2642 y
Fv(k)1077 2627 y Fx(\\)20 b(fj)p FG(y)1276 2589 y Ft(0)1320
2627 y Fx(\000)g FG(y)1459 2589 y Ft(0)1456 2649 y Fy(0)1495
2627 y Fx(j)25 b FG(<)1651 2565 y(\021)1696 2579 y Fy(0)p
1651 2606 85 4 v 1671 2689 a FA(2)1745 2627 y Fx(g)h(\022)f(fj)p
FG(y)2027 2641 y Fv(N)2115 2627 y Fx(\000)20 b FG(y)2251
2641 y Fy(0)11 b Fv(N)2365 2627 y Fx(j)25 b FG(<)g FA(2\(1)d
Fx(\000)d FG(\021)2793 2641 y Fy(0)2833 2627 y FA(\))p
Fx(g)c FG(:)456 2799 y FA(By)30 b(iterating,)g(w)m(e)h(get)456
2987 y(\(7.3\))256 b FG(A)965 3002 y Fv(k)1028 2987 y
Fx(\\)20 b(fj)p FG(y)1227 2950 y Ft(0)1271 2987 y Fx(\000)g
FG(y)1410 2950 y Ft(0)1407 3010 y Fy(0)1446 2987 y Fx(j)26
b FG(<)1602 2926 y(\021)1650 2893 y Fv(m)1647 2950 y
Fy(0)p 1602 2967 115 4 v 1637 3050 a FA(2)1727 2987 y
Fx(g)g(\022)f(fj)p FG(y)2009 3001 y Fv(N)2097 2987 y
Fx(\000)20 b FG(y)2233 3001 y Fy(0)11 b Fv(N)2347 2987
y Fx(j)25 b FG(<)g FA(2\(1)c Fx(\000)f FG(\021)2775 3001
y Fy(0)2815 2987 y FA(\))2850 2950 y Fv(m)2917 2987 y
Fx(g)15 b FG(;)456 3161 y FA(pro)m(vided)456 3351 y(\(7.4\))1308
3289 y(4)p FG(\022)1396 3304 y Fv(k)p 1308 3330 131 4
v 1338 3413 a FG(l)1365 3428 y Fv(k)1474 3351 y Fx(\024)25
b FG(\021)1618 3313 y Fv(m)p Ft(\000)p Fy(1)1615 3377
y(0)1775 3351 y FG(")1817 3365 y Fy(0)1857 3250 y Fu(\020)1911
3351 y FA(2\(1)d Fx(\000)e FG(\021)2194 3365 y Fy(0)2233
3351 y FA(\))2268 3313 y Fv(m)p Ft(\000)p Fy(1)2425 3351
y FG(\022)2468 3365 y Fy(0)2507 3250 y Fu(\021)2577 3351
y FG(:)456 3544 y FA(W)-8 b(e)44 b(no)m(w)g(\014x)f FG(m)1050
3558 y Fy(0)1136 3544 y Fx(2)k Fw(N)56 b FA(and)43 b(consider)f
FG(m)47 b Fx(\024)g FG(m)2235 3558 y Fy(0)2317 3544 y
FA(\(later)d(on,)j(during)41 b(a)j(limiting)456 3652
y(pro)s(cedure)35 b(p)s(erformed)g(on)i(page)h(69,)h(w)m(e)e(let)g
FG(m)2189 3666 y Fy(0)2264 3652 y Fx(\000)-15 b(!)36
b FA(+)p Fx(1)p FA(\).)60 b(Note)38 b(that,)h(in)c(this)456
3760 y(setting,)h(\(7.4\))h(\(and)e(therefore)h(\(7.3\)\))h(is)d
(ful\014lled)e(for)i FG(k)39 b FA(suitably)33 b(large,)k(sa)m(y)e
FG(k)i Fx(\025)456 3868 y FG(k)506 3835 y Fv(?)545 3868
y FA(\()p FG(m)660 3882 y Fy(0)700 3868 y FA(\).)k(W)-8
b(e)31 b(claim)f(that)h FG(A)1465 3883 y Fv(k)1528 3868
y Fx(\\)20 b(fj)p FG(y)1727 3835 y Ft(0)1750 3868 y Fx(j)26
b(\024)f FA(1)p FG(=)p FA(2)p Fx(g)32 b FA(is)d(ab)s(o)m(v)m(e)j(the)f
(graph)e(of)456 4022 y(\(7.5\))445 b(\011)1157 4037 y
Fv(y)1192 4046 y Fq(0)1226 4037 y Fv(;k)1289 4022 y FA(\()p
FG(y)1372 3984 y Ft(0)1395 4022 y FA(\))26 b(=)f FG(y)1597
4036 y Fy(0)11 b Fv(N)1731 4022 y Fx(\000)20 b FA(2\(1)h
Fx(\000)f FG(\021)2104 4036 y Fy(0)2144 4022 y FA(\))2179
3984 y Fv(m)2241 3993 y Fq(0)2300 4022 y Fx(\000)g FG(\013)p
Fx(j)p FG(y)2522 3984 y Ft(0)2566 4022 y Fx(\000)g FG(y)2705
3984 y Ft(0)2702 4044 y Fy(0)2741 4022 y Fx(j)2766 3984
y Fv(\014)456 4169 y FA(where)29 b FG(\013)i FA(and)f
FG(\014)h(>)24 b FA(0)31 b(dep)s(end)e(only)g(on)i FG(\021)1924
4183 y Fy(0)1963 4169 y FA(.)456 4277 y(T)-8 b(o)32 b(pro)m(v)m(e)h
(this,)e(let)h(\()p FG(y)1258 4244 y Ft(0)1281 4277 y
FG(;)15 b(y)1366 4291 y Fv(N)1434 4277 y FA(\))28 b Fx(2)f
FG(A)1653 4292 y Fv(k)1717 4277 y Fx(\\)21 b(fj)p FG(y)1917
4244 y Ft(0)1941 4277 y Fx(j)28 b(\024)f FA(1)p FG(=)p
FA(2)p Fx(g)p FA(.)48 b(Since)30 b Fx(j)p FG(y)2656 4244
y Ft(0)2653 4302 y Fy(0)2693 4277 y Fx(j)e(\024)2854
4242 y Fy(1)p 2854 4257 36 4 v 2854 4309 a(2)2931 4277
y FA(w)m(e)33 b(ha)m(v)m(e)g(that)456 4426 y Fx(j)p FG(y)529
4393 y Ft(0)560 4426 y Fx(\000)8 b FG(y)687 4393 y Ft(0)684
4450 y Fy(0)722 4426 y Fx(j)25 b(\024)g FA(1.)39 b(No)m(w,)27
b(w)m(e)d(consider)f(three)h(di\013eren)m(t)g(cases:)38
b(the)24 b(case)i Fx(j)p FG(y)2929 4393 y Ft(0)2959 4426
y Fx(\000)8 b FG(y)3086 4393 y Ft(0)3083 4450 y Fy(0)3122
4426 y Fx(j)25 b(\024)3278 4376 y Fv(\021)3315 4338 y
Fs(m)3369 4353 y Fq(0)3313 4397 y(0)p 3278 4405 132 4
v 3326 4457 a Fy(2)3419 4426 y FA(,)456 4574 y(the)30
b(case)815 4525 y Fv(\021)852 4487 y Fs(m)906 4502 y
Fq(0)850 4546 y(0)p 815 4553 V 862 4605 a Fy(2)981 4574
y Fx(\024)25 b(j)p FG(y)1150 4541 y Ft(0)1193 4574 y
Fx(\000)20 b FG(y)1332 4541 y Ft(0)1329 4598 y Fy(0)1368
4574 y Fx(j)26 b(\024)1525 4538 y Fy(1)p 1525 4553 36
4 v 1525 4605 a(2)1570 4574 y FA(,)k(and)g(the)h(case)2161
4538 y Fy(1)p 2161 4553 V 2161 4605 a(2)2232 4574 y Fx(\024)25
b(j)p FG(y)2401 4541 y Ft(0)2444 4574 y Fx(\000)20 b
FG(y)2583 4541 y Ft(0)2580 4598 y Fy(0)2619 4574 y Fx(j)26
b(\024)f FA(1.)456 4722 y(In)k(case)i Fx(j)p FG(y)834
4689 y Ft(0)877 4722 y Fx(\000)19 b FG(y)1015 4689 y
Ft(0)1012 4747 y Fy(0)1051 4722 y Fx(j)25 b(\024)1207
4673 y Fv(\021)1244 4635 y Fs(m)1298 4650 y Fq(0)1242
4694 y(0)p 1207 4701 132 4 v 1255 4754 a Fy(2)1348 4722
y FA(,)30 b(\(7.5\))i(follo)m(ws)d(immediately)g(from)g(\(7.3\))r(,)h
(with)f FG(m)c FA(=)g FG(m)3380 4736 y Fy(0)3419 4722
y FA(.)456 4870 y(If,)37 b(on)e(the)h(other)g(hand,)1385
4821 y Fv(\021)1422 4783 y Fs(m)1476 4798 y Fq(0)1420
4842 y(0)p 1385 4850 V 1433 4902 a Fy(2)1560 4870 y Fx(\024)e(j)p
FG(y)1738 4837 y Ft(0)1785 4870 y Fx(\000)24 b FG(y)1928
4837 y Ft(0)1925 4895 y Fy(0)1964 4870 y Fx(j)34 b(\024)2138
4835 y Fy(1)p 2138 4850 36 4 v 2138 4902 a(2)2184 4870
y FA(,)j(then)e(w)m(e)i(argue)f(as)g(follo)m(ws.)56 b(W)-8
b(e)456 4978 y(\014rst)29 b(note)i(that,)g(in)f(this)f(case,)i(there)g
(exists)f FG(m)g FA(with)f(0)d Fx(\024)f FG(m)g Fx(\024)g
FG(m)2817 4992 y Fy(0)2856 4978 y FA(,)31 b(suc)m(h)f(that)456
5186 y(\(7.6\))1480 5125 y FG(\021)1528 5087 y Fv(m)p
Fy(+1)1525 5151 y(0)p 1480 5166 206 4 v 98 x FA(2)1721
5186 y Fx(\024)25 b(j)p FG(y)1890 5149 y Ft(0)1933 5186
y Fx(\000)20 b FG(y)2072 5149 y Ft(0)2069 5209 y Fy(0)2108
5186 y Fx(j)26 b(\024)2264 5125 y FG(\021)2312 5092 y
Fv(m)2309 5149 y Fy(0)p 2264 5166 115 4 v 2299 5249 a
FA(2)2405 5186 y FG(:)p eop
%%Page: 68 68
68 67 bop 456 251 a Fy(68)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(Consequen)m(tly)g(,)30 b(from)g(\(7.3\))r(,)g(w)m(e)
h(ha)m(v)m(e)h(that)456 611 y(\(7.7\))813 b(2\(1)21 b
Fx(\000)f FG(\021)1736 625 y Fy(0)1776 611 y FA(\))1811
574 y Fv(m)1903 611 y Fx(\025)25 b(j)p FG(y)2069 625
y Fv(N)2156 611 y Fx(\000)20 b FG(y)2292 625 y Fy(0)12
b Fv(N)2406 611 y Fx(j)456 773 y FA(By)32 b(\(7.6\))g(and)d(the)i(fact)
g(that)g(0)26 b FG(<)f(\021)1738 787 y Fy(0)1803 773
y FG(<)g FA(1,)31 b(w)m(e)g(also)f(get)1343 989 y FG(m)25
b Fx(\024)1554 927 y(\000)15 b FA(ln)o(\(2)p Fx(j)p FG(y)1869
894 y Ft(0)1913 927 y Fx(\000)20 b FG(y)2052 894 y Ft(0)2049
951 y Fy(0)2088 927 y Fx(j)p FA(\))p 1554 968 595 4 v
1733 1057 a(ln)o(\()1872 1021 y Fy(1)p 1854 1036 71 4
v 1854 1089 a Fv(\021)1889 1098 y Fq(0)1934 1057 y FA(\))2184
989 y Fx(\024)25 b FG(m)20 b FA(+)g(1)15 b FG(:)456 1226
y FA(In)29 b(particular,)g(it)h(follo)m(ws)g(that)1200
1482 y(\(1)21 b Fx(\000)f FG(\021)1437 1496 y Fy(0)1476
1482 y FA(\))1511 1444 y Fv(m)1603 1482 y Fx(\024)25
b FA(\(1)c Fx(\000)f FG(\021)1936 1496 y Fy(0)1976 1482
y FA(\))2011 1323 y Fu(\000)2063 1354 y Fp(\000)10 b
Fq(ln)o(\(2)p Fp(j)p Fs(y)2279 1333 y Fp(0)2302 1354
y(\000)p Fs(y)2384 1333 y Fp(0)2382 1375 y Fq(0)2416
1354 y Fp(j)p Fq(\))p 2063 1381 396 3 v 2168 1436 a(ln)o(\()2271
1416 y(1)p 2253 1424 67 3 v 2253 1457 a Fs(\021)2285
1472 y Fq(0)2329 1436 y(\))2468 1396 y Ft(\000)p Fy(1)2558
1323 y Fu(\001)2630 1482 y FA(=)1169 1723 y(=)1408 1661
y(1)p 1275 1702 312 4 v 1275 1785 a(\(1)21 b Fx(\000)e
FG(\021)1511 1799 y Fy(0)1551 1785 y FA(\))1596 1723
y FG(e)1638 1685 y Fv(\014)c Fy(ln\(2)p Ft(j)p Fv(y)1871
1662 y Fp(0)1894 1685 y Ft(\000)p Fv(y)1986 1662 y Fp(0)1984
1706 y Fq(0)2019 1685 y Ft(j)p Fy(\))2095 1723 y FA(=)2201
1661 y(\(2)p Fx(j)p FG(y)2354 1628 y Ft(0)2398 1661 y
Fx(\000)20 b FG(y)2537 1628 y Ft(0)2534 1685 y Fy(0)2573
1661 y Fx(j)p FA(\))2633 1628 y Fv(\014)p 2201 1702 481
4 v 2285 1785 a FA(\(1)h Fx(\000)f FG(\021)2522 1799
y Fy(0)2562 1785 y FA(\))2706 1723 y FG(;)456 1945 y
FA(where)29 b FG(\014)46 b FA(:=)961 1900 y Ft(\000)12
b Fy(ln\(1)p Ft(\000)p Fv(\021)1239 1909 y Fq(0)1275
1900 y Fy(\))p 961 1924 341 4 v 1032 1987 a(ln)o(\()1146
1960 y Fq(1)p 1128 1972 67 3 v 1128 2013 a Fs(\021)1160
2028 y Fq(0)1204 1987 y Fy(\))1312 1945 y FA(.)456 2105
y(Therefore,)30 b(recalling)f(\(7.7\),)j(it)e(follo)m(ws)1325
2330 y Fx(j)p FG(y)1395 2344 y Fv(N)1483 2330 y Fx(\000)20
b FG(y)1619 2344 y Fy(0)11 b Fv(N)1732 2330 y Fx(j)26
b(\024)1953 2268 y FA(2)1998 2235 y Fv(\014)s Fy(+1)p
1889 2309 312 4 v 1889 2392 a FA(\(1)21 b Fx(\000)f FG(\021)2126
2406 y Fy(0)2165 2392 y FA(\))2211 2330 y Fx(j)p FG(y)2284
2292 y Ft(0)2327 2330 y Fx(\000)g FG(y)2466 2292 y Ft(0)2463
2352 y Fy(0)2502 2330 y Fx(j)2527 2292 y Fv(\014)456
2556 y FA(whic)m(h)29 b(is)g(the)i(desired)e(result,)g(with)g
FG(\013)d FA(:=)f(2)2008 2523 y Fv(\014)s Fy(+1)2146
2556 y FG(=)p FA(\(1)c Fx(\000)f FG(\021)2428 2570 y
Fy(0)2468 2556 y FA(\).)456 2665 y(Finally)-8 b(,)32
b(ev)m(en)m(tually)h(adding)1492 2629 y Fy(12)1595 2665
y FA(a)g(constan)m(t)i(to)f FG(\013)p FA(,)g(the)g(result)e(also)h
(follo)m(ws)f(for)h(the)456 2773 y(case)e Fx(j)p FG(y)721
2740 y Ft(0)765 2773 y Fx(\000)20 b FG(y)904 2740 y Ft(0)901
2797 y Fy(0)940 2773 y Fx(j)25 b(2)g FA([1)p FG(=)p FA(2)p
FG(;)15 b FA(1].)43 b(This)29 b(ends)g(the)i(pro)s(of)f(of)g(\(7.5\).)
555 2881 y(Note)36 b(no)m(w)e(that,)j(as)d FG(y)1360
2895 y Fy(0)1434 2881 y FA(v)-5 b(aries,)35 b(\011)1791
2896 y Fv(y)1826 2905 y Fq(0)1860 2896 y Fv(;k)1957 2881
y FA(are)g(H\177)-45 b(older)34 b(con)m(tin)m(uous)g(functions)f(with)
456 2996 y(H\177)-45 b(older)29 b(mo)s(dulus)f(of)i(con)m(tin)m(uit)m
(y)g(b)s(ounded)e(via)i(the)g(function)f FG(\013t)2773
2963 y Fv(\014)2851 2996 y FA(\(recall)g(that)i FG(m)3405
3010 y Fy(0)456 3104 y FA(is)h(\014xed)g(for)h(the)h(momen)m(t,)g(and)f
(that)h FG(\013)f FA(and)g FG(\014)38 b FA(dep)s(end)32
b(only)g(on)h FG(\021)2888 3118 y Fy(0)2928 3104 y FA(\).)49
b(Therefore,)456 3212 y(if)29 b(w)m(e)i(set)1464 3335
y FG( )1523 3350 y Fv(k)1566 3335 y FA(\()p FG(y)1649
3298 y Ft(0)1672 3335 y FA(\))26 b(:=)68 b(sup)1864 3420
y Fp(j)p Fs(y)1917 3402 y Fp(0)1915 3444 y Fq(0)1949
3420 y Fp(j\024)2026 3400 y Fq(1)p 2026 3408 31 3 v 2026
3443 a(2)1869 3487 y Fs(y)1901 3502 y Fq(0)1936 3487
y Fp(2)p Fs(A)2024 3505 y(k)2076 3335 y FA(\011)2147
3350 y Fv(y)2182 3359 y Fq(0)2217 3350 y Fv(;k)2279 3335
y FA(\()p FG(y)2362 3298 y Ft(0)2385 3335 y FA(\))456
3625 y(then,)26 b FG( )743 3640 y Fv(k)811 3625 y FA(is)e(a)h(H\177)-45
b(older)25 b(con)m(tin)m(uous)f(function)g(\(with)g(H\177)-45
b(older)25 b(mo)s(dulus)d(of)j(con)m(tin)m(uit)m(y)456
3735 y(b)s(ounded)j(via)i(the)h(function)e FG(\013t)1580
3702 y Fv(\014)1627 3735 y FA(\),)i(and)f FG(A)1963 3750
y Fv(k)2026 3735 y Fx(\\)20 b(fj)p FG(y)2225 3702 y Ft(0)2249
3735 y Fx(j)25 b(\024)g FA(1)p FG(=)p FA(2)p Fx(g)33
b FA(is)c(ab)s(o)m(v)m(e)j(the)e(graph)g(of)456 3843
y FG( )515 3858 y Fv(k)557 3843 y FA(.)555 3951 y(Arguing)c(in)g(the)i
(same)g(w)m(a)m(y)-8 b(,)29 b(p)s(ossibly)c(taking)i
FG(\013)h FA(and)f FG(\014)32 b FA(larger)27 b(\(dep)s(ending)e(only)
456 4059 y(on)30 b FG(\021)627 4073 y Fy(0)666 4059 y
FA(\),)h(w)m(e)g(also)g(get)g(that,)g(if)f(w)m(e)g(de\014ne)1056
4227 y(\010)1122 4242 y Fv(y)1157 4251 y Fq(0)1191 4242
y Fv(;k)1253 4227 y FA(\()p FG(y)1336 4189 y Ft(0)1360
4227 y FA(\))25 b(:=)h FG(y)1587 4241 y Fy(0)11 b Fv(N)1721
4227 y FA(+)20 b(2\(1)h Fx(\000)f FG(\021)2094 4241 y
Fy(0)2134 4227 y FA(\))2169 4189 y Fv(m)2231 4198 y Fq(0)2290
4227 y FA(+)g FG(\013)p Fx(j)p FG(y)2512 4189 y Ft(0)2556
4227 y Fx(\000)g FG(y)2695 4189 y Ft(0)2692 4249 y Fy(0)2731
4227 y Fx(j)2756 4189 y Fv(\014)2819 4227 y FG(;)456
4388 y FA(then)35 b FG(A)736 4403 y Fv(k)802 4388 y Fx(\\)23
b(fj)p FG(y)1004 4355 y Ft(0)1028 4388 y Fx(j)34 b(\024)f
FA(1)p FG(=)p FA(2)p Fx(g)38 b FA(is)c(b)s(elo)m(w)h(the)g(graph)g(of)h
(\010)2367 4403 y Fv(y)2402 4412 y Fq(0)2436 4403 y Fv(;k)2498
4388 y FA(.)56 b(Arguing)34 b(as)i(ab)s(o)m(v)m(e,)i(w)m(e)456
4496 y(de\014ne)1457 4620 y FG(\036)1511 4635 y Fv(k)1553
4620 y FA(\()p FG(y)1636 4582 y Ft(0)1660 4620 y FA(\))25
b(:=)81 b(inf)1852 4686 y Fp(j)p Fs(y)1905 4669 y Fp(0)1903
4711 y Fq(0)1937 4686 y Fp(j\024)2013 4667 y Fq(1)p 2013
4675 V 2013 4710 a(2)1857 4754 y Fs(y)1889 4769 y Fq(0)1923
4754 y Fp(2)p Fs(A)2011 4771 y(k)2064 4620 y FA(\010)2130
4635 y Fv(y)2165 4644 y Fq(0)2199 4635 y Fv(;k)2261 4620
y FA(\()p FG(y)2344 4582 y Ft(0)2368 4620 y FA(\))15
b FG(;)p 456 4843 499 4 v 555 4914 a Fy(12)626 4943 y
FF(Notice)26 b(indeed)f(that,)h(b)n(y)e FC(\(C2\))p FF(,)j(w)n(e)f(ha)n
(v)n(e)f(that)1421 5079 y Fr(j)p FD(y)1480 5087 y Fs(N)1555
5079 y Fr(\000)17 b FD(y)1670 5087 y Fq(0)11 b Fs(N)1769
5079 y Fr(j)21 b(\024)g(j)p FD(y)1951 5087 y Fs(N)2009
5079 y Fr(j)d FF(+)e Fr(j)p FD(y)2183 5087 y Fq(0)c Fs(N)2282
5079 y Fr(j)22 b(\024)f FF(2)13 b FD(:)p eop
%%Page: 69 69
69 68 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(69)456
450 y FA(so)26 b(that)g FG(\036)809 465 y Fv(k)878 450
y FA(is)f(a)i(H\177)-45 b(older)25 b(con)m(tin)m(uous)h(function)e
(\(with)h(H\177)-45 b(older)26 b(mo)s(dulus)d(of)j(con)m(tin)m(u-)456
560 y(it)m(y)i(b)s(ounded)e(via)i(the)g(function)f FG(\013t)1705
527 y Fv(\014)1752 560 y FA(\),)j(and)d FG(A)2084 575
y Fv(k)2143 560 y Fx(\\)16 b(fj)p FG(y)2338 527 y Ft(0)2361
560 y Fx(j)26 b(\024)f FA(1)p FG(=)p FA(2)p Fx(g)30 b
FA(is)d(b)s(elo)m(w)g(the)i(graph)456 668 y(of)h FG(\036)613
683 y Fv(k)656 668 y FA(.)555 821 y(In)38 b(particular,)h
FG(A)1200 836 y Fv(k)1268 821 y Fx(\\)25 b(fj)p FG(y)1472
788 y Ft(0)1496 821 y Fx(j)38 b(\024)h FA(1)p FG(=)p
FA(2)p Fx(g)h FA(lies)d(b)s(et)m(w)m(een)h FG( )2471
836 y Fv(k)2514 821 y FA(\()p FG(y)2597 788 y Ft(0)2621
821 y FA(\))g(and)g FG(\036)2933 836 y Fv(k)2976 821
y FA(\()p FG(y)3059 788 y Ft(0)3082 821 y FA(\))h(for)f(an)m(y)456
929 y FG(k)28 b Fx(\025)d FG(k)677 896 y Fv(?)717 929
y FA(\()p FG(m)832 943 y Fy(0)871 929 y FA(\))31 b(and,)f(b)m(y)g
(construction)456 1083 y(\(7.8\))619 b(0)26 b Fx(\024)f
FG(\036)1481 1098 y Fv(k)1523 1083 y FA(\()p FG(y)1606
1045 y Ft(0)1630 1083 y FA(\))20 b Fx(\000)g FG( )1835
1098 y Fv(k)1878 1083 y FA(\()p FG(y)1961 1045 y Ft(0)1985
1083 y FA(\))25 b Fx(\024)g FA(4\(1)d Fx(\000)e FG(\021)2424
1097 y Fy(0)2463 1083 y FA(\))2498 1045 y Fv(m)2560 1054
y Fq(0)2615 1083 y FG(:)456 1236 y FA(Also,)k(for)f FG(m)892
1250 y Fy(0)954 1236 y FA(\014xed,)h(b)m(y)f(Ascoli-Arzel\022)-45
b(a)23 b(Theorem,,)i(letting)e FG(k)28 b Fx(!)d(1)p FA(,)g(it)e(follo)m
(ws)f(that)456 1344 y FG( )515 1359 y Fv(k)557 1344 y
FA(\()p FG(y)640 1311 y Ft(0)664 1344 y FA(\))33 b(uniformly)c(con)m(v)
m(erges)34 b(to)f(a)g(H\177)-45 b(older)32 b(con)m(tin)m(uous)g
(function)f(whic)m(h)g(dep)s(ends)456 1452 y(on)f FG(m)662
1466 y Fy(0)701 1452 y FA(,)h(Sa)m(y)1512 1567 y(lim)1437
1629 y Fv(k)r Ft(\000)-13 b(!)p Fy(+)p Ft(1)1730 1567
y FG( )1789 1582 y Fv(k)1831 1567 y FA(\()p FG(y)1914
1530 y Ft(0)1938 1567 y FA(\))26 b Fx(!)f FG(w)2182 1530
y Ft(\000)2180 1590 y Fv(m)2242 1599 y Fq(0)2281 1567
y FA(\()p FG(y)2364 1530 y Ft(0)2387 1567 y FA(\))15
b FG(:)456 1753 y FA(Analogously)-8 b(,)30 b(w)m(e)h(\014nd)e(a)h
(H\177)-45 b(older)30 b(con)m(tin)m(uous)g(function)f
FG(w)2554 1720 y Fy(+)2552 1775 y Fv(m)2614 1784 y Fq(0)2653
1753 y FA(,)i(suc)m(h)f(that)1535 1916 y(lim)1460 1977
y Fv(k)r Ft(\000)-13 b(!)p Fy(+)p Ft(1)1752 1916 y FG(\036)1806
1931 y Fv(k)1849 1916 y FA(\()p FG(y)1932 1878 y Ft(0)1956
1916 y FA(\))25 b Fx(!)g FG(w)2199 1878 y Fy(+)2197 1938
y Fv(m)2259 1947 y Fq(0)2299 1916 y FA(\()p FG(y)2382
1878 y Ft(0)2405 1916 y FA(\))456 2120 y(uniformly)-8
b(.)37 b(Also,)28 b(b)m(y)g(construction,)g(w)m(e)g(ha)m(v)m(e)h(that)f
FG(w)2392 2087 y Ft(\000)2390 2142 y Fv(m)2452 2151 y
Fq(0)2516 2120 y Fx(\024)d FG(w)2679 2087 y Fy(+)2677
2142 y Fv(m)2739 2151 y Fq(0)2806 2120 y FA(and)i FG(A)3048
2135 y Fv(k)3106 2120 y Fx(\\)15 b(fj)p FG(y)3300 2087
y Ft(0)3323 2120 y Fx(j)26 b(\024)456 2233 y FA(1)p FG(=)p
FA(2)p Fx(g)32 b FA(lies)d(b)s(et)m(w)m(een)i FG(w)1243
2200 y Ft(\000)1241 2255 y Fv(m)1303 2264 y Fq(0)1372
2233 y FA(and)f FG(w)1616 2200 y Fy(+)1614 2255 y Fv(m)1676
2264 y Fq(0)1715 2233 y FA(.)555 2396 y(Let)43 b(no)m(w)g
FG(m)1012 2410 y Fy(0)1097 2396 y Fx(!)i(1)p FA(.)78
b(In)41 b(this)h(case,)47 b(b)m(y)42 b(Ascoli-Arzel\022)-45
b(a)43 b(Theorem)3036 2361 y Fy(13)3106 2396 y FA(,)j(w)m(e)d(get)456
2504 y(that)35 b(there)f(exists)g(a)h(H\177)-45 b(older)34
b(con)m(tin)m(uous)g(function)f FG(w)k FA(suc)m(h)e(that)g
FG(w)2927 2471 y Ft(\000)2925 2527 y Fv(m)2987 2536 y
Fq(0)3060 2504 y FA(uniformly)456 2617 y(con)m(v)m(erges)h(to)g
FG(w)r FA(.)55 b(By)36 b(\(7.8\))r(,)h(also)d FG(w)1780
2584 y Fy(+)1778 2640 y Fv(m)1840 2649 y Fq(0)1915 2617
y FA(uniformly)d(con)m(v)m(erges)37 b(to)f FG(w)r FA(.)55
b(Since)34 b FG(A)3318 2632 y Fv(k)3384 2617 y Fx(\\)456
2730 y(fj)p FG(y)574 2697 y Ft(0)597 2730 y Fx(j)26 b(\024)f
FA(1)p FG(=)p FA(2)p Fx(g)32 b FA(lies)d(b)s(et)m(w)m(een)i
FG(w)1531 2697 y Ft(\000)1529 2753 y Fv(m)1591 2762 y
Fq(0)1661 2730 y FA(and)e FG(w)1904 2697 y Fy(+)1902
2753 y Fv(m)1964 2762 y Fq(0)2004 2730 y FA(,)h(the)h(claim)e(is)h(pro)
m(v)m(ed.)555 2884 y FH(STEP)35 b(2:)41 b FA(The)30 b(function)f
FG(w)k FA(constructed)d(in)f(the)i(\014rst)e(step)i(is)e(harmonic.)456
3037 y Fo(Pr)-5 b(o)g(of)34 b(of)f(step)g(2.)456 3145
y FA(W)-8 b(e)32 b(pro)m(v)m(e)h(that)f FG(w)i FA(is)c(harmonic)h(in)f
(the)i(viscosit)m(y)f(sense.)44 b(Then)31 b(it)g(follo)m(ws)g(that)h
(it)456 3253 y(is)d(harmonic)g(in)g(the)i(classic)f(sense)g(\(see,)i
(e.g.,)g(Theorem)e(6.6)h(in)e([8)q(]\).)456 3361 y(F)-8
b(or)31 b(this,)e(let)i FG(P)43 b FA(b)s(e)30 b(the)g(quadratic)g(p)s
(olynomial)1411 3552 y FG(P)13 b FA(\()p FG(y)1565 3515
y Ft(0)1588 3552 y FA(\))41 b(:=)1810 3491 y(1)p 1810
3531 46 4 v 1810 3614 a(2)1866 3552 y FG(y)1914 3515
y Ft(0)p Fv(T)1988 3552 y FG(M)10 b(y)2134 3515 y Ft(0)2177
3552 y FA(+)20 b FG(\030)k Fx(\001)d FG(y)2426 3515 y
Ft(0)2464 3552 y FG(:)456 3731 y FA(Assume,)27 b(b)m(y)g(con)m
(tradiction,)h(that)g(\001)p FG(P)38 b(>)25 b FA(0,)j(that)g
FG(P)40 b FA(touc)m(hes)28 b(the)f(graph)g(of)g FG(w)r
FA(,)i(sa)m(y)456 3839 y(at)e(0)h(for)f(simplicit)m(y)d(and)i(that)i
FG(P)40 b FA(sta)m(ys)28 b(b)s(elo)m(w)e(it)h(in)e Fx(j)p
FG(y)2390 3806 y Ft(0)2414 3839 y Fx(j)g FG(<)g FA(2)p
FG(r)s FA(,)j(for)f(some)g FG(r)h Fx(2)d FA(\(0)p FG(;)15
b FA(1\).)456 3947 y(Let)35 b(no)m(w)g FG(\016)857 3961
y Fy(0)930 3947 y FG(>)e FA(0)i(b)s(e)g(the)g(univ)m(ersal)e(constan)m
(t)k(of)e(Lemma)g(9.3)h(in)e([30)q(])i(and)e(let)h(us)456
4055 y(de\014ne)898 4208 y FG(\016)29 b FA(:=)c(min)1255
4080 y Fu(\032)1323 4108 y(\020)1387 4147 y FA(\001)p
FG(P)p 1387 4188 147 4 v 1397 4271 a FA(2)p FG(\022)1485
4285 y Fy(0)1544 4108 y Fu(\021)1608 4103 y Fq(1)p 1608
4115 31 3 v 1608 4156 a(2)1653 4208 y FG(;)1872 4147
y FA(1)p 1728 4188 332 4 v 1728 4271 a(2)p FG(\022)1816
4285 y Fy(0)1871 4271 y Fx(k)p FG(M)10 b Fx(k)2070 4208
y FG(;)2242 4147 y FA(1)p 2146 4188 238 4 v 2146 4271
a(2)p FG(\022)2234 4285 y Fy(0)2288 4271 y Fx(j)p FG(\030)t
Fx(j)2393 4208 y FG(;)2459 4108 y Fu(\020)2547 4147 y
FG(\016)2587 4161 y Fy(0)p 2523 4188 128 4 v 2523 4271
a FA(2)p FG(\022)2611 4285 y Fy(0)2660 4108 y Fu(\021)2725
4103 y Fq(1)p 2725 4115 31 3 v 2725 4156 a(2)2769 4208
y FG(;)41 b(r)2879 4080 y Fu(\033)2977 4208 y FG(:)456
4393 y FA(Th)m(us,)29 b FG(\016)34 b FA(is)c(suc)m(h)g(that)832
4584 y(\001)p FG(P)38 b(>)25 b FA(2)p FG(\016)1188 4547
y Fy(2)1229 4584 y FG(\022)1272 4598 y Fy(0)1493 4584
y FG(;)197 b Fx(k)p FG(M)10 b Fx(k)26 b(\024)2098 4523
y FA(1)p 2035 4563 172 4 v 2035 4647 a(2)p FG(\016)s(\022)2166
4661 y Fy(0)2398 4584 y FG(;)198 b Fx(j)p FG(\030)t Fx(j)25
b(\024)2909 4523 y FA(1)p 2846 4563 V 2846 4647 a(2)p
FG(\016)s(\022)2977 4661 y Fy(0)3043 4584 y FG(;)456
4916 y FA(\(7.9\))1080 b FG(\016)1764 4878 y Fy(2)1805
4916 y FG(\022)1848 4930 y Fy(0)1927 4916 y Fx(\024)2048
4854 y FG(\016)2088 4868 y Fy(0)p 2048 4895 80 4 v 2066
4978 a FA(2)2153 4916 y FG(:)p 456 5021 499 4 v 555 5092
a Fy(13)648 5121 y FF(W)-6 b(e)23 b(remark)e(that,)j(b)n(y)d(the)h
(construction)h(of)h FD(\013)f FF(and)f FD(\014)27 b
FF(ab)r(o)n(v)n(e,)c(the)f(H\177)-38 b(older)23 b(constan)n(ts)g(of)g
FD(w)3355 5089 y Fp(\006)3353 5134 y Fs(m)3407 5144 y
Fn(0)456 5216 y FF(dep)r(end)h(on)i FD(\021)867 5224
y Fq(0)902 5216 y FF(,)g(but)f(are)h(indep)r(enden)n(t)e(of)j
FD(m)1809 5224 y Fq(0)1843 5216 y FF(.)p eop
%%Page: 70 70
70 69 bop 456 251 a Fy(70)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)555 450 y FA(Note)26 b(that,)g(ev)m(en)m(tually)e(replacing)f
FG(\016)28 b FA(with)22 b(2)p FG(\016)29 b FA(and)23
b FG(P)13 b FA(\()p FG(y)2503 417 y Ft(0)2527 450 y FA(\))24
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450 y FA(\))8 b Fx(\000)g FG(\016)s Fx(j)p FG(y)3202
417 y Ft(0)3226 450 y Fx(j)3251 417 y Fy(2)3290 450 y
FA(,)26 b(w)m(e)456 558 y(ma)m(y)31 b(assume,)g(with)e(no)h(lose)h(of)g
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b(sta)m(ys)h(strictly)f(b)s(elo)m(w)f(it)h(in)g Fx(j)p
FG(y)1671 633 y Ft(0)1694 666 y Fx(j)h FG(<)f FA(2)p
FG(\016)k(<)c FA(2.)40 b(therefore,)27 b(since)e FG(A)2850
681 y Fv(k)2903 666 y Fx(\\)11 b(fj)p FG(y)3093 633 y
Ft(0)3116 666 y Fx(j)26 b(\024)f FA(1)p FG(=)p FA(2)p
Fx(g)456 774 y FA(uniformly)30 b(con)m(v)m(erges)36 b(to)f(the)f(graph)
f(of)h FG(w)r FA(,)h(it)f(follo)m(ws)f(that,)i(for)f
FG(k)j FA(large,)e(w)m(e)f(\014nd)456 882 y(p)s(oin)m(ts)20
b FG(y)765 897 y Fv(k)833 882 y FA(=)25 b(\()p FG(y)1012
849 y Ft(0)1009 909 y Fv(k)1051 882 y FG(;)15 b(y)1136
897 y Fv(k)e(N)1254 882 y FA(\))22 b(close)g(to)g(0,)i(suc)m(h)e(that)g
FG(P)13 b FA(\()p FG(y)2255 849 y Ft(0)2278 882 y FA(\))s
Fx(\000)s FG(K)2467 897 y Fv(k)2531 882 y FA(touc)m(hes)23
b FG(A)2917 897 y Fv(k)2982 882 y FA(at)f(\()p FG(y)3167
849 y Ft(0)3164 909 y Fv(k)3207 882 y FG(;)15 b(y)3292
897 y Fv(k)e(N)3409 882 y FA(\))456 990 y(and)25 b(sta)m(ys)i(b)s(elo)m
(w)f(it)g(in)f Fx(j)p FG(y)1368 957 y Ft(0)1403 990 y
Fx(\000)12 b FG(y)1534 957 y Ft(0)1531 1017 y Fv(k)1573
990 y Fx(j)26 b(\024)f FG(\016)s FA(,)j(for)e(an)g(appropriate)f
FG(K)2636 1005 y Fv(k)2704 990 y Fx(2)g Fw(R)s FA(.)45
b(In)26 b(particular,)456 1098 y(w)m(e)k(ha)m(v)m(e)456
1287 y(\(7.10\))646 b FG(y)1377 1302 y Fv(k)13 b(N)1515
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1225 y(1)p 1857 1266 46 4 v 1857 1349 a(2)1912 1287 y
FG(y)1960 1249 y Ft(0)p Fv(T)1957 1310 y(k)2034 1287
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1249 y Ft(0)2485 1310 y Fv(k)2542 1287 y FG(:)456 1481
y FA(Let)30 b(us)g(no)m(w)h(consider)e(the)h(follo)m(wing)f
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FG(z)1945 1664 y Fv(N)2038 1650 y FA(=)25 b FG(y)2179
1664 y Fv(N)2266 1650 y Fx(\000)20 b FA(\()p FG(y)2437
1665 y Fv(k)13 b(N)2575 1650 y FA(+)19 b FG(K)2742 1665
y Fv(k)2785 1650 y FA(\))456 1818 y(Exploiting)29 b(\(7.10\))k(w)m(e)e
(\014nd)d(a)j(surface)1420 1956 y Fu(\010)1473 2030 y
FG(z)1515 2044 y Fv(N)1608 2030 y FA(=)1714 1968 y(1)p
1714 2009 V 1714 2092 a(2)1769 2030 y FG(z)1815 1992
y Ft(0)p Fv(T)1890 2030 y FG(M)10 b(z)2034 1992 y Ft(0)2078
2030 y FA(+)20 b FG(\030)2209 2045 y Fv(k)2271 2030 y
Fx(\001)h FG(z)2363 1992 y Ft(0)2386 1956 y Fu(\011)2454
2030 y FG(;)456 2224 y FA(with)1650 2355 y FG(\030)1690
2370 y Fv(k)1773 2355 y FA(:=)40 b FG(M)10 b(y)2055 2317
y Ft(0)2052 2378 y Fv(k)2115 2355 y FA(+)20 b FG(\030)456
2504 y FA(that)27 b(touc)m(hes)g FG(A)1039 2519 y Fv(k)1109
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b(\024)2034 2758 y FA(1)p 1994 2798 126 4 v 1994 2881
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3027 y FA(Rescaling)j(bac)m(k,)j(w)m(e)f(get)g(that)g(the)g(surface)
1314 3137 y Fu(n)1375 3238 y FG(x)1427 3252 y Fv(N)1519
3238 y FA(=)1625 3177 y FG(\022)1668 3192 y Fv(k)p 1625
3217 86 4 v 1633 3302 a FG(l)1662 3271 y Fy(2)1660 3332
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a(2)1786 3238 y FG(x)1838 3201 y Ft(0)p Fv(T)1912 3238
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3177 y FG(\022)2249 3192 y Fv(k)p 2206 3217 86 4 v 2214
3301 a FG(l)2241 3316 y Fv(k)2302 3238 y FG(\030)2342
3253 y Fv(k)2404 3238 y Fx(\001)21 b FG(x)2502 3201 y
Ft(0)2525 3137 y Fu(o)456 3470 y FA(touc)m(hes)31 b Fx(f)p
FG(u)879 3485 y Fv(k)947 3470 y FA(=)25 b(0)p Fx(g)31
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3684 y Fu(n)1212 3785 y FG(x)1264 3799 y Fv(N)1356 3785
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46 4 v 2411 3847 a FG(\016)2465 3785 y(\030)2505 3800
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%%Page: 71 71
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Fo(and)2203 5110 y FG(\022)2246 5125 y Fv(k)p 2203 5151
86 4 v 2211 5234 a FG(l)2238 5249 y Fv(k)2323 5172 y
Fx(!)25 b FA(0)15 b FG(;)p eop
%%Page: 72 72
72 71 bop 456 251 a Fy(72)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
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620 y Fx(g)2485 519 y Fu(\021)2565 620 y Fx(\032)g(fj)p
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y(Pr)-5 b(o)g(of.)43 b FA(Let)28 b(\014x)g FG(\022)1067
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1320 y(\(8.3\))1627 1258 y FG(\022)1670 1273 y Fv(k)p
1627 1299 86 4 v 1635 1382 a FG(l)1662 1397 y Fv(k)1748
1320 y Fx(\024)25 b FG(")h Fx(\024)f FG(")2050 1334 y
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1633 y(in\014nitely)27 b(man)m(y)j FG(k)s FA('s)h FG(\022)1269
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1633 y FA(.)555 1797 y(In)j(the)g(\014rst)f(case,)j(w)m(e)f(tak)m(e)h
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FG(\022)2893 1812 y Fv(k)2960 1797 y Fx(\024)e FG(\022)3099
1811 y Fy(0)3166 1797 y FA(and)j(w)m(e)456 1905 y(assume,)33
b(b)m(y)f(p)s(ossibly)e(extracting)j(a)g(further)e(subsequence,)i(that)
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2229 y(system)30 b(of)h(co)s(ordinates,)1471 2394 y Fx(f)p
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2408 y Fv(N)2115 2394 y Fx(j)25 b(\024)g FG(\022)2304
2408 y Fy(0)2343 2394 y Fx(g)15 b FG(:)456 2558 y FA(Thence,)30
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y Fy(\()p Fv(h)p Fy(\))1799 3704 y Fv(k)1926 3675 y Fx(\024)25
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y FA(More)i(precisely)-8 b(,)26 b(w)m(e)h(stop)g(the)f(iterativ)m(e)h
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4055 y FG(>)25 b(\022)1977 4069 y Fy(0)2041 4055 y Fx(\025)g
FG(\022)2183 4007 y Fy(\()p Fv(h)p Fy(\))2180 4084 y
Fv(k)2297 4055 y FG(:)456 4201 y FA(F)-8 b(or)31 b(suc)m(h)f
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y Fv(k)1985 4381 y Fx(\025)g FG(\021)2126 4395 y Fy(1)2166
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y FG(\022)1579 4640 y Fy(\()p Fv(h)p Fy(\))1576 4717
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4830 y Fv(k)2198 4753 y Fx(\024)25 b FG(")15 b(:)456
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%%Page: 73 73
73 72 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
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450 y FA(What)37 b(is)e(more,)j(the)e(rep)s(eated)h(use)e(of)i(Theorem)
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y(that,)31 b(in)e(some)i(system)f(of)h(co)s(ordinates,)796
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1658 y Fy(0)805 1644 y FA(w)m(as)i(arbitrary)-8 b(,)30
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2504 y Fv(")1493 2490 y FA(\()p FG(x)p FA(\))c(:=)f FG(u)p
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FG(u)2992 2720 y Fv(")3067 2706 y FA(con)m(v)m(erges)456
2814 y(almost)22 b(ev)m(erywhere)h(and)f(in)f FG(L)1535
2781 y Fy(1)1535 2842 y(lo)r(c)1650 2814 y FA(to)i(the)f(step)h
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2939 y(set)31 b FG(E)f Fx(\022)25 b Fw(R)851 2906 y Fv(N)954
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3434 y Fv(N)1302 3467 y FA(and)k(p)s(oin)m(ts)h FG(x)1804
3482 y Fv(k)1846 3467 y FA(,)h(suc)m(h)f(that)896 3623
y FG(x)948 3638 y Fv(k)1015 3623 y Fx(2)25 b(f)p FG(u)1198
3637 y Fv(")1231 3649 y Fs(k)1299 3623 y FA(=)g(0)p Fx(g)c(\\)f
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b(\016)s FA(\))93 b(with)c FG(B)5 b FA(\()p FG(z)2406
3637 y Fy(0)2446 3623 y FG(;)15 b FA(2)p FG(\016)s FA(\))22
b Fx(\\)e FG(@)5 b(E)31 b FA(=)25 b Fx(;)456 3779 y FA(Assume)g(e.g.)41
b FG(B)5 b FA(\()p FG(z)1115 3793 y Fy(0)1154 3779 y
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b FA(and)25 b(note)i(that)g(in)e(this)g(case,)j(exploiting)d(the)h
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FG(u)2986 3901 y Fv(")3067 3887 y FA(con)m(v)m(erges)456
3995 y(almost)26 b(ev)m(erywhere)g(and)f(in)g FG(L)1549
3962 y Fy(1)1549 4023 y(lo)r(c)1667 3995 y FA(to)h(the)g(step)g
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b FG(\037)2732 4016 y Fj(R)2780 3998 y Fs(N)2837 4016
y Ft(n)p Fv(E)2958 3995 y FA(\(in)25 b(the)h(same)456
4120 y(w)m(a)m(y)31 b(w)m(e)g(get)g(a)g(con)m(tradiction)f(if)g
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b FA(2)p FG(\016)s FA(\))28 b Fx(\032)d Fw(R)2174 4087
y Fv(N)2268 4120 y Fx(n)20 b FG(E)5 b FA(\).)555 4229
y(Since)44 b FG(@)5 b(E)51 b FA(is)44 b(a)h(minimal)d(surface)j(in)f
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b(Also,)456 4445 y(since)33 b FG(u)734 4459 y Fv(")767
4471 y Fs(k)809 4445 y FA(\(0\))g(=)e(0)k(and)e Fx(f)p
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b Fx(f)p FG(u)1457 4831 y Fv(")1490 4843 y Fs(k)1558
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y Fx(j)g(\024)g FG(\016)2451 4832 y Fv(k)2495 4817 y
Fx(g)456 4973 y FA(with)k FG(\016)703 4988 y Fv(k)771
4973 y Fx(!)c FA(0.)41 b(Rescaling)30 b(bac)m(k)h(w)m(e)g(get)h(that)
456 5172 y(\(8.6\))719 b Fx(f)p FG(u)26 b FA(=)f(0)p
Fx(g)c(\\)f FG(B)1868 5173 y Fq(1)p 1850 5185 68 3 v
1850 5226 a Fs(")1879 5243 y(k)1957 5172 y Fx(\032)25
b(fj)p FG(x)2175 5186 y Fv(N)2243 5172 y Fx(j)g(\024)2400
5110 y FG(\016)2440 5125 y Fv(k)p 2399 5151 86 4 v 2399
5234 a FG(")2441 5249 y Fv(k)2494 5172 y Fx(g)p eop
%%Page: 74 74
74 73 bop 456 251 a Fy(74)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(Tw)m(o)28 b(cases)h(are)g(no)m(w)g(p)s(ossible:)37
b(either)28 b FG(\016)1894 465 y Fv(k)1937 450 y FG(=")2024
465 y Fv(k)2096 450 y FA(is)f(b)s(ounded)f(a)m(w)m(a)m(y)31
b(from)d(zero,)h(or,)g(up)456 558 y(to)k(subsequences,)g
FG(\016)1187 573 y Fv(k)1230 558 y FG(=")1317 573 y Fv(k)1390
558 y Fx(\000)-16 b(!)29 b FA(0.)48 b(In)32 b(the)h(latter)g(case,)i(w)
m(e)e(pass)g(to)g(the)g(limit)e(\(8.5\))456 666 y(b)m(y)25
b(sending)f FG(k)29 b Fx(\000)-16 b(!)25 b FA(+)p Fx(1)p
FA(,)i(getting)f(that)g Fx(f)p FG(u)g FA(=)f(0)p Fx(g)h
FA(is)f(a)h(h)m(yp)s(erplane.)37 b(If,)27 b(on)e(the)h(other)456
774 y(hand,)j FG(\016)748 789 y Fv(k)791 774 y FG(=")878
789 y Fv(k)947 774 y Fx(\025)c FG(\022)1086 788 y Fy(0)1125
774 y FA(,)31 b(for)f(some)h FG(\022)1591 788 y Fy(0)1655
774 y FG(>)25 b FA(0,)31 b(w)m(e)g(de\014ne)1457 978
y FG(l)1484 993 y Fv(k)1567 978 y FA(:=)1756 917 y(1)p
1713 957 131 4 v 1713 1040 a(2)p FG(")1800 1055 y Fv(k)1854
978 y FG(;)197 b(\022)2119 993 y Fv(k)2202 978 y FA(:=)2349
917 y FG(\016)2389 932 y Fv(k)p 2348 957 86 4 v 2348
1040 a FG(")2390 1055 y Fv(k)456 1175 y FA(and)29 b(w)m(e)i(observ)m(e)
g(that)1654 1257 y FG(\022)1697 1272 y Fv(k)p 1654 1298
V 1662 1381 a FG(l)1689 1396 y Fv(k)1774 1319 y FA(=)1880
1257 y FG(\016)1920 1272 y Fv(k)p 1880 1298 84 4 v 1899
1381 a FA(2)1999 1319 y Fx(\000)-16 b(!)25 b FA(0)15
b(;)456 1497 y(then,)32 b(it)g(follo)m(ws)g(that)g(the)h(assumptions)d
(of)j(Lemma)f(8.1)i(are)e(ful\014lled.)43 b(Th)m(us,)32
b(the)456 1605 y(application)27 b(of)j(Lemma)g(8.1)g(pro)m(v)m(es)g
(that)g Fx(f)p FG(u)c FA(=)f(0)p Fx(g)30 b FA(is)f(a)h(h)m(yp)s
(erplane,)d(whic)m(h)i(is)f(the)456 1713 y(desired)g(result.)1376
1927 y(9.)46 b FB(Pr)n(oof)34 b(of)f(Theorem)h(1.4)555
2089 y FA(First,)c(w)m(e)h(pro)m(v)m(e)g(the)g(minimalit)m(y)c(of)k
FG(u)p FA(:)456 2221 y FH(Lemma)37 b(9.1.)45 b Fo(L)-5
b(et)37 b FG(h)1251 2235 y Fy(0)1327 2221 y Fo(satisfy)46
b FA(\(1.1\))r Fo(,)37 b FA(\(1.2\))r Fo(,)h FA(\(1.3\))g
Fo(and)47 b FA(\(1.4\))r Fo(.)54 b(L)-5 b(et)37 b FG(u)g
Fo(b)-5 b(e)36 b(a)i(we)-5 b(ak)456 2330 y(Sob)g(olev)35
b(solution)h(of)f(\(1.5\))h(in)e(the)h(whole)h Fw(R)2042
2297 y Fv(N)2115 2330 y Fo(,)f(satisfying)h Fx(j)p FG(u)p
Fx(j)29 b(\024)g FA(1)p Fo(,)35 b FG(@)2972 2344 y Fv(N)3040
2330 y FG(u)29 b(>)f FA(0)35 b Fo(and)456 2438 y FA(lim)582
2452 y Fv(x)622 2463 y Fs(N)679 2452 y Ft(!)p Fy(+)p
Ft(1)895 2438 y FG(u)25 b FA(=)g Fx(\006)p FA(1)p Fo(.)42
b(Then,)33 b FG(u)f Fo(is)h(a)g(class)g(A)f(minimizer.)456
2614 y(Pr)-5 b(o)g(of.)43 b FA(Since)32 b FG(u)i FA(is)e(strictly)h
(increasing,)g Fx(j)p FG(u)p Fx(j)f FG(<)e FA(1.)51 b(Let)34
b FG(B)i Fx(\032)30 b Fw(R)2714 2581 y Fv(N)2820 2614
y FA(b)s(e)j(a)h(closed)g(ball)456 2721 y(and)26 b(let)h
FG(v)k FA(b)s(e)26 b(a)i(minimizer)c(for)j Fx(F)1647
2735 y Fv(B)1736 2721 y FA(with)e FG(v)k FA(=)c FG(u)i
FA(on)g FG(@)5 b(B)g FA(.)40 b(Our)26 b(aim)g(is)h(to)h(sho)m(w)f(that)
456 2829 y FG(u)e FA(=)g FG(v)33 b FA(in)d FG(B)5 b FA(.)40
b(Let)31 b(us)e(argue)i(b)m(y)f(con)m(tradiction)h(and)e(assume,)i(sa)m
(y)-8 b(,)31 b(that)456 2986 y(\(9.1\))1001 b FG(v)s
FA(\()p FG(x)1776 2948 y Fv(?)1816 2986 y FA(\))41 b
FG(>)f(u)p FA(\()p FG(x)2142 2948 y Fv(?)2182 2986 y
FA(\))15 b FG(;)456 3142 y FA(for)28 b(some)h FG(x)871
3109 y Fv(?)936 3142 y Fx(2)c FG(B)5 b FA(.)40 b(P)m(ossibly)27
b(cutting)h FG(v)33 b FA(on)28 b(the)h Fx(\006)p FA(1-lev)m(els)g
(\(whic)m(h)f(decreases)i Fx(F)3323 3156 y Fv(B)3384
3142 y FA(\),)456 3250 y(w)m(e)36 b(ma)m(y)h(and)f(do)g(assume)g(that)h
Fx(j)p FG(v)s Fx(j)f(\024)e FA(1.)59 b(More)37 b(precisely)-8
b(,)37 b(as)f(men)m(tioned)g(in)f(the)456 3358 y(fo)s(otnote)c(on)f
(page)i(3,)f(b)m(y)g(\(1.4\))h(it)e(follo)m(ws)g(that)h
Fx(j)p FG(v)s Fx(j)26 b FG(<)f FA(1.)456 3466 y(Then,)k(since)h
Fx(jr)p FG(u)p Fx(j)25 b(\024)41 b FA(const)30 b(thanks)g(to)h([15)q(])
g(or)f([34)q(],)h(and)f(lim)2683 3480 y Fv(x)2723 3491
y Fs(N)2781 3480 y Ft(!)p Fy(+)p Ft(1)2996 3466 y FG(u)25
b FA(=)g Fx(\006)p FA(1,)31 b(w)m(e)456 3574 y(deduce)f(that)456
3730 y(\(9.2\))949 b FG(u)p FA(\()p FG(x)21 b FA(+)f
FG(te)1916 3744 y Fv(N)1983 3730 y FA(\))26 b Fx(\025)f
FG(v)s FA(\()p FG(x)p FA(\))456 3887 y(for)h(an)m(y)h
FG(x)e Fx(2)g FG(B)5 b FA(,)27 b(pro)m(vided)e(that)j
FG(t)e FA(is)g(large)g(enough.)40 b(Indeed,)26 b(to)i(pro)m(v)m(e)f
(\(9.2\),)i(let)e(us)456 3995 y(argue)e(b)m(y)g(con)m(tradiction)g(and)
g(assume)f(that)i FG(u)p FA(\()p FG(x)2178 4009 y Fv(t)2218
3995 y FA(+)10 b FG(te)2374 4009 y Fv(N)2441 3995 y FA(\))25
b FG(<)g(v)s FA(\()p FG(x)2731 4009 y Fv(t)2761 3995
y FA(\))h(for)f(some)g FG(x)3230 4009 y Fv(t)3285 3995
y Fx(2)g FG(B)456 4103 y FA(and)j(a)h(div)m(erging)e(sequence)i(of)g
FG(t)p FA(;)g(let)g(also)g FG(\013)c(>)g FA(0)30 b(so)e(that)i
FG(v)e Fx(\024)d FA(1)18 b Fx(\000)e FG(\013)p FA(.)41
b(Then,)28 b(up)g(to)456 4210 y(subsequence,)h(w)m(e)i(ma)m(y)g(assume)
f(that)h FG(x)1888 4224 y Fv(t)1948 4210 y FA(con)m(v)m(erges)i(to)e
FG(x)2519 4224 y Ft(1)2619 4210 y Fx(2)25 b FG(B)5 b
FA(;)30 b(but)f(then)1042 4367 y(1)83 b(=)130 b(lim)1324
4423 y Fv(t)p Ft(!)p Fy(+)p Ft(1)1560 4367 y FG(u)p FA(\()p
FG(x)1699 4381 y Ft(1)1795 4367 y FA(+)20 b FG(te)1961
4381 y Fv(N)2028 4367 y FA(\))26 b Fx(\024)1170 4542
y(\024)130 b FA(lim)1324 4598 y Fv(t)p Ft(!)p Fy(+)p
Ft(1)1560 4542 y FG(u)p FA(\()p FG(x)1699 4556 y Fv(t)1750
4542 y FA(+)20 b FG(te)1916 4556 y Fv(N)1983 4542 y FA(\))h(+)35
b(const)15 b Fx(j)p FG(x)2444 4556 y Fv(t)2494 4542 y
Fx(\000)20 b FG(x)2637 4556 y Ft(1)2712 4542 y Fx(j)25
b FA(=)1170 4718 y(=)130 b(lim)1324 4774 y Fv(t)p Ft(!)p
Fy(+)p Ft(1)1560 4718 y FG(u)p FA(\()p FG(x)1699 4732
y Fv(t)1750 4718 y FA(+)20 b FG(te)1916 4732 y Fv(N)1983
4718 y FA(\))26 b Fx(\024)1170 4893 y(\024)130 b FA(lim)1324
4949 y Fv(t)p Ft(!)p Fy(+)p Ft(1)1560 4893 y FG(v)s FA(\()p
FG(x)1694 4907 y Fv(t)1725 4893 y FA(\))25 b Fx(\024)1170
5059 y(\024)83 b FA(1)20 b Fx(\000)g FG(\013)c(:)456
5216 y FA(This)28 b(con)m(tradiction)i(pro)m(v)m(es)i(\(9.2\).)p
eop
%%Page: 75 75
75 74 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(75)555
450 y FA(Thanks)33 b(to)h(\(9.2\),)i(w)m(e)f(th)m(us)e(slide)f
FG(u)p FA(\()p Fx(\001)22 b FA(+)h FG(te)2100 464 y Fv(N)2167
450 y FA(\))34 b(to)m(w)m(ards)g(the)g FG(e)2780 464
y Fv(N)2847 450 y FA(-direction)f(un)m(til)456 558 y(w)m(e)j(touc)m(h)g
FG(v)j FA(from)c(ab)s(o)m(v)m(e.)58 b(Sa)m(y)36 b(this)f(happ)s(en)f
(at)42 b(\026)-51 b FG(x)34 b Fx(2)g FG(B)40 b FA(for)35
b FG(t)f FA(=)2871 541 y(\026)2870 558 y FG(t)p FA(.)56
b(In)35 b(the)h(ligh)m(t)456 666 y(of)30 b(\(9.1\),)j(w)m(e)d(ha)m(v)m
(e)i(that)1432 825 y FG(u)p FA(\()p FG(x)1571 788 y Fv(?)1631
825 y FA(+)1724 809 y(\026)1722 825 y FG(te)1797 839
y Fv(N)1865 825 y FA(\))83 b Fx(\025)g FG(v)s FA(\()p
FG(x)2271 788 y Fv(?)2311 825 y FA(\))26 b FG(>)1983
960 y(>)83 b(u)p FA(\()p FG(x)2276 923 y Fv(?)2316 960
y FA(\))15 b FG(;)456 1120 y FA(thence,)31 b(since)e
FG(u)i FA(is)e(strictly)h(increasing)f(in)g(the)h FG(e)2206
1134 y Fv(N)2274 1120 y FA(-direction,)1831 1262 y(\026)1830
1279 y FG(t)25 b(>)g FA(0)15 b FG(:)456 1438 y FA(Since)25
b(no)m(w)i FG(@)923 1452 y Fv(N)991 1438 y FG(u)e(>)g
FA(0)i(w)m(e)h(ha)m(v)m(e)g(that)f Fx(r)p FG(u)p FA(\()p
Fx(\001)13 b FA(+)2052 1422 y(\026)2051 1438 y FG(t)q(e)2127
1452 y Fv(N)2194 1438 y FA(\))26 b Fx(6)p FA(=)f(0.)40
b(Therefore,)27 b(it)g(follo)m(ws)f(that)456 1546 y(the)e(assumptions)f
(of)h(the)g(Strong)g(Comparison)f(Principle)e(for)j FG(p)p
FA(-Laplace)h(equations)456 1654 y(in)c([9)q(])i(\(see)h(Corollary)e
(B.5)i(here\))f(applies)e(to)j FG(u)p FA(\()p Fx(\001)5
b FA(+)2221 1638 y(\026)2219 1654 y FG(t)q(e)2295 1668
y Fv(N)2362 1654 y FA(\))23 b(and)g FG(v)j FA(and)c(so)i(this)d(touc)m
(hing)456 1762 y(p)s(oin)m(t)27 b(m)m(ust)h(o)s(ccur)h(on)f
FG(@)5 b(B)g FA(,)29 b(that)g(is)k(\026)-50 b FG(x)25
b Fx(2)g FG(@)5 b(B)g FA(.)39 b(Since)28 b FG(u)d FA(=)g
FG(v)32 b FA(on)c FG(@)5 b(B)g FA(,)29 b(it)f(follo)m(ws)g(that)456
1870 y FG(v)s FA(\()6 b(\026)-51 b FG(x)p FA(\))26 b(=)f
FG(u)p FA(\()6 b(\026)-51 b FG(x)p FA(\).)40 b(Consequen)m(tly)-8
b(,)27 b(since)e FG(u)i FA(is)e(strictly)g(increasing)g(in)g(the)i
FG(e)2973 1884 y Fv(N)3040 1870 y FA(-direction,)1274
2030 y FG(u)p FA(\()6 b(\026)-51 b FG(x)q FA(\))25 b(=)g
FG(v)s FA(\()6 b(\026)-51 b FG(x)q FA(\))25 b(=)g FG(u)p
FA(\()6 b(\026)-51 b FG(x)21 b FA(+)2113 2013 y(\026)2112
2030 y FG(te)2187 2044 y Fv(N)2254 2030 y FA(\))26 b
FG(>)f(u)p FA(\()6 b(\026)-51 b FG(x)p FA(\))15 b FG(:)456
2189 y FA(This)26 b(con)m(tradiction)j(sho)m(ws)f(that)h(\(9.1\))i
(cannot)e(hold,)f(hence)h FG(v)f Fx(\024)d FG(u)p FA(.)40
b(Analogously)-8 b(,)456 2297 y(one)30 b(sees)h(that)g
FG(v)e Fx(\025)c FG(u)p FA(,)30 b(thence)h FG(v)e FA(=)c
FG(u)p FA(.)1562 b Fl(\003)555 2473 y FA(By)43 b(means)g(of)g(Lemma)f
(9.1,)48 b(w)m(e)43 b(are)g(no)m(w)g(in)e(the)i(p)s(osition)e(to)i
(complete)g(the)456 2581 y(pro)s(of)30 b(of)i(Theorem)e(1.4,)j(b)m(y)e
(arguing)g(as)g(follo)m(ws.)43 b(With)31 b(no)g(loss)f(of)i(generalit)m
(y)-8 b(,)32 b(w)m(e)456 2689 y(assume)i(that)h FG(u)p
FA(\(0\))f(=)f(0.)54 b(Then,)35 b(for)f FG(")f(>)g FA(0,)j(setting)f
FG(u)2455 2703 y Fv(")2492 2689 y FA(\()p FG(x)p FA(\))e(:=)g
FG(u)p FA(\()p FG(x=")p FA(\),)k(w)m(e)e(kno)m(w)456
2796 y(from)27 b(Lemma)g(9.1)i(and)e(the)h(results)e(of)i([7)q(])g
(that)g FG(u)2218 2810 y Fv(")2282 2796 y FG(L)2344 2763
y Fy(1)2344 2824 y(lo)r(c)2436 2796 y FA(-con)m(v)m(erges)i(\(and)e(th)
m(us)f(a.e.-)456 2904 y(con)m(v)m(erges\),)g(up)22 b(to)j(subsequence,)
f(to)g FG(\037)1841 2918 y Fv(E)1907 2904 y Fx(\000)6
b FG(\037)2041 2926 y Fj(R)2089 2907 y Fs(N)2147 2926
y Ft(n)p Fv(E)2242 2904 y FA(,)25 b(for)e(a)h(suitable)e
FG(E)29 b FA(with)22 b(minimal)456 3012 y(p)s(erimeter.)555
3172 y(Since)30 b FG(@)841 3186 y Fv(N)909 3172 y FG(u)d(>)f
FA(0)32 b(and)e(lim)1466 3186 y Fv(x)1506 3197 y Fs(N)1563
3186 y Ft(!)p Fy(+)p Ft(1)1778 3172 y FG(u)d FA(=)f Fx(\006)p
FA(1,)32 b(w)m(e)g(ha)m(v)m(e)g(that)g(the)f(zero)h(lev)m(el)f(set)h
(of)456 3283 y FG(u)26 b FA(is)g(a)h(graph)f(in)f(the)i
FG(e)1245 3297 y Fv(N)1312 3283 y FA(-direction;)g(more)g(precisely)-8
b(,)27 b(there)g(exists)f FG(\015)k FA(:)c Fw(R)3018
3250 y Fv(N)7 b Ft(\000)p Fy(1)3207 3283 y Fx(\000)-15
b(!)25 b Fw(R)456 3391 y FA(so)30 b(that)1435 3513 y
Fx(f)p FG(u)25 b(<)g FA(0)p Fx(g)42 b FA(=)e Fx(f)p FG(x)1993
3527 y Fv(N)2086 3513 y FG(<)25 b(\015)5 b FA(\()p FG(x)2321
3475 y Ft(0)2344 3513 y FA(\))p Fx(g)15 b FG(:)456 3653
y FA(By)30 b(scaling,)g(w)m(e)h(th)m(us)f(deduce)g(that)456
3813 y(\(9.3\))759 b Fx(f)p FG(u)1497 3827 y Fv(")1560
3813 y FG(<)25 b FA(0)p Fx(g)41 b FA(=)f Fx(f)p FG(x)1995
3827 y Fv(N)2088 3813 y FG(<)25 b(\015)2231 3827 y Fv(")2268
3813 y FA(\()p FG(x)2355 3775 y Ft(0)2378 3813 y FA(\))p
Fx(g)15 b FG(;)456 3972 y FA(with)1562 4094 y FG(\015)1609
4108 y Fv(")1646 4094 y FA(\()p FG(x)1733 4056 y Ft(0)1757
4094 y FA(\))41 b(:=)f FG("\015)5 b FA(\()p FG(x)2150
4056 y Ft(0)2174 4094 y FG(=")p FA(\))15 b FG(:)456 4234
y FA(W)-8 b(e)31 b(no)m(w)g(claim)e(that)456 4394 y(\(9.4\))513
b FG(\037)1211 4412 y Ft(f)p Fv(x)1286 4423 y Fs(N)1343
4412 y Fv(<\015)1434 4420 y Fs(")1468 4412 y Fy(\()p
Fv(x)1535 4393 y Fp(0)1557 4412 y Fy(\))p Ft(g)1655 4394
y FA(con)m(v)m(erges)32 b(in)d FG(L)2230 4361 y Fy(1)2230
4421 y(lo)r(c)2353 4394 y FA(to)i FG(\037)2521 4415 y
Fj(R)2569 4396 y Fs(N)2626 4415 y Ft(n)p Fv(E)2721 4394
y FA(.)456 4570 y(Indeed:)38 b(w)m(e)29 b(kno)m(w)g(that)g
FG(u)1400 4584 y Fv(")1465 4570 y FA(con)m(v)m(erges)i(to)e
FG(\037)2037 4584 y Fv(E)2113 4570 y Fx(\000)16 b FG(\037)2257
4592 y Fj(R)2305 4573 y Fs(N)2363 4592 y Ft(n)p Fv(E)2486
4570 y FA(in)28 b Fw(R)2650 4537 y Fv(N)2740 4570 y Fx(n)17
b FG(Z)7 b FA(,)29 b(for)f(a)h(suitable)456 4695 y(set)i
FG(Z)36 b FA(with)29 b Fi(L)958 4662 y Fv(N)1026 4695
y FA(\()p FG(Z)7 b FA(\))26 b(=)e(0;)31 b(th)m(us,)g(if)e
FG(x)c Fx(2)g FG(E)h Fx(n)20 b FG(Z)7 b FA(,)1244 4854
y(lim)1191 4920 y Fv(")p Ft(\000)-12 b(!)p Fy(0)1373
4901 y Fq(+)1439 4854 y FG(u)1491 4868 y Fv(")1527 4854
y FA(\()p FG(x)p FA(\))26 b(=)f FG(\037)1828 4868 y Fv(E)1888
4854 y FA(\()p FG(x)p FA(\))c Fx(\000)f FG(\037)2179
4876 y Fj(R)2227 4857 y Fs(N)2284 4876 y Ft(n)p Fv(E)2379
4854 y FA(\()p FG(x)p FA(\))26 b(=)f(1)15 b FG(;)456
5052 y FA(therefore)1316 5174 y(lim)1263 5240 y Fv(")p
Ft(\000)-13 b(!)p Fy(0)1444 5221 y Fq(+)1510 5174 y FG(\037)1567
5192 y Ft(f)p Fv(u)1643 5200 y Fs(")1677 5192 y Fv(<)p
Fy(0)p Ft(g)1806 5174 y FA(\()p FG(x)p FA(\))26 b(=)f(0)h(=)f
FG(\037)2274 5195 y Fj(R)2322 5177 y Fs(N)2379 5195 y
Ft(n)p Fv(E)2474 5174 y FA(\()p FG(x)p FA(\))15 b FG(;)p
eop
%%Page: 76 76
76 75 bop 456 251 a Fy(76)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
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Fv(")p Ft(\000)-12 b(!)p Fy(0)1338 668 y Fq(+)1403 621
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Fs(N)2249 643 y Ft(n)p Fv(E)2344 621 y FA(\()p FG(x)p
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964 y(lim)1263 1030 y Fv(")p Ft(\000)-13 b(!)p Fy(0)1444
1011 y Fq(+)1510 964 y FG(\037)1567 982 y Ft(f)p Fv(u)1643
990 y Fs(")1677 982 y Fv(<)p Fy(0)p Ft(g)1806 964 y FA(\()p
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967 y Fs(N)2379 986 y Ft(n)p Fv(E)2474 964 y FA(\()p
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1178 y Fv(<)p Fy(0)p Ft(g)1424 1160 y FA(con)m(v)m(erges)i(almost)d(ev)
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y(subsequences,)c(and)e(whic)m(h)g(ma)m(y)h(attain)g(the)g(v)-5
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1738 y Fv(N)7 b Ft(\000)p Fy(1)866 1771 y Fx(\000)-15
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y Fw(R)1486 2286 y Fv(N)1580 2323 y Fx(n)20 b FG(E)46
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2881 y(ing)h(to)j(our)e(h)m(yp)s(otheses.)40 b(If)29
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3313 y(most)f(linear)f(gro)m(wth,)i(then)1282 3484 y
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%%Page: 77 77
77 76 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
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4607 y Fy(0)1962 4593 y FA(\))1997 4519 y Fu(\001)2039
4488 y(\014)2039 4543 y(\014)2039 4598 y(\014)2095 4593
y Fx(\024)2201 4532 y FA(1)p 2201 4572 V 2201 4655 a(4)2276
4593 y(+)2377 4532 y(1)p 2377 4572 V 2377 4655 a(2)2458
4593 y FG(<)25 b FA(1)15 b FG(:)456 4771 y FA(Therefore,)30
b(from)g(\(A.1\),)i(\(A.2\))g(and)e(\(4.2\),)1442 4918
y Fx(j)p FA(\()p FG(x)1554 4932 y Fy(0)1614 4918 y Fx(\000)20
b FG(y)s FA(\))h Fx(\001)f FG(e)1896 4932 y Fv(N)1964
4918 y Fx(j)25 b(\025)40 b FA(const)16 b FG(R)g(;)456
5069 y FA(pro)m(vided)p 830 4995 72 4 v 29 w FG(C)901
5083 y Fy(0)971 5069 y FA(is)29 b(large)i(enough,)f(and)g(therefore)456
5216 y(\(A.3\))840 b Fx(j)p FG(x)1581 5230 y Fv(N)1669
5216 y Fx(\000)19 b FG(y)1804 5230 y Fv(N)1871 5216 y
Fx(j)41 b(\025)55 b FA(const)16 b FG(R)g(;)p eop
%%Page: 78 78
78 77 bop 456 251 a Fy(78)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(for)30 b(an)m(y)g FG(X)j Fx(2)25 b
FG(B)1029 464 y Fy(3)p Fv(a)1106 450 y FA(\()p FG(X)1216
464 y Fy(0)1256 450 y FA(\),)31 b(if)e FG(R)i FA(is)f(large)g(enough.)
555 558 y(W)-8 b(e)32 b(no)m(w)e(p)s(oin)m(t)g(out)g(that,)h(if)f
FG(X)i Fx(2)25 b FG(B)1868 572 y Fy(3)p Fv(a)1945 558
y FA(\()p FG(X)2055 572 y Fy(0)2095 558 y FA(\),)456
756 y(\(A.4\))1374 651 y Fu(\014)1374 706 y(\014)1374
761 y(\014)1404 756 y FG(@)1452 770 y Fv(N)1520 756 y
FG(g)1562 773 y Fk(e)1563 790 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p
Fy(\))1777 756 y FA(\()p FG(x)p FA(\))1899 651 y Fu(\014)1899
706 y(\014)1899 761 y(\014)1970 756 y Fx(\025)40 b FA(const)h
FG(>)f FA(0)15 b FG(:)456 949 y FA(In)29 b(order)h(to)h(pro)m(v)m(e)g
(the)f(ab)s(o)m(v)m(e)h(inequalit)m(y)-8 b(,)29 b(\014rst)h(notice)g
(that,)h(b)m(y)f(a)h(direct)e(compu-)456 1057 y(tation,)i(using)d
(\(2.35\),)33 b(\(2.33\))g(and)d(\(2.37\),)j(one)d(gets)i(that)456
1280 y(\(A.5\))1258 1175 y Fu(\014)1258 1230 y(\014)1258
1284 y(\014)1289 1280 y FG(@)1337 1294 y Fv(N)1404 1280
y FG(g)1446 1297 y Fk(e)1447 1314 y Fj(S)-12 b Fy(\()p
Fv(Y)t(;R)p Fy(\))1661 1280 y FA(\()p FG(x)p FA(\))1783
1175 y Fu(\014)1783 1230 y(\014)1783 1284 y(\014)1855
1280 y Fx(\025)40 b FA(const)2199 1218 y Fx(j)p FG(x)2276
1232 y Fv(N)2363 1218 y Fx(\000)20 b FG(y)2499 1232 y
Fv(N)2566 1218 y Fx(j)p 2199 1259 393 4 v 2264 1342 a(j)p
FG(x)h Fx(\000)f FG(y)s Fx(j)2616 1280 y FG(:)456 1499
y FA(Also,)30 b(if)f FG(x)i FA(is)e(in)g(the)i(domain)e(of)h
FG(g)1674 1515 y Fk(e)1675 1533 y Fj(S)-11 b Fy(\()p
Fv(Y)t(;R)p Fy(\))1890 1499 y FA(,)30 b(then)1585 1688
y Fx(j)p FG(x)20 b Fx(\000)g FG(y)s Fx(j)25 b(\024)40
b FA(const)16 b FG(R)g(:)456 1858 y FA(The)23 b(latter)i(inequalit)m(y)
-8 b(,)24 b(together)h(with)e(\(A.5\))j(and)d(\(A.3\),)k(ends)d(the)g
(pro)s(of)f(of)i(\(A.4\).)555 2029 y(Let)31 b(no)m(w)938
2199 y Fx(R)p FA(\()p FG(x)1102 2213 y Fy(1)1142 2199
y FG(;)15 b(:)g(:)g(:)h(;)f(x)1395 2213 y Fv(N)7 b Ft(\000)p
Fy(1)1553 2199 y FG(;)15 b(x)1645 2213 y Fv(N)7 b Fy(+1)1803
2199 y FA(\))41 b(:=)759 2399 y(:=)938 2298 y Fu(h\020)1035
2399 y FG(H)1111 2413 y Fy(0)1150 2399 y FA(\()p FG(x)1237
2413 y Fv(N)7 b Fy(+1)1395 2399 y FA(\))21 b Fx(\000)p
1554 2264 72 4 v 1554 2337 a FG(C)1625 2351 y Fy(0)p
1551 2378 116 4 v 1551 2461 a FA(2)p FG(R)1677 2399 y
FA(\()p FG(x)1764 2413 y Fv(N)7 b Fy(+1)1941 2399 y Fx(\000)20
b FG(y)2077 2413 y Fv(N)7 b Fy(+1)2234 2399 y FA(\))2269
2361 y Fy(2)2329 2399 y FA(+)20 b FG(R)h Fx(\000)f FG(H)2677
2413 y Fy(0)2716 2399 y FA(\()p FG(y)2796 2413 y Fv(N)7
b Fy(+1)2953 2399 y FA(\))2988 2298 y Fu(\021)3043 2320
y Fy(2)3118 2399 y Fx(\000)938 2664 y(\000)1024 2551
y Fv(N)g Ft(\000)p Fy(1)1035 2578 y Fu(X)1039 2774 y
Fv(j)t Fy(=1)1192 2664 y Fx(j)p FG(x)1269 2678 y Fv(i)1318
2664 y Fx(\000)20 b FG(y)1454 2678 y Fv(i)1482 2664 y
Fx(j)1507 2627 y Fy(2)1546 2564 y Fu(i)1589 2586 y Fy(1)p
Fv(=)p Fy(2)1714 2664 y FG(:)456 2945 y FA(Let)30 b(also)h
FG(\031)853 2959 y Fv(N)945 2945 y FA(:)26 b Fw(R)1055
2912 y Fv(N)7 b Fy(+1)1244 2945 y Fx(\000)-16 b(!)26
b(f)p FG(x)1513 2959 y Fv(N)1606 2945 y FA(=)f(0)p Fx(g)31
b FA(b)s(e)e(the)i(natural)e(pro)5 b(jection,)31 b(i.e.,)1041
3115 y FG(\031)1093 3129 y Fv(N)1160 3115 y FA(\()p FG(x)1247
3129 y Fy(1)1286 3115 y FG(;)15 b(:)g(:)g(:)i(;)e(x)1540
3129 y Fv(N)7 b Fy(+1)1698 3115 y FA(\))41 b(:=)f(\()p
FG(x)1997 3129 y Fy(1)2037 3115 y FG(;)15 b(:)g(:)g(:)h(;)f(x)2290
3129 y Fv(N)7 b Ft(\000)p Fy(1)2448 3115 y FG(;)15 b
FA(0)p FG(;)g(x)2625 3129 y Fv(N)7 b Fy(+1)2784 3115
y FA(\))15 b FG(:)456 3285 y FA(W)-8 b(e)31 b(no)m(w)g(sho)m(w)f(that)
456 3460 y(\(A.6\))530 b FG(\031)1246 3474 y Fv(N)1313
3356 y Fu(\014)1313 3410 y(\014)1313 3465 y(\014)1343
3506 y Fk(e)1344 3524 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p
Fy(\))p Ft(\\)p Fv(B)1653 3533 y Fq(3)p Fs(a)1722 3524
y Fy(\()p Fv(X)1807 3533 y Fq(0)1842 3524 y Fy(\))1904
3460 y FA(is)29 b(a)i(di\013eomorphism.)456 3674 y(F)-8
b(or)31 b(pro)m(ving)e(this,)h(tak)m(e)i(an)m(y)1121
3867 y FG(X)h FA(=)1325 3766 y Fu(\020)1379 3867 y FG(x;)15
b(g)1513 3884 y Fk(e)1514 3901 y Fj(S)-11 b Fy(\()p Fv(Y)t(;R)p
Fy(\))1729 3867 y FA(\()p FG(x)p FA(\))1851 3766 y Fu(\021)1931
3867 y Fx(2)2017 3843 y Fu(e)2017 3867 y Fw(S)j FA(\()p
FG(Y)5 b(;)15 b(R)q FA(\))21 b Fx(\\)f FG(B)2476 3881
y Fy(3)p Fv(a)2553 3867 y FA(\()p FG(X)2663 3881 y Fy(0)2703
3867 y FA(\))15 b FG(;)456 4060 y FA(and)29 b(consider)1174
4197 y FG(\031)1226 4211 y Fv(N)1293 4197 y FA(\()p FG(X)7
b FA(\))27 b(=)1568 4096 y Fu(\020)1622 4197 y FG(x)1674
4211 y Fy(1)1713 4197 y FG(;)15 b(:)g(:)g(:)i(;)e(x)1967
4211 y Fv(N)7 b Ft(\000)p Fy(1)2125 4197 y FG(;)15 b
FA(0)p FG(;)g(g)2292 4214 y Fk(e)2293 4231 y Fj(S)-11
b Fy(\()p Fv(Y)t(;R)p Fy(\))2508 4197 y FA(\()p FG(x)p
FA(\))2630 4096 y Fu(\021)2700 4197 y FG(:)456 4376 y
FA(Then,)29 b(b)m(y)h(\(2.31\))j(and)d(\(2.33\),)1194
4574 y FG(\031)1249 4536 y Ft(\000)p Fy(1)1246 4603 y
Fv(N)1344 4473 y Fu(\020)1398 4574 y FG(x)1450 4588 y
Fy(1)1490 4574 y FG(;)15 b(:)g(:)g(:)h(;)f(x)1743 4588
y Fv(N)7 b Ft(\000)p Fy(1)1901 4574 y FG(;)15 b FA(0)p
FG(;)g(g)2068 4591 y Fk(e)2069 4608 y Fj(S)-11 b Fy(\()p
Fv(Y)t(;R)p Fy(\))2285 4574 y FA(\()p FG(x)p FA(\))2407
4473 y Fu(\021)2487 4574 y FA(=)25 b FG(X)d(;)456 4767
y FA(with)29 b FG(x)715 4781 y Fv(N)812 4767 y FA(so)i(that)1223
4937 y Fx(j)p FG(x)1300 4951 y Fv(N)1387 4937 y Fx(\000)20
b FG(y)1523 4951 y Fv(N)1590 4937 y Fx(j)26 b FA(=)f
Fx(R)p FA(\()p FG(x)1901 4951 y Fy(1)1940 4937 y FG(;)15
b(:)g(:)g(:)i(;)e(x)2194 4951 y Fv(N)7 b Ft(\000)p Fy(1)2352
4937 y FG(;)15 b(x)2444 4951 y Fv(N)7 b Fy(+1)2602 4937
y FA(\))15 b FG(;)456 5108 y FA(where)47 b Fx(R)h FA(has)g(b)s(een)f
(de\014ned)f(here)i(ab)s(o)m(v)m(e.)95 b(F)-8 b(rom)48
b(\(A.3\),)54 b(w)m(e)49 b(get)g(that)f Fx(R)55 b(\025)456
5216 y FA(const)15 b FG(R)33 b(>)f FA(0,)k(th)m(us)e
Fx(R)h FA(is)e(smo)s(oth)i(for)f FG(X)39 b Fx(2)32 b
FG(B)2145 5230 y Fy(3)p Fv(a)2222 5216 y FA(\()p FG(X)2332
5230 y Fy(0)2372 5216 y FA(\).)54 b(Also,)35 b(using)e(again)i
(\(A.3\),)p eop
%%Page: 79 79
79 78 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(79)456
456 y FA(w)m(e)35 b(see)h(that,)h(either)e FG(x)1289
470 y Fv(N)1390 456 y FG(>)e(y)1539 470 y Fv(N)1641 456
y FA(or)i FG(x)1809 470 y Fv(N)1910 456 y FG(<)e(y)2059
470 y Fv(N)2161 456 y FA(for)i(an)m(y)g FG(X)41 b Fx(2)2691
433 y Fu(e)2691 456 y Fw(S)-7 b FA(\()p FG(Y)5 b(;)15
b(R)q FA(\))24 b Fx(\\)f FG(B)3157 470 y Fy(3)p Fv(a)3234
456 y FA(\()p FG(X)3344 470 y Fy(0)3384 456 y FA(\):)456
564 y(that)31 b(is,)e(in)g(the)i(\014rst)e(case,)1021
754 y FG(\031)1076 715 y Ft(\000)p Fy(1)1073 782 y Fv(N)1170
653 y Fu(\020)1224 754 y FG(x)1276 768 y Fy(1)1316 754
y FG(;)15 b(:)g(:)g(:)h(;)f(x)1569 768 y Fv(N)7 b Ft(\000)p
Fy(1)1727 754 y FG(;)15 b FA(0)p FG(;)g(g)1894 770 y
Fk(e)1895 788 y Fj(S)-10 b Fy(\()p Fv(Y)t(;R)p Fy(\))2111
754 y FA(\()p FG(x)p FA(\))2233 653 y Fu(\021)2313 754
y FA(=)867 953 y(=)1021 852 y Fu(\020)1075 953 y FG(x)1127
967 y Fy(1)1166 953 y FG(;)15 b(:)g(:)g(:)i(;)e(x)1420
967 y Fv(N)7 b Ft(\000)p Fy(1)1578 953 y FG(;)30 b(y)1678
967 y Fv(N)1765 953 y FA(+)20 b Fx(R)p FA(\()p FG(x)2020
967 y Fy(1)2060 953 y FG(;)15 b(:)g(:)g(:)i(;)e(x)2314
967 y Fv(N)7 b Ft(\000)p Fy(1)2471 953 y FG(;)15 b(x)2563
967 y Fv(N)7 b Fy(+1)2721 953 y FA(\))p FG(;)31 b(x)2864
967 y Fv(N)7 b Fy(+1)3021 852 y Fu(\021)3091 953 y FG(;)456
1142 y FA(while,)28 b(in)h(the)i(second)f(case,)1021
1332 y FG(\031)1076 1293 y Ft(\000)p Fy(1)1073 1360 y
Fv(N)1170 1231 y Fu(\020)1224 1332 y FG(x)1276 1346 y
Fy(1)1316 1332 y FG(;)15 b(:)g(:)g(:)h(;)f(x)1569 1346
y Fv(N)7 b Ft(\000)p Fy(1)1727 1332 y FG(;)15 b FA(0)p
FG(;)g(g)1894 1348 y Fk(e)1895 1366 y Fj(S)-10 b Fy(\()p
Fv(Y)t(;R)p Fy(\))2111 1332 y FA(\()p FG(x)p FA(\))2233
1231 y Fu(\021)2313 1332 y FA(=)867 1531 y(=)1021 1430
y Fu(\020)1075 1531 y FG(x)1127 1545 y Fy(1)1166 1531
y FG(;)15 b(:)g(:)g(:)i(;)e(x)1420 1545 y Fv(N)7 b Ft(\000)p
Fy(1)1578 1531 y FG(;)30 b(y)1678 1545 y Fv(N)1765 1531
y Fx(\000)20 b(R)p FA(\()p FG(x)2020 1545 y Fy(1)2060
1531 y FG(;)15 b(:)g(:)g(:)i(;)e(x)2314 1545 y Fv(N)7
b Ft(\000)p Fy(1)2471 1531 y FG(;)15 b(x)2563 1545 y
Fv(N)7 b Fy(+1)2721 1531 y FA(\))p FG(;)31 b(x)2864 1545
y Fv(N)7 b Fy(+1)3021 1430 y Fu(\021)3091 1531 y FG(:)456
1725 y FA(In)29 b(an)m(y)i(case,)h(\(A.6\))g(is)d(pro)m(v)m(ed.)555
1894 y(Let)i(no)m(w)g FG(\031)960 1908 y Fv(N)7 b Fy(+1)1142
1894 y FA(:)25 b Fw(R)1252 1861 y Fv(N)7 b Fy(+1)1441
1894 y Fx(\000)-16 b(!)26 b(f)p FG(x)1710 1908 y Fv(N)7
b Fy(+1)1893 1894 y FA(=)25 b(0)p Fx(g)31 b FA(b)s(e)f(the)g(natural)g
(pro)5 b(jection,)30 b(i.e.,)1165 2061 y FG(\031)1217
2075 y Fv(N)7 b Fy(+1)1375 2061 y FA(\()p FG(x)1462 2075
y Fy(1)1501 2061 y FG(;)15 b(:)g(:)g(:)i(;)e(x)1755 2075
y Fv(N)7 b Fy(+1)1913 2061 y FA(\))41 b(:=)f(\()p FG(x)2212
2075 y Fy(1)2252 2061 y FG(;)15 b(:)g(:)g(:)h(;)f(x)2505
2075 y Fv(N)2573 2061 y FG(;)g FA(0\))g FG(:)456 2240
y FA(Notice)31 b(that,)g(if)f FG(X)i FA(=)25 b(\()p FG(x;)15
b(g)1419 2256 y Fk(e)1420 2274 y Fj(S)-11 b Fy(\()p Fv(Y)t(;R)p
Fy(\))1635 2240 y FA(\()p FG(x)p FA(\)\))26 b Fx(2)1904
2216 y Fu(e)1904 2240 y Fw(S)-7 b FA(\()p FG(Y)5 b(;)15
b(R)q FA(\),)31 b(then)f FG(\031)2508 2254 y Fv(N)7 b
Fy(+1)2665 2240 y FA(\()p FG(X)g FA(\))27 b(=)e(\()p
FG(x;)15 b FA(0\))32 b(and)d(so)1355 2462 y FG(\031)1410
2424 y Ft(\000)p Fy(1)1407 2491 y Fv(N)7 b Fy(+1)1564
2462 y FA(\()p FG(x;)15 b FA(0\))42 b(=)1924 2361 y Fu(\020)1978
2462 y FG(x;)15 b(g)2112 2479 y Fk(e)2113 2496 y Fj(S)-11
b Fy(\()p Fv(Y)t(;R)p Fy(\))2328 2462 y FA(\()p FG(x)p
FA(\))2450 2361 y Fu(\021)2520 2462 y FG(;)456 2652 y
FA(sho)m(wing)29 b(that)456 2841 y(\(A.7\))485 b FG(\031)1201
2855 y Fv(N)7 b Fy(+1)1358 2737 y Fu(\014)1358 2791 y(\014)1358
2846 y(\014)1388 2887 y Fk(e)1389 2904 y Fj(S)-12 b Fy(\()p
Fv(Y)t(;R)p Fy(\))p Ft(\\)p Fv(B)1698 2913 y Fq(3)p Fs(a)1767
2904 y Fy(\()p Fv(X)1852 2913 y Fq(0)1887 2904 y Fy(\))1949
2841 y FA(is)29 b(a)i(di\013eomorphism.)555 3052 y(Th)m(us,)f(w)m(e)h
(de\014ne)1039 3241 y FG(T)53 b FA(:=)40 b FG(\031)1333
3255 y Fv(N)7 b Fy(+1)1490 3137 y Fu(\014)1490 3191 y(\014)1490
3246 y(\014)1520 3287 y Fk(e)1521 3304 y Fj(S)-12 b Fy(\()p
Fv(Y)t(;R)p Fy(\))p Ft(\\)p Fv(B)1830 3313 y Fq(3)p Fs(a)1899
3304 y Fy(\()p Fv(X)1984 3313 y Fq(0)2019 3304 y Fy(\))2086
3241 y Fx(\016)21 b FG(\031)2207 3203 y Ft(\000)p Fy(1)2204
3270 y Fv(N)2301 3137 y Fu(\014)2301 3191 y(\014)2301
3246 y(\014)2330 3287 y Fk(e)2331 3304 y Fj(S)-12 b Fy(\()p
Fv(Y)t(;R)p Fy(\))p Ft(\\)p Fv(B)2640 3313 y Fq(3)p Fs(a)2710
3304 y Fy(\()p Fv(X)2795 3313 y Fq(0)2830 3304 y Fy(\))456
3451 y FA(Let)30 b(also)h(in)m(tro)s(duce)e(the)i(follo)m(wing)e
(domains:)776 3641 y FG(O)845 3655 y Fy(1)967 3641 y
FA(:=)83 b FG(T)1212 3540 y Fu(\020)1267 3641 y Fx(f)p
FG(x)1364 3655 y Fv(N)1456 3641 y FA(=)25 b(0)p Fx(g)c(\\)f(fj)p
FG(x)1866 3655 y Fv(N)7 b Fy(+1)2024 3641 y Fx(j)26 b
FG(<)f FA(3)p FG(=)p FA(4)p Fx(g)d(\\)e FG(B)2523 3655
y Fv(a)p Fy(+2)2654 3641 y FA(\()p FG(\031)2741 3655
y Fv(N)2809 3641 y FA(\()p FG(X)2919 3655 y Fy(0)2959
3641 y FA(\)\))3029 3540 y Fu(\021)776 3840 y FG(O)845
3854 y Fy(2)967 3840 y FA(:=)83 b FG(T)1212 3739 y Fu(\020)1267
3840 y Fx(f)p FG(x)1364 3854 y Fv(N)1456 3840 y FA(=)25
b(0)p Fx(g)c(\\)f(fj)p FG(x)1866 3854 y Fv(N)7 b Fy(+1)2024
3840 y Fx(j)26 b FG(<)f FA(5)p FG(=)p FA(8)p Fx(g)d(\\)e
FG(B)2523 3854 y Fv(a)p Fy(+1)2654 3840 y FA(\()p FG(\031)2741
3854 y Fv(N)2809 3840 y FA(\()p FG(X)2919 3854 y Fy(0)2959
3840 y FA(\)\))3029 3739 y Fu(\021)3099 3840 y FG(:)456
4034 y FA(Of)29 b(course,)i FG(x)940 4048 y Fy(0)1005
4034 y Fx(2)25 b FG(O)1160 4048 y Fy(2)1225 4034 y Fx(\032)g
FG(O)1390 4048 y Fy(1)1460 4034 y FA(and,)30 b(more)g(precisely)-8
b(,)30 b(b)m(y)g(\(A.6\))i(and)e(\(A.7\),)456 4201 y(\(A.8\))814
b(dist)14 b(\()p FG(O)1744 4215 y Fy(2)1784 4201 y FG(;)h(@)5
b(O)1946 4215 y Fy(1)1987 4201 y FA(\))41 b Fx(\025)f
FA(const)16 b FG(:)555 4368 y FA(Let)31 b(us)f(no)m(w)g(notice)h(that,)
g(in)e(the)i(ligh)m(t)e(of)i(Prop)s(osition)d(2.18,)849
4571 y(\001)925 4585 y Fv(p)964 4571 y FG(g)1006 4587
y Fk(e)1007 4605 y Fj(S)-11 b Fy(\()p Fv(Y)t(;R)p Fy(\))1242
4571 y Fx(\000)20 b FA(\001)1409 4585 y Fv(p)1448 4571
y FG(u)83 b Fx(\024)1737 4470 y Fu(\020)1791 4571 y FG(h)1843
4533 y Ft(0)1843 4593 y Fy(0)1883 4571 y FA(\()p FG(g)1960
4587 y Fk(e)1961 4605 y Fj(S)-11 b Fy(\()p Fv(Y)t(;R)p
Fy(\))2176 4571 y FA(\))20 b Fx(\000)g FG(h)2374 4533
y Ft(0)2374 4593 y Fy(0)2414 4571 y FA(\()p FG(u)p FA(\))2536
4470 y Fu(\021)2611 4571 y FA(+)2712 4509 y(const)p 2712
4550 208 4 v 2781 4633 a FG(R)2955 4571 y Fx(\024)1583
4789 y(\024)83 b FA(\003)1800 4688 y Fu(\020)1855 4789
y FA(\()p FG(g)1932 4805 y Fk(e)1933 4822 y Fj(S)-12
b Fy(\()p Fv(Y)t(;R)p Fy(\))2167 4789 y FA(+)2280 4727
y(1)p 2268 4768 70 4 v 2268 4851 a FG(R)2348 4789 y FA(\))20
b Fx(\000)g FG(u)2546 4688 y Fu(\021)2616 4789 y FG(:)456
4981 y FA(Hence,)31 b(if)e FG(X)k Fx(2)25 b FG(O)1097
4995 y Fy(1)1137 4981 y FA(,)883 5186 y Fx(\000)p FA(\001)1030
5200 y Fv(p)1069 5186 y FG(u)20 b FA(+)g(\003)p FG(u)25
b Fx(\024)g(\000)p FA(\001)1615 5200 y Fv(p)1654 5086
y Fu(\020)1709 5186 y FG(g)1751 5203 y Fk(e)1752 5220
y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))1986 5186 y FA(+)2099
5125 y(1)p 2087 5166 V 2087 5249 a FG(R)2167 5086 y Fu(\021)2241
5186 y FA(+)20 b(\003)2410 5086 y Fu(\020)2465 5186 y
FG(g)2507 5203 y Fk(e)2508 5220 y Fj(S)-12 b Fy(\()p
Fv(Y)t(;R)p Fy(\))2742 5186 y FA(+)2855 5125 y(1)p 2843
5166 V 2843 5249 a FG(R)2923 5086 y Fu(\021)2992 5186
y FG(;)p eop
%%Page: 80 80
80 79 bop 456 251 a Fy(80)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(where)41 b(\003)j(:=)g(sup)1114 472
y Fy([)p Ft(\000)p Fy(3)p Fv(=)p Fy(4)p Fv(;)12 b Fy(3)p
Fv(=)p Fy(4])1471 450 y Fx(j)p FG(h)1548 417 y Ft(0)1548
474 y Fy(0)1588 450 y Fx(j)p FA(.)75 b(Hence,)46 b(from)41
b(the)h(Harnac)m(k-t)m(yp)s(e)h(comparison)456 575 y(inequalit)m(y)28
b(for)986 540 y Fy(15)1087 575 y FG(p)p FA(-Laplacian)h(\(see,)j(for)e
(instance)g([9)q(],)h([35)q(])f(or)h([10)q(]\),)g(w)m(e)g(get)h(that)
456 798 y(\(A.9\))834 b(sup)1522 877 y Fv(O)1576 886
y Fq(2)1635 798 y FA(\()p FG(g)1712 815 y Fk(e)1713 832
y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))1948 798 y Fx(\000)19
b FG(u)p FA(\))26 b Fx(\024)2257 737 y FG(C)2329 704
y Ft(0)p 2257 778 95 4 v 2270 861 a FG(R)2377 798 y(;)456
1034 y FA(for)k(a)h(suitable)d FG(C)1081 1001 y Ft(0)1129
1034 y FG(>)d FA(1,)31 b(whic)m(h)e(ma)m(y)i(also)g(dep)s(end)d(on)j
FG(a)p FA(.)555 1204 y(Fix)43 b(no)m(w)g FG(Z)53 b Fx(2)46
b FG(L)29 b Fx(\\)f FG(B)1404 1218 y Fv(a)1446 1204 y
FA(\()p FG(\031)1533 1218 y Fv(N)1600 1204 y FG(X)1675
1218 y Fy(0)1715 1204 y FA(\).)79 b(Then,)46 b(from)c(\(A.7\),)48
b(there)c(exists)e FG(X)3232 1171 y Fy(\(1\))3374 1204
y FA(=)456 1333 y(\()p FG(x)543 1300 y Fy(\(1\))637 1333
y FG(;)15 b(x)729 1285 y Fy(\(1\))729 1362 y Fv(N)7 b
Fy(+1)887 1333 y FA(\))26 b Fx(2)1033 1310 y Fu(e)1033
1333 y Fw(S)-7 b FA(\()p FG(Y)5 b(;)15 b(R)q FA(\))21
b Fx(\\)f FG(B)1493 1347 y Fy(3)p Fv(a)1570 1333 y FA(\()p
FG(X)1680 1347 y Fy(0)1720 1333 y FA(\))30 b(so)h(that)1372
1543 y FG(\031)1424 1557 y Fv(N)1491 1438 y Fu(\014)1491
1493 y(\014)1491 1547 y(\014)1520 1589 y Fk(e)1521 1606
y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))p Ft(\\)p Fv(B)1830
1615 y Fq(3)p Fs(a)1899 1606 y Fy(\()p Fv(X)1984 1615
y Fq(0)2019 1606 y Fy(\))2051 1543 y FA(\()p FG(X)2168
1505 y Fy(\(1\))2263 1543 y FA(\))26 b(=)f FG(Z)c(;)456
1755 y FA(that)31 b(is)456 1924 y(\(A.10\))884 b FG(X)1675
1886 y Fy(\(1\))1795 1924 y FA(=)25 b FG(Z)i FA(+)19
b FG(t)2103 1886 y Fy(\(1\))2198 1924 y FG(e)2240 1938
y Fv(N)456 2099 y FA(for)30 b(some)h FG(t)856 2066 y
Fy(\(1\))975 2099 y Fx(2)25 b Fw(R)r FA(,)37 b(and)456
2275 y(\(A.11\))917 b FG(X)1708 2238 y Fy(\(1\))1843
2275 y Fx(2)1944 2252 y Fu(e)1944 2275 y Fw(S)-7 b FA(\()p
FG(Y)5 b(;)15 b(R)q FA(\))g FG(:)456 2444 y FA(Also,)30
b(b)m(y)g(\(A.6\),)1029 2658 y Fx(j)p FG(X)1136 2620
y Fy(\(1\))1251 2658 y Fx(\000)20 b FG(X)1417 2672 y
Fy(0)1457 2658 y Fx(j)25 b(\024)1603 2553 y Fu(\014)1603
2608 y(\014)1603 2662 y(\014)1634 2557 y(\020)1688 2658
y FG(\031)1740 2672 y Fv(N)1807 2553 y Fu(\014)1807 2608
y(\014)1807 2662 y(\014)1836 2704 y Fk(e)1837 2721 y
Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))p Ft(\\)p Fv(B)2146
2730 y Fq(3)p Fs(a)2215 2721 y Fy(\()p Fv(X)2300 2730
y Fq(0)2336 2721 y Fy(\))2367 2557 y Fu(\021)2421 2579
y Ft(\000)p Fy(1)2516 2658 y FA(\()p FG(Z)27 b Fx(\000)20
b FG(\031)2783 2672 y Fv(N)2850 2658 y FG(X)2925 2672
y Fy(0)2965 2658 y FA(\))3000 2553 y Fu(\014)3000 2608
y(\014)3000 2662 y(\014)3055 2658 y Fx(\024)1507 2841
y(\024)40 b FA(const)16 b Fx(j)p FG(Z)27 b Fx(\000)20
b FG(\031)2098 2855 y Fv(N)2165 2841 y FG(X)2240 2855
y Fy(0)2280 2841 y Fx(j)25 b(\024)1507 2976 y(\024)40
b FA(const)16 b FG(a)f(:)456 2798 y FA(\(A.12\))456 3150
y(Moreo)m(v)m(er,)32 b(from)e(\(A.4\),)i(w)m(e)f(ha)m(v)m(e)h(that,)f
(for)f(an)m(y)h FG(t)25 b Fx(\025)g FA(0,)972 3330 y
FG(g)1014 3346 y Fk(e)1015 3363 y Fj(S)-12 b Fy(\()p
Fv(Y)t(;R)p Fy(\))1229 3330 y FA(\()p FG(x)1316 3292
y Fy(\(1\))1431 3330 y FA(+)20 b FG(te)1597 3344 y Fv(N)1664
3330 y FA(\))83 b Fx(\025)98 b FA(const)16 b FG(t)k FA(+)g
FG(g)2360 3346 y Fk(e)2361 3363 y Fj(S)-12 b Fy(\()p
Fv(Y)t(;R)p Fy(\))2575 3330 y FA(\()p FG(x)2662 3292
y Fy(\(1\))2757 3330 y FA(\))41 b(=)1782 3513 y(=)98
b(const)16 b FG(t)k FA(+)g FG(x)2370 3465 y Fy(\(1\))2370
3542 y Fv(N)7 b Fy(+1)2568 3513 y FA(=)1782 3648 y(=)98
b(const)16 b FG(t)k FA(+)g FG(z)2360 3662 y Fv(N)7 b
Fy(+1)2533 3648 y FG(:)456 3817 y FA(Therefore,)30 b(from)g(\(A.9\),)
456 3996 y(\(A.13\))799 b FG(u)p FA(\()p FG(x)1647 3959
y Fy(\(1\))1762 3996 y FA(+)20 b FG(te)1928 4010 y Fv(N)1995
3996 y FA(\))26 b FG(>)f(z)2194 4010 y Fv(N)7 b Fy(+1)2367
3996 y FG(;)456 4166 y FA(pro)m(vided)36 b FG(t)h Fx(\025)g
FG(C)1087 4133 y Ft(00)1129 4166 y FG(=R)q FA(,)j(for)d(a)h(suitable)e
FG(C)1956 4133 y Ft(00)2035 4166 y FG(>)h FA(1,)k(whic)m(h)36
b(ma)m(y)i(also)g(dep)s(end)e(on)h FG(a)p FA(.)456 4273
y(Analogously)-8 b(,)456 4448 y(\(A.14\))799 b FG(u)p
FA(\()p FG(x)1647 4410 y Fy(\(1\))1762 4448 y Fx(\000)20
b FG(te)1928 4462 y Fv(N)1995 4448 y FA(\))26 b FG(<)f(z)2194
4462 y Fv(N)7 b Fy(+1)2367 4448 y FG(;)456 4617 y FA(pro)m(vided)35
b FG(t)h Fx(\025)g FG(C)1084 4584 y Ft(00)1126 4617 y
FG(=R)q FA(.)60 b(F)-8 b(rom)37 b(\(A.13\))i(and)d(\(A.14\),)41
b(w)m(e)c(deduce)g(the)g(existence)g(of)456 4731 y FG(t)489
4698 y Fy(\(2\))608 4731 y Fx(2)25 b FA([)p Fx(\000)p
FG(C)862 4698 y Ft(00)904 4731 y FG(=R)q(;)30 b(C)1146
4698 y Ft(00)1188 4731 y FG(=R)q FA(])h(so)g(that)1461
4910 y FG(u)p FA(\()p FG(x)1600 4873 y Fy(\(1\))1715
4910 y FA(+)20 b FG(t)1839 4873 y Fy(\(2\))1933 4910
y FG(e)1975 4924 y Fv(N)2042 4910 y FA(\))26 b(=)f FG(z)2241
4924 y Fv(N)7 b Fy(+1)2414 4910 y FG(:)p 456 5025 499
4 v 555 5095 a Fy(15)649 5124 y FF(W)-6 b(e)24 b(recall)h(that)e(the)g
(gradien)n(t)h(of)h FD(u)f FF(do)r(es)g(not)f(v)l(anish)h(in)f(the)g
(region)i(w)n(e)f(are)g(considering,)i(so)456 5216 y(that)f(the)g
(assumptions)h(needed)f(in)g([9)q(,)h(35)q(,)g(10)q(])g(are)g
(ful\014lled.)p eop
%%Page: 81 81
81 80 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(81)456
450 y FA(Let)36 b(us)f(de\014ne)g FG(X)1096 417 y Fy(\(2\))1225
450 y FA(:=)g FG(X)1438 417 y Fy(\(1\))1556 450 y FA(+)24
b FG(t)1684 417 y Fy(\(2\))1778 450 y FG(e)1820 464 y
Fv(N)1888 450 y FA(.)57 b(The)35 b(p)s(oin)m(t)g FG(x)2457
417 y Fy(\(2\))2585 450 y FA(=)f FG(x)2742 417 y Fy(\(2\))2837
450 y FA(\()p FG(Z)7 b FA(\))36 b(will)d(b)s(e)i(the)456
558 y(one)27 b(satisfying)e(the)i(thesis)f(of)g(Lemma)h(4.1,)i(as)e(w)m
(e)g(are)g(no)m(w)g(going)g(to)g(sho)m(w.)39 b(Notice)456
666 y(that,)31 b(b)m(y)f(construction,)1479 821 y Fu(\014)1479
876 y(\014)1479 930 y(\014)1509 926 y FG(X)1591 888 y
Fy(\(1\))1706 926 y Fx(\000)20 b FG(X)1879 888 y Fy(\(2\))1974
821 y Fu(\014)1974 876 y(\014)1974 930 y(\014)2029 926
y Fx(\024)2135 864 y FG(C)2207 831 y Ft(00)p 2135 905
114 4 v 2157 988 a FG(R)2441 926 y FA(and)-2132 b(\(A.15\))1479
1121 y FG(x)1531 1073 y Fy(\(2\))1531 1150 y Fv(N)7 b
Fy(+1)1713 1121 y FA(=)25 b FG(z)1851 1135 y Fv(N)7 b
Fy(+1)2034 1121 y FA(=)25 b FG(u)p FA(\()p FG(x)2269
1083 y Fy(\(2\))2364 1121 y FA(\))15 b FG(:)456 1331
y FA(In)29 b(particular,)1607 1564 y FG(\031)1659 1578
y Fv(N)1726 1463 y Fu(\020)1780 1564 y FG(x)1832 1526
y Fy(\(2\))1927 1564 y FG(;)h(u)p FA(\()p FG(x)2121 1526
y Fy(\(2\))2216 1564 y FA(\))2251 1463 y Fu(\021)2331
1564 y FA(=)25 b FG(Z)1405 1746 y FA(and)55 b Fx(j)p
FG(x)1684 1708 y Fy(\(2\))1799 1746 y Fx(\000)20 b FG(x)1942
1760 y Fy(0)1981 1746 y Fx(j)25 b(\024)1536 1949 y(\024j)p
FG(x)1684 1911 y Fy(\(1\))1799 1949 y Fx(\000)20 b FG(x)1942
1963 y Fy(0)1981 1949 y Fx(j)g FA(+)2127 1887 y FG(C)2199
1854 y Ft(00)p 2127 1928 V 2149 2011 a FG(R)2276 1949
y Fx(\024)1536 2178 y(\024)15 b FA(const)h FG(a)k FA(+)2014
2116 y FG(C)2086 2083 y Ft(00)p 2014 2157 V 2036 2240
a FG(R)2163 2178 y Fx(\024)1536 2339 y(\024)15 b FA(const)h
FG(a)f(;)456 1940 y FA(\(A.16\))456 2554 y(thanks)30
b(to)h(\(A.12\).)555 2764 y(W)-8 b(e)32 b(no)m(w)e(sho)m(w)g(that)456
3006 y(\(A.17\))960 2905 y Fu(\020)1014 3006 y FG(x)1066
2968 y Fy(\(1\))1180 3006 y Fx(\000)20 b FG(x)1323 3020
y Fy(0)1363 2905 y Fu(\021)1437 3006 y Fx(\001)1518 2944
y(r)p FG(u)p FA(\()p FG(x)1733 2958 y Fy(0)1772 2944
y FA(\))p 1493 2985 341 4 v 1493 3068 a Fx(jr)p FG(u)p
FA(\()p FG(x)1733 3082 y Fy(0)1772 3068 y FA(\))p Fx(j)1883
3006 y(\024)41 b FG(H)2071 3020 y Fy(0)2110 3006 y FA(\()p
FG(z)2187 3020 y Fv(N)7 b Fy(+1)2345 3006 y FA(\))20
b Fx(\000)g FG(H)2567 3020 y Fy(0)2606 3006 y FA(\()p
FG(u)p FA(\()p FG(x)2780 3020 y Fy(0)2820 3006 y FA(\)\))h(+)3012
2944 y FG(C)3084 2911 y Ft(000)p 3012 2985 134 4 v 3044
3068 a FG(R)3170 3006 y(;)456 3271 y FA(for)30 b(some)h
FG(C)895 3238 y Ft(000)981 3271 y FG(>)25 b FA(0)31 b(whic)m(h)e(ma)m
(y)i(dep)s(end)d(on)j FG(a)p FA(.)40 b(T)-8 b(o)31 b(pro)m(v)m(e)g
(\(A.17\),)i(let)d(us)g(de\014ne)1499 3482 y FG(w)r FA(\()p
FG(x)p FA(\))c(:=)f FG(H)1911 3496 y Fy(0)1951 3482 y
FA(\()p FG(g)2028 3498 y Fk(e)2029 3515 y Fj(S)-12 b
Fy(\()p Fv(Y)t(;R)p Fy(\))2243 3482 y FA(\()p FG(x)p
FA(\)\))456 3711 y(and)29 b(notice)i(that,)g(from)f(\(A.10\))j(and)c
(\(A.11\),)1387 3942 y FG(z)1429 3956 y Fv(N)7 b Fy(+1)1612
3942 y FA(=)25 b FG(x)1760 3894 y Fy(\(1\))1760 3970
y Fv(N)7 b Fy(+1)1943 3942 y FA(=)25 b FG(g)2081 3958
y Fk(e)2082 3975 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))2296
3942 y FA(\()p FG(x)2383 3904 y Fy(\(1\))2478 3942 y
FA(\))456 4192 y(and)29 b(so,)i(since)f FG(X)1067 4206
y Fy(0)1137 4192 y FA(is)f(a)i(p)s(oin)m(t)f(where)f(the)i(graph)f(of)g
FG(u)h FA(and)2581 4168 y Fu(e)2581 4192 y Fw(S)-7 b
FA(\()p FG(Y)5 b(;)15 b(R)q FA(\))31 b(touc)m(hes,)1084
4402 y FG(H)1160 4416 y Fy(0)1199 4402 y FA(\()p FG(z)1276
4416 y Fv(N)7 b Fy(+1)1434 4402 y FA(\))21 b Fx(\000)f
FG(H)1657 4416 y Fy(0)1696 4402 y FA(\()p FG(u)p FA(\()p
FG(x)1870 4416 y Fy(0)1910 4402 y FA(\)\))26 b(=)930
4564 y(=)83 b FG(H)1160 4578 y Fy(0)1199 4464 y Fu(\020)1253
4564 y FG(g)1295 4581 y Fk(e)1296 4598 y Fj(S)-11 b Fy(\()p
Fv(Y)t(;R)p Fy(\))1511 4564 y FA(\()p FG(x)1598 4527
y Fy(\(1\))1692 4564 y FA(\))1727 4464 y Fu(\021)1802
4564 y Fx(\000)20 b FG(H)1969 4578 y Fy(0)2008 4464 y
Fu(\020)2062 4564 y FG(g)2104 4581 y Fk(e)2105 4598 y
Fj(S)-11 b Fy(\()p Fv(Y)t(;R)p Fy(\))2320 4564 y FA(\()p
FG(x)2407 4578 y Fy(0)2446 4564 y FA(\))2481 4464 y Fu(\021)2561
4564 y FA(=)930 4746 y(=)83 b FG(w)r FA(\()p FG(x)1238
4709 y Fy(\(1\))1333 4746 y FA(\))21 b Fx(\000)f FG(w)r
FA(\()p FG(x)1634 4760 y Fy(0)1674 4746 y FA(\))26 b
Fx(\025)930 4892 y(\025)83 b(r)p FG(w)r FA(\()p FG(x)1314
4906 y Fy(0)1354 4892 y FA(\))21 b Fx(\001)f FA(\()p
FG(x)1542 4854 y Fy(\(1\))1657 4892 y Fx(\000)g FG(x)1800
4906 y Fy(0)1839 4892 y FA(\))h Fx(\000)35 b FA(const)16
b Fx(j)p FG(D)2327 4854 y Fy(2)2366 4892 y FG(w)r FA(\()p
FG(\030)t FA(\))p Fx(jj)p FG(x)2649 4854 y Fy(\(1\))2765
4892 y Fx(\000)k FG(x)2908 4906 y Fy(0)2948 4892 y Fx(j)2973
4854 y Fy(2)3027 4892 y FG(;)456 5108 y FA(for)35 b(some)h
FG(\030)j FA(lying)34 b(on)i(the)g(segmen)m(t)g(joining)e
FG(x)2150 5075 y Fy(\(1\))2280 5108 y FA(and)h FG(x)2514
5122 y Fy(0)2553 5108 y FA(.)57 b(Notice)36 b(no)m(w)g(that,)h(b)m(y)
456 5216 y(the)31 b(de\014nition)e(of)j FG(w)i FA(and)d(the)g(fact)h
(that)g FG(X)2010 5230 y Fy(0)2082 5216 y FA(is)e(a)i(p)s(oin)m(t)e(of)
i(touc)m(hing)f(b)s(et)m(w)m(een)h(the)p eop
%%Page: 82 82
82 81 bop 456 251 a Fy(82)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 456 y FA(graph)29 b(of)i FG(u)f FA(and)1076
433 y Fu(e)1076 456 y Fw(S)-7 b FA(\()p FG(Y)5 b(;)15
b(R)q FA(\),)870 618 y Fx(r)p FG(w)r FA(\()p FG(x)1100
632 y Fy(0)1140 618 y FA(\))21 b Fx(\001)f FA(\()p FG(x)1328
580 y Fy(\(1\))1443 618 y Fx(\000)g FG(x)1586 632 y Fy(0)1626
618 y FA(\))25 b(=)717 780 y(=)82 b FG(H)953 743 y Ft(0)946
803 y Fy(0)985 680 y Fu(\020)1040 780 y FG(g)1082 797
y Fk(e)1083 814 y Fj(S)-12 b Fy(\()p Fv(Y)t(;R)p Fy(\))1297
780 y FA(\()p FG(x)1384 794 y Fy(0)1424 780 y FA(\))1459
680 y Fu(\021)1529 780 y Fx(r)p FG(g)1647 797 y Fk(e)1648
814 y Fj(S)g Fy(\()p Fv(Y)t(;R)p Fy(\))1862 780 y FA(\()p
FG(x)1949 794 y Fy(0)1988 780 y FA(\))21 b Fx(\001)f
FA(\()p FG(x)2176 743 y Fy(\(1\))2291 780 y Fx(\000)g
FG(x)2434 794 y Fy(0)2473 780 y FA(\))26 b(=)717 1029
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5125 y FG(l)p 1666 5166 46 4 v 1666 5249 a FA(8)1742
5186 y(+)35 b(const)41 b Fx(\024)25 b FG(R)c Fx(\000)35
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70 4 v 81 x FG(R)p eop
%%Page: 86 86
86 85 bop 456 251 a Fy(86)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(and)29 b(th)m(us,)i(exploiting)d(\(2.15\),)1257
598 y FG(u)p FA(\()p Fi(x)p FA(\))e Fx(\024)f FG(g)1576
616 y Fj(S)-6 b Fy(\()p Fv(Y)t(;R)p Fy(\))1790 598 y
FA(\()p Fi(x)p FA(\))26 b Fx(\024)1437 774 y(\024)f FG(g)1576
788 y Fv(y)1611 799 y Fs(N)5 b Fq(+1)1746 788 y Fv(;R)1823
673 y Fu(\020)1893 774 y FA(const)15 b(\(1)21 b Fx(\000)2332
724 y Fq(3)2312 693 y Fx(p)p 2388 693 70 4 v 81 x FG(R)q
FA(\))2493 673 y Fu(\021)2573 774 y FA(=)1437 931 y(=)k
FG(s)1576 945 y Fv(R)1648 931 y FG(:)456 772 y FA(\(A.27\))456
1088 y(More)31 b(precisely)-8 b(,)29 b(w)m(e)i(ha)m(v)m(e)h(that)456
1244 y(\(A.28\))1053 b FG(u)p FA(\()p Fi(x)p FA(\))26
b FG(<)f(s)2081 1258 y Fv(R)456 1400 y FA(for)31 b(an)m(y)h
Fi(x)g FA(so)g(that)g Fx(j)p Fi(x)1201 1367 y Ft(0)1225
1400 y Fx(j)27 b(\024)g FG(l)r(=)p FA(8)32 b(and)f Fi(x)1736
1414 y Fv(N)1831 1400 y Fx(2)c FA([)p Fx(\000)p FG(l)r(=)p
FA(2)p FG(;)k Fx(\000)p FG(l)r(=)p FA(4]:)44 b(indeed,)31
b(if)f(not,)j(b)m(y)e(\(A.27\))456 1508 y(there)j(w)m(ould)e(b)s(e)i(a)
g(p)s(oin)m(t)f(for)h(whic)m(h)e(the)i(graph)g(of)g FG(u)g
FA(touc)m(hes)g(the)h FG(s)2962 1522 y Fv(R)3019 1508
y FA(-lev)m(el)f(from)456 1616 y(b)s(elo)m(w)41 b(and)i(then)f(a)h(con)
m(tradiction)g(follo)m(ws)e(b)m(y)i(applying)e(Theorem)h(B.6)i(to)f
(the)456 1724 y(function)29 b FG(s)855 1738 y Fv(R)932
1724 y Fx(\000)20 b FG(u)30 b FA(\(see,)i(e.g.,)g(the)f(argumen)m(t)f
(on)h(page)g(19\).)555 1880 y(W)-8 b(e)31 b(no)m(w)e(\014x)f
FG(C)1091 1894 y Ft(\003)1156 1880 y FG(>)d FA(1,)30
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b(The)28 b(\014rst)h(step)g(of)g(the)456 1988 y(pro)s(of)e(of)h(Lemma)g
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FG(Y)3071 2002 y Ft(\003)3135 1988 y Fx(2)f Fw(R)3281
1955 y Fv(N)7 b Fy(+1)456 2096 y FA(and)28 b FG(R)700
2110 y Ft(\003)765 2096 y FG(>)d(R)q(=)15 b FA(const)q(,)30
b(so)f(that)h FG(\031)1612 2110 y Fv(e)1645 2121 y Fs(N)1707
2096 y FG(Y)1760 2110 y Ft(\003)1824 2096 y FA(=)25 b
FG(\031)1972 2110 y Fv(e)2005 2121 y Fs(N)2066 2096 y
FG(Y)50 b FA(and)28 b Fw(S)o FA(\()p FG(Y)2489 2110 y
Ft(\003)2523 2096 y FG(;)15 b(R)2632 2110 y Ft(\003)2671
2096 y FA(\))30 b(touc)m(hes)g(the)g(graph)456 2215 y(of)36
b FG(u)g FA(from)f(ab)s(o)m(v)m(e,)k(in)c(the)h(region)g
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2192 y FA(\025)2028 2215 y FG(C)6 b(q)s Fx(g)p FA(,)38
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2215 y FG(;)30 b(u)p FA(\()p FG(x)3094 2229 y Ft(\003)3134
2215 y FA(\)\),)39 b(with)456 2333 y Fx(j)p FG(x)533
2300 y Ft(0)533 2355 y(\003)572 2333 y Fx(j)26 b(\024)739
2310 y FA(\025)719 2333 y FG(C)6 b(q)33 b FA(and)456
2549 y(\(A.29\))353 b FG(T)1115 2563 y Fv(Y)1156 2571
y Fp(\003)1192 2563 y Fv(;R)1265 2571 y Fp(\003)1304
2549 y FG(x)1356 2563 y Ft(\003)1421 2549 y Fx(2)1507
2448 y Fu(n)1568 2549 y FG(x)1620 2563 y Fv(N)1712 2549
y Fx(\024)1818 2488 y FA(const)16 b FG(q)2085 2455 y
Fy(2)p 1818 2528 307 4 v 1936 2612 a FG(R)2134 2448 y
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Fx(j)p FG(x)2443 2512 y Ft(0)2467 2549 y Fx(j)25 b FG(<)2653
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y FA(W)-8 b(e)37 b(will)c(pro)m(v)m(e)j(\(A.29\))i(b)m(y)e(iteration.)
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2967 y(so)37 b(that)g FG(\031)829 2981 y Fv(e)862 2992
y Fs(N)924 2967 y FG(Y)977 2982 y Fv(k)r Fy(+1)1146 2967
y FA(=)e FG(\031)1304 2981 y Fv(e)1337 2992 y Fs(N)1399
2967 y FG(Y)1452 2982 y Fv(k)1531 2967 y FA(and)h Fw(S)o
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Fv(k)r Fy(+1)1766 3087 y FA(\)\),)g(with)e Fx(j)p FG(x)2176
3054 y Ft(0)2176 3114 y Fv(k)r Fy(+1)2310 3087 y Fx(j)c(\024)2477
3064 y FA(\025)2456 3087 y FG(C)6 b(q)34 b FA(and)456
3315 y(\(A.30\))370 b FG(T)1132 3329 y Fv(Y)1173 3341
y Fs(k)q Fq(+1)1288 3329 y Fv(;R)1361 3341 y Fs(k)q Fq(+1)1480
3315 y FG(x)1532 3330 y Fv(k)r Fy(+1)1690 3315 y Fx(2)1776
3214 y Fu(n)1837 3315 y FG(x)1889 3329 y Fv(N)1981 3315
y Fx(\024)2087 3254 y FA(const)16 b FG(q)2354 3221 y
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y Fx(j)26 b FG(<)f(\021)s(q)2975 3214 y Fu(o)3051 3315
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b(The)28 b(pro)s(of)f(of)i(\(A.30\))h(is)d(actually)h(quite)g(non)g
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4105 y(F)-8 b(or)46 b(pro)m(ving)e(\(A.30\),)52 b(let)45
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eop
%%Page: 87 87
87 86 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
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450 y FA(Therefore,)456 643 y(\(A.31\))357 b Fx(j)p FG(h)1143
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826 y(Moreo)m(v)m(er,)32 b(from)e(De\014nition)f(2.5,)456
1052 y(\(A.32\))231 b FG(h)992 1019 y Ft(0)992 1079 y
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1532 y(\(A.33\))484 b FG(!)28 b FA(:=)d(2)1444 1499 y
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2559 5231 a FG(\015)5 b(r)2681 5068 y Fu(i)2739 5169
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y(ab)s(o)m(v)m(e.)41 b(By)29 b(its)g(minimizing)c(prop)s(ert)m(y)-8
b(,)29 b FG(b)1910 1582 y Ft(\003)1967 1615 y Fx(\000)17
b FG(a)2103 1582 y Ft(\003)2171 1615 y FA(is)28 b(orthogonal)i(to)f
FG(@)5 b(B)2946 1629 y Fv(r)2978 1637 y Fs(!)3027 1615
y FA(\()p FG(y)3107 1629 y Fv(!)3158 1615 y FA(\))29
b(at)h FG(a)3380 1582 y Ft(\003)3419 1615 y FA(,)456
1723 y(thence,)456 1882 y(\(A.38\))728 b Fx(j)p FG(a)1510
1844 y Ft(\003)1570 1882 y Fx(\000)19 b FG(b)1699 1844
y Ft(\003)1739 1882 y Fx(j)25 b FA(=)g Fx(j)p FG(b)1949
1844 y Ft(\003)2009 1882 y Fx(\000)20 b FG(y)2145 1896
y Fv(!)2195 1882 y Fx(j)h(\000)f FG(r)2373 1896 y Fv(!)2438
1882 y FG(:)456 2041 y FA(W)-8 b(e)22 b(notice)g(that)g
FG(b)1089 2008 y Ft(\003)1149 2041 y FA(cannot)g(lie)f(in)f
Fx(f)p FG(x)1745 2055 y Fv(N)1838 2041 y FG(<)k FA(0)p
Fx(g)p FA(,)h(otherwise)c(its)f(minimizing)e(prop)s(erties)456
2149 y(w)m(ould)31 b(imply)g(that)i FG(b)1219 2116 y
Ft(\003)1280 2149 y Fx(\000)22 b FG(a)1421 2116 y Ft(\003)1493
2149 y FA(is)32 b(orthogonal)h(to)h(\000)2215 2163 y
Fy(1)2287 2149 y FA(\(and)f(thence)g(to)g FG(@)5 b(B)3027
2163 y Fv(r)3066 2149 y FA(\()p FG(y)s FA(\)\))33 b(at)h
FG(b)3405 2116 y Ft(\003)456 2256 y FA(and)j(therefore,)j(b)m(y)e
(these)g(orthogonalit)m(y)h(prop)s(erties,)f(the)g(line)e(passing)h
(through)456 2364 y FG(a)504 2331 y Ft(\003)573 2364
y FA(and)29 b FG(b)788 2331 y Ft(\003)857 2364 y FA(should)f(also)h
(con)m(tain)i FG(y)h FA(and)d FG(y)1945 2378 y Fv(!)1995
2364 y FA(,)h(whic)m(h)f(is)f(imp)s(ossible.)37 b(This)28
b(sa)m(ys)j(that)456 2472 y FG(b)495 2439 y Ft(\003)495
2499 y Fv(N)587 2472 y FA(=)25 b(0)31 b(and)f Fx(j)p
FA(\()p FG(b)1035 2439 y Ft(\003)1075 2472 y FA(\))1110
2439 y Ft(0)1133 2472 y Fx(j)c FA(=)f FG(q)s FA(.)40
b(This)29 b(and)h(\(A.38\))i(yield)d(that)1516 2631 y
Fx(j)p FG(a)1589 2594 y Ft(\003)1649 2631 y Fx(\000)20
b FG(b)1779 2594 y Ft(\003)1818 2631 y Fx(j)26 b FA(=)1362
2800 y(=)1516 2697 y Fu(q)p 1607 2697 373 4 v 103 x FG(y)1655
2768 y Fy(2)1652 2828 y Fv(!)r(;N)1805 2800 y FA(+)20
b FG(q)1940 2773 y Fy(2)1999 2800 y Fx(\000)g FG(r)2131
2814 y Fv(!)2207 2800 y FA(=)1362 3070 y(=)83 b Fx(j)p
FG(y)1586 3084 y Fv(!)r(;N)1719 3070 y Fx(j)1759 2899
y Fu(s)p 1850 2899 528 4 v 171 x FA(1)21 b(+)2007 2942
y Fu(\022)2151 3008 y FG(q)p 2084 3049 178 4 v 2084 3132
a(y)2129 3146 y Fv(!)r(;N)2271 2942 y Fu(\023)2338 2964
y Fy(2)2398 3070 y Fx(\000)f FG(r)2530 3084 y Fv(!)2595
3070 y FG(:)456 3296 y FA(Th)m(us,)29 b(from)h(\(A.35\))i(and)e
(\(A.36\),)739 3455 y Fx(j)p FG(a)812 3417 y Ft(\003)872
3455 y Fx(\000)20 b FG(b)1002 3417 y Ft(\003)1042 3455
y Fx(j)25 b(\025)586 3650 y(\025)82 b(j)p FG(y)809 3664
y Fv(!)r(;N)942 3650 y Fx(j)21 b FA(+)1207 3588 y FG(q)1251
3555 y Fy(2)p 1104 3629 289 4 v 1104 3712 a FA(2)15 b
Fx(j)p FG(y)1234 3726 y Fv(!)r(;N)1367 3712 y Fx(j)1423
3650 y(\000)20 b FG(r)1555 3664 y Fv(!)1625 3650 y Fx(\000)35
b FA(const)1964 3588 y FG(q)2008 3555 y Fy(4)p 1964 3629
84 4 v 1964 3712 a FG(r)2008 3686 y Fy(3)2083 3650 y
Fx(\025)586 3912 y(\025)749 3850 y FA(1)p 749 3891 46
4 v 749 3974 a(2)805 3827 y Fu(p)p 896 3827 278 4 v 85
x FG(r)940 3886 y Fy(2)999 3912 y Fx(\000)20 b FG(q)1134
3886 y Fy(2)1193 3912 y FA(+)1554 3850 y FG(q)1598 3817
y Fy(2)p 1294 3891 604 4 v 1294 3913 a Fu(p)p 1385 3913
278 4 v 81 x FG(r)1429 3968 y Fy(2)1488 3994 y Fx(\000)g
FG(q)1623 3968 y Fy(2)1683 3994 y FA(+)1783 3953 y Fy(4)p
Fv(q)1852 3934 y Fq(2)p 1783 3973 104 4 v 1798 4025 a
Fv(\015)t(r)1927 3912 y Fx(\000)2028 3850 y FA(1)p 2028
3891 46 4 v 2028 3974 a(2)2084 3829 y Fu(p)p 2175 3829
650 4 v 83 x FG(r)2219 3886 y Fy(2)2278 3912 y FA(+)g
FG(q)2413 3886 y Fy(2)2452 3912 y FA(\(4)p FG(!)2592
3886 y Fy(2)2652 3912 y Fx(\000)g FA(1\))h Fx(\000)35
b FA(const)3183 3850 y FG(q)3227 3817 y Fy(4)p 3183 3891
84 4 v 3183 3974 a FG(r)3227 3948 y Fy(3)3301 3912 y
Fx(\025)586 4211 y(\025)82 b(\000)820 4149 y FG(q)864
4116 y Fy(2)p 820 4190 V 840 4273 a FG(r)913 4211 y(!)973
4173 y Fy(2)1033 4211 y FA(+)1134 4149 y FG(q)1178 4116
y Fy(2)p 1134 4190 V 1154 4273 a FG(r)1532 4149 y FA(1)p
1252 4190 607 4 v 1252 4273 a(1)21 b(+)35 b(const)15
b FG(q)1690 4247 y Fy(2)1730 4273 y FG(=r)1819 4247 y
Fy(2)1888 4211 y Fx(\000)35 b FA(const)2227 4149 y FG(q)2271
4116 y Fy(4)p 2227 4190 84 4 v 2227 4273 a FG(r)2271
4247 y Fy(3)2346 4211 y Fx(\025)586 4468 y(\025)749 4407
y FG(q)793 4374 y Fy(2)p 749 4448 V 769 4531 a FG(r)843
4468 y FA(\(1)21 b Fx(\000)f FG(!)1095 4431 y Fy(2)1134
4468 y FA(\))g Fx(\000)35 b FA(const)1528 4407 y FG(q)1572
4374 y Fy(4)p 1528 4448 V 1528 4531 a FG(r)1572 4505
y Fy(3)1637 4468 y FG(:)456 4658 y FA(This)28 b(pro)m(v)m(es)j
(\(A.37\).)555 4819 y(W)-8 b(e)32 b(no)m(w)e(de\014ne)g(the)g(follo)m
(wing)f(h)m(yp)s(ersurface)h(in)f Fw(R)2401 4786 y Fv(N)7
b Fy(+1)2565 4819 y FA(:)751 5000 y(\011)40 b(:=)998
4899 y Fu(n)1059 5000 y FG(X)32 b Fx(2)25 b Fw(R)1312
4963 y Fv(N)7 b Fy(+1)1501 5000 y Fx(j)25 b FG(x)1603
5014 y Fv(N)7 b Fy(+1)1786 5000 y FA(=)25 b FG(g)1925
5014 y Fv(y)1960 5025 y Fs(N)5 b Fq(+1)2095 5014 y Fv(;R)2172
4899 y Fu(\020)2226 5000 y FG(d)2273 5014 y Fy(\000)2322
5000 y FA(\()p FG(x)p FA(\))21 b(+)e FG(H)2631 5014 y
Fv(y)2666 5025 y Fs(N)5 b Fq(+1)2801 5014 y Fv(;R)2878
5000 y FA(\(0\))2993 4899 y Fu(\021)q(o)3124 5000 y FG(:)p
456 5116 499 4 v 555 5187 a Fy(19)626 5216 y FF(This)26
b(is)g(the)g(reason)g(also)h(for)g(in)n(tro)r(ducing)e(\006\()p
FD(Y)2061 5224 y Fs(!)2106 5216 y FD(;)13 b(R)2198 5224
y Fq(1)2233 5216 y FF(\))25 b(later)i(on)e(\(see)i(page)f(100\).)p
eop
%%Page: 90 90
90 89 bop 456 251 a Fy(90)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(F)g(or)31 b(the)f(sak)m(e)i(of)e(simplicit)m(y)-8
b(,)29 b(w)m(e)h(will)e(set)1161 622 y FG(g)1204 636
y Fy(\011)1263 622 y FA(\()p FG(x)p FA(\))e(:=)f FG(g)1575
636 y Fv(y)1610 647 y Fs(N)5 b Fq(+1)1745 636 y Fv(;R)1822
521 y Fu(\020)1877 622 y FG(d)1924 636 y Fy(\000)1972
622 y FA(\()p FG(x)p FA(\))21 b(+)f FG(H)2282 636 y Fv(y)2317
647 y Fs(N)5 b Fq(+1)2451 636 y Fv(;R)2529 622 y FA(\(0\))2644
521 y Fu(\021)2714 622 y FG(;)456 798 y FA(so)30 b(that)h(\011)25
b(=)g Fx(f)p FG(x)1053 812 y Fv(N)7 b Fy(+1)1236 798
y FA(=)25 b FG(g)1375 812 y Fy(\011)1435 798 y FA(\()p
FG(x)p FA(\))p Fx(g)p FA(.)555 906 y(Notice)32 b(that)456
1055 y(\(A.39\))81 b(the)31 b(sets)g Fx(f)p FG(d)1217
1069 y Fy(\000)1296 1055 y FA(attained)g(on)f(\000)1838
1069 y Fy(1)1877 1055 y Fx(g)h FA(and)f Fx(f)p FG(d)2222
1069 y Fy(\000)2301 1055 y FA(attained)h(on)f(\000)2843
1069 y Fy(3)2882 1055 y Fx(g)h FA(are)g(op)s(en.)456
1204 y(W)-8 b(e)31 b(pro)m(v)m(e)h(the)e(\014rst)g(claim)g(in)f
(\(A.39\),)k(the)d(other)h(one)g(b)s(eing)e(analogous.)42
b(Namely)-8 b(,)456 1312 y(w)m(e)30 b(sho)m(w)h(that)456
1460 y(\(A.40\))270 b(the)31 b(set)g(where)e FG(d)1587
1474 y Fy(\000)1666 1460 y FA(is)h(attained)g(on)h(\000)2300
1474 y Fy(2)2359 1460 y Fx([)20 b FA(\000)2497 1474 y
Fy(3)2566 1460 y FA(is)30 b(closed.)456 1609 y(Indeed,)25
b(tak)m(e)i FG(x)1016 1624 y Fv(k)1083 1609 y FA(in)d(the)h(ab)s(o)m(v)
m(e)h(set,)h(with)d FG(x)2004 1624 y Fv(k)2071 1609 y
FA(con)m(v)m(erging)i(to)g FG(x)2676 1623 y Ft(1)2776
1609 y FA(as)f FG(k)j FA(go)s(es)e(to)g(+)p Fx(1)p FA(:)456
1717 y(then,)k(since)g(\000)968 1731 y Fy(2)1027 1717
y Fx([)20 b FA(\000)1165 1731 y Fy(3)1234 1717 y FA(is)30
b(closed,)g(there)h(exist)f FG(z)2109 1732 y Fv(k)2177
1717 y Fx(2)25 b FA(\000)2320 1731 y Fy(2)2380 1717 y
Fx([)19 b FA(\000)2517 1731 y Fy(3)2587 1717 y FA(so)31
b(that)1266 1866 y Fx(j)p FG(x)1343 1881 y Fv(k)1406
1866 y Fx(\000)20 b FG(z)1539 1881 y Fv(k)1581 1866 y
Fx(j)26 b FA(=)f FG(d)1775 1880 y Fy(\000)1819 1889 y
Fq(2)1854 1880 y Ft(\\)p Fy(\000)1945 1889 y Fq(3)1984
1866 y FA(\()p FG(x)2071 1881 y Fv(k)2114 1866 y FA(\))g
Fx(\024)g(j)p FG(x)2347 1881 y Fv(k)2410 1866 y Fx(\000)20
b FG(w)r Fx(j)15 b FG(;)456 2015 y FA(for)33 b(an)m(y)h
FG(w)f Fx(2)e FA(\000)1020 2029 y Fy(1)1059 2015 y FA(.)51
b(Since)32 b(\000)1432 2029 y Fy(2)1494 2015 y Fx([)22
b FA(\000)1634 2029 y Fy(3)1707 2015 y FA(is)33 b(compact,)j(w)m(e)e
(ma)m(y)g(and)f(do)h(assume)f(that)h FG(z)3401 2030 y
Fv(k)456 2123 y FA(con)m(v)m(erges)e(to)f FG(z)1016 2137
y Ft(1)1116 2123 y Fx(2)25 b FA(\000)1259 2137 y Fy(2)1319
2123 y Fx([)19 b FA(\000)1456 2137 y Fy(3)1526 2123 y
FA(and,)30 b(therefore,)1489 2272 y Fx(j)p FG(x)1566
2286 y Ft(1)1661 2272 y Fx(\000)20 b FG(z)1794 2286 y
Ft(1)1869 2272 y Fx(j)25 b(\024)g(j)p FG(x)2092 2286
y Ft(1)2187 2272 y Fx(\000)20 b FG(w)r Fx(j)15 b FG(;)456
2421 y FA(for)30 b(an)m(y)g FG(w)e Fx(2)d FA(\000)1002
2435 y Fy(1)1041 2421 y FA(.)41 b(Hence,)1216 2570 y
FG(d)1263 2584 y Fy(\000)1307 2593 y Fq(2)1342 2584 y
Ft([)p Fy(\000)1433 2593 y Fq(3)1471 2570 y FA(\()p FG(x)1558
2584 y Ft(1)1633 2570 y FA(\))26 b Fx(\024)f(j)p FG(x)1867
2584 y Ft(1)1962 2570 y Fx(\000)20 b FG(z)2095 2584 y
Ft(1)2170 2570 y Fx(j)26 b(\024)f FG(d)2364 2584 y Fy(\000)2408
2593 y Fq(1)2447 2570 y FA(\()p FG(x)2534 2584 y Ft(1)2609
2570 y FA(\))15 b FG(:)456 2718 y FA(This)28 b(pro)m(v)m(es)j(\(A.40\))
i(and)c(th)m(us)h(\(A.39\).)555 2867 y(In)g(analogy)h(with)e(\(A.39\))j
(the)f(reader)f(will)e(easily)h(con)m(vince)i(herself)e(that)456
3016 y(\(A.41\))149 b(the)31 b(set)f(where)g FG(d)1466
3030 y Fy(\000)1545 3016 y FA(is)g(attained)g(on)g(\000)2178
3030 y Fy(2)2238 3016 y Fx(\\)20 b(fj)p FG(x)2441 2983
y Ft(0)2465 3016 y Fx(j)25 b(2)g FA(\()p FG(!)s(q)s(;)31
b(q)s FA(\))p Fx(g)g FA(is)e(op)s(en.)555 3165 y(Let)42
b(us)e(no)m(w)h(sho)m(w)g(some)g(prop)s(erties)e(of)i(\011.)72
b(First)40 b(of)i(all,)g(\011)f(coincides)f(with)456
3273 y Fw(S)n FA(\()p FG(Y)5 b(;)15 b(R)q FA(\))25 b(at)31
b(an)m(y)g(p)s(oin)m(ts)e(for)h(whic)m(h)f FG(d)1777
3287 y Fy(\000)1856 3273 y FA(is)h(realized)f(on)h(\000)2464
3287 y Fy(1)2504 3273 y FA(;)g(more)h(precisely)-8 b(,)456
3422 y(\(A.42\))376 b(if)29 b FG(d)1215 3436 y Fy(\000)1264
3422 y FA(\()p FG(x)p FA(\))d(=)f FG(d)1555 3436 y Fy(\000)1599
3445 y Fq(1)1638 3422 y FA(\()p FG(x)p FA(\),)31 b(then)f
FG(g)2066 3436 y Fy(\011)2126 3422 y FA(\()p FG(x)p FA(\))25
b(=)g FG(g)2412 3440 y Fj(S)-5 b Fy(\()p Fv(Y)t(;R)p
Fy(\))2627 3422 y FA(\()p FG(x)p FA(\).)456 3574 y(Indeed,)30
b(if)g FG(d)908 3588 y Fy(\000)957 3574 y FA(\()p FG(x)p
FA(\))d(=)f FG(d)1250 3588 y Fy(\000)1294 3597 y Fq(1)1333
3574 y FA(\()p FG(x)p FA(\),)32 b(b)m(y)f(some)g(geometric)h
(considerations)e(and)g(\(2.39\),)k(w)m(e)456 3682 y(ha)m(v)m(e)d(that)
1058 3836 y FG(g)1101 3850 y Fy(\011)1160 3836 y FA(\()p
FG(x)p FA(\))84 b(=)f FG(g)1563 3850 y Fv(y)1598 3861
y Fs(N)5 b Fq(+1)1733 3850 y Fv(;R)1810 3735 y Fu(\020)1864
3836 y FG(d)1911 3850 y Fy(\000)1955 3859 y Fq(1)1994
3836 y FA(\()p FG(x)p FA(\))21 b(+)f FG(H)2304 3850 y
Fv(y)2339 3861 y Fs(N)5 b Fq(+1)2473 3850 y Fv(;R)2551
3836 y FA(\(0\))2666 3735 y Fu(\021)2746 3836 y FA(=)1366
4072 y(=)83 b FG(g)1563 4086 y Fv(y)1598 4097 y Fs(N)5
b Fq(+1)1733 4086 y Fv(;R)1810 3971 y Fu(\020)1864 4072
y FG(d)1911 4086 y Fy(\000)1955 4095 y Fq(1)1994 4072
y FA(\()p FG(x)p FA(\))21 b Fx(\000)p 2240 3938 72 4
v 2240 4011 a FG(C)2312 4025 y Fy(0)p 2238 4051 116 4
v 2238 4134 a FA(2)p FG(R)2378 4072 y(y)2426 4035 y Fy(2)2423
4095 y Fv(N)7 b Fy(+1)2580 3971 y Fu(\021)2660 4072 y
FA(=)1366 4312 y(=)83 b FG(g)1563 4326 y Fv(y)1598 4337
y Fs(N)5 b Fq(+1)1733 4326 y Fv(;R)1810 4211 y Fu(\020)1864
4312 y Fx(j)p FG(x)21 b Fx(\000)e FG(y)s Fx(j)i(\000)f
FG(r)i Fx(\000)p 2403 4177 72 4 v 2403 4250 a FG(C)2475
4264 y Fy(0)p 2401 4291 116 4 v 2401 4374 a FA(2)p FG(R)2542
4312 y(y)2590 4274 y Fy(2)2587 4334 y Fv(N)7 b Fy(+1)2744
4211 y Fu(\021)2813 4312 y FG(:)456 4491 y FA(This,)28
b(\(A.18\))33 b(and)d(\(2.20\))i(end)e(the)g(pro)s(of)g(of)h(\(A.42\).)
555 4640 y(In)f(the)g(ligh)m(t)g(of)h(\(A.42\),)h(\(A.39\))h(and)c
(Prop)s(osition)g(2.13,)j(w)m(e)f(deduce)f(that)1109
4782 y(\001)1185 4796 y Fv(p)1224 4782 y FG(g)1267 4796
y Fy(\011)1327 4782 y FA(\()p FG(x)p FA(\))c FG(<)f(h)1623
4749 y Ft(0)1623 4806 y Fy(0)1662 4782 y FA(\()p FG(g)1740
4796 y Fy(\011)1800 4782 y FA(\()p FG(x)p FA(\)\))32
b(in)d(the)h(viscosit)m(y)g(sense)1109 4932 y(at)h(an)m(y)g
FG(x)25 b Fx(2)g Fw(R)1614 4899 y Fv(N)1718 4932 y FA(for)30
b(whic)m(h)1109 5072 y FG(d)1156 5086 y Fy(\000)1204
5072 y FA(\()p FG(x)p FA(\))h(is)f(attained)g(on)h(\000)1991
5086 y Fy(1)2060 5072 y FA(and)1109 5212 y FG(g)1152
5226 y Fy(\011)1211 5212 y FA(\()p FG(x)p FA(\))26 b
Fx(2)f FA([)p FG(s)1513 5226 y Fv(R)1571 5212 y FG(;)30
b Fx(\000)p FA(1)p FG(=)p FA(2])22 b Fx(\\)e FA([1)p
FG(=)p FA(2)p FG(;)32 b FA(1\).)456 5004 y(\(A.43\))p
eop
%%Page: 91 91
91 90 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(91)555
450 y FA(F)-8 b(urthermore,)39 b(in)c(an)i(appropriate)f(system)h(of)g
(co)s(ordinates)f(\(see)i Fx(x)p FA(14.6)h(of)e([19)q(])456
558 y(for)30 b(details)f(on)h(the)h(distance)f(function\),)1461
719 y Fx(r)p FG(g)1580 734 y Fv( )1633 719 y FA(\()p
FG(x)p FA(\))41 b(=)f FG(g)1953 682 y Ft(0)1950 742 y
Fv(y)1985 753 y Fs(N)5 b Fq(+1)2120 742 y Fv(;R)2198
719 y FA(\()p FG(?)p FA(\))15 b FG(e)2370 733 y Fv(N)456
898 y FA(and)26 b FG(D)707 865 y Fy(2)746 898 y FG(g)789
913 y Fv( )842 898 y FA(\()p FG(x)p FA(\))h(is)f(the)h
FG(N)22 b Fx(\002)13 b FG(N)36 b FA(diagonal)26 b(matrix)g(with)f(the)i
(follo)m(wing)e(en)m(tries)i(on)f(the)456 1005 y(diagonal:)828
1124 y FG(\024)880 1138 y Fy(1)p 641 1165 467 4 v 641
1248 a FG(\024)693 1262 y Fy(1)732 1248 y FG(d)779 1262
y Fy(\000)828 1248 y FA(\()p FG(x)p FA(\))21 b Fx(\000)f
FA(1)1132 1186 y FG(g)1178 1148 y Ft(0)1175 1208 y Fv(y)1210
1219 y Fs(N)5 b Fq(+1)1345 1208 y Fv(;R)1423 1186 y FA(\()p
FG(?)p FA(\))p FG(;)15 b(:)g(:)g(:)i(;)1938 1124 y(\024)1990
1138 y Fv(N)7 b Ft(\000)p Fy(1)p 1751 1165 585 4 v 1751
1248 a FG(\024)1803 1262 y Fv(N)g Ft(\000)p Fy(1)1961
1248 y FG(d)2008 1262 y Fy(\000)2056 1248 y FA(\()p FG(x)p
FA(\))21 b Fx(\000)f FA(1)2360 1186 y FG(g)2406 1148
y Ft(0)2403 1208 y Fv(y)2438 1219 y Fs(N)5 b Fq(+1)2573
1208 y Fv(;R)2651 1186 y FA(\()p FG(?)p FA(\))p FG(;)31
b(g)2868 1148 y Ft(00)2865 1208 y Fv(y)2900 1219 y Fs(N)5
b Fq(+1)3035 1208 y Fv(;R)3113 1186 y FA(\()p FG(?)p
FA(\))15 b FG(:)456 1395 y FA(Here)31 b(ab)s(o)m(v)m(e,)g(w)m(e)g
(denoted)1435 1557 y FG(?)41 b FA(:=)f FG(d)1704 1571
y Fy(\000)1752 1557 y FA(\()p FG(x)p FA(\))21 b(+)f FG(H)2062
1571 y Fv(y)2097 1582 y Fs(N)5 b Fq(+1)2231 1571 y Fv(;R)2309
1557 y FA(\(0\))15 b FG(;)456 1718 y FA(while,)32 b(as)i(standard,)g
FG(\024)1298 1732 y Fy(1)1338 1718 y FG(;)15 b(:)g(:)g(:)h(;)f(\024)
1591 1732 y Fv(N)7 b Fy(+1)1783 1718 y FA(represen)m(t)33
b(the)h(principal)c(curv)-5 b(atures)34 b(of)f(\000)h(at)456
1826 y(the)29 b(p)s(oin)m(t)g(where)g FG(d)1157 1840
y Fy(\000)1206 1826 y FA(\()p FG(x)p FA(\))h(is)f(realized.)40
b(Hence,)30 b(from)g(De\014nitions)e(2.8)j(and)e(2.11)i(and)456
1934 y(Lemma)f(B.3,)i(w)m(e)f(get)g(that)456 2169 y(\(A.44\))489
b Fx(r)p FG(g)1317 2183 y Fy(\011)1376 2169 y FA(\()p
FG(x)p FA(\))41 b(=)1650 2041 y Fu(\022)1805 2107 y FG(p)p
1727 2148 203 4 v 1727 2231 a(p)20 b Fx(\000)g FA(1)1939
2169 y FG(h)1991 2183 y Fv(y)2026 2194 y Fs(N)5 b Fq(+1)2161
2183 y Fv(;R)2239 2169 y FA(\()p FG(])p FA(\))2344 2041
y Fu(\023)2412 2063 y Fy(1)p Fv(=p)2552 2169 y FG(e)2594
2183 y Fv(N)2677 2169 y FG(;)456 2393 y FA(where)27 b(w)m(e)i(denoted)f
FG(])e FA(:=)f FG(g)1414 2407 y Fv(y)1449 2418 y Fs(N)5
b Fq(+1)1584 2407 y Fv(;R)1662 2393 y FA(\()p FG(?)p
FA(\),)30 b(and)d(that)i FG(D)2279 2360 y Fy(2)2318 2393
y FG(g)2361 2408 y Fv( )2414 2393 y FA(\()p FG(x)p FA(\))g(ma)m(y)g(b)s
(e)f(represen)m(ted)g(as)456 2501 y(the)i FG(N)g Fx(\002)20
b FG(N)40 b FA(diagonal)30 b(matrix)g(with)f(the)h(follo)m(wing)f(en)m
(tries)h(on)h(the)f(diagonal:)1394 2674 y FG(\024)1446
2688 y Fy(1)p 1207 2715 467 4 v 1207 2798 a FG(\024)1259
2812 y Fy(1)1299 2798 y FG(d)1346 2812 y Fy(\000)1394
2798 y FA(\()p FG(x)p FA(\))21 b Fx(\000)f FA(1)1714
2607 y Fu(\022)1869 2674 y FG(p)p 1790 2715 203 4 v 1790
2798 a(p)g Fx(\000)g FA(1)2003 2735 y FG(h)2055 2749
y Fv(y)2090 2760 y Fs(N)5 b Fq(+1)2225 2749 y Fv(;R)2302
2735 y FA(\()p FG(])p FA(\))2407 2607 y Fu(\023)2475
2630 y Fy(1)p Fv(=p)1197 2917 y FA(.)1197 2950 y(.)1197
2983 y(.)1394 3149 y FG(\024)1446 3163 y Fv(N)i Ft(\000)p
Fy(1)p 1207 3189 585 4 v 1207 3272 a FG(\024)1259 3286
y Fv(N)g Ft(\000)p Fy(1)1417 3272 y FG(d)1464 3286 y
Fy(\000)1512 3272 y FA(\()p FG(x)p FA(\))21 b Fx(\000)f
FA(1)1832 3082 y Fu(\022)1987 3149 y FG(p)p 1908 3189
203 4 v 1908 3272 a(p)g Fx(\000)g FA(1)2121 3210 y FG(h)2173
3224 y Fv(y)2208 3235 y Fs(N)5 b Fq(+1)2343 3224 y Fv(;R)2420
3210 y FA(\()p FG(])p FA(\))2525 3082 y Fu(\023)2593
3105 y Fy(1)p Fv(=p)1207 3415 y FA(\()p FG(ph)1340 3429
y Fv(y)1375 3440 y Fs(N)g Fq(+1)1510 3429 y Fv(;R)1587
3415 y FA(\()p FG(])p FA(\)\))1727 3382 y Fy(\(2)p Ft(\000)p
Fv(p)p Fy(\))p Fv(=p)p 1207 3461 778 4 v 1404 3548 a
FA(\()p FG(p)20 b Fx(\000)g FA(1\))1676 3522 y Fy(2)p
Fv(=p)2009 3482 y FG(h)2061 3445 y Ft(0)2061 3505 y Fv(y)2096
3516 y Fs(N)5 b Fq(+1)2231 3505 y Fv(;R)2308 3482 y FA(\()p
FG(])p FA(\))15 b FG(:)456 3099 y FA(\(A.45\))456 3701
y(F)-8 b(rom)30 b(\(A.44\))j(and)d(\(A.45\),)1032 3862
y(\001)1108 3876 y Fv(p)1147 3862 y FG(g)1190 3876 y
Fy(\011)1333 3862 y FA(=)83 b FG(h)1539 3825 y Ft(0)1539
3885 y Fv(y)1574 3896 y Fs(N)5 b Fq(+1)1708 3885 y Fv(;R)1806
3862 y FA(+)1333 4107 y(+)1487 3978 y Fu(\022)1642 4045
y FG(p)p 1563 4086 203 4 v 1563 4169 a(p)20 b Fx(\000)g
FA(1)1776 4107 y FG(h)1828 4121 y Fv(y)1863 4132 y Fs(N)5
b Fq(+1)1998 4121 y Fv(;R)2075 3978 y Fu(\023)2152 3970
y Fs(p)p Fp(\000)p Fq(1)p 2152 3986 111 3 v 2191 4027
a Fs(p)2307 3993 y Fv(N)i Ft(\000)p Fy(1)2318 4020 y
Fu(X)2326 4216 y Fv(i)p Fy(=1)2611 4045 y FG(\024)2663
4059 y Fv(i)p 2485 4086 333 4 v 2485 4169 a FG(\024)2537
4183 y Fv(i)2566 4169 y FG(d)2613 4183 y Fy(\000)2681
4169 y Fx(\000)20 b FA(1)2843 4107 y FG(;)-2412 b FA(\(A.46\))456
4355 y(outside)30 b Fx(fj)p FG(g)883 4369 y Fy(\011)943
4355 y Fx(j)d FA(=)f(1)p FG(=)p FA(2)p Fx(g)c([)e(fr)p
FG(g)1539 4369 y Fy(\011)1625 4355 y FA(=)26 b(0)p Fx(g)p
FA(,)33 b(where)d(w)m(e)i(dropp)s(ed)d(the)i FG(])p FA(-dep)s(endence)g
(for)456 4472 y(the)i(sak)m(e)g(of)g(simplicit)m(y)-8
b(.)46 b(Let)33 b(us)g(no)m(w)f(compute)h(the)g(principal)d(curv)-5
b(atures)32 b FG(\024)3255 4431 y Fy(\000)3299 4440 y
Fq(2)3255 4499 y Fv(i)3371 4472 y FA(of)456 4580 y(the)e(h)m(yp)s
(ersurface)f(\000)1200 4594 y Fy(2)1240 4580 y FA(:)40
b(exploiting)29 b(Lemma)i(B.14,)h(w)m(e)f(ha)m(v)m(e)g(that)692
4796 y Fx(\000)20 b FG(\024)835 4756 y Fy(\000)879 4765
y Fq(2)835 4822 y Fy(1)943 4796 y FA(=)25 b Fx(\001)15
b(\001)g(\001)27 b FA(=)e Fx(\000)p FG(\024)1390 4756
y Fy(\000)1434 4765 y Fq(2)1390 4825 y Fv(N)7 b Ft(\000)p
Fy(2)1630 4796 y FA(=)2305 4735 y FG(q)2349 4702 y Fv(\015)t
Fy(+2)p 1794 4775 1201 4 v 1794 4872 a Fx(j)p FG(x)1871
4845 y Ft(0)1894 4872 y Fx(j)1934 4794 y Fu(p)p 2026
4794 969 4 v 2026 4872 a FA(\()p FG(r)2105 4845 y Fy(2)2164
4872 y Fx(\000)20 b FG(q)2299 4845 y Fy(2)2338 4872 y
FA(\))p Fx(j)p FG(x)2450 4845 y Ft(0)2474 4872 y Fx(j)2499
4845 y Fy(2)p Fv(\015)t Fy(+2)2689 4872 y FA(+)g FG(q)2824
4845 y Fy(2)p Fv(\015)t Fy(+4)3044 4796 y Fx(\025)25
b FA(0)1337 5072 y FG(\024)1389 5031 y Fy(\000)1433 5040
y Fq(2)1389 5100 y Fv(N)7 b Ft(\000)p Fy(1)1630 5072
y FA(=)1821 5010 y(\()p FG(\015)26 b FA(+)20 b(1\))15
b FG(q)2159 4977 y Fv(\015)t Fy(+2)2310 5010 y FA(\()p
FG(r)2389 4977 y Fy(2)2448 5010 y Fx(\000)20 b FG(q)2583
4977 y Fy(2)2622 5010 y FA(\))15 b Fx(j)p FG(x)2749 4977
y Ft(0)2773 5010 y Fx(j)2798 4977 y Fy(2)p Fv(\015)t
Fy(+1)p 1794 5051 1203 4 v 1794 5089 a Fu(\020)1848 5190
y FA(\()p FG(r)1927 5163 y Fy(2)1987 5190 y Fx(\000)k
FG(q)2121 5163 y Fy(2)2161 5190 y FA(\))c Fx(j)p FG(x)2288
5163 y Ft(0)2312 5190 y Fx(j)2337 5163 y Fy(2)p Fv(\015)t
Fy(+2)2527 5190 y FA(+)20 b FG(q)2662 5163 y Fy(2)p Fv(\015)t
Fy(+4)2832 5089 y Fu(\021)2886 5111 y Fy(3)p Fv(=)p Fy(2)3046
5072 y Fx(\025)25 b FA(0)15 b FG(:)-2771 b FA(\(A.47\))p
eop
%%Page: 92 92
92 91 bop 456 251 a Fy(92)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(In)29 b(particular,)g(since)h Fx(j)p
FG(x)1314 417 y Ft(0)1337 450 y Fx(j)c(\025)f FG(!)s(q)33
b FA(on)d(\000)1801 464 y Fy(2)1840 450 y FA(,)h(w)m(e)g(infer)e(from)h
(the)g(ab)s(o)m(v)m(e)i(relations)d(that)456 686 y(\(A.48\))1002
b Fx(\000)p FG(\024)1834 645 y Fy(\000)1878 654 y Fq(2)1834
713 y Fv(i)1942 686 y Fx(\024)2071 624 y FA(1)p 2048
665 92 4 v 2048 748 a FG(r)2089 762 y Fv(!)2164 686 y
FG(;)456 923 y FA(for)28 b FG(i)e FA(=)f(1)p FG(;)15
b(:)g(:)g(:)i(;)e(N)28 b Fx(\000)17 b FA(2.)41 b(Th)m(us,)29
b(w)m(e)g(deduce)g(from)g(\(A.34\),)i(\(A.48\))g(and)e(\(A.18\))i(that)
456 1159 y(\(A.49\))1012 b Fx(\000)p FG(\024)1844 1118
y Fy(\000)1888 1127 y Fq(2)1844 1186 y Fv(i)1953 1159
y Fx(\024)2071 1097 y FA(3)p 2058 1138 70 4 v 2058 1221
a FG(R)2153 1159 y(;)456 1384 y FA(for)31 b FG(i)26 b
FA(=)h(1)p FG(;)15 b(:)g(:)g(:)i(;)e(N)31 b Fx(\000)20
b FA(2,)32 b(if)e FG(R)i FA(is)e(large)i(enough.)42 b(F)-8
b(urthermore,)32 b(since)e FG(q)g Fx(\025)c(j)p FG(x)3168
1351 y Ft(0)3192 1384 y Fx(j)g(\025)h FG(!)s(q)456 1492
y FA(on)j(\000)639 1506 y Fy(2)678 1492 y FA(,)h(\(A.47\))h(giv)m(es)
1212 1745 y FG(\024)1264 1704 y Fy(\000)1308 1713 y Fq(2)1264
1773 y Fv(N)7 b Ft(\000)p Fy(1)1505 1745 y Fx(\025)1669
1683 y FA(\()p FG(\015)26 b FA(+)19 b(1\))c(\()p FG(r)2041
1650 y Fy(2)2102 1683 y Fx(\000)20 b FG(q)2237 1650 y
Fy(2)2276 1683 y FA(\))15 b FG(!)2386 1650 y Fy(2)p Fv(\015)t
Fy(+1)p 1669 1724 888 4 v 2071 1807 a FG(r)2115 1781
y Fy(3)2592 1745 y Fx(\025)1505 1979 y(\025)1692 1918
y FA(9)p 1669 1959 91 4 v 1669 2042 a(10)1795 1918 y(\()p
FG(\015)26 b FA(+)19 b(1\))p FG(!)2133 1885 y Fy(2)p
Fv(\015)t Fy(+1)p 1795 1959 510 4 v 2028 2042 a FG(r)2339
1979 y Fx(\025)1505 2207 y(\025)1692 2145 y FA(9)p 1669
2186 91 4 v 1669 2269 a(40)1795 2145 y(\()p FG(\015)26
b FA(+)19 b(1\))p 1795 2186 280 4 v 1913 2269 a FG(r)2099
2207 y(;)456 2622 y FA(\(A.50\))779 b FG(\024)1540 2581
y Fy(\000)1584 2590 y Fq(2)1540 2650 y Fv(N)7 b Ft(\000)p
Fy(1)1738 2622 y Fx(\025)40 b FA(const)2082 2560 y(\()p
FG(\015)26 b FA(+)20 b(1\))p 2082 2601 V 2187 2684 a
FG(R)2387 2622 y(:)456 2840 y FA(Analogously)-8 b(,)30
b(one)h(sees)f(that)456 3080 y(\(A.51\))779 b FG(\024)1540
3040 y Fy(\000)1584 3049 y Fq(2)1540 3109 y Fv(N)7 b
Ft(\000)p Fy(1)1738 3080 y Fx(\024)40 b FA(const)2082
3019 y(\()p FG(\015)26 b FA(+)20 b(1\))p 2082 3060 V
2187 3143 a FG(R)2387 3080 y(:)456 3299 y FA(W)-8 b(e)31
b(no)m(w)g(claim)e(that)931 3509 y(\001)1007 3523 y Fv(p)1046
3509 y FG(g)1089 3523 y Fy(\011)1148 3509 y FA(\()p FG(x)p
FA(\))d Fx(\024)f FG(h)1444 3476 y Ft(0)1444 3533 y Fy(0)1484
3509 y FA(\()p FG(g)1562 3523 y Fy(\011)1622 3509 y FA(\()p
FG(x)p FA(\)\))c Fx(\000)1901 3448 y FA(const)16 b FG(\015)p
1901 3488 275 4 v 2003 3571 a(R)931 3685 y FA(at)31 b(an)m(y)f
FG(x)c Fx(2)e Fw(R)1436 3652 y Fv(N)1540 3685 y FA(for)30
b(whic)m(h)f Fx(j)p FG(g)2007 3699 y Fy(\011)2067 3685
y FA(\()p FG(x)p FA(\))p Fx(j)d(6)p FA(=)f(1)p FG(=)p
FA(2,)931 3825 y Fx(r)p FG(g)1050 3839 y Fy(\011)1109
3825 y FA(\()p FG(x)p FA(\))h Fx(6)p FA(=)f(0,)900 3981
y Fx(j)p FG(d)972 3995 y Fy(\000)1021 3981 y FA(\()p
FG(x)p FA(\))p Fx(j)h(\024)f FA(2)1335 3904 y Fx(p)p
1411 3904 70 4 v 77 x FG(R)31 b FA(and)f FG(d)1735 3995
y Fy(\000)1814 3981 y FA(is)f(attained)i(on)f(\000)2447
3995 y Fy(2)2507 3981 y Fx(\\)20 b(fj)p FG(x)2710 3948
y Ft(0)2734 3981 y Fx(j)25 b(2)g FA(\()p FG(!)s(q)s(;)30
b(q)s FA(\))p Fx(g)p FA(.)456 3728 y(\(A.52\))456 4179
y(T)-8 b(o)23 b(pro)m(v)m(e)h(this,)g(tak)m(e)h FG(x)e
FA(as)g(requested)g(here)g(ab)s(o)m(v)m(e:)38 b(then,)25
b(thanks)e(to)h(\(A.49\),)i(\(A.51\))456 4294 y(and)j(the)i(fact)g
(that)g Fx(j)p FG(d)1237 4308 y Fy(\000)1286 4294 y FA(\()p
FG(x)p FA(\))p Fx(j)26 b(\024)f FA(2)1600 4217 y Fx(p)p
1676 4217 V 77 x FG(R)q FA(,)31 b(that)1268 4473 y Fu(\014)1268
4528 y(\014)1268 4582 y(\014)1298 4578 y FG(\024)1350
4537 y Fy(\000)1394 4546 y Fq(2)1350 4606 y Fv(N)7 b
Ft(\000)p Fy(1)1508 4578 y FG(d)1555 4592 y Fy(\000)1603
4578 y FA(\()p FG(x)p FA(\))1725 4473 y Fu(\014)1725
4528 y(\014)1725 4582 y(\014)1776 4578 y FA(+)1867 4464
y Fv(N)g Ft(\000)p Fy(2)1878 4491 y Fu(X)1887 4687 y
Fv(i)p Fy(=1)2036 4473 y Fu(\014)2036 4528 y(\014)2036
4582 y(\014)2066 4578 y FG(\024)2118 4537 y Fy(\000)2162
4546 y Fq(2)2118 4605 y Fv(i)2201 4578 y FG(d)2248 4592
y Fy(\000)2297 4578 y FA(\()p FG(x)p FA(\))2419 4473
y Fu(\014)2419 4528 y(\014)2419 4582 y(\014)2475 4578
y Fx(\024)2581 4516 y FA(const)p 2581 4557 208 4 v 2612
4575 a Fx(p)p 2688 4575 70 4 v 77 x FG(R)456 4852 y FA(and)29
b(so,)i(if)f FG(R)h FA(is)e(large)h(enough,)1200 5017
y Fv(N)7 b Ft(\000)p Fy(1)1211 5044 y Fu(X)1220 5240
y Fv(i)p Fy(=1)1530 5068 y Fx(\000)p FG(\024)1653 5027
y Fy(\000)1697 5036 y Fq(2)1653 5095 y Fv(i)p 1378 5110
510 4 v 1378 5207 a FA(1)21 b Fx(\000)f FG(\024)1587
5166 y Fy(\000)1631 5175 y Fq(2)1587 5234 y Fv(i)1670
5207 y FG(d)1717 5221 y Fy(\000)1766 5207 y FA(\()p FG(x)p
FA(\))1924 5131 y Fx(\024)25 b(\000)p FA(2)2151 5017
y Fv(N)7 b Ft(\000)p Fy(2)2162 5044 y Fu(X)2170 5240
y Fv(i)p Fy(=1)2319 5131 y FG(\024)2371 5090 y Fy(\000)2415
5099 y Fq(2)2371 5158 y Fv(i)2490 5131 y Fx(\000)2606
5060 y FG(\024)2658 5019 y Fy(\000)2702 5028 y Fq(2)2658
5088 y Fv(N)g Ft(\000)p Fy(1)p 2606 5110 210 4 v 2688
5193 a FA(2)2841 5131 y FG(:)-2410 b FA(\(A.53\))p eop
%%Page: 93 93
93 92 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(93)456
450 y FA(Hence,)30 b(using)c(the)j(regularit)m(y)f(of)g(the)h
(functions)e(in)m(v)m(olv)m(ed)h(in)f(our)h(domain,)g(\(A.46\),)456
558 y(\(A.53\),)k(\(A.31\))h(and)c(\(A.32\),)572 816
y(\001)648 830 y Fv(p)687 816 y FG(g)730 830 y Fy(\011)873
816 y FA(=)82 b FG(h)1078 779 y Ft(0)1078 839 y Fv(y)1113
850 y Fs(N)5 b Fq(+1)1248 839 y Fv(;R)1346 816 y FA(+)1437
688 y Fu(\022)1592 755 y FG(p)p 1514 795 203 4 v 1514
879 a(p)20 b Fx(\000)g FA(1)1726 816 y FG(h)1778 830
y Fv(y)1813 841 y Fs(N)5 b Fq(+1)1948 830 y Fv(;R)2025
688 y Fu(\023)2102 680 y Fs(p)p Fp(\000)p Fq(1)p 2102
696 111 3 v 2141 737 a Fs(p)2257 703 y Fv(N)i Ft(\000)p
Fy(1)2268 730 y Fu(X)2277 925 y Fv(i)p Fy(=1)2561 754
y FG(\024)2613 713 y Fy(\000)2657 722 y Fq(2)2613 781
y Fv(i)p 2435 795 388 4 v 2435 892 a FG(\024)2487 852
y Fy(\000)2531 861 y Fq(2)2487 919 y Fv(i)2571 892 y
FG(d)2618 906 y Fy(\000)2686 892 y Fx(\000)20 b FA(1)2858
816 y Fx(\024)873 1138 y(\024)82 b FG(h)1078 1100 y Ft(0)1078
1160 y Fv(y)1113 1171 y Fs(N)5 b Fq(+1)1248 1160 y Fv(;R)1346
1138 y FA(+)1437 1010 y Fu(\022)1592 1077 y FG(p)p 1514
1117 203 4 v 1514 1200 a(p)20 b Fx(\000)g FA(1)1726 1138
y FG(h)1778 1152 y Fv(y)1813 1163 y Fs(N)5 b Fq(+1)1948
1152 y Fv(;R)2025 1010 y Fu(\023)2102 1002 y Fs(p)p Fp(\000)p
Fq(1)p 2102 1017 111 3 v 2141 1059 a Fs(p)2257 983 y
Fu( )2329 1138 y Fx(\000)p FA(2)2460 1024 y Fv(N)i Ft(\000)p
Fy(2)2471 1052 y Fu(X)2480 1247 y Fv(i)p Fy(=1)2629 1138
y FG(\024)2681 1097 y Fy(\000)2725 1106 y Fq(2)2681 1165
y Fv(i)2799 1138 y Fx(\000)2915 1067 y FG(\024)2967 1026
y Fy(\000)3011 1035 y Fq(2)2967 1096 y Fv(N)g Ft(\000)p
Fy(1)p 2915 1117 210 4 v 2997 1200 a FA(2)3135 983 y
Fu(!)3232 1138 y Fx(\024)873 1409 y(\024)82 b FG(h)1078
1371 y Ft(0)1078 1431 y Fy(0)1138 1409 y FA(+)1239 1347
y(const)p 1239 1388 208 4 v 1308 1471 a FG(R)1477 1409
y FA(+)1026 1675 y(+)1112 1547 y Fu(\022)1267 1613 y
FG(p)p 1189 1654 203 4 v 1189 1737 a(p)20 b Fx(\000)g
FA(1)1401 1675 y FG(h)1453 1689 y Fv(y)1488 1700 y Fs(N)5
b Fq(+1)1623 1689 y Fv(;R)1701 1547 y Fu(\023)1778 1539
y Fs(p)p Fp(\000)p Fq(1)p 1778 1554 111 3 v 1817 1595
a Fs(p)1932 1519 y Fu( )2004 1675 y Fx(\000)p FA(2)2135
1561 y Fv(N)i Ft(\000)p Fy(2)2147 1589 y Fu(X)2155 1784
y Fv(i)p Fy(=1)2304 1675 y FG(\024)2356 1634 y Fy(\000)2400
1643 y Fq(2)2356 1702 y Fv(i)2475 1675 y Fx(\000)2591
1604 y FG(\024)2643 1563 y Fy(\000)2687 1572 y Fq(2)2643
1633 y Fv(N)g Ft(\000)p Fy(1)p 2591 1654 210 4 v 2673
1737 a FA(2)2810 1519 y Fu(!)2913 1675 y FG(:)456 1928
y FA(W)-8 b(e)31 b(th)m(us)f(deduce,)g(b)m(y)h(\(A.49\))h(and)e
(\(A.50\),)i(that)663 2135 y(\001)739 2149 y Fv(p)778
2135 y FG(g)821 2149 y Fy(\011)964 2135 y Fx(\024)83
b FG(h)1170 2098 y Ft(0)1170 2158 y Fy(0)1230 2135 y
FA(+)1331 2074 y(const)p 1331 2115 208 4 v 1399 2198
a FG(R)1568 2135 y FA(+)1118 2401 y(+)1204 2273 y Fu(\022)1359
2340 y FG(p)p 1280 2381 203 4 v 1280 2464 a(p)20 b Fx(\000)g
FA(1)1493 2401 y FG(h)1545 2415 y Fv(y)1580 2426 y Fs(N)5
b Fq(+1)1715 2415 y Fv(;R)1792 2273 y Fu(\023)1869 2265
y Fs(p)p Fp(\000)p Fq(1)p 1869 2281 111 3 v 1908 2322
a Fs(p)2024 2273 y Fu(\022)2101 2340 y FA(6\()p FG(N)30
b Fx(\000)20 b FA(2\))p 2101 2381 356 4 v 2244 2464 a
FG(R)2486 2401 y Fx(\000)2602 2340 y FA(const)c(\()p
FG(\015)26 b FA(+)20 b(1\))p 2587 2381 518 4 v 2788 2464
a(2)p FG(R)3114 2273 y Fu(\023)3212 2401 y FG(:)456 2618
y FA(Th)m(us,)29 b(if)g FG(\015)36 b FA(is)29 b(large)i(enough,)803
2871 y(\001)879 2885 y Fv(p)919 2871 y FG(g)962 2885
y Fy(\011)1046 2871 y Fx(\024)25 b FG(h)1194 2834 y Ft(0)1194
2894 y Fy(0)1254 2871 y FA(+)1355 2810 y(const)p 1355
2850 208 4 v 1424 2933 a FG(R)1593 2871 y Fx(\000)35
b FA(const)1937 2743 y Fu(\022)2092 2810 y FG(p)p 2014
2850 203 4 v 2014 2933 a(p)20 b Fx(\000)g FA(1)2226 2871
y FG(h)2278 2885 y Fv(y)2313 2896 y Fs(N)5 b Fq(+1)2448
2885 y Fv(;R)2525 2743 y Fu(\023)2602 2735 y Fs(p)p Fp(\000)p
Fq(1)p 2602 2750 111 3 v 2641 2792 a Fs(p)2767 2810 y
FA(\()p FG(\015)26 b FA(+)20 b(1\))p 2767 2850 280 4
v 2849 2933 a(4)p FG(R)3071 2871 y(:)456 3093 y FA(Therefore,)30
b(since)g(w)m(e)h(are)f(ev)-5 b(aluating)30 b FG(h)1890
3107 y Fv(y)1925 3118 y Fs(N)5 b Fq(+1)2060 3107 y Fv(;R)2168
3093 y FA(at)31 b FG(g)2322 3107 y Fy(\011)2381 3093
y FA(\()p FG(x)p FA(\))26 b Fx(2)f FA([)p Fx(\000)p FA(1)p
FG(=)p FA(2)p FG(;)15 b FA(1)p FG(=)p FA(2],)1228 3322
y(\001)1304 3336 y Fv(p)1344 3322 y FG(g)1387 3336 y
Fy(\011)1472 3322 y Fx(\024)25 b FG(h)1620 3284 y Ft(0)1620
3344 y Fy(0)1680 3322 y FA(+)1780 3260 y(const)p 1780
3301 208 4 v 1849 3384 a FG(R)2018 3322 y Fx(\000)2119
3260 y FA(const)16 b(\()p FG(\015)25 b FA(+)20 b(1\))p
2119 3301 503 4 v 2335 3384 a FG(R)2646 3322 y(;)456
3519 y FA(whic)m(h)29 b(pro)m(v)m(es)i(\(A.52\),)h(if)e
FG(\015)35 b FA(is)30 b(c)m(hosen)g(to)i(b)s(e)d(con)m(v)m(enien)m(tly)
i(large.)555 3686 y(W)-8 b(e)32 b(no)m(w)e(sho)m(w)g(that)1109
3823 y(\001)1185 3837 y Fv(p)1224 3823 y FG(g)1267 3837
y Fy(\011)1327 3823 y FA(\()p FG(x)p FA(\))c FG(<)f(h)1623
3790 y Ft(0)1623 3847 y Fy(0)1662 3823 y FA(\()p FG(g)1740
3837 y Fy(\011)1800 3823 y FA(\()p FG(x)p FA(\)\))32
b(in)d(the)h(viscosit)m(y)g(sense)1109 3973 y(at)h(an)m(y)g
FG(x)25 b Fx(2)g Fw(R)1614 3940 y Fv(N)1718 3973 y FA(for)30
b(whic)m(h)1109 4108 y FG(d)1156 4122 y Fy(\000)1235
4108 y FA(is)f(attained)i(on)f(\000)1868 4122 y Fy(2)1907
4108 y FA(.)456 3972 y(\(A.54\))456 4274 y(F)-8 b(or)31
b(pro)m(ving)e(this,)h(w)m(e)h(\014rst)e(p)s(oin)m(t)h(out)g(that)h(w)m
(e)g(ma)m(y)g(assume)456 4441 y(\(A.55\))1018 b FG(g)1770
4455 y Fy(\011)1829 4441 y FA(\()p FG(x)p FA(\))26 b
FG(>)f(s)2116 4455 y Fv(R)456 4608 y FA(Indeed,)k(if)h
FG(g)903 4622 y Fy(\011)962 4608 y FA(\()p FG(x)p FA(\))c(=)f
FG(s)1249 4622 y Fv(R)1306 4608 y FA(,)31 b(arguing)e(as)i(in)e(Prop)s
(osition)f(2.13)1230 4774 y(\001)1306 4788 y Fv(p)1345
4774 y FG(g)1388 4788 y Fy(\011)1448 4774 y FA(\()p FG(x)p
FA(\))e(=)e(0)i FG(<)f(h)1910 4737 y Ft(0)1910 4797 y
Fy(0)1950 4774 y FA(\()p FG(s)2028 4788 y Fv(R)2086 4774
y FA(\))g(=)g FG(h)2294 4737 y Ft(0)2294 4797 y Fy(0)2334
4774 y FA(\()p FG(g)2412 4788 y Fy(\011)2472 4774 y FA(\()p
FG(x)p FA(\)\))15 b FG(;)456 4941 y FA(in)29 b(the)h(viscosit)m(y)g
(sense,)h(giving)e(the)i(desired)e(claim.)555 5049 y(F)-8
b(or)31 b(pro)m(ving)f(\(A.54\),)j(w)m(e)d(ma)m(y)h(also)g(assume)456
5216 y(\(A.56\))987 b Fx(r)p FG(g)1815 5230 y Fy(\011)1874
5216 y FA(\()p FG(x)p FA(\))26 b Fx(6)p FA(=)f(0)15 b
FG(;)p eop
%%Page: 94 94
94 93 bop 456 251 a Fy(94)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(otherwise,)35 b(w)m(e)g(w)m(ould)e(ha)m(v)m(e)j
FG(g)1555 464 y Fy(\011)1615 450 y FA(\()p FG(x)p FA(\))d(=)f
FG(s)1916 464 y Fv(R)2007 450 y FA(and)i(w)m(e)h(go)h(bac)m(k)f(to)h
(\(A.55\).)55 b(W)-8 b(e)36 b(ma)m(y)456 558 y(also)30
b(assume)g(that)456 707 y(\(A.57\))940 b Fx(j)p FG(d)1721
721 y Fy(\000)1770 707 y FA(\()p FG(x)p FA(\))p Fx(j)26
b(\024)2039 625 y(p)p 2115 625 70 4 v 82 x FG(R)41 b
FA(:)456 863 y(indeed,)29 b(if)g Fx(j)p FG(d)924 877
y Fy(\000)973 863 y FA(\()p FG(x)p FA(\))p Fx(j)d(\025)1242
786 y(p)p 1318 786 V 77 x FG(R)p FA(,)31 b(w)m(e)g(ha)m(v)m(e)g(that)
1133 1023 y Fx(j)p FG(d)1205 1037 y Fy(\000)1254 1023
y FA(\()p FG(x)p FA(\))21 b(+)e FG(H)1563 1037 y Fv(y)1598
1048 y Fs(N)5 b Fq(+1)1733 1037 y Fv(;R)1810 1023 y FA(\(0\))p
Fx(j)27 b(\025)2073 941 y(p)p 2148 941 V 2148 1023 a
FG(R)q(=)p FA(2)f FG(>)f(C)2495 1037 y Fy(1)2565 1023
y FA(log)16 b FG(R)456 1171 y FA(and)29 b(so,)i(b)m(y)g(Lemma)f(2.10,)i
(either)1266 1320 y FG(d)1313 1334 y Fy(\000)1361 1320
y FA(\()p FG(x)p FA(\))21 b(+)f FG(H)1671 1334 y Fv(y)1706
1345 y Fs(N)5 b Fq(+1)1840 1334 y Fv(;R)1918 1320 y FA(\(0\))26
b FG(>)f(H)2231 1334 y Fv(y)2266 1345 y Fs(N)5 b Fq(+1)2400
1334 y Fv(;R)2478 1320 y FA(\(1\))15 b FG(;)456 1469
y FA(in)29 b(whic)m(h)g(case,)i(due)f(to)h(De\014nition)e(2.11,)j
FG(g)2005 1483 y Fy(\011)2065 1469 y FA(\()p FG(x)p FA(\))f(is)e(not)i
(ev)m(en)g(de\014ned,)e(or)1238 1618 y FG(d)1285 1632
y Fy(\000)1334 1618 y FA(\()p FG(x)p FA(\))21 b(+)f FG(H)1644
1632 y Fv(y)1679 1643 y Fs(N)5 b Fq(+1)1813 1632 y Fv(;R)1890
1618 y FA(\(0\))27 b FG(<)e(H)2204 1632 y Fv(y)2239 1643
y Fs(N)5 b Fq(+1)2373 1632 y Fv(;R)2450 1618 y FA(\()p
FG(s)2528 1632 y Fv(R)2586 1618 y FA(\))15 b FG(;)456
1767 y FA(in)29 b(whic)m(h)g(case)i FG(g)1057 1781 y
Fy(\011)1147 1767 y FA(is)e(constan)m(tly)i(equal)f(to)h
FG(s)2072 1781 y Fv(R)2160 1767 y FA(in)e(a)h(neigh)m(b)s(orho)s(o)s(d)
e(of)j FG(x)f FA(and)g(th)m(us)456 1874 y(w)m(e)38 b(go)g(bac)m(k)g
(again)g(to)g(\(A.55\).)64 b(Also,)40 b(in)c(the)i(pro)s(of)e(of)i
(\(A.54\),)k(w)m(e)c(can)g(restrict)456 1982 y(ourselv)m(es)30
b(to)h(the)f(case)i(in)d(whic)m(h)456 2131 y(\(A.58\))1016
b Fx(j)p FG(g)1793 2145 y Fy(\011)1852 2131 y Fx(j)26
b(6)p FA(=)f(1)p FG(=)p FA(2)15 b FG(;)456 2280 y FA(since,)30
b(b)m(y)g(Prop)s(osition)f(2.13,)j(no)e(smo)s(oth)h(function)e(can)i
(touc)m(h)g FG(g)2801 2294 y Fy(\011)2891 2280 y FA(from)f(b)s(elo)m(w)
g(at)456 2387 y(lev)m(el)j Fx(\006)p FA(1)p FG(=)p FA(2.)50
b(With)33 b(these)h(further)e(\(non)h(restrictiv)m(e\))g(assumptions,)g
(w)m(e)g(no)m(w)h(deal)456 2495 y(with)29 b(the)h(pro)s(of)g(of)g
(\(A.54\).)456 2603 y(T)-8 b(ak)m(e)40 b FG(x)f FA(as)h(in)e(\(A.54\))j
(and)d(consider)h(a)g(sequence)h FG(x)2376 2618 y Fv(k)2458
2603 y FA(con)m(v)m(erging)g(to)g FG(x)p FA(,)h(so)f(that)456
2711 y FG(d)503 2725 y Fy(\000)551 2711 y FA(\()p FG(x)638
2726 y Fv(k)681 2711 y FA(\))d(is)e(attained)i(in)e(\000)1384
2725 y Fy(2)1448 2711 y Fx(\\)24 b(fj)p FG(x)1655 2678
y Ft(0)1679 2711 y Fx(j)35 b(2)g FA(\()p FG(!)s(q)s(;)15
b(q)s FA(\))p Fx(g)p FA(.)60 b(Due)37 b(to)g(\(A.58\),)k
Fx(r)p FG(g)2975 2725 y Fy(\011)3034 2711 y FA(\()p FG(x)p
FA(\))c(is)e(w)m(ell)456 2819 y(de\014ned)25 b(and)i(it)f(is)g(nonzero)
h(thanks)g(to)g(\(A.56\).)42 b(F)-8 b(rom)27 b(this,)g(\(A.58\))i(and)d
(\(A.57\))j(w)m(e)456 2927 y(ma)m(y)j(also)g(assume)f(that)i
Fx(j)p FG(g)1420 2941 y Fy(\011)1479 2927 y FA(\()p FG(x)1566
2942 y Fv(k)1609 2927 y FA(\))p Fx(j)c(6)p FA(=)e(1)p
FG(=)p FA(2,)34 b Fx(r)p FG(g)2109 2941 y Fy(\011)2168
2927 y FA(\()p FG(x)2255 2942 y Fv(k)2298 2927 y FA(\))e(is)f(w)m(ell)f
(de\014ned)h(and)g(nonzero)456 3042 y(and)38 b Fx(j)p
FG(d)713 3056 y Fy(\000)761 3042 y FA(\()p FG(x)848 3057
y Fv(k)891 3042 y FA(\))p Fx(j)i(\024)f FA(2)1146 2965
y Fx(p)p 1222 2965 V 77 x FG(R)p FA(.)66 b(Then,)40 b(the)e(h)m(yp)s
(otheses)h(of)g(\(A.52\))h(are)f(ful\014lled)c(b)m(y)k
FG(x)3377 3057 y Fv(k)3419 3042 y FA(:)456 3150 y(hence,)i(from)d
(\(A.52\))i(and)e(the)h(con)m(tin)m(uit)m(y)f(of)h(the)g(functions)e
(in)m(v)m(olv)m(ed,)j(whic)m(h)d(is)456 3258 y(assured)29
b(b)m(y)h(\(A.58\),)j(w)m(e)e(ha)m(v)m(e)g(that)1058
3407 y(\001)1134 3421 y Fv(p)1173 3407 y FG(g)1216 3421
y Fy(\011)1276 3407 y FA(\()p FG(x)p FA(\))83 b(=)137
b(lim)1635 3468 y Fv(k)r Ft(!)p Fy(+)p Ft(1)1885 3407
y FA(\001)1961 3421 y Fv(p)2000 3407 y FG(g)2043 3421
y Fy(\011)2103 3407 y FA(\()p FG(x)2190 3422 y Fv(k)2233
3407 y FA(\))25 b Fx(\024)1481 3637 y(\024)137 b FA(lim)1635
3698 y Fv(k)r Ft(!)p Fy(+)p Ft(1)1885 3637 y FG(h)1937
3599 y Ft(0)1937 3659 y Fy(0)1977 3637 y FA(\()p FG(g)2055
3651 y Fy(\011)2114 3637 y FA(\()p FG(x)2201 3652 y Fv(k)2244
3637 y FA(\)\))21 b Fx(\000)2436 3575 y FA(const)16 b
FG(\015)p 2436 3616 275 4 v 2539 3699 a(R)2746 3637 y
FA(=)1481 3867 y(=)83 b FG(h)1687 3829 y Ft(0)1687 3889
y Fy(0)1727 3867 y FA(\()p FG(g)1805 3881 y Fy(\011)1865
3867 y FA(\()p FG(x)p FA(\)\))21 b Fx(\000)2144 3805
y FA(const)15 b FG(\015)p 2144 3846 V 2246 3929 a(R)2444
3867 y(;)456 4046 y FA(th)m(us)30 b(pro)m(ving)f(\(A.54\).)555
4195 y(W)-8 b(e)32 b(no)m(w)e(pro)m(v)m(e)h(that)1106
4331 y(\001)1182 4345 y Fv(p)1221 4331 y FG(g)1264 4345
y Fy(\011)1324 4331 y FA(\()p FG(x)p FA(\))26 b FG(<)f(h)1620
4298 y Ft(0)1620 4356 y Fy(0)1660 4331 y FA(\()p FG(g)1738
4345 y Fy(\011)1797 4331 y FA(\()p FG(x)p FA(\)\))32
b(in)d(the)h(viscosit)m(y)g(sense)1106 4482 y(at)h(an)m(y)g
FG(x)25 b Fx(2)g Fw(R)1612 4449 y Fv(N)1715 4482 y FA(for)30
b(whic)m(h)f Fx(j)p FG(g)2182 4496 y Fy(\011)2242 4482
y FA(\()p FG(x)p FA(\))p Fx(j)d(\025)f FA(1)p FG(=)p
FA(2)32 b(and)1106 4622 y FG(d)1153 4636 y Fy(\000)1202
4622 y FA(\()p FG(x)p FA(\))f(is)e(attained)i(on)f(\000)1988
4636 y Fy(3)2027 4622 y FA(.)456 4483 y(\(A.59\))456
4770 y(T)-8 b(o)42 b(pro)m(v)m(e)h(the)f(ab)s(o)m(v)m(e)h(claim,)h
(notice)e(that,)j(as)e(remark)m(ed)f(here)f(ab)s(o)m(v)m(e,)46
b(w)m(e)d(ma)m(y)456 4878 y(restrict)34 b(ourselv)m(es)g(to)h(the)f
(case)h(in)e(whic)m(h)g(\(A.58\))k(and)c(\(A.57\))k(hold.)51
b(Also,)35 b(since)456 4986 y(\000)513 5000 y Fy(3)582
4986 y FA(is)30 b(a)g(p)s(ortion)f(of)i(sphere,)1348
5173 y FG(\024)1400 5132 y Fy(\000)1444 5141 y Fq(3)1400
5199 y Fy(1)1509 5173 y FA(=)25 b Fx(\001)15 b(\001)g(\001)26
b FA(=)f FG(\024)1884 5132 y Fy(\000)1928 5141 y Fq(3)1884
5201 y Fv(N)7 b Ft(\000)p Fy(1)2067 5173 y FA(=)25 b
Fx(\000)2266 5111 y FA(1)p 2244 5152 92 4 v 2244 5235
a FG(r)2285 5249 y Fv(!)2370 5173 y FG(<)g FA(0)15 b
FG(;)p eop
%%Page: 95 95
95 94 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(95)456
450 y FA(so,)30 b(from)g(\(A.57\),)j(w)m(e)e(get)g(that,)h(for)e
FG(i)25 b FA(=)g(1)p FG(;)15 b(:)g(:)g(:)i(;)e(N)31 b
Fx(\000)20 b FA(1,)1630 550 y Fu(\014)1630 604 y(\014)1630
659 y(\014)1660 654 y FG(\024)1712 614 y Fy(\000)1756
623 y Fq(3)1712 681 y Fv(i)1795 654 y FG(d)1842 668 y
Fy(\000)1891 550 y Fu(\014)1891 604 y(\014)1891 659 y(\014)1947
654 y Fx(\024)2052 593 y FA(const)p 2052 633 208 4 v
2084 652 a Fx(p)p 2159 652 70 4 v 2159 729 a FG(R)456
871 y FA(whic)m(h)29 b(is)g(small;)g(therefore,)i(from)f(\(A.32\),)1105
1035 y(\001)1181 1049 y Fv(p)1220 1035 y FG(g)1263 1049
y Fy(\011)1348 1035 y FA(=)951 1265 y(=)83 b FG(h)1157
1227 y Ft(0)1157 1287 y Fv(y)1192 1298 y Fs(N)5 b Fq(+1)1327
1287 y Fv(;R)1424 1265 y FA(+)1515 1137 y Fu(\022)1670
1203 y FG(p)p 1592 1244 203 4 v 1592 1327 a(p)20 b Fx(\000)g
FA(1)1804 1265 y FG(h)1856 1279 y Fv(y)1891 1290 y Fs(N)5
b Fq(+1)2026 1279 y Fv(;R)2104 1137 y Fu(\023)2180 1129
y Fs(p)p Fp(\000)p Fq(1)p 2180 1144 111 3 v 2220 1185
a Fs(p)2335 1151 y Fv(N)i Ft(\000)p Fy(1)2346 1179 y
Fu(X)2355 1374 y Fv(i)p Fy(=1)2640 1202 y FG(\024)2692
1162 y Fy(\000)2736 1171 y Fq(3)2692 1229 y Fv(i)p 2514
1244 388 4 v 2514 1341 a FG(\024)2566 1300 y Fy(\000)2610
1309 y Fq(3)2566 1368 y Fv(i)2649 1341 y FG(d)2696 1355
y Fy(\000)2765 1341 y Fx(\000)19 b FA(1)2936 1265 y Fx(\024)951
1586 y(\024)83 b FG(h)1157 1549 y Ft(0)1157 1609 y Fv(y)1192
1620 y Fs(N)5 b Fq(+1)1327 1609 y Fv(;R)1424 1586 y FA(+)35
b(const)1768 1473 y Fv(N)7 b Ft(\000)p Fy(1)1779 1500
y Fu(X)1788 1696 y Fv(i)p Fy(=1)1937 1586 y Fx(j)p FG(\024)2014
1546 y Fy(\000)2058 1555 y Fq(3)2014 1614 y Fv(i)2097
1586 y Fx(j)25 b(\024)951 1857 y(\024)83 b FG(h)1157
1820 y Ft(0)1157 1880 y Fv(y)1192 1891 y Fs(N)5 b Fq(+1)1327
1880 y Fv(;R)1424 1857 y FA(+)1525 1796 y(const)p 1525
1836 208 4 v 1594 1920 a FG(R)1768 1857 y Fx(\024)951
2105 y(\024)83 b FG(h)1157 2068 y Ft(0)1157 2128 y Fy(0)1217
2105 y Fx(\000)1336 2021 y Fu(b)1318 2044 y FG(C)1383
2058 y Fy(0)p 1318 2084 105 4 v 1335 2167 a FG(R)1452
2105 y FA(+)1553 2044 y(const)p 1553 2084 208 4 v 1622
2167 a FG(R)1796 2105 y(<)951 2277 y(<)g(h)1157 2240
y Ft(0)1157 2300 y Fy(0)1212 2277 y FG(;)456 2452 y FA(pro)m(vided)29
b(that)1045 2429 y Fu(b)1027 2452 y FG(C)1092 2466 y
Fy(0)1161 2452 y FA(is)h(c)m(hosen)h(suitably)d(large,)j(th)m(us)f(pro)
m(ving)f(\(A.59\).)555 2616 y(In)h(the)g(ligh)m(t)g(of)h(\(A.43\),)h
(\(A.54\))h(and)c(\(A.59\),)k(w)m(e)e(deduce)f(that)769
2771 y FG(g)812 2785 y Fy(\011)902 2771 y FA(is)f(a)i(strict)f(sup)s
(ersolution)d(of)k(\(1.5\))h(ev)m(erywhere)769 2947 y(p)s(ossibly)c
(except)j(the)f(set)h Fx(fj)p FG(g)1815 2961 y Fy(\011)1875
2947 y Fx(j)26 b FG(<)f FA(1)p FG(=)p FA(2)p Fx(g)d(\\)2305
2846 y Fu(n)2381 2947 y FG(d)2428 2961 y Fy(\000)2506
2947 y FA(realized)30 b(on)g(\000)3023 2961 y Fy(1)3083
2947 y Fx([)19 b FA(\000)3220 2961 y Fy(3)3260 2846 y
Fu(o)3320 2947 y FA(.)456 2878 y(\(A.60\))555 3234 y(W)-8
b(e)32 b(no)m(w)e(sho)m(w)g(that,)i(if)1453 3211 y(\025)1432
3234 y FG(C)37 b FA(is)29 b(large)i(enough,)f(then)2363
3199 y Fy(20)804 3421 y FA(if)g FG(x)25 b Fx(2)g FA([)p
Fx(\000)p FG(l)r(;)15 b(l)r FA(])1270 3388 y Fv(N)1368
3421 y FA(is)29 b(in)g(the)i(domain)e(of)i Fw(S)n FA(\()p
FG(Y)5 b(;)15 b(R)q FA(\))25 b(and)30 b Fx(j)p FG(x)2722
3388 y Ft(0)2745 3421 y Fx(j)c(\025)2917 3362 y Fq(4)2897
3323 y Fu(p)p 2988 3323 72 4 v 3008 3398 a FA(\025)2988
3421 y FG(C)c(q)s FA(,)30 b(then)804 3561 y Fx(j)p FG(d)876
3575 y Fy(\000)920 3584 y Fq(1)960 3561 y FA(\()p FG(x)p
FA(\))p Fx(j)c FG(<)f FA(min)n Fx(fj)p FG(d)1497 3575
y Fy(\000)1541 3584 y Fq(2)1581 3561 y FA(\()p FG(x)p
FA(\))p Fx(j)p FG(;)32 b Fx(j)p FG(d)1857 3575 y Fy(\000)1901
3584 y Fq(3)1940 3561 y FA(\()p FG(x)p FA(\))p Fx(jg)15
b FG(:)456 3480 y FA(\(A.61\))456 3730 y(T)-8 b(o)33
b(pro)m(v)m(e)g(\(A.61\),)j(w)m(e)d(pro)s(ceed)f(as)h(follo)m(ws.)47
b(Let)33 b FG(\027)6 b FA(\()p FG(p)p FA(\))33 b(b)s(e)f(the)h(outer)g
(unit)e(normal)456 3838 y(at)g(a)f(giv)m(en)h FG(p)24
b Fx(2)h FA(\000.)41 b(By)30 b(con)m(v)m(exit)m(y)i(and)e(conca)m(vit)m
(y)i(considerations,)d(one)i(sees)f(that,)456 3946 y(for)e
FG(p)d Fx(2)g FA(\000)807 3960 y Fy(2)862 3946 y Fx([)16
b FA(\000)996 3960 y Fy(3)1035 3946 y FA(,)29 b(the)g(minim)m(um)c(of)k
FG(\027)1795 3960 y Fv(N)1862 3946 y FA(\()p FG(p)p FA(\))p
FG(=)p Fx(j)p FG(\027)2099 3913 y Ft(0)2123 3946 y FA(\()p
FG(p)p FA(\))p Fx(j)g FA(is)f(attained)g(for)g Fx(j)p
FG(p)2947 3913 y Ft(0)2971 3946 y Fx(j)d FA(=)g FG(!)s(q)s
FA(,)k(th)m(us)456 4267 y(\(A.62\))737 b(min)1378 4328
y Fv(p)p Ft(2)p Fy(\000)1505 4337 y Fq(2)1540 4328 y
Ft([)p Fy(\000)1631 4337 y Fq(3)1697 4205 y FG(\027)1742
4219 y Fv(N)1809 4205 y FA(\()p FG(p)p FA(\))p 1691 4246
241 4 v 1691 4329 a Fx(j)p FG(\027)1767 4303 y Ft(0)1790
4329 y FA(\()p FG(p)p FA(\))p Fx(j)1982 4267 y FA(=)2103
4125 y Fu(p)p 2194 4125 278 4 v 80 x FG(r)2238 4179 y
Fy(2)2297 4205 y Fx(\000)20 b FG(q)2432 4179 y Fy(2)p
2103 4246 369 4 v 2212 4329 a FA(2)p FG(!)s(q)2496 4267
y(:)456 4521 y FA(T)-8 b(ak)m(e)39 b(no)m(w)f FG(x)g
FA(in)f(the)h(domain)f(of)h Fw(S)o FA(\()p FG(Y)5 b(;)15
b(R)q FA(\))32 b(with)37 b Fx(j)p FG(x)2309 4488 y Ft(0)2332
4521 y Fx(j)i(\025)2530 4463 y Fq(4)2510 4423 y Fu(p)p
2601 4423 72 4 v 2621 4498 a FA(\025)2601 4521 y FG(C)21
b(q)s FA(.)64 b(Let)38 b FG(p)g Fx(2)g FA(\000)g(b)s(e)f(a)456
4629 y(p)s(oin)m(t)31 b(realizing)g FG(d)1108 4643 y
Fy(\000)1156 4629 y FA(\()p FG(x)p FA(\).)47 b(Then,)33
b FG(x)21 b Fx(\000)g FG(p)32 b FA(is)g(orthogonal)g(to)h(\000)f(at)i
FG(p)p FA(.)46 b(Let)33 b(assume,)f(b)m(y)456 4737 y(con)m(tradicting)e
(\(A.61\),)i(that)f FG(p)25 b Fx(2)g FA(\000)1730 4751
y Fy(2)1790 4737 y Fx([)20 b FA(\000)1928 4751 y Fy(3)1967
4737 y FA(;)30 b(then,)1652 4901 y FG(x)25 b FA(=)g FG(p)20
b FA(+)g FG(\013\027)6 b FA(\()p FG(p)p FA(\))15 b FG(;)p
456 5002 499 4 v 555 5073 a Fy(20)653 5102 y FF(Notational)29
b(remark:)37 b(w)n(e)28 b(ha)n(v)n(e)f(that)g Fr(j)p
FD(d)1856 5110 y Fs(S)1901 5102 y FF(\()p FD(x)p FF(\))p
Fr(j)d FF(:=)g(inf)2249 5110 y Fs(y)r Fp(2)p Fs(S)2381
5102 y Fr(j)p FD(x)18 b Fr(\000)g FD(y)s Fr(j)27 b FF(is)h(the)f
(absolute)i(distance.)456 5216 y(Then,)c(\(A.61\))i(sa)n(ys)e(in)h
(particular)h(that,)e(if)i Fr(j)p FD(x)1849 5184 y Fp(0)1871
5216 y Fr(j)22 b(\025)2016 5167 y Fn(4)2002 5135 y Fg(p)p
2078 5135 61 4 v 2096 5197 a FF(\025)2078 5216 y FD(C)c(q)s
FF(,)26 b FD(d)2275 5224 y Fq(\000)2317 5216 y FF(\()p
FD(x)p FF(\))f(is)h(realized)h(on)e(\000)2959 5224 y
Fq(1)2994 5216 y FF(.)p eop
%%Page: 96 96
96 95 bop 456 251 a Fy(96)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(for)30 b(some)h FG(\013)25 b(>)g FA(0.)41
b(Th)m(us,)30 b(from)g(\(A.62\),)1528 581 y Fq(4)1508
541 y Fu(p)p 1599 541 72 4 v 1619 616 a FA(\025)1599
639 y FG(C)6 b(q)23 b Fx(\000)d FG(q)28 b Fx(\024)d(j)p
FG(x)2067 606 y Ft(0)2091 639 y Fx(j)20 b(\000)g(j)p
FG(p)2298 606 y Ft(0)2321 639 y Fx(j)26 b(\024)1349 827
y(\024)96 b(j)p FG(x)1593 794 y Ft(0)1637 827 y Fx(\000)20
b FG(p)1774 794 y Ft(0)1797 827 y Fx(j)25 b(\024)1953
766 y FA(2)p FG(\013!)s(q)19 b(\027)2221 780 y Fv(N)2288
766 y FA(\()p FG(p)p FA(\))p 1953 806 452 4 v 1995 824
a Fu(p)p 2085 824 278 4 v 2085 905 a FG(r)2129 879 y
Fy(2)2189 905 y Fx(\000)h FG(q)2324 879 y Fy(2)2430 827
y FG(;)456 1054 y FA(hence)1494 1187 y FG(\013\027)1597
1201 y Fv(N)1664 1187 y FA(\()p FG(p)p FA(\))26 b Fx(\025)40
b FA(const)2165 1123 y Fq(4)2145 1085 y Fu(p)p 2236 1085
72 4 v 2256 1164 a FA(\025)2236 1187 y FG(C)21 b(r)d(:)456
1329 y FA(Therefore,)1261 1480 y FG(x)1313 1494 y Fv(N)1406
1480 y FA(=)25 b FG(p)1548 1494 y Fv(N)1635 1480 y FA(+)20
b FG(\013\027)1829 1494 y Fv(N)1896 1480 y FA(\()p FG(p)p
FA(\))26 b Fx(\025)40 b FA(const)2397 1416 y Fq(4)2377
1378 y Fu(p)p 2468 1378 V 2489 1457 a FA(\025)2468 1480
y FG(C)22 b(r)17 b(;)456 1622 y FA(from)30 b(whic)m(h)1466
1755 y FG(x)1518 1769 y Fv(N)1605 1755 y Fx(\000)20 b
FG(y)1741 1769 y Fv(N)1833 1755 y Fx(\025)41 b FA(const)2193
1691 y Fq(4)2172 1653 y Fu(p)p 2263 1653 V 2284 1732
a FA(\025)2263 1755 y FG(C)22 b(r)c(;)456 1901 y FA(pro)m(ving)38
b(that)j FG(x)e FA(is)g(outside)g(the)h(domain)e(of)i
Fw(S)o FA(\()p FG(Y)5 b(;)15 b(R)q FA(\))-6 b(,)42 b(if)2596
1878 y(\025)2575 1901 y FG(C)k FA(is)39 b(large)h(enough,)i(in)456
2009 y(con)m(tradiction)30 b(with)f(our)h(assumptions.)39
b(This)28 b(pro)m(v)m(es)j(\(A.61\).)555 2179 y(In)h(the)h(ligh)m(t)f
(of)h(\(A.61\),)j(w)m(e)d(already)f(kno)m(w)h(that,)h(if)e
Fx(j)p FG(x)2549 2146 y Ft(0)2573 2179 y Fx(j)d(\025)2748
2156 y FA(\025)2727 2179 y FG(C)7 b(q)35 b FA(and)3075
2156 y(\025)3054 2179 y FG(C)k FA(is)32 b(large)456 2287
y(enough,)e(then)1503 2410 y Fx(j)p FG(d)1575 2424 y
Fy(\000)1619 2433 y Fq(1)1658 2410 y FA(\()p FG(x)p FA(\))p
Fx(j)d FG(<)e Fx(j)p FG(d)2000 2424 y Fy(\000)2044 2433
y Fq(2)2079 2424 y Ft([)p Fy(\000)2170 2433 y Fq(3)2209
2410 y FA(\()p FG(x)p FA(\))p Fx(j)15 b FG(:)456 2554
y FA(W)-8 b(e)42 b(no)m(w)g(impro)m(v)m(e)f(this)f(estimate)j(b)m(y)e
(sho)m(wing)g(that,)k(if)40 b FG(x)k Fx(2)f FA([)p Fx(\000)p
FG(l)r(;)15 b(l)r FA(])2989 2521 y Fv(N)3098 2554 y FA(is)41
b(in)f(the)456 2671 y(domain)29 b(of)h Fw(S)o FA(\()p
FG(Y)5 b(;)15 b(R)q FA(\))25 b(and)30 b Fx(j)p FG(x)1456
2638 y Ft(0)1479 2671 y Fx(j)c(\025)1646 2648 y FA(\025)1626
2671 y FG(C)6 b(q)s FA(,)30 b(then)456 2903 y(\(A.63\))512
b Fx(j)p FG(d)1293 2917 y Fy(\000)1337 2926 y Fq(2)1373
2917 y Ft([)p Fy(\000)1464 2926 y Fq(3)1503 2903 y FA(\()p
FG(x)p FA(\))p Fx(j)21 b(\000)f(j)p FG(d)1834 2917 y
Fy(\000)1878 2926 y Fq(1)1917 2903 y FA(\()p FG(x)p FA(\))p
Fx(j)26 b(\025)40 b FA(const)2455 2818 y(\025)2434 2841
y FG(C)2506 2808 y Fy(2)2545 2841 y FG(q)2589 2808 y
Fy(2)p 2434 2882 195 4 v 2496 2965 a FG(R)2653 2903 y(:)456
3095 y FA(T)-8 b(o)30 b(pro)m(v)m(e)i(\(A.63\),)g(w)m(e)f(ma)m(y)g(and)
f(do)g(assume)g(that)456 3265 y(\(A.64\))905 b Fx(j)p
FG(d)1686 3279 y Fy(\000)1730 3288 y Fq(1)1770 3265 y
FA(\()p FG(x)p FA(\))p Fx(j)26 b(\025)2060 3242 y FA(\025)2039
3265 y FG(C)6 b(q)s(=)p FA(2)15 b FG(;)456 3426 y FA(since)1540
3549 y Fx(j)p FG(d)1612 3563 y Fy(\000)1656 3572 y Fq(2)1691
3563 y Ft([)p Fy(\000)1782 3572 y Fq(3)1821 3549 y FA(\()p
FG(x)p FA(\))p Fx(j)26 b(\025)2111 3526 y FA(\025)2090
3549 y FG(C)6 b(q)23 b Fx(\000)d FG(q)456 3691 y FA(and)29
b(so)i(\(A.63\))h(w)m(ould)d(b)s(e)h(ob)m(viously)f(ful\014lled)e(if)i
(\(A.64\))k(w)m(ere)e(violated.)40 b(Also,)1124 3875
y Fx(j)p FG(x)20 b Fx(\000)g FG(y)s Fx(j)26 b(\025)f
FG(x)1559 3889 y Fv(N)1646 3875 y Fx(\000)20 b FG(y)1782
3889 y Fv(N)1874 3875 y Fx(\025)25 b(\000)p FG(l)d FA(+)2181
3790 y Fu(p)p 2272 3790 278 4 v 85 x FG(r)2316 3849 y
Fy(2)2375 3875 y Fx(\000)e FG(q)2510 3849 y Fy(2)2574
3875 y Fx(\025)2681 3814 y FG(r)p 2680 3854 46 4 v 2680
3938 a FA(2)2751 3875 y FG(;)456 4062 y FA(if)29 b FG(r)k
FA(is)c(large)i(enough.)40 b(This)29 b(sa)m(ys)i(that)456
4270 y(\(A.65\))1691 4209 y Fx(j)p FG(x)20 b Fx(\000)g
FG(y)s Fx(j)p 1598 4249 446 4 v 1598 4333 a(j)p FG(d)1670
4351 y Fv(@)t(B)1764 4359 y Fs(r)1800 4351 y Fy(\()p
Fv(y)r Fy(\))1896 4333 y FA(\()p FG(x)p FA(\))p Fx(j)2095
4270 y(\025)2216 4209 y FA(1)p 2216 4249 46 4 v 2216
4333 a(2)2286 4270 y FG(:)456 4497 y FA(Indeed,)29 b(if)h
FG(x)g FA(is)f(inside)f FG(B)1360 4511 y Fv(r)1398 4497
y FA(\()p FG(y)s FA(\),)j(the)g(ab)s(o)m(v)m(e)h(estimate)f(giv)m(es)f
(that)1505 4649 y Fx(j)p FG(x)20 b Fx(\000)g FG(y)s Fx(j)p
1412 4690 446 4 v 1412 4773 a(j)p FG(d)1484 4792 y Fv(@)t(B)1578
4800 y Fs(r)1614 4792 y Fy(\()p Fv(y)r Fy(\))1710 4773
y FA(\()p FG(x)p FA(\))p Fx(j)1893 4711 y(\025)1999 4649
y(j)p FG(x)h Fx(\000)f FG(y)s Fx(j)p 1999 4690 262 4
v 2108 4773 a FG(r)2296 4711 y Fx(\025)2402 4649 y FA(1)p
2402 4690 46 4 v 2402 4773 a(2)2472 4711 y FG(;)456 4938
y FA(while)28 b(if)h FG(x)i FA(is)e(outside)h FG(B)1333
4952 y Fv(r)1371 4938 y FA(\()p FG(y)s FA(\))1437 5090
y Fx(j)p FG(x)21 b Fx(\000)e FG(y)s Fx(j)p 1345 5130
446 4 v 1345 5214 a(j)p FG(d)1417 5232 y Fv(@)t(B)1511
5240 y Fs(r)1547 5232 y Fy(\()p Fv(y)r Fy(\))1643 5214
y FA(\()p FG(x)p FA(\))p Fx(j)1826 5151 y FA(=)2009 5090
y Fx(j)p FG(x)i Fx(\000)f FG(y)s Fx(j)p 1932 5130 417
4 v 1932 5214 a(j)p FG(x)g Fx(\000)g FG(y)s Fx(j)g(\000)g
FG(r)2383 5151 y Fx(\025)25 b FA(1)15 b FG(;)p eop
%%Page: 97 97
97 96 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(97)456
453 y FA(whic)m(h)33 b(pro)m(v)m(es)h(\(A.65\).)55 b(Let)34
b(no)m(w)g FG(x)g FA(b)s(e)g(so)g(that)h Fx(j)p FG(x)2303
420 y Ft(0)2327 453 y Fx(j)d(\025)2506 430 y FA(\025)2486
453 y FG(C)6 b(q)37 b FA(and)d(let)g FG(p)d Fx(2)h FA(\000)3178
467 y Fy(2)3240 453 y Fx([)22 b FA(\000)3380 467 y Fy(3)3419
453 y FA(.)456 561 y(Let)30 b(us)g(de\014ne)g(the)g(cone)456
836 y(\(A.66\))379 b Fi(C)41 b FA(:=)1316 680 y Fu(\()1389
836 y FG(x)25 b Fx(2)g Fw(R)1612 798 y Fv(N)1736 836
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741 y Ft(0)2081 774 y Fx(\000)20 b FG(y)2220 741 y Ft(0)2243
774 y Fx(j)p 1918 815 393 4 v 1918 898 a(j)p FG(x)1995
912 y Fv(N)2082 898 y Fx(\000)g FG(y)2218 912 y Fv(N)2285
898 y Fx(j)2346 836 y(\024)2452 676 y Fu(p)p 2542 676
72 4 v 2563 751 a FA(\025)2542 774 y FG(C)i(q)p 2452
815 222 4 v 2540 898 a(r)2683 680 y Fu(\))2786 836 y
FG(:)456 1144 y FA(Notice)31 b(that,)g(since)f FG(x)g
FA(is)g(in)f(the)h(domain)f(of)i Fw(S)n FA(\()p FG(Y)5
b(;)15 b(R)q FA(\))25 b(and)30 b Fx(j)p FG(x)2622 1111
y Ft(0)2646 1144 y Fx(j)25 b(\025)2813 1121 y FA(\025)2792
1144 y FG(C)7 b(q)s FA(,)30 b(then)456 1422 y(\(A.67\))1559
1360 y Fx(j)p FG(x)1636 1327 y Ft(0)1680 1360 y Fx(\000)20
b FG(y)1819 1327 y Ft(0)1842 1360 y Fx(j)p 1517 1401
393 4 v 1517 1484 a(j)p FG(x)1594 1498 y Fv(N)1681 1484
y Fx(\000)g FG(y)1817 1498 y Fv(N)1884 1484 y Fx(j)1945
1422 y(\025)2160 1337 y FA(\025)2139 1360 y FG(C)6 b(q)p
2050 1401 293 4 v 2050 1484 a FA(const)16 b FG(R)2368
1422 y(;)456 1701 y FA(if)564 1678 y(\025)543 1701 y
FG(C)42 b FA(is)33 b(c)m(hosen)j(large)f(enough.)53 b(This)33
b(sho)m(ws)i(also)g(that)g FG(x)g FA(is)f(outside)g(the)h(cone)g
Fi(C)p FA(.)456 1809 y(On)26 b(the)h(con)m(trary)-8 b(,)29
b(b)m(y)e(construction,)h(\000)1860 1823 y Fy(2)1913
1809 y Fx([)14 b FA(\000)2045 1823 y Fy(3)2111 1809 y
FA(lies)26 b(inside)e Fi(C)p FA(.)40 b(Th)m(us,)27 b(w)m(e)g(can)h
(denote)456 1917 y(b)m(y)34 b(^)-52 b FG(p)26 b FA(the)h(in)m
(tersection)g(b)s(et)m(w)m(een)h(the)f(segmen)m(t)h(joining)d
FG(x)i FA(and)g FG(p)f FA(with)g FG(@)5 b Fi(C)p FA(.)40
b(Let)27 b(also)463 2025 y(~)-52 b FG(p)25 b Fx(2)f FA(\000)669
2039 y Fy(1)738 2025 y FA(b)s(e)k(the)h(pro)5 b(jection)29
b(of)37 b(^)-53 b FG(p)29 b FA(on)g(\000.)40 b(By)29
b(construction,)37 b(~)-53 b FG(p)25 b Fx(2)g FG(@)5
b Fi(C)17 b Fx(\\)h FG(@)5 b(B)3035 2039 y Fv(r)3073
2025 y FA(\()p FG(y)s FA(\),)30 b(th)m(us)456 2322 y(\(A.68\))1618
2261 y Fx(j)8 b FA(~)-53 b FG(p)1689 2228 y Ft(0)1732
2261 y Fx(\000)20 b FG(y)1871 2228 y Ft(0)1894 2261 y
Fx(j)p 1576 2301 387 4 v 1576 2385 a(j)8 b FA(~)-53 b
FG(p)1647 2399 y Fv(N)1734 2385 y Fx(\000)20 b FG(y)1870
2399 y Fv(N)1937 2385 y Fx(j)1997 2322 y FA(=)2103 2163
y Fu(p)p 2194 2163 72 4 v 2215 2238 a FA(\025)2194 2261
y FG(C)i(q)p 2103 2301 222 4 v 2192 2385 a(r)456 2578
y FA(and)29 b(so)1022 2749 y(\025)1001 2772 y FG(C)7
b(q)1117 2739 y Fy(2)1176 2772 y FA(+)20 b FG(r)1311
2739 y Fy(2)p 1001 2813 349 4 v 1119 2887 a FA(\025)1098
2910 y FG(C)7 b(q)1214 2884 y Fy(2)1360 2834 y Fx(j)h
FA(^)-53 b FG(p)1431 2796 y Ft(0)1454 2834 y Fx(j)1479
2796 y Fy(2)1602 2834 y FA(=)83 b Fx(j)8 b FA(^)-53 b
FG(p)1827 2796 y Ft(0)1850 2834 y Fx(j)1875 2796 y Fy(2)1935
2834 y FA(+)2072 2772 y FG(r)2116 2739 y Fy(2)p 2036
2813 155 4 v 2056 2887 a FA(\025)2036 2910 y FG(C)6 b(q)2151
2884 y Fy(2)2200 2834 y Fx(j)i FA(^)-53 b FG(p)2271 2796
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FA(=)1602 3100 y(=)83 b Fx(j)8 b FA(^)-53 b FG(p)1827
3063 y Ft(0)1870 3100 y Fx(\000)20 b FG(y)2009 3063 y
Ft(0)2032 3100 y Fx(j)2057 3063 y Fy(2)2117 3100 y FA(+)2254
3039 y FG(r)2298 3006 y Fy(2)p 2218 3079 V 2239 3153
a FA(\025)2218 3176 y FG(C)6 b(q)2333 3150 y Fy(2)2383
3100 y Fx(j)i FA(^)-53 b FG(p)2454 3063 y Ft(0)2497 3100
y Fx(\000)20 b FG(y)2636 3063 y Ft(0)2659 3100 y Fx(j)2684
3063 y Fy(2)2749 3100 y FA(=)1602 3310 y(=)83 b Fx(j)8
b FA(^)-53 b FG(p)1827 3272 y Ft(0)1870 3310 y Fx(\000)20
b FG(y)2009 3272 y Ft(0)2032 3310 y Fx(j)2057 3272 y
Fy(2)2117 3310 y FA(+)g Fx(j)8 b FA(^)-53 b FG(p)2279
3324 y Fv(N)2366 3310 y Fx(\000)20 b FG(y)2502 3324 y
Fv(N)2569 3310 y Fx(j)2594 3272 y Fy(2)2674 3310 y FA(=)40
b FG(r)2829 3272 y Fy(2)2884 3310 y FG(:)456 3517 y FA(F)-8
b(rom)30 b(the)h(latter)g(estimate,)g(w)m(e)g(get)h(that)1681
3753 y Fx(j)8 b FA(^)-53 b FG(p)1752 3715 y Ft(0)1775
3753 y Fx(j)41 b(\025)1977 3688 y Fq(4)1957 3650 y Fu(p)p
2048 3650 72 4 v 2069 3730 a FA(\025)2048 3753 y FG(C)22
b(q)c(;)456 3969 y FA(if)29 b FG(r)k FA(and)810 3946
y(\025)790 3969 y FG(C)j FA(are)31 b(large)g(enough;)f(therefore,)h(in)
e(the)i(ligh)m(t)e(of)i(\(A.61\),)h(w)m(e)f(kno)m(w)g(that)1444
4177 y Fx(j)8 b FA(^)-53 b FG(p)21 b Fx(\000)27 b FA(~)-52
b FG(p)o Fx(j)26 b FA(=)f Fx(j)p FG(d)1891 4191 y Fy(\000)1940
4177 y FA(\()8 b(^)-53 b FG(p)p FA(\))p Fx(j)25 b(\024)g(j)8
b FA(^)-53 b FG(p)21 b Fx(\000)f FG(p)p Fx(j)456 4384
y FA(and,)30 b(therefore,)782 4592 y Fx(j)p FG(x)21 b
Fx(\000)e FG(p)p Fx(j)26 b FA(=)f Fx(j)p FG(x)20 b Fx(\000)28
b FA(^)-53 b FG(p)p Fx(j)20 b FA(+)g Fx(j)8 b FA(^)-53
b FG(p)20 b Fx(\000)g FG(p)p Fx(j)25 b(\025)g(j)p FG(x)c
Fx(\000)27 b FA(^)-52 b FG(p)o Fx(j)21 b FA(+)f Fx(j)8
b FA(^)-53 b FG(p)20 b Fx(\000)28 b FA(~)-53 b FG(p)p
Fx(j)25 b(\025)g(j)p FG(x)c Fx(\000)27 b FA(~)-53 b FG(p)p
Fx(j)15 b FG(:)-2495 b FA(\(A.69\))456 4799 y(Let)30
b(no)m(w)h FG(\014)k FA(b)s(e)30 b(the)h(angle)f(b)s(et)m(w)m(een)h
FG(x)20 b Fx(\000)g FG(y)33 b FA(and)38 b(~)-53 b FG(p)20
b Fx(\000)g FG(y)s FA(.)p eop
%%Page: 98 98
98 97 bop 456 251 a Fy(98)175 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
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Fi(C)555 2543 y FA(By)h(\(A.67\))h(and)e(\(A.68\),)j(w)m(e)d(ha)m(v)m
(e)i(that)1647 2947 y FG(\014)46 b Fx(\025)40 b FA(const)2108
2862 y(\025)2087 2885 y FG(C)7 b(q)p 2087 2926 116 4
v 2110 3009 a(R)2228 2947 y(;)456 3328 y FA(pro)m(vided)850
3305 y(\025)830 3328 y FG(C)36 b FA(is)30 b(large)g(enough,)h
(therefore)738 3661 y Fx(j)p FG(x)20 b Fx(\000)28 b FA(~)-53
b FG(p)p Fx(j)997 3624 y Fy(2)1120 3661 y FA(=)83 b Fx(j)p
FG(y)23 b Fx(\000)k FA(~)-52 b FG(p)o Fx(j)1528 3624
y Fy(2)1588 3661 y FA(+)20 b Fx(j)p FG(x)h Fx(\000)f
FG(y)s Fx(j)1941 3624 y Fy(2)2000 3661 y Fx(\000)g FA(2)15
b Fx(j)p FG(y)24 b Fx(\000)k FA(~)-53 b FG(p)p Fx(j)15
b(j)p FG(x)21 b Fx(\000)e FG(y)s Fx(j)31 b FA(cos)16
b FG(\014)30 b Fx(\025)1120 3840 y(\025)1274 3739 y Fu(\020)1328
3840 y Fx(j)p FG(x)20 b Fx(\000)g FG(y)s Fx(j)g(\000)g(j)p
FG(y)k Fx(\000)j FA(~)-52 b FG(p)o Fx(j)1955 3739 y Fu(\021)2010
3762 y Fy(2)2069 3840 y FA(+)35 b(const)16 b Fx(j)p FG(y)24
b Fx(\000)j FA(~)-53 b FG(p)p Fx(j)15 b(j)p FG(x)21 b
Fx(\000)f FG(y)s Fx(j)15 b FG(\014)3001 3803 y Fy(2)3066
3840 y FA(=)1120 4056 y(=)1274 3955 y Fu(\020)1328 4056
y FA(\()p FG(r)23 b FA(+)d FG(d)1565 4070 y Fy(\000)1613
4056 y FA(\()p FG(x)p FA(\)\))i Fx(\000)d FG(r)1926 3955
y Fu(\021)1980 3977 y Fy(2)2040 4056 y FA(+)35 b(const)16
b FG(r)h Fx(j)p FG(x)k Fx(\000)f FG(y)s Fx(j)15 b FG(\014)2760
4018 y Fy(2)2825 4056 y Fx(\025)-2440 b FA(\(A.70\))1120
4298 y Fx(\025)1274 4197 y Fu(\020)1328 4298 y FG(d)1375
4312 y Fy(\000)1423 4298 y FA(\()p FG(x)p FA(\))1545
4197 y Fu(\021)1600 4219 y Fy(2)1660 4298 y FA(+)35 b(const)16
b FG(r)h Fx(j)p FG(x)k Fx(\000)f FG(y)s Fx(j)2355 4213
y FA(\025)2334 4236 y FG(C)2406 4203 y Fy(2)2445 4236
y FG(q)2489 4203 y Fy(2)p 2334 4277 195 4 v 2376 4360
a FG(R)2446 4334 y Fy(2)2553 4298 y FG(:)456 4661 y FA(Also,)33
b FG(d)736 4680 y Fv(@)t(B)830 4688 y Fs(r)866 4680 y
Fy(\()p Fv(y)r Fy(\))962 4661 y FA(\()p FG(x)p FA(\))h(m)m(ust)f(b)s(e)
f(attained)i(on)f(\000)2020 4675 y Fy(1)2059 4661 y FA(,)h(since)e(the)
i(straigh)m(t)f(lines)e(normal)i(to)456 4773 y FG(@)5
b(B)578 4787 y Fv(r)616 4773 y FA(\()p FG(y)s FA(\))13
b Fx(n)g FA(\000)862 4787 y Fy(1)929 4773 y FA(are)28
b(con)m(tained)f(in)e Fi(C)p FA(,)j(while)d FG(x)i FA(is)f(outside)g
Fi(C)p FA(.)39 b(Therefore,)28 b(from)e(\(A.61\),)456
5214 y(\(A.71\))591 b Fx(j)p FG(d)1372 5228 y Fy(\000)1420
5214 y FA(\()p FG(x)p FA(\))p Fx(j)27 b FA(=)e Fx(j)p
FG(d)1762 5228 y Fy(\000)1806 5237 y Fq(1)1845 5214 y
FA(\()p FG(x)p FA(\))p Fx(j)h FA(=)f Fx(j)p FG(d)2186
5232 y Fv(@)t(B)2280 5240 y Fs(r)2316 5232 y Fy(\()p
Fv(y)r Fy(\))2412 5214 y FA(\()p FG(x)p FA(\))p Fx(j)15
b FG(;)p eop
%%Page: 99 99
99 98 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)f
Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)175 b(99)456
453 y FA(if)19 b FG(x)26 b Fx(2)e FA([)p Fx(\000)p FG(l)r(;)15
b(l)r FA(])911 420 y Fv(N)1000 453 y FA(with)k Fx(j)p
FG(x)1274 420 y Ft(0)1298 453 y Fx(j)25 b(\025)1465 430
y FA(\025)1444 453 y FG(C)7 b(q)s FA(.)37 b(Hence,)24
b(from)c(\(A.71\),)25 b(\(A.70\),)g(\(A.69\))e(and)d(\(A.65\),)456
561 y(w)m(e)30 b(gather)i(that)970 645 y Fu(\022)1047
712 y FG(d)1094 726 y Fy(\000)1138 735 y Fq(2)1173 726
y Ft([)p Fy(\000)1264 735 y Fq(3)1303 712 y FA(\()p FG(x)p
FA(\))p 1047 753 379 4 v 1110 836 a FG(d)1157 850 y Fy(\000)1201
859 y Fq(1)1240 836 y FA(\()p FG(x)p FA(\))1436 645 y
Fu(\023)1503 668 y Fy(2)1625 773 y Fx(\025)1840 712 y(j)p
FG(x)20 b Fx(\000)28 b FA(~)-53 b FG(p)p Fx(j)2099 679
y Fy(2)p 1789 753 401 4 v 1789 784 a Fu(\020)1843 885
y FG(d)1890 899 y Fy(\000)1934 908 y Fq(1)1973 885 y
FA(\()p FG(x)p FA(\))2095 784 y Fu(\021)2150 806 y Fy(2)2225
773 y Fx(\025)-1840 b FA(\(A.72\))1625 1127 y Fx(\025)83
b FA(1)21 b(+)35 b(const)2224 1065 y FG(r)17 b Fx(j)p
FG(x)k Fx(\000)f FG(y)s Fx(j)p 2183 1106 V 2183 1137
a Fu(\020)2238 1238 y FG(d)2285 1252 y Fy(\000)2329 1261
y Fq(1)2368 1238 y FA(\()p FG(x)p FA(\))2490 1137 y Fu(\021)2545
1159 y Fy(2)2640 1042 y FA(\025)2619 1065 y FG(C)2691
1032 y Fy(2)2730 1065 y FG(q)2774 1032 y Fy(2)p 2619
1106 195 4 v 2662 1189 a FG(R)2732 1163 y Fy(2)2849 1127
y Fx(\025)1625 1480 y(\025)83 b FA(1)21 b(+)35 b(const)2253
1418 y Fx(j)p FG(x)20 b Fx(\000)g FG(y)s Fx(j)p 2183
1459 401 4 v 2183 1490 a Fu(\020)2238 1591 y FG(d)2285
1605 y Fy(\000)2329 1614 y Fq(1)2368 1591 y FA(\()p FG(x)p
FA(\))2490 1490 y Fu(\021)2545 1512 y Fy(2)2640 1395
y FA(\025)2619 1418 y FG(C)2691 1385 y Fy(2)2730 1418
y FG(q)2774 1385 y Fy(2)p 2619 1459 195 4 v 2682 1542
a FG(R)2849 1480 y Fx(\025)1625 1833 y(\025)83 b FA(1)21
b(+)35 b(const)2312 1771 y(1)p 2183 1812 304 4 v 2183
1895 a Fx(j)p FG(d)2255 1909 y Fy(\000)2299 1918 y Fq(1)2339
1895 y FA(\()p FG(x)p FA(\))p Fx(j)2542 1748 y FA(\025)2522
1771 y FG(C)2594 1738 y Fy(2)2633 1771 y FG(q)2677 1738
y Fy(2)p 2522 1812 195 4 v 2584 1895 a FG(R)2741 1833
y(:)456 2027 y FA(Let)30 b(no)m(w)1423 2183 y FG(\017)1460
2145 y Ft(\003)1525 2183 y FA(:=)40 b(const)2023 2121
y(1)p 1894 2162 304 4 v 1894 2245 a Fx(j)p FG(d)1966
2259 y Fy(\000)2010 2268 y Fq(1)2049 2245 y FA(\()p FG(x)p
FA(\))p Fx(j)2253 2098 y FA(\025)2232 2121 y FG(C)2304
2088 y Fy(2)2343 2121 y FG(q)2387 2088 y Fy(2)p 2232
2162 195 4 v 2294 2245 a FG(R)2452 2183 y(;)456 2382
y FA(so)d(that,)j(b)m(y)e(\(A.64\),)j FG(\017)1295 2349
y Ft(\003)1372 2382 y Fx(2)36 b FA([0)p FG(;)47 b FA(const)1854
2359 y(\025)1833 2382 y FG(C)7 b(q)s(=R)q FA(])37 b(is)g(small)f(and)h
(so,)i(from)e(\(A.72\),)42 b(w)m(e)456 2490 y(ha)m(v)m(e)31
b(that)903 2641 y Fx(j)p FG(d)975 2655 y Fy(\000)1019
2664 y Fq(2)1054 2655 y Ft([)p Fy(\000)1145 2664 y Fq(3)1184
2641 y FA(\()p FG(x)p FA(\))p Fx(j)21 b(\000)f(j)p FG(d)1515
2655 y Fy(\000)1559 2664 y Fq(1)1599 2641 y FA(\()p FG(x)p
FA(\))p Fx(j)84 b(\025)e(j)p FG(d)2055 2655 y Fy(\000)2099
2664 y Fq(1)2139 2641 y FA(\()p FG(x)p FA(\))p Fx(j)2301
2540 y Fu(\020)2356 2565 y Fx(p)p 2432 2565 233 4 v 76
x FA(1)21 b(+)e FG(\017)2625 2615 y Ft(\003)2685 2641
y Fx(\000)h FA(1)2821 2540 y Fu(\021)2901 2641 y Fx(\025)1830
2804 y(\025)82 b FA(const)16 b FG(d)2253 2818 y Fy(\000)2297
2827 y Fq(1)2336 2804 y FA(\()p FG(x)p FA(\))f FG(\017)2510
2766 y Ft(\003)2576 2804 y Fx(\025)1830 3010 y(\025)98
b FA(const)2252 2926 y(\025)2231 2949 y FG(C)2303 2916
y Fy(2)2342 2949 y FG(q)2386 2916 y Fy(2)p 2231 2989
195 4 v 2293 3072 a FG(R)2450 3010 y(;)456 3187 y FA(th)m(us)30
b(pro)m(ving)f(\(A.63\).)555 3342 y(With)d(similar)d(argumen)m(ts,)28
b(w)m(e)e(no)m(w)g(sho)m(w)g(that,)i(if)d FG(x)g Fx(2)g
FA([)p Fx(\000)p FG(l)r(;)15 b(l)r FA(])2760 3309 y Fv(N)2854
3342 y FA(with)25 b Fx(j)p FG(x)3134 3309 y Ft(0)3157
3342 y Fx(j)h(\025)3324 3319 y FA(\025)3304 3342 y FG(C)6
b(q)s FA(,)456 3450 y(then)456 3634 y(\(A.73\))327 b
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3652 y Fv(;R)1321 3660 y Fs(!)1365 3652 y Fy(\))1397
3634 y FA(\()p FG(x)p FA(\))21 b Fx(\000)f FG(d)1678
3652 y Fy(\006\()p Fv(Y)t(;R)p Fy(\))1906 3634 y FA(\()p
FG(x)p FA(\))41 b Fx(\025)55 b FA(const)16 b(\(1)21 b
Fx(\000)f FG(!)s FA(\))2730 3572 y FG(q)2774 3539 y Fy(2)p
2730 3613 84 4 v 2737 3696 a FG(R)2839 3634 y(:)456 3811
y FA(Indeed,)35 b(if)e FG(x)h FA(is)g(as)h(ab)s(o)m(v)m(e,)i(then,)e(b)
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Fi(C)g FA(de\014ned)456 3919 y(in)27 b(\(A.66\))j(and,)e(b)m(y)g(the)h
(argumen)m(ts)f(ab)s(o)m(v)m(e)i(\(A.71\),)h FG(d)2374
3937 y Fy(\006\()p Fv(Y)t(;R)p Fy(\))2602 3919 y FA(\()p
FG(x)p FA(\))e(is)e(attained)i(on)f(\000)3380 3933 y
Fy(1)3419 3919 y FA(,)456 4030 y(sa)m(y)j(at)g(a)g(p)s(oin)m(t)e
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y FG(d)1421 4190 y Fy(\000)1470 4176 y FA(\()p FG(x)p
FA(\))c(=)f FG(d)1761 4190 y Fy(\000)1805 4199 y Fq(1)1844
4176 y FA(\()p FG(x)p FA(\))h(=)f FG(d)2135 4195 y Fy(\006\()p
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b FG(:)456 4327 y FA(Let)28 b(also)f FG(p)841 4294 y
Ft(0)892 4327 y FA(b)s(e)g(the)g(p)s(oin)m(t)g(on)g(\006\()p
FG(Y)1678 4341 y Fv(!)1729 4327 y FG(;)15 b(R)1838 4341
y Fv(!)1888 4327 y FA(\))28 b(realizing)e FG(d)2359 4346
y Fy(\006\()p Fv(Y)2478 4354 y Fs(!)2524 4346 y Fv(;R)2597
4354 y Fs(!)2641 4346 y Fy(\))2673 4327 y FA(\()p FG(x)p
FA(\).)40 b(W)-8 b(e)29 b(distinguish)456 4439 y(di\013eren)m(t)34
b(cases:)51 b(\014rst,)36 b(if)e FG(x)h FA(is)f(outside)g(\000)h(\(and)
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4439 y FG(;)15 b(R)2956 4453 y Fv(!)3007 4439 y FA(\)\),)37
b(w)m(e)f(tak)m(e)463 4547 y(^)-52 b FG(p)28 b Fx(2)g
FA(\000)k(on)g(the)h(segmen)m(t)g(joining)d FG(x)j FA(and)e
FG(p)1959 4514 y Ft(0)2014 4547 y FA(\(it)i(ma)m(y)f(happ)s(en)f(that)
41 b(^)-53 b FG(p)28 b FA(=)g FG(p)3082 4514 y Ft(0)3105
4547 y FA(\).)47 b(Then,)456 4655 y(if)37 b(^)-53 b FG(p)25
b Fx(2)g FA(\000)753 4669 y Fy(1)792 4655 y FA(,)31 b(w)m(e)f(ha)m(v)m
(e)i(that)1576 4801 y FG(d)1623 4819 y Fy(\006\()p Fv(Y)1742
4827 y Fs(!)1788 4819 y Fv(;R)1861 4827 y Fs(!)1905 4819
y Fy(\))1937 4801 y FA(\()p FG(x)p FA(\))21 b Fx(\000)f(j)p
FG(d)2243 4815 y Fy(\000)2291 4801 y FA(\()p FG(x)p FA(\))p
Fx(j)27 b FA(=)1423 4946 y(=)82 b Fx(j)p FG(x)21 b Fx(\000)f
FG(p)1811 4908 y Ft(0)1834 4946 y Fx(j)g(\000)g(j)p FG(x)g
Fx(\000)g FG(p)p Fx(j)26 b(\025)1423 5081 y(\025)82 b(j)p
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Fx(j)g(\000)g(j)p FG(x)g Fx(\000)28 b FA(^)-53 b FG(p)p
Fx(j)26 b FA(=)1423 5216 y(=)82 b Fx(j)p FG(p)1647 5178
y Ft(0)1691 5216 y Fx(\000)27 b FA(^)-52 b FG(p)o Fx(j)15
b FG(:)p eop
%%Page: 100 100
100 99 bop 456 251 a Fy(100)140 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(Therefore,)32 b(if)f FG(x)h FA(is)g(outside)f(\000)h
(and)39 b(^)-52 b FG(p)28 b Fx(2)f FA(\000)1960 464 y
Fy(1)2000 450 y FA(,)32 b(b)m(y)g(using)f(\(A.37\),)k(one)d(pro)m(v)m
(es)h(\(A.73\))456 558 y(in)c(this)g(case;)j(if,)d(con)m(v)m(ersely)-8
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558 y Fx([)c FA(\000)2764 572 y Fy(3)2803 558 y FA(,)30
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746 y Fs(!)1806 738 y Fy(\))1838 719 y FA(\()p FG(x)p
FA(\))21 b Fx(\000)e(j)p FG(d)2143 733 y Fy(\000)2192
719 y FA(\()p FG(x)p FA(\))p Fx(j)26 b FA(=)1323 864
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864 y Fx(j)h(\000)g(j)p FG(d)1943 878 y Fy(\000)1987
887 y Fq(1)2026 864 y FA(\()p FG(x)p FA(\))p Fx(j)26
b FA(=)1323 999 y(=)83 b Fx(j)p FG(x)21 b Fx(\000)27
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Fx(j)g(\000)g(j)p FG(d)2307 1013 y Fy(\000)2351 1022
y Fq(1)2390 999 y FA(\()p FG(x)p FA(\))p Fx(j)27 b(\025)1323
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1157 y Fq(1)2003 1134 y FA(\()p FG(x)p FA(\))p Fx(j)26
b(\025)1323 1269 y(\025)83 b(j)p FG(d)1549 1283 y Fy(\000)1593
1292 y Fq(2)1628 1283 y Ft([)p Fy(\000)1719 1292 y Fq(3)1758
1269 y FA(\()p FG(x)p FA(\))p Fx(j)21 b(\000)f(j)p FG(d)2089
1283 y Fy(\000)2133 1292 y Fq(1)2173 1269 y FA(\()p FG(x)p
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1552 y Fv(!)2808 1538 y FA(\),)31 b(w)m(e)f(ha)m(v)m(e)i(that)1551
1700 y FG(d)1598 1718 y Fy(\006\()p Fv(Y)1717 1726 y
Fs(!)1762 1718 y Fv(;R)1835 1726 y Fs(!)1880 1718 y Fy(\))1912
1700 y FA(\()p FG(x)p FA(\))20 b Fx(\000)g FG(d)2192
1714 y Fy(\000)2241 1700 y FA(\()p FG(x)p FA(\))26 b(=)1397
1838 y(=)83 b Fx(j)p FG(d)1623 1857 y Fy(\006\()p Fv(Y)1742
1865 y Fs(!)1788 1857 y Fv(;R)1861 1865 y Fs(!)1905 1857
y Fy(\))1937 1838 y FA(\()p FG(x)p FA(\))p Fx(j)21 b
FA(+)f Fx(j)p FG(d)2268 1852 y Fy(\000)2317 1838 y FA(\()p
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FG(x)20 b Fx(\000)g FG(p)1785 1946 y Ft(0)1808 1983 y
Fx(j)h FA(+)f Fx(j)p FG(x)g Fx(\000)g FG(p)p Fx(j)25
b(\025)1397 2118 y(\025)83 b(j)p FG(p)1622 2081 y Ft(0)1665
2118 y Fx(\000)20 b FG(p)p Fx(j)15 b FG(;)456 2280 y
FA(th)m(us)32 b(\(A.37\))i(giv)m(es)f(\(A.73\))h(in)d(this)g(case.)48
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b Fx(2)g FA(\006\()p FG(Y)1743 2401 y Fv(!)1793 2387
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2495 y(observ)m(e)i(that)1516 2657 y FG(d)1563 2675 y
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2814 y Fv(;R)1896 2822 y Fs(!)1941 2814 y Fy(\))1972
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Fx(j)26 b(\025)1362 2934 y(\025)83 b(\000j)p FG(x)20
b Fx(\000)27 b FA(\025)-52 b FG(p)o Fx(j)21 b FA(+)f
Fx(j)p FG(x)g Fx(\000)g FG(p)p Fx(j)25 b FA(=)1362 3069
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3647 y FA(\025)1550 3670 y FG(C)7 b(q)s FA(,)30 b(then)g
FG(g)1971 3684 y Fy(\011)2031 3670 y FA(\()p FG(x)p FA(\))c(=)f
FG(g)2318 3689 y Fj(S)-6 b Fy(\()p Fv(Y)t(;R)p Fy(\))2532
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1335 4155 y FG(Y)1388 4169 y Fv(!)1521 4155 y FA(:=)83
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4169 y Fv(N)7 b Fy(+1)2073 4155 y FA(\))15 b FG(;)1330
4354 y(R)1399 4368 y Fy(1)1521 4354 y FA(:=)83 b FG(r)1741
4368 y Fv(!)1812 4354 y FA(+)20 b FG(H)1979 4368 y Fy(0)2018
4354 y FA(\()p FG(y)2098 4368 y Fv(N)7 b Fy(+1)2255 4354
y FA(\))21 b(+)2412 4293 y(5)p 2472 4220 72 4 v 15 w
FG(C)2544 4307 y Fy(0)p 2412 4334 172 4 v 2463 4417 a
FG(R)2608 4354 y(;)1358 4594 y(r)1399 4608 y Fy(1)1521
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2450 4459 72 4 v 2450 4532 a FG(C)2522 4546 y Fy(0)p
2429 4573 154 4 v 2429 4656 a FA(2)p FG(R)2543 4670 y
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4773 y FA(\))21 b Fx(\\)f(f)p FG(x)2272 4787 y Fv(N)7
b Fy(+1)2455 4773 y FA(=)25 b(0)p Fx(g)15 b FG(:)456
4935 y FA(By)30 b(de\014nition,)1596 5077 y FG(R)1665
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b(const)41 b Fx(\024)1526 5212 y(\024)o FG(!)s(R)21 b
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eop
%%Page: 101 101
101 100 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)
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653 y Fx(2)1867 525 y Fu(\024)1925 592 y FA(3)p 1970
519 72 4 v FG(C)2042 606 y Fy(0)p 1925 632 157 4 v 1968
716 a FG(R)2091 653 y(;)2157 592 y FA(5)p 2202 519 72
4 v FG(C)2274 606 y Fy(0)p 2157 632 157 4 v 2200 716
a FG(R)2323 525 y Fu(\025)2402 653 y FG(:)456 864 y FA(F)-8
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1048 y Fq(1)2198 1025 y FA(\()p FG(y)2278 1039 y Fv(!)2329
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Fv(!)1445 1338 y FG(;)15 b(R)1554 1352 y Fy(1)1594 1338
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Fx(g)31 b FA(is)e(at)i(distance)f(at)i(least)2699 1821
y(8)p 2744 1748 V FG(C)2816 1835 y Fy(0)p 2699 1862 157
4 v 2742 1945 a FG(R)2896 1883 y FA(inside)c(\000.)456
2075 y(The)36 b(pro)s(of)g(of)h(this)f(fact)i(will)c(tak)m(e)k(some)g
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Fs(!)1825 2182 y FA(\()p FG(y)1905 2196 y Fv(!)1956 2182
y FA(\))15 b Fx(\\)g(fj)p FG(x)2204 2149 y Ft(0)2228
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2451 y FG(w)g Fx(2)d FA(\000)1747 2465 y Fy(3)1811 2451
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2734 y Fy(1)3419 2720 y FA(,)456 2828 y FG(z)498 2842
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456 3097 y(p)s(ossible)e(to)j FG(w)r FA(.)41 b(Of)30
b(course,)456 3258 y(\(A.80\))852 b Fx(j)p FG(z)1628
3272 y Fy(1)1688 3258 y Fx(\000)20 b FG(w)r Fx(j)41 b(\025)g(j)p
FG(p)20 b Fx(\000)g FG(w)r Fx(j)15 b FG(:)456 3419 y
FA(If)30 b FG(p)593 3433 y Fv(N)685 3419 y FA(=)25 b(0,)31
b(w)m(e)g(ha)m(v)m(e)g(that)901 3580 y Fx(j)p FG(w)23
b Fx(\000)d FG(p)p Fx(j)83 b(\025)f(j)p FG(w)1502 3594
y Fv(N)1590 3580 y Fx(\000)20 b FG(p)1727 3594 y Fv(N)1794
3580 y Fx(j)26 b FA(=)1259 3715 y(=)82 b Fx(j)p FG(w)1502
3729 y Fv(N)1570 3715 y Fx(j)26 b(\025)1259 3850 y(\025)82
b(j)p FG(y)1482 3864 y Fv(!)r(N)1596 3850 y Fx(j)20 b(\000)g(j)p
FG(w)1822 3864 y Fv(N)1910 3850 y Fx(\000)g FG(y)2046
3864 y Fv(!)r(N)2159 3850 y Fx(j)25 b FA(=)1259 3998
y(=)82 b Fx(j)p FG(y)1482 4012 y Fv(!)r(N)1596 3998 y
Fx(j)20 b(\000)1732 3915 y Fu(p)p 1823 3915 384 4 v 83
x FG(r)1867 3972 y Fy(2)1864 4020 y Fv(!)1935 3998 y
Fx(\000)g(j)p FG(w)2118 3972 y Ft(0)2142 3998 y Fx(j)2167
3972 y Fy(2)2232 3998 y Fx(\025)1259 4146 y(\025)82 b(j)p
FG(y)1482 4160 y Fv(!)r(N)1596 4146 y Fx(j)20 b(\000)1732
4063 y Fu(p)p 1823 4063 374 4 v 83 x FG(r)1867 4120 y
Fy(2)1864 4169 y Fv(!)1935 4146 y Fx(\000)g FG(\021)2074
4120 y Fy(2)2113 4146 y FG(q)2157 4120 y Fy(2)2222 4146
y FA(=)1259 4341 y(=)1422 4280 y(1)p 1422 4320 46 4 v
1422 4404 a(2)1478 4256 y Fu(p)p 1569 4256 278 4 v 85
x FG(r)1613 4315 y Fy(2)1672 4341 y Fx(\000)g FG(q)1807
4315 y Fy(2)1866 4341 y Fx(\000)2143 4280 y FG(q)2187
4247 y Fy(2)p 1967 4320 436 4 v 1967 4419 a FG(\015)2034
4339 y Fu(p)p 2125 4339 278 4 v 80 x FG(r)2169 4393 y
Fy(2)2229 4419 y Fx(\000)f FG(q)2363 4393 y Fy(2)2443
4213 y Fu(\022)2549 4280 y FA(1)p 2520 4320 105 4 v 2520
4404 a FG(!)2580 4377 y Fv(\015)2654 4341 y Fx(\000)h
FA(1)2790 4213 y Fu(\023)2878 4341 y Fx(\000)1412 4651
y(\000)1483 4487 y Fu(s)p 1574 4487 1304 4 v 1584 4589
a FA(1)p 1584 4630 46 4 v 1584 4713 a(4)1655 4522 y Fu(\022)1722
4651 y FG(r)1766 4624 y Fy(2)1825 4651 y FA(+)g FG(q)1960
4624 y Fy(2)2014 4550 y Fu(\020)2170 4589 y FA(1)p 2078
4630 230 4 v 2078 4713 a FG(!)2138 4687 y Fy(2)p Fv(\015)t
Fy(+2)2338 4651 y Fx(\000)g FA(1)2474 4550 y Fu(\021)2529
4522 y(\023)2616 4651 y Fx(\000)g FG(\021)2755 4624 y
Fy(2)2795 4651 y FG(q)2839 4624 y Fy(2)2903 4651 y FA(=)1259
4928 y(=)1422 4867 y(1)p 1422 4907 46 4 v 1422 4991 a(2)1478
4843 y Fu(p)p 1569 4843 278 4 v 85 x FG(r)1613 4902 y
Fy(2)1672 4928 y Fx(\000)g FG(q)1807 4902 y Fy(2)1866
4928 y Fx(\000)2143 4867 y FG(q)2187 4834 y Fy(2)p 1967
4907 436 4 v 1967 5006 a FG(\015)2034 4926 y Fu(p)p 2125
4926 278 4 v 80 x FG(r)2169 4980 y Fy(2)2229 5006 y Fx(\000)f
FG(q)2363 4980 y Fy(2)2443 4855 y Fu(\000)2485 4928 y
FA(2)p FG(!)2590 4891 y Fy(2)2650 4928 y Fx(\000)h FA(1)2786
4855 y Fu(\001)2848 4928 y Fx(\000)1412 5186 y(\000)1493
5125 y FA(1)p 1493 5166 46 4 v 1493 5249 a(2)1549 5104
y Fu(p)p 1639 5104 909 4 v 1639 5186 a FG(r)1683 5160
y Fy(2)1743 5186 y Fx(\000)g FG(q)1878 5160 y Fy(2)1932
5186 y FA(\(1)h Fx(\000)f FA(4)p FG(!)2229 5160 y Fy(2)2289
5186 y FA(+)g(4)p FG(\021)2473 5160 y Fy(2)2513 5186
y FA(\))c FG(:)p eop
%%Page: 102 102
102 101 bop 456 251 a Fy(102)140 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(Let)31 b(us)f(no)m(w)h(\014x)g(an)g(auxiliary)d
(small)i(parameter)h FG(\016)f(>)c FA(0.)43 b(F)-8 b(rom)32
b(the)f(estimate)g(here)456 558 y(ab)s(o)m(v)m(e,)41
b(\(A.33\))e(and)e(\(A.35\),w)m(e)k(deduce,)e(if)e FG(\015)43
b FA(and)37 b FG(r)s(=q)j FA(are)e(large)g(enough)3190
523 y Fy(21)3298 558 y FA(and)456 666 y FG(p)502 680
y Fv(N)594 666 y FA(=)25 b(0,)31 b(that)539 880 y Fx(j)p
FG(w)23 b Fx(\000)c FG(p)p Fx(j)83 b(\025)1060 818 y
FG(q)1104 785 y Fy(2)p 1060 859 84 4 v 1080 942 a FG(r)1184
751 y Fu(\024)1232 880 y FG(\021)1280 842 y Fy(2)1340
880 y Fx(\000)20 b FG(!)1491 842 y Fy(2)1550 880 y Fx(\000)1651
818 y FA(\(1)h(+)f FG(\016)s FA(\))15 b(\(2)p FG(!)2076
785 y Fy(2)2138 818 y Fx(\000)k FA(1\))p 1651 859 659
4 v 1876 942 a FG(\015)25 b FA(+)20 b(2)2339 880 y Fx(\000)35
b FA(const)2668 779 y Fu(\020)2732 818 y FG(q)p 2732
859 44 4 v 2732 942 a(r)2786 779 y Fu(\021)2841 801 y
Fy(2)2880 751 y Fu(\025)2953 880 y FA(=)896 1138 y(=)1060
1077 y FG(q)1104 1044 y Fy(2)p 1060 1118 84 4 v 1080
1201 a FG(r)1184 1010 y Fu(\024)1232 1138 y FA(\()p FG(\021)24
b FA(+)c FG(!)s FA(\))15 b(\(1)21 b Fx(\000)f FG(\017)p
FA(\))15 b(\(1)21 b Fx(\000)f FG(!)s FA(\))h Fx(\000)2224
1077 y FA(\(1)g(+)f FG(\016)s FA(\))15 b(\(2)p FG(!)2649
1044 y Fy(2)2711 1077 y Fx(\000)20 b FA(1\))31 b(log)q(\(1)p
FG(=!)s FA(\))p 2224 1118 1028 4 v 2649 1201 a(log)17
b(2)3262 1138 y Fx(\000)1050 1392 y(\000)40 b FA(const)1384
1291 y Fu(\020)1448 1330 y FG(q)p 1448 1371 44 4 v 1448
1454 a(r)1502 1291 y Fu(\021)1556 1314 y Fy(2)1596 1264
y Fu(\025)1669 1392 y Fx(\025)896 1651 y(\025)1060 1589
y FG(q)1104 1556 y Fy(2)p 1060 1630 84 4 v 1080 1713
a FG(r)1184 1523 y Fu(\024)1232 1651 y FA(\()p FG(\021)24
b FA(+)c FG(!)s FA(\))15 b(\(1)21 b Fx(\000)f FG(\017)p
FA(\))15 b(\(1)21 b Fx(\000)f FG(!)s FA(\))h Fx(\000)2224
1589 y FA(\(1)g(+)f(2)p FG(\016)s FA(\))15 b(\(2)p FG(!)2694
1556 y Fy(2)2756 1589 y Fx(\000)20 b FA(1\))15 b(\(1)22
b Fx(\000)e FG(!)s FA(\))p 2224 1630 1007 4 v 2638 1713
a(log)d(2)3271 1651 y Fx(\000)1060 1904 y(\000)e FA(const)1369
1803 y Fu(\020)1433 1843 y FG(q)p 1433 1883 44 4 v 1433
1967 a(r)1487 1803 y Fu(\021)1541 1826 y Fy(2)1581 1776
y Fu(\025)1654 1904 y FA(=)896 2163 y(=)1060 2102 y FG(q)1104
2069 y Fy(2)1159 2102 y FA(\(1)21 b Fx(\000)e FG(!)s
FA(\))p 1060 2142 386 4 v 1231 2226 a FG(r)1486 2035
y Fu(\024)1534 2163 y FA(\()p FG(\021)24 b FA(+)c FG(!)s
FA(\))15 b(\(1)21 b Fx(\000)f FG(\017)p FA(\))h Fx(\000)2224
2102 y FA(\(1)g(+)f(2)p FG(\016)s FA(\))15 b(\(2)p FG(!)2694
2069 y Fy(2)2756 2102 y Fx(\000)20 b FA(1\))p 2224 2142
704 4 v 2487 2226 a(log)d(2)2938 2163 y Fx(\000)1050
2417 y(\000)40 b FA(const)1384 2316 y Fu(\020)1448 2355
y FG(q)p 1448 2396 44 4 v 1448 2479 a(r)1502 2316 y Fu(\021)1556
2338 y Fy(2)1596 2289 y Fu(\025)1674 2417 y FG(:)456
2625 y FA(When)30 b FG(\017)g FA(and)g FG(\016)k FA(are)d(close)f(to)h
(0)g(and)f FG(\021)k FA(and)29 b FG(!)34 b FA(are)c(close)h(to)g(1,)g
(the)g(quan)m(tit)m(y)1248 2838 y(\()p FG(\021)24 b FA(+)c
FG(!)s FA(\))15 b(\(1)21 b Fx(\000)f FG(\017)p FA(\))h
Fx(\000)1938 2777 y FA(\(1)g(+)f(2)p FG(\016)s FA(\))15
b(\(2)p FG(!)2408 2744 y Fy(2)2470 2777 y Fx(\000)20
b FA(1\))p 1938 2817 704 4 v 2201 2901 a(log)d(2)456
3040 y(is)29 b(close)i(to)1548 3181 y(\024)-47 b FG(c)41
b FA(:=)f(2)21 b Fx(\000)1995 3119 y FA(1)p 1929 3160
179 4 v 1929 3243 a(log)16 b(2)2157 3181 y FG(>)40 b
FA(0)15 b FG(;)456 3364 y FA(hence)30 b(the)h(ab)s(o)m(v)m(e)g
(estimate)g(implies)d(that,)j(if)f FG(p)2147 3378 y Fv(N)2239
3364 y FA(=)25 b(0,)456 3577 y(\(A.81\))778 b Fx(j)p
FG(w)23 b Fx(\000)d FG(p)p Fx(j)40 b(\025)1925 3516 y
FA(\024)-47 b FG(c)15 b FA(\(1)21 b Fx(\000)f FG(!)s
FA(\))15 b FG(q)2323 3483 y Fy(2)p 1923 3556 441 4 v
2098 3640 a FA(2)p FG(r)2388 3577 y(:)456 3761 y FA(On)33
b(the)i(other)f(hand,)h(if)e FG(p)1404 3775 y Fv(N)1503
3761 y Fx(6)p FA(=)f(0,)k(its)d(minimizing)e(prop)s(ert)m(y)j(giv)m(es)
g(that)h FG(w)26 b Fx(\000)c FG(p)34 b FA(is)456 3869
y(orthogonal)e(to)h FG(@)5 b(B)1147 3883 y Fv(r)1186
3869 y FA(\()p FG(y)s FA(\))32 b(at)i FG(p)p FA(,)e(whic)m(h)f(sa)m(ys)
i(that)g(the)g(line)d(joining)h FG(p)h FA(and)g FG(w)i
FA(passes)456 3977 y(through)29 b FG(y)k FA(and,)d(therefore,)1458
4135 y Fx(j)p FG(p)20 b Fx(\000)g FG(w)r Fx(j)84 b(\025)f
FG(r)23 b Fx(\000)c(j)p FG(w)k Fx(\000)d FG(y)s Fx(j)15
b FG(:)-1985 b FA(\(A.82\))456 4294 y(Let)30 b(no)m(w)1439
4414 y FG(\014)g FA(=)25 b FG(\014)1667 4428 y Fv(w)1750
4414 y FA(=)g Fl(\\)p FA(\()p FG(w)d Fx(\000)e FG(y)2170
4428 y Fv(!)2220 4414 y FG(;)31 b(e)2318 4428 y Fv(N)2385
4414 y FA(\))15 b FG(:)456 4553 y FA(Of)29 b(course,)456
4712 y(\(A.83\))689 b Fl(\\)p FA(\()p FG(w)23 b Fx(\000)d
FG(y)1723 4726 y Fv(!)1773 4712 y FG(;)30 b(y)23 b Fx(\000)d
FG(y)2032 4726 y Fv(!)2082 4712 y FA(\))26 b(=)f FG(\031)e
Fx(\000)d FG(\014)g(:)456 4870 y FA(Elemen)m(tary)30
b(trigonometry)g(giv)m(es)h(that)1514 5028 y FG(\021)s(q)e
Fx(\024)c(j)p FG(w)1820 4990 y Ft(0)1844 5028 y Fx(j)g
FA(=)g FG(r)2031 5042 y Fv(!)2097 5028 y Fx(j)15 b FA(sin)f
FG(\014)5 b Fx(j)15 b FG(;)p 456 5116 499 4 v 555 5187
a Fy(21)626 5216 y FF(Notation)20 b(remark:)31 b(\\log)r(")21
b(here)e(and)h(in)g(what)g(follo)n(ws)i(is)f(in)n(tended)e(as)h(the)g
Fc(natur)l(al)h FF(logarithm.)p eop
%%Page: 103 103
103 102 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)
f Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)140 b(103)456
450 y FA(that)31 b(is)456 1271 y(\(A.84\))777 b Fx(j)15
b FA(cos)h FG(\014)5 b Fx(j)26 b(\024)1866 1135 y Fu(r)p
1957 1135 417 4 v 136 x FA(1)21 b Fx(\000)2113 1170 y
Fu(\020)2178 1210 y FG(\021)s(q)p 2178 1250 93 4 v 2178
1334 a(r)2219 1348 y Fv(!)2280 1170 y Fu(\021)2334 1193
y Fy(2)2389 1271 y FG(:)456 2095 y FA(In)29 b(the)i(ligh)m(t)e(of)i
(\(A.83\),)i(\(A.84\))f(and)e(\(A.35\),)i(w)m(e)f(th)m(us)f(gather)h
(that)541 2876 y Fx(j)p FG(w)23 b Fx(\000)d FG(y)s Fx(j)818
2838 y Fy(2)941 2876 y FA(=)82 b Fx(j)p FA(\()p FG(w)24
b Fx(\000)c FG(y)1379 2890 y Fv(!)1428 2876 y FA(\))h
Fx(\000)f FA(\()p FG(y)j Fx(\000)d FG(y)1814 2890 y Fv(!)1864
2876 y FA(\))p Fx(j)1924 2838 y Fy(2)1990 2876 y FA(=)941
3014 y(=)82 b Fx(j)p FG(w)23 b Fx(\000)d FG(y)1343 3028
y Fv(!)1393 3014 y Fx(j)1418 2976 y Fy(2)1478 3014 y
FA(+)g Fx(j)p FG(y)j Fx(\000)d FG(y)1798 3028 y Fv(!)1848
3014 y Fx(j)1873 2976 y Fy(2)1933 3014 y Fx(\000)1094
3176 y(\000)p FA(2)15 b Fx(j)p FG(w)24 b Fx(\000)19 b
FG(y)1474 3190 y Fv(!)1524 3176 y Fx(j)c(j)p FG(y)24
b Fx(\000)c FG(y)1794 3190 y Fv(!)1844 3176 y Fx(j)15
b FA(cos)2021 3076 y Fu(\020)2076 3176 y Fl(\\)p FA(\()p
FG(w)22 b Fx(\000)e FG(y)2400 3190 y Fv(!)2450 3176 y
FG(;)31 b(y)23 b Fx(\000)d FG(y)2710 3190 y Fv(!)2760
3176 y FA(\))2795 3076 y Fu(\021)2875 3176 y FA(=)941
3351 y(=)82 b FG(r)1138 3314 y Fy(2)1135 3374 y Fv(!)1206
3351 y FA(+)20 b Fx(j)p FG(y)j Fx(\000)d FG(y)1526 3365
y Fv(!)1576 3351 y Fx(j)1601 3314 y Fy(2)1661 3351 y
FA(+)g(2)p FG(r)1838 3365 y Fv(!)1889 3351 y Fx(j)p FG(y)j
Fx(\000)d FG(y)2118 3365 y Fv(!)2168 3351 y Fx(j)30 b
FA(cos)16 b FG(\014)31 b Fx(\024)941 3553 y(\024)82 b
FG(r)1138 3515 y Fy(2)1135 3575 y Fv(!)1206 3553 y FA(+)20
b Fx(j)p FG(y)j Fx(\000)d FG(y)1526 3567 y Fv(!)1576
3553 y Fx(j)1601 3515 y Fy(2)1661 3553 y FA(+)g(2)p FG(r)1838
3567 y Fv(!)1889 3553 y Fx(j)p FG(y)j Fx(\000)d FG(y)2118
3567 y Fv(!)2168 3553 y Fx(j)2208 3416 y Fu(r)p 2299
3416 417 4 v 137 x FA(1)h Fx(\000)2456 3452 y Fu(\020)2520
3491 y FG(\021)s(q)p 2520 3532 93 4 v 2520 3615 a(r)2561
3629 y Fv(!)2622 3452 y Fu(\021)2676 3475 y Fy(2)2741
3553 y Fx(\024)941 3747 y(\024)82 b FG(r)1138 3709 y
Fy(2)1135 3769 y Fv(!)1206 3747 y FA(+)20 b Fx(j)p FG(y)j
Fx(\000)d FG(y)1526 3761 y Fv(!)1576 3747 y Fx(j)1601
3709 y Fy(2)1661 3747 y FA(+)1094 3926 y(+2)p FG(r)1251
3940 y Fv(!)1302 3926 y Fx(j)p FG(y)j Fx(\000)d FG(y)1531
3940 y Fv(!)1581 3926 y Fx(j)1621 3825 y Fu(h)1664 3926
y FA(1)h Fx(\000)1831 3864 y FA(1)p 1831 3905 46 4 v
1831 3988 a(2)1901 3825 y Fu(\020)1966 3864 y FG(\021)s(q)p
1966 3905 93 4 v 1966 3988 a(r)2007 4002 y Fv(!)2068
3825 y Fu(\021)2122 3847 y Fy(2)2182 3926 y FA(+)35 b(const)2511
3825 y Fu(\020)2599 3864 y FG(q)p 2575 3905 92 4 v 2575
3988 a(r)2616 4002 y Fv(!)2676 3825 y Fu(\021)2731 3847
y Fy(4)2770 3825 y Fu(i)2838 3926 y FA(=)941 4174 y(=)1094
4073 y Fu(\020)1149 4174 y FG(r)1190 4188 y Fv(!)1260
4174 y FA(+)20 b Fx(j)p FG(y)j Fx(\000)d FG(y)1580 4188
y Fv(!)1630 4174 y Fx(j)1655 4073 y Fu(\021)1710 4096
y Fy(2)1769 4174 y Fx(\000)g FG(r)1901 4188 y Fv(!)1967
4174 y Fx(j)p FG(y)j Fx(\000)d FG(y)2196 4188 y Fv(!)2246
4174 y Fx(j)2286 4073 y Fu(\020)2351 4113 y FG(\021)s(q)p
2351 4153 93 4 v 2351 4237 a(r)2392 4251 y Fv(!)2453
4073 y Fu(\021)2507 4096 y Fy(2)2567 4174 y FA(+)35 b(const)2906
4113 y Fx(j)p FG(y)23 b Fx(\000)d FG(y)3135 4127 y Fv(!)3185
4113 y Fx(j)15 b FG(q)3269 4080 y Fy(4)p 2906 4153 403
4 v 3061 4237 a FG(r)3105 4210 y Fy(3)3102 4259 y Fv(!)3334
4174 y FG(:)456 5000 y FA(W)-8 b(e)33 b(no)m(w)f(p)s(erform)e(a)i(mo)s
(di\014cation)f(of)h(the)g(argumen)m(t)g(done)g(for)f(the)h(case)h
FG(p)3180 5014 y Fv(N)3275 5000 y FA(=)28 b(0:)456 5108
y(w)m(e)i(\014x)f(a)h(small)e(parameter)j FG(\016)e(>)c
FA(0,)30 b(to)h(b)s(e)e(appropriately)f(c)m(hosen,)i(and)g(w)m(e)g
(observ)m(e)456 5216 y(that,)i(if)e(1)21 b Fx(\000)f
FG(!)s FA(,)31 b(1)p FG(=\015)37 b FA(and)30 b FG(q)s(=r)k
FA(are)d(small)f(enough)h(\(p)s(ossibly)d(in)i(dep)s(endence)g(of)h
FG(\016)s FA(\),)p eop
%%Page: 104 104
104 103 bop 456 251 a Fy(104)140 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(then)30 b(the)g(ab)s(o)m(v)m(e)i(estimate,)f
(\(A.36\))i(and)c(\(A.35\))k(yield)28 b(that)726 1127
y Fx(j)p FG(w)23 b Fx(\000)d FG(y)s Fx(j)1003 1089 y
Fy(2)1068 1127 y Fx(\024)572 1322 y(\024)726 1221 y Fu(\020)780
1322 y FG(r)821 1336 y Fv(!)892 1322 y FA(+)g Fx(j)p
FG(y)j Fx(\000)d FG(y)1212 1336 y Fv(!)1262 1322 y Fx(j)1287
1221 y Fu(\021)1342 1244 y Fy(2)1417 1322 y Fx(\000)1533
1261 y FG(\021)1581 1228 y Fy(2)1620 1261 y FG(q)1664
1228 y Fy(2)1719 1261 y Fx(j)p FG(y)j Fx(\000)d FG(y)1948
1275 y Fv(!)1998 1261 y Fx(j)p 1533 1301 491 4 v 1732
1384 a FG(r)1773 1398 y Fv(!)2053 1322 y FA(+)35 b(const)2392
1261 y FG(q)2436 1228 y Fy(4)p 2392 1301 84 4 v 2392
1384 a FG(r)2436 1358 y Fy(2)2511 1322 y Fx(\024)572
1571 y(\024)726 1470 y Fu(\020)790 1509 y FA(1)p 790
1550 46 4 v 790 1633 a(2)861 1488 y Fu(p)p 952 1488 650
4 v 83 x FG(r)996 1544 y Fy(2)1055 1571 y FA(+)20 b FG(q)1190
1544 y Fy(2)1229 1571 y FA(\(4)p FG(!)1369 1544 y Fy(2)1430
1571 y Fx(\000)f FA(1\))i(+)1722 1509 y(1)p 1722 1550
46 4 v 1722 1633 a(2)1793 1486 y Fu(p)p 1884 1486 278
4 v 85 x FG(r)1928 1544 y Fy(2)1987 1571 y Fx(\000)f
FG(q)2122 1544 y Fy(2)2181 1571 y FA(+)2282 1509 y(\(1)h(+)f
FG(\016)s FA(\)\()p FG(!)2647 1476 y Ft(\000)p Fv(\015)2768
1509 y Fx(\000)g FA(1\))15 b FG(q)2998 1476 y Fy(2)p
2282 1550 756 4 v 2612 1633 a FG(\015)5 b(r)3048 1470
y Fu(\021)3103 1492 y Fy(2)3162 1571 y Fx(\000)726 1835
y(\000)900 1774 y FG(\021)948 1741 y Fy(2)988 1774 y
FG(q)1032 1741 y Fy(2)1086 1693 y Fu(p)p 1177 1693 278
4 v 81 x FG(r)1221 1748 y Fy(2)1280 1774 y Fx(\000)20
b FG(q)1415 1748 y Fy(2)p 807 1815 741 4 v 807 1833 a
Fu(p)p 898 1833 650 4 v 78 x FG(r)942 1885 y Fy(2)1001
1911 y FA(+)g FG(q)1136 1885 y Fy(2)1175 1911 y FA(\(4)p
FG(!)1315 1885 y Fy(2)1375 1911 y Fx(\000)g FA(1\))1577
1835 y(+)35 b(const)1916 1774 y FG(q)1960 1741 y Fy(4)p
1916 1815 84 4 v 1916 1898 a FG(r)1960 1872 y Fy(2)2035
1835 y FA(=)572 2129 y(=)83 b FG(r)770 2091 y Fy(2)824
2028 y Fu(\020)889 2067 y FA(1)p 889 2108 46 4 v 889
2191 a(2)959 1981 y Fu(r)p 1050 1981 632 4 v 148 x FA(1)21
b(+)1217 2067 y FG(q)1261 2041 y Fy(2)1300 2067 y FA(\(4)p
FG(!)1440 2041 y Fy(2)1500 2067 y Fx(\000)f FA(1\))p
1217 2108 456 4 v 1403 2191 a FG(r)1447 2165 y Fy(2)1702
2129 y FA(+)1803 2067 y(1)p 1803 2108 46 4 v 1803 2191
a(2)1873 1981 y Fu(r)p 1964 1981 260 4 v 148 x FA(1)h
Fx(\000)2131 2067 y FG(q)2175 2041 y Fy(2)p 2131 2108
84 4 v 2131 2191 a FG(r)2175 2165 y Fy(2)2244 2129 y
FA(+)2345 2067 y(\(1)g(+)f FG(\016)s FA(\)\()p FG(!)2710
2034 y Ft(\000)p Fv(\015)2831 2067 y Fx(\000)g FA(1\))15
b FG(q)3061 2034 y Fy(2)p 2345 2108 756 4 v 2656 2191
a FG(\015)5 b(r)2752 2165 y Fy(2)3111 2028 y Fu(\021)3165
2051 y Fy(2)3225 2129 y Fx(\000)726 2415 y(\000)p FG(\021)845
2378 y Fy(2)885 2415 y FG(q)929 2378 y Fy(2)983 2251
y Fu(s)p 1074 2251 826 4 v 164 x FA(1)21 b Fx(\000)1451
2354 y FA(4)p FG(!)1556 2328 y Fy(2)1596 2354 y FG(q)1640
2328 y Fy(2)p 1241 2395 650 4 v 1241 2478 a FG(r)1285
2452 y Fy(2)1344 2478 y FA(+)f FG(q)1479 2452 y Fy(2)1518
2478 y FA(\(4)p FG(!)1658 2452 y Fy(2)1718 2478 y Fx(\000)g
FA(1\))1920 2415 y(+)35 b(const)2259 2354 y FG(q)2303
2321 y Fy(4)p 2259 2395 84 4 v 2259 2478 a FG(r)2303
2452 y Fy(2)2377 2415 y Fx(\024)572 2693 y(\024)83 b
FG(r)770 2656 y Fy(2)824 2592 y Fu(\020)879 2693 y FA(1)20
b(+)1068 2632 y FG(q)1112 2599 y Fy(2)p 1045 2672 129
4 v 1045 2755 a FA(2)p FG(r)1134 2729 y Fy(2)1199 2693
y FA(\(2)p FG(!)1339 2656 y Fy(2)1399 2693 y Fx(\000)g
FA(1\))h(+)1692 2632 y(\(1)g(+)f FG(\016)s FA(\)\()p
FG(!)2057 2599 y Ft(\000)p Fv(\015)2178 2632 y Fx(\000)f
FA(1\))c FG(q)2407 2599 y Fy(2)p 1692 2672 756 4 v 2002
2755 a FG(\015)5 b(r)2098 2729 y Fy(2)2458 2592 y Fu(\021)2512
2615 y Fy(2)2572 2693 y Fx(\000)726 2946 y(\000)p FG(\021)845
2908 y Fy(2)885 2946 y FG(q)929 2908 y Fy(2)988 2946
y FA(+)35 b(const)1327 2884 y FG(q)1371 2851 y Fy(4)p
1327 2925 84 4 v 1327 3008 a FG(r)1371 2982 y Fy(2)1445
2946 y Fx(\024)572 3181 y(\024)83 b FG(r)770 3143 y Fy(2)824
3080 y Fu(\020)879 3181 y FA(1)20 b(+)1045 3119 y FG(q)1089
3086 y Fy(2)p 1045 3160 V 1045 3243 a FG(r)1089 3217
y Fy(2)1154 3181 y FA(\(2)p FG(!)1294 3143 y Fy(2)1354
3181 y Fx(\000)g FA(1\))h(+)1647 3119 y(2\(1)g(+)f FG(\016)s
FA(\)\()p FG(!)2057 3086 y Ft(\000)p Fv(\015)2178 3119
y Fx(\000)g FA(1\))15 b FG(q)2408 3086 y Fy(2)p 1647
3160 802 4 v 1980 3243 a FG(\015)5 b(r)2076 3217 y Fy(2)2458
3080 y Fu(\021)2532 3181 y Fx(\000)20 b FG(\021)2671
3143 y Fy(2)2711 3181 y FG(q)2755 3143 y Fy(2)2815 3181
y FA(+)35 b(const)3153 3119 y FG(q)3197 3086 y Fy(4)p
3153 3160 84 4 v 3153 3243 a FG(r)3197 3217 y Fy(2)3272
3181 y FA(=)572 3433 y(=)83 b FG(r)770 3396 y Fy(2)824
3332 y Fu(h)867 3433 y FA(1)21 b(+)1034 3372 y FG(q)1078
3339 y Fy(2)p 1034 3412 V 1034 3496 a FG(r)1078 3469
y Fy(2)1142 3433 y FA(\(2)p FG(!)1282 3396 y Fy(2)1342
3433 y Fx(\000)f FA(1\))h(+)1635 3372 y(2\(1)h(+)e FG(\016)s
FA(\)\()p FG(!)2046 3339 y Ft(\000)p Fv(\015)2167 3372
y Fx(\000)f FA(1\))c FG(q)2396 3339 y Fy(2)p 1635 3412
802 4 v 1968 3496 a FG(\015)5 b(r)2064 3469 y Fy(2)2467
3433 y Fx(\000)2568 3372 y FG(\021)2616 3339 y Fy(2)2656
3372 y FG(q)2700 3339 y Fy(2)p 2568 3412 172 4 v 2612
3496 a FG(r)2656 3469 y Fy(2)2769 3433 y FA(+)35 b(const)3108
3372 y FG(q)3152 3339 y Fy(4)p 3108 3412 84 4 v 3108
3496 a FG(r)3152 3469 y Fy(4)3201 3332 y Fu(i)3269 3433
y Fx(\024)572 3686 y(\024)83 b FG(r)770 3648 y Fy(2)824
3585 y Fu(h)867 3686 y FA(1)21 b(+)1034 3624 y FG(q)1078
3591 y Fy(2)p 1034 3665 V 1034 3748 a FG(r)1078 3722
y Fy(2)1142 3686 y FA(\(2)p FG(!)1282 3648 y Fy(2)1342
3686 y Fx(\000)f FA(1\))h(+)1635 3624 y(2\(1)h(+)e(2)p
FG(\016)s FA(\)\()p FG(!)2091 3591 y Ft(\000)p Fv(\015)2212
3624 y Fx(\000)g FA(1\))15 b FG(q)2442 3591 y Fy(2)p
1635 3665 847 4 v 1877 3748 a FA(\()p FG(\015)26 b FA(+)20
b(2\))p FG(r)2200 3722 y Fy(2)2512 3686 y Fx(\000)2613
3624 y FG(\021)2661 3591 y Fy(2)2701 3624 y FG(q)2745
3591 y Fy(2)p 2613 3665 172 4 v 2657 3748 a FG(r)2701
3722 y Fy(2)2814 3686 y FA(+)36 b(const)3153 3624 y FG(q)3197
3591 y Fy(4)p 3153 3665 84 4 v 3153 3748 a FG(r)3197
3722 y Fy(4)3246 3585 y Fu(i)3315 3686 y FA(=)572 3944
y(=)83 b FG(r)770 3906 y Fy(2)824 3843 y Fu(h)867 3944
y FA(1)21 b(+)1034 3882 y FG(q)1078 3849 y Fy(2)p 1034
3923 V 1034 4006 a FG(r)1078 3980 y Fy(2)1142 3944 y
FA(\(2)p FG(!)1282 3906 y Fy(2)1342 3944 y Fx(\000)f
FA(1\))h(+)1635 3882 y(2\(1)h(+)e(2)p FG(\016)s FA(\)\()p
FG(!)2091 3849 y Ft(\000)p Fv(\015)2212 3882 y Fx(\000)g
FA(1\))31 b(log)q(\(1)p FG(=!)s FA(\))15 b FG(q)2810
3849 y Fy(2)p 1635 3923 1217 4 v 2070 4006 a FA(\(log)i(2\))e
FG(r)2377 3980 y Fy(2)2881 3944 y Fx(\000)2982 3882 y
FG(\021)3030 3849 y Fy(2)3070 3882 y FG(q)3114 3849 y
Fy(2)p 2982 3923 172 4 v 3026 4006 a FG(r)3070 3980 y
Fy(2)3184 3944 y FA(+)726 4201 y(+)g(const)1045 4140
y FG(q)1089 4107 y Fy(4)p 1045 4180 84 4 v 1045 4264
a FG(r)1089 4237 y Fy(4)1138 4100 y Fu(i)1206 4201 y
Fx(\024)572 4436 y(\024)83 b FG(r)770 4399 y Fy(2)824
4335 y Fu(h)867 4436 y FA(1)21 b(+)1034 4375 y FG(q)1078
4342 y Fy(2)p 1034 4415 V 1034 4499 a FG(r)1078 4472
y Fy(2)1142 4436 y FA(\(2)p FG(!)1282 4399 y Fy(2)1342
4436 y Fx(\000)f FA(1\))h(+)1635 4375 y(2\(1)h(+)e(3)p
FG(\016)s FA(\)\()p FG(!)2091 4342 y Ft(\000)p Fv(\015)2212
4375 y Fx(\000)g FA(1\)\(1)i Fx(\000)e FG(!)s FA(\))p
FG(q)2715 4342 y Fy(2)p 1635 4415 1119 4 v 2021 4499
a FA(\(log)d(2\))e FG(r)2328 4472 y Fy(2)2784 4436 y
Fx(\000)2885 4375 y FG(\021)2933 4342 y Fy(2)2973 4375
y FG(q)3017 4342 y Fy(2)p 2885 4415 172 4 v 2929 4499
a FG(r)2973 4472 y Fy(2)3086 4436 y FA(+)726 4694 y(+)g(const)1045
4632 y FG(q)1089 4599 y Fy(4)p 1045 4673 84 4 v 1045
4756 a FG(r)1089 4730 y Fy(4)1138 4593 y Fu(i)1206 4694
y FA(=)572 4929 y(=)83 b FG(r)770 4891 y Fy(2)824 4828
y Fu(h)867 4929 y FA(1)21 b(+)1034 4867 y FG(q)1078 4834
y Fy(2)p 1034 4908 V 1034 4991 a FG(r)1078 4965 y Fy(2)1142
4929 y FA(\(2)p FG(!)1282 4891 y Fy(2)1342 4929 y Fx(\000)f
FA(1)h Fx(\000)f FG(\021)1638 4891 y Fy(2)1678 4929 y
FA(\))g(+)1834 4867 y(2\(1)i(+)e(3)p FG(\016)s FA(\)\()p
FG(!)2290 4834 y Ft(\000)p Fv(\015)2411 4867 y Fx(\000)g
FA(1\)\(1)i Fx(\000)e FG(!)s FA(\))p FG(q)2914 4834 y
Fy(2)p 1834 4908 1119 4 v 2220 4991 a FA(\(log)d(2\))e
FG(r)2527 4965 y Fy(2)2983 4929 y FA(+)726 5186 y(+)g(const)1045
5125 y FG(q)1089 5092 y Fy(4)p 1045 5166 84 4 v 1045
5249 a FG(r)1089 5223 y Fy(4)1138 5086 y Fu(i)1196 5186
y FG(:)p eop
%%Page: 105 105
105 104 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)
f Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)140 b(105)456
450 y FA(Note)31 b(also)g(that,)g(b)m(y)f(\(A.33\),)758
603 y(2)p FG(!)863 565 y Fy(2)922 603 y Fx(\000)20 b
FA(1)h Fx(\000)f FG(\021)1218 565 y Fy(2)1341 603 y FA(=)83
b FG(!)1555 565 y Fy(2)1614 603 y Fx(\000)20 b FA(1)h(+)e
FG(!)1921 565 y Fy(2)1981 603 y Fx(\000)h FG(\021)2120
565 y Fy(2)2185 603 y FA(=)1341 738 y(=)83 b(\()p FG(!)23
b Fx(\000)d FA(1\)\()p FG(!)k FA(+)c(1\))h(+)f(\()p FG(!)j
Fx(\000)d FG(\021)s FA(\)\()p FG(!)k FA(+)c FG(\021)s
FA(\))26 b(=)1341 873 y(=)83 b(\()p FG(!)23 b Fx(\000)d
FA(1\)\()p FG(!)k FA(+)c(1\))h(+)f(\()p FG(!)j Fx(\000)d
FA(1\)\(1)i Fx(\000)e FG(\017)p FA(\)\()p FG(!)k FA(+)19
b FG(\021)s FA(\))27 b(=)1341 1007 y(=)83 b(\()p FG(!)23
b Fx(\000)d FA(1\))15 b FG($)k(;)456 1157 y FA(where)1362
1268 y FG($)43 b FA(:=)d FG(!)23 b FA(+)d(1)h(+)f(\(1)h
Fx(\000)f FG(\017)p FA(\)\()p FG(!)j FA(+)d FG(\021)s
FA(\))15 b FG(:)456 1398 y FA(Thence,)30 b(from)g(the)h(estimates)g
(here)f(ab)s(o)m(v)m(e,)737 1548 y Fx(j)p FG(w)23 b Fx(\000)d
FG(y)s Fx(j)1014 1510 y Fy(2)1079 1548 y Fx(\024)584
1743 y(\024)82 b FG(r)781 1705 y Fy(2)835 1642 y Fu(h)878
1743 y FA(1)21 b Fx(\000)1045 1681 y FA(\(1)g Fx(\000)f
FG(!)s FA(\))15 b FG($)j(q)1484 1648 y Fy(2)p 1045 1722
479 4 v 1243 1805 a FG(r)1287 1779 y Fy(2)1553 1743 y
FA(+)1654 1681 y(2\(1)k(+)e(3)p FG(\016)s FA(\)\()p FG(!)2110
1648 y Ft(\000)p Fv(\015)2231 1681 y Fx(\000)g FA(1\)\(1)i
Fx(\000)e FG(!)s FA(\))p FG(q)2734 1648 y Fy(2)p 1654
1722 1119 4 v 2040 1805 a FA(\(log)d(2\))e FG(r)2347
1779 y Fy(2)2803 1743 y FA(+)35 b(const)3142 1681 y FG(q)3186
1648 y Fy(4)p 3142 1722 84 4 v 3142 1805 a FG(r)3186
1779 y Fy(4)3235 1642 y Fu(i)3304 1743 y FA(=)584 2001
y(=)82 b FG(r)781 1963 y Fy(2)835 1900 y Fu(h)878 2001
y FA(1)21 b Fx(\000)1045 1939 y FA(\(1)g Fx(\000)f FG(!)s
FA(\))15 b FG($)j(q)1484 1906 y Fy(2)p 1045 1980 479
4 v 1243 2063 a FG(r)1287 2037 y Fy(2)1553 2001 y FA(+)1654
1939 y(2\(1)k(+)e(3)p FG(\016)s FA(\)\(2)p FG(!)2155
1906 y Fy(2)2217 1939 y Fx(\000)g FA(1\)\(1)h Fx(\000)f
FG(!)s FA(\))p FG(q)2719 1906 y Fy(2)p 1654 1980 1105
4 v 2033 2063 a FA(\(log)d(2\))e FG(r)2340 2037 y Fy(2)2789
2001 y FA(+)35 b(const)3128 1939 y FG(q)3172 1906 y Fy(4)p
3128 1980 84 4 v 3128 2063 a FG(r)3172 2037 y Fy(4)3221
1900 y Fu(i)3289 2001 y FA(=)584 2258 y(=)82 b FG(r)781
2221 y Fy(2)835 2157 y Fu(h)878 2258 y FA(1)21 b Fx(\000)1045
2197 y FA(\(1)g Fx(\000)f FG(!)s FA(\))15 b FG(q)1391
2164 y Fy(2)p 1045 2237 386 4 v 1196 2321 a FG(r)1240
2294 y Fy(2)1455 2258 y FA(\002)20 b(+)35 b(const)1885
2197 y FG(q)1929 2164 y Fy(4)p 1885 2237 84 4 v 1885
2321 a FG(r)1929 2294 y Fy(4)1978 2157 y Fu(i)2036 2258
y FG(;)456 2433 y FA(where)29 b(w)m(e)i(used)f(the)g(short)h(hand)e
(notation)1335 2620 y(\002)40 b(:=)g FG($)23 b Fx(\000)1781
2559 y FA(2\(1)e(+)f(3)p FG(\016)s FA(\)\(2)p FG(!)2281
2526 y Fy(2)2343 2559 y Fx(\000)g FA(1\))p 1781 2599
735 4 v 2059 2683 a(log)d(2)2540 2620 y FG(:)456 2813
y FA(Note)38 b(that)f(when)f FG(\021)41 b FA(and)36 b
FG(!)k FA(are)d(close)g(to)h(1,)h(and)d FG(\017)h FA(and)g
FG(\016)j FA(are)e(small,)f(then)g FG($)i FA(is)456 2921
y(close)30 b(to)h(4,)g(and)f(th)m(us)g(\002)g(is)g(close)g(to)1654
3108 y(4)21 b Fx(\000)1887 3047 y FA(2)p 1821 3087 179
4 v 1821 3171 a(log)16 b(2)2049 3108 y FG(>)40 b FA(0)15
b FG(:)456 3301 y FA(Hence,)31 b(for)f(some)h(small)g(\024)-47
b FG(c)25 b(>)g FA(0,)1117 3506 y Fx(j)p FG(w)e Fx(\000)d
FG(y)s Fx(j)25 b(\024)g FG(r)1559 3468 y Fy(2)1613 3405
y Fu(h)1656 3506 y FA(1)c Fx(\000)f FA(2)r(\024)-47 b
FG(c)1908 3444 y(q)1952 3411 y Fy(2)1991 3444 y FA(\(1)21
b Fx(\000)f FG(!)s FA(\))p 1908 3485 371 4 v 2051 3568
a FG(r)2095 3542 y Fy(2)2308 3506 y FA(+)35 b(const)2647
3444 y FG(q)2691 3411 y Fy(4)p 2647 3485 84 4 v 2647
3568 a FG(r)2691 3542 y Fy(4)2740 3405 y Fu(i)456 3681
y FA(and)29 b(so)1346 3830 y Fx(j)p FG(w)23 b Fx(\000)c
FG(y)s Fx(j)26 b(\024)f FG(r)1788 3792 y Fy(2)1842 3729
y Fu(h)1885 3830 y FA(1)20 b Fx(\000)i FA(\024)-47 b
FG(c)2091 3768 y(q)2135 3735 y Fy(2)2174 3768 y FA(\(1)21
b Fx(\000)f FG(!)s FA(\))p 2091 3809 371 4 v 2234 3892
a FG(r)2278 3866 y Fy(2)2471 3729 y Fu(i)2529 3830 y
FG(:)456 3991 y FA(Therefore,)30 b(b)m(y)g(\(A.35\),)1062
4219 y FG(r)23 b Fx(\000)d(j)p FG(w)j Fx(\000)d FG(y)s
Fx(j)83 b(\025)g FG(r)1775 4118 y Fu(\020)1829 4219 y
FA(1)20 b Fx(\000)1985 4071 y Fu(r)p 2076 4071 587 4
v 148 x FA(1)h Fx(\000)h FA(\024)-47 b FG(c)2282 4157
y(q)2326 4131 y Fy(2)2365 4157 y FA(\(1)21 b Fx(\000)f
FG(!)s FA(\))p 2282 4198 371 4 v 2426 4281 a FG(r)2470
4255 y Fy(2)2662 4118 y Fu(\021)2742 4219 y Fx(\025)1577
4458 y(\025)1741 4396 y FA(const)15 b(\(1)21 b Fx(\000)f
FG(!)s FA(\))15 b FG(q)2309 4363 y Fy(2)p 1741 4437 609
4 v 2023 4520 a FG(r)2374 4458 y(;)456 4638 y FA(in)34
b(case)i FG(p)810 4652 y Fv(N)910 4638 y Fx(6)p FA(=)d(0.)56
b(In)34 b(the)i(ligh)m(t)e(of)i(the)f(latter)h(estimate,)h(\(A.81\))h
(and)c(\(A.80\),)39 b(w)m(e)456 4746 y(deduce)30 b(that)456
4933 y(\(A.85\))827 b Fx(j)p FG(z)1603 4947 y Fy(1)1663
4933 y Fx(\000)20 b FG(w)r Fx(j)41 b(\025)2008 4871 y
FA(const)16 b FG(q)2275 4838 y Fy(2)p 2008 4912 307 4
v 2139 4995 a FG(r)2339 4933 y(:)456 5108 y FA(This)24
b(will)g(con)m(trol)i(the)h(distance)f(of)g FG(w)j FA(from)d(\000)2092
5122 y Fy(1)2131 5108 y FA(;)i(w)m(e)f(will)c(no)m(w)j(con)m(trol)h
(the)g(distance)456 5216 y(from)34 b(\000)732 5230 y
Fy(2)806 5216 y FA(and)g(\000)1044 5230 y Fy(3)1083 5216
y FA(,)j(in)c(order)h(to)i(complete)f(the)g(pro)s(of)f(of)h(\(A.79\).)
57 b(F)-8 b(or)35 b(this,)g(let)g(us)p eop
%%Page: 106 106
106 105 bop 456 251 a Fy(106)140 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(consider)40 b(the)i(cone)h Fx(C)k
FA(whic)m(h)41 b(is)g(tangen)m(t)i(to)g FG(@)5 b(B)2265
464 y Fv(r)2297 472 y Fs(!)2346 450 y FA(\()p FG(y)2426
464 y Fv(!)2476 450 y FA(\))42 b(\(from)g(outside\))f(and)h(to)456
558 y(\000)513 572 y Fy(2)588 558 y FA(\(from)36 b(inside\))e(at)i
Fx(fj)p FG(x)1381 525 y Ft(0)1405 558 y Fx(j)f FA(=)g
FG(!)s(q)s Fx(g)p FA(.)57 b(Let)37 b(also)f(\005)g(b)s(e)f(the)h(t)m(w)
m(o-dimensional)f(plane)456 666 y(spanned)29 b(b)m(y)h(the)g(v)m
(ectors)i FG(w)23 b Fx(\000)d FG(y)1625 680 y Fv(!)1705
666 y FA(and)30 b FG(e)1924 680 y Fv(N)2022 666 y FA(and)g(passing)f
(through)h FG(y)2907 680 y Fv(!)2957 666 y FA(,)g(i.e.,)1129
889 y(\005)41 b(:=)1374 788 y Fu(n)1435 889 y FG(y)1480
903 y Fv(!)1550 889 y FA(+)20 b(\()p FG(w)j Fx(\000)d
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3817 y Fx(\000)20 b FG(q)2686 3791 y Fy(2)2725 3716 y
Fu(\021)2800 3817 y FA(+)1356 4057 y(+)1451 3995 y(const)c
FG(q)1718 3962 y Fy(2)p 1437 4036 322 4 v 1575 4119 a
FG(r)1788 4057 y FA(+)1904 3995 y(const)p 2126 3922 72
4 v 15 w FG(C)2198 4009 y Fy(0)p 1888 4036 349 4 v 2028
4119 a FG(R)2272 4057 y Fx(\024)1202 4296 y(\024)1381
4235 y FA(const)f FG(q)1647 4202 y Fy(2)p 1366 4275 322
4 v 1504 4358 a FG(r)1717 4296 y FA(+)1833 4235 y(const)p
2056 4162 72 4 v 16 w FG(C)2127 4249 y Fy(0)p 1818 4275
349 4 v 1957 4358 a FG(R)2202 4296 y Fx(\024)1202 4531
y(\024)1381 4470 y FA(const)g FG(q)1647 4437 y Fy(2)p
1366 4510 322 4 v 1491 4593 a FG(R)1712 4531 y(;)456
4758 y FA(thence,)456 5014 y(\(A.91\))584 b(\006\()p
FG(Y)1447 5028 y Fv(!)1498 5014 y FG(;)15 b(R)1607 5028
y Fy(1)1646 5014 y FA(\))41 b Fx(\022)1833 4886 y Fu(\032)1901
5014 y FG(x)1953 5028 y Fv(N)2046 5014 y Fx(\024)2167
4953 y FA(const)16 b FG(q)2434 4920 y Fy(2)p 2152 4994
V 2278 5077 a FG(R)2483 4886 y Fu(\033)2581 5014 y FG(:)p
eop
%%Page: 108 108
108 107 bop 456 251 a Fy(108)140 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)555 450 y FA(W)g(e)31 b(no)m(w)e(in)m(v)m(estigate)h(the)g(m)m
(utual)e(p)s(osition)g(of)h(\011)g(and)g Fw(S)n FA(\()p
FG(Y)2683 464 y Fv(!)2728 450 y FG(;)15 b(R)2837 464
y Fy(1)2877 450 y FA(\).)40 b(F)-8 b(or)30 b(this,)f(w)m(e)456
558 y(start)i(b)m(y)f(claiming)e(that)948 762 y(The)i(region)1408
661 y Fu(n)1468 762 y FG(X)j Fx(2)25 b FA(\011)30 b(where)g
Fx(j)p FG(x)2103 776 y Fv(N)7 b Fy(+1)2260 762 y Fx(j)26
b FG(<)f FA(1)p FG(=)p FA(2)31 b(and)948 961 y FG(d)995
975 y Fy(\000)1043 961 y FA(\()p FG(x)p FA(\))g(is)f(realized)f(on)i
(\000)1805 975 y Fy(3)1844 860 y Fu(o)1935 961 y FA(is)e(ab)s(o)m(v)m
(e)j Fw(S)n FA(\()p FG(Y)2428 975 y Fv(!)2473 961 y FG(;)15
b(R)2582 975 y Fy(1)2622 961 y FA(\))948 1123 y(\(where)30
b(ab)s(o)m(v)m(e)i(means)e(with)f(resp)s(ect)h(to)h(the)g
FG(e)2610 1137 y Fv(N)7 b Fy(+1)2768 1123 y FA(-direction\).)456
932 y(\(A.92\))456 1309 y(T)-8 b(o)21 b(pro)m(v)m(e)g(\(A.92\),)k(note)
c(that,)i(if)d FG(d)1645 1323 y Fy(\000)1693 1309 y FA(\()p
FG(x)p FA(\))h(is)f(realized)g(on)g(\000)2415 1323 y
Fy(3)2454 1309 y FA(,)j(then)d FG(x)q Fx(\000)q FG(y)2869
1323 y Fv(!)2938 1309 y FA(is)g(orthogonal)456 1417 y(to)31
b(\000)624 1431 y Fy(3)688 1417 y Fx(\032)25 b FG(@)5
b(B)906 1431 y Fv(r)938 1439 y Fs(!)987 1417 y FA(\()p
FG(y)1067 1431 y Fv(!)1118 1417 y FA(\),)30 b(therefore,)i(recalling)c
(\(A.76\))33 b(and)d(\(2.39\),)754 1598 y FG(d)801 1612
y Fy(\000)849 1598 y FA(\()p FG(x)p FA(\))84 b(=)f FG(d)1256
1612 y Fy(\000)1300 1621 y Fq(3)1339 1598 y FA(\()p FG(x)p
FA(\))26 b(=)f FG(d)1630 1617 y Fy(\006\()p Fv(Y)1749
1625 y Fs(!)1794 1617 y Fv(;R)1867 1626 y Fq(1)1901 1617
y Fy(\))1933 1598 y FA(\()p FG(x)p FA(\))c(+)f FG(r)2208
1612 y Fy(1)2267 1598 y Fx(\000)g FG(r)2399 1612 y Fv(!)2475
1598 y Fx(\025)1055 1810 y(\025)83 b FG(d)1256 1829 y
Fy(\006\()p Fv(Y)1375 1837 y Fs(!)1420 1829 y Fv(;R)1493
1838 y Fq(1)1527 1829 y Fy(\))1559 1810 y FA(\()p FG(x)p
FA(\))21 b(+)1803 1749 y(3)p 1848 1676 72 4 v FG(C)1920
1763 y Fy(0)p 1803 1789 157 4 v 1846 1873 a FG(R)1994
1810 y FA(=)1055 2050 y(=)83 b Fx(j)p FG(x)20 b Fx(\000)g
FG(y)1442 2064 y Fv(!)1492 2050 y Fx(j)g(\000)g FG(r)1669
2064 y Fy(1)1729 2050 y FA(+)1830 1988 y(3)p 1875 1915
72 4 v FG(C)1947 2002 y Fy(0)p 1830 2029 157 4 v 1873
2112 a FG(R)2021 2050 y FA(=)1055 2289 y(=)83 b Fx(j)p
FG(x)20 b Fx(\000)g FG(y)1442 2303 y Fv(!)1492 2289 y
Fx(j)g(\000)g FG(R)1697 2303 y Fy(1)1757 2289 y FA(+)g
FG(H)1924 2303 y Fy(0)1963 2289 y FA(\()p FG(y)2043 2303
y Fv(N)7 b Fy(+1)2200 2289 y FA(\))21 b(+)p 2378 2155
72 4 v 2378 2228 a FG(C)2450 2242 y Fy(0)p 2357 2268
154 4 v 2357 2352 a FA(2)p FG(R)2471 2366 y Fy(1)2536
2289 y FG(y)2584 2252 y Fy(2)2581 2312 y Fv(N)7 b Fy(+1)2758
2289 y FA(+)2859 2228 y(3)p 2904 2155 72 4 v FG(C)2976
2242 y Fy(0)p 2859 2268 157 4 v 2902 2352 a FG(R)3050
2289 y FA(=)1055 2542 y(=)83 b Fx(j)p FG(x)20 b Fx(\000)g
FG(y)1442 2556 y Fv(!)1492 2542 y Fx(j)g(\000)g FG(R)1697
2556 y Fy(1)1757 2542 y FA(+)g FG(H)1924 2556 y Fy(0)1963
2542 y FA(\()p FG(y)2043 2556 y Fv(N)7 b Fy(+1)2200 2542
y FA(\))21 b(+)p 2378 2408 72 4 v 2378 2481 a FG(C)2450
2495 y Fy(0)p 2357 2521 154 4 v 2357 2605 a FA(2)p FG(R)2471
2619 y Fy(1)2536 2542 y FG(y)2584 2505 y Fy(2)2581 2565
y Fv(N)7 b Fy(+1)2758 2542 y FA(+)2859 2481 y(3)p 2904
2408 72 4 v FG(C)2976 2495 y Fy(0)p 2859 2521 157 4 v
2902 2605 a FG(R)3045 2542 y Fx(\000)1209 2795 y(\000)p
FG(H)1356 2809 y Fv(y)1391 2820 y Fs(N)e Fq(+1)1525 2809
y Fv(;R)1602 2795 y FA(\(0\))21 b Fx(\000)p 1841 2661
72 4 v 1841 2734 a FG(C)1913 2748 y Fy(0)p 1839 2774
116 4 v 1839 2858 a FA(2)p FG(R)1964 2795 y(y)2012 2758
y Fy(2)2009 2818 y Fv(N)7 b Fy(+1)2181 2795 y FG(:)456
3007 y FA(This,)28 b(recalling)h(\(2.39\))k(and)d(\(2.38\),)j(giv)m(es)
d(that)916 3253 y FG(H)992 3267 y Fv(y)1027 3278 y Fs(N)5
b Fq(+1)1161 3267 y Fv(;R)1234 3276 y Fq(1)1273 3152
y Fu(\020)1327 3253 y FG(g)1370 3267 y Fy(\011)1429 3253
y FA(\()p FG(x)p FA(\))1551 3152 y Fu(\021)1626 3253
y Fx(\000)p 1727 3119 72 4 v 1727 3192 a FG(C)1799 3206
y Fy(0)p 1727 3232 111 4 v 1760 3316 a FA(2)1863 3152
y Fu(\020)1918 3253 y FG(g)1961 3267 y Fy(\011)2020 3253
y FA(\()p FG(x)p FA(\))21 b Fx(\000)f FG(y)2299 3267
y Fv(N)7 b Fy(+1)2456 3152 y Fu(\021)2510 3175 y Fy(2)2580
3125 y Fu(\022)2669 3192 y FA(1)p 2657 3232 70 4 v 2657
3316 a FG(R)2757 3253 y Fx(\000)2889 3192 y FA(1)p 2857
3232 109 4 v 2857 3316 a FG(R)2926 3330 y Fy(1)2976 3125
y Fu(\023)3068 3253 y FA(=)762 3517 y(=)937 3494 y Fu(e)916
3517 y FG(H)992 3531 y Fv(y)1027 3542 y Fs(N)e Fq(+1)1161
3531 y Fv(;R)1234 3540 y Fq(1)1273 3416 y Fu(\020)1327
3517 y FG(g)1370 3531 y Fy(\011)1429 3517 y FA(\()p FG(x)p
FA(\))1551 3416 y Fu(\021)1626 3517 y Fx(\000)p 1727
3382 72 4 v 1727 3455 a FG(C)1799 3469 y Fy(0)p 1727
3496 111 4 v 1760 3579 a FA(2)1863 3416 y Fu(\020)1918
3517 y FG(g)1961 3531 y Fy(\011)2020 3517 y FA(\()p FG(x)p
FA(\))21 b Fx(\000)f FG(y)2299 3531 y Fv(N)7 b Fy(+1)2456
3416 y Fu(\021)2510 3438 y Fy(2)2580 3388 y Fu(\022)2669
3455 y FA(1)p 2657 3496 70 4 v 2657 3579 a FG(R)2757
3517 y Fx(\000)2889 3455 y FA(1)p 2857 3496 109 4 v 2857
3579 a FG(R)2926 3593 y Fy(1)2976 3388 y Fu(\023)3068
3517 y FA(=)762 3743 y(=)937 3720 y Fu(e)916 3743 y FG(H)992
3757 y Fv(y)1027 3768 y Fs(N)e Fq(+1)1161 3757 y Fv(;R)1238
3642 y Fu(\020)1293 3743 y FG(g)1336 3757 y Fy(\011)1395
3743 y FA(\()p FG(x)p FA(\))1517 3642 y Fu(\021)1597
3743 y FA(=)762 3942 y(=)83 b FG(H)992 3956 y Fv(y)1027
3967 y Fs(N)5 b Fq(+1)1161 3956 y Fv(;R)1238 3841 y Fu(\020)1293
3942 y FG(g)1336 3956 y Fy(\011)1395 3942 y FA(\()p FG(x)p
FA(\))1517 3841 y Fu(\021)1597 3942 y FA(=)762 4141 y(=)83
b FG(H)992 4155 y Fv(y)1027 4166 y Fs(N)5 b Fq(+1)1161
4155 y Fv(;R)1254 4040 y Fu(\020)1308 4141 y FG(g)1351
4155 y Fv(y)1386 4166 y Fs(N)g Fq(+1)1521 4155 y Fv(;R)1598
4040 y Fu(\020)1652 4141 y FG(d)1699 4155 y Fy(\000)1748
4141 y FA(\()p FG(x)p FA(\))21 b(+)f FG(H)2058 4155 y
Fv(y)2093 4166 y Fs(N)5 b Fq(+1)2227 4155 y Fv(;R)2304
4141 y FA(\(0\))2419 4040 y Fu(\021)r(\021)2554 4141
y FA(=)762 4303 y(=)83 b FG(d)963 4317 y Fy(\000)1011
4303 y FA(\()p FG(x)p FA(\))21 b(+)f FG(H)1321 4317 y
Fv(y)1356 4328 y Fs(N)5 b Fq(+1)1490 4317 y Fv(;R)1568
4303 y FA(\(0\))26 b Fx(\025)762 4512 y(\025)83 b(j)p
FG(x)20 b Fx(\000)g FG(y)1149 4526 y Fv(!)1199 4512 y
Fx(j)h(\000)f FG(R)1405 4526 y Fy(1)1464 4512 y FA(+)g
FG(H)1631 4526 y Fy(0)1670 4512 y FA(\()p FG(y)1750 4526
y Fv(N)7 b Fy(+1)1907 4512 y FA(\))21 b(+)p 2085 4378
72 4 v 2085 4451 a FG(C)2157 4465 y Fy(0)p 2064 4492
154 4 v 2064 4575 a FA(2)p FG(R)2178 4589 y Fy(1)2243
4512 y FG(y)2291 4475 y Fy(2)2288 4535 y Fv(N)7 b Fy(+1)2465
4512 y FA(+)2566 4451 y(3)p 2611 4378 72 4 v FG(C)2683
4465 y Fy(0)p 2566 4492 157 4 v 2609 4575 a FG(R)2752
4512 y Fx(\000)p 2855 4378 72 4 v 2855 4451 a FG(C)2927
4465 y Fy(0)p 2853 4492 116 4 v 2853 4575 a FA(2)p FG(R)2979
4512 y(y)3027 4475 y Fy(2)3024 4535 y Fv(N)g Fy(+1)3196
4512 y FG(:)456 4733 y FA(Therefore,)851 4937 y FG(H)927
4951 y Fv(y)962 4962 y Fs(N)e Fq(+1)1096 4951 y Fv(;R)1169
4960 y Fq(1)1208 4836 y Fu(\020)1262 4937 y FG(g)1305
4951 y Fy(\011)1365 4937 y FA(\()p FG(x)p FA(\))1487
4836 y Fu(\021)1567 4937 y Fx(\025)25 b(j)p FG(x)20 b
Fx(\000)g FG(y)1896 4951 y Fv(!)1946 4937 y Fx(j)h(\000)e
FG(R)2151 4951 y Fy(1)2211 4937 y FA(+)h FG(H)2378 4951
y Fy(0)2417 4937 y FA(\()p FG(y)2497 4951 y Fv(N)7 b
Fy(+1)2654 4937 y FA(\)+)1562 5173 y(+)1663 5111 y(3)p
1708 5038 72 4 v FG(C)1780 5125 y Fy(0)p 1663 5152 157
4 v 1706 5235 a FG(R)1849 5173 y Fx(\000)p 1952 5038
72 4 v 1952 5111 a FG(C)2024 5125 y Fy(0)p 1950 5152
116 4 v 1950 5235 a FA(2)p FG(R)2075 5173 y(y)2123 5135
y Fy(2)2120 5195 y Fv(N)g Fy(+1)2297 5173 y Fx(\000)p
2420 5038 72 4 v 2420 5111 a FG(C)2491 5125 y Fy(0)p
2398 5152 154 4 v 2398 5235 a FA(2)p FG(R)2512 5249 y
Fy(1)2577 5072 y Fu(\020)2632 5173 y FG(g)2675 5187 y
Fy(\011)2734 5173 y FA(\()p FG(x)p FA(\))21 b Fx(\000)f
FG(y)3013 5187 y Fv(N)7 b Fy(+1)3170 5072 y Fu(\021)3224
5095 y Fy(2)3279 5173 y FG(:)456 5064 y FA(\(A.93\))p
eop
%%Page: 109 109
109 108 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)
f Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)140 b(109)456
450 y FA(Note)41 b(ho)m(w)m(ev)m(er)g(that)f FG(R)1318
464 y Fy(1)1398 450 y Fx(\030)h FG(R)q(=)p FA(2)f(for)g(large)g
FG(R)q FA(,)i(and)d(so,)j(in)d(particular,)i(w)m(e)f(ma)m(y)456
558 y(assume)30 b(that)h FG(R)1036 572 y Fy(1)1100 558
y Fx(\025)25 b FG(R)q(=)p FA(3.)42 b(Thence,)1212 711
y(3)p 1257 638 72 4 v FG(C)1329 725 y Fy(0)p 1212 751
157 4 v 1256 835 a FG(R)1399 772 y Fx(\000)p 1502 638
72 4 v 1502 711 a FG(C)1573 725 y Fy(0)p 1500 751 116
4 v 1500 835 a FA(2)p FG(R)1625 772 y(y)1673 735 y Fy(2)1670
795 y Fv(N)7 b Fy(+1)1847 772 y Fx(\000)p 1969 638 72
4 v 1969 711 a FG(C)2041 725 y Fy(0)p 1948 751 154 4
v 1948 835 a FA(2)p FG(R)2062 849 y Fy(1)2127 671 y Fu(\020)2181
772 y FG(g)2224 786 y Fy(\011)2284 772 y FA(\()p FG(x)p
FA(\))21 b Fx(\000)e FG(y)2562 786 y Fv(N)7 b Fy(+1)2719
671 y Fu(\021)2774 694 y Fy(2)2838 772 y Fx(\025)1049
1032 y(\025)1212 970 y FA(3)p 1257 897 72 4 v FG(C)1329
984 y Fy(0)p 1212 1011 157 4 v 1256 1094 a FG(R)1399
1032 y Fx(\000)p 1502 897 72 4 v 1502 970 a FG(C)1573
984 y Fy(0)p 1500 1011 116 4 v 1500 1094 a FA(2)p FG(R)1640
904 y Fu(\022)1717 970 y FA(1)p 1717 1011 46 4 v 1717
1094 a(4)1772 904 y Fu(\023)1839 926 y Fy(2)1899 1032
y Fx(\000)2000 970 y FA(3)p 2045 897 72 4 v FG(C)2117
984 y Fy(0)p 2000 1011 157 4 v 2020 1094 a FA(2)p FG(R)2196
904 y Fu(\022)2273 970 y FA(3)p 2273 1011 46 4 v 2273
1094 a(4)2329 904 y Fu(\023)2396 926 y Fy(2)2460 1032
y FG(>)1049 1212 y(>)82 b FA(0)15 b FG(:)456 1361 y FA(This)28
b(and)i(\(A.93\))i(yield)d(that)995 1538 y FG(H)1071
1552 y Fv(y)1106 1563 y Fs(N)5 b Fq(+1)1241 1552 y Fv(;R)1314
1561 y Fq(1)1352 1437 y Fu(\020)1406 1538 y FG(g)1449
1552 y Fy(\011)1509 1538 y FA(\()p FG(x)p FA(\))1631
1437 y Fu(\021)1726 1538 y FG(>)40 b Fx(j)p FG(x)21 b
Fx(\000)f FG(y)2071 1552 y Fv(!)2121 1538 y Fx(j)g(\000)g
FG(R)2326 1552 y Fy(1)2386 1538 y FA(+)f FG(H)2552 1552
y Fy(0)2591 1538 y FA(\()p FG(y)2671 1552 y Fv(N)7 b
Fy(+1)2829 1538 y FA(\))15 b FG(;)456 1710 y FA(and)29
b(so)739 1864 y FG(g)782 1878 y Fy(\011)841 1864 y FA(\()p
FG(x)p FA(\))d FG(>)f(g)1128 1878 y Fv(y)1163 1889 y
Fs(N)5 b Fq(+1)1298 1878 y Fv(;R)1371 1887 y Fq(1)1409
1763 y Fu(\020)1464 1864 y Fx(j)p FG(x)20 b Fx(\000)g
FG(y)1697 1878 y Fv(!)1747 1864 y Fx(j)g(\000)g FG(R)1952
1878 y Fy(1)2012 1864 y FA(+)g FG(H)2179 1878 y Fy(0)2218
1864 y FA(\()p FG(y)2298 1878 y Fv(N)7 b Fy(+1)2455 1864
y FA(\))2490 1763 y Fu(\021)2570 1864 y FA(=)25 b FG(g)2709
1883 y Fj(S)-6 b Fy(\()p Fv(Y)2820 1891 y Fs(!)2859 1883
y Fv(;R)2932 1892 y Fq(1)2967 1883 y Fy(\))2998 1864
y FA(\()p FG(x)p FA(\))15 b FG(;)456 2041 y FA(whic)m(h)29
b(pro)m(v)m(es)i(\(A.92\).)555 2190 y(W)-8 b(e)32 b(no)m(w)e(sho)m(w)g
(that)1040 2344 y(The)g(region)1500 2244 y Fu(n)1561
2344 y FG(X)i Fx(2)25 b Fw(S)o FA(\()p FG(Y)1899 2358
y Fv(!)1943 2344 y FG(;)15 b(R)2052 2358 y Fy(1)2092
2344 y FA(\))31 b(where)f Fx(j)p FG(x)2498 2358 y Fv(N)7
b Fy(+1)2655 2344 y Fx(j)26 b FG(<)f FA(1)p FG(=)p FA(2)31
b(and)1040 2543 y FG(d)1087 2562 y Fy(\006\()p Fv(Y)1206
2570 y Fs(!)1252 2562 y Fv(;R)1325 2571 y Fq(1)1359 2562
y Fy(\))1391 2543 y FA(\()p FG(x)p FA(\))g(is)e(realized)h(at)h(a)f(p)s
(oin)m(t)g FG(z)k FA(with)29 b Fx(j)p FG(z)2747 2510
y Ft(0)2771 2543 y Fx(j)d(\025)f FG(\021)s(q)3010 2442
y Fu(o)1040 2706 y FA(is)30 b(strictly)f(ab)s(o)m(v)m(e)j(\011)e(\(in)f
(the)i FG(e)2142 2720 y Fv(N)7 b Fy(+1)2299 2706 y FA(-direction\).)456
2514 y(\(A.94\))456 2860 y(In)23 b(order)g(to)h(pro)m(v)m(e)h
(\(A.94\),)i(tak)m(e)e FG(X)33 b Fx(2)25 b Fw(S)n FA(\()p
FG(Y)1970 2874 y Fv(!)2015 2860 y FG(;)15 b(R)2124 2874
y Fy(1)2164 2860 y FA(\),)25 b(with)d Fx(j)p FG(x)2526
2874 y Fv(N)7 b Fy(+1)2684 2860 y Fx(j)26 b FG(<)f FA(1)p
FG(=)p FA(2)g(and)e(assume)456 2968 y(that)1458 3085
y FG(\033)44 b FA(:=)c(dist)1852 2984 y Fu(\020)1907
3085 y FG(x;)15 b FA(\006\()p FG(Y)2153 3099 y Fv(!)2203
3085 y FG(;)g(R)2312 3099 y Fy(1)2352 3085 y FA(\))2387
2984 y Fu(\021)456 3244 y FA(is)29 b(attained)i(at)g
FG(z)e Fx(2)c FA(\006\()p FG(Y)1328 3258 y Fv(!)1378
3244 y FG(;)15 b(R)1487 3258 y Fy(1)1527 3244 y FA(\))31
b(and)f Fx(j)p FG(z)1841 3211 y Ft(0)1864 3244 y Fx(j)c(\025)f
FG(\021)s(q)s FA(.)41 b(Then,)29 b(b)m(y)i(construction,)1522
3394 y FG(x)1574 3408 y Fv(N)7 b Fy(+1)1772 3394 y FA(=)40
b FG(g)1926 3412 y Fj(S)-6 b Fy(\()p Fv(Y)2037 3420 y
Fs(!)2076 3412 y Fv(;R)2149 3421 y Fq(1)2183 3412 y Fy(\))2215
3394 y FA(\()p FG(x)p FA(\))15 b FG(;)456 3546 y FA(th)m(us)30
b(from)g(\(2.20\))i(and)e(De\014nition)f(2.11,)677 3761
y Fx(j)p FG(x)20 b Fx(\000)g FG(y)910 3775 y Fv(!)960
3761 y Fx(j)41 b FA(=)f FG(H)1213 3775 y Fy(0)1252 3761
y FA(\()p FG(x)1339 3775 y Fv(N)7 b Fy(+1)1496 3761 y
FA(\))21 b Fx(\000)p 1674 3626 72 4 v 1674 3699 a FG(C)1746
3713 y Fy(0)p 1653 3740 154 4 v 1653 3823 a FA(2)p FG(R)1767
3837 y Fy(1)1832 3761 y FA(\()p FG(x)1919 3775 y Fv(N)7
b Fy(+1)2097 3761 y Fx(\000)20 b FG(y)2233 3775 y Fv(N)7
b Fy(+1)2390 3761 y FA(\))2425 3723 y Fy(2)2485 3761
y FA(+)20 b FG(R)2645 3775 y Fy(1)2704 3761 y Fx(\000)g
FG(H)2871 3775 y Fy(0)2910 3761 y FA(\()p FG(y)2990 3775
y Fv(N)7 b Fy(+1)3148 3761 y FA(\))15 b FG(:)456 3949
y FA(Therefore,)929 4121 y(dist)1091 4020 y Fu(\020)1146
4121 y FG(x;)g FA(\006\()p FG(Y)1392 4135 y Fv(!)1442
4121 y FG(;)g(R)1551 4135 y Fy(1)1591 4121 y FA(\))1626
4020 y Fu(\021)1721 4121 y FA(=)40 b Fx(j)p FG(x)21 b
Fx(\000)f FG(y)2066 4135 y Fv(!)2116 4121 y Fx(j)g(\000)g
FG(r)2293 4135 y Fy(1)2358 4121 y FA(=)776 4357 y(=)82
b FG(H)1005 4371 y Fy(0)1044 4357 y FA(\()p FG(x)1131
4371 y Fv(N)7 b Fy(+1)1289 4357 y FA(\))21 b Fx(\000)p
1467 4223 72 4 v 1467 4296 a FG(C)1538 4310 y Fy(0)p
1445 4336 154 4 v 1445 4420 a FA(2)p FG(R)1559 4434 y
Fy(1)1624 4357 y FA(\()p FG(x)1711 4371 y Fv(N)7 b Fy(+1)1889
4357 y Fx(\000)20 b FG(y)2025 4371 y Fv(N)7 b Fy(+1)2182
4357 y FA(\))2217 4320 y Fy(2)2277 4357 y FA(+)20 b FG(R)2437
4371 y Fy(1)2497 4357 y Fx(\000)g FG(H)2664 4371 y Fy(0)2703
4357 y FA(\()p FG(y)2783 4371 y Fv(N)7 b Fy(+1)2940 4357
y FA(\))21 b Fx(\000)f FG(r)3128 4371 y Fy(1)3182 4357
y FG(;)456 4546 y FA(whic)m(h)29 b(implies)e(that)1158
4755 y FG(\033)43 b Fx(\024)d FG(H)1440 4769 y Fv(y)1475
4780 y Fs(N)5 b Fq(+1)1609 4769 y Fv(;R)1687 4755 y FA(\()p
FG(x)1774 4769 y Fv(N)i Fy(+1)1931 4755 y FA(\))21 b(+)p
2088 4621 72 4 v 2088 4694 a FG(C)2159 4708 y Fy(0)p
2088 4734 111 4 v 2121 4818 a FA(2)2224 4627 y Fu(\022)2313
4694 y FA(1)p 2301 4734 70 4 v 2301 4818 a FG(R)2401
4755 y FA(+)2533 4694 y(1)p 2502 4734 109 4 v 2502 4818
a FG(R)2571 4832 y Fy(1)2620 4627 y Fu(\023)2717 4755
y FG(:)456 4953 y FA(Hence,)456 5162 y(\(A.95\))293 b
FG(g)1045 5181 y Fj(S)-6 b Fy(\()p Fv(Y)1156 5189 y Fs(!)1195
5181 y Fv(;R)1268 5190 y Fq(1)1302 5181 y Fy(\))1334
5162 y FA(\()p FG(x)p FA(\))26 b(=)f FG(x)1630 5176 y
Fv(N)7 b Fy(+1)1813 5162 y Fx(\025)24 b FG(g)1951 5176
y Fv(y)1986 5187 y Fs(N)5 b Fq(+1)2121 5176 y Fv(;R)2214
5034 y Fu(\022)2281 5162 y FG(\033)23 b Fx(\000)p 2457
5028 72 4 v 2457 5101 a FG(C)2529 5115 y Fy(0)p 2457
5142 111 4 v 2490 5225 a FA(2)2593 5061 y Fu(\020)2670
5101 y FA(1)p 2657 5142 70 4 v 2657 5225 a FG(R)2757
5162 y FA(+)2890 5101 y(1)p 2858 5142 109 4 v 2858 5225
a FG(R)2927 5239 y Fy(1)2977 5061 y Fu(\021)3031 5034
y(\023)3128 5162 y FG(:)p eop
%%Page: 110 110
110 109 bop 456 251 a Fy(110)140 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(W)g(e)35 b(consider)k(\025)-51 b FG(y)35
b Fx(2)c FA(\000)j(on)g(the)h(half-line)c(from)j FG(y)2146
464 y Fv(!)2230 450 y FA(to)m(w)m(ards)h FG(z)t FA(.)53
b(Note)36 b(that,)g(b)m(y)e(con-)456 558 y(struction,)j
FG(y)918 572 y Fv(!)968 558 y FA(,)h FG(z)t FA(,)h FG(x)d
FA(and)43 b(\025)-52 b FG(y)40 b FA(lie)35 b(on)h(the)h(same)g
(half-line.)57 b(Let)37 b(also)43 b(~)-52 b FG(y)39 b
Fx(2)c FA(\000)h(b)s(e)g(the)456 666 y(p)s(oin)m(t)29
b(realizing)g FG(d)1104 680 y Fy(\000)1152 666 y FA(\()p
FG(x)p FA(\).)42 b(By)30 b(means)h(of)f(\(A.90\),)j(w)m(e)e(ha)m(v)m(e)
g(that)456 890 y(\(A.96\))768 b Fx(j)p FG(z)25 b Fx(\000)h
FA(\025)-51 b FG(y)s Fx(j)15 b FG(;)56 b Fx(j)p FG(z)25
b Fx(\000)h FA(~)-51 b FG(y)r Fx(j)26 b(\025)2216 829
y FA(3)p 2261 756 72 4 v FG(C)2333 843 y Fy(0)p 2216
869 157 4 v 2259 953 a FG(R)2397 890 y(:)456 1075 y FA(W)-8
b(e)31 b(claim)f(that)h(this)e(implies)e(that)456 1295
y(\(A.97\))865 b FG(\033)43 b Fx(\025)d FG(d)1827 1309
y Fy(\000)1876 1295 y FA(\()p FG(x)p FA(\))21 b(+)2120
1233 y(3)p 2165 1160 72 4 v FG(C)2237 1247 y Fy(0)p 2120
1274 157 4 v 2163 1357 a FG(R)2301 1295 y(:)456 1480
y FA(T)-8 b(o)35 b(pro)m(v)m(e)g(this,)g(w)m(e)g(distingush)c(three)k
(cases,)i(according)d(to)i(the)e(m)m(utual)g(p)s(osition)456
1588 y(of)27 b FG(x)p FA(,)h FG(z)k FA(and)h(\025)-51
b FG(y)s FA(.)40 b(Namely)-8 b(,)28 b(if)f FG(x)g FA(is)g(outside)f
(\000)i(and)f(outside)f(\006\()p FG(Y)2652 1602 y Fv(!)2703
1588 y FG(;)15 b(R)2812 1602 y Fy(1)2851 1588 y FA(\),)29
b(w)m(e)f(ha)m(v)m(e)h(that)1259 1747 y FG(\033)23 b
Fx(\000)d FG(d)1472 1761 y Fy(\000)1521 1747 y FA(\()p
FG(x)p FA(\))83 b(=)g Fx(j)p FG(x)21 b Fx(\000)f FG(z)t
Fx(j)g(\000)g(j)p FG(d)2323 1761 y Fy(\000)2372 1747
y FA(\()p FG(x)p FA(\))p Fx(j)26 b(\025)1726 1882 y(\025)83
b(j)p FG(x)21 b Fx(\000)f FG(z)t Fx(j)g(\000)g(j)p FG(x)h
Fx(\000)26 b FA(\025)-52 b FG(y)s Fx(j)26 b FA(=)1726
2017 y(=)83 b Fx(j)6 b FA(\025)-51 b FG(y)23 b Fx(\000)d
FG(z)t Fx(j)15 b FG(;)456 2176 y FA(and)35 b(so)h(\(A.97\))i(follo)m
(ws)d(from)h(\(A.96\))i(in)d(this)g(case;)40 b(if,)c(on)g(the)h(other)f
(hand,)g FG(x)g FA(is)456 2284 y(inside)28 b(\000)i(but)f(outside)h
(\006\()p FG(Y)1435 2298 y Fv(!)1485 2284 y FG(;)15 b(R)1594
2298 y Fy(1)1634 2284 y FA(\),)1259 2443 y FG(\033)23
b Fx(\000)d FG(d)1472 2457 y Fy(\000)1521 2443 y FA(\()p
FG(x)p FA(\))83 b(=)g Fx(j)p FG(x)21 b Fx(\000)f FG(z)t
Fx(j)g FA(+)g Fx(j)p FG(d)2323 2457 y Fy(\000)2372 2443
y FA(\()p FG(x)p FA(\))p Fx(j)26 b FA(=)1726 2578 y(=)83
b Fx(j)p FG(x)21 b Fx(\000)f FG(z)t Fx(j)g FA(+)g Fx(j)p
FG(x)h Fx(\000)26 b FA(~)-52 b FG(y)s Fx(j)26 b(\025)1726
2713 y(\025)83 b(j)6 b FA(~)-51 b FG(y)23 b Fx(\000)d
FG(z)t Fx(j)15 b FG(;)456 2873 y FA(whic)m(h)28 b(giv)m(es)h(\(A.97\))j
(via)d(\(A.96\))i(in)d(this)g(case.)42 b(The)29 b(case)h(in)e(whic)m(h)
g FG(x)i FA(is)e(outside)h(\000)456 2981 y(and)j(inside)g(\006\()p
FG(Y)1051 2995 y Fv(!)1101 2981 y FG(;)15 b(R)1210 2995
y Fy(1)1250 2981 y FA(\))33 b(do)s(es)g(not)h(hold:)46
b(indeed,)33 b(if)f FG(x)h FA(is)g(inside)e(\006\()p
FG(Y)2933 2995 y Fv(!)2983 2981 y FG(;)15 b(R)3092 2995
y Fy(1)3132 2981 y FA(\),)35 b(since,)456 3089 y(due)25
b(to)h(\(A.90\),)j FG(z)h FA(is)25 b(inside)e(\000,)k(then)e
FG(x)h FA(lies,)f(in)g(this)f(case,)k(on)e(the)g(segmen)m(t)h(b)s(et)m
(w)m(een)456 3196 y FG(y)501 3210 y Fv(!)582 3196 y FA(and)j(a)i(p)s
(oin)m(t)e(inside)f(\000)i(\(that)h(is,)e FG(z)36 b FA(itself)7
b(\):)41 b(th)m(us,)32 b(since)e(\000)h(is)f(star-shap)s(ed)h(with)456
3304 y(resp)s(ect)41 b(to)h FG(y)945 3318 y Fv(!)995
3304 y FA(,)i FG(x)d FA(is)f(inside)f(\000.)72 b(Let)42
b(us)e(\014nally)f(consider)h(the)i(case)g(in)e(whic)m(h)f
FG(x)456 3412 y FA(is)g(inside)f(\006\()p FG(Y)979 3426
y Fv(!)1029 3412 y FG(;)15 b(R)1138 3426 y Fy(1)1178
3412 y FA(\))41 b(and)f(inside)e(\000.)70 b(In)40 b(this)f(case,)44
b(note)d(that)47 b(~)-51 b FG(y)43 b FA(cannot)e(lie)e(on)456
3520 y(\006\()p FG(Y)610 3534 y Fv(!)660 3520 y FG(;)15
b(R)769 3534 y Fv(!)820 3520 y FA(\))31 b(\(that)h(is,)37
b(~)-51 b FG(y)30 b Fx(62)c FA(\000)1456 3534 y Fy(3)1495
3520 y FA(\),)32 b(otherwise)37 b(~)-51 b FG(y)34 b FA(w)m(ould)c(b)s
(e)h(on)g(the)g(radius)f(from)g FG(y)3282 3534 y Fv(!)3364
3520 y FA(to)456 3628 y FG(z)k FA(and)c(so)1529 3751
y(~)-52 b FG(y)44 b FA(=)c FG(y)1767 3765 y Fv(!)1837
3751 y FA(+)1938 3689 y FG(r)1979 3703 y Fv(!)p 1938
3730 92 4 v 1943 3813 a FG(r)1984 3827 y Fy(1)2054 3751
y FA(\()p FG(z)25 b Fx(\000)20 b FG(y)2292 3765 y Fv(!)2342
3751 y FA(\))456 3931 y(whic)m(h)29 b(giv)m(es)h(that)1403
4071 y Fx(j)6 b FA(~)-51 b FG(y)1476 4033 y Ft(0)1499
4071 y Fx(j)26 b FA(=)1655 4009 y FG(r)1696 4023 y Fv(!)p
1655 4050 V 1660 4133 a FG(r)1701 4147 y Fy(1)1772 4071
y Fx(j)p FG(z)1843 4033 y Ft(0)1867 4071 y Fx(j)f(\025)2023
4009 y FG(r)2064 4023 y Fv(!)p 2023 4050 V 2028 4133
a FG(r)2069 4147 y Fy(1)2140 4071 y FG(\021)s(q)j(>)d(!)s(q)18
b(;)456 4256 y FA(if)29 b FG(R)i FA(is)e(large,)i(thanks)f(to)h
(\(A.76\),)i(in)c(con)m(trast)j(with)d(the)h(fact)i(that)37
b(~)-51 b FG(y)28 b Fx(2)c FA(\000)3109 4270 y Fy(3)3149
4256 y FA(.)555 4364 y(Also,)j(if)f FG(x)g FA(is)f(inside)f(\006\()p
FG(Y)1435 4378 y Fv(!)1486 4364 y FG(;)15 b(R)1595 4378
y Fy(1)1634 4364 y FA(\))27 b(and)f(inside)e(\000)i(and)32
b(~)-51 b FG(y)29 b FA(lies)c(on)h(\000)2785 4378 y Fy(1)2824
4364 y FA(,)i(then)e(note)h(that)456 4472 y(it)33 b(m)m(ust)g(b)s(e)g
(outside)g(\006\()p FG(Y)1376 4486 y Fv(!)1426 4472 y
FG(;)15 b(R)1535 4486 y Fy(1)1575 4472 y FA(\),)35 b(thanks)e(to)h
(\(A.37\))h(and)e(\(A.76\).)52 b(Th)m(us,)34 b(w)m(e)g(tak)m(e)461
4580 y(~)-50 b FG(z)29 b Fx(2)c FA(\006\()p FG(Y)767
4594 y Fv(!)817 4580 y FG(;)15 b(R)926 4594 y Fy(1)966
4580 y FA(\))31 b(on)f(the)h(segmen)m(t)g(joining)e FG(x)h
FA(and)36 b(~)-51 b FG(y)33 b FA(and)c(w)m(e)i(de\014ne)1498
4763 y(~)-63 b FG(w)43 b FA(:=)d FG(y)1769 4777 y Fv(!)1839
4763 y FA(+)1940 4701 y FG(r)1981 4715 y Fv(!)p 1940
4742 V 1945 4825 a FG(r)1986 4839 y Fy(1)2057 4763 y
FA(\()5 b(~)-50 b FG(z)25 b Fx(\000)19 b FG(y)2294 4777
y Fv(!)2344 4763 y FA(\))c FG(:)456 4967 y FA(Note)31
b(that)50 b(~)-64 b FG(w)28 b Fx(2)d FA(\006\()p FG(Y)1205
4981 y Fv(!)1255 4967 y FG(;)15 b(R)1364 4981 y Fv(!)1415
4967 y FA(\))31 b(and)e(therefore,)i(b)m(y)g(\(A.37\),)1409
5186 y Fx(j)19 b FA(~)-64 b FG(w)23 b Fx(\000)j FA(~)-51
b FG(y)s Fx(j)25 b(\025)40 b FA(const)16 b(\(1)21 b Fx(\000)f
FG(!)s FA(\))2357 5125 y FG(q)2401 5092 y Fy(2)p 2357
5166 84 4 v 2377 5249 a FG(r)2466 5186 y(:)p eop
%%Page: 111 111
111 110 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)
f Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)140 b(111)456
450 y FA(F)-8 b(urthermore,)30 b(b)m(y)g(\(A.76\),)1169
650 y Fx(j)19 b FA(~)-64 b FG(w)23 b Fx(\000)i FA(~)-50
b FG(z)t Fx(j)83 b FA(=)1681 546 y Fu(\014)1681 600 y(\014)1681
655 y(\014)1721 589 y FG(r)1762 603 y Fv(!)p 1721 629
92 4 v 1726 712 a FG(r)1767 726 y Fy(1)1823 650 y FA(\()5
b(~)-50 b FG(z)24 b Fx(\000)c FG(y)2060 664 y Fv(!)2110
650 y FA(\))h Fx(\000)f FA(\()5 b(~)-50 b FG(z)25 b Fx(\000)19
b FG(y)2494 664 y Fv(!)2544 650 y FA(\))2579 546 y Fu(\014)2579
600 y(\014)2579 655 y(\014)2635 650 y FA(=)1527 866 y(=)1681
761 y Fu(\014)1681 816 y(\014)1681 870 y(\014)1721 804
y FG(r)1762 818 y Fv(!)p 1721 845 V 1726 928 a FG(r)1767
942 y Fy(1)1843 866 y Fx(\000)h FA(1)1979 761 y Fu(\014)1979
816 y(\014)1979 870 y(\014)2025 866 y FG(r)2066 880 y
Fy(1)2130 866 y FA(=)1527 1119 y(=)83 b Fx(j)p FG(r)1747
1133 y Fv(!)1818 1119 y Fx(\000)20 b FG(r)1950 1133 y
Fy(1)1989 1119 y Fx(j)26 b(\024)2146 1058 y FA(5)p 2206
985 72 4 v 15 w FG(C)2278 1072 y Fy(0)p 2146 1098 172
4 v 2196 1181 a FG(R)2342 1119 y(:)456 1317 y FA(Therefore,)1154
1490 y FG(\033)d Fx(\000)d FG(d)1367 1504 y Fy(\000)1416
1490 y FA(\()p FG(x)p FA(\))83 b(=)g Fx(\000j)p FG(x)20
b Fx(\000)g FG(z)t Fx(j)h FA(+)f Fx(j)p FG(d)2289 1504
y Fy(\000)2338 1490 y FA(\()p FG(x)p FA(\))p Fx(j)26
b(\025)1621 1625 y(\025)83 b(\000j)p FG(x)20 b Fx(\000)26
b FA(~)-51 b FG(z)t Fx(j)21 b(\000)f(j)p FG(x)g Fx(\000)26
b FA(~)-51 b FG(y)s Fx(j)25 b FA(=)1621 1759 y(=)83 b
Fx(j)6 b FA(~)-51 b FG(y)23 b Fx(\000)j FA(~)-51 b FG(z)5
b Fx(j)25 b(\025)1621 1959 y(\025)83 b(j)6 b FA(~)-51
b FG(y)23 b Fx(\000)39 b FA(~)-64 b FG(w)s Fx(j)20 b(\000)2173
1898 y FA(5)p 2233 1825 72 4 v 15 w FG(C)2305 1912 y
Fy(0)p 2173 1938 172 4 v 2224 2022 a FG(R)2380 1959 y
Fx(\025)1621 2199 y(\025)83 b FA(const)16 b(\(1)21 b
Fx(\000)f FG(!)s FA(\))2310 2137 y FG(q)2354 2104 y Fy(2)p
2310 2178 84 4 v 2317 2261 a FG(R)2423 2199 y Fx(\000)2524
2137 y FA(5)p 2584 2064 72 4 v 15 w FG(C)2656 2151 y
Fy(0)p 2524 2178 172 4 v 2575 2261 a FG(R)2721 2199 y(;)456
2402 y FA(whic)m(h)g(pro)m(v)m(es)j(\(A.97\))g(also)f(in)e(this)h
(case,)k(b)m(y)d(taking)f FG(q)k FA(con)m(v)m(enien)m(tly)d(large)g
(\(p)s(ossibly)456 2513 y(in)29 b(dep)s(endence)g(of)i(1)p
FG(=)p FA(\(1)21 b Fx(\000)f FG(!)s FA(\))31 b(and)p
1738 2440 72 4 v 30 w FG(C)1809 2527 y Fy(0)1849 2513
y FA(\).)555 2621 y(Then,)38 b(to)g(end)f(the)g(pro)s(of)g(of)g
(\(A.97\),)42 b(w)m(e)37 b(only)g(need)g(to)h(consider)e(the)h(case)i
(in)456 2729 y(whic)m(h)33 b FG(x)h FA(is)f(inside)f(\006\()p
FG(Y)1317 2743 y Fv(!)1368 2729 y FG(;)15 b(R)1477 2743
y Fy(1)1516 2729 y FA(\))35 b(and)f(inside)e(\000)i(and)40
b(~)-51 b FG(y)37 b FA(lies)c(on)h(\000)2731 2743 y Fy(2)2770
2729 y FA(.)53 b(F)-8 b(or)35 b(this,)g(let)f(us)456
2837 y(de\014ne)1528 3014 y(~)-51 b FG(x)25 b FA(=)g
FG(x)20 b Fx(\000)1868 2952 y FA(6)p 1913 2879 V FG(C)1985
2966 y Fy(0)p 1868 2993 157 4 v 1911 3076 a FG(R)2085
2952 y(x)g Fx(\000)g FG(y)2293 2966 y Fv(!)p 2060 2993
309 4 v 2060 3076 a Fx(j)p FG(x)g Fx(\000)g FG(y)2293
3090 y Fv(!)2343 3076 y Fx(j)456 3216 y FA(and)29 b(observ)m(e)i(that)
1260 3393 y FG(r)1301 3407 y Fy(1)1356 3393 y FA(sin)1482
3292 y Fu(\020)1537 3393 y Fl(\\)p FA(\()p FG(z)24 b
Fx(\000)c FG(y)1840 3407 y Fv(!)1890 3393 y FG(;)31 b(e)1988
3407 y Fv(N)2055 3393 y FA(\))2090 3292 y Fu(\021)2170
3393 y FA(=)25 b Fx(j)p FG(z)2337 3356 y Ft(0)2361 3393
y Fx(j)g(\025)g FG(\021)s(q)19 b(;)456 3593 y FA(th)m(us,)30
b(recalling)f(\(A.76\),)1209 3766 y Fl(\\)p FA(\()6 b(~)-51
b FG(x)20 b Fx(\000)g FG(y)1518 3780 y Fv(!)1568 3766
y FG(;)31 b(e)1666 3780 y Fv(N)1733 3766 y FA(\))84 b(=)e
Fl(\\)p FA(\()p FG(x)21 b Fx(\000)e FG(y)2314 3780 y
Fv(!)2364 3766 y FG(;)31 b(e)2462 3780 y Fv(N)2530 3766
y FA(\))25 b(=)1852 3901 y(=)82 b Fl(\\)p FA(\()p FG(z)25
b Fx(\000)20 b FG(y)2309 3915 y Fv(!)2359 3901 y FG(;)30
b(e)2456 3915 y Fv(N)2524 3901 y FA(\))25 b Fx(\025)1852
4096 y(\025)2015 4034 y FG(\021)s(q)p 2015 4075 93 4
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5190 y FA(=)25 b FG(!)s(q)p eop
%%Page: 112 112
112 111 bop 456 251 a Fy(112)140 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
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2901 y Fy(2)2714 2934 y FG(q)2758 2901 y Fy(2)p 2134
2975 V 2424 3058 a FG(r)2468 3032 y Fy(2)2807 2868 y
Fu(\023)2894 2996 y Fx(\000)2995 2934 y FA(7)p 3040 2861
72 4 v FG(C)3112 2948 y Fy(0)p 2995 2975 157 4 v 3038
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3552 y(joining)d(~)-51 b FG(y)33 b FA(to)k(~)-51 b FG(x)p
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eop
%%Page: 113 113
113 112 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)
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3348 y Fs(!)1462 3340 y Fy(\))1493 3321 y FA(\()p FG(x)p
FA(\))21 b(+)f FG(r)1768 3335 y Fv(!)1839 3321 y Fx(\000)g
FG(d)1977 3340 y Fy(\006\()p Fv(Y)t(;R)p Fy(\))2205 3321
y FA(\()p FG(x)p FA(\))h Fx(\000)f FG(r)j FA(+)c FG(R)i
Fx(\000)f FG(R)2843 3335 y Fy(1)2908 3321 y FA(=)979
3533 y(=)83 b FG(d)1180 3552 y Fy(\006\()p Fv(Y)1299
3560 y Fs(!)1344 3552 y Fv(;R)1417 3560 y Fs(!)1462 3552
y Fy(\))1493 3533 y FA(\()p FG(x)p FA(\))21 b Fx(\000)f
FG(d)1774 3552 y Fy(\006\()p Fv(Y)t(;R)p Fy(\))2002 3533
y FA(\()p FG(x)p FA(\))h(+)p 2248 3399 72 4 v 2248 3472
a FG(C)2320 3486 y Fy(0)p 2246 3512 116 4 v 2246 3596
a FA(2)p FG(R)2371 3533 y(y)2419 3496 y Fy(2)2416 3556
y Fv(N)7 b Fy(+1)2594 3533 y Fx(\000)2694 3472 y FA(5)p
2739 3399 72 4 v FG(C)2811 3486 y Fy(0)p 2694 3512 157
4 v 2715 3596 a FA(2)p FG(R)2886 3533 y Fx(\025)979 3773
y(\025)83 b FG(d)1180 3791 y Fy(\006\()p Fv(Y)1299 3799
y Fs(!)1344 3791 y Fv(;R)1417 3799 y Fs(!)1462 3791 y
Fy(\))1493 3773 y FA(\()p FG(x)p FA(\))21 b Fx(\000)f
FG(d)1774 3791 y Fy(\006\()p Fv(Y)t(;R)p Fy(\))2002 3773
y FA(\()p FG(x)p FA(\))h Fx(\000)2246 3711 y FA(5)p 2291
3638 72 4 v FG(C)2363 3725 y Fy(0)p 2246 3752 157 4 v
2267 3835 a FA(2)p FG(R)2438 3773 y Fx(\025)979 4012
y(\025)98 b FA(const)1381 3951 y FG(q)1425 3918 y Fy(2)p
1381 3991 84 4 v 1388 4074 a FG(R)1494 4012 y Fx(\000)1595
3951 y FA(5)p 1640 3878 72 4 v FG(C)1712 3965 y Fy(0)p
1595 3991 157 4 v 1616 4074 a FA(2)p FG(R)1787 4012 y(>)25
b FA(0)15 b FG(;)456 4192 y FA(if)29 b FG(q)k FA(is)c(large)i(enough.)
40 b(Th)m(us,)30 b(recalling)f(\(A.75\))j(and)e(\(2.14\),)j(for)d(an)m
(y)h FG(\034)k Fx(2)25 b FA([)p FG(s)3185 4206 y Fv(R)3238
4215 y Fq(1)3277 4192 y FG(;)15 b FA(1],)456 4342 y(\(A.100\))209
b Fx(j)p FG(x)20 b Fx(\000)g FG(y)1196 4356 y Fv(!)1246
4342 y Fx(j)h(\000)e(j)p FG(x)i Fx(\000)f FG(y)s Fx(j)g
FA(+)g FG(R)h Fx(\000)f FG(R)2005 4356 y Fy(1)2070 4342
y FG(>)25 b(H)2242 4356 y Fv(y)2277 4367 y Fs(N)5 b Fq(+1)2411
4356 y Fv(;R)2484 4365 y Fq(1)2522 4342 y FA(\()p FG(\034)10
b FA(\))21 b Fx(\000)f FG(H)2830 4356 y Fv(y)2865 4367
y Fs(N)5 b Fq(+1)2999 4356 y Fv(;R)3077 4342 y FA(\()p
FG(\034)10 b FA(\))15 b FG(:)456 4491 y FA(Let)30 b(no)m(w)911
4646 y FG(\034)35 b FA(:=)26 b FG(g)1151 4660 y Fv(y)1186
4671 y Fs(N)5 b Fq(+1)1320 4660 y Fv(;R)1398 4545 y Fu(\020)1452
4646 y FG(H)1528 4660 y Fy(0)1567 4646 y FA(\()p FG(y)1647
4660 y Fv(N)i Fy(+1)1804 4646 y FA(\))21 b(+)f Fx(j)p
FG(x)g Fx(\000)g FG(y)s Fx(j)g(\000)g FG(R)2393 4545
y Fu(\021)2473 4646 y FA(=)25 b FG(g)2612 4664 y Fj(S)-6
b Fy(\()p Fv(Y)t(;R)p Fy(\))2826 4646 y FA(\()p FG(x)p
FA(\))15 b FG(:)456 4823 y FA(If)25 b FG(\034)35 b(<)25
b(s)756 4837 y Fv(R)809 4846 y Fq(1)847 4823 y FA(,)i(then)e(\(A.99\))j
(holds)c(true)h(and)g(w)m(e)h(are)g(done;)i(if)c FG(\034)36
b Fx(\025)24 b FG(s)2782 4837 y Fv(R)2835 4846 y Fq(1)2874
4823 y FA(,)j(w)m(e)f(infer)e(from)456 4931 y(\(A.100\))32
b(that)714 5080 y FG(H)790 5094 y Fv(y)825 5105 y Fs(N)5
b Fq(+1)960 5094 y Fv(;R)1033 5103 y Fq(1)1071 5080 y
FA(\()p FG(\034)10 b FA(\))84 b FG(<)f Fx(j)p FG(x)20
b Fx(\000)g FG(y)1662 5094 y Fv(!)1712 5080 y Fx(j)g(\000)g(j)p
FG(x)h Fx(\000)f FG(y)s Fx(j)g FA(+)g FG(R)h Fx(\000)f
FG(R)2471 5094 y Fy(1)2530 5080 y FA(+)g FG(H)2697 5094
y Fv(y)2732 5105 y Fs(N)5 b Fq(+1)2866 5094 y Fv(;R)2944
5080 y FA(\()p FG(\034)10 b FA(\))26 b(=)1275 5216 y(=)83
b Fx(j)p FG(x)20 b Fx(\000)g FG(y)1662 5230 y Fv(!)1712
5216 y Fx(j)g(\000)g FG(R)1917 5230 y Fy(1)1977 5216
y FA(+)g FG(H)2144 5230 y Fy(0)2183 5216 y FA(\()p FG(y)2263
5230 y Fv(N)7 b Fy(+1)2420 5216 y FA(\))15 b FG(;)p eop
%%Page: 114 114
114 113 bop 456 251 a Fy(114)140 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(hence,)768 631 y FG(\034)93 b(<)83
b(g)1098 645 y Fv(y)1133 656 y Fs(N)5 b Fq(+1)1268 645
y Fv(;R)1341 654 y Fq(1)1380 530 y Fu(\020)1434 631 y
Fx(j)p FG(x)20 b Fx(\000)g FG(y)1667 645 y Fv(!)1717
631 y Fx(j)h(\000)f FG(R)1923 645 y Fy(1)1982 631 y FA(+)g
FG(H)2149 645 y Fy(0)2188 631 y FA(\()p FG(y)2268 645
y Fv(N)7 b Fy(+1)2425 631 y FA(\))2460 530 y Fu(\021)2540
631 y FA(=)25 b FG(g)2679 649 y Fj(S)-5 b Fy(\()p Fv(Y)2791
657 y Fs(!)2830 649 y Fv(;R)2903 658 y Fq(1)2937 649
y Fy(\))2969 631 y FA(\()p FG(x)p FA(\))15 b FG(:)456
816 y FA(F)-8 b(rom)35 b(this)f(and)h(the)g(de\014nition)e(of)i
FG(\034)10 b FA(,)36 b(\(A.99\))h(follo)m(ws.)54 b(W)-8
b(e)37 b(ha)m(v)m(e)f(therefore)f(com-)456 924 y(pleted)29
b(the)i(pro)s(of)f(of)g(\(A.98\).)555 1091 y(W)-8 b(e)40
b(no)m(w)e(consider)f(the)h(domain)f Fx(fj)p FG(x)1896
1058 y Ft(0)1920 1091 y Fx(j)i(\024)2113 1068 y FA(\025)2093
1091 y FG(C)6 b(q)s Fx(g)26 b(\002)f(fj)p FG(x)2497 1105
y Fv(N)2565 1091 y Fx(j)39 b(\024)f FG(l)r(=)p FA(2)p
Fx(g)h FA(and)e(w)m(e)i(slide)456 1199 y(\011)45 b(from)f
Fx(\0001)h FA(in)f(the)i FG(e)1343 1213 y Fv(N)1455 1199
y FA(direction,)i(un)m(til)c(w)m(e)i(touc)m(h)f FG(u)h
FA(for)f(the)g(\014rst)g(time)g(in)456 1307 y(suc)m(h)c(domain)g
(\(this)g(m)m(ust)g(happ)s(en)f(since)h Fx(j)p FG(u)p
FA(\()p FG(x)2197 1321 y Fy(0)2237 1307 y FA(\))p Fx(j)k
FG(<)f FA(1)p FG(=)p FA(2\):)65 b(sa)m(y)-8 b(,)46 b(for)c(\014xing)e
(the)456 1415 y(notations,)27 b(that)g(for)e(some)i FG(\014)j
Fx(2)25 b Fw(R)s FA(,)33 b(\011)12 b Fx(\000)g FG(\014)5
b(e)1977 1429 y Fv(N)2069 1415 y FA(touc)m(hes)27 b(for)f(the)h
(\014rst)e(time)h(the)g(graph)456 1523 y(of)32 b FG(u)h
FA(b)m(y)g(ab)s(o)m(v)m(e)h(at)f(a)g(p)s(oin)m(t)f FG(Z)7
b FA(.)47 b(Notice)34 b(that,)g(b)m(y)e(the)h(h)m(yp)s(otheses)g(of)g
(Lemma)g(A.1)456 1631 y(and)c(\(A.42\),)k(w)m(e)e(ha)m(v)m(e)g(that)456
1789 y(\(A.101\))606 b FG(u)p FA(\()p FG(x)1499 1803
y Fy(0)1538 1789 y FA(\))26 b(=)f FG(g)1738 1807 y Fj(S)-6
b Fy(\()p Fv(Y)t(;R)p Fy(\))1952 1789 y FA(\()p FG(x)2039
1803 y Fy(0)2079 1789 y FA(\))26 b(=)f FG(g)2279 1803
y Fy(\011)2338 1789 y FA(\()p FG(x)2425 1803 y Fy(0)2465
1789 y FA(\))15 b FG(;)456 1946 y FA(therefore)30 b(w)m(e)h(ha)m(v)m(e)
h(that)f FG(\014)f Fx(\025)25 b FA(0.)41 b(More)31 b(precisely)-8
b(,)30 b(it)g(holds)f(that)456 2104 y(\(A.102\))1049
b FG(\014)46 b(>)40 b FA(0)15 b FG(:)456 2262 y FA(Indeed,)33
b(if,)g(b)m(y)g(con)m(tradiction,)h FG(\014)i FA(=)29
b(0,)35 b(\(A.101\))h(sa)m(ys)e(that)f(\011)g(touc)m(hes)h(the)g(graph)
456 2370 y(of)28 b FG(u)g FA(from)g(ab)s(o)m(v)m(e)i(at)f
FG(x)1267 2384 y Fy(0)1334 2370 y FA(\(whic)m(h)f(is,)g(b)m(y)g
(construction,)g(an)g(in)m(terior)g(p)s(oin)m(t\):)39
b(thence,)456 2478 y(b)m(y)27 b(\(A.60\),)j FG(d)934
2492 y Fy(\000)983 2478 y FA(\()p FG(x)1070 2492 y Fy(0)1110
2478 y FA(\))d(m)m(ust)h(b)s(e)e(realized)h(on)g(\000)2026
2492 y Fy(1)2080 2478 y Fx([)14 b FA(\000)2212 2492 y
Fy(3)2251 2478 y FA(.)40 b(But,)29 b(b)m(y)e(construction,)h
FG(d)3234 2492 y Fy(\000)3282 2478 y FA(\()p FG(x)3369
2492 y Fy(0)3409 2478 y FA(\))456 2586 y(is)h(realized)h(at)h
FG(T)1045 2600 y Fv(Y)t(;R)1167 2586 y FA(\()p FG(x)1254
2600 y Fy(0)1294 2586 y FA(\))25 b Fx(2)g FA(\000)1497
2600 y Fy(2)1536 2586 y FA(:)41 b(this)29 b(con)m(tradiction)i(pro)m(v)
m(es)g(\(A.102\).)555 2744 y(Note)h(no)m(w)e(that,)i(if)d
Fi(x)h FA(is)g(in)f(the)h(domain)g(of)g(\011,)h(then,)f(b)m(y)g
(\(2.16\),)j(w)m(e)e(ha)m(v)m(e)g(that)507 2902 y FG(d)554
2916 y Fy(\000)603 2902 y FA(\()p Fi(x)p FA(\))26 b Fx(\024)f
FG(H)903 2916 y Fv(y)938 2927 y Fs(N)5 b Fq(+1)1072 2916
y Fv(;R)1150 2902 y FA(\(1\))21 b Fx(\000)f FG(H)1453
2916 y Fv(y)1488 2927 y Fs(N)5 b Fq(+1)1622 2916 y Fv(;R)1699
2902 y FA(\(0\))27 b Fx(\024)40 b FA(const)16 b(\(1)21
b(+)f(log)c FG(R)q FA(\))26 b Fx(\024)40 b FA(const)15
b(\(1)21 b(+)f(log)d FG(l)r FA(\))e FG(:)456 3060 y FA(So,)30
b(if)f Fi(x)722 3074 y Fv(N)815 3060 y Fx(\025)c FA(1,)31
b(the)g(fact)g(that)g(\000)f(is)g(b)s(elo)m(w)f Fx(f)p
FG(x)2078 3074 y Fv(N)2171 3060 y FA(=)c(1)p Fx(g)31
b FA(implies)d(that)1662 3218 y FG(d)1709 3232 y Fy(\000)1758
3218 y FA(\()p Fi(x)p FA(\))e Fx(\025)f Fi(x)2014 3232
y Fv(N)2102 3218 y Fx(\000)19 b FA(1)456 3376 y(and)29
b(so)1506 3496 y Fi(x)1538 3510 y Fv(N)1631 3496 y Fx(\024)40
b FA(const)16 b(\(1)21 b(+)f(log)c FG(l)r FA(\))f FG(:)456
3635 y FA(This)28 b(sa)m(ys)j(that)g(the)g(domain)e(of)h(\011)g(is)g(b)
s(elo)m(w)f(the)i(h)m(yp)s(erplane)1456 3715 y Fu(n)1516
3816 y FG(x)1568 3830 y Fv(N)1661 3816 y FA(=)40 b(const)16
b(\(1)21 b(+)f(log)c FG(l)r FA(\))2383 3715 y Fu(o)456
3996 y FA(with)30 b(resp)s(ect)h(to)h(the)g FG(e)1288
4010 y Fv(N)1356 3996 y FA(-direction.)43 b(Since)30
b FG(\014)i Fx(\025)27 b FA(0,)32 b(also)g(the)f(domain)g(of)g(\011)21
b Fx(\000)g FG(\014)5 b(e)3377 4010 y Fv(N)456 4104 y
FA(is)29 b(b)s(elo)m(w)h(the)g(h)m(yp)s(erplane)456 4284
y(\(A.103\))1436 4184 y Fu(n)1496 4284 y FG(x)1548 4298
y Fv(N)1641 4284 y FA(=)40 b(const)16 b(\(1)21 b(+)e(log)e
FG(l)r FA(\))2363 4184 y Fu(o)2439 4284 y FG(:)555 4600
y FA(What)31 b(is)f(more,)g(since)g Fx(j)p FG(z)1444
4614 y Fv(N)1512 4600 y Fx(j)25 b(\024)g FG(l)r FA(,)31
b(b)m(y)f(\(A.21\),)j(w)m(e)e(gather)g(that)456 4758
y(\(A.104\))744 b FG(z)1540 4772 y Fv(N)1632 4758 y Fx(\025)25
b(\000)p FG(l)i Fx(\025)e(\000)p FG(cR)h Fx(\025)f FG(y)2295
4772 y Fv(N)2377 4758 y FG(;)456 4916 y FA(pro)m(vided)k
FG(c)h FA(in)f(\(A.21\))k(is)c(small)g(enough.)40 b(Thence,)31
b(from)f(\(A.102\))j(and)c(\(A.104\),)918 5077 y Fx(j)p
FG(z)c FA(+)20 b FG(\014)5 b(e)1199 5091 y Fv(N)1287
5077 y Fx(\000)20 b FG(y)s Fx(j)1451 5040 y Fy(2)1574
5077 y FA(=)82 b Fx(j)p FG(z)25 b Fx(\000)20 b FG(y)s
Fx(j)1983 5040 y Fy(2)2043 5077 y FA(+)g FG(\014)2190
5040 y Fy(2)2249 5077 y FA(+)g(2)p FG(\014)h FA(\()p
FG(z)2534 5091 y Fv(N)2622 5077 y Fx(\000)f FG(y)2758
5091 y Fv(N)2825 5077 y FA(\))26 b FG(>)1574 5216 y(>)82
b Fx(j)p FG(z)25 b Fx(\000)20 b FG(y)s Fx(j)1983 5178
y Fy(2)2038 5216 y FG(;)p eop
%%Page: 115 115
115 114 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)
f Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)140 b(115)456
450 y FA(that)31 b(is)456 606 y(\(A.105\))706 b Fx(j)p
FG(z)25 b Fx(\000)20 b FG(y)s Fx(j)40 b FG(<)g Fx(j)p
FG(z)25 b FA(+)20 b FG(\014)5 b(e)2148 620 y Fv(N)2236
606 y Fx(\000)20 b FG(y)s Fx(j)15 b FG(:)456 762 y FA(W)-8
b(e)31 b(no)m(w)g(pro)m(v)m(e)g(that)456 922 y(\(A.106\))982
b Fx(j)p FG(z)1807 884 y Ft(0)1831 922 y Fx(j)41 b FG(<)2029
899 y FA(\025)2008 922 y FG(C)6 b(q)18 b(:)456 1078 y
FA(Indeed,)29 b(if)h(\(A.106\))i(w)m(ere)f(false,)g(then)f(\(A.74\))i
(and)e(\(A.105\))j(w)m(ould)c(yield)f(that)855 1233 y
FG(u)p FA(\()p FG(z)t FA(\))85 b(=)d FG(g)1304 1247 y
Fy(\011)1364 1233 y FA(\()p FG(z)24 b FA(+)c FG(\014)5
b(e)1654 1247 y Fv(N)1722 1233 y FA(\))26 b(=)1108 1368
y(=)82 b FG(g)1304 1387 y Fj(S)-6 b Fy(\()p Fv(Y)t(;R)p
Fy(\))1519 1368 y FA(\()p FG(z)24 b FA(+)c FG(\014)5
b(e)1809 1382 y Fv(N)1877 1368 y FA(\))26 b(=)1108 1543
y(=)82 b FG(g)1304 1557 y Fv(y)1339 1568 y Fs(N)5 b Fq(+1)1474
1557 y Fv(;R)1552 1442 y Fu(\020)1606 1543 y Fx(j)p FG(z)25
b FA(+)19 b FG(\014)5 b(e)1886 1557 y Fv(N)1974 1543
y Fx(\000)20 b FG(y)s Fx(j)h(\000)f FG(R)g FA(+)g FG(H)2506
1557 y Fv(y)2541 1568 y Fs(N)5 b Fq(+1)2676 1557 y Fv(;R)2753
1543 y FA(\(0\))2868 1442 y Fu(\021)2949 1543 y FG(>)1108
1742 y(>)82 b(g)1304 1756 y Fv(y)1339 1767 y Fs(N)5 b
Fq(+1)1474 1756 y Fv(;R)1552 1641 y Fu(\020)1606 1742
y Fx(j)p FG(z)25 b Fx(\000)19 b FG(y)s Fx(j)i(\000)f
FG(R)h FA(+)e FG(H)2229 1756 y Fv(y)2264 1767 y Fs(N)5
b Fq(+1)2399 1756 y Fv(;R)2476 1742 y FA(\(0\))2591 1641
y Fu(\021)2672 1742 y FA(=)1108 1905 y(=)82 b FG(g)1304
1923 y Fj(S)-6 b Fy(\()p Fv(Y)t(;R)p Fy(\))1519 1905
y FA(\()p FG(z)t FA(\))26 b Fx(\025)1108 2043 y(\025)82
b FG(u)p FA(\()p FG(z)t FA(\))15 b FG(:)456 2199 y FA(This)28
b(con)m(tradiction)i(pro)m(v)m(es)i(\(A.106\).)555 2355
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b(w)m(e)c(ha)m(v)m(e)g(that)g FG(z)j FA(lies)42 b(in)h(the)456
2472 y(in)m(terior)29 b(of)h Fx(fj)p FG(x)1002 2439 y
Ft(0)1026 2472 y Fx(j)c(\024)1193 2449 y FA(\025)1173
2472 y FG(C)6 b(q)s Fx(g)21 b(\002)f(fj)p FG(x)1567 2486
y Fv(N)1635 2472 y Fx(j)25 b(\024)g FG(l)r(=)p FA(2)p
Fx(g)p FA(.)555 2628 y(W)-8 b(e)32 b(no)m(w)e(claim)g(that)456
2784 y(\(A.107\))408 b Fx(j)p FG(z)1229 2798 y Fv(N)7
b Fy(+1)1387 2784 y Fx(j)25 b FG(<)g FA(1)p FG(=)p FA(2)32
b(and)e FG(d)1924 2798 y Fy(\000)1972 2784 y FA(\()p
FG(z)25 b FA(+)20 b FG(\014)5 b(e)2263 2798 y Fv(N)2331
2784 y FA(\))30 b(is)g(realized)f(on)i(\000)3005 2798
y Fy(3)3044 2784 y FA(.)456 2940 y(Indeed,)g(since)h(\011)g(touc)m(hes)
h(b)m(y)f(ab)s(o)m(v)m(e)h FG(u)21 b FA(+)h FG(\014)5
b(e)2085 2954 y Fv(N)2184 2940 y FA(at)p 2297 2890 47
4 v 33 w FG(z)33 b FA(:=)28 b FG(z)d FA(+)c FG(\014)5
b(e)2753 2954 y Fv(N)2821 2940 y FA(,)33 b(\(A.60\))h(implies)456
3048 y(that)1508 3166 y(1)p FG(=)p FA(2)27 b FG(>)e Fx(j)p
1791 3116 V FG(z)1838 3180 y Fv(N)7 b Fy(+1)1995 3166
y Fx(j)25 b FA(=)g Fx(j)p FG(z)2208 3180 y Fv(N)7 b Fy(+1)2366
3166 y Fx(j)456 3303 y FA(and)22 b(that)i FG(d)862 3317
y Fy(\000)911 3303 y FA(\()p 946 3253 V FG(z)t FA(\))g(is)e(realized)g
(on)h(\000)1637 3317 y Fy(1)1683 3303 y Fx([)6 b FA(\000)1807
3317 y Fy(3)1845 3303 y FA(.)39 b(T)-8 b(o)23 b(pro)m(v)m(e)h
(\(A.107\),)k(w)m(e)23 b(th)m(us)g(ha)m(v)m(e)i(to)e(sho)m(w)456
3411 y(that)28 b FG(d)697 3425 y Fy(\000)746 3411 y FA(\()p
781 3361 V FG(z)t FA(\))h(is)e(not)h(realized)f(on)h(\000)1651
3425 y Fy(1)1690 3411 y FA(.)40 b(If,)28 b(b)m(y)g(con)m(tradiction,)g
FG(d)2616 3425 y Fy(\000)2665 3411 y FA(\()p 2700 3361
V FG(z)5 b FA(\))28 b(w)m(ere)g(realized)f(on)456 3519
y(\000)513 3533 y Fy(1)552 3519 y FA(,)j(from)g(\(A.42\))j(and)c
(\(A.105\),)k(w)m(e)e(w)m(ould)e(ha)m(v)m(e)j(that)855
3674 y FG(u)p FA(\()p FG(z)t FA(\))85 b(=)d FG(g)1304
3688 y Fy(\011)1364 3674 y FA(\()p FG(z)24 b FA(+)c FG(\014)5
b(e)1654 3688 y Fv(N)1722 3674 y FA(\))26 b(=)1108 3809
y(=)82 b FG(g)1304 3828 y Fj(S)-6 b Fy(\()p Fv(Y)t(;R)p
Fy(\))1519 3809 y FA(\()p FG(z)24 b FA(+)c FG(\014)5
b(e)1809 3823 y Fv(N)1877 3809 y FA(\))26 b(=)1108 3984
y(=)82 b FG(g)1304 3998 y Fv(y)1339 4009 y Fs(N)5 b Fq(+1)1474
3998 y Fv(;R)1552 3883 y Fu(\020)1606 3984 y Fx(j)p FG(z)25
b FA(+)19 b FG(\014)5 b(e)1886 3998 y Fv(N)1974 3984
y Fx(\000)20 b FG(y)s Fx(j)h(\000)f FG(R)g FA(+)g FG(H)2506
3998 y Fv(y)2541 4009 y Fs(N)5 b Fq(+1)2676 3998 y Fv(;R)2753
3984 y FA(\(0\))2868 3883 y Fu(\021)2949 3984 y FG(>)1108
4183 y(>)82 b(g)1304 4197 y Fv(y)1339 4208 y Fs(N)5 b
Fq(+1)1474 4197 y Fv(;R)1552 4082 y Fu(\020)1606 4183
y Fx(j)p FG(z)25 b Fx(\000)19 b FG(y)s Fx(j)i(\000)f
FG(R)h FA(+)e FG(H)2229 4197 y Fv(y)2264 4208 y Fs(N)5
b Fq(+1)2399 4197 y Fv(;R)2476 4183 y FA(\(0\))2591 4082
y Fu(\021)2672 4183 y FA(=)1108 4346 y(=)82 b FG(g)1304
4364 y Fj(S)-6 b Fy(\()p Fv(Y)t(;R)p Fy(\))1519 4346
y FA(\()p FG(z)t FA(\))26 b Fx(\025)1108 4484 y(\025)82
b FG(u)p FA(\()p FG(z)t FA(\))15 b FG(:)456 4640 y FA(This)28
b(con)m(tradiction)i(completes)h(the)g(pro)s(of)e(of)i(\(A.107\).)555
4796 y(W)-8 b(e)32 b(no)m(w)e(claim)g(that)456 4952 y(\(A.108\))840
b Fx(j)p FG(Y)40 b Fx(\000)20 b FG(Y)1856 4966 y Fv(!)1906
4952 y Fx(j)h(\000)f FG(\014)30 b Fx(\025)25 b FA(0)15
b FG(:)456 5108 y FA(In)38 b(order)g(to)h(get)h(this,)g(\014rst)e
(observ)m(e)h(that)h(if)d FG(\014)45 b Fx(\024)39 b FA(5)p
FG(l)r FA(,)i(then)d(\(A.108\))k(follo)m(ws)37 b(b)m(y)456
5216 y(noticing)20 b(that)i Fx(j)p FG(Y)g Fx(\000)r FG(Y)1209
5230 y Fv(!)1259 5216 y Fx(j)k(\025)40 b FA(const)16
b FG(r)23 b FA(and)e(b)m(y)g(taking)g FG(l)r(=R)i FA(suitably)c(small.)
36 b(Therefore,)p eop
%%Page: 116 116
116 115 bop 456 251 a Fy(116)140 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(w)m(e)23 b(restrict)g(ourselv)m(es)g(to)h(the)g(pro)
s(of)e(of)i(\(A.108\))i(under)21 b(the)j(additional)d(assumption)456
558 y(that)456 762 y(\(A.109\))1035 b FG(\014)45 b Fx(\025)40
b FA(5)p FG(l)18 b(:)456 967 y FA(F)-8 b(or)31 b(this)e(scop)s(e,)i
(note)g(that)g(\(A.109\))i(implies)27 b(that)1412 1171
y(\()p FG(z)e FA(+)20 b FG(\014)5 b(e)1703 1185 y Fv(N)1771
1171 y FA(\))1806 1185 y Fv(N)1914 1171 y Fx(\025)40
b FG(\014)25 b Fx(\000)20 b FG(l)43 b Fx(\025)d FA(4)p
FG(l)17 b(;)456 1376 y FA(hence,)30 b(recalling)f(\(A.107\),)456
1580 y(\(A.110\))495 b FG(d)1296 1594 y Fy(\000)1340
1603 y Fq(3)1379 1580 y FA(\()p FG(z)25 b FA(+)20 b FG(\014)5
b(e)1670 1594 y Fv(N)1737 1580 y FA(\))41 b(=)f FG(d)1971
1594 y Fy(\000)2020 1580 y FA(\()p FG(z)25 b FA(+)20
b FG(\014)5 b(e)2311 1594 y Fv(N)2378 1580 y FA(\))41
b FG(>)f FA(0)15 b FG(:)456 1784 y FA(Let)30 b(us)g(observ)m(e)h(no)m
(w)g(that,)g(b)m(y)f(\(A.107\),)1314 2016 y FG(g)1357
2030 y Fv(y)1392 2041 y Fs(N)5 b Fq(+1)1527 2030 y Fv(;R)1605
1915 y Fu(\020)1659 2016 y FG(H)1735 2030 y Fy(0)1774
2016 y FA(\()p FG(y)1854 2030 y Fv(N)i Fy(+1)2011 2016
y FA(\))21 b(+)f Fx(j)p FG(z)k Fx(\000)c FG(y)s Fx(j)h(\000)e
FG(R)2594 1915 y Fu(\021)2674 2016 y FA(=)1160 2179 y(=)83
b FG(g)1357 2197 y Fj(S)-6 b Fy(\()p Fv(Y)t(;R)p Fy(\))1572
2179 y FA(\()p FG(z)t FA(\))26 b Fx(\025)1160 2317 y(\025)83
b FG(u)p FA(\()p FG(z)t FA(\))27 b(=)1160 2452 y(=)83
b FG(g)1357 2466 y Fy(\011)1417 2452 y FA(\()p FG(z)24
b FA(+)c FG(\014)5 b(e)1707 2466 y Fv(N)1775 2452 y FA(\))26
b(=)1160 2615 y(=)83 b FG(g)1357 2629 y Fv(y)1392 2640
y Fs(N)5 b Fq(+1)1527 2629 y Fv(;R)1605 2514 y Fu(\020)1659
2615 y FG(H)1735 2629 y Fy(0)1774 2615 y FA(\()p FG(y)1854
2629 y Fv(N)i Fy(+1)2011 2615 y FA(\))21 b(+)f FG(d)2205
2629 y Fy(\000)2253 2615 y FA(\()p FG(z)25 b FA(+)20
b FG(\014)5 b(e)2544 2629 y Fv(N)2612 2615 y FA(\))2647
2514 y Fu(\021)2727 2615 y FA(=)1160 2814 y(=)83 b FG(g)1357
2828 y Fv(y)1392 2839 y Fs(N)5 b Fq(+1)1527 2828 y Fv(;R)1605
2713 y Fu(\020)1659 2814 y FG(H)1735 2828 y Fy(0)1774
2814 y FA(\()p FG(y)1854 2828 y Fv(N)i Fy(+1)2011 2814
y FA(\))21 b(+)f FG(d)2205 2828 y Fy(\000)2249 2837 y
Fq(3)2288 2814 y FA(\()p FG(z)25 b FA(+)19 b FG(\014)5
b(e)2578 2828 y Fv(N)2646 2814 y FA(\))2681 2713 y Fu(\021)2751
2814 y FG(;)456 3041 y FA(thence)456 3245 y(\(A.111\))620
b Fx(j)p FG(z)24 b Fx(\000)c FG(y)s Fx(j)h(\000)f FG(R)41
b Fx(\025)f FG(d)2009 3259 y Fy(\000)2053 3268 y Fq(3)2092
3245 y FA(\()p FG(z)25 b FA(+)20 b FG(\014)5 b(e)2383
3259 y Fv(N)2450 3245 y FA(\))15 b FG(:)456 3450 y FA(If)30
b(w)m(e)g(no)m(w)h(tak)m(e)h(an)m(y)e FG(x)c Fx(2)e FA(\000)1458
3464 y Fy(3)1498 3450 y FA(,)30 b(w)m(e)h(ha)m(v)m(e)h(that)f
FG(x)2146 3464 y Fv(N)2238 3450 y Fx(\024)25 b FG(l)32
b FA(and)e(th)m(us)1421 3654 y(\()p FG(z)25 b FA(+)20
b FG(\014)5 b(e)1712 3668 y Fv(N)1800 3654 y Fx(\000)20
b FG(x)p FA(\))1978 3668 y Fv(N)2086 3654 y Fx(\025)40
b FG(\014)25 b Fx(\000)20 b FA(2)p FG(l)e(:)456 3858
y FA(This)28 b(and)i(\(A.110\))j(imply)28 b(that)1471
4063 y FG(d)1518 4077 y Fy(\000)1562 4086 y Fq(3)1601
4063 y FA(\()p FG(z)d FA(+)20 b FG(\014)5 b(e)1892 4077
y Fv(N)1960 4063 y FA(\))41 b Fx(\025)f FG(\014)25 b
Fx(\000)20 b FA(2)p FG(l)e(:)456 4267 y FA(F)-8 b(or)31
b(this)e(reason,)i(recalling)e(\(A.111\),)k(w)m(e)e(deduce)f(that)1191
4472 y FG(\014)88 b Fx(\024)83 b FA(2)p FG(l)22 b FA(+)e
FG(d)1716 4486 y Fy(\000)1760 4495 y Fq(3)1799 4472 y
FA(\()p FG(z)25 b FA(+)20 b FG(\014)5 b(e)2090 4486 y
Fv(N)2158 4472 y FA(\))26 b Fx(\024)1330 4606 y(\024)83
b FA(2)p FG(l)22 b FA(+)e Fx(j)p FG(z)25 b Fx(\000)20
b FG(y)s Fx(j)g(\000)g FG(R)26 b FA(=)1330 4741 y(=)83
b(2)p FG(l)22 b FA(+)e(dist)o(\()p FG(z)t(;)15 b(@)5
b(B)2059 4755 y Fv(r)2099 4741 y FA(\()p FG(y)s FA(\)\))21
b(+)f FG(r)i Fx(\000)e FG(R)26 b Fx(\024)1330 4876 y(\024)83
b FA(2)p FG(l)22 b FA(+)e Fx(j)p FG(z)25 b Fx(\000)20
b FG(T)1905 4890 y Fv(y)r(;R)2020 4876 y FG(x)2072 4890
y Fy(0)2111 4876 y Fx(j)h FA(+)e FG(r)k Fx(\000)d FG(R)26
b Fx(\024)1330 5011 y(\024)83 b FA(5)p FG(l)17 b(:)456
5216 y FA(This)28 b(ends)i(the)g(pro)s(of)g(of)g(\(A.108\),)k(b)m(y)c
(taking)g FG(l)r(=R)i FA(suitably)c(small.)p eop
%%Page: 117 117
117 116 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)
f Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)140 b(117)555
450 y FA(Let)27 b(us)e(no)m(w)h(consider,)g(for)f FG(t)g
Fx(\025)g FA(0,)j(the)e(surface)g Fw(S)n FA(\()p FG(Y)35
b FA(+)20 b FG(te)2572 464 y Fv(N)2639 450 y FG(;)15
b(R)2748 464 y Fy(1)2788 450 y FA(\).)39 b(F)-8 b(or)27
b FG(t)e FA(=)g(0,)i(this)456 558 y(surface)j(is)f(ab)s(o)m(v)m(e)j
Fw(S)n FA(\()p FG(Y)5 b(;)15 b(R)q FA(\))25 b(since,)30
b(thanks)g(to)h(\(A.75\))i(and)d(\(2.14\),)875 765 y
FG(g)918 783 y Fj(S)-6 b Fy(\()p Fv(Y)t(;R)p Fy(\))1132
765 y FA(\()p FG(x)p FA(\))84 b(=)e FG(g)1534 779 y Fv(y)1569
790 y Fs(N)5 b Fq(+1)1704 779 y Fv(;R)1782 664 y Fu(\020)1836
765 y FG(H)1912 779 y Fy(0)1951 765 y FA(\()p FG(y)2031
779 y Fv(N)i Fy(+1)2188 765 y FA(\))21 b(+)f Fx(j)p FG(x)g
Fx(\000)g FG(y)s Fx(j)g(\000)g FG(R)2777 664 y Fu(\021)2856
765 y Fx(\024)1338 964 y(\024)82 b FG(g)1534 978 y Fv(y)1569
989 y Fs(N)5 b Fq(+1)1704 978 y Fv(;R)1782 863 y Fu(\020)1836
964 y FG(H)1912 978 y Fy(0)1951 964 y FA(\()p FG(y)2031
978 y Fv(N)i Fy(+1)2188 964 y FA(\))21 b(+)f Fx(j)p FG(x)g
Fx(\000)g FG(y)s Fx(j)g(\000)g FG(R)2776 978 y Fy(1)2816
863 y Fu(\021)2895 964 y Fx(\024)1338 1163 y(\024)82
b FG(g)1534 1177 y Fv(y)1569 1188 y Fs(N)5 b Fq(+1)1704
1177 y Fv(;R)1777 1186 y Fq(1)1816 1062 y Fu(\020)1870
1163 y FG(H)1946 1177 y Fy(0)1985 1163 y FA(\()p FG(y)2065
1177 y Fv(N)i Fy(+1)2222 1163 y FA(\))21 b(+)f Fx(j)p
FG(x)g Fx(\000)g FG(y)s Fx(j)g(\000)g FG(R)2810 1177
y Fy(1)2850 1062 y Fu(\021)2929 1163 y FA(=)1338 1325
y(=)82 b FG(g)1534 1344 y Fj(S)-5 b Fy(\()p Fv(Y)t(;)o(R)1722
1353 y Fq(1)1751 1344 y Fy(\))1783 1325 y FA(\()p FG(x)p
FA(\))15 b FG(:)456 1526 y FA(Since)34 b Fw(S)o FA(\()p
FG(Y)5 b(;)15 b(R)q FA(\))30 b(is)k(ab)s(o)m(v)m(e)j(the)e(graph)g(of)h
FG(u)f FA(b)m(y)h(our)f(h)m(yp)s(otheses)g(in)f Fx(fj)p
FG(x)3002 1493 y Ft(0)3026 1526 y Fx(j)g(\024)3210 1503
y FA(\025)3189 1526 y FG(C)7 b(q)s Fx(g)24 b(\002)456
1634 y(fj)p FG(x)578 1648 y Fv(N)646 1634 y Fx(j)32 b(\024)h
FG(l)r(=)p FA(2)p Fx(g)p FA(,)k(w)m(e)e(th)m(us)f(deduce)h(that)g
Fw(S)o FA(\()p FG(Y)f FA(+)20 b FG(te)2229 1648 y Fv(N)2297
1634 y FG(;)15 b(R)2406 1648 y Fy(1)2445 1634 y FA(\))35
b(is,)h(for)e FG(t)e FA(=)h(0,)j(ab)s(o)m(v)m(e)g(the)456
1751 y(graph)h(of)i FG(u)f FA(in)g Fx(fj)p FG(x)1160
1718 y Ft(0)1183 1751 y Fx(j)h(\024)1377 1728 y FA(\025)1357
1751 y FG(C)6 b(q)s Fx(g)26 b(\002)f(fj)p FG(x)1761 1765
y Fv(N)1829 1751 y Fx(j)39 b(\024)f FG(l)r(=)p FA(2)p
Fx(g)p FA(.)67 b(Hence,)41 b(w)m(e)e(ma)m(y)g(increase)f
FG(t)g FA(till)456 1868 y(w)m(e)33 b(touc)m(h)g(the)g(graph)f(of)h
FG(u)f FA(in)f Fx(fj)p FG(x)1683 1835 y Ft(0)1707 1868
y Fx(j)f(\024)1882 1845 y FA(\025)1861 1868 y FG(C)7
b(q)s Fx(g)22 b(\002)f(fj)p FG(x)2258 1882 y Fv(N)2326
1868 y Fx(j)29 b(\024)g FG(l)r(=)p FA(2)p Fx(g)p FA(.)49
b(In)32 b(order)g(to)h(\014x)f(the)456 1976 y(notation,)d(sa)m(y)g
(this)f(happ)s(en)f(for)h FG(t)e FA(=)f FG(t)1818 1990
y Fy(1)1882 1976 y Fx(\025)g FA(0)k(and)f(let)h FG(X)2432
1990 y Fy(1)2500 1976 y FA(b)s(e)f(the)h(ab)s(o)m(v)m(e)h(men)m(tioned)
456 2084 y(touc)m(hing)g(p)s(oin)m(t.)40 b(Set)30 b(also)1612
2225 y FG(Y)1665 2239 y Fy(1)1745 2225 y FA(:=)40 b FG(Y)h
FA(+)19 b FG(t)2098 2239 y Fy(1)2138 2225 y FG(e)2180
2239 y Fv(N)2262 2225 y FG(:)456 2385 y FA(Let)30 b(also)456
2564 y(\(A.112\))375 b(~)-49 b FG(r)1166 2578 y Fy(1)1246
2564 y FA(:=)40 b FG(R)1451 2578 y Fy(1)1511 2564 y FA(+)20
b FG(H)1678 2578 y Fv(y)1713 2589 y Fs(N)5 b Fq(+1)1847
2578 y Fv(;R)1920 2587 y Fq(1)1958 2564 y FA(\()p Fx(\000)p
FA(1)p FG(=)p FA(2\))22 b Fx(\000)e FG(H)2423 2578 y
Fy(0)2462 2564 y FA(\()p FG(y)2542 2578 y Fv(N)7 b Fy(+1)2699
2564 y FA(\))15 b FG(;)456 2744 y FA(so)30 b(that)1330
2804 y Fu(n)1391 2905 y FG(g)1434 2924 y Fj(S)-6 b Fy(\()p
Fv(Y)1545 2933 y Fq(1)1574 2924 y Fv(;R)1647 2933 y Fq(1)1681
2924 y Fy(\))1738 2905 y FA(=)25 b Fx(\000)1915 2844
y FA(1)p 1915 2884 46 4 v 1915 2968 a(2)1970 2804 y Fu(o)2071
2905 y FA(=)40 b FG(@)5 b(B)2307 2919 y Fy(~)-38 b Fv(r)2336
2928 y Fq(1)2375 2905 y FA(\()p FG(y)2455 2919 y Fy(1)2494
2905 y FA(\))15 b FG(:)456 3096 y FA(Then,)29 b(since)g
FG(u)p FA(\()p FG(x)1079 3110 y Fy(0)1118 3096 y FA(\))d
FG(>)f Fx(\000)p FA(1)p FG(=)p FA(2)31 b(b)m(y)e(assumption,)g(the)g
(\014rst)g(touc)m(hing)h(prop)s(ert)m(y)f(of)h FG(X)3405
3110 y Fy(1)456 3204 y FA(implies)d(that)k FG(x)1013
3218 y Fy(0)1083 3204 y FA(is)e(ab)s(o)m(v)m(e)j FG(@)5
b(B)1557 3218 y Fy(~)-38 b Fv(r)1586 3227 y Fq(1)1625
3204 y FA(\()p FG(y)1705 3218 y Fy(1)1744 3204 y FA(\).)41
b(In)30 b(particular,)f(recalling)g(\(A.26\),)j(since)1128
3425 y FG(y)1173 3439 y Fy(1)1232 3425 y FA(+)20 b(\()p
FG(x)1410 3387 y Ft(0)1410 3447 y Fy(0)1450 3425 y FG(;)15
b FA(0\))21 b(+)1682 3314 y Fu(q)p 1773 3314 376 4 v
1777 3425 a FA(~)-49 b FG(r)1817 3393 y Fy(2)1814 3451
y(1)1876 3425 y Fx(\000)20 b(j)p FG(x)2044 3393 y Ft(0)2044
3451 y Fy(0)2084 3425 y Fx(j)2109 3398 y Fy(2)2148 3425
y FG(e)2190 3439 y Fv(N)2298 3425 y Fx(2)25 b FG(@)5
b(B)2509 3439 y Fy(~)-38 b Fv(r)2538 3448 y Fq(1)2577
3425 y FA(\()p FG(y)2657 3439 y Fy(1)2696 3425 y FA(\))15
b FG(;)456 3622 y FA(w)m(e)30 b(ha)m(v)m(e,)i(using)d(\(A.24\),)k(that)
1101 3842 y FG(y)1146 3856 y Fy(1)p Fv(;N)1288 3842 y
FA(+)1379 3731 y Fu(q)p 1470 3731 323 4 v 1474 3842 a
FA(~)-49 b FG(r)1514 3810 y Fy(2)1511 3868 y(1)1573 3842
y Fx(\000)20 b FA(4)p FG(q)1753 3815 y Fy(2)1833 3842
y Fx(\024)40 b FG(y)1989 3856 y Fy(1)p Fv(;N)2131 3842
y FA(+)2222 3731 y Fu(q)p 2313 3731 376 4 v 2317 3842
a FA(~)-49 b FG(r)2357 3810 y Fy(2)2354 3868 y(1)2416
3842 y Fx(\000)20 b(j)p FG(x)2584 3810 y Ft(0)2584 3868
y Fy(0)2623 3842 y Fx(j)2648 3815 y Fy(2)2728 3842 y
Fx(\024)1833 3989 y(\024)40 b FG(x)1996 4003 y Fy(0)p
Fv(;N)2158 3989 y Fx(\024)1833 4124 y(\024)g FA(const)15
b FG(:)456 3965 y FA(\(A.113\))456 4308 y(By)30 b(construction,)g(the)h
(domain)e(of)i Fw(S)n FA(\()p FG(Y)1872 4322 y Fy(1)1906
4308 y FG(;)15 b(R)2015 4322 y Fy(1)2055 4308 y FA(\))30
b(lies)f(b)s(elo)m(w)h(the)h(h)m(yp)s(erplane)1014 4414
y Fu(n)1075 4515 y FG(x)1127 4529 y Fv(N)1220 4515 y
FA(=)24 b FG(y)1360 4529 y Fy(1)p Fv(;N)1502 4515 y FA(+)c
FG(R)1662 4529 y Fy(1)1722 4515 y Fx(\000)g FG(H)1889
4529 y Fy(0)1928 4515 y FA(\()p FG(y)2008 4529 y Fv(N)7
b Fy(+1)2165 4515 y FA(\))21 b(+)f FG(H)2388 4529 y Fv(y)2423
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%%Page: 118 118
118 117 bop 456 251 a Fy(118)140 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
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119 118 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)
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y Ft(\003)1549 4842 y Fx(\000)19 b FG(x)1691 4856 y Ft(\003)1751
4842 y FA(+)1877 4780 y Fx(r)p FG(u)p FA(\()p FG(x)2092
4794 y Ft(\003)2132 4780 y FA(\))p 1852 4821 341 4 v
1852 4904 a Fx(jr)p FG(u)p FA(\()p FG(x)2092 4918 y Ft(\003)2132
4904 y FA(\))p Fx(j)2217 4842 y FG(H)2293 4856 y Fy(0)2332
4842 y FA(\()p FG(u)p FA(\()p FG(x)2506 4856 y Ft(\003)2547
4842 y FA(\)\))2617 4737 y Fu(\014)2617 4792 y(\014)2617
4846 y(\014)2673 4842 y Fx(\024)2779 4780 y FA(const)p
2779 4821 208 4 v 2848 4904 a FG(R)3011 4842 y(:)456
5047 y FA(Note)31 b(that)g(\(A.124\))i(is)c(ob)m(viously)h(ful\014lled)
c(if)k FG(T)2159 5061 y Fv(Y)2200 5069 y Fp(\003)2236
5061 y Fv(;R)2309 5069 y Fp(\003)2348 5047 y FG(x)2400
5061 y Ft(\003)2465 5047 y FA(=)25 b FG(x)2613 5061 y
Ft(\003)2652 5047 y FA(,)31 b(since,)f(in)f(this)g(case)555
5198 y FG(H)631 5212 y Fy(0)670 5198 y FA(\()p FG(u)p
FA(\()p FG(x)844 5212 y Ft(\003)884 5198 y FA(\)\))d(=)f
FG(H)1152 5212 y Fy(0)1191 5198 y FA(\()p FG(g)1268 5215
y Fk(e)1269 5232 y Fj(S)-11 b Fy(\()p Fv(Y)1375 5240
y Fp(\003)1411 5232 y Fv(;R)1484 5240 y Fp(\003)1520
5232 y Fy(\))1551 5198 y FA(\()p FG(x)1638 5212 y Ft(\003)1678
5198 y FA(\)\))26 b(=)f FG(H)1946 5212 y Fy(0)1985 5198
y FA(\()p FG(g)2062 5215 y Fk(e)2063 5232 y Fj(S)-11
b Fy(\()p Fv(Y)2169 5240 y Fp(\003)2205 5232 y Fv(;R)2278
5240 y Fp(\003)2314 5232 y Fy(\))2345 5198 y FA(\()p
FG(T)2433 5212 y Fv(Y)2474 5220 y Fp(\003)2511 5212 y
Fv(;R)2584 5220 y Fp(\003)2623 5198 y FG(x)2675 5212
y Ft(\003)2715 5198 y FA(\)\))26 b(=)f FG(H)2983 5212
y Fy(0)3022 5198 y FA(\(0\))h(=)f(0)15 b FG(:)p eop
%%Page: 120 120
120 119 bop 456 251 a Fy(120)140 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(Hence,)36 b(w)m(e)e(fo)s(cus)f(on)h(the)g(pro)s(of)g
(of)g(\(A.124\))i(under)d(the)h(assumption)e FG(T)3061
464 y Fv(Y)3102 472 y Fp(\003)3138 464 y Fv(;R)3211 472
y Fp(\003)3251 450 y FG(x)3303 464 y Ft(\003)3374 450
y Fx(6)p FA(=)456 558 y FG(x)508 572 y Ft(\003)547 558
y FA(.)51 b(F)-8 b(or)35 b(this)e(scop)s(e,)i(\014rst)f(notice)g(that)g
FG(x)1961 572 y Ft(\003)2023 558 y Fx(\000)23 b FG(y)2162
572 y Ft(\003)2235 558 y FA(is,)34 b(b)m(y)g(construction,)g(parallel)f
(to)456 666 y FG(x)508 680 y Ft(\003)567 666 y Fx(\000)20
b FG(T)711 680 y Fv(Y)752 688 y Fp(\003)788 680 y Fv(;R)861
688 y Fp(\003)901 666 y FG(x)953 680 y Ft(\003)992 666
y FA(,)31 b(that)g(is)456 1145 y(\(A.125\))1407 1083
y FG(x)1459 1097 y Ft(\003)1519 1083 y Fx(\000)20 b FG(y)1655
1097 y Ft(\003)p 1382 1124 338 4 v 1382 1207 a Fx(j)p
FG(x)1459 1221 y Ft(\003)1519 1207 y Fx(\000)g FG(y)1655
1221 y Ft(\003)1694 1207 y Fx(j)1769 1145 y FA(=)1916
1082 y FG(x)1968 1096 y Ft(\003)2027 1082 y Fx(\000)g
FG(T)2171 1096 y Fv(Y)2212 1104 y Fp(\003)2248 1096 y
Fv(;R)2321 1104 y Fp(\003)2361 1082 y FG(x)2413 1096
y Ft(\003)p 1890 1124 588 4 v 1890 1207 a Fx(j)p FG(x)1967
1221 y Ft(\003)2027 1207 y Fx(\000)g FG(T)2171 1221 y
Fv(Y)2212 1229 y Fp(\003)2248 1221 y Fv(;R)2321 1229
y Fp(\003)2361 1207 y FG(x)2413 1221 y Ft(\003)2452 1207
y Fx(j)2503 1145 y FG(:)456 1630 y FA(Also,)30 b(from)g(\(A.19\),)456
2084 y(\(A.126\))1459 1980 y Fu(\014)1459 2034 y(\014)1459
2089 y(\014)1489 2084 y FG(x)1541 2098 y Ft(\003)1601
2084 y Fx(\000)20 b FG(T)1745 2098 y Fv(Y)1786 2106 y
Fp(\003)1822 2098 y Fv(;R)1895 2106 y Fp(\003)1935 2084
y FG(x)1987 2098 y Ft(\003)2026 1980 y Fu(\014)2026 2034
y(\014)2026 2089 y(\014)2082 2084 y Fx(\024)40 b FA(const)16
b FG(:)456 2539 y FA(In)29 b(the)i(ligh)m(t)e(of)i(\(2.57\))i(and)c
(\(A.125\),)456 3043 y(\(A.127\))1165 2957 y Fx(r)p FG(g)1283
2973 y Fk(e)1284 2991 y Fj(S)-12 b Fy(\()p Fv(Y)1389
2999 y Fp(\003)1425 2991 y Fv(;R)1498 2999 y Fp(\003)1534
2991 y Fy(\))1566 2957 y FA(\()p FG(x)1653 2971 y Ft(\003)1692
2957 y FA(\))p 1140 3022 614 4 v 1140 3105 a Fx(jr)p
FG(g)1283 3122 y Fk(e)1284 3139 y Fj(S)g Fy(\()p Fv(Y)1389
3147 y Fp(\003)1425 3139 y Fv(;R)1498 3147 y Fp(\003)1534
3139 y Fy(\))1566 3105 y FA(\()p FG(x)1653 3119 y Ft(\003)1692
3105 y FA(\))p Fx(j)1803 3043 y FA(=)1950 2981 y FG(x)2002
2995 y Ft(\003)2061 2981 y Fx(\000)20 b FG(y)2197 2995
y Ft(\003)p 1925 3022 338 4 v 1925 3105 a Fx(j)p FG(x)2002
3119 y Ft(\003)2061 3105 y Fx(\000)g FG(y)2197 3119 y
Ft(\003)2236 3105 y Fx(j)2312 3043 y FA(=)2458 2980 y
FG(x)2510 2994 y Ft(\003)2570 2980 y Fx(\000)g FG(T)2714
2994 y Fv(Y)2755 3002 y Fp(\003)2791 2994 y Fv(;R)2864
3002 y Fp(\003)2904 2980 y FG(x)2956 2994 y Ft(\003)p
2433 3022 588 4 v 2433 3105 a Fx(j)p FG(x)2510 3119 y
Ft(\003)2570 3105 y Fx(\000)g FG(T)2714 3119 y Fv(Y)2755
3127 y Fp(\003)2791 3119 y Fv(;R)2864 3127 y Fp(\003)2904
3105 y FG(x)2956 3119 y Ft(\003)2995 3105 y Fx(j)3045
3043 y FG(:)456 3551 y FA(Let)36 b(us)f(no)m(w)g(de\014ne)g
FG(W)13 b FA(\()p FG(x)p FA(\))34 b(:=)g FG(H)1669 3565
y Fy(0)1708 3551 y FA(\()p FG(g)1785 3567 y Fk(e)1786
3585 y Fj(S)-11 b Fy(\()p Fv(Y)1892 3593 y Fp(\003)1928
3585 y Fv(;R)2001 3593 y Fp(\003)2037 3585 y Fy(\))2069
3551 y FA(\()p FG(x)p FA(\)\).)57 b(By)36 b(the)g(touc)m(hing)f(prop)s
(ert)m(y)g(of)456 3678 y FG(x)508 3692 y Ft(\003)547
3678 y FA(,)c(the)f(fact)h(that)g FG(T)1188 3692 y Fv(Y)1229
3700 y Fp(\003)1265 3692 y Fv(;R)1338 3700 y Fp(\003)1378
3678 y FG(x)1430 3692 y Ft(\003)1495 3678 y Fx(2)25 b
FA(\006\()p FG(Y)1735 3692 y Ft(\003)1774 3678 y FG(;)15
b(R)1883 3692 y Ft(\003)1923 3678 y FA(\))31 b(and)e(\(A.127\),)k(w)m
(e)e(ha)m(v)m(e)h(that)1147 4057 y Fu(\014)1147 4112
y(\014)1147 4166 y(\014)1178 4162 y FG(T)1231 4176 y
Fv(Y)1272 4184 y Fp(\003)1308 4176 y Fv(;R)1381 4184
y Fp(\003)1421 4162 y FG(x)1473 4176 y Ft(\003)1532 4162
y Fx(\000)20 b FG(x)1675 4176 y Ft(\003)1735 4162 y FA(+)1861
4100 y Fx(r)p FG(u)p FA(\()p FG(x)2076 4114 y Ft(\003)2115
4100 y FA(\))p 1836 4141 341 4 v 1836 4224 a Fx(jr)p
FG(u)p FA(\()p FG(x)2076 4238 y Ft(\003)2115 4224 y FA(\))p
Fx(j)2201 4162 y FG(H)2277 4176 y Fy(0)2316 4162 y FA(\()p
FG(u)p FA(\()p FG(x)2490 4176 y Ft(\003)2530 4162 y FA(\)\))2600
4057 y Fu(\014)2600 4112 y(\014)2600 4166 y(\014)2657
4162 y FA(=)1033 4436 y(=)1129 4331 y Fu(\014)1129 4386
y(\014)1129 4441 y(\014)1159 4436 y FG(T)1212 4450 y
Fv(Y)1253 4458 y Fp(\003)1289 4450 y Fv(;R)1362 4458
y Fp(\003)1402 4436 y FG(x)1454 4450 y Ft(\003)1513 4436
y Fx(\000)g FG(x)1656 4450 y Ft(\003)1716 4436 y FA(+)1842
4350 y Fx(r)p FG(g)1960 4367 y Fk(e)1961 4384 y Fj(S)-12
b Fy(\()p Fv(Y)2066 4392 y Fp(\003)2102 4384 y Fv(;R)2175
4392 y Fp(\003)2211 4384 y Fy(\))2243 4350 y FA(\()p
FG(x)2330 4364 y Ft(\003)2369 4350 y FA(\))p 1817 4415
614 4 v 1817 4498 a Fx(jr)p FG(g)1960 4515 y Fk(e)1961
4532 y Fj(S)g Fy(\()p Fv(Y)2066 4540 y Fp(\003)2102 4532
y Fv(;R)2175 4540 y Fp(\003)2211 4532 y Fy(\))2243 4498
y FA(\()p FG(x)2330 4512 y Ft(\003)2369 4498 y FA(\))p
Fx(j)2455 4436 y FG(W)13 b FA(\()p FG(x)2641 4450 y Ft(\003)2680
4436 y FA(\))2715 4331 y Fu(\014)2715 4386 y(\014)2715
4441 y(\014)2771 4436 y FA(=)965 4738 y(=)1060 4634 y
Fu(\014)1060 4688 y(\014)1060 4743 y(\014)1126 4652 y
Fx(r)p FG(g)1244 4669 y Fk(e)1245 4686 y Fj(S)-12 b Fy(\()p
Fv(Y)1350 4694 y Fp(\003)1386 4686 y Fv(;R)1459 4694
y Fp(\003)1495 4686 y Fy(\))1527 4652 y FA(\()p FG(x)1614
4666 y Ft(\003)1653 4652 y FA(\))p 1101 4717 V 1101 4801
a Fx(jr)p FG(g)1244 4817 y Fk(e)1245 4834 y Fj(S)g Fy(\()p
Fv(Y)1350 4842 y Fp(\003)1386 4834 y Fv(;R)1459 4842
y Fp(\003)1495 4834 y Fy(\))1527 4801 y FA(\()p FG(x)1614
4815 y Ft(\003)1653 4801 y FA(\))p Fx(j)1739 4637 y Fu(h)1782
4738 y Fx(j)p FG(T)1860 4752 y Fv(Y)1901 4760 y Fp(\003)1937
4752 y Fv(;R)2010 4760 y Fp(\003)2050 4738 y FG(x)2102
4752 y Ft(\003)2162 4738 y Fx(\000)20 b FG(x)2305 4752
y Ft(\003)2344 4738 y Fx(j)h FA(+)e FG(W)13 b FA(\()p
FG(x)2666 4752 y Ft(\003)2706 4738 y FA(\))2741 4637
y Fu(i)2784 4634 y(\014)2784 4688 y(\014)2784 4743 y(\014)2840
4738 y FA(=)1332 4991 y(=)1428 4886 y Fu(\014)1428 4941
y(\014)1428 4995 y(\014)1458 4991 y Fx(j)p FG(T)1536
5005 y Fv(Y)1577 5013 y Fp(\003)1613 5005 y Fv(;R)1686
5013 y Fp(\003)1726 4991 y FG(x)1778 5005 y Ft(\003)1838
4991 y Fx(\000)20 b FG(x)1981 5005 y Ft(\003)2020 4991
y Fx(j)g FA(+)g FG(W)13 b FA(\()p FG(x)2342 5005 y Ft(\003)2382
4991 y FA(\))2417 4886 y Fu(\014)2417 4941 y(\014)2417
4995 y(\014)2472 4991 y FA(=)1065 5190 y(=)1161 5085
y Fu(\014)1161 5140 y(\014)1161 5194 y(\014)1191 5190
y Fx(j)p FG(T)1269 5204 y Fv(Y)1310 5212 y Fp(\003)1346
5204 y Fv(;R)1419 5212 y Fp(\003)1459 5190 y FG(x)1511
5204 y Ft(\003)1571 5190 y Fx(\000)20 b FG(x)1714 5204
y Ft(\003)1753 5190 y Fx(j)g FA(+)g FG(W)13 b FA(\()p
FG(x)2075 5204 y Ft(\003)2115 5190 y FA(\))20 b Fx(\000)g
FG(W)13 b FA(\()p FG(T)2448 5204 y Fv(Y)2489 5212 y Fp(\003)2525
5204 y Fv(;R)2598 5212 y Fp(\003)2638 5190 y FG(x)2690
5204 y Ft(\003)2729 5190 y FA(\))2764 5085 y Fu(\014)2764
5140 y(\014)2764 5194 y(\014)2810 5190 y FG(:)p eop
%%Page: 121 121
121 120 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)
f Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)140 b(121)456
450 y FA(Therefore,)34 b(a)g(second)f(order)g(T)-8 b(a)m(ylor)34
b(expansion)e(of)i FG(W)13 b FA(,)34 b(\(2.54\),)j(\(A.126\),)g
(\(A.127\))456 558 y(and)29 b(\(2.53\))k(giv)m(e)954
750 y Fu(\014)954 804 y(\014)954 859 y(\014)984 854 y
FG(T)1037 868 y Fv(Y)1078 876 y Fp(\003)1114 868 y Fv(;R)1187
876 y Fp(\003)1227 854 y FG(x)1279 868 y Ft(\003)1338
854 y Fx(\000)20 b FG(x)1481 868 y Ft(\003)1541 854 y
FA(+)1667 793 y Fx(r)p FG(u)p FA(\()p FG(x)1882 807 y
Ft(\003)1922 793 y FA(\))p 1642 833 341 4 v 1642 917
a Fx(jr)p FG(u)p FA(\()p FG(x)1882 931 y Ft(\003)1922
917 y FA(\))p Fx(j)2007 854 y FG(H)2083 868 y Fy(0)2122
854 y FA(\()p FG(u)p FA(\()p FG(x)2296 868 y Ft(\003)2337
854 y FA(\)\))2407 750 y Fu(\014)2407 804 y(\014)2407
859 y(\014)2463 854 y Fx(\024)800 1079 y(\024)954 975
y Fu(\014)954 1029 y(\014)954 1084 y(\014)984 1079 y
Fx(j)p FG(T)1062 1093 y Fv(Y)1103 1101 y Fp(\003)1139
1093 y Fv(;R)1212 1101 y Fp(\003)1252 1079 y FG(x)1304
1093 y Ft(\003)1364 1079 y Fx(\000)g FG(x)1507 1093 y
Ft(\003)1546 1079 y Fx(j)h(\000)954 1278 y(\000r)p FG(W)13
b FA(\()p FG(x)1287 1292 y Ft(\003)1325 1278 y FA(\)\()p
FG(T)1448 1292 y Fv(Y)1489 1300 y Fp(\003)1526 1292 y
Fv(;R)1599 1300 y Fp(\003)1639 1278 y FG(x)1691 1292
y Ft(\003)1751 1278 y Fx(\000)20 b FG(x)1894 1292 y Ft(\003)1933
1278 y FA(\))1968 1174 y Fu(\014)1968 1228 y(\014)1968
1283 y(\014)2019 1278 y FA(+)954 1453 y(+)15 b(const)g
Fx(j)p FG(D)1365 1415 y Fy(2)1405 1453 y FG(W)e Fx(j)i(j)p
FG(T)1622 1467 y Fv(Y)1663 1475 y Fp(\003)1699 1467 y
Fv(;R)1772 1475 y Fp(\003)1812 1453 y FG(x)1864 1467
y Ft(\003)1923 1453 y Fx(\000)20 b FG(x)2066 1467 y Ft(\003)2106
1453 y Fx(j)2131 1415 y Fy(2)2196 1453 y Fx(\024)800
1620 y(\024)954 1515 y Fu(\014)954 1570 y(\014)954 1624
y(\014)984 1620 y Fx(j)p FG(T)1062 1634 y Fv(Y)1103 1642
y Fp(\003)1139 1634 y Fv(;R)1212 1642 y Fp(\003)1252
1620 y FG(x)1304 1634 y Ft(\003)1364 1620 y Fx(\000)g
FG(x)1507 1634 y Ft(\003)1546 1620 y Fx(j)h(\000)954
1819 y(\000)p FG(H)1108 1781 y Ft(0)1101 1841 y Fy(0)1139
1819 y FA(\()p FG(g)1216 1835 y Fk(e)1217 1852 y Fj(S)-11
b Fy(\()p Fv(Y)1323 1860 y Fp(\003)1359 1852 y Fv(;R)1432
1860 y Fp(\003)1468 1852 y Fy(\))1500 1819 y FA(\()p
FG(x)1587 1833 y Ft(\003)1626 1819 y FA(\)\))p Fx(r)p
FG(g)1814 1835 y Fk(e)1815 1852 y Fj(S)g Fy(\()p Fv(Y)1921
1860 y Fp(\003)1957 1852 y Fv(;R)2030 1860 y Fp(\003)2066
1852 y Fy(\))2098 1819 y FA(\()p FG(x)2185 1833 y Ft(\003)2225
1819 y FA(\))20 b Fx(\001)h FA(\()p FG(T)2414 1833 y
Fv(Y)2455 1841 y Fp(\003)2491 1833 y Fv(;R)2564 1841
y Fp(\003)2604 1819 y FG(x)2656 1833 y Ft(\003)2715 1819
y Fx(\000)f FG(x)2858 1833 y Ft(\003)2898 1819 y FA(\))2933
1714 y Fu(\014)2933 1769 y(\014)2933 1823 y(\014)2984
1819 y FA(+)954 2030 y(+)1035 1969 y(const)p 1035 2009
208 4 v 1103 2093 a FG(R)1277 2030 y Fx(\024)800 2253
y(\024)954 2148 y Fu(\014)954 2203 y(\014)954 2257 y(\014)1019
2190 y FG(T)1072 2204 y Fv(Y)1113 2212 y Fp(\003)1149
2204 y Fv(;R)1222 2212 y Fp(\003)1262 2190 y FG(x)1314
2204 y Ft(\003)1374 2190 y Fx(\000)g FG(x)1517 2204 y
Ft(\003)p 994 2232 588 4 v 994 2315 a Fx(j)p FG(T)1072
2329 y Fv(Y)1113 2337 y Fp(\003)1149 2329 y Fv(;R)1222
2337 y Fp(\003)1262 2315 y FG(x)1314 2329 y Ft(\003)1374
2315 y Fx(\000)g FG(x)1517 2329 y Ft(\003)1556 2315 y
Fx(j)1606 2152 y Fu(h)1649 2253 y FG(T)1702 2267 y Fv(Y)1743
2275 y Fp(\003)1779 2267 y Fv(;R)1852 2275 y Fp(\003)1892
2253 y FG(x)1944 2267 y Ft(\003)2004 2253 y Fx(\000)g
FG(x)2147 2267 y Ft(\003)2206 2253 y Fx(\000)954 2482
y(\000)p FG(H)1108 2444 y Ft(0)1101 2504 y Fy(0)1139
2482 y FA(\()p FG(g)1216 2498 y Fk(e)1217 2516 y Fj(S)-11
b Fy(\()p Fv(Y)1323 2524 y Fp(\003)1359 2516 y Fv(;R)1432
2524 y Fp(\003)1468 2516 y Fy(\))1500 2482 y FA(\()p
FG(x)1587 2496 y Ft(\003)1626 2482 y FA(\)\))15 b Fx(jr)p
FG(g)1854 2498 y Fk(e)1855 2516 y Fj(S)-10 b Fy(\()p
Fv(Y)1962 2524 y Fp(\003)1998 2516 y Fv(;R)2071 2524
y Fp(\003)2107 2516 y Fy(\))2138 2482 y FA(\()p FG(x)2225
2496 y Ft(\003)2265 2482 y FA(\))p Fx(j)21 b(\001)f FA(\()p
FG(T)2479 2496 y Fv(Y)2520 2504 y Fp(\003)2557 2496 y
Fv(;R)2630 2504 y Fp(\003)2669 2482 y FG(x)2721 2496
y Ft(\003)2781 2482 y Fx(\000)g FG(x)2924 2496 y Ft(\003)2963
2482 y FA(\))2998 2381 y Fu(i)3042 2377 y(\014)3042 2432
y(\014)3042 2486 y(\014)3092 2482 y FA(+)954 2694 y(+)1035
2632 y(const)p 1035 2673 208 4 v 1103 2756 a FG(R)1277
2694 y FA(=)800 2896 y(=)954 2791 y Fu(\014)954 2846
y(\014)954 2900 y(\014)984 2896 y FG(T)1037 2910 y Fv(Y)1078
2918 y Fp(\003)1114 2910 y Fv(;R)1187 2918 y Fp(\003)1227
2896 y FG(x)1279 2910 y Ft(\003)1338 2896 y Fx(\000)g
FG(x)1481 2910 y Ft(\003)1541 2896 y Fx(\000)954 3095
y(\000)p FG(H)1108 3057 y Ft(0)1101 3117 y Fy(0)1139
3095 y FA(\()p FG(g)1216 3111 y Fk(e)1217 3128 y Fj(S)-11
b Fy(\()p Fv(Y)1323 3136 y Fp(\003)1359 3128 y Fv(;R)1432
3136 y Fp(\003)1468 3128 y Fy(\))1500 3095 y FA(\()p
FG(x)1587 3109 y Ft(\003)1626 3095 y FA(\)\))15 b Fx(jr)p
FG(g)1854 3111 y Fk(e)1855 3128 y Fj(S)-10 b Fy(\()p
Fv(Y)1962 3136 y Fp(\003)1998 3128 y Fv(;R)2071 3136
y Fp(\003)2107 3128 y Fy(\))2138 3095 y FA(\()p FG(x)2225
3109 y Ft(\003)2265 3095 y FA(\))p Fx(j)21 b(\001)f FA(\()p
FG(T)2479 3109 y Fv(Y)2520 3117 y Fp(\003)2557 3109 y
Fv(;R)2630 3117 y Fp(\003)2669 3095 y FG(x)2721 3109
y Ft(\003)2781 3095 y Fx(\000)g FG(x)2924 3109 y Ft(\003)2963
3095 y FA(\))2998 2990 y Fu(\014)2998 3045 y(\014)2998
3099 y(\014)3049 3095 y FA(+)954 3306 y(+)1035 3245 y(const)p
1035 3286 V 1103 3369 a FG(R)1277 3306 y Fx(\024)800
3521 y(\024)954 3417 y Fu(\014)954 3471 y(\014)954 3526
y(\014)984 3521 y FG(T)1037 3535 y Fv(Y)1078 3543 y Fp(\003)1114
3535 y Fv(;R)1187 3543 y Fp(\003)1227 3521 y FG(x)1279
3535 y Ft(\003)1338 3521 y Fx(\000)g FG(x)1481 3535 y
Ft(\003)1541 3521 y Fx(\000)g FA(\()p FG(T)1720 3535
y Fv(Y)1761 3543 y Fp(\003)1797 3535 y Fv(;R)1870 3543
y Fp(\003)1910 3521 y FG(x)1962 3535 y Ft(\003)2022 3521
y Fx(\000)g FG(x)2165 3535 y Ft(\003)2204 3521 y FA(\))2239
3417 y Fu(\014)2239 3471 y(\014)2239 3526 y(\014)2290
3521 y FA(+)2391 3460 y(const)p 2391 3501 V 2460 3584
a FG(R)2623 3521 y(;)456 3796 y FA(that)31 b(is)e(\(A.124\).)555
4039 y(W)-8 b(e)32 b(no)m(w)e(sho)m(w)g(that)456 4315
y(\(A.128\))587 b Fl(\\)1422 4187 y Fu(\022)1524 4253
y Fx(r)p FG(u)p FA(\()p FG(x)1739 4267 y Ft(\003)1778
4253 y FA(\))p 1498 4294 341 4 v 1498 4377 a Fx(jr)p
FG(u)p FA(\()p FG(x)1738 4391 y Ft(\003)1778 4377 y FA(\))p
Fx(j)1849 4315 y FG(;)30 b(e)1946 4329 y Fv(N)2014 4187
y Fu(\023)2121 4315 y Fx(\024)2242 4253 y FA(const)16
b FG(q)p 2242 4294 267 4 v 2341 4377 a(R)2534 4315 y(:)456
4608 y FA(F)-8 b(or)31 b(pro)m(ving)e(this,)h(let)g(us)g(note)h(that)
1166 4838 y Fx(j)p FG(T)1244 4852 y Fv(Y)1285 4860 y
Fp(\003)1322 4852 y Fv(;R)1395 4860 y Fp(\003)1435 4838
y FG(x)1487 4852 y Ft(\003)1546 4838 y Fx(\000)20 b FG(y)1682
4852 y Ft(\003)1721 4838 y Fx(j)26 b FA(=)f FG(r)s FA(\()p
FG(Y)2000 4852 y Ft(\003)2039 4838 y FG(;)15 b(R)2148
4852 y Ft(\003)2187 4838 y FA(\))26 b(=)1772 5052 y(=)f
FG(R)1937 5066 y Ft(\003)1996 5052 y Fx(\000)20 b FG(H)2163
5066 y Fy(0)2202 5052 y FA(\()p FG(y)2282 5066 y Fv(N)7
b Fy(+1)2439 5052 y FA(\))21 b Fx(\000)p 2598 4917 72
4 v 2598 4990 a FG(C)2669 5004 y Fy(0)p 2596 5031 116
4 v 2596 5114 a FA(2)p FG(R)2736 5052 y(y)2784 5014 y
Fy(2)2781 5074 y Fv(N)7 b Fy(+1)2963 5052 y Fx(\025)1772
5212 y(\025)40 b FA(const)15 b FG(R)2174 5226 y Ft(\003)2229
5212 y FG(;)456 5032 y FA(\(A.129\))p eop
%%Page: 122 122
122 121 bop 456 251 a Fy(122)140 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(hence,)1406 589 y(sin)13 b([)q Fl(\\)p
FA(\()p FG(T)1712 603 y Fv(Y)1753 611 y Fp(\003)1789
603 y Fv(;R)1862 611 y Fp(\003)1901 589 y FG(x)1953 603
y Ft(\003)2013 589 y Fx(\000)20 b FG(y)2149 603 y Ft(\003)2188
589 y FG(;)30 b(e)2285 603 y Fv(N)2353 589 y FA(\)])c(=)1416
793 y(=)1522 730 y Fx(j)p FA(\()p FG(T)1635 744 y Fv(Y)1676
752 y Fp(\003)1712 744 y Fv(;R)1785 752 y Fp(\003)1825
730 y FG(x)1877 744 y Ft(\003)1937 730 y Fx(\000)20 b
FG(y)2073 744 y Ft(\003)2112 730 y FA(\))2147 697 y Ft(0)2170
730 y Fx(j)p 1522 772 674 4 v 1569 855 a(j)p FG(T)1647
869 y Fv(Y)1688 877 y Fp(\003)1724 869 y Fv(;R)1797 877
y Fp(\003)1837 855 y FG(x)1889 869 y Ft(\003)1948 855
y Fx(\000)g FG(y)2084 869 y Ft(\003)2123 855 y Fx(j)2231
793 y FA(=)1416 1050 y(=)1572 987 y Fx(j)p FA(\()p FG(T)1685
1001 y Fv(Y)1726 1009 y Fp(\003)1763 1001 y Fv(;R)1836
1009 y Fp(\003)1876 987 y FG(x)1928 1001 y Ft(\003)1967
987 y FA(\))2002 954 y Ft(0)2026 987 y Fx(j)p 1522 1029
580 4 v 1522 1112 a(j)p FG(T)1600 1126 y Fv(Y)1641 1134
y Fp(\003)1677 1126 y Fv(;R)1750 1134 y Fp(\003)1790
1112 y FG(x)1842 1126 y Ft(\003)1901 1112 y Fx(\000)g
FG(y)2037 1126 y Ft(\003)2076 1112 y Fx(j)2137 1050 y(\024)1416
1292 y(\024)1537 1230 y FA(const)15 b FG(q)p 1522 1271
282 4 v 1628 1354 a(R)1828 1292 y(:)456 961 y FA(\(A.130\))456
1475 y(Also,)30 b(in)f(analogy)i(with)e(\(A.127\),)k(w)m(e)e(ha)m(v)m
(e)g(that)1055 1643 y Fx(r)p FG(u)p FA(\()p FG(x)1270
1657 y Ft(\003)1309 1643 y FA(\))p 1029 1684 341 4 v
1029 1767 a Fx(jr)p FG(u)p FA(\()p FG(x)1269 1781 y Ft(\003)1309
1767 y FA(\))p Fx(j)1420 1705 y FA(=)1567 1619 y FG(g)1609
1636 y Fk(e)1610 1653 y Fj(S)-12 b Fy(\()p Fv(Y)1715
1661 y Fp(\003)1751 1653 y Fv(;R)1824 1661 y Fp(\003)1860
1653 y Fy(\))1891 1619 y FA(\()p FG(x)1978 1633 y Ft(\003)2018
1619 y FA(\))p 1541 1684 538 4 v 1541 1767 a Fx(j)p FG(g)1608
1784 y Fk(e)1609 1801 y Fj(S)h Fy(\()p Fv(Y)1715 1809
y Fp(\003)1751 1801 y Fv(;R)1824 1809 y Fp(\003)1860
1801 y Fy(\))1891 1767 y FA(\()p FG(x)1978 1781 y Ft(\003)2018
1767 y FA(\))p Fx(j)2129 1705 y FA(=)2275 1642 y FG(T)2328
1656 y Fv(Y)2369 1664 y Fp(\003)2405 1656 y Fv(;R)2478
1664 y Fp(\003)2518 1642 y FG(x)2570 1656 y Ft(\003)2630
1642 y Fx(\000)20 b FG(y)2766 1656 y Ft(\003)p 2250 1684
580 4 v 2250 1767 a Fx(j)p FG(T)2328 1781 y Fv(Y)2369
1789 y Fp(\003)2405 1781 y Fv(;R)2478 1789 y Fp(\003)2518
1767 y FG(x)2570 1781 y Ft(\003)2630 1767 y Fx(\000)g
FG(y)2766 1781 y Ft(\003)2805 1767 y Fx(j)2855 1705 y
FG(;)456 1939 y FA(whic)m(h,)29 b(together)j(with)d(\(A.130\),)k(pro)m
(v)m(es)e(\(A.128\).)555 2091 y(W)-8 b(e)32 b(no)m(w)e(observ)m(e)h
(that,)g(thanks)f(to)h(\(A.124\))i(and)d(\(A.128\),)j(w)m(e)e(ha)m(v)m
(e)1001 2230 y FG(x)1053 2244 y Ft(\003)1113 2230 y Fx(\001)20
b FG(e)1200 2244 y Fv(N)1293 2230 y Fx(\024)1027 2431
y(\024)25 b FA(\()p FG(T)1211 2445 y Fv(Y)1252 2453 y
Fp(\003)1288 2445 y Fv(;R)1361 2453 y Fp(\003)1401 2431
y FG(x)1453 2445 y Ft(\003)1492 2431 y FA(\))c Fx(\001)f
FG(e)1635 2445 y Fv(N)1723 2431 y FA(+)1849 2370 y Fx(r)p
FG(u)p FA(\()p FG(x)2064 2384 y Ft(\003)2104 2370 y FA(\))p
1824 2410 341 4 v 1824 2494 a Fx(jr)p FG(u)p FA(\()p
FG(x)2064 2508 y Ft(\003)2104 2494 y FA(\))p Fx(j)2194
2431 y(\001)h FG(e)2282 2445 y Fv(N)2365 2431 y FG(H)2441
2445 y Fy(0)2480 2431 y FA(\()p FG(u)p FA(\()p FG(x)2654
2445 y Ft(\003)2694 2431 y FA(\)\))g(+)2886 2370 y(const)p
2886 2410 208 4 v 2955 2494 a FG(R)3128 2431 y Fx(\024)1027
2689 y(\024)k FA(\()p FG(T)1211 2703 y Fv(Y)1252 2711
y Fp(\003)1288 2703 y Fv(;R)1361 2711 y Fp(\003)1401
2689 y FG(x)1453 2703 y Ft(\003)1492 2689 y FA(\))c Fx(\001)f
FG(e)1635 2703 y Fv(N)1723 2689 y FA(+)g FG(H)1890 2703
y Fy(0)1929 2689 y FA(\()p FG(u)p FA(\()p FG(x)2103 2703
y Ft(\003)2143 2689 y FA(\)\))h(+)2335 2627 y(const)16
b FG(q)2602 2594 y Fy(2)p 2335 2668 307 4 v 2433 2751
a FG(R)2503 2725 y Fy(2)2671 2689 y FA(+)2772 2627 y(const)p
2772 2668 208 4 v 2841 2751 a FG(R)3005 2689 y(:)456
2482 y FA(\(A.131\))555 2957 y(Let)31 b(no)m(w)g(consider)e(the)h(set)
1007 3114 y(\004)1068 3128 y Ft(\003)1190 3114 y FA(:=)1369
3013 y Fu(n)1443 3091 y FA(\026)1430 3114 y FG(Y)45 b
FA(=)25 b(\()6 b(\026)-51 b FG(y)1707 3076 y Ft(0)1730
3114 y FG(;)15 b FA(0)p FG(;)23 b FA(\026)-53 b FG(y)1900
3128 y Fv(N)7 b Fy(+1)2059 3114 y FA(\))25 b Fx(2)g Fw(R)2265
3076 y Fv(N)7 b Fy(+1)2479 3114 y FA(s)p FG(:)p FA(t)p
FG(:)-2144 b FA(\(A.132\))1369 3313 y Fx(j)p FA(\()6
b(\026)-51 b FG(y)24 b Fx(\000)c FG(T)1642 3327 y Fv(Y)1683
3335 y Fp(\003)1719 3327 y Fv(;R)1792 3335 y Fp(\003)1832
3313 y FG(x)1884 3327 y Ft(\003)1923 3313 y FA(\))1958
3275 y Ft(0)1982 3313 y Fx(j)25 b(\024)g FG(c)15 b(q)j(;)56
b Fx(j)6 b FA(\026)-51 b FG(y)2392 3327 y Fv(N)7 b Fy(+1)2550
3313 y Fx(j)25 b(\024)g FA(1)p FG(=)p FA(4)2831 3212
y Fu(o)456 3488 y FA(and)k(let)456 3640 y(\(A.133\))1807
3617 y(\025)1787 3640 y FG(R)d FA(:=)g FG(cR)456 3793
y FA(with)j FG(c)d(>)g FA(0)31 b(suitably)e(small.)41
b(Recalling)29 b(the)i(de\014nition)e(of)i FG(r)s FA(\()p
Fx(\001)p FG(;)15 b Fx(\001)p FA(\))32 b(giv)m(en)f(in)e(\(A.18\),)456
3910 y(w)m(e)h(also)h(denote)k(\025)-49 b FG(r)27 b FA(:=)f
FG(r)s FA(\()1348 3887 y(\025)1334 3910 y FG(Y)20 b(;)1467
3887 y FA(\025)1447 3910 y FG(R)q FA(\).)555 4072 y(F)-8
b(or)33 b(an)m(y)907 4049 y(\026)893 4072 y FG(Y)48 b
Fx(2)28 b FA(\004)1144 4086 y Ft(\003)1183 4072 y FA(,)k(let)g(us)f
(slide)f Fw(S)o FA(\()1806 4049 y(\026)1792 4072 y FG(Y)14
b(;)1919 4049 y FA(\025)1899 4072 y FG(R)q FA(\))33 b(from)e
Fx(\0001)h FA(in)e(the)i FG(e)2754 4086 y Fv(N)2854 4072
y FA(direction,)f(un)m(til)456 4189 y(it)h(touc)m(hes)i(the)f(graph)f
(of)h FG(u)g FA(b)m(y)g(ab)s(o)m(v)m(e)h(for)f(the)g(\014rst)f(time,)i
(and)e(let)2931 4166 y(\025)2917 4189 y FG(Y)53 b FA(denote)33
b(the)456 4301 y(cen)m(ter)d(of)g(the)f(corresp)s(onding)e(barrier:)39
b(more)30 b(explicitly)-8 b(,)2551 4278 y(\025)2537 4301
y FG(Y)46 b FA(=)2745 4278 y(\026)2732 4301 y FG(Y)38
b Fx(\000)18 b FG(te)2987 4315 y Fv(N)3054 4301 y FA(,)30
b(for)f(some)456 4413 y FG(t)43 b Fx(2)g Fw(R)50 b FA(and)41
b Fw(S)n FA(\()1036 4390 y(\025)1022 4413 y FG(Y)14 b(;)1149
4390 y FA(\025)1129 4413 y FG(R)q FA(\))42 b(touc)m(hes)g(the)g(graph)e
(of)i FG(u)f FA(from)g(ab)s(o)m(v)m(e)i(for)e(the)g(\014rst)g(time)456
4521 y(coming)31 b(from)g Fx(\0001)g FA(in)f(the)i FG(e)1486
4535 y Fv(N)1585 4521 y FA(direction.)43 b(Sa)m(y)-8
b(,)33 b(also,)f(to)g(\014x)f(notations,)h(that)h(suc)m(h)456
4633 y(touc)m(hing)28 b(o)s(ccurs)g(at)h(some)g(p)s(oin)m(t)1701
4610 y(\025)1675 4633 y FG(X)7 b FA(.)40 b(W)-8 b(e)30
b(will)c(denote)j(b)m(y)2570 4610 y(\025)2562 4633 y(\004)f(the)h(set)g
(collecting)f(all)456 4745 y(the)j(p)s(oin)m(ts)913 4722
y(\025)887 4745 y FG(X)39 b FA(whic)m(h)30 b(lie)g(in)g(the)h(in)m
(terior)f(of)i(the)f(domain)f(of)i FG(u)p FA(,)g(when)3022
4722 y(\026)3008 4745 y FG(Y)51 b FA(v)-5 b(aries)31
b(in)456 4853 y(\004)517 4867 y Ft(\003)555 5005 y FA(W)-8
b(e)32 b(kno)m(w)e(from)g(\(A.29\))i(that)456 5186 y(\(A.134\))1487
5085 y Fu(\020)1541 5186 y FG(T)1594 5200 y Fv(Y)1635
5208 y Fp(\003)1671 5200 y Fv(;R)1744 5208 y Fp(\003)1784
5186 y FG(x)1836 5200 y Ft(\003)1875 5085 y Fu(\021)1930
5249 y Fv(N)2022 5186 y Fx(\024)25 b FG(C)2183 5201 y
Fv(])2215 5186 y FG(q)2259 5148 y Fy(2)2298 5186 y FG(=R)p
eop
%%Page: 123 123
123 122 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)
f Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)140 b(123)456
450 y FA(for)30 b(some)h(suitably)d(large)i(constan)m(t)i
FG(C)1823 465 y Fv(])1855 450 y FA(.)40 b(W)-8 b(e)32
b(no)m(w)e(claim)g(that,)h(if)2828 427 y(\026)2814 450
y FG(Y)46 b Fx(2)24 b FA(\004)3059 464 y Ft(\003)3098
450 y FA(,)31 b(then)456 684 y(\(A.135\))1089 556 y Fu(\022)1156
583 y(\020)1211 684 y FG(T)1264 698 y Fv(Y)1305 706 y
Fp(\003)1341 698 y Fv(;R)1414 706 y Fp(\003)1453 684
y FG(x)1505 698 y Ft(\003)1545 583 y Fu(\021)1599 606
y Ft(0)1622 684 y FG(;)15 b FA(2)p FG(C)1772 699 y Fv(])1830
623 y FG(q)1874 590 y Fy(2)p 1830 663 84 4 v 1837 747
a FG(R)1923 556 y Fu(\023)2020 684 y FA(is)30 b(outside)f(\006\()2540
661 y(\025)2526 684 y FG(Y)20 b(;)2660 661 y FA(\025)2639
684 y FG(R)r FA(\).)456 908 y(The)35 b(pro)s(of)g(of)h(\(A.135\))i
(\(whic)m(h)d(is)g(prett)m(y)i(long)e(and)g(will)e(b)s(e)j(completed)f
(only)g(on)456 1016 y(page)24 b(126\))i(is)c(b)m(y)i(con)m(tradiction.)
39 b(If)23 b(\(A.135\))j(w)m(ere)e(false,)h(Lemma)f(B.7)h(w)m(ould)e
(imply)456 1124 y(that)1149 1279 y(\006\()1264 1256 y(\025)1250
1279 y FG(Y)d(;)1383 1256 y FA(\025)1363 1279 y FG(R)q
FA(\))h Fx(\\)f(fj)p FA(\()p FG(x)h Fx(\000)f FG(T)1892
1293 y Fv(Y)1933 1301 y Fp(\003)1969 1293 y Fv(;R)2042
1301 y Fp(\003)2082 1279 y FG(x)2134 1293 y Ft(\003)2173
1279 y FA(\))2208 1241 y Ft(0)2232 1279 y Fx(j)25 b(\024)g
FG(cq)s Fx(g)c(\\)f(f)p FG(x)2705 1293 y Fv(N)2798 1279
y Fx(\025)31 b FA(\025)-51 b FG(y)2939 1293 y Fv(N)3006
1279 y Fx(g)1180 1426 y FA(is)29 b(ab)s(o)m(v)m(e)j(the)e(h)m(yp)s
(erplane)f Fx(f)p FG(x)2250 1440 y Fv(N)2343 1426 y FA(=)c(3)p
FG(C)2549 1441 y Fv(])2581 1426 y FG(q)2625 1394 y Fy(2)2664
1426 y FG(=)p FA(\(2)p FG(R)q FA(\))p Fx(g)p FA(,)456
1351 y(\(A.136\))456 1610 y(pro)m(vided)k FG(c)h FA(is)g(small)f
(enough)1512 1575 y Fy(22)1582 1610 y FA(.)41 b(W)-8
b(e)31 b(no)m(w)g(sho)m(w)f(that)1287 1765 y FG(d)1334
1797 y Fy(\006\()1423 1780 y(\025)1412 1797 y Fv(Y)16
b(;)1504 1780 y Fy(\025)1489 1797 y Fv(R)p Fy(\))1574
1765 y FA(\()p FG(x)1661 1779 y Ft(\003)1701 1765 y FA(\))25
b Fx(\024)1312 1988 y(\024)g FG(H)1484 2002 y Fy(0)1523
1988 y FA(\()p FG(u)p FA(\()p FG(x)1697 2002 y Ft(\003)1737
1988 y FA(\)\))c(+)35 b(const)2180 1926 y FG(q)p 2167
1967 70 4 v 2167 2050 a(R)2267 1988 y Fx(\000)2368 1924
y FG(C)2433 1939 y Fv(])2479 1924 y FG(q)2523 1891 y
Fy(2)p 2368 1967 196 4 v 2408 2050 a FA(2)p FG(R)2588
1988 y(:)456 1897 y FA(\(A.137\))456 2187 y(T)-8 b(o)39
b(pro)m(v)m(e)h(this)d(\(and)i(some)g(e\013ort)h(will)c(b)s(e)i
(needed\),)k(w)m(e)d(distinguish)c(t)m(w)m(o)40 b(cases:)456
2304 y(either)29 b FG(x)765 2318 y Ft(\003)835 2304 y
FA(is)h(in)f(the)h(exterior)h(or)f(it)g(is)f(in)g(the)i(in)m(terior)e
(of)i(\006\()2622 2281 y(\025)2608 2304 y FG(Y)20 b(;)2741
2281 y FA(\025)2721 2304 y FG(R)q FA(\).)555 2482 y(Let)25
b(us)f(\014rst)f(assume)h(that)h FG(x)1555 2496 y Ft(\003)1619
2482 y FA(is)e(not)h(in)f(the)i(in)m(terior)e(of)h(\006\()2637
2459 y(\025)2623 2482 y FG(Y)c(;)2756 2459 y FA(\025)2736
2482 y FG(R)q FA(\).)39 b(W)-8 b(e)26 b(\014rst)d(p)s(oin)m(t)456
2590 y(out)30 b(that)456 2798 y(\(A.138\))844 b FG(x)1650
2812 y Ft(\003)1710 2798 y Fx(\001)20 b FG(e)1797 2812
y Fv(N)1905 2798 y Fx(\025)2026 2735 y FA(3)p FG(C)2136
2750 y Fv(])2168 2735 y FG(q)2212 2702 y Fy(2)p 2026
2777 226 4 v 2081 2861 a FA(2)p FG(R)2277 2798 y(:)456
2998 y FA(F)-8 b(or)31 b(this,)e(note)i(that,)h(b)m(y)e(\(A.126\),)1481
3090 y Fu(\014)1481 3144 y(\014)1481 3199 y(\014)1511
3121 y(\000)1553 3194 y FG(x)1605 3208 y Ft(\003)1664
3194 y Fx(\000)20 b FG(T)1808 3208 y Fv(Y)1849 3216 y
Fp(\003)1885 3208 y Fv(;R)1958 3216 y Fp(\003)1998 3194
y FG(x)2050 3208 y Ft(\003)2089 3121 y Fu(\001)2131 3143
y Ft(0)2154 3090 y Fu(\014)2154 3144 y(\014)2154 3199
y(\014)2225 3194 y Fx(\024)40 b FG(cq)456 3391 y FA(if)e
FG(q)k FA(is)c(large)h(enough.)67 b(This,)40 b(\(A.136\))i(and)d(the)g
(assumption)f(that)h FG(x)3019 3405 y Ft(\003)3098 3391
y FA(is)f(not)i(in)456 3508 y(the)27 b(in)m(terior)g(of)g(\006\()1143
3485 y(\025)1129 3508 y FG(Y)20 b(;)1262 3485 y FA(\025)1242
3508 y FG(R)q FA(\))28 b(w)m(ould)e(imply)f(that)k(either)d(\(A.138\))k
(holds)c(or)i FG(x)3062 3522 y Ft(\003)3129 3508 y FA(is)e(b)s(elo)m(w)
456 3625 y(\006\()571 3602 y(\025)557 3625 y FG(Y)20
b(;)690 3602 y FA(\025)670 3625 y FG(R)q FA(\).)67 b(Th)m(us,)41
b(to)f(complete)f(the)h(pro)s(of)e(of)h(\(A.138\),)44
b(w)m(e)c(sho)m(w)f(no)m(w)g(that)h(the)456 3733 y(latter)30
b(p)s(ossibilit)m(y)c(cannot)31 b(hold.)39 b(Indeed,)30
b(note)g(that)h(the)f(\014rst)f(touc)m(hing)h(prop)s(ert)m(y)456
3845 y(of)g Fw(S)o FA(\()665 3822 y(\025)651 3845 y FG(Y)14
b(;)778 3822 y FA(\025)758 3845 y FG(R)q FA(\))31 b(and)f(Lemma)g(2.25)
i(imply)c(that)1671 4014 y(\025)-52 b FG(y)23 b Fx(\001)e
FG(e)1820 4028 y Fv(N)1913 4014 y Fx(\024)30 b FA(\025)-50
b FG(x)20 b Fx(\001)g FG(e)2168 4028 y Fv(N)456 4182
y FA(while)28 b(\(A.19\))k(sa)m(ys)f(that)1577 4322 y
Fx(j)6 b FA(\025)-51 b FG(x)21 b Fx(\000)26 b FA(\025)-52
b FG(y)s Fx(j)41 b(\025)f FA(const)2232 4299 y(\025)2213
4322 y FG(R)16 b(:)456 4472 y FA(F)-8 b(urthermore,)30
b(in)f(analogy)i(with)e(\(A.128\),)k(w)m(e)e(ha)m(v)m(e)h(that)456
4681 y(\(A.139\))707 b Fl(\\)1527 4608 y Fu(\000)1574
4681 y FA(\025)-51 b FG(x)20 b Fx(\000)27 b FA(\025)-52
b FG(y)s(;)31 b(e)1877 4695 y Fv(N)1944 4608 y Fu(\001)2026
4681 y Fx(\024)2163 4620 y FA(const)15 b FG(q)p 2147
4661 282 4 v 2253 4744 a(R)p 456 4824 499 4 v 555 4895
a Fy(22)626 4924 y FF(The)26 b(reader)g(will)h(indeed)e(notice)h(that,)
g(due)f(to)h(\(A.132\),)850 5073 y(\025)-44 b FD(y)20
b FF(+)d FD(r)r(e)1052 5081 y Fs(N)1143 5073 y Fr(2)34
b FF(\006\()1325 5054 y(\025)1313 5073 y FD(Y)17 b(;)1426
5054 y FF(\025)1410 5073 y FD(R)p FF(\))g Fr(\\)g(fj)p
FF(\()p FD(x)h Fr(\000)e FD(T)1855 5081 y Fs(Y)1891 5089
y Fb(\003)1926 5081 y Fs(;R)1991 5089 y Fb(\003)2030
5073 y FD(x)2074 5081 y Fp(\003)2109 5073 y FF(\))2139
5037 y Fp(0)2161 5073 y Fr(j)22 b(\024)f FD(cq)s Fr(g)d(\\)f(f)p
FD(x)2561 5081 y Fs(N)2640 5073 y Fr(\025)k FD(y)2759
5081 y Fs(N)2817 5073 y Fr(g)34 b(6)p FF(=)g Fr(;)13
b FD(:)p eop
%%Page: 124 124
124 123 bop 456 251 a Fy(124)140 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(Then,)29 b(the)i(ab)s(o)m(v)m(e)g(estimates)g(yield)
e(that)1146 585 y(\()p FG(x)1233 599 y Ft(\003)1293 585
y Fx(\000)d FA(\025)-52 b FG(y)s FA(\))21 b Fx(\001)f
FG(e)1574 599 y Fv(N)1667 585 y Fx(\025)25 b FA(\()6
b(\025)-51 b FG(x)21 b Fx(\000)26 b FA(\025)-51 b FG(y)r
FA(\))21 b Fx(\001)f FG(e)2152 599 y Fv(N)2240 585 y
Fx(\000)35 b FA(const)16 b FG(l)27 b FA(=)1667 734 y(=)1763
657 y Fu(\014)1763 711 y(\014)1793 734 y FA(\()6 b(\025)-51
b FG(x)21 b Fx(\000)26 b FA(\025)-51 b FG(y)s FA(\))20
b Fx(\001)h FG(e)2183 748 y Fv(N)2250 657 y Fu(\014)2250
711 y(\014)2301 734 y Fx(\000)35 b FA(const)16 b FG(l)27
b Fx(\025)1667 926 y(\025)1773 864 y FA(1)p 1773 905
46 4 v 1773 988 a(2)1844 848 y Fu(\014)1844 903 y(\014)1880
926 y FA(\025)-51 b FG(x)20 b Fx(\000)26 b FA(\025)-51
b FG(y)2085 848 y Fu(\014)2085 903 y(\014)2135 926 y
Fx(\000)35 b FA(const)16 b FG(l)27 b Fx(\025)1667 1109
y(\025)40 b FA(const)2021 1086 y(\025)2001 1109 y FG(R)21
b Fx(\000)35 b FA(const)16 b FG(l)h(;)456 854 y FA(\(A.140\))456
1281 y(th)m(us)30 b(sho)m(wing)f(that)i(\()p FG(x)1286
1295 y Ft(\003)1346 1281 y Fx(\000)26 b FA(\025)-51 b
FG(y)s FA(\))21 b Fx(\001)f FG(e)1628 1295 y Fv(N)1721
1281 y FG(>)25 b FA(0)31 b(and)f(th)m(us)g(that)h FG(x)2519
1295 y Ft(\003)2589 1281 y FA(is)e(not)i(b)s(elo)m(w)f(\006\()3215
1258 y(\025)3201 1281 y FG(Y)20 b(;)3334 1258 y FA(\025)3314
1281 y FG(R)q FA(\).)456 1389 y(This)28 b(ends)i(the)g(pro)s(of)g(of)g
(\(A.138\).)555 1497 y(With)41 b(this,)i(w)m(e)f(no)m(w)g(go)g(bac)m(k)
g(to)g(the)g(pro)s(of)e(of)i(\(A.137\))i(when)c FG(x)3013
1511 y Ft(\003)3094 1497 y FA(is)g(not)i(in)456 1614
y(the)k(in)m(terior)g(of)g(\006\()1200 1591 y(\025)1186
1614 y FG(Y)20 b(;)1320 1591 y FA(\025)1299 1614 y FG(R)r
FA(\).)89 b(By)47 b(means)g(of)53 b(\(A.136\))s(,)e(w)m(e)c(ha)m(v)m(e)
g(that)h(the)e(p)s(oin)m(t)456 1649 y Fu(\020)510 1749
y FA(\()p FG(T)598 1763 y Fv(Y)639 1771 y Fp(\003)675
1763 y Fv(;R)748 1771 y Fp(\003)788 1749 y FG(x)840 1763
y Ft(\003)880 1749 y FA(\))915 1716 y Ft(0)938 1749 y
FG(;)31 b FA(3)p FG(C)1104 1764 y Fv(])1136 1749 y FG(q)1180
1716 y Fy(2)1219 1749 y FG(=)p FA(\(2)p FG(R)q FA(\))1449
1649 y Fu(\021)1535 1749 y FA(lies)e(inside)f(\006\()2065
1726 y(\025)2051 1749 y FG(Y)20 b(;)2184 1726 y FA(\025)2164
1749 y FG(R)q FA(\),)31 b(th)m(us)1048 1996 y FG(d)1095
2028 y Fy(\006\()1184 2011 y(\025)1173 2028 y Fv(Y)16
b(;)1264 2011 y Fy(\025)1250 2028 y Fv(R)p Fy(\))1335
1996 y FA(\()p FG(x)1422 2010 y Ft(\003)1461 1996 y FA(\))26
b Fx(\024)1618 1864 y Fu(\014)1618 1919 y(\014)1618 1974
y(\014)1618 2028 y(\014)1648 1996 y FG(x)1700 2010 y
Ft(\003)1775 1996 y Fx(\000)1881 1868 y Fu(\022)1948
1895 y(\020)2002 1996 y FG(T)2055 2010 y Fv(Y)2096 2018
y Fp(\003)2132 2010 y Fv(;R)2205 2018 y Fp(\003)2245
1996 y FG(x)2297 2010 y Ft(\003)2336 1895 y Fu(\021)2391
1918 y Ft(0)2414 1996 y FG(;)2464 1933 y FA(3)p FG(C)2574
1948 y Fv(])2606 1933 y FG(q)2650 1900 y Fy(2)p 2464
1975 226 4 v 2520 2059 a FA(2)p FG(R)2700 1868 y Fu(\023)2767
1864 y(\014)2767 1919 y(\014)2767 1974 y(\014)2767 2028
y(\014)2827 1996 y FG(;)456 2200 y FA(and)j(therefore,)i(recalling)e
(\(A.138\),)937 2410 y FG(d)984 2442 y Fy(\006\()1073
2425 y(\025)1062 2442 y Fv(Y)16 b(;)1154 2425 y Fy(\025)1139
2442 y Fv(R)p Fy(\))1224 2410 y FA(\()p FG(x)1311 2424
y Ft(\003)1351 2410 y FA(\))25 b Fx(\024)g(j)p FA(\()p
FG(x)1619 2424 y Ft(\003)1680 2410 y Fx(\000)19 b FG(T)1823
2424 y Fv(Y)1864 2432 y Fp(\003)1900 2424 y Fv(;R)1973
2432 y Fp(\003)2013 2410 y FG(x)2065 2424 y Ft(\003)2105
2410 y FA(\))2140 2373 y Ft(0)2163 2410 y Fx(j)i FA(+)f
FG(x)2352 2373 y Ft(\003)2411 2410 y Fx(\001)h FG(e)2499
2424 y Fv(N)2587 2410 y Fx(\000)2687 2347 y FA(3)p FG(C)2797
2362 y Fv(])2845 2347 y FG(q)2889 2314 y Fy(2)p 2687
2389 241 4 v 2750 2472 a FA(2)p FG(R)2938 2410 y(:)456
2589 y FA(W)-8 b(e)31 b(recall)f(that,)h(b)m(y)h(\(A.128\))r(,)849
2793 y Fl(\\)15 b FA(\(\()p FG(x)1052 2807 y Ft(\003)1112
2793 y Fx(\000)20 b FG(T)1256 2807 y Fv(Y)1297 2815 y
Fp(\003)1333 2807 y Fv(;R)1406 2815 y Fp(\003)1446 2793
y FA(\))p FG(;)30 b(e)1578 2807 y Fv(N)1646 2793 y FA(\))41
b(=)f Fl(\\)1914 2665 y Fu(\022)2016 2731 y Fx(r)p FG(u)p
FA(\()p FG(x)2231 2745 y Ft(\003)2270 2731 y FA(\))p
1991 2772 341 4 v 1991 2855 a Fx(jr)p FG(u)p FA(\()p
FG(x)2231 2869 y Ft(\003)2270 2855 y FA(\))p Fx(j)2341
2793 y FG(;)30 b(e)2438 2807 y Fv(N)2506 2665 y Fu(\023)2613
2793 y Fx(\024)2734 2731 y FA(const)16 b FG(q)p 2734
2772 267 4 v 2833 2855 a(R)3026 2793 y(;)456 2996 y FA(and)29
b(so,)i(b)m(y)g(means)f(of)37 b(\(A.126\))s(,)30 b(w)m(e)h(ha)m(v)m(e)h
(that)456 3172 y(\(A.141\))1318 3068 y Fu(\014)1318 3122
y(\014)1318 3177 y(\014)1348 3098 y(\000)1390 3172 y
FA(\()p FG(T)1478 3186 y Fv(Y)1519 3194 y Fp(\003)1555
3186 y Fv(;R)1628 3194 y Fp(\003)1668 3172 y FG(x)1720
3186 y Ft(\003)1760 3172 y FA(\))20 b Fx(\000)g FG(x)1958
3186 y Ft(\003)1998 3098 y Fu(\001)2039 3121 y Ft(0)2062
3068 y Fu(\014)2062 3122 y(\014)2062 3177 y(\014)2118
3172 y Fx(\024)40 b FA(const)2475 3111 y FG(q)p 2462
3151 70 4 v 2462 3234 a(R)2557 3172 y(:)456 3351 y FA(Therefore,)30
b(recalling)g(\(A.126\))j(and)d(taking)g(in)m(to)g(accoun)m(t)i
(\(A.131\))s(,)738 3562 y FG(d)785 3593 y Fy(\006\()874
3577 y(\025)863 3593 y Fv(Y)16 b(;)955 3577 y Fy(\025)940
3593 y Fv(R)p Fy(\))1025 3562 y FA(\()p FG(x)1112 3576
y Ft(\003)1152 3562 y FA(\))25 b Fx(\024)g FA(const)1541
3500 y FG(q)1585 3467 y Fy(2)p 1541 3541 84 4 v 1548
3624 a FG(R)1654 3562 y FA(+)20 b FG(T)1798 3576 y Fv(Y)1839
3584 y Fp(\003)1875 3576 y Fv(;R)1948 3584 y Fp(\003)1988
3562 y FG(x)2040 3576 y Ft(\003)2100 3562 y Fx(\001)g
FG(e)2187 3576 y Fv(N)2275 3562 y FA(+)g FG(H)2442 3576
y Fy(0)2481 3562 y FA(\()p FG(u)p FA(\()p FG(x)2655 3576
y Ft(\003)2695 3562 y FA(\)\))h Fx(\000)2887 3499 y FA(3)p
FG(C)2997 3514 y Fv(])3044 3499 y FG(q)3088 3466 y Fy(2)p
2887 3541 241 4 v 2949 3624 a FA(2)p FG(R)3137 3562 y(:)456
3759 y FA(Thence,)36 b(if)f FG(x)946 3773 y Ft(\003)1020
3759 y FA(is)g(not)g(in)f(the)i(in)m(terior)e(of)i(\006\()2106
3736 y(\025)2092 3759 y FG(Y)20 b(;)2225 3736 y FA(\025)2205
3759 y FG(R)q FA(\),)37 b(\(A.137\))h(follo)m(ws)c(b)m(y)j(\(A.134\))
456 3876 y(and)24 b(b)m(y)i(taking)f FG(C)1089 3891 y
Fv(])1146 3876 y FA(suitably)e(large.)39 b(If)25 b(otherwise)g
FG(x)2281 3890 y Ft(\003)2346 3876 y FA(is)f(in)g(the)i(in)m(terior)e
(of)i(\006\()3215 3853 y(\025)3201 3876 y FG(Y)20 b(;)3334
3853 y FA(\025)3314 3876 y FG(R)q FA(\),)456 4025 y(that)31
b(is,)e(if)456 4173 y(\(A.142\))853 b FG(d)1654 4205
y Fy(\006\()1743 4188 y(\025)1732 4205 y Fv(Y)16 b(;)1824
4188 y Fy(\025)1809 4205 y Fv(R)p Fy(\))1894 4173 y FA(\()p
FG(x)1981 4187 y Ft(\003)2021 4173 y FA(\))40 b FG(<)g
FA(0)15 b FG(;)456 4338 y FA(in)34 b(order)h(to)h(pro)m(v)m(e)g
(\(A.137\))s(,)h(w)m(e)f(argue)f(as)h(follo)m(ws.)55
b(W)-8 b(e)37 b(\014rst)e(note)h(that,)h(in)d(this)456
4446 y(case,)1483 4557 y FG(u)p FA(\()p FG(x)1622 4571
y Ft(\003)1662 4557 y FA(\))26 b Fx(\024)f FG(g)1861
4573 y Fk(e)1862 4591 y Fj(S)-12 b Fy(\()1937 4574 y(\025)1926
4591 y Fv(Y)16 b(;)2018 4574 y Fy(\025)2003 4591 y Fv(R)p
Fy(\))2088 4557 y FA(\()p FG(x)2175 4571 y Ft(\003)2215
4557 y FA(\))25 b FG(<)g FA(0)456 4700 y(and)k(consequen)m(tly)1435
4811 y FG(g)1477 4827 y Fk(e)1478 4845 y Fj(S)-11 b Fy(\()p
Fv(Y)1584 4853 y Fp(\003)1620 4845 y Fv(;R)1693 4853
y Fp(\003)1729 4845 y Fy(\))1760 4811 y FA(\()p FG(x)1847
4825 y Ft(\003)1887 4811 y FA(\))26 b(=)f FG(u)p FA(\()p
FG(x)2183 4825 y Ft(\003)2222 4811 y FA(\))h FG(<)f FA(0)15
b FG(;)456 4959 y FA(so)32 b(that)h FG(x)820 4973 y Ft(\003)891
4959 y FA(is)f(also)g(in)f(the)h(in)m(terior)f(of)h(\006\()p
FG(Y)2017 4973 y Ft(\003)2057 4959 y FG(;)15 b(R)2166
4973 y Ft(\003)2206 4959 y FA(\).)46 b(Therefore,)33
b(recalling)f(\(A.128\))r(,)456 5067 y(it)e(follo)m(ws)f(that)456
5216 y(\(A.143\))692 b FG(x)1498 5230 y Ft(\003)1557
5216 y Fx(\001)21 b FG(e)1645 5230 y Fv(N)1738 5216 y
FG(<)k FA(\()p FG(T)1922 5230 y Fv(Y)1963 5238 y Fp(\003)1999
5230 y Fv(;R)2072 5238 y Fp(\003)2112 5216 y FG(x)2164
5230 y Ft(\003)2203 5216 y FA(\))c Fx(\001)f FG(e)2346
5230 y Fv(N)2429 5216 y FG(:)p eop
%%Page: 125 125
125 124 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)
f Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)140 b(125)456
453 y FA(W)-8 b(e)31 b(no)m(w)g(deduce)f(that,)h(when)e
FG(x)1616 467 y Ft(\003)1686 453 y FA(is)g(in)g(the)i(in)m(terior)e(of)
i(\006\()2580 430 y(\025)2566 453 y FG(Y)20 b(;)2699
430 y FA(\025)2679 453 y FG(R)q FA(\),)31 b(one)f(has)g(that)456
663 y(\(A.144\))413 b Fx(\000)p FG(d)1285 694 y Fy(\006\()1374
677 y(\025)1363 694 y Fv(Y)16 b(;)1454 677 y Fy(\025)1440
694 y Fv(R)p Fy(\))1524 663 y FA(\()p FG(x)1611 677 y
Ft(\003)1651 663 y FA(\))41 b Fx(\025)f FA(min)2005 534
y Fu(\032)2083 601 y FG(cq)p 2083 642 84 4 v 2102 725
a FA(4)2191 663 y FG(;)2267 599 y FA(3)p FG(C)2377 614
y Fv(])2424 599 y FG(q)2468 566 y Fy(2)p 2267 642 241
4 v 2329 725 a FA(2)p FG(R)2537 663 y Fx(\000)20 b FG(x)2680
677 y Ft(\003)2740 663 y Fx(\001)g FG(e)2827 677 y Fv(N)2910
534 y Fu(\033)3009 663 y FG(:)456 883 y FA(T)-8 b(o)25
b(pro)m(v)m(e)h(\(A.144\),)j(tak)m(e)e FG(x)e Fx(2)g
FA(\006\()1646 860 y(\025)1632 883 y FG(Y)20 b(;)1765
860 y FA(\025)1745 883 y FG(R)q FA(\).)40 b(Ob)m(viously)-8
b(,)24 b(w)m(e)i(ma)m(y)g(and)f(do)g(assume)g(that)456
1138 y(\(A.145\))931 b Fx(j)p FG(x)p Fx(j)25 b(\024)40
b FA(const)16 b FG(l)h(;)456 1285 y FA(otherwise)1611
1394 y Fx(j)p FG(x)1688 1408 y Ft(\003)1748 1394 y Fx(\000)j
FG(x)p Fx(j)25 b(\025)g FA(const)16 b FG(l)456 1523 y
FA(and)31 b(so)h(\(A.144\))i(trivially)29 b(follo)m(ws)i(from)g
(\(A.142\).)48 b(There)31 b(are)h(no)m(w)g(t)m(w)m(o)h(p)s(ossibili-)
456 1631 y(ties:)44 b(either)32 b FG(x)h FA(is)e(ab)s(o)m(v)m(e)j(the)f
(h)m(yp)s(erplane)d Fx(f)p FG(x)2085 1645 y Fv(N)2181
1631 y FA(=)f(3)p FG(C)2391 1646 y Fv(])2423 1631 y FG(q)2467
1598 y Fy(2)2506 1631 y FG(=)p FA(\(2)p FG(R)q FA(\))p
Fx(g)34 b FA(or)f(the)g(con)m(v)m(erse.)456 1743 y(If)d
FG(x)g FA(is)f(ab)s(o)m(v)m(e)j(the)e(h)m(yp)s(erplane)f
Fx(f)p FG(x)1699 1757 y Fv(N)1792 1743 y FA(=)c(3)p FG(C)1998
1758 y Fv(])2030 1743 y FG(q)2074 1710 y Fy(2)2113 1743
y FG(=)p FA(\(2)p FG(R)q FA(\))p Fx(g)p FA(,)32 b(then)1096
1957 y Fx(j)p FG(x)21 b Fx(\000)e FG(x)1336 1971 y Ft(\003)1376
1957 y Fx(j)25 b(\025)g FG(x)1574 1971 y Fv(N)1662 1957
y Fx(\000)20 b FG(x)1805 1971 y Ft(\003)1864 1957 y Fx(\001)h
FG(e)1952 1971 y Fv(N)2045 1957 y Fx(\025)2150 1894 y
FA(3)p FG(C)2260 1909 y Fv(])2292 1894 y FG(q)2336 1861
y Fy(2)p 2150 1936 226 4 v 2205 2019 a FA(2)p FG(R)2406
1957 y Fx(\000)f FG(x)2549 1971 y Ft(\003)2608 1957 y
Fx(\001)h FG(e)2696 1971 y Fv(N)2779 1957 y FG(;)456
2135 y FA(whic)m(h,)32 b(together)i(with)d(\(A.142\),)36
b(pro)m(v)m(es)d(\(A.144\))i(in)d(case)h FG(x)g FA(is)e(ab)s(o)m(v)m(e)
j(the)f(h)m(yp)s(er-)456 2243 y(plane)c Fx(f)p FG(x)795
2257 y Fv(N)888 2243 y FA(=)c(3)p FG(C)1094 2258 y Fv(])1126
2243 y FG(q)1170 2210 y Fy(2)1209 2243 y FG(=)p FA(\(2)p
FG(R)q FA(\))p Fx(g)p FA(.)43 b(If,)30 b(on)g(the)g(other)h(hand,)e
FG(x)h FA(is)g(not)g(ab)s(o)m(v)m(e)i(the)e(h)m(yp)s(er-)456
2355 y(plane)19 b Fx(f)p FG(x)785 2369 y Fv(N)878 2355
y FA(=)25 b(3)p FG(C)1084 2370 y Fv(])1116 2355 y FG(q)1160
2322 y Fy(2)1199 2355 y FG(=)p FA(\(2)p FG(R)q FA(\))p
Fx(g)p FA(,)g(b)m(y)20 b(\(A.136\),)25 b(w)m(e)c(ha)m(v)m(e)h(that)f
(either)e Fx(j)p FA(\()p FG(x)p Fx(\000)p FG(T)2982 2369
y Fv(Y)3023 2377 y Fp(\003)3060 2369 y Fv(;R)3133 2377
y Fp(\003)3173 2355 y FG(x)3225 2369 y Ft(\003)3265 2355
y FA(\))3300 2322 y Ft(0)3323 2355 y Fx(j)26 b(\025)456
2463 y FG(cq)41 b FA(or)e FG(x)749 2477 y Fv(N)855 2463
y Fx(\024)45 b FA(\025)-52 b FG(y)1009 2477 y Fv(N)1076
2463 y FA(.)65 b(Ho)m(w)m(ev)m(er,)43 b(the)c(latter)g(cannot)g(hold,)g
(since,)h(b)m(y)f(Lemma)g(2.16,)456 2571 y(\(A.145\),)33
b(\(A.139\))f(and)e(\(A.140\),)j(w)m(e)e(ha)m(v)m(e)h(that)1173
2718 y FG(x)1225 2732 y Fv(N)1313 2718 y Fx(\000)26 b
FA(\025)-51 b FG(y)1449 2732 y Fv(N)1598 2718 y FA(=)83
b Fx(j)6 b FA(\025)-51 b FG(x)1829 2732 y Fv(N)1917 2718
y Fx(\000)26 b FA(\025)-51 b FG(y)2053 2732 y Fv(N)2120
2718 y Fx(j)20 b FA(+)g FG(x)2308 2732 y Fv(N)2395 2718
y Fx(\000)26 b FA(\025)-51 b FG(x)2538 2732 y Fv(N)2631
2718 y Fx(\025)1598 2896 y(\025)1762 2835 y FA(1)p 1762
2875 46 4 v 1762 2958 a(2)1833 2896 y Fx(j)6 b FA(\025)-51
b FG(x)20 b Fx(\000)26 b FA(\025)-51 b FG(y)s Fx(j)20
b FA(+)g FG(x)2257 2910 y Fv(N)2345 2896 y Fx(\000)25
b FA(\025)-50 b FG(x)2488 2910 y Fv(N)2580 2896 y Fx(\025)1598
3057 y(\025)98 b FA(const)16 b FG(R)21 b Fx(\000)35 b
FA(const)16 b FG(l)42 b(>)e FA(0)15 b FG(:)456 3204 y
FA(Therefore,)44 b(from)e(these)g(considerations,)i(w)m(e)e(ha)m(v)m(e)
h(that)g(if)d FG(x)i FA(is)f(not)h(ab)s(o)m(v)m(e)h(the)456
3312 y(h)m(yp)s(erplane)21 b Fx(f)p FG(x)1013 3326 y
Fv(N)1106 3312 y FA(=)k(3)p FG(C)1312 3327 y Fv(])1344
3312 y FG(q)1388 3279 y Fy(2)1428 3312 y FG(=)p FA(\(2)p
FG(R)q FA(\))p Fx(g)p FA(,)i(then)c Fx(j)p FA(\()p FG(x)7
b Fx(\000)g FG(T)2205 3326 y Fv(Y)2246 3334 y Fp(\003)2283
3326 y Fv(;R)2356 3334 y Fp(\003)2395 3312 y FG(x)2447
3326 y Ft(\003)2487 3312 y FA(\))2522 3279 y Ft(0)2545
3312 y Fx(j)26 b(\025)f FG(cq)s FA(.)39 b(F)-8 b(or)24
b(this)f(reason,)456 3420 y(recalling)28 b(\(A.141\),)756
3518 y Fu(\014)756 3572 y(\014)786 3595 y FA(\()p FG(x)21
b Fx(\000)f FG(x)1037 3609 y Ft(\003)1076 3595 y FA(\))1111
3557 y Ft(0)1135 3518 y Fu(\014)1135 3572 y(\014)1248
3595 y Fx(\025)1402 3490 y Fu(\014)1402 3545 y(\014)1402
3599 y(\014)1432 3521 y(\000)1474 3595 y FG(x)g Fx(\000)g
FA(\()p FG(T)1725 3609 y Fv(Y)1766 3617 y Fp(\003)1802
3609 y Fv(;R)1875 3617 y Fp(\003)1915 3595 y FG(x)1967
3609 y Ft(\003)2006 3595 y FA(\))2041 3521 y Fu(\001)2083
3544 y Ft(0)2107 3490 y Fu(\014)2107 3545 y(\014)2107
3599 y(\014)2157 3595 y Fx(\000)2248 3490 y Fu(\014)2248
3545 y(\014)2248 3599 y(\014)2278 3521 y(\000)2320 3595
y FA(\()p FG(T)2408 3609 y Fv(Y)2449 3617 y Fp(\003)2485
3609 y Fv(;R)2558 3617 y Fp(\003)2598 3595 y FG(x)2650
3609 y Ft(\003)2690 3595 y FA(\))g Fx(\000)g FG(x)2888
3609 y Ft(\003)2928 3521 y Fu(\001)2969 3544 y Ft(0)2993
3490 y Fu(\014)2993 3545 y(\014)2993 3599 y(\014)3048
3595 y Fx(\025)1248 3807 y(\025)83 b FG(cq)23 b Fx(\000)1621
3745 y FA(const)16 b FG(q)p 1606 3786 282 4 v 1712 3869
a(R)1938 3807 y Fx(\025)2059 3745 y FG(cq)p 2059 3786
84 4 v 2078 3869 a FA(2)2167 3807 y FG(;)456 3985 y FA(whic)m(h,)33
b(together)i(with)e(\(A.143\))j(and)d(\(A.134\),)k(yields)32
b(the)i(pro)s(of)e(of)i(\(A.144\))i(also)456 4093 y(when)i
FG(x)h FA(is)g(not)h(ab)s(o)m(v)m(e)g(the)g(h)m(yp)s(erplane)e
Fx(f)p FG(x)2071 4107 y Fv(N)2179 4093 y FA(=)i(3)p FG(C)2400
4108 y Fv(])2432 4093 y FG(q)2476 4060 y Fy(2)2515 4093
y FG(=)p FA(\(2)p FG(R)q FA(\))p Fx(g)p FA(.)70 b(This)37
b(ends)i(the)456 4201 y(pro)s(of)29 b(of)i(\(A.144\).)555
4348 y(W)-8 b(e)32 b(no)m(w)e(exploit)g(\(A.131\))r(,)h(\(A.134\))i
(and)c(\(A.144\))k(to)e(get)h(that)992 4557 y Fx(\000)p
FG(d)1110 4589 y Fy(\006\()1199 4572 y(\025)1188 4589
y Fv(Y)16 b(;)1279 4572 y Fy(\025)1265 4589 y Fv(R)p
Fy(\))1349 4557 y FA(\()p FG(x)1436 4571 y Ft(\003)1476
4557 y FA(\))26 b Fx(\025)f(\000)p FG(H)1780 4571 y Fy(0)1818
4557 y FA(\()p FG(u)p FA(\()p FG(x)1992 4571 y Ft(\003)2033
4557 y FA(\)\))20 b Fx(\000)36 b FA(const)2475 4496 y
FG(q)p 2462 4536 70 4 v 2462 4620 a(R)2562 4557 y FA(+)2663
4494 y FG(C)2728 4509 y Fv(])2775 4494 y FG(q)2819 4461
y Fy(2)p 2663 4536 196 4 v 2703 4620 a FA(2)p FG(R)2883
4557 y(:)456 4735 y FA(This)26 b(ends)i(the)g(pro)s(of)g(of)36
b(\(A.137\))30 b(also)f(in)e(the)h(case)i(in)d(whic)m(h)g
FG(x)2738 4749 y Ft(\003)2805 4735 y FA(is)h(in)f(the)h(in)m(terior)456
4852 y(of)i(\006\()674 4829 y(\025)660 4852 y FG(Y)20
b(;)793 4829 y FA(\025)773 4852 y FG(R)q FA(\).)555 5108
y(Ha)m(ving)29 b(completed)g(the)g(pro)s(of)f(of)h(\(A.137\),)j(w)m(e)d
(no)m(w)g(deduce)f(the)h(con)m(tradiction)456 5216 y(that)42
b(will)d(\014nish)g(the)j(pro)s(of)f(of)g(\(A.135\).)77
b(T)-8 b(o)42 b(this)f(end,)j(note)e(that,)k(b)m(y)41
b(\(A.19\),)p eop
%%Page: 126 126
126 125 bop 456 251 a Fy(126)140 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(letting)35 b FG(s)792 464 y Ft(\003)866
450 y FA(:=)f FG(u)p FA(\()p FG(x)1135 464 y Ft(\003)1175
450 y FA(\),)k(w)m(e)f(ha)m(v)m(e)g(that)g(the)f(signed)f(distance)g(b)
s(et)m(w)m(een)i(the)f FG(s)3198 464 y Ft(\003)3237 450
y FA(-lev)m(el)456 567 y(set)31 b(of)f FG(g)744 599 y
Fj(S)-6 b Fy(\()825 582 y(\025)814 599 y Fv(Y)10 b(;)900
582 y Fy(\025)885 599 y Fv(R)q Fy(\))1000 567 y FA(and)30
b(\006\()1292 544 y(\025)1278 567 y FG(Y)20 b(;)1411
544 y FA(\025)1391 567 y FG(R)q FA(\))31 b(is)e(bigger)h(than)1561
794 y FG(H)1637 808 y Fy(0)1676 794 y FA(\()p FG(u)p
FA(\()p FG(x)1850 808 y Ft(\003)1890 794 y FA(\)\))21
b Fx(\000)2081 733 y FA(const)p 2081 774 208 4 v 2150
857 a FG(R)2314 794 y(:)456 987 y FA(Th)m(us,)29 b(from)h(\(A.137\),)j
(b)m(y)d(taking)h FG(q)i FA(and)d FG(C)2005 1002 y Fv(])2066
987 y FA(suitably)f(large,)i(w)m(e)g(ha)m(v)m(e)g(that)1434
1148 y FG(d)1481 1179 y Fy(\006\()1570 1163 y(\025)1559
1179 y Fv(Y)17 b(;)1651 1163 y Fy(\025)1637 1179 y Fv(R)p
Fy(\))1722 1148 y FA(\()p Fi(x)p FA(\))41 b FG(>)f(d)2023
1179 y Fy(\006\()2112 1163 y(\025)2101 1179 y Fv(Y)16
b(;)2193 1163 y Fy(\025)2178 1179 y Fv(R)p Fy(\))2263
1148 y FA(\()p FG(x)2350 1162 y Ft(\003)2390 1148 y FA(\))f
FG(;)456 1326 y FA(for)32 b(an)m(y)i Fi(x)f FA(so)g(that)g
FG(g)1193 1357 y Fj(S)-5 b Fy(\()1275 1340 y(\025)1264
1357 y Fv(Y)10 b(;)1349 1340 y Fy(\025)1335 1357 y Fv(R)p
Fy(\))1419 1326 y FA(\()p Fi(x)p FA(\))31 b(=)e FG(s)1695
1340 y Ft(\003)1734 1326 y FA(.)48 b(Therefore,)34 b
FG(x)2300 1340 y Ft(\003)2372 1326 y FA(is)e(strictly)g(in)f(the)i(in)m
(terior)f(of)456 1445 y(the)e FG(s)655 1459 y Ft(\003)694
1445 y FA(-lev)m(el)h(set)g(of)f FG(g)1220 1477 y Fj(S)-6
b Fy(\()1301 1460 y(\025)1290 1477 y Fv(Y)10 b(;)1376
1460 y Fy(\025)1361 1477 y Fv(R)p Fy(\))1446 1445 y FA(,)31
b(that)g(is)1445 1623 y FG(g)1488 1655 y Fj(S)-6 b Fy(\()1569
1638 y(\025)1558 1655 y Fv(Y)10 b(;)1644 1638 y Fy(\025)1629
1655 y Fv(R)p Fy(\))1714 1623 y FA(\()p FG(x)1801 1637
y Ft(\003)1841 1623 y FA(\))25 b FG(<)g(s)2040 1637 y
Ft(\003)2105 1623 y FA(=)g FG(u)p FA(\()p FG(x)2340 1637
y Ft(\003)2379 1623 y FA(\))15 b FG(:)456 1796 y FA(This)31
b(con)m(tradicts)j(the)f(fact)i(that)e FG(g)1721 1828
y Fj(S)-5 b Fy(\()1803 1811 y(\025)1792 1828 y Fv(Y)10
b(;)1877 1811 y Fy(\025)1863 1828 y Fv(R)p Fy(\))1981
1796 y FA(touc)m(hes)34 b FG(u)f FA(b)m(y)g(ab)s(o)m(v)m(e)i(and,)e(th)
m(us,)h(yields)456 1921 y(the)c(pro)s(of)g(of)g(\(A.135\).)555
2082 y(In)22 b(the)h(ligh)m(t)e(of)i(\(A.135\),)k(some)c(elemen)m(tary)
g(trigonometry)f(implies)e(that)j FG(y)8 b FA(+)g(\025)-49
b FG(r)r(e)3377 2096 y Fv(N)456 2190 y FA(is)29 b(b)s(elo)m(w)h(the)g
(h)m(yp)s(erplane)f Fx(f)p FG(x)1526 2204 y Fv(N)1618
2190 y FA(=)c(4)p FG(C)1824 2205 y Fv(])1856 2190 y FG(q)1900
2157 y Fy(2)1940 2190 y FG(=R)q Fx(g)31 b FA(and)e(therefore)456
2365 y(\(A.146\))478 b(\006\()1347 2342 y(\025)1333 2365
y FG(Y)20 b(;)1466 2342 y FA(\025)1446 2365 y FG(R)q
FA(\))31 b(is)e(b)s(elo)m(w)h Fx(f)p FG(x)2028 2379 y
Fv(N)2121 2365 y FA(=)25 b(4)p FG(C)2327 2380 y Fv(])2359
2365 y FG(q)2403 2332 y Fy(2)2442 2365 y FG(=R)q Fx(g)p
FA(.)456 2527 y(Let)42 b(us)f(no)m(w)h(\014x)f(a)h(small)f(constan)m(t)
i FG(c)1853 2541 y Ft(\003)1937 2527 y FG(>)h FA(0.)75
b(If)41 b FG(q)s(=R)i FA(is)e(assumed)g(to)i(b)s(e)e(small)456
2634 y(enough)33 b(\(p)s(ossibly)e(in)h(dep)s(endence)g(of)h
FG(c)1904 2648 y Ft(\003)1944 2634 y FG(=C)2054 2649
y Fv(])2086 2634 y FA(\),)i(then)e(\(A.135\))j(and)c(Corollary)g(B.9)
456 2742 y(imply)c(that)456 2908 y(\(A.147\))345 b(\006\()1214
2885 y(\025)1200 2908 y FG(Y)20 b(;)1333 2885 y FA(\025)1313
2908 y FG(R)q FA(\))g Fx(\\)g(fj)p FG(x)1641 2875 y Ft(0)1665
2908 y Fx(j)26 b(\025)f FG(c)1851 2922 y Ft(\003)1890
2908 y FG(q)s Fx(g)31 b FA(is)e(b)s(elo)m(w)h Fx(f)p
FG(x)2456 2922 y Fv(N)2549 2908 y FA(=)25 b(0)p Fx(g)p
FA(.)555 3069 y(W)-8 b(e)32 b(no)m(w)e(sho)m(w)g(that)h(the)g(ab)s(o)m
(v)m(e)h(geometric)f(considerations)e(imply)f(that)1079
3218 y(outside)i Fx(fj)p FG(x)1515 3185 y Ft(0)1539 3218
y Fx(j)25 b(\024)g FG(c)1724 3232 y Ft(\003)1764 3218
y FG(q)s Fx(g)20 b(\002)g(f)p FG(x)2061 3232 y Fv(N)2154
3218 y FG(>)25 b FA(0)p Fx(g)p FA(,)1079 3376 y(\006\()1194
3353 y(\025)1180 3376 y FG(Y)20 b(;)1313 3353 y FA(\025)1293
3376 y FG(R)q FA(\))31 b(is)e(at)i(distance)g(greater)g(than)f
FG(q)2550 3343 y Fy(2)2589 3376 y FG(=)p FA(\(4)p FG(R)q
FA(\))1079 3516 y(in)f(the)i(in)m(terior)e(of)i(\006\()p
FG(Y)5 b(;)15 b(R)q FA(\).)456 3374 y(\(A.148\))456 3682
y(F)-8 b(or)31 b(the)f(pro)s(of)g(of)g(\(A.148\),)k(\014rst)29
b(notice)i(that)456 3843 y(\(A.149\))693 b(\025)-51 b
FG(y)1486 3857 y Fv(N)1573 3843 y Fx(\000)24 b FA(\025)-49
b FG(r)43 b Fx(\025)d(\000)p FA(const)19 b(\025)-49 b
FG(r)43 b Fx(\025)d FG(y)2392 3857 y Fv(N)456 4005 y
FA(if)29 b FG(c)i FA(in)e(\(A.133\))j(is)e(c)m(hosen)h(suitably)d
(small.)39 b(Let)31 b(us)f(no)m(w)g(consider)g(the)g(surface)1401
4175 y(\006)1467 4189 y Fv(t)1496 4175 y FA(\()1545 4152
y(\025)1531 4175 y FG(Y)20 b(;)1664 4152 y FA(\025)1644
4175 y FG(R)q FA(\))41 b(:=)f(\006\()2041 4152 y(\025)2027
4175 y FG(Y)20 b(;)2161 4152 y FA(\025)2140 4175 y FG(R)r
FA(\))g(+)g FG(te)2432 4189 y Fv(N)456 4345 y FA(with)33
b FG(t)h Fx(\025)f FA(0.)55 b(By)37 b(\(A.149\))r(,)g(it)e(follo)m(ws)f
(that,)j(giv)m(en)e Fi(x)f Fx(2)f FA(\006\()2618 4322
y(\025)2604 4345 y FG(Y)20 b(;)2737 4322 y FA(\025)2717
4345 y FG(R)q FA(\),)37 b(then)e Fi(x)3128 4359 y Fv(N)3229
4345 y Fx(\025)e FG(y)3378 4359 y Fv(N)456 4453 y FA(and)c(so)1465
4577 y Fx(j)p FA(\()p Fi(x)21 b FA(+)f FG(te)1744 4591
y Fv(N)1812 4577 y FA(\))g Fx(\000)g FG(y)s Fx(j)26 b(\025)f(j)p
Fi(x)20 b Fx(\000)g FG(y)s Fx(j)15 b FG(:)456 4719 y
FA(Hence,)456 4880 y(\(A.150\))610 b FG(d)1411 4899 y
Fy(\006\()p Fv(Y)t(;R)p Fy(\))1639 4880 y FA(\()p Fi(x)p
FA(\))27 b Fx(\024)e FG(d)1911 4899 y Fy(\006\()p Fv(Y)t(;R)p
Fy(\))2139 4880 y FA(\()p Fi(x)c FA(+)f FG(te)2393 4894
y Fv(N)2460 4880 y FA(\))15 b FG(:)456 5045 y FA(Recalling)30
b(\(A.147\))s(,)g(w)m(e)h(no)m(w)f(select)h(the)g(\014rst)f
FG(t)25 b Fx(\025)g FA(0)30 b(for)g(whic)m(h)456 5216
y(\(A.151\))411 b Fi(S)40 b FA(:=)g(\006)1476 5230 y
Fv(t)1505 5216 y FA(\()1554 5193 y(\025)1540 5216 y FG(Y)21
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5230 y Ft(\003)2231 5216 y FG(q)s Fx(g)c(\\)f(f)p FG(x)2519
5230 y Fv(N)2612 5216 y FA(=)25 b(0)p Fx(g)41 b(6)p FA(=)f
Fx(;)15 b FG(:)p eop
%%Page: 127 127
127 126 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)
f Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)140 b(127)456
453 y FA(W)-8 b(e)39 b(also)g(denote)g(b)m(y)g Fi(S)1316
420 y Ft(\000)1413 453 y FA(the)g(p)s(ortion)e(of)i(\006)2085
467 y Fv(t)2114 453 y FA(\()2163 430 y(\025)2149 453
y FG(Y)20 b(;)2282 430 y FA(\025)2262 453 y FG(R)r FA(\))38
b(whic)m(h)g(is)f(b)s(elo)m(w)h Fx(f)p FG(x)3137 467
y Fv(N)3244 453 y FA(=)g(0)p Fx(g)456 570 y FA(\(with)30
b(resp)s(ect)i(to)h(the)f FG(e)1325 584 y Fv(N)1392 570
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1141 y(\(A.152\))914 b(\025)-52 b FG(y)1706 1155 y Fv(N)1793
1141 y FA(+)20 b FG(t)g FA(+)k(\025)-49 b FG(r)28 b Fx(\025)d
FA(0)456 1603 y(and,)30 b(b)m(y)h(\(A.132\))i(and)d(\(A.29\))r(,)g
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b FG(y)1680 2028 y Ft(0)1703 2065 y Fx(j)26 b(\024)1453
2201 y(\024)1607 2123 y Fu(\014)1607 2178 y(\014)1637
2201 y FA(\()6 b(\025)-51 b FG(y)23 b Fx(\000)d FG(T)1884
2215 y Fv(Y)1925 2223 y Fp(\003)1961 2215 y Fv(;R)2034
2223 y Fp(\003)2074 2201 y FG(x)2126 2215 y Ft(\003)2166
2201 y FA(\))2201 2163 y Ft(0)2224 2123 y Fu(\014)2224
2178 y(\014)2275 2201 y FA(+)2366 2123 y Fu(\014)2366
2178 y(\014)2396 2201 y FA(\()p FG(T)2484 2215 y Fv(Y)2525
2223 y Fp(\003)2561 2215 y Fv(;R)2634 2223 y Fp(\003)2674
2201 y FG(x)2726 2215 y Ft(\003)2766 2201 y FA(\))2801
2163 y Ft(0)2824 2123 y Fu(\014)2824 2178 y(\014)2880
2201 y Fx(\024)1453 2368 y(\024)83 b FG(cq)23 b FA(+)1841
2307 y FG(q)p 1811 2347 105 4 v 1811 2431 a(C)1876 2445
y Ft(\003)1951 2368 y FA(=)-1566 b(\(A.153\))1453 2540
y(=)85 b(~)-47 b FG(c)1646 2554 y Ft(\003)1686 2540 y
FG(q)18 b(;)456 3003 y FA(where)456 3485 y(\(A.154\))933
b(~)-47 b FG(c)1724 3499 y Ft(\003)1804 3485 y FA(:=)40
b FG(c)21 b FA(+)2130 3424 y(1)p 2101 3464 V 2101 3548
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3627 2733 M 3632 2715 I 3633 2713 3635 2712 3636 2713 -c 3638 2713 3639 2715 3638 2716 -c 3633 2734 I 3632 2736 3630 2737 3629 2736 -c 3627 2736 3626 2734 3627 2733 -c C 
3640 2691 M 3645 2673 I 3646 2671 3647 2670 3649 2671 -c 3651 2671 3652 2673 3651 2675 -c 3646 2693 I 3645 2694 3643 2695 3642 2695 -c 3640 2694 3639 2692 3640 2691 -c C 
3653 2649 M 3658 2631 I 3659 2630 3660 2629 3662 2629 -c 3664 2630 3665 2631 3664 2633 -c 3659 2651 I 3658 2653 3656 2654 3655 2653 -c 3653 2652 3652 2651 3653 2649 -c C 
3666 2607 M 3671 2590 I 3672 2588 3673 2587 3675 2587 -c 3677 2588 3678 2590 3677 2591 -c 3672 2609 I 3671 2611 3669 2612 3668 2611 -c 3666 2611 3665 2609 3666 2607 -c C 
3679 2566 M 3684 2548 I 3685 2546 3686 2545 3688 2546 -c 3690 2546 3691 2548 3690 2550 -c 3685 2567 I 3684 2569 3682 2570 3681 2570 -c 3679 2569 3678 2567 3679 2566 -c C 
3692 2524 M 3697 2506 I 3697 2504 3699 2504 3701 2504 -c 3703 2505 3703 2506 3703 2508 -c 3697 2526 I 3697 2527 3695 2528 3693 2528 -c 3692 2527 3691 2526 3692 2524 -c C 
3704 2482 M 3710 2464 I 3710 2463 3712 2462 3714 2462 -c 3716 2463 3716 2465 3716 2466 -c 3710 2484 I 3710 2486 3708 2487 3706 2486 -c 3705 2486 3704 2484 3704 2482 -c C 
3717 2441 M 3723 2423 I 3723 2421 3725 2420 3727 2421 -c 3728 2421 3729 2423 3729 2424 -c 3723 2442 I 3723 2444 3721 2445 3719 2444 -c 3718 2444 3717 2442 3717 2441 -c C 
3730 2399 M 3736 2381 I 3736 2379 3738 2378 3740 2379 -c 3741 2379 3742 2381 3742 2383 -c 3736 2401 I 3736 2402 3734 2403 3732 2403 -c 3731 2402 3730 2401 3730 2399 -c C 
3743 2357 M 3749 2339 I 3749 2338 3751 2337 3753 2337 -c 3754 2338 3755 2340 3755 2341 -c 3749 2359 I 3749 2361 3747 2362 3745 2361 -c 3744 2361 3743 2359 3743 2357 -c C 
3756 2315 M 3762 2298 I 3762 2296 3764 2295 3766 2296 -c 3767 2296 3768 2298 3768 2299 -c 3762 2317 I 3762 2319 3760 2320 3758 2319 -c 3757 2319 3756 2317 3756 2315 -c C 
3769 2274 M 3775 2256 I 3775 2254 3777 2253 3779 2254 -c 3780 2254 3781 2256 3781 2258 -c 3775 2276 I 3775 2277 3773 2278 3771 2278 -c 3770 2277 3769 2275 3769 2274 -c C 
3782 2232 M 3788 2214 I 3788 2213 3790 2212 3792 2212 -c 3793 2213 3794 2214 3794 2216 -c 3788 2234 I 3788 2236 3786 2236 3784 2236 -c 3783 2235 3782 2234 3782 2232 -c C 
3795 2190 M 3801 2173 I 3801 2171 3803 2170 3805 2171 -c 3806 2171 3807 2173 3807 2174 -c 3801 2192 I 3801 2194 3799 2195 3797 2194 -c 3796 2194 3795 2192 3795 2190 -c C 
3808 2149 M 3814 2131 I 3814 2129 3816 2128 3818 2129 -c 3819 2129 3820 2131 3820 2133 -c 3814 2151 I 3814 2152 3812 2153 3810 2153 -c 3809 2152 3808 2150 3808 2149 -c C 
3821 2107 M 3827 2089 I 3827 2087 3829 2087 3831 2087 -c 3832 2088 3833 2089 3833 2091 -c 3827 2109 I 3827 2111 3825 2111 3823 2111 -c 3822 2110 3821 2109 3821 2107 -c C 
3834 2065 M 3840 2047 I 3840 2046 3842 2045 3844 2045 -c 3845 2046 3846 2048 3846 2049 -c 3840 2067 I 3840 2069 3838 2070 3836 2069 -c 3835 2069 3834 2067 3834 2065 -c C 
3847 2024 M 3853 2006 I 3853 2004 3855 2003 3857 2004 -c 3858 2004 3859 2006 3859 2008 -c 3853 2025 I 3853 2027 3851 2028 3849 2028 -c 3848 2027 3847 2025 3847 2024 -c C 
3860 1982 M 3866 1964 I 3866 1962 3868 1961 3870 1962 -c 3871 1962 3872 1964 3872 1966 -c 3866 1984 I 3866 1985 3864 1986 3862 1986 -c 3861 1985 3860 1984 3860 1982 -c C 
3873 1940 M 3879 1922 I 3879 1921 3881 1920 3883 1920 -c 3884 1921 3885 1923 3885 1924 -c 3879 1942 I 3879 1944 3877 1945 3875 1944 -c 3873 1944 3873 1942 3873 1940 -c C 
3886 1899 M 3892 1881 I 3892 1879 3894 1878 3896 1879 -c 3897 1879 3898 1881 3898 1882 -c 3892 1900 I 3891 1902 3890 1903 3888 1902 -c 3886 1902 3886 1900 3886 1899 -c C 
3899 1857 M 3905 1839 I 3905 1837 3907 1836 3908 1837 -c 3910 1837 3911 1839 3911 1841 -c 3905 1859 I 3904 1860 3903 1861 3901 1861 -c 3899 1860 3899 1858 3899 1857 -c C 
3912 1815 M 3918 1797 I 3918 1796 3920 1795 3921 1795 -c 3923 1796 3924 1797 3924 1799 -c 3918 1817 I 3917 1819 3916 1819 3914 1819 -c 3912 1818 3911 1817 3912 1815 -c C 
3925 1773 M 3931 1756 I 3931 1754 3933 1753 3934 1753 -c 3936 1754 3937 1756 3936 1757 -c 3931 1775 I 3930 1777 3929 1778 3927 1777 -c 3925 1777 3924 1775 3925 1773 -c C 
3938 1732 M 3943 1714 I 3944 1712 3946 1711 3947 1712 -c 3949 1712 3950 1714 3949 1716 -c 3944 1734 I 3943 1735 3942 1736 3940 1736 -c 3938 1735 3937 1733 3938 1732 -c C 
3951 1690 M 3956 1672 I 3957 1671 3959 1670 3960 1670 -c 3962 1671 3963 1672 3962 1674 -c 3957 1692 I 3956 1693 3954 1694 3953 1694 -c 3951 1693 3950 1692 3951 1690 -c C 
3964 1648 M 3969 1630 I 3970 1629 3972 1628 3973 1628 -c 3975 1629 3976 1631 3975 1632 -c 3970 1650 I 3969 1652 3967 1653 3966 1652 -c 3964 1652 3963 1650 3964 1648 -c C 
3977 1607 M 3982 1589 I 3983 1587 3985 1586 3986 1587 -c 3988 1587 3989 1589 3988 1591 -c 3983 1609 I 3982 1610 3980 1611 3979 1611 -c 3977 1610 3976 1608 3977 1607 -c C 
3990 1565 M 3995 1547 I 3996 1545 3998 1545 3999 1545 -c 4001 1546 4002 1547 4001 1549 -c 3996 1567 I 3995 1568 3993 1569 3992 1569 -c 3990 1568 3989 1567 3990 1565 -c C 
4003 1523 M 4008 1505 I 4009 1504 4011 1503 4012 1503 -c 4014 1504 4015 1506 4014 1507 -c 4009 1525 I 4008 1527 4006 1528 4005 1527 -c 4003 1527 4002 1525 4003 1523 -c C 
4016 1482 M 4021 1464 I 4022 1462 4023 1461 4025 1462 -c 4027 1462 4028 1464 4027 1466 -c 4022 1483 I 4021 1485 4019 1486 4018 1485 -c 4016 1485 4015 1483 4016 1482 -c C 
4029 1440 M 4034 1422 I 4035 1420 4036 1419 4038 1420 -c 4040 1420 4041 1422 4040 1424 -c 4035 1442 I 4034 1443 4032 1444 4031 1444 -c 4029 1443 4028 1441 4029 1440 -c C 
4042 1398 M 4047 1380 I 4048 1379 4049 1378 4051 1378 -c 4053 1379 4054 1380 4053 1382 -c 4048 1400 I 4047 1402 4045 1403 4044 1402 -c 4042 1402 4041 1400 4042 1398 -c C 
4055 1356 M 4060 1339 I 4061 1337 4062 1336 4064 1337 -c 4066 1337 4067 1339 4066 1340 -c 4061 1358 I 4060 1360 4058 1361 4057 1360 -c 4055 1360 4054 1358 4055 1356 -c C 
4068 1315 M 4073 1297 I 4074 1295 4075 1294 4077 1295 -c 4079 1295 4080 1297 4079 1299 -c 4074 1317 I 4073 1318 4071 1319 4070 1319 -c 4068 1318 4067 1316 4068 1315 -c C 
4081 1273 M 4086 1255 I 4087 1253 4088 1253 4090 1253 -c 4092 1254 4093 1255 4092 1257 -c 4087 1275 I 4086 1277 4084 1277 4083 1277 -c 4081 1276 4080 1275 4081 1273 -c C 
4094 1231 M 4099 1213 I 4100 1212 4101 1211 4103 1211 -c 4105 1212 4106 1214 4105 1215 -c 4099 1233 I 4099 1235 4097 1236 4096 1235 -c 4094 1235 4093 1233 4094 1231 -c C 
4106 1190 M 4112 1172 I 4113 1170 4114 1169 4116 1170 -c 4118 1170 4119 1172 4118 1174 -c 4112 1191 I 4112 1193 4110 1194 4109 1193 -c 4107 1193 4106 1191 4106 1190 -c C 
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4145 1064 M 4151 1047 I 4151 1045 4153 1044 4155 1045 -c 4157 1045 4157 1047 4157 1049 -c 4151 1066 I 4151 1068 4149 1069 4147 1068 -c 4146 1068 4145 1066 4145 1064 -c C 
4158 1023 M 4164 1005 I 4164 1003 4166 1002 4168 1003 -c 4170 1003 4170 1005 4170 1007 -c 4164 1025 I 4164 1026 4162 1027 4160 1027 -c 4159 1026 4158 1024 4158 1023 -c C 
4171 981 M 4177 963 I 4177 962 4179 961 4181 961 -c 4182 962 4183 963 4183 965 -c 4177 983 I 4177 985 4175 986 4173 985 -c 4172 985 4171 983 4171 981 -c C 
4184 939 M 4190 922 I 4190 920 4192 919 4194 920 -c 4195 920 4196 922 4196 923 -c 4190 941 I 4190 943 4188 944 4186 943 -c 4185 943 4184 941 4184 939 -c C 
4197 898 M 4203 880 I 4203 878 4205 877 4207 878 -c 4208 878 4209 880 4209 882 -c 4203 900 I 4203 901 4201 902 4199 902 -c 4198 901 4197 899 4197 898 -c C 
4210 856 M 4216 838 I 4216 837 4218 836 4220 836 -c 4221 837 4222 838 4222 840 -c 4216 858 I 4216 860 4214 861 4212 860 -c 4211 859 4210 858 4210 856 -c C 
4223 814 M 4229 797 I 4229 795 4231 794 4233 794 -c 4234 795 4235 797 4235 798 -c 4229 816 I 4229 818 4227 819 4225 818 -c 4224 818 4223 816 4223 814 -c C 
4236 773 M 4242 755 I 4242 753 4244 752 4246 753 -c 4247 753 4248 755 4248 757 -c 4242 774 I 4242 776 4240 777 4238 777 -c 4237 776 4236 774 4236 773 -c C 
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4262 689 M 4268 671 I 4268 670 4270 669 4272 669 -c 4273 670 4274 672 4274 673 -c 4268 691 I 4268 693 4266 694 4264 693 -c 4262 693 4262 691 4262 689 -c C 
4275 648 M 4281 630 I 4281 628 4283 627 4285 628 -c 4286 628 4287 630 4287 631 -c 4281 649 I 4280 651 4279 652 4277 651 -c 4275 651 4275 649 4275 648 -c C 
4288 606 M 4294 588 I 4294 586 4296 585 4297 586 -c 4299 586 4300 588 4300 590 -c 4294 608 I 4293 609 4292 610 4290 610 -c 4288 609 4287 608 4288 606 -c C 
4301 564 M 4307 546 I 4307 545 4309 544 4310 544 -c 4312 545 4313 547 4313 548 -c 4307 566 I 4306 568 4305 569 4303 568 -c 4301 568 4300 566 4301 564 -c C 
4314 522 M 4320 505 I 4320 503 4322 502 4323 502 -c 4325 503 4326 505 4325 506 -c 4320 524 I 4319 526 4318 527 4316 526 -c 4314 526 4313 524 4314 522 -c C 
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4353 397 M 4358 379 I 4359 378 4361 377 4362 377 -c 4364 378 4365 380 4364 381 -c 4359 399 I 4358 401 4357 402 4355 401 -c 4353 401 4352 399 4353 397 -c C 
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784 3090 M 802 3090 I 804 3090 805 3091 805 3093 -c 805 3095 804 3096 802 3096 -c 784 3096 I 782 3096 780 3095 780 3093 -c 780 3091 782 3090 784 3090 -c C 
827 3090 M 846 3090 I 848 3090 849 3091 849 3093 -c 849 3095 848 3096 846 3096 -c 827 3096 I 826 3096 824 3095 824 3093 -c 824 3091 826 3090 827 3090 -c C 
871 3090 M 890 3090 I 892 3090 893 3091 893 3093 -c 893 3095 892 3096 890 3096 -c 871 3096 I 869 3096 868 3095 868 3093 -c 868 3091 869 3090 871 3090 -c C 
915 3090 M 933 3090 I 935 3090 936 3091 936 3093 -c 936 3095 935 3096 933 3096 -c 915 3096 I 913 3096 912 3095 912 3093 -c 912 3091 913 3090 915 3090 -c C 
958 3090 M 977 3090 I 979 3090 980 3091 980 3093 -c 980 3095 979 3096 977 3096 -c 958 3096 I 957 3096 955 3095 955 3093 -c 955 3091 957 3090 958 3090 -c C 
1002 3090 M 1021 3090 I 1023 3090 1024 3091 1024 3093 -c 1024 3095 1023 3096 1021 3096 -c 1002 3096 I 1000 3096 999 3095 999 3093 -c 999 3091 1000 3090 1002 3090 -c C 
1046 3090 M 1065 3090 I 1066 3090 1068 3091 1068 3093 -c 1068 3095 1066 3096 1065 3096 -c 1046 3096 I 1044 3096 1043 3095 1043 3093 -c 1043 3091 1044 3090 1046 3090 -c C 
1090 3090 M 1108 3090 I 1110 3090 1111 3091 1111 3093 -c 1111 3095 1110 3096 1108 3096 -c 1090 3096 I 1088 3096 1086 3095 1086 3093 -c 1086 3091 1088 3090 1090 3090 -c C 
1133 3090 M 1152 3090 I 1154 3090 1155 3091 1155 3093 -c 1155 3095 1154 3096 1152 3096 -c 1133 3096 I 1132 3096 1130 3095 1130 3093 -c 1130 3091 1132 3090 1133 3090 -c C 
1177 3090 M 1196 3090 I 1198 3090 1199 3091 1199 3093 -c 1199 3095 1198 3096 1196 3096 -c 1177 3096 I 1175 3096 1174 3095 1174 3093 -c 1174 3091 1175 3090 1177 3090 -c C 
1221 3090 M 1240 3090 I 1241 3090 1243 3091 1243 3093 -c 1243 3095 1241 3096 1240 3096 -c 1221 3096 I 1219 3096 1218 3095 1218 3093 -c 1218 3091 1219 3090 1221 3090 -c C 
1265 3090 M 1283 3090 I 1285 3090 1286 3091 1286 3093 -c 1286 3095 1285 3096 1283 3096 -c 1265 3096 I 1263 3096 1261 3095 1261 3093 -c 1261 3091 1263 3090 1265 3090 -c C 
1308 3090 M 1327 3090 I 1329 3090 1330 3091 1330 3093 -c 1330 3095 1329 3096 1327 3096 -c 1308 3096 I 1307 3096 1305 3095 1305 3093 -c 1305 3091 1307 3090 1308 3090 -c C 
1352 3090 M 1371 3090 I 1373 3090 1374 3091 1374 3093 -c 1374 3095 1373 3096 1371 3096 -c 1352 3096 I 1350 3096 1349 3095 1349 3093 -c 1349 3091 1350 3090 1352 3090 -c C 
1396 3090 M 1415 3090 I 1416 3090 1418 3091 1418 3093 -c 1418 3095 1416 3096 1415 3096 -c 1396 3096 I 1394 3096 1393 3095 1393 3093 -c 1393 3091 1394 3090 1396 3090 -c C 
1439 3090 M 1458 3090 I 1460 3090 1461 3091 1461 3093 -c 1461 3095 1460 3096 1458 3096 -c 1439 3096 I 1438 3096 1436 3095 1436 3093 -c 1436 3091 1438 3090 1439 3090 -c C 
1483 3090 M 1502 3090 I 1504 3090 1505 3091 1505 3093 -c 1505 3095 1504 3096 1502 3096 -c 1483 3096 I 1481 3096 1480 3095 1480 3093 -c 1480 3091 1481 3090 1483 3090 -c C 
1527 3090 M 1546 3090 I 1547 3090 1549 3091 1549 3093 -c 1549 3095 1547 3096 1546 3096 -c 1527 3096 I 1525 3096 1524 3095 1524 3093 -c 1524 3091 1525 3090 1527 3090 -c C 
1571 3090 M 1589 3090 I 1591 3090 1592 3091 1592 3093 -c 1592 3095 1591 3096 1589 3096 -c 1571 3096 I 1569 3096 1567 3095 1567 3093 -c 1567 3091 1569 3090 1571 3090 -c C 
1614 3090 M 1633 3090 I 1635 3090 1636 3091 1636 3093 -c 1636 3095 1635 3096 1633 3096 -c 1614 3096 I 1613 3096 1611 3095 1611 3093 -c 1611 3091 1613 3090 1614 3090 -c C 
1658 3090 M 1677 3090 I 1679 3090 1680 3091 1680 3093 -c 1680 3095 1679 3096 1677 3096 -c 1658 3096 I 1656 3096 1655 3095 1655 3093 -c 1655 3091 1656 3090 1658 3090 -c C 
1702 3090 M 1721 3090 I 1722 3090 1724 3091 1724 3093 -c 1724 3095 1722 3096 1721 3096 -c 1702 3096 I 1700 3096 1699 3095 1699 3093 -c 1699 3091 1700 3090 1702 3090 -c C 
1746 3090 M 1764 3090 I 1766 3090 1767 3091 1767 3093 -c 1767 3095 1766 3096 1764 3096 -c 1746 3096 I 1744 3096 1742 3095 1742 3093 -c 1742 3091 1744 3090 1746 3090 -c C 
1789 3090 M 1808 3090 I 1810 3090 1811 3091 1811 3093 -c 1811 3095 1810 3096 1808 3096 -c 1789 3096 I 1788 3096 1786 3095 1786 3093 -c 1786 3091 1788 3090 1789 3090 -c C 
1833 3090 M 1852 3090 I 1854 3090 1855 3091 1855 3093 -c 1855 3095 1854 3096 1852 3096 -c 1833 3096 I 1831 3096 1830 3095 1830 3093 -c 1830 3091 1831 3090 1833 3090 -c C 
1877 3090 M 1896 3090 I 1897 3090 1899 3091 1899 3093 -c 1899 3095 1897 3096 1896 3096 -c 1877 3096 I 1875 3096 1874 3095 1874 3093 -c 1874 3091 1875 3090 1877 3090 -c C 
1921 3090 M 1939 3090 I 1941 3090 1942 3091 1942 3093 -c 1942 3095 1941 3096 1939 3096 -c 1921 3096 I 1919 3096 1917 3095 1917 3093 -c 1917 3091 1919 3090 1921 3090 -c C 
1964 3090 M 1983 3090 I 1985 3090 1986 3091 1986 3093 -c 1986 3095 1985 3096 1983 3096 -c 1964 3096 I 1962 3096 1961 3095 1961 3093 -c 1961 3091 1962 3090 1964 3090 -c C 
2008 3090 M 2027 3090 I 2028 3090 2030 3091 2030 3093 -c 2030 3095 2028 3096 2027 3096 -c 2008 3096 I 2006 3096 2005 3095 2005 3093 -c 2005 3091 2006 3090 2008 3090 -c C 
2052 3090 M 2070 3090 I 2072 3090 2073 3091 2073 3093 -c 2073 3095 2072 3096 2070 3096 -c 2052 3096 I 2050 3096 2048 3095 2048 3093 -c 2048 3091 2050 3090 2052 3090 -c C 
2095 3090 M 2114 3090 I 2116 3090 2117 3091 2117 3093 -c 2117 3095 2116 3096 2114 3096 -c 2095 3096 I 2094 3096 2092 3095 2092 3093 -c 2092 3091 2094 3090 2095 3090 -c C 
2139 3090 M 2158 3090 I 2160 3090 2161 3091 2161 3093 -c 2161 3095 2160 3096 2158 3096 -c 2139 3096 I 2137 3096 2136 3095 2136 3093 -c 2136 3091 2137 3090 2139 3090 -c C 
2183 3090 M 2202 3090 I 2203 3090 2205 3091 2205 3093 -c 2205 3095 2203 3096 2202 3096 -c 2183 3096 I 2181 3096 2180 3095 2180 3093 -c 2180 3091 2181 3090 2183 3090 -c C 
2227 3090 M 2245 3090 I 2247 3090 2248 3091 2248 3093 -c 2248 3095 2247 3096 2245 3096 -c 2227 3096 I 2225 3096 2223 3095 2223 3093 -c 2223 3091 2225 3090 2227 3090 -c C 
2270 3090 M 2289 3090 I 2291 3090 2292 3091 2292 3093 -c 2292 3095 2291 3096 2289 3096 -c 2270 3096 I 2269 3096 2267 3095 2267 3093 -c 2267 3091 2269 3090 2270 3090 -c C 
2314 3090 M 2333 3090 I 2335 3090 2336 3091 2336 3093 -c 2336 3095 2335 3096 2333 3096 -c 2314 3096 I 2312 3096 2311 3095 2311 3093 -c 2311 3091 2312 3090 2314 3090 -c C 
2358 3090 M 2377 3090 I 2378 3090 2380 3091 2380 3093 -c 2380 3095 2378 3096 2377 3096 -c 2358 3096 I 2356 3096 2355 3095 2355 3093 -c 2355 3091 2356 3090 2358 3090 -c C 
2402 3090 M 2420 3090 I 2422 3090 2423 3091 2423 3093 -c 2423 3095 2422 3096 2420 3096 -c 2402 3096 I 2400 3096 2398 3095 2398 3093 -c 2398 3091 2400 3090 2402 3090 -c C 
2445 3090 M 2464 3090 I 2466 3090 2467 3091 2467 3093 -c 2467 3095 2466 3096 2464 3096 -c 2445 3096 I 2443 3096 2442 3095 2442 3093 -c 2442 3091 2443 3090 2445 3090 -c C 
2489 3090 M 2508 3090 I 2509 3090 2511 3091 2511 3093 -c 2511 3095 2509 3096 2508 3096 -c 2489 3096 I 2487 3096 2486 3095 2486 3093 -c 2486 3091 2487 3090 2489 3090 -c C 
2533 3090 M 2551 3090 I 2553 3090 2554 3091 2554 3093 -c 2554 3095 2553 3096 2551 3096 -c 2533 3096 I 2531 3096 2529 3095 2529 3093 -c 2529 3091 2531 3090 2533 3090 -c C 
2576 3090 M 2595 3090 I 2597 3090 2598 3091 2598 3093 -c 2598 3095 2597 3096 2595 3096 -c 2576 3096 I 2575 3096 2573 3095 2573 3093 -c 2573 3091 2575 3090 2576 3090 -c C 
2620 3090 M 2639 3090 I 2641 3090 2642 3091 2642 3093 -c 2642 3095 2641 3096 2639 3096 -c 2620 3096 I 2618 3096 2617 3095 2617 3093 -c 2617 3091 2618 3090 2620 3090 -c C 
2664 3090 M 2683 3090 I 2684 3090 2686 3091 2686 3093 -c 2686 3095 2684 3096 2683 3096 -c 2664 3096 I 2662 3096 2661 3095 2661 3093 -c 2661 3091 2662 3090 2664 3090 -c C 
2708 3090 M 2726 3090 I 2728 3090 2729 3091 2729 3093 -c 2729 3095 2728 3096 2726 3096 -c 2708 3096 I 2706 3096 2704 3095 2704 3093 -c 2704 3091 2706 3090 2708 3090 -c C 
2751 3090 M 2770 3090 I 2772 3090 2773 3091 2773 3093 -c 2773 3095 2772 3096 2770 3096 -c 2751 3096 I 2750 3096 2748 3095 2748 3093 -c 2748 3091 2750 3090 2751 3090 -c C 
2795 3090 M 2814 3090 I 2816 3090 2817 3091 2817 3093 -c 2817 3095 2816 3096 2814 3096 -c 2795 3096 I 2793 3096 2792 3095 2792 3093 -c 2792 3091 2793 3090 2795 3090 -c C 
2839 3090 M 2858 3090 I 2859 3090 2861 3091 2861 3093 -c 2861 3095 2859 3096 2858 3096 -c 2839 3096 I 2837 3096 2836 3095 2836 3093 -c 2836 3091 2837 3090 2839 3090 -c C 
2883 3090 M 2901 3090 I 2903 3090 2904 3091 2904 3093 -c 2904 3095 2903 3096 2901 3096 -c 2883 3096 I 2881 3096 2879 3095 2879 3093 -c 2879 3091 2881 3090 2883 3090 -c C 
2926 3090 M 2945 3090 I 2947 3090 2948 3091 2948 3093 -c 2948 3095 2947 3096 2945 3096 -c 2926 3096 I 2925 3096 2923 3095 2923 3093 -c 2923 3091 2925 3090 2926 3090 -c C 
2970 3090 M 2989 3090 I 2990 3090 2992 3091 2992 3093 -c 2992 3095 2990 3096 2989 3096 -c 2970 3096 I 2968 3096 2967 3095 2967 3093 -c 2967 3091 2968 3090 2970 3090 -c C 
3014 3090 M 3032 3090 I 3034 3090 3035 3091 3035 3093 -c 3035 3095 3034 3096 3032 3096 -c 3014 3096 I 3012 3096 3010 3095 3010 3093 -c 3010 3091 3012 3090 3014 3090 -c C 
3057 3090 M 3076 3090 I 3078 3090 3079 3091 3079 3093 -c 3079 3095 3078 3096 3076 3096 -c 3057 3096 I 3056 3096 3054 3095 3054 3093 -c 3054 3091 3056 3090 3057 3090 -c C 
3101 3090 M 3120 3090 I 3122 3090 3123 3091 3123 3093 -c 3123 3095 3122 3096 3120 3096 -c 3101 3096 I 3099 3096 3098 3095 3098 3093 -c 3098 3091 3099 3090 3101 3090 -c C 
3145 3090 M 3164 3090 I 3165 3090 3167 3091 3167 3093 -c 3167 3095 3165 3096 3164 3096 -c 3145 3096 I 3143 3096 3142 3095 3142 3093 -c 3142 3091 3143 3090 3145 3090 -c C 
3189 3090 M 3207 3090 I 3209 3090 3210 3091 3210 3093 -c 3210 3095 3209 3096 3207 3096 -c 3189 3096 I 3187 3096 3185 3095 3185 3093 -c 3185 3091 3187 3090 3189 3090 -c C 
3232 3090 M 3251 3090 I 3253 3090 3254 3091 3254 3093 -c 3254 3095 3253 3096 3251 3096 -c 3232 3096 I 3231 3096 3229 3095 3229 3093 -c 3229 3091 3231 3090 3232 3090 -c C 
3276 3090 M 3295 3090 I 3297 3090 3298 3091 3298 3093 -c 3298 3095 3297 3096 3295 3096 -c 3276 3096 I 3274 3096 3273 3095 3273 3093 -c 3273 3091 3274 3090 3276 3090 -c C 
3320 3090 M 3339 3090 I 3340 3090 3342 3091 3342 3093 -c 3342 3095 3340 3096 3339 3096 -c 3320 3096 I 3318 3096 3317 3095 3317 3093 -c 3317 3091 3318 3090 3320 3090 -c C 
3364 3090 M 3382 3090 I 3384 3090 3385 3091 3385 3093 -c 3385 3095 3384 3096 3382 3096 -c 3364 3096 I 3362 3096 3360 3095 3360 3093 -c 3360 3091 3362 3090 3364 3090 -c C 
3407 3090 M 3426 3090 I 3428 3090 3429 3091 3429 3093 -c 3429 3095 3428 3096 3426 3096 -c 3407 3096 I 3406 3096 3404 3095 3404 3093 -c 3404 3091 3406 3090 3407 3090 -c C 
3451 3090 M 3470 3090 I 3471 3090 3473 3091 3473 3093 -c 3473 3095 3471 3096 3470 3096 -c 3451 3096 I 3449 3096 3448 3095 3448 3093 -c 3448 3091 3449 3090 3451 3090 -c C 
3495 3090 M 3513 3090 I 3515 3090 3516 3091 3516 3093 -c 3516 3095 3515 3096 3513 3096 -c 3495 3096 I 3493 3096 3491 3095 3491 3093 -c 3491 3091 3493 3090 3495 3090 -c C 
3538 3090 M 3557 3090 I 3559 3090 3560 3091 3560 3093 -c 3560 3095 3559 3096 3557 3096 -c 3538 3096 I 3537 3096 3535 3095 3535 3093 -c 3535 3091 3537 3090 3538 3090 -c C 
3582 3090 M 3601 3090 I 3603 3090 3604 3091 3604 3093 -c 3604 3095 3603 3096 3601 3096 -c 3582 3096 I 3580 3096 3579 3095 3579 3093 -c 3579 3091 3580 3090 3582 3090 -c C 
3626 3090 M 3645 3090 I 3646 3090 3648 3091 3648 3093 -c 3648 3095 3646 3096 3645 3096 -c 3626 3096 I 3624 3096 3623 3095 3623 3093 -c 3623 3091 3624 3090 3626 3090 -c C 
3670 3090 M 3688 3090 I 3690 3090 3691 3091 3691 3093 -c 3691 3095 3690 3096 3688 3096 -c 3670 3096 I 3668 3096 3666 3095 3666 3093 -c 3666 3091 3668 3090 3670 3090 -c C 
3713 3090 M 3732 3090 I 3734 3090 3735 3091 3735 3093 -c 3735 3095 3734 3096 3732 3096 -c 3713 3096 I 3712 3096 3710 3095 3710 3093 -c 3710 3091 3712 3090 3713 3090 -c C 
3757 3090 M 3776 3090 I 3778 3090 3779 3091 3779 3093 -c 3779 3095 3778 3096 3776 3096 -c 3757 3096 I 3755 3096 3754 3095 3754 3093 -c 3754 3091 3755 3090 3757 3090 -c C 
3801 3090 M 3820 3090 I 3821 3090 3823 3091 3823 3093 -c 3823 3095 3821 3096 3820 3096 -c 3801 3096 I 3799 3096 3798 3095 3798 3093 -c 3798 3091 3799 3090 3801 3090 -c C 
3845 3090 M 3863 3090 I 3865 3090 3866 3091 3866 3093 -c 3866 3095 3865 3096 3863 3096 -c 3845 3096 I 3843 3096 3841 3095 3841 3093 -c 3841 3091 3843 3090 3845 3090 -c C 
3888 3090 M 3907 3090 I 3909 3090 3910 3091 3910 3093 -c 3910 3095 3909 3096 3907 3096 -c 3888 3096 I 3887 3096 3885 3095 3885 3093 -c 3885 3091 3887 3090 3888 3090 -c C 
3932 3090 M 3951 3090 I 3952 3090 3954 3091 3954 3093 -c 3954 3095 3952 3096 3951 3096 -c 3932 3096 I 3930 3096 3929 3095 3929 3093 -c 3929 3091 3930 3090 3932 3090 -c C 
3976 3090 M 3994 3090 I 3996 3090 3997 3091 3997 3093 -c 3997 3095 3996 3096 3994 3096 -c 3976 3096 I 3974 3096 3972 3095 3972 3093 -c 3972 3091 3974 3090 3976 3090 -c C 
4019 3090 M 4038 3090 I 4040 3090 4041 3091 4041 3093 -c 4041 3095 4040 3096 4038 3096 -c 4019 3096 I 4018 3096 4016 3095 4016 3093 -c 4016 3091 4018 3090 4019 3090 -c C 
4063 3090 M 4082 3090 I 4084 3090 4085 3091 4085 3093 -c 4085 3095 4084 3096 4082 3096 -c 4063 3096 I 4061 3096 4060 3095 4060 3093 -c 4060 3091 4061 3090 4063 3090 -c C 
4107 3090 M 4126 3090 I 4127 3090 4129 3091 4129 3093 -c 4129 3095 4127 3096 4126 3096 -c 4107 3096 I 4105 3096 4104 3095 4104 3093 -c 4104 3091 4105 3090 4107 3090 -c C 
4151 3090 M 4169 3090 I 4171 3090 4172 3091 4172 3093 -c 4172 3095 4171 3096 4169 3096 -c 4151 3096 I 4149 3096 4147 3095 4147 3093 -c 4147 3091 4149 3090 4151 3090 -c C 
4194 3090 M 4213 3090 I 4215 3090 4216 3091 4216 3093 -c 4216 3095 4215 3096 4213 3096 -c 4194 3096 I 4193 3096 4191 3095 4191 3093 -c 4191 3091 4193 3090 4194 3090 -c C 
4238 3090 M 4257 3090 I 4259 3090 4260 3091 4260 3093 -c 4260 3095 4259 3096 4257 3096 -c 4238 3096 I 4236 3096 4235 3095 4235 3093 -c 4235 3091 4236 3090 4238 3090 -c C 
4282 3090 M 4301 3090 I 4302 3090 4304 3091 4304 3093 -c 4304 3095 4302 3096 4301 3096 -c 4282 3096 I 4280 3096 4279 3095 4279 3093 -c 4279 3091 4280 3090 4282 3090 -c C 
4326 3090 M 4344 3090 I 4346 3090 4347 3091 4347 3093 -c 4347 3095 4346 3096 4344 3096 -c 4326 3096 I 4324 3096 4322 3095 4322 3093 -c 4322 3091 4324 3090 4326 3090 -c C 
4369 3090 M 4388 3090 I 4390 3090 4391 3091 4391 3093 -c 4391 3095 4390 3096 4388 3096 -c 4369 3096 I 4368 3096 4366 3095 4366 3093 -c 4366 3091 4368 3090 4369 3090 -c C 
4413 3090 M 4432 3090 I 4434 3090 4435 3091 4435 3093 -c 4435 3095 4434 3096 4432 3096 -c 4413 3096 I 4411 3096 4410 3095 4410 3093 -c 4410 3091 4411 3090 4413 3090 -c C 
4457 3090 M 4475 3090 I 4477 3090 4478 3091 4478 3093 -c 4478 3095 4477 3096 4475 3096 -c 4457 3096 I 4455 3096 4453 3095 4453 3093 -c 4453 3091 4455 3090 4457 3090 -c C 
4500 3090 M 4519 3090 I 4521 3090 4522 3091 4522 3093 -c 4522 3095 4521 3096 4519 3096 -c 4500 3096 I 4499 3096 4497 3095 4497 3093 -c 4497 3091 4499 3090 4500 3090 -c C 
4544 3090 M 4563 3090 I 4565 3090 4566 3091 4566 3093 -c 4566 3095 4565 3096 4563 3096 -c 4544 3096 I 4542 3096 4541 3095 4541 3093 -c 4541 3091 4542 3090 4544 3090 -c C 
4588 3090 M 4607 3090 I 4608 3090 4610 3091 4610 3093 -c 4610 3095 4608 3096 4607 3096 -c 4588 3096 I 4586 3096 4585 3095 4585 3093 -c 4585 3091 4586 3090 4588 3090 -c C 
4632 3090 M 4650 3090 I 4652 3090 4653 3091 4653 3093 -c 4653 3095 4652 3096 4650 3096 -c 4632 3096 I 4630 3096 4628 3095 4628 3093 -c 4628 3091 4630 3090 4632 3090 -c C 
4675 3090 M 4694 3090 I 4696 3090 4697 3091 4697 3093 -c 4697 3095 4696 3096 4694 3096 -c 4675 3096 I 4674 3096 4672 3095 4672 3093 -c 4672 3091 4674 3090 4675 3090 -c C 
4719 3090 M 4738 3090 I 4740 3090 4741 3091 4741 3093 -c 4741 3095 4740 3096 4738 3096 -c 4719 3096 I 4717 3096 4716 3095 4716 3093 -c 4716 3091 4717 3090 4719 3090 -c C 
4763 3090 M 4782 3090 I 4783 3090 4785 3091 4785 3093 -c 4785 3095 4783 3096 4782 3096 -c 4763 3096 I 4761 3096 4760 3095 4760 3093 -c 4760 3091 4761 3090 4763 3090 -c C 
4807 3090 M 4825 3090 I 4827 3090 4828 3091 4828 3093 -c 4828 3095 4827 3096 4825 3096 -c 4807 3096 I 4805 3096 4803 3095 4803 3093 -c 4803 3091 4805 3090 4807 3090 -c C 
4850 3090 M 4869 3090 I 4871 3090 4872 3091 4872 3093 -c 4872 3095 4871 3096 4869 3096 -c 4850 3096 I 4849 3096 4847 3095 4847 3093 -c 4847 3091 4849 3090 4850 3090 -c C 
4894 3090 M 4913 3090 I 4915 3090 4916 3091 4916 3093 -c 4916 3095 4915 3096 4913 3096 -c 4894 3096 I 4892 3096 4891 3095 4891 3093 -c 4891 3091 4892 3090 4894 3090 -c C 
4938 3090 M 4956 3090 I 4958 3090 4959 3091 4959 3093 -c 4959 3095 4958 3096 4956 3096 -c 4938 3096 I 4936 3096 4935 3095 4935 3093 -c 4935 3091 4936 3090 4938 3090 -c C 
4981 3090 M 5000 3090 I 5002 3090 5003 3091 5003 3093 -c 5003 3095 5002 3096 5000 3096 -c 4981 3096 I 4980 3096 4978 3095 4978 3093 -c 4978 3091 4980 3090 4981 3090 -c C 
5025 3090 M 5044 3090 I 5046 3090 5047 3091 5047 3093 -c 5047 3095 5046 3096 5044 3096 -c 5025 3096 I 5023 3096 5022 3095 5022 3093 -c 5022 3091 5023 3090 5025 3090 -c C 
5069 3090 M 5088 3090 I 5089 3090 5091 3091 5091 3093 -c 5091 3095 5089 3096 5088 3096 -c 5069 3096 I 5067 3096 5066 3095 5066 3093 -c 5066 3091 5067 3090 5069 3090 -c C 
5113 3090 M 5131 3090 I 5133 3090 5134 3091 5134 3093 -c 5134 3095 5133 3096 5131 3096 -c 5113 3096 I 5111 3096 5109 3095 5109 3093 -c 5109 3091 5111 3090 5113 3090 -c C 
5156 3090 M 5175 3090 I 5177 3090 5178 3091 5178 3093 -c 5178 3095 5177 3096 5175 3096 -c 5156 3096 I 5155 3096 5153 3095 5153 3093 -c 5153 3091 5155 3090 5156 3090 -c C 
5200 3090 M 5219 3090 I 5221 3090 5222 3091 5222 3093 -c 5222 3095 5221 3096 5219 3096 -c 5200 3096 I 5198 3096 5197 3095 5197 3093 -c 5197 3091 5198 3090 5200 3090 -c C 
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5288 3090 M 5306 3090 I 5308 3090 5309 3091 5309 3093 -c 5309 3095 5308 3096 5306 3096 -c 5288 3096 I 5286 3096 5284 3095 5284 3093 -c 5284 3091 5286 3090 5288 3090 -c C 
5331 3090 M 5350 3090 I 5352 3090 5353 3091 5353 3093 -c 5353 3095 5352 3096 5350 3096 -c 5331 3096 I 5330 3096 5328 3095 5328 3093 -c 5328 3091 5330 3090 5331 3090 -c C 
5375 3090 M 5394 3090 I 5396 3090 5397 3091 5397 3093 -c 5397 3095 5396 3096 5394 3096 -c 5375 3096 I 5373 3096 5372 3095 5372 3093 -c 5372 3091 5373 3090 5375 3090 -c C 
5419 3090 M 5438 3090 I 5439 3090 5441 3091 5441 3093 -c 5441 3095 5439 3096 5438 3096 -c 5419 3096 I 5417 3096 5416 3095 5416 3093 -c 5416 3091 5417 3090 5419 3090 -c C 
5462 3090 M 5481 3090 I 5483 3090 5484 3091 5484 3093 -c 5484 3095 5483 3096 5481 3096 -c 5462 3096 I 5461 3096 5459 3095 5459 3093 -c 5459 3091 5461 3090 5462 3090 -c C 
5506 3090 M 5525 3090 I 5527 3090 5528 3091 5528 3093 -c 5528 3095 5527 3096 5525 3096 -c 5506 3096 I 5504 3096 5503 3095 5503 3093 -c 5503 3091 5504 3090 5506 3090 -c C 
5550 3090 M 5569 3090 I 5570 3090 5572 3091 5572 3093 -c 5572 3095 5570 3096 5569 3096 -c 5550 3096 I 5548 3096 5547 3095 5547 3093 -c 5547 3091 5548 3090 5550 3090 -c C 
5594 3090 M 5612 3090 I 5614 3090 5615 3091 5615 3093 -c 5615 3095 5614 3096 5612 3096 -c 5594 3096 I 5592 3096 5590 3095 5590 3093 -c 5590 3091 5592 3090 5594 3090 -c C 
5637 3090 M 5656 3090 I 5658 3090 5659 3091 5659 3093 -c 5659 3095 5658 3096 5656 3096 -c 5637 3096 I 5636 3096 5634 3095 5634 3093 -c 5634 3091 5636 3090 5637 3090 -c C 
5681 3090 M 5700 3090 I 5702 3090 5703 3091 5703 3093 -c 5703 3095 5702 3096 5700 3096 -c 5681 3096 I 5679 3096 5678 3095 5678 3093 -c 5678 3091 5679 3090 5681 3090 -c C 
5725 3090 M 5744 3090 I 5745 3090 5747 3091 5747 3093 -c 5747 3095 5745 3096 5744 3096 -c 5725 3096 I 5723 3096 5722 3095 5722 3093 -c 5722 3091 5723 3090 5725 3090 -c C 
5769 3090 M 5787 3090 I 5789 3090 5790 3091 5790 3093 -c 5790 3095 5789 3096 5787 3096 -c 5769 3096 I 5767 3096 5765 3095 5765 3093 -c 5765 3091 5767 3090 5769 3090 -c C 
5812 3090 M 5831 3090 I 5833 3090 5834 3091 5834 3093 -c 5834 3095 5833 3096 5831 3096 -c 5812 3096 I 5811 3096 5809 3095 5809 3093 -c 5809 3091 5811 3090 5812 3090 -c C 
5856 3090 M 5875 3090 I 5877 3090 5878 3091 5878 3093 -c 5878 3095 5877 3096 5875 3096 -c 5856 3096 I 5854 3096 5853 3095 5853 3093 -c 5853 3091 5854 3090 5856 3090 -c C 
5900 3090 M 5919 3090 I 5920 3090 5922 3091 5922 3093 -c 5922 3095 5920 3096 5919 3096 -c 5900 3096 I 5898 3096 5897 3095 5897 3093 -c 5897 3091 5898 3090 5900 3090 -c C 
5944 3090 M 5962 3090 I 5964 3090 5965 3091 5965 3093 -c 5965 3095 5964 3096 5962 3096 -c 5944 3096 I 5942 3096 5940 3095 5940 3093 -c 5940 3091 5942 3090 5944 3090 -c C 
5987 3090 M 6006 3090 I 6008 3090 6009 3091 6009 3093 -c 6009 3095 6008 3096 6006 3096 -c 5987 3096 I 5985 3096 5984 3095 5984 3093 -c 5984 3091 5985 3090 5987 3090 -c C 
6031 3090 M 6050 3090 I 6051 3090 6053 3091 6053 3093 -c 6053 3095 6051 3096 6050 3096 -c 6031 3096 I 6029 3096 6028 3095 6028 3093 -c 6028 3091 6029 3090 6031 3090 -c C 
6075 3090 M 6093 3090 I 6095 3090 6096 3091 6096 3093 -c 6096 3095 6095 3096 6093 3096 -c 6075 3096 I 6073 3096 6071 3095 6071 3093 -c 6071 3091 6073 3090 6075 3090 -c C 
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6206 3090 M 6225 3090 I 6226 3090 6228 3091 6228 3093 -c 6228 3095 6226 3096 6225 3096 -c 6206 3096 I 6204 3096 6203 3095 6203 3093 -c 6203 3091 6204 3090 6206 3090 -c C 
6250 3090 M 6265 3090 I 6267 3090 6268 3091 6268 3093 -c 6268 3095 6267 3096 6265 3096 -c 6250 3096 I 6248 3096 6246 3095 6246 3093 -c 6246 3091 6248 3090 6250 3090 -c C 
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; N 3983 1432 M 3964 1431 I 3963 1431 3961 1430 3961 1428 -c 3961 1426 3963 1425 3965 1425 -c 3983 1425 I 3985 1425 3986 1427 3986 1429 -c 3986 1430 3985 1432 3983 1432 -c C 
3940 1431 M 3921 1431 I 3919 1431 3918 1430 3918 1428 -c 3918 1427 3919 1425 3921 1425 -c 3940 1425 I 3941 1425 3943 1426 3943 1428 -c 3943 1429 3942 1431 3940 1431 -c C 
3896 1433 M 3878 1434 I 3876 1434 3874 1433 3874 1431 -c 3874 1429 3875 1428 3877 1428 -c 3896 1426 I 3898 1426 3899 1427 3899 1429 -c 3899 1431 3898 1432 3896 1433 -c C 
3853 1436 M 3834 1439 I 3833 1439 3831 1438 3831 1436 -c 3831 1434 3832 1433 3833 1432 -c 3852 1430 I 3854 1430 3855 1431 3856 1433 -c 3856 1435 3855 1436 3853 1436 -c C 
3810 1442 M 3791 1446 I 3790 1446 3788 1445 3788 1443 -c 3787 1441 3788 1440 3790 1440 -c 3809 1436 I 3810 1436 3812 1437 3812 1439 -c 3813 1440 3811 1442 3810 1442 -c C 
3767 1451 M 3749 1455 I 3747 1455 3745 1454 3745 1453 -c 3745 1451 3746 1449 3747 1449 -c 3766 1445 I 3767 1444 3769 1445 3769 1447 -c 3770 1449 3769 1450 3767 1451 -c C 
3725 1461 M 3707 1467 I 3705 1467 3703 1466 3703 1464 -c 3702 1463 3703 1461 3705 1461 -c 3723 1455 I 3725 1455 3726 1456 3727 1458 -c 3727 1459 3726 1461 3725 1461 -c C 
3683 1474 M 3665 1480 I 3664 1481 3662 1480 3661 1478 -c 3661 1477 3662 1475 3663 1474 -c 3681 1468 I 3683 1468 3684 1469 3685 1470 -c 3686 1472 3685 1474 3683 1474 -c C 
3642 1489 M 3627 1495 I 3625 1496 I 3623 1497 3622 1496 3621 1495 -c 3620 1493 3621 1491 3622 1490 -c 3625 1489 I 3640 1484 I 3641 1483 3643 1484 3644 1485 -c 3645 1487 3644 1489 3642 1489 -c C 
3602 1506 M 3591 1512 I 3585 1514 I 3584 1515 3582 1515 3581 1513 -c 3580 1511 3581 1510 3582 1509 -c 3588 1506 I 3600 1501 I 3601 1500 3603 1501 3604 1502 -c 3604 1504 3604 1506 3602 1506 -c C 
3563 1526 M 3555 1530 I 3547 1535 I 3545 1535 3543 1535 3542 1533 -c 3542 1532 3542 1530 3544 1529 -c 3552 1524 I 3560 1520 I 3562 1519 3564 1520 3564 1522 -c 3565 1523 3565 1525 3563 1526 -c C 
3525 1547 M 3521 1550 I 3509 1557 I 3508 1558 3506 1557 3505 1556 -c 3504 1555 3504 1553 3506 1552 -c 3518 1544 I 3522 1542 I 3523 1541 3525 1541 3526 1543 -c 3527 1544 3527 1546 3525 1547 -c C 
3488 1570 M 3487 1571 I 3473 1581 I 3472 1582 3470 1582 3469 1580 -c 3468 1579 3468 1577 3469 1576 -c 3484 1566 I 3485 1565 I 3486 1564 3488 1565 3489 1566 -c 3490 1568 3490 1570 3488 1570 -c C 
3453 1596 M 3438 1607 I 3437 1608 3435 1608 3434 1607 -c 3433 1605 3433 1603 3435 1602 -c 3449 1591 I 3450 1590 3452 1590 3453 1591 -c 3454 1593 3454 1595 3453 1596 -c C 
3419 1623 M 3405 1635 I 3404 1636 3402 1636 3401 1635 -c 3399 1633 3400 1631 3401 1630 -c 3415 1618 I 3416 1617 3418 1617 3419 1618 -c 3421 1620 3420 1621 3419 1623 -c C 
3386 1652 M 3373 1665 I 3372 1666 3370 1666 3368 1664 -c 3367 1663 3367 1661 3369 1660 -c 3382 1647 I 3383 1646 3385 1646 3387 1647 -c 3388 1648 3388 1650 3386 1652 -c C 
3355 1682 M 3343 1696 I 3341 1697 3339 1697 3338 1696 -c 3337 1695 3337 1693 3338 1691 -c 3351 1678 I 3352 1677 3354 1677 3355 1678 -c 3357 1679 3357 1681 3355 1682 -c C 
3326 1714 M 3314 1729 I 3313 1730 3311 1730 3309 1729 -c 3308 1728 3308 1726 3309 1724 -c 3321 1710 I 3322 1709 3324 1709 3326 1710 -c 3327 1711 3327 1713 3326 1714 -c C 
3298 1748 M 3288 1760 I 3287 1763 I 3286 1764 3284 1764 3283 1763 -c 3281 1762 3281 1760 3282 1759 -c 3284 1757 I 3293 1744 I 3295 1743 3296 1742 3298 1743 -c 3299 1744 3299 1746 3298 1748 -c C 
3272 1783 M 3266 1793 I 3262 1798 I 3261 1800 3259 1800 3258 1799 -c 3256 1798 3256 1796 3257 1795 -c 3260 1789 I 3267 1779 I 3268 1778 3270 1778 3272 1779 -c 3273 1780 3273 1781 3272 1783 -c C 
3248 1819 M 3244 1826 I 3239 1835 I 3238 1837 3236 1837 3235 1836 -c 3233 1835 3233 1834 3233 1832 -c 3239 1823 I 3243 1816 I 3244 1814 3246 1814 3248 1815 -c 3249 1816 3249 1818 3248 1819 -c C 
3226 1857 M 3224 1861 I 3218 1873 I 3217 1875 3215 1875 3214 1874 -c 3212 1874 3211 1872 3212 1870 -c 3219 1857 I 3221 1854 I 3222 1852 3224 1852 3225 1853 -c 3227 1853 3227 1855 3226 1857 -c C 
3206 1895 M 3206 1896 I 3199 1912 I 3198 1914 3196 1915 3195 1914 -c 3193 1913 3192 1911 3193 1910 -c 3201 1893 I 3201 1892 I 3202 1891 3203 1890 3205 1891 -c 3207 1892 3207 1894 3206 1895 -c C 
3189 1935 M 3182 1952 I 3181 1954 3179 1955 3178 1954 -c 3176 1954 3175 1952 3176 1950 -c 3183 1933 I 3183 1931 3185 1930 3187 1931 -c 3188 1932 3189 1933 3189 1935 -c C 
3173 1976 M 3167 1993 I 3166 1995 3165 1996 3163 1995 -c 3161 1995 3160 1993 3161 1991 -c 3167 1974 I 3168 1972 3169 1971 3171 1972 -c 3173 1972 3174 1974 3173 1976 -c C 
3159 2017 M 3154 2035 I 3154 2037 3152 2037 3150 2037 -c 3149 2037 3148 2035 3148 2033 -c 3153 2015 I 3154 2013 3156 2012 3157 2013 -c 3159 2013 3160 2015 3159 2017 -c C 
3148 2059 M 3144 2077 I 3144 2079 3142 2080 3140 2080 -c 3138 2079 3137 2077 3138 2076 -c 3142 2058 I 3142 2056 3144 2055 3146 2055 -c 3147 2056 3149 2057 3148 2059 -c C 
3139 2102 M 3136 2120 I 3135 2121 3134 2123 3132 2122 -c 3130 2122 3129 2120 3130 2119 -c 3133 2100 I 3133 2099 3135 2098 3137 2098 -c 3138 2098 3139 2100 3139 2102 -c C 
3132 2144 M 3130 2163 I 3130 2165 3128 2166 3126 2166 -c 3125 2165 3124 2164 3124 2162 -c 3126 2144 I 3126 2142 3128 2141 3130 2141 -c 3131 2141 3133 2143 3132 2144 -c C 
3128 2188 M 3126 2206 I 3126 2208 3125 2209 3123 2209 -c 3121 2209 3120 2208 3120 2206 -c 3122 2187 I 3122 2185 3123 2184 3125 2184 -c 3127 2184 3128 2186 3128 2188 -c C 
3126 2231 M 3125 2250 I 3125 2251 3124 2253 3122 2253 -c 3120 2253 3119 2251 3119 2250 -c 3119 2231 I 3119 2229 3121 2228 3123 2228 -c 3124 2228 3126 2229 3126 2231 -c C 
3126 2275 M 3126 2293 I 3126 2295 3125 2296 3123 2296 -c 3121 2297 3120 2295 3120 2293 -c 3119 2275 I 3119 2273 3121 2272 3123 2271 -c 3124 2271 3126 2273 3126 2275 -c C 
3128 2318 M 3129 2336 I 3129 2337 I 3130 2338 3128 2340 3127 2340 -c 3125 2340 3124 2339 3123 2337 -c 3123 2336 I 3122 2319 I 3122 2317 3123 2315 3125 2315 -c 3126 2315 3128 2316 3128 2318 -c C 
3133 2361 M 3135 2377 I 3135 2380 I 3135 2381 3134 2383 3133 2383 -c 3131 2383 3129 2382 3129 2381 -c 3128 2377 I 3126 2362 I 3126 2360 3127 2359 3129 2359 -c 3131 2358 3132 2360 3133 2361 -c C 
3140 2404 M 3142 2417 I 3143 2422 I 3143 2424 3142 2426 3141 2426 -c 3139 2427 3137 2426 3137 2424 -c 3136 2418 I 3133 2405 I 3133 2404 3134 2402 3136 2402 -c 3138 2401 3139 2402 3140 2404 -c C 
3149 2447 M 3151 2457 I 3153 2465 I 3154 2466 3153 2468 3151 2469 -c 3149 2469 3148 2468 3147 2466 -c 3145 2458 I 3143 2448 I 3142 2447 3143 2445 3145 2444 -c 3147 2444 3148 2445 3149 2447 -c C 
3160 2489 M 3162 2496 I 3166 2506 I 3166 2508 3165 2510 3164 2510 -c 3162 2511 3160 2510 3160 2508 -c 3156 2497 I 3154 2490 I 3154 2489 3155 2487 3156 2486 -c 3158 2486 3160 2487 3160 2489 -c C 
3174 2530 M 3175 2534 I 3180 2547 I 3181 2549 3180 2551 3179 2551 -c 3177 2552 3175 2551 3175 2550 -c 3169 2536 I 3168 2532 I 3167 2530 3168 2529 3170 2528 -c 3171 2527 3173 2528 3174 2530 -c C 
3190 2571 M 3190 2571 I 3197 2588 I 3198 2589 3197 2591 3196 2592 -c 3194 2592 3192 2592 3192 2590 -c 3184 2573 I 3184 2573 I 3183 2571 3184 2569 3186 2569 -c 3187 2568 3189 2569 3190 2571 -c C 
3208 2610 M 3216 2627 I 3217 2628 3216 2630 3215 2631 -c 3213 2632 3211 2631 3211 2629 -c 3202 2613 I 3201 2611 3202 2609 3203 2609 -c 3205 2608 3207 2609 3208 2610 -c C 
3228 2649 M 3237 2665 I 3238 2666 3237 2668 3236 2669 -c 3234 2670 3233 2669 3232 2668 -c 3222 2652 I 3221 2650 3222 2648 3223 2647 -c 3225 2647 3227 2647 3228 2649 -c C 
3250 2686 M 3260 2702 I 3261 2703 3261 2705 3259 2706 -c 3258 2707 3256 2707 3255 2705 -c 3245 2689 I 3244 2688 3244 2686 3246 2685 -c 3247 2684 3249 2685 3250 2686 -c C 
3274 2722 M 3285 2737 I 3286 2739 3286 2741 3284 2742 -c 3283 2743 3281 2742 3280 2741 -c 3269 2726 I 3268 2725 3268 2723 3270 2722 -c 3271 2721 3273 2721 3274 2722 -c C 
3300 2757 M 3312 2772 I 3313 2773 3312 2775 3311 2776 -c 3310 2777 3308 2777 3307 2776 -c 3295 2761 I 3294 2760 3294 2758 3296 2757 -c 3297 2756 3299 2756 3300 2757 -c C 
3328 2791 M 3339 2803 I 3340 2805 I 3341 2806 3341 2808 3340 2809 -c 3339 2810 3337 2810 3336 2809 -c 3334 2808 I 3323 2795 I 3322 2794 3322 2792 3323 2790 -c 3325 2789 3327 2789 3328 2791 -c C 
3357 2823 M 3366 2832 I 3370 2836 I 3372 2837 3372 2839 3370 2841 -c 3369 2842 3367 2842 3366 2841 -c 3361 2836 I 3353 2827 I 3352 2826 3352 2824 3353 2823 -c 3354 2822 3356 2822 3357 2823 -c C 
3388 2853 M 3395 2859 I 3402 2866 I 3404 2867 3404 2869 3403 2870 -c 3401 2871 3399 2871 3398 2870 -c 3390 2864 I 3384 2858 I 3383 2857 3383 2855 3384 2853 -c 3385 2852 3387 2852 3388 2853 -c C 
3421 2882 M 3424 2885 I 3436 2894 I 3437 2895 3437 2897 3436 2898 -c 3435 2899 3433 2900 3432 2899 -c 3420 2890 I 3417 2887 I 3416 2886 3416 2884 3417 2882 -c 3418 2881 3420 2881 3421 2882 -c C 
3455 2909 M 3455 2909 I 3470 2920 I 3472 2921 3472 2923 3471 2924 -c 3470 2926 3468 2926 3467 2925 -c 3451 2914 I 3451 2914 I 3450 2913 3450 2911 3451 2910 -c 3452 2908 3454 2908 3455 2909 -c C 
3490 2934 M 3506 2944 I 3508 2945 3508 2947 3507 2949 -c 3506 2950 3504 2950 3503 2949 -c 3487 2939 I 3486 2938 3485 2936 3486 2935 -c 3487 2934 3489 2933 3490 2934 -c C 
3527 2957 M 3544 2967 I 3545 2968 3546 2969 3545 2971 -c 3544 2972 3542 2973 3540 2972 -c 3524 2963 I 3523 2962 3522 2960 3523 2958 -c 3524 2957 3526 2956 3527 2957 -c C 
3565 2979 M 3582 2987 I 3584 2988 3584 2990 3583 2991 -c 3583 2993 3581 2993 3579 2992 -c 3563 2984 I 3561 2983 3560 2981 3561 2980 -c 3562 2978 3564 2978 3565 2979 -c C 
3605 2998 M 3622 3005 I 3623 3006 3624 3008 3623 3009 -c 3622 3011 3621 3012 3619 3011 -c 3602 3003 I 3600 3003 3600 3001 3600 2999 -c 3601 2998 3603 2997 3605 2998 -c C 
3645 3015 M 3662 3022 I 3664 3022 3664 3024 3664 3026 -c 3663 3027 3661 3028 3660 3027 -c 3642 3020 I 3641 3020 3640 3018 3641 3016 -c 3641 3015 3643 3014 3645 3015 -c C 
3686 3030 M 3703 3035 I 3703 3036 I 3705 3036 3706 3038 3705 3039 -c 3705 3041 3703 3042 3701 3042 -c 3701 3041 I 3684 3036 I 3682 3035 3681 3033 3682 3032 -c 3682 3030 3684 3029 3686 3030 -c C 
3727 3042 M 3742 3046 I 3745 3047 I 3747 3048 3748 3049 3748 3051 -c 3747 3053 3745 3054 3744 3053 -c 3740 3052 I 3726 3048 I 3724 3048 3723 3046 3723 3045 -c 3724 3043 3726 3042 3727 3042 -c C 
3770 3053 M 3782 3056 I 3788 3057 I 3789 3057 3791 3059 3790 3060 -c 3790 3062 3788 3063 3787 3063 -c 3780 3062 I 3768 3059 I 3766 3059 3765 3057 3766 3055 -c 3766 3054 3768 3053 3770 3053 -c C 
3812 3061 M 3822 3063 I 3831 3064 I 3832 3064 3834 3066 3833 3068 -c 3833 3069 3832 3070 3830 3070 -c 3821 3069 I 3811 3067 I 3810 3067 3808 3065 3809 3064 -c 3809 3062 3811 3061 3812 3061 -c C 
3855 3067 M 3863 3068 I 3874 3069 I 3876 3069 3877 3071 3877 3072 -c 3877 3074 3875 3075 3874 3075 -c 3862 3074 I 3855 3073 I 3853 3073 3852 3072 3852 3070 -c 3852 3068 3854 3067 3855 3067 -c C 
3899 3071 M 3905 3071 I 3917 3072 I 3919 3072 3921 3073 3921 3075 -c 3920 3077 3919 3078 3917 3078 -c 3905 3077 I 3898 3077 I 3897 3077 3895 3075 3896 3074 -c 3896 3072 3897 3071 3899 3071 -c C 
3942 3072 M 3947 3072 I 3961 3072 I 3963 3072 3964 3073 3964 3075 -c 3964 3077 3963 3078 3961 3078 -c 3947 3079 I 3942 3078 I 3941 3078 3939 3077 3939 3075 -c 3939 3074 3941 3072 3942 3072 -c C 
3986 3071 M 3989 3071 I 4004 3070 I 4006 3070 4008 3071 4008 3073 -c 4008 3075 4007 3076 4005 3076 -c 3990 3077 I 3986 3078 I 3984 3078 3983 3076 3983 3075 -c 3983 3073 3984 3071 3986 3071 -c C 
4029 3068 M 4031 3068 I 4048 3066 I 4049 3066 4051 3067 4051 3069 -c 4051 3070 4050 3072 4049 3072 -c 4032 3074 I 4030 3074 I 4028 3075 4027 3073 4026 3072 -c 4026 3070 4028 3068 4029 3068 -c C 
4072 3063 M 4091 3060 I 4093 3059 4094 3060 4094 3062 -c 4095 3064 4094 3065 4092 3066 -c 4073 3069 I 4072 3069 4070 3068 4070 3067 -c 4070 3065 4071 3063 4072 3063 -c C 
4115 3055 M 4134 3051 I 4135 3050 4137 3052 4137 3053 -c 4138 3055 4137 3057 4135 3057 -c 4117 3061 I 4115 3062 4113 3061 4113 3059 -c 4113 3057 4114 3055 4115 3055 -c C 
4158 3045 M 4176 3040 I 4177 3039 4179 3040 4180 3042 -c 4180 3044 4179 3045 4177 3046 -c 4159 3051 I 4158 3052 4156 3051 4156 3049 -c 4155 3047 4156 3045 4158 3045 -c C 
4199 3033 M 4217 3027 I 4219 3026 4221 3027 4221 3029 -c 4222 3030 4221 3032 4219 3033 -c 4201 3039 I 4200 3039 4198 3038 4198 3037 -c 4197 3035 4198 3033 4199 3033 -c C 
4241 3018 M 4258 3011 I 4260 3011 4261 3011 4262 3013 -c 4263 3015 4262 3017 4260 3017 -c 4243 3024 I 4241 3025 4239 3024 4239 3022 -c 4238 3021 4239 3019 4241 3018 -c C 
4281 3002 M 4298 2994 I 4300 2993 4301 2994 4302 2995 -c 4303 2997 4302 2999 4301 3000 -c 4283 3007 I 4282 3008 4280 3007 4279 3006 -c 4279 3004 4279 3002 4281 3002 -c C 
4320 2983 M 4337 2974 I 4338 2974 4340 2974 4341 2976 -c 4342 2977 4341 2979 4340 2980 -c 4323 2988 I 4322 2989 4320 2988 4319 2987 -c 4318 2985 4319 2984 4320 2983 -c C 
4358 2962 M 4374 2954 I 4375 2953 I 4376 2952 4378 2952 4379 2954 -c 4380 2955 4379 2957 4378 2958 -c 4377 2959 I 4362 2968 I 4360 2968 4358 2968 4357 2966 -c 4356 2965 4357 2963 4358 2962 -c C 
4396 2939 M 4407 2932 I 4411 2929 I 4413 2928 4414 2928 4415 2930 -c 4416 2931 4416 2933 4415 2934 -c 4410 2937 I 4399 2945 I 4398 2946 4396 2945 4395 2944 -c 4394 2942 4394 2940 4396 2939 -c C 
4432 2915 M 4439 2909 I 4446 2904 I 4448 2902 4449 2903 4451 2904 -c 4452 2905 4451 2907 4450 2908 -c 4443 2914 I 4435 2920 I 4434 2921 4432 2920 4431 2919 -c 4430 2918 4430 2916 4432 2915 -c C 
4466 2888 M 4470 2885 I 4480 2876 I 4481 2875 4483 2875 4484 2876 -c 4486 2878 4485 2880 4484 2881 -c 4474 2890 I 4470 2893 I 4468 2894 4466 2894 4465 2892 -c 4464 2891 4464 2889 4466 2888 -c C 
4499 2860 M 4500 2859 I 4512 2847 I 4514 2846 4516 2846 4517 2847 -c 4518 2848 4518 2850 4517 2852 -c 4504 2864 I 4503 2864 I 4502 2866 4500 2865 4499 2864 -c 4497 2863 4498 2861 4499 2860 -c C 
4530 2830 M 4543 2816 I 4544 2815 4546 2815 4548 2816 -c 4549 2817 4549 2819 4548 2821 -c 4535 2834 I 4534 2835 4532 2836 4530 2834 -c 4529 2833 4529 2831 4530 2830 -c C 
4560 2798 M 4572 2784 I 4574 2783 4576 2782 4577 2784 -c 4578 2785 4578 2787 4577 2788 -c 4565 2802 I 4564 2803 4562 2804 4560 2803 -c 4559 2801 4559 2799 4560 2798 -c C 
4588 2765 M 4600 2750 I 4601 2749 4603 2748 4604 2750 -c 4606 2751 4606 2753 4605 2754 -c 4593 2769 I 4592 2770 4590 2770 4589 2769 -c 4587 2768 4587 2766 4588 2765 -c C 
4615 2730 M 4625 2715 I 4626 2714 4628 2713 4630 2714 -c 4631 2715 4632 2717 4631 2719 -c 4620 2734 I 4619 2735 4617 2735 4615 2734 -c 4614 2733 4614 2732 4615 2730 -c C 
4639 2694 M 4649 2678 I 4650 2677 4652 2677 4654 2678 -c 4655 2678 4655 2680 4654 2682 -c 4644 2698 I 4643 2699 4641 2699 4640 2699 -c 4639 2698 4638 2696 4639 2694 -c C 
4662 2657 M 4670 2643 I 4671 2641 I 4672 2639 4674 2639 4675 2639 -c 4677 2640 4677 2642 4676 2644 -c 4675 2646 I 4667 2660 I 4666 2662 4664 2662 4663 2661 -c 4661 2660 4661 2658 4662 2657 -c C 
4682 2619 M 4688 2607 I 4690 2602 I 4691 2601 4693 2600 4695 2601 -c 4696 2601 4697 2603 4696 2605 -c 4694 2610 I 4688 2621 I 4687 2623 4685 2624 4684 2623 -c 4682 2622 4681 2620 4682 2619 -c C 
4701 2579 M 4705 2571 I 4708 2562 I 4709 2561 4710 2560 4712 2560 -c 4714 2561 4714 2563 4714 2564 -c 4710 2574 I 4706 2582 I 4706 2583 4704 2584 4702 2583 -c 4701 2583 4700 2581 4701 2579 -c C 
4717 2539 M 4719 2534 I 4723 2521 I 4724 2520 4726 2519 4727 2519 -c 4729 2520 4730 2522 4729 2523 -c 4725 2536 I 4723 2541 I 4722 2543 4720 2544 4719 2543 -c 4717 2542 4716 2541 4717 2539 -c C 
4731 2498 M 4732 2495 I 4737 2480 I 4737 2478 4739 2478 4740 2478 -c 4742 2478 4743 2480 4743 2482 -c 4738 2497 I 4737 2500 I 4737 2501 4735 2502 4733 2502 -c 4732 2501 4731 2499 4731 2498 -c C 
4743 2456 M 4748 2438 I 4748 2436 4750 2435 4751 2436 -c 4753 2436 4754 2438 4754 2439 -c 4749 2458 I 4749 2459 4747 2460 4746 2460 -c 4744 2460 4743 2458 4743 2456 -c C 
4753 2414 M 4756 2395 I 4757 2394 4758 2392 4760 2393 -c 4762 2393 4763 2395 4762 2396 -c 4759 2415 I 4759 2417 4757 2418 4756 2417 -c 4754 2417 4753 2415 4753 2414 -c C 
4760 2371 M 4763 2352 I 4763 2351 4765 2350 4766 2350 -c 4768 2350 4769 2352 4769 2353 -c 4767 2372 I 4766 2373 4765 2374 4763 2374 -c 4761 2374 4760 2372 4760 2371 -c C 
4765 2328 M 4767 2309 I 4767 2307 4768 2306 4770 2306 -c 4772 2306 4773 2308 4773 2310 -c 4772 2328 I 4772 2330 4770 2331 4768 2331 -c 4767 2331 4765 2329 4765 2328 -c C 
4768 2284 M 4769 2266 I 4769 2264 4770 2262 4772 2262 -c 4774 2263 4775 2264 4775 2266 -c 4774 2284 I 4774 2286 4773 2287 4771 2287 -c 4770 2287 4768 2286 4768 2284 -c C 
4769 2241 M 4768 2222 I 4768 2220 4770 2219 4771 2219 -c 4773 2219 4775 2220 4775 2222 -c 4775 2241 I 4775 2242 4774 2244 4772 2244 -c 4770 2244 4769 2242 4769 2241 -c C 
4767 2197 M 4766 2179 I 4766 2177 4767 2175 4769 2175 -c 4770 2175 4772 2176 4772 2178 -c 4773 2197 I 4773 2199 4772 2200 4770 2200 -c 4769 2200 4767 2199 4767 2197 -c C 
4763 2154 M 4761 2135 I 4761 2134 4762 2132 4763 2132 -c 4765 2132 4767 2133 4767 2135 -c 4769 2153 I 4770 2155 4768 2156 4767 2157 -c 4765 2157 4763 2156 4763 2154 -c C 
4757 2111 M 4754 2093 I 4753 2091 4754 2089 4756 2089 -c 4758 2089 4759 2090 4760 2092 -c 4763 2110 I 4763 2111 4762 2113 4760 2113 -c 4759 2114 4757 2113 4757 2111 -c C 
4748 2068 M 4744 2050 I 4744 2048 4745 2047 4746 2046 -c 4748 2046 4750 2047 4750 2049 -c 4754 2067 I 4755 2069 4754 2070 4752 2071 -c 4750 2071 4749 2070 4748 2068 -c C 
4737 2026 M 4732 2008 I 4732 2006 4733 2005 4734 2004 -c 4736 2004 4738 2005 4738 2006 -c 4743 2024 I 4744 2026 4743 2028 4741 2028 -c 4740 2029 4738 2028 4737 2026 -c C 
4724 1985 M 4719 1969 I 4718 1967 I 4718 1966 4718 1964 4720 1963 -c 4722 1963 4724 1963 4724 1965 -c 4725 1967 I 4730 1983 I 4731 1984 4730 1986 4728 1987 -c 4727 1987 4725 1986 4724 1985 -c C 
4709 1944 M 4704 1932 I 4702 1927 I 4701 1925 4702 1923 4704 1923 -c 4705 1922 4707 1923 4708 1924 -c 4710 1930 I 4715 1942 I 4716 1943 4715 1945 4713 1946 -c 4712 1946 4710 1946 4709 1944 -c C 
4692 1904 M 4688 1896 I 4684 1887 I 4683 1886 4684 1884 4685 1883 -c 4687 1882 4689 1883 4689 1884 -c 4694 1893 I 4698 1901 I 4698 1903 4698 1905 4696 1906 -c 4694 1906 4693 1906 4692 1904 -c C 
4672 1865 M 4670 1860 I 4663 1849 I 4662 1848 4663 1846 4664 1845 -c 4666 1844 4668 1844 4669 1846 -c 4675 1858 I 4678 1862 I 4679 1864 4678 1866 4677 1866 -c 4675 1867 4673 1867 4672 1865 -c C 
4651 1827 M 4650 1826 I 4641 1812 I 4640 1810 4640 1808 4642 1807 -c 4643 1806 4645 1807 4646 1808 -c 4655 1823 I 4656 1824 I 4657 1826 4657 1828 4655 1828 -c 4654 1829 4652 1829 4651 1827 -c C 
4627 1791 M 4616 1776 I 4615 1774 4616 1772 4617 1771 -c 4619 1770 4621 1770 4622 1772 -c 4632 1787 I 4633 1789 4633 1791 4632 1792 -c 4630 1793 4628 1792 4627 1791 -c C 
4602 1755 M 4590 1741 I 4589 1739 4589 1737 4591 1737 -c 4592 1735 4594 1736 4595 1737 -c 4607 1752 I 4608 1753 4608 1755 4606 1756 -c 4605 1757 4603 1757 4602 1755 -c C 
4575 1722 M 4562 1708 I 4561 1706 4561 1704 4563 1703 -c 4564 1702 4566 1702 4567 1703 -c 4579 1718 I 4580 1719 4580 1721 4579 1722 -c 4578 1723 4576 1723 4575 1722 -c C 
4546 1689 M 4533 1676 I 4531 1674 4531 1672 4533 1671 -c 4534 1670 4536 1670 4537 1671 -c 4550 1685 I 4551 1686 4551 1688 4550 1689 -c 4549 1691 4547 1690 4546 1689 -c C 
4515 1658 M 4501 1645 I 4500 1644 4500 1642 4501 1641 -c 4502 1640 4504 1640 4506 1641 -c 4519 1654 I 4520 1655 4520 1657 4519 1658 -c 4518 1659 4516 1659 4515 1658 -c C 
4483 1629 M 4470 1618 I 4468 1617 I 4467 1616 4467 1614 4468 1613 -c 4469 1611 4471 1611 4472 1612 -c 4474 1613 I 4487 1624 I 4488 1625 4488 1627 4487 1629 -c 4486 1630 4484 1630 4483 1629 -c C 
4449 1602 M 4439 1594 I 4434 1590 I 4433 1589 4432 1587 4433 1586 -c 4434 1585 4436 1584 4438 1585 -c 4443 1589 I 4453 1597 I 4454 1598 4454 1600 4453 1601 -c 4452 1602 4450 1603 4449 1602 -c C 
4414 1576 M 4407 1571 I 4398 1566 I 4397 1565 4397 1563 4397 1561 -c 4398 1560 4400 1559 4402 1560 -c 4410 1566 I 4417 1571 I 4419 1572 4419 1574 4418 1575 -c 4417 1577 4415 1577 4414 1576 -c C 
4377 1552 M 4373 1549 I 4361 1543 I 4360 1542 4359 1540 4360 1538 -c 4361 1537 4363 1536 4364 1537 -c 4377 1544 I 4381 1547 I 4382 1548 4383 1550 4382 1551 -c 4381 1553 4379 1553 4377 1552 -c C 
4340 1530 M 4339 1530 I 4323 1522 I 4322 1521 4321 1519 4322 1517 -c 4323 1516 4324 1515 4326 1516 -c 4342 1524 I 4343 1525 I 4344 1526 4345 1527 4344 1529 -c 4343 1530 4341 1531 4340 1530 -c C 
4301 1510 M 4284 1503 I 4282 1502 4282 1500 4282 1499 -c 4283 1497 4285 1496 4286 1497 -c 4304 1505 I 4305 1505 4306 1507 4305 1509 -c 4304 1510 4303 1511 4301 1510 -c C 
4261 1493 M 4244 1486 I 4242 1486 4241 1484 4242 1482 -c 4243 1480 4244 1480 4246 1480 -c 4263 1487 I 4265 1488 4266 1489 4265 1491 -c 4265 1493 4263 1493 4261 1493 -c C 
4220 1477 M 4203 1471 I 4201 1471 4200 1469 4201 1468 -c 4201 1466 4203 1465 4205 1466 -c 4222 1472 I 4224 1472 4225 1474 4224 1476 -c 4224 1477 4222 1478 4220 1477 -c C 
4179 1464 M 4161 1459 I 4159 1459 4158 1457 4159 1455 -c 4159 1453 4161 1453 4163 1453 -c 4181 1458 I 4182 1459 4183 1460 4183 1462 -c 4182 1464 4181 1465 4179 1464 -c C 
4137 1453 M 4118 1449 I 4117 1448 4116 1447 4116 1445 -c 4117 1443 4118 1442 4120 1443 -c 4138 1447 I 4140 1447 4141 1449 4141 1451 -c 4140 1452 4138 1453 4137 1453 -c C 
4094 1444 M 4076 1441 I 4074 1441 4073 1439 4073 1437 -c 4073 1435 4075 1434 4077 1435 -c 4095 1438 I 4097 1438 4098 1440 4098 1442 -c 4097 1443 4096 1444 4094 1444 -c C 
4051 1438 M 4032 1435 I 4031 1435 4029 1433 4030 1432 -c 4030 1430 4031 1429 4033 1429 -c 4052 1431 I 4054 1432 4055 1433 4055 1435 -c 4054 1437 4053 1438 4051 1438 -c C 
4008 1433 M 3989 1432 I 3988 1432 3986 1430 3986 1429 -c 3987 1427 3988 1425 3990 1426 -c 4008 1427 I 4010 1427 4011 1429 4011 1430 -c 4011 1432 4009 1433 4008 1433 -c C 
:  L ; : 0.246 0.246 +S K 
; 3932 2315 M <B7>S 
Pscript_WinNT_Incr begin
%%BeginResource: file Pscript_Encoding256 5.0 0
/CharCol256Encoding[/.notdef/breve/caron/dotaccent/dotlessi/fi/fl/fraction
/hungarumlaut/Lslash/lslash/minus/ogonek/ring/Zcaron/zcaron/.notdef/.notdef
/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef
/.notdef/.notdef/.notdef/.notdef/.notdef/space/exclam/quotedbl/numbersign
/dollar/percent/ampersand/quotesingle/parenleft/parenright/asterisk/plus/comma
/hyphen/period/slash/zero/one/two/three/four/five/six/seven/eight/nine/colon
/semicolon/less/equal/greater/question/at/A/B/C/D/E/F/G/H/I/J/K/L/M/N/O/P/Q/R/S
/T/U/V/W/X/Y/Z/bracketleft/backslash/bracketright/asciicircum/underscore/grave
/a/b/c/d/e/f/g/h/i/j/k/l/m/n/o/p/q/r/s/t/u/v/w/x/y/z/braceleft/bar/braceright
/asciitilde/.notdef/Euro/.notdef/quotesinglbase/florin/quotedblbase/ellipsis
/dagger/daggerdbl/circumflex/perthousand/Scaron/guilsinglleft/OE/.notdef
/.notdef/.notdef/.notdef/quoteleft/quoteright/quotedblleft/quotedblright/bullet
/endash/emdash/tilde/trademark/scaron/guilsinglright/oe/.notdef/.notdef
/Ydieresis/.notdef/exclamdown/cent/sterling/currency/yen/brokenbar/section
/dieresis/copyright/ordfeminine/guillemotleft/logicalnot/.notdef/registered
/macron/degree/plusminus/twosuperior/threesuperior/acute/mu/paragraph
/periodcentered/cedilla/onesuperior/ordmasculine/guillemotright/onequarter
/onehalf/threequarters/questiondown/Agrave/Aacute/Acircumflex/Atilde/Adieresis
/Aring/AE/Ccedilla/Egrave/Eacute/Ecircumflex/Edieresis/Igrave/Iacute
/Icircumflex/Idieresis/Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis
/multiply/Oslash/Ugrave/Uacute/Ucircumflex/Udieresis/Yacute/Thorn/germandbls
/agrave/aacute/acircumflex/atilde/adieresis/aring/ae/ccedilla/egrave/eacute
/ecircumflex/edieresis/igrave/iacute/icircumflex/idieresis/eth/ntilde/ograve
/oacute/ocircumflex/otilde/odieresis/divide/oslash/ugrave/uacute/ucircumflex
/udieresis/yacute/thorn/ydieresis]def EncodeDict/256 CharCol256Encoding put
%%EndResource
end reinitialize

%%IncludeResource: font Times-Italic
Pscript_WinNT_Incr begin
%%BeginResource: file Pscript_Win_Euro_L2 5.0 0
/UseT3EuroFont{/currentdistillerparams where{pop currentdistillerparams
/CoreDistVersion get 4000 le}{false}ifelse}bind def/NewEuroT3Font?{dup/FontType
get 3 eq{dup/EuroFont known exch/BaseFont known and}{pop false}ifelse}bind def
/T1FontHasEuro{dup/CharStrings known not{dup NewEuroT3Font?{dup/EuroGlyphName
get exch/EuroFont get/CharStrings get exch known{true}{false}ifelse}{pop false}
ifelse}{dup/FontType get 1 eq{/CharStrings get/Euro known}{dup/InfoDict known{
/InfoDict get/Euro known}{/CharStrings get/Euro known}ifelse}ifelse}ifelse}bind
def/FontHasEuro{findfont dup/Blend known{pop true}{T1FontHasEuro}ifelse}bind
def/EuroEncodingIdx 1 def/EuroFontHdr{12 dict begin/FontInfo 10 dict dup begin
/version(001.000)readonly def/Notice(Copyright (c)1999 Adobe Systems
Incorporated. All Rights Reserved.)readonly def/FullName(Euro)readonly def
/FamilyName(Euro)readonly def/Weight(Regular)readonly def/isFixedPitch false
def/ItalicAngle 0 def/UnderlinePosition -100 def/UnderlineThickness 50 def end
readonly def/FontName/Euro def/Encoding 256 array 0 1 255{1 index exch/.notdef
put}for def/PaintType 0 def/FontType 1 def/FontMatrix[0.001 0 0 0.001 0 0]def
/FontBBox{-25 -23 1500 804}readonly def currentdict end dup/Private 20 dict dup
begin/ND{def}def/NP{put}def/lenIV -1 def/RD{string currentfile exch
readhexstring pop}def/-|{string currentfile exch readstring pop}executeonly def
/|-{def}executeonly def/|{put}executeonly def/BlueValues[-20 0 706 736 547 572]
|-/OtherBlues[-211 -203]|-/BlueScale 0.0312917 def/MinFeature{16 16}|-/StdHW
[60]|-/StdVW[71]|-/ForceBold false def/password 5839 def/Erode{8.5 dup 3 -1
roll 0.1 mul exch 0.5 sub mul cvi sub dup mul 71 0 dtransform dup mul exch dup
mul add le{pop pop 1.0 1.0}{pop pop 0.0 1.5}ifelse}def/OtherSubrs[{}{}{}
{systemdict/internaldict known not{pop 3}{1183615869 systemdict/internaldict
get exec dup/startlock known{/startlock get exec}{dup/strtlck known{/strtlck
get exec}{pop 3}ifelse}ifelse}ifelse}executeonly]|-/Subrs 5 array dup 0
<8E8B0C100C110C110C210B>put dup 1<8B8C0C100B>put dup 2<8B8D0C100B>put dup 3<0B>
put dup 4<8E8C8E0C100C110A0B>put |- 2 index/CharStrings 256 dict dup begin
/.notdef<8b8b0d0e>def end end put put dup/FontName get exch definefont pop}bind
def/AddEuroGlyph{2 index exch EuroEncodingIdx 1 eq{EuroFontHdr}if systemdict
begin/Euro findfont dup dup/Encoding get 5 1 roll/Private get begin/CharStrings
get dup 3 index known{pop pop pop pop end end}{begin 1 index exch def end end
end EuroEncodingIdx dup 1 add/EuroEncodingIdx exch def exch put}ifelse}bind def
/GetNewXUID{currentdict/XUID known{[7 XUID aload pop]true}{currentdict/UniqueID
known{[7 UniqueID]true}{false}ifelse}ifelse}bind def/BuildT3EuroFont{exch 16
dict begin dup/FontName exch def findfont dup/Encoding get/Encoding exch def
dup length 1 add dict copy dup/FID undef begin dup dup/FontName exch def
/Encoding 256 array 0 1 255{1 index exch/.notdef put}for def GetNewXUID{/XUID
exch def}if currentdict end definefont pop/BaseFont exch findfont 1000
scalefont def/EuroFont exch findfont 1000 scalefont def pop/EuroGlyphName exch
def/FontType 3 def/FontMatrix[.001 0 0 .001 0 0]def/FontBBox BaseFont/FontBBox
get def/Char 1 string def/BuildChar{exch dup begin/Encoding get 1 index get
/Euro eq{BaseFont T1FontHasEuro{false}{true}ifelse}{false}ifelse{EuroFont
setfont pop userdict/Idx 0 put EuroFont/Encoding get{EuroGlyphName eq{exit}
{userdict/Idx Idx 1 add put}ifelse}forall userdict/Idx get}{dup dup Encoding
exch get BaseFont/Encoding get 3 1 roll put BaseFont setfont}ifelse Char 0 3 -1
roll put Char stringwidth newpath 0 0 moveto Char true charpath flattenpath
pathbbox setcachedevice 0 0 moveto Char show end}bind def currentdict end dup
/FontName get exch definefont pop}bind def/AddEuroToT1Font{dup findfont dup
length 10 add dict copy dup/FID undef begin/EuroFont 3 -1 roll findfont 1000
scalefont def CharStrings dup length 1 add dict copy begin/Euro{EuroFont
setfont pop EuroGBBox aload pop setcachedevice 0 0 moveto EuroGName glyphshow}
bind def currentdict end/CharStrings exch def GetNewXUID{/XUID exch def}if 3 1
roll/EuroGBBox exch def/EuroGName exch def currentdict end definefont pop}bind
def/BuildNewFont{UseT3EuroFont{BuildT3EuroFont}{pop AddEuroToT1Font}ifelse}bind
def/UseObliqueEuro{findfont/FontMatrix get dup 2 get 0 eq exch dup 0 get exch 3
get eq and UseT3EuroFont or}bind def
%%EndResource
end reinitialize
/Times-Italic FontHasEuro not
{
/Euro.Times-Italic
 [500 0 23 -7 578 676 ] 
<A2F8880DDBDB03F7F8A003F88D9D0384A701F79DBB01F802BB01F91CA7018F0ADBDB03F7
F8A003F88D9D0384A701F79DBB01F802BB01F89CF71301F8BFF93315770680758A89818B
08848B79947592088F0ADBDB03F7F8A003F88D9D0384A701F79DBB01F802BB01F91CA701
75916F926D8B08FB0E8B27FB0A5BFB24084606735B05D6068879857A8A79084A06745B05
DD068A7D8A7C8B7C08FB01B3FB0BF7101EA28BA4919E94089D94948F928B08928B8D858D
7508A006B0F72205780679674E3D438B08477BD1D11F8BB290B28FA508F76306A2BB05FB
71068D9D929C8F9D08F77E06A3BB05FB88069FDEC8F72BEC8B08D9B757461F84078F0ADB
DB03F7F8A003F88D9D0384A701F79DBB01F802BB01F89CF713019D06090E>
AddEuroGlyph
/Euro /Times-Italic /Times-Italic-Copy BuildNewFont
} if
F /F1 0 /256 T /Times-Italic mF 
/F1S10DYFFFFFEF2 F1 [269.496 0 0 -269.457 0 0 ] mFS
F1S10DYFFFFFEF2 Ji 
3271 3273 M (y)S 
/F1SCCYFFFFFF33 F1 [204.539 0 0 -204.512 0 0 ] mFS
F1SCCYFFFFFF33 Ji 
3904 2470 M (y)S 
11 dict begin
/FontName /TT76Do00 def
/FontMatrix [1 1000 div 0 0 1 1000 div 0 0 ] def
/Encoding  256 array 0 1 255 {1 index exch /.notdef put} for  def
/PaintType 0 def
/FontType 1 def
/FontBBox { 0 0 0 0 } def
AddFontInfoBegin
AddFontInfo
AddFontInfoEnd
currentdict
end

systemdict begin
dup /Private 7 dict dup begin
/BlueValues [] def
/MinFeature {16 16} def
/password 5839 def
/ND {def} def
/NP {put} def
/lenIV -1 def
/RD {string currentfile exch readhexstring pop} def

2 index /CharStrings 256 dict dup begin
/.notdef 4 RD 
8b8b0d0e ND 
/parenleft 101 RD 
8bf82f0df82ff704158c85817e7776087776777c7780086a7a64835e8b08
5f8b6593699c087696789b779f0878a081988b9108938f059f739c7b9984
0898849d85a08508a383ab88b18b08b28bab8ea293089f909c919a93089a
939c9a9ea308948705090e ND 
end
end
put
put
dup /FontName get exch definefont pop
end
/TT76Do00 findfont /Encoding get
dup 1 /parenleft put
pop
F /F2 0 /0 F /TT76Do00 mF 
/F2SCCYFFFFFF33 F2 [204.539 0 0 -204.512 0 0 ] mFS
F2SCCYFFFFFF33 Ji 
3911 2344 M <01>S 
/F1S72YFFFFFF8D F1 [114.598 0 0 -114.582 0 0 ] mFS
F1S72YFFFFFF8D Ji 
4060 1700 M (N)S 
/F1SC5YFFFFFF3A F1 [197.629 0 0 -197.602 0 0 ] mFS
F1SC5YFFFFFF3A Ji 
3637 1651 M (yte)[279 56  0]xS 
/F0SC5YFFFFFF3A F0 [197.629 0 0 -197.602 0 0 ] mFS
F0SC5YFFFFFF3A Ji 
3771 1651 M (+)S 
11 dict begin
/FontName /TT771o00 def
/FontMatrix [1 1000 div 0 0 1 1000 div 0 0 ] def
/Encoding  256 array 0 1 255 {1 index exch /.notdef put} for  def
/PaintType 0 def
/FontType 1 def
/FontBBox { 0 0 0 0 } def
AddFontInfoBegin
AddFontInfo
AddFontInfoEnd
currentdict
end

systemdict begin
dup /Private 7 dict dup begin
/BlueValues [] def
/MinFeature {16 16} def
/password 5839 def
/ND {def} def
/NP {put} def
/lenIV -1 def
/RD {string currentfile exch readhexstring pop} def

2 index /CharStrings 256 dict dup begin
/.notdef 4 RD 
8b8b0d0e ND 
/parenleft 101 RD 
8bf82e0df82ef706158c84827e7676087776777c788008697a65825e8b08
5f8b64946a9c087696779b78a008789f81988b9208938f059f729c7c9984
0898849d84a08508a384aa87b28b08b28baa8fa392089e909c929a93089b
929c9b9ea308938705090e ND 
end
end
put
put
dup /FontName get exch definefont pop
end
/TT771o00 findfont /Encoding get
dup 1 /parenleft put
pop
F /F3 0 /0 F /TT771o00 mF 
/F3SC5YFFFFFF3A F3 [197.629 0 0 -197.602 0 0 ] mFS
F3SC5YFFFFFF3A Ji 
3644 1529 M <01>S 
F1SC5YFFFFFF3A Ji 
4542 982 M (cq)[143  0]xS 
/F0S72YFFFFFF8D F0 [114.598 0 0 -114.582 0 0 ] mFS
F0S72YFFFFFF8D Ji 
4623 1031 M (*)S 
F0S95YFFFFFF6A Ji 
3970 1084 M <B7>S 
F1SC5YFFFFFF3A Ji 
3838 1202 M (cq)[143  0]xS 
F0S72YFFFFFF8D Ji 
3919 1251 M (*)S 
11 dict begin
/FontName /TT774o00 def
/FontMatrix [1 1000 div 0 0 1 1000 div 0 0 ] def
/Encoding  256 array 0 1 255 {1 index exch /.notdef put} for  def
/PaintType 0 def
/FontType 1 def
/FontBBox { 0 0 0 0 } def
AddFontInfoBegin
AddFontInfo
AddFontInfoEnd
currentdict
end

systemdict begin
dup /Private 7 dict dup begin
/BlueValues [] def
/MinFeature {16 16} def
/password 5839 def
/ND {def} def
/NP {put} def
/lenIV -1 def
/RD {string currentfile exch readhexstring pop} def

2 index /CharStrings 256 dict dup begin
/.notdef 4 RD 
8b8b0d0e ND 
/percent 106 RD 
8bf7e20df7e2f9111589708073767508777473806f8c087b8a7c8e7d9208
7d90809181920882908290828f08828e808d7d8b087d8b7f868082088081
84808980086d0693a998a49d9d089d9ea394a98c08a18ab57cc76e089688
968c958f089e92989b91a308a806090e ND 
end
end
put
put
dup /FontName get exch definefont pop
end
/TT774o00 findfont /Encoding get
dup 1 /percent put
pop
F /F4 0 /0 F /TT774o00 mF 
/F4SC5YFFFFFF3A F4 [197.629 0 0 -197.602 0 0 ] mFS
F4SC5YFFFFFF3A Ji 
3861 1184 M <01>S 

%%IncludeResource: font Times-Roman
/Times-Roman FontHasEuro not
{
/Euro.Times-Roman
 [500 0 24 -14 493 676 ] 
<A3F8880DC0EB03F854F743950C0C037DA501F7A0BB01F805BB01F91EA501F869F8911584
F73A0576068478827E7F8B087F8B70AC438B082A8BFB0D6471FB700863067C5B05BE068A
7E8B878B7D08750764067D5B05C206A0FB7BF71858E88B088F0AC0EB03F84BF789950C0C
037DA501F7A0BB01F805BB01F91EA501CA8BACAC998B08988B95809076088F0AC0EB03F8
54F743950C0C037DA501F7A0BB01F805BB01F91EA501A00692F73705750675FB09417667
8B083A8B43C182F75E08F7740699BB05FB84069C078B978C978B9708F7950699BB05FBA1
069AF755D5B3D18B08AB8BDD71A2FB0708090E>
AddEuroGlyph
/Euro /Times-Roman /Times-Roman-Copy BuildNewFont
} if
F /F5 0 /256 T /Times-Roman mF 
/F5SA2YFFFFFF5D F5 [162.754 0 0 -162.73 0 0 ] mFS
F5SA2YFFFFFF5D Ji 
5023 1492 M (\(,\))[159 167  0]xS 
/F1S5EYFFFFFFA1 F1 [94.375 0 0 -94.359 0 0 ] mFS
F1S5EYFFFFFFA1 Ji 
4973 1560 M (t)S 
/F1SA2YFFFFFF5D F1 [162.754 0 0 -162.73 0 0 ] mFS
F1SA2YFFFFFF5D Ji 
5071 1492 M (YR)[173  0]xS 
11 dict begin
/FontName /TT778o00 def
/FontMatrix [1 2048 div 0 0 1 2048 div 0 0 ] def
/Encoding  256 array 0 1 255 {1 index exch /.notdef put} for  def
/PaintType 0 def
/FontType 1 def
/FontBBox { 0 0 0 0 } def
AddFontInfoBegin
AddFontInfo
AddFontInfoEnd
currentdict
end

systemdict begin
dup /Private 7 dict dup begin
/BlueValues [] def
/MinFeature {16 16} def
/password 5839 def
/ND {def} def
/NP {put} def
/lenIV -1 def
/RD {string currentfile exch readhexstring pop} def

2 index /CharStrings 256 dict dup begin
/.notdef 4 RD 
8b8b0d0e ND 
/summation 107 RD 
bdff000005b40dff00000522fb5d15fffffffadb06b407f930f9c305fd30
f9cd05b207ff0000050306a4fbec056a0680e771ce62b30861b3529f408c
08fd7906f8c6fd4b05fd0ffd9a05f9c906bb8bbe94c29d08b197aba1a4ac
08a4aba0bd9cd008b5840548fc2905090e ND 
end
end
put
put
dup /FontName get exch definefont pop
end
/TT778o00 findfont /Encoding get
dup 1 /summation put
pop
F /F6 0 /0 F /TT778o00 mF 
/F6SF4YFFFFFF0B F6 [244.129 0 0 -244.098 0 0 ] mFS
F6SF4YFFFFFF0B Ji 
4789 1530 M <01>S 
11 dict begin
/FontName /TT779o00 def
/FontMatrix [1 1000 div 0 0 1 1000 div 0 0 ] def
/Encoding  256 array 0 1 255 {1 index exch /.notdef put} for  def
/PaintType 0 def
/FontType 1 def
/FontBBox { 0 0 0 0 } def
AddFontInfoBegin
AddFontInfo
AddFontInfoEnd
currentdict
end

systemdict begin
dup /Private 7 dict dup begin
/BlueValues [] def
/MinFeature {16 16} def
/password 5839 def
/ND {def} def
/NP {put} def
/lenIV -1 def
/RD {string currentfile exch readhexstring pop} def

2 index /CharStrings 256 dict dup begin
/.notdef 4 RD 
8b8b0d0e ND 
/parenleft 101 RD 
8bf8300df830f708158c84827e7676087776777b7781086a7864825e8b08
5f8b6494699e087696779b78a008789f81988b9208938f059f739c7b9984
0899849c84a08608a483aa88b28b08b28baa8ea393089f909c919a93089a
939c9a9fa308938705090e ND 
end
end
put
put
dup /FontName get exch definefont pop
end
/TT779o00 findfont /Encoding get
dup 1 /parenleft put
pop
F /F7 0 /0 F /TT779o00 mF 
/F7SA2YFFFFFF5D F7 [162.754 0 0 -162.73 0 0 ] mFS
F7SA2YFFFFFF5D Ji 
5110 1356 M <0101>[162  0]xS 
F5SA2YFFFFFF5D Ji 
4979 2700 M (\(,\))[159 167  0]xS 
F1SA2YFFFFFF5D Ji 
5027 2700 M (YR)[174  0]xS 
11 dict begin
/FontName /TT77Co00 def
/FontMatrix [1 2048 div 0 0 1 2048 div 0 0 ] def
/Encoding  256 array 0 1 255 {1 index exch /.notdef put} for  def
/PaintType 0 def
/FontType 1 def
/FontBBox { 0 0 0 0 } def
AddFontInfoBegin
AddFontInfo
AddFontInfoEnd
currentdict
end

systemdict begin
dup /Private 7 dict dup begin
/BlueValues [] def
/MinFeature {16 16} def
/password 5839 def
/ND {def} def
/NP {put} def
/lenIV -1 def
/RD {string currentfile exch readhexstring pop} def

2 index /CharStrings 256 dict dup begin
/.notdef 4 RD 
8b8b0d0e ND 
/summation 107 RD 
bdff000005b40dff00000522fb5d15fffffffadb06b407f930f9c305fd30
f9cd05b207ff0000050306a4fbec056a0680e771ce62b30861b3529f408c
08fd7906f8c6fd4b05fd0ffd9a05f9c906bb8bbe94c29d08b197aba1a4ac
08a4aba0bd9cd008b5840548fc2905090e ND 
end
end
put
put
dup /FontName get exch definefont pop
end
/TT77Co00 findfont /Encoding get
dup 1 /summation put
pop
F /F8 0 /0 F /TT77Co00 mF 
/F8SF4YFFFFFF0B F8 [244.129 0 0 -244.098 0 0 ] mFS
F8SF4YFFFFFF0B Ji 
4789 2737 M <01>S 
11 dict begin
/FontName /TT77Do00 def
/FontMatrix [1 1000 div 0 0 1 1000 div 0 0 ] def
/Encoding  256 array 0 1 255 {1 index exch /.notdef put} for  def
/PaintType 0 def
/FontType 1 def
/FontBBox { 0 0 0 0 } def
AddFontInfoBegin
AddFontInfo
AddFontInfoEnd
currentdict
end

systemdict begin
dup /Private 7 dict dup begin
/BlueValues [] def
/MinFeature {16 16} def
/password 5839 def
/ND {def} def
/NP {put} def
/lenIV -1 def
/RD {string currentfile exch readhexstring pop} def

2 index /CharStrings 256 dict dup begin
/.notdef 4 RD 
8b8b0d0e ND 
/parenleft 101 RD 
8bf8300df830f708158c84827e7676087776777b7781086a7864825e8b08
5f8b6494699e087696779b78a008789f81988b9208938f059f739c7b9984
0899849c84a08608a483aa88b28b08b28baa8ea393089f909c919a93089a
939c9a9fa308938705090e ND 
end
end
put
put
dup /FontName get exch definefont pop
end
/TT77Do00 findfont /Encoding get
dup 1 /parenleft put
pop
F /F9 0 /0 F /TT77Do00 mF 
/F9SA2YFFFFFF5D F9 [162.754 0 0 -162.73 0 0 ] mFS
F9SA2YFFFFFF5D Ji 
5066 2564 M <0101>[162  0]xS 
N 3524 4246 M 3524 109 I 3524 107 3525 105 3527 105 -c 3529 105 3530 107 3530 109 -c 3530 4246 I 3530 4248 3529 4249 3527 4249 -c 3525 4249 3524 4248 3524 4246 -c C 
3502 117 M 3527 67 I 3552 117 I C 
:  L ; : 0.246 0.246 +S K 
; 0 Lj 1 Lc 8 Lm 75 Lw : 472 0 6025 4413 rc N 480 8 M 480 4329 I 6477 4329 I 6477 8 I C 
: 0.246 0.246 +S K 
; ; 1 Lj 10 Lm : 729 322 5556 4153 rc N 3506 332 M 1978 332 740 1569 740 3094 -c 740 4620 1978 5856 3506 5856 -c 5034 5856 6273 4620 6273 3094 -c 6273 1569 5034 332 3506 332 -c : 0.246 0.246 +S K 
; ; : N 646 3633 5668 842 rp C 
1 1 1 1 scol  L ; 2 Lj 0 Lc 8 Lw N 564 1033 M 6307 1033 I 6309 1033 6310 1034 6310 1036 -c 6310 1037 6309 1039 6307 1039 -c 564 1039 I 562 1039 561 1037 561 1036 -c 561 1034 562 1033 564 1033 -c C 
6299 1011 M 6349 1036 I 6299 1061 I C 
:  L ; : 0.246 0.246 +S K 
; F0S95YFFFFFF6A Ji 
3494 3142 M <B7>S 
4428 1082 M <B7>S 
1 Lj 1 Lc 25 Lw N 3362 3457 M 4835 237 I : 0.246 0.246 +S K 
; 50 Lw N 3694 3444 M 2068 231 I : 0.121 0.121 +S K 
; 2447 1084 M <B7>S 
75 Lw N 3939 885 M 3483 885 3114 1254 3114 1709 -c 3114 2164 3483 2533 3939 2533 -c 4395 2533 4764 2164 4764 1709 -c 4764 1254 4395 885 3939 885 -c : 0.246 0.246 +S K 
; 3924 1752 M <B7>S 
2 Lj 0 Lc 16 Lw N 3523 3066 M 3528 3048 I 3529 3046 3531 3046 3532 3046 -c 3534 3047 3535 3048 3534 3050 -c 3529 3068 I 3528 3070 3527 3070 3525 3070 -c 3523 3069 3522 3068 3523 3066 -c C 
3536 3024 M 3541 3006 I 3542 3005 3544 3004 3545 3004 -c 3547 3005 3548 3007 3547 3008 -c 3542 3026 I 3541 3028 3540 3029 3538 3028 -c 3536 3028 3535 3026 3536 3024 -c C 
3549 2983 M 3554 2965 I 3555 2963 3557 2962 3558 2963 -c 3560 2963 3561 2965 3560 2967 -c 3555 2984 I 3554 2986 3553 2987 3551 2986 -c 3549 2986 3548 2984 3549 2983 -c C 
3562 2941 M 3567 2923 I 3568 2921 3570 2920 3571 2921 -c 3573 2921 3574 2923 3573 2925 -c 3568 2943 I 3567 2944 3566 2945 3564 2945 -c 3562 2944 3561 2943 3562 2941 -c C 
3575 2899 M 3580 2881 I 3581 2880 3583 2879 3584 2879 -c 3586 2880 3587 2882 3586 2883 -c 3581 2901 I 3580 2903 3578 2904 3577 2903 -c 3575 2903 3574 2901 3575 2899 -c C 
3588 2858 M 3593 2840 I 3594 2838 3596 2837 3597 2838 -c 3599 2838 3600 2840 3599 2842 -c 3594 2859 I 3593 2861 3591 2862 3590 2861 -c 3588 2861 3587 2859 3588 2858 -c C 
3601 2816 M 3606 2798 I 3607 2796 3609 2795 3610 2796 -c 3612 2796 3613 2798 3612 2800 -c 3607 2818 I 3606 2819 3604 2820 3603 2820 -c 3601 2819 3600 2817 3601 2816 -c C 
3614 2774 M 3619 2756 I 3620 2755 3622 2754 3623 2754 -c 3625 2755 3626 2756 3625 2758 -c 3620 2776 I 3619 2778 3617 2779 3616 2778 -c 3614 2778 3613 2776 3614 2774 -c C 
3627 2733 M 3632 2715 I 3633 2713 3635 2712 3636 2713 -c 3638 2713 3639 2715 3638 2716 -c 3633 2734 I 3632 2736 3630 2737 3629 2736 -c 3627 2736 3626 2734 3627 2733 -c C 
3640 2691 M 3645 2673 I 3646 2671 3647 2670 3649 2671 -c 3651 2671 3652 2673 3651 2675 -c 3646 2693 I 3645 2694 3643 2695 3642 2695 -c 3640 2694 3639 2692 3640 2691 -c C 
3653 2649 M 3658 2631 I 3659 2630 3660 2629 3662 2629 -c 3664 2630 3665 2631 3664 2633 -c 3659 2651 I 3658 2653 3656 2654 3655 2653 -c 3653 2652 3652 2651 3653 2649 -c C 
3666 2607 M 3671 2590 I 3672 2588 3673 2587 3675 2587 -c 3677 2588 3678 2590 3677 2591 -c 3672 2609 I 3671 2611 3669 2612 3668 2611 -c 3666 2611 3665 2609 3666 2607 -c C 
3679 2566 M 3684 2548 I 3685 2546 3686 2545 3688 2546 -c 3690 2546 3691 2548 3690 2550 -c 3685 2567 I 3684 2569 3682 2570 3681 2570 -c 3679 2569 3678 2567 3679 2566 -c C 
3692 2524 M 3697 2506 I 3697 2504 3699 2504 3701 2504 -c 3703 2505 3703 2506 3703 2508 -c 3697 2526 I 3697 2527 3695 2528 3693 2528 -c 3692 2527 3691 2526 3692 2524 -c C 
3704 2482 M 3710 2464 I 3710 2463 3712 2462 3714 2462 -c 3716 2463 3716 2465 3716 2466 -c 3710 2484 I 3710 2486 3708 2487 3706 2486 -c 3705 2486 3704 2484 3704 2482 -c C 
3717 2441 M 3723 2423 I 3723 2421 3725 2420 3727 2421 -c 3728 2421 3729 2423 3729 2424 -c 3723 2442 I 3723 2444 3721 2445 3719 2444 -c 3718 2444 3717 2442 3717 2441 -c C 
3730 2399 M 3736 2381 I 3736 2379 3738 2378 3740 2379 -c 3741 2379 3742 2381 3742 2383 -c 3736 2401 I 3736 2402 3734 2403 3732 2403 -c 3731 2402 3730 2401 3730 2399 -c C 
3743 2357 M 3749 2339 I 3749 2338 3751 2337 3753 2337 -c 3754 2338 3755 2340 3755 2341 -c 3749 2359 I 3749 2361 3747 2362 3745 2361 -c 3744 2361 3743 2359 3743 2357 -c C 
3756 2315 M 3762 2298 I 3762 2296 3764 2295 3766 2296 -c 3767 2296 3768 2298 3768 2299 -c 3762 2317 I 3762 2319 3760 2320 3758 2319 -c 3757 2319 3756 2317 3756 2315 -c C 
3769 2274 M 3775 2256 I 3775 2254 3777 2253 3779 2254 -c 3780 2254 3781 2256 3781 2258 -c 3775 2276 I 3775 2277 3773 2278 3771 2278 -c 3770 2277 3769 2275 3769 2274 -c C 
3782 2232 M 3788 2214 I 3788 2213 3790 2212 3792 2212 -c 3793 2213 3794 2214 3794 2216 -c 3788 2234 I 3788 2236 3786 2236 3784 2236 -c 3783 2235 3782 2234 3782 2232 -c C 
3795 2190 M 3801 2173 I 3801 2171 3803 2170 3805 2171 -c 3806 2171 3807 2173 3807 2174 -c 3801 2192 I 3801 2194 3799 2195 3797 2194 -c 3796 2194 3795 2192 3795 2190 -c C 
3808 2149 M 3814 2131 I 3814 2129 3816 2128 3818 2129 -c 3819 2129 3820 2131 3820 2133 -c 3814 2151 I 3814 2152 3812 2153 3810 2153 -c 3809 2152 3808 2150 3808 2149 -c C 
3821 2107 M 3827 2089 I 3827 2087 3829 2087 3831 2087 -c 3832 2088 3833 2089 3833 2091 -c 3827 2109 I 3827 2111 3825 2111 3823 2111 -c 3822 2110 3821 2109 3821 2107 -c C 
3834 2065 M 3840 2047 I 3840 2046 3842 2045 3844 2045 -c 3845 2046 3846 2048 3846 2049 -c 3840 2067 I 3840 2069 3838 2070 3836 2069 -c 3835 2069 3834 2067 3834 2065 -c C 
3847 2024 M 3853 2006 I 3853 2004 3855 2003 3857 2004 -c 3858 2004 3859 2006 3859 2008 -c 3853 2025 I 3853 2027 3851 2028 3849 2028 -c 3848 2027 3847 2025 3847 2024 -c C 
3860 1982 M 3866 1964 I 3866 1962 3868 1961 3870 1962 -c 3871 1962 3872 1964 3872 1966 -c 3866 1984 I 3866 1985 3864 1986 3862 1986 -c 3861 1985 3860 1984 3860 1982 -c C 
3873 1940 M 3879 1922 I 3879 1921 3881 1920 3883 1920 -c 3884 1921 3885 1923 3885 1924 -c 3879 1942 I 3879 1944 3877 1945 3875 1944 -c 3873 1944 3873 1942 3873 1940 -c C 
3886 1899 M 3892 1881 I 3892 1879 3894 1878 3896 1879 -c 3897 1879 3898 1881 3898 1882 -c 3892 1900 I 3891 1902 3890 1903 3888 1902 -c 3886 1902 3886 1900 3886 1899 -c C 
3899 1857 M 3905 1839 I 3905 1837 3907 1836 3908 1837 -c 3910 1837 3911 1839 3911 1841 -c 3905 1859 I 3904 1860 3903 1861 3901 1861 -c 3899 1860 3899 1858 3899 1857 -c C 
3912 1815 M 3918 1797 I 3918 1796 3920 1795 3921 1795 -c 3923 1796 3924 1797 3924 1799 -c 3918 1817 I 3917 1819 3916 1819 3914 1819 -c 3912 1818 3911 1817 3912 1815 -c C 
3925 1773 M 3931 1756 I 3931 1754 3933 1753 3934 1753 -c 3936 1754 3937 1756 3936 1757 -c 3931 1775 I 3930 1777 3929 1778 3927 1777 -c 3925 1777 3924 1775 3925 1773 -c C 
3938 1732 M 3943 1714 I 3944 1712 3946 1711 3947 1712 -c 3949 1712 3950 1714 3949 1716 -c 3944 1734 I 3943 1735 3942 1736 3940 1736 -c 3938 1735 3937 1733 3938 1732 -c C 
3951 1690 M 3956 1672 I 3957 1671 3959 1670 3960 1670 -c 3962 1671 3963 1672 3962 1674 -c 3957 1692 I 3956 1693 3954 1694 3953 1694 -c 3951 1693 3950 1692 3951 1690 -c C 
3964 1648 M 3969 1630 I 3970 1629 3972 1628 3973 1628 -c 3975 1629 3976 1631 3975 1632 -c 3970 1650 I 3969 1652 3967 1653 3966 1652 -c 3964 1652 3963 1650 3964 1648 -c C 
3977 1607 M 3982 1589 I 3983 1587 3985 1586 3986 1587 -c 3988 1587 3989 1589 3988 1591 -c 3983 1609 I 3982 1610 3980 1611 3979 1611 -c 3977 1610 3976 1608 3977 1607 -c C 
3990 1565 M 3995 1547 I 3996 1545 3998 1545 3999 1545 -c 4001 1546 4002 1547 4001 1549 -c 3996 1567 I 3995 1568 3993 1569 3992 1569 -c 3990 1568 3989 1567 3990 1565 -c C 
4003 1523 M 4008 1505 I 4009 1504 4011 1503 4012 1503 -c 4014 1504 4015 1506 4014 1507 -c 4009 1525 I 4008 1527 4006 1528 4005 1527 -c 4003 1527 4002 1525 4003 1523 -c C 
4016 1482 M 4021 1464 I 4022 1462 4023 1461 4025 1462 -c 4027 1462 4028 1464 4027 1466 -c 4022 1483 I 4021 1485 4019 1486 4018 1485 -c 4016 1485 4015 1483 4016 1482 -c C 
4029 1440 M 4034 1422 I 4035 1420 4036 1419 4038 1420 -c 4040 1420 4041 1422 4040 1424 -c 4035 1442 I 4034 1443 4032 1444 4031 1444 -c 4029 1443 4028 1441 4029 1440 -c C 
4042 1398 M 4047 1380 I 4048 1379 4049 1378 4051 1378 -c 4053 1379 4054 1380 4053 1382 -c 4048 1400 I 4047 1402 4045 1403 4044 1402 -c 4042 1402 4041 1400 4042 1398 -c C 
4055 1356 M 4060 1339 I 4061 1337 4062 1336 4064 1337 -c 4066 1337 4067 1339 4066 1340 -c 4061 1358 I 4060 1360 4058 1361 4057 1360 -c 4055 1360 4054 1358 4055 1356 -c C 
4068 1315 M 4073 1297 I 4074 1295 4075 1294 4077 1295 -c 4079 1295 4080 1297 4079 1299 -c 4074 1317 I 4073 1318 4071 1319 4070 1319 -c 4068 1318 4067 1316 4068 1315 -c C 
4081 1273 M 4086 1255 I 4087 1253 4088 1253 4090 1253 -c 4092 1254 4093 1255 4092 1257 -c 4087 1275 I 4086 1277 4084 1277 4083 1277 -c 4081 1276 4080 1275 4081 1273 -c C 
4094 1231 M 4099 1213 I 4100 1212 4101 1211 4103 1211 -c 4105 1212 4106 1214 4105 1215 -c 4099 1233 I 4099 1235 4097 1236 4096 1235 -c 4094 1235 4093 1233 4094 1231 -c C 
4106 1190 M 4112 1172 I 4113 1170 4114 1169 4116 1170 -c 4118 1170 4119 1172 4118 1174 -c 4112 1191 I 4112 1193 4110 1194 4109 1193 -c 4107 1193 4106 1191 4106 1190 -c C 
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4145 1064 M 4151 1047 I 4151 1045 4153 1044 4155 1045 -c 4157 1045 4157 1047 4157 1049 -c 4151 1066 I 4151 1068 4149 1069 4147 1068 -c 4146 1068 4145 1066 4145 1064 -c C 
4158 1023 M 4164 1005 I 4164 1003 4166 1002 4168 1003 -c 4170 1003 4170 1005 4170 1007 -c 4164 1025 I 4164 1026 4162 1027 4160 1027 -c 4159 1026 4158 1024 4158 1023 -c C 
4171 981 M 4177 963 I 4177 962 4179 961 4181 961 -c 4182 962 4183 963 4183 965 -c 4177 983 I 4177 985 4175 986 4173 985 -c 4172 985 4171 983 4171 981 -c C 
4184 939 M 4190 922 I 4190 920 4192 919 4194 920 -c 4195 920 4196 922 4196 923 -c 4190 941 I 4190 943 4188 944 4186 943 -c 4185 943 4184 941 4184 939 -c C 
4197 898 M 4203 880 I 4203 878 4205 877 4207 878 -c 4208 878 4209 880 4209 882 -c 4203 900 I 4203 901 4201 902 4199 902 -c 4198 901 4197 899 4197 898 -c C 
4210 856 M 4216 838 I 4216 837 4218 836 4220 836 -c 4221 837 4222 838 4222 840 -c 4216 858 I 4216 860 4214 861 4212 860 -c 4211 859 4210 858 4210 856 -c C 
4223 814 M 4229 797 I 4229 795 4231 794 4233 794 -c 4234 795 4235 797 4235 798 -c 4229 816 I 4229 818 4227 819 4225 818 -c 4224 818 4223 816 4223 814 -c C 
4236 773 M 4242 755 I 4242 753 4244 752 4246 753 -c 4247 753 4248 755 4248 757 -c 4242 774 I 4242 776 4240 777 4238 777 -c 4237 776 4236 774 4236 773 -c C 
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4262 689 M 4268 671 I 4268 670 4270 669 4272 669 -c 4273 670 4274 672 4274 673 -c 4268 691 I 4268 693 4266 694 4264 693 -c 4262 693 4262 691 4262 689 -c C 
4275 648 M 4281 630 I 4281 628 4283 627 4285 628 -c 4286 628 4287 630 4287 631 -c 4281 649 I 4280 651 4279 652 4277 651 -c 4275 651 4275 649 4275 648 -c C 
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827 3090 M 846 3090 I 848 3090 849 3091 849 3093 -c 849 3095 848 3096 846 3096 -c 827 3096 I 826 3096 824 3095 824 3093 -c 824 3091 826 3090 827 3090 -c C 
871 3090 M 890 3090 I 892 3090 893 3091 893 3093 -c 893 3095 892 3096 890 3096 -c 871 3096 I 869 3096 868 3095 868 3093 -c 868 3091 869 3090 871 3090 -c C 
915 3090 M 933 3090 I 935 3090 936 3091 936 3093 -c 936 3095 935 3096 933 3096 -c 915 3096 I 913 3096 912 3095 912 3093 -c 912 3091 913 3090 915 3090 -c C 
958 3090 M 977 3090 I 979 3090 980 3091 980 3093 -c 980 3095 979 3096 977 3096 -c 958 3096 I 957 3096 955 3095 955 3093 -c 955 3091 957 3090 958 3090 -c C 
1002 3090 M 1021 3090 I 1023 3090 1024 3091 1024 3093 -c 1024 3095 1023 3096 1021 3096 -c 1002 3096 I 1000 3096 999 3095 999 3093 -c 999 3091 1000 3090 1002 3090 -c C 
1046 3090 M 1065 3090 I 1066 3090 1068 3091 1068 3093 -c 1068 3095 1066 3096 1065 3096 -c 1046 3096 I 1044 3096 1043 3095 1043 3093 -c 1043 3091 1044 3090 1046 3090 -c C 
1090 3090 M 1108 3090 I 1110 3090 1111 3091 1111 3093 -c 1111 3095 1110 3096 1108 3096 -c 1090 3096 I 1088 3096 1086 3095 1086 3093 -c 1086 3091 1088 3090 1090 3090 -c C 
1133 3090 M 1152 3090 I 1154 3090 1155 3091 1155 3093 -c 1155 3095 1154 3096 1152 3096 -c 1133 3096 I 1132 3096 1130 3095 1130 3093 -c 1130 3091 1132 3090 1133 3090 -c C 
1177 3090 M 1196 3090 I 1198 3090 1199 3091 1199 3093 -c 1199 3095 1198 3096 1196 3096 -c 1177 3096 I 1175 3096 1174 3095 1174 3093 -c 1174 3091 1175 3090 1177 3090 -c C 
1221 3090 M 1240 3090 I 1241 3090 1243 3091 1243 3093 -c 1243 3095 1241 3096 1240 3096 -c 1221 3096 I 1219 3096 1218 3095 1218 3093 -c 1218 3091 1219 3090 1221 3090 -c C 
1265 3090 M 1283 3090 I 1285 3090 1286 3091 1286 3093 -c 1286 3095 1285 3096 1283 3096 -c 1265 3096 I 1263 3096 1261 3095 1261 3093 -c 1261 3091 1263 3090 1265 3090 -c C 
1308 3090 M 1327 3090 I 1329 3090 1330 3091 1330 3093 -c 1330 3095 1329 3096 1327 3096 -c 1308 3096 I 1307 3096 1305 3095 1305 3093 -c 1305 3091 1307 3090 1308 3090 -c C 
1352 3090 M 1371 3090 I 1373 3090 1374 3091 1374 3093 -c 1374 3095 1373 3096 1371 3096 -c 1352 3096 I 1350 3096 1349 3095 1349 3093 -c 1349 3091 1350 3090 1352 3090 -c C 
1396 3090 M 1415 3090 I 1416 3090 1418 3091 1418 3093 -c 1418 3095 1416 3096 1415 3096 -c 1396 3096 I 1394 3096 1393 3095 1393 3093 -c 1393 3091 1394 3090 1396 3090 -c C 
1439 3090 M 1458 3090 I 1460 3090 1461 3091 1461 3093 -c 1461 3095 1460 3096 1458 3096 -c 1439 3096 I 1438 3096 1436 3095 1436 3093 -c 1436 3091 1438 3090 1439 3090 -c C 
1483 3090 M 1502 3090 I 1504 3090 1505 3091 1505 3093 -c 1505 3095 1504 3096 1502 3096 -c 1483 3096 I 1481 3096 1480 3095 1480 3093 -c 1480 3091 1481 3090 1483 3090 -c C 
1527 3090 M 1546 3090 I 1547 3090 1549 3091 1549 3093 -c 1549 3095 1547 3096 1546 3096 -c 1527 3096 I 1525 3096 1524 3095 1524 3093 -c 1524 3091 1525 3090 1527 3090 -c C 
1571 3090 M 1589 3090 I 1591 3090 1592 3091 1592 3093 -c 1592 3095 1591 3096 1589 3096 -c 1571 3096 I 1569 3096 1567 3095 1567 3093 -c 1567 3091 1569 3090 1571 3090 -c C 
1614 3090 M 1633 3090 I 1635 3090 1636 3091 1636 3093 -c 1636 3095 1635 3096 1633 3096 -c 1614 3096 I 1613 3096 1611 3095 1611 3093 -c 1611 3091 1613 3090 1614 3090 -c C 
1658 3090 M 1677 3090 I 1679 3090 1680 3091 1680 3093 -c 1680 3095 1679 3096 1677 3096 -c 1658 3096 I 1656 3096 1655 3095 1655 3093 -c 1655 3091 1656 3090 1658 3090 -c C 
1702 3090 M 1721 3090 I 1722 3090 1724 3091 1724 3093 -c 1724 3095 1722 3096 1721 3096 -c 1702 3096 I 1700 3096 1699 3095 1699 3093 -c 1699 3091 1700 3090 1702 3090 -c C 
1746 3090 M 1764 3090 I 1766 3090 1767 3091 1767 3093 -c 1767 3095 1766 3096 1764 3096 -c 1746 3096 I 1744 3096 1742 3095 1742 3093 -c 1742 3091 1744 3090 1746 3090 -c C 
1789 3090 M 1808 3090 I 1810 3090 1811 3091 1811 3093 -c 1811 3095 1810 3096 1808 3096 -c 1789 3096 I 1788 3096 1786 3095 1786 3093 -c 1786 3091 1788 3090 1789 3090 -c C 
1833 3090 M 1852 3090 I 1854 3090 1855 3091 1855 3093 -c 1855 3095 1854 3096 1852 3096 -c 1833 3096 I 1831 3096 1830 3095 1830 3093 -c 1830 3091 1831 3090 1833 3090 -c C 
1877 3090 M 1896 3090 I 1897 3090 1899 3091 1899 3093 -c 1899 3095 1897 3096 1896 3096 -c 1877 3096 I 1875 3096 1874 3095 1874 3093 -c 1874 3091 1875 3090 1877 3090 -c C 
1921 3090 M 1939 3090 I 1941 3090 1942 3091 1942 3093 -c 1942 3095 1941 3096 1939 3096 -c 1921 3096 I 1919 3096 1917 3095 1917 3093 -c 1917 3091 1919 3090 1921 3090 -c C 
1964 3090 M 1983 3090 I 1985 3090 1986 3091 1986 3093 -c 1986 3095 1985 3096 1983 3096 -c 1964 3096 I 1962 3096 1961 3095 1961 3093 -c 1961 3091 1962 3090 1964 3090 -c C 
2008 3090 M 2027 3090 I 2028 3090 2030 3091 2030 3093 -c 2030 3095 2028 3096 2027 3096 -c 2008 3096 I 2006 3096 2005 3095 2005 3093 -c 2005 3091 2006 3090 2008 3090 -c C 
2052 3090 M 2070 3090 I 2072 3090 2073 3091 2073 3093 -c 2073 3095 2072 3096 2070 3096 -c 2052 3096 I 2050 3096 2048 3095 2048 3093 -c 2048 3091 2050 3090 2052 3090 -c C 
2095 3090 M 2114 3090 I 2116 3090 2117 3091 2117 3093 -c 2117 3095 2116 3096 2114 3096 -c 2095 3096 I 2094 3096 2092 3095 2092 3093 -c 2092 3091 2094 3090 2095 3090 -c C 
2139 3090 M 2158 3090 I 2160 3090 2161 3091 2161 3093 -c 2161 3095 2160 3096 2158 3096 -c 2139 3096 I 2137 3096 2136 3095 2136 3093 -c 2136 3091 2137 3090 2139 3090 -c C 
2183 3090 M 2202 3090 I 2203 3090 2205 3091 2205 3093 -c 2205 3095 2203 3096 2202 3096 -c 2183 3096 I 2181 3096 2180 3095 2180 3093 -c 2180 3091 2181 3090 2183 3090 -c C 
2227 3090 M 2245 3090 I 2247 3090 2248 3091 2248 3093 -c 2248 3095 2247 3096 2245 3096 -c 2227 3096 I 2225 3096 2223 3095 2223 3093 -c 2223 3091 2225 3090 2227 3090 -c C 
2270 3090 M 2289 3090 I 2291 3090 2292 3091 2292 3093 -c 2292 3095 2291 3096 2289 3096 -c 2270 3096 I 2269 3096 2267 3095 2267 3093 -c 2267 3091 2269 3090 2270 3090 -c C 
2314 3090 M 2333 3090 I 2335 3090 2336 3091 2336 3093 -c 2336 3095 2335 3096 2333 3096 -c 2314 3096 I 2312 3096 2311 3095 2311 3093 -c 2311 3091 2312 3090 2314 3090 -c C 
2358 3090 M 2377 3090 I 2378 3090 2380 3091 2380 3093 -c 2380 3095 2378 3096 2377 3096 -c 2358 3096 I 2356 3096 2355 3095 2355 3093 -c 2355 3091 2356 3090 2358 3090 -c C 
2402 3090 M 2420 3090 I 2422 3090 2423 3091 2423 3093 -c 2423 3095 2422 3096 2420 3096 -c 2402 3096 I 2400 3096 2398 3095 2398 3093 -c 2398 3091 2400 3090 2402 3090 -c C 
2445 3090 M 2464 3090 I 2466 3090 2467 3091 2467 3093 -c 2467 3095 2466 3096 2464 3096 -c 2445 3096 I 2443 3096 2442 3095 2442 3093 -c 2442 3091 2443 3090 2445 3090 -c C 
2489 3090 M 2508 3090 I 2509 3090 2511 3091 2511 3093 -c 2511 3095 2509 3096 2508 3096 -c 2489 3096 I 2487 3096 2486 3095 2486 3093 -c 2486 3091 2487 3090 2489 3090 -c C 
2533 3090 M 2551 3090 I 2553 3090 2554 3091 2554 3093 -c 2554 3095 2553 3096 2551 3096 -c 2533 3096 I 2531 3096 2529 3095 2529 3093 -c 2529 3091 2531 3090 2533 3090 -c C 
2576 3090 M 2595 3090 I 2597 3090 2598 3091 2598 3093 -c 2598 3095 2597 3096 2595 3096 -c 2576 3096 I 2575 3096 2573 3095 2573 3093 -c 2573 3091 2575 3090 2576 3090 -c C 
2620 3090 M 2639 3090 I 2641 3090 2642 3091 2642 3093 -c 2642 3095 2641 3096 2639 3096 -c 2620 3096 I 2618 3096 2617 3095 2617 3093 -c 2617 3091 2618 3090 2620 3090 -c C 
2664 3090 M 2683 3090 I 2684 3090 2686 3091 2686 3093 -c 2686 3095 2684 3096 2683 3096 -c 2664 3096 I 2662 3096 2661 3095 2661 3093 -c 2661 3091 2662 3090 2664 3090 -c C 
2708 3090 M 2726 3090 I 2728 3090 2729 3091 2729 3093 -c 2729 3095 2728 3096 2726 3096 -c 2708 3096 I 2706 3096 2704 3095 2704 3093 -c 2704 3091 2706 3090 2708 3090 -c C 
2751 3090 M 2770 3090 I 2772 3090 2773 3091 2773 3093 -c 2773 3095 2772 3096 2770 3096 -c 2751 3096 I 2750 3096 2748 3095 2748 3093 -c 2748 3091 2750 3090 2751 3090 -c C 
2795 3090 M 2814 3090 I 2816 3090 2817 3091 2817 3093 -c 2817 3095 2816 3096 2814 3096 -c 2795 3096 I 2793 3096 2792 3095 2792 3093 -c 2792 3091 2793 3090 2795 3090 -c C 
2839 3090 M 2858 3090 I 2859 3090 2861 3091 2861 3093 -c 2861 3095 2859 3096 2858 3096 -c 2839 3096 I 2837 3096 2836 3095 2836 3093 -c 2836 3091 2837 3090 2839 3090 -c C 
2883 3090 M 2901 3090 I 2903 3090 2904 3091 2904 3093 -c 2904 3095 2903 3096 2901 3096 -c 2883 3096 I 2881 3096 2879 3095 2879 3093 -c 2879 3091 2881 3090 2883 3090 -c C 
2926 3090 M 2945 3090 I 2947 3090 2948 3091 2948 3093 -c 2948 3095 2947 3096 2945 3096 -c 2926 3096 I 2925 3096 2923 3095 2923 3093 -c 2923 3091 2925 3090 2926 3090 -c C 
2970 3090 M 2989 3090 I 2990 3090 2992 3091 2992 3093 -c 2992 3095 2990 3096 2989 3096 -c 2970 3096 I 2968 3096 2967 3095 2967 3093 -c 2967 3091 2968 3090 2970 3090 -c C 
3014 3090 M 3032 3090 I 3034 3090 3035 3091 3035 3093 -c 3035 3095 3034 3096 3032 3096 -c 3014 3096 I 3012 3096 3010 3095 3010 3093 -c 3010 3091 3012 3090 3014 3090 -c C 
3057 3090 M 3076 3090 I 3078 3090 3079 3091 3079 3093 -c 3079 3095 3078 3096 3076 3096 -c 3057 3096 I 3056 3096 3054 3095 3054 3093 -c 3054 3091 3056 3090 3057 3090 -c C 
3101 3090 M 3120 3090 I 3122 3090 3123 3091 3123 3093 -c 3123 3095 3122 3096 3120 3096 -c 3101 3096 I 3099 3096 3098 3095 3098 3093 -c 3098 3091 3099 3090 3101 3090 -c C 
3145 3090 M 3164 3090 I 3165 3090 3167 3091 3167 3093 -c 3167 3095 3165 3096 3164 3096 -c 3145 3096 I 3143 3096 3142 3095 3142 3093 -c 3142 3091 3143 3090 3145 3090 -c C 
3189 3090 M 3207 3090 I 3209 3090 3210 3091 3210 3093 -c 3210 3095 3209 3096 3207 3096 -c 3189 3096 I 3187 3096 3185 3095 3185 3093 -c 3185 3091 3187 3090 3189 3090 -c C 
3232 3090 M 3251 3090 I 3253 3090 3254 3091 3254 3093 -c 3254 3095 3253 3096 3251 3096 -c 3232 3096 I 3231 3096 3229 3095 3229 3093 -c 3229 3091 3231 3090 3232 3090 -c C 
3276 3090 M 3295 3090 I 3297 3090 3298 3091 3298 3093 -c 3298 3095 3297 3096 3295 3096 -c 3276 3096 I 3274 3096 3273 3095 3273 3093 -c 3273 3091 3274 3090 3276 3090 -c C 
3320 3090 M 3339 3090 I 3340 3090 3342 3091 3342 3093 -c 3342 3095 3340 3096 3339 3096 -c 3320 3096 I 3318 3096 3317 3095 3317 3093 -c 3317 3091 3318 3090 3320 3090 -c C 
3364 3090 M 3382 3090 I 3384 3090 3385 3091 3385 3093 -c 3385 3095 3384 3096 3382 3096 -c 3364 3096 I 3362 3096 3360 3095 3360 3093 -c 3360 3091 3362 3090 3364 3090 -c C 
3407 3090 M 3426 3090 I 3428 3090 3429 3091 3429 3093 -c 3429 3095 3428 3096 3426 3096 -c 3407 3096 I 3406 3096 3404 3095 3404 3093 -c 3404 3091 3406 3090 3407 3090 -c C 
3451 3090 M 3470 3090 I 3471 3090 3473 3091 3473 3093 -c 3473 3095 3471 3096 3470 3096 -c 3451 3096 I 3449 3096 3448 3095 3448 3093 -c 3448 3091 3449 3090 3451 3090 -c C 
3495 3090 M 3513 3090 I 3515 3090 3516 3091 3516 3093 -c 3516 3095 3515 3096 3513 3096 -c 3495 3096 I 3493 3096 3491 3095 3491 3093 -c 3491 3091 3493 3090 3495 3090 -c C 
3538 3090 M 3557 3090 I 3559 3090 3560 3091 3560 3093 -c 3560 3095 3559 3096 3557 3096 -c 3538 3096 I 3537 3096 3535 3095 3535 3093 -c 3535 3091 3537 3090 3538 3090 -c C 
3582 3090 M 3601 3090 I 3603 3090 3604 3091 3604 3093 -c 3604 3095 3603 3096 3601 3096 -c 3582 3096 I 3580 3096 3579 3095 3579 3093 -c 3579 3091 3580 3090 3582 3090 -c C 
3626 3090 M 3645 3090 I 3646 3090 3648 3091 3648 3093 -c 3648 3095 3646 3096 3645 3096 -c 3626 3096 I 3624 3096 3623 3095 3623 3093 -c 3623 3091 3624 3090 3626 3090 -c C 
3670 3090 M 3688 3090 I 3690 3090 3691 3091 3691 3093 -c 3691 3095 3690 3096 3688 3096 -c 3670 3096 I 3668 3096 3666 3095 3666 3093 -c 3666 3091 3668 3090 3670 3090 -c C 
3713 3090 M 3732 3090 I 3734 3090 3735 3091 3735 3093 -c 3735 3095 3734 3096 3732 3096 -c 3713 3096 I 3712 3096 3710 3095 3710 3093 -c 3710 3091 3712 3090 3713 3090 -c C 
3757 3090 M 3776 3090 I 3778 3090 3779 3091 3779 3093 -c 3779 3095 3778 3096 3776 3096 -c 3757 3096 I 3755 3096 3754 3095 3754 3093 -c 3754 3091 3755 3090 3757 3090 -c C 
3801 3090 M 3820 3090 I 3821 3090 3823 3091 3823 3093 -c 3823 3095 3821 3096 3820 3096 -c 3801 3096 I 3799 3096 3798 3095 3798 3093 -c 3798 3091 3799 3090 3801 3090 -c C 
3845 3090 M 3863 3090 I 3865 3090 3866 3091 3866 3093 -c 3866 3095 3865 3096 3863 3096 -c 3845 3096 I 3843 3096 3841 3095 3841 3093 -c 3841 3091 3843 3090 3845 3090 -c C 
3888 3090 M 3907 3090 I 3909 3090 3910 3091 3910 3093 -c 3910 3095 3909 3096 3907 3096 -c 3888 3096 I 3887 3096 3885 3095 3885 3093 -c 3885 3091 3887 3090 3888 3090 -c C 
3932 3090 M 3951 3090 I 3952 3090 3954 3091 3954 3093 -c 3954 3095 3952 3096 3951 3096 -c 3932 3096 I 3930 3096 3929 3095 3929 3093 -c 3929 3091 3930 3090 3932 3090 -c C 
3976 3090 M 3994 3090 I 3996 3090 3997 3091 3997 3093 -c 3997 3095 3996 3096 3994 3096 -c 3976 3096 I 3974 3096 3972 3095 3972 3093 -c 3972 3091 3974 3090 3976 3090 -c C 
4019 3090 M 4038 3090 I 4040 3090 4041 3091 4041 3093 -c 4041 3095 4040 3096 4038 3096 -c 4019 3096 I 4018 3096 4016 3095 4016 3093 -c 4016 3091 4018 3090 4019 3090 -c C 
4063 3090 M 4082 3090 I 4084 3090 4085 3091 4085 3093 -c 4085 3095 4084 3096 4082 3096 -c 4063 3096 I 4061 3096 4060 3095 4060 3093 -c 4060 3091 4061 3090 4063 3090 -c C 
4107 3090 M 4126 3090 I 4127 3090 4129 3091 4129 3093 -c 4129 3095 4127 3096 4126 3096 -c 4107 3096 I 4105 3096 4104 3095 4104 3093 -c 4104 3091 4105 3090 4107 3090 -c C 
4151 3090 M 4169 3090 I 4171 3090 4172 3091 4172 3093 -c 4172 3095 4171 3096 4169 3096 -c 4151 3096 I 4149 3096 4147 3095 4147 3093 -c 4147 3091 4149 3090 4151 3090 -c C 
4194 3090 M 4213 3090 I 4215 3090 4216 3091 4216 3093 -c 4216 3095 4215 3096 4213 3096 -c 4194 3096 I 4193 3096 4191 3095 4191 3093 -c 4191 3091 4193 3090 4194 3090 -c C 
4238 3090 M 4257 3090 I 4259 3090 4260 3091 4260 3093 -c 4260 3095 4259 3096 4257 3096 -c 4238 3096 I 4236 3096 4235 3095 4235 3093 -c 4235 3091 4236 3090 4238 3090 -c C 
4282 3090 M 4301 3090 I 4302 3090 4304 3091 4304 3093 -c 4304 3095 4302 3096 4301 3096 -c 4282 3096 I 4280 3096 4279 3095 4279 3093 -c 4279 3091 4280 3090 4282 3090 -c C 
4326 3090 M 4344 3090 I 4346 3090 4347 3091 4347 3093 -c 4347 3095 4346 3096 4344 3096 -c 4326 3096 I 4324 3096 4322 3095 4322 3093 -c 4322 3091 4324 3090 4326 3090 -c C 
4369 3090 M 4388 3090 I 4390 3090 4391 3091 4391 3093 -c 4391 3095 4390 3096 4388 3096 -c 4369 3096 I 4368 3096 4366 3095 4366 3093 -c 4366 3091 4368 3090 4369 3090 -c C 
4413 3090 M 4432 3090 I 4434 3090 4435 3091 4435 3093 -c 4435 3095 4434 3096 4432 3096 -c 4413 3096 I 4411 3096 4410 3095 4410 3093 -c 4410 3091 4411 3090 4413 3090 -c C 
4457 3090 M 4475 3090 I 4477 3090 4478 3091 4478 3093 -c 4478 3095 4477 3096 4475 3096 -c 4457 3096 I 4455 3096 4453 3095 4453 3093 -c 4453 3091 4455 3090 4457 3090 -c C 
4500 3090 M 4519 3090 I 4521 3090 4522 3091 4522 3093 -c 4522 3095 4521 3096 4519 3096 -c 4500 3096 I 4499 3096 4497 3095 4497 3093 -c 4497 3091 4499 3090 4500 3090 -c C 
4544 3090 M 4563 3090 I 4565 3090 4566 3091 4566 3093 -c 4566 3095 4565 3096 4563 3096 -c 4544 3096 I 4542 3096 4541 3095 4541 3093 -c 4541 3091 4542 3090 4544 3090 -c C 
4588 3090 M 4607 3090 I 4608 3090 4610 3091 4610 3093 -c 4610 3095 4608 3096 4607 3096 -c 4588 3096 I 4586 3096 4585 3095 4585 3093 -c 4585 3091 4586 3090 4588 3090 -c C 
4632 3090 M 4650 3090 I 4652 3090 4653 3091 4653 3093 -c 4653 3095 4652 3096 4650 3096 -c 4632 3096 I 4630 3096 4628 3095 4628 3093 -c 4628 3091 4630 3090 4632 3090 -c C 
4675 3090 M 4694 3090 I 4696 3090 4697 3091 4697 3093 -c 4697 3095 4696 3096 4694 3096 -c 4675 3096 I 4674 3096 4672 3095 4672 3093 -c 4672 3091 4674 3090 4675 3090 -c C 
4719 3090 M 4738 3090 I 4740 3090 4741 3091 4741 3093 -c 4741 3095 4740 3096 4738 3096 -c 4719 3096 I 4717 3096 4716 3095 4716 3093 -c 4716 3091 4717 3090 4719 3090 -c C 
4763 3090 M 4782 3090 I 4783 3090 4785 3091 4785 3093 -c 4785 3095 4783 3096 4782 3096 -c 4763 3096 I 4761 3096 4760 3095 4760 3093 -c 4760 3091 4761 3090 4763 3090 -c C 
4807 3090 M 4825 3090 I 4827 3090 4828 3091 4828 3093 -c 4828 3095 4827 3096 4825 3096 -c 4807 3096 I 4805 3096 4803 3095 4803 3093 -c 4803 3091 4805 3090 4807 3090 -c C 
4850 3090 M 4869 3090 I 4871 3090 4872 3091 4872 3093 -c 4872 3095 4871 3096 4869 3096 -c 4850 3096 I 4849 3096 4847 3095 4847 3093 -c 4847 3091 4849 3090 4850 3090 -c C 
4894 3090 M 4913 3090 I 4915 3090 4916 3091 4916 3093 -c 4916 3095 4915 3096 4913 3096 -c 4894 3096 I 4892 3096 4891 3095 4891 3093 -c 4891 3091 4892 3090 4894 3090 -c C 
4938 3090 M 4956 3090 I 4958 3090 4959 3091 4959 3093 -c 4959 3095 4958 3096 4956 3096 -c 4938 3096 I 4936 3096 4935 3095 4935 3093 -c 4935 3091 4936 3090 4938 3090 -c C 
4981 3090 M 5000 3090 I 5002 3090 5003 3091 5003 3093 -c 5003 3095 5002 3096 5000 3096 -c 4981 3096 I 4980 3096 4978 3095 4978 3093 -c 4978 3091 4980 3090 4981 3090 -c C 
5025 3090 M 5044 3090 I 5046 3090 5047 3091 5047 3093 -c 5047 3095 5046 3096 5044 3096 -c 5025 3096 I 5023 3096 5022 3095 5022 3093 -c 5022 3091 5023 3090 5025 3090 -c C 
5069 3090 M 5088 3090 I 5089 3090 5091 3091 5091 3093 -c 5091 3095 5089 3096 5088 3096 -c 5069 3096 I 5067 3096 5066 3095 5066 3093 -c 5066 3091 5067 3090 5069 3090 -c C 
5113 3090 M 5131 3090 I 5133 3090 5134 3091 5134 3093 -c 5134 3095 5133 3096 5131 3096 -c 5113 3096 I 5111 3096 5109 3095 5109 3093 -c 5109 3091 5111 3090 5113 3090 -c C 
5156 3090 M 5175 3090 I 5177 3090 5178 3091 5178 3093 -c 5178 3095 5177 3096 5175 3096 -c 5156 3096 I 5155 3096 5153 3095 5153 3093 -c 5153 3091 5155 3090 5156 3090 -c C 
5200 3090 M 5219 3090 I 5221 3090 5222 3091 5222 3093 -c 5222 3095 5221 3096 5219 3096 -c 5200 3096 I 5198 3096 5197 3095 5197 3093 -c 5197 3091 5198 3090 5200 3090 -c C 
5244 3090 M 5263 3090 I 5264 3090 5266 3091 5266 3093 -c 5266 3095 5264 3096 5263 3096 -c 5244 3096 I 5242 3096 5241 3095 5241 3093 -c 5241 3091 5242 3090 5244 3090 -c C 
5288 3090 M 5306 3090 I 5308 3090 5309 3091 5309 3093 -c 5309 3095 5308 3096 5306 3096 -c 5288 3096 I 5286 3096 5284 3095 5284 3093 -c 5284 3091 5286 3090 5288 3090 -c C 
5331 3090 M 5350 3090 I 5352 3090 5353 3091 5353 3093 -c 5353 3095 5352 3096 5350 3096 -c 5331 3096 I 5330 3096 5328 3095 5328 3093 -c 5328 3091 5330 3090 5331 3090 -c C 
5375 3090 M 5394 3090 I 5396 3090 5397 3091 5397 3093 -c 5397 3095 5396 3096 5394 3096 -c 5375 3096 I 5373 3096 5372 3095 5372 3093 -c 5372 3091 5373 3090 5375 3090 -c C 
5419 3090 M 5438 3090 I 5439 3090 5441 3091 5441 3093 -c 5441 3095 5439 3096 5438 3096 -c 5419 3096 I 5417 3096 5416 3095 5416 3093 -c 5416 3091 5417 3090 5419 3090 -c C 
5462 3090 M 5481 3090 I 5483 3090 5484 3091 5484 3093 -c 5484 3095 5483 3096 5481 3096 -c 5462 3096 I 5461 3096 5459 3095 5459 3093 -c 5459 3091 5461 3090 5462 3090 -c C 
5506 3090 M 5525 3090 I 5527 3090 5528 3091 5528 3093 -c 5528 3095 5527 3096 5525 3096 -c 5506 3096 I 5504 3096 5503 3095 5503 3093 -c 5503 3091 5504 3090 5506 3090 -c C 
5550 3090 M 5569 3090 I 5570 3090 5572 3091 5572 3093 -c 5572 3095 5570 3096 5569 3096 -c 5550 3096 I 5548 3096 5547 3095 5547 3093 -c 5547 3091 5548 3090 5550 3090 -c C 
5594 3090 M 5612 3090 I 5614 3090 5615 3091 5615 3093 -c 5615 3095 5614 3096 5612 3096 -c 5594 3096 I 5592 3096 5590 3095 5590 3093 -c 5590 3091 5592 3090 5594 3090 -c C 
5637 3090 M 5656 3090 I 5658 3090 5659 3091 5659 3093 -c 5659 3095 5658 3096 5656 3096 -c 5637 3096 I 5636 3096 5634 3095 5634 3093 -c 5634 3091 5636 3090 5637 3090 -c C 
5681 3090 M 5700 3090 I 5702 3090 5703 3091 5703 3093 -c 5703 3095 5702 3096 5700 3096 -c 5681 3096 I 5679 3096 5678 3095 5678 3093 -c 5678 3091 5679 3090 5681 3090 -c C 
5725 3090 M 5744 3090 I 5745 3090 5747 3091 5747 3093 -c 5747 3095 5745 3096 5744 3096 -c 5725 3096 I 5723 3096 5722 3095 5722 3093 -c 5722 3091 5723 3090 5725 3090 -c C 
5769 3090 M 5787 3090 I 5789 3090 5790 3091 5790 3093 -c 5790 3095 5789 3096 5787 3096 -c 5769 3096 I 5767 3096 5765 3095 5765 3093 -c 5765 3091 5767 3090 5769 3090 -c C 
5812 3090 M 5831 3090 I 5833 3090 5834 3091 5834 3093 -c 5834 3095 5833 3096 5831 3096 -c 5812 3096 I 5811 3096 5809 3095 5809 3093 -c 5809 3091 5811 3090 5812 3090 -c C 
5856 3090 M 5875 3090 I 5877 3090 5878 3091 5878 3093 -c 5878 3095 5877 3096 5875 3096 -c 5856 3096 I 5854 3096 5853 3095 5853 3093 -c 5853 3091 5854 3090 5856 3090 -c C 
5900 3090 M 5919 3090 I 5920 3090 5922 3091 5922 3093 -c 5922 3095 5920 3096 5919 3096 -c 5900 3096 I 5898 3096 5897 3095 5897 3093 -c 5897 3091 5898 3090 5900 3090 -c C 
5944 3090 M 5962 3090 I 5964 3090 5965 3091 5965 3093 -c 5965 3095 5964 3096 5962 3096 -c 5944 3096 I 5942 3096 5940 3095 5940 3093 -c 5940 3091 5942 3090 5944 3090 -c C 
5987 3090 M 6006 3090 I 6008 3090 6009 3091 6009 3093 -c 6009 3095 6008 3096 6006 3096 -c 5987 3096 I 5985 3096 5984 3095 5984 3093 -c 5984 3091 5985 3090 5987 3090 -c C 
6031 3090 M 6050 3090 I 6051 3090 6053 3091 6053 3093 -c 6053 3095 6051 3096 6050 3096 -c 6031 3096 I 6029 3096 6028 3095 6028 3093 -c 6028 3091 6029 3090 6031 3090 -c C 
6075 3090 M 6093 3090 I 6095 3090 6096 3091 6096 3093 -c 6096 3095 6095 3096 6093 3096 -c 6075 3096 I 6073 3096 6071 3095 6071 3093 -c 6071 3091 6073 3090 6075 3090 -c C 
6118 3090 M 6137 3090 I 6139 3090 6140 3091 6140 3093 -c 6140 3095 6139 3096 6137 3096 -c 6118 3096 I 6117 3096 6115 3095 6115 3093 -c 6115 3091 6117 3090 6118 3090 -c C 
6162 3090 M 6181 3090 I 6183 3090 6184 3091 6184 3093 -c 6184 3095 6183 3096 6181 3096 -c 6162 3096 I 6160 3096 6159 3095 6159 3093 -c 6159 3091 6160 3090 6162 3090 -c C 
6206 3090 M 6225 3090 I 6226 3090 6228 3091 6228 3093 -c 6228 3095 6226 3096 6225 3096 -c 6206 3096 I 6204 3096 6203 3095 6203 3093 -c 6203 3091 6204 3090 6206 3090 -c C 
6250 3090 M 6265 3090 I 6267 3090 6268 3091 6268 3093 -c 6268 3095 6267 3096 6265 3096 -c 6250 3096 I 6248 3096 6246 3095 6246 3093 -c 6246 3091 6248 3090 6250 3090 -c C 
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; N 3983 1432 M 3964 1431 I 3963 1431 3961 1430 3961 1428 -c 3961 1426 3963 1425 3965 1425 -c 3983 1425 I 3985 1425 3986 1427 3986 1429 -c 3986 1430 3985 1432 3983 1432 -c C 
3940 1431 M 3921 1431 I 3919 1431 3918 1430 3918 1428 -c 3918 1427 3919 1425 3921 1425 -c 3940 1425 I 3941 1425 3943 1426 3943 1428 -c 3943 1429 3942 1431 3940 1431 -c C 
3896 1433 M 3878 1434 I 3876 1434 3874 1433 3874 1431 -c 3874 1429 3875 1428 3877 1428 -c 3896 1426 I 3898 1426 3899 1427 3899 1429 -c 3899 1431 3898 1432 3896 1433 -c C 
3853 1436 M 3834 1439 I 3833 1439 3831 1438 3831 1436 -c 3831 1434 3832 1433 3833 1432 -c 3852 1430 I 3854 1430 3855 1431 3856 1433 -c 3856 1435 3855 1436 3853 1436 -c C 
3810 1442 M 3791 1446 I 3790 1446 3788 1445 3788 1443 -c 3787 1441 3788 1440 3790 1440 -c 3809 1436 I 3810 1436 3812 1437 3812 1439 -c 3813 1440 3811 1442 3810 1442 -c C 
3767 1451 M 3749 1455 I 3747 1455 3745 1454 3745 1453 -c 3745 1451 3746 1449 3747 1449 -c 3766 1445 I 3767 1444 3769 1445 3769 1447 -c 3770 1449 3769 1450 3767 1451 -c C 
3725 1461 M 3707 1467 I 3705 1467 3703 1466 3703 1464 -c 3702 1463 3703 1461 3705 1461 -c 3723 1455 I 3725 1455 3726 1456 3727 1458 -c 3727 1459 3726 1461 3725 1461 -c C 
3683 1474 M 3665 1480 I 3664 1481 3662 1480 3661 1478 -c 3661 1477 3662 1475 3663 1474 -c 3681 1468 I 3683 1468 3684 1469 3685 1470 -c 3686 1472 3685 1474 3683 1474 -c C 
3642 1489 M 3627 1495 I 3625 1496 I 3623 1497 3622 1496 3621 1495 -c 3620 1493 3621 1491 3622 1490 -c 3625 1489 I 3640 1484 I 3641 1483 3643 1484 3644 1485 -c 3645 1487 3644 1489 3642 1489 -c C 
3602 1506 M 3591 1512 I 3585 1514 I 3584 1515 3582 1515 3581 1513 -c 3580 1511 3581 1510 3582 1509 -c 3588 1506 I 3600 1501 I 3601 1500 3603 1501 3604 1502 -c 3604 1504 3604 1506 3602 1506 -c C 
3563 1526 M 3555 1530 I 3547 1535 I 3545 1535 3543 1535 3542 1533 -c 3542 1532 3542 1530 3544 1529 -c 3552 1524 I 3560 1520 I 3562 1519 3564 1520 3564 1522 -c 3565 1523 3565 1525 3563 1526 -c C 
3525 1547 M 3521 1550 I 3509 1557 I 3508 1558 3506 1557 3505 1556 -c 3504 1555 3504 1553 3506 1552 -c 3518 1544 I 3522 1542 I 3523 1541 3525 1541 3526 1543 -c 3527 1544 3527 1546 3525 1547 -c C 
3488 1570 M 3487 1571 I 3473 1581 I 3472 1582 3470 1582 3469 1580 -c 3468 1579 3468 1577 3469 1576 -c 3484 1566 I 3485 1565 I 3486 1564 3488 1565 3489 1566 -c 3490 1568 3490 1570 3488 1570 -c C 
3453 1596 M 3438 1607 I 3437 1608 3435 1608 3434 1607 -c 3433 1605 3433 1603 3435 1602 -c 3449 1591 I 3450 1590 3452 1590 3453 1591 -c 3454 1593 3454 1595 3453 1596 -c C 
3419 1623 M 3405 1635 I 3404 1636 3402 1636 3401 1635 -c 3399 1633 3400 1631 3401 1630 -c 3415 1618 I 3416 1617 3418 1617 3419 1618 -c 3421 1620 3420 1621 3419 1623 -c C 
3386 1652 M 3373 1665 I 3372 1666 3370 1666 3368 1664 -c 3367 1663 3367 1661 3369 1660 -c 3382 1647 I 3383 1646 3385 1646 3387 1647 -c 3388 1648 3388 1650 3386 1652 -c C 
3355 1682 M 3343 1696 I 3341 1697 3339 1697 3338 1696 -c 3337 1695 3337 1693 3338 1691 -c 3351 1678 I 3352 1677 3354 1677 3355 1678 -c 3357 1679 3357 1681 3355 1682 -c C 
3326 1714 M 3314 1729 I 3313 1730 3311 1730 3309 1729 -c 3308 1728 3308 1726 3309 1724 -c 3321 1710 I 3322 1709 3324 1709 3326 1710 -c 3327 1711 3327 1713 3326 1714 -c C 
3298 1748 M 3288 1760 I 3287 1763 I 3286 1764 3284 1764 3283 1763 -c 3281 1762 3281 1760 3282 1759 -c 3284 1757 I 3293 1744 I 3295 1743 3296 1742 3298 1743 -c 3299 1744 3299 1746 3298 1748 -c C 
3272 1783 M 3266 1793 I 3262 1798 I 3261 1800 3259 1800 3258 1799 -c 3256 1798 3256 1796 3257 1795 -c 3260 1789 I 3267 1779 I 3268 1778 3270 1778 3272 1779 -c 3273 1780 3273 1781 3272 1783 -c C 
3248 1819 M 3244 1826 I 3239 1835 I 3238 1837 3236 1837 3235 1836 -c 3233 1835 3233 1834 3233 1832 -c 3239 1823 I 3243 1816 I 3244 1814 3246 1814 3248 1815 -c 3249 1816 3249 1818 3248 1819 -c C 
3226 1857 M 3224 1861 I 3218 1873 I 3217 1875 3215 1875 3214 1874 -c 3212 1874 3211 1872 3212 1870 -c 3219 1857 I 3221 1854 I 3222 1852 3224 1852 3225 1853 -c 3227 1853 3227 1855 3226 1857 -c C 
3206 1895 M 3206 1896 I 3199 1912 I 3198 1914 3196 1915 3195 1914 -c 3193 1913 3192 1911 3193 1910 -c 3201 1893 I 3201 1892 I 3202 1891 3203 1890 3205 1891 -c 3207 1892 3207 1894 3206 1895 -c C 
3189 1935 M 3182 1952 I 3181 1954 3179 1955 3178 1954 -c 3176 1954 3175 1952 3176 1950 -c 3183 1933 I 3183 1931 3185 1930 3187 1931 -c 3188 1932 3189 1933 3189 1935 -c C 
3173 1976 M 3167 1993 I 3166 1995 3165 1996 3163 1995 -c 3161 1995 3160 1993 3161 1991 -c 3167 1974 I 3168 1972 3169 1971 3171 1972 -c 3173 1972 3174 1974 3173 1976 -c C 
3159 2017 M 3154 2035 I 3154 2037 3152 2037 3150 2037 -c 3149 2037 3148 2035 3148 2033 -c 3153 2015 I 3154 2013 3156 2012 3157 2013 -c 3159 2013 3160 2015 3159 2017 -c C 
3148 2059 M 3144 2077 I 3144 2079 3142 2080 3140 2080 -c 3138 2079 3137 2077 3138 2076 -c 3142 2058 I 3142 2056 3144 2055 3146 2055 -c 3147 2056 3149 2057 3148 2059 -c C 
3139 2102 M 3136 2120 I 3135 2121 3134 2123 3132 2122 -c 3130 2122 3129 2120 3130 2119 -c 3133 2100 I 3133 2099 3135 2098 3137 2098 -c 3138 2098 3139 2100 3139 2102 -c C 
3132 2144 M 3130 2163 I 3130 2165 3128 2166 3126 2166 -c 3125 2165 3124 2164 3124 2162 -c 3126 2144 I 3126 2142 3128 2141 3130 2141 -c 3131 2141 3133 2143 3132 2144 -c C 
3128 2188 M 3126 2206 I 3126 2208 3125 2209 3123 2209 -c 3121 2209 3120 2208 3120 2206 -c 3122 2187 I 3122 2185 3123 2184 3125 2184 -c 3127 2184 3128 2186 3128 2188 -c C 
3126 2231 M 3125 2250 I 3125 2251 3124 2253 3122 2253 -c 3120 2253 3119 2251 3119 2250 -c 3119 2231 I 3119 2229 3121 2228 3123 2228 -c 3124 2228 3126 2229 3126 2231 -c C 
3126 2275 M 3126 2293 I 3126 2295 3125 2296 3123 2296 -c 3121 2297 3120 2295 3120 2293 -c 3119 2275 I 3119 2273 3121 2272 3123 2271 -c 3124 2271 3126 2273 3126 2275 -c C 
3128 2318 M 3129 2336 I 3129 2337 I 3130 2338 3128 2340 3127 2340 -c 3125 2340 3124 2339 3123 2337 -c 3123 2336 I 3122 2319 I 3122 2317 3123 2315 3125 2315 -c 3126 2315 3128 2316 3128 2318 -c C 
3133 2361 M 3135 2377 I 3135 2380 I 3135 2381 3134 2383 3133 2383 -c 3131 2383 3129 2382 3129 2381 -c 3128 2377 I 3126 2362 I 3126 2360 3127 2359 3129 2359 -c 3131 2358 3132 2360 3133 2361 -c C 
3140 2404 M 3142 2417 I 3143 2422 I 3143 2424 3142 2426 3141 2426 -c 3139 2427 3137 2426 3137 2424 -c 3136 2418 I 3133 2405 I 3133 2404 3134 2402 3136 2402 -c 3138 2401 3139 2402 3140 2404 -c C 
3149 2447 M 3151 2457 I 3153 2465 I 3154 2466 3153 2468 3151 2469 -c 3149 2469 3148 2468 3147 2466 -c 3145 2458 I 3143 2448 I 3142 2447 3143 2445 3145 2444 -c 3147 2444 3148 2445 3149 2447 -c C 
3160 2489 M 3162 2496 I 3166 2506 I 3166 2508 3165 2510 3164 2510 -c 3162 2511 3160 2510 3160 2508 -c 3156 2497 I 3154 2490 I 3154 2489 3155 2487 3156 2486 -c 3158 2486 3160 2487 3160 2489 -c C 
3174 2530 M 3175 2534 I 3180 2547 I 3181 2549 3180 2551 3179 2551 -c 3177 2552 3175 2551 3175 2550 -c 3169 2536 I 3168 2532 I 3167 2530 3168 2529 3170 2528 -c 3171 2527 3173 2528 3174 2530 -c C 
3190 2571 M 3190 2571 I 3197 2588 I 3198 2589 3197 2591 3196 2592 -c 3194 2592 3192 2592 3192 2590 -c 3184 2573 I 3184 2573 I 3183 2571 3184 2569 3186 2569 -c 3187 2568 3189 2569 3190 2571 -c C 
3208 2610 M 3216 2627 I 3217 2628 3216 2630 3215 2631 -c 3213 2632 3211 2631 3211 2629 -c 3202 2613 I 3201 2611 3202 2609 3203 2609 -c 3205 2608 3207 2609 3208 2610 -c C 
3228 2649 M 3237 2665 I 3238 2666 3237 2668 3236 2669 -c 3234 2670 3233 2669 3232 2668 -c 3222 2652 I 3221 2650 3222 2648 3223 2647 -c 3225 2647 3227 2647 3228 2649 -c C 
3250 2686 M 3260 2702 I 3261 2703 3261 2705 3259 2706 -c 3258 2707 3256 2707 3255 2705 -c 3245 2689 I 3244 2688 3244 2686 3246 2685 -c 3247 2684 3249 2685 3250 2686 -c C 
3274 2722 M 3285 2737 I 3286 2739 3286 2741 3284 2742 -c 3283 2743 3281 2742 3280 2741 -c 3269 2726 I 3268 2725 3268 2723 3270 2722 -c 3271 2721 3273 2721 3274 2722 -c C 
3300 2757 M 3312 2772 I 3313 2773 3312 2775 3311 2776 -c 3310 2777 3308 2777 3307 2776 -c 3295 2761 I 3294 2760 3294 2758 3296 2757 -c 3297 2756 3299 2756 3300 2757 -c C 
3328 2791 M 3339 2803 I 3340 2805 I 3341 2806 3341 2808 3340 2809 -c 3339 2810 3337 2810 3336 2809 -c 3334 2808 I 3323 2795 I 3322 2794 3322 2792 3323 2790 -c 3325 2789 3327 2789 3328 2791 -c C 
3357 2823 M 3366 2832 I 3370 2836 I 3372 2837 3372 2839 3370 2841 -c 3369 2842 3367 2842 3366 2841 -c 3361 2836 I 3353 2827 I 3352 2826 3352 2824 3353 2823 -c 3354 2822 3356 2822 3357 2823 -c C 
3388 2853 M 3395 2859 I 3402 2866 I 3404 2867 3404 2869 3403 2870 -c 3401 2871 3399 2871 3398 2870 -c 3390 2864 I 3384 2858 I 3383 2857 3383 2855 3384 2853 -c 3385 2852 3387 2852 3388 2853 -c C 
3421 2882 M 3424 2885 I 3436 2894 I 3437 2895 3437 2897 3436 2898 -c 3435 2899 3433 2900 3432 2899 -c 3420 2890 I 3417 2887 I 3416 2886 3416 2884 3417 2882 -c 3418 2881 3420 2881 3421 2882 -c C 
3455 2909 M 3455 2909 I 3470 2920 I 3472 2921 3472 2923 3471 2924 -c 3470 2926 3468 2926 3467 2925 -c 3451 2914 I 3451 2914 I 3450 2913 3450 2911 3451 2910 -c 3452 2908 3454 2908 3455 2909 -c C 
3490 2934 M 3506 2944 I 3508 2945 3508 2947 3507 2949 -c 3506 2950 3504 2950 3503 2949 -c 3487 2939 I 3486 2938 3485 2936 3486 2935 -c 3487 2934 3489 2933 3490 2934 -c C 
3527 2957 M 3544 2967 I 3545 2968 3546 2969 3545 2971 -c 3544 2972 3542 2973 3540 2972 -c 3524 2963 I 3523 2962 3522 2960 3523 2958 -c 3524 2957 3526 2956 3527 2957 -c C 
3565 2979 M 3582 2987 I 3584 2988 3584 2990 3583 2991 -c 3583 2993 3581 2993 3579 2992 -c 3563 2984 I 3561 2983 3560 2981 3561 2980 -c 3562 2978 3564 2978 3565 2979 -c C 
3605 2998 M 3622 3005 I 3623 3006 3624 3008 3623 3009 -c 3622 3011 3621 3012 3619 3011 -c 3602 3003 I 3600 3003 3600 3001 3600 2999 -c 3601 2998 3603 2997 3605 2998 -c C 
3645 3015 M 3662 3022 I 3664 3022 3664 3024 3664 3026 -c 3663 3027 3661 3028 3660 3027 -c 3642 3020 I 3641 3020 3640 3018 3641 3016 -c 3641 3015 3643 3014 3645 3015 -c C 
3686 3030 M 3703 3035 I 3703 3036 I 3705 3036 3706 3038 3705 3039 -c 3705 3041 3703 3042 3701 3042 -c 3701 3041 I 3684 3036 I 3682 3035 3681 3033 3682 3032 -c 3682 3030 3684 3029 3686 3030 -c C 
3727 3042 M 3742 3046 I 3745 3047 I 3747 3048 3748 3049 3748 3051 -c 3747 3053 3745 3054 3744 3053 -c 3740 3052 I 3726 3048 I 3724 3048 3723 3046 3723 3045 -c 3724 3043 3726 3042 3727 3042 -c C 
3770 3053 M 3782 3056 I 3788 3057 I 3789 3057 3791 3059 3790 3060 -c 3790 3062 3788 3063 3787 3063 -c 3780 3062 I 3768 3059 I 3766 3059 3765 3057 3766 3055 -c 3766 3054 3768 3053 3770 3053 -c C 
3812 3061 M 3822 3063 I 3831 3064 I 3832 3064 3834 3066 3833 3068 -c 3833 3069 3832 3070 3830 3070 -c 3821 3069 I 3811 3067 I 3810 3067 3808 3065 3809 3064 -c 3809 3062 3811 3061 3812 3061 -c C 
3855 3067 M 3863 3068 I 3874 3069 I 3876 3069 3877 3071 3877 3072 -c 3877 3074 3875 3075 3874 3075 -c 3862 3074 I 3855 3073 I 3853 3073 3852 3072 3852 3070 -c 3852 3068 3854 3067 3855 3067 -c C 
3899 3071 M 3905 3071 I 3917 3072 I 3919 3072 3921 3073 3921 3075 -c 3920 3077 3919 3078 3917 3078 -c 3905 3077 I 3898 3077 I 3897 3077 3895 3075 3896 3074 -c 3896 3072 3897 3071 3899 3071 -c C 
3942 3072 M 3947 3072 I 3961 3072 I 3963 3072 3964 3073 3964 3075 -c 3964 3077 3963 3078 3961 3078 -c 3947 3079 I 3942 3078 I 3941 3078 3939 3077 3939 3075 -c 3939 3074 3941 3072 3942 3072 -c C 
3986 3071 M 3989 3071 I 4004 3070 I 4006 3070 4008 3071 4008 3073 -c 4008 3075 4007 3076 4005 3076 -c 3990 3077 I 3986 3078 I 3984 3078 3983 3076 3983 3075 -c 3983 3073 3984 3071 3986 3071 -c C 
4029 3068 M 4031 3068 I 4048 3066 I 4049 3066 4051 3067 4051 3069 -c 4051 3070 4050 3072 4049 3072 -c 4032 3074 I 4030 3074 I 4028 3075 4027 3073 4026 3072 -c 4026 3070 4028 3068 4029 3068 -c C 
4072 3063 M 4091 3060 I 4093 3059 4094 3060 4094 3062 -c 4095 3064 4094 3065 4092 3066 -c 4073 3069 I 4072 3069 4070 3068 4070 3067 -c 4070 3065 4071 3063 4072 3063 -c C 
4115 3055 M 4134 3051 I 4135 3050 4137 3052 4137 3053 -c 4138 3055 4137 3057 4135 3057 -c 4117 3061 I 4115 3062 4113 3061 4113 3059 -c 4113 3057 4114 3055 4115 3055 -c C 
4158 3045 M 4176 3040 I 4177 3039 4179 3040 4180 3042 -c 4180 3044 4179 3045 4177 3046 -c 4159 3051 I 4158 3052 4156 3051 4156 3049 -c 4155 3047 4156 3045 4158 3045 -c C 
4199 3033 M 4217 3027 I 4219 3026 4221 3027 4221 3029 -c 4222 3030 4221 3032 4219 3033 -c 4201 3039 I 4200 3039 4198 3038 4198 3037 -c 4197 3035 4198 3033 4199 3033 -c C 
4241 3018 M 4258 3011 I 4260 3011 4261 3011 4262 3013 -c 4263 3015 4262 3017 4260 3017 -c 4243 3024 I 4241 3025 4239 3024 4239 3022 -c 4238 3021 4239 3019 4241 3018 -c C 
4281 3002 M 4298 2994 I 4300 2993 4301 2994 4302 2995 -c 4303 2997 4302 2999 4301 3000 -c 4283 3007 I 4282 3008 4280 3007 4279 3006 -c 4279 3004 4279 3002 4281 3002 -c C 
4320 2983 M 4337 2974 I 4338 2974 4340 2974 4341 2976 -c 4342 2977 4341 2979 4340 2980 -c 4323 2988 I 4322 2989 4320 2988 4319 2987 -c 4318 2985 4319 2984 4320 2983 -c C 
4358 2962 M 4374 2954 I 4375 2953 I 4376 2952 4378 2952 4379 2954 -c 4380 2955 4379 2957 4378 2958 -c 4377 2959 I 4362 2968 I 4360 2968 4358 2968 4357 2966 -c 4356 2965 4357 2963 4358 2962 -c C 
4396 2939 M 4407 2932 I 4411 2929 I 4413 2928 4414 2928 4415 2930 -c 4416 2931 4416 2933 4415 2934 -c 4410 2937 I 4399 2945 I 4398 2946 4396 2945 4395 2944 -c 4394 2942 4394 2940 4396 2939 -c C 
4432 2915 M 4439 2909 I 4446 2904 I 4448 2902 4449 2903 4451 2904 -c 4452 2905 4451 2907 4450 2908 -c 4443 2914 I 4435 2920 I 4434 2921 4432 2920 4431 2919 -c 4430 2918 4430 2916 4432 2915 -c C 
4466 2888 M 4470 2885 I 4480 2876 I 4481 2875 4483 2875 4484 2876 -c 4486 2878 4485 2880 4484 2881 -c 4474 2890 I 4470 2893 I 4468 2894 4466 2894 4465 2892 -c 4464 2891 4464 2889 4466 2888 -c C 
4499 2860 M 4500 2859 I 4512 2847 I 4514 2846 4516 2846 4517 2847 -c 4518 2848 4518 2850 4517 2852 -c 4504 2864 I 4503 2864 I 4502 2866 4500 2865 4499 2864 -c 4497 2863 4498 2861 4499 2860 -c C 
4530 2830 M 4543 2816 I 4544 2815 4546 2815 4548 2816 -c 4549 2817 4549 2819 4548 2821 -c 4535 2834 I 4534 2835 4532 2836 4530 2834 -c 4529 2833 4529 2831 4530 2830 -c C 
4560 2798 M 4572 2784 I 4574 2783 4576 2782 4577 2784 -c 4578 2785 4578 2787 4577 2788 -c 4565 2802 I 4564 2803 4562 2804 4560 2803 -c 4559 2801 4559 2799 4560 2798 -c C 
4588 2765 M 4600 2750 I 4601 2749 4603 2748 4604 2750 -c 4606 2751 4606 2753 4605 2754 -c 4593 2769 I 4592 2770 4590 2770 4589 2769 -c 4587 2768 4587 2766 4588 2765 -c C 
4615 2730 M 4625 2715 I 4626 2714 4628 2713 4630 2714 -c 4631 2715 4632 2717 4631 2719 -c 4620 2734 I 4619 2735 4617 2735 4615 2734 -c 4614 2733 4614 2732 4615 2730 -c C 
4639 2694 M 4649 2678 I 4650 2677 4652 2677 4654 2678 -c 4655 2678 4655 2680 4654 2682 -c 4644 2698 I 4643 2699 4641 2699 4640 2699 -c 4639 2698 4638 2696 4639 2694 -c C 
4662 2657 M 4670 2643 I 4671 2641 I 4672 2639 4674 2639 4675 2639 -c 4677 2640 4677 2642 4676 2644 -c 4675 2646 I 4667 2660 I 4666 2662 4664 2662 4663 2661 -c 4661 2660 4661 2658 4662 2657 -c C 
4682 2619 M 4688 2607 I 4690 2602 I 4691 2601 4693 2600 4695 2601 -c 4696 2601 4697 2603 4696 2605 -c 4694 2610 I 4688 2621 I 4687 2623 4685 2624 4684 2623 -c 4682 2622 4681 2620 4682 2619 -c C 
4701 2579 M 4705 2571 I 4708 2562 I 4709 2561 4710 2560 4712 2560 -c 4714 2561 4714 2563 4714 2564 -c 4710 2574 I 4706 2582 I 4706 2583 4704 2584 4702 2583 -c 4701 2583 4700 2581 4701 2579 -c C 
4717 2539 M 4719 2534 I 4723 2521 I 4724 2520 4726 2519 4727 2519 -c 4729 2520 4730 2522 4729 2523 -c 4725 2536 I 4723 2541 I 4722 2543 4720 2544 4719 2543 -c 4717 2542 4716 2541 4717 2539 -c C 
4731 2498 M 4732 2495 I 4737 2480 I 4737 2478 4739 2478 4740 2478 -c 4742 2478 4743 2480 4743 2482 -c 4738 2497 I 4737 2500 I 4737 2501 4735 2502 4733 2502 -c 4732 2501 4731 2499 4731 2498 -c C 
4743 2456 M 4748 2438 I 4748 2436 4750 2435 4751 2436 -c 4753 2436 4754 2438 4754 2439 -c 4749 2458 I 4749 2459 4747 2460 4746 2460 -c 4744 2460 4743 2458 4743 2456 -c C 
4753 2414 M 4756 2395 I 4757 2394 4758 2392 4760 2393 -c 4762 2393 4763 2395 4762 2396 -c 4759 2415 I 4759 2417 4757 2418 4756 2417 -c 4754 2417 4753 2415 4753 2414 -c C 
4760 2371 M 4763 2352 I 4763 2351 4765 2350 4766 2350 -c 4768 2350 4769 2352 4769 2353 -c 4767 2372 I 4766 2373 4765 2374 4763 2374 -c 4761 2374 4760 2372 4760 2371 -c C 
4765 2328 M 4767 2309 I 4767 2307 4768 2306 4770 2306 -c 4772 2306 4773 2308 4773 2310 -c 4772 2328 I 4772 2330 4770 2331 4768 2331 -c 4767 2331 4765 2329 4765 2328 -c C 
4768 2284 M 4769 2266 I 4769 2264 4770 2262 4772 2262 -c 4774 2263 4775 2264 4775 2266 -c 4774 2284 I 4774 2286 4773 2287 4771 2287 -c 4770 2287 4768 2286 4768 2284 -c C 
4769 2241 M 4768 2222 I 4768 2220 4770 2219 4771 2219 -c 4773 2219 4775 2220 4775 2222 -c 4775 2241 I 4775 2242 4774 2244 4772 2244 -c 4770 2244 4769 2242 4769 2241 -c C 
4767 2197 M 4766 2179 I 4766 2177 4767 2175 4769 2175 -c 4770 2175 4772 2176 4772 2178 -c 4773 2197 I 4773 2199 4772 2200 4770 2200 -c 4769 2200 4767 2199 4767 2197 -c C 
4763 2154 M 4761 2135 I 4761 2134 4762 2132 4763 2132 -c 4765 2132 4767 2133 4767 2135 -c 4769 2153 I 4770 2155 4768 2156 4767 2157 -c 4765 2157 4763 2156 4763 2154 -c C 
4757 2111 M 4754 2093 I 4753 2091 4754 2089 4756 2089 -c 4758 2089 4759 2090 4760 2092 -c 4763 2110 I 4763 2111 4762 2113 4760 2113 -c 4759 2114 4757 2113 4757 2111 -c C 
4748 2068 M 4744 2050 I 4744 2048 4745 2047 4746 2046 -c 4748 2046 4750 2047 4750 2049 -c 4754 2067 I 4755 2069 4754 2070 4752 2071 -c 4750 2071 4749 2070 4748 2068 -c C 
4737 2026 M 4732 2008 I 4732 2006 4733 2005 4734 2004 -c 4736 2004 4738 2005 4738 2006 -c 4743 2024 I 4744 2026 4743 2028 4741 2028 -c 4740 2029 4738 2028 4737 2026 -c C 
4724 1985 M 4719 1969 I 4718 1967 I 4718 1966 4718 1964 4720 1963 -c 4722 1963 4724 1963 4724 1965 -c 4725 1967 I 4730 1983 I 4731 1984 4730 1986 4728 1987 -c 4727 1987 4725 1986 4724 1985 -c C 
4709 1944 M 4704 1932 I 4702 1927 I 4701 1925 4702 1923 4704 1923 -c 4705 1922 4707 1923 4708 1924 -c 4710 1930 I 4715 1942 I 4716 1943 4715 1945 4713 1946 -c 4712 1946 4710 1946 4709 1944 -c C 
4692 1904 M 4688 1896 I 4684 1887 I 4683 1886 4684 1884 4685 1883 -c 4687 1882 4689 1883 4689 1884 -c 4694 1893 I 4698 1901 I 4698 1903 4698 1905 4696 1906 -c 4694 1906 4693 1906 4692 1904 -c C 
4672 1865 M 4670 1860 I 4663 1849 I 4662 1848 4663 1846 4664 1845 -c 4666 1844 4668 1844 4669 1846 -c 4675 1858 I 4678 1862 I 4679 1864 4678 1866 4677 1866 -c 4675 1867 4673 1867 4672 1865 -c C 
4651 1827 M 4650 1826 I 4641 1812 I 4640 1810 4640 1808 4642 1807 -c 4643 1806 4645 1807 4646 1808 -c 4655 1823 I 4656 1824 I 4657 1826 4657 1828 4655 1828 -c 4654 1829 4652 1829 4651 1827 -c C 
4627 1791 M 4616 1776 I 4615 1774 4616 1772 4617 1771 -c 4619 1770 4621 1770 4622 1772 -c 4632 1787 I 4633 1789 4633 1791 4632 1792 -c 4630 1793 4628 1792 4627 1791 -c C 
4602 1755 M 4590 1741 I 4589 1739 4589 1737 4591 1737 -c 4592 1735 4594 1736 4595 1737 -c 4607 1752 I 4608 1753 4608 1755 4606 1756 -c 4605 1757 4603 1757 4602 1755 -c C 
4575 1722 M 4562 1708 I 4561 1706 4561 1704 4563 1703 -c 4564 1702 4566 1702 4567 1703 -c 4579 1718 I 4580 1719 4580 1721 4579 1722 -c 4578 1723 4576 1723 4575 1722 -c C 
4546 1689 M 4533 1676 I 4531 1674 4531 1672 4533 1671 -c 4534 1670 4536 1670 4537 1671 -c 4550 1685 I 4551 1686 4551 1688 4550 1689 -c 4549 1691 4547 1690 4546 1689 -c C 
4515 1658 M 4501 1645 I 4500 1644 4500 1642 4501 1641 -c 4502 1640 4504 1640 4506 1641 -c 4519 1654 I 4520 1655 4520 1657 4519 1658 -c 4518 1659 4516 1659 4515 1658 -c C 
4483 1629 M 4470 1618 I 4468 1617 I 4467 1616 4467 1614 4468 1613 -c 4469 1611 4471 1611 4472 1612 -c 4474 1613 I 4487 1624 I 4488 1625 4488 1627 4487 1629 -c 4486 1630 4484 1630 4483 1629 -c C 
4449 1602 M 4439 1594 I 4434 1590 I 4433 1589 4432 1587 4433 1586 -c 4434 1585 4436 1584 4438 1585 -c 4443 1589 I 4453 1597 I 4454 1598 4454 1600 4453 1601 -c 4452 1602 4450 1603 4449 1602 -c C 
4414 1576 M 4407 1571 I 4398 1566 I 4397 1565 4397 1563 4397 1561 -c 4398 1560 4400 1559 4402 1560 -c 4410 1566 I 4417 1571 I 4419 1572 4419 1574 4418 1575 -c 4417 1577 4415 1577 4414 1576 -c C 
4377 1552 M 4373 1549 I 4361 1543 I 4360 1542 4359 1540 4360 1538 -c 4361 1537 4363 1536 4364 1537 -c 4377 1544 I 4381 1547 I 4382 1548 4383 1550 4382 1551 -c 4381 1553 4379 1553 4377 1552 -c C 
4340 1530 M 4339 1530 I 4323 1522 I 4322 1521 4321 1519 4322 1517 -c 4323 1516 4324 1515 4326 1516 -c 4342 1524 I 4343 1525 I 4344 1526 4345 1527 4344 1529 -c 4343 1530 4341 1531 4340 1530 -c C 
4301 1510 M 4284 1503 I 4282 1502 4282 1500 4282 1499 -c 4283 1497 4285 1496 4286 1497 -c 4304 1505 I 4305 1505 4306 1507 4305 1509 -c 4304 1510 4303 1511 4301 1510 -c C 
4261 1493 M 4244 1486 I 4242 1486 4241 1484 4242 1482 -c 4243 1480 4244 1480 4246 1480 -c 4263 1487 I 4265 1488 4266 1489 4265 1491 -c 4265 1493 4263 1493 4261 1493 -c C 
4220 1477 M 4203 1471 I 4201 1471 4200 1469 4201 1468 -c 4201 1466 4203 1465 4205 1466 -c 4222 1472 I 4224 1472 4225 1474 4224 1476 -c 4224 1477 4222 1478 4220 1477 -c C 
4179 1464 M 4161 1459 I 4159 1459 4158 1457 4159 1455 -c 4159 1453 4161 1453 4163 1453 -c 4181 1458 I 4182 1459 4183 1460 4183 1462 -c 4182 1464 4181 1465 4179 1464 -c C 
4137 1453 M 4118 1449 I 4117 1448 4116 1447 4116 1445 -c 4117 1443 4118 1442 4120 1443 -c 4138 1447 I 4140 1447 4141 1449 4141 1451 -c 4140 1452 4138 1453 4137 1453 -c C 
4094 1444 M 4076 1441 I 4074 1441 4073 1439 4073 1437 -c 4073 1435 4075 1434 4077 1435 -c 4095 1438 I 4097 1438 4098 1440 4098 1442 -c 4097 1443 4096 1444 4094 1444 -c C 
4051 1438 M 4032 1435 I 4031 1435 4029 1433 4030 1432 -c 4030 1430 4031 1429 4033 1429 -c 4052 1431 I 4054 1432 4055 1433 4055 1435 -c 4054 1437 4053 1438 4051 1438 -c C 
4008 1433 M 3989 1432 I 3988 1432 3986 1430 3986 1429 -c 3987 1427 3988 1425 3990 1426 -c 4008 1427 I 4010 1427 4011 1429 4011 1430 -c 4011 1432 4009 1433 4008 1433 -c C 
:  L ; : 0.246 0.246 +S K 
; 3932 2315 M <B7>S 
F1S10DYFFFFFEF2 Ji 
3271 3273 M (y)S 
F1SCCYFFFFFF33 Ji 
3904 2470 M (y)S 
F2SCCYFFFFFF33 Ji 
3911 2344 M <01>S 
F1S72YFFFFFF8D Ji 
4060 1700 M (N)S 
F1SC5YFFFFFF3A Ji 
3637 1651 M (yte)[279 56  0]xS 
F0SC5YFFFFFF3A Ji 
3771 1651 M (+)S 
F3SC5YFFFFFF3A Ji 
3644 1529 M <01>S 
F1SC5YFFFFFF3A Ji 
4542 982 M (cq)[143  0]xS 
F0S72YFFFFFF8D Ji 
4623 1031 M (*)S 
F0S95YFFFFFF6A Ji 
3970 1084 M <B7>S 
F1SC5YFFFFFF3A Ji 
3838 1202 M (cq)[143  0]xS 
F0S72YFFFFFF8D Ji 
3919 1251 M (*)S 
F4SC5YFFFFFF3A Ji 
3861 1184 M <01>S 
F5SA2YFFFFFF5D Ji 
5023 1492 M (\(,\))[159 167  0]xS 
F1S5EYFFFFFFA1 Ji 
4973 1560 M (t)S 
F1SA2YFFFFFF5D Ji 
5071 1492 M (YR)[173  0]xS 
F6SF4YFFFFFF0B Ji 
4789 1530 M <01>S 
F7SA2YFFFFFF5D Ji 
5110 1356 M <0101>[162  0]xS 
F5SA2YFFFFFF5D Ji 
4979 2700 M (\(,\))[159 167  0]xS 
F1SA2YFFFFFF5D Ji 
5027 2700 M (YR)[174  0]xS 
F8SF4YFFFFFF0B Ji 
4789 2737 M <01>S 
F9SA2YFFFFFF5D Ji 
5066 2564 M <0101>[162  0]xS 
N 3524 4246 M 3524 109 I 3524 107 3525 105 3527 105 -c 3529 105 3530 107 3530 109 -c 3530 4246 I 3530 4248 3529 4249 3527 4249 -c 3525 4249 3524 4248 3524 4246 -c C 
3502 117 M 3527 67 I 3552 117 I C 
:  L ; : 0.246 0.246 +S K 
; 1 Lj 1 Lc 75 Lw : 729 322 5556 4153 rc N 3506 332 M 1978 332 740 1569 740 3094 -c 740 4620 1978 5856 3506 5856 -c 5034 5856 6273 4620 6273 3094 -c 6273 1569 5034 332 3506 332 -c : 0.246 0.246 +S K 
; ; : N 646 3633 5668 842 rp C 
1 1 1 1 scol  L ; 2 Lj 0 Lc 8 Lw N 564 1033 M 6307 1033 I 6309 1033 6310 1034 6310 1036 -c 6310 1037 6309 1039 6307 1039 -c 564 1039 I 562 1039 561 1037 561 1036 -c 561 1034 562 1033 564 1033 -c C 
6299 1011 M 6349 1036 I 6299 1061 I C 
:  L ; : 0.246 0.246 +S K 
; F0S95YFFFFFF6A Ji 
3494 3142 M <B7>S 
4428 1082 M <B7>S 
1 Lj 1 Lc 25 Lw N 3362 3457 M 4835 237 I : 0.246 0.246 +S K 
; 50 Lw N 3694 3444 M 2068 231 I : 0.121 0.121 +S K 
; 2447 1084 M <B7>S 
75 Lw N 3939 885 M 3483 885 3114 1254 3114 1709 -c 3114 2164 3483 2533 3939 2533 -c 4395 2533 4764 2164 4764 1709 -c 4764 1254 4395 885 3939 885 -c : 0.246 0.246 +S K 
; 3924 1752 M <B7>S 
N 3939 885 M 3483 885 3114 1254 3114 1709 -c 3114 2164 3483 2533 3939 2533 -c 4395 2533 4764 2164 4764 1709 -c 4764 1254 4395 885 3939 885 -c : 0.246 0.246 +S K 
; 3924 1752 M <B7>S 
2 Lj 0 Lc 16 Lw N 3523 3066 M 3528 3048 I 3529 3046 3531 3046 3532 3046 -c 3534 3047 3535 3048 3534 3050 -c 3529 3068 I 3528 3070 3527 3070 3525 3070 -c 3523 3069 3522 3068 3523 3066 -c C 
3536 3024 M 3541 3006 I 3542 3005 3544 3004 3545 3004 -c 3547 3005 3548 3007 3547 3008 -c 3542 3026 I 3541 3028 3540 3029 3538 3028 -c 3536 3028 3535 3026 3536 3024 -c C 
3549 2983 M 3554 2965 I 3555 2963 3557 2962 3558 2963 -c 3560 2963 3561 2965 3560 2967 -c 3555 2984 I 3554 2986 3553 2987 3551 2986 -c 3549 2986 3548 2984 3549 2983 -c C 
3562 2941 M 3567 2923 I 3568 2921 3570 2920 3571 2921 -c 3573 2921 3574 2923 3573 2925 -c 3568 2943 I 3567 2944 3566 2945 3564 2945 -c 3562 2944 3561 2943 3562 2941 -c C 
3575 2899 M 3580 2881 I 3581 2880 3583 2879 3584 2879 -c 3586 2880 3587 2882 3586 2883 -c 3581 2901 I 3580 2903 3578 2904 3577 2903 -c 3575 2903 3574 2901 3575 2899 -c C 
3588 2858 M 3593 2840 I 3594 2838 3596 2837 3597 2838 -c 3599 2838 3600 2840 3599 2842 -c 3594 2859 I 3593 2861 3591 2862 3590 2861 -c 3588 2861 3587 2859 3588 2858 -c C 
3601 2816 M 3606 2798 I 3607 2796 3609 2795 3610 2796 -c 3612 2796 3613 2798 3612 2800 -c 3607 2818 I 3606 2819 3604 2820 3603 2820 -c 3601 2819 3600 2817 3601 2816 -c C 
3614 2774 M 3619 2756 I 3620 2755 3622 2754 3623 2754 -c 3625 2755 3626 2756 3625 2758 -c 3620 2776 I 3619 2778 3617 2779 3616 2778 -c 3614 2778 3613 2776 3614 2774 -c C 
3627 2733 M 3632 2715 I 3633 2713 3635 2712 3636 2713 -c 3638 2713 3639 2715 3638 2716 -c 3633 2734 I 3632 2736 3630 2737 3629 2736 -c 3627 2736 3626 2734 3627 2733 -c C 
3640 2691 M 3645 2673 I 3646 2671 3647 2670 3649 2671 -c 3651 2671 3652 2673 3651 2675 -c 3646 2693 I 3645 2694 3643 2695 3642 2695 -c 3640 2694 3639 2692 3640 2691 -c C 
3653 2649 M 3658 2631 I 3659 2630 3660 2629 3662 2629 -c 3664 2630 3665 2631 3664 2633 -c 3659 2651 I 3658 2653 3656 2654 3655 2653 -c 3653 2652 3652 2651 3653 2649 -c C 
3666 2607 M 3671 2590 I 3672 2588 3673 2587 3675 2587 -c 3677 2588 3678 2590 3677 2591 -c 3672 2609 I 3671 2611 3669 2612 3668 2611 -c 3666 2611 3665 2609 3666 2607 -c C 
3679 2566 M 3684 2548 I 3685 2546 3686 2545 3688 2546 -c 3690 2546 3691 2548 3690 2550 -c 3685 2567 I 3684 2569 3682 2570 3681 2570 -c 3679 2569 3678 2567 3679 2566 -c C 
3692 2524 M 3697 2506 I 3697 2504 3699 2504 3701 2504 -c 3703 2505 3703 2506 3703 2508 -c 3697 2526 I 3697 2527 3695 2528 3693 2528 -c 3692 2527 3691 2526 3692 2524 -c C 
3704 2482 M 3710 2464 I 3710 2463 3712 2462 3714 2462 -c 3716 2463 3716 2465 3716 2466 -c 3710 2484 I 3710 2486 3708 2487 3706 2486 -c 3705 2486 3704 2484 3704 2482 -c C 
3717 2441 M 3723 2423 I 3723 2421 3725 2420 3727 2421 -c 3728 2421 3729 2423 3729 2424 -c 3723 2442 I 3723 2444 3721 2445 3719 2444 -c 3718 2444 3717 2442 3717 2441 -c C 
3730 2399 M 3736 2381 I 3736 2379 3738 2378 3740 2379 -c 3741 2379 3742 2381 3742 2383 -c 3736 2401 I 3736 2402 3734 2403 3732 2403 -c 3731 2402 3730 2401 3730 2399 -c C 
3743 2357 M 3749 2339 I 3749 2338 3751 2337 3753 2337 -c 3754 2338 3755 2340 3755 2341 -c 3749 2359 I 3749 2361 3747 2362 3745 2361 -c 3744 2361 3743 2359 3743 2357 -c C 
3756 2315 M 3762 2298 I 3762 2296 3764 2295 3766 2296 -c 3767 2296 3768 2298 3768 2299 -c 3762 2317 I 3762 2319 3760 2320 3758 2319 -c 3757 2319 3756 2317 3756 2315 -c C 
3769 2274 M 3775 2256 I 3775 2254 3777 2253 3779 2254 -c 3780 2254 3781 2256 3781 2258 -c 3775 2276 I 3775 2277 3773 2278 3771 2278 -c 3770 2277 3769 2275 3769 2274 -c C 
3782 2232 M 3788 2214 I 3788 2213 3790 2212 3792 2212 -c 3793 2213 3794 2214 3794 2216 -c 3788 2234 I 3788 2236 3786 2236 3784 2236 -c 3783 2235 3782 2234 3782 2232 -c C 
3795 2190 M 3801 2173 I 3801 2171 3803 2170 3805 2171 -c 3806 2171 3807 2173 3807 2174 -c 3801 2192 I 3801 2194 3799 2195 3797 2194 -c 3796 2194 3795 2192 3795 2190 -c C 
3808 2149 M 3814 2131 I 3814 2129 3816 2128 3818 2129 -c 3819 2129 3820 2131 3820 2133 -c 3814 2151 I 3814 2152 3812 2153 3810 2153 -c 3809 2152 3808 2150 3808 2149 -c C 
3821 2107 M 3827 2089 I 3827 2087 3829 2087 3831 2087 -c 3832 2088 3833 2089 3833 2091 -c 3827 2109 I 3827 2111 3825 2111 3823 2111 -c 3822 2110 3821 2109 3821 2107 -c C 
3834 2065 M 3840 2047 I 3840 2046 3842 2045 3844 2045 -c 3845 2046 3846 2048 3846 2049 -c 3840 2067 I 3840 2069 3838 2070 3836 2069 -c 3835 2069 3834 2067 3834 2065 -c C 
3847 2024 M 3853 2006 I 3853 2004 3855 2003 3857 2004 -c 3858 2004 3859 2006 3859 2008 -c 3853 2025 I 3853 2027 3851 2028 3849 2028 -c 3848 2027 3847 2025 3847 2024 -c C 
3860 1982 M 3866 1964 I 3866 1962 3868 1961 3870 1962 -c 3871 1962 3872 1964 3872 1966 -c 3866 1984 I 3866 1985 3864 1986 3862 1986 -c 3861 1985 3860 1984 3860 1982 -c C 
3873 1940 M 3879 1922 I 3879 1921 3881 1920 3883 1920 -c 3884 1921 3885 1923 3885 1924 -c 3879 1942 I 3879 1944 3877 1945 3875 1944 -c 3873 1944 3873 1942 3873 1940 -c C 
3886 1899 M 3892 1881 I 3892 1879 3894 1878 3896 1879 -c 3897 1879 3898 1881 3898 1882 -c 3892 1900 I 3891 1902 3890 1903 3888 1902 -c 3886 1902 3886 1900 3886 1899 -c C 
3899 1857 M 3905 1839 I 3905 1837 3907 1836 3908 1837 -c 3910 1837 3911 1839 3911 1841 -c 3905 1859 I 3904 1860 3903 1861 3901 1861 -c 3899 1860 3899 1858 3899 1857 -c C 
3912 1815 M 3918 1797 I 3918 1796 3920 1795 3921 1795 -c 3923 1796 3924 1797 3924 1799 -c 3918 1817 I 3917 1819 3916 1819 3914 1819 -c 3912 1818 3911 1817 3912 1815 -c C 
3925 1773 M 3931 1756 I 3931 1754 3933 1753 3934 1753 -c 3936 1754 3937 1756 3936 1757 -c 3931 1775 I 3930 1777 3929 1778 3927 1777 -c 3925 1777 3924 1775 3925 1773 -c C 
3938 1732 M 3943 1714 I 3944 1712 3946 1711 3947 1712 -c 3949 1712 3950 1714 3949 1716 -c 3944 1734 I 3943 1735 3942 1736 3940 1736 -c 3938 1735 3937 1733 3938 1732 -c C 
3951 1690 M 3956 1672 I 3957 1671 3959 1670 3960 1670 -c 3962 1671 3963 1672 3962 1674 -c 3957 1692 I 3956 1693 3954 1694 3953 1694 -c 3951 1693 3950 1692 3951 1690 -c C 
3964 1648 M 3969 1630 I 3970 1629 3972 1628 3973 1628 -c 3975 1629 3976 1631 3975 1632 -c 3970 1650 I 3969 1652 3967 1653 3966 1652 -c 3964 1652 3963 1650 3964 1648 -c C 
3977 1607 M 3982 1589 I 3983 1587 3985 1586 3986 1587 -c 3988 1587 3989 1589 3988 1591 -c 3983 1609 I 3982 1610 3980 1611 3979 1611 -c 3977 1610 3976 1608 3977 1607 -c C 
3990 1565 M 3995 1547 I 3996 1545 3998 1545 3999 1545 -c 4001 1546 4002 1547 4001 1549 -c 3996 1567 I 3995 1568 3993 1569 3992 1569 -c 3990 1568 3989 1567 3990 1565 -c C 
4003 1523 M 4008 1505 I 4009 1504 4011 1503 4012 1503 -c 4014 1504 4015 1506 4014 1507 -c 4009 1525 I 4008 1527 4006 1528 4005 1527 -c 4003 1527 4002 1525 4003 1523 -c C 
4016 1482 M 4021 1464 I 4022 1462 4023 1461 4025 1462 -c 4027 1462 4028 1464 4027 1466 -c 4022 1483 I 4021 1485 4019 1486 4018 1485 -c 4016 1485 4015 1483 4016 1482 -c C 
4029 1440 M 4034 1422 I 4035 1420 4036 1419 4038 1420 -c 4040 1420 4041 1422 4040 1424 -c 4035 1442 I 4034 1443 4032 1444 4031 1444 -c 4029 1443 4028 1441 4029 1440 -c C 
4042 1398 M 4047 1380 I 4048 1379 4049 1378 4051 1378 -c 4053 1379 4054 1380 4053 1382 -c 4048 1400 I 4047 1402 4045 1403 4044 1402 -c 4042 1402 4041 1400 4042 1398 -c C 
4055 1356 M 4060 1339 I 4061 1337 4062 1336 4064 1337 -c 4066 1337 4067 1339 4066 1340 -c 4061 1358 I 4060 1360 4058 1361 4057 1360 -c 4055 1360 4054 1358 4055 1356 -c C 
4068 1315 M 4073 1297 I 4074 1295 4075 1294 4077 1295 -c 4079 1295 4080 1297 4079 1299 -c 4074 1317 I 4073 1318 4071 1319 4070 1319 -c 4068 1318 4067 1316 4068 1315 -c C 
4081 1273 M 4086 1255 I 4087 1253 4088 1253 4090 1253 -c 4092 1254 4093 1255 4092 1257 -c 4087 1275 I 4086 1277 4084 1277 4083 1277 -c 4081 1276 4080 1275 4081 1273 -c C 
4094 1231 M 4099 1213 I 4100 1212 4101 1211 4103 1211 -c 4105 1212 4106 1214 4105 1215 -c 4099 1233 I 4099 1235 4097 1236 4096 1235 -c 4094 1235 4093 1233 4094 1231 -c C 
4106 1190 M 4112 1172 I 4113 1170 4114 1169 4116 1170 -c 4118 1170 4119 1172 4118 1174 -c 4112 1191 I 4112 1193 4110 1194 4109 1193 -c 4107 1193 4106 1191 4106 1190 -c C 
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4145 1064 M 4151 1047 I 4151 1045 4153 1044 4155 1045 -c 4157 1045 4157 1047 4157 1049 -c 4151 1066 I 4151 1068 4149 1069 4147 1068 -c 4146 1068 4145 1066 4145 1064 -c C 
4158 1023 M 4164 1005 I 4164 1003 4166 1002 4168 1003 -c 4170 1003 4170 1005 4170 1007 -c 4164 1025 I 4164 1026 4162 1027 4160 1027 -c 4159 1026 4158 1024 4158 1023 -c C 
4171 981 M 4177 963 I 4177 962 4179 961 4181 961 -c 4182 962 4183 963 4183 965 -c 4177 983 I 4177 985 4175 986 4173 985 -c 4172 985 4171 983 4171 981 -c C 
4184 939 M 4190 922 I 4190 920 4192 919 4194 920 -c 4195 920 4196 922 4196 923 -c 4190 941 I 4190 943 4188 944 4186 943 -c 4185 943 4184 941 4184 939 -c C 
4197 898 M 4203 880 I 4203 878 4205 877 4207 878 -c 4208 878 4209 880 4209 882 -c 4203 900 I 4203 901 4201 902 4199 902 -c 4198 901 4197 899 4197 898 -c C 
4210 856 M 4216 838 I 4216 837 4218 836 4220 836 -c 4221 837 4222 838 4222 840 -c 4216 858 I 4216 860 4214 861 4212 860 -c 4211 859 4210 858 4210 856 -c C 
4223 814 M 4229 797 I 4229 795 4231 794 4233 794 -c 4234 795 4235 797 4235 798 -c 4229 816 I 4229 818 4227 819 4225 818 -c 4224 818 4223 816 4223 814 -c C 
4236 773 M 4242 755 I 4242 753 4244 752 4246 753 -c 4247 753 4248 755 4248 757 -c 4242 774 I 4242 776 4240 777 4238 777 -c 4237 776 4236 774 4236 773 -c C 
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4262 689 M 4268 671 I 4268 670 4270 669 4272 669 -c 4273 670 4274 672 4274 673 -c 4268 691 I 4268 693 4266 694 4264 693 -c 4262 693 4262 691 4262 689 -c C 
4275 648 M 4281 630 I 4281 628 4283 627 4285 628 -c 4286 628 4287 630 4287 631 -c 4281 649 I 4280 651 4279 652 4277 651 -c 4275 651 4275 649 4275 648 -c C 
4288 606 M 4294 588 I 4294 586 4296 585 4297 586 -c 4299 586 4300 588 4300 590 -c 4294 608 I 4293 609 4292 610 4290 610 -c 4288 609 4287 608 4288 606 -c C 
4301 564 M 4307 546 I 4307 545 4309 544 4310 544 -c 4312 545 4313 547 4313 548 -c 4307 566 I 4306 568 4305 569 4303 568 -c 4301 568 4300 566 4301 564 -c C 
4314 522 M 4320 505 I 4320 503 4322 502 4323 502 -c 4325 503 4326 505 4325 506 -c 4320 524 I 4319 526 4318 527 4316 526 -c 4314 526 4313 524 4314 522 -c C 
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4353 397 M 4358 379 I 4359 378 4361 377 4362 377 -c 4364 378 4365 380 4364 381 -c 4359 399 I 4358 401 4357 402 4355 401 -c 4353 401 4352 399 4353 397 -c C 
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784 3090 M 802 3090 I 804 3090 805 3091 805 3093 -c 805 3095 804 3096 802 3096 -c 784 3096 I 782 3096 780 3095 780 3093 -c 780 3091 782 3090 784 3090 -c C 
827 3090 M 846 3090 I 848 3090 849 3091 849 3093 -c 849 3095 848 3096 846 3096 -c 827 3096 I 826 3096 824 3095 824 3093 -c 824 3091 826 3090 827 3090 -c C 
871 3090 M 890 3090 I 892 3090 893 3091 893 3093 -c 893 3095 892 3096 890 3096 -c 871 3096 I 869 3096 868 3095 868 3093 -c 868 3091 869 3090 871 3090 -c C 
915 3090 M 933 3090 I 935 3090 936 3091 936 3093 -c 936 3095 935 3096 933 3096 -c 915 3096 I 913 3096 912 3095 912 3093 -c 912 3091 913 3090 915 3090 -c C 
958 3090 M 977 3090 I 979 3090 980 3091 980 3093 -c 980 3095 979 3096 977 3096 -c 958 3096 I 957 3096 955 3095 955 3093 -c 955 3091 957 3090 958 3090 -c C 
1002 3090 M 1021 3090 I 1023 3090 1024 3091 1024 3093 -c 1024 3095 1023 3096 1021 3096 -c 1002 3096 I 1000 3096 999 3095 999 3093 -c 999 3091 1000 3090 1002 3090 -c C 
1046 3090 M 1065 3090 I 1066 3090 1068 3091 1068 3093 -c 1068 3095 1066 3096 1065 3096 -c 1046 3096 I 1044 3096 1043 3095 1043 3093 -c 1043 3091 1044 3090 1046 3090 -c C 
1090 3090 M 1108 3090 I 1110 3090 1111 3091 1111 3093 -c 1111 3095 1110 3096 1108 3096 -c 1090 3096 I 1088 3096 1086 3095 1086 3093 -c 1086 3091 1088 3090 1090 3090 -c C 
1133 3090 M 1152 3090 I 1154 3090 1155 3091 1155 3093 -c 1155 3095 1154 3096 1152 3096 -c 1133 3096 I 1132 3096 1130 3095 1130 3093 -c 1130 3091 1132 3090 1133 3090 -c C 
1177 3090 M 1196 3090 I 1198 3090 1199 3091 1199 3093 -c 1199 3095 1198 3096 1196 3096 -c 1177 3096 I 1175 3096 1174 3095 1174 3093 -c 1174 3091 1175 3090 1177 3090 -c C 
1221 3090 M 1240 3090 I 1241 3090 1243 3091 1243 3093 -c 1243 3095 1241 3096 1240 3096 -c 1221 3096 I 1219 3096 1218 3095 1218 3093 -c 1218 3091 1219 3090 1221 3090 -c C 
1265 3090 M 1283 3090 I 1285 3090 1286 3091 1286 3093 -c 1286 3095 1285 3096 1283 3096 -c 1265 3096 I 1263 3096 1261 3095 1261 3093 -c 1261 3091 1263 3090 1265 3090 -c C 
1308 3090 M 1327 3090 I 1329 3090 1330 3091 1330 3093 -c 1330 3095 1329 3096 1327 3096 -c 1308 3096 I 1307 3096 1305 3095 1305 3093 -c 1305 3091 1307 3090 1308 3090 -c C 
1352 3090 M 1371 3090 I 1373 3090 1374 3091 1374 3093 -c 1374 3095 1373 3096 1371 3096 -c 1352 3096 I 1350 3096 1349 3095 1349 3093 -c 1349 3091 1350 3090 1352 3090 -c C 
1396 3090 M 1415 3090 I 1416 3090 1418 3091 1418 3093 -c 1418 3095 1416 3096 1415 3096 -c 1396 3096 I 1394 3096 1393 3095 1393 3093 -c 1393 3091 1394 3090 1396 3090 -c C 
1439 3090 M 1458 3090 I 1460 3090 1461 3091 1461 3093 -c 1461 3095 1460 3096 1458 3096 -c 1439 3096 I 1438 3096 1436 3095 1436 3093 -c 1436 3091 1438 3090 1439 3090 -c C 
1483 3090 M 1502 3090 I 1504 3090 1505 3091 1505 3093 -c 1505 3095 1504 3096 1502 3096 -c 1483 3096 I 1481 3096 1480 3095 1480 3093 -c 1480 3091 1481 3090 1483 3090 -c C 
1527 3090 M 1546 3090 I 1547 3090 1549 3091 1549 3093 -c 1549 3095 1547 3096 1546 3096 -c 1527 3096 I 1525 3096 1524 3095 1524 3093 -c 1524 3091 1525 3090 1527 3090 -c C 
1571 3090 M 1589 3090 I 1591 3090 1592 3091 1592 3093 -c 1592 3095 1591 3096 1589 3096 -c 1571 3096 I 1569 3096 1567 3095 1567 3093 -c 1567 3091 1569 3090 1571 3090 -c C 
1614 3090 M 1633 3090 I 1635 3090 1636 3091 1636 3093 -c 1636 3095 1635 3096 1633 3096 -c 1614 3096 I 1613 3096 1611 3095 1611 3093 -c 1611 3091 1613 3090 1614 3090 -c C 
1658 3090 M 1677 3090 I 1679 3090 1680 3091 1680 3093 -c 1680 3095 1679 3096 1677 3096 -c 1658 3096 I 1656 3096 1655 3095 1655 3093 -c 1655 3091 1656 3090 1658 3090 -c C 
1702 3090 M 1721 3090 I 1722 3090 1724 3091 1724 3093 -c 1724 3095 1722 3096 1721 3096 -c 1702 3096 I 1700 3096 1699 3095 1699 3093 -c 1699 3091 1700 3090 1702 3090 -c C 
1746 3090 M 1764 3090 I 1766 3090 1767 3091 1767 3093 -c 1767 3095 1766 3096 1764 3096 -c 1746 3096 I 1744 3096 1742 3095 1742 3093 -c 1742 3091 1744 3090 1746 3090 -c C 
1789 3090 M 1808 3090 I 1810 3090 1811 3091 1811 3093 -c 1811 3095 1810 3096 1808 3096 -c 1789 3096 I 1788 3096 1786 3095 1786 3093 -c 1786 3091 1788 3090 1789 3090 -c C 
1833 3090 M 1852 3090 I 1854 3090 1855 3091 1855 3093 -c 1855 3095 1854 3096 1852 3096 -c 1833 3096 I 1831 3096 1830 3095 1830 3093 -c 1830 3091 1831 3090 1833 3090 -c C 
1877 3090 M 1896 3090 I 1897 3090 1899 3091 1899 3093 -c 1899 3095 1897 3096 1896 3096 -c 1877 3096 I 1875 3096 1874 3095 1874 3093 -c 1874 3091 1875 3090 1877 3090 -c C 
1921 3090 M 1939 3090 I 1941 3090 1942 3091 1942 3093 -c 1942 3095 1941 3096 1939 3096 -c 1921 3096 I 1919 3096 1917 3095 1917 3093 -c 1917 3091 1919 3090 1921 3090 -c C 
1964 3090 M 1983 3090 I 1985 3090 1986 3091 1986 3093 -c 1986 3095 1985 3096 1983 3096 -c 1964 3096 I 1962 3096 1961 3095 1961 3093 -c 1961 3091 1962 3090 1964 3090 -c C 
2008 3090 M 2027 3090 I 2028 3090 2030 3091 2030 3093 -c 2030 3095 2028 3096 2027 3096 -c 2008 3096 I 2006 3096 2005 3095 2005 3093 -c 2005 3091 2006 3090 2008 3090 -c C 
2052 3090 M 2070 3090 I 2072 3090 2073 3091 2073 3093 -c 2073 3095 2072 3096 2070 3096 -c 2052 3096 I 2050 3096 2048 3095 2048 3093 -c 2048 3091 2050 3090 2052 3090 -c C 
2095 3090 M 2114 3090 I 2116 3090 2117 3091 2117 3093 -c 2117 3095 2116 3096 2114 3096 -c 2095 3096 I 2094 3096 2092 3095 2092 3093 -c 2092 3091 2094 3090 2095 3090 -c C 
2139 3090 M 2158 3090 I 2160 3090 2161 3091 2161 3093 -c 2161 3095 2160 3096 2158 3096 -c 2139 3096 I 2137 3096 2136 3095 2136 3093 -c 2136 3091 2137 3090 2139 3090 -c C 
2183 3090 M 2202 3090 I 2203 3090 2205 3091 2205 3093 -c 2205 3095 2203 3096 2202 3096 -c 2183 3096 I 2181 3096 2180 3095 2180 3093 -c 2180 3091 2181 3090 2183 3090 -c C 
2227 3090 M 2245 3090 I 2247 3090 2248 3091 2248 3093 -c 2248 3095 2247 3096 2245 3096 -c 2227 3096 I 2225 3096 2223 3095 2223 3093 -c 2223 3091 2225 3090 2227 3090 -c C 
2270 3090 M 2289 3090 I 2291 3090 2292 3091 2292 3093 -c 2292 3095 2291 3096 2289 3096 -c 2270 3096 I 2269 3096 2267 3095 2267 3093 -c 2267 3091 2269 3090 2270 3090 -c C 
2314 3090 M 2333 3090 I 2335 3090 2336 3091 2336 3093 -c 2336 3095 2335 3096 2333 3096 -c 2314 3096 I 2312 3096 2311 3095 2311 3093 -c 2311 3091 2312 3090 2314 3090 -c C 
2358 3090 M 2377 3090 I 2378 3090 2380 3091 2380 3093 -c 2380 3095 2378 3096 2377 3096 -c 2358 3096 I 2356 3096 2355 3095 2355 3093 -c 2355 3091 2356 3090 2358 3090 -c C 
2402 3090 M 2420 3090 I 2422 3090 2423 3091 2423 3093 -c 2423 3095 2422 3096 2420 3096 -c 2402 3096 I 2400 3096 2398 3095 2398 3093 -c 2398 3091 2400 3090 2402 3090 -c C 
2445 3090 M 2464 3090 I 2466 3090 2467 3091 2467 3093 -c 2467 3095 2466 3096 2464 3096 -c 2445 3096 I 2443 3096 2442 3095 2442 3093 -c 2442 3091 2443 3090 2445 3090 -c C 
2489 3090 M 2508 3090 I 2509 3090 2511 3091 2511 3093 -c 2511 3095 2509 3096 2508 3096 -c 2489 3096 I 2487 3096 2486 3095 2486 3093 -c 2486 3091 2487 3090 2489 3090 -c C 
2533 3090 M 2551 3090 I 2553 3090 2554 3091 2554 3093 -c 2554 3095 2553 3096 2551 3096 -c 2533 3096 I 2531 3096 2529 3095 2529 3093 -c 2529 3091 2531 3090 2533 3090 -c C 
2576 3090 M 2595 3090 I 2597 3090 2598 3091 2598 3093 -c 2598 3095 2597 3096 2595 3096 -c 2576 3096 I 2575 3096 2573 3095 2573 3093 -c 2573 3091 2575 3090 2576 3090 -c C 
2620 3090 M 2639 3090 I 2641 3090 2642 3091 2642 3093 -c 2642 3095 2641 3096 2639 3096 -c 2620 3096 I 2618 3096 2617 3095 2617 3093 -c 2617 3091 2618 3090 2620 3090 -c C 
2664 3090 M 2683 3090 I 2684 3090 2686 3091 2686 3093 -c 2686 3095 2684 3096 2683 3096 -c 2664 3096 I 2662 3096 2661 3095 2661 3093 -c 2661 3091 2662 3090 2664 3090 -c C 
2708 3090 M 2726 3090 I 2728 3090 2729 3091 2729 3093 -c 2729 3095 2728 3096 2726 3096 -c 2708 3096 I 2706 3096 2704 3095 2704 3093 -c 2704 3091 2706 3090 2708 3090 -c C 
2751 3090 M 2770 3090 I 2772 3090 2773 3091 2773 3093 -c 2773 3095 2772 3096 2770 3096 -c 2751 3096 I 2750 3096 2748 3095 2748 3093 -c 2748 3091 2750 3090 2751 3090 -c C 
2795 3090 M 2814 3090 I 2816 3090 2817 3091 2817 3093 -c 2817 3095 2816 3096 2814 3096 -c 2795 3096 I 2793 3096 2792 3095 2792 3093 -c 2792 3091 2793 3090 2795 3090 -c C 
2839 3090 M 2858 3090 I 2859 3090 2861 3091 2861 3093 -c 2861 3095 2859 3096 2858 3096 -c 2839 3096 I 2837 3096 2836 3095 2836 3093 -c 2836 3091 2837 3090 2839 3090 -c C 
2883 3090 M 2901 3090 I 2903 3090 2904 3091 2904 3093 -c 2904 3095 2903 3096 2901 3096 -c 2883 3096 I 2881 3096 2879 3095 2879 3093 -c 2879 3091 2881 3090 2883 3090 -c C 
2926 3090 M 2945 3090 I 2947 3090 2948 3091 2948 3093 -c 2948 3095 2947 3096 2945 3096 -c 2926 3096 I 2925 3096 2923 3095 2923 3093 -c 2923 3091 2925 3090 2926 3090 -c C 
2970 3090 M 2989 3090 I 2990 3090 2992 3091 2992 3093 -c 2992 3095 2990 3096 2989 3096 -c 2970 3096 I 2968 3096 2967 3095 2967 3093 -c 2967 3091 2968 3090 2970 3090 -c C 
3014 3090 M 3032 3090 I 3034 3090 3035 3091 3035 3093 -c 3035 3095 3034 3096 3032 3096 -c 3014 3096 I 3012 3096 3010 3095 3010 3093 -c 3010 3091 3012 3090 3014 3090 -c C 
3057 3090 M 3076 3090 I 3078 3090 3079 3091 3079 3093 -c 3079 3095 3078 3096 3076 3096 -c 3057 3096 I 3056 3096 3054 3095 3054 3093 -c 3054 3091 3056 3090 3057 3090 -c C 
3101 3090 M 3120 3090 I 3122 3090 3123 3091 3123 3093 -c 3123 3095 3122 3096 3120 3096 -c 3101 3096 I 3099 3096 3098 3095 3098 3093 -c 3098 3091 3099 3090 3101 3090 -c C 
3145 3090 M 3164 3090 I 3165 3090 3167 3091 3167 3093 -c 3167 3095 3165 3096 3164 3096 -c 3145 3096 I 3143 3096 3142 3095 3142 3093 -c 3142 3091 3143 3090 3145 3090 -c C 
3189 3090 M 3207 3090 I 3209 3090 3210 3091 3210 3093 -c 3210 3095 3209 3096 3207 3096 -c 3189 3096 I 3187 3096 3185 3095 3185 3093 -c 3185 3091 3187 3090 3189 3090 -c C 
3232 3090 M 3251 3090 I 3253 3090 3254 3091 3254 3093 -c 3254 3095 3253 3096 3251 3096 -c 3232 3096 I 3231 3096 3229 3095 3229 3093 -c 3229 3091 3231 3090 3232 3090 -c C 
3276 3090 M 3295 3090 I 3297 3090 3298 3091 3298 3093 -c 3298 3095 3297 3096 3295 3096 -c 3276 3096 I 3274 3096 3273 3095 3273 3093 -c 3273 3091 3274 3090 3276 3090 -c C 
3320 3090 M 3339 3090 I 3340 3090 3342 3091 3342 3093 -c 3342 3095 3340 3096 3339 3096 -c 3320 3096 I 3318 3096 3317 3095 3317 3093 -c 3317 3091 3318 3090 3320 3090 -c C 
3364 3090 M 3382 3090 I 3384 3090 3385 3091 3385 3093 -c 3385 3095 3384 3096 3382 3096 -c 3364 3096 I 3362 3096 3360 3095 3360 3093 -c 3360 3091 3362 3090 3364 3090 -c C 
3407 3090 M 3426 3090 I 3428 3090 3429 3091 3429 3093 -c 3429 3095 3428 3096 3426 3096 -c 3407 3096 I 3406 3096 3404 3095 3404 3093 -c 3404 3091 3406 3090 3407 3090 -c C 
3451 3090 M 3470 3090 I 3471 3090 3473 3091 3473 3093 -c 3473 3095 3471 3096 3470 3096 -c 3451 3096 I 3449 3096 3448 3095 3448 3093 -c 3448 3091 3449 3090 3451 3090 -c C 
3495 3090 M 3513 3090 I 3515 3090 3516 3091 3516 3093 -c 3516 3095 3515 3096 3513 3096 -c 3495 3096 I 3493 3096 3491 3095 3491 3093 -c 3491 3091 3493 3090 3495 3090 -c C 
3538 3090 M 3557 3090 I 3559 3090 3560 3091 3560 3093 -c 3560 3095 3559 3096 3557 3096 -c 3538 3096 I 3537 3096 3535 3095 3535 3093 -c 3535 3091 3537 3090 3538 3090 -c C 
3582 3090 M 3601 3090 I 3603 3090 3604 3091 3604 3093 -c 3604 3095 3603 3096 3601 3096 -c 3582 3096 I 3580 3096 3579 3095 3579 3093 -c 3579 3091 3580 3090 3582 3090 -c C 
3626 3090 M 3645 3090 I 3646 3090 3648 3091 3648 3093 -c 3648 3095 3646 3096 3645 3096 -c 3626 3096 I 3624 3096 3623 3095 3623 3093 -c 3623 3091 3624 3090 3626 3090 -c C 
3670 3090 M 3688 3090 I 3690 3090 3691 3091 3691 3093 -c 3691 3095 3690 3096 3688 3096 -c 3670 3096 I 3668 3096 3666 3095 3666 3093 -c 3666 3091 3668 3090 3670 3090 -c C 
3713 3090 M 3732 3090 I 3734 3090 3735 3091 3735 3093 -c 3735 3095 3734 3096 3732 3096 -c 3713 3096 I 3712 3096 3710 3095 3710 3093 -c 3710 3091 3712 3090 3713 3090 -c C 
3757 3090 M 3776 3090 I 3778 3090 3779 3091 3779 3093 -c 3779 3095 3778 3096 3776 3096 -c 3757 3096 I 3755 3096 3754 3095 3754 3093 -c 3754 3091 3755 3090 3757 3090 -c C 
3801 3090 M 3820 3090 I 3821 3090 3823 3091 3823 3093 -c 3823 3095 3821 3096 3820 3096 -c 3801 3096 I 3799 3096 3798 3095 3798 3093 -c 3798 3091 3799 3090 3801 3090 -c C 
3845 3090 M 3863 3090 I 3865 3090 3866 3091 3866 3093 -c 3866 3095 3865 3096 3863 3096 -c 3845 3096 I 3843 3096 3841 3095 3841 3093 -c 3841 3091 3843 3090 3845 3090 -c C 
3888 3090 M 3907 3090 I 3909 3090 3910 3091 3910 3093 -c 3910 3095 3909 3096 3907 3096 -c 3888 3096 I 3887 3096 3885 3095 3885 3093 -c 3885 3091 3887 3090 3888 3090 -c C 
3932 3090 M 3951 3090 I 3952 3090 3954 3091 3954 3093 -c 3954 3095 3952 3096 3951 3096 -c 3932 3096 I 3930 3096 3929 3095 3929 3093 -c 3929 3091 3930 3090 3932 3090 -c C 
3976 3090 M 3994 3090 I 3996 3090 3997 3091 3997 3093 -c 3997 3095 3996 3096 3994 3096 -c 3976 3096 I 3974 3096 3972 3095 3972 3093 -c 3972 3091 3974 3090 3976 3090 -c C 
4019 3090 M 4038 3090 I 4040 3090 4041 3091 4041 3093 -c 4041 3095 4040 3096 4038 3096 -c 4019 3096 I 4018 3096 4016 3095 4016 3093 -c 4016 3091 4018 3090 4019 3090 -c C 
4063 3090 M 4082 3090 I 4084 3090 4085 3091 4085 3093 -c 4085 3095 4084 3096 4082 3096 -c 4063 3096 I 4061 3096 4060 3095 4060 3093 -c 4060 3091 4061 3090 4063 3090 -c C 
4107 3090 M 4126 3090 I 4127 3090 4129 3091 4129 3093 -c 4129 3095 4127 3096 4126 3096 -c 4107 3096 I 4105 3096 4104 3095 4104 3093 -c 4104 3091 4105 3090 4107 3090 -c C 
4151 3090 M 4169 3090 I 4171 3090 4172 3091 4172 3093 -c 4172 3095 4171 3096 4169 3096 -c 4151 3096 I 4149 3096 4147 3095 4147 3093 -c 4147 3091 4149 3090 4151 3090 -c C 
4194 3090 M 4213 3090 I 4215 3090 4216 3091 4216 3093 -c 4216 3095 4215 3096 4213 3096 -c 4194 3096 I 4193 3096 4191 3095 4191 3093 -c 4191 3091 4193 3090 4194 3090 -c C 
4238 3090 M 4257 3090 I 4259 3090 4260 3091 4260 3093 -c 4260 3095 4259 3096 4257 3096 -c 4238 3096 I 4236 3096 4235 3095 4235 3093 -c 4235 3091 4236 3090 4238 3090 -c C 
4282 3090 M 4301 3090 I 4302 3090 4304 3091 4304 3093 -c 4304 3095 4302 3096 4301 3096 -c 4282 3096 I 4280 3096 4279 3095 4279 3093 -c 4279 3091 4280 3090 4282 3090 -c C 
4326 3090 M 4344 3090 I 4346 3090 4347 3091 4347 3093 -c 4347 3095 4346 3096 4344 3096 -c 4326 3096 I 4324 3096 4322 3095 4322 3093 -c 4322 3091 4324 3090 4326 3090 -c C 
4369 3090 M 4388 3090 I 4390 3090 4391 3091 4391 3093 -c 4391 3095 4390 3096 4388 3096 -c 4369 3096 I 4368 3096 4366 3095 4366 3093 -c 4366 3091 4368 3090 4369 3090 -c C 
4413 3090 M 4432 3090 I 4434 3090 4435 3091 4435 3093 -c 4435 3095 4434 3096 4432 3096 -c 4413 3096 I 4411 3096 4410 3095 4410 3093 -c 4410 3091 4411 3090 4413 3090 -c C 
4457 3090 M 4475 3090 I 4477 3090 4478 3091 4478 3093 -c 4478 3095 4477 3096 4475 3096 -c 4457 3096 I 4455 3096 4453 3095 4453 3093 -c 4453 3091 4455 3090 4457 3090 -c C 
4500 3090 M 4519 3090 I 4521 3090 4522 3091 4522 3093 -c 4522 3095 4521 3096 4519 3096 -c 4500 3096 I 4499 3096 4497 3095 4497 3093 -c 4497 3091 4499 3090 4500 3090 -c C 
4544 3090 M 4563 3090 I 4565 3090 4566 3091 4566 3093 -c 4566 3095 4565 3096 4563 3096 -c 4544 3096 I 4542 3096 4541 3095 4541 3093 -c 4541 3091 4542 3090 4544 3090 -c C 
4588 3090 M 4607 3090 I 4608 3090 4610 3091 4610 3093 -c 4610 3095 4608 3096 4607 3096 -c 4588 3096 I 4586 3096 4585 3095 4585 3093 -c 4585 3091 4586 3090 4588 3090 -c C 
4632 3090 M 4650 3090 I 4652 3090 4653 3091 4653 3093 -c 4653 3095 4652 3096 4650 3096 -c 4632 3096 I 4630 3096 4628 3095 4628 3093 -c 4628 3091 4630 3090 4632 3090 -c C 
4675 3090 M 4694 3090 I 4696 3090 4697 3091 4697 3093 -c 4697 3095 4696 3096 4694 3096 -c 4675 3096 I 4674 3096 4672 3095 4672 3093 -c 4672 3091 4674 3090 4675 3090 -c C 
4719 3090 M 4738 3090 I 4740 3090 4741 3091 4741 3093 -c 4741 3095 4740 3096 4738 3096 -c 4719 3096 I 4717 3096 4716 3095 4716 3093 -c 4716 3091 4717 3090 4719 3090 -c C 
4763 3090 M 4782 3090 I 4783 3090 4785 3091 4785 3093 -c 4785 3095 4783 3096 4782 3096 -c 4763 3096 I 4761 3096 4760 3095 4760 3093 -c 4760 3091 4761 3090 4763 3090 -c C 
4807 3090 M 4825 3090 I 4827 3090 4828 3091 4828 3093 -c 4828 3095 4827 3096 4825 3096 -c 4807 3096 I 4805 3096 4803 3095 4803 3093 -c 4803 3091 4805 3090 4807 3090 -c C 
4850 3090 M 4869 3090 I 4871 3090 4872 3091 4872 3093 -c 4872 3095 4871 3096 4869 3096 -c 4850 3096 I 4849 3096 4847 3095 4847 3093 -c 4847 3091 4849 3090 4850 3090 -c C 
4894 3090 M 4913 3090 I 4915 3090 4916 3091 4916 3093 -c 4916 3095 4915 3096 4913 3096 -c 4894 3096 I 4892 3096 4891 3095 4891 3093 -c 4891 3091 4892 3090 4894 3090 -c C 
4938 3090 M 4956 3090 I 4958 3090 4959 3091 4959 3093 -c 4959 3095 4958 3096 4956 3096 -c 4938 3096 I 4936 3096 4935 3095 4935 3093 -c 4935 3091 4936 3090 4938 3090 -c C 
4981 3090 M 5000 3090 I 5002 3090 5003 3091 5003 3093 -c 5003 3095 5002 3096 5000 3096 -c 4981 3096 I 4980 3096 4978 3095 4978 3093 -c 4978 3091 4980 3090 4981 3090 -c C 
5025 3090 M 5044 3090 I 5046 3090 5047 3091 5047 3093 -c 5047 3095 5046 3096 5044 3096 -c 5025 3096 I 5023 3096 5022 3095 5022 3093 -c 5022 3091 5023 3090 5025 3090 -c C 
5069 3090 M 5088 3090 I 5089 3090 5091 3091 5091 3093 -c 5091 3095 5089 3096 5088 3096 -c 5069 3096 I 5067 3096 5066 3095 5066 3093 -c 5066 3091 5067 3090 5069 3090 -c C 
5113 3090 M 5131 3090 I 5133 3090 5134 3091 5134 3093 -c 5134 3095 5133 3096 5131 3096 -c 5113 3096 I 5111 3096 5109 3095 5109 3093 -c 5109 3091 5111 3090 5113 3090 -c C 
5156 3090 M 5175 3090 I 5177 3090 5178 3091 5178 3093 -c 5178 3095 5177 3096 5175 3096 -c 5156 3096 I 5155 3096 5153 3095 5153 3093 -c 5153 3091 5155 3090 5156 3090 -c C 
5200 3090 M 5219 3090 I 5221 3090 5222 3091 5222 3093 -c 5222 3095 5221 3096 5219 3096 -c 5200 3096 I 5198 3096 5197 3095 5197 3093 -c 5197 3091 5198 3090 5200 3090 -c C 
5244 3090 M 5263 3090 I 5264 3090 5266 3091 5266 3093 -c 5266 3095 5264 3096 5263 3096 -c 5244 3096 I 5242 3096 5241 3095 5241 3093 -c 5241 3091 5242 3090 5244 3090 -c C 
5288 3090 M 5306 3090 I 5308 3090 5309 3091 5309 3093 -c 5309 3095 5308 3096 5306 3096 -c 5288 3096 I 5286 3096 5284 3095 5284 3093 -c 5284 3091 5286 3090 5288 3090 -c C 
5331 3090 M 5350 3090 I 5352 3090 5353 3091 5353 3093 -c 5353 3095 5352 3096 5350 3096 -c 5331 3096 I 5330 3096 5328 3095 5328 3093 -c 5328 3091 5330 3090 5331 3090 -c C 
5375 3090 M 5394 3090 I 5396 3090 5397 3091 5397 3093 -c 5397 3095 5396 3096 5394 3096 -c 5375 3096 I 5373 3096 5372 3095 5372 3093 -c 5372 3091 5373 3090 5375 3090 -c C 
5419 3090 M 5438 3090 I 5439 3090 5441 3091 5441 3093 -c 5441 3095 5439 3096 5438 3096 -c 5419 3096 I 5417 3096 5416 3095 5416 3093 -c 5416 3091 5417 3090 5419 3090 -c C 
5462 3090 M 5481 3090 I 5483 3090 5484 3091 5484 3093 -c 5484 3095 5483 3096 5481 3096 -c 5462 3096 I 5461 3096 5459 3095 5459 3093 -c 5459 3091 5461 3090 5462 3090 -c C 
5506 3090 M 5525 3090 I 5527 3090 5528 3091 5528 3093 -c 5528 3095 5527 3096 5525 3096 -c 5506 3096 I 5504 3096 5503 3095 5503 3093 -c 5503 3091 5504 3090 5506 3090 -c C 
5550 3090 M 5569 3090 I 5570 3090 5572 3091 5572 3093 -c 5572 3095 5570 3096 5569 3096 -c 5550 3096 I 5548 3096 5547 3095 5547 3093 -c 5547 3091 5548 3090 5550 3090 -c C 
5594 3090 M 5612 3090 I 5614 3090 5615 3091 5615 3093 -c 5615 3095 5614 3096 5612 3096 -c 5594 3096 I 5592 3096 5590 3095 5590 3093 -c 5590 3091 5592 3090 5594 3090 -c C 
5637 3090 M 5656 3090 I 5658 3090 5659 3091 5659 3093 -c 5659 3095 5658 3096 5656 3096 -c 5637 3096 I 5636 3096 5634 3095 5634 3093 -c 5634 3091 5636 3090 5637 3090 -c C 
5681 3090 M 5700 3090 I 5702 3090 5703 3091 5703 3093 -c 5703 3095 5702 3096 5700 3096 -c 5681 3096 I 5679 3096 5678 3095 5678 3093 -c 5678 3091 5679 3090 5681 3090 -c C 
5725 3090 M 5744 3090 I 5745 3090 5747 3091 5747 3093 -c 5747 3095 5745 3096 5744 3096 -c 5725 3096 I 5723 3096 5722 3095 5722 3093 -c 5722 3091 5723 3090 5725 3090 -c C 
5769 3090 M 5787 3090 I 5789 3090 5790 3091 5790 3093 -c 5790 3095 5789 3096 5787 3096 -c 5769 3096 I 5767 3096 5765 3095 5765 3093 -c 5765 3091 5767 3090 5769 3090 -c C 
5812 3090 M 5831 3090 I 5833 3090 5834 3091 5834 3093 -c 5834 3095 5833 3096 5831 3096 -c 5812 3096 I 5811 3096 5809 3095 5809 3093 -c 5809 3091 5811 3090 5812 3090 -c C 
5856 3090 M 5875 3090 I 5877 3090 5878 3091 5878 3093 -c 5878 3095 5877 3096 5875 3096 -c 5856 3096 I 5854 3096 5853 3095 5853 3093 -c 5853 3091 5854 3090 5856 3090 -c C 
5900 3090 M 5919 3090 I 5920 3090 5922 3091 5922 3093 -c 5922 3095 5920 3096 5919 3096 -c 5900 3096 I 5898 3096 5897 3095 5897 3093 -c 5897 3091 5898 3090 5900 3090 -c C 
5944 3090 M 5962 3090 I 5964 3090 5965 3091 5965 3093 -c 5965 3095 5964 3096 5962 3096 -c 5944 3096 I 5942 3096 5940 3095 5940 3093 -c 5940 3091 5942 3090 5944 3090 -c C 
5987 3090 M 6006 3090 I 6008 3090 6009 3091 6009 3093 -c 6009 3095 6008 3096 6006 3096 -c 5987 3096 I 5985 3096 5984 3095 5984 3093 -c 5984 3091 5985 3090 5987 3090 -c C 
6031 3090 M 6050 3090 I 6051 3090 6053 3091 6053 3093 -c 6053 3095 6051 3096 6050 3096 -c 6031 3096 I 6029 3096 6028 3095 6028 3093 -c 6028 3091 6029 3090 6031 3090 -c C 
6075 3090 M 6093 3090 I 6095 3090 6096 3091 6096 3093 -c 6096 3095 6095 3096 6093 3096 -c 6075 3096 I 6073 3096 6071 3095 6071 3093 -c 6071 3091 6073 3090 6075 3090 -c C 
6118 3090 M 6137 3090 I 6139 3090 6140 3091 6140 3093 -c 6140 3095 6139 3096 6137 3096 -c 6118 3096 I 6117 3096 6115 3095 6115 3093 -c 6115 3091 6117 3090 6118 3090 -c C 
6162 3090 M 6181 3090 I 6183 3090 6184 3091 6184 3093 -c 6184 3095 6183 3096 6181 3096 -c 6162 3096 I 6160 3096 6159 3095 6159 3093 -c 6159 3091 6160 3090 6162 3090 -c C 
6206 3090 M 6225 3090 I 6226 3090 6228 3091 6228 3093 -c 6228 3095 6226 3096 6225 3096 -c 6206 3096 I 6204 3096 6203 3095 6203 3093 -c 6203 3091 6204 3090 6206 3090 -c C 
6250 3090 M 6265 3090 I 6267 3090 6268 3091 6268 3093 -c 6268 3095 6267 3096 6265 3096 -c 6250 3096 I 6248 3096 6246 3095 6246 3093 -c 6246 3091 6248 3090 6250 3090 -c C 
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; N 3983 1432 M 3964 1431 I 3963 1431 3961 1430 3961 1428 -c 3961 1426 3963 1425 3965 1425 -c 3983 1425 I 3985 1425 3986 1427 3986 1429 -c 3986 1430 3985 1432 3983 1432 -c C 
3940 1431 M 3921 1431 I 3919 1431 3918 1430 3918 1428 -c 3918 1427 3919 1425 3921 1425 -c 3940 1425 I 3941 1425 3943 1426 3943 1428 -c 3943 1429 3942 1431 3940 1431 -c C 
3896 1433 M 3878 1434 I 3876 1434 3874 1433 3874 1431 -c 3874 1429 3875 1428 3877 1428 -c 3896 1426 I 3898 1426 3899 1427 3899 1429 -c 3899 1431 3898 1432 3896 1433 -c C 
3853 1436 M 3834 1439 I 3833 1439 3831 1438 3831 1436 -c 3831 1434 3832 1433 3833 1432 -c 3852 1430 I 3854 1430 3855 1431 3856 1433 -c 3856 1435 3855 1436 3853 1436 -c C 
3810 1442 M 3791 1446 I 3790 1446 3788 1445 3788 1443 -c 3787 1441 3788 1440 3790 1440 -c 3809 1436 I 3810 1436 3812 1437 3812 1439 -c 3813 1440 3811 1442 3810 1442 -c C 
3767 1451 M 3749 1455 I 3747 1455 3745 1454 3745 1453 -c 3745 1451 3746 1449 3747 1449 -c 3766 1445 I 3767 1444 3769 1445 3769 1447 -c 3770 1449 3769 1450 3767 1451 -c C 
3725 1461 M 3707 1467 I 3705 1467 3703 1466 3703 1464 -c 3702 1463 3703 1461 3705 1461 -c 3723 1455 I 3725 1455 3726 1456 3727 1458 -c 3727 1459 3726 1461 3725 1461 -c C 
3683 1474 M 3665 1480 I 3664 1481 3662 1480 3661 1478 -c 3661 1477 3662 1475 3663 1474 -c 3681 1468 I 3683 1468 3684 1469 3685 1470 -c 3686 1472 3685 1474 3683 1474 -c C 
3642 1489 M 3627 1495 I 3625 1496 I 3623 1497 3622 1496 3621 1495 -c 3620 1493 3621 1491 3622 1490 -c 3625 1489 I 3640 1484 I 3641 1483 3643 1484 3644 1485 -c 3645 1487 3644 1489 3642 1489 -c C 
3602 1506 M 3591 1512 I 3585 1514 I 3584 1515 3582 1515 3581 1513 -c 3580 1511 3581 1510 3582 1509 -c 3588 1506 I 3600 1501 I 3601 1500 3603 1501 3604 1502 -c 3604 1504 3604 1506 3602 1506 -c C 
3563 1526 M 3555 1530 I 3547 1535 I 3545 1535 3543 1535 3542 1533 -c 3542 1532 3542 1530 3544 1529 -c 3552 1524 I 3560 1520 I 3562 1519 3564 1520 3564 1522 -c 3565 1523 3565 1525 3563 1526 -c C 
3525 1547 M 3521 1550 I 3509 1557 I 3508 1558 3506 1557 3505 1556 -c 3504 1555 3504 1553 3506 1552 -c 3518 1544 I 3522 1542 I 3523 1541 3525 1541 3526 1543 -c 3527 1544 3527 1546 3525 1547 -c C 
3488 1570 M 3487 1571 I 3473 1581 I 3472 1582 3470 1582 3469 1580 -c 3468 1579 3468 1577 3469 1576 -c 3484 1566 I 3485 1565 I 3486 1564 3488 1565 3489 1566 -c 3490 1568 3490 1570 3488 1570 -c C 
3453 1596 M 3438 1607 I 3437 1608 3435 1608 3434 1607 -c 3433 1605 3433 1603 3435 1602 -c 3449 1591 I 3450 1590 3452 1590 3453 1591 -c 3454 1593 3454 1595 3453 1596 -c C 
3419 1623 M 3405 1635 I 3404 1636 3402 1636 3401 1635 -c 3399 1633 3400 1631 3401 1630 -c 3415 1618 I 3416 1617 3418 1617 3419 1618 -c 3421 1620 3420 1621 3419 1623 -c C 
3386 1652 M 3373 1665 I 3372 1666 3370 1666 3368 1664 -c 3367 1663 3367 1661 3369 1660 -c 3382 1647 I 3383 1646 3385 1646 3387 1647 -c 3388 1648 3388 1650 3386 1652 -c C 
3355 1682 M 3343 1696 I 3341 1697 3339 1697 3338 1696 -c 3337 1695 3337 1693 3338 1691 -c 3351 1678 I 3352 1677 3354 1677 3355 1678 -c 3357 1679 3357 1681 3355 1682 -c C 
3326 1714 M 3314 1729 I 3313 1730 3311 1730 3309 1729 -c 3308 1728 3308 1726 3309 1724 -c 3321 1710 I 3322 1709 3324 1709 3326 1710 -c 3327 1711 3327 1713 3326 1714 -c C 
3298 1748 M 3288 1760 I 3287 1763 I 3286 1764 3284 1764 3283 1763 -c 3281 1762 3281 1760 3282 1759 -c 3284 1757 I 3293 1744 I 3295 1743 3296 1742 3298 1743 -c 3299 1744 3299 1746 3298 1748 -c C 
3272 1783 M 3266 1793 I 3262 1798 I 3261 1800 3259 1800 3258 1799 -c 3256 1798 3256 1796 3257 1795 -c 3260 1789 I 3267 1779 I 3268 1778 3270 1778 3272 1779 -c 3273 1780 3273 1781 3272 1783 -c C 
3248 1819 M 3244 1826 I 3239 1835 I 3238 1837 3236 1837 3235 1836 -c 3233 1835 3233 1834 3233 1832 -c 3239 1823 I 3243 1816 I 3244 1814 3246 1814 3248 1815 -c 3249 1816 3249 1818 3248 1819 -c C 
3226 1857 M 3224 1861 I 3218 1873 I 3217 1875 3215 1875 3214 1874 -c 3212 1874 3211 1872 3212 1870 -c 3219 1857 I 3221 1854 I 3222 1852 3224 1852 3225 1853 -c 3227 1853 3227 1855 3226 1857 -c C 
3206 1895 M 3206 1896 I 3199 1912 I 3198 1914 3196 1915 3195 1914 -c 3193 1913 3192 1911 3193 1910 -c 3201 1893 I 3201 1892 I 3202 1891 3203 1890 3205 1891 -c 3207 1892 3207 1894 3206 1895 -c C 
3189 1935 M 3182 1952 I 3181 1954 3179 1955 3178 1954 -c 3176 1954 3175 1952 3176 1950 -c 3183 1933 I 3183 1931 3185 1930 3187 1931 -c 3188 1932 3189 1933 3189 1935 -c C 
3173 1976 M 3167 1993 I 3166 1995 3165 1996 3163 1995 -c 3161 1995 3160 1993 3161 1991 -c 3167 1974 I 3168 1972 3169 1971 3171 1972 -c 3173 1972 3174 1974 3173 1976 -c C 
3159 2017 M 3154 2035 I 3154 2037 3152 2037 3150 2037 -c 3149 2037 3148 2035 3148 2033 -c 3153 2015 I 3154 2013 3156 2012 3157 2013 -c 3159 2013 3160 2015 3159 2017 -c C 
3148 2059 M 3144 2077 I 3144 2079 3142 2080 3140 2080 -c 3138 2079 3137 2077 3138 2076 -c 3142 2058 I 3142 2056 3144 2055 3146 2055 -c 3147 2056 3149 2057 3148 2059 -c C 
3139 2102 M 3136 2120 I 3135 2121 3134 2123 3132 2122 -c 3130 2122 3129 2120 3130 2119 -c 3133 2100 I 3133 2099 3135 2098 3137 2098 -c 3138 2098 3139 2100 3139 2102 -c C 
3132 2144 M 3130 2163 I 3130 2165 3128 2166 3126 2166 -c 3125 2165 3124 2164 3124 2162 -c 3126 2144 I 3126 2142 3128 2141 3130 2141 -c 3131 2141 3133 2143 3132 2144 -c C 
3128 2188 M 3126 2206 I 3126 2208 3125 2209 3123 2209 -c 3121 2209 3120 2208 3120 2206 -c 3122 2187 I 3122 2185 3123 2184 3125 2184 -c 3127 2184 3128 2186 3128 2188 -c C 
3126 2231 M 3125 2250 I 3125 2251 3124 2253 3122 2253 -c 3120 2253 3119 2251 3119 2250 -c 3119 2231 I 3119 2229 3121 2228 3123 2228 -c 3124 2228 3126 2229 3126 2231 -c C 
3126 2275 M 3126 2293 I 3126 2295 3125 2296 3123 2296 -c 3121 2297 3120 2295 3120 2293 -c 3119 2275 I 3119 2273 3121 2272 3123 2271 -c 3124 2271 3126 2273 3126 2275 -c C 
3128 2318 M 3129 2336 I 3129 2337 I 3130 2338 3128 2340 3127 2340 -c 3125 2340 3124 2339 3123 2337 -c 3123 2336 I 3122 2319 I 3122 2317 3123 2315 3125 2315 -c 3126 2315 3128 2316 3128 2318 -c C 
3133 2361 M 3135 2377 I 3135 2380 I 3135 2381 3134 2383 3133 2383 -c 3131 2383 3129 2382 3129 2381 -c 3128 2377 I 3126 2362 I 3126 2360 3127 2359 3129 2359 -c 3131 2358 3132 2360 3133 2361 -c C 
3140 2404 M 3142 2417 I 3143 2422 I 3143 2424 3142 2426 3141 2426 -c 3139 2427 3137 2426 3137 2424 -c 3136 2418 I 3133 2405 I 3133 2404 3134 2402 3136 2402 -c 3138 2401 3139 2402 3140 2404 -c C 
3149 2447 M 3151 2457 I 3153 2465 I 3154 2466 3153 2468 3151 2469 -c 3149 2469 3148 2468 3147 2466 -c 3145 2458 I 3143 2448 I 3142 2447 3143 2445 3145 2444 -c 3147 2444 3148 2445 3149 2447 -c C 
3160 2489 M 3162 2496 I 3166 2506 I 3166 2508 3165 2510 3164 2510 -c 3162 2511 3160 2510 3160 2508 -c 3156 2497 I 3154 2490 I 3154 2489 3155 2487 3156 2486 -c 3158 2486 3160 2487 3160 2489 -c C 
3174 2530 M 3175 2534 I 3180 2547 I 3181 2549 3180 2551 3179 2551 -c 3177 2552 3175 2551 3175 2550 -c 3169 2536 I 3168 2532 I 3167 2530 3168 2529 3170 2528 -c 3171 2527 3173 2528 3174 2530 -c C 
3190 2571 M 3190 2571 I 3197 2588 I 3198 2589 3197 2591 3196 2592 -c 3194 2592 3192 2592 3192 2590 -c 3184 2573 I 3184 2573 I 3183 2571 3184 2569 3186 2569 -c 3187 2568 3189 2569 3190 2571 -c C 
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3228 2649 M 3237 2665 I 3238 2666 3237 2668 3236 2669 -c 3234 2670 3233 2669 3232 2668 -c 3222 2652 I 3221 2650 3222 2648 3223 2647 -c 3225 2647 3227 2647 3228 2649 -c C 
3250 2686 M 3260 2702 I 3261 2703 3261 2705 3259 2706 -c 3258 2707 3256 2707 3255 2705 -c 3245 2689 I 3244 2688 3244 2686 3246 2685 -c 3247 2684 3249 2685 3250 2686 -c C 
3274 2722 M 3285 2737 I 3286 2739 3286 2741 3284 2742 -c 3283 2743 3281 2742 3280 2741 -c 3269 2726 I 3268 2725 3268 2723 3270 2722 -c 3271 2721 3273 2721 3274 2722 -c C 
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3328 2791 M 3339 2803 I 3340 2805 I 3341 2806 3341 2808 3340 2809 -c 3339 2810 3337 2810 3336 2809 -c 3334 2808 I 3323 2795 I 3322 2794 3322 2792 3323 2790 -c 3325 2789 3327 2789 3328 2791 -c C 
3357 2823 M 3366 2832 I 3370 2836 I 3372 2837 3372 2839 3370 2841 -c 3369 2842 3367 2842 3366 2841 -c 3361 2836 I 3353 2827 I 3352 2826 3352 2824 3353 2823 -c 3354 2822 3356 2822 3357 2823 -c C 
3388 2853 M 3395 2859 I 3402 2866 I 3404 2867 3404 2869 3403 2870 -c 3401 2871 3399 2871 3398 2870 -c 3390 2864 I 3384 2858 I 3383 2857 3383 2855 3384 2853 -c 3385 2852 3387 2852 3388 2853 -c C 
3421 2882 M 3424 2885 I 3436 2894 I 3437 2895 3437 2897 3436 2898 -c 3435 2899 3433 2900 3432 2899 -c 3420 2890 I 3417 2887 I 3416 2886 3416 2884 3417 2882 -c 3418 2881 3420 2881 3421 2882 -c C 
3455 2909 M 3455 2909 I 3470 2920 I 3472 2921 3472 2923 3471 2924 -c 3470 2926 3468 2926 3467 2925 -c 3451 2914 I 3451 2914 I 3450 2913 3450 2911 3451 2910 -c 3452 2908 3454 2908 3455 2909 -c C 
3490 2934 M 3506 2944 I 3508 2945 3508 2947 3507 2949 -c 3506 2950 3504 2950 3503 2949 -c 3487 2939 I 3486 2938 3485 2936 3486 2935 -c 3487 2934 3489 2933 3490 2934 -c C 
3527 2957 M 3544 2967 I 3545 2968 3546 2969 3545 2971 -c 3544 2972 3542 2973 3540 2972 -c 3524 2963 I 3523 2962 3522 2960 3523 2958 -c 3524 2957 3526 2956 3527 2957 -c C 
3565 2979 M 3582 2987 I 3584 2988 3584 2990 3583 2991 -c 3583 2993 3581 2993 3579 2992 -c 3563 2984 I 3561 2983 3560 2981 3561 2980 -c 3562 2978 3564 2978 3565 2979 -c C 
3605 2998 M 3622 3005 I 3623 3006 3624 3008 3623 3009 -c 3622 3011 3621 3012 3619 3011 -c 3602 3003 I 3600 3003 3600 3001 3600 2999 -c 3601 2998 3603 2997 3605 2998 -c C 
3645 3015 M 3662 3022 I 3664 3022 3664 3024 3664 3026 -c 3663 3027 3661 3028 3660 3027 -c 3642 3020 I 3641 3020 3640 3018 3641 3016 -c 3641 3015 3643 3014 3645 3015 -c C 
3686 3030 M 3703 3035 I 3703 3036 I 3705 3036 3706 3038 3705 3039 -c 3705 3041 3703 3042 3701 3042 -c 3701 3041 I 3684 3036 I 3682 3035 3681 3033 3682 3032 -c 3682 3030 3684 3029 3686 3030 -c C 
3727 3042 M 3742 3046 I 3745 3047 I 3747 3048 3748 3049 3748 3051 -c 3747 3053 3745 3054 3744 3053 -c 3740 3052 I 3726 3048 I 3724 3048 3723 3046 3723 3045 -c 3724 3043 3726 3042 3727 3042 -c C 
3770 3053 M 3782 3056 I 3788 3057 I 3789 3057 3791 3059 3790 3060 -c 3790 3062 3788 3063 3787 3063 -c 3780 3062 I 3768 3059 I 3766 3059 3765 3057 3766 3055 -c 3766 3054 3768 3053 3770 3053 -c C 
3812 3061 M 3822 3063 I 3831 3064 I 3832 3064 3834 3066 3833 3068 -c 3833 3069 3832 3070 3830 3070 -c 3821 3069 I 3811 3067 I 3810 3067 3808 3065 3809 3064 -c 3809 3062 3811 3061 3812 3061 -c C 
3855 3067 M 3863 3068 I 3874 3069 I 3876 3069 3877 3071 3877 3072 -c 3877 3074 3875 3075 3874 3075 -c 3862 3074 I 3855 3073 I 3853 3073 3852 3072 3852 3070 -c 3852 3068 3854 3067 3855 3067 -c C 
3899 3071 M 3905 3071 I 3917 3072 I 3919 3072 3921 3073 3921 3075 -c 3920 3077 3919 3078 3917 3078 -c 3905 3077 I 3898 3077 I 3897 3077 3895 3075 3896 3074 -c 3896 3072 3897 3071 3899 3071 -c C 
3942 3072 M 3947 3072 I 3961 3072 I 3963 3072 3964 3073 3964 3075 -c 3964 3077 3963 3078 3961 3078 -c 3947 3079 I 3942 3078 I 3941 3078 3939 3077 3939 3075 -c 3939 3074 3941 3072 3942 3072 -c C 
3986 3071 M 3989 3071 I 4004 3070 I 4006 3070 4008 3071 4008 3073 -c 4008 3075 4007 3076 4005 3076 -c 3990 3077 I 3986 3078 I 3984 3078 3983 3076 3983 3075 -c 3983 3073 3984 3071 3986 3071 -c C 
4029 3068 M 4031 3068 I 4048 3066 I 4049 3066 4051 3067 4051 3069 -c 4051 3070 4050 3072 4049 3072 -c 4032 3074 I 4030 3074 I 4028 3075 4027 3073 4026 3072 -c 4026 3070 4028 3068 4029 3068 -c C 
4072 3063 M 4091 3060 I 4093 3059 4094 3060 4094 3062 -c 4095 3064 4094 3065 4092 3066 -c 4073 3069 I 4072 3069 4070 3068 4070 3067 -c 4070 3065 4071 3063 4072 3063 -c C 
4115 3055 M 4134 3051 I 4135 3050 4137 3052 4137 3053 -c 4138 3055 4137 3057 4135 3057 -c 4117 3061 I 4115 3062 4113 3061 4113 3059 -c 4113 3057 4114 3055 4115 3055 -c C 
4158 3045 M 4176 3040 I 4177 3039 4179 3040 4180 3042 -c 4180 3044 4179 3045 4177 3046 -c 4159 3051 I 4158 3052 4156 3051 4156 3049 -c 4155 3047 4156 3045 4158 3045 -c C 
4199 3033 M 4217 3027 I 4219 3026 4221 3027 4221 3029 -c 4222 3030 4221 3032 4219 3033 -c 4201 3039 I 4200 3039 4198 3038 4198 3037 -c 4197 3035 4198 3033 4199 3033 -c C 
4241 3018 M 4258 3011 I 4260 3011 4261 3011 4262 3013 -c 4263 3015 4262 3017 4260 3017 -c 4243 3024 I 4241 3025 4239 3024 4239 3022 -c 4238 3021 4239 3019 4241 3018 -c C 
4281 3002 M 4298 2994 I 4300 2993 4301 2994 4302 2995 -c 4303 2997 4302 2999 4301 3000 -c 4283 3007 I 4282 3008 4280 3007 4279 3006 -c 4279 3004 4279 3002 4281 3002 -c C 
4320 2983 M 4337 2974 I 4338 2974 4340 2974 4341 2976 -c 4342 2977 4341 2979 4340 2980 -c 4323 2988 I 4322 2989 4320 2988 4319 2987 -c 4318 2985 4319 2984 4320 2983 -c C 
4358 2962 M 4374 2954 I 4375 2953 I 4376 2952 4378 2952 4379 2954 -c 4380 2955 4379 2957 4378 2958 -c 4377 2959 I 4362 2968 I 4360 2968 4358 2968 4357 2966 -c 4356 2965 4357 2963 4358 2962 -c C 
4396 2939 M 4407 2932 I 4411 2929 I 4413 2928 4414 2928 4415 2930 -c 4416 2931 4416 2933 4415 2934 -c 4410 2937 I 4399 2945 I 4398 2946 4396 2945 4395 2944 -c 4394 2942 4394 2940 4396 2939 -c C 
4432 2915 M 4439 2909 I 4446 2904 I 4448 2902 4449 2903 4451 2904 -c 4452 2905 4451 2907 4450 2908 -c 4443 2914 I 4435 2920 I 4434 2921 4432 2920 4431 2919 -c 4430 2918 4430 2916 4432 2915 -c C 
4466 2888 M 4470 2885 I 4480 2876 I 4481 2875 4483 2875 4484 2876 -c 4486 2878 4485 2880 4484 2881 -c 4474 2890 I 4470 2893 I 4468 2894 4466 2894 4465 2892 -c 4464 2891 4464 2889 4466 2888 -c C 
4499 2860 M 4500 2859 I 4512 2847 I 4514 2846 4516 2846 4517 2847 -c 4518 2848 4518 2850 4517 2852 -c 4504 2864 I 4503 2864 I 4502 2866 4500 2865 4499 2864 -c 4497 2863 4498 2861 4499 2860 -c C 
4530 2830 M 4543 2816 I 4544 2815 4546 2815 4548 2816 -c 4549 2817 4549 2819 4548 2821 -c 4535 2834 I 4534 2835 4532 2836 4530 2834 -c 4529 2833 4529 2831 4530 2830 -c C 
4560 2798 M 4572 2784 I 4574 2783 4576 2782 4577 2784 -c 4578 2785 4578 2787 4577 2788 -c 4565 2802 I 4564 2803 4562 2804 4560 2803 -c 4559 2801 4559 2799 4560 2798 -c C 
4588 2765 M 4600 2750 I 4601 2749 4603 2748 4604 2750 -c 4606 2751 4606 2753 4605 2754 -c 4593 2769 I 4592 2770 4590 2770 4589 2769 -c 4587 2768 4587 2766 4588 2765 -c C 
4615 2730 M 4625 2715 I 4626 2714 4628 2713 4630 2714 -c 4631 2715 4632 2717 4631 2719 -c 4620 2734 I 4619 2735 4617 2735 4615 2734 -c 4614 2733 4614 2732 4615 2730 -c C 
4639 2694 M 4649 2678 I 4650 2677 4652 2677 4654 2678 -c 4655 2678 4655 2680 4654 2682 -c 4644 2698 I 4643 2699 4641 2699 4640 2699 -c 4639 2698 4638 2696 4639 2694 -c C 
4662 2657 M 4670 2643 I 4671 2641 I 4672 2639 4674 2639 4675 2639 -c 4677 2640 4677 2642 4676 2644 -c 4675 2646 I 4667 2660 I 4666 2662 4664 2662 4663 2661 -c 4661 2660 4661 2658 4662 2657 -c C 
4682 2619 M 4688 2607 I 4690 2602 I 4691 2601 4693 2600 4695 2601 -c 4696 2601 4697 2603 4696 2605 -c 4694 2610 I 4688 2621 I 4687 2623 4685 2624 4684 2623 -c 4682 2622 4681 2620 4682 2619 -c C 
4701 2579 M 4705 2571 I 4708 2562 I 4709 2561 4710 2560 4712 2560 -c 4714 2561 4714 2563 4714 2564 -c 4710 2574 I 4706 2582 I 4706 2583 4704 2584 4702 2583 -c 4701 2583 4700 2581 4701 2579 -c C 
4717 2539 M 4719 2534 I 4723 2521 I 4724 2520 4726 2519 4727 2519 -c 4729 2520 4730 2522 4729 2523 -c 4725 2536 I 4723 2541 I 4722 2543 4720 2544 4719 2543 -c 4717 2542 4716 2541 4717 2539 -c C 
4731 2498 M 4732 2495 I 4737 2480 I 4737 2478 4739 2478 4740 2478 -c 4742 2478 4743 2480 4743 2482 -c 4738 2497 I 4737 2500 I 4737 2501 4735 2502 4733 2502 -c 4732 2501 4731 2499 4731 2498 -c C 
4743 2456 M 4748 2438 I 4748 2436 4750 2435 4751 2436 -c 4753 2436 4754 2438 4754 2439 -c 4749 2458 I 4749 2459 4747 2460 4746 2460 -c 4744 2460 4743 2458 4743 2456 -c C 
4753 2414 M 4756 2395 I 4757 2394 4758 2392 4760 2393 -c 4762 2393 4763 2395 4762 2396 -c 4759 2415 I 4759 2417 4757 2418 4756 2417 -c 4754 2417 4753 2415 4753 2414 -c C 
4760 2371 M 4763 2352 I 4763 2351 4765 2350 4766 2350 -c 4768 2350 4769 2352 4769 2353 -c 4767 2372 I 4766 2373 4765 2374 4763 2374 -c 4761 2374 4760 2372 4760 2371 -c C 
4765 2328 M 4767 2309 I 4767 2307 4768 2306 4770 2306 -c 4772 2306 4773 2308 4773 2310 -c 4772 2328 I 4772 2330 4770 2331 4768 2331 -c 4767 2331 4765 2329 4765 2328 -c C 
4768 2284 M 4769 2266 I 4769 2264 4770 2262 4772 2262 -c 4774 2263 4775 2264 4775 2266 -c 4774 2284 I 4774 2286 4773 2287 4771 2287 -c 4770 2287 4768 2286 4768 2284 -c C 
4769 2241 M 4768 2222 I 4768 2220 4770 2219 4771 2219 -c 4773 2219 4775 2220 4775 2222 -c 4775 2241 I 4775 2242 4774 2244 4772 2244 -c 4770 2244 4769 2242 4769 2241 -c C 
4767 2197 M 4766 2179 I 4766 2177 4767 2175 4769 2175 -c 4770 2175 4772 2176 4772 2178 -c 4773 2197 I 4773 2199 4772 2200 4770 2200 -c 4769 2200 4767 2199 4767 2197 -c C 
4763 2154 M 4761 2135 I 4761 2134 4762 2132 4763 2132 -c 4765 2132 4767 2133 4767 2135 -c 4769 2153 I 4770 2155 4768 2156 4767 2157 -c 4765 2157 4763 2156 4763 2154 -c C 
4757 2111 M 4754 2093 I 4753 2091 4754 2089 4756 2089 -c 4758 2089 4759 2090 4760 2092 -c 4763 2110 I 4763 2111 4762 2113 4760 2113 -c 4759 2114 4757 2113 4757 2111 -c C 
4748 2068 M 4744 2050 I 4744 2048 4745 2047 4746 2046 -c 4748 2046 4750 2047 4750 2049 -c 4754 2067 I 4755 2069 4754 2070 4752 2071 -c 4750 2071 4749 2070 4748 2068 -c C 
4737 2026 M 4732 2008 I 4732 2006 4733 2005 4734 2004 -c 4736 2004 4738 2005 4738 2006 -c 4743 2024 I 4744 2026 4743 2028 4741 2028 -c 4740 2029 4738 2028 4737 2026 -c C 
4724 1985 M 4719 1969 I 4718 1967 I 4718 1966 4718 1964 4720 1963 -c 4722 1963 4724 1963 4724 1965 -c 4725 1967 I 4730 1983 I 4731 1984 4730 1986 4728 1987 -c 4727 1987 4725 1986 4724 1985 -c C 
4709 1944 M 4704 1932 I 4702 1927 I 4701 1925 4702 1923 4704 1923 -c 4705 1922 4707 1923 4708 1924 -c 4710 1930 I 4715 1942 I 4716 1943 4715 1945 4713 1946 -c 4712 1946 4710 1946 4709 1944 -c C 
4692 1904 M 4688 1896 I 4684 1887 I 4683 1886 4684 1884 4685 1883 -c 4687 1882 4689 1883 4689 1884 -c 4694 1893 I 4698 1901 I 4698 1903 4698 1905 4696 1906 -c 4694 1906 4693 1906 4692 1904 -c C 
4672 1865 M 4670 1860 I 4663 1849 I 4662 1848 4663 1846 4664 1845 -c 4666 1844 4668 1844 4669 1846 -c 4675 1858 I 4678 1862 I 4679 1864 4678 1866 4677 1866 -c 4675 1867 4673 1867 4672 1865 -c C 
4651 1827 M 4650 1826 I 4641 1812 I 4640 1810 4640 1808 4642 1807 -c 4643 1806 4645 1807 4646 1808 -c 4655 1823 I 4656 1824 I 4657 1826 4657 1828 4655 1828 -c 4654 1829 4652 1829 4651 1827 -c C 
4627 1791 M 4616 1776 I 4615 1774 4616 1772 4617 1771 -c 4619 1770 4621 1770 4622 1772 -c 4632 1787 I 4633 1789 4633 1791 4632 1792 -c 4630 1793 4628 1792 4627 1791 -c C 
4602 1755 M 4590 1741 I 4589 1739 4589 1737 4591 1737 -c 4592 1735 4594 1736 4595 1737 -c 4607 1752 I 4608 1753 4608 1755 4606 1756 -c 4605 1757 4603 1757 4602 1755 -c C 
4575 1722 M 4562 1708 I 4561 1706 4561 1704 4563 1703 -c 4564 1702 4566 1702 4567 1703 -c 4579 1718 I 4580 1719 4580 1721 4579 1722 -c 4578 1723 4576 1723 4575 1722 -c C 
4546 1689 M 4533 1676 I 4531 1674 4531 1672 4533 1671 -c 4534 1670 4536 1670 4537 1671 -c 4550 1685 I 4551 1686 4551 1688 4550 1689 -c 4549 1691 4547 1690 4546 1689 -c C 
4515 1658 M 4501 1645 I 4500 1644 4500 1642 4501 1641 -c 4502 1640 4504 1640 4506 1641 -c 4519 1654 I 4520 1655 4520 1657 4519 1658 -c 4518 1659 4516 1659 4515 1658 -c C 
4483 1629 M 4470 1618 I 4468 1617 I 4467 1616 4467 1614 4468 1613 -c 4469 1611 4471 1611 4472 1612 -c 4474 1613 I 4487 1624 I 4488 1625 4488 1627 4487 1629 -c 4486 1630 4484 1630 4483 1629 -c C 
4449 1602 M 4439 1594 I 4434 1590 I 4433 1589 4432 1587 4433 1586 -c 4434 1585 4436 1584 4438 1585 -c 4443 1589 I 4453 1597 I 4454 1598 4454 1600 4453 1601 -c 4452 1602 4450 1603 4449 1602 -c C 
4414 1576 M 4407 1571 I 4398 1566 I 4397 1565 4397 1563 4397 1561 -c 4398 1560 4400 1559 4402 1560 -c 4410 1566 I 4417 1571 I 4419 1572 4419 1574 4418 1575 -c 4417 1577 4415 1577 4414 1576 -c C 
4377 1552 M 4373 1549 I 4361 1543 I 4360 1542 4359 1540 4360 1538 -c 4361 1537 4363 1536 4364 1537 -c 4377 1544 I 4381 1547 I 4382 1548 4383 1550 4382 1551 -c 4381 1553 4379 1553 4377 1552 -c C 
4340 1530 M 4339 1530 I 4323 1522 I 4322 1521 4321 1519 4322 1517 -c 4323 1516 4324 1515 4326 1516 -c 4342 1524 I 4343 1525 I 4344 1526 4345 1527 4344 1529 -c 4343 1530 4341 1531 4340 1530 -c C 
4301 1510 M 4284 1503 I 4282 1502 4282 1500 4282 1499 -c 4283 1497 4285 1496 4286 1497 -c 4304 1505 I 4305 1505 4306 1507 4305 1509 -c 4304 1510 4303 1511 4301 1510 -c C 
4261 1493 M 4244 1486 I 4242 1486 4241 1484 4242 1482 -c 4243 1480 4244 1480 4246 1480 -c 4263 1487 I 4265 1488 4266 1489 4265 1491 -c 4265 1493 4263 1493 4261 1493 -c C 
4220 1477 M 4203 1471 I 4201 1471 4200 1469 4201 1468 -c 4201 1466 4203 1465 4205 1466 -c 4222 1472 I 4224 1472 4225 1474 4224 1476 -c 4224 1477 4222 1478 4220 1477 -c C 
4179 1464 M 4161 1459 I 4159 1459 4158 1457 4159 1455 -c 4159 1453 4161 1453 4163 1453 -c 4181 1458 I 4182 1459 4183 1460 4183 1462 -c 4182 1464 4181 1465 4179 1464 -c C 
4137 1453 M 4118 1449 I 4117 1448 4116 1447 4116 1445 -c 4117 1443 4118 1442 4120 1443 -c 4138 1447 I 4140 1447 4141 1449 4141 1451 -c 4140 1452 4138 1453 4137 1453 -c C 
4094 1444 M 4076 1441 I 4074 1441 4073 1439 4073 1437 -c 4073 1435 4075 1434 4077 1435 -c 4095 1438 I 4097 1438 4098 1440 4098 1442 -c 4097 1443 4096 1444 4094 1444 -c C 
4051 1438 M 4032 1435 I 4031 1435 4029 1433 4030 1432 -c 4030 1430 4031 1429 4033 1429 -c 4052 1431 I 4054 1432 4055 1433 4055 1435 -c 4054 1437 4053 1438 4051 1438 -c C 
4008 1433 M 3989 1432 I 3988 1432 3986 1430 3986 1429 -c 3987 1427 3988 1425 3990 1426 -c 4008 1427 I 4010 1427 4011 1429 4011 1430 -c 4011 1432 4009 1433 4008 1433 -c C 
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3940 1431 M 3921 1431 I 3919 1431 3918 1430 3918 1428 -c 3918 1427 3919 1425 3921 1425 -c 3940 1425 I 3941 1425 3943 1426 3943 1428 -c 3943 1429 3942 1431 3940 1431 -c C 
3896 1433 M 3878 1434 I 3876 1434 3874 1433 3874 1431 -c 3874 1429 3875 1428 3877 1428 -c 3896 1426 I 3898 1426 3899 1427 3899 1429 -c 3899 1431 3898 1432 3896 1433 -c C 
3853 1436 M 3834 1439 I 3833 1439 3831 1438 3831 1436 -c 3831 1434 3832 1433 3833 1432 -c 3852 1430 I 3854 1430 3855 1431 3856 1433 -c 3856 1435 3855 1436 3853 1436 -c C 
3810 1442 M 3791 1446 I 3790 1446 3788 1445 3788 1443 -c 3787 1441 3788 1440 3790 1440 -c 3809 1436 I 3810 1436 3812 1437 3812 1439 -c 3813 1440 3811 1442 3810 1442 -c C 
3767 1451 M 3749 1455 I 3747 1455 3745 1454 3745 1453 -c 3745 1451 3746 1449 3747 1449 -c 3766 1445 I 3767 1444 3769 1445 3769 1447 -c 3770 1449 3769 1450 3767 1451 -c C 
3725 1461 M 3707 1467 I 3705 1467 3703 1466 3703 1464 -c 3702 1463 3703 1461 3705 1461 -c 3723 1455 I 3725 1455 3726 1456 3727 1458 -c 3727 1459 3726 1461 3725 1461 -c C 
3683 1474 M 3665 1480 I 3664 1481 3662 1480 3661 1478 -c 3661 1477 3662 1475 3663 1474 -c 3681 1468 I 3683 1468 3684 1469 3685 1470 -c 3686 1472 3685 1474 3683 1474 -c C 
3642 1489 M 3627 1495 I 3625 1496 I 3623 1497 3622 1496 3621 1495 -c 3620 1493 3621 1491 3622 1490 -c 3625 1489 I 3640 1484 I 3641 1483 3643 1484 3644 1485 -c 3645 1487 3644 1489 3642 1489 -c C 
3602 1506 M 3591 1512 I 3585 1514 I 3584 1515 3582 1515 3581 1513 -c 3580 1511 3581 1510 3582 1509 -c 3588 1506 I 3600 1501 I 3601 1500 3603 1501 3604 1502 -c 3604 1504 3604 1506 3602 1506 -c C 
3563 1526 M 3555 1530 I 3547 1535 I 3545 1535 3543 1535 3542 1533 -c 3542 1532 3542 1530 3544 1529 -c 3552 1524 I 3560 1520 I 3562 1519 3564 1520 3564 1522 -c 3565 1523 3565 1525 3563 1526 -c C 
3525 1547 M 3521 1550 I 3509 1557 I 3508 1558 3506 1557 3505 1556 -c 3504 1555 3504 1553 3506 1552 -c 3518 1544 I 3522 1542 I 3523 1541 3525 1541 3526 1543 -c 3527 1544 3527 1546 3525 1547 -c C 
3488 1570 M 3487 1571 I 3473 1581 I 3472 1582 3470 1582 3469 1580 -c 3468 1579 3468 1577 3469 1576 -c 3484 1566 I 3485 1565 I 3486 1564 3488 1565 3489 1566 -c 3490 1568 3490 1570 3488 1570 -c C 
3453 1596 M 3438 1607 I 3437 1608 3435 1608 3434 1607 -c 3433 1605 3433 1603 3435 1602 -c 3449 1591 I 3450 1590 3452 1590 3453 1591 -c 3454 1593 3454 1595 3453 1596 -c C 
3419 1623 M 3405 1635 I 3404 1636 3402 1636 3401 1635 -c 3399 1633 3400 1631 3401 1630 -c 3415 1618 I 3416 1617 3418 1617 3419 1618 -c 3421 1620 3420 1621 3419 1623 -c C 
3386 1652 M 3373 1665 I 3372 1666 3370 1666 3368 1664 -c 3367 1663 3367 1661 3369 1660 -c 3382 1647 I 3383 1646 3385 1646 3387 1647 -c 3388 1648 3388 1650 3386 1652 -c C 
3355 1682 M 3343 1696 I 3341 1697 3339 1697 3338 1696 -c 3337 1695 3337 1693 3338 1691 -c 3351 1678 I 3352 1677 3354 1677 3355 1678 -c 3357 1679 3357 1681 3355 1682 -c C 
3326 1714 M 3314 1729 I 3313 1730 3311 1730 3309 1729 -c 3308 1728 3308 1726 3309 1724 -c 3321 1710 I 3322 1709 3324 1709 3326 1710 -c 3327 1711 3327 1713 3326 1714 -c C 
3298 1748 M 3288 1760 I 3287 1763 I 3286 1764 3284 1764 3283 1763 -c 3281 1762 3281 1760 3282 1759 -c 3284 1757 I 3293 1744 I 3295 1743 3296 1742 3298 1743 -c 3299 1744 3299 1746 3298 1748 -c C 
3272 1783 M 3266 1793 I 3262 1798 I 3261 1800 3259 1800 3258 1799 -c 3256 1798 3256 1796 3257 1795 -c 3260 1789 I 3267 1779 I 3268 1778 3270 1778 3272 1779 -c 3273 1780 3273 1781 3272 1783 -c C 
3248 1819 M 3244 1826 I 3239 1835 I 3238 1837 3236 1837 3235 1836 -c 3233 1835 3233 1834 3233 1832 -c 3239 1823 I 3243 1816 I 3244 1814 3246 1814 3248 1815 -c 3249 1816 3249 1818 3248 1819 -c C 
3226 1857 M 3224 1861 I 3218 1873 I 3217 1875 3215 1875 3214 1874 -c 3212 1874 3211 1872 3212 1870 -c 3219 1857 I 3221 1854 I 3222 1852 3224 1852 3225 1853 -c 3227 1853 3227 1855 3226 1857 -c C 
3206 1895 M 3206 1896 I 3199 1912 I 3198 1914 3196 1915 3195 1914 -c 3193 1913 3192 1911 3193 1910 -c 3201 1893 I 3201 1892 I 3202 1891 3203 1890 3205 1891 -c 3207 1892 3207 1894 3206 1895 -c C 
3189 1935 M 3182 1952 I 3181 1954 3179 1955 3178 1954 -c 3176 1954 3175 1952 3176 1950 -c 3183 1933 I 3183 1931 3185 1930 3187 1931 -c 3188 1932 3189 1933 3189 1935 -c C 
3173 1976 M 3167 1993 I 3166 1995 3165 1996 3163 1995 -c 3161 1995 3160 1993 3161 1991 -c 3167 1974 I 3168 1972 3169 1971 3171 1972 -c 3173 1972 3174 1974 3173 1976 -c C 
3159 2017 M 3154 2035 I 3154 2037 3152 2037 3150 2037 -c 3149 2037 3148 2035 3148 2033 -c 3153 2015 I 3154 2013 3156 2012 3157 2013 -c 3159 2013 3160 2015 3159 2017 -c C 
3148 2059 M 3144 2077 I 3144 2079 3142 2080 3140 2080 -c 3138 2079 3137 2077 3138 2076 -c 3142 2058 I 3142 2056 3144 2055 3146 2055 -c 3147 2056 3149 2057 3148 2059 -c C 
3139 2102 M 3136 2120 I 3135 2121 3134 2123 3132 2122 -c 3130 2122 3129 2120 3130 2119 -c 3133 2100 I 3133 2099 3135 2098 3137 2098 -c 3138 2098 3139 2100 3139 2102 -c C 
3132 2144 M 3130 2163 I 3130 2165 3128 2166 3126 2166 -c 3125 2165 3124 2164 3124 2162 -c 3126 2144 I 3126 2142 3128 2141 3130 2141 -c 3131 2141 3133 2143 3132 2144 -c C 
3128 2188 M 3126 2206 I 3126 2208 3125 2209 3123 2209 -c 3121 2209 3120 2208 3120 2206 -c 3122 2187 I 3122 2185 3123 2184 3125 2184 -c 3127 2184 3128 2186 3128 2188 -c C 
3126 2231 M 3125 2250 I 3125 2251 3124 2253 3122 2253 -c 3120 2253 3119 2251 3119 2250 -c 3119 2231 I 3119 2229 3121 2228 3123 2228 -c 3124 2228 3126 2229 3126 2231 -c C 
3126 2275 M 3126 2293 I 3126 2295 3125 2296 3123 2296 -c 3121 2297 3120 2295 3120 2293 -c 3119 2275 I 3119 2273 3121 2272 3123 2271 -c 3124 2271 3126 2273 3126 2275 -c C 
3128 2318 M 3129 2336 I 3129 2337 I 3130 2338 3128 2340 3127 2340 -c 3125 2340 3124 2339 3123 2337 -c 3123 2336 I 3122 2319 I 3122 2317 3123 2315 3125 2315 -c 3126 2315 3128 2316 3128 2318 -c C 
3133 2361 M 3135 2377 I 3135 2380 I 3135 2381 3134 2383 3133 2383 -c 3131 2383 3129 2382 3129 2381 -c 3128 2377 I 3126 2362 I 3126 2360 3127 2359 3129 2359 -c 3131 2358 3132 2360 3133 2361 -c C 
3140 2404 M 3142 2417 I 3143 2422 I 3143 2424 3142 2426 3141 2426 -c 3139 2427 3137 2426 3137 2424 -c 3136 2418 I 3133 2405 I 3133 2404 3134 2402 3136 2402 -c 3138 2401 3139 2402 3140 2404 -c C 
3149 2447 M 3151 2457 I 3153 2465 I 3154 2466 3153 2468 3151 2469 -c 3149 2469 3148 2468 3147 2466 -c 3145 2458 I 3143 2448 I 3142 2447 3143 2445 3145 2444 -c 3147 2444 3148 2445 3149 2447 -c C 
3160 2489 M 3162 2496 I 3166 2506 I 3166 2508 3165 2510 3164 2510 -c 3162 2511 3160 2510 3160 2508 -c 3156 2497 I 3154 2490 I 3154 2489 3155 2487 3156 2486 -c 3158 2486 3160 2487 3160 2489 -c C 
3174 2530 M 3175 2534 I 3180 2547 I 3181 2549 3180 2551 3179 2551 -c 3177 2552 3175 2551 3175 2550 -c 3169 2536 I 3168 2532 I 3167 2530 3168 2529 3170 2528 -c 3171 2527 3173 2528 3174 2530 -c C 
3190 2571 M 3190 2571 I 3197 2588 I 3198 2589 3197 2591 3196 2592 -c 3194 2592 3192 2592 3192 2590 -c 3184 2573 I 3184 2573 I 3183 2571 3184 2569 3186 2569 -c 3187 2568 3189 2569 3190 2571 -c C 
3208 2610 M 3216 2627 I 3217 2628 3216 2630 3215 2631 -c 3213 2632 3211 2631 3211 2629 -c 3202 2613 I 3201 2611 3202 2609 3203 2609 -c 3205 2608 3207 2609 3208 2610 -c C 
3228 2649 M 3237 2665 I 3238 2666 3237 2668 3236 2669 -c 3234 2670 3233 2669 3232 2668 -c 3222 2652 I 3221 2650 3222 2648 3223 2647 -c 3225 2647 3227 2647 3228 2649 -c C 
3250 2686 M 3260 2702 I 3261 2703 3261 2705 3259 2706 -c 3258 2707 3256 2707 3255 2705 -c 3245 2689 I 3244 2688 3244 2686 3246 2685 -c 3247 2684 3249 2685 3250 2686 -c C 
3274 2722 M 3285 2737 I 3286 2739 3286 2741 3284 2742 -c 3283 2743 3281 2742 3280 2741 -c 3269 2726 I 3268 2725 3268 2723 3270 2722 -c 3271 2721 3273 2721 3274 2722 -c C 
3300 2757 M 3312 2772 I 3313 2773 3312 2775 3311 2776 -c 3310 2777 3308 2777 3307 2776 -c 3295 2761 I 3294 2760 3294 2758 3296 2757 -c 3297 2756 3299 2756 3300 2757 -c C 
3328 2791 M 3339 2803 I 3340 2805 I 3341 2806 3341 2808 3340 2809 -c 3339 2810 3337 2810 3336 2809 -c 3334 2808 I 3323 2795 I 3322 2794 3322 2792 3323 2790 -c 3325 2789 3327 2789 3328 2791 -c C 
3357 2823 M 3366 2832 I 3370 2836 I 3372 2837 3372 2839 3370 2841 -c 3369 2842 3367 2842 3366 2841 -c 3361 2836 I 3353 2827 I 3352 2826 3352 2824 3353 2823 -c 3354 2822 3356 2822 3357 2823 -c C 
3388 2853 M 3395 2859 I 3402 2866 I 3404 2867 3404 2869 3403 2870 -c 3401 2871 3399 2871 3398 2870 -c 3390 2864 I 3384 2858 I 3383 2857 3383 2855 3384 2853 -c 3385 2852 3387 2852 3388 2853 -c C 
3421 2882 M 3424 2885 I 3436 2894 I 3437 2895 3437 2897 3436 2898 -c 3435 2899 3433 2900 3432 2899 -c 3420 2890 I 3417 2887 I 3416 2886 3416 2884 3417 2882 -c 3418 2881 3420 2881 3421 2882 -c C 
3455 2909 M 3455 2909 I 3470 2920 I 3472 2921 3472 2923 3471 2924 -c 3470 2926 3468 2926 3467 2925 -c 3451 2914 I 3451 2914 I 3450 2913 3450 2911 3451 2910 -c 3452 2908 3454 2908 3455 2909 -c C 
3490 2934 M 3506 2944 I 3508 2945 3508 2947 3507 2949 -c 3506 2950 3504 2950 3503 2949 -c 3487 2939 I 3486 2938 3485 2936 3486 2935 -c 3487 2934 3489 2933 3490 2934 -c C 
3527 2957 M 3544 2967 I 3545 2968 3546 2969 3545 2971 -c 3544 2972 3542 2973 3540 2972 -c 3524 2963 I 3523 2962 3522 2960 3523 2958 -c 3524 2957 3526 2956 3527 2957 -c C 
3565 2979 M 3582 2987 I 3584 2988 3584 2990 3583 2991 -c 3583 2993 3581 2993 3579 2992 -c 3563 2984 I 3561 2983 3560 2981 3561 2980 -c 3562 2978 3564 2978 3565 2979 -c C 
3605 2998 M 3622 3005 I 3623 3006 3624 3008 3623 3009 -c 3622 3011 3621 3012 3619 3011 -c 3602 3003 I 3600 3003 3600 3001 3600 2999 -c 3601 2998 3603 2997 3605 2998 -c C 
3645 3015 M 3662 3022 I 3664 3022 3664 3024 3664 3026 -c 3663 3027 3661 3028 3660 3027 -c 3642 3020 I 3641 3020 3640 3018 3641 3016 -c 3641 3015 3643 3014 3645 3015 -c C 
3686 3030 M 3703 3035 I 3703 3036 I 3705 3036 3706 3038 3705 3039 -c 3705 3041 3703 3042 3701 3042 -c 3701 3041 I 3684 3036 I 3682 3035 3681 3033 3682 3032 -c 3682 3030 3684 3029 3686 3030 -c C 
3727 3042 M 3742 3046 I 3745 3047 I 3747 3048 3748 3049 3748 3051 -c 3747 3053 3745 3054 3744 3053 -c 3740 3052 I 3726 3048 I 3724 3048 3723 3046 3723 3045 -c 3724 3043 3726 3042 3727 3042 -c C 
3770 3053 M 3782 3056 I 3788 3057 I 3789 3057 3791 3059 3790 3060 -c 3790 3062 3788 3063 3787 3063 -c 3780 3062 I 3768 3059 I 3766 3059 3765 3057 3766 3055 -c 3766 3054 3768 3053 3770 3053 -c C 
3812 3061 M 3822 3063 I 3831 3064 I 3832 3064 3834 3066 3833 3068 -c 3833 3069 3832 3070 3830 3070 -c 3821 3069 I 3811 3067 I 3810 3067 3808 3065 3809 3064 -c 3809 3062 3811 3061 3812 3061 -c C 
3855 3067 M 3863 3068 I 3874 3069 I 3876 3069 3877 3071 3877 3072 -c 3877 3074 3875 3075 3874 3075 -c 3862 3074 I 3855 3073 I 3853 3073 3852 3072 3852 3070 -c 3852 3068 3854 3067 3855 3067 -c C 
3899 3071 M 3905 3071 I 3917 3072 I 3919 3072 3921 3073 3921 3075 -c 3920 3077 3919 3078 3917 3078 -c 3905 3077 I 3898 3077 I 3897 3077 3895 3075 3896 3074 -c 3896 3072 3897 3071 3899 3071 -c C 
3942 3072 M 3947 3072 I 3961 3072 I 3963 3072 3964 3073 3964 3075 -c 3964 3077 3963 3078 3961 3078 -c 3947 3079 I 3942 3078 I 3941 3078 3939 3077 3939 3075 -c 3939 3074 3941 3072 3942 3072 -c C 
3986 3071 M 3989 3071 I 4004 3070 I 4006 3070 4008 3071 4008 3073 -c 4008 3075 4007 3076 4005 3076 -c 3990 3077 I 3986 3078 I 3984 3078 3983 3076 3983 3075 -c 3983 3073 3984 3071 3986 3071 -c C 
4029 3068 M 4031 3068 I 4048 3066 I 4049 3066 4051 3067 4051 3069 -c 4051 3070 4050 3072 4049 3072 -c 4032 3074 I 4030 3074 I 4028 3075 4027 3073 4026 3072 -c 4026 3070 4028 3068 4029 3068 -c C 
4072 3063 M 4091 3060 I 4093 3059 4094 3060 4094 3062 -c 4095 3064 4094 3065 4092 3066 -c 4073 3069 I 4072 3069 4070 3068 4070 3067 -c 4070 3065 4071 3063 4072 3063 -c C 
4115 3055 M 4134 3051 I 4135 3050 4137 3052 4137 3053 -c 4138 3055 4137 3057 4135 3057 -c 4117 3061 I 4115 3062 4113 3061 4113 3059 -c 4113 3057 4114 3055 4115 3055 -c C 
4158 3045 M 4176 3040 I 4177 3039 4179 3040 4180 3042 -c 4180 3044 4179 3045 4177 3046 -c 4159 3051 I 4158 3052 4156 3051 4156 3049 -c 4155 3047 4156 3045 4158 3045 -c C 
4199 3033 M 4217 3027 I 4219 3026 4221 3027 4221 3029 -c 4222 3030 4221 3032 4219 3033 -c 4201 3039 I 4200 3039 4198 3038 4198 3037 -c 4197 3035 4198 3033 4199 3033 -c C 
4241 3018 M 4258 3011 I 4260 3011 4261 3011 4262 3013 -c 4263 3015 4262 3017 4260 3017 -c 4243 3024 I 4241 3025 4239 3024 4239 3022 -c 4238 3021 4239 3019 4241 3018 -c C 
4281 3002 M 4298 2994 I 4300 2993 4301 2994 4302 2995 -c 4303 2997 4302 2999 4301 3000 -c 4283 3007 I 4282 3008 4280 3007 4279 3006 -c 4279 3004 4279 3002 4281 3002 -c C 
4320 2983 M 4337 2974 I 4338 2974 4340 2974 4341 2976 -c 4342 2977 4341 2979 4340 2980 -c 4323 2988 I 4322 2989 4320 2988 4319 2987 -c 4318 2985 4319 2984 4320 2983 -c C 
4358 2962 M 4374 2954 I 4375 2953 I 4376 2952 4378 2952 4379 2954 -c 4380 2955 4379 2957 4378 2958 -c 4377 2959 I 4362 2968 I 4360 2968 4358 2968 4357 2966 -c 4356 2965 4357 2963 4358 2962 -c C 
4396 2939 M 4407 2932 I 4411 2929 I 4413 2928 4414 2928 4415 2930 -c 4416 2931 4416 2933 4415 2934 -c 4410 2937 I 4399 2945 I 4398 2946 4396 2945 4395 2944 -c 4394 2942 4394 2940 4396 2939 -c C 
4432 2915 M 4439 2909 I 4446 2904 I 4448 2902 4449 2903 4451 2904 -c 4452 2905 4451 2907 4450 2908 -c 4443 2914 I 4435 2920 I 4434 2921 4432 2920 4431 2919 -c 4430 2918 4430 2916 4432 2915 -c C 
4466 2888 M 4470 2885 I 4480 2876 I 4481 2875 4483 2875 4484 2876 -c 4486 2878 4485 2880 4484 2881 -c 4474 2890 I 4470 2893 I 4468 2894 4466 2894 4465 2892 -c 4464 2891 4464 2889 4466 2888 -c C 
4499 2860 M 4500 2859 I 4512 2847 I 4514 2846 4516 2846 4517 2847 -c 4518 2848 4518 2850 4517 2852 -c 4504 2864 I 4503 2864 I 4502 2866 4500 2865 4499 2864 -c 4497 2863 4498 2861 4499 2860 -c C 
4530 2830 M 4543 2816 I 4544 2815 4546 2815 4548 2816 -c 4549 2817 4549 2819 4548 2821 -c 4535 2834 I 4534 2835 4532 2836 4530 2834 -c 4529 2833 4529 2831 4530 2830 -c C 
4560 2798 M 4572 2784 I 4574 2783 4576 2782 4577 2784 -c 4578 2785 4578 2787 4577 2788 -c 4565 2802 I 4564 2803 4562 2804 4560 2803 -c 4559 2801 4559 2799 4560 2798 -c C 
4588 2765 M 4600 2750 I 4601 2749 4603 2748 4604 2750 -c 4606 2751 4606 2753 4605 2754 -c 4593 2769 I 4592 2770 4590 2770 4589 2769 -c 4587 2768 4587 2766 4588 2765 -c C 
4615 2730 M 4625 2715 I 4626 2714 4628 2713 4630 2714 -c 4631 2715 4632 2717 4631 2719 -c 4620 2734 I 4619 2735 4617 2735 4615 2734 -c 4614 2733 4614 2732 4615 2730 -c C 
4639 2694 M 4649 2678 I 4650 2677 4652 2677 4654 2678 -c 4655 2678 4655 2680 4654 2682 -c 4644 2698 I 4643 2699 4641 2699 4640 2699 -c 4639 2698 4638 2696 4639 2694 -c C 
4662 2657 M 4670 2643 I 4671 2641 I 4672 2639 4674 2639 4675 2639 -c 4677 2640 4677 2642 4676 2644 -c 4675 2646 I 4667 2660 I 4666 2662 4664 2662 4663 2661 -c 4661 2660 4661 2658 4662 2657 -c C 
4682 2619 M 4688 2607 I 4690 2602 I 4691 2601 4693 2600 4695 2601 -c 4696 2601 4697 2603 4696 2605 -c 4694 2610 I 4688 2621 I 4687 2623 4685 2624 4684 2623 -c 4682 2622 4681 2620 4682 2619 -c C 
4701 2579 M 4705 2571 I 4708 2562 I 4709 2561 4710 2560 4712 2560 -c 4714 2561 4714 2563 4714 2564 -c 4710 2574 I 4706 2582 I 4706 2583 4704 2584 4702 2583 -c 4701 2583 4700 2581 4701 2579 -c C 
4717 2539 M 4719 2534 I 4723 2521 I 4724 2520 4726 2519 4727 2519 -c 4729 2520 4730 2522 4729 2523 -c 4725 2536 I 4723 2541 I 4722 2543 4720 2544 4719 2543 -c 4717 2542 4716 2541 4717 2539 -c C 
4731 2498 M 4732 2495 I 4737 2480 I 4737 2478 4739 2478 4740 2478 -c 4742 2478 4743 2480 4743 2482 -c 4738 2497 I 4737 2500 I 4737 2501 4735 2502 4733 2502 -c 4732 2501 4731 2499 4731 2498 -c C 
4743 2456 M 4748 2438 I 4748 2436 4750 2435 4751 2436 -c 4753 2436 4754 2438 4754 2439 -c 4749 2458 I 4749 2459 4747 2460 4746 2460 -c 4744 2460 4743 2458 4743 2456 -c C 
4753 2414 M 4756 2395 I 4757 2394 4758 2392 4760 2393 -c 4762 2393 4763 2395 4762 2396 -c 4759 2415 I 4759 2417 4757 2418 4756 2417 -c 4754 2417 4753 2415 4753 2414 -c C 
4760 2371 M 4763 2352 I 4763 2351 4765 2350 4766 2350 -c 4768 2350 4769 2352 4769 2353 -c 4767 2372 I 4766 2373 4765 2374 4763 2374 -c 4761 2374 4760 2372 4760 2371 -c C 
4765 2328 M 4767 2309 I 4767 2307 4768 2306 4770 2306 -c 4772 2306 4773 2308 4773 2310 -c 4772 2328 I 4772 2330 4770 2331 4768 2331 -c 4767 2331 4765 2329 4765 2328 -c C 
4768 2284 M 4769 2266 I 4769 2264 4770 2262 4772 2262 -c 4774 2263 4775 2264 4775 2266 -c 4774 2284 I 4774 2286 4773 2287 4771 2287 -c 4770 2287 4768 2286 4768 2284 -c C 
4769 2241 M 4768 2222 I 4768 2220 4770 2219 4771 2219 -c 4773 2219 4775 2220 4775 2222 -c 4775 2241 I 4775 2242 4774 2244 4772 2244 -c 4770 2244 4769 2242 4769 2241 -c C 
4767 2197 M 4766 2179 I 4766 2177 4767 2175 4769 2175 -c 4770 2175 4772 2176 4772 2178 -c 4773 2197 I 4773 2199 4772 2200 4770 2200 -c 4769 2200 4767 2199 4767 2197 -c C 
4763 2154 M 4761 2135 I 4761 2134 4762 2132 4763 2132 -c 4765 2132 4767 2133 4767 2135 -c 4769 2153 I 4770 2155 4768 2156 4767 2157 -c 4765 2157 4763 2156 4763 2154 -c C 
4757 2111 M 4754 2093 I 4753 2091 4754 2089 4756 2089 -c 4758 2089 4759 2090 4760 2092 -c 4763 2110 I 4763 2111 4762 2113 4760 2113 -c 4759 2114 4757 2113 4757 2111 -c C 
4748 2068 M 4744 2050 I 4744 2048 4745 2047 4746 2046 -c 4748 2046 4750 2047 4750 2049 -c 4754 2067 I 4755 2069 4754 2070 4752 2071 -c 4750 2071 4749 2070 4748 2068 -c C 
4737 2026 M 4732 2008 I 4732 2006 4733 2005 4734 2004 -c 4736 2004 4738 2005 4738 2006 -c 4743 2024 I 4744 2026 4743 2028 4741 2028 -c 4740 2029 4738 2028 4737 2026 -c C 
4724 1985 M 4719 1969 I 4718 1967 I 4718 1966 4718 1964 4720 1963 -c 4722 1963 4724 1963 4724 1965 -c 4725 1967 I 4730 1983 I 4731 1984 4730 1986 4728 1987 -c 4727 1987 4725 1986 4724 1985 -c C 
4709 1944 M 4704 1932 I 4702 1927 I 4701 1925 4702 1923 4704 1923 -c 4705 1922 4707 1923 4708 1924 -c 4710 1930 I 4715 1942 I 4716 1943 4715 1945 4713 1946 -c 4712 1946 4710 1946 4709 1944 -c C 
4692 1904 M 4688 1896 I 4684 1887 I 4683 1886 4684 1884 4685 1883 -c 4687 1882 4689 1883 4689 1884 -c 4694 1893 I 4698 1901 I 4698 1903 4698 1905 4696 1906 -c 4694 1906 4693 1906 4692 1904 -c C 
4672 1865 M 4670 1860 I 4663 1849 I 4662 1848 4663 1846 4664 1845 -c 4666 1844 4668 1844 4669 1846 -c 4675 1858 I 4678 1862 I 4679 1864 4678 1866 4677 1866 -c 4675 1867 4673 1867 4672 1865 -c C 
4651 1827 M 4650 1826 I 4641 1812 I 4640 1810 4640 1808 4642 1807 -c 4643 1806 4645 1807 4646 1808 -c 4655 1823 I 4656 1824 I 4657 1826 4657 1828 4655 1828 -c 4654 1829 4652 1829 4651 1827 -c C 
4627 1791 M 4616 1776 I 4615 1774 4616 1772 4617 1771 -c 4619 1770 4621 1770 4622 1772 -c 4632 1787 I 4633 1789 4633 1791 4632 1792 -c 4630 1793 4628 1792 4627 1791 -c C 
4602 1755 M 4590 1741 I 4589 1739 4589 1737 4591 1737 -c 4592 1735 4594 1736 4595 1737 -c 4607 1752 I 4608 1753 4608 1755 4606 1756 -c 4605 1757 4603 1757 4602 1755 -c C 
4575 1722 M 4562 1708 I 4561 1706 4561 1704 4563 1703 -c 4564 1702 4566 1702 4567 1703 -c 4579 1718 I 4580 1719 4580 1721 4579 1722 -c 4578 1723 4576 1723 4575 1722 -c C 
4546 1689 M 4533 1676 I 4531 1674 4531 1672 4533 1671 -c 4534 1670 4536 1670 4537 1671 -c 4550 1685 I 4551 1686 4551 1688 4550 1689 -c 4549 1691 4547 1690 4546 1689 -c C 
4515 1658 M 4501 1645 I 4500 1644 4500 1642 4501 1641 -c 4502 1640 4504 1640 4506 1641 -c 4519 1654 I 4520 1655 4520 1657 4519 1658 -c 4518 1659 4516 1659 4515 1658 -c C 
4483 1629 M 4470 1618 I 4468 1617 I 4467 1616 4467 1614 4468 1613 -c 4469 1611 4471 1611 4472 1612 -c 4474 1613 I 4487 1624 I 4488 1625 4488 1627 4487 1629 -c 4486 1630 4484 1630 4483 1629 -c C 
4449 1602 M 4439 1594 I 4434 1590 I 4433 1589 4432 1587 4433 1586 -c 4434 1585 4436 1584 4438 1585 -c 4443 1589 I 4453 1597 I 4454 1598 4454 1600 4453 1601 -c 4452 1602 4450 1603 4449 1602 -c C 
4414 1576 M 4407 1571 I 4398 1566 I 4397 1565 4397 1563 4397 1561 -c 4398 1560 4400 1559 4402 1560 -c 4410 1566 I 4417 1571 I 4419 1572 4419 1574 4418 1575 -c 4417 1577 4415 1577 4414 1576 -c C 
4377 1552 M 4373 1549 I 4361 1543 I 4360 1542 4359 1540 4360 1538 -c 4361 1537 4363 1536 4364 1537 -c 4377 1544 I 4381 1547 I 4382 1548 4383 1550 4382 1551 -c 4381 1553 4379 1553 4377 1552 -c C 
4340 1530 M 4339 1530 I 4323 1522 I 4322 1521 4321 1519 4322 1517 -c 4323 1516 4324 1515 4326 1516 -c 4342 1524 I 4343 1525 I 4344 1526 4345 1527 4344 1529 -c 4343 1530 4341 1531 4340 1530 -c C 
4301 1510 M 4284 1503 I 4282 1502 4282 1500 4282 1499 -c 4283 1497 4285 1496 4286 1497 -c 4304 1505 I 4305 1505 4306 1507 4305 1509 -c 4304 1510 4303 1511 4301 1510 -c C 
4261 1493 M 4244 1486 I 4242 1486 4241 1484 4242 1482 -c 4243 1480 4244 1480 4246 1480 -c 4263 1487 I 4265 1488 4266 1489 4265 1491 -c 4265 1493 4263 1493 4261 1493 -c C 
4220 1477 M 4203 1471 I 4201 1471 4200 1469 4201 1468 -c 4201 1466 4203 1465 4205 1466 -c 4222 1472 I 4224 1472 4225 1474 4224 1476 -c 4224 1477 4222 1478 4220 1477 -c C 
4179 1464 M 4161 1459 I 4159 1459 4158 1457 4159 1455 -c 4159 1453 4161 1453 4163 1453 -c 4181 1458 I 4182 1459 4183 1460 4183 1462 -c 4182 1464 4181 1465 4179 1464 -c C 
4137 1453 M 4118 1449 I 4117 1448 4116 1447 4116 1445 -c 4117 1443 4118 1442 4120 1443 -c 4138 1447 I 4140 1447 4141 1449 4141 1451 -c 4140 1452 4138 1453 4137 1453 -c C 
4094 1444 M 4076 1441 I 4074 1441 4073 1439 4073 1437 -c 4073 1435 4075 1434 4077 1435 -c 4095 1438 I 4097 1438 4098 1440 4098 1442 -c 4097 1443 4096 1444 4094 1444 -c C 
4051 1438 M 4032 1435 I 4031 1435 4029 1433 4030 1432 -c 4030 1430 4031 1429 4033 1429 -c 4052 1431 I 4054 1432 4055 1433 4055 1435 -c 4054 1437 4053 1438 4051 1438 -c C 
4008 1433 M 3989 1432 I 3988 1432 3986 1430 3986 1429 -c 3987 1427 3988 1425 3990 1426 -c 4008 1427 I 4010 1427 4011 1429 4011 1430 -c 4011 1432 4009 1433 4008 1433 -c C 
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; 3932 2315 M <B7>S 
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3271 3273 M (y)S 
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3904 2470 M (y)S 
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3911 2344 M <01>S 
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4060 1700 M (N)S 
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3637 1651 M (yte)[279 56  0]xS 
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3771 1651 M (+)S 
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3644 1529 M <01>S 
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4542 982 M (cq)[143  0]xS 
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4623 1031 M (*)S 
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3970 1084 M <B7>S 
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3838 1202 M (cq)[143  0]xS 
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3919 1251 M (*)S 
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3861 1184 M <01>S 
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5023 1492 M (\(,\))[159 167  0]xS 
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4973 1560 M (t)S 
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5071 1492 M (YR)[173  0]xS 
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4789 1530 M <01>S 
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5110 1356 M <0101>[162  0]xS 
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4979 2700 M (\(,\))[159 167  0]xS 
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5027 2700 M (YR)[174  0]xS 
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4789 2737 M <01>S 
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3058 y FA(Indeed,)g(b)m(y)g(\(A.152\))j(and)d(\(A.153\),)1424
3166 y FG(c)1463 3180 y Ft(\003)1503 3166 y FG(q)p 1301
3207 369 4 v 1301 3225 a Fu(p)p 1392 3225 278 4 v 81
x FG(r)1436 3280 y Fy(2)1495 3306 y Fx(\000)20 b FG(q)1630
3280 y Fy(2)1694 3154 y Fu(\000)1742 3228 y FA(\025)-51
b FG(y)1781 3242 y Fv(N)1868 3228 y FA(+)20 b FG(t)g
FA(+)2103 3143 y Fu(p)p 2194 3143 V 85 x FG(r)2238 3202
y Fy(2)2297 3228 y Fx(\000)g FG(q)2432 3202 y Fy(2)2471
3154 y Fu(\001)2538 3228 y Fx(\025)1316 3467 y(\025)1545
3405 y FG(c)1584 3419 y Ft(\003)1624 3405 y FG(q)p 1422
3446 369 4 v 1422 3464 a Fu(p)p 1513 3464 278 4 v 81
x FG(r)1557 3518 y Fy(2)1616 3545 y Fx(\000)g FG(q)1751
3518 y Fy(2)1816 3393 y Fu(\000)1857 3381 y(p)p 1948
3381 V 86 x FG(r)1992 3440 y Fy(2)2051 3467 y Fx(\000)g
FG(q)2186 3440 y Fy(2)2246 3467 y Fx(\000)k FA(\025)-49
b FG(r)2380 3393 y Fu(\001)2447 3467 y FA(=)1316 3729
y(=)25 b FG(c)1451 3743 y Ft(\003)1491 3729 y FG(q)1550
3628 y Fu(\020)1604 3729 y FA(1)c Fx(\000)1937 3668 y
FA(\025)-49 b FG(r)p 1771 3708 369 4 v 1771 3726 a Fu(p)p
1862 3726 278 4 v 81 x FG(r)1906 3781 y Fy(2)1965 3807
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3729 y Fx(\025)1316 3968 y(\025)1422 3906 y FG(c)1461
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4308 y Ft(\003)1507 4294 y FA(\(and)g(th)m(us)i(~)-47
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y FA(,)30 b(and)g FG(x)1311 4672 y Fv(])1368 4657 y Fx(2)25
b Fi(S)p FA(,)30 b(then)456 4809 y(\(A.155\))1387 4732
y Fu(\014)1387 4786 y(\014)1418 4809 y FG(d)1465 4827
y Fy(\006\()p Fv(Y)t(;R)p Fy(\))1693 4809 y FA(\()p FG(x)p
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y Fx(\025)1967 4732 y Fu(\014)1967 4786 y(\014)1997 4809
y FG(d)2044 4827 y Fy(\006\()p Fv(Y)t(;R)p Fy(\))2273
4809 y FA(\()p FG(x)2360 4824 y Fv(])2392 4809 y FA(\))2427
4732 y Fu(\014)2427 4786 y(\014)2488 4809 y FG(:)456
4959 y FA(By)g(construction,)g(since)f FG(c)1405 4973
y Ft(\003)1470 4959 y FG(<)c FA(1,)30 b(if)f FG(x)1801
4974 y Fv(])1858 4959 y Fx(2)c Fi(S)p FA(,)30 b(then)g
FG(x)2327 4974 y Fv(])2388 4959 y FA(lies)f(inside)e(\006\()p
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5232 y Fy(\006\()p Fv(Y)t(;R)p Fy(\))1892 5214 y FA(\()p
FG(x)1979 5229 y Fv(])2011 5214 y FA(\))41 b FG(<)f FA(0)15
b FG(:)p eop
%%Page: 129 129
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450 y FA(Also,)31 b(from)g(\(A.151\))i(and)e(the)h(fact)g(that)k(\025)
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FA(is)f(also)i(inside)456 558 y(\006\()p FG(Y)5 b(;)15
b(R)q FA(\))30 b(and)g(therefore)456 708 y(\(A.157\))878
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b FG(x)1145 1184 y Fv(])1202 1169 y Fx(2)24 b Fi(S)p
FA(.)555 1319 y(Also,)31 b(since)e FG(y)1056 1286 y Ft(0)1105
1319 y FA(=)c(0)30 b(b)m(y)h(its)e(de\014nition,)g(w)m(e)i(gather)g
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Fi(S)p FA(,)30 b(then)1144 1477 y Fx(j)p FG(x)1221 1492
y Fv(])1273 1477 y Fx(\000)20 b FG(y)s Fx(j)1437 1439
y Fy(2)1502 1477 y FA(=)25 b Fx(j)p FG(y)1668 1491 y
Fv(N)1735 1477 y Fx(j)1760 1439 y Fy(2)1820 1477 y FA(+)20
b Fx(j)p FG(x)1988 1439 y Ft(0)1988 1500 y Fv(])2019
1477 y Fx(j)2044 1439 y Fy(2)2109 1477 y FA(=)25 b FG(r)2249
1439 y Fy(2)2309 1477 y Fx(\000)19 b FG(q)2443 1439 y
Fy(2)2503 1477 y FA(+)h FG(c)2633 1439 y Fy(2)2633 1499
y Ft(\003)2673 1477 y FG(q)2717 1439 y Fy(2)456 1626
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1795 y Fy(\006\()p Fv(Y)t(;R)p Fy(\))1397 1776 y FA(\()p
FG(x)1484 1791 y Fv(])1516 1776 y FA(\))26 b(=)f Fx(j)p
FG(x)1750 1791 y Fv(])1802 1776 y Fx(\000)20 b FG(y)s
Fx(j)g(\000)g FG(r)28 b FA(=)2242 1693 y Fu(p)p 2333
1693 551 4 v 83 x FG(r)2377 1750 y Fy(2)2436 1776 y Fx(\000)20
b FG(q)2571 1750 y Fy(2)2630 1776 y FA(+)g FG(c)2760
1750 y Fy(2)2760 1799 y Ft(\003)2800 1776 y FG(q)2844
1750 y Fy(2)2903 1776 y Fx(\000)g FG(r)e(:)456 1934 y
FA(Th)m(us,)32 b(taking)h FG(x)c Fx(2)g Fi(S)1237 1901
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1934 y Fx(2)c Fi(S)p FA(,)34 b(making)e(use)g(of)h(\(A.158\))j(and)c
(\(A.159\),)k(and)456 2042 y(recalling)28 b(also)j(\(A.35\))r(,)f(w)m
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2202 y Fy(\006\()p Fv(Y)t(;R)p Fy(\))1271 2183 y FA(\()p
FG(x)p FA(\))26 b Fx(\025)20 b(\000)g FG(d)1648 2202
y Fy(\006\()p Fv(Y)t(;R)p Fy(\))1877 2183 y FA(\()p FG(x)1964
2198 y Fv(])1995 2183 y FA(\))26 b(=)1419 2410 y(=)p
FG(r)d Fx(\000)d FG(r)1703 2262 y Fu(r)p 1794 2262 553
4 v 148 x FA(1)h Fx(\000)1961 2349 y FG(q)2005 2323 y
Fy(2)p 1961 2390 84 4 v 1961 2473 a FG(r)2005 2447 y
Fy(2)2074 2410 y FA(+)2175 2349 y FG(c)2214 2323 y Fy(2)2214
2371 y Ft(\003)2254 2349 y FG(q)2298 2323 y Fy(2)p 2175
2390 163 4 v 2215 2473 a FG(r)2259 2447 y Fy(2)2372 2410
y Fx(\025)1419 2650 y(\025)p FG(r)i Fx(\000)d FG(r)1719
2521 y Fu(\024)1767 2650 y FA(1)g Fx(\000)1956 2588 y
FG(q)2000 2555 y Fy(2)p 1933 2629 129 4 v 1933 2712 a
FA(2)p FG(r)2022 2686 y Fy(2)2092 2650 y FA(+)2193 2588
y FG(c)2232 2555 y Fy(2)2232 2611 y Ft(\003)2271 2588
y FG(q)2315 2555 y Fy(2)p 2193 2629 163 4 v 2209 2712
a FA(2)p FG(r)2298 2686 y Fy(2)2385 2650 y FA(+)2476
2549 y Fu(\020)2550 2650 y Fx(\000)2651 2588 y FG(q)2695
2555 y Fy(2)p 2651 2629 84 4 v 2651 2712 a FG(r)2695
2686 y Fy(2)2764 2650 y FA(+)2865 2588 y FG(c)2904 2555
y Fy(2)2904 2611 y Ft(\003)2944 2588 y FG(q)2988 2555
y Fy(2)p 2865 2629 163 4 v 2905 2712 a FG(r)2949 2686
y Fy(2)3037 2549 y Fu(\021)3092 2571 y Fy(2)3131 2521
y Fu(\025)3204 2650 y Fx(\025)1419 2908 y(\025)1503 2847
y FG(q)1547 2814 y Fy(2)p 1500 2888 89 4 v 1500 2971
a FA(2)p FG(r)1599 2808 y Fu(\020)1653 2908 y FA(1)h
Fx(\000)f FG(c)1849 2871 y Fy(2)1849 2931 y Ft(\003)1909
2908 y Fx(\000)35 b FA(const)2261 2847 y FG(q)2305 2814
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y Fy(2)2367 2808 y Fu(\021)2446 2908 y FG(>)1419 3143
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3324 y(pro)m(vided)29 b FG(c)869 3338 y Ft(\003)939 3324
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3483 y(T)-8 b(o)34 b(complete)h(the)e(pro)s(of)g(of)41
b(\(A.148\))s(,)34 b(tak)m(e)i(no)m(w)d Fi(x)e Fx(2)g
FA(\006\()2617 3460 y(\025)2603 3483 y FG(Y)20 b(;)2736
3460 y FA(\025)2716 3483 y FG(R)q FA(\))34 b(outside)f
Fx(fj)p FG(x)3294 3450 y Ft(0)3318 3483 y Fx(j)e(\024)456
3591 y FG(c)495 3605 y Ft(\003)534 3591 y FG(q)s Fx(g)13
b(\002)g(f)p FG(x)817 3605 y Fv(N)911 3591 y FG(>)25
b FA(0)p Fx(g)i FA(and)f(let)h FG(x)e FA(:=)h Fi(x)13
b FA(+)g FG(te)1828 3605 y Fv(N)1922 3591 y FA(Then)26
b FG(x)f Fx(2)g Fi(S)2388 3558 y Ft(\000)2474 3591 y
FA(b)m(y)h(the)h(c)m(hoice)h(in)e(\(A.151\),)456 3699
y(th)m(us,)k(from)g(\(A.150\))j(and)c(\(A.160\),)k(w)m(e)e(gather)g
(that)1252 3909 y Fx(\000)p FG(d)1370 3927 y Fy(\006\()p
Fv(Y)t(;R)p Fy(\))1598 3909 y FA(\()p Fi(x)p FA(\))41
b Fx(\025)f(\000)p FG(d)1970 3927 y Fy(\006\()p Fv(Y)t(;R)p
Fy(\))2198 3909 y FA(\()p FG(x)p FA(\))h FG(>)2499 3847
y(q)2543 3814 y Fy(2)p 2483 3888 V 2483 3971 a FA(4)p
FG(R)2623 3909 y(:)456 4089 y FA(This)28 b(ends)i(the)g(pro)s(of)g(of)
38 b(\(A.148\))r(.)555 4281 y(Let)31 b(us)f(no)m(w)g(pro)m(v)m(e)h
(that)1120 4412 y(the)f FG(s)p FA(-lev)m(el)g(surface)h(of)f
FG(g)2009 4430 y Fj(S)-5 b Fy(\()p Fv(Y)t(;)o(R)p Fy(\))2254
4412 y FA(is)29 b(at)i(distance)1120 4552 y(greater)g(than)g
FG(H)1717 4583 y Fv(y)1752 4594 y Fs(N)5 b Fq(+1)1886
4583 y Fv(;)1920 4567 y Fy(\025)1906 4583 y Fv(R)1963
4552 y FA(\()p FG(s)p FA(\))31 b(from)f(\006\()p FG(Y)5
b(;)15 b(R)q FA(\).)456 4498 y(\(A.161\))456 4715 y(In)36
b(order)h(to)g(pro)m(v)m(e)h(this,)g(tak)m(e)h(an)m(y)k(^)-51
b FG(x)37 b FA(in)f(the)h FG(s)p FA(-lev)m(el)g(surface)g(of)g
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4715 y FA(,)39 b(that)e(is)456 4822 y(assume)30 b(that)1197
4934 y FG(g)1240 4948 y Fv(y)1275 4959 y Fs(N)5 b Fq(+1)1410
4948 y Fv(;R)1487 4934 y FA(\()p FG(H)1598 4948 y Fy(0)1638
4934 y FA(\()p FG(y)1718 4948 y Fv(N)i Fy(+1)1875 4934
y FA(\))20 b(+)g Fx(j)6 b FA(^)-51 b FG(x)21 b Fx(\000)f
FG(y)s Fx(j)g(\000)g FG(R)q FA(\))25 b(=)g FG(s)15 b(:)456
5066 y FA(Let)30 b(also)37 b(\026)-51 b FG(x)25 b Fx(2)g
FA(\006\()p FG(Y)5 b(;)15 b(R)q FA(\),)31 b(that)g(is)1195
5216 y FG(g)1238 5230 y Fv(y)1273 5241 y Fs(N)5 b Fq(+1)1408
5230 y Fv(;R)1486 5216 y FA(\()p FG(H)1597 5230 y Fy(0)1636
5216 y FA(\()p FG(y)1716 5230 y Fv(N)i Fy(+1)1873 5216
y FA(\))21 b(+)f Fx(j)6 b FA(\026)-51 b FG(x)20 b Fx(\000)g
FG(y)s Fx(j)g(\000)g FG(R)q FA(\))26 b(=)f(0)15 b FG(;)p
eop
%%Page: 130 130
130 129 bop 456 251 a Fy(130)140 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(and)29 b(assume)i(that)36 b(\026)-51
b FG(x)31 b FA(lies)e(on)h(the)h(half-line)d(from)i FG(y)j
FA(to)m(w)m(ards)k(^)-51 b FG(x)p FA(.)40 b(Then,)490
596 y FG(d)537 615 y Fy(\006\()p Fv(Y)t(;R)p Fy(\))765
596 y FA(\()6 b(^)-51 b FG(x)p FA(\))84 b(=)e Fx(j)6
b FA(^)-51 b FG(x)21 b Fx(\000)f FG(y)s Fx(j)g(\000)g(j)6
b FA(\026)-51 b FG(x)20 b Fx(\000)g FG(y)s Fx(j)25 b
FA(=)971 735 y(=)82 b(\()p FG(H)1235 749 y Fy(0)1275
735 y FA(\()p FG(y)1355 749 y Fv(N)7 b Fy(+1)1512 735
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Fs(N)5 b Fq(+1)1916 884 y Fv(;R)1994 870 y FA(\(0\))15
b FG(:)456 1034 y FA(Hence,)31 b(b)m(y)f(the)h(fact)g(that)1430
1011 y(\025)1410 1034 y FG(R)26 b Fx(\024)f FG(R)q FA(,)30
b(\(2.14\))j(and)d(\(2.39\),)j(w)m(e)d(get)i(that)1079
1181 y FG(d)1126 1199 y Fy(\006\()p Fv(Y)t(;R)p Fy(\))1355
1181 y FA(\()6 b(^)-51 b FG(x)p FA(\))83 b Fx(\025)g
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1212 y Fv(;)1994 1195 y Fy(\025)1979 1212 y Fv(R)2036
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1181 y FA(\(0\))27 b(=)1560 1407 y(=)83 b FG(H)1790 1439
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1422 y Fy(\025)1979 1439 y Fv(R)2036 1407 y FA(\()p FG(s)p
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1360 y Fy(0)p 2271 1386 116 4 v 2271 1470 a FA(2)p FG(R)2411
1407 y(y)2459 1370 y Fy(2)2456 1430 y Fv(N)7 b Fy(+1)2638
1407 y Fx(\025)1560 1573 y(\025)83 b FG(H)1790 1605 y
Fv(y)1825 1616 y Fs(N)5 b Fq(+1)1959 1605 y Fv(;)1994
1588 y Fy(\025)1979 1605 y Fv(R)2036 1573 y FA(\()p FG(s)p
FA(\))15 b FG(;)456 1738 y FA(and)29 b(this)h(ends)f(the)i(pro)s(of)e
(of)i(\(A.161\).)555 1884 y(W)-8 b(e)32 b(no)m(w)e(sho)m(w)g(that)1019
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FA(2,)32 b(the)f FG(s)p FA(-lev)m(el)f(surface)g(of)g
FG(g)2398 2030 y Fj(S)-5 b Fy(\()2480 2013 y(\025)2469
2030 y Fv(Y)10 b(;)2554 2013 y Fy(\025)2540 2030 y Fv(R)p
Fy(\))2655 1998 y FA(is)29 b(at)i(distance)1019 2168
y(less)f(than)g FG(H)1475 2200 y Fv(y)1510 2211 y Fs(N)5
b Fq(+1)1644 2200 y Fv(;)1679 2183 y Fy(\025)1664 2200
y Fv(R)1722 2168 y FA(\()p FG(s)p FA(\))20 b(+)35 b(const)p
2184 2095 72 4 v 16 w FG(C)2256 2182 y Fy(0)2295 2168
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2715 y FG(d)531 2747 y Fy(\006\()620 2730 y(\025)609
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2885 y(=)83 b(\()p FG(H)1241 2899 y Fy(0)1281 2885 y
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b(\000)3210 2862 y FA(\025)3190 2885 y FG(R)q FA(\))25
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Fv(y)1241 3062 y Fs(N)5 b Fq(+1)1375 3051 y Fv(;)1410
3034 y Fy(\025)1395 3051 y Fv(R)1453 3020 y FA(\()p FG(s)p
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y Fv(R)1453 3246 y FA(\()p FG(s)p FA(\))20 b(+)p 1689
3112 V 1689 3185 a FG(C)1761 3199 y Fy(0)p 1687 3226
116 4 v 1687 3323 a FA(2)1752 3300 y(\025)1732 3323 y
FG(R)1834 3246 y FA(\025)-52 b FG(y)1875 3209 y Fy(2)1872
3269 y Fv(N)7 b Fy(+1)2055 3246 y Fx(\024)976 3500 y(\024)83
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b Fq(+1)1375 3531 y Fv(;)1410 3514 y Fy(\025)1395 3531
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1925 3479 116 4 v 1925 3562 a FA(2)p FG(R)2065 3500 y(:)456
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3820 y Fv(;)2219 3803 y Fy(\025)2205 3820 y Fv(R)2262
3788 y FA(\()p FG(s)p FA(\))18 b(+)p 2481 3715 72 4 v
17 w FG(C)2552 3802 y Fy(0)2592 3788 y FG(=R)q FA(,)29
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4060 y(W)-8 b(e)32 b(no)m(w)e(deduce)g(from)g(the)h(ab)s(o)m(v)m(e)g
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1178 4310 y Fx(j)p FG(g)1246 4341 y Fj(S)-6 b Fy(\()1327
4325 y(\025)1316 4341 y Fv(Y)10 b(;)1402 4325 y Fy(\025)1387
4341 y Fv(R)p Fy(\))1472 4310 y FA(\()p FG(x)p FA(\))p
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y FG(d)1225 4505 y Fy(\006\()1314 4488 y(\025)1303 4505
y Fv(Y)16 b(;)1395 4488 y Fy(\025)1380 4505 y Fv(R)p
Fy(\))1465 4474 y FA(\()p FG(x)p FA(\))21 b Fx(\000)f
FG(d)1746 4492 y Fy(\006\()p Fv(Y)t(;R)p Fy(\))1974 4474
y FA(\()p FG(x)p FA(\))26 b Fx(\025)40 b FA(const)p 2456
4401 V 16 w FG(C)2527 4488 y Fy(0)2567 4474 y FG(=R)16
b(;)1178 4620 y FA(w)m(e)31 b(ha)m(v)m(e)g(that)1178
4760 y FG(g)1221 4792 y Fj(S)-6 b Fy(\()1302 4775 y(\025)1291
4792 y Fv(Y)10 b(;)1377 4775 y Fy(\025)1362 4792 y Fv(R)p
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4779 y Fj(S)-6 b Fy(\()p Fv(Y)t(;R)p Fy(\))1978 4760
y FA(\()p FG(x)p FA(\))15 b FG(:)456 4481 y FA(\(A.163\))456
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b FG(:)456 5144 y FA(\(A.164\))p eop
%%Page: 131 131
131 130 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)
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450 y FA(Then,)29 b(b)m(y)h(\(A.161\))j(and)d(\(A.162\),)1120
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652 y Fv(Y)16 b(;)1337 635 y Fy(\025)1322 652 y Fv(R)p
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Fx(\\)g(fj)p FG(x)1705 1650 y Fv(N)7 b Fy(+1)1863 1636
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FA(\))-5 b(,)456 1705 y(\(A.166\))456 1948 y(pro)m(vided)29
b FG(q)k FA(is)c(large)i(enough.)40 b(Hence,)32 b(since)d
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Fx(\\)g(fj)p FG(x)1705 2151 y Fv(N)7 b Fy(+1)1863 2137
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2203 y(\(A.167\))555 2439 y(W)-8 b(e)36 b(no)m(w)e(apply)f(the)h
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3884 y(\014)2029 3879 y Fx(\024)40 b FA(const)16 b FG(;)456
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FA(is)c(large.)76 b(F)-8 b(rom)43 b(this)e(circumstance,)456
4182 y(recalling)28 b(the)j(touc)m(hing)f(prop)s(erties)f(of)36
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1020 4494 a Fx(jr)p FG(u)p FA(\()6 b(\025)-51 b FG(x)p
FA(\))p Fx(j)1346 4432 y FG(;)41 b(e)1454 4446 y Fv(N)1521
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y(\025)2452 4380 y Fv(Y)10 b(;)2538 4363 y Fy(\025)2523
4380 y Fv(R)p Fy(\))2608 4348 y FA(\()c(\025)-51 b FG(x)p
FA(\))p 2238 4411 518 4 v 2238 4494 a Fx(jr)p FG(g)2382
4526 y Fj(S)-6 b Fy(\()2463 4509 y(\025)2452 4526 y Fv(Y)10
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4494 y FA(\()c(\025)-51 b FG(x)p FA(\))p Fx(j)2781 4432
y FG(;)41 b(e)2889 4446 y Fv(N)2956 4331 y Fu(\021)3026
4276 y(!)3123 4432 y FA(=)1740 4712 y(=)83 b(sin)2021
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4712 y FG(;)41 b(e)2627 4726 y Fv(N)2695 4611 y Fu(\021)2764
4584 y(\023)2856 4712 y FA(=)1740 4965 y(=)1904 4904
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4965 y(\024)1740 5186 y(\024)98 b FA(const)2142 5125
y FG(q)p 2142 5166 44 4 v 2142 5249 a(r)2211 5186 y(:)p
eop
%%Page: 132 132
132 131 bop 456 251 a Fy(132)140 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
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1959 y Fu(\020)1347 2060 y FG(\031)1399 2074 y Fv(N)1466
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2511 y Fv(Y)15 b(;)1354 2494 y Fy(\025)1339 2511 y Fv(R)1403
2479 y FA(\025)-51 b FG(x)20 b Fx(\000)26 b FA(\025)-51
b FG(x)20 b FA(+)1758 2418 y Fx(r)p FG(u)p FA(\()6 b(\025)-51
b FG(x)p FA(\))p 1733 2458 301 4 v 1733 2542 a Fx(jr)p
FG(u)p FA(\()6 b(\025)-51 b FG(x)p FA(\))p Fx(j)2059
2479 y FG(H)2135 2493 y Fy(0)2174 2479 y FA(\()p FG(u)p
FA(\()6 b(\025)-51 b FG(x)q FA(\)\))2419 2375 y Fu(\014)2419
2429 y(\014)2419 2484 y(\014)2475 2479 y Fx(\024)2581
2418 y FA(const)p 2581 2458 208 4 v 2650 2542 a FG(R)3021
2479 y FA(and)1033 2625 y Fu(\014)1033 2679 y(\014)1033
2734 y(\014)1064 2729 y FG(T)1117 2743 y Fv(Y)t(;R)1239
2729 y FG(x)1291 2743 y Fy(0)1351 2729 y Fx(\000)20 b
FG(x)1494 2743 y Fy(0)1553 2729 y FA(+)1679 2668 y Fx(r)p
FG(u)p FA(\()p FG(x)1894 2682 y Fy(0)1934 2668 y FA(\))p
1654 2708 341 4 v 1654 2792 a Fx(jr)p FG(u)p FA(\()p
FG(x)1894 2806 y Fy(0)1934 2792 y FA(\))p Fx(j)2020 2729
y FG(H)2096 2743 y Fy(0)2135 2729 y FA(\()p FG(u)p FA(\()p
FG(x)2309 2743 y Fy(0)2349 2729 y FA(\)\))2419 2625 y
Fu(\014)2419 2679 y(\014)2419 2734 y(\014)2475 2729 y
Fx(\024)2581 2668 y FA(const)p 2581 2708 208 4 v 2650
2792 a FG(R)2814 2729 y(:)456 2605 y FA(\(A.175\))555
2953 y(Hence,)28 b(from)e(\(A.175\),)j(\(A.170\))f(and)e(the)g(fact)h
(that)f FG(T)2481 2967 y Fv(Y)t(;R)2603 2953 y FG(x)2655
2967 y Fy(0)2706 2953 y Fx(\001)11 b FG(e)2784 2967 y
Fv(N)2877 2953 y FA(=)25 b(0)i(\(recall)e(the)456 3060
y(assumptions)j(in)h(Lemma)i(A.1\),)h(w)m(e)e(ha)m(v)m(e)i(that)670
3258 y(\()6 b(\025)-51 b FG(x)21 b Fx(\000)f FG(x)921
3272 y Fy(0)960 3258 y FA(\))h Fx(\001)f FG(e)1103 3272
y Fv(N)1196 3258 y Fx(\024)1292 3157 y Fu(\020)1346 3258
y FG(T)1410 3273 y Fy(\025)1399 3289 y Fv(Y)c(;)1491
3273 y Fy(\025)1476 3289 y Fv(R)1539 3258 y FA(\025)-51
b FG(x)21 b Fx(\000)e FG(T)1749 3272 y Fv(Y)t(;R)1872
3258 y FG(x)1924 3272 y Fy(0)1963 3258 y FA(+)1191 3482
y(+)1317 3420 y Fx(r)p FG(u)p FA(\()6 b(\025)-51 b FG(x)p
FA(\))p 1292 3461 301 4 v 1292 3544 a Fx(jr)p FG(u)p
FA(\()6 b(\025)-51 b FG(x)p FA(\))p Fx(j)1618 3482 y
FG(H)1694 3496 y Fy(0)1733 3482 y FA(\()p FG(u)p FA(\()6
b(\025)-51 b FG(x)q FA(\)\))21 b Fx(\000)2125 3420 y(r)p
FG(u)p FA(\()p FG(x)2340 3434 y Fy(0)2379 3420 y FA(\))p
2100 3461 341 4 v 2100 3544 a Fx(jr)p FG(u)p FA(\()p
FG(x)2340 3558 y Fy(0)2379 3544 y FA(\))p Fx(j)2465 3482
y FG(H)2541 3496 y Fy(0)2580 3482 y FA(\()p FG(u)p FA(\()p
FG(x)2754 3496 y Fy(0)2794 3482 y FA(\)\))2864 3381 y
Fu(\021)2939 3482 y Fx(\001)g FG(e)3027 3496 y Fv(N)3094
3482 y FA(+)1191 3720 y(+)1292 3658 y(const)p 1292 3699
208 4 v 1361 3782 a FG(R)1535 3720 y Fx(\024)1196 3947
y(\024)1292 3846 y Fu(\020)1382 3885 y Fx(r)p FG(u)p
FA(\()6 b(\025)-51 b FG(x)p FA(\))p 1356 3926 301 4 v
1356 4009 a Fx(jr)p FG(u)p FA(\()6 b(\025)-51 b FG(x)q
FA(\))p Fx(j)1682 3947 y FG(H)1758 3961 y Fy(0)1797 3947
y FA(\()p FG(u)p FA(\()6 b(\025)-51 b FG(x)q FA(\)\))21
b Fx(\000)2189 3885 y(r)p FG(u)p FA(\()p FG(x)2404 3899
y Fy(0)2444 3885 y FA(\))p 2164 3926 341 4 v 2164 4009
a Fx(jr)p FG(u)p FA(\()p FG(x)2404 4023 y Fy(0)2444 4009
y FA(\))p Fx(j)2529 3947 y FG(H)2605 3961 y Fy(0)2644
3947 y FA(\()p FG(u)p FA(\()p FG(x)2818 3961 y Fy(0)2859
3947 y FA(\)\))2929 3846 y Fu(\021)3004 3947 y Fx(\001)f
FG(e)3091 3961 y Fv(N)3159 3947 y FA(+)1191 4206 y(+)1292
4143 y(4)15 b FG(C)1417 4158 y Fv(])1464 4143 y FG(q)1508
4110 y Fy(2)p 1292 4185 256 4 v 1385 4269 a FG(R)1578
4206 y FA(+)1679 4145 y(const)p 1679 4185 208 4 v 1748
4269 a FG(R)1911 4206 y(;)456 4407 y FA(and)29 b(so,)i(thanks)f(to)h
(\(A.168\),)i(\(A.171\))g(and)d(\(A.172\),)975 4568 y(\()6
b(\025)-51 b FG(x)20 b Fx(\000)g FG(x)1225 4582 y Fy(0)1265
4568 y FA(\))g Fx(\001)h FG(e)1408 4582 y Fv(N)1500 4568
y Fx(\024)1000 4778 y(\024)k FG(H)1172 4792 y Fy(0)1211
4778 y FA(\()p FG(u)p FA(\()6 b(\025)-51 b FG(x)q FA(\)\))21
b Fx(\000)f FG(H)1644 4792 y Fy(0)1683 4778 y FA(\()p
FG(u)p FA(\()p FG(x)1857 4792 y Fy(0)1897 4778 y FA(\)\))h(+)2088
4717 y(const)16 b FG(q)2355 4684 y Fy(2)p 2088 4758 307
4 v 2187 4841 a FG(R)2257 4815 y Fy(2)2425 4778 y FA(+)2526
4715 y(4)f FG(C)2651 4730 y Fv(])2698 4715 y FG(q)2742
4682 y Fy(2)p 2526 4758 256 4 v 2618 4841 a FG(R)2811
4778 y FA(+)2912 4717 y(const)p 2912 4758 208 4 v 2981
4841 a FG(R)3155 4778 y Fx(\024)1000 5015 y(\024)25 b
FG(H)1172 5029 y Fy(0)1211 5015 y FA(\()p FG(u)p FA(\()6
b(\025)-51 b FG(x)q FA(\)\))21 b Fx(\000)f FG(H)1644
5029 y Fy(0)1683 5015 y FA(\()p FG(u)p FA(\()p FG(x)1857
5029 y Fy(0)1897 5015 y FA(\)\))h(+)2088 4952 y(5)15
b FG(C)2213 4967 y Fv(])2261 4952 y FG(q)2305 4919 y
Fy(2)p 2088 4994 256 4 v 2181 5077 a FG(R)2369 5015 y(;)456
4812 y FA(\(A.176\))456 5216 y(b)m(y)30 b(assuming)f
FG(C)1042 5231 y Fv(])1073 5216 y FA(,)i FG(q)i FA(and)d
FG(R)q(=q)j FA(suitably)c(large.)p eop
%%Page: 133 133
133 132 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)
f Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)140 b(133)555
450 y FA(Moreo)m(v)m(er,)34 b(in)29 b(the)j(ligh)m(t)e(of)h(\(A.26\),)i
(\(A.169\))h(and)c(\(A.176\))s(,)h(and)f(recalling)g(that)456
558 y Fx(j)p FG(u)p FA(\()6 b(\025)-51 b FG(x)p FA(\))p
Fx(j)26 b FG(<)f FA(1)p FG(=)p FA(2)32 b(and)d Fx(j)p
FG(u)p FA(\()p FG(x)1309 572 y Fy(0)1350 558 y FA(\))p
Fx(j)c FG(<)g FA(1)p FG(=)p FA(2,)32 b(w)m(e)f(ha)m(v)m(e)456
717 y(\(A.177\))48 b Fx(j)6 b FA(\025)-51 b FG(x)11 b
Fx(\000)g FG(x)1024 731 y Fy(0)1063 717 y Fx(j)25 b(\024)g(j)p
FA(\()6 b(\025)-51 b FG(x)12 b Fx(\000)f FG(x)1467 731
y Fy(0)1506 717 y FA(\))1541 680 y Ft(0)1565 717 y Fx(j)g
FA(+)g Fx(j)p FA(\()6 b(\025)-51 b FG(x)11 b Fx(\000)g
FG(x)1940 731 y Fy(0)1979 717 y FA(\))2014 731 y Fv(N)2082
717 y Fx(j)25 b(\024)g FA(\(1)11 b(+)g(2)p FG(c)2485
731 y Ft(\003)2526 717 y FA(\))p FG(q)j FA(+)d Fx(j)p
FA(\()6 b(\025)-51 b FG(x)11 b Fx(\000)g FG(x)2955 731
y Fy(0)2994 717 y FA(\))3029 731 y Fv(N)3097 717 y Fx(j)25
b(\024)3264 694 y FA(\025)3243 717 y FG(C)7 b(q)18 b(;)456
876 y FA(where)737 853 y(\025)716 876 y FG(C)35 b FA(is)27
b(a)h(con)m(v)m(enien)m(tly)g(large)g(constan)m(t)i(as)e(in)f(the)h
(statemen)m(t)i(of)e(the)g(Lemma.)456 984 y(By)i(means)h(of)f
(\(A.176\))j(and)d(\(A.172\),)j(w)m(e)e(conclude)e(that)456
1196 y(\(A.178\))231 b(\()6 b(\025)-51 b FG(x)20 b Fx(\000)g
FG(x)1235 1210 y Fy(0)1274 1196 y FA(\))36 b Fx(\001)1441
1134 y(r)p FG(u)p FA(\()p FG(x)1656 1148 y Fy(0)1695
1134 y FA(\))p 1416 1175 341 4 v 1416 1258 a Fx(jr)p
FG(u)p FA(\()p FG(x)1656 1272 y Fy(0)1695 1258 y FA(\))p
Fx(j)1806 1196 y(\024)41 b FG(H)1994 1210 y Fy(0)2033
1196 y FA(\()p FG(u)p FA(\()6 b(\025)-51 b FG(x)p FA(\)\))21
b Fx(\000)f FG(H)2465 1210 y Fy(0)2504 1196 y FA(\()p
FG(u)p FA(\()p FG(x)2678 1210 y Fy(0)2718 1196 y FA(\)\))h(+)2910
1133 y(6)15 b FG(C)3035 1148 y Fv(])3082 1133 y FG(q)3126
1100 y Fy(2)p 2910 1175 256 4 v 3003 1258 a FG(R)3191
1196 y(:)456 1394 y FA(Let)33 b(us)f(no)m(w)h(consider)f(the)h(set)g
(\004)f(de\014ned)g(in)g(the)h(statemen)m(t)h(of)f(Lemma)g(A.1)h(and)
456 1502 y(tak)m(e)24 b FG(c)684 1516 y Ft(\003)747 1502
y FA(con)m(v)m(enien)m(tly)g(small:)35 b(then,)25 b(the)e(reader)g(ma)m
(y)h(easily)e(con)m(vince)i(herself)d(that,)456 1610
y(thanks)30 b(to)h(\(A.169\),)i(\(A.168\),)g(\(A.177\),)g(\(A.173\))g
(and)c(\(A.178\),)k(w)m(e)e(ha)m(v)m(e)h(that)1793 1769
y(\004)40 b Fx(\023)2013 1746 y FA(\025)2006 1769 y(\004)14
b(;)456 1919 y(the)30 b(pro)s(of)g(of)g(\(A.22\){\(A.23\))35
b(is)30 b(th)m(us)g(ended)f(b)m(y)h(means)h(of)f(\(A.174\).)555
2069 y(T)-8 b(o)38 b(\014nish)d(the)j(pro)s(of)f(of)h(Lemma)g(A.1,)i(w)
m(e)e(need)f(no)m(w)h(to)g(c)m(hec)m(k)i(the)d Fo(Lipschitz)456
2181 y(gr)-5 b(aph)41 b FA(prop)s(ert)m(y)e(of)h(\004)1265
2195 y Fv(s)1342 2181 y FA(:=)1487 2158 y(\025)1479 2181
y(\004)26 b Fx(\\)g(f)p FG(x)1750 2195 y Fv(N)7 b Fy(+1)1949
2181 y FA(=)40 b FG(s)p Fx(g)p FA(,)j(for)c(an)m(y)h
Fx(j)p FG(s)p Fx(j)h FG(<)g FA(1)p FG(=)p FA(2.)70 b(The)39
b(graph)456 2289 y(prop)s(ert)m(y)d(is)h(a)g(straigh)m(tforw)m(ard)h
(consequence)g(of)f(the)h Fo(\014rst)i(o)-5 b(c)g(curr)g(enc)g(e)38
b FA(touc)m(hing)456 2397 y(p)s(oin)m(t)h(prop)s(ert)m(y)-8
b(,)44 b(hence)d(w)m(e)g(fo)s(cus)f(on)h(the)g(Lipsc)m(hitz)e
(estimate.)73 b(F)-8 b(or)41 b(this,)i(tak)m(e)482 2486
y(\025)456 2509 y FG(X)7 b(;)630 2486 y FA(\026)604 2509
y FG(X)32 b Fx(2)25 b FA(\004)858 2523 y Fv(s)895 2509
y FA(;)30 b(then,)1100 2659 y FG(s)25 b FA(=)f FG(u)p
FA(\()6 b(\026)-51 b FG(x)q FA(\))25 b(=)g FG(g)1602
2690 y Fj(S)-5 b Fy(\()1684 2674 y(\026)1673 2690 y Fv(Y)10
b(;)1758 2674 y Fy(\025)1744 2690 y Fv(R)p Fy(\))1828
2659 y FA(\()c(\026)-51 b FG(x)q FA(\))25 b(=)g FG(g)2115
2690 y Fj(S)-5 b Fy(\()2197 2674 y(\025)2186 2690 y Fv(Y)10
b(;)2271 2674 y Fy(\025)2257 2690 y Fv(R)p Fy(\))2341
2659 y FA(\()c(\025)-51 b FG(x)q FA(\))25 b(=)g FG(u)p
FA(\()6 b(\025)-51 b FG(x)q FA(\))15 b FG(;)456 2843
y FA(for)30 b(suitable)947 2820 y(\025)934 2843 y FG(Y)20
b(;)1061 2820 y FA(\026)1047 2843 y FG(Y)45 b Fx(2)25
b Fw(R)1291 2810 y Fv(N)7 b Fy(+1)1454 2843 y FA(,)31
b(so)f(that)h FG(\031)1870 2857 y Fv(N)1951 2820 y FA(\025)1937
2843 y FG(Y)20 b(;)31 b(\031)2118 2857 y Fv(N)2199 2820
y FA(\026)2185 2843 y FG(Y)45 b Fx(2)25 b FA(\004)2430
2857 y Ft(\003)2469 2843 y FA(.)41 b(W)-8 b(e)31 b(will)d(pro)m(v)m(e)j
(that)456 3029 y(\(A.179\))592 b Fx(j)6 b FA(\025)-51
b FG(x)1423 3043 y Fv(N)1511 3029 y Fx(\000)26 b FA(\026)-51
b FG(x)1654 3043 y Fv(N)1721 3029 y Fx(j)25 b(\024)1877
2967 y FA(const)16 b FG(q)p 1877 3008 267 4 v 1976 3091
a(R)2154 3029 y Fx(j)p FA(\()6 b(\025)-51 b FG(x)21 b
Fx(\000)k FA(\026)-50 b FG(x)p FA(\))2465 2991 y Ft(0)2488
3029 y Fx(j)15 b FG(;)456 3209 y FA(whic)m(h)27 b(implies)e(the)k
(desired)e(Lipsc)m(hitz)g(estimate)i(\(with)f(constan)m(t)h(const)16
b FG(q)s(=R)26 b(<)f FA(1\).)456 3317 y(With)k(no)i(loss)e(of)i
(generalit)m(y)-8 b(,)31 b(w)m(e)g(ma)m(y)g(and)f(do)g(assume)g(that)
456 3467 y(\(A.180\))1002 b(\026)-51 b FG(x)1802 3481
y Fv(N)1894 3467 y FG(>)31 b FA(\025)-51 b FG(x)2042
3481 y Fv(N)2125 3467 y FG(:)456 3617 y FA(Let)30 b(us)g(de\014ne)1223
3767 y(\026)-49 b FG(r)86 b FA(:=)1545 3744 y(\025)1525
3767 y FG(R)21 b FA(+)f FG(H)1787 3798 y Fy(\026)-40
b Fv(y)1817 3809 y Fs(N)5 b Fq(+1)1951 3798 y Fv(;)1985
3782 y Fy(\025)1971 3798 y Fv(R)2028 3767 y FA(\()p FG(s)p
FA(\))21 b Fx(\000)f FG(H)2329 3781 y Fy(0)2368 3767
y FA(\()6 b(\026)-51 b FG(y)2448 3781 y Fv(N)7 b Fy(+1)2605
3767 y FA(\))15 b FG(;)456 3930 y FA(so)30 b(that)1410
4080 y Fx(f)p FG(g)1498 4111 y Fj(S)-5 b Fy(\()1580 4094
y(\026)1569 4111 y Fv(Y)10 b(;)1655 4094 y Fy(\025)1640
4111 y Fv(R)p Fy(\))1750 4080 y FA(=)25 b FG(s)p Fx(g)83
b FA(=)g FG(@)5 b(B)2296 4094 y Fy(\026)-38 b Fv(r)2331
4080 y FA(\()6 b(\026)-51 b FG(y)s FA(\))15 b FG(:)456
4246 y FA(By)30 b(\(2.57\))j(and)d(\(A.171\),)j(w)m(e)e(ha)m(v)m(e)g
(that)456 4396 y(\(A.181\))813 b Fx(j)6 b FA(\026)-51
b FG(x)1644 4358 y Ft(0)1687 4396 y Fx(\000)27 b FA(\026)-52
b FG(y)1826 4358 y Ft(0)1849 4396 y Fx(j)41 b(\024)f
FA(const)16 b FG(q)i(;)456 4546 y FA(and)29 b(therefore,)i(from)f
(\(A.181\),)j(w)m(e)e(get)h(that)1094 4693 y Fx(j)6 b
FA(\025)-51 b FG(x)1171 4656 y Ft(0)1215 4693 y Fx(\000)26
b FA(\026)-51 b FG(y)1354 4656 y Ft(0)1377 4693 y Fx(j)25
b(\024)g(j)6 b FA(\025)-51 b FG(x)1600 4656 y Ft(0)1644
4693 y Fx(\000)20 b FG(x)1787 4656 y Ft(0)1787 4716 y
Fy(0)1826 4693 y Fx(j)h FA(+)f Fx(j)p FG(x)2040 4656
y Ft(0)2040 4716 y Fy(0)2099 4693 y Fx(\000)26 b FA(\026)-51
b FG(x)2242 4656 y Ft(0)2266 4693 y Fx(j)20 b FA(+)g
Fx(j)6 b FA(\026)-51 b FG(x)2479 4656 y Ft(0)2523 4693
y Fx(\000)26 b FA(\026)-51 b FG(y)2662 4656 y Ft(0)2685
4693 y Fx(j)25 b(\024)1427 4828 y(\024)40 b FA(const)16
b FG(q)i(;)456 4765 y FA(\(A.182\))456 4983 y(whic)m(h)33
b(is)g(less)h(then)k(\026)-49 b FG(r)r FA(,)36 b(if)d
FG(q)s(=R)i FA(is)e(small)g(enough.)52 b(Th)m(us,)35
b(b)m(y)f(the)h(\014rst)e(o)s(ccurrence)456 5091 y(touc)m(hing)24
b(prop)s(ert)m(y)g(of)h FG(g)1328 5122 y Fj(S)-5 b Fy(\()1410
5106 y(\026)1399 5122 y Fv(Y)10 b(;)1484 5106 y Fy(\025)1470
5122 y Fv(R)p Fy(\))1555 5091 y FA(,)31 b(\025)-50 b
FG(x)24 b FA(m)m(ust)h(b)s(e)f(ab)s(o)m(v)m(e)i Fx(f)p
FG(g)2360 5122 y Fj(S)-5 b Fy(\()2442 5106 y(\026)2431
5122 y Fv(Y)10 b(;)2517 5106 y Fy(\025)2502 5122 y Fv(R)p
Fy(\))2612 5091 y FA(=)25 b FG(s)p Fx(g)g FA(\(with)f(resp)s(ect)h(to)
456 5216 y(the)j FG(e)652 5230 y Fv(N)748 5216 y FA(direction\).)39
b(This,)27 b(\(A.182\))k(and)c(\(A.180\))k(imply)26 b(that)35
b(\025)-51 b FG(x)28 b FA(is)g(trapp)s(ed)f(inside)p
eop
%%Page: 134 134
134 133 bop 456 251 a Fy(134)140 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(a)22 b(cone)h(with)e(v)m(ertex)i(in)k(\026)-51
b FG(x)22 b FA(and)g(slop)s(e)f(b)s(ounded)f(b)m(y)38
b(const)15 b FG(q)s(=)t FA(\026)-49 b FG(r)s FA(,)24
b(whic)m(h)d(giv)m(es)i(\(A.179\).)456 558 y(This)28
b(ends)i(the)g(pro)s(of)g(of)g(Lemma)h(A.1.)1518 b Fl(\003)750
2620 y @beginspecial 0 @llx 0 @lly 367 @urx 274 @ury
2880 @rwi @setspecial
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4166 y Fx(\000)11 b FG(o)p FA(\))g Fx(\001)g FG(\030)t
Fx(j)27 b FG(<)e(l)r(=)p FA(4)p Fo(,)30 b Fx(j)p FA(\()p
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FA(\))g Fx(\000)g FA(\(\()p FG(x)2938 4180 y Fy(0)2990
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b Fw(R)42 b Fo(satisfying)456 558 y(the)33 b(fol)5 b(lowing)33
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FA(\()p Fi(x)p FA(\))p Fx(j)27 b FG(<)e FA(1)p FG(=)p
FA(2)p Fo(,)34 b Fx(j)p Fi(x)21 b Fx(\000)f FG(x)2283
720 y Fy(0)2322 706 y Fx(j)25 b FG(<)2489 683 y FA(\025)2468
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b(e)25 b(exists)1230 800 y FA(^)1216 823 y FG(Y)45 b
Fx(2)25 b Fw(R)1460 790 y Fv(N)7 b Fy(+1)1647 823 y Fo(such)24
b(that)h Fw(S)o FA(\()2128 800 y(^)2114 823 y FG(Y)14
b(;)h(R)q(=C)2401 837 y Fy(2)2441 823 y FA(\))24 b Fo(is)g(ab)-5
b(ove)24 b FG(u)g Fo(and)h(it)f(touches)758 931 y FG(u)33
b Fo(at)g FA(\()p Fi(x)p FG(;)e(u)p FA(\()p Fi(x)p FA(\)\))p
Fo(;)671 1094 y Fx(\017)42 b Fl(\\)839 966 y Fu(\022)941
1033 y Fx(r)p FG(u)p FA(\()p Fi(x)p FA(\))p 916 1073
282 4 v 916 1157 a Fx(jr)p FG(u)p FA(\()p Fi(x)p FA(\))p
Fx(j)1222 1094 y FG(;)1298 1033 y Fx(r)p FG(u)p FA(\()p
FG(x)1513 1047 y Fy(0)1553 1033 y FA(\))p 1273 1073 341
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1094 y Fx(\024)1821 1033 y FG(C)1886 1047 y Fy(1)1941
1033 y FG(q)p 1821 1073 164 4 v 1868 1157 a(R)1995 1094
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FG(x)989 1340 y Fy(0)1028 1326 y FA(\))h Fx(\001)1165
1265 y(r)p FG(u)p FA(\()p FG(x)1380 1279 y Fy(0)1419
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1265 y FG(q)1815 1232 y Fy(2)p 1651 1305 203 4 v 1718
1388 a FG(R)1884 1326 y FA(+)f FG(H)2051 1340 y Fy(0)2090
1326 y FA(\()p FG(u)p FA(\()p Fi(x)p FA(\)\))i Fx(\000)e
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1519 y(Then,)1508 1684 y Fi(L)1563 1646 y Fv(N)1630 1583
y Fu(\020)1685 1684 y FG(\031)1737 1699 y Fv(\030)1774
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b FG(cq)2194 1646 y Fv(N)7 b Ft(\000)p Fy(1)2367 1684
y FG(:)456 1873 y Fo(Mor)-5 b(e)27 b(pr)-5 b(e)g(cisely,)29
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FG(=)p FA(2)p FG(;)32 b FA(1)p FG(=)p FA(2\))p Fo(,)d(ther)-5
b(e)28 b(exists)g(a)f(set)g FA(\004)2798 1887 y Fv(s)2860
1873 y Fx(\022)e FA(\004)8 b Fx(\\)g(f)p FG(x)3191 1887
y Fv(N)f Fy(+1)3374 1873 y FA(=)456 1981 y FG(s)p Fx(g)p
Fo(,)30 b(which)g(is)g(c)-5 b(ontaine)g(d)32 b(in)d(a)i(Lipschitz)f(gr)
-5 b(aph)32 b(in)d(the)h FG(\030)t Fo(-dir)-5 b(e)g(ction,)32
b(with)e(Lipschitz)456 2089 y(c)-5 b(onstant)34 b(less)f(than)h
FA(1)p Fo(,)f(and)g(so)g(that,)h(if)1561 2259 y FA(\025)1554
2282 y(\004)25 b(:=)1927 2196 y Fu([)1761 2397 y Fv(s)p
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(the)h(assumptions)d(of)j(Lemma)f(4.2)i(and)d(\(2.57\),)995
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y Fx(\000)20 b FG(T)2079 5200 y Fv(Y)2120 5209 y Fq(0)2155
5200 y Fv(;R)2232 5186 y FG(x)2284 5200 y Fy(0)2324 5186
y Fx(j)30 b FA(sin)14 b FG(#)25 b FA(=)g FG(r)e Fx(\001)2800
5125 y FG(q)p 2800 5166 44 4 v 2800 5249 a(r)2879 5186
y FA(=)i FG(q)p eop
%%Page: 136 136
136 135 bop 456 251 a Fy(136)140 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(and)29 b(so)1087 598 y Fx(j)p FG(T)1165
612 y Fv(Y)1206 621 y Fq(0)1241 612 y Fv(;R)1318 598
y FG(x)1370 612 y Fy(0)1429 598 y Fx(\000)20 b FG(y)1565
612 y Fy(0)1604 598 y Fx(j)1629 560 y Fy(2)1694 598 y
FA(=)25 b FG(r)1834 560 y Fy(2)1898 598 y FA(=)1694 750
y(=)g Fx(j)p FG(o)c Fx(\000)f FG(y)2016 764 y Fy(0)2055
750 y Fx(j)2080 713 y Fy(2)2140 750 y FA(+)f FG(q)2274
713 y Fy(2)2339 750 y FA(=)1694 902 y(=)25 b Fx(j)p FG(o)c
Fx(\000)f FG(y)2016 916 y Fy(0)2055 902 y Fx(j)2080 865
y Fy(2)2140 902 y FA(+)f Fx(j)p FG(T)2308 916 y Fv(Y)2349
925 y Fq(0)2384 916 y Fv(;R)2462 902 y FG(x)2514 916
y Fy(0)2573 902 y Fx(\000)h FG(o)p Fx(j)2733 865 y Fy(2)2788
902 y FG(:)456 751 y FA(\(A.185\))456 1074 y(Then,)31
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FA(and)g FG(T)3030 1088 y Fv(Y)3071 1097 y Fq(0)3106
1088 y Fv(;R)3183 1074 y FG(x)3235 1088 y Fy(0)3306 1074
y FA(is)g(a)456 1182 y(righ)m(t)f(triangle)f(in)g FG(o)p
FA(,)i(that)g(is)456 1349 y(\(A.186\))761 b(\()p FG(T)1603
1363 y Fv(Y)1644 1372 y Fq(0)1679 1363 y Fv(;R)1756 1349
y FG(x)1808 1363 y Fy(0)1868 1349 y Fx(\000)20 b FG(o)p
FA(\))g Fx(\001)h FG(\030)44 b FA(=)c(0)15 b FG(:)456
1517 y FA(Also,)30 b(if)1110 1646 y(\024)-52 b FG(q)44
b FA(:=)c Fx(j)p FA(\()p FG(T)1437 1660 y Fv(Y)1478 1669
y Fq(0)1513 1660 y Fv(;R)1590 1646 y FG(x)1642 1660 y
Fy(0)1702 1646 y Fx(\000)20 b FG(o)p FA(\))h Fx(\000)f
FA(\(\()p FG(T)2107 1660 y Fv(Y)2148 1669 y Fq(0)2183
1660 y Fv(;R)2260 1646 y FG(x)2312 1660 y Fy(0)2372 1646
y Fx(\000)g FG(o)p FA(\))g Fx(\001)h FG(\030)t FA(\))p
FG(\030)t Fx(j)15 b FG(;)456 1794 y FA(then)20 b(\(A.183\))i(and)e
(\(A.19\))i(giv)m(e)f(that)g FG(q)28 b Fx(\030)k FA(\024)-52
b FG(q)s FA(.)37 b(Then,)22 b(the)e(h)m(yp)s(otheses)g(of)h(Lemma)f
(A.2)456 1902 y(b)s(eing)36 b(ful\014lled)d(\(with)43
b(\024)-51 b FG(q)40 b FA(replacing)c FG(q)s FA(,)j(p)s(ossibly)34
b(scaling)j FG(l)i FA(to)53 b(const)16 b FG(l)r FA(\))38
b(thanks)f(to)456 2010 y(\(A.183\),)c(\(A.184\))f(and)e(\(A.186\).)43
b(Thence,)31 b(w)m(e)g(deduce)f(that,)456 2183 y(\(A.187\))673
b Fi(L)1482 2146 y Fv(N)1550 2183 y FA(\()p FG(\031)1637
2198 y Fv(\030)1675 2183 y FA(\(\004\)\))41 b Fx(\025)55
b FA(const)16 b FG(q)2275 2146 y Fv(N)7 b Ft(\000)p Fy(1)2447
2183 y FG(:)456 2350 y FA(Also,)36 b(in)f(the)g(ligh)m(t)g(of)h(the)g
(Lipsc)m(hitz)e(graph)h(prop)s(ert)m(y)g(in)f(Lemma)i(A.2,)i(w)m(e)e
(ha)m(v)m(e)456 2462 y(that)31 b(\004)24 b Fx(\023)842
2439 y FA(\025)834 2462 y(\004,)31 b(for)f(an)g(appropriate)f(set)1856
2439 y(\025)1848 2462 y(\004,)h(with)1293 2611 y(\025)1285
2634 y(\004)20 b Fx(\\)g(f)p FG(x)1544 2648 y Fv(N)7
b Fy(+1)1727 2634 y FA(=)25 b FG(s)p Fx(g)40 b FA(=)g(\004)2123
2648 y Fv(s)2200 2634 y FA(=)g FG(F)2369 2648 y Fv(s)2406
2634 y FA(\(\004)2502 2596 y Fv(])2502 2656 y(s)2539
2634 y FA(\))15 b FG(;)456 2801 y FA(for)30 b(an)m(y)g
Fx(j)p FG(s)p Fx(j)c FG(<)f FA(1)p FG(=)p FA(2,)32 b(for)e(some)1349
2974 y(\004)1410 2937 y Fv(])1410 2997 y(s)1471 2974
y Fx(\022)25 b(f)p FG(x)c Fx(\001)f FG(\030)29 b FA(=)c(0)p
Fx(g)d(\\)d(f)p FG(x)2184 2988 y Fv(N)7 b Fy(+1)2367
2974 y FA(=)25 b FG(s)p Fx(g)456 3155 y FA(and)h Fx(j)p
FG(F)712 3169 y Fv(s)749 3155 y Fx(j)774 3169 y Fy(Lip)907
3155 y Fx(\024)f FA(1.)39 b(Note)28 b(that)f(the)g(\004)1735
3111 y Fv(])1735 3167 y(s)1772 3155 y FA('s)f(are)h(all)f(disjoin)m(t)f
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3263 y(lying)k(on)j Fx(f)p FG(x)904 3277 y Fv(N)7 b Fy(+1)1087
3263 y FA(=)25 b FG(s)p Fx(g)p FA(.)555 3444 y(Giv)m(en)31
b FG(a;)g(b)26 b Fx(2)g FA(\004)1136 3399 y Fv(])1136
3455 y(s)1173 3444 y FA(,)31 b(let)g(no)m(w)g FG(a)1599
3411 y Ft(0)1649 3444 y FA(:=)26 b FG(F)1829 3458 y Fv(s)1866
3444 y FA(\()p FG(a)p FA(\))32 b(and)e FG(b)2232 3411
y Ft(0)2282 3444 y FA(:=)c FG(F)2462 3458 y Fv(s)2500
3444 y FA(\()p FG(b)p FA(\).)43 b(Then,)31 b(the)g(fact)h(that)456
3551 y FG(F)514 3565 y Fv(s)581 3551 y FA(giv)m(es)f(a)g(Lipsc)m(hitz)e
(graph)h(in)f(the)h FG(\030)t FA(-direction)g(with)f
Fx(j)p FG(F)2532 3565 y Fv(s)2569 3551 y Fx(j)2594 3565
y Fy(Lip)2727 3551 y Fx(\024)c FA(1)30 b(implies)e(that)1570
3747 y Fl(\\)1636 3673 y Fu(\000)1677 3747 y FG(a)1725
3709 y Ft(0)1769 3747 y Fx(\000)20 b FG(b)1899 3709 y
Ft(0)1922 3747 y FG(;)31 b(\030)2022 3673 y Fu(\001)2104
3747 y Fx(\025)2225 3685 y FG(\031)p 2225 3726 56 4 v
2230 3809 a FA(4)2305 3747 y FG(:)456 3945 y FA(Therefore,)f(b)m(y)g
(means)h(of)f(\(A.183\),)1328 4135 y Fl(\\)1394 4062
y Fu(\000)1435 4135 y FG(a)1483 4098 y Ft(0)1527 4135
y Fx(\000)20 b FG(b)1657 4098 y Ft(0)1680 4135 y FG(;)31
b(e)1778 4149 y Fv(N)1845 4062 y Fu(\001)1927 4135 y
Fx(\025)2048 4074 y FG(\031)p 2048 4114 V 2053 4198 a
FA(4)2134 4135 y Fx(\000)2234 4074 y FG(\031)p 2234 4114
V 2239 4198 a FA(6)2325 4135 y(=)2449 4074 y FG(\031)p
2431 4114 91 4 v 2431 4198 a FA(12)2547 4135 y FG(:)456
4370 y FA(This)22 b(sa)m(ys)k(that)f FG(\031)1088 4384
y Fv(N)1155 4265 y Fu(\014)1155 4320 y(\014)1155 4374
y(\014)1185 4433 y Fy(\004)1232 4441 y Fs(s)1294 4370
y FA(is)f(in)m(v)m(ertible)f(and)h(that)i(its)e(in)m(v)m(erse)g(is)g(a)
h(Lipsc)m(hitz)e(function,)456 4518 y(with)29 b(Lipsc)m(hitz)g(constan)
m(t)i(b)s(ounded)e(b)m(y)h(1)p FG(=)15 b FA(tan)q(\()p
FG(\031)s(=)p FA(12\).)43 b(Therefore,)31 b(if)1481 4729
y FG(G)1552 4743 y Fv(s)1630 4729 y FA(:=)40 b FG(\031)1818
4744 y Fv(\030)1876 4729 y Fx(\016)1942 4628 y Fu(\020)1996
4729 y FG(\031)2048 4743 y Fv(N)2115 4625 y Fu(\014)2115
4679 y(\014)2115 4734 y(\014)2145 4792 y Fy(\004)2192
4800 y Fs(s)2230 4628 y Fu(\021)2284 4651 y Ft(\000)p
Fy(1)2394 4729 y FG(;)456 4958 y FA(w)m(e)30 b(ha)m(v)m(e)i(that)f
FG(G)1067 4972 y Fv(s)1135 4958 y FA(is)e(a)i(Lipsc)m(hitz)e(function)g
(whose)h(range)h(is)e(\004)2714 4914 y Fv(])2714 4970
y(s)2751 4958 y FA(,)h(with)1327 5164 y Fx(j)p FG(G)1423
5178 y Fv(s)1461 5164 y Fx(j)1486 5178 y Fy(Lip)1634
5164 y Fx(\024)1929 5102 y FA(1)p 1755 5143 394 4 v 1755
5226 a(tan\()p FG(\031)s(=)p FA(12\))2198 5164 y Fx(\024)56
b FA(const)15 b FG(:)p eop
%%Page: 137 137
137 136 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)
f Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)140 b(137)456
450 y FA(Hence,)32 b(using)d(the)j(c)m(hange)g(of)f(v)-5
b(ariables)30 b(form)m(ula)g(\(see,)i(e.g.,)h(page)f(99)g(in)e([18)q
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b FG(q)1140 680 y Fv(N)7 b Ft(\000)p Fy(1)1381 718 y
Fx(\024)82 b Fi(L)1589 680 y Fv(N)1657 617 y Fu(\020)1711
718 y FG(\031)1763 733 y Fv(\030)1801 718 y FA(\()1844
695 y(\025)1836 718 y(\004\))1932 617 y Fu(\021)2012
718 y FA(=)1381 917 y(=)g Fi(L)1589 879 y Fv(N)1657 816
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917 y FA(\004)2223 879 y Fv(])2223 939 y(s)2260 816 y
Fu(\021)2339 917 y FA(=)1381 1240 y(=)1534 1116 y Fu(Z)1625
1143 y Fy(1)p Fv(=)p Fy(2)1585 1323 y Ft(\000)p Fy(1)p
Fv(=)p Fy(2)1765 1240 y Fi(L)1820 1203 y Fv(N)7 b Ft(\000)p
Fy(1)1978 1240 y FA(\(\004)2074 1203 y Fv(])2074 1263
y(s)2110 1240 y FA(\))15 b FG(ds)26 b FA(=)1381 1522
y(=)1534 1398 y Fu(Z)1625 1424 y Fy(1)p Fv(=)p Fy(2)1585
1604 y Ft(\000)p Fy(1)p Fv(=)p Fy(2)1765 1398 y Fu(Z)1815
1604 y Fy(\004)1862 1572 y Fs(])1862 1615 y(s)1930 1522
y FG(dy)19 b(ds)25 b Fx(\024)1381 1803 y(\024)1534 1680
y Fu(Z)1625 1706 y Fy(1)p Fv(=)p Fy(2)1585 1886 y Ft(\000)p
Fy(1)p Fv(=)p Fy(2)1765 1680 y Fu(Z)1815 1886 y Fv(G)1870
1856 y Fp(\000)p Fq(1)1870 1896 y Fs(s)1953 1886 y Fy(\(\004)2027
1853 y Fs(])2027 1896 y(s)2061 1886 y Fy(\))2108 1803
y Fx(j)15 b FA(det)g FG(G)2360 1766 y Ft(0)2360 1826
y Fv(s)2398 1803 y FA(\()p FG(x)p FA(\))p Fx(j)g FG(dx)g(ds)26
b Fx(\024)1381 2085 y(\024)98 b FA(const)1787 1961 y
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Fy(1)p Fv(=)p Fy(2)2018 1961 y Fu(Z)2069 2167 y Fv(G)2124
2137 y Fp(\000)p Fq(1)2124 2178 y Fs(s)2206 2167 y Fy(\(\004)2280
2135 y Fs(])2280 2178 y(s)2314 2167 y Fy(\))2376 2085
y FG(dx)15 b(ds)25 b FA(=)1381 2366 y(=)98 b(const)1787
2243 y Fu(Z)1878 2269 y Fy(1)p Fv(=)p Fy(2)1838 2449
y Ft(\000)p Fy(1)p Fv(=)p Fy(2)2018 2366 y Fi(L)2073
2329 y Fv(N)7 b Ft(\000)p Fy(1)2231 2293 y Fu(\000)2272
2366 y FG(G)2343 2329 y Ft(\000)p Fy(1)2343 2389 y Fv(s)2438
2366 y FA(\(\004)2534 2329 y Fv(])2534 2389 y(s)2571
2366 y FA(\))2606 2293 y Fu(\001)2663 2366 y FG(ds)25
b FA(=)1381 2605 y(=)98 b(const)15 b Fi(L)1827 2568 y
Fv(N)1895 2504 y Fu(\020)2124 2519 y([)1964 2720 y Fv(s)p
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Fy(1)2471 2628 y Fv(s)2566 2605 y FA(\(\004)2662 2568
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2605 y FA(=)1381 2886 y(=)98 b(const)15 b Fi(L)1827 2848
y Fv(N)1895 2785 y Fu(\020)2124 2799 y([)1964 3001 y
Fv(s)p Ft(2)p Fy(\()p Ft(\000)p Fy(1)p Fv(=)p Fy(2)p
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2900 y Fv(N)2519 2812 y Fu(\000)2561 2886 y FG(F)2619
2900 y Fv(s)2656 2886 y FA(\(\004)2752 2848 y Fv(])2752
2908 y(s)2789 2886 y FA(\))2824 2812 y Fu(\001)2866 2785
y(\021)2945 2886 y FA(=)1381 3166 y(=)98 b(const)15 b
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3281 y Fv(s)p Ft(2)p Fy(\()p Ft(\000)p Fy(1)p Fv(=)p
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3180 y Fv(N)2519 3166 y FA(\(\004)2615 3180 y Fv(s)2652
3166 y FA(\))2687 3065 y Fu(\021)2767 3166 y Fx(\024)1381
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3425 y FA(\(\004\))2187 3352 y Fu(\001)2244 3425 y FG(;)456
3580 y FA(completing)29 b(the)i(pro)s(of)e(of)i(Lemma)f(4.2.)456
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(de\014ned)e(as)1129 3916 y FG(F)1187 3931 y Fv(k)1271
3916 y FA(:=)40 b Fx(f)p FG(Z)32 b Fx(2)25 b FG(L)g Fx(j)h
FA(dist)14 b(\()p FG(Z)q(;)h(D)2145 3931 y Fv(k)2209
3916 y Fx(\\)20 b FG(Q)2362 3931 y Fv(l)q Fy(+)p Fv(a)2480
3916 y FA(\))26 b Fx(\024)f FG(a)p Fx(g)15 b FG(:)456
4071 y FA(If)30 b FG(Q)619 4086 y Fv(l)665 4071 y Fx(n)20
b FG(F)788 4086 y Fv(k)857 4071 y FA(=)25 b Fx(;)p FA(,)1303
4194 y Fi(L)1358 4156 y Fv(N)1426 4194 y FA(\()p FG(Q)1533
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4194 y FA(\))k Fx(\024)g Fi(L)1966 4156 y Fv(N)2034 4194
y FA(\()p FG(Q)2141 4209 y Fv(l)2187 4194 y Fx(n)c FG(F)2311
4209 y Fv(k)2354 4194 y FA(\))26 b(=)f(0)15 b FG(;)456
4330 y FA(pro)m(ving)29 b(the)i(claim,)e(hence)i(w)m(e)g(ma)m(y)f(and)g
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y Fx(n)21 b FG(F)2847 4345 y Fv(k)2915 4330 y Fx(6)p
FA(=)k Fx(;)p FA(.)41 b(Let)31 b(no)m(w)456 4438 y FG(Z)g
Fx(2)25 b FG(Q)707 4453 y Fv(l)753 4438 y Fx(n)c FG(F)877
4453 y Fv(k)950 4438 y FA(and)30 b(tak)m(e)i FG(Z)1386
4452 y Ft(\003)1451 4438 y Fx(2)25 b FG(F)1595 4453 y
Fv(k)1668 4438 y FA(b)s(e)30 b(so)h(that)456 4593 y(\(A.188\))635
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y FA(\))26 b(=)e Fx(j)p FG(Z)k Fx(\000)19 b FG(Z)2215
4607 y Ft(\003)2255 4593 y Fx(j)25 b FA(=:)h FG(r)17
b(:)456 4748 y FA(W)-8 b(e)31 b(use)f(the)h(notation)f
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y FG(;)15 b FA(0)p FG(;)g(z)1751 4762 y Fv(N)7 b Fy(+1)1910
4748 y FA(\))26 b Fx(2)f FG(L)30 b FA(and)f(w)m(e)i(claim)f(that)456
4927 y(\(A.189\))874 b FG(r)28 b Fx(\024)d FG(l)d FA(+)e
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4865 y FG(a)p 2174 4906 49 4 v 2175 4989 a FA(2)2247
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b(pro)s(ceed)g(as)h(follo)m(ws.)45 b(First)32 b(of)g(all,)g(notice)h
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%%Page: 138 138
138 137 bop 456 251 a Fy(138)140 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
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Fx(2)37 b FG(D)2417 468 y Fv(k)2485 453 y Fx(\\)24 b
FG(Q)2642 468 y Fv(l)2668 453 y FA(.)62 b(Let)38 b FG(Z)2987
468 y Fv(])3056 453 y FA(lie)f(on)g(the)456 575 y(segmen)m(t)31
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730 y Fx(j)26 b FA(=)f Fx(j)p FG(Z)i Fx(\000)2168 707
y FA(\025)2148 730 y FG(Z)7 b Fx(j)21 b(\000)e FG(a)456
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b Fx(\024)1582 1355 y(\024)1736 1272 y Fu(p)p 1827 1272
527 4 v 83 x FA(\()p Fx(j)p FG(z)1933 1329 y Ft(0)1957
1355 y Fx(j)20 b FA(+)g FG(l)r FA(\))2157 1329 y Fy(2)2217
1355 y FA(+)g(1)h Fx(\000)e FG(a)c(:)-2096 b FA(\(A.190\))456
1506 y(Also,)30 b(b)m(y)g(construction,)g FG(Z)1420 1521
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1506 y FA(and)k(th)m(us)456 1656 y(\(A.191\))932 b FG(r)27
b Fx(\024)e(j)p FG(Z)i Fx(\000)20 b FG(Z)2117 1671 y
Fv(])2149 1656 y Fx(j)15 b FG(:)456 1807 y FA(The)25
b(pro)s(of)f(of)i(\(A.189\))i(no)m(w)e(follo)m(ws)e(from)i(\(A.190\))i
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2109 y(Notice)32 b(no)m(w)e(that,)h(since)f FG(Z)i Fx(2)25
b FG(Q)1727 2124 y Fv(l)1752 2109 y FA(,)31 b(\(A.189\))i(implies)27
b(that)456 2283 y(\(A.192\))967 b FG(r)27 b Fx(\024)e
FA(2)p FG(l)e Fx(\000)2081 2222 y FG(a)p 2081 2262 49
4 v 2082 2345 a FA(2)2154 2283 y FG(:)555 2460 y FA(W)-8
b(e)32 b(no)m(w)e(claim)g(that)456 2615 y(\(A.193\))401
b Fi(L)1210 2578 y Fv(N)1277 2514 y Fu(\020)1332 2615
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b FG(Q)1696 2630 y Fv(l)1742 2615 y Fx(\\)g FG(B)1892
2629 y Fv(r)1930 2615 y FA(\()p FG(Z)7 b FA(\))2069 2514
y Fu(\021)2164 2615 y Fx(\025)42 b FA(\026)-47 b FG(c)15
b Fi(L)2384 2578 y Fv(N)2452 2514 y Fu(\020)2506 2615
y FG(Q)2578 2630 y Fv(l)2624 2615 y Fx(\\)20 b FG(B)2774
2629 y Fv(r)2812 2615 y FA(\()p FG(Z)7 b FA(\))2951 2514
y Fu(\021)3021 2615 y FG(;)456 2794 y FA(for)30 b(a)h(suitable)h(\026)
-47 b FG(c)26 b Fx(2)g FA(\(0)p FG(;)15 b FA(1\),)33
b(whic)m(h)d(ma)m(y)h(dep)s(end)e(on)i(the)g(quan)m(tit)m(y)g
FG(c)g FA(in)m(tro)s(duced)f(in)456 2902 y(\(P2\))h(during)d(the)j
(statemen)m(t)h(of)e(Lemma)h(4.3.)555 3010 y(W)-8 b(e)44
b(no)m(w)f(b)s(egin)e(with)h(the)h(pro)s(of)f(of)h(\(A.193\),)48
b(whic)m(h)42 b(will)e(b)s(e)i(completed)g(on)456 3118
y(page)31 b(142.)555 3225 y(Since)e FG(Z)854 3239 y Ft(\003)919
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b(there)g(exists)f FG(Z)1705 3239 y Fy(0)1770 3225 y
Fx(2)24 b FG(D)1930 3240 y Fv(k)1992 3225 y Fx(\\)19
b FG(Q)2144 3240 y Fv(l)q Fy(+)p Fv(a)2291 3225 y FA(b)s(e)29
b(so)h(that)h Fx(j)p FG(Z)2809 3239 y Ft(\003)2867 3225
y Fx(\000)19 b FG(Z)3019 3239 y Fy(0)3058 3225 y Fx(j)26
b(\024)f FG(a)p FA(.)40 b(W)-8 b(e)456 3333 y(p)s(oin)m(t)29
b(out)i(that,)g(in)e(fact,)456 3484 y(\(A.194\))909 b
Fx(j)p FG(Z)1750 3498 y Ft(\003)1810 3484 y Fx(\000)20
b FG(Z)1963 3498 y Fy(0)2002 3484 y Fx(j)26 b FA(=)f
FG(a)15 b(:)456 3635 y FA(Indeed,)25 b(if,)h(b)m(y)f(con)m(tradiction,)
i Fx(j)p FG(Z)1660 3649 y Ft(\003)1710 3635 y Fx(\000)10
b FG(Z)1853 3649 y Fy(0)1892 3635 y Fx(j)26 b FG(<)e(a)p
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4119 y Fx(\000)g FG(Z)2333 4133 y Fy(0)2373 4119 y Fx(j)25
b FA(=)g FG(r)e FA(+)d FG(a)15 b(:)456 4269 y FA(W)-8
b(e)31 b(notice)g(that,)g(in)e(fact,)456 4420 y(\(A.196\))848
b Fx(j)p FG(Z)27 b Fx(\000)20 b FG(Z)1869 4434 y Fy(0)1908
4420 y Fx(j)26 b FA(=)f FG(r)e FA(+)d FG(a)15 b(:)456
4578 y FA(Indeed,)27 b(if)f(it)h(holded)f(that)i Fx(j)p
FG(Z)20 b Fx(\000)14 b FG(Z)1680 4592 y Fy(0)1720 4578
y Fx(j)25 b FG(<)g(r)17 b FA(+)d FG(a)p FA(,)28 b(tak)m(e)2323
4555 y(~)2304 4578 y FG(Z)j Fx(2)25 b FG(@)5 b(B)2605
4592 y Fv(a)2647 4578 y FA(\()p FG(Z)2744 4592 y Fy(0)2784
4578 y FA(\))28 b(on)f(the)g(segmen)m(t)456 4692 y(joining)h
FG(Z)37 b FA(and)30 b FG(Z)1094 4706 y Fy(0)1133 4692
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Fx(2)25 b FG(F)1644 4707 y Fv(k)1687 4692 y FA(,)30 b(since)g
FG(Z)2027 4706 y Fy(0)2092 4692 y Fx(2)25 b FG(D)2253
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(of)j FG(r)i FA(and)c(pro)m(v)m(es)i(\(A.196\).)555 5108
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30.000 slw
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/Times-Italic-iso ff 405.00 scf sf
480 3150 m
gs 1 -1 sc (L) col0 sh gr
/Times-Roman-iso ff 315.00 scf sf
5430 2610 m
gs 1 -1 sc (1/2) col0 sh gr
/Times-Roman-iso ff 315.00 scf sf
5505 7320 m
gs 1 -1 sc (-1/2) col0 sh gr
/Times-Roman-iso ff 315.00 scf sf
10620 5025 m
gs 1 -1 sc (e') col0 sh gr
/Times-Italic-iso ff 375.00 scf sf
2490 3150 m
gs 1 -1 sc (Q) col0 sh gr
/Times-Italic-iso ff 300.00 scf sf
2760 3360 m
gs 1 -1 sc (l) col0 sh gr
/Times-Italic-iso ff 300.00 scf sf
8385 3120 m
gs 1 -1 sc (a) col0 sh gr
/Times-Italic-iso ff 240.00 scf sf
5475 1980 m
gs 1 -1 sc (N+1) col0 sh gr
/Times-Roman-iso ff 315.00 scf sf
5280 1845 m
gs 1 -1 sc (e) col0 sh gr
/Times-Italic-iso ff 315.00 scf sf
8475 4725 m
gs 1 -1 sc (l) col0 sh gr
/Times-Italic-iso ff 315.00 scf sf
2175 4725 m
gs 1 -1 sc (-l) col0 sh gr
/Times-Italic-iso ff 285.00 scf sf
900 4725 m
gs 1 -1 sc (-l-a) col0 sh gr
/Times-Italic-iso ff 285.00 scf sf
9000 4725 m
gs 1 -1 sc (l+a) col0 sh gr
/Times-Roman-iso ff 315.00 scf sf
7875 6225 m
gs 1 -1 sc (F) col0 sh gr
/Times-Italic-iso ff 300.00 scf sf
8100 6300 m
gs 1 -1 sc (k) col0 sh gr
/Times-Roman-iso ff 315.00 scf sf
7575 5175 m
gs 1 -1 sc (D) col0 sh gr
/Times-Italic-iso ff 300.00 scf sf
7800 5325 m
gs 1 -1 sc (k) col0 sh gr
/Times-Roman-iso ff 270.00 scf sf
7800 4500 m
gs 1 -1 sc (Z) col0 sh gr
/Times-Roman-iso ff 270.00 scf sf
7050 3750 m
gs 1 -1 sc (Z) col0 sh gr
/Times-Roman-iso ff 270.00 scf sf
5700 3075 m
gs 1 -1 sc (Z) col0 sh gr
/Times-Italic-iso ff 210.00 scf sf
6375 3450 m
gs 1 -1 sc (r/2) col0 sh gr
/Times-Roman-iso ff 210.00 scf sf
7275 3825 m
gs 1 -1 sc (*) col0 sh gr
/Times-Roman-iso ff 210.00 scf sf
8025 4575 m
gs 1 -1 sc (0) col0 sh gr
/Times-Roman-iso ff 270.00 scf sf
4560 3225 m
gs 1 -1 sc (Z) col0 sh gr
/Times-Roman-iso ff 210.00 scf sf
4755 3315 m
gs 1 -1 sc (1) col0 sh gr
% here ends figure;
$F2psEnd
rs
showpage

%%EndDocument
 @endspecial 1465 2671 a FC(The)29 b(pro)r(of)h(of)g(Lemma)f(4.3)555
2944 y FA(T)-8 b(o)38 b(con)m(tin)m(ue)f(with)f(the)h(pro)s(of)g(of)g
(\(A.193\),)k(w)m(e)d(need)f(no)m(w)g(to)h(distinguish)33
b(t)m(w)m(o)456 3052 y(cases:)47 b(either)32 b FG(a)e
Fx(\024)g FA(2)p FG(r)36 b FA(or)d FG(a)d(>)g FA(2)p
FG(r)s FA(.)49 b(Let)34 b(us)e(\014rst)g(deal)h(with)f(the)h(case)i
FG(a)29 b Fx(\024)h FA(2)p FG(r)s FA(.)49 b(W)-8 b(e)456
3160 y(claim)29 b(that)i(there)g(exists)f FG(Z)1441 3174
y Fy(1)1505 3160 y Fx(2)25 b FG(Q)1663 3175 y Fv(l)1719
3160 y FA(so)31 b(that)456 3327 y(\(A.197\))446 b Fx(j)p
FG(Z)27 b Fx(\000)20 b FG(Z)1467 3341 y Fy(1)1506 3327
y Fx(j)26 b FA(=)1664 3266 y FG(r)p 1663 3306 46 4 v
1663 3389 a FA(2)1900 3327 y(and)181 b FG(B)2297 3346
y Fv(r)r(=)p Fy(2)2406 3327 y FA(\()p FG(Z)2503 3341
y Fy(1)2543 3327 y FA(\))20 b Fx(\\)g FG(L)25 b Fx(\022)g
FG(Q)2934 3342 y Fv(l)2975 3327 y FG(:)456 3508 y FA(T)-8
b(o)41 b(pro)m(v)m(e)i(\(A.197\),)j(w)m(e)c(need)f(to)h(distinguish)c
(t)m(w)m(o)k(sub-cases.)74 b(If)41 b FG(z)2982 3475 y
Ft(0)3049 3508 y FA(=)i(0,)i(tak)m(e)456 3616 y FG(Z)518
3630 y Fy(1)582 3616 y FA(:=)26 b FG(Z)g FA(+)20 b FG(r)s(e)969
3630 y Fy(1)1009 3616 y FG(=)p FA(2.)41 b(Then,)1518
3807 y Fx(j)p FG(z)1585 3821 y Fy(1)p Fv(;N)7 b Fy(+1)1798
3807 y Fx(j)26 b FA(=)f Fx(j)p FG(z)2012 3821 y Fv(N)7
b Fy(+1)2170 3807 y Fx(j)25 b(\024)2326 3746 y FA(1)p
2326 3786 V 2326 3870 a(2)456 3986 y(and,)30 b(recalling)f(\(A.192\),)
1695 4121 y Fx(j)p FG(z)1766 4083 y Ft(0)1762 4143 y
Fy(1)1802 4121 y Fx(j)d FA(=)1960 4059 y FG(r)p 1959
4100 V 1959 4183 a FA(2)2039 4121 y FG(<)f(l)17 b(;)456
4281 y FA(sho)m(wing)47 b(that)h FG(Z)1096 4295 y Fy(1)1190
4281 y Fx(2)54 b FG(Q)1377 4296 y Fv(l)1451 4281 y FA(in)47
b(this)g(case.)94 b(Also,)52 b(exploiting)47 b(\(A.189\),)55
b(if)47 b FG(W)67 b FA(=)456 4389 y(\()p FG(w)558 4356
y Ft(0)582 4389 y FG(;)15 b FA(0)p FG(;)g(w)772 4403
y Fv(N)7 b Fy(+1)931 4389 y FA(\))25 b Fx(2)g FG(B)1146
4407 y Fv(r)r(=)p Fy(2)1254 4389 y FA(\()p FG(Z)1351
4403 y Fy(1)1391 4389 y FA(\),)1702 4509 y Fx(j)p FG(w)1794
4471 y Ft(0)1818 4509 y Fx(j)g(\024)g FG(r)j Fx(\024)d
FG(l)17 b(;)456 4638 y FA(pro)m(ving)29 b(\(A.197\))k(in)c(this)g
(sub-case.)41 b(If,)30 b(on)h(the)f(other)h(hand,)e FG(z)2728
4605 y Ft(0)2777 4638 y Fx(6)p FA(=)c(0,)31 b(tak)m(e)1616
4841 y FG(Z)1678 4855 y Fy(1)1742 4841 y FA(:=)26 b FG(Z)g
Fx(\000)2079 4780 y FG(r)18 b(z)2184 4747 y Ft(0)p 2053
4820 181 4 v 2053 4903 a FA(2)d Fx(j)p FG(z)2184 4877
y Ft(0)2209 4903 y Fx(j)2259 4841 y FA(;)456 5038 y(th)m(us,)1518
5186 y Fx(j)p FG(z)1585 5200 y Fy(1)p Fv(;N)7 b Fy(+1)1798
5186 y Fx(j)26 b FA(=)f Fx(j)p FG(z)2012 5200 y Fv(N)7
b Fy(+1)2170 5186 y Fx(j)25 b(\024)2326 5125 y FA(1)p
2326 5166 46 4 v 2326 5249 a(2)p eop
%%Page: 140 140
140 139 bop 456 251 a Fy(140)140 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(and,)30 b(since)f FG(Z)j Fx(2)25 b
FG(Q)1132 465 y Fv(l)1158 450 y FA(,)30 b(recalling)f(also)i
(\(A.189\),)i(w)m(e)e(ha)m(v)m(e)g(that)1223 657 y Fx(j)p
FG(z)1294 620 y Ft(0)1290 680 y Fy(1)1330 657 y Fx(j)83
b FA(=)1592 525 y Fu(\014)1592 580 y(\014)1592 635 y(\014)1592
689 y(\014)1657 596 y FG(z)1703 563 y Ft(0)p 1632 636
121 4 v 1632 720 a Fx(j)p FG(z)1703 693 y Ft(0)1727 720
y Fx(j)1762 556 y Fu(\020)1816 657 y Fx(j)p FG(z)1887
620 y Ft(0)1911 657 y Fx(j)21 b(\000)2058 596 y FG(r)p
2057 636 46 4 v 2057 720 a FA(2)2113 556 y Fu(\021)2167
525 y(\014)2167 580 y(\014)2167 635 y(\014)2167 689 y(\014)2223
657 y FA(=)1438 883 y(=)1592 779 y Fu(\014)1592 833 y(\014)1592
888 y(\014)1622 883 y Fx(j)p FG(z)1693 846 y Ft(0)1717
883 y Fx(j)f(\000)1864 822 y FG(r)p 1863 863 V 1863 946
a FA(2)1919 779 y Fu(\014)1919 833 y(\014)1919 888 y(\014)1974
883 y FA(=)1438 1086 y(=)83 b(max)1776 985 y Fu(n)1837
1086 y Fx(j)p FG(z)1908 1048 y Ft(0)1931 1086 y Fx(j)21
b(\000)2079 1024 y FG(r)p 2078 1065 V 2078 1148 a FA(2)2133
1086 y FG(;)2210 1024 y(r)p 2209 1065 V 2209 1148 a FA(2)2284
1086 y Fx(\000)f(j)p FG(z)2446 1048 y Ft(0)2470 1086
y Fx(j)2495 985 y Fu(o)2581 1086 y Fx(\024)1438 1315
y(\024)83 b FA(max)1776 1187 y Fu(\032)1844 1315 y FG(l)22
b Fx(\000)1995 1254 y FG(r)p 1994 1294 V 1994 1377 a
FA(2)2050 1315 y FG(;)2134 1254 y(l)p 2125 1294 V 2125
1377 a FA(2)2181 1187 y Fu(\033)2274 1315 y FG(<)1438
1500 y(<)83 b(l)17 b(;)456 1652 y FA(whic)m(h)29 b(sho)m(ws)h(that)h
FG(Z)1236 1666 y Fy(1)1301 1652 y Fx(2)24 b FG(Q)1458
1667 y Fv(l)1515 1652 y FA(in)29 b(this)g(case.)555 1760
y(Let)i(also)f FG(W)38 b FA(=)25 b(\()p FG(w)1222 1727
y Ft(0)1246 1760 y FG(;)15 b FA(0)p FG(;)g(w)1436 1774
y Fv(N)7 b Fy(+1)1595 1760 y FA(\))26 b Fx(2)f FG(B)1811
1779 y Fv(r)r(=)p Fy(2)1919 1760 y FA(\()p FG(Z)2016
1774 y Fy(1)2056 1760 y FA(\))20 b Fx(\\)g FG(L)p FA(;)31
b(then,)f(w)m(e)h(ha)m(v)m(e)g(that)1107 1948 y Fx(j)p
FG(w)1199 1911 y Ft(0)1223 1948 y Fx(j)84 b(\024)e(j)p
FG(z)1556 1911 y Ft(0)1552 1971 y Fy(1)1592 1948 y Fx(j)21
b FA(+)1740 1887 y FG(r)p 1739 1927 V 1739 2011 a FA(2)1819
1948 y(=)1332 2177 y(=)1485 2073 y Fu(\014)1485 2127
y(\014)1485 2182 y(\014)1551 2116 y FG(z)1597 2083 y
Ft(0)p 1526 2157 121 4 v 1526 2240 a Fx(j)p FG(z)1597
2214 y Ft(0)1620 2240 y Fx(j)1671 2077 y Fu(\020)1725
2177 y Fx(j)p FG(z)1796 2140 y Ft(0)1820 2177 y Fx(j)f(\000)1967
2116 y FG(r)p 1966 2157 46 4 v 1966 2240 a FA(2)2022
2077 y Fu(\021)2076 2073 y(\014)2076 2127 y(\014)2076
2182 y(\014)2126 2177 y FA(+)2228 2116 y FG(r)p 2227
2157 V 2227 2240 a FA(2)2308 2177 y(=)1332 2402 y(=)1485
2298 y Fu(\014)1485 2352 y(\014)1485 2407 y(\014)1516
2402 y Fx(j)p FG(z)1587 2365 y Ft(0)1610 2402 y Fx(j)h(\000)1758
2341 y FG(r)p 1757 2381 V 1757 2465 a FA(2)1812 2298
y Fu(\014)1812 2352 y(\014)1812 2407 y(\014)1863 2402
y FA(+)1964 2341 y FG(r)p 1964 2381 V 1964 2465 a FA(2)2044
2402 y(=)1332 2605 y(=)82 b(max)1670 2504 y Fu(n)1730
2605 y Fx(j)p FG(z)1801 2567 y Ft(0)1825 2605 y Fx(j)21
b(\000)1972 2543 y FG(r)p 1971 2584 V 1971 2667 a FA(2)2042
2605 y FG(;)2119 2543 y(r)p 2118 2584 V 2118 2667 a FA(2)2193
2605 y Fx(\000)f(j)p FG(z)2355 2567 y Ft(0)2379 2605
y Fx(j)2404 2504 y Fu(o)2500 2605 y FA(+)2617 2543 y
FG(r)p 2616 2584 V 2616 2667 a FA(2)2697 2605 y(=)1332
2779 y(=)82 b(max)1670 2706 y Fu(\010)1723 2779 y Fx(j)p
FG(z)1794 2742 y Ft(0)1817 2779 y Fx(j)15 b FG(;)42 b(r)22
b Fx(\000)e(j)p FG(z)2149 2742 y Ft(0)2173 2779 y Fx(j)2198
2706 y Fu(\011)2282 2779 y FG(:)456 2932 y FA(Hence,)31
b(using)e(that)i FG(Z)h Fx(2)25 b FG(Q)1438 2947 y Fv(l)1494
2932 y FA(and)30 b(\(A.189\),)j(w)m(e)d(deduce)g(from)g(the)h(ab)s(o)m
(v)m(e)g(that)1784 3084 y Fx(j)p FG(w)1876 3047 y Ft(0)1900
3084 y Fx(j)26 b(\024)f FG(l)17 b(;)456 3236 y FA(pro)m(ving)29
b(\(A.197\))k(in)c(this)g(sub-case.)555 3344 y(This)g(completes)i(the)f
(pro)s(of)g(of)g(\(A.197\).)555 3541 y(As)42 b(a)g(consequence)g(of)g
(\(A.197\),)48 b(w)m(e)42 b(immediately)e(infer)g(from)h(the)h(fact)h
(that)456 3649 y FG(B)525 3667 y Fv(r)r(=)p Fy(2)633
3649 y FA(\()p FG(Z)730 3663 y Fy(1)770 3649 y FA(\))20
b Fx(\\)g FG(L)25 b Fx(\022)g FG(Q)1161 3664 y Fv(l)1218
3649 y FA(that)1527 3845 y(^)1500 3868 y FG(W)38 b FA(:=)25
b FG(Z)1807 3882 y Fy(1)1867 3868 y FA(+)1993 3806 y
FG(z)2039 3773 y Ft(0)2035 3831 y Fy(1)p 1968 3847 133
4 v 1968 3930 a Fx(j)p FG(z)2039 3899 y Ft(0)2035 3956
y Fy(1)2075 3930 y Fx(j)2136 3806 y FG(r)p 2135 3847
46 4 v 2135 3930 a FA(2)2216 3868 y Fx(2)g FG(Q)2374
3883 y Fv(l)456 4072 y FA(and)k(therefore)1324 4191 y
FG(l)e Fx(\025)e(j)19 b FA(^)-64 b FG(w)1566 4154 y Ft(0)1590
4191 y Fx(j)26 b FA(=)1737 4087 y Fu(\014)1737 4141 y(\014)1737
4196 y(\014)1767 4191 y Fx(j)p FG(z)1838 4154 y Ft(0)1834
4214 y Fy(1)1874 4191 y Fx(j)20 b FA(+)2021 4130 y FG(r)p
2020 4171 V 2020 4254 a FA(2)2076 4087 y Fu(\014)2076
4141 y(\014)2076 4196 y(\014)2131 4191 y FA(=)25 b Fx(j)p
FG(z)2298 4154 y Ft(0)2294 4214 y Fy(1)2334 4191 y Fx(j)20
b FA(+)2481 4130 y FG(r)p 2480 4171 V 2480 4254 a FA(2)2551
4191 y FG(;)456 4351 y FA(that)31 b(is)456 4504 y(\(A.198\))946
b Fx(j)p FG(z)1771 4466 y Ft(0)1767 4526 y Fy(1)1807
4504 y Fx(j)26 b(\024)f FG(l)d Fx(\000)2105 4442 y FG(r)p
2104 4483 V 2104 4566 a FA(2)2174 4504 y FG(:)456 4682
y FA(W)-8 b(e)31 b(sho)m(w)f(that)h(this)f(yields)e(that)456
4834 y(\(A.199\))798 b Fx(j)p FG(z)1623 4797 y Ft(0)1619
4857 y Fy(1)1679 4834 y Fx(\000)20 b FG(z)1816 4797 y
Ft(0)1812 4857 y Fy(0)1852 4834 y Fx(j)40 b(\024)h FA(2)p
FG(l)22 b Fx(\000)e FA(2)p FG(a)15 b(:)456 4986 y FA(Indeed,)29
b(if)h FG(r)d(>)e FA(2\()p FG(l)e Fx(\000)d FA(3)p FG(a)p
FA(\))p FG(=)p FA(3,)33 b(\(A.198\))f(and)e(the)h(fact)g(that)g
FG(Z)2623 5000 y Fy(0)2688 4986 y Fx(2)25 b FG(Q)2846
5001 y Fv(l)q Fy(+)p Fv(a)2994 4986 y FA(imply)j(that)1043
5186 y Fx(j)p FG(z)1114 5149 y Ft(0)1110 5209 y Fy(1)1170
5186 y Fx(\000)20 b FG(z)1307 5149 y Ft(0)1303 5209 y
Fy(0)1343 5186 y Fx(j)26 b(\024)e(j)p FG(z)1560 5149
y Ft(0)1556 5209 y Fy(1)1596 5186 y Fx(j)d FA(+)f Fx(j)p
FG(z)1804 5149 y Ft(0)1800 5209 y Fy(0)1840 5186 y Fx(j)25
b(\024)g FA(2)p FG(l)e Fx(\000)2183 5125 y FG(r)p 2182
5166 V 2182 5249 a FA(2)2257 5186 y(+)d FG(a)26 b Fx(\024)2527
5125 y FA(5)p FG(l)p 2527 5166 75 4 v 2541 5249 a FA(3)2632
5186 y(+)20 b(2)p FG(a)15 b(;)p eop
%%Page: 141 141
141 140 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)
f Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)140 b(141)456
450 y FA(whic)m(h)27 b(pro)m(v)m(es)i(\(A.199\))i(in)c(this)g(case;)j
(if,)e(on)g(the)h(other)g(hand,)f FG(r)f Fx(\024)e FA(2\()p
FG(l)19 b Fx(\000)d FA(3)p FG(a)p FA(\))p FG(=)p FA(3,)31
b(w)m(e)456 558 y(ha)m(v)m(e,)g(b)m(y)g(\(A.196\))i(and)c(\(A.197\),)k
(that)483 736 y Fx(j)p FG(z)554 698 y Ft(0)550 758 y
Fy(1)610 736 y Fx(\000)20 b FG(z)747 698 y Ft(0)743 758
y Fy(0)783 736 y Fx(j)25 b(\024)g(j)p FG(z)1000 698 y
Ft(0)996 758 y Fy(1)1056 736 y Fx(\000)20 b FG(z)1193
698 y Ft(0)1217 736 y Fx(j)g FA(+)g Fx(j)p FG(z)1424
698 y Ft(0)1468 736 y Fx(\000)g FG(z)1605 698 y Ft(0)1601
758 y Fy(0)1641 736 y Fx(j)25 b(\024)g(j)p FG(Z)1874
750 y Fy(1)1934 736 y Fx(\000)20 b FG(Z)7 b Fx(j)20 b
FA(+)g Fx(j)p FG(Z)27 b Fx(\000)20 b FG(Z)2497 750 y
Fy(0)2537 736 y Fx(j)25 b FA(=)2694 674 y FG(r)p 2693
715 46 4 v 2693 798 a FA(2)2769 736 y(+)20 b FG(r)i FA(+)e
FG(a)25 b Fx(\024)g FG(l)e Fx(\000)c FA(2)p FG(a)456
921 y FA(in)29 b(this)g(case.)42 b(This)28 b(ends)i(the)g(pro)s(of)g
(of)h(\(A.199\).)555 1075 y(W)-8 b(e)33 b(no)m(w)f(complete)h(the)f
(pro)s(of)f(of)h(\(A.193\))i(in)d(the)h(case)h FG(a)28
b Fx(\024)g FA(2)p FG(r)s FA(,)k(b)m(y)g(arguing)f(as)456
1183 y(follo)m(ws.)38 b(W)-8 b(e)27 b(notice)g(that,)h(b)m(y)e
(construction,)g(and)g(recalling)f(\(A.196\))j(and)e(\(A.197\),)1272
1361 y FG(a)20 b FA(+)1442 1299 y FG(r)p 1441 1340 V
1441 1423 a FA(2)1579 1361 y(=)83 b Fx(j)p FG(Z)27 b
Fx(\000)20 b FG(Z)2000 1375 y Fy(0)2039 1361 y Fx(j)h(\000)f(j)p
FG(Z)2263 1375 y Fy(1)2323 1361 y Fx(\000)f FG(Z)7 b
Fx(j)26 b(\024)1579 1526 y(\024)83 b(j)p FG(Z)1820 1540
y Fy(1)1880 1526 y Fx(\000)20 b FG(Z)2033 1540 y Fy(0)2072
1526 y Fx(j)26 b(\024)1579 1661 y(\024)83 b(j)p FG(Z)1820
1675 y Fy(1)1880 1661 y Fx(\000)20 b FG(Z)7 b Fx(j)20
b FA(+)g Fx(j)p FG(Z)27 b Fx(\000)20 b FG(Z)2443 1675
y Fy(0)2482 1661 y Fx(j)26 b FA(=)1579 1820 y(=)1744
1758 y FG(r)p 1743 1799 V 1743 1882 a FA(2)1818 1820
y(+)20 b FG(r)j FA(+)d FG(a)25 b Fx(\024)1579 1986 y(\024)83
b FA(5)p FG(r)18 b(:)-1406 b FA(\(A.200\))456 2141 y(F)-8
b(urthermore,)978 2106 y Fy(23)1079 2141 y FA(using)29
b(\(A.199\),)k(w)m(e)d(ha)m(v)m(e)i(that)1341 2334 y
Fx(j)p FG(Z)1428 2348 y Fy(1)1488 2334 y Fx(\000)20 b
FG(Z)1641 2348 y Fy(0)1680 2334 y Fx(j)26 b(\024)1827
2225 y Fu(q)p 1918 2225 522 4 v 109 x Fx(j)p FG(z)1989
2302 y Ft(0)1985 2360 y Fy(1)2045 2334 y Fx(\000)20 b
FG(z)2182 2302 y Ft(0)2178 2360 y Fy(0)2217 2334 y Fx(j)2242
2307 y Fy(2)2302 2334 y FA(+)g(1)26 b Fx(\024)1731 2511
y(\024)1827 2428 y Fu(p)p 1918 2428 546 4 v 83 x FA(\(2)p
FG(l)c Fx(\000)e FA(2)p FG(a)p FA(\))2266 2484 y Fy(2)2327
2511 y FA(+)g(1)25 b Fx(\024)1731 2646 y(\024)g FA(2)p
FG(l)17 b(:)456 2473 y FA(\(A.201\))456 2805 y(No)m(w)31
b(notice)f(that,)i(from)d(\(A.200\),)456 2986 y(\(A.202\))245
b Fi(L)1054 2949 y Fv(N)1122 2885 y Fu(\020)1176 2986
y FG(D)1251 3001 y Fv(k)r Fy(+1)1404 2986 y Fx(\\)20
b FG(B)1554 3005 y Fv(r)r(=)p Fy(2)1663 2986 y FA(\()p
FG(Z)1760 3000 y Fy(1)1800 2986 y FA(\))1835 2885 y Fu(\021)1914
2986 y Fx(\025)25 b Fi(L)2065 2949 y Fv(N)2133 2885 y
Fu(\020)2187 2986 y FG(D)2262 3001 y Fv(k)r Fy(+1)2416
2986 y Fx(\\)19 b FG(B)2565 3005 y Ft(j)p Fv(Z)2633 3014
y Fq(1)2667 3005 y Ft(\000)p Fv(Z)2770 3014 y Fq(0)2805
3005 y Ft(j)p Fv(=)p Fy(10)2934 2986 y FA(\()p FG(Z)3031
3000 y Fy(1)3071 2986 y FA(\))3106 2885 y Fu(\021)3176
2986 y FG(:)456 3168 y FA(F)-8 b(urthermore,)35 b(b)m(y)e(\(A.197\),)38
b FG(B)1572 3187 y Fv(r)r(=)p Fy(2)1680 3168 y FA(\()p
FG(Z)1777 3182 y Fy(1)1817 3168 y FA(\))23 b Fx(\\)f
FG(L)34 b FA(con)m(tains)g(a)g(circular)e(sector)j(of)f(heigh)m(t)456
3280 y(1)d(of)g(a)g(ball)e(of)i(radius)e FG(r)s(=)p FA(2,)i(while,)e
(on)i(the)g(other)g(hand,)f FG(B)2564 3294 y Fv(r)2602
3280 y FA(\()p FG(Z)7 b FA(\))21 b Fx(\\)f FG(Q)2915
3295 y Fv(l)2971 3280 y FA(is)30 b(con)m(tained)456 3388
y(in)f(a)h(circular)f(sector)j(of)e(heigh)m(t)g(1)h(of)g(a)f(ball)f(of)
i(radius)d FG(r)s FA(.)40 b(Therefore,)1101 3564 y Fi(L)1156
3527 y Fv(N)1224 3463 y Fu(\020)1278 3564 y FG(B)1347
3583 y Fv(r)r(=)p Fy(2)1456 3564 y FA(\()p FG(Z)1553
3578 y Fy(1)1593 3564 y FA(\))20 b Fx(\\)g FG(L)1791
3463 y Fu(\021)1928 3564 y Fx(\025)98 b FA(const)16 b
FG(r)2364 3527 y Fv(N)7 b Ft(\000)p Fy(1)2561 3564 y
Fx(\025)1928 3763 y(\025)83 b Fi(L)2137 3726 y Fv(N)2204
3662 y Fu(\020)2259 3763 y FG(B)2328 3777 y Fv(r)2366
3763 y FA(\()p FG(Z)7 b FA(\))20 b Fx(\\)g FG(Q)2678
3778 y Fv(l)2704 3662 y Fu(\021)2773 3763 y FG(:)-2342
b FA(\(A.203\))456 3945 y(Also,)42 b(b)m(y)e(construction,)j
FG(B)1462 3963 y Fv(r)r(=)p Fy(2)1570 3945 y FA(\()p
FG(Z)1667 3959 y Fy(1)1707 3945 y FA(\))f Fx(\022)g FG(B)1966
3959 y Fv(r)2003 3945 y FA(\()p FG(Z)7 b FA(\),)44 b
FG(D)2286 3960 y Fv(k)r Fy(+1)2460 3945 y Fx(\022)e FG(F)2631
3960 y Fv(k)r Fy(+1)2791 3945 y Fx(\\)26 b FG(Q)2950
3960 y Fv(l)q Fy(+)p Fv(a)3109 3945 y FA(and)39 b FG(L)27
b Fx(\\)456 4056 y FG(B)525 4075 y Fv(r)r(=)p Fy(2)633
4056 y FA(\()p FG(Z)730 4070 y Fy(1)770 4056 y FA(\))f
Fx(\022)e FG(Q)998 4071 y Fv(l)1024 4056 y FA(,)31 b(therefore,)456
4251 y(\(A.204\))321 b Fi(L)1130 4213 y Fv(N)1198 4150
y Fu(\020)1252 4251 y FG(F)1310 4266 y Fv(k)r Fy(+1)1463
4251 y Fx(\\)20 b FG(Q)1616 4266 y Fv(l)1662 4251 y Fx(\\)g
FG(B)1812 4265 y Fv(r)1850 4251 y FA(\()p FG(Z)7 b FA(\))1989
4150 y Fu(\021)2084 4251 y Fx(\025)40 b Fi(L)2250 4213
y Fv(N)2318 4150 y Fu(\020)2372 4251 y FG(D)2447 4266
y Fv(k)r Fy(+1)2600 4251 y Fx(\\)20 b FG(B)2750 4269
y Fv(r)r(=)p Fy(2)2859 4251 y FA(\()p FG(Z)2956 4265
y Fy(1)2995 4251 y FA(\))3030 4150 y Fu(\021)3100 4251
y FG(:)456 4432 y FA(Finally)-8 b(,)42 b(from)f(\(P2\))h(of)f(Lemma)h
(4.3)g(\(whic)m(h)e(ma)m(y)i(b)s(e)f(used)f(thanks)h(to)h(\(A.201\))456
4540 y(and)29 b(\(A.200\)\),)34 b(w)m(e)d(ha)m(v)m(e)g(that)712
4722 y Fi(L)767 4684 y Fv(N)834 4621 y Fu(\020)889 4722
y FG(D)964 4737 y Fv(k)r Fy(+1)1117 4722 y Fx(\\)20 b
FG(B)1267 4740 y Ft(j)p Fv(Z)1335 4749 y Fq(1)1369 4740
y Ft(\000)p Fv(Z)1472 4749 y Fq(0)1506 4740 y Ft(j)p
Fv(=)p Fy(10)1636 4722 y FA(\()p FG(Z)1733 4736 y Fy(1)1773
4722 y FA(\))1808 4621 y Fu(\021)1945 4722 y Fx(\025)83
b FG(c)15 b Fi(L)2208 4684 y Fv(N)2276 4621 y Fu(\020)2330
4722 y FG(L)20 b Fx(\\)g FG(B)2562 4740 y Ft(j)p Fv(Z)2630
4749 y Fq(1)2664 4740 y Ft(\000)p Fv(Z)2767 4749 y Fq(0)2801
4740 y Ft(j)2825 4722 y FA(\()p FG(Z)2922 4736 y Fy(1)2962
4722 y FA(\))2997 4621 y Fu(\021)3092 4722 y Fx(\025)1945
4921 y(\025)83 b FG(c)15 b Fi(L)2208 4883 y Fv(N)2276
4820 y Fu(\020)2330 4921 y FG(L)20 b Fx(\\)g FG(B)2562
4939 y Fv(r)r(=)p Fy(2)2671 4921 y FA(\()p FG(Z)2768
4935 y Fy(1)2808 4921 y FA(\))2843 4820 y Fu(\021)2912
4921 y FG(;)-2481 b FA(\(A.205\))p 456 5025 499 4 v 555
5095 a Fy(23)626 5124 y FF(The)40 b(reader)h(will)h(notice)f(that)f
(\(A.201\))h(is)g(needed)f(in)g(order)h(to)g(use)f(prop)r(ert)n(y)g
(\(P2\))h(of)456 5216 y(Lemma)24 b(4.3)i(here)g(in)g(the)f(sequel.)p
eop
%%Page: 142 142
142 141 bop 456 251 a Fy(142)140 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y FA(where,)33 b(in)f(the)h(latter)h(estimate,)g(w)m(e)g
(used)e(again)h(\(A.200\).)52 b(Then,)33 b(\(A.193\))i(easily)456
558 y(follo)m(ws)29 b(in)g(this)g(case)j(from)e(\(A.204\),)j
(\(A.202\),)g(\(A.205\))g(and)c(\(A.203\).)555 768 y(Let)i(us)f(no)m(w)
g(deal)g(with)f(the)i(case)g(in)e(whic)m(h)g FG(a)d(>)f
FA(2)p FG(r)s FA(.)40 b(In)30 b(this)f(case,)456 959
y(\(A.206\))1744 898 y FG(r)23 b FA(+)c FG(a)p 1744 938
203 4 v 1800 1022 a FA(10)1982 959 y FG(<)25 b(a)15 b(:)456
1144 y FA(Since)29 b FG(Z)j Fx(2)25 b FG(L)30 b FA(and)g
FG(a)25 b Fx(\025)40 b FA(const)h FG(>)25 b FA(0,)1301
1325 y Fi(L)1356 1288 y Fv(N)1423 1225 y Fu(\020)1477
1325 y FG(B)1546 1339 y Fv(r)r Fy(+)p Fv(a)1677 1325
y FA(\()p FG(Z)7 b FA(\))20 b Fx(\\)g FG(L)1979 1225
y Fu(\021)2058 1325 y Fx(\025)41 b FA(const)g FG(>)24
b FA(0)15 b FG(:)456 1512 y FA(In)37 b(particular,)j(b)m(y)e(\(P2\))h
(of)g(Lemma)g(4.3,)j(\(A.196\))f(and)d(\(A.206\),)43
b(w)m(e)c(infer)e(from)456 1620 y(this)29 b(that)778
1784 y Fi(L)833 1746 y Fv(N)900 1683 y Fu(\020)955 1784
y FG(B)1024 1802 y Fy(\()p Fv(r)r Fy(+)p Fv(a)p Fy(\))p
Fv(=)p Fy(10)1315 1784 y FA(\()p FG(Z)7 b FA(\))20 b
Fx(\\)g FG(D)1630 1799 y Fv(k)r Fy(+1)1763 1683 y Fu(\021)1843
1784 y FA(=)25 b Fi(L)1994 1746 y Fv(N)2061 1683 y Fu(\020)2115
1784 y FG(B)2184 1802 y Ft(j)p Fv(Z)5 b Ft(\000)p Fv(Z)2360
1811 y Fq(0)2394 1802 y Ft(j)p Fv(=)p Fy(10)2523 1784
y FA(\()p FG(Z)i FA(\))21 b Fx(\\)f FG(D)2839 1799 y
Fv(k)r Fy(+1)2972 1683 y Fu(\021)3051 1784 y Fx(\025)1843
1983 y(\025)40 b FA(const)16 b Fi(L)2232 1945 y Fv(N)2299
1882 y Fu(\020)2353 1983 y FG(B)2422 2001 y Ft(j)p Fv(Z)5
b Ft(\000)p Fv(Z)2598 2010 y Fq(0)2632 2001 y Ft(j)2655
1983 y FA(\()p FG(Z)i FA(\))21 b Fx(\\)f FG(L)2958 1882
y Fu(\021)3037 1983 y FA(=)1843 2182 y(=)40 b(const)16
b Fi(L)2232 2144 y Fv(N)2299 2081 y Fu(\020)2353 2182
y FG(B)2422 2196 y Fv(r)r Fy(+)p Fv(a)2553 2182 y FA(\()p
FG(Z)7 b FA(\))20 b Fx(\\)g FG(L)2855 2081 y Fu(\021)2934
2182 y Fx(\025)1843 2339 y(\025)40 b FA(const)56 b FG(>)40
b FA(0)456 2498 y(thence,)31 b(there)f(exists)1423 2619
y FG(Z)1485 2634 y Fv(])1542 2619 y Fx(2)25 b FG(D)1703
2634 y Fv(k)r Fy(+1)1856 2619 y Fx(\\)20 b FG(B)2006
2638 y Fy(\()p Fv(r)r Fy(+)p Fv(a)p Fy(\))p Fv(=)p Fy(10)2297
2619 y FA(\()p FG(Z)7 b FA(\))15 b FG(:)456 2763 y FA(Also,)30
b(the)g(fact)i(that)f FG(Z)h Fx(2)24 b FG(Q)1470 2778
y Fv(l)1527 2763 y FA(and)29 b(\(A.206\))k(giv)m(e)e(that)1370
2922 y FG(D)1445 2937 y Fv(k)r Fy(+1)1598 2922 y Fx(\\)20
b FG(B)1748 2940 y Fy(\()p Fv(r)r Fy(+)p Fv(a)p Fy(\))p
Fv(=)p Fy(10)2039 2922 y FA(\()p FG(Z)7 b FA(\))26 b
Fx(\022)e FG(Q)2371 2937 y Fv(l)q Fy(+)p Fv(a)2505 2922
y FG(;)456 3084 y FA(thence)30 b FG(Z)805 3099 y Fv(])862
3084 y Fx(2)25 b FG(Q)1020 3099 y Fv(l)q Fy(+)p Fv(a)1138
3084 y FA(.)41 b(Hence,)32 b(the)e(fact)h(that)g FG(Z)2094
3099 y Fv(])2151 3084 y Fx(2)25 b FG(B)2306 3103 y Fy(\()p
Fv(r)r Fy(+)p Fv(a)p Fy(\))p Fv(=)p Fy(10)2597 3084 y
FA(\()p FG(Z)7 b FA(\))31 b(implies)c(that)456 3247 y(\(A.207\))656
b FG(Q)1482 3262 y Fv(l)1528 3247 y Fx(\\)20 b FG(B)1678
3261 y Fv(r)1716 3247 y FA(\()p FG(Z)7 b FA(\))25 b Fx(\022)g
FG(Q)2048 3262 y Fv(l)2094 3247 y Fx(\\)20 b FG(B)2244
3261 y Fv(a)2285 3247 y FA(\()p FG(Z)2382 3262 y Fv(])2414
3247 y FA(\))15 b FG(;)456 3405 y FA(while)28 b(the)j(fact)g(that)g
FG(Z)1288 3420 y Fv(])1345 3405 y Fx(2)25 b FG(D)1506
3420 y Fv(k)r Fy(+1)1669 3405 y FA(implies)j(that)456
3564 y(\(A.208\))783 b FG(Q)1609 3579 y Fv(l)1655 3564
y Fx(\\)20 b FG(B)1805 3578 y Fv(a)1846 3564 y FA(\()p
FG(Z)1943 3579 y Fv(])1975 3564 y FA(\))26 b Fx(\022)f
FG(F)2190 3579 y Fv(k)r Fy(+1)2338 3564 y FG(:)456 3723
y FA(Then,)k(from)h(\(A.207\))j(and)d(\(A.208\),)1571
3882 y FG(Q)1643 3897 y Fv(l)1689 3882 y Fx(\\)20 b FG(B)1839
3896 y Fv(r)1877 3882 y FA(\()p FG(Z)7 b FA(\))26 b Fx(\022)e
FG(F)2195 3897 y Fv(k)r Fy(+1)456 4041 y FA(and,)30 b(therefore,)1278
4162 y FG(F)1336 4177 y Fv(k)r Fy(+1)1489 4162 y Fx(\\)20
b FG(Q)1642 4177 y Fv(l)1688 4162 y Fx(\\)g FG(B)1838
4176 y Fv(r)1876 4162 y FA(\()p FG(Z)7 b FA(\))26 b(=)f
FG(Q)2209 4177 y Fv(l)2255 4162 y Fx(\\)19 b FG(B)2404
4176 y Fv(r)2442 4162 y FA(\()p FG(Z)7 b FA(\))15 b FG(:)456
4302 y FA(This)28 b(pro)m(v)m(es)j(\(A.193\))i(in)c(this)g(case)j
(\(with)f(\026)-47 b FG(c)26 b FA(=)f(1\).)555 4512 y(Ha)m(ving)46
b(completed)g(the)h(pro)s(of)e(of)h(\(A.193\))i(w)m(e)f(no)m(w)f(tak)m
(e)h(a)g(\014nite)d(o)m(v)m(erlap-)456 4620 y(ping)39
b(co)m(v)m(er)j Fi(C)e FA(of)g FG(Q)1189 4635 y Fv(l)1242
4620 y Fx(n)27 b FG(F)1372 4635 y Fv(k)1456 4620 y FA(with)39
b(balls)f(of)j(radius)d FG(r)s FA(,)43 b(in)c(order)h(to)h(end)f(the)g
(pro)s(of)456 4728 y(of)30 b(Lemma)h(4.3.)41 b(Th)m(us,)30
b(using)f(suc)m(h)h(co)m(v)m(er,)619 4909 y Fi(L)674
4872 y Fv(N)742 4808 y Fu(\020)796 4909 y FG(F)854 4924
y Fv(k)r Fy(+1)1007 4909 y Fx(\\)20 b FA(\()p FG(Q)1195
4924 y Fv(l)1242 4909 y Fx(n)g FG(F)1365 4924 y Fv(k)1408
4909 y FA(\))1443 4808 y Fu(\021)1581 4909 y FA(=)83
b Fi(L)1790 4872 y Fv(N)1857 4808 y Fu(\020)1912 4909
y FG(F)1970 4924 y Fv(k)r Fy(+1)2123 4909 y Fx(\\)20
b FG(Q)2276 4924 y Fv(l)2322 4909 y Fx(\\)g FA(\()p FG(Q)2510
4924 y Fv(l)2556 4909 y Fx(n)h FG(F)2680 4924 y Fv(k)2723
4909 y FA(\))2758 4808 y Fu(\021)2853 4909 y Fx(\025)1581
5108 y(\025)83 b FA(const)2049 5022 y Fu(X)1973 5223
y Fv(B)2026 5231 y Fs(r)2061 5223 y Fy(\()p Fv(Z)5 b
Fy(\))p Ft(2)p Ff(C)2272 5108 y Fi(L)2327 5071 y Fv(N)2395
5007 y Fu(\020)2449 5108 y FG(F)2507 5123 y Fv(k)r Fy(+1)2660
5108 y Fx(\\)20 b FG(Q)2813 5123 y Fv(l)2859 5108 y Fx(\\)g
FG(B)3009 5122 y Fv(r)3047 5108 y FA(\()p FG(Z)7 b FA(\))3186
5007 y Fu(\021)3256 5108 y FG(:)p eop
%%Page: 143 143
143 142 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)
f Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)140 b(143)456
450 y FA(Then,)29 b(using)g(\(A.193\),)k(w)m(e)e(deduce)f(from)g(the)h
(ab)s(o)m(v)m(e)g(that)589 633 y(const)17 b(\026)-47
b FG(c)850 596 y Ft(\000)p Fy(1)960 633 y Fi(L)1015 596
y Fv(N)1083 532 y Fu(\020)1137 633 y FG(F)1195 648 y
Fv(k)r Fy(+1)1348 633 y Fx(\\)20 b FA(\()p FG(Q)1536
648 y Fv(l)1583 633 y Fx(n)g FG(F)1706 648 y Fv(k)1749
633 y FA(\))1784 532 y Fu(\021)1922 633 y Fx(\025)2152
547 y Fu(X)2076 748 y Fv(B)2129 756 y Fs(r)2164 748 y
Fy(\()p Fv(Z)5 b Fy(\))p Ft(2)p Ff(C)2375 633 y Fi(L)2430
596 y Fv(N)2498 532 y Fu(\020)2552 633 y FG(Q)2624 648
y Fv(l)2670 633 y Fx(\\)20 b FG(B)2820 647 y Fv(r)2858
633 y FA(\()p FG(Z)7 b FA(\))2997 532 y Fu(\021)3092
633 y Fx(\025)1922 914 y(\025)83 b Fi(L)2131 876 y Fv(N)2198
813 y Fu(\020)2253 914 y FG(Q)2325 929 y Fv(l)2371 914
y Fx(\\)2452 813 y Fu(\020)2613 827 y([)2521 1029 y Fv(B)2574
1037 y Fs(r)2609 1029 y Fy(\()p Fv(Z)5 b Fy(\))p Ft(2)p
Ff(C)2820 914 y FG(B)2889 928 y Fv(r)2927 914 y FA(\()p
FG(Z)i FA(\))3066 813 y Fu(\021)q(\021)3216 914 y Fx(\025)1922
1194 y(\025)83 b Fi(L)2131 1157 y Fv(N)2198 1093 y Fu(\020)2253
1194 y FG(Q)2325 1209 y Fv(l)2371 1194 y Fx(\\)20 b FA(\()p
FG(Q)2559 1209 y Fv(l)2605 1194 y Fx(n)h FG(F)2729 1209
y Fv(k)2772 1194 y FA(\))2807 1093 y Fu(\021)2902 1194
y FA(=)1922 1371 y(=)83 b Fi(L)2131 1334 y Fv(N)2198
1371 y FA(\()p FG(Q)2305 1386 y Fv(l)2352 1371 y Fx(n)20
b FG(F)2475 1386 y Fv(k)2519 1371 y FA(\))15 b FG(:)-2138
b FA(\(A.209\))456 1527 y(F)-8 b(urthermore,)30 b(since)g
FG(D)1306 1542 y Fv(k)1374 1527 y Fx(\022)25 b FG(D)1545
1542 y Fv(k)r Fy(+1)1678 1527 y FA(,)31 b(w)m(e)f(ha)m(v)m(e)i(that)f
FG(F)2332 1542 y Fv(k)2400 1527 y Fx(\022)25 b FG(F)2554
1542 y Fv(k)r Fy(+1)2718 1527 y FA(and)30 b(so)456 1709
y(\(A.210\))310 b FG(Q)1136 1724 y Fv(l)1182 1709 y Fx(n)21
b FG(F)1306 1724 y Fv(k)r Fy(+1)1464 1709 y Fx(\022)1560
1608 y Fu(\020)1614 1709 y FG(Q)1686 1724 y Fv(l)1732
1709 y Fx(n)g FG(F)1856 1724 y Fv(k)1899 1608 y Fu(\021)1974
1709 y Fx(n)2039 1608 y Fu(\020)2094 1709 y FG(F)2152
1724 y Fv(k)r Fy(+1)2305 1709 y Fx(\\)f FA(\()p FG(Q)2493
1724 y Fv(l)2539 1709 y Fx(n)h FG(F)2663 1724 y Fv(k)2706
1709 y FA(\))2741 1608 y Fu(\021)2811 1709 y FG(:)456
1892 y FA(Hence,)31 b(b)m(y)f(using)f(\(A.210\))k(and)d(\(A.209\),)745
2075 y Fi(L)800 2037 y Fv(N)868 2075 y FA(\()p FG(Q)975
2090 y Fv(l)1021 2075 y Fx(n)21 b FG(F)1145 2090 y Fv(k)r
Fy(+1)1278 2075 y FA(\))84 b Fx(\024)e Fi(L)1605 2037
y Fv(N)1673 2075 y FA(\()p FG(Q)1780 2090 y Fv(l)1826
2075 y Fx(n)21 b FG(F)1950 2090 y Fv(k)1993 2075 y FA(\))g
Fx(\000)e Fi(L)2194 2037 y Fv(N)2262 1974 y Fu(\020)2316
2075 y FG(F)2374 2090 y Fv(k)r Fy(+1)2528 2075 y Fx(\\)h
FA(\()p FG(Q)2716 2090 y Fv(l)2762 2075 y Fx(n)h FG(F)2886
2090 y Fv(k)2929 2075 y FA(\))2964 1974 y Fu(\021)3059
2075 y Fx(\024)1397 2252 y(\024)82 b FA(\(1)21 b Fx(\000)h
FA(^)-47 b FG(c)p FA(\))15 b Fi(L)1886 2214 y Fv(N)1955
2252 y FA(\()p FG(Q)2062 2267 y Fv(l)2108 2252 y Fx(n)21
b FG(F)2232 2267 y Fv(k)2275 2252 y FA(\))15 b FG(;)456
2407 y FA(for)30 b(a)h(suitable)f(^)-47 b FG(c)q FA(.)40
b(Therefore,)31 b(iterating)f(the)g(ab)s(o)m(v)m(e)i(estimate,)1322
2564 y Fi(L)1377 2527 y Fv(N)1445 2564 y FA(\()p FG(Q)1552
2579 y Fv(l)1598 2564 y Fx(n)21 b FG(F)1722 2579 y Fv(k)1765
2564 y FA(\))26 b Fx(\024)f FA(\(1)c Fx(\000)h FA(^)-47
b FG(c)p FA(\))2188 2527 y Fv(k)2246 2564 y Fi(L)2301
2527 y Fv(N)2369 2564 y FA(\()p FG(Q)2476 2579 y Fv(l)2502
2564 y FA(\))15 b FG(:)456 2720 y FA(This)28 b(completes)j(the)f(pro)s
(of)g(of)h(Lemma)f(4.3)i(since)d FG(E)2341 2735 y Fv(k)2409
2720 y Fx(\023)c FG(F)2563 2735 y Fv(k)2637 2720 y FA(b)m(y)30
b(construction.)888 2931 y FB(Appendix)j FA(B.)46 b FB(Summar)-6
b(y)33 b(of)g(element)-6 b(ar)g(y)32 b(lemma)-6 b(t)g(a)555
3093 y FA(W)e(e)39 b(collect)f(here)f(some)h(lemmata)f(that)h(are)g(in)
e(use)h(during)f(the)h(pro)s(ofs)g(of)g(the)456 3200
y(main)31 b(results.)45 b(W)-8 b(e)34 b(will)c(skip)g(the)j(pro)s(ofs)e
(of)i(most)g(of)f(these)h(lemmata,)g(since)f(they)456
3308 y(are)j(quite)g(elemen)m(tary)h(\(a)h(detailed)d(pro)s(of)h(of)g
(them,)i(ho)m(w)m(ev)m(er,)i(ma)m(y)d(b)s(e)f(found)f(in)456
3416 y([30)q(]\).)456 3548 y FH(Lemma)e(B.1.)43 b Fo(F)-7
b(or)33 b(any)g FA(0)26 b Fx(\024)f FG(s)g Fx(\024)g
FG(t)g Fx(\024)g FG(\022)1927 3515 y Fv(?)1966 3548 y
Fo(,)1198 3703 y FG(h)1250 3717 y Fy(0)1290 3703 y FA(\()p
Fx(\000)p FA(1)c(+)f FG(t)p FA(\))g Fx(\000)g FG(h)1784
3717 y Fy(0)1824 3703 y FA(\()p Fx(\000)p FA(1)g(+)g
FG(s)p FA(\))26 b Fx(\025)e FG(c)p FA(\()p FG(t)2392
3666 y Fv(p)2453 3703 y Fx(\000)c FG(s)2587 3666 y Fv(p)2626
3703 y FA(\))15 b FG(;)456 3859 y Fo(for)33 b(a)g(suitable)g(universal)
g(c)-5 b(onstant)34 b FG(c)26 b(>)f FA(0)p Fo(.)456 3990
y FH(Lemma)39 b(B.2.)46 b Fo(L)-5 b(et)38 b FG(U)48 b
Fo(b)-5 b(e)38 b(an)g(op)-5 b(en)39 b(subset)f(of)h Fw(R)r
Fo(.)64 b(L)-5 b(et)39 b FG(g)f Fx(2)d FG(C)2757 3957
y Fy(2)2796 3990 y FA(\()p FG(U)10 b FA(\))39 b Fo(and)g(assume)456
4098 y(that)33 b FG(g)k Fo(has)c(no)g(critic)-5 b(al)34
b(p)-5 b(oints.)43 b(De\014ne)456 4255 y FA(\(B.1\))804
b(\011)1535 4218 y Fv(y)r(;l)1618 4255 y FA(\()p FG(x)p
FA(\))41 b(:=)f FG(g)s FA(\()p Fx(j)p FG(x)22 b Fx(\000)d
FG(y)s Fx(j)i(\000)f FG(l)r FA(\))456 4411 y Fo(Then,)32
b(for)i FG(t)25 b FA(=)g Fx(j)p FG(x)20 b Fx(\000)g FG(y)s
Fx(j)g(\000)g FG(l)27 b Fx(2)e FG(U)43 b Fo(and)33 b
FG(x)26 b Fx(6)p FA(=)e FG(y)s Fo(,)33 b(we)f(have)456
4613 y FA(\(B.2\))211 b(\001)947 4627 y Fv(p)986 4613
y FA(\(\011)1092 4575 y Fv(y)r(;l)1175 4613 y FA(\()p
FG(x)p FA(\)\))27 b(=)e(\()p FG(p)20 b Fx(\000)g FA(1\))p
FG(g)1773 4575 y Ft(00)1817 4613 y FA(\()p FG(t)p FA(\))p
FG(g)1966 4575 y Ft(0)1990 4564 y Fy(\()p Fv(p)p Ft(\000)p
Fy(2\))2175 4613 y FA(\()p FG(t)p FA(\))h(+)e FG(g)2435
4575 y Ft(0)2459 4564 y Fy(\()p Fv(p)p Ft(\000)p Fy(1\))2644
4613 y FA(\()p FG(t)p FA(\))2769 4551 y FG(N)30 b Fx(\000)20
b FA(1)p 2757 4592 262 4 v 2757 4675 a Fx(j)p FG(x)h
Fx(\000)f FG(y)s Fx(j)456 4829 y FH(Lemma)48 b(B.3.)i
Fo(L)-5 b(et)45 b FG(I)55 b Fx(3)48 b FA(0)d Fo(b)-5
b(e)45 b(an)h(interval)g(of)f Fw(R)54 b Fo(and)46 b(let)f
FG(h)j Fx(2)g FG(C)2992 4796 y Fy(1)3031 4829 y FA(\()p
FG(I)7 b FA(\))45 b Fo(satisfy)456 4937 y FG(h)p FA(\()p
FG(s)p FA(\))26 b FG(>)f FA(0)32 b Fo(for)i(any)f FG(s)25
b Fx(2)g FG(I)7 b Fo(.)41 b(L)-5 b(et)1169 5160 y FG(H)7
b FA(\()p FG(s)p FA(\))41 b(:=)1542 5036 y Fu(Z)1633
5062 y Fv(s)1593 5242 y Fy(0)1695 5098 y FA(\()p FG(p)21
b Fx(\000)e FA(1\))1967 5065 y Fy(1)p Fv(=p)2094 5098
y FG(d\020)p 1695 5139 493 4 v 1736 5226 a FA(\()p FG(p)c(h)p
FA(\()p FG(\020)7 b FA(\)\))2036 5200 y Fy(1)p Fv(=p)2213
5160 y FG(;)201 b Fx(8)p FG(s)24 b Fx(2)h FG(I)d(:)p
eop
%%Page: 144 144
144 143 bop 456 251 a Fy(144)140 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y Fo(De\014ne)33 b(also)i FG(g)i Fo(as)d(the)g(inverse)g
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FA(\))28 b(:=)f FG(H)2515 417 y Ft(\000)p Fy(1)2609 450
y FA(\()p FG(t)p FA(\))34 b Fo(for)h(any)f FG(t)27 b
Fx(2)g FG(H)7 b FA(\()p FG(I)g FA(\))p Fo(.)456 558 y(Then,)32
b FG(g)d Fx(2)c FG(C)950 525 y Fy(2)1004 558 y FA(\()q
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b(=)1630 684 y Fu(\020)1772 724 y FG(p)p 1694 764 203
4 v 1694 847 a(p)20 b Fx(\000)g FA(1)1921 785 y FG(h)p
FA(\()p FG(g)s FA(\()p FG(t)p FA(\)\))2192 684 y Fu(\021)2249
707 y Fy(1)p Fv(=p)1200 1123 y FG(g)1246 1086 y Ft(00)1289
1123 y FA(\()p FG(t)p FA(\))84 b(=)1639 928 y Fu(\020)1694
1029 y FG(p)15 b(h)p FA(\()p FG(g)s FA(\()p FG(t)p FA(\)\))2026
928 y Fu(\021)2082 950 y Fy(\(2)p Ft(\000)p Fv(p)p Fy(\))p
Fv(=p)p 1639 1102 699 4 v 1797 1189 a FA(\()p FG(p)20
b Fx(\000)g FA(1\))2069 1163 y Fy(2)p Fv(=p)2363 1123
y FG(h)2415 1086 y Ft(0)2439 1123 y FA(\()p FG(g)s FA(\()p
FG(t)p FA(\)\))15 b FG(;)456 1356 y Fo(for)33 b(any)g
FG(t)25 b Fx(2)g FG(H)7 b FA(\()p FG(I)g FA(\))p Fo(.)555
1498 y FA(W)-8 b(e)40 b(recall)d(no)m(w)i(the)f(maxim)m(um)f(and)h
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y(purp)s(oses.)53 b(First)35 b(of)g(all,)h(in)e([9)q(])h(\(see)h(in)e
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1714 y(lo)m(wing)29 b(result)g(w)m(as)i(obtained:)456
1856 y FH(Theorem)47 b(B.4)i FA(\(Strong)42 b(Comparison)e(Principle)f
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b(e)43 b(an)h(op)-5 b(en)44 b(\(not)456 1966 y(ne)-5
b(c)g(essarily)37 b(b)-5 b(ounde)g(d)38 b(nor)e(c)-5
b(onne)g(cte)g(d\))38 b(subset)e(of)g Fw(R)2329 1933
y Fv(N)2402 1966 y Fo(,)h FA(\003)31 b Fx(2)g Fw(R)44
b Fo(and)37 b FG(u;)15 b(v)35 b Fx(2)c FG(C)3269 1933
y Fy(1)3308 1966 y FA(\(\012\))456 2073 y Fo(satisfy)456
2249 y FA(\(B.3\))393 b Fx(\000)p FA(\001)1200 2263 y
Fv(p)1240 2249 y FG(u)20 b FA(+)g(\003)p FG(u)25 b Fx(\024)g(\000)p
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FG(v)s(;)202 b(u)25 b Fx(\024)g FG(v)36 b Fo(in)d FA(\012)p
FG(:)456 2425 y Fo(De\014ne)g FG(Z)806 2439 y Fv(u;v)936
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FG(D)s(v)s FA(\()p FG(x)p FA(\))p Fx(j)29 b FA(=)f(0)p
Fx(g)36 b Fo(if)e FG(p)28 b Fx(6)p FA(=)g(2)p Fo(,)35
b FG(Z)2799 2439 y Fv(u;v)2929 2425 y FA(=)28 b Fx(;)35
b Fo(if)f FG(p)28 b FA(=)g(2)p Fo(.)456 2533 y(If)33
b FG(x)604 2547 y Fy(0)669 2533 y Fx(2)26 b FA(\012)20
b Fx(n)i FG(Z)971 2547 y Fv(u;v)1105 2533 y Fo(and)34
b FG(u)p FA(\()p FG(x)1421 2547 y Fy(0)1461 2533 y FA(\))27
b(=)f FG(v)s FA(\()p FG(x)1754 2547 y Fy(0)1794 2533
y FA(\))p Fo(,)33 b(then)h FG(u)26 b FA(=)g FG(v)36 b
Fo(in)e(the)f(c)-5 b(onne)g(cte)g(d)35 b(c)-5 b(omp)g(onent)456
2641 y(of)32 b FA(\012)20 b Fx(n)h FG(Z)776 2655 y Fv(u;v)910
2641 y Fo(c)-5 b(ontaining)34 b FG(x)1406 2655 y Fy(0)1445
2641 y Fo(.)555 2783 y FA(An)j(easy)g(consequence)h(of)f(the)g(ab)s(o)m
(v)m(e)h(result)d(is)h(the)h(follo)m(wing)e(one)i(\(see)h
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b(e)38 b(an)g(op)-5 b(en)39 b(\(not)456 3142 y(ne)-5
b(c)g(essarily)34 b(b)-5 b(ounde)g(d)34 b(nor)f(c)-5
b(onne)g(cte)g(d\))35 b(subset)d(of)h Fw(R)2309 3109
y Fv(N)2382 3142 y Fo(,)f(and)i FG(u;)15 b(v)29 b Fx(2)c
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3318 y FA(\(B.4\))324 b Fx(\000)p FA(\001)1131 3332 y
Fv(p)1170 3318 y FG(u)20 b FA(+)g FG(f)10 b FA(\()p FG(u)p
FA(\))26 b Fx(\024)e(\000)p FA(\001)1778 3332 y Fv(p)1818
3318 y FA(\()p FG(v)s FA(\))d(+)f FG(f)10 b FA(\()p FG(v)s
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FA(\012)15 b FG(;)456 3494 y Fo(with)45 b FG(f)54 b Fo(lo)-5
b(c)g(al)5 b(ly)46 b(Lipschitz)g(c)-5 b(ontinuous.)78
b(De\014ne)44 b FG(Z)2340 3508 y Fv(u;v)2488 3494 y FA(=)j
Fx(f)p FG(x)g Fx(2)f FA(\012)h(:)g Fx(j)p FG(D)s(u)p
FA(\()p FG(x)p FA(\))p Fx(j)30 b FA(+)456 3602 y Fx(j)p
FG(D)s(v)s FA(\()p FG(x)p FA(\))p Fx(j)c FA(=)f(0)p Fx(g)30
b Fo(if)f FG(p)c Fx(6)p FA(=)g(2)p Fo(,)30 b FG(Z)1412
3616 y Fv(u;v)1539 3602 y FA(=)25 b Fx(;)k Fo(if)g FG(p)c
FA(=)g(2)p Fo(.)41 b(If)29 b FG(x)2219 3616 y Fy(0)2284
3602 y Fx(2)c FA(\012)13 b Fx(n)g FG(Z)2569 3616 y Fv(u;v)2699
3602 y Fo(and)30 b FG(u)p FA(\()p FG(x)3011 3616 y Fy(0)3051
3602 y FA(\))25 b(=)g FG(v)s FA(\()p FG(x)3341 3616 y
Fy(0)3381 3602 y FA(\))p Fo(,)456 3710 y(then)33 b FG(u)25
b FA(=)g FG(v)36 b Fo(in)c(the)h(c)-5 b(onne)g(cte)g(d)35
b(c)-5 b(omp)g(onent)35 b(of)d FA(\012)20 b Fx(n)h FG(Z)2358
3724 y Fv(u;v)2492 3710 y Fo(c)-5 b(ontaining)34 b FG(x)2988
3724 y Fy(0)3027 3710 y Fo(.)555 3852 y FA(As)e(w)m(ell)g(kno)m(wn,)g
(the)h(\\dangerous")g(p)s(oin)m(ts)e(in)g(dealing)g(with)g
FG(p)p FA(-Laplace)h(op)s(era-)456 3960 y(tors)j(are)h(the)g(ones)g(in)
e(whic)m(h)h(the)g(gradien)m(t)h(v)-5 b(anishes,)36 b(due)f(to)h(lac)m
(k)g(of)g(ellipticit)m(y)-8 b(.)456 4068 y(Next)32 b(result,)f(pro)m(v)
m(ed)h(in)e([37)q(])i(\(see)g(also)f([30)r(])g(for)g(details\),)h(will)
d(help)h(us)g(in)h(dealing)456 4176 y(with)e(this)g(circumstance.)456
4318 y FH(Theorem)34 b(B.6)j FA(\(Strong)31 b(Maxim)m(um)g(Principle)d
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b FA(\012)e Fo(b)-5 b(e)456 4427 y(an)43 b(op)-5 b(en)45
b(c)-5 b(onne)g(cte)g(d)45 b(\(not)f(ne)-5 b(c)g(essarily\))45
b(b)-5 b(ounde)g(d)45 b(set)e(in)h Fw(R)2644 4394 y Fv(N)2761
4427 y Fo(and)g(supp)-5 b(ose)45 b(that)456 4535 y FG(u)25
b Fx(2)g FG(C)691 4502 y Fy(1)730 4535 y FA(\(\012\))p
Fo(,)32 b FG(u)26 b Fx(\025)f FA(0)33 b Fo(in)f FA(\012)p
Fo(,)g(we)-5 b(akly)34 b(solves)1362 4711 y Fx(\000)p
FA(\001)1509 4725 y Fv(p)1548 4711 y FG(u)20 b FA(+)g
FG(cu)1802 4673 y Fv(q)1866 4711 y FA(=)25 b FG(g)k Fx(\025)c
FA(0)93 b Fo(in)125 b FA(\012)456 4887 y Fo(with)40 b
FG(q)g Fx(\025)c FG(p)25 b Fx(\000)g FA(1)p Fo(,)41 b
FG(c)c Fx(\025)g FA(0)i Fo(and)h FG(g)h Fx(2)c FG(L)1824
4854 y Ft(1)1824 4914 y Fv(l)q(oc)1914 4887 y FA(\(\012\))p
Fo(.)62 b(If)39 b FG(u)g Fo(is)g(not)h(identic)-5 b(al)5
b(ly)40 b(zer)-5 b(o,)42 b(then)456 4995 y FG(u)e(>)g
FA(0)h Fo(in)g FA(\012)p Fo(.)66 b(Mor)-5 b(e)g(over,)44
b(for)e(any)f(p)-5 b(oint)43 b FG(x)2090 5009 y Fy(0)2169
4995 y Fx(2)d FG(@)5 b FA(\012)41 b Fo(wher)-5 b(e)42
b(the)f(interior)h(spher)-5 b(e)456 5103 y(c)g(ondition)37
b(is)e(satis\014e)-5 b(d,)37 b(and)f(such)g(that)g FG(u)f
Fo(is)h FG(C)2170 5070 y Fy(1)2244 5103 y Fo(in)f(a)g(neighb)-5
b(orho)g(o)g(d)39 b(of)d FA(\012)21 b Fx([)h(f)p FG(x)3359
5117 y Fy(0)3399 5103 y Fx(g)456 5216 y Fo(and)33 b FG(u)p
FA(\()p FG(x)771 5230 y Fy(0)811 5216 y FA(\))26 b(=)f(0)p
Fo(,)32 b(we)h(have)g(that)1613 5180 y Fv(@)t(u)p 1613
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b Fo(for)g(any)h(inwar)-5 b(d)34 b(dir)-5 b(e)g(ctional)35
b(derivative.)p eop
%%Page: 145 145
145 144 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)
f Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)140 b(145)555
450 y FA(F)-8 b(ollo)m(wing)43 b(are)h(some)g(easy)g(result)e(on)h(the)
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b(elemen)m(tary)-8 b(,)30 b(w)m(e)e(giv)m(e)g(full)d(details)i(of)h
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666 y(of)g(the)h(constan)m(ts)g(in)m(v)m(olv)m(ed.)456
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y Fo(and)h FG(c)2755 808 y Fy(2)2830 794 y Fo(b)-5 b(e)36
b(non-ne)-5 b(gative)456 902 y(and)33 b(so)g(that)456
1050 y FA(\(B.5\))644 b(max)p Fx(f)p FG(c)1557 1064 y
Fy(1)1597 1050 y FG(;)41 b(c)1702 1064 y Fy(2)1741 1050
y Fx(g)26 b(\024)f FA(min)o Fx(f)2105 980 y(p)p 2181
980 37 4 v 70 x FG(\017=)p FA(3)p FG(;)41 b FA(1)p FG(=)p
FA(2)p Fx(g)15 b FG(:)456 1199 y Fo(Supp)-5 b(ose)768
1164 y Fy(24)872 1199 y Fo(that)34 b FG(z)c Fx(2)25 b
FG(B)1285 1213 y Fv(r)1322 1199 y FA(\()p FG(y)s FA(\))h
Fx(\032)f Fw(R)1622 1166 y Fv(N)1695 1199 y Fo(,)33 b(with)g
Fx(j)p FG(z)2024 1166 y Ft(0)2068 1199 y Fx(\000)20 b
FG(y)2207 1166 y Ft(0)2230 1199 y Fx(j)26 b(\024)f FG(c)2416
1213 y Fy(1)2455 1199 y FG(q)s Fo(.)42 b(Then,)1125 1375
y FG(@)5 b(B)1247 1389 y Fv(r)1285 1375 y FA(\()p FG(y)s
FA(\))20 b Fx(\\)1504 1274 y Fu(n)1565 1375 y Fx(j)p
FG(x)1642 1337 y Ft(0)1686 1375 y Fx(\000)g FG(z)1823
1337 y Ft(0)1846 1375 y Fx(j)26 b(\024)f FG(c)2032 1389
y Fy(2)2071 1375 y FG(q)2115 1274 y Fu(o)2196 1375 y
Fx(\\)2277 1274 y Fu(n)2337 1375 y FG(x)2389 1389 y Fv(N)2482
1375 y Fx(\025)g FG(y)2623 1389 y Fv(N)2690 1274 y Fu(o)2775
1375 y Fx(\022)1054 1607 y(\022)1125 1506 y Fu(n)1185
1607 y FG(x)1237 1621 y Fv(N)1330 1607 y Fx(\025)g FG(z)1468
1621 y Fv(N)1555 1607 y Fx(\000)1656 1545 y FG(\017)15
b(q)1752 1512 y Fy(2)p 1656 1586 136 4 v 1702 1669 a
FG(r)1801 1506 y Fu(o)1877 1607 y FG(:)456 1493 y FA(\(B.6\))456
1835 y Fo(Pr)-5 b(o)g(of.)43 b FA(T)-8 b(ak)m(e)29 b
FG(w)f Fx(2)c FG(@)5 b(B)1254 1849 y Fv(r)1293 1835 y
FA(\()p FG(y)s FA(\))15 b Fx(\\)1502 1734 y Fu(n)1563
1835 y Fx(j)p FG(x)1640 1802 y Ft(0)1679 1835 y Fx(\000)g
FG(z)1811 1802 y Ft(0)1834 1835 y Fx(j)26 b(\024)f FG(c)2020
1849 y Fy(2)2059 1835 y FG(q)2103 1734 y Fu(o)2179 1835
y Fx(\\)2255 1734 y Fu(n)2316 1835 y FG(x)2368 1849 y
Fv(N)2460 1835 y Fx(\025)g FG(y)2601 1849 y Fv(N)2668
1734 y Fu(o)2728 1835 y FA(.)40 b(F)-8 b(or)29 b(an)m(y)f
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2393 y(^)-51 b FG(z)30 b Fx(2)25 b FG(B)1958 2407 y Fy(1)1997
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b FG(z)1642 2519 y Ft(0)1666 2552 y Fx(j)26 b(\024)1822
2491 y FG(c)1861 2505 y Fy(1)1916 2491 y FG(q)p 1822
2531 138 4 v 1870 2614 a(r)2182 2552 y FA(and)1070 2754
y(^)-64 b FG(w)28 b Fx(2)d FG(@)5 b(B)1352 2768 y Fy(1)1392
2754 y FA(\(0\))21 b Fx(\\)1609 2653 y Fu(n)1669 2754
y Fx(j)6 b FA(^)-51 b FG(x)1746 2721 y Ft(0)1790 2754
y Fx(\000)25 b FA(^)-50 b FG(z)1927 2721 y Ft(0)1951
2754 y Fx(j)25 b(\024)2107 2693 y FG(c)2146 2707 y Fy(2)2186
2693 y FG(q)p 2107 2733 123 4 v 2147 2817 a(r)2240 2653
y Fu(o)2320 2754 y Fx(\\)2401 2653 y Fu(n)2468 2754 y
FA(^)-51 b FG(x)2514 2768 y Fv(N)2606 2754 y Fx(\025)25
b FA(0)2747 2653 y Fu(o)2823 2754 y FG(:)456 2928 y FA(Let)30
b(also)h FG(t)25 b Fx(\025)g FA(0)31 b(so)f(that)1489
3014 y(^)1493 3038 y FG(b)25 b FA(:=)31 b(^)-51 b FG(z)25
b FA(+)20 b FG(te)1911 3052 y Fv(N)2003 3038 y Fx(2)25
b FG(@)5 b(B)2211 3052 y Fy(1)2251 3038 y FA(\(0\))15
b FG(:)456 3167 y FA(W)-8 b(e)31 b(claim)f(that)456 3353
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y Fx(\025)1867 3329 y FA(^)1870 3353 y FG(b)1909 3367
y Fv(N)1997 3353 y Fx(\000)2098 3292 y FG(\017)15 b(q)2194
3259 y Fy(2)p 2098 3332 136 4 v 2124 3415 a FG(r)2168
3389 y Fy(2)2258 3353 y FG(:)456 3551 y FA(T)-8 b(o)33
b(pro)m(v)m(e)g(this,)f(\014rst)g(note)h(that,)g(if)50
b(^)-63 b FG(w)1813 3565 y Fv(N)1909 3551 y Fx(\025)2005
3527 y FA(^)2008 3551 y FG(b)2047 3565 y Fv(N)2115 3551
y FA(,)33 b(\(B.7\))h(is)d(ob)m(vious;)i(hence,)h(w)m(e)f(ma)m(y)456
3659 y(assume)d(that)456 3807 y(\(B.8\))1093 b(^)-63
b FG(w)1800 3821 y Fv(N)1907 3807 y FG(<)2015 3783 y
FA(^)2019 3807 y FG(b)2058 3821 y Fv(N)2140 3807 y FG(:)456
3965 y FA(Also,)29 b(if)763 3941 y(^)766 3965 y FG(b)805
3979 y Fv(N)898 3965 y Fx(\024)c FA(0,)30 b(\(B.7\))g(w)m(ould)e(follo)
m(w)g(from)g(the)h(fact)h(that)47 b(^)-63 b FG(w)2659
3979 y Fv(N)2751 3965 y Fx(\025)25 b FA(0,)30 b(th)m(us)f(w)m(e)g(ma)m
(y)456 4073 y(also)h(supp)s(ose)f(that)456 4226 y(\(B.9\))1790
4202 y(^)1793 4226 y FG(b)1832 4240 y Fv(N)1925 4226
y FG(>)c FA(0)15 b FG(:)456 4374 y FA(Also,)1504 4527
y(1)83 b(=)g Fx(j)1808 4503 y FA(^)1811 4527 y FG(b)1850
4490 y Ft(0)1873 4527 y Fx(j)1898 4490 y Fy(2)1958 4527
y FA(+)20 b Fx(j)2071 4503 y FA(^)2074 4527 y FG(b)2113
4541 y Fv(N)2181 4527 y Fx(j)2206 4490 y Fy(2)2271 4527
y FA(=)1632 4673 y(=)83 b Fx(j)5 b FA(^)-50 b FG(z)1857
4635 y Ft(0)1881 4673 y Fx(j)1906 4635 y Fy(2)1966 4673
y FA(+)19 b Fx(j)2078 4649 y FA(^)2081 4673 y FG(b)2120
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y Fx(\024)1632 4868 y(\024)1796 4806 y FG(c)1835 4773
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1796 4847 163 4 v 1835 4930 a FG(r)1879 4904 y Fy(2)1988
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4882 y Fv(N)2210 4868 y Fx(j)2235 4830 y Fy(2)2300 4868
y Fx(\024)1632 5053 y(\024)83 b FG(c)1825 5015 y Fy(2)1825
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5053 y FG(b)2040 5067 y Fv(N)2107 5053 y Fx(j)2132 5015
y Fy(2)2187 5053 y FG(;)p 456 5116 499 4 v 555 5187 a
Fy(24)646 5216 y FF(Notation)i(remark:)31 b(in)20 b(Lemma)g(B.7,)j(for)
e(de\014niteness,)h(the)f(balls)g(are)g(assumed)g(to)f(b)r(e)h(closed.)
p eop
%%Page: 146 146
146 145 bop 456 251 a Fy(146)140 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
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c(that)456 648 y(\(B.10\))1780 624 y(^)1783 648 y FG(b)1822
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627 46 4 v 2021 710 a(2)2091 648 y FG(;)456 833 y FA(thanks)i(to)h
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Fy(2)1937 997 y FA(+)20 b Fx(j)2050 973 y FA(^)2053 997
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b FG(z)2323 1692 y Ft(0)2347 1730 y Fx(j)2372 1629 y
Fu(\021)2426 1651 y Fy(2)2486 1730 y FA(+)20 b Fx(j)p
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Fx(\000)20 b(j)f FA(^)-64 b FG(w)1408 2090 y Fv(N)1476
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Fx(\024)1624 2506 y(\024)1787 2445 y FG(\017)15 b(q)1883
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4485 y Fy(2)1708 4518 y FG(=r)s FA(\))28 b Fx(2)f Fw(R)2008
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FA(\))21 b Fx(\\)1525 5056 y Fu(n)1585 5156 y Fx(j)p
FG(x)1662 5119 y Ft(0)1706 5156 y Fx(\000)f FG(y)1845
5119 y Ft(0)1868 5156 y Fx(j)25 b(\025)g FA(4)p FG(\024q)2155
5056 y Fu(o)2242 5156 y Fx(\022)g(f)p FG(x)2435 5170
y Fv(N)2528 5156 y FG(<)g FA(0)p Fx(g)15 b FG(:)p eop
%%Page: 147 147
147 146 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)
f Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)140 b(147)456
472 y Fo(Pr)-5 b(o)g(of.)43 b FA(T)-8 b(ak)m(e)31 b FG(w)d
Fx(2)d FG(@)5 b(B)1257 486 y Fv(r)1295 472 y FA(\()p
FG(y)s FA(\))21 b Fx(\\)1515 371 y Fu(n)1575 472 y Fx(j)p
FG(x)1652 439 y Ft(0)1696 472 y Fx(\000)f FG(y)1835 439
y Ft(0)1858 472 y Fx(j)25 b(\025)g FA(4)p FG(\024q)2145
371 y Fu(o)2207 472 y FA(.)40 b(Let)31 b(also)g FG(t)25
b Fx(\025)g FA(0)30 b(b)s(e)g(so)h(that)456 665 y(\(B.13\))494
b FG(p)25 b FA(:=)g FG(v)f Fx(\000)c FG(te)1625 679 y
Fv(N)1717 665 y Fx(2)25 b FG(@)5 b(B)1925 679 y Fv(r)1964
665 y FA(\()p FG(y)s FA(\))30 b(with)f FG(p)2365 679
y Fv(N)2458 665 y Fx(\025)c FG(y)2599 679 y Fv(N)2665
665 y FA(.)456 830 y(Note)31 b(that)456 995 y(\(B.14\))729
b Fx(j)p FG(p)1505 958 y Ft(0)1549 995 y Fx(\000)20 b
FG(y)1688 958 y Ft(0)1711 995 y Fx(j)40 b FA(=)g Fx(j)p
FG(v)1959 958 y Ft(0)2003 995 y Fx(\000)20 b FG(y)2142
958 y Ft(0)2166 995 y Fx(j)40 b(\024)g FG(cq)19 b(:)456
1161 y FA(Also,)30 b(b)m(y)g(our)g(assumptions,)1340
1312 y FG(w)1405 1326 y Fv(N)1497 1312 y Fx(\024)25 b
FG(y)1638 1326 y Fv(N)1725 1312 y FA(+)20 b Fx(j)p FG(w)1906
1326 y Fv(N)1994 1312 y Fx(\000)g FG(y)2130 1326 y Fv(N)2197
1312 y Fx(j)25 b FA(=)1497 1474 y(=)g FG(y)1638 1488
y Fv(N)1725 1474 y FA(+)1816 1391 y Fu(p)p 1907 1391
558 4 v 83 x FG(r)1951 1448 y Fy(2)2010 1474 y Fx(\000)20
b(j)p FG(w)2193 1448 y Ft(0)2238 1474 y Fx(\000)f FG(y)2376
1448 y Ft(0)2400 1474 y Fx(j)2425 1448 y Fy(2)2489 1474
y Fx(\024)1497 1638 y(\024)25 b FG(y)1638 1652 y Fv(N)1725
1638 y FA(+)1816 1553 y Fu(p)p 1907 1553 461 4 v 85 x
FG(r)1951 1612 y Fy(2)2010 1638 y Fx(\000)20 b FA(16)p
FG(\024)2243 1612 y Fy(2)2284 1638 y FG(q)2328 1612 y
Fy(2)2382 1638 y FG(:)456 1483 y FA(\(B.15\))456 1804
y(Let)31 b FG(z)e FA(:=)d FG(y)d FA(+)d FG(r)s(e)1057
1818 y Fv(N)1124 1804 y FA(.)41 b(W)-8 b(e)32 b(no)m(w)f(apply)e(Lemma)
i(B.7)g(with)e FG(\017)d FA(:=)f FG(\024)31 b FA(and)f
FG(c)2967 1818 y Fy(1)3032 1804 y FA(:=)c FG(c)3193 1818
y Fy(2)3258 1804 y FA(:=)g FG(c)p FA(,)456 1912 y(recalling)i
(\(B.12\).)43 b(Indeed,)30 b(b)m(y)g(\(B.13\),)j(\(B.14\))g(and)d
(Lemma)g(B.7,)1023 2105 y FG(p)25 b Fx(2)p FG(@)5 b(B)1277
2119 y Fv(r)1315 2105 y FA(\()p FG(y)s FA(\))21 b Fx(\\)1535
2004 y Fu(n)1595 2105 y Fx(j)p FG(x)1672 2067 y Ft(0)1716
2105 y Fx(\000)f FG(z)1853 2067 y Ft(0)1877 2105 y Fx(j)25
b(\024)g FG(c)2062 2119 y Fy(2)2102 2105 y FG(q)2146
2004 y Fu(o)2226 2105 y Fx(\\)2307 2004 y Fu(n)2368 2105
y FG(x)2420 2119 y Fv(N)2512 2105 y Fx(\025)g FG(y)2653
2119 y Fv(N)2720 2004 y Fu(o)2806 2105 y Fx(\022)1180
2336 y(\022)1276 2235 y Fu(n)1337 2336 y FG(x)1389 2350
y Fv(N)1481 2336 y Fx(\025)g FG(z)1619 2350 y Fv(N)1707
2336 y Fx(\000)1808 2275 y FG(\024)15 b(q)1919 2242 y
Fy(2)p 1808 2315 151 4 v 1861 2399 a FG(r)1969 2235 y
Fu(o)2054 2336 y FA(=)1180 2571 y(=)1276 2470 y Fu(n)1337
2571 y FG(x)1389 2585 y Fv(N)1481 2571 y Fx(\025)25 b
FG(y)1622 2585 y Fv(N)1709 2571 y FA(+)20 b FG(r)j Fx(\000)1965
2510 y FG(\024)15 b(q)2076 2477 y Fy(2)p 1965 2550 V
2018 2634 a FG(r)2126 2470 y Fu(o)2201 2571 y FG(:)456
2340 y FA(\(B.16\))456 2768 y(Exploiting)28 b(\(B.15\))k(and)e
(\(B.16\),)j(w)m(e)e(get)1107 2993 y FG(w)1172 3007 y
Fv(N)1322 2993 y Fx(\024)83 b FG(p)1522 3007 y Fv(N)1609
2993 y Fx(\000)20 b FG(r)j FA(+)1864 2932 y FG(\024q)1960
2899 y Fy(2)p 1864 2972 136 4 v 1910 3056 a FG(r)2030
2993 y FA(+)2121 2908 y Fu(p)p 2212 2908 461 4 v 85 x
FG(r)2256 2967 y Fy(2)2315 2993 y Fx(\000)d FA(16)p FG(\024)2548
2967 y Fy(2)2589 2993 y FG(q)2633 2967 y Fy(2)2697 2993
y Fx(\024)1322 3228 y(\024)83 b FG(v)1520 3242 y Fv(N)1607
3228 y Fx(\000)20 b FG(r)j FA(+)1863 3167 y FG(\024q)1959
3134 y Fy(2)p 1863 3207 136 4 v 1909 3291 a FG(r)2029
3228 y FA(+)2119 3143 y Fu(p)p 2210 3143 461 4 v 85 x
FG(r)2254 3202 y Fy(2)2314 3228 y Fx(\000)c FA(16)p FG(\024)2546
3202 y Fy(2)2587 3228 y FG(q)2631 3202 y Fy(2)2696 3228
y FA(=)1322 3463 y(=)1486 3402 y(2)p FG(\024q)1627 3369
y Fy(2)p 1486 3442 182 4 v 1554 3526 a FG(r)1697 3463
y Fx(\000)h FG(r)j FA(+)1943 3378 y Fu(p)p 2033 3378
461 4 v 2033 3463 a FG(r)2077 3437 y Fy(2)2137 3463 y
Fx(\000)d FA(16)p FG(\024)2370 3437 y Fy(2)2410 3463
y FG(q)2454 3437 y Fy(2)2519 3463 y Fx(\024)1322 3698
y(\024)1486 3637 y FA(2)p FG(\024q)1627 3604 y Fy(2)p
1486 3677 182 4 v 1554 3761 a FG(r)1697 3698 y Fx(\000)g
FG(r)j FA(+)d FG(r)i Fx(\000)2107 3637 y FA(8)p FG(\024)2204
3604 y Fy(2)2244 3637 y FG(q)2288 3604 y Fy(2)p 2107
3677 221 4 v 2196 3761 a FG(r)2378 3698 y(<)40 b FA(0)15
b FG(;)456 3889 y FA(whic)m(h)29 b(is)g(the)i(desired)e(result.)1852
b Fl(\003)456 4073 y FH(Corollary)36 b(B.9.)42 b Fo(L)-5
b(et)33 b FG(a)26 b(>)f FA(0)p Fo(,)32 b FG(r)c(>)d(q)j
Fx(\025)d FA(0)33 b Fo(so)g(that)456 4280 y FA(\(B.17\))1799
4219 y FG(q)p 1799 4259 44 4 v 1799 4343 a(r)1893 4280
y Fx(\024)2016 4219 y FG(K)p 2014 4259 88 4 v 2014 4343
a(a)2062 4316 y Fy(2)456 4471 y Fo(and)456 4638 y FA(\(B.18\))1758
4576 y FG(q)p 1758 4617 44 4 v 1758 4700 a(r)1852 4638
y Fx(\024)2013 4576 y FG(a)p 1973 4617 130 4 v 1973 4700
a FA(8)p FG(K)2127 4638 y(:)555 4844 y Fo(L)-5 b(et)36
b(us)e(supp)-5 b(ose)37 b(that)f FG(v)d FA(=)c(\()p FG(v)1617
4811 y Ft(0)1641 4844 y FG(;)15 b(K)7 b(q)1809 4811 y
Fy(2)1849 4844 y FG(=r)s FA(\))30 b Fx(2)f Fw(R)2152
4811 y Fv(N)2260 4844 y Fo(is)35 b(ab)-5 b(ove)36 b FG(B)2678
4858 y Fv(r)2716 4844 y FA(\()p FG(y)s FA(\))f Fo(with)h(r)-5
b(esp)g(e)g(ct)37 b(to)456 4952 y(the)c FG(e)649 4966
y Fv(N)749 4952 y Fo(dir)-5 b(e)g(ction.)43 b(L)-5 b(et)33
b(us)f(assume)i(also)g(that)f Fx(j)p FG(v)2198 4919 y
Ft(0)2243 4952 y Fx(\000)19 b FG(y)2381 4919 y Ft(0)2405
4952 y Fx(j)25 b(\024)i FA(^)-47 b FG(cq)s Fo(,)33 b(with)456
5186 y FA(\(B.19\))790 b(^)-47 b FG(c)41 b Fx(\024)f
FA(min)1851 5086 y Fu(n)1921 5048 y Fx(p)p 1997 5048
84 4 v 77 x FG(K)p 1921 5166 160 4 v 1978 5249 a FA(3)2091
5186 y FG(;)2167 5125 y FA(2)p FG(K)p 2167 5166 130 4
v 2207 5249 a(a)2306 5086 y Fu(o)2381 5186 y FG(:)p eop
%%Page: 148 148
148 147 bop 456 251 a Fy(148)140 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y Fo(Then,)456 642 y FA(\(B.20\))465 b
FG(@)5 b(B)1292 656 y Fv(r)1330 642 y FA(\()p FG(y)s
FA(\))21 b Fx(\\)1550 541 y Fu(n)1610 642 y Fx(j)p FG(x)1687
605 y Ft(0)1731 642 y Fx(\000)f FG(y)1870 605 y Ft(0)1893
642 y Fx(j)25 b(\025)g FG(aq)2131 541 y Fu(o)2217 642
y Fx(\022)g(f)p FG(x)2410 656 y Fv(N)2503 642 y FG(<)g
FA(0)p Fx(g)15 b FG(:)456 867 y Fo(Pr)-5 b(o)g(of.)43
b FA(Let)38 b(\026)-52 b FG(q)30 b FA(:=)c(4)p FG(K)7
b(q)s(=a)p FA(,)32 b FG(\024)27 b FA(:=)g FG(a)1666 834
y Fy(2)1705 867 y FG(=)p FA(\(16)p FG(K)7 b FA(\),)34
b FG(c)27 b FA(:=)f FG(a)r FA(^)-47 b FG(c)q(=)p FA(\(4)p
FG(K)7 b FA(\).)44 b(Note)33 b(that)38 b(\026)-51 b FG(q)29
b(<)d(r)34 b FA(due)456 975 y(to)j(\(B.18\).)60 b(What)37
b(is)e(more,)j FG(v)g FA(=)d(\()p FG(v)1795 942 y Ft(0)1819
975 y FG(;)15 b(\024)7 b FA(\026)-52 b FG(q)1955 942
y Fy(2)1995 975 y FG(=r)s FA(\))36 b(and)g Fx(j)p FG(v)2410
942 y Ft(0)2458 975 y Fx(\000)23 b FG(y)2600 942 y Ft(0)2624
975 y Fx(j)35 b(\024)f FG(c)7 b FA(\026)-52 b FG(q)t
FA(.)58 b(Also,)37 b(\(B.11\))456 1083 y(and)25 b(\(B.12\))j(are)f
(ful\014lled)22 b(thanks)k(to)g(\(B.17\))j(and)c(\(B.19\).)42
b(Th)m(us,)26 b(b)m(y)g(Corollary)e(B.8)456 1191 y(\(applied)k(with)36
b(\026)-52 b FG(q)33 b FA(instead)d(of)h FG(q)s FA(\),)1145
1388 y FG(@)5 b(B)1267 1402 y Fv(r)1305 1388 y FA(\()p
FG(y)s FA(\))21 b Fx(\\)1525 1287 y Fu(n)1585 1388 y
Fx(j)p FG(x)1662 1351 y Ft(0)1706 1388 y Fx(\000)f FG(y)1845
1351 y Ft(0)1868 1388 y Fx(j)25 b(\025)g FA(4)p FG(\024)7
b FA(\026)-52 b FG(q)2156 1287 y Fu(o)2242 1388 y Fx(\022)25
b(f)p FG(x)2435 1402 y Fv(N)2528 1388 y FG(<)g FA(0)p
Fx(g)15 b FG(;)456 1586 y FA(whic)m(h)29 b(is)g(\(B.20\).)2293
b Fl(\003)555 1776 y FA(W)-8 b(e)32 b(no)m(w)e(p)s(oin)m(t)g(out)h
(some)f(observ)-5 b(ations)30 b(on)h(rotation)g(h)m(yp)s(ersurfaces)e
(in)g Fw(R)3256 1743 y Fv(N)7 b Fy(+1)3419 1776 y FA(.)456
1884 y(First,)40 b(the)e(normal)g(of)h(a)g(rotation)g(surface)f(is)g
(in)f(the)h(space)2651 1849 y Fy(25)2761 1884 y FA(generated)i(b)m(y)e
(the)456 1992 y(radial)28 b(direction)h(and)h FG(e)1311
2006 y Fv(N)7 b Fy(+1)1469 1992 y FA(,)31 b(as)f(sho)m(w)m(ed)h(b)m(y)f
(the)h(next)f(result:)456 2131 y FH(Lemma)i(B.10.)43
b Fo(Fix)33 b FG(y)28 b Fx(2)d Fw(R)1485 2098 y Fv(N)1559
2131 y Fo(.)41 b(L)-5 b(et)33 b FG(f)i Fx(2)25 b FG(C)2023
2098 y Fy(1)2061 2131 y FA(\()p Fw(R)t FG(;)15 b Fw(R)9
b FA(\))38 b Fo(and)c(de\014ne)1569 2301 y FA(\010\()p
FG(x)p FA(\))26 b(:=)f FG(f)10 b FA(\()p Fx(j)p FG(x)20
b Fx(\000)g FG(y)s Fx(j)p FA(\))15 b FG(:)456 2471 y
Fo(L)-5 b(et)44 b FG(\027)6 b FA(\()p FG(x)p FA(\))45
b Fo(b)-5 b(e)44 b(a)h(normal)h(ve)-5 b(ctor)46 b(to)e(the)h(surfac)-5
b(e)45 b Fx(f)p FG(x)2363 2485 y Fv(N)7 b Fy(+1)2568
2471 y FA(=)46 b(\010\()p FG(x)p FA(\))p Fx(g)f Fo(at)g(the)g(p)-5
b(oint)456 2578 y FA(\()p FG(x;)15 b FA(\010\()p FG(x)p
FA(\)\))p Fo(.)53 b(Then,)38 b(if)d FG(x)d Fx(6)p FA(=)f
FG(y)s Fo(,)37 b FG(\027)6 b FA(\()p FG(x)p FA(\))36
b Fo(b)-5 b(elongs)37 b(to)g(the)f(sp)-5 b(ac)g(e)38
b(sp)-5 b(anne)g(d)38 b(by)e FG(x)23 b Fx(\000)f FG(y)39
b Fo(and)456 2686 y FG(e)498 2700 y Fv(N)7 b Fy(+1)655
2686 y Fo(.)42 b(If)32 b FG(x)26 b FA(=)f FG(y)35 b Fo(and)f
FG(f)1306 2653 y Ft(0)1328 2686 y FA(\(0\))27 b(=)e(0)p
Fo(,)32 b(the)h(same)h(r)-5 b(esult)33 b(holds.)456 2876
y(Pr)-5 b(o)g(of.)43 b FA(Assume)29 b FG(x)d Fx(6)p FA(=)f
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b FA(\()p FG(x)p FA(\))26 b(=)f FG(\013)1834 2967 y Fu(\020)1888
3068 y Fx(r)p FA(\010\()p FG(x)p FA(\))p FG(;)31 b Fx(\000)p
FA(1)2324 2967 y Fu(\021)2394 3068 y FG(;)456 3260 y
FA(for)f(some)h FG(\013)25 b Fx(2)g Fw(R)s FA(.)46 b(Therefore,)1140
3480 y FG(\027)6 b FA(\()p FG(x)p FA(\))83 b(=)1560 3418
y FG(\013)16 b(f)1689 3385 y Ft(0)1711 3418 y FA(\()p
Fx(j)p FG(x)21 b Fx(\000)f FG(y)s Fx(j)p FA(\))p 1560
3459 484 4 v 1671 3542 a Fx(j)p FG(x)h Fx(\000)e FG(y)s
Fx(j)2069 3480 y FA(\()p FG(x)h Fx(\000)g FG(y)s FA(\))g
Fx(\000)g FG(\013e)2561 3494 y Fv(N)7 b Fy(+1)2735 3480
y FG(;)456 3698 y FA(th)m(us)30 b(pro)m(ving)f(the)i(claim)e(if)g
FG(x)d Fx(6)p FA(=)f FG(y)s FA(.)555 3806 y(If,)30 b(on)f(the)g(other)h
(hand,)e FG(x)e FA(=)e FG(y)33 b FA(and)28 b FG(f)1920
3773 y Ft(0)1943 3806 y FA(\(0\))e(=)f(0,)30 b(then)f
FG(\027)6 b FA(\()p FG(x)p FA(\))26 b(=)f FG(\013)15
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3914 y FG(\013)c Fx(2)g Fw(R)s FA(,)36 b(hence)31 b(completing)f(the)g
(pro)s(of)g(of)g(the)h(claim.)1018 b Fl(\003)555 4103
y FA(The)36 b(next)h(result)e(will)f(relate)i(the)h(\\cen)m(ter")h(of)f
(a)g(rotation)f(h)m(yp)s(ersurface)g(with)456 4211 y(the)30
b(normal)g(at)h(an)m(y)f(p)s(oin)m(t:)456 4350 y FH(Lemma)i(B.11.)43
b Fo(Fix)33 b FG(y)28 b Fx(2)d Fw(R)1485 4317 y Fv(N)1559
4350 y Fo(.)41 b(L)-5 b(et)33 b FG(f)i Fx(2)25 b FG(C)2023
4317 y Fy(1)2061 4350 y FA(\()p Fw(R)t FG(;)15 b Fw(R)9
b FA(\))38 b Fo(and)c(de\014ne)1569 4519 y FA(\010\()p
FG(x)p FA(\))26 b(:=)f FG(f)10 b FA(\()p Fx(j)p FG(x)20
b Fx(\000)g FG(y)s Fx(j)p FA(\))15 b FG(:)456 4691 y
Fo(L)-5 b(et)33 b(us)f(de\014ne)h(the)g(fol)5 b(lowing)34
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b Fy(+1)2390 4691 y Fo(:)1365 4883 y FA(\006)40 b(:=)1607
4782 y Fu(n\020)1722 4883 y FG(x;)15 b FA(\010\()p FG(x)p
FA(\))2002 4782 y Fu(\021)2083 4778 y(\014)2083 4833
y(\014)2083 4888 y(\014)2138 4883 y FG(x)25 b Fx(2)g
Fw(R)2361 4845 y Fv(N)2434 4782 y Fu(o)2510 4883 y FG(:)p
456 5025 499 4 v 555 5095 a Fy(25)626 5124 y FF(And,)j(in)g(fact,)i
(this)e(prop)r(ert)n(y)g(c)n(haracterizes)i(the)d(rotation)j(surfaces,)
g(as)f(p)r(oin)n(ted)f(out)g(to)g(us)456 5216 y(b)n(y)c(Ra)t(jk)n(o)i
(Quarta)g(Marcon,)h(an)f(undergraduate)f(in)h(T)-6 b(or)26
b(V)-6 b(ergata.)p eop
%%Page: 149 149
149 148 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)
f Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)140 b(149)456
472 y Fo(L)-5 b(et)33 b(us)f(c)-5 b(onsider)34 b(the)f(normal)h
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b(oint)2177 371 y Fu(\020)2232 472 y FG(x;)15 b FA(\010\()p
FG(x)p FA(\))2512 371 y Fu(\021)2592 472 y Fx(2)25 b
FA(\006)p Fo(,)32 b(given)g(by)947 811 y FG(\027)6 b
FA(\()p FG(x)p FA(\))41 b(=)1272 710 y Fu(\020)1326 811
y FG(\027)1371 825 y Fy(1)1410 811 y FA(\()p FG(x)p FA(\))p
FG(;)15 b(:)g(:)g(:)j(;)d(\027)1780 825 y Fv(N)7 b Fy(+1)1937
811 y FA(\()p FG(x)p FA(\))2059 710 y Fu(\021)2155 811
y FA(:=)2304 615 y Fu(\020)2379 716 y Fx(\000)20 b(r)p
FA(\010\()p FG(x)p FA(\))p FG(;)41 b FA(1)2845 615 y
Fu(\021)p 2301 790 602 4 v 2301 808 a(p)p 2392 808 511
4 v 78 x FA(1)21 b(+)f Fx(jr)p FA(\010\()p FG(x)p FA(\))p
Fx(j)2863 860 y Fy(2)2928 811 y FG(:)456 1073 y Fo(Then,)32
b(for)i(any)f FG(x)25 b Fx(2)g Fw(R)1263 1040 y Fv(N)1356
1073 y Fx(n)c(f)p FG(y)s Fx(g)p Fo(,)33 b(the)g(ve)-5
b(ctors)1210 1281 y FG(x)21 b Fx(\000)f FG(y)188 b Fo(and)1937
1180 y Fu(\020)1991 1281 y FG(\027)2036 1295 y Fy(1)2076
1281 y FA(\()p FG(x)p FA(\))p FG(;)15 b(:)g(:)g(:)i(;)e(\027)2445
1295 y Fv(N)2513 1281 y FA(\()p FG(x)p FA(\))2635 1180
y Fu(\021)456 1485 y Fo(ar)-5 b(e)33 b(p)-5 b(ar)g(al)5
b(lel.)456 1688 y(Pr)-5 b(o)g(of.)43 b FA(If)21 b(\()p
FG(\027)899 1702 y Fy(1)938 1688 y FA(\()p FG(x)p FA(\))p
FG(;)15 b(:)g(:)g(:)j(;)d(\027)1308 1702 y Fv(N)1375
1688 y FA(\()p FG(x)p FA(\)\))26 b(=)f(0,)f(there)e(is)f(nothing)f(to)i
(pro)m(v)m(e,)j(so)c(w)m(e)h(ma)m(y)h(assume)456 1796
y(\()p FG(\027)536 1810 y Fy(1)575 1796 y FA(\()p FG(x)p
FA(\))p FG(;)15 b(:)g(:)g(:)j(;)d(\027)945 1810 y Fv(N)1012
1796 y FA(\()p FG(x)p FA(\)\))26 b Fx(6)p FA(=)f(0.)41
b(F)-8 b(or)32 b(this)d(reason,)1365 2015 y Fx(r)p FA(\010\()p
FG(x)p FA(\))c(=)g FG(f)1805 1977 y Ft(0)1828 1914 y
Fu(\020)1882 2015 y Fx(j)p FG(x)c Fx(\000)f FG(y)s Fx(j)2144
1914 y Fu(\021)2248 1953 y FG(x)h Fx(\000)e FG(y)p 2223
1994 262 4 v 2223 2077 a Fx(j)p FG(x)i Fx(\000)e FG(y)s
Fx(j)2510 2015 y FG(;)456 2244 y FA(th)m(us)1623 2392
y FG(f)1678 2354 y Ft(0)1700 2291 y Fu(\020)1755 2392
y Fx(j)p FG(x)h Fx(\000)g FG(y)s Fx(j)2016 2291 y Fu(\021)2096
2392 y Fx(6)p FA(=)k(0)15 b FG(:)456 2575 y FA(Let)1219
2766 y FG(a)p FA(\()p FG(x)p FA(\))26 b(:=)1546 2704
y Fx(j)p FG(x)20 b Fx(\000)g FG(y)s Fx(j)1822 2626 y
Fu(p)p 1913 2626 511 4 v 78 x FA(1)h(+)f Fx(jr)p FA(\010\()p
FG(x)p FA(\))p Fx(j)2384 2677 y Fy(2)p 1546 2746 879
4 v 1761 2861 a FG(f)1816 2835 y Ft(0)1839 2760 y Fu(\020)1893
2861 y Fx(j)p FG(x)g Fx(\000)g FG(y)s Fx(j)2154 2760
y Fu(\021)2474 2766 y Fx(2)40 b Fw(R)18 b FG(:)456 3045
y FA(Then,)1212 3249 y FG(a)p FA(\()p FG(x)p FA(\))1397
3148 y Fu(\020)1452 3249 y FG(\027)1497 3263 y Fy(1)1536
3249 y FA(\()p FG(x)p FA(\))p FG(;)d(:)g(:)g(:)i(;)e(\027)1905
3263 y Fv(N)1973 3249 y FA(\()p FG(x)p FA(\))2095 3148
y Fu(\021)2175 3249 y FA(=)25 b FG(x)20 b Fx(\000)g FG(y)e(;)456
3452 y FA(pro)m(ving)29 b(the)i(claim.)2195 b Fl(\003)555
3656 y FA(Next)31 b(result)e(is)g(an)g(explicit)f(computation)i(on)g
(the)g(di\013eren)m(tial)e(of)i(the)g(unit)f(nor-)456
3764 y(mal)i(of)h(a)g(h)m(yp)s(ersurface)e(\(up)h(to)i(a)f(sign,)f(suc)
m(h)g(quan)m(tit)m(y)h(is)f(sometimes)h(referred)f(to)456
3872 y(as)f(Second)g(F)-8 b(undamen)m(tal)31 b(F)-8 b(orm)30
b(or)h(Shap)s(e)e(Op)s(erator\):)456 4017 y FH(Lemma)j(B.12.)43
b Fo(L)-5 b(et)33 b FA(\011)25 b Fx(2)g FG(C)1516 3984
y Fy(1)1554 4017 y FA(\()p Fw(R)1649 3984 y Fv(N)1723
4017 y FG(;)15 b Fw(R)s FA(\))p Fo(.)48 b(L)-5 b(et)33
b FA(\006)f Fo(b)-5 b(e)32 b(the)h(hyp)-5 b(ersurfac)g(e)35
b(de\014ne)-5 b(d)34 b(by)1378 4226 y FA(\006)24 b(:=)1590
4125 y Fu(n)o(\020)1705 4226 y FG(x;)15 b FA(\011\()p
FG(x)p FA(\))1990 4125 y Fu(\021)2070 4121 y(\014)2070
4176 y(\014)2070 4230 y(\014)2125 4226 y FG(x)26 b Fx(2)f
Fw(R)2348 4188 y Fv(N)2421 4125 y Fu(o)2497 4226 y FG(:)456
4434 y Fo(L)-5 b(et)34 b FG(X)i FA(=)29 b FG(X)7 b FA(\()p
FG(x)p FA(\))29 b(:=)g(\()p FG(x;)15 b FA(\011\()p FG(x)p
FA(\)\))36 b Fo(and)f(c)-5 b(onsider)36 b(the)e(unit)h(normal)h(to)f
FA(\006)f Fo(at)h(the)g(p)-5 b(oint)456 4542 y FG(X)7
b Fo(,)32 b(given)g(by)1312 4802 y FA(^)-50 b FG(\027)5
b FA(\()p FG(x)p FA(\))41 b(:=)1670 4606 y Fu(\020)1745
4707 y Fx(\000)19 b(r)p FA(\011\()p FG(x)p FA(\))p FG(;)41
b FA(1)2215 4606 y Fu(\021)p 1667 4781 607 4 v 1667 4799
a(p)p 1758 4799 516 4 v 78 x FA(1)20 b(+)g Fx(jr)p FA(\011\()p
FG(x)p FA(\))p Fx(j)2233 4851 y Fy(2)2324 4802 y Fx(2)40
b FG(S)2486 4764 y Fv(N)2568 4802 y FG(:)456 5035 y Fo(F)-7
b(or)33 b FG(X)g FA(=)25 b(\()p FG(x;)15 b FA(\011\()p
FG(x)p FA(\)\))p Fo(,)34 b(let)f(also)1404 5216 y FG(\027)6
b FA(\()p FG(X)h FA(\))26 b(=)f FG(\027)6 b FA(\()p FG(x;)15
b FA(\011\()p FG(x)p FA(\)\))26 b(:=)31 b(^)-51 b FG(\027)6
b FA(\()p FG(x)p FA(\))15 b FG(:)p eop
%%Page: 150 150
150 149 bop 456 251 a Fy(150)140 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)456 450 y Fo(L)j(et)580 415 y Fy(26)689 450 y FG(D)764
464 y Fv(X)831 450 y FG(\027)42 b FA(:)36 b FG(T)1032
464 y Fv(X)1099 450 y FA(\006)f Fx(\000)-15 b(!)36 b
Fw(R)1442 417 y Fv(N)7 b Fy(+1)1644 450 y Fo(b)-5 b(e)38
b(the)h(di\013er)-5 b(ential)39 b(map.)61 b(Then,)40
b(for)f(any)g FG(W)48 b FA(=)456 558 y(\()p FG(w)r(;)15
b(w)663 572 y Fv(N)7 b Fy(+1)821 558 y FA(\))26 b Fx(2)f
FG(T)1021 572 y Fv(X)1088 558 y FA(\006)p Fo(,)756 718
y FG(D)831 732 y Fv(X)899 718 y FG(\027)6 b FA(\()p FG(X)h
FA(\)[)p FG(W)13 b FA(])41 b(=)-907 b(\(B.21\))602 1270
y(=)756 787 y Fu(0)756 947 y(B)756 1002 y(B)756 1056
y(B)756 1111 y(B)756 1165 y(B)756 1220 y(B)756 1274 y(B)756
1329 y(B)756 1383 y(B)756 1438 y(B)756 1493 y(B)756 1551
y(@)873 852 y Fx(\000)p FA(\(1)21 b(+)f Fx(jr)p FA(\011\()p
FG(x)p FA(\))p Fx(j)1455 819 y Fy(2)1495 852 y FA(\))p
FG(@)1578 866 y Fy(1)p Fv(j)1651 852 y FA(\011\()p FG(X)7
b FA(\))p FG(w)1939 866 y Fv(j)1996 852 y FA(+)20 b FG(@)2135
866 y Fy(1)2175 852 y FA(\011\()p FG(X)7 b FA(\))15 b
FG(@)2461 867 y Fv(k)2505 852 y FA(\011\()p FG(X)7 b
FA(\))15 b FG(@)2791 867 y Fv(k)r(j)2867 852 y FA(\011\()p
FG(X)7 b FA(\))15 b FG(w)3170 866 y Fv(j)p 873 894 2335
4 v 1692 981 a FA(\(1)21 b(+)f Fx(jr)p FA(\011\()p FG(x)p
FA(\))p Fx(j)2203 955 y Fy(2)2243 981 y FA(\))2278 955
y Fy(3)p Fv(=)p Fy(2)2027 1063 y Fo(.)2027 1097 y(.)2027
1130 y(.)846 1238 y Fx(\000)p FA(\(1)g(+)g Fx(jr)p FA(\011\()p
FG(x)p FA(\))p Fx(j)1427 1205 y Fy(2)1467 1238 y FA(\))p
FG(@)1550 1252 y Fv(N)7 b(j)1651 1238 y FA(\011\()p FG(X)g
FA(\))p FG(w)1939 1252 y Fv(j)1996 1238 y FA(+)20 b FG(@)2135
1252 y Fv(N)2203 1238 y FA(\011\()p FG(X)7 b FA(\))15
b FG(@)2489 1253 y Fv(k)2533 1238 y FA(\011\()p FG(X)7
b FA(\))15 b FG(@)2819 1253 y Fv(k)r(j)2895 1238 y FA(\011\()p
FG(X)7 b FA(\))15 b FG(w)3198 1252 y Fv(j)p 846 1280
2390 4 v 1692 1367 a FA(\(1)21 b(+)f Fx(jr)p FA(\011\()p
FG(x)p FA(\))p Fx(j)2203 1341 y Fy(2)2243 1367 y FA(\))2278
1341 y Fy(3)p Fv(=)p Fy(2)1608 1566 y Fx(\000)p FG(@)1727
1581 y Fv(k)1770 1566 y FA(\011\()p FG(X)7 b FA(\))15
b FG(@)2056 1581 y Fv(k)r(j)2132 1566 y FA(\011\()p FG(X)7
b FA(\))15 b FG(w)2435 1580 y Fv(j)p 1608 1608 865 4
v 1692 1696 a FA(\(1)21 b(+)f Fx(jr)p FA(\011\()p FG(x)p
FA(\))p Fx(j)2203 1669 y Fy(2)2243 1696 y FA(\))2278
1669 y Fy(3)p Fv(=)p Fy(2)3245 787 y Fu(1)3245 947 y(C)3245
1002 y(C)3245 1056 y(C)3245 1111 y(C)3245 1165 y(C)3245
1220 y(C)3245 1274 y(C)3245 1329 y(C)3245 1383 y(C)3245
1438 y(C)3245 1493 y(C)3245 1551 y(A)3355 1270 y FG(:)456
1874 y Fo(Pr)-5 b(o)g(of.)43 b FA(Let)30 b FG(W)38 b
Fx(2)25 b FG(T)1162 1888 y Fv(X)1229 1874 y FA(\006.)41
b(Then,)29 b(for)i(some)f FG(v)f Fx(2)c Fw(R)2208 1841
y Fv(N)2282 1874 y FA(,)936 2083 y FG(W)38 b FA(=)1180
2022 y FG(d)p 1176 2062 81 4 v 1176 2146 a(dt)1266 1982
y Fu(\020)1321 2083 y FG(x)20 b FA(+)g FG(tv)s(;)30 b
FA(\011\()p FG(x)21 b FA(+)e FG(tv)s FA(\))2003 1982
y Fu(\021)2058 1952 y(\014)2058 2006 y(\014)2058 2061
y(\014)2058 2115 y(\014)2089 2174 y Fv(t)p Fy(=0)2234
2083 y FA(=)2330 1982 y Fu(\020)2384 2083 y FG(v)s(;)31
b Fx(r)p FA(\011\()p FG(x)p FA(\))20 b Fx(\001)g FG(v)2868
1982 y Fu(\021)2938 2083 y FG(;)456 2302 y FA(that)31
b(is)e FG(w)809 2316 y Fv(N)7 b Fy(+1)992 2302 y FA(=)25
b Fx(r)p FA(\011\()p FG(x)p FA(\))20 b Fx(\001)g FG(w)r
FA(,)32 b(or,)e(equiv)-5 b(alen)m(tly)456 2468 y(\(B.22\))542
b FG(T)1300 2482 y Fv(X)1367 2468 y FA(\006)25 b(=)g
Fx(f)p FA(\()p FG(w)r(;)15 b Fx(r)p FA(\011\()p FG(x)p
FA(\))22 b Fx(\001)e FG(w)r FA(\))27 b Fx(j)e FG(w)j
Fx(2)d Fw(R)2494 2431 y Fv(N)2568 2468 y Fx(g)15 b FG(:)456
2628 y FA(Because)31 b(of)g(this,)952 2838 y FG(D)1027
2852 y Fv(X)1094 2838 y FG(\027)6 b FA(\()p FG(X)h FA(\)[)p
FG(W)13 b FA(])84 b(=)1708 2777 y FG(d)p 1704 2817 V
1704 2901 a(dt)1794 2838 y(\027)1845 2737 y Fu(\020)1899
2838 y FG(x)20 b FA(+)g FG(tw)r(;)31 b FA(\011\()p FG(x)20
b FA(+)g FG(tw)r FA(\))2622 2737 y Fu(\021)2677 2706
y(\014)2677 2761 y(\014)2677 2815 y(\014)2677 2870 y(\014)2707
2929 y Fv(t)p Fy(=0)2852 2838 y FA(=)1530 3096 y(=)1708
3034 y FG(d)p 1704 3075 V 1704 3158 a(dt)1799 3096 y
FA(^)-50 b FG(\027)6 b FA(\()p FG(x)20 b FA(+)g FG(tw)r
FA(\))2178 2964 y Fu(\014)2178 3019 y(\014)2178 3073
y(\014)2178 3128 y(\014)2209 3186 y Fv(t)p Fy(=0)2359
3096 y FG(;)456 3310 y FA(from)30 b(whic)m(h)f(a)h(straigh)m(tforw)m
(ard)h(calculation)e(giv)m(es)i(the)f(claim.)651 b Fl(\003)456
3486 y Fo(R)-5 b(emark)52 b FA(B.13)p Fo(.)c FA(In)39
b(relation)g(with)g(Lemma)h(B.12)h(ab)s(o)m(v)m(e,)j(w)m(e)c(notice)g
(that,)j(since)456 3594 y FG(\027)6 b FA(\(\006\))27
b Fx(\022)f FG(S)828 3561 y Fv(N)927 3594 y FA(and)31
b FG(T)1158 3613 y Fv(\027)t Fy(\()p Fv(X)5 b Fy(\))1319
3594 y FG(S)1380 3561 y Fv(N)1474 3594 y FA(=)27 b FG(T)1625
3608 y Fv(X)1692 3594 y FA(\006,)32 b(w)m(e)g(ma)m(y)g(think)e
FG(D)2464 3608 y Fv(X)2531 3594 y FG(\027)37 b FA(as)32
b(a)g(linear)e(map)h(from)456 3706 y FG(T)509 3720 y
Fv(X)576 3706 y FA(\006)21 b(to)i(itself.)36 b(Using)21
b(\(B.21\))j(and)d(\(B.22\),)26 b(the)c(reader)g(will)d(easily)i(con)m
(vince)h(herself)456 3814 y(that)31 b FG(D)728 3828 y
Fv(X)795 3814 y FG(\027)g FA(:)41 b FG(T)990 3828 y Fv(X)1057
3814 y FA(\006)25 b Fx(\000)-15 b(!)25 b FG(T)1373 3828
y Fv(X)1440 3814 y FA(\006)30 b(ma)m(y)h(th)m(us)f(b)s(e)g(represen)m
(ted)g(in)g(matrix)f(form)h(as)456 4031 y(\(B.23\))1012
3930 y Fu(\020)1066 4031 y FG(D)1141 4045 y Fv(X)1209
4031 y FG(\027)1260 3930 y Fu(\021)1313 4094 y Fv(ij)1414
4031 y FA(=)1536 3968 y Fx(\000)p FA(\(1)20 b(+)g Fx(jr)p
FA(\011)p Fx(j)1995 3935 y Fy(2)2035 3968 y FA(\))p FG(@)2118
3982 y Fv(ij)2179 3968 y FA(\011)g(+)g FG(@)2409 3982
y Fv(i)2437 3968 y FA(\011)p FG(@)2556 3983 y Fv(k)2599
3968 y FA(\011)p FG(@)2718 3983 y Fv(k)r(j)2793 3968
y FA(\011)p 1536 4010 1328 4 v 1913 4097 a(\(1)h(+)e
Fx(jr)p FA(\011)p Fx(j)2301 4071 y Fy(2)2341 4097 y FA(\))2376
4071 y Fy(3)p Fv(=)p Fy(2)456 4248 y FA(for)43 b(1)k
Fx(\024)g FG(i;)15 b(j)54 b Fx(\024)47 b FG(N)53 b FA(\(where,)47
b(of)d(course,)j(the)d(summation)e(o)m(v)m(er)j(the)f(index)e
FG(k)k FA(is)456 4356 y(understo)s(o)s(d)28 b(here)i(ab)s(o)m(v)m(e\).)
555 4477 y(With)e(this,)g(w)m(e)h(no)m(w)f(p)s(oin)m(t)f(out)i(an)f
(explicit)f(computation)h(of)h(the)f(curv)-5 b(atures)28
b(of)456 4585 y(the)i(rotation)h(surfaces:)456 4719 y
FH(Lemma)h(B.14.)43 b Fo(L)-5 b(et)33 b FA(\010)25 b
Fx(2)f FG(C)1510 4686 y Fy(2)1549 4719 y FA(\(\(0)p FG(;)15
b FA(+)p Fx(1)p FA(\))p FG(;)33 b Fw(R)r FA(\))39 b Fo(and)1350
4907 y FA(\006)25 b(:=)1562 4806 y Fu(n\020)1677 4907
y FG(x;)30 b FA(\010\()p Fx(j)p FG(x)p Fx(j)p FA(\))2022
4806 y Fu(\021)2103 4907 y Fx(j)25 b FG(x)g Fx(2)g Fw(R)2376
4869 y Fv(N)2449 4806 y Fu(o)2525 4907 y FG(:)p 456 5025
499 4 v 555 5095 a Fy(26)626 5124 y FF(As)19 b(standard,)h(giv)n(en)f
(a)g(manifold)h FD(M)27 b FF(and)18 b(a)i(p)r(oin)n(t)f
FD(X)27 b Fr(2)22 b FD(M)8 b FF(,)21 b(w)n(e)e(denote)g(b)n(y)f
FD(T)2906 5132 y Fs(X)2964 5124 y FD(M)27 b FF(the)19
b(tangen)n(t)456 5216 y(space)26 b(at)g FD(X)6 b FF(.)34
b(Also,)27 b(as)f(usual,)h FD(S)1442 5184 y Fs(N)1521
5216 y FF(:=)21 b Fr(f)p FD(X)28 b Fr(2)22 b Fd(R)1876
5184 y Fs(N)5 b Fq(+1)2035 5216 y Fr(j)21 b(j)p FD(X)6
b Fr(j)23 b FF(=)e(1)p Fr(g)p FF(.)p eop
%%Page: 151 151
151 150 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)
f Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)140 b(151)456
450 y Fo(Then,)32 b(the)h(princip)-5 b(al)35 b(curvatur)-5
b(es)1657 415 y Fy(27)1761 450 y Fo(of)33 b FA(\006)f
Fo(ar)-5 b(e)34 b(given)d(by)1317 665 y FG(\024)1369
679 y Fy(1)1450 665 y FA(=)40 b FG(:)15 b(:)g(:)41 b
FA(=)25 b FG(\024)1855 679 y Fv(N)7 b Ft(\000)p Fy(1)2053
665 y FA(=)2412 603 y(\010)2478 570 y Ft(0)p 2174 644
565 4 v 2174 740 a Fx(j)p FG(x)p Fx(j)2291 662 y Fu(p)p
2383 662 356 4 v 2383 740 a FA(1)21 b(+)f(\(\010)2641
714 y Ft(0)2664 740 y FA(\))2699 714 y Fy(2)1317 934
y FG(\024)1369 948 y Fv(N)1477 934 y FA(=)1803 872 y(\010)1869
839 y Ft(00)p 1599 913 517 4 v 1599 1000 a FA(\(1)h(+)e
Fx(j)p FA(\010)1881 974 y Ft(0)1905 1000 y Fx(j)1930
974 y Fy(2)1969 1000 y FA(\))2004 974 y Fy(3)p Fv(=)p
Fy(2)2140 934 y FG(:)456 1175 y Fo(Pr)-5 b(o)g(of.)43
b FA(W)-8 b(e)38 b(set)g(\011\()p FG(x)p FA(\))e(:=)h(\010\()p
Fx(j)p FG(x)p Fx(j)p FA(\))h(and,)g(after)g(some)f(easy)h(computation,)
h(w)m(e)f(infer)456 1283 y(from)30 b(\(B.23\))i(that)1022
1407 y Fu(\020)1076 1508 y FG(D)1151 1522 y Fv(X)1219
1508 y FG(\027)1270 1407 y Fu(\021)1324 1572 y Fv(ij)1425
1508 y FA(=)40 b(\(1)21 b(+)f(\(\010)1829 1471 y Ft(0)1852
1508 y FA(\))1887 1471 y Fy(2)1927 1508 y FA(\))1962
1471 y Ft(\000)p Fy(3)p Fv(=)p Fy(2)2127 1407 y Fu(h)2191
1508 y Fx(\000)2291 1446 y FA(\(1)h(+)f(\(\010)2584 1413
y Ft(0)2608 1446 y FA(\))2643 1413 y Fy(2)2682 1446 y
FA(\)\010)2783 1413 y Ft(0)2822 1446 y FG(\016)2862 1460
y Fv(ij)p 2291 1487 632 4 v 2556 1571 a Fx(j)p FG(x)p
Fx(j)2953 1508 y FA(+)1022 1766 y(+)1103 1705 y(\(1)h(+)f(\(\010)1396
1672 y Ft(0)1419 1705 y FA(\))1454 1672 y Fy(2)1494 1705
y FA(\)\010)1595 1672 y Ft(0)1638 1705 y Fx(\000)g(j)p
FG(x)p Fx(j)p FA(\010)1897 1672 y Ft(00)p 1103 1745 837
4 v 1450 1828 a Fx(j)p FG(x)p Fx(j)1552 1802 y Fy(3)1965
1766 y FG(x)2017 1780 y Fv(i)2045 1766 y FG(x)2097 1780
y Fv(j)2134 1665 y Fu(i)2192 1766 y FG(:)-1761 b FA(\(B.24\))456
1980 y(Notice)49 b(no)m(w)g(that,)54 b(up)48 b(to)i(rotation,)k(w)m(e)
49 b(ma)m(y)g(assume)g(that)g(the)g(p)s(oin)m(t)f FG(X)64
b FA(=)456 2088 y(\()p FG(x;)15 b FA(\010\()p Fx(j)p
FG(x)p Fx(j)p FA(\),)49 b(in)43 b(whic)m(h)g(w)m(e)h(compute)h(the)f
(principal)d(curv)-5 b(atures,)47 b(is)c(of)h(the)h(form)456
2195 y FG(x)25 b FA(=)g Fx(j)p FG(x)p Fx(j)15 b FG(e)788
2209 y Fv(N)856 2195 y FA(;)31 b(hence,)f(from)g(\(B.24\))1022
2320 y Fu(\020)1076 2421 y FG(D)1151 2435 y Fv(X)1219
2421 y FG(\027)1270 2320 y Fu(\021)1324 2485 y Fv(ij)1425
2421 y FA(=)40 b(\(1)21 b(+)f(\(\010)1829 2384 y Ft(0)1852
2421 y FA(\))1887 2384 y Fy(2)1927 2421 y FA(\))1962
2384 y Ft(\000)p Fy(3)p Fv(=)p Fy(2)2127 2320 y Fu(h)2191
2421 y Fx(\000)2291 2359 y FA(\(1)h(+)f(\(\010)2584 2326
y Ft(0)2608 2359 y FA(\))2643 2326 y Fy(2)2682 2359 y
FA(\)\010)2783 2326 y Ft(0)2822 2359 y FG(\016)2862 2373
y Fv(ij)p 2291 2400 632 4 v 2556 2484 a Fx(j)p FG(x)p
Fx(j)2953 2421 y FA(+)1022 2679 y(+)1103 2618 y(\(1)h(+)f(\(\010)1396
2585 y Ft(0)1419 2618 y FA(\))1454 2585 y Fy(2)1494 2618
y FA(\)\010)1595 2585 y Ft(0)1638 2618 y Fx(\000)g(j)p
FG(x)p Fx(j)p FA(\010)1897 2585 y Ft(00)p 1103 2658 837
4 v 1470 2741 a Fx(j)p FG(x)p Fx(j)1965 2679 y FG(\016)2005
2693 y Fv(iN)2097 2679 y FG(\016)2137 2693 y Fv(j)t(N)2237
2578 y Fu(i)2295 2679 y FG(;)456 2893 y FA(and)29 b(so)i(the)g(desired)
d(claim)i(easily)f(follo)m(ws.)1378 b Fl(\003)1685 3131
y FB(References)494 3276 y FF([1])39 b(G.)30 b(Alb)r(erti,)h(L.)e(Am)n
(brosio,)h(X.)f(Cabr)n(\023)-36 b(e,)31 b Fc(On)g(a)g(long-standing)h
(c)l(onje)l(ctur)l(e)h(of)d(E.)h(De)g(Gior)l(gi:)613
3368 y(symmetry)23 b(in)e(3D)g(for)h(gener)l(al)g(nonline)l(arities)g
(and)g(a)g(lo)l(c)l(al)f(minimality)f(pr)l(op)l(erty)p
FF(,)j(Acta)c(Appl.)613 3459 y(Math.)27 b(65)f(\(2001\),)h(no.)f(1-3,)h
(9{33.)494 3550 y([2])39 b(L.)25 b(Am)n(brosio,)g(X.)f(Cabr)n(\023)-36
b(e,)52 b Fc(Entir)l(e)27 b(solutions)h(of)e(semiline)l(ar)g(el)t
(liptic)g(e)l(quations)i(in)e Fd(R)3194 3519 y Fq(3)3257
3550 y Fc(and)h(a)613 3642 y(c)l(onje)l(ctur)l(e)j(of)d(De)h(Gior)l(gi)
p FF(,)e(J.)h(Amer.)d(Math.)j(So)r(c.)f(13)g(\(2000\),)h(no.)f(4,)h
(725{739.)494 3733 y([3])39 b(S.)21 b(S.)g(An)n(tman,)e
Fc(Nonuniqueness)25 b(of)e(e)l(quilibrium)f(states)j(for)e(b)l(ars)h
(in)f(tension)p FF(,)g(J.)e(Math.)g(Anal.)613 3824 y(Appl.)26
b(\(1973\),)h(333{349.)494 3916 y([4])39 b(M.)44 b(T.)f(Barlo)n(w,)49
b(R.)42 b(F.)h(Bass,)48 b(C.)c(Gui,)j Fc(The)c(Liouvil)t(le)f(pr)l(op)l
(erty)k(and)d(a)g(c)l(onje)l(ctur)l(e)j(of)613 4007 y(De)28
b(Gior)l(gi)p FF(,)e(Comm.)f(Pure)h(Appl.)f(Math.)h(53)h(\(2000\),)g
(no.)f(8,)g(1007{1038.)494 4098 y([5])39 b(H.)22 b(Berest)n(yc)n(ki,)h
(L.)f(A.)g(Ca\013arelli,)j(L.)d(Niren)n(b)r(erg,)h Fc(F)-6
b(urther)26 b(qualitative)f(pr)l(op)l(erties)h(for)e(el)t(liptic)613
4190 y(e)l(quations)29 b(in)d(unb)l(ounde)l(d)i(domains)p
FF(,)e(Ann.)e(Sc.)g(Norm.)g(Sup)r(er.)g(Pisa,)j(Cl.)f(Sci.)f(\(4\))f
(25)i(\(1997\),)613 4281 y(no.)g(1-2,)h(69{94.)494 4372
y([6])39 b(H.)26 b(Berest)n(yc)n(ki,)g(F.)f(Hamel,)h(R.)f(Monneau,)h
Fc(One-dimensional)i(symmetry)g(of)f(b)l(ounde)l(d)i(entir)l(e)613
4464 y(solutions)g(of)e(some)h(el)t(liptic)f(e)l(quations)p
FF(,)h(Duk)n(e)c(Math.)i(J.)h(103)g(\(2000\),)g(no.)f(3,)g(375{396.)494
4555 y([7])39 b(G.)20 b(Bouc)n(hitt)n(\023)-36 b(e,)42
b Fc(Singular)22 b(p)l(erturb)l(ations)i(of)c(variational)i(pr)l
(oblems)g(arising)f(fr)l(om)g(a)g(two-phase)613 4646
y(tr)l(ansition)29 b(mo)l(del)p FF(,)d(Appl.)g(Math.)g(Optim.)e(21)j
(\(1990\),)g(no.)f(3,)g(289{314.)494 4738 y([8])39 b(L.A.)26
b(Ca\013arelli,)i(X.)d(Cabr)n(\023)-36 b(e,)26 b Fc(F)-6
b(ul)t(ly)27 b(nonline)l(ar)h(el)t(liptic)f(e)l(quations)p
FF(,)g(American)d(Mathematical)613 4829 y(So)r(ciet)n(y)34
b(Collo)r(quium)f(Publications,)k(43.)d(American)f(Mathematical)h(So)r
(ciet)n(y)-6 b(,)35 b(Pro)n(vidence,)613 4920 y(RI,)26
b(1995.)p 456 5025 499 4 v 555 5095 a Fy(27)626 5124
y FF(As)h(standard,)g(the)g(principal)g(curv)l(ature)f(of)i(a)g
(surface)g(\006)f(at)g(the)g(p)r(oin)n(t)2834 5105 y(\026)2812
5124 y FD(X)33 b FF(are)27 b(here)g(de\014ned)456 5216
y(as)f(the)f(eigen)n(v)l(alues)h(of)h Fr(\000)p FD(D)1290
5224 y Fs(X)1348 5216 y FD(\027)5 b FF(\()1442 5197 y(\026)1421
5216 y FD(X)h FF(\))21 b(:)h FD(T)1645 5219 y Fq(\026)1629
5232 y Fs(X)1686 5216 y FF(\006)g Fr(\000)-13 b(!)21
b FD(T)1969 5219 y Fq(\026)1953 5232 y Fs(X)2011 5216
y FF(\006.)p eop
%%Page: 152 152
152 151 bop 456 251 a Fy(152)140 b(ENRICO)24 b(V)-8 b(ALDINOCI,)23
b(BERARDINO)g(SCIUNZI)i(AND)e(V)-8 b(ASILE)24 b(O)n(VIDIU)g(SA)-8
b(VIN)494 450 y FF([9])39 b(L.)31 b(Damascelli,)i Fc(Comp)l(arison)f
(the)l(or)l(ems)i(for)d(some)i(quasiline)l(ar)f(de)l(gener)l(ate)j(el)t
(liptic)c(op)l(er)l(a-)613 541 y(tors)f(and)e(applic)l(ations)h(to)g
(symmetry)g(and)f(monotonicity)h(r)l(esults)p FF(,)g(Ann.)c(Inst.)h(H.)
g(P)n(oincar)n(\023)-36 b(e.)613 633 y(Analyse)26 b(non)f(lin)n(\023)
-36 b(eaire,)28 b(15\(4\),)f(1998,)g(493)g({)f(516.)456
724 y([10])39 b(L.)22 b(Damascelli)g(and)e(B.)i(Sciunzi,)g
Fc(Harnack)j(Ine)l(qualities,)f(Maximum)e(and)i(Comp)l(arison)g(Prin-)
613 815 y(ciples,)i(and)f(R)l(e)l(gularity)h(of)e(p)l(ositive)i
(solutions)g(of)f FD(m)p Fc(-L)l(aplac)l(e)h(e)l(quations)p
FF(,)f(to)e(app)r(ear)g(in)g(Calc.)613 907 y(V)-6 b(ar.)26
b(P)n(artial)h(Di\013eren)n(tial)f(Equations,)456 998
y([11])39 b(D.)22 b(Danielli,)j(N.)c(Garofalo,)26 b Fc(Pr)l(op)l
(erties)g(of)e(entir)l(e)h(solutions)g(of)f(non-uniformly)g(el)t
(liptic)g(e)l(qua-)613 1089 y(tions)39 b(arising)e(in)h(ge)l(ometry)h
(and)f(in)f(phase)i(tr)l(ansitions)p FF(,)i(Calc.)d(V)-6
b(ar.)37 b(P)n(artial)h(Di\013eren)n(tial)613 1181 y(Equations)26
b(15)h(\(2002\),)g(no.)f(4,)g(451{491.)456 1272 y([12])39
b(A.)32 b(F)-6 b(arina,)34 b Fc(One-dimensional)g(symmetry)g(for)f
(solutions)h(of)f(quasiline)l(ar)h(e)l(quations)h(in)e
Fd(R)3386 1240 y Fq(2)3423 1272 y FF(,)613 1363 y(Bollettino)28
b(UMI)d(8)h(6-B)g(\(2003\),)h(685{692.)456 1455 y([13])39
b(A.)31 b(F)-6 b(arina,)34 b Fc(R)n(igidity)e(and)h(one-dimensional)h
(symmetry)g(for)e(semiline)l(ar)h(el)t(liptic)f(e)l(quations)613
1547 y(in)25 b(the)i(whole)e(of)g Fd(R)1179 1516 y Fs(N)1265
1547 y Fc(and)h(in)f(half)g(sp)l(ac)l(es)p FF(,)h(Adv.)d(Math.)h(Sci.)f
(Appl.)g(13)h(\(2003\),)i(no.)e(1,)g(65{82.)456 1639
y([14])39 b(E.)d(De)e(Giorgi,)39 b Fc(Conver)l(genc)l(e)f(pr)l(oblems)f
(for)e(functionals)h(and)h(op)l(er)l(ators)p FF(,)i(Pro)r(ceedings)e
(of)613 1730 y(the)30 b(In)n(ternational)f(Meeting)i(on)e(Recen)n(t)g
(Metho)r(ds)h(in)f(Nonlinear)h(Analysis)g(\(Rome,)g(1978\),)613
1821 y(131{188,)f(Pitagora,)f(Bologna,)g(1979.)456 1913
y([15])39 b(E.)32 b(DiBenedetto,)g FD(C)1251 1881 y Fq(1+)p
Fs(\013)1405 1913 y Fc(lo)l(c)l(al)g(r)l(e)l(gularity)i(of)d(we)l(ak)i
(solutions)h(of)d(de)l(gener)l(ate)k(el)t(liptic)d(e)l(qua-)613
2004 y(tions)p FF(,)27 b(Nonlinear)f(Anal.)g(7)g(\(1983\),)h(no.)f(8,)h
(827{850.)456 2095 y([16])39 b(J.)34 b(Dolb)r(eault,)h(R.)e(Monneau,)i
Fc(On)f(a)g(Liouvil)t(le)f(typ)l(e)i(the)l(or)l(em)h(for)e(isotr)l
(opic)h(homo)l(gene)l(ous)613 2187 y(ful)t(ly)30 b(nonline)l(ar)i(el)t
(liptic)e(e)l(quations)j(in)d(dimension)h(two)p FF(,)g(Ann.)e(Sc.)g
(Norm.)g(Sup)r(er.)g(Pisa)h(Cl.)613 2278 y(Sci.)d(\(5\))e(2)h
(\(2003\),)h(no.)f(1,)h(181{197.)456 2369 y([17])39 b(Y.)d(Du,)h(L.)f
(Ma,)j Fc(Some)e(R)l(emarks)h(on)e(De)h(Gior)l(gi)g(c)l(onje)l(ctur)l
(e)p FF(,)j(Pro)r(c.)d(Amer.)d(Math.)i(So)r(c.,)613 2461
y(131\(2002\),)29 b(2415{2422.)456 2552 y([18])39 b(L.C.)24
b(Ev)l(ans,)g(R.F.)f(Gariep)n(y)-6 b(,)24 b Fc(Me)l(asur)l(e)i(the)l
(ory)h(and)e(\014ne)h(pr)l(op)l(erties)h(of)d(functions)p
FF(,)h(Studies)d(in)613 2643 y(Adv)l(anced)j(Mathematics.)h(CR)n(C)g
(Press,)h(Bo)r(ca)g(Raton,)f(FL,)g(1992.)456 2735 y([19])39
b(D.)34 b(Gilbarg,)j(N.S.)c(T)-6 b(rudinger,)72 b Fc(El)t(liptic)34
b(p)l(artial)h(di\013er)l(ential)h(e)l(quations)g(of)e(se)l(c)l(ond)j
(or)l(der)p FF(,)613 2826 y(Classics)29 b(in)c(Mathematics.)i
(Springer-V)-6 b(erlag,)26 b(Berlin,)g(2001.)456 2917
y([20])39 b(E.)g(Giusti,)83 b Fc(Minimal)37 b(surfac)l(es)k(and)e
(functions)g(of)g(b)l(ounde)l(d)h(variation)p FF(,)i(Monographs)d(in)
613 3009 y(Mathematics,)27 b(80.)g(Birkh\177)-38 b(auser)26
b(V)-6 b(erlag,)26 b(Basel,)i(1984.)456 3100 y([21])39
b(N.A.)28 b(Ghoussoub,)g(C.)g(Gui,)57 b Fc(On)29 b(a)g(c)l(onje)l(ctur)
l(e)i(of)e(De)g(Gior)l(gi)g(and)h(some)f(r)l(elate)l(d)i(pr)l(oblems)p
FF(,)613 3191 y(Math.)c(Ann.)e(311)h(\(1998\),)i(no.)e(3,)g(481{491.)
456 3283 y([22])39 b(N.)26 b(A.)f(Ghoussoub,)h(C.)h(Gui,)f
Fc(On)h(the)i(De)e(Gior)l(gi)h(c)l(onje)l(ctur)l(e)h(in)e(dimensions)h
FF(4)g Fc(and)g FF(5,)e(Ann.)613 3374 y(Math.)h(157)f(\(2003\),)i(no.)e
(1,)g(313{334.)456 3465 y([23])39 b(M.E.)24 b(Gurtin,)f
Fc(On)h(a)h(the)l(ory)h(of)e(phase)i(tr)l(ansitions)g(with)e
(interfacial)h(ener)l(gy)p FF(,)g(Arc)n(h.)c(Rational)613
3557 y(Mec)n(h.)27 b(Anal.)f(\(1985\),)h(no.)f(3,)g(187{212.)456
3648 y([24])39 b(D.)c(Jerison,)k(R.)c(Monneau,)j Fc(The)f(existenc)l(e)
h(of)d(a)h(symmetric)h(glob)l(al)f(minimizer)f(on)i Fd(R)3321
3616 y Fs(n)p Fp(\000)p Fq(1)613 3739 y Fc(implies)26
b(the)h(existenc)l(e)h(of)e(a)g(c)l(ounter-example)j(to)e(a)f(c)l(onje)
l(ctur)l(e)i(of)e(De)g(Gior)l(gi)h(in)e Fd(R)3148 3707
y Fs(n)3220 3739 y FF(,)g(C.)g(R.)613 3831 y(Acad.)h(Sci.)g(P)n(aris)h
(333)g(\(2001\),)g(no.)f(5,)g(427{431.)456 3922 y([25])39
b(L.)24 b(Mo)r(dica,)i(S.)d(Mortola,)51 b Fc(Un)26 b(esempio)g(di)f
FF(\000)1977 3890 y Fp(\000)2029 3922 y Fc(-c)l(onver)l(genza)p
FF(,)i(Boll.)e(Un.)e(Mat.)i(Ital.)f(B)f(\(5\))h(14)613
4013 y(\(1977\),)k(no.)e(1,)g(285{299.)456 4105 y([26])39
b(L.)29 b(Mo)r(dica,)61 b Fc(The)30 b(gr)l(adient)h(the)l(ory)h(of)d
(phase)i(tr)l(ansitions)h(and)e(the)h(minimal)d(interfac)l(e)j(cri-)613
4196 y(terion)p FF(,)c(Arc)n(h.)f(Rational)g(Mec)n(h.)g(Anal.)g(98)h
(\(1987\),)g(no.)f(2,)g(123{142.)456 4287 y([27])39 b(A.)22
b(P)n(etrosy)n(an,)h(E.)f(V)-6 b(aldino)r(ci,)24 b Fc(Ge)l(ometric)h
(pr)l(op)l(erties)h(of)e(Bernoul)t(li-typ)l(e)h(minimizers)p
FF(,)d(In)n(ter-)613 4378 y(faces)28 b(F)-6 b(ree)25
b(Bound.)h(7)g(\(2005\),)h(55{78.)456 4470 y([28])39
b(A.)j(P)n(etrosy)n(an,)j(E.)d(V)-6 b(aldino)r(ci,)92
b Fc(Density)43 b(estimates)g(for)f(a)g(de)l(gener)l(ate/singular)k
(phase-)613 4561 y(tr)l(ansition)29 b(mo)l(del)p FF(,)d(to)g(app)r(ear)
g(in)g(SIAM)f(J.)h(Math.)g(Anal.)456 4652 y([29])39 b(J.)c(S.)e(Ro)n
(wlinson,)j Fc(T)-6 b(r)l(anslation)36 b(of)e(J.)h(D.)f(van)h(der)g
(Waals')f(\\The)h(thermo)l(dynamic)i(the)l(ory)613 4744
y(of)32 b(c)l(apil)t(larity)g(under)h(the)g(hyp)l(othesis)h(of)e(a)g(c)
l(ontinuous)h(variation)g(of)f(density")p FF(,)i(J.)d(Statist.)613
4835 y(Ph)n(ys.)26 b(\(1979\),)h(no.)f(2,)h(197{244.)456
4926 y([30])39 b(B.)27 b(Sciunzi,)f(E.)g(V)-6 b(aldino)r(ci,)55
b Fc(Me)l(an)28 b(curvatur)l(e)i(pr)l(op)l(erties)g(for)e
FD(p)p Fc(-L)l(aplac)l(e)g(phase)h(tr)l(ansitions)p FF(,)613
5018 y(to)d(app)r(ear)g(in)g(J.)g(Eur.)g(Math.)g(So)r(c.)h(\(JEMS\).)
456 5109 y([31])39 b(V.)31 b(O.)f(Sa)n(vin,)64 b Fc(Phase)32
b(T)-6 b(r)l(ansitions:)45 b(R)l(e)l(gularity)33 b(of)e(Flat)h(L)l
(evel)g(Sets)p FF(,)h(Ph.D.)e(thesis)f(at)h(the)613 5200
y(Univ)n(ersit)n(y)25 b(of)i(T)-6 b(exas)26 b(at)f(Austin,)h(2003.)p
eop
%%Page: 153 153
153 152 bop 700 251 a Fy(FLA)-6 b(T)31 b(LEVEL)f(SET)h(REGULARITY)g(OF)
f Fv(p)p Fy(-LAPLA)n(CE)g(PHASE)g(TRANSITIONS)140 b(153)456
450 y FF([32])39 b(V.)26 b(O.)g(Sa)n(vin,)f(man)n(uscript,)g(2004.)456
541 y([33])39 b(P)-6 b(.)19 b(Stern)n(b)r(erg,)41 b Fc(The)21
b(e\013e)l(ct)h(of)e(a)g(singular)h(p)l(erturb)l(ation)j(on)c(nonc)l
(onvex)j(variational)e(pr)l(oblems)p FF(,)613 633 y(Arc)n(h.)26
b(Ration.)g(Mec)n(h.)g(Anal.)g(101)h(\(1988\),)g(no.)f(3,)g(209{260.)
456 724 y([34])39 b(P)-6 b(.)29 b(T)-6 b(olksdorf,)31
b Fc(R)l(e)l(gularity)g(for)g(a)f(mor)l(e)h(gener)l(al)g(class)g(of)f
(quasiline)l(ar)h(el)t(liptic)e(e)l(quations)p FF(,)j(J.)613
815 y(Di\013eren)n(tial)27 b(Equations)f(51)g(\(1984\),)h(no.)f(1,)h
(126{150.)456 907 y([35])39 b(P)-6 b(.)24 b(T)-6 b(olksdorf,)26
b Fc(On)f(the)i(dirichlet)f(pr)l(oblem)g(for)g(quasiline)l(ar)g(e)l
(quations)h(in)f(domains)g(with)f(c)l(on-)613 998 y(ic)l(al)j(b)l
(oundary)h(p)l(oints)p FF(,)e(Comm.)e(P)n(artial)i(Di\013eren)n(tial)f
(Equations)g(\(1983\),)h(no.)f(7,)g(773{817.)456 1089
y([36])39 b(E.)i(V)-6 b(aldino)r(ci,)90 b Fc(Plane-like)41
b(minimizers)g(in)g(p)l(erio)l(dic)h(me)l(dia:)62 b(jet)41
b(\015ows)h(and)f(Ginzbur)l(g-)613 1181 y(L)l(andau-typ)l(e)31
b(functionals)p FF(,)26 b(J.)h(Reine)e(Angew.)h(Math)g(574)h(\(2004\),)
g(147{185.)456 1272 y([37])39 b(J.)32 b(L.)e(V)-6 b(azquez,)31
b Fc(A)h(str)l(ong)h(maximum)f(principle)g(for)f(some)i(quasiline)l(ar)
f(el)t(liptic)f(e)l(quations)p FF(,)613 1363 y(Appl.)26
b(Math.)g(Optim.,)f(1984,)i(191)g({)f(202.)1680 1822
y FG(?)202 b(?)h(?)555 2331 y FE(Dip)-5 b(ar)g(timento)34
b(di)f(Ma)-5 b(tema)g(tica,)35 b(Universit)2082 2325
y(\022)2080 2331 y(a)e(di)h(R)n(oma)g(Tor)h(Ver)n(ga)-5
b(t)g(a,)36 b(Via)e(della)456 2422 y(Ricer)n(ca)27 b(Scientifica,)g(1,)
h(I-00133)i(R)n(oma)e(\(It)-5 b(al)g(y\))555 2513 y Fc(E-mail)27
b(addr)l(ess)6 b FF(:)37 b Fa(sciunzi,)k(valdinoci@mat.uniroma2.it)555
2668 y FE(Dep)-5 b(ar)g(tment)27 b(of)i(Ma)-5 b(thema)g(tics,)27
b(UC)h(Berkeley,)f(Berkeley,)h(CA)g(94720,)h(USA)555
2760 y Fc(E-mail)e(addr)l(ess)6 b FF(:)115 b Fa
(osavin@math.berkeley.edu)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF
