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b(de\014ned)e(b)o(y)h (the)h(relation)563 427 y Fo(\031)591 434 y Fl(1)627 427 y Fp(!)d Fo(\031)719 434 y Fl(2)855 427 y Fp(\()-8 b(\))1070 414 y Fk(e)1060 427 y Fo(B)r Fu(\()p Fo(\031)1147 434 y Fl(1)1170 427 y Fu(\))12 b Fp(\\)f Fo(\031)1273 434 y Fl(2)1310 427 y Fp(6)p Fu(=)i Fp(;)0 526 y Fu(i.e.)34 b(w)o(e)20 b(determine)g(the)h(orbit)f (of)g(the)h(cycle)g Fp(f)p Fo(\033)960 508 y Fj(0)975 526 y Fp(g)f Fu(=)g Fo(\031)r Fu(\()p Fo(\033)1158 508 y Fj(0)1173 526 y Fo(;)8 b Fp(f\000)p Fu(1)p 1220 538 V Fo(;)g Fu(+1)p 1306 538 V Fp(g)1395 508 y Fn(c)1415 526 y Fu(\))21 b(in)f(this)g(graph)g (\(i.e.)34 b(the)0 586 y(minimal)23 b(stable)h(subgraph)e(con)o(taining)h Fp(f)p Fo(\033)883 567 y Fj(0)897 586 y Fp(g)p Fu(\).)46 b(The)24 b(go)q(o)q(d)g(cycle)h(paths)f(are)g(the)h(cycle)g(paths)0 645 y(included)15 b(in)h(this)h(graph)e(whic)o(h)g(end)h(at)h(a)g(cycle)g (whose)f(principal)f(b)q(oundary)g(con)o(tains)g(+1)p 1732 657 V(.)604 756 y Fv(5.)27 b(The)18 b(bottom)h(of)e(the)h(cycles)60 846 y Fu(The)g(results)f(of)i(this)e(section)i(are)e(v)m(alid)i(in)e(an)o(y)h (dimension)f Fo(d)p Fu(.)27 b(They)18 b(describ)q(e)g(some)f(essen)o(tial)0 905 y(facts)k(concerning)e(the)i(geometry)f(of)h(the)g(cycles)f(of)h(the)g (energy)f(landscap)q(e)g(of)g(the)h(Ising)f(mo)q(del.)0 965 y(Theorem)14 b(5)p Fo(:)p Fu(3)h(concerns)e(the)j(p)q(oin)o(ts)e(of)h(the)g (cycle)h(w)o(e)e(migh)o(t)g(reac)o(h)g(starting)h(from)f(a)h(con\014guration) 0 1025 y(b)q(elonging)k(to)i(the)g(b)q(ottom)f(of)h(the)f(cycle.)34 b(Theorem)19 b(5)p Fo(:)p Fu(5)i(giv)o(es)e(a)i(condition)e(implying)g(that)i (the)0 1085 y(b)q(ottom)c(of)g(the)f(cycle)i(is)e(reduced)f(to)i(one)f(p)q (oin)o(t.)22 b(These)16 b(theorems)g(will)g(b)q(e)h(crucial)e(to)i(handle)f (the)0 1144 y(t)o(w)o(o)g(and)g(three)g(dimensional)f(cases.)21 b(Their)16 b(pro)q(ofs)g(rely)g(hea)o(vily)g(on)g(the)h(follo)o(wing)f (inequalit)o(y)l(.)0 1230 y Fw(Theorem)h(5.1.)28 b Fh(\(an)17 b(energy)f(inequalit)o(y\))0 1289 y(F)l(or)f(an)o(y)h(con\014gurations)f Fo(\021)r(;)h(\033)o(;)h(\030)h Fh(of)f Fo(X)k Fh(suc)o(h)15 b(that)i Fo(\021)f Fp(\032)d Fo(\033)j Fp(\032)e Fo(\030)r Fh(,)i(w)o(e)h(ha)o(v)o(e)f(\(for)g(p)q(ositiv)o(e)h Fo(h)p Fh(\))556 1389 y Fo(E)s Fu(\()p Fo(\033)c Fp(n)e Fo(\021)r Fu(\))20 b Fp(\000)f Fo(E)s Fu(\()p Fo(\033)r Fu(\))31 b Fp(\024)f Fo(E)s Fu(\()p Fo(\030)13 b Fp(n)e Fo(\021)r Fu(\))19 b Fp(\000)h Fo(E)s Fu(\()p Fo(\030)r Fu(\))p Fo(:)0 1513 y Fe(R)m(emark.)27 b Fu(This)13 b(inequalit)o(y)h(migh)o(t)f(b)q(e)h(in)o(terpreted)f(as)h (follo)o(ws.)20 b(The)14 b(v)m(ariation)f(of)i(energy)f(when)f(w)o(e)0 1573 y(turn)j(do)o(wn)f(a)i(\014xed)f(set)h(of)g(spins)e(of)h(a)h (con\014guration)e Fo(\033)k Fu(increases)c(when)h(w)o(e)g(enlarge)g Fo(\033)r Fu(.)0 1658 y Fe(Pr)m(o)m(of.)28 b Fu(W)l(e)16 b(compute)g Fo(E)s Fu(\()p Fo(\033)e Fp(n)d Fo(\021)r Fu(\):)29 1783 y Fo(E)s Fu(\()p Fo(\033)i Fp(n)e Fo(\021)r Fu(\))30 b(=)h Fp(\000)356 1750 y Fu(1)p 356 1772 25 2 v 356 1818 a(2)450 1736 y Fk(X)395 1845 y Fj(f)p Fn(x;y)q Fj(g)p Fl(:)p Fn(x)p Fj(\030)p Fn(y)578 1783 y Fu(\()p Fo(\033)13 b Fp(n)e Fo(\021)r Fu(\)\()p Fo(x)p Fu(\)\()p Fo(\033)k Fp(n)c Fo(\021)r Fu(\)\()p Fo(y)r Fu(\))20 b Fp(\000)1078 1750 y Fo(h)p 1078 1772 29 2 v 1080 1818 a Fu(2)1124 1736 y Fk(X)1121 1843 y Fn(x)p Fj(2)p Fl(\003)1198 1783 y Fu(\()p Fo(\033)14 b Fp(n)d Fo(\021)r Fu(\)\()p Fo(x)p Fu(\))241 1951 y(=)31 b Fp(\000)356 1917 y Fu(1)p 356 1940 25 2 v 356 1985 a(2)386 1881 y Fk(\022)516 1904 y(X)471 2006 y Ff(f)p Fg(x;y)q Ff(g)p Fd(:)p Fg(x)p Ff(\030)p Fg(y)437 2036 y(\033)q Fd(\()p Fg(x)p Fd(\)=)p Fg(\033)q Fd(\()p Fg(y)q Fd(\)=)p Ff(\000)p Fd(1)649 1951 y Fu(1)19 b(+)850 1904 y Fk(X)805 2006 y Ff(f)p Fg(x;y)q Ff(g)p Fd(:)p Fg(x)p Ff(\030)p Fg(y)757 2036 y(\033)q Fd(\()p Fg(x)p Fd(\)=1)p Fg(;\033)q Fd(\()p Fg(y)q Fd(\)=)p Ff(\000)p Fd(1)997 1951 y Fo(\021)r Fu(\()p Fo(x)p Fu(\))h(+)1240 1904 y Fk(X)1194 2006 y Ff(f)p Fg(x;y)q Ff(g)p Fd(:)p Fg(x)p Ff(\030)p Fg(y)1174 2036 y(\033)q Fd(\()p 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Fd(\)=)p Ff(\000)p Fd(1)698 703 y Fu(\(1)12 b(+)f Fo(\021)r Fu(\()p Fo(x)p Fu(\)\))20 b(+)1000 670 y(1)p 1000 692 V 1000 738 a(2)1078 656 y Fk(X)1032 758 y Ff(f)p Fg(x;y)q Ff(g)p Fd(:)p Fg(x)p Ff(\030)p Fg(y)1012 788 y(\033)q Fd(\()p Fg(x)p Fd(\)=)p Fg(\033)q Fd(\()p Fg(y)q Fd(\)=1)1188 703 y Fu(\(1)11 b Fp(\000)g Fo(\021)r Fu(\()p Fo(x)p Fu(\))p Fo(\021)r Fu(\()p Fo(y)r Fu(\)\))21 b(+)1581 670 y Fo(h)p 1581 692 29 2 v 1583 738 a Fu(2)1624 656 y Fk(X)1649 761 y Fn(x)1696 703 y Fu(\(1)12 b(+)e Fo(\021)r Fu(\()p Fo(x)p Fu(\)\))p Fo(:)0 909 y Fu(Only)18 b(the)h(\014rst)f(t)o(w)o(o)g(terms)g(of)h(the)g(righ)o (thand)e(side)h(dep)q(end)g(on)g Fo(\033)r Fu(.)29 b(Y)l(et)19 b(for)g(an)o(y)f(con\014guration)f Fo(\030)0 969 y Fu(suc)o(h)e(that)i Fo(\033)f Fp(\032)e Fo(\030)19 b Fu(w)o(e)d(ha)o(v)o(e)486 1120 y Fp(\000)531 1087 y Fu(1)p 531 1109 25 2 v 531 1154 a(2)604 1073 y Fk(X)559 1175 y Ff(f)p Fg(x;y)q Ff(g)p Fd(:)p Fg(x)p Ff(\030)p Fg(y)572 1205 y(\033)q Fd(\()p Fg(y)q Fd(\)=)p Ff(\000)p Fd(1)711 1120 y Fu(\(1)11 b(+)g Fo(\021)r Fu(\()p Fo(x)p Fu(\)\))32 b Fp(\024)e(\000)1074 1087 y Fu(1)p 1074 1109 V 1074 1154 a(2)1148 1073 y Fk(X)1102 1175 y Ff(f)p Fg(x;y)q Ff(g)p Fd(:)p Fg(x)p Ff(\030)p Fg(y)1117 1205 y(\030)p Fd(\()p Fg(y)q Fd(\)=)p Ff(\000)p Fd(1)1254 1120 y Fu(\(1)12 b(+)e Fo(\021)r Fu(\()p Fo(x)p Fu(\)\))e Fo(;)432 1268 y Fu(1)p 432 1290 V 432 1336 a(2)509 1254 y Fk(X)464 1356 y Ff(f)p Fg(x;y)q Ff(g)p Fd(:)p Fg(x)p Ff(\030)p Fg(y)443 1386 y(\033)q Fd(\()p Fg(x)p Fd(\)=)p Fg(\033)q Fd(\()p Fg(y)q Fd(\)=1)620 1302 y Fu(\(1)j Fp(\000)g Fo(\021)r Fu(\()p Fo(x)p Fu(\))p Fo(\021)r Fu(\()p Fo(y)r Fu(\)\))32 b Fp(\024)1035 1268 y Fu(1)p 1035 1290 V 1035 1336 a(2)1108 1254 y Fk(X)1063 1356 y Ff(f)p Fg(x;y)q Ff(g)p Fd(:)p Fg(x)p Ff(\030)p Fg(y)1047 1386 y(\030)p Fd(\()p Fg(x)p Fd(\)=)p Fg(\030)p Fd(\()p Fg(y)q Fd(\)=1)1215 1302 y Fu(\(1)11 b Fp(\000)g Fo(\021)r Fu(\()p Fo(x)p Fu(\))p Fo(\021)r Fu(\()p Fo(y)r Fu(\)\))0 1510 y(so)16 b(that)h(\014nally)f Fo(E)s Fu(\()p Fo(\033)d Fp(n)e Fo(\021)r Fu(\))g Fp(\000)g Fo(E)s Fu(\()p Fo(\033)r Fu(\))j Fp(\024)g Fo(E)s Fu(\()p Fo(\030)f Fp(n)e Fo(\021)r Fu(\))g Fp(\000)g Fo(E)s Fu(\()p Fo(\030)r Fu(\).)50 b Fc(\003)0 1611 y Fw(Notations)23 b(5.2.)28 b Fu(Let)21 b Fo(x)f Fu(b)q(e)h(a)f(site)g(in)g(\003.)32 b(W)l(e)20 b(de\014ne)g(the)g(spin)f(\015ip)g(op)q(erator)h Fo(F)7 b Fu(\()p Fo(x)p Fu(\))21 b(:)e Fo(X)24 b Fp(!)c Fo(X)0 1671 y Fu(b)o(y)c Fo(F)7 b Fu(\()p Fo(x)p Fu(\)\()p Fo(\033)r Fu(\))16 b(=)e Fo(\033)341 1653 y Fn(x)383 1671 y Fu(where)i(w)o(e)g(recall)g (that)664 1834 y Fo(\033)694 1813 y Fn(x)719 1834 y Fu(\()p Fo(y)r Fu(\))23 b(=)867 1763 y Fk(\032)921 1796 y Fu(+)p Fo(\033)r Fu(\()p Fo(y)r Fu(\))51 b(if)17 b Fo(y)f Fp(6)p Fu(=)d Fo(x)921 1871 y Fp(\000)p Fo(\033)r Fu(\()p Fo(y)r Fu(\))51 b(if)17 b Fo(y)f Fu(=)d Fo(x)0 1997 y Fu(Let)j Fo(x)116 2004 y Fl(1)139 1997 y Fo(;)8 b Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)286 2004 y Fn(r)324 1997 y Fu(b)q(e)16 b(a)f(sequence)h(of)g(sites)f(in)g(\003.)21 b(W)l(e)16 b(de\014ne)f(the)h(op)q(erator)f Fo(F)7 b Fu(\()p Fo(x)1529 2004 y Fl(1)1552 1997 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1699 2004 y Fn(r)1721 1997 y Fu(\))14 b(:)g Fo(X)k Fp(!)c Fo(X)0 2056 y Fu(b)o(y)i Fo(F)7 b Fu(\()p Fo(x)155 2063 y Fl(1)178 2056 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)325 2063 y Fn(r)348 2056 y Fu(\))14 b(=)g Fo(F)7 b Fu(\()p Fo(x)520 2063 y Fn(r)542 2056 y Fu(\))12 b Fp(\016)f(\001)d(\001)g(\001)j (\016)g Fo(F)c Fu(\()p Fo(x)800 2063 y Fl(1)823 2056 y Fu(\).)0 2116 y(Let)19 b Fo(\033)i Fu(b)q(elong)d(to)h Fo(X)t Fu(.)28 b(T)l(o)18 b Fo(\033)j Fu(w)o(e)d(asso)q(ciate)h(t)o(w)o(o)f(op)q(erators)f Fo(S)1211 2123 y Fl(+)1244 2116 y Fu(\()p Fo(\033)r Fu(\))i(and)f Fo(S)1461 2123 y Fj(\000)1494 2116 y Fu(\()p Fo(\033)r Fu(\))i(on)e(the)h (sequences)0 2176 y(of)25 b(sites.)48 b(The)25 b(sequence)g Fo(S)579 2183 y Fj(\000)612 2176 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)727 2183 y Fl(1)751 2176 y Fo(;)8 b Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)898 2183 y Fn(r)920 2176 y Fu(\))26 b(\(resp)q(ectiv)o(ely)g Fo(S)1296 2183 y Fl(+)1337 2176 y Fp(\001)8 b(\001)g(\001)g Fu(\)\))26 b(is)e(the)i(subsequence)e(of)0 2236 y(\()p Fo(x)47 2243 y Fl(1)70 2236 y Fo(;)8 b Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)217 2243 y Fn(r)240 2236 y Fu(\))17 b(consisting)e(of)i(the)f(sites)h Fo(x)788 2243 y Fn(i)821 2236 y Fu(suc)o(h)f(that)h Fo(\033)r Fu(\()p Fo(x)1119 2243 y Fn(i)1136 2236 y Fu(\))e(=)e Fp(\000)p Fu(1)j(\(resp.)22 b Fo(\033)r Fu(\()p Fo(x)1524 2243 y Fn(i)1541 2236 y Fu(\))15 b(=)e(+1\)\).)0 2336 y Fe(R)m(emark.)27 b Fu(F)l(or)15 b(an)o(y)h(sequence)g(of)h(sites)f(\()p Fo(x)812 2343 y Fl(1)836 2336 y Fo(;)8 b Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)983 2343 y Fn(r)1005 2336 y Fu(\))17 b(and)f(an)o(y)g(con\014guration)f Fo(\033)k Fu(w)o(e)d(ha)o(v)o(e)315 2467 y Fo(F)7 b Fu(\()p Fo(S)404 2474 y Fl(+)437 2467 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)552 2474 y Fl(1)576 2467 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)723 2474 y Fn(r)745 2467 y Fu(\)\)\()p Fo(\033)r Fu(\))32 b Fp(\032)f Fo(\033)h Fp(\032)f Fo(F)7 b Fu(\()p Fo(S)1172 2474 y Fj(\000)1205 2467 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)1320 2474 y Fl(1)1344 2467 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1491 2474 y Fn(r)1514 2467 y Fu(\)\)\()p Fo(\033)r Fu(\))p Fo(:)954 2670 y Fs(10)p eop %%Page: 11 11 11 10 bop 0 200 a Fw(Theorem)17 b(5.3.)50 b Fh(Let)17 b Fo(\031)i Fh(b)q(e)e(a)f(cycle)i(and)e(let)g Fo(\033)j Fh(b)q(e)e(a)g(con\014guration)d (of)j(the)g(b)q(ottom)g Fo(F)7 b Fu(\()p Fo(\031)r Fu(\))17 b Fh(of)g Fo(\031)r Fh(.)0 260 y(Let)j Fu(\()p Fo(x)139 267 y Fl(1)162 260 y Fo(;)8 b Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)309 267 y Fn(r)332 260 y Fu(\))20 b Fh(b)q(e)f(a)h(sequence)f(of)h(sites) f(suc)o(h)f(that)i Fo(F)7 b Fu(\()p Fo(x)1181 267 y Fl(1)1204 260 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1351 267 y Fn(j)1372 260 y Fu(\)\()p Fo(\033)r Fu(\))21 b Fh(b)q(elongs)e(to)h Fo(\031)h Fh(for)f(all)f Fo(j)0 319 y Fh(in)14 b Fp(f)p Fu(1)8 b Fp(\001)g(\001)g(\001)g Fo(r)q Fp(g)p Fh(.)22 b(Then)14 b Fo(F)7 b Fu(\()p Fo(S)481 326 y Fj(\000)515 319 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)630 326 y Fl(1)654 319 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)801 326 y Fn(r)823 319 y Fu(\)\)\()p Fo(\033)r Fu(\))16 b Fh(and)e Fo(F)7 b Fu(\()p Fo(S)1129 326 y Fl(+)1162 319 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)1277 326 y Fl(1)1301 319 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1448 326 y Fn(r)1471 319 y Fu(\)\)\()p Fo(\033)r Fu(\))16 b Fh(also)d(b)q(elong)h(to)h Fo(\031)r Fh(.)0 379 y(Equiv)m(alen)o(tly)l(,)h(if)h(w)o(e)f(let)h Fo(\021)e Fu(=)f Fo(F)7 b Fu(\()p Fo(x)668 386 y Fl(1)691 379 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)838 386 y Fn(r)861 379 y Fu(\)\()p Fo(\033)r Fu(\))p Fh(,)18 b(then)e Fo(\021)d Fp([)e Fo(\033)19 b Fh(and)d Fo(\021)c Fp(\\)f Fo(\033)19 b Fh(are)d(in)h Fo(\031)r Fh(.)0 468 y Fe(Pr)m(o)m(of.)28 b Fu(W)l(e)11 b(pro)o(v)o(e)f(the)i(result)e(b)o(y)h(induction)f(on)h(the)h(length)e Fo(r)k Fu(of)d(the)h(sequence)f(of)g(sites)g(\()p Fo(x)1724 475 y Fl(1)1747 468 y Fo(;)d Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1894 475 y Fn(r)1917 468 y Fu(\).)0 528 y(The)21 b(result)f(is)h(ob)o (viously)f(true)h(for)g Fo(r)i Fu(=)f(0)f(and)f Fo(r)k Fu(=)d(1.)36 b(Assume)20 b(it)h(is)g(true)g(at)g(rank)g Fo(r)i Fu(and)e(let)0 587 y(\()p Fo(x)47 594 y Fl(1)70 587 y Fo(;)8 b Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)217 594 y Fn(r)q Fl(+1)290 587 y Fu(\))23 b(b)q(e)g(a)f (sequence)g(of)h(sites)f(suc)o(h)f(that)i Fo(F)7 b Fu(\()p Fo(x)1163 594 y Fl(1)1186 587 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1333 594 y Fn(j)1355 587 y Fu(\)\()p Fo(\033)r Fu(\))23 b(b)q(elongs)f(to)h Fo(\031)h Fu(for)f(all)f Fo(j)0 647 y Fu(in)16 b Fp(f)p Fu(1)8 b Fp(\001)g(\001)g(\001)h Fo(r)j Fu(+)f(1)p Fp(g)p Fu(.)22 b(Let)17 b Fo(\030)f Fu(=)d Fo(F)7 b Fu(\()p Fo(S)621 654 y Fj(\000)655 647 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)770 654 y Fl(1)794 647 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)941 654 y Fn(r)q Fl(+1)1014 647 y Fu(\)\)\()p Fo(\033)r Fu(\))18 b(and)e(let)h Fo(\021)h Fu(b)q(e)e(the)h(unique)f (con\014guration)0 707 y(included)k(in)h Fo(\033)i Fu(suc)o(h)d(that)h Fo(\033)c Fp(n)d Fo(\021)31 b Fu(=)f Fo(F)7 b Fu(\()p Fo(S)845 714 y Fl(+)878 707 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)993 714 y Fl(1)1017 707 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1164 714 y Fn(r)q Fl(+1)1237 707 y Fu(\)\)\()p Fo(\033)r Fu(\).)37 b(W)l(e)22 b(ha)o(v)o(e)e Fo(\021)j Fp(\032)e Fo(\033)j Fp(\032)d Fo(\030)j Fu(and)0 767 y(theorem)16 b(5)p Fo(:)p Fu(1)g(yields)g Fo(E)s Fu(\()p Fo(\030)r Fu(\))c Fp(\000)e Fo(E)s Fu(\()p Fo(\030)k Fp(n)d Fo(\021)r Fu(\))22 b Fp(\024)g Fo(E)s Fu(\()p Fo(\033)r Fu(\))11 b Fp(\000)g Fo(E)s Fu(\()p Fo(\033)i Fp(n)e Fo(\021)r Fu(\))17 b(or)f(equiv)m(alen)o(tly)0 882 y Fo(E)s Fu(\()p Fo(F)7 b Fu(\()p Fo(S)148 889 y Fj(\000)182 882 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)297 889 y Fl(1)321 882 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)468 889 y Fn(r)q Fl(+1)541 882 y Fu(\)\)\()p Fo(\033)r Fu(\)\))21 b(+)e Fo(E)s Fu(\()p Fo(F)7 b Fu(\()p Fo(S)893 889 y Fl(+)926 882 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)1041 889 y Fl(1)1065 882 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1212 889 y Fn(r)q Fl(+1)1285 882 y Fu(\)\)\()p Fo(\033)r Fu(\)\))33 b Fp(\024)595 957 y Fu(\(1\))596 b Fo(E)s Fu(\()p Fo(F)7 b Fu(\()p Fo(x)1399 964 y Fl(1)1422 957 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1569 964 y Fn(r)q Fl(+1)1642 957 y Fu(\)\()p Fo(\033)r Fu(\)\))21 b(+)f Fo(E)s Fu(\()p Fo(\033)r Fu(\))p Fo(:)17 1073 y Fu(By)d(h)o(yp)q(othesis,)e Fo(\030)e Fp(n)e Fo(\021)24 b Fu(=)e Fo(F)7 b Fu(\()p Fo(x)621 1080 y Fl(1)644 1073 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)791 1080 y Fn(r)q Fl(+1)864 1073 y Fu(\)\()p Fo(\033)r Fu(\))15 b Fp(2)f Fo(\031)19 b Fu(whence)d Fo(E)s Fu(\()p Fo(\030)d Fp(n)e Fo(\021)r Fu(\))j Fp(\024)g Fo(\025)p Fu(\()p Fo(\031)r Fu(\).)0 1133 y Fp(\017)50 b Fu(First)15 b(case:)22 b Fo(\033)r Fu(\()p Fo(x)398 1140 y Fn(r)q Fl(+1)472 1133 y Fu(\))14 b(=)f Fp(\000)p Fu(1.)22 b(In)16 b(this)g(situation,)g(w)o(e) g(ha)o(v)o(e)428 1248 y Fo(S)459 1255 y Fj(\000)492 1248 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)607 1255 y Fl(1)632 1248 y Fo(;)8 b Fp(\001)g(\001)g(\001)16 b Fo(;)8 b(x)778 1255 y Fn(r)q Fl(+1)851 1248 y Fu(\))23 b(=)f(\()p Fo(S)1004 1255 y Fj(\000)1037 1248 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)1152 1255 y Fl(1)1177 1248 y Fo(;)8 b Fp(\001)g(\001)g(\001)16 b Fo(;)8 b(x)1323 1255 y Fn(r)1346 1248 y Fu(\))p Fo(;)g(x)1415 1255 y Fn(r)q Fl(+1)1489 1248 y Fu(\))p Fo(;)429 1326 y(S)460 1333 y Fl(+)492 1326 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)607 1333 y Fl(1)632 1326 y Fo(;)g Fp(\001)g(\001)g(\001)16 b Fo(;)8 b(x)778 1333 y Fn(r)q Fl(+1)851 1326 y Fu(\))23 b(=)f Fo(S)985 1333 y Fl(+)1018 1326 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)1133 1333 y Fl(1)1157 1326 y Fo(;)8 b Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1304 1333 y Fn(r)1326 1326 y Fu(\))p Fo(:)0 1441 y Fu(By)17 b(the)g(induction)e(h) o(yp)q(othesis,)h(w)o(e)g(kno)o(w)g(that)252 1557 y Fo(F)7 b Fu(\()p Fo(S)341 1564 y Fj(\000)375 1557 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)490 1564 y Fl(1)514 1557 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)661 1564 y Fn(r)683 1557 y Fu(\)\)\()p Fo(\033)r Fu(\))16 b Fp(2)e Fo(\031)r(;)108 b(F)7 b Fu(\()p Fo(S)1093 1564 y Fl(+)1126 1557 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)1241 1564 y Fl(1)1265 1557 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1412 1564 y Fn(r)q Fl(+1)1485 1557 y Fu(\)\)\()p Fo(\033)r Fu(\))16 b Fp(2)e Fo(\031)r(:)0 1673 y Fu(No)o(w)19 b Fo(\030)g Fu(=)e Fo(F)7 b Fu(\()p Fo(x)299 1680 y Fn(r)q Fl(+1)372 1673 y Fu(\))13 b Fp(\016)g Fo(F)7 b Fu(\()p Fo(S)531 1680 y Fj(\000)564 1673 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)679 1680 y Fl(1)703 1673 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)850 1680 y Fn(r)873 1673 y Fu(\)\)\()p Fo(\033)r Fu(\))20 b(i.e.)f Fo(\030)i Fu(comm)o(unicates)16 b(b)o(y)j(one)f(spin)g(\015ip)g(with)g(a)0 1732 y(con\014guration)10 b(of)i Fo(\031)r Fu(.)21 b(Moreo)o(v)o(er)10 b Fo(\033)t Fp(n)r Fo(\021)k Fu(b)q(elongs)d(to)h Fo(\031)i Fu(whence)e Fo(E)s Fu(\()p Fo(\033)r Fu(\))i Fp(\024)g Fo(E)s Fu(\()p Fo(\033)t Fp(n)r Fo(\021)r Fu(\))e(\(b)q(ecause)g Fo(\033)k Fp(2)e Fo(F)7 b Fu(\()p Fo(\031)r Fu(\)\).)0 1792 y(Inequalit)o(y)16 b(\(1\))h(sho)o(ws)f(that)g Fo(E)s Fu(\()p Fo(\030)r Fu(\))i(is)e(less)g (than)g(the)h(lev)o(el)f(of)h Fo(\031)i Fu(so)d(that)h Fo(\030)h Fu(is)e(in)h Fo(\031)r Fu(.)0 1852 y Fp(\017)50 b Fu(Second)16 b(case:)21 b Fo(\033)r Fu(\()p Fo(x)445 1859 y Fn(r)q Fl(+1)519 1852 y Fu(\))14 b(=)g(+1.)21 b(In)c(this)f(situation,)f(w)o(e)i(ha)o(v)o(e) 428 1968 y Fo(S)459 1975 y Fj(\000)493 1968 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)608 1975 y Fl(1)632 1968 y Fo(;)8 b Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)779 1975 y Fn(r)q Fl(+1)852 1968 y Fu(\))22 b(=)g Fo(S)985 1975 y Fj(\000)1018 1968 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)1133 1975 y Fl(1)1158 1968 y Fo(;)8 b Fp(\001)g(\001)g(\001)16 b Fo(;)8 b(x)1304 1975 y Fn(r)1327 1968 y Fu(\))p Fo(;)429 2045 y(S)460 2052 y Fl(+)493 2045 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)608 2052 y Fl(1)632 2045 y Fo(;)g Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)779 2052 y Fn(r)q Fl(+1)852 2045 y Fu(\))22 b(=)g(\()p Fo(S)1004 2052 y Fl(+)1037 2045 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)1152 2052 y Fl(1)1176 2045 y Fo(;)8 b Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1323 2052 y Fn(r)1346 2045 y Fu(\))p Fo(;)g(x)1415 2052 y Fn(r)q Fl(+1)1488 2045 y Fu(\))p Fo(:)0 2160 y Fu(By)17 b(the)g(induction)e(h)o(yp)q(othesis,)h(w)o(e)g(kno)o(w)g(that)252 2276 y Fo(F)7 b Fu(\()p Fo(S)341 2283 y Fj(\000)375 2276 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)490 2283 y Fl(1)514 2276 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)661 2283 y Fn(r)q Fl(+1)734 2276 y Fu(\)\)\()p Fo(\033)r Fu(\))15 b Fp(2)g Fo(\031)r(;)107 b(F)7 b Fu(\()p Fo(S)1143 2283 y Fl(+)1176 2276 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)1291 2283 y Fl(1)1316 2276 y Fo(;)h Fp(\001)g(\001)g(\001)16 b Fo(;)8 b(x)1462 2283 y Fn(r)1485 2276 y Fu(\)\)\()p Fo(\033)r Fu(\))16 b Fp(2)e Fo(\031)r(:)0 2392 y Fu(Inequalit)o(y)g(\(1\))i(sho)o(ws)d (that)i Fo(E)s Fu(\()p Fo(\033)10 b Fp(n)d Fo(\021)r Fu(\))15 b(is)f(less)g(than)h(the)g(lev)o(el)f(of)h Fo(\031)r Fu(.)22 b(Since)14 b Fo(\033)c Fp(n)d Fo(\021)17 b Fu(di\013ers)c(b)o(y)i(one)f(spin) 0 2452 y(\015ip)i(\(at)h(site)g Fo(x)286 2459 y Fn(r)q Fl(+1)359 2452 y Fu(\))g(from)e Fo(F)7 b Fu(\()p Fo(S)601 2459 y Fl(+)634 2452 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)749 2459 y Fl(1)773 2452 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)920 2459 y Fn(r)943 2452 y Fu(\)\)\()p Fo(\033)r Fu(\))18 b(whic)o(h)e(is)g(in)g Fo(\031)r Fu(,)h Fo(\033)c Fp(n)e Fo(\021)18 b Fu(is)e(also)g(in)g Fo(\031)r Fu(.)0 2511 y(Th)o(us)f(the)i(induction)e(is)h(completed.)50 b Fc(\003)954 2670 y Fs(11)p eop %%Page: 12 12 12 11 bop 0 200 a Fw(Corollary)17 b(5.4.)28 b Fh(Let)14 b Fo(\031)i Fh(b)q(e)e(a)g(cycle)h(an)e(let)h Fo(\033)j Fh(b)q(e)d(an)f(elemen)o(t)h(of)g Fo(F)7 b Fu(\()p Fo(\031)r Fu(\))p Fh(.)22 b(If)14 b Fo(\021)h Fh(is)f(a)f(minimal)f(\(resp)q(ec-)0 260 y(tiv)o(ely)k(maximal\))f(elemen)o (t)f(of)i Fo(\031)i Fh(with)d(resp)q(ect)h(to)f(the)h(inclusion,)e(then)h (there)h(exists)f(a)h(sequence)f(of)0 319 y(sites)j Fu(\()p Fo(x)160 326 y Fl(1)183 319 y Fo(;)8 b Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)330 326 y Fn(r)353 319 y Fu(\))19 b Fh(suc)o(h)e(that)i Fo(\021)g Fu(=)d Fo(F)7 b Fu(\()p Fo(x)800 326 y Fl(1)823 319 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)970 326 y Fn(r)993 319 y Fu(\)\()p Fo(\033)r Fu(\))p Fh(,)20 b Fo(F)7 b Fu(\()p Fo(x)1200 326 y Fl(1)1223 319 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1370 326 y Fn(j)1392 319 y Fu(\)\()p Fo(\033)r Fu(\))18 b Fp(2)g Fo(\031)i Fh(and)e Fo(\033)r Fu(\()p Fo(x)1772 326 y Fn(j)1794 319 y Fu(\))g(=)e(+1)0 379 y Fh(\(resp.)21 b Fo(\033)r Fu(\()p Fo(x)221 386 y Fn(j)243 379 y Fu(\))15 b(=)e Fp(\000)p Fu(1)p Fh(\))k(for)f(all)g Fo(j)j Fh(in)d Fp(f)p Fu(1)8 b Fp(\001)g(\001)g(\001)h Fo(r)q Fp(g)p Fh(.)23 b(In)16 b(particular,)f Fo(\021)g Fp(\032)f Fo(\033)k Fh(\(resp.)k Fo(\033)16 b Fp(\032)d Fo(\021)r Fh(\).)0 465 y Fw(Theorem)k(5.5.)50 b Fh(Let)17 b Fo(\031)i Fh(b)q(e)e(a)f(cycle)i(and)e(let)g Fo(\033)j Fh(b)q(e)e(a)g(con\014guration)d(of)j Fo(\031)i Fh(satisfying:)0 525 y(for)d(eac)o(h)g Fo(\017)h Fh(in)f Fp(f\000)p Fu(1)p Fo(;)8 b Fu(+1)p Fp(g)16 b Fh(and)g(eac)o(h)g(sequence)h(of)f(sites)g Fu(\()p Fo(x)1129 532 y Fl(1)1153 525 y Fo(;)8 b Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1300 532 y Fn(l)1315 525 y Fu(\))17 b Fh(in)g Fu(\003)1445 507 y Fn(l)1476 525 y Fh(suc)o(h)e(that)141 629 y Fo(F)7 b Fu(\()p Fo(x)227 636 y Fl(1)250 629 y Fo(;)h Fp(\001)g(\001)g (\001)17 b Fo(;)8 b(x)397 636 y Fn(l)413 629 y Fu(\)\()p Fo(\033)r Fu(\))24 b Fp(6)p Fu(=)e Fo(\033)o(;)108 b Fp(8)p Fo(j)15 b Fp(2)f(f)p Fu(1)8 b Fp(\001)g(\001)g(\001)g Fo(l)q Fp(g)50 b Fo(\033)r Fu(\()p Fo(x)1138 636 y Fn(j)1160 629 y Fu(\))14 b(=)g Fo(\017;)58 b(F)7 b Fu(\()p Fo(x)1424 636 y Fl(1)1447 629 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1594 636 y Fn(j)1615 629 y Fu(\)\()p Fo(\033)r Fu(\))16 b Fp(2)e Fo(\031)r(;)0 733 y Fh(w)o(e)i(ha)o(v)o(e)g Fo(E)s Fu(\()p Fo(F)7 b Fu(\()p Fo(x)333 740 y Fl(1)356 733 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)503 740 y Fn(l)519 733 y Fu(\)\))23 b Fo(>)f(E)s Fu(\()p Fo(\033)r Fu(\))p Fh(.)0 793 y(Then)16 b(the)h(con\014guration)e Fo(\033)j Fh(also)e(satis\014es:)0 853 y(for)i(eac)o(h)h(sequence)f(of)h (sites)g Fu(\()p Fo(x)621 860 y Fl(1)644 853 y Fo(;)8 b Fp(\001)g(\001)g (\001)17 b Fo(;)8 b(x)791 860 y Fn(r)814 853 y Fu(\))19 b Fh(in)f Fu(\003)947 835 y Fn(r)988 853 y Fh(suc)o(h)f(that)i Fo(F)7 b Fu(\()p Fo(x)1298 860 y Fl(1)1322 853 y Fo(;)h Fp(\001)g(\001)g(\001)16 b Fo(;)8 b(x)1468 860 y Fn(j)1490 853 y Fu(\)\()p Fo(\033)r Fu(\))20 b Fh(b)q(elongs)e(to)h Fo(\031)i Fh(for)0 913 y(all)14 b Fo(j)i Fh(in)e Fp(f)p Fu(1)8 b Fp(\001)g(\001)g(\001)g Fo(r)q Fp(g)p Fh(,)16 b(the)e(con\014gurations)e Fo(F)7 b Fu(\()p Fo(S)853 920 y Fj(\000)887 913 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)1002 920 y Fl(1)1026 913 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1173 920 y Fn(r)1195 913 y Fu(\)\)\()p Fo(\033)r Fu(\))16 b Fh(and)e Fo(F)7 b Fu(\()p Fo(S)1501 920 y Fl(+)1534 913 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)1649 920 y Fl(1)1673 913 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1820 920 y Fn(r)1842 913 y Fu(\)\)\()p Fo(\033)r Fu(\))0 973 y Fh(are)16 b(in)g Fo(\031)j Fh(and)299 1077 y Fo(E)s Fu(\()p Fo(F)7 b Fu(\()p Fo(x)444 1084 y Fl(1)468 1077 y Fo(;)h Fp(\001)g(\001)g(\001)16 b Fo(;)8 b(x)614 1084 y Fn(r)637 1077 y Fu(\)\()p Fo(\033)r Fu(\)\))24 b Fp(\025)e Fo(E)s Fu(\()p Fo(F)7 b Fu(\()p Fo(S)976 1084 y Fj(\000)1009 1077 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)1124 1084 y Fl(1)1148 1077 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1295 1084 y Fn(r)1318 1077 y Fu(\)\)\()p Fo(\033)r Fu(\)\))24 b Fp(\025)e Fo(E)s Fu(\()p Fo(\033)r Fu(\))p Fo(;)299 1154 y(E)s Fu(\()p Fo(F)7 b Fu(\()p Fo(x)444 1161 y Fl(1)468 1154 y Fo(;)h Fp(\001)g(\001)g(\001)16 b Fo(;)8 b(x)614 1161 y Fn(r)637 1154 y Fu(\)\()p Fo(\033)r Fu(\)\))24 b Fp(\025)e Fo(E)s Fu(\()p Fo(F)7 b Fu(\()p Fo(S)976 1161 y Fl(+)1009 1154 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)1124 1161 y Fl(1)1148 1154 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1295 1161 y Fn(r)1317 1154 y Fu(\)\)\()p Fo(\033)r Fu(\)\))25 b Fp(\025)d Fo(E)s Fu(\()p Fo(\033)r Fu(\))p Fo(;)299 1231 y(E)s Fu(\()p Fo(F)7 b Fu(\()p Fo(x)444 1238 y Fl(1)468 1231 y Fo(;)h Fp(\001)g(\001)g(\001)16 b Fo(;)8 b(x)614 1238 y Fn(r)637 1231 y Fu(\)\()p Fo(\033)r Fu(\)\))24 b(=)e Fo(E)s Fu(\()p Fo(\033)r Fu(\))31 b(=)-8 b Fp(\))30 b Fo(F)7 b Fu(\()p Fo(x)1164 1238 y Fl(1)1187 1231 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1334 1238 y Fn(r)1356 1231 y Fu(\)\()p Fo(\033)r Fu(\))24 b(=)e Fo(\033)o(:)0 1338 y Fh(The)16 b(b)q(ottom)h Fo(F)7 b Fu(\()p Fo(\031)r Fu(\))18 b Fh(of)e(the)h(cycle)g Fo(\031)i Fh(is)d(then)h(reduced)e(to)i(this)f(single)g(con\014guration)f Fo(\033)r Fh(.)0 1424 y Fe(Pr)m(o)m(of.)28 b Fu(W)l(e)12 b(pro)o(v)o(e)f(the) h(result)g(b)o(y)g(induction)f(on)g(the)i(length)f Fo(r)i Fu(of)e(the)h (sequence)f(of)g(sites)g(\()p Fo(x)1738 1431 y Fl(1)1761 1424 y Fo(;)c Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1908 1431 y Fn(r)1931 1424 y Fu(\))0 1484 y(app)q(earing)j(in)g(the)i(conclusion)d(of)j (the)f(theorem.)20 b(F)l(or)11 b Fo(r)k Fu(=)f(0)e(and)g Fo(r)j Fu(=)f(1)e(it)g(is)g(true.)20 b(Assume)11 b(it)i(is)e(true)0 1544 y(un)o(til)g(rank)g Fo(r)j Fu(and)d(let)h(\()p Fo(x)465 1551 y Fl(1)488 1544 y Fo(;)c Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)635 1551 y Fn(r)q Fl(+1)708 1544 y Fu(\))k(b)q(e)g(a)g(sequence)g(of)g (sites)f(suc)o(h)g(that)h Fo(F)7 b Fu(\()p Fo(x)1496 1551 y Fl(1)1519 1544 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1666 1551 y Fn(j)1688 1544 y Fu(\)\()p Fo(\033)r Fu(\))13 b(b)q(elongs)0 1603 y(to)j Fo(\031)i Fu(for)e(all)f Fo(j)j Fu(in)d Fp(f)p Fu(1)8 b Fp(\001)g(\001)g(\001)h Fo(r)i Fu(+)f(1)p Fp(g)p Fu(.)21 b(The)16 b(induction)e(h)o(yp)q(othesis)h(yields)g(that)h(for)g(all)f Fo(j)j Fu(in)e Fp(f)p Fu(1)8 b Fp(\001)g(\001)g(\001)g Fo(r)q Fp(g)17 b Fu(the)0 1663 y(con\014gurations)12 b Fo(F)7 b Fu(\()p Fo(S)408 1670 y Fj(\000)442 1663 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)557 1670 y Fl(1)581 1663 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)728 1670 y Fn(j)749 1663 y Fu(\)\)\()p Fo(\033)r Fu(\)\),)17 b Fo(F)7 b Fu(\()p Fo(S)994 1670 y Fl(+)1027 1663 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)1142 1670 y Fl(1)1166 1663 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1313 1670 y Fn(j)1335 1663 y Fu(\)\)\()p Fo(\033)r Fu(\)\))16 b(are)e(in)g Fo(\031)i Fu(and)e(moreo)o(v)o(er)310 1767 y Fo(E)s Fu(\()p Fo(F)7 b Fu(\()p Fo(x)455 1774 y Fl(1)478 1767 y Fo(;)h Fp(\001)g(\001)g (\001)17 b Fo(;)8 b(x)625 1774 y Fn(j)647 1767 y Fu(\)\()p Fo(\033)r Fu(\))23 b Fp(\025)f Fo(E)s Fu(\()p Fo(F)7 b Fu(\()p Fo(S)966 1774 y Fj(\000)1000 1767 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)1115 1774 y Fl(1)1139 1767 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1286 1774 y Fn(j)1307 1767 y Fu(\)\)\()p Fo(\033)r Fu(\)\))25 b Fp(\025)c Fo(E)s Fu(\()p Fo(\033)r Fu(\))p Fo(;)310 1844 y(E)s Fu(\()p Fo(F)7 b Fu(\()p Fo(x)455 1851 y Fl(1)478 1844 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)625 1851 y Fn(j)647 1844 y Fu(\)\()p Fo(\033)r Fu(\))23 b Fp(\025)f Fo(E)s Fu(\()p Fo(F)7 b Fu(\()p Fo(S)966 1851 y Fl(+)999 1844 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)1114 1851 y Fl(1)1138 1844 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1285 1851 y Fn(j)1307 1844 y Fu(\)\)\()p Fo(\033)r Fu(\)\))24 b Fp(\025)e Fo(E)s Fu(\()p Fo(\033)r Fu(\))p Fo(;)310 1922 y(E)s Fu(\()p Fo(F)7 b Fu(\()p Fo(x)455 1929 y Fl(1)478 1922 y Fo(;)h Fp(\001)g(\001)g (\001)17 b Fo(;)8 b(x)625 1929 y Fn(j)647 1922 y Fu(\)\()p Fo(\033)r Fu(\))23 b(=)f Fo(E)s Fu(\()p Fo(\033)r Fu(\))31 b(=)-8 b Fp(\))30 b Fo(F)7 b Fu(\()p Fo(x)1154 1929 y Fl(1)1177 1922 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1324 1929 y Fn(j)1345 1922 y Fu(\)\()p Fo(\033)r Fu(\))24 b(=)e Fo(\033)o(:)0 2026 y Fu(No)o(w)16 b(theorem)g(5)p Fo(:)p Fu(1)h(yields)0 2130 y Fo(E)s Fu(\()p Fo(F)7 b Fu(\()p Fo(x)145 2137 y Fl(1)168 2130 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)315 2137 y Fn(r)q Fl(+1)388 2130 y Fu(\)\()p Fo(\033)r Fu(\)\))21 b Fp(\000)e Fo(E)s Fu(\()p Fo(F)7 b Fu(\()p Fo(S)721 2137 y Fj(\000)755 2130 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)870 2137 y Fl(1)894 2130 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1041 2137 y Fn(r)q Fl(+1)1114 2130 y Fu(\)\)\()p Fo(\033)r Fu(\)\))33 b Fp(\025)508 2205 y Fu(\(2\))526 b Fo(E)s Fu(\()p Fo(F)7 b Fu(\()p Fo(S)1245 2212 y Fl(+)1278 2205 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)1393 2212 y Fl(1)1417 2205 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1564 2212 y Fn(r)q Fl(+1)1637 2205 y Fu(\)\)\()p Fo(\033)r Fu(\)\))21 b Fp(\000)e Fo(E)s Fu(\()p Fo(\033)r Fu(\))17 2309 y(By)e(h)o(yp)q(othesis,)e Fo(F)7 b Fu(\()p Fo(x)440 2316 y Fl(1)463 2309 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)610 2316 y Fn(r)q Fl(+1)683 2309 y Fu(\)\()p Fo(\033)r Fu(\))18 b(is)e(in)g Fo(\031)j Fu(so)d(that)h Fo(E)s Fu(\()p Fo(F)7 b Fu(\()p Fo(x)1257 2316 y Fl(1)1280 2309 y Fo(;)h Fp(\001)g(\001)g (\001)17 b Fo(;)8 b(x)1427 2316 y Fn(r)q Fl(+1)1500 2309 y Fu(\)\()p Fo(\033)r Fu(\))16 b Fp(\024)d Fo(\025)p Fu(\()p Fo(\031)r Fu(\).)0 2369 y Fp(\017)50 b Fu(First)15 b(case:)22 b Fo(\033)r Fu(\()p Fo(x)398 2376 y Fn(r)q Fl(+1)472 2369 y Fu(\))14 b(=)f Fp(\000)p Fu(1.)22 b(In)16 b(this)g(situation,)g(w)o(e)g(ha)o (v)o(e)428 2473 y Fo(S)459 2480 y Fj(\000)492 2473 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)607 2480 y Fl(1)632 2473 y Fo(;)8 b Fp(\001)g(\001)g(\001)16 b Fo(;)8 b(x)778 2480 y Fn(r)q Fl(+1)851 2473 y Fu(\))23 b(=)f(\()p Fo(S)1004 2480 y Fj(\000)1037 2473 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)1152 2480 y Fl(1)1177 2473 y Fo(;)8 b Fp(\001)g(\001)g(\001)16 b Fo(;)8 b(x)1323 2480 y Fn(r)1346 2473 y Fu(\))p Fo(;)g(x)1415 2480 y Fn(r)q Fl(+1)1489 2473 y Fu(\))p Fo(;)429 2550 y(S)460 2557 y Fl(+)492 2550 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)607 2557 y Fl(1)632 2550 y Fo(;)g Fp(\001)g(\001)g(\001)16 b Fo(;)8 b(x)778 2557 y Fn(r)q Fl(+1)851 2550 y Fu(\))23 b(=)f Fo(S)985 2557 y Fl(+)1018 2550 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)1133 2557 y Fl(1)1157 2550 y Fo(;)8 b Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1304 2557 y Fn(r)1326 2550 y Fu(\))p Fo(:)954 2670 y Fs(12)p eop %%Page: 13 13 13 12 bop 0 200 a Fu(By)19 b(the)f(induction)g(h)o(yp)q(othesis,)f(w)o(e)h (kno)o(w)g(that)g Fo(F)7 b Fu(\()p Fo(S)1057 207 y Fl(+)1090 200 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)1205 207 y Fl(1)1230 200 y Fo(;)h Fp(\001)g(\001)g(\001)16 b Fo(;)8 b(x)1376 207 y Fn(r)q Fl(+1)1450 200 y Fu(\)\)\()p Fo(\033)r Fu(\))20 b(is)d(in)h Fo(\031)r Fu(.)28 b(Since)17 b(for)0 260 y(all)j Fo(j)k Fu(in)c Fp(f)p Fu(1)8 b Fp(\001)g(\001)g(\001)g Fo(r)16 b Fu(+)e(1)p Fp(g)p Fu(,)21 b Fo(F)7 b Fu(\()p Fo(S)569 267 y Fl(+)602 260 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)717 267 y Fl(1)742 260 y Fo(;)h Fp(\001)g(\001)g(\001)16 b Fo(;)8 b(x)888 267 y Fn(j)910 260 y Fu(\)\)\()p Fo(\033)r Fu(\))23 b(is)d(in)g Fo(\031)r Fu(,)i(the)f(h)o(yp)q(othesis)f(on)g Fo(\033)j Fu(implies)d(that)0 319 y Fo(E)s 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Fl(+1)852 869 y Fu(\))22 b(=)g Fo(S)985 876 y Fj(\000)1018 869 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)1133 876 y Fl(1)1158 869 y Fo(;)8 b Fp(\001)g(\001)g(\001)16 b Fo(;)8 b(x)1304 876 y Fn(r)1327 869 y Fu(\))p Fo(;)429 946 y(S)460 953 y Fl(+)493 946 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)608 953 y Fl(1)632 946 y Fo(;)g Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)779 953 y Fn(r)q Fl(+1)852 946 y Fu(\))22 b(=)g(\()p Fo(S)1004 953 y Fl(+)1037 946 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)1152 953 y Fl(1)1176 946 y Fo(;)8 b Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1323 953 y Fn(r)1346 946 y Fu(\))p Fo(;)g(x)1415 953 y Fn(r)q Fl(+1)1488 946 y Fu(\))p Fo(:)0 1072 y Fu(By)19 b(the)f(induction)f(h) o(yp)q(othesis,)h(w)o(e)g(kno)o(w)g(that)g Fo(F)7 b Fu(\()p Fo(S)1057 1079 y Fj(\000)1091 1072 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)1206 1079 y Fl(1)1230 1072 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1377 1079 y Fn(r)q Fl(+1)1450 1072 y Fu(\)\)\()p Fo(\033)r Fu(\))20 b(is)e(in)f Fo(\031)r Fu(.)28 b(Since)17 b(for)0 1132 y(all)j Fo(j)k Fu(in)c Fp(f)p Fu(1)8 b Fp(\001)g(\001)g(\001)g Fo(r)16 b Fu(+)e(1)p Fp(g)p Fu(,)21 b Fo(F)7 b Fu(\()p Fo(S)569 1139 y Fj(\000)603 1132 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)718 1139 y Fl(1)742 1132 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)889 1139 y Fn(j)910 1132 y Fu(\)\)\()p Fo(\033)r Fu(\))23 b(is)d(in)g Fo(\031)r Fu(,)i(the)f(h)o(yp)q(othesis)f(on)g Fo(\033)j Fu(implies)d(that)0 1192 y Fo(E)s Fu(\()p Fo(F)7 b Fu(\()p Fo(S)148 1199 y Fj(\000)182 1192 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)297 1199 y Fl(1)321 1192 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)468 1199 y Fn(r)q Fl(+1)541 1192 y Fu(\)\)\()p Fo(\033)r Fu(\)\))16 b Fp(\025)d Fo(E)s Fu(\()p Fo(\033)r Fu(\).)23 b(Inequalit)o(y)16 b(\(2\))h(implies)261 1315 y Fo(E)s Fu(\()p Fo(F)7 b Fu(\()p Fo(S)409 1322 y Fl(+)442 1315 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)557 1322 y Fl(1)582 1315 y Fo(;)h Fp(\001)g(\001)g(\001)16 b Fo(;)8 b(x)728 1322 y Fn(r)q Fl(+1)802 1315 y Fu(\)\)\()p Fo(\033)r Fu(\)\))24 b Fp(\024)e Fo(E)s Fu(\()p Fo(F)7 b Fu(\()p Fo(x)1157 1322 y Fl(1)1180 1315 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1327 1322 y Fn(r)q Fl(+1)1400 1315 y Fu(\)\()p Fo(\033)r Fu(\)\))24 b Fp(\024)e Fo(\025)p Fu(\()p Fo(\031)r Fu(\))0 1439 y(and)17 b Fo(F)7 b Fu(\()p Fo(S)187 1446 y Fl(+)220 1439 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)335 1446 y Fl(1)359 1439 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)506 1446 y Fn(r)q Fl(+1)579 1439 y Fu(\)\)\()p Fo(\033)r Fu(\)\))20 b(is)d(in)g Fo(\031)r Fu(,)h(since)f(its)h(energy)f(is)g(less)g(than)h(the)g (lev)o(el)f(of)h Fo(\031)i Fu(and)d(it)0 1498 y(di\013ers)c(b)o(y)h(one)f (spin)g(\015ip)h(\(at)g(site)g Fo(x)682 1505 y Fn(r)q Fl(+1)755 1498 y Fu(\))h(from)e(a)h(con\014guration)e(of)j Fo(\031)r Fu(.)21 b(The)14 b(h)o(yp)q(othesis)f(on)g Fo(\033)k Fu(\014nally)0 1558 y(implies)e(that)i Fo(E)s Fu(\()p Fo(F)7 b Fu(\()p Fo(S)425 1565 y Fl(+)458 1558 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)573 1565 y Fl(1)597 1558 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)744 1565 y Fn(r)q Fl(+1)817 1558 y Fu(\)\)\()p Fo(\033)r Fu(\)\))16 b Fp(\025)e Fo(E)s Fu(\()p Fo(\033)r Fu(\).)0 1618 y(In)i(addition)g(the)h(case)f(of)h(equalit)o(y)f(in)g(inequalit)o(y)g(\(2\)) i(is)e(the)h(case)f(where)167 1741 y Fo(E)s Fu(\()p Fo(F)7 b Fu(\()p Fo(S)315 1748 y Fj(\000)349 1741 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)464 1748 y Fl(1)488 1741 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)635 1748 y Fn(r)q Fl(+1)708 1741 y Fu(\)\)\()p Fo(\033)r Fu(\)\))24 b(=)14 b Fo(E)s Fu(\()p Fo(F)7 b Fu(\()p Fo(S)1058 1748 y Fl(+)1091 1741 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)1206 1748 y Fl(1)1230 1741 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1377 1748 y Fn(r)q Fl(+1)1450 1741 y Fu(\)\)\()p Fo(\033)r Fu(\)\))24 b(=)e Fo(E)s Fu(\()p Fo(\033)r Fu(\))p Fo(:)0 1865 y Fu(The)16 b(h)o(yp)q(othesis)g(on)g Fo(\033)j Fu(then)d(implies)f(that)292 1988 y Fo(F)7 b Fu(\()p Fo(S)381 1995 y Fj(\000)414 1988 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)529 1995 y Fl(1)554 1988 y Fo(;)h Fp(\001)g(\001)g(\001)16 b Fo(;)8 b(x)700 1995 y Fn(r)q Fl(+1)774 1988 y Fu(\)\)\()p Fo(\033)r Fu(\))23 b(=)14 b Fo(F)7 b Fu(\()p Fo(S)1045 1995 y Fl(+)1078 1988 y Fu(\()p Fo(\033)r Fu(\)\()p Fo(x)1193 1995 y Fl(1)1217 1988 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1364 1995 y Fn(r)q Fl(+1)1437 1988 y Fu(\)\)\()p Fo(\033)r Fu(\))24 b(=)e Fo(\033)0 2112 y Fu(or)16 b(equiv)m(alen)o(tly)h Fo(F)7 b Fu(\()p Fo(x)424 2119 y Fl(1)447 2112 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)594 2119 y Fn(r)q Fl(+1)667 2112 y Fu(\)\()p Fo(\033)r Fu(\))23 b(=)f Fo(\033)r Fu(.)g(The)17 b(induction)e(is)h(completed.)49 b Fc(\003)754 2238 y Fv(6.)27 b(Dimension)17 b(tw)o(o)60 2329 y Fu(A)h(p)q(oly)o(omino)e(asso)q(ciated)i(to)g(a)g(con\014guration)e(is)h(a) h(\014nite)f(union)g(of)h(squares.)25 b(F)l(or)16 b Fo(\033)k Fu(a)e(con\014g-)0 2388 y(uration)e(w)o(e)g(will)h(denote)f(b)o(y)h Fo(p)p Fu(\()p Fo(\033)r Fu(\))h(the)f(p)q(erimeter)f(of)h(the)g(asso)q (ciated)f(p)q(oly)o(omino)f(and)i(b)o(y)f Fo(a)p Fu(\()p Fo(\033)r Fu(\))i(its)0 2448 y(area.)29 b(W)l(e)19 b(will)f(rely)h(hea)o(vily)g(on)f (the)h(notations)g(and)f(results)g(of)23 b([1].)28 b(Let)20 b(us)e(recall)h(brie\015y)f(some)0 2508 y(essen)o(tial)d(p)q(oin)o(ts.)954 2670 y Fs(13)p eop %%Page: 14 14 14 13 bop 0 200 a Fw(Summary)16 b(of)i(the)i(com)n(binatorial)g(results.)60 273 y Fu(The)15 b(rectangle)g(of)h(sides)e Fo(l)562 280 y Fl(1)600 273 y Fu(and)h Fo(l)711 280 y Fl(2)749 273 y Fu(is)g(denoted)g(b)o(y)g Fo(l)1068 280 y Fl(1)1099 273 y Fp(\002)9 b Fo(l)1162 280 y Fl(2)1184 273 y Fu(.)21 b(By)16 b Fo(l)1311 280 y Fl(1)1343 273 y Fp(\002)8 b Fo(l)1405 280 y Fl(2)1436 273 y Fu(+)1475 280 y Fl(1)1506 273 y Fo(k)17 b Fu(\(resp.)k Fo(l)1708 280 y Fl(1)1739 273 y Fp(\002)9 b Fo(l)1802 280 y Fl(2)1833 273 y Fu(+)1872 280 y Fl(2)1903 273 y Fo(k)r Fu(\))0 333 y(w)o(e)19 b(denote)h(the)g(rectangle)f Fo(l)560 340 y Fl(1)595 333 y Fp(\002)13 b Fo(l)662 340 y Fl(2)704 333 y Fu(plus)19 b(a)g(v)o(ertical)h (\(resp.)30 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y(that)g(the)g(p)q(oly)o(omino)e(is)h(alw)o(a)o(ys)f(included)h(in)g(\()p Fo(l)958 818 y Fl(1)993 811 y Fu(+)13 b(1\))h Fp(\002)e Fo(l)1169 818 y Fl(2)1192 811 y Fu(\).)31 b(F)l(or)18 b Fo(A)i Fu(a)g(set)g(of)f(p)q (oly)o(omino)q(es,)g(w)o(e)0 873 y(denote)i(b)o(y)p 239 833 38 2 v 21 w Fo(A)g Fu(its)g(orbit)g(under)f(the)h(action)g(of)g(the)g(planar) f(isometries)g(whic)o(h)g(lea)o(v)o(e)g(the)i(in)o(teger)0 933 y(lattice)d Fm(Z)191 915 y Fl(2)229 933 y Fu(in)o(v)m(arian)o(t.)27 b(By)p 545 893 V 20 w Fo(A)582 901 y Fl(12)643 933 y Fu(w)o(e)19 b(denote)g(its)f(orbit)h(under)e(the)i(action)g(of)g(the)g(t)o(w)o(o)f (symmetries)0 992 y(with)e(resp)q(ect)h(to)g(the)g(axis.)0 1080 y Fw(Prop)r(osition)i(6.1.)27 b Fh(F)l(or)16 b(eac)o(h)g(in)o(teger)g Fo(n)g Fh(there)h(exists)f(a)h(unique)f Fu(3)p Fh({uple)f Fu(\()p Fo(l)q(;)8 b(k)r(;)g(\017)p Fu(\))18 b Fh(suc)o(h)d(that)432 1190 y Fo(\017)f Fp(2)g(f)p Fu(0)p Fo(;)8 b Fu(1)p Fp(g)p Fo(;)57 b Fu(0)14 b Fp(\024)g Fo(k)h(<)f(l)d Fu(+)g Fo(\017)50 b Fh(and)f Fo(n)14 b Fu(=)g Fo(l)q Fu(\()p Fo(l)e Fu(+)f Fo(\017)p Fu(\))g(+)g Fo(k)r(:)0 1300 y Fu(The)j(set)h(of)f(the)h(p)q(oly)o(omino)q(es)e(of)i(area) f Fo(n)g Fu(is)g Fo(C)886 1307 y Fn(n)913 1300 y Fu(;)h(the)f(set)h Fo(M)1149 1307 y Fn(n)1191 1300 y Fu(of)g(the)f(minimal)f(p)q(oly)o(omino)q (es)g(of)i(area)0 1360 y Fo(n)j Fu(is)g(the)g(set)g(of)g(the)h(p)q(oly)o (omino)q(es)e(of)h Fo(C)793 1367 y Fn(n)838 1360 y Fu(ha)o(ving)f(minimal)f (p)q(erimeter.)26 b(Let)18 b Fo(n)f Fu(=)f Fo(l)q Fu(\()p Fo(l)d Fu(+)f Fo(\017)p Fu(\))g(+)g Fo(k)20 b Fu(b)q(e)0 1419 y(the)d(decomp)q (osition)e(of)i Fo(n)p Fu(.)22 b(The)16 b(canonical)g(p)q(oly)o(omino)f(of)i (area)f Fo(n)g Fu(is)562 1559 y Fo(m)606 1566 y Fn(n)647 1559 y Fu(=)699 1489 y Fk(\032)753 1522 y Fo(l)c Fp(\002)f Fo(l)h Fu(+)896 1529 y Fl(1)929 1522 y Fu(1)f Fp(\002)g Fo(k)176 b Fu(if)16 b Fo(\017)e Fu(=)g(0)753 1596 y(\()p Fo(l)e Fu(+)f(1\))h Fp(\002)e Fo(l)i Fu(+)1020 1603 y Fl(2)1053 1596 y Fo(k)h Fp(\002)e Fu(1)50 b(if)16 b Fo(\017)e Fu(=)g(1)0 1698 y Fw(Theorem)j(6.2.)28 b Fh(The)16 b(p)q(oly)o(omino)g Fo(m)750 1705 y Fn(n)793 1698 y Fh(is)g(minimal.)0 1786 y Fu(W)l(e)h(de\014ne)f(sev)o(eral)f(imp)q(ortan)o (t)h(subsets)f(of)i Fo(M)906 1793 y Fn(n)933 1786 y Fu(.)22 b(The)17 b(set)f Fo(S)1180 1793 y Fn(n)1224 1786 y Fu(of)g(the)h(standard)e (p)q(oly)o(omino)q(es)h(is)552 1927 y Fo(S)583 1934 y Fn(n)640 1927 y Fu(=)709 1856 y Fk(\032)p 763 1850 290 2 v 763 1890 a Fo(l)c Fp(\002)f Fo(l)h Fp(\010)906 1897 y Fl(1)939 1890 y Fu(1)f Fp(\002)g Fo(k)176 b Fu(if)16 b Fo(\017)e Fu(=)g(0)p 763 1922 414 2 v 763 1965 a(\()p Fo(l)e Fu(+)f(1\))h Fp(\002)e Fo(l)i Fp(\010)1030 1972 y Fl(2)1063 1965 y Fo(k)h Fp(\002)e Fu(1)50 b(if)16 b Fo(\017)e Fu(=)g(1)0 2083 y(The)i(set)186 2070 y Fk(f)180 2083 y Fo(M)228 2090 y Fn(n)272 2083 y Fu(of)h(the)g (principal)d(p)q(oly)o(omino)q(es)i(is)433 2187 y Fk(f)427 2200 y Fo(M)475 2207 y Fn(n)533 2200 y Fu(=)p 602 2156 409 2 v 30 w Fo(l)c Fp(\002)f Fu(\()p Fo(l)h Fu(+)f Fo(\017)p Fu(\))h Fp(\010)865 2207 y Fl(1)898 2200 y Fu(1)f Fp(\002)g Fo(k)1028 2152 y Fk([)p 1100 2156 V 1100 2200 a Fo(l)h Fp(\002)f Fu(\()p Fo(l)h Fu(+)f Fo(\017)p Fu(\))g Fp(\010)1362 2207 y Fl(2)1396 2200 y Fo(k)h Fp(\002)f Fu(1)p Fo(:)0 2323 y Fw(Theorem)17 b(6.3.)50 b Fh(The)17 b(set)f Fo(M)611 2330 y Fn(n)655 2323 y Fh(is)g(reduced)g(to)h Fp(f)8 b Fo(m)1029 2330 y Fn(n)1064 2323 y Fp(g)17 b Fh(if)f(and)g(only)h(if)f Fo(n)h Fh(is)f(of)h(the)f(form)g Fo(l)1776 2305 y Fl(2)1798 2323 y Fh(.)0 2383 y(The)k(set)g Fo(M)235 2390 y Fn(n)283 2383 y Fh(coincides)f(with)h Fo(S)645 2390 y Fn(n)692 2383 y Fh(if)g(and)g(only)g(if)g Fo(n)g Fh(is)g(of)g(the)g (form)g Fo(l)1395 2365 y Fl(2)1417 2383 y Fo(;)8 b(l)q Fu(\()p Fo(l)14 b Fu(+)g(1\))f Fp(\000)h Fu(1)p Fo(;)8 b(l)q Fu(\()p Fo(l)14 b Fu(+)f(1\))p Fh(,)21 b(in)0 2450 y(whic)o(h)c(case)h Fo(S)282 2457 y Fn(n)324 2450 y Fu(=)p 379 2423 44 2 v 16 w Fo(m)423 2457 y Fn(n)450 2450 y Fh(.)26 b(The)18 b(set)g Fo(M)721 2457 y Fn(n)767 2450 y Fh(coincide)f(with)1080 2437 y Fk(f)1074 2450 y Fo(M)1122 2457 y Fn(n)1167 2450 y Fh(if)h(and)f(only)h(if)g(the)g(in)o (teger)f Fo(n)h Fh(is)g(of)g(the)0 2510 y(form)e Fo(l)134 2492 y Fl(2)156 2510 y Fo(;)8 b(l)q Fu(\()p Fo(l)k Fu(+)f(1\))g Fp(\000)g Fu(1)p Fo(;)d(l)q Fu(\()p Fo(l)k Fu(+)f(1\))p Fo(;)d Fu(\()p Fo(l)13 b Fu(+)e(1\))761 2492 y Fl(2)794 2510 y Fp(\000)g Fu(1)p Fh(.)954 2670 y Fs(14)p eop %%Page: 15 15 15 14 bop 0 200 a Fw(Lemma)17 b(6.4.)50 b Fh(F)l(or)15 b Fo(n)i Fh(of)g(the)f(form)g Fo(l)755 182 y Fl(2)777 200 y Fo(;)8 b(l)q Fu(\()p Fo(l)13 b Fu(+)d(1\))p Fh(,)17 b(w)o(e)g(ha)o(v)o(e)576 317 y Fp(f)8 b Fo(c)13 b Fp(2)h Fo(M)739 324 y Fn(n)p Fj(\000)p Fl(1)831 317 y Fu(:)g Fo(q)r Fu(\()p Fo(M)950 324 y Fn(n)978 317 y Fo(;)8 b(c)p Fu(\))14 b Fo(>)f Fu(0)8 b Fp(g)23 b Fu(=)f Fo(S)1280 324 y Fn(n)p Fj(\000)p Fl(1)1357 317 y Fo(;)576 406 y Fp(f)8 b Fo(c)14 b Fp(2)g Fo(M)740 413 y Fn(n)p Fl(+1)831 406 y Fu(:)g Fo(q)r Fu(\()p Fo(M)950 413 y Fn(n)978 406 y Fo(;)8 b(c)p Fu(\))14 b Fo(>)f Fu(0)8 b Fp(g)23 b Fu(=)1255 394 y Fk(f)1249 406 y Fo(M)1297 413 y Fn(n)p Fl(+1)0 526 y Fh(\(where)16 b Fo(q)r Fu(\()p Fo(M)254 533 y Fn(n)282 526 y Fo(;)8 b(c)p Fu(\))14 b(=)g(min)o Fp(f)p Fo(q)r Fu(\()p Fo(d;)8 b(c)p Fu(\))14 b(:)g Fo(q)h Fp(2)f Fo(M)826 533 y Fn(n)854 526 y Fp(g)p Fh(\).)0 615 y Fw(Theorem)i(6.5.)50 b Fh(The)15 b(principal)e(p)q(oly)o(omino)q(es)i (can)g(b)q(e)h(completely)f(shrink)o(ed)e(through)i(the)g(min-)0 679 y(imal)g(p)q(oly)o(omino)q(es:)20 b(for)15 b(an)o(y)g(in)o(teger)g Fo(n)g Fh(and)g(for)g(an)o(y)h(principal)d(p)q(oly)o(omino)h Fo(c)h Fh(in)1604 667 y Fk(f)1598 679 y Fo(M)1646 686 y Fn(n)1673 679 y Fh(,)h(there)f(exists)0 739 y(an)f(increasing)f(sequence)h Fo(c)524 746 y Fl(0)546 739 y Fo(;)8 b Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(c)687 746 y Fn(n)728 739 y Fh(of)15 b(minimal)e(p)q(oly)o(omino)q (es)g(suc)o(h)g(that)i Fo(c)1496 746 y Fl(0)1532 739 y Fu(=)e Fp(;)i Fh(and)f Fo(c)1741 746 y Fn(n)1781 739 y Fu(=)g Fo(c)g Fh(and)0 799 y Fo(q)r Fu(\()p Fo(c)65 806 y Fn(i)p Fj(\000)p Fl(1)133 799 y Fo(;)8 b(c)177 806 y Fn(i)193 799 y Fu(\))14 b Fo(>)g Fu(0)i Fh(for)g Fo(i)h Fh(in)f Fp(f)p Fu(1)8 b Fp(\001)g(\001)g (\001)h Fo(n)p Fp(g)p Fh(.)0 888 y Fw(Theorem)26 b(6.6.)74 b Fh(The)24 b(standard)f(p)q(oly)o(omino)q(es)g(can)h(b)q(e)h(gro)o(wn)e(or)h (shrink)o(ed)f(arbitrarily)f(far)0 948 y(through)12 b(minimal)f(p)q(oly)o (omino)q(es:)19 b(for)12 b(an)o(y)h(in)o(tegers)f Fo(m)h Fp(\024)h Fo(n)f Fh(and)f(for)h(an)o(y)f(standard)f(p)q(oly)o(omino)h Fo(c)g Fh(in)0 1008 y Fo(S)31 1015 y Fn(m)68 1008 y Fh(,)h(there)f(exists)g (an)f(increasing)f(sequence)i Fo(c)867 1015 y Fl(0)889 1008 y Fo(;)c Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(c)1030 1015 y Fn(n)1068 1008 y Fh(of)k(minimal)e(p)q(oly)o(omino)q(es)h(suc)o(h)g(that)h Fo(c)1823 1015 y Fl(0)1859 1008 y Fu(=)h Fp(;)p Fh(,)0 1067 y Fo(c)22 1074 y Fn(m)73 1067 y Fu(=)h Fo(c)i Fh(and)g Fo(q)r Fu(\()p Fo(c)326 1074 y Fn(i)p Fj(\000)p Fl(1)393 1067 y Fo(;)8 b(c)437 1074 y Fn(i)454 1067 y Fu(\))14 b Fo(>)f Fu(0)k Fh(for)f Fo(i)h Fh(in)f Fp(f)p Fu(1)8 b Fp(\001)g(\001)g(\001)h Fo(n)p Fp(g)p Fh(.)0 1156 y Fw(Notation)20 b(6.7.)50 b Fu(W)l(e)17 b(ha)o(v)o(e)f(also)h(some)f(results)f(concerning)h(the)h(b)q(est)g(w)o(a)o (y)f(to)h(shrink)f(or)g(to)h(gro)o(w)0 1216 y(a)g(rectangle,)f(that)h(is)f (on)g(the)h(sets)325 1333 y Fo(M)5 b Fu(\()p Fo(l)412 1340 y Fl(1)446 1333 y Fp(\002)11 b Fo(l)511 1340 y Fl(2)534 1333 y Fo(;)d Fp(\000)p Fo(k)r Fu(\))21 b(=)h Fp(f)8 b Fo(c)14 b Fp(2)g Fo(C)876 1340 y Fn(l)889 1345 y Fd(1)908 1340 y Fn(l)921 1345 y Fd(2)940 1340 y Fj(\000)p Fn(k)1009 1333 y Fu(:)g Fo(c)f Fp(\032)h Fo(l)1140 1340 y Fl(1)1173 1333 y Fp(\002)d Fo(l)1238 1340 y Fl(2)1260 1333 y Fo(;)25 b(p)p Fu(\()p Fo(c)p Fu(\))17 b(minimal)6 b Fp(g)p Fo(;)325 1410 y(M)f Fu(\()p Fo(l)412 1417 y Fl(1)446 1410 y Fp(\002)11 b Fo(l)511 1417 y Fl(2)534 1410 y Fo(;)d Fu(+)p Fo(k)r Fu(\))21 b(=)h Fp(f)8 b Fo(c)14 b Fp(2)g Fo(C)876 1417 y Fn(l)889 1422 y Fd(1)908 1417 y Fn(l)921 1422 y Fd(2)940 1417 y Fl(+)p Fn(k)1009 1410 y Fu(:)f Fo(l)1051 1417 y Fl(1)1084 1410 y Fp(\002)e Fo(l)1149 1417 y Fl(2)1185 1410 y Fp(\032)j Fo(c;)24 b(p)p Fu(\()p Fo(c)p Fu(\))17 b(minimal)7 b Fp(g)p Fo(:)0 1529 y Fu(W)l(e)17 b(denote)f(b)o(y)g Fo(E)s Fu(\()p Fo(l)389 1536 y Fl(1)423 1529 y Fp(\002)11 b Fo(l)488 1536 y Fl(2)510 1529 y Fo(;)d(k)r Fu(\))16 b(the)h(energy)f(of)h(an)f(elemen) o(t)g(of)h Fo(M)5 b Fu(\()p Fo(l)1294 1536 y Fl(1)1329 1529 y Fp(\002)10 b Fo(l)1393 1536 y Fl(2)1416 1529 y Fo(;)e(k)r Fu(\))16 b(for)g Fo(k)j Fu(in)d Fm(Z)-11 b Fu(.)0 1589 y(A)17 b(natural)g(w)o(a)o(y)f(to)h(remo)o(v)o(e)f(\(add\))h Fo(k)i Fu(squares)d(\(for)h Fo(k)f(<)f(l)1128 1596 y Fl(1)1150 1589 y Fo(;)8 b(k)17 b(<)d(l)1283 1596 y Fl(2)1305 1589 y Fu(\))k(is)e(to)i(remo)o (v)o(e)e(\(add\))h(a)g(line)f(on)0 1649 y(a)h(side)e(of)i(the)g(rectangle;)f (th)o(us)g(w)o(e)g(de\014ne)95 1778 y Fo(S)s Fu(\()p Fo(l)163 1785 y Fl(1)196 1778 y Fp(\002)11 b Fo(l)261 1785 y Fl(2)283 1778 y Fo(;)d Fp(\000)p Fo(k)r Fu(\))22 b(=)p 474 1735 594 2 v 22 w(\()p Fo(l)508 1785 y Fl(1)541 1778 y Fp(\000)11 b Fu(1\))h Fp(\002)e Fo(l)711 1785 y Fl(2)745 1778 y Fp(\010)784 1785 y Fl(1)817 1778 y Fu(1)h Fp(\002)g Fu(\()p Fo(l)937 1785 y Fl(2)970 1778 y Fp(\000)g Fo(k)r Fu(\))1067 1743 y Fl(12)1126 1731 y Fk([)p 1198 1735 V 1198 1778 a Fo(l)1213 1785 y Fl(1)1246 1778 y Fp(\002)g Fu(\()p Fo(l)1330 1785 y Fl(2)1363 1778 y Fp(\000)g Fu(1\))h Fp(\010)1508 1785 y Fl(2)1541 1778 y Fu(\()p Fo(l)1575 1785 y Fl(1)1608 1778 y Fp(\000)f Fo(k)r Fu(\))g Fp(\002)g Fu(1)1791 1743 y Fl(12)1841 1778 y Fo(;)397 1883 y(S)s Fu(\()p Fo(l)465 1890 y Fl(1)498 1883 y Fp(\002)g Fo(l)563 1890 y Fl(2)585 1883 y Fo(;)d(k)r Fu(\))22 b(=)p 737 1840 761 2 v 22 w Fp(f)8 b Fo(l)785 1890 y Fl(1)818 1883 y Fp(\002)j Fo(l)883 1890 y Fl(2)916 1883 y Fp(\010)955 1890 y Fl(2)989 1883 y Fo(k)h Fp(\002)f Fu(1)p Fo(;)16 b(l)1147 1890 y Fl(1)1181 1883 y Fp(\002)10 b Fo(l)1245 1890 y Fl(2)1279 1883 y Fp(\010)1318 1890 y Fl(1)1351 1883 y Fu(1)h Fp(\002)g Fo(k)e Fp(g)1497 1848 y Fl(12)1539 1883 y Fo(:)0 2029 y Fw(Theorem)17 b(6.8.)50 b Fh(Let)17 b Fo(l)487 2036 y Fl(1)510 2029 y Fo(;)8 b(l)547 2036 y Fl(2)569 2029 y Fo(;)g(k)18 b Fh(b)q(e)f(p)q(ositiv)o(e)f(in)o(tegers) g(suc)o(h)f(that)i Fo(k)e(<)f(l)1401 2036 y Fl(1)1423 2029 y Fo(;)j(k)e(<)f(l)1563 2036 y Fl(2)1585 2029 y Fh(.)0 2089 y(The)j(set)g Fo(M)5 b Fu(\()p Fo(l)268 2096 y Fl(1)303 2089 y Fp(\002)11 b Fo(l)368 2096 y Fl(2)390 2089 y Fo(;)d Fp(\000)p Fo(k)r Fu(\))17 b Fh(is)g(the)g(set)g(of)h(the)f(p)q(oly)o(omino)q(es)f (obtained)g(b)o(y)h(remo)o(ving)e(successiv)o(ely)h Fo(k)0 2149 y Fh(corner)f(squares)h(from)g Fo(l)462 2156 y Fl(1)495 2149 y Fp(\002)11 b Fo(l)560 2156 y Fl(2)582 2149 y Fh(.)22 b(In)16 b(particular,)f Fo(S)s Fu(\()p Fo(l)992 2156 y Fl(1)1025 2149 y Fp(\002)c Fo(l)1090 2156 y Fl(2)1112 2149 y Fo(;)d Fp(\000)p Fo(k)r Fu(\))16 b Fh(is)h(included)e(in)h Fo(M)5 b Fu(\()p Fo(l)1631 2156 y Fl(1)1665 2149 y Fp(\002)11 b Fo(l)1730 2156 y Fl(2)1752 2149 y Fo(;)d Fp(\000)p Fo(k)r Fu(\))p Fh(.)0 2238 y Fw(Theorem)17 b(6.9.)50 b Fh(Let)17 b Fo(l)487 2245 y Fl(1)510 2238 y Fo(;)8 b(l)547 2245 y Fl(2)569 2238 y Fo(;)g(k)18 b Fh(b)q(e)f(p)q(ositiv)o(e)f(in)o(tegers)g(suc)o(h)f(that)i Fo(k)e(<)f(l)1401 2245 y Fl(1)1423 2238 y Fo(;)j(k)e(<)f(l)1563 2245 y Fl(2)1585 2238 y Fh(.)0 2297 y(The)i(set)h Fo(M)5 b Fu(\()p Fo(l)267 2304 y Fl(1)302 2297 y Fp(\002)10 b Fo(l)366 2304 y Fl(2)389 2297 y Fo(;)e Fu(+)p Fo(k)r Fu(\))16 b Fh(is)g(equal)g(to)h (the)g(set)g Fo(S)s Fu(\()p Fo(l)987 2304 y Fl(1)1020 2297 y Fp(\002)11 b Fo(l)1085 2304 y Fl(2)1107 2297 y Fo(;)d(k)r Fu(\))p Fh(.)0 2416 y Fw(Application)22 b(to)d(the)g(t)n(w)n(o)i(dimensional) e(Ising)g(mo)r(del.)60 2490 y Fu(W)l(e)f(\014rst)g(express)g(the)g(energy)g (of)h(the)g(Ising)e(mo)q(del)h(with)h(the)f(help)g(of)h(the)f(p)q(erimeter)g (and)g(the)0 2550 y(area)e(of)h(the)g(p)q(oly)o(omino)e(asso)q(ciated)h(to)h (the)g(con\014guration.)954 2670 y Fs(15)p eop %%Page: 16 16 16 15 bop 0 200 a Fw(Lemma)17 b(6.10.)28 b Fh(F)l(or)15 b(an)o(y)h (con\014guration)f Fo(\033)k Fh(in)d Fo(X)t Fh(,)g(w)o(e)h(ha)o(v)o(e)100 319 y Fo(E)s Fu(\()p Fo(\033)r Fu(\))31 b(=)f Fp(\000)353 285 y Fu(1)p 353 307 25 2 v 353 353 a(2)447 272 y Fk(X)392 381 y Fj(f)p Fn(x;y)q Fj(g)p Fl(:)p Fn(x)p Fj(\030)p Fn(y)583 319 y Fo(\033)r Fu(\()p Fo(x)p Fu(\))p Fo(\033)r Fu(\()p Fo(y)r Fu(\))22 b Fp(\000)859 285 y Fo(h)p 859 307 29 2 v 861 353 a Fu(2)905 272 y Fk(X)902 378 y Fn(x)p Fj(2)p Fl(\003)987 319 y Fo(\033)r Fu(\()p Fo(x)p Fu(\))32 b(=)e Fo(p)p Fu(\()p Fo(\033)r Fu(\))21 b Fp(\000)e Fo(ha)p Fu(\()p Fo(\033)r Fu(\))h(+)f Fo(N)1602 298 y Fl(2)1625 319 y Fu(\()p Fo(h=)p Fu(2)11 b Fp(\000)g Fu(1\))d Fo(:)60 470 y Fu(W)l(e)13 b(do)f(not)h(c)o(hange)f(the)h(dynamics)f (nor)g(the)h(cycle)h(decomp)q(osition)d(b)o(y)i(adding)e(a)i(constan)o(t)g (to)g(the)0 530 y(energy)g Fo(E)s Fu(.)20 b(In)12 b(dimension)f(t)o(w)o(o,)i (w)o(e)g(will)f(w)o(ork)g(with)h(the)g(energy)f Fo(E)s Fu(\()p Fo(\033)r Fu(\))j(=)e 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13 30 14 30 13 29 13 30 14 29 13 30 13 29 14 30 13 30 14 29 13 30 13 29 14 -252 13 30 14 29 13 30 13 30 14 29 13 30 14 29 13 30 13 29 14 30 13 30 14 29 13 30 13 29 14 30 13 -252 14 29 13 30 13 30 14 29 13 30 14 29 13 30 13 29 14 30 13 30 14 29 13 30 13 29 14 30 13 30 14 -253 13 30 13 30 14 29 13 30 14 29 13 30 13 29 14 30 13 30 14 29 13 30 13 29 14 30 13 30 14 -253 13 30 13 30 14 29 13 30 14 29 13 30 13 29 14 30 13 30 14 29 13 30 13 29 14 30 13 30 14 -253 13 30 13 30 14 29 13 30 14 29 13 30 13 29 14 30 3605 1716 100 MP stroke 13 30 13 29 14 30 13 29 14 30 13 -252 13 29 14 30 13 30 14 29 13 30 13 29 14 30 13 29 14 30 13 30 13 29 14 30 13 29 14 -252 13 30 13 29 14 30 13 30 14 29 13 30 13 29 14 30 13 29 14 30 13 30 13 29 14 -252 13 30 14 29 13 30 13 29 14 30 13 30 14 29 13 30 13 29 14 30 13 29 14 30 13 -252 13 29 14 30 13 30 14 29 13 30 13 29 14 30 13 30 14 29 13 30 13 29 14 -252 13 30 14 29 13 30 13 29 14 30 13 30 14 29 13 30 13 29 14 30 13 30 14 -253 13 30 13 30 14 29 13 30 14 29 13 30 13 29 14 30 13 30 14 29 13 -252 13 30 14 29 13 30 13 29 14 30 13 30 14 29 13 30 13 29 14 30 13 -252 14 29 13 30 13 29 14 30 13 30 14 29 13 30 2279 1041 100 MP stroke 13 29 14 30 13 -252 14 29 13 30 13 30 14 29 13 30 14 29 13 30 13 29 14 30 13 -252 14 29 13 30 13 30 14 29 13 30 14 29 13 30 13 30 14 -253 13 30 14 30 13 29 13 30 14 29 13 30 14 29 13 30 13 -252 14 29 13 30 14 30 13 29 13 30 14 29 13 30 14 -252 13 29 13 30 14 30 13 29 14 30 13 29 13 30 14 -252 13 29 14 30 13 29 13 30 14 30 13 29 14 -252 13 30 13 29 14 30 13 29 14 30 13 30 13 -253 14 30 13 30 13 29 14 30 13 29 14 -252 13 30 13 29 14 30 13 30 14 29 13 -252 13 30 14 29 13 30 14 29 13 -252 13 30 14 29 13 30 14 29 13 -252 13 30 14 29 13 30 14 -252 13 29 13 30 14 30 13 -253 14 30 13 30 13 -253 14 30 13 30 14 -252 13 29 13 -252 953 2902 100 MP stroke 14 30 13 -253 14 -252 912 3377 4 MP stroke gr 2991 274 mt (2D energy, h=0.21) s 3435 4944 mt (area) s 646 2990 mt -90 rotate (minimal energy) s 90 rotate end eplot %%EndObject graph 1 epage end showpage %%Trailer %%EOF %%EndDocument endTexFig 60 1846 a Fu(T)l(o)k(compute)g Fo(E)s Fu(\()p Fo(n)p Fu(\))h(w)o(e)f(apply)g (theorem)g(6)p Fo(:)p Fu(2.)28 b(The)18 b(minimal)f(p)q(erimeter)h Fo(p)p Fu(\()p Fo(n)p Fu(\))h(of)g(a)f(p)q(oly)o(omino)0 1906 y(of)f(area)f Fo(n)g Fu(is)0 2014 y(\(3\))555 b Fo(p)p Fu(\()p Fo(n)p Fu(\))15 b(=)778 1944 y Fk(\032)832 1977 y Fu(2\(2)p Fo(l)d Fu(+)f Fo(\017)p Fu(\))136 b(if)17 b Fo(k)e Fu(=)f(0)832 2052 y(2\(2)p Fo(l)e Fu(+)f Fo(\017)p Fu(\))h(+)e(2)50 b(if)17 b Fo(k)e(>)f Fu(0)17 2138 y(where)i(\()p Fo(l)q(;)8 b(k)r(;)g(\017)p Fu(\))17 b(is)f(the)h(unique)f(3{uple)f(in)h Fm(N)872 2120 y Fl(3)912 2138 y Fu(satisfying)g Fo(n)e Fu(=)g Fo(l)q Fu(\()p Fo(l)e Fu(+)f Fo(\017)p Fu(\))g(+)g Fo(k)r(;)16 b(\017)e Fp(2)g(f)p Fu(0)p Fo(;)8 b Fu(1)p Fp(g)p Fo(;)16 b(k)g(<)d(l)f Fu(+)f Fo(\017)p Fu(.)0 2198 y(Let)25 b Fo(E)134 2205 y Fl(1)180 2198 y Fu(b)q(e)f(the)g(corresp)q(onding)d(one)j(dimensional)d(energy)j(i.e.)44 b Fo(E)1367 2205 y Fl(1)1389 2198 y Fu(\()p Fo(k)r Fu(\))26 b(=)g(2)16 b Fp(\001)f Fu(1\()p Fo(k)28 b(>)e Fu(0\))16 b Fp(\000)g Fo(hk)0 2258 y Fu(\(where)d(1\()p Fo(A)p Fu(\))i(denotes)e(the)h(c)o (haracteristic)e(function)h(of)h Fo(A)p Fu(\).)21 b(W)l(e)14 b(ha)o(v)o(e)f Fo(E)s Fu(\()p Fo(n)p Fu(\))h(=)g Fo(E)s Fu(\()p Fo(l)q Fu(\()p Fo(l)6 b Fu(+)f Fo(\017)p Fu(\)\))g(+)g Fo(E)1848 2265 y Fl(1)1870 2258 y Fu(\()p Fo(k)r Fu(\).)0 2318 y(Starting)16 b(with)g Fo(E)344 2325 y Fl(1)383 2318 y Fu(w)o(e)g(\014nd)526 2478 y(max)490 2510 y Fn(k)511 2515 y Fd(1)531 2510 y Fj(\024)p Fn(k)q Fj(\024)p Fn(k)636 2515 y Fd(2)664 2478 y Fo(E)701 2485 y Fl(1)723 2478 y Fu(\()p Fo(k)r Fu(\))22 b(=)872 2361 y Fk(8)872 2406 y(>)872 2420 y(<)872 2510 y(>)872 2525 y(:)933 2403 y Fu(2)11 b Fp(\000)g Fo(hk)1074 2410 y Fl(1)1146 2403 y Fu(if)16 b Fo(k)1217 2410 y Fl(1)1253 2403 y Fp(\025)e Fu(2)933 2478 y(0)188 b(if)16 b Fo(k)1217 2485 y Fl(2)1253 2478 y Fo(<)e Fu(1)933 2553 y(2)d Fp(\000)g Fo(h)98 b Fu(if)16 b Fo(k)1217 2560 y Fl(1)1253 2553 y Fp(\024)e Fu(1)g Fp(\024)f Fo(k)1423 2560 y Fl(2)954 2670 y Fs(17)p eop %%Page: 18 18 18 17 bop 0 200 a Fu(W)l(e)17 b(next)g(compute)f(the)h(energy)f(barrier)f(b)q (et)o(w)o(een)h(t)o(w)o(o)g(consecutiv)o(e)g(quasisquares:)457 303 y(max)390 338 y Fn(l)403 328 y Fd(2)422 338 y Fj(\024)p Fn(n)p Fj(\024)p Fn(l)p Fl(\()p Fn(l)p Fl(+1\))625 303 y Fo(E)s Fu(\()p Fo(n)p Fu(\))22 b(=)g Fo(E)s Fu(\()p Fo(l)891 283 y Fl(2)913 303 y Fu(\))12 b(+)23 b(max)994 336 y Fl(0)p Fj(\024)p Fn(k)q Fj(\024)p Fn(l)1119 303 y Fo(E)1156 310 y Fl(1)1178 303 y Fu(\()p Fo(k)r Fu(\))f(=)g Fo(E)s Fu(\()p Fo(l)1402 283 y Fl(2)1424 303 y Fu(\))12 b(+)f(2)g Fp(\000)f Fo(h)e(;)416 409 y Fu(max)309 443 y Fn(l)p Fl(\()p Fn(l)p Fl(+1\))p Fj(\024)p Fn(n)p Fj(\024)p Fl(\()p Fn(l)p Fl(+1\))600 433 y Fd(2)625 409 y Fo(E)s Fu(\()p Fo(n)p Fu(\))22 b(=)g Fo(E)s Fu(\()p Fo(l)q Fu(\()p Fo(l)12 b Fu(+)f(1\)\))h(+)f(2)g Fp(\000)f Fo(h)e(:)0 540 y Fu(Th)o(us)693 643 y(max)585 677 y Fn(l)p Fl(\()p Fn(l)p Fl(+1\))p Fj(\024)p 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b(this,)h(w)o(e)g(deduce)g(the)h(energy)f(barrier)f(b)q(et)o(w)o(een)h(t)o(w) o(o)g(remote)g(quasisquares:)234 1117 y(max)75 1152 y Fn(l)88 1157 y Fd(1)107 1152 y Fl(\()p Fn(l)136 1157 y Fd(1)155 1152 y Fl(+)p Fn(\017)203 1157 y Fd(1)222 1152 y Fl(\))p Fj(\024)p Fn(n)p Fj(\024)p Fn(l)338 1157 y Fd(2)356 1152 y Fl(\()p Fn(l)385 1157 y Fd(2)404 1152 y Fl(+)p Fn(\017)452 1157 y Fd(2)470 1152 y Fl(\))494 1117 y Fo(E)s Fu(\()p Fo(n)p Fu(\))31 b(=)702 1000 y Fk(8)702 1045 y(>)702 1060 y(<)702 1150 y(>)702 1165 y(:)763 1043 y Fo(E)s Fu(\()p Fo(l)837 1050 y Fl(2)859 1043 y Fu(\()p Fo(l)893 1050 y Fl(2)927 1043 y Fu(+)11 b Fo(\017)997 1050 y Fl(2)1030 1043 y Fp(\000)g Fu(1\))h(+)e(1\))51 b(if)16 b(\()p Fo(l)1359 1050 y Fl(2)1382 1043 y Fo(;)8 b(\017)1424 1050 y Fl(2)1446 1043 y Fu(\))15 b Fp(\024)e Fu(\()p Fo(l)1566 1050 y Fn(c)1587 1043 y Fo(;)8 b Fu(1\))763 1117 y Fo(E)s Fu(\()p Fo(l)837 1124 y Fn(c)857 1117 y Fu(\()p Fo(l)891 1124 y Fn(c)923 1117 y Fu(+)j(1\))g(+)g(1\))158 b(if)16 b(\()p 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Fp(\000)p Fu(1)p 142 1557 V Fo(;)8 b Fu(+1)p 228 1557 V -1 w(\))31 b(=)f Fo(E)s Fu(\()p Fo(n)499 1552 y Fn(c)520 1545 y Fu(\))g(=)14 b Fo(E)s Fu(\()p Fo(l)696 1552 y Fn(c)716 1545 y Fu(\()p Fo(l)750 1552 y Fn(c)782 1545 y Fu(+)d(1\))g(+)g(1\))30 b(=)h(\(4)11 b Fp(\000)g Fo(h)p Fp(b)p Fu(2)p Fo(=h)p Fp(c)p Fu(\))p Fp(b)p Fu(2)p Fo(=h)p Fp(c)f(\000)h Fo(h)p Fp(b)p Fu(2)p Fo(=h)p Fp(c)g Fu(+)g(4)g Fp(\000)f Fo(h)0 1648 y Fh(where)16 b(the)h(critical)f(length)g Fo(l)563 1655 y Fn(c)600 1648 y Fh(is)g Fp(b)p Fu(2)p Fo(=h)p Fp(c)g Fh(and)g(the)h (critical)f(area)g Fo(n)1276 1655 y Fn(c)1313 1648 y Fh(is)g Fo(l)1378 1655 y Fn(c)1398 1648 y Fu(\()p Fo(l)1432 1655 y Fn(c)1464 1648 y Fu(+)11 b(1\))g(+)g(1)p Fh(.)0 1734 y Fe(R)m(emark.)27 b Fu(In)21 b(particular,)g Fo(E)s Fu(\()p Fp(\000)p Fu(1)p 584 1746 V Fo(;)8 b Fu(+1)p 670 1746 V(\))31 b Fp(\030)g Fu(4)p Fo(=h)21 b Fu(when)h Fo(h)f Fu(go)q(es)h(to)g(zero:)32 b(the)22 b(energy)g(barrier)e(the)0 1794 y(system)c(has)g(to)h(o)o(v)o(ercome)e(go)q (es)i(to)g(in\014nit)o(y)e(lik)o(e)i Fo(h)995 1776 y Fj(\000)p Fl(1)1048 1794 y Fu(.)60 1880 y(F)l(rom)e(this)h(w)o(e)g(deduce)g(the)h(lev)o (el)f(of)h(the)g(greatest)g(cycle)g(con)o(taining)e Fp(\000)p Fu(1)p 1423 1892 V 16 w(and)h(not)h(+1)p 1689 1892 V -1 w(.)0 1966 y Fw(Corollary)j(6.16.)27 b Fh(The)17 b(lev)o(el)f(of)h(the)g(cycle)g Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 932 1978 V Fo(;)8 b Fu(+1)p 1018 1978 V -20 x Fn(c)1102 1966 y Fu(\))17 b Fh(is)f(pred)7 b Fo(E)s Fu(\()p Fo(n)1382 1973 y Fn(c)1403 1966 y Fu(\))p Fh(.)0 2052 y Fe(R)m(emark.)27 b Fu(W)l(e)16 b(recall)g(that)h(pred)8 b Fo(\025)13 b Fu(=)h(max)p Fp(f)8 b Fo(E)s Fu(\()p Fo(\033)r Fu(\))14 b(:)g Fo(E)s Fu(\()p Fo(\033)r Fu(\))h Fo(<)e(\025;)k(\033)f Fp(2)e Fo(X)e Fp(g)p Fu(,)17 b(see)f(notation)g(3)p Fo(:)p Fu(4.)0 2138 y Fw(Notation)i(6.17.)27 b Fu(If)16 b Fo(Y)26 b Fu(is)15 b(a)h(subset)e(of)i Fo(X)t Fu(,)g(its)f(minimal)e(and)i(maximal)f (areas)h Fo(a)p 1547 2146 27 2 v Fu(\()p Fo(Y)c Fu(\))16 b(and)p 1763 2110 V 15 w Fo(a)p Fu(\()p Fo(Y)c Fu(\))j(are)313 2241 y Fo(a)p 313 2249 V 1 w Fu(\()p Fo(Y)c Fu(\))23 b(=)f(min)o Fp(f)8 b Fo(a)p Fu(\()p Fo(\033)r Fu(\))15 b(:)f Fo(\033)i Fp(2)e Fo(Y)19 b Fp(g)8 b Fo(;)p 999 2214 V 58 w(a)q Fu(\()p Fo(Y)j Fu(\))22 b(=)g(max)p Fp(f)8 b Fo(a)p Fu(\()p Fo(\033)r Fu(\))15 b(:)f Fo(\033)i Fp(2)e Fo(Y)19 b Fp(g)8 b Fo(:)60 2371 y Fu(Since)17 b Fo(E)s Fu(\()p Fp(\000)p Fu(1)p 250 2383 64 2 v -1 w Fo(;)8 b(C)371 2378 y Fn(n)398 2371 y Fu(\))16 b Fp(\025)f Fo(E)s Fu(\()p Fo(n)576 2378 y Fn(c)596 2371 y Fu(\))j(for)f(an)o(y)g Fo(n)e Fp(\025)g Fo(n)934 2378 y Fn(c)971 2371 y Fu(\(b)o(y)j(prop)q(osition)e(6)p Fo(:)p Fu(11\),)h(then)g(all)g(the)h (con\014gura-)0 2430 y(tions)d(of)g(the)g(cycle)h Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 431 2443 V Fo(;)8 b Fu(+1)p 517 2443 V -19 x Fn(c)601 2430 y Fu(\))16 b(ha)o(v)o(e)e(an)h(area)g(less)f(then)h Fo(n)1156 2437 y Fn(c)1185 2430 y Fp(\000)8 b Fu(1)15 b(i.e.)p 1357 2403 27 2 v 21 w Fo(a)q Fu(\()p Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 1452 2443 64 2 v Fo(;)8 b Fu(+1)p 1538 2443 V -19 x Fn(c)1622 2430 y Fu(\)\))15 b Fp(\024)e Fo(n)1757 2437 y Fn(c)1786 2430 y Fp(\000)8 b Fu(1.)21 b(T)l(o)0 2490 y(complete)d(step)f Fo(i)p Fu(\))i(of)f(the)g(general)f(strategy)l(,)h(w)o(e)g(determine)f(the)h (con\014gurations)e(of)i(the)g(principal)0 2550 y(b)q(oundary)d(of)i(the)g (cycle)g Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 539 2562 V Fo(;)8 b Fu(+1)p 625 2562 V -20 x Fn(c)709 2550 y Fu(\).)954 2670 y Fs(18)p eop %%Page: 19 19 19 18 bop 0 200 a Fw(Theorem)23 b(6.18.)k Fh(The)22 b(principal)e(b)q (oundary)956 187 y Fk(e)945 200 y Fo(B)r Fu(\()p Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 1053 212 64 2 v 1 w Fo(;)8 b Fu(+1)p 1140 212 V -20 x Fn(c)1224 200 y Fu(\)\))22 b Fh(of)g(the)g(cycle)g Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 1614 212 V 1 w Fo(;)8 b Fu(+1)p 1701 212 V 1764 180 a Fn(c)1785 200 y Fu(\))22 b Fh(is)f(the)0 260 y(set)88 247 y Fk(f)82 260 y Fo(M)135 267 y Fn(n)160 272 y Fg(c)202 260 y Fh(of)g(the)f(principal)f(con\014gurations)f(of)j(area)f Fo(n)1092 267 y Fn(c)1112 260 y Fh(.)34 b(F)l(or)19 b Fo(\033)k Fh(in)1372 247 y Fk(f)1366 260 y Fo(M)1420 267 y Fn(n)1445 272 y Fg(c)1486 260 y Fh(there)e(exists)f(a)h(unique)0 319 y(con\014guration)16 b(in)g Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 410 331 V 1 w Fo(;)8 b Fu(+1)p 497 331 V 560 300 a Fn(c)581 319 y Fu(\))18 b Fh(comm)o(unicating)d(with)i Fo(\033)r Fh(,)h(whic)o(h)e(is) h(the)g(quasisquare)f Fo(l)1707 326 y Fn(c)1739 319 y Fp(\002)11 b Fu(\()p Fo(l)1823 326 y Fn(c)1855 319 y Fu(+)h(1\))0 379 y Fh(or)k Fu(\()p Fo(l)95 386 y Fn(c)127 379 y Fu(+)10 b(1\))i Fp(\002)f Fo(l)297 386 y Fn(c)333 379 y Fh(included)16 b(in)g Fo(\033)r Fh(.)0 475 y Fe(Pr)m(o)m(of.)28 b Fu(Let)17 b Fo(\033)j Fu(b)q(elong)c(to)521 462 y Fk(f)514 475 y Fo(M)562 482 y Fn(n)587 487 y Fg(c)609 475 y Fu(.)23 b(Then)16 b Fo(E)s Fu(\()p Fo(\033)r Fu(\))g(=)e Fo(E)s Fu(\()p Fo(n)1042 482 y Fn(c)1062 475 y Fu(\))j(and)g(theorem)f(6)p Fo(:)p Fu(5)h(yields)f(the)h(existence)h(of)0 535 y(a)f(sequence)f Fo(\033)276 542 y Fl(0)299 535 y Fo(;)8 b Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(\033)446 542 y Fn(n)471 547 y Fg(c)509 535 y Fu(suc)o(h)15 b(that)209 648 y Fo(\033)237 655 y Fl(0)274 648 y Fu(=)f Fp(\000)p Fu(1)p 327 660 V -1 w Fo(;)58 b(\033)490 655 y Fn(n)515 660 y Fg(c)550 648 y Fu(=)14 b Fo(\033)o(;)58 b Fp(8)p Fo(j)15 b Fp(2)f(f)p Fu(0)8 b Fp(\001)g(\001)g (\001)h Fo(n)969 655 y Fn(c)1000 648 y Fp(\000)i Fu(1)p Fp(g)50 b Fo(\033)1178 655 y Fn(j)1213 648 y Fp(2)14 b Fo(M)1308 655 y Fn(j)1329 648 y Fo(;)58 b(q)r Fu(\()p Fo(\033)1472 655 y Fn(j)1494 648 y Fo(;)8 b(\033)1544 655 y Fn(j)r Fl(+1)1616 648 y Fu(\))14 b Fo(>)g Fu(0)p Fo(:)0 761 y Fu(In)25 b(particular,)i(w)o(e)e (ha)o(v)o(e)g(max)p Fp(f)8 b Fo(E)s Fu(\()p Fo(\033)746 768 y Fn(n)774 761 y Fu(\))29 b(:)g(0)h Fp(\024)e Fo(n)i(<)e(n)1144 768 y Fn(c)1173 761 y Fp(g)h Fo(<)g(E)s Fu(\()p Fo(n)1384 768 y Fn(c)1404 761 y Fu(\))d(so)g(that)g Fo(\033)1665 768 y Fn(n)1690 773 y Fg(c)1709 768 y Fj(\000)p Fl(1)1788 761 y Fu(b)q(elongs)0 821 y(to)16 b Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 109 833 V Fo(;)8 b Fu(+1)p 195 833 V -20 x Fn(c)279 821 y Fu(\).)22 b(Since)15 b Fo(q)r Fu(\()p Fo(\033)534 828 y Fn(n)559 833 y Fg(c)579 828 y Fj(\000)p Fl(1)632 821 y Fo(;)8 b(\033)682 828 y Fn(n)707 833 y Fg(c)728 821 y Fu(\))15 b Fo(>)e Fu(0,)j(the)g(con\014guration)d Fo(\033)1282 828 y Fn(n)1307 833 y Fg(c)1344 821 y Fu(is)j(in)f(the)h (principal)d(b)q(oundary)0 890 y(of)k Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 106 902 V Fo(;)8 b Fu(+1)p 192 902 V -20 x Fn(c)276 890 y Fu(\).)22 b(Th)o(us)464 877 y Fk(f)458 890 y Fo(M)506 897 y Fn(n)531 902 y Fg(c)566 890 y Fp(\032)629 877 y Fk(e)618 890 y Fo(B)r Fu(\()p Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 726 902 V 1 w Fo(;)8 b Fu(+1)p 813 902 V -20 x Fn(c)897 890 y Fu(\)\).)0 957 y(Con)o(v)o(ersely)l(,)23 b(let)g Fo(\033)i Fu(b)q(elong)e(to)644 945 y Fk(e)634 957 y Fo(B)r Fu(\()p Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 742 969 V 1 w Fo(;)8 b Fu(+1)p 829 969 V -19 x Fn(c)913 957 y Fu(\)\).)42 b(Necessarily)22 b Fo(E)s Fu(\()p Fo(\033)r Fu(\))j(=)f Fo(E)s Fu(\()p Fo(n)1556 964 y Fn(c)1576 957 y Fu(\))g(so)e(that)h Fo(\033)i Fu(is)e(of)0 1017 y(area)c Fo(n)141 1024 y Fn(c)181 1017 y Fu(and)g(it)h(is)g(a)f(minimal)f(con\014guration.)30 b(In)20 b(addition)e(there)i(m)o(ust)f(exist)h Fo(\021)h Fu(in)e Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 1760 1029 V 1 w Fo(;)8 b Fu(+1)p 1847 1029 V 1910 997 a Fn(c)1931 1017 y Fu(\))0 1077 y(comm)o(unicating)21 b(with)i Fo(\033)r Fu(.)43 b(This)23 b Fo(\021)i Fu(is)e(in)g Fo(C)890 1084 y Fn(n)915 1089 y Fg(c)932 1084 y Fj(\000)p Fl(1)1009 1077 y Fu(and)g(satis\014es)f Fo(E)s Fu(\()p Fo(\021)r Fu(\))k Fo(<)f(E)s Fu(\()p Fo(\033)r Fu(\).)43 b(Th)o(us)22 b Fo(p)p Fu(\()p Fo(\021)r Fu(\))j Fp(\024)0 1137 y Fo(p)p Fu(\()p Fo(\033)r Fu(\))13 b Fp(\000)e Fo(h)k Fu(=)g Fo(p)p Fu(\()p Fo(n)328 1144 y Fn(c)349 1137 y Fu(\))d Fp(\000)f Fo(h)k Fu(=)g(4)p Fo(l)568 1144 y Fn(c)600 1137 y Fu(+)c(4)h Fp(\000)f Fo(h)18 b Fu(and)e Fo(p)p Fu(\()p Fo(\021)r Fu(\))g Fp(\025)f Fo(p)p Fu(\()p Fo(n)1115 1144 y Fn(c)1147 1137 y Fp(\000)d Fu(1\))j(=)g(4)p Fo(l)1351 1144 y Fn(c)1383 1137 y Fu(+)c(2.)25 b(Ho)o(w)o(ev)o(er)17 b(the)g(p)q(erime-)0 1196 y(ter)j(is)g(an)g(ev)o(en)g(in)o(teger.)32 b(The)20 b(only)g(p)q(ossibilit)o (y)f(is)g Fo(p)p Fu(\()p Fo(\021)r Fu(\))h(=)g(4)p Fo(l)1242 1203 y Fn(c)1276 1196 y Fu(+)13 b(2)20 b(whence)g Fo(\021)i Fu(is)d(minimal.)31 b(Y)l(et)0 1256 y Fo(n)30 1263 y Fn(c)63 1256 y Fp(\000)12 b Fu(1)18 b(=)f Fo(l)228 1263 y Fn(c)249 1256 y Fu(\()p Fo(l)283 1263 y Fn(c)316 1256 y Fu(+)12 b(1\))19 b(and)f(b)o(y)h(theorem)f(6)p Fo(:)p Fu(3,)h Fo(M)942 1263 y Fn(n)967 1268 y Fg(c)986 1263 y Fj(\000)p Fl(1)1057 1256 y Fu(=)f Fo(S)1145 1263 y Fn(n)1170 1268 y Fg(c)1187 1263 y Fj(\000)p Fl(1)1260 1256 y Fu(so)g(that)i Fo(\021)g Fu(b)q(elongs)e(to)h Fo(S)1754 1263 y Fn(n)1779 1268 y Fg(c)1797 1263 y Fj(\000)p Fl(1)1870 1256 y Fu(and)0 1316 y(it)f(is)f(a)g(quasisquare)f Fo(l)429 1323 y Fn(c)461 1316 y Fp(\002)c Fu(\()p Fo(l)546 1323 y Fn(c)578 1316 y Fu(+)f(1\))18 b(or)f(\()p Fo(l)786 1323 y Fn(c)818 1316 y Fu(+)12 b(1\))g Fp(\002)f Fo(l)990 1323 y Fn(c)1010 1316 y Fu(.)25 b(Lemma)17 b(6)p Fo(:)p Fu(4)g(sho)o(ws)f(that)i (the)g(only)f(p)q(oin)o(ts)g(of)0 1383 y Fo(M)48 1390 y Fn(n)73 1395 y Fg(c)110 1383 y Fu(whic)o(h)d(comm)o(unicate)g(with)h Fo(M)713 1390 y Fn(n)738 1395 y Fg(c)757 1390 y Fj(\000)p Fl(1)826 1383 y Fu(are)g(the)h(con\014gurations)d(of)1375 1371 y Fk(f)1369 1383 y Fo(M)1417 1390 y Fn(n)1442 1395 y Fg(c)1463 1383 y Fu(.)22 b(Th)o(us)14 b Fo(\033)j Fu(is)e(a)h(principal)0 1451 y(con\014guration)f(of) i(area)f Fo(n)497 1458 y Fn(c)533 1451 y Fu(and)641 1438 y Fk(e)630 1451 y Fo(B)s Fu(\()p Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 739 1463 V Fo(;)8 b Fu(+1)p 825 1463 V -20 x Fn(c)909 1451 y Fu(\)\))15 b Fp(\032)1020 1438 y Fk(f)1014 1451 y Fo(M)1062 1458 y Fn(n)1087 1463 y Fg(c)1108 1451 y Fu(.)50 b Fc(\003)60 1539 y Fu(W)l(e)19 b(no)o(w)e(pro)q(ceed)i(to)f(steps)h Fo(ii)p Fu(\))g(and)f Fo(iii)p Fu(\))h(of)g(the)g(general)f(strategy)l(.)28 b(On)18 b(the)g(set)h(of)g(the)g(cycles)0 1599 y(included)c(in)h Fp(f\000)p Fu(1)p 282 1611 V Fo(;)8 b Fu(+1)p 368 1611 V Fp(g)457 1581 y Fn(c)477 1599 y Fu(,)16 b(w)o(e)h(consider)e(the)i(graph)e Fo(G)i Fu(de\014ned)e(b)o(y)486 1720 y(\()p Fo(\031)533 1727 y Fl(1)570 1720 y Fp(!)f Fo(\031)662 1727 y Fl(2)685 1720 y Fu(\))g Fp(2)g Fo(G)114 b Fp(\()-8 b(\))1133 1707 y Fk(e)1122 1720 y Fo(B)s Fu(\()p Fo(\031)1210 1727 y Fl(1)1233 1720 y Fu(\))11 b Fp(\\)h Fo(\031)1336 1727 y Fl(2)1372 1720 y Fp(6)p Fu(=)i Fp(;)p Fo(:)0 1833 y Fu(W)l(e)i(denote)g(b)o(y)g Fo(G)353 1815 y Fl(+)402 1833 y Fu(the)h(restriction)e(of)h Fo(G)g Fu(to)h Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 947 1845 V Fo(;)8 b Fu(+1)p 1033 1845 V -20 x Fn(c)1117 1833 y Fu(\))1136 1815 y Fn(c)1173 1833 y Fu(\(obtained)15 b(b)o(y)h(deleting)g(all)g(the)g(arro)o(ws)0 1892 y(whose)23 b(starting)g(cycle)h(is)f(included)f(in)h Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 854 1905 V Fo(;)8 b Fu(+1)p 940 1905 V -19 x Fn(c)1024 1892 y Fu(\)\).)43 b(According)23 b(to)h(the)g(general)e (metho)q(d)h(for)0 1952 y(determining)16 b(the)h(saddle)g(paths)g(of)g (minimal)f(cost)i(exiting)f(from)g Fp(\000)p Fu(1)p 1295 1964 V 17 w(and)g(reac)o(hing)f(+1)p 1673 1964 V -1 w(,)i(the)g(next)0 2012 y(step)d(is)g(to)h(\014nd,)e(for)i(eac)o(h)e(p)q(oin)o(t)h Fo(\033)670 1994 y Fj(0)700 2012 y Fu(in)g(the)h(principal)d(b)q(oundary)h (of)i Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 1376 2024 V Fo(;)8 b Fu(+1)p 1462 2024 V -20 x Fn(c)1546 2012 y Fu(\),)16 b(the)g(orbit)e(of)i Fp(f)p Fo(\033)1911 1994 y Fj(0)1925 2012 y Fp(g)0 2072 y Fu(in)i Fo(G)99 2054 y Fl(+)151 2072 y Fu(\(i.e.)29 b(the)19 b(minimal)e(stable)h (subgraph)f(of)i Fo(G)1010 2054 y Fl(+)1062 2072 y Fu(con)o(taining)e Fp(f)p Fo(\033)1361 2054 y Fj(0)1375 2072 y Fp(g)p Fu(\).)29 b(If)19 b(there)f(do)q(es)h(not)g(exist)0 2132 y(a)j(path)f(in)g(this)h (graph)e(leading)h(from)g Fp(f)p Fo(\033)835 2113 y Fj(0)849 2132 y Fp(g)h Fu(to)g(+1)p 962 2144 V(,)g(then)g Fo(\033)1211 2113 y Fj(0)1247 2132 y Fu(is)f(a)h(dead{end)f(and)g(the)h(last)f(exit)0 2191 y(from)j Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 175 2203 V Fo(;)8 b Fu(+1)p 261 2203 V -19 x Fn(c)345 2191 y Fu(\))25 b(b)q(efore)g(reac)o(hing)e(+1)p 753 2203 V 24 w(w)o(on't)h(tak)o(e)h(place)g (through)e Fo(\033)1464 2173 y Fj(0)1478 2191 y Fu(.)46 b(Otherwise,)26 b Fo(\033)1828 2173 y Fj(0)1867 2191 y Fu(is)e(a)0 2251 y(p)q(ossible)16 b(global)h(saddle)f(b)q(et)o(w)o(een)h Fp(\000)p Fu(1)p 682 2263 V 17 w(and)g(+1)p 861 2263 V 17 w(and)g(w)o(e)g(ha)o(v)o(e)g(to)h (determine)e(all)h(the)h(paths)f(in)g Fo(G)1917 2233 y Fl(+)0 2311 y Fu(leading)22 b(from)h Fp(f)p Fo(\033)358 2293 y Fj(0)372 2311 y Fp(g)g Fu(to)g(+1)p 487 2323 V(;)j(then)d(w)o(e)g(m)o(ust)f(searc)o(h) g(for)h(all)g(the)g(cycle)h(paths)e(in)h Fp(f\000)p Fu(1)p 1692 2323 V Fo(;)8 b Fu(+1)p 1778 2323 V -1 w Fp(g)1866 2293 y Fn(c)1910 2311 y Fu(of)0 2371 y(n)o(ull)19 b(cost)h(realizing)f(the)h(exit) h(of)f Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 711 2383 V Fo(;)8 b Fu(+1)p 797 2383 V -20 x Fn(c)881 2371 y Fu(\))20 b(at)h Fo(\033)1015 2353 y Fj(0)1029 2371 y Fu(.)32 b(More)20 b(precisely)l(,)f(if)i (w)o(e)e(denote)h(b)o(y)g Fo(G)1827 2353 y Fj(\000)1881 2371 y Fu(the)0 2430 y(restriction)e(of)g Fo(G)h Fu(to)g Fo(\031)r Fu(\(+1)p 469 2443 V Fo(;)8 b Fp(\000)p Fu(1)p 555 2443 V -19 x Fn(c)640 2430 y Fu(\))659 2412 y Fn(c)698 2430 y Fu(\(obtained)18 b(b)o(y)g(deleting)h(all)f(the)h(arro)o(ws)e(whose)h(starting)f(cycle)0 2490 y(is)f(in)g Fo(\031)r Fu(\(+1)p 157 2502 V Fo(;)8 b Fp(\000)p Fu(1)p 243 2502 V -20 x Fn(c)327 2490 y Fu(\)\),)18 b(w)o(e)e(ha)o(v)o(e)g (to)g(determine)g(the)h(minimal)d(stable)i(subgraph)f(of)h Fo(G)1613 2472 y Fj(\000)1664 2490 y Fu(con)o(taining)f Fo(\033)1936 2472 y Fj(0)0 2550 y Fu(\(i.e.)23 b(the)17 b(orbit)f(of)h Fo(\033)401 2532 y Fj(0)432 2550 y Fu(in)f(the)h(graph)e Fo(G)756 2532 y Fj(\000)790 2550 y Fu(\))i(and)f(all)h(the)g(paths)f(in)g(this)g(graph)g (starting)g(at)h Fp(f)p Fo(\033)1814 2532 y Fj(0)1828 2550 y Fp(g)g Fu(and)954 2670 y Fs(19)p eop %%Page: 20 20 20 19 bop 0 200 a Fu(ending)15 b(at)h Fp(\000)p Fu(1)p 220 212 64 2 v(.)22 b(T)l(o)16 b(ac)o(hiev)o(e)f(these)h(goals,)f(w)o(e)h(will)f (next)i(describ)q(e)e(a)h(list)g(of)g(relev)m(an)o(t)g(cycles,)g(whic)o(h)0 260 y(are)i(included)f(in)g Fp(f\000)p Fu(1)p 370 272 V Fo(;)8 b Fu(+1)p 456 272 V Fp(g)545 242 y Fn(c)583 260 y Fu(and)18 b(w)o(e)g(will)f(precise)h(their)f(principal)g(b)q(oundaries.)25 b(Notice)19 b(that)f(w)o(e)0 319 y(don't)c(kno)o(w)g Fe(a)j(priori)f Fu(that)f(these)g(cycles)g(are)f(in)h Fp(M)p Fu(\()p Fp(f\000)p Fu(1)p 1051 331 V Fo(;)8 b Fu(+1)p 1137 331 V -1 w Fp(g)1225 301 y Fn(c)1245 319 y Fu(\).)22 b(W)l(e)15 b(will)f(then)h(use)f(the)h(follo) o(wing)0 379 y(simple)g(result.)0 479 y Fw(Lemma)20 b(6.19.)58 b Fh(Let)19 b Fo(\031)502 486 y Fl(0)525 479 y Fo(;)8 b Fp(\001)g(\001)g (\001)17 b Fo(;)8 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Fo(E)s Fu(\()p Fo(l)946 1089 y Fl(1)979 1082 y Fp(\002)j Fo(l)1044 1089 y Fl(2)1066 1082 y Fo(;)d(k)r Fu(\))14 b(:)g Fo(a)p Fu(\()p Fo(\021)1247 1089 y Fn(r)1269 1082 y Fu(\))e Fp(\000)e Fo(l)1364 1089 y Fl(1)1387 1082 y Fo(l)1402 1089 y Fl(2)1438 1082 y Fp(\024)j Fo(k)j Fp(\024)d Fu(0)8 b Fp(g)p Fo(:)0 1200 y Fu(Theorem)16 b(6)p Fo(:)p Fu(8)i(sho)o(ws)e (that)h(for)h(0)d Fp(\024)g Fo(k)i Fp(\024)e Fo(l)829 1207 y Fl(1)851 1200 y Fu(,)j Fo(E)s Fu(\()p Fo(l)957 1207 y Fl(1)991 1200 y Fp(\002)11 b Fo(l)1056 1207 y Fl(2)1078 1200 y Fo(;)d Fp(\000)p Fo(k)r Fu(\))24 b(=)f Fo(E)s Fu(\()p Fo(l)1346 1207 y Fl(1)1380 1200 y Fp(\002)12 b Fu(\()p Fo(l)1465 1207 y Fl(2)1499 1200 y Fp(\000)f Fu(1\))h Fp(\010)1644 1207 y Fl(2)1678 1200 y Fu(\()p Fo(l)1712 1207 y Fl(1)1746 1200 y Fp(\000)g Fo(k)r Fu(\))f Fp(\002)h Fu(1\))0 1260 y(so)k(that)g Fo(E)s Fu(\()p Fo(l)242 1267 y Fl(1)274 1260 y Fp(\002)10 b Fo(l)338 1267 y Fl(2)370 1260 y Fu(+)f(1\))h Fp(\000)g Fo(E)s Fu(\()p Fo(l)595 1267 y Fl(1)627 1260 y Fp(\002)f Fo(l)690 1267 y Fl(2)713 1260 y Fo(;)f Fp(\000)p Fo(k)r Fu(\))13 b(=)h(2)c Fp(\000)f Fo(h)p Fu(\()p Fo(k)j Fu(+)d(1\).)22 b(F)l(or)15 b(this)h(quan)o(tit)o(y)f(to)h(b)q (e)h(p)q(ositiv)o(e,)e(w)o(e)0 1320 y(m)o(ust)g(ha)o(v)o(e)g Fo(k)h Fp(\024)d Fo(l)346 1327 y Fn(c)377 1320 y Fp(\000)d Fu(1)16 b(whence)g Fo(a)p Fu(\()p Fo(\021)711 1327 y Fn(r)733 1320 y Fu(\))e Fp(\025)g Fo(l)834 1327 y Fl(1)856 1320 y Fo(l)871 1327 y Fl(2)904 1320 y Fp(\000)c Fu(\()p Fo(l)987 1327 y Fn(c)1017 1320 y Fp(\000)g Fu(1\).)23 b(In)15 b(addition)h(for)f(an)o(y)h Fo(k)r Fu(,)g(0)d Fp(\024)h Fo(k)h Fp(\024)f Fo(l)1832 1327 y Fn(c)1862 1320 y Fp(\000)c Fu(1,)0 1380 y Fo(E)s Fu(\()p Fo(l)74 1387 y Fl(1)107 1380 y Fp(\002)h Fo(l)172 1387 y Fl(2)194 1380 y Fo(;)d Fp(\000)p Fo(k)r Fu(\))j Fp(\000)g Fo(E)s Fu(\()p Fo(l)437 1387 y Fl(1)470 1380 y Fp(\002)g Fo(l)535 1387 y Fl(2)557 1380 y Fu(\))j(=)g Fo(hk)h(>)f Fu(0)i(so)g(that)h Fo(E)s Fu(\()p Fo(\021)1060 1387 y Fn(r)1082 1380 y Fu(\))d Fo(>)g(E)s Fu(\()p Fo(l)1242 1387 y Fl(1)1275 1380 y Fp(\002)d Fo(l)1340 1387 y Fl(2)1362 1380 y Fu(\))17 b(whenev)o(er)f Fo(\021)1642 1387 y Fn(r)1678 1380 y Fp(6)p Fu(=)d Fo(l)1745 1387 y Fl(1)1778 1380 y Fp(\002)e Fo(l)1843 1387 y Fl(2)1865 1380 y Fu(.)0 1440 y(W)l(e)g(ha)o(v)o(e)g(pro)o(v)o(ed)e(that)j Fo(l)464 1447 y Fl(1)486 1440 y Fp(\002)p Fo(l)540 1447 y Fl(2)573 1440 y Fu(and)f(the)g(cycle)h Fo(\031)r Fu(\()p Fo(l)927 1447 y Fl(1)950 1440 y Fp(\002)p Fo(l)1004 1447 y Fl(2)1026 1440 y Fo(;)c Fu(pred)f Fo(E)s Fu(\()p Fo(l)1227 1447 y Fl(1)1250 1440 y Fp(\002)p Fo(l)1304 1447 y Fl(2)1326 1440 y Fu(+1\)\))k(satisfy)h(the)f(h)o(yp)q (othesis)f(of)0 1499 y(theorem)i(5)p Fo(:)p Fu(5.)21 b(Th)o(us)12 b(the)h(b)q(ottom)g(of)g(the)h(cycle)f(is)g Fp(f)p Fo(l)1008 1506 y Fl(1)1034 1499 y Fp(\002)t Fo(l)1092 1506 y Fl(2)1114 1499 y Fp(g)p Fu(.)21 b(W)l(e)13 b(ha)o(v)o(e)g(also)f(pro)o(v)o(ed)g(the)h (inequalities)0 1559 y(on)p 67 1532 27 2 v 14 w Fo(a)j Fu(and)e Fo(a)p 204 1567 V Fu(.)21 b(That)15 b Fo(M)5 b Fu(\()p Fo(l)475 1566 y Fl(1)506 1559 y Fp(\002)i Fo(l)567 1566 y Fl(2)589 1559 y Fo(;)h Fp(\000)p Fo(k)r Fu(\))15 b(is)f(included)g(in)g(the)h(cycle)g(for)g (0)f Fp(\024)f Fo(k)i Fp(\024)f Fo(l)1494 1566 y Fn(c)1522 1559 y Fp(\000)7 b Fu(1)15 b(is)f(ob)o(vious:)20 b(eac)o(h)0 1619 y(con\014guration)15 b(of)j(this)f(set)g(comm)o(unicates)e(with)j Fo(l)989 1626 y Fl(1)1022 1619 y Fp(\002)11 b Fo(l)1087 1626 y Fl(2)1127 1619 y Fu(under)16 b(the)h(lev)o(el)g(pred)8 b Fo(E)s Fu(\()p Fo(l)1649 1626 y Fl(1)1682 1619 y Fp(\002)k Fo(l)1748 1626 y Fl(2)1781 1619 y Fu(+)g(1\))17 b(\(cf)0 1679 y(theorem)i(6)p Fo(:)p Fu(8:)27 b(these)20 b(con\014gurations)d(are)i (obtained)g(b)o(y)g(deleting)g(successiv)o(ely)g Fo(k)i Fu(corner)d(squares)0 1738 y(from)d Fo(l)132 1745 y Fl(1)164 1738 y Fp(\002)9 b Fo(l)227 1745 y Fl(2)250 1738 y Fu(\).)22 b(Finally)14 b(a)i(con\014guration)e Fo(\030)k Fu(of)e(the)g(principal)e(b)q(oundary)h(of)h(this)f(cycle)i(is)e (of)h(energy)0 1798 y Fo(E)s Fu(\()p Fo(\030)r Fu(\))e(=)g Fo(E)s Fu(\()p Fo(l)243 1805 y Fl(1)272 1798 y Fp(\002)7 b Fo(l)333 1805 y Fl(2)362 1798 y Fu(+)g(1\))15 b(so)f(that)h(its)f(area)g(is)g (equal)h(to)g Fo(l)1057 1805 y Fl(1)1079 1798 y Fo(l)1094 1805 y Fl(2)1123 1798 y Fu(+)7 b(1,)15 b(and)e(its)i(p)q(erimeter)f(to)h(2\()p Fo(l)1727 1805 y Fl(1)1756 1798 y Fu(+)7 b Fo(l)1817 1805 y Fl(2)1839 1798 y Fu(\))g(+)g(2.)0 1858 y(Let)20 b Fo(\021)h Fu(b)q(e)f(a)f(con\014guration)f(of)i(the)f(cycle)h(suc)o(h)f(that)h Fo(q)r Fu(\()p Fo(\021)r(;)8 b(\030)r Fu(\))19 b Fo(>)f Fu(0.)31 b(Necessarily)l(,)19 b(the)h(area)f(of)g Fo(\021)j Fu(is)0 1918 y Fo(l)15 1925 y Fl(1)37 1918 y Fo(l)52 1925 y Fl(2)74 1918 y Fu(.)29 b(Th)o(us)17 b Fo(\021)k Fu(is)d(a)h(con\014guration)e(of)i (maximal)e(area)i(of)g(the)g(cycle)g(and)f(as)h(suc)o(h)e(it)i(is)f(a)h (maximal)0 1978 y(con\014guration)d(of)i(the)h(cycle)f(for)g(the)g(inclusion) f(relation.)25 b(By)19 b(theorem)e(5)p Fo(:)p Fu(3)h(the)g(rectangle)g Fo(l)1828 1985 y Fl(1)1862 1978 y Fp(\002)12 b Fo(l)1928 1985 y Fl(2)0 2037 y Fu(is)j(included)f(in)h Fo(\021)i Fu(whence)e(in)g(fact)h Fo(\021)f Fu(=)f Fo(l)781 2044 y Fl(1)811 2037 y Fp(\002)9 b Fo(l)874 2044 y Fl(2)896 2037 y Fu(.)21 b(Th)o(us)14 b Fo(\030)k Fu(b)q(elongs)c(to)i Fo(M)5 b Fu(\()p Fo(l)1420 2044 y Fl(1)1452 2037 y Fp(\002)j Fo(l)1514 2044 y Fl(2)1536 2037 y Fo(;)g Fu(+1\).)22 b(Theorem)14 b(6)p Fo(:)p Fu(9)0 2097 y(sho)o(ws)22 b(that)i(this)e(set)i(is) f(equal)g(to)p 716 2056 222 2 v 24 w Fo(l)731 2104 y Fl(1)764 2097 y Fp(\002)11 b Fo(l)829 2104 y Fl(2)862 2097 y Fp(\010)g Fu(1)937 2065 y Fl(12)979 2097 y Fu(.)42 b(Con)o(v)o(ersely)22 b(eac)o(h)h(con\014guration)e(of)j(this)f(set)0 2157 y(comm)o(unicates)15 b(with)h Fo(l)451 2164 y Fl(1)484 2157 y Fp(\002)11 b Fo(l)549 2164 y Fl(2)588 2157 y Fu(and)16 b(b)q(elongs)g(to)h(the)f(principal)f(b)q (oundary)g(of)i(the)g(cycle.)50 b Fc(\003)0 2246 y Fw(Lemma)17 b(6.21.)50 b Fh(Let)17 b Fo(l)476 2253 y Fl(1)498 2246 y Fo(;)8 b(l)535 2253 y Fl(2)557 2246 y Fo(;)g(k)r(;)g(i)17 b Fh(b)q(e)g(in)o(tegers)e (with)i Fo(l)1044 2253 y Fl(1)1080 2246 y Fo(>)c Fu(0)p Fo(;)k(l)1203 2253 y Fl(2)1239 2246 y Fo(>)c Fu(0)p Fo(;)k(k)e(>)f Fu(1)p Fo(;)i(i)e Fp(\025)g Fu(0)p Fo(;)i(l)1650 2253 y Fl(1)1686 2246 y Fo(>)d(k)g Fu(+)e Fo(i)p Fh(.)0 2309 y(The)16 b(cycle)i Fp(f)p Fo(l)266 2316 y Fl(1)299 2309 y Fp(\002)11 b Fo(l)364 2316 y Fl(2)397 2309 y Fu(+)436 2291 y Fn(i)436 2321 y Fl(2)469 2309 y Fo(k)r Fp(g)16 b Fh(has)g(one)g(or)g(t)o(w)o(o)g(con\014gurations)f (in)h(its)g(principal)f(b)q(oundary)1765 2296 y Fk(e)1754 2309 y Fo(B)s Fh(:)60 2375 y Fp(\017)49 b Fh(if)17 b Fo(i)d Fu(=)g(0)i Fh(then)429 2362 y Fk(e)419 2375 y Fo(B)g Fu(=)d Fp(f)p Fo(l)565 2382 y Fl(1)599 2375 y Fp(\002)d Fo(l)663 2382 y Fl(2)697 2375 y Fu(+)736 2357 y Fn(i)736 2387 y Fl(2)769 2375 y Fu(\()p Fo(k)j Fu(+)e(1\))p Fp(g)p Fh(,)60 2441 y Fp(\017)49 b Fh(if)17 b Fo(i)d Fu(=)g Fo(l)279 2448 y Fl(1)312 2441 y Fp(\000)d Fu(1)g Fp(\000)g Fo(k)18 b Fh(then)616 2428 y Fk(e)605 2441 y Fo(B)e Fu(=)e Fp(f)p Fo(l)752 2448 y Fl(1)785 2441 y Fp(\002)d Fo(l)850 2448 y Fl(2)883 2441 y Fu(+)922 2420 y Fn(i)p Fj(\000)p Fl(1)922 2454 y(2)1001 2441 y Fu(\()p Fo(k)h Fu(+)f(1\))p Fp(g)p Fh(,)60 2508 y Fp(\017)49 b Fh(if)17 b Fu(0)d Fo(<)f(i)h(<)g(l)370 2515 y Fl(1)403 2508 y Fp(\000)d Fu(1)g Fp(\000)g Fo(k)18 b Fh(then)707 2495 y Fk(e)697 2508 y Fo(B)e Fu(=)d Fp(f)p Fo(l)843 2515 y Fl(1)876 2508 y Fp(\002)e Fo(l)941 2515 y Fl(2)974 2508 y Fu(+)1013 2486 y Fn(i)p Fj(\000)p Fl(1)1013 2521 y(2)1092 2508 y Fu(\()p Fo(k)i Fu(+)e(1\))p Fo(;)16 b(l)1289 2515 y Fl(1)1323 2508 y Fp(\002)10 b Fo(l)1387 2515 y Fl(2)1421 2508 y Fu(+)1460 2490 y Fn(i)1460 2520 y Fl(2)1493 2508 y Fu(\()p Fo(k)j Fu(+)e(1\))p Fp(g)p Fh(.)954 2670 y Fs(21)p eop %%Page: 22 22 22 21 bop 0 200 a Fw(Lemma)17 b(6.22.)50 b Fh(Let)17 b Fo(l)476 207 y Fl(1)498 200 y Fo(;)8 b(l)535 207 y Fl(2)557 200 y Fo(;)g(i)17 b Fh(b)q(e)g(in)o(tegers)f(with)g Fo(l)994 207 y Fl(1)1030 200 y Fo(>)e Fu(0)p Fo(;)i(l)1153 207 y Fl(2)1189 200 y Fo(>)e Fu(0)p Fo(;)i(i)e Fp(\025)g Fu(0)p Fo(;)i(l)1451 207 y Fl(1)1487 200 y Fo(>)e(i)d Fu(+)g(1)p Fh(.)0 263 y(The)16 b(cycle)i Fp(f)p Fo(l)266 270 y Fl(1)299 263 y Fp(\002)11 b Fo(l)364 270 y Fl(2)397 263 y Fu(+)436 245 y Fn(i)436 275 y Fl(2)469 263 y Fu(1)p Fp(g)16 b Fh(has)g(t)o(w)o(o)g(or)g(three)h(con\014gurations)d(in)i(its)h(principal)d (b)q(oundary)1798 250 y Fk(e)1788 263 y Fo(B)r Fh(:)60 329 y Fp(\017)49 b Fh(if)17 b Fo(i)d Fu(=)g(0)i Fh(then)429 316 y Fk(e)419 329 y Fo(B)g Fu(=)d Fp(f)p Fo(l)565 336 y Fl(1)599 329 y Fp(\002)d Fo(l)663 336 y Fl(2)686 329 y Fo(;)16 b(l)731 336 y Fl(1)764 329 y Fp(\002)11 b Fo(l)829 336 y Fl(2)862 329 y Fu(+)901 311 y Fn(i)901 341 y Fl(2)935 329 y Fu(2)p Fp(g)p Fh(,)60 395 y Fp(\017)49 b Fh(if)17 b Fo(i)d Fu(=)g Fo(l)279 402 y Fl(1)312 395 y Fp(\000)d Fu(1)g Fp(\000)g Fo(k)18 b Fh(then)616 382 y Fk(e)605 395 y Fo(B)e Fu(=)e Fp(f)p Fo(l)752 402 y Fl(1)785 395 y Fp(\002)d Fo(l)850 402 y Fl(2)872 395 y Fo(;)17 b(l)918 402 y Fl(1)951 395 y Fp(\002)11 b Fo(l)1016 402 y Fl(2)1049 395 y Fu(+)1088 374 y Fn(i)p Fj(\000)p Fl(1)1088 408 y(2)1166 395 y Fu(2)p Fp(g)p Fh(,)60 462 y Fp(\017)49 b Fh(if)17 b Fu(0)d Fo(<)f(i)h(<)g(l)370 469 y Fl(1)403 462 y Fp(\000)d Fu(1)g Fp(\000)g Fo(k)18 b Fh(then)707 449 y Fk(e)697 462 y Fo(B)e Fu(=)d Fp(f)p Fo(l)843 469 y Fl(1)876 462 y Fp(\002)e Fo(l)941 469 y Fl(2)963 462 y Fo(;)17 b(l)1009 469 y Fl(1)1042 462 y Fp(\002)11 b Fo(l)1107 469 y Fl(2)1140 462 y Fu(+)1179 441 y Fn(i)p Fj(\000)p Fl(1)1179 475 y(2)1258 462 y Fu(2)p Fo(;)16 b(l)1328 469 y Fl(1)1361 462 y Fp(\002)11 b Fo(l)1426 469 y Fl(2)1459 462 y Fu(+)1498 444 y Fn(i)1498 474 y Fl(2)1531 462 y Fu(2)p Fp(g)p Fh(.)0 550 y Fe(R)m(emark.)27 b Fu(Results)17 b(similar)g(to)i(those)g(stated)g(in)f(lemmas)f(6)p Fo(:)p Fu(21)h(and)g(6)p Fo(:)p Fu(22)h(hold)f(for)g(the)h(con\014gura-)0 610 y(tions)d Fo(l)137 617 y Fl(1)170 610 y Fp(\002)11 b Fo(l)235 617 y Fl(2)268 610 y Fu(+)307 592 y Fn(i)307 622 y Fl(1)341 610 y Fu(1)16 b(and)g(also)g(for)g(an)o(y)g(con\014guration)f(in)p 1109 569 222 2 v 16 w Fo(l)1124 617 y Fl(1)1157 610 y Fp(\002)c Fo(l)1222 617 y Fl(2)1255 610 y Fp(\010)g Fu(1.)0 698 y Fw(Corollary)24 b(6.23.)59 b Fh(The)20 b(follo)o(wing)e(cycles)j(are)e(maximal)g(cycles)h(of) g Fp(f\000)p Fu(1)p 1427 710 64 2 v -1 w Fo(;)8 b Fu(+1)p 1512 710 V Fp(g)1601 680 y Fn(c)1621 698 y Fh(:)29 b(\(with)20 b Fo(l)1815 705 y Fl(1)1857 698 y Fo(>)f(l)1930 705 y Fn(c)0 757 y Fh(and)d Fo(l)112 764 y Fl(2)148 757 y Fo(>)d(l)215 764 y Fn(c)236 757 y Fh(\))431 872 y Fp(f)p Fo(\021)r Fp(g)8 b Fo(;)107 b(\021)16 b Fp(2)p 725 831 247 2 v 14 w Fo(l)740 879 y Fl(1)773 872 y Fp(\002)11 b Fo(l)838 879 y Fl(2)871 872 y Fp(\010)910 879 y Fl(2)943 872 y Fo(k)r(;)108 b Fu(0)13 b Fo(<)h(k)h(<)f(l) 1293 879 y Fl(1)1326 872 y Fp(\000)d Fo(l)1391 879 y Fn(c)1422 872 y Fu(+)g(1)d Fo(;)431 956 y Fp(f)p Fo(\021)r Fp(g)g Fo(;)107 b(\021)16 b Fp(2)p 725 916 V 14 w Fo(l)740 963 y Fl(1)773 956 y Fp(\002)11 b Fo(l)838 963 y Fl(2)871 956 y Fp(\010)910 963 y Fl(1)943 956 y Fo(k)r(;)108 b Fu(0)13 b Fo(<)h(k)h(<)f(l)1293 963 y Fl(2)1326 956 y Fp(\000)d Fo(l)1391 963 y Fn(c)1422 956 y Fu(+)g(1)d Fo(;)431 1033 y(\031)r Fu(\()p Fo(l)495 1040 y Fl(1)529 1033 y Fp(\002)j Fo(l)594 1040 y Fl(2)616 1033 y Fo(;)d Fh(pred)f Fo(E)s Fu(\()p Fo(l)817 1040 y Fl(1)851 1033 y Fp(\002)j Fo(l)915 1040 y Fl(2)949 1033 y Fu(+)g(1\)\))e Fo(:)0 1174 y Fe(Pr)m(o)m(of.)28 b Fu(This)23 b(corollary)g(is)g(a)h(consequence)g(of)g (lemma)f(6)p Fo(:)p Fu(19)g(together)h(with)g(theorem)f(6)p Fo(:)p Fu(20)h(and)0 1233 y(lemmas)16 b(6)p Fo(:)p Fu(21)h(and)f(6)p Fo(:)p Fu(22.)24 b(Notice)18 b(that)g(w)o(e)f(ha)o(v)o(e)g(to)g(put)g (together)h(the)f(descriptions)f(of)h(the)h(cycles)0 1293 y(of)d(theorem)f(6) p Fo(:)p Fu(20,)g(lemmas)g(6)p Fo(:)p Fu(21)p Fo(;)8 b Fu(6)p Fo(:)p Fu(22)14 b(in)g(order)g(to)h(c)o(hec)o(k)f(that)h(for)f(eac)o(h)h (cycle)g Fo(\031)i Fu(in)d(the)h(ab)q(o)o(v)o(e)f(list,)0 1356 y(there)i(is)g(a)h(sequence)f(of)h(cycles)f Fo(\031)652 1363 y Fl(0)675 1356 y Fo(;)8 b Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(\031)822 1363 y Fn(r)861 1356 y Fu(suc)o(h)15 b(that)i Fo(\031)1109 1363 y Fl(0)1146 1356 y Fu(=)c Fo(\031)r Fu(,)1269 1344 y Fk(e)1259 1356 y Fo(B)r Fu(\()p Fo(\031)1346 1363 y Fn(i)1363 1356 y Fu(\))e Fp(\\)g Fo(\031)1465 1363 y Fn(i)p Fl(+1)1547 1356 y Fp(6)p Fu(=)i Fp(;)p Fu(,)k(0)c Fp(\024)h Fo(i)g(<)f(r)19 b Fu(and)0 1422 y(+1)p 0 1435 64 2 v 13 w Fp(2)135 1410 y Fk(e)125 1422 y Fo(B)r Fu(\()p Fo(\031)212 1429 y Fn(r)235 1422 y Fu(\).)50 b Fc(\003)0 1510 y Fw(Corollary)24 b(6.24.)61 b Fh(Let)21 b Fo(\033)i Fh(b)q(elong)c(to)i Fo(S)841 1517 y Fn(n)866 1522 y Fg(c)884 1517 y Fl(+1)937 1510 y Fh(.)33 b(The)20 b(minimal)f(stable)h (subgraph)e Fo(G)1695 1492 y Fl(+)1728 1510 y Fu(\()p Fo(\033)r Fu(\))j Fh(of)g Fo(G)1917 1492 y Fl(+)0 1570 y Fh(con)o(taining)f Fo(\033)j Fh(is)e(the)h(restriction)e(of)h Fo(G)h Fh(to)f(the)h(v)o(ertices)f (listed)f(in)h(corollary)f Fu(6)p Fo(:)p Fu(23)p Fh(.)36 b(The)21 b(arro)o(ws)0 1630 y(of)c Fo(G)96 1612 y Fl(+)129 1630 y Fu(\()p Fo(\033)r Fu(\))h Fh(are)e(\(where)g(the)h(rectangles)f Fo(l)795 1637 y Fl(1)828 1630 y Fp(\002)11 b Fo(l)893 1637 y Fl(2)932 1630 y Fh(are)16 b(con\014gurations)e(con)o(taining)h Fo(\033)r Fh(\))143 1744 y Fp(f)p Fo(l)183 1751 y Fl(1)216 1744 y Fp(\002)c Fo(l)281 1751 y Fl(2)315 1744 y Fu(+)354 1724 y Fn(i)354 1756 y Fl(2)387 1744 y Fo(k)r Fp(g)i(!)h(f)p Fo(l)557 1751 y Fl(1)590 1744 y Fp(\002)d Fo(l)655 1751 y Fl(2)688 1744 y Fu(+)727 1723 y Fn(i)727 1757 y Fl(2)805 1744 y Fo(k)i Fu(+)e(1)p Fp(g)p Fo(;)107 b Fu(0)14 b Fp(\024)g Fo(i)g(<)f(l)1255 1751 y Fl(1)1288 1744 y Fp(\000)e Fo(k)r(;)58 b Fu(0)13 b Fo(<)h(k)h(<)f(l)1638 1751 y Fl(1)1671 1744 y Fp(\000)d Fo(l)1736 1751 y Fn(c)1764 1744 y Fo(;)143 1828 y Fp(f)p Fo(l)183 1835 y Fl(1)216 1828 y Fp(\002)g Fo(l)281 1835 y Fl(2)315 1828 y Fu(+)354 1808 y Fn(i)354 1841 y Fl(2)387 1828 y Fo(k)r Fp(g)i(!)h(f)p Fo(l)557 1835 y Fl(1)590 1828 y Fp(\002)d Fo(l)655 1835 y Fl(2)688 1828 y Fu(+)727 1807 y Fn(i)p Fj(\000)p Fl(1)727 1842 y(2)805 1828 y Fo(k)i Fu(+)e(1)p Fp(g)p Fo(;)107 b Fu(0)14 b Fo(<)g(i)g Fp(\024)f Fo(l)1255 1835 y Fl(1)1288 1828 y Fp(\000)e Fo(k)r(;)58 b Fu(0)13 b Fo(<)h(k)h(<)f(l)1638 1835 y Fl(1)1671 1828 y Fp(\000)d Fo(l)1736 1835 y Fn(c)1764 1828 y Fo(;)143 1913 y Fp(f)p Fo(l)183 1920 y Fl(1)216 1913 y Fp(\002)g Fo(l)281 1920 y Fl(2)315 1913 y Fu(+)354 1892 y Fn(i)354 1925 y Fl(1)387 1913 y Fo(k)r Fp(g)i(!)h(f)p Fo(l)557 1920 y Fl(1)590 1913 y Fp(\002)d Fo(l)655 1920 y Fl(2)688 1913 y Fu(+)727 1891 y Fn(i)727 1926 y Fl(1)805 1913 y Fo(k)i Fu(+)e(1)p Fp(g)p Fo(;)107 b Fu(0)14 b Fp(\024)g Fo(i)g(<)f(l)1255 1920 y Fl(2)1288 1913 y Fp(\000)e Fo(k)r(;)58 b Fu(0)13 b Fo(<)h(k)h(<)f(l) 1638 1920 y Fl(2)1671 1913 y Fp(\000)d Fo(l)1736 1920 y Fn(c)1764 1913 y Fo(;)143 1997 y Fp(f)p Fo(l)183 2004 y Fl(1)216 1997 y Fp(\002)g Fo(l)281 2004 y Fl(2)315 1997 y Fu(+)354 1976 y Fn(i)354 2009 y Fl(1)387 1997 y Fo(k)r Fp(g)i(!)h(f)p Fo(l)557 2004 y Fl(1)590 1997 y Fp(\002)d Fo(l)655 2004 y Fl(2)688 1997 y Fu(+)727 1976 y Fn(i)p Fj(\000)p Fl(1)727 2010 y(1)805 1997 y Fo(k)i Fu(+)e(1)p Fp(g)p Fo(;)107 b Fu(0)14 b Fo(<)g(i)g Fp(\024)f Fo(l)1255 2004 y Fl(2)1288 1997 y Fp(\000)e Fo(k)r(;)58 b Fu(0)13 b Fo(<)h(k)h(<)f(l)1638 2004 y Fl(2)1671 1997 y Fp(\000)d Fo(l)1736 2004 y Fn(c)1764 1997 y Fo(;)143 2080 y Fp(f)p Fo(l)183 2087 y Fl(1)216 2080 y Fp(\002)g Fo(l)281 2087 y Fl(2)315 2080 y Fu(+)354 2060 y Fn(i)354 2093 y Fl(2)387 2080 y Fo(l)402 2087 y Fl(1)435 2080 y Fp(\000)g Fo(l)500 2087 y Fn(c)520 2080 y Fp(g)j(!)f Fo(\031)r Fu(\()p Fo(l)686 2087 y Fl(1)720 2080 y Fp(\002)e Fu(\()p Fo(l)804 2087 y Fl(2)838 2080 y Fu(+)f(1\))p Fo(;)e Fh(pred)g Fo(E)s Fu(\()p Fo(l)1133 2087 y Fl(1)1166 2080 y Fp(\002)j Fu(\()p Fo(l)1250 2087 y Fl(2)1284 2080 y Fu(+)g(1\))g(+)g(1\)\))p Fo(;)59 b Fu(0)13 b Fp(\024)h Fo(i)g Fp(\024)f Fo(l)1764 2087 y Fn(c)1793 2080 y Fo(;)143 2164 y Fp(f)p Fo(l)183 2171 y Fl(1)216 2164 y Fp(\002)e Fo(l)281 2171 y Fl(2)315 2164 y Fu(+)354 2143 y Fn(i)354 2176 y Fl(1)387 2164 y Fo(l)402 2171 y Fl(2)435 2164 y Fp(\000)g Fo(l)500 2171 y Fn(c)520 2164 y Fp(g)j(!)f Fo(\031)r Fu(\(\()p Fo(l)705 2171 y Fl(1)740 2164 y Fu(+)d(1\))i Fp(\002)f Fo(l)910 2171 y Fl(2)932 2164 y Fo(;)d Fh(pred)f Fo(E)s Fu(\(\()p Fo(l)1152 2171 y Fl(1)1186 2164 y Fu(+)k(1\))g Fp(\002)g Fo(l)1356 2171 y Fl(2)1389 2164 y Fu(+)g(1\)\))p Fo(;)59 b Fu(0)13 b Fp(\024)h Fo(i)g Fp(\024)f Fo(l)1764 2171 y Fn(c)1793 2164 y Fo(;)143 2241 y(\031)r Fu(\()p Fo(l)207 2248 y Fl(1)241 2241 y Fp(\002)e Fo(l)306 2248 y Fl(2)328 2241 y Fo(;)d Fh(pred)g Fo(E)s Fu(\()p Fo(l)530 2248 y Fl(1)563 2241 y Fp(\002)j Fo(l)628 2248 y Fl(2)661 2241 y Fu(+)g(1\)\))j Fp($)g(f)p Fo(\021)r Fp(g)8 b Fo(;)107 b(\021)16 b Fp(2)e Fo(l)1161 2248 y Fl(1)1194 2241 y Fp(\002)d Fo(l)1259 2248 y Fl(2)1292 2241 y Fp(\010)1331 2248 y Fl(1)1364 2241 y Fu(1)d Fo(;)143 2318 y(\031)r Fu(\()p Fo(l)207 2325 y Fl(1)241 2318 y Fp(\002)j Fo(l)306 2325 y Fl(2)328 2318 y Fo(;)d Fh(pred)g Fo(E)s Fu(\()p Fo(l)530 2325 y Fl(1)563 2318 y Fp(\002)j Fo(l)628 2325 y Fl(2)661 2318 y Fu(+)g(1\)\))j Fp($)g(f)p Fo(\021)r Fp(g)8 b Fo(;)107 b(\021)16 b Fp(2)e Fo(l)1161 2325 y Fl(1)1194 2318 y Fp(\002)d Fo(l)1259 2325 y Fl(2)1292 2318 y Fp(\010)1331 2325 y Fl(2)1364 2318 y Fu(1)d Fo(:)0 2430 y Fh(The)18 b(sym)o(b)q(ol)f Fp($)i Fh(means)e(that)h(b)q(oth)h(arro)o(ws)d Fp(!)i Fh(and)g Fp( )g Fh(are)g(presen)o(t.)26 b(The)18 b(ab)q(o)o(v)o(e)g(list)g(should)f(b)q(e)0 2490 y(completed)f(with)g(all)g(the)h(isometric)e(arro)o(ws)f(\(obtained)i(b) o(y)g(applying)g(the)g(same)g(isometry)f(to)i(b)q(oth)0 2550 y(ends)f(of)g(an)h(arro)o(w\).)954 2670 y Fs(22)p eop %%Page: 23 23 23 22 bop 60 200 a Fu(The)16 b(only)g(lo)q(ops)g(in)h(the)f(graph)g Fo(G)725 182 y Fl(+)758 200 y Fu(\()p Fo(\033)r Fu(\))h(are)192 320 y Fo(\031)r Fu(\()p Fo(l)256 327 y Fl(1)289 320 y Fp(\002)11 b Fo(l)354 327 y Fl(2)376 320 y Fo(;)d Fu(pred)g Fo(E)s Fu(\()p Fo(l)578 327 y Fl(1)611 320 y Fp(\002)j Fo(l)676 327 y Fl(2)709 320 y Fu(+)g(1\)\))k Fp($)e(f)p Fo(\021)r Fp(g)8 b Fo(;)108 b(\021)15 b Fp(2)f Fo(l)1209 327 y Fl(1)1242 320 y Fp(\002)d Fo(l)1307 327 y Fl(2)1340 320 y Fp(\010)1379 327 y Fl(1)1413 320 y Fu(1)g Fp([)g Fo(l)1508 327 y Fl(1)1541 320 y Fp(\002)g Fo(l)1606 327 y Fl(2)1639 320 y Fp(\010)1678 327 y Fl(2)1711 320 y Fu(1)d Fo(:)0 440 y Fu(An)o(y)19 b(other)f(arro)o(w)g Fo(\031)412 447 y Fl(1)452 440 y Fp(!)g Fo(\031)548 447 y Fl(2)589 440 y Fu(of)h Fo(G)687 422 y Fl(+)721 440 y Fu(\()p Fo(\033)r Fu(\))g(satis\014es)p 994 413 27 2 v 18 w Fo(a)q Fu(\()p Fo(\031)1068 447 y Fl(1)1091 440 y Fu(\))f Fo(<)f(a)p 1184 448 V 1 w Fu(\()p Fo(\031)1258 447 y Fl(2)1281 440 y Fu(\).)29 b(As)19 b(a)g(consequence)f(a)h (path)f(in)0 500 y Fo(G)39 482 y Fl(+)72 500 y Fu(\()p Fo(\033)r Fu(\))g(starting)e(at)h Fp(f)p Fo(\033)r Fp(g)f Fu(with)h(no)f(lo)q(op)g (ends)g(in)g Fp(f)p Fu(+1)p 990 512 64 2 v Fp(g)p Fu(.)60 590 y(W)l(e)23 b(are)g(done)g(with)h(the)f(standard)f(con\014gurations.)40 b(W)l(e)24 b(ha)o(v)o(e)f(to)g(examine)g(the)h(remaining)0 650 y(con\014gurations)c(of)i(the)g(principal)e(b)q(oundary)g(of)i(the)g (cycle)h Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 1253 662 V Fo(;)8 b Fu(+1)p 1339 662 V -20 x Fn(c)1423 650 y Fu(\))22 b(i.e.)38 b(the)22 b(principal)e(non)0 710 y(standard)15 b(con\014gurations.)0 800 y Fw(The)g(principal)h(non)e(standard)f(con\014gurations.)28 b Fu(W)l(e)13 b(no)o(w)f(do)g(the)h(same)f(w)o(ork)g(for)g(the)h(con\014g-)0 865 y(urations)g(in)255 852 y Fk(f)248 865 y Fo(M)296 872 y Fn(n)321 877 y Fg(c)350 865 y Fp(n)7 b Fo(S)413 872 y Fn(n)438 877 y Fg(c)458 865 y Fu(.)22 b(The)14 b(relev)m(an)o(t)h(cycles)f(are)h(the)g (cycles)g(around)e(the)h(rectangles)g Fo(l)1724 872 y Fn(c)1752 865 y Fp(\002)7 b Fu(\()p Fo(l)1832 872 y Fn(c)1860 865 y Fu(+)g(2\))0 932 y(and)16 b(around)f(the)i(con\014gurations)d(of)736 920 y Fk(f)730 932 y Fo(M)778 939 y Fn(n)803 944 y Fg(c)821 939 y Fl(+1)885 932 y Fp(n)d Fo(S)952 939 y Fn(n)977 944 y Fg(c)995 939 y Fl(+1)1048 932 y Fu(.)0 1023 y Fw(Theorem)21 b(6.25.)58 b Fh(The)20 b(cycle)g Fo(\031)r Fu(\()p Fo(l)720 1030 y Fn(c)754 1023 y Fp(\002)13 b Fu(\()p Fo(l)840 1030 y Fn(c)873 1023 y Fu(+)g(2\))p Fo(;)8 b Fh(pred)g Fo(E)s Fu(\()p Fo(n)1186 1030 y Fn(c)1206 1023 y Fu(\)\))21 b Fh(is)e(the)h(greatest)f(cycle)h(con)o (taining)0 1082 y Fo(l)15 1089 y Fn(c)46 1082 y Fp(\002)11 b Fu(\()p Fo(l)130 1089 y Fn(c)162 1082 y Fu(+)f(2\))17 b Fh(included)f(in)g Fp(f\000)p Fu(1)p 555 1095 V -1 w Fo(;)8 b Fu(+1)p 640 1095 V Fp(g)729 1064 y Fn(c)749 1082 y Fh(.)22 b(Moreo)o(v)o(er,)448 1203 y Fo(a)p 448 1211 27 2 v Fu(\()p Fo(\031)r Fu(\()p Fo(l)557 1210 y Fn(c)589 1203 y Fp(\002)11 b Fu(\()p Fo(l)673 1210 y Fn(c)705 1203 y Fu(+)f(2\))p Fo(;)e Fh(pred)g Fo(E)s Fu(\()p Fo(n)1015 1210 y Fn(c)1035 1203 y Fu(\)\)\))24 b(=)d Fo(l)1191 1210 y Fn(c)1212 1203 y Fu(\()p Fo(l)1246 1210 y Fn(c)1277 1203 y Fu(+)11 b(1\))g(+)g(2)d Fo(;)171 1280 y Fp(8)p Fo(k)14 b Fp(2)g(f)p Fu(1)8 b Fp(\001)g(\001)g(\001)g Fo(l)426 1287 y Fn(c)458 1280 y Fp(\000)i Fu(2)p Fp(g)100 b Fo(M)5 b Fu(\()p Fo(l)744 1287 y Fn(c)776 1280 y Fp(\002)11 b Fu(\()p Fo(l)860 1287 y Fn(c)892 1280 y Fu(+)f(2\))p Fo(;)e Fp(\000)p Fo(k)r Fu(\))23 b Fp(\032)e Fo(\031)r Fu(\()p Fo(l)1240 1287 y Fn(c)1272 1280 y Fp(\002)11 b Fu(\()p Fo(l)1356 1287 y Fn(c)1388 1280 y Fu(+)f(2\))p Fo(;)e Fh(pred)g Fo(E)s Fu(\()p Fo(n)1698 1287 y Fn(c)1718 1280 y Fu(\)\))g Fo(;)p 448 1330 V 448 1357 a(a)p Fu(\()p Fo(\031)r Fu(\()p Fo(l)557 1364 y Fn(c)589 1357 y Fp(\002)j Fu(\()p Fo(l)673 1364 y Fn(c)705 1357 y Fu(+)f(2\))p Fo(;)e Fh(pred)g Fo(E)s Fu(\()p Fo(n)1015 1364 y Fn(c)1035 1357 y Fu(\)\)\))24 b(=)d Fo(l)1191 1364 y Fn(c)1212 1357 y Fu(\()p Fo(l)1246 1364 y Fn(c)1277 1357 y Fu(+)11 b(2\))d Fo(:)0 1480 y Fh(The)13 b(b)q(ottom)g(of)g(this)f(cycle)i(is)e Fp(f)p Fo(l)623 1487 y Fn(c)647 1480 y Fp(\002)t Fu(\()p Fo(l)724 1487 y Fn(c)748 1480 y Fu(+)t(2\))p Fp(g)p Fh(;)i(its)f(principal)e(b)q(oundary)g(is)i Fo(M)5 b Fu(\()p Fo(l)1512 1487 y Fn(c)1537 1480 y Fp(\002)t Fu(\()p Fo(l)1614 1487 y Fn(c)1638 1480 y Fu(+)t(2\))p Fo(;)j Fp(\000)p Fu(\()p Fo(l)1820 1487 y Fn(c)1844 1480 y Fp(\000)t Fu(1\)\))0 1540 y Fh(and)16 b(th)o(us)f(con)o(tains)408 1527 y Fk(f)402 1540 y Fo(M)450 1547 y Fn(n)475 1552 y Fg(c)510 1540 y Fu(=)p 562 1496 364 2 v 13 w Fo(l)577 1547 y Fn(c)609 1540 y Fp(\002)10 b Fu(\()p Fo(l)692 1547 y Fn(c)724 1540 y Fu(+)h(1\))g Fp(\010)868 1547 y Fl(2)901 1540 y Fu(1)926 1505 y Fl(12)968 1540 y Fh(.)0 1630 y Fe(R)m(emark.)27 b Fu(Ob)o(viously)l(,)14 b(similar)h(statemen)o(ts)h(are)g(true)h(for)f(the)h(rectangle)f(\()p Fo(l)1464 1637 y Fn(c)1495 1630 y Fu(+)11 b(2\))h Fp(\002)e Fo(l)1665 1637 y Fn(c)1686 1630 y Fu(.)0 1720 y Fe(Pr)m(o)m(of.)28 b Fu(W)l(e)17 b(c)o(hec)o(k)g(that)g(the)g(rectangle)g Fo(l)801 1727 y Fn(c)833 1720 y Fp(\002)11 b Fu(\()p Fo(l)917 1727 y Fn(c)949 1720 y Fu(+)g(2\))18 b(and)e(the)h(cycle)h Fo(\031)r Fu(\()p Fo(l)1432 1727 y Fn(c)1464 1720 y Fp(\002)11 b Fu(\()p Fo(l)1548 1727 y Fn(c)1580 1720 y Fu(+)g(2\))p Fo(;)d Fu(pred)g Fo(E)s Fu(\()p Fo(n)1891 1727 y Fn(c)1911 1720 y Fu(\)\))0 1780 y(satisfy)23 b(the)g(h)o(yp)q(othesis)e(of)i(theorem)f(5)p Fo(:)p Fu(5.)41 b(Let)23 b Fo(x)1010 1787 y Fl(1)1033 1780 y Fo(;)8 b Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1180 1787 y Fn(r)1225 1780 y Fu(b)q(e)23 b(a)g(sequence)g(of)g(sites)f(suc)o(h)g(that)0 1840 y Fo(F)7 b Fu(\()p Fo(x)86 1847 y Fl(1)109 1840 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)256 1847 y Fn(j)278 1840 y Fu(\)\()p Fo(l)331 1847 y Fn(c)358 1840 y Fp(\002)f Fu(\()p Fo(l)438 1847 y Fn(c)466 1840 y Fu(+)g(2\)\))15 b(is)f(in)g Fo(\031)r Fu(\()p Fo(l)758 1847 y Fn(c)785 1840 y Fp(\002)7 b Fu(\()p Fo(l)865 1847 y Fn(c)893 1840 y Fu(+)g(2\))p Fo(;)h Fu(pred)f Fo(E)s Fu(\()p Fo(n)1199 1847 y Fn(c)1219 1840 y Fu(\)\))16 b(for)e Fo(j)j Fu(in)d Fp(f)p Fu(1)8 b Fp(\001)g(\001)g(\001)g Fo(r)q Fp(g)16 b Fu(\(i.e.)21 b(these)15 b(con-)0 1900 y(\014gurations)e(ha)o (v)o(e)h(an)h(energy)f(less)h(or)f(equal)h(than)f(pred)8 b Fo(E)s Fu(\()p Fo(n)1172 1907 y Fn(c)1192 1900 y Fu(\)\).)22 b(W)l(e)15 b(put)g Fo(\021)1465 1907 y Fn(j)1499 1900 y Fu(=)f Fo(F)7 b Fu(\()p Fo(x)1638 1907 y Fl(1)1661 1900 y Fo(;)h Fp(\001)g(\001)g (\001)17 b Fo(;)8 b(x)1808 1907 y Fn(j)1830 1900 y Fu(\)\()p Fo(l)1883 1907 y Fn(c)1911 1900 y Fp(\002)0 1959 y Fu(\()p Fo(l)34 1966 y Fn(c)66 1959 y Fu(+)i(2\)\))18 b(for)e Fo(j)j Fu(in)d Fp(f)p Fu(0)8 b Fp(\001)g(\001)g(\001)h Fo(r)q Fp(g)p Fu(.)0 2019 y Fp(\017)50 b Fu(First)14 b(case:)21 b(all)15 b(the)g(sites)g(are)g(outside)g Fo(l)849 2026 y Fn(c)877 2019 y Fp(\002)9 b Fu(\()p Fo(l)959 2026 y Fn(c)988 2019 y Fu(+)f(2\).)22 b(W)l(e)15 b(ha)o(v)o(e)g(that)g Fo(l)1434 2026 y Fn(c)1463 2019 y Fp(\002)8 b Fu(\()p Fo(l)1544 2026 y Fn(c)1573 2019 y Fu(+)h(2\))14 b Fp(\032)f Fo(\021)1756 2026 y Fn(j)1792 2019 y Fu(whence)0 2079 y Fo(E)s Fu(\()p Fo(\021)84 2086 y Fn(j)105 2079 y Fu(\))22 b Fp(\025)g Fo(E)s Fu(\()p Fo(l)281 2086 y Fn(c)312 2079 y Fp(\002)11 b Fu(\()p Fo(l)396 2086 y Fn(c)428 2079 y Fu(+)g(2\))p Fo(;)d(a)p Fu(\()p Fo(\021)614 2086 y Fn(j)635 2079 y Fu(\))k Fp(\000)f Fo(l)731 2086 y Fn(c)751 2079 y Fu(\()p Fo(l)785 2086 y Fn(c)816 2079 y Fu(+)g(2\)\))18 b(and)200 2199 y(pred)7 b Fo(E)s Fu(\()p Fo(n)394 2206 y Fn(c)414 2199 y Fu(\))31 b Fp(\025)44 b Fu(max)533 2231 y Fl(0)p Fj(\024)p Fn(j)r Fj(\024)p Fn(r)662 2199 y Fo(E)s Fu(\()p Fo(\021)746 2206 y Fn(j)766 2199 y Fu(\))31 b Fp(\025)44 b Fu(max)885 2231 y Fl(0)p Fj(\024)p Fn(j)r Fj(\024)p Fn(r)1014 2199 y Fo(E)s Fu(\()p Fo(l)1088 2206 y Fn(c)1119 2199 y Fp(\002)11 b Fu(\()p Fo(l)1203 2206 y Fn(c)1234 2199 y Fu(+)g(2\))p Fo(;)d(a)p Fu(\()p Fo(\021)1420 2206 y Fn(j)1442 2199 y Fu(\))k Fp(\000)e Fo(l)1537 2206 y Fn(c)1558 2199 y Fu(\()p Fo(l)1592 2206 y Fn(c)1623 2199 y Fu(+)h(2\)\))p Fo(:)0 2346 y Fu(Since)16 b(the)g(sequence)h Fo(\021)447 2353 y Fl(0)469 2346 y Fo(;)8 b Fp(\001)g(\001)g(\001)16 b Fo(;)8 b(\021)612 2353 y Fn(r)651 2346 y Fu(is)16 b(a)g(sequence)g(of)g (spin)g(\015ips)f(w)o(e)h(ha)o(v)o(e)g Fp(j)p Fo(a)p Fu(\()p Fo(\021)1487 2353 y Fn(j)r Fl(+1)1558 2346 y Fu(\))11 b Fp(\000)f Fo(a)p Fu(\()p Fo(\021)1707 2353 y Fn(j)1729 2346 y Fu(\))p Fp(j)k(\024)f Fu(1)k(and)0 2406 y(\()p Fo(a)p Fu(\()p Fo(\021)89 2413 y Fn(j)111 2406 y Fu(\))p Fo(;)g Fu(0)c Fp(\024)h Fo(j)i Fp(\024)e Fo(r)q Fu(\))k(tak)o(es)e(all)g(the)h(v)m(alues)f(b)q(et)o(w)o(een) h Fo(l)1040 2413 y Fn(c)1060 2406 y Fu(\()p Fo(l)1094 2413 y Fn(c)1126 2406 y Fu(+)10 b(2\))17 b(and)f Fo(a)p Fu(\()p Fo(\021)1403 2413 y Fn(r)1426 2406 y Fu(\).)22 b(Henceforth)14 2526 y(max)0 2558 y Fl(0)p Fj(\024)p Fn(j)r Fj(\024)p Fn(r)128 2526 y Fo(E)s Fu(\()p Fo(l)202 2533 y Fn(c)225 2526 y Fp(\002)r Fu(\()p Fo(l)300 2533 y Fn(c)323 2526 y Fu(+)r(2\))p Fo(;)8 b(a)p Fu(\()p Fo(\021)500 2533 y Fn(j)522 2526 y Fu(\))r Fp(\000)r Fo(l)599 2533 y Fn(c)620 2526 y Fu(\()p Fo(l)654 2533 y Fn(c)677 2526 y Fu(+)r(2\)\))32 b Fp(\025)e Fu(max)p Fp(f)8 b Fo(E)s Fu(\()p Fo(l)1082 2533 y Fn(c)1104 2526 y Fp(\002)r Fu(\()p Fo(l)1179 2533 y Fn(c)1203 2526 y Fu(+)r(2\))p Fo(;)g(k)r Fu(\))14 b(:)f(0)h Fp(\024)g Fo(k)h Fp(\024)f Fo(a)p Fu(\()p Fo(\021)1654 2533 y Fn(r)1676 2526 y Fu(\))r Fp(\000)r Fo(l)1753 2533 y Fn(c)1774 2526 y Fu(\()p Fo(l)1808 2533 y Fn(c)1831 2526 y Fu(+)r(2\))8 b Fp(g)954 2670 y Fs(23)p eop %%Page: 24 24 24 23 bop 0 200 a Fu(and)16 b(the)h(area)f(of)g Fo(\021)372 207 y Fn(r)411 200 y Fu(m)o(ust)f(satisfy)233 309 y(max)o Fp(f)8 b Fo(E)s Fu(\()p Fo(l)432 316 y Fn(c)464 309 y Fp(\002)j Fu(\()p Fo(l)548 316 y Fn(c)579 309 y Fu(+)g(2\))p Fo(;)d(k)r Fu(\))14 b(:)g(0)g Fp(\024)f Fo(k)j Fp(\024)d Fo(a)p Fu(\()p Fo(\021)1039 316 y Fn(r)1062 309 y Fu(\))e Fp(\000)g Fo(l)1157 316 y Fn(c)1177 309 y Fu(\()p Fo(l)1211 316 y Fn(c)1243 309 y Fu(+)f(2\))e Fp(g)31 b(\024)g Fu(pred)7 b Fo(E)s Fu(\()p Fo(n)1664 316 y Fn(c)1684 309 y Fu(\))p Fo(:)0 418 y Fu(By)21 b(theorem)f(6)p Fo(:)p Fu(9,)h(w)o(e)g(ha)o(v)o(e)f Fo(E)s Fu(\()p Fo(l)650 425 y Fn(c)683 418 y Fp(\002)14 b Fu(\()p Fo(l)770 425 y Fn(c)804 418 y Fu(+)g(2\))p Fo(;)8 b Fu(1\))21 b(=)f Fo(E)s Fu(\()p Fo(l)1121 425 y Fn(c)1155 418 y Fp(\002)14 b Fu(\()p Fo(l)1242 425 y Fn(c)1276 418 y Fu(+)g(2\))g(+)f(1\))21 b(whence)g Fo(E)s Fu(\()p Fo(l)1757 425 y Fn(c)1790 418 y Fp(\002)14 b Fu(\()p Fo(l)1877 425 y Fn(c)1911 418 y Fu(+)0 477 y(2\))p Fo(;)8 b Fu(1\))j Fp(\000)e Fo(E)s Fu(\()p Fo(n)258 484 y Fn(c)278 477 y Fu(\))14 b(=)g(2)c Fp(\000)f Fo(hl)491 484 y Fn(c)527 477 y Fu(whic)o(h)15 b(is)h(strictly)g(p)q(ositiv)o(e)f(since)h Fo(l)1209 484 y Fn(c)1243 477 y Fu(=)d Fp(b)p Fu(2)p Fo(=h)p Fp(c)p Fu(.)22 b(Th)o(us)14 b Fo(a)p Fu(\()p Fo(\021)1649 484 y Fn(r)1672 477 y Fu(\))g(=)g Fo(l)1773 484 y Fn(c)1793 477 y Fu(\()p Fo(l)1827 484 y Fn(c)1857 477 y Fu(+)c(2\))0 537 y(and)16 b Fo(\021)122 544 y Fn(r)157 537 y Fu(=)e Fo(l)225 544 y Fn(c)256 537 y Fp(\002)d Fu(\()p Fo(l)340 544 y Fn(c)372 537 y Fu(+)g(2\).)0 597 y Fp(\017)50 b Fu(Second)16 b(case:)21 b(all)c(the)f(sites)h(are)f(inside)f Fo(l)874 604 y Fn(c)905 597 y Fp(\002)c Fu(\()p Fo(l)989 604 y Fn(c)1021 597 y Fu(+)g(2\).)22 b(No)o(w)16 b Fo(\021)1289 604 y Fn(j)1324 597 y Fp(\032)d Fo(l)1391 604 y Fn(c)1423 597 y Fp(\002)d Fu(\()p Fo(l)1506 604 y Fn(c)1538 597 y Fu(+)h(2\))17 b(so)f(that)206 706 y(pred)8 b Fo(E)s Fu(\()p Fo(n)401 713 y Fn(c)421 706 y Fu(\))31 b Fp(\025)44 b Fu(max)540 738 y Fl(0)p Fj(\024)p Fn(j)r Fj(\024)p Fn(r)668 706 y Fo(E)s Fu(\()p Fo(\021)752 713 y Fn(j)773 706 y Fu(\))31 b Fp(\025)44 b Fu(max)892 738 y Fl(0)p Fj(\024)p Fn(j)r Fj(\024)p Fn(r)1021 706 y Fo(E)s Fu(\()p Fo(l)1095 713 y Fn(c)1126 706 y Fp(\002)11 b Fu(\()p Fo(l)1210 713 y Fn(c)1241 706 y Fu(+)g(2\))p Fo(;)d(a)p Fu(\()p Fo(\021)1427 713 y Fn(j)1449 706 y Fu(\))j Fp(\000)g Fo(l)1544 713 y Fn(c)1564 706 y Fu(\()p Fo(l)1598 713 y Fn(c)1630 706 y Fu(+)g(2\)\))471 807 y Fp(\025)30 b Fu(max)p Fp(f)8 b Fo(E)s Fu(\()p Fo(l)740 814 y Fn(c)771 807 y Fp(\002)j Fu(\()p Fo(l)855 814 y Fn(c)887 807 y Fu(+)f(2\))p Fo(;)e(k)r Fu(\))15 b(:)e Fo(a)p Fu(\()p Fo(\021)1161 814 y Fn(r)1184 807 y Fu(\))e Fp(\000)g Fo(l)1279 814 y Fn(c)1299 807 y Fu(\()p Fo(l)1333 814 y Fn(c)1365 807 y Fu(+)f(2\))15 b Fp(\024)e Fo(k)j Fp(\024)d Fu(0)8 b Fp(g)p Fo(:)0 919 y Fu(Theorem)13 b(6)p Fo(:)p Fu(8)g(sho)o(ws)g(that)h(for)g(0)f Fp(\024)h Fo(k)h Fp(\024)f Fo(l)805 926 y Fn(c)825 919 y Fu(,)g Fo(E)s Fu(\()p Fo(l)927 926 y Fn(c)953 919 y Fp(\002)6 b Fu(\()p Fo(l)1032 926 y Fn(c)1058 919 y Fu(+)g(2\))p Fo(;)i Fp(\000)p Fo(k)r Fu(\))22 b(=)g Fo(E)s Fu(\()p Fo(l)1412 926 y Fn(c)1438 919 y Fp(\002)6 b Fu(\()p Fo(l)1517 926 y Fn(c)1542 919 y Fu(+)g(1\))g Fp(\010)1676 926 y Fl(2)1704 919 y Fu(\()p Fo(l)1738 926 y Fn(c)1764 919 y Fp(\000)g Fo(k)r Fu(\))g Fp(\002)g Fu(1\))0 978 y(so)21 b(that)g Fo(E)s Fu(\()p Fo(n)267 985 y Fn(c)288 978 y Fu(\))14 b Fp(\000)g Fo(E)s Fu(\()p Fo(l)448 985 y Fn(c)483 978 y Fp(\002)f Fu(\()p Fo(l)569 985 y Fn(c)604 978 y Fu(+)h(2\))p Fo(;)8 b Fp(\000)p Fo(k)r Fu(\))22 b(=)f Fo(h)p Fu(\()p Fo(l)954 985 y Fn(c)989 978 y Fp(\000)14 b Fo(k)h Fp(\000)f Fu(1\).)37 b(F)l(or)20 b(this)g(quan)o(tit)o(y)h(to)h(b)q(e)f(p)q(ositiv)o(e,)0 1038 y(w)o(e)h(m)o(ust)f(ha)o(v)o(e)g Fo(k)k(<)d(l)455 1045 y Fn(c)490 1038 y Fp(\000)15 b Fu(1)22 b(whence)g Fo(a)p Fu(\()p Fo(\021)841 1045 y Fn(r)863 1038 y Fu(\))i Fp(\025)e Fo(l)982 1045 y Fn(c)1003 1038 y Fu(\()p Fo(l)1037 1045 y Fn(c)1072 1038 y Fu(+)14 b(2\))h Fp(\000)g Fu(\()p Fo(l)1272 1045 y Fn(c)1307 1038 y Fp(\000)g Fu(2\).)38 b(In)22 b(addition)f(for)h(an)o(y)g Fo(k)r Fu(,)0 1098 y(0)14 b Fp(\024)f Fo(k)j Fp(\024)d Fo(l)200 1105 y Fn(c)224 1098 y Fp(\000)t Fu(2,)g Fo(E)s Fu(\()p Fo(l)393 1105 y Fn(c)417 1098 y Fp(\002)t Fu(\()p Fo(l)494 1105 y Fn(c)518 1098 y Fu(+)t(2\))p Fo(;)8 b Fp(\000)p Fo(k)r Fu(\))t Fp(\000)t Fo(E)s Fu(\()p Fo(l)834 1105 y Fn(c)857 1098 y Fp(\002)t Fu(\()p Fo(l)934 1105 y Fn(c)958 1098 y Fu(+)t(2\)\))14 b(=)g Fo(hk)h(>)e Fu(0)g(so)g(that)g Fo(E)s Fu(\()p Fo(\021)1537 1105 y Fn(r)1559 1098 y Fu(\))h Fo(>)g(E)s Fu(\()p Fo(l)1719 1105 y Fn(c)1743 1098 y Fp(\002)t Fu(\()p Fo(l)1820 1105 y Fn(c)1844 1098 y Fu(+)t(2\)\))0 1158 y(whenev)o(er)i Fo(\021)244 1165 y Fn(r)279 1158 y Fp(6)p Fu(=)e Fo(l)347 1165 y Fn(c)378 1158 y Fp(\002)d Fu(\()p Fo(l)462 1165 y Fn(c)494 1158 y Fu(+)f(2\).)0 1218 y(W)l(e)22 b(ha)o(v)o(e)f(pro)o(v)o(ed)f(that)i Fo(l)506 1225 y Fn(c)541 1218 y Fp(\002)14 b Fu(\()p Fo(l)628 1225 y Fn(c)663 1218 y Fu(+)h(2\))22 b(and)f(the)h(cycle)g Fo(\031)r Fu(\()p Fo(l)1168 1225 y Fn(c)1203 1218 y Fp(\002)14 b Fu(\()p Fo(l)1290 1225 y Fn(c)1325 1218 y Fu(+)h(2\))p Fo(;)8 b Fu(pred)f Fo(E)s Fu(\()p Fo(n)1639 1225 y Fn(c)1660 1218 y Fu(\)\))22 b(satisfy)g(the)0 1277 y(h)o(yp)q(othesis)17 b(of)h(theorem)f(5)p Fo(:)p Fu(5.)25 b(It)18 b(follo)o(ws)f(that)h(the)g(b)q(ottom)g(of)f(the)h(cycle)h(is)e Fp(f)p Fo(l)1556 1284 y Fn(c)1588 1277 y Fp(\002)12 b Fu(\()p Fo(l)1673 1284 y Fn(c)1705 1277 y Fu(+)g(2\))p Fp(g)p Fu(.)25 b(The)0 1337 y(inequalities)14 b(on)p 328 1310 27 2 v 15 w Fo(a)h Fu(and)g Fo(a)p 465 1345 V 15 w Fu(ha)o(v)o(e)g(also)f(b)q(een)h(pro)o (v)o(ed.)20 b(That)15 b Fo(M)5 b Fu(\()p Fo(l)1225 1344 y Fn(c)1254 1337 y Fp(\002)j Fu(\()p Fo(l)1335 1344 y Fn(c)1364 1337 y Fu(+)g(2\))p Fo(;)g Fp(\000)p Fo(k)r Fu(\))15 b(is)f(included)g(in)h(the)0 1397 y(cycle)d(for)g(0)h Fp(\024)h Fo(k)h Fp(\024)f Fo(l)390 1404 y Fn(c)412 1397 y Fp(\000)q Fu(2)d(is)h(ob)o(vious:)18 b(eac)o(h)11 b(con\014guration)f(of)i(this)f(set)h(comm)o(unicates)e(with)i Fo(l)1794 1404 y Fn(c)1815 1397 y Fp(\002)q Fu(\()p Fo(l)1889 1404 y Fn(c)1911 1397 y Fu(+)0 1457 y(2\))i(under)f(the)h(lev)o(el)f(pred)8 b Fo(E)s Fu(\()p Fo(n)585 1464 y Fn(c)605 1457 y Fu(\))14 b(\(theorem)g(6)p Fo(:)p Fu(8)f(sho)o(ws)f(that)j(all)e(these)h(con\014gurations)d(are)j (obtained)0 1516 y(b)o(y)k(deleting)g(successiv)o(ely)f Fo(k)i Fu(corner)f(squares)f(from)g Fo(l)1045 1523 y Fn(c)1077 1516 y Fp(\002)12 b Fu(\()p Fo(l)1162 1523 y Fn(c)1195 1516 y Fu(+)g(2\)\).)27 b(Finally)17 b(a)h(con\014guration)f Fo(\030)j Fu(of)0 1576 y(the)d(principal)e(b)q(oundary)g(of)i(this)f(cycle)i(is)e(of)h(energy)f Fo(E)s Fu(\()p Fo(\030)r Fu(\))f(=)e Fo(E)s Fu(\()p Fo(n)1320 1583 y Fn(c)1341 1576 y Fu(\))k(so)f(that)h(its)g(area)f(is)g(equal)h(to)0 1636 y Fo(n)30 1643 y Fn(c)50 1636 y Fu(,)f(and)g(its)g(p)q(erimeter)g(to)g Fo(p)p Fu(\()p Fo(n)606 1643 y Fn(c)627 1636 y Fu(\).)22 b(Let)17 b Fo(\021)h Fu(b)q(e)f(a)f(con\014guration)f(of)h(the)h(cycle)g(suc)o(h)e (that)h Fo(q)r Fu(\()p Fo(\021)r(;)8 b(\030)r Fu(\))15 b Fo(>)e Fu(0.)0 1696 y(Necessarily)l(,)f(the)g(area)f(of)h Fo(\021)i Fu(is)d Fo(n)614 1703 y Fn(c)636 1696 y Fu(+)r(1.)19 b(Th)o(us)11 b Fo(\021)i Fu(is)e(a)h(con\014guration)e(of)i(minimal)e(area)h(of)h(the)g (cycle)h(and)0 1756 y(as)h(suc)o(h)f(it)i(is)f(a)g(minimal)e(con\014guration) h(of)i(the)f(cycle)h(for)f(the)h(inclusion)d(relation.)21 b(By)15 b(theorem)e(5)p Fo(:)p Fu(3)0 1815 y(it)k(is)g(included)f(in)h(the)g (rectangle)g Fo(l)675 1822 y Fn(c)707 1815 y Fp(\002)11 b Fu(\()p Fo(l)791 1822 y Fn(c)823 1815 y Fu(+)g(2\).)25 b(Th)o(us)15 b Fo(\030)20 b Fu(b)q(elongs)c(to)i Fo(M)5 b Fu(\()p Fo(l)1452 1822 y Fn(c)1485 1815 y Fp(\002)11 b Fu(\()p Fo(l)1569 1822 y Fn(c)1601 1815 y Fu(+)g(2\))p Fo(;)d Fp(\000)p Fu(\()p Fo(l)1790 1822 y Fn(c)1822 1815 y Fp(\000)k Fu(1\)\).)0 1875 y(Con)o(v)o(ersely)h(eac)o (h)h(con\014guration)e(of)i(this)g(set)h(comm)o(unicates)d(with)i Fo(M)5 b Fu(\()p Fo(l)1402 1882 y Fn(c)1430 1875 y Fp(\002)h Fu(\()p Fo(l)1509 1882 y Fn(c)1536 1875 y Fu(+)g(2\))p Fo(;)i Fp(\000)p Fu(\()p Fo(l)1720 1882 y Fn(c)1747 1875 y Fp(\000)e Fu(1\)\))15 b(and)0 1935 y(b)q(elongs)k(to)g(the)h(principal)d(b)q(oundary)h (of)i(the)f(cycle.)31 b(Finally)l(,)19 b(theorem)f(6)p Fo(:)p Fu(8)h(sho)o(ws)f(that)i(this)f(set)0 1999 y(con)o(tains)202 1987 y Fk(f)195 1999 y Fo(M)243 2006 y Fn(n)268 2011 y Fg(c)289 1999 y Fu(.)50 b Fc(\003)0 2094 y Fw(Corollary)18 b(6.26.)27 b Fh(Let)16 b Fo(\033)h Fh(b)q(elong)d(to)755 2082 y Fk(f)749 2094 y Fo(M)797 2101 y Fn(n)822 2106 y Fg(c)840 2101 y Fl(+1)901 2094 y Fp(n)8 b Fo(S)965 2101 y Fn(n)990 2106 y Fg(c)1007 2101 y Fl(+1)1060 2094 y Fh(.)21 b(Supp)q(ose)14 b Fo(l)1304 2101 y Fn(c)1332 2094 y Fp(\002)8 b Fu(\()p Fo(l)1413 2101 y Fn(c)1440 2094 y Fu(+)g(1\))14 b Fp(\032)f Fo(\033)k Fp(\032)c Fo(l)1709 2101 y Fn(c)1737 2094 y Fp(\002)8 b Fu(\()p Fo(l)1818 2101 y Fn(c)1845 2094 y Fu(+)g(2\))p Fh(.)0 2154 y(Let)20 b Fo(G)131 2136 y Fl(+)164 2154 y Fu(\()p Fo(\033)r Fu(\))h Fh(b)q(e)e(the)h(minimal)d (stable)i(subgraph)e(of)i Fo(G)1072 2136 y Fl(+)1125 2154 y Fh(con)o(taining)f Fo(\031)r Fu(\()p Fo(\033)o(;)8 b Fp(f\000)p Fu(1)p 1493 2166 64 2 v 1 w Fo(;)g Fu(+1)p 1580 2166 V -1 w Fp(g)1668 2136 y Fn(c)1689 2154 y Fu(\))p Fh(.)30 b(The)19 b(only)0 2214 y(arro)o(ws)c(of)h Fo(G)255 2196 y Fl(+)289 2214 y Fu(\()p Fo(\033)r Fu(\))h Fh(en)o(tering)f Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 616 2226 V Fo(;)8 b Fu(+1)p 702 2226 V -20 x Fn(c)786 2214 y Fu(\))17 b Fh(are)213 2332 y Fp(f)p Fo(\030)r Fp(g)d(!)f Fo(\031)r Fu(\()p Fo(l)428 2339 y Fn(c)460 2332 y Fp(\002)e Fu(\()p Fo(l)544 2339 y Fn(c)576 2332 y Fu(+)f(1\))p Fo(;)e Fp(f\000)p Fu(1)p 716 2344 V Fo(;)g Fu(+1)p 802 2344 V Fp(g)891 2312 y Fn(c)911 2332 y Fu(\))p Fo(;)59 b(\030)16 b Fp(2)1094 2320 y Fk(f)1088 2332 y Fo(M)1136 2339 y Fn(n)1161 2344 y Fg(c)1193 2332 y Fp(n)11 b Fo(S)1260 2339 y Fn(n)1285 2344 y Fg(c)1305 2332 y Fo(;)58 b(l)1392 2339 y Fn(c)1423 2332 y Fp(\002)11 b Fu(\()p Fo(l)1507 2339 y Fn(c)1539 2332 y Fu(+)f(1\))15 b Fp(\032)e Fo(\030)r(:)0 2441 y Fh(The)j(remaining)f(arro)o(ws)g(of)h Fo(G)590 2423 y Fl(+)624 2441 y Fu(\()p Fo(\033)r Fu(\))h Fh(are)358 2550 y Fo(\031)r Fu(\()p Fo(\033)o(;)8 b Fp(f\000)p Fu(1)p 481 2562 V 1 w Fo(;)g Fu(+1)p 568 2562 V Fp(g)657 2529 y Fn(c)677 2550 y Fu(\))14 b Fp($)g(f)p Fo(\021)r Fp(g)p Fo(;)57 b(\021)15 b Fp(2)f Fo(M)5 b Fu(\()p Fo(l)1096 2557 y Fn(c)1129 2550 y Fp(\002)10 b Fu(\()p Fo(l)1212 2557 y Fn(c)1244 2550 y Fu(+)h(2\))p Fo(;)d Fp(\000)p Fu(\()p Fo(l)1433 2557 y Fn(c)1465 2550 y Fp(\000)i Fu(1\)\))p Fo(:)954 2670 y Fs(24)p eop %%Page: 25 25 25 24 bop 0 200 a Fh(There)16 b(is)g(no)g(arro)o(w)f(in)h Fo(G)500 182 y Fl(+)534 200 y Fu(\()p Fo(\033)r Fu(\))h Fh(ending)f(at)h Fp(f)p Fu(+1)p 866 212 64 2 v -1 w Fp(g)p Fh(.)0 285 y Fe(R)m(emark.)27 b Fu(In)18 b(this)h(statemen)o(t,)g(the)g(rectangle)g Fo(l)939 292 y Fn(c)972 285 y Fp(\002)12 b Fu(\()p Fo(l)1057 292 y Fn(c)1090 285 y Fu(+)h(1\))19 b(should)e(b)q(e)j(understo)q(o)q(d)e(as)g(a)h(con\014g-) 0 345 y(uration)f(and)h(not)h(as)f(a)g(p)q(oly)o(omino.)29 b(That)19 b(is,)h Fo(l)957 352 y Fn(c)990 345 y Fp(\002)13 b Fu(\()p Fo(l)1076 352 y Fn(c)1109 345 y Fu(+)g(1\))19 b Fp(\032)f Fo(\030)k Fu(means)c(that)i(the)g(same)e(\(\014xed\))0 405 y(rectangle)e(is)g(included)g(in)g Fo(\033)j Fu(and)c Fo(\030)r Fu(.)0 490 y Fe(Pr)m(o)m(of.)28 b Fu(That)11 b(these)h(arro)o(ws)e(b)q(elong) h(to)h Fo(G)803 472 y Fl(+)836 490 y Fu(\()p Fo(\033)r Fu(\))h(is)e(a)h (straigh)o(tforw)o(ard)c(consequence)k(of)f(theorem)g(6)p Fo(:)p Fu(25)0 550 y(whic)o(h)j(implies)f(in)i(particular)e(that)i Fo(\031)r Fu(\()p Fo(\033)o(;)8 b Fp(f\000)p Fu(1)p 822 562 V 1 w Fo(;)g Fu(+1)p 909 562 V Fp(g)998 532 y Fn(c)1018 550 y Fu(\))22 b(=)g Fo(\031)r Fu(\()p Fo(l)1184 557 y Fn(c)1213 550 y Fp(\002)8 b Fu(\()p Fo(l)1294 557 y Fn(c)1321 550 y Fu(+)g(2\))p Fo(;)g Fp(f\000)p Fu(1)p 1459 562 V Fo(;)g Fu(+1)p 1545 562 V -1 w Fp(g)1633 532 y Fn(c)1654 550 y Fu(\).)21 b(W)l(e)16 b(ha)o(v)o(e)e(to)0 610 y(c)o(hec)o(k)g(that)h(there)g(are)f(no)g(other)h (arro)o(w.)20 b(Let)15 b Fo(\021)h Fu(b)q(elong)e(to)h(the)g(principal)e(b)q (oundary)h(of)g Fo(\031)r Fu(\()p Fo(l)1775 617 y Fn(c)1803 610 y Fp(\002)7 b Fu(\()p Fo(l)1883 617 y Fn(c)1911 610 y Fu(+)0 677 y(2\))p Fo(;)h Fu(pred)g Fo(E)s Fu(\()p Fo(n)261 684 y Fn(c)281 677 y Fu(\)\).)27 b(If)18 b Fo(\021)h Fu(is)f(in)572 665 y Fk(f)566 677 y Fo(M)614 684 y Fn(n)639 689 y Fg(c)660 677 y Fu(,)g(then)g(all)f(arro)o(ws)f(of)i Fo(G)h Fu(starting)e(at)h(the)g (cycle)g Fp(f)p Fo(\021)r Fp(g)g Fu(are)f(presen)o(t)0 745 y(in)f(the)h(ab)q(o)o(v)o(e)f(list)g(\(lemma)g(6)p Fo(:)p Fu(22\).)22 b(If)17 b Fo(\021)h Fu(is)e(not)h(in)988 732 y Fk(f)982 745 y Fo(M)1030 752 y Fn(n)1055 757 y Fg(c)1076 745 y Fu(,)f(w)o(e)h(claim)f (that)608 845 y Fp(f)p Fo(\021)r Fp(g)d(!)h Fo(\031)r Fu(\()p Fo(l)826 852 y Fn(c)857 845 y Fp(\002)d Fu(\()p Fo(l)941 852 y Fn(c)973 845 y Fu(+)g(2\))p Fo(;)d Fu(pred)f Fo(E)s Fu(\()p Fo(n)1283 852 y Fn(c)1304 845 y Fu(\)\))0 945 y(is)14 b(the)g(unique)f(arro)o (w)g(of)h Fo(G)g Fu(starting)f(at)i Fp(f)p Fo(\021)r Fp(g)p Fu(.)20 b(Let)15 b Fo(\030)h Fu(b)q(e)e(a)g(p)q(oin)o(t)g(suc)o(h)e(that)j Fo(q)r Fu(\()p Fo(\021)r(;)8 b(\030)r Fu(\))14 b Fo(>)g Fu(0)g(and)f Fo(E)s Fu(\()p Fo(\030)r Fu(\))h Fp(\024)0 1005 y Fo(E)s Fu(\()p Fo(\021)r Fu(\).)33 b(Since)20 b Fo(a)p Fu(\()p Fo(\021)r Fu(\))g(=)g Fo(n)486 1012 y Fn(c)506 1005 y Fu(,)h(then)g Fo(a)p Fu(\()p Fo(\030)r Fu(\))g(is)f(equal)g(to)h Fo(n)1052 1012 y Fn(c)1085 1005 y Fp(\000)14 b Fu(1)20 b(or)g Fo(n)1278 1012 y Fn(c)1311 1005 y Fu(+)14 b(1.)33 b(Moreo)o(v)o(er)18 b Fo(\021)k Fu(is)e(minimal,)0 1065 y(and)i(the)g(inequalit)o(y)g Fo(E)s Fu(\()p Fo(\030)r Fu(\))j Fp(\024)e Fo(E)s Fu(\()p Fo(\021)r Fu(\))f(implies)f(in)h(b)q(oth)h (cases)f(that)h Fo(\030)h Fu(is)e(also)g(minimal.)38 b(Th)o(us)21 b Fo(\030)0 1125 y Fu(cannot)e(b)q(e)h(of)g(area)f Fo(n)438 1132 y Fn(c)471 1125 y Fp(\000)13 b Fu(1)19 b(\(b)o(y)h(lemma)e(6)p Fo(:)p Fu(4,)i(the)g(only)f(con\014gurations)f(of)i Fo(M)1554 1132 y Fn(n)1579 1137 y Fg(c)1619 1125 y Fu(comm)o(unicating)0 1185 y(with)g Fo(M)165 1192 y Fn(n)190 1197 y Fg(c)209 1192 y Fj(\000)p Fl(1)282 1185 y Fu(are)g(the)g(principal)e(con\014gurations)1000 1172 y Fk(f)994 1185 y Fo(M)1042 1192 y Fn(n)1067 1197 y Fg(c)1088 1185 y Fu(\).)33 b(Th)o(us)19 b Fo(a)p Fu(\()p Fo(\030)r Fu(\))i(=)e Fo(n)1481 1192 y Fn(c)1515 1185 y Fu(+)13 b(1.)32 b(W)l(e)21 b(next)f(sho)o(w)0 1244 y(that)15 b Fo(\030)i Fu(b)q(elongs)d(to)h Fo(\031)r Fu(\()p Fo(l)445 1251 y Fn(c)473 1244 y Fp(\002)7 b Fu(\()p Fo(l)553 1251 y Fn(c)582 1244 y Fu(+)g(2\))p Fo(;)h Fu(pred)f Fo(E)s Fu(\()p Fo(n)888 1251 y Fn(c)909 1244 y Fu(\)\).)22 b(Let)15 b Fo(x)1098 1251 y Fl(1)1121 1244 y Fo(;)8 b Fp(\001)g(\001)g(\001) 17 b Fo(;)8 b(x)1268 1251 y Fn(r)1305 1244 y Fu(b)q(e)15 b(a)g(sequence)g(of) g(sites)f(inside)g Fo(\033)0 1304 y Fu(suc)o(h)h(that)h Fo(F)7 b Fu(\()p Fo(x)305 1311 y Fl(1)328 1304 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)475 1311 y Fn(j)497 1304 y Fu(\)\()p Fo(l)550 1311 y Fn(c)580 1304 y Fp(\002)i Fu(\()p Fo(l)663 1311 y Fn(c)694 1304 y Fu(+)f(2\)\))17 b(is)e(in)h Fo(\031)r Fu(\()p Fo(l)993 1311 y Fn(c)1023 1304 y Fp(\002)10 b Fu(\()p Fo(l)1106 1311 y Fn(c)1136 1304 y Fu(+)g(2\))p Fo(;)e Fu(pred)g Fo(E)s Fu(\()p Fo(n)1446 1311 y Fn(c)1466 1304 y Fu(\)\))17 b(for)e Fo(j)k Fu(in)c Fp(f)p Fu(1)8 b Fp(\001)g(\001)g(\001)h Fo(r)j Fp(\000)d Fu(1)p Fp(g)0 1364 y Fu(and)16 b Fo(\021)f Fu(=)f Fo(F)7 b Fu(\()p Fo(x)276 1371 y Fl(1)299 1364 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)446 1371 y Fn(r)469 1364 y Fu(\)\()p Fo(l)522 1371 y Fn(c)554 1364 y Fp(\002)j Fu(\()p Fo(l)638 1371 y Fn(c)669 1364 y Fu(+)g(2\)\).)23 b(W)l(e)17 b(put)f Fo(\021)1021 1371 y Fn(j)1056 1364 y Fu(=)e Fo(F)7 b Fu(\()p Fo(x)1195 1371 y Fl(1)1218 1364 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1365 1371 y Fn(j)1386 1364 y Fu(\)\()p Fo(l)1439 1371 y Fn(c)1471 1364 y Fp(\002)j Fu(\()p Fo(l)1555 1371 y Fn(c)1587 1364 y Fu(+)g(2\)\),)17 b(0)d Fp(\024)f Fo(j)k Fp(\024)c Fo(r)q Fu(.)0 1424 y(Since)22 b Fo(\021)i Fu(is)f(in)f(the)h(b)q(oundary)e(of)i(the)g (cycle,)i(w)o(e)d(ha)o(v)o(e)g Fo(E)s Fu(\()p Fo(\021)1212 1431 y Fn(r)q Fj(\000)p Fl(1)1285 1424 y Fu(\))j Fo(<)e(E)s Fu(\()p Fo(\021)1475 1431 y Fn(r)1497 1424 y Fu(\).)41 b(Moreo)o(v)o(er)21 b Fo(\021)1821 1431 y Fn(r)q Fj(\000)p Fl(1)1917 1424 y Fu(is)0 1483 y(a)j(minimal)d(con\014guration)h(of)i(the)g(cycle,)i(of)d(area)h Fo(n)1070 1490 y Fn(c)1106 1483 y Fu(+)15 b(1,)25 b(so)e(that)h(the)g(last)g (spin)e(\015ip)h(at)h(site)0 1543 y Fo(x)28 1550 y Fn(r)71 1543 y Fu(has)c(decreased)f(the)i(area.)33 b(Let)21 b Fo(x)742 1550 y Fn(r)q Fl(+1)835 1543 y Fu(b)q(e)g(the)g(unique)e(site)i(suc)o(h)e (that)i Fo(\030)h Fu(=)e Fo(F)7 b Fu(\()p Fo(x)1675 1550 y Fn(r)q Fl(+1)1749 1543 y Fu(\)\()p Fo(\021)r Fu(\).)34 b(W)l(e)0 1603 y(ha)o(v)o(e)18 b Fo(\030)g Fu(=)f Fo(F)7 b Fu(\()p Fo(x)299 1610 y Fn(r)q Fl(+1)372 1603 y Fu(\)\()p Fo(\021)435 1610 y Fn(r)458 1603 y Fu(\))17 b(=)f Fo(F)7 b Fu(\()p Fo(x)635 1610 y Fn(r)q Fl(+1)709 1603 y Fo(;)h(x)759 1610 y Fn(r)781 1603 y Fu(\)\()p Fo(\021)844 1610 y Fn(r)q Fj(\000)p Fl(1)918 1603 y Fu(\))17 b(=)g Fo(F)7 b Fu(\()p Fo(x)1096 1610 y Fn(r)1118 1603 y Fo(;)h(x)1168 1610 y Fn(r)q Fl(+1)1242 1603 y Fu(\)\()p Fo(\021)1305 1610 y Fn(r)q Fj(\000)p Fl(1)1378 1603 y Fu(\))17 b(=)g Fo(F)7 b Fu(\()p Fo(x)1556 1610 y Fn(r)1579 1603 y Fu(\)\()p Fo(\021)1644 1585 y Fj(0)1642 1615 y Fn(r)1664 1603 y Fu(\))19 b(where)f Fo(\021)1875 1585 y Fj(0)1873 1615 y Fn(r)1911 1603 y Fu(=)0 1663 y Fo(F)7 b Fu(\()p Fo(x)86 1670 y Fn(r)q Fl(+1)159 1663 y Fu(\)\()p Fo(\021)222 1670 y Fn(r)q Fj(\000)p Fl(1)296 1663 y Fu(\).)25 b(The)17 b(energy)h(inequalit)o(y)f(5)p Fo(:)p Fu(1)g(yields)g Fo(E)s Fu(\()p Fo(\021)1158 1645 y Fj(0)1156 1675 y Fn(r)1178 1663 y Fu(\))12 b Fp(\000)f Fo(E)s Fu(\()p Fo(\021)1343 1670 y Fn(r)q Fj(\000)p Fl(1)1416 1663 y Fu(\))16 b Fp(\024)f Fo(E)s Fu(\()p Fo(\030)r Fu(\))d Fp(\000)g Fo(E)s Fu(\()p Fo(\021)r Fu(\))17 b(whence)0 1723 y Fo(E)s Fu(\()p Fo(\021)86 1704 y Fj(0)84 1735 y Fn(r)106 1723 y Fu(\))e Fp(\024)g Fo(E)s Fu(\()p Fo(\021)278 1730 y Fn(r)q Fj(\000)p Fl(1)351 1723 y Fu(\).)24 b(It)18 b(follo)o(ws)e(that)i Fo(\021)763 1704 y Fj(0)761 1735 y Fn(r)800 1723 y Fu(is)f(in)f(the)i(cycle,)g(as)f(w)o (ell)f(as)h Fo(\030)j Fu(\(their)d(energies)f(are)h(less)g(or)0 1782 y(equal)f(than)h Fo(E)s Fu(\()p Fo(n)337 1789 y Fn(c)357 1782 y Fu(\))g(and)f(they)h(b)q(oth)f(comm)o(unicate)f(with)i(a)f (con\014guration)f(of)i(the)g(cycle\).)50 b Fc(\003)0 1875 y Fw(Corollary)18 b(6.27.)28 b Fh(The)15 b(principal)e(non)i(standard)e (con\014gurations)g(of)j(area)f Fo(n)1522 1882 y Fn(c)1557 1875 y Fh(\(i.e.)22 b(the)15 b(set)1829 1863 y Fk(f)1823 1875 y Fo(M)1871 1882 y Fn(n)1896 1887 y Fg(c)1925 1875 y Fp(n)0 1935 y Fo(S)31 1942 y Fn(n)56 1947 y Fg(c)76 1935 y Fh(\))24 b(are)g(dead{ends:)34 b(there)24 b(is)f(no)g(saddle)g(path)g(of)h(n)o(ull)e (cost)i(b)q(et)o(w)o(een)f Fp(\000)p Fu(1)p 1513 1947 V 23 w Fh(and)g Fu(+1)p 1704 1947 V 24 w Fh(passing)0 1995 y(through)15 b(them.)0 2080 y Fw(Corollary)21 b(6.28.)28 b Fh(The)17 b(set)h(of)g(the)g (global)e(saddle)h(p)q(oin)o(ts)g(b)q(et)o(w)o(een)g Fp(\000)p Fu(1)p 1375 2092 V 18 w Fh(and)g Fu(+1)p 1555 2092 V 17 w Fh(is)g(exactly)i Fo(S)1891 2087 y Fn(n)1916 2092 y Fg(c)1936 2080 y Fh(.)0 2140 y(These)d(con\014gurations)f(are)h(the)h(critical)f(t)o(w)o(o)g(dimensional)e (con\014gurations.)60 2225 y Fu(Steps)i Fo(ii)p Fu(\))h(and)f Fo(iii)p Fu(\))i(are)e(no)o(w)f(completed)h(and)g(w)o(e)h(pro)q(ceed)f(to)h (steps)f Fo(iv)r Fu(\))h(and)f Fo(v)r Fu(\).)0 2311 y Fw(The)j(ascending)g (part.)28 b Fu(F)l(or)16 b(eac)o(h)g(con\014guration)f Fo(\033)1062 2293 y Fj(0)1093 2311 y Fu(in)h Fo(S)1182 2318 y Fn(n)1207 2323 y Fg(c)1227 2311 y Fu(,)h(w)o(e)f(m)o(ust)f(determine)h(the)h(minimal)0 2371 y(stable)g(subgraph)f(of)i Fo(G)460 2353 y Fj(\000)511 2371 y Fu(con)o(taining)f Fo(\033)785 2353 y Fj(0)817 2371 y Fu(and)g(all)g(the)h(paths)f(in)g(this)g(graph)g(starting)g(at)h Fp(f)p Fo(\033)1813 2353 y Fj(0)1827 2371 y Fp(g)g Fu(and)0 2430 y(ending)13 b(at)i Fp(\000)p Fu(1)p 217 2443 V(.)21 b(Our)13 b(exp)q(osition)h(is)g(similar)e(as)i(b)q(efore:)21 b(w)o(e)14 b(\014rst)g(list)g(the)g(set)h(of)f(the)h(relev)m(an)o(t)f(cycles)0 2490 y(and)k(w)o(e)h(use)f(lemma)f(6)p Fo(:)p Fu(19)i(to)g(\014nd)f(those)g (b)q(elonging)g(to)h Fp(M)p Fu(\()p Fp(f\000)p Fu(1)p 1225 2502 V Fo(;)8 b Fu(+1)p 1311 2502 V -1 w Fp(g)1399 2472 y Fn(c)1420 2490 y Fu(\).)28 b(W)l(e)19 b(\014nally)f(c)o(hec)o(k)h(that)0 2550 y(w)o(e)d(ha)o(v)o(e)g(in)g(hand)g(all)g(the)h(v)o(ertices)f(of)h Fo(G)802 2532 y Fj(\000)836 2550 y Fu(.)954 2670 y Fs(25)p eop %%Page: 26 26 26 25 bop 0 200 a Fw(Theorem)17 b(6.29.)50 b Fh(Let)17 b Fo(l)g Fh(b)q(e)g(an)f(in)o(teger)g(less)g(or)g(equal)h(than)f Fo(l)1251 207 y Fn(c)1271 200 y Fh(.)0 260 y(The)g(cycle)i Fo(\031)r Fu(\()p Fo(l)12 b Fp(\002)f Fo(l)q(;)d Fh(pred)f Fo(E)s Fu(\()p Fo(l)q Fu(\()p Fo(l)12 b Fp(\000)f Fu(1\))g(+)g(1\)\))17 b Fh(do)q(es)g(not)f(con)o(tain)g Fp(\000)p Fu(1)p 1228 272 64 2 v 16 w Fh(and)g Fu(+1)p 1405 272 V Fh(.)22 b(Moreo)o(v)o(er)313 383 y Fo(a)p 313 391 27 2 v 1 w Fu(\()p Fo(\031)r Fu(\()p Fo(l)13 b Fp(\002)d Fo(l)q(;)e Fh(pred)g Fo(E)s Fu(\()p Fo(l)q Fu(\()p Fo(l)k Fp(\000)f Fu(1\))g(+)g(1\)\)\))23 b(=)f Fo(l)q Fu(\()p Fo(l)12 b Fp(\000)f Fu(1\))g(+)g(2)d Fo(;)248 460 y Fp(8)p Fo(k)14 b Fp(2)g(f)p Fu(1)8 b Fp(\001)g(\001)g(\001)g Fo(l)k Fp(\000)f Fu(2)p Fp(g)99 b Fo(M)5 b Fu(\()p Fo(l)13 b Fp(\002)e Fo(l)q(;)d Fp(\000)p Fo(k)r Fu(\))22 b Fp(\032)g Fo(\031)r Fu(\()p Fo(l)12 b Fp(\002)f Fo(l)q(;)d Fh(pred)f Fo(E)s Fu(\()p Fo(l)q Fu(\()p Fo(l)13 b Fp(\000)d Fu(1\))i(+)f(1\)\))d Fo(;)p 313 515 V 313 543 a(a)q Fu(\()p Fo(\031)r Fu(\()p Fo(l)13 b Fp(\002)d Fo(l)q(;)e Fh(pred)g Fo(E)s Fu(\()p Fo(l)q Fu(\()p Fo(l)k Fp(\000)f Fu(1\))g(+)g(1\)\)\))23 b(=)f Fo(l)1087 522 y Fl(2)1117 543 y Fo(:)0 669 y Fh(The)c(b)q(ottom)g(of)h(this)e(cycle)i(is)f Fp(f)p Fo(l)13 b Fp(\002)e Fo(l)q Fp(g)p Fh(;)19 b(its)f(principal)e(b)q (oundary)h(is)h Fo(M)5 b Fu(\()p Fo(l)14 b Fp(\002)e Fo(l)q(;)c Fp(\000)p Fu(\()p Fo(l)13 b Fp(\000)f Fu(1\)\))19 b Fh(and)e(th)o(us)0 728 y(con)o(tains)e Fo(S)226 737 y Fn(l)p Fl(\()p Fn(l)p Fj(\000)p Fl(1\)+1)387 728 y Fh(.)0 822 y Fe(Pr)m(o)m(of.)28 b Fu(W)l(e)16 b(apply)g(corollary)g(6)p Fo(:)p Fu(13)g(and)g(form)o(ula)f(\(3\).)22 b(F)l(or)16 b(an)o(y)g Fo(n)g Fu(greater)g(than)h Fo(l)1602 804 y Fl(2)1624 822 y Fu(,)f(w)o(e)h(ha)o(v)o(e)503 945 y Fo(E)s Fu(\()p Fo(l)12 b Fp(\002)e Fo(l)q(;)e(C)712 952 y Fn(n)739 945 y Fu(\))23 b Fp(\025)f Fo(E)s Fu(\()p Fo(l)917 925 y Fl(2)950 945 y Fu(+)11 b(1\))22 b Fp(\025)g Fo(E)s Fu(\()p Fo(l)q Fu(\()p Fo(l)12 b Fp(\000)f Fu(1\))g(+)g(1\))0 1069 y(whence)p 174 1041 V 16 w Fo(a)p Fu(\()p Fo(\031)r Fu(\()p Fo(l)h Fp(\002)e Fo(l)q(;)e Fu(pred)g Fo(E)s Fu(\()p Fo(l)q Fu(\()p Fo(l)j Fp(\000)f Fu(1\))g(+)g(1\)\)\))15 b(=)f Fo(l)926 1051 y Fl(2)948 1069 y Fu(.)22 b(Analogously)l(,)15 b(w)o(e)h(ha)o(v)o(e)f(for)h(an)o(y)g Fo(n)g Fu(smaller)e(than)0 1128 y Fo(l)q Fu(\()p Fo(l)e Fp(\000)f Fu(1\))g(+)g(2,)16 b Fo(E)s Fu(\()p Fo(l)c Fp(\002)f Fo(l)q(;)d(C)482 1135 y Fn(n)509 1128 y Fu(\))22 b Fp(\025)g Fo(E)s Fu(\()p Fo(l)q Fu(\()p Fo(l)12 b Fp(\000)f Fu(1\))h(+)e(1\);)17 b(moreo)o(v)o(er,)387 1252 y Fo(E)s Fu(\()p Fo(l)12 b Fp(\002)e Fo(l)q(;)e(C)596 1261 y Fn(l)p Fl(\()p Fn(l)p Fj(\000)p Fl(1\)+2)757 1252 y Fu(\))22 b(=)g Fo(E)s Fu(\()p Fo(l)q Fu(\()p Fo(l)12 b Fp(\000)f Fu(1\))g(+)g(2\))22 b Fo(<)g(E)s Fu(\()p Fo(l)q Fu(\()p Fo(l)13 b Fp(\000)d Fu(1\))i(+)e(1)0 1375 y(so)k(that)h Fo(a)p 165 1383 V Fu(\()p Fo(\031)r Fu(\()p Fo(l)8 b Fp(\002)f Fo(l)q(;)h Fu(pred)f Fo(E)s Fu(\()p Fo(l)q Fu(\()p Fo(l)h Fp(\000)f Fu(1\))g(+)g (1\)\)\))14 b(=)f Fo(l)q Fu(\()p Fo(l)8 b Fp(\000)f Fu(1\))g(+)g(2.)20 b(T)l(o)14 b(pro)o(v)o(e)f(that)i(the)g(b)q(ottom)f(of)h(the)f(cycle)0 1435 y(is)g Fp(f)p Fo(l)7 b Fp(\002)f Fo(l)q Fp(g)p Fu(,)13 b(w)o(e)h(could)f(pro)q(ceed)h(as)g(b)q(efore)g(and)f(use)h(theorem)f(5)p Fo(:)p Fu(5.)21 b(Ho)o(w)o(ev)o(er,)13 b(a)h(direct)g(application)f(of)0 1495 y(our)j(geometrical)g(results)g(\(theorem)h(6)p Fo(:)p Fu(2\))g(yields)f(that)i(the)f(minim)o(um)d(min)o Fp(f)8 b Fo(E)s Fu(\()p Fo(n)p Fu(\))15 b(:)g Fo(l)q Fu(\()p Fo(l)d Fp(\000)f Fu(1\))h(+)f(2)j Fp(\024)0 1554 y Fo(n)21 b Fp(\024)f Fo(l)126 1536 y Fl(2)156 1554 y Fp(g)h Fu(is)f(equal)h(to)g Fo(E)s Fu(\()p Fo(l)532 1536 y Fl(2)554 1554 y Fu(\);)i(the)e(unique)f (con\014guration)f(of)i(energy)f Fo(E)s Fu(\()p Fo(l)1471 1536 y Fl(2)1493 1554 y Fu(\))h(is)f(the)h(square)f Fo(l)15 b Fp(\002)e Fo(l)q Fu(.)0 1614 y(Theorem)f(5)p Fo(:)p Fu(3)i(implies)e(also)h(that)h(the) g(altitude)f(of)h(comm)o(unication)d(b)q(et)o(w)o(een)i(t)o(w)o(o)h (di\013eren)o(t)e(squares)0 1674 y Fo(l)i Fp(\002)f Fo(l)20 b Fu(is)f(greater)g(than)g Fo(E)s Fu(\()p Fo(l)535 1656 y Fl(2)571 1674 y Fu(+)12 b(1\))20 b(\(one)g(has)f(to)h(mak)o(e)f(a)g(spin{\015ip)f (outside)g(the)i(initial)f(square)g(to)0 1734 y(create)e(another)f(square\).) 49 b Fc(\003)0 1827 y Fw(Theorem)17 b(6.30.)50 b Fh(Let)17 b Fo(l)g Fh(b)q(e)g(an)f(in)o(teger)g(less)g(or)g(equal)h(than)f Fo(l)1251 1834 y Fn(c)1271 1827 y Fh(.)0 1887 y(The)g(cycle)i Fo(\031)r Fu(\()p Fo(l)12 b Fp(\002)f Fu(\()p Fo(l)h Fu(+)f(1\))p Fo(;)d Fh(pred)g Fo(E)s Fu(\()p Fo(l)695 1869 y Fl(2)728 1887 y Fu(+)j(1\)\))17 b Fh(do)q(es)f(not)h(con)o(tain)f Fp(\000)p Fu(1)p 1235 1899 64 2 v 16 w Fh(and)g Fu(+1)p 1412 1899 V -1 w Fh(.)22 b(Moreo)o(v)o(er)366 2012 y Fo(a)p 366 2020 27 2 v Fu(\()p Fo(\031)r Fu(\()p Fo(l)13 b Fp(\002)e Fu(\()p Fo(l)h Fu(+)f(1\))p Fo(;)d Fh(pred)g Fo(E)s Fu(\()p Fo(l)881 1992 y Fl(2)914 2012 y Fu(+)j(1\)\)\))23 b(=)f Fo(l)1146 1992 y Fl(2)1179 2012 y Fu(+)11 b(2)d Fo(;)182 2095 y Fp(8)p Fo(k)14 b Fp(2)g(f)p Fu(1)8 b Fp(\001)g(\001)g(\001)h Fo(l)j Fp(\000)e Fu(2)p Fp(g)100 b Fo(M)5 b Fu(\()p Fo(l)13 b Fp(\002)e Fu(\()p Fo(l)h Fu(+)f(1\))p Fo(;)d Fp(\000)p Fo(k)r Fu(\))22 b Fp(\032)g Fo(\031)r Fu(\()p Fo(l)12 b Fp(\002)f Fu(\()p Fo(l)h Fu(+)f(1\))p Fo(;)d Fh(pred)g Fo(E)s Fu(\()p Fo(l)1599 2075 y Fl(2)1632 2095 y Fu(+)j(1\)\))d Fo(;)p 366 2151 V 366 2178 a(a)p Fu(\()p Fo(\031)r Fu(\()p Fo(l)13 b Fp(\002)e Fu(\()p Fo(l)h Fu(+)f(1\))p Fo(;)d Fh(pred)g Fo(E)s Fu(\()p Fo(l)881 2157 y Fl(2)914 2178 y Fu(+)j(1\)\)\))23 b(=)f Fo(l)q Fu(\()p Fo(l)12 b Fu(+)f(1\))d Fo(:)0 2304 y Fh(The)16 b(b)q(ottom)g(of)g(this)g(cycle)h(is)e Fp(f)p Fo(l)c Fp(\002)f Fu(\()p Fo(l)h Fu(+)f(1\))p Fp(g)p Fh(;)16 b(its)g(principal)f(b)q(oundary)f(is)i Fo(M)5 b Fu(\()p Fo(l)12 b Fp(\002)e Fu(\()p Fo(l)h Fu(+)f(1\))p Fo(;)e Fp(\000)p Fu(\()p Fo(l)k Fp(\000)d Fu(1\)\))0 2363 y Fh(and)16 b(th)o(us)f(con)o(tains) h Fo(S)433 2372 y Fn(l)446 2362 y Fd(2)465 2372 y Fl(+1)518 2363 y Fh(.)0 2457 y Fe(Pr)m(o)m(of.)28 b Fu(The)16 b(pro)q(of)g(is)g (similar)f(as)h(the)h(pro)q(of)f(of)h(theorem)f(6)p Fo(:)p Fu(29.)0 2550 y Fe(R)m(emark.)27 b Fu(Similar)14 b(statemen)o(ts)i(hold)g (for)g(the)h(quasisquares)e(\()p Fo(l)d Fu(+)f(1\))g Fp(\002)g Fo(l)q Fu(.)954 2670 y Fs(26)p eop %%Page: 27 27 27 26 bop 0 200 a Fw(Corollary)20 b(6.31.)49 b Fh(The)17 b(follo)o(wing)e (cycles)i(are)f(maximal)f(cycles)i(of)g Fp(f\000)p Fu(1)p 1390 212 64 2 v Fo(;)8 b Fu(+1)p 1476 212 V -1 w Fp(g)1564 182 y Fn(c)1601 200 y Fh(\(where)17 b Fo(l)d Fp(\024)g Fo(l)1862 207 y Fn(c)1882 200 y Fh(\):)566 318 y Fp(f)p Fo(\021)r Fp(g)p Fo(;)107 b(\021)15 b Fp(2)f Fo(M)5 b Fu(\()p Fo(l)13 b Fp(\002)e Fu(\()p Fo(l)h Fu(+)f(1\))p Fo(;)d Fp(\000)p Fu(\()p Fo(l)13 b Fp(\000)d Fu(1\)\))e Fo(;)566 395 y Fp(f)p Fo(\021)r Fp(g)p Fo(;)107 b(\021)15 b Fp(2)f Fo(M)5 b Fu(\(\()p Fo(l)14 b Fu(+)c(1\))i Fp(\002)f Fo(l)q(;)d Fp(\000)p Fu(\()p Fo(l)k Fp(\000)e Fu(1\)\))e Fo(;)566 473 y Fp(f)p Fo(\021)r Fp(g)p Fo(;)107 b(\021)15 b Fp(2)f Fo(M)5 b Fu(\()p Fo(l)13 b Fp(\002)e Fo(l)q(;)d Fp(\000)p Fu(\()p Fo(l)k Fp(\000)f Fu(1\)\))d Fo(;)566 550 y(\031)r Fu(\()p Fo(l)k Fp(\002)f Fo(l)q(;)d Fh(pred)f Fo(E)s Fu(\()p Fo(l)q Fu(\()p Fo(l)12 b Fp(\000)f Fu(1\))h(+)e(1\)\))e Fo(;)566 633 y(\031)r Fu(\()p Fo(l)k Fp(\002)f Fu(\()p Fo(l)h Fu(+)f(1\))p Fo(;)d Fh(pred)g Fo(E)s Fu(\()p Fo(l)1035 612 y Fl(2)1068 633 y Fu(+)j(1\)\))d Fo(;)566 716 y(\031)r Fu(\(\()p Fo(l)13 b Fu(+)d(1\))i Fp(\002)f Fo(l)q(;)d Fh(pred)f Fo(E)s Fu(\()p Fo(l)1035 695 y Fl(2)1068 716 y Fu(+)k(1\)\))d Fo(:)0 864 y Fe(Pr)m(o)m(of.)28 b Fu(This)12 b(corollary)f(is)h(a)h(consequence)f(of)h (lemma)f(6)p Fo(:)p Fu(19)g(together)h(with)g(theorems)e(6)p Fo(:)p Fu(29)i(and)f(6)p Fo(:)p Fu(30.)0 924 y(Notice)k(that)f(w)o(e)g(ha)o (v)o(e)f(to)h(put)g(together)g(the)g(descriptions)e(of)i(the)g(cycles)g(of)g (theorems)f(6)p Fo(:)p Fu(29)p Fo(;)8 b Fu(6)p Fo(:)p Fu(30)14 b(in)0 983 y(order)d(to)i(c)o(hec)o(k)g(that)g(for)f(eac)o(h)g(cycle)i Fo(\031)h Fu(in)d(the)h(ab)q(o)o(v)o(e)f(list,)h(there)g(is)f(a)g(sequence)h (of)g(cycles)g Fo(\031)1758 990 y Fl(0)1781 983 y Fo(;)8 b Fp(\001)g(\001)g(\001)16 b Fo(;)8 b(\031)1927 990 y Fn(r)0 1047 y Fu(suc)o(h)15 b(that)i Fo(\031)248 1054 y Fl(0)285 1047 y Fu(=)c Fo(\031)r Fu(,)409 1034 y Fk(e)398 1047 y Fo(B)r Fu(\()p Fo(\031)485 1054 y Fn(i)503 1047 y Fu(\))e Fp(\\)g Fo(\031)605 1054 y Fn(i)p Fl(+1)687 1047 y Fp(6)p Fu(=)i Fp(;)p Fu(,)k(0)c Fp(\024)h Fo(i)g(<)g(r)k Fu(and)e Fp(\000)p Fu(1)p 1107 1059 V 13 w Fp(2)1242 1034 y Fk(e)1231 1047 y Fo(B)s Fu(\()p Fo(\031)1319 1054 y Fn(r)1341 1047 y Fu(\).)50 b Fc(\003)0 1136 y Fw(Corollary)24 b(6.32.)61 b Fh(Let)20 b Fo(\033)j Fh(b)q(elong)d(to)g Fo(S)840 1143 y Fn(n)865 1148 y Fg(c)883 1143 y Fj(\000)p Fl(1)937 1136 y Fh(.)33 b(The)20 b(minimal)e(stable)i(subgraph)e Fo(G)1694 1118 y Fj(\000)1728 1136 y Fu(\()p Fo(\033)r Fu(\))j Fh(of)f Fo(G)1916 1118 y Fj(\000)0 1196 y Fh(con)o(taining)g Fo(\033)j Fh(is)e(the)h(restriction)e(of)h Fo(G)h Fh(to)f(the)h(v)o(ertices)f(listed)f (in)h(corollary)f Fu(6)p Fo(:)p Fu(31)p Fh(.)36 b(The)21 b(arro)o(ws)0 1256 y(of)c Fo(G)96 1237 y Fj(\000)130 1256 y Fu(\()p Fo(\033)r Fu(\))h Fh(are)e(\(where)g(eac)o(h)h(square)f Fo(l)c Fp(\002)f Fo(l)17 b Fh(or)g(quasisquare)e Fo(l)d Fp(\002)f Fu(\()p Fo(l)h Fu(+)f(1\))17 b Fh(app)q(earing)f(on)g(a)h(line)f(m)o(ust)0 1315 y(b)q(e)h(included)e(in)h Fo(\033)r Fh(\);)215 1434 y Fp(f)p Fo(\021)r Fp(g)d($)h Fo(\031)r Fu(\()p Fo(l)e Fp(\002)f Fu(\()p Fo(l)h Fu(+)f(1\))p Fo(;)d Fh(pred)g Fo(E)s Fu(\()p Fo(l)838 1414 y Fl(2)871 1434 y Fu(+)j(1\)\))d Fo(;)109 b(\021)15 b Fp(2)f Fo(M)5 b Fu(\()p Fo(l)13 b Fp(\002)e Fu(\()p Fo(l)h Fu(+)f(1\))p Fo(;)d Fp(\000)p Fu(\()p Fo(l)k Fp(\000)f Fu(1\)\))d Fo(;)215 1517 y Fp(f)p Fo(\021)r Fp(g)13 b($)h Fo(\031)r Fu(\(\()p Fo(l)f Fu(+)d(1\))i Fp(\002)f Fo(l)q(;)d Fh(pred)f Fo(E)s Fu(\()p Fo(l)838 1497 y Fl(2)871 1517 y Fu(+)k(1\)\))d Fo(;)109 b(\021)15 b Fp(2)f Fo(M)5 b Fu(\(\()p Fo(l)13 b Fu(+)e(1\))h Fp(\002)e Fo(l)q(;)e Fp(\000)p Fu(\()p Fo(l)k Fp(\000)f Fu(1\)\))d Fo(;)215 1595 y Fp(f)p Fo(\021)r Fp(g)13 b($)h Fo(\031)r Fu(\()p Fo(l)e Fp(\002)f Fo(l)q(;)d Fh(pred)f Fo(E)s Fu(\()p Fo(l)q Fu(\()p Fo(l)12 b Fp(\000)f Fu(1\))h(+)e(1\)\))e Fo(;)109 b(\021)16 b Fp(2)e Fo(M)5 b Fu(\()p Fo(l)13 b Fp(\002)d Fo(l)q(;)e Fp(\000)p Fu(\()p Fo(l)k Fp(\000)f Fu(1\)\))d Fo(;)215 1684 y Fp(f)p Fo(\021)r Fp(g)13 b(!)h Fo(\031)r Fu(\()p Fo(l)e Fp(\002)f Fo(l)q(;)d Fh(pred)f Fo(E)s Fu(\()p Fo(l)q Fu(\()p Fo(l)12 b Fp(\000)f Fu(1\))h(+)e(1\)\))e Fo(;)109 b(\021)16 b Fp(2)1202 1672 y Fk(f)1196 1684 y Fo(M)1244 1693 y Fn(l)1257 1683 y Fd(2)1276 1693 y Fl(+1)1337 1684 y Fo(;)215 1772 y Fp(f)p Fo(\021)r Fp(g)d(!)h Fo(\031)r Fu(\()p Fo(l)e Fp(\002)f Fu(\()p Fo(l)h Fu(+)f(1\))p Fo(;)d Fh(pred)g Fo(E)s Fu(\()p Fo(l)838 1751 y Fl(2)871 1772 y Fu(+)j(1\)\))d Fo(;)109 b(\021)15 b Fp(2)p 1202 1728 326 2 v 14 w Fo(l)d Fp(\002)f Fu(\()p Fo(l)h Fu(+)f(1\))g Fp(\010)1469 1779 y Fl(2)1502 1772 y Fu(1)e Fo(;)215 1859 y Fp(f)p Fo(\021)r Fp(g)k(!)h Fo(\031)r Fu(\(\()p Fo(l)f Fu(+)d(1\))i Fp(\002)f Fo(l)q(;)d Fh(pred)f Fo(E)s Fu(\()p Fo(l)838 1838 y Fl(2)871 1859 y Fu(+)k(1\)\))d Fo(;)109 b(\021)15 b Fp(2)p 1202 1815 V 14 w Fu(\()p Fo(l)d Fu(+)f(1\))h Fp(\002)e Fo(l)i Fp(\010)1469 1866 y Fl(1)1502 1859 y Fu(1)d Fo(;)60 1982 y Fu(The)17 b(only)h(lo)q(ops)f (in)h(the)g(graph)f Fo(G)733 1964 y Fj(\000)766 1982 y Fu(\()p Fo(\033)r Fu(\))i(are)f(lo)q(ops)f(around)f(t)o(w)o(o)i(cycles)g(\(corresp)q (onding)e(to)i(the)0 2042 y(arro)o(ws)f(describ)q(ed)h(in)g(the)h(\014rst)f (three)h(lines)f(of)h(the)g(list\).)29 b(An)o(y)19 b(other)f(arro)o(w)g Fo(\031)1572 2049 y Fl(1)1612 2042 y Fp(!)g Fo(\031)1708 2049 y Fl(2)1749 2042 y Fu(of)h Fo(G)1847 2024 y Fj(\000)1881 2042 y Fu(\()p Fo(\033)r Fu(\))0 2102 y(satis\014es)p 186 2074 27 2 v 18 w Fo(a)p Fu(\()p Fo(\031)259 2109 y Fl(1)282 2102 y Fu(\))f Fo(>)f(a)p 375 2110 V 1 w Fu(\()p Fo(\031)449 2109 y Fl(2)472 2102 y Fu(\).)29 b(As)19 b(a)f(consequence)h(a)g(path)f(in)g Fo(G)1198 2084 y Fj(\000)1232 2102 y Fu(\()p Fo(\033)r Fu(\))i(starting)e(at) h Fp(f)p Fo(\033)r Fp(g)g Fu(with)g(no)f(lo)q(op)0 2161 y(ends)e(in)g Fp(f\000)p Fu(1)p 197 2173 64 2 v -1 w Fp(g)p Fu(.)0 2251 y Fw(The)25 b(exit)i(path.)h Fu(W)l(e)22 b(ha)o(v)o(e)f(\014nally)g(reac)o(hed) g(the)h(last)f(step)h Fo(v)r(i)p Fu(\))g(whic)o(h)f(consists)g(in)g(doing)g (the)0 2311 y(syn)o(thesis)16 b(of)i(all)f(the)h(results)e(gathered)h(in)g (the)h(previous)e(steps)i Fo(i)p Fu(\))12 b Fp(\000)f Fo(v)r Fu(\).)26 b(The)17 b(set)h(of)f(all)h(the)f(cycle)0 2370 y(paths)c(of)i(n)o (ull)e(cost)h(is)g(obtained)f(b)o(y)h(glueing)f(together)h(an)g(ascending)f (part)g(and)h(a)g(descending)f(part.)0 2430 y(W)l(e)k(de\014ne)f(a)g(graph)f Fo(H)22 b 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Fo(\022)q Fu(\()p Fp(\000)p Fu(1)p 761 1845 V 1 w Fo(;)d(\034)e Fu(\(+1)p 895 1845 V -20 x Fn(c)979 1833 y Fu(\)\))23 b Fp(\024)f Fu(exp)8 b Fo(\014)s Fu(\(2)j Fp(\000)g Fo(h)g Fu(+)g Fo(\017)p Fu(\))d(;)0 1892 y Fp(\017)50 b Fh(during)13 b(the)i(exit)g(path)g Fo(\033)554 1899 y Fn(n)581 1892 y Fo(;)8 b(\022)16 b Fp(\024)e Fo(n)f Fp(\024)h Fo(\034)6 b Fh(,)14 b(the)h(pro)q(cess)f(crosses)g(the)h(set)g(of)g (critical)f(con\014gurations)0 1952 y Fo(S)31 1959 y Fn(n)56 1964 y Fg(c)93 1952 y Fh(at)j(exactly)g(one)g(p)q(oin)o(t)f Fo(\033)575 1959 y Fn(c)596 1952 y Fh(;)g(it)h(do)q(es)f(not)h(cross)e Fo(C)1035 1959 y Fn(n)1060 1964 y Fg(c)1092 1952 y Fp(n)c Fo(S)1159 1959 y Fn(n)1184 1964 y Fg(c)1204 1952 y Fh(;)0 2012 y Fp(\017)50 b Fh(if)17 b(w)o(e)g(let)h Fo(n)298 2019 y Fj(\003)336 2012 y Fu(=)d(min)o Fp(f)p Fo(n)g Fp(\025)g Fo(\022)i Fu(:)e Fo(a)p Fu(\()p Fo(\033)739 2019 y Fn(n)767 2012 y Fu(\))h(=)f Fo(n)886 2019 y Fn(c)906 2012 y Fp(g)p Fh(,)i Fo(n)992 1994 y Fj(\003)1030 2012 y 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Fl(1)477 439 y Fo(;)8 b(l)514 446 y Fl(2)537 439 y Fu(\))15 b Fo(>)g(l)640 446 y Fn(c)660 439 y Fh(\);)j(he)g(visits)f(eac)o(h)g(of)g (them)g Fu(exp)9 b Fo(\014)s Fu(\(2)i Fp(\000)h Fo(h)p Fu(\))17 b Fh(times;)g(when)g(it)h(lea)o(v)o(es)0 499 y(de\014nitely)d(a)h(rectangle,) f(he)h(c)o(hoses)e(randomly)g(the)i(next)g(rectangle)f(it)h(creates,)g (either)f Fo(l)1713 506 y Fl(1)1744 499 y Fp(\002)9 b Fu(\()p Fo(l)1826 506 y Fl(2)1858 499 y Fu(+)g(1\))0 558 y Fh(or)16 b Fo(l)76 565 y Fl(1)109 558 y Fp(\002)11 b Fu(\()p Fo(l)193 565 y Fl(2)227 558 y Fu(+)g(1\))p Fh(;)0 618 y Fp(\017)50 b Fh(the)25 b(pro)q(cess)f(reac)o(hes)g(the)h(thermal)f(equilibrium)e(within)j (eac)o(h)f(cycle)i(of)f Fp(M)p Fu(\()p Fp(f\000)p Fu(1)p 1678 630 64 2 v -1 w Fo(;)8 b Fu(+1)p 1763 630 V Fp(g)1852 600 y Fn(c)1872 618 y Fu(\))26 b Fh(it)0 678 y(crosses:)31 b(the)21 b(distribution)f(of)i(the)f(pro)q(cess)g(b)q(efore)h(the)f(exit)i(of)e(the)h (cycle)g(is)f(v)o(ery)h(close)f(to)h(the)0 738 y(Gibbs)16 b(distribution)e (at)j(in)o(v)o(erse)e(temp)q(erature)h Fo(\014)j Fh(restricted)d(to)h(the)g (cycle.)732 858 y Fv(7.)27 b(Dimension)17 b(three)60 948 y Fu(A)f(p)q(oly)o(omino)e(asso)q(ciated)i(to)g(a)f(con\014guration)f(is)i(a)f (\014nite)h(union)e(of)i(cub)q(es.)22 b(F)l(or)14 b Fo(\033)k Fu(a)e(con\014gura-)0 1008 y(tion)h(w)o(e)g(will)f(denote)h(b)o(y)g Fo(a)p Fu(\()p Fo(\033)r Fu(\))h(the)f(area)g(of)g(the)g(asso)q(ciated)g(p)q (oly)o(omino)e(and)i(b)o(y)f Fo(v)r Fu(\()p Fo(\033)r Fu(\))i(its)f(v)o (olume.)0 1067 y(W)l(e)e(will)g(rely)g(hea)o(vily)f(on)h(the)g(notations)g (and)f(results)g(of)19 b([1].)i(Let)16 b(us)e(recall)h(brie\015y)f(some)g (essen)o(tial)0 1127 y(p)q(oin)o(ts.)0 1217 y Fw(Summary)i(of)i(the)i(com)n (binatorial)g(results.)60 1292 y Fu(The)h(parallelepip)q(ed)e(of)i(sides)f Fo(j)697 1299 y Fl(1)718 1292 y Fo(;)8 b(j)761 1299 y Fl(2)783 1292 y Fo(;)g(j)826 1299 y Fl(3)869 1292 y Fu(is)21 b(denoted)f(b)o(y)h Fo(j)1211 1299 y Fl(1)1247 1292 y Fp(\002)13 b Fo(j)1320 1299 y Fl(2)1356 1292 y Fp(\002)h Fo(j)1430 1299 y Fl(3)1452 1292 y Fu(.)35 b(By)21 b Fo(j)1604 1299 y Fl(1)1640 1292 y Fp(\002)14 b Fo(j)1714 1299 y Fl(2)1750 1292 y Fp(\002)f Fo(j)1823 1299 y Fl(3)1859 1292 y Fu(+)1898 1299 y Fn(i)1928 1292 y Fo(c)0 1352 y Fu(w)o(e)j(denote)h(the)g(parallelepip)q(ed)e Fo(j)661 1359 y Fl(1)694 1352 y Fp(\002)c Fo(j)765 1359 y Fl(2)798 1352 y Fp(\002)g Fo(j)869 1359 y Fl(3)907 1352 y Fu(with)17 b(the)g(p)q(oly)o (omino)e Fo(c)h Fu(stic)o(k)o(ed)g(against)g(the)h(side)f(of)0 1412 y Fo(j)21 1419 y Fl(1)55 1412 y Fp(\002)c Fo(j)127 1419 y Fl(2)160 1412 y Fp(\002)g Fo(j)232 1419 y Fl(3)272 1412 y Fu(whose)17 b(out)o(w)o(ard)f(normal)h(is)g(the)h(v)o(ector)g Fo(e)1098 1419 y Fn(i)1133 1412 y Fu(\(where)g Fo(e)1321 1419 y Fl(1)1343 1412 y Fo(;)8 b(e)1388 1419 y Fl(2)1411 1412 y Fo(;)g(e)1456 1419 y Fl(3)1496 1412 y Fu(is)18 b(the)g(standard)e(basis)0 1472 y(of)k Fm(Z)96 1453 y Fl(3)115 1472 y Fu(\).)31 b(The)19 b(p)q(oly)o(omino)f Fo(c)h Fu(is)g(stic)o(k)o(ed)f(in)h(the)h(p)q(osition)f (whic)o(h)f(minimizes)f(its)j(co)q(ordinates)e(with)0 1531 y(resp)q(ect)j(to)g(the)g(other)f(v)o(ectors)g(\(for)h(the)g(precise)f (de\014nition,)g(see)h([1]\).)35 b(When)20 b(w)o(e)g(don't)h(sp)q(ecify)0 1591 y(the)f(direction,)g(i.e.)32 b(w)o(e)19 b(write)h Fo(j)639 1598 y Fl(1)674 1591 y Fp(\002)13 b Fo(j)747 1598 y Fl(2)782 1591 y Fp(\002)g Fo(j)855 1598 y Fl(3)891 1591 y Fu(+)g Fo(c)p Fu(,)20 b(it)g(means)f(that)h(the)g(direction)f(is)h(not)f(relev)m(an)o(t,)0 1651 y(and)f(the)h(statemen)o(t)g(holds)e(for)i(the)g(three)g(p)q(ossible)e (p)q(oly)o(omino)q(es.)28 b(Most)18 b(often,)i Fo(c)e Fu(will)h(b)q(e)g(a)g (t)o(w)o(o)0 1711 y(dimensional)f(p)q(oly)o(omino,)j(for)f(instance)g Fo(l)833 1718 y Fl(1)869 1711 y Fp(\002)14 b Fo(l)937 1718 y Fl(2)973 1711 y Fu(+)g Fo(k)r Fu(.)34 b(In)20 b(this)g(case,)i(w)o(e)f (\014rst)f(transform)f(it)i(in)o(to)0 1770 y(a)e(three)h(dimensional)d(p)q (oly)o(omino)h(b)o(y)h(replacing)f(its)i(unit)f(squares)f(b)o(y)h(unit)g(cub) q(es.)31 b(In)19 b(this)g(w)o(a)o(y)l(,)0 1830 y(t)o(w)o(o)g(dimensional)e(p) q(oly)o(omino)q(es)h(migh)o(t)g(b)q(e)i(seen)f(as)g(three)g(dimensional)e ("planar")g(p)q(oly)o(omino)q(es.)0 1890 y(W)l(e)j(mak)o(e)f(also)h(the)g (follo)o(wing)e(con)o(v)o(en)o(tion:)27 b(when)20 b(w)o(e)f(sp)q(eak)h(of)g Fo(j)1336 1897 y Fl(1)1371 1890 y Fp(\002)13 b Fo(j)1444 1897 y Fl(2)1479 1890 y Fp(\002)g Fo(j)1552 1897 y Fl(3)1587 1890 y Fu(+)1626 1897 y Fl(3)1662 1890 y Fo(c)p Fu(,)20 b(w)o(e)f(assume)0 1950 y(implicitly)14 b(that)h(the)f(t)o(w)o(o)g(dimensional)e(p)q(oly)o (omino)i Fo(c)g Fu(is)g(included)f(in)h(the)h(rectangle)f Fo(j)1668 1957 y Fl(1)1697 1950 y Fp(\002)7 b Fo(j)1764 1957 y Fl(2)1800 1950 y Fu(so)14 b(that)0 2010 y(the)h(resulting)e(p)q(oly)o(omino)g(is)g (included)g(in)h Fo(j)848 2017 y Fl(1)877 2010 y Fp(\002)7 b Fo(j)944 2017 y Fl(2)972 2010 y Fp(\002)g Fu(\()p Fo(j)1058 2017 y Fl(3)1086 2010 y Fu(+)g(1\).)21 b(By)15 b Fo(j)1308 2017 y Fl(1)1337 2010 y Fp(\002)7 b Fo(j)1404 2017 y Fl(2)1431 2010 y Fp(\002)g Fo(j)1498 2017 y Fl(3)1526 2010 y Fp(\010)1565 2017 y Fl(3)1594 2010 y Fo(l)1609 2017 y Fl(1)1638 2010 y Fp(\002)g Fo(l)1699 2017 y Fl(2)1735 2010 y Fu(w)o(e)14 b(denote)0 2069 y(the)j(set)h(of)f(all)g(the)h(p)q(oly)o(omino)q(es)e(obtained)g(b)o(y)h (translating)f(the)h(rectangle)g Fo(l)1508 2076 y Fl(1)1542 2069 y Fp(\002)11 b Fo(l)1607 2076 y Fl(2)1647 2069 y Fu(along)16 b(the)i(side)0 2129 y Fo(j)21 2136 y Fl(1)51 2129 y Fp(\002)8 b Fo(j)119 2136 y Fl(2)155 2129 y Fu(\(in)15 b(suc)o(h)f(a)h(w)o(a)o(y)g (that)g(the)g(p)q(oly)o(omino)f(is)g(alw)o(a)o(ys)g(included)g(in)h Fo(j)1395 2136 y Fl(1)1424 2129 y Fp(\002)8 b Fo(j)1492 2136 y Fl(2)1522 2129 y Fp(\002)g Fu(\()p Fo(j)1609 2136 y Fl(3)1639 2129 y Fu(+)g(1\)\).)22 b(F)l(or)14 b Fo(A)h Fu(a)0 2191 y(set)g(of)g(p)q (oly)o(omino,)f(w)o(e)g(denote)h(b)o(y)p 683 2151 38 2 v 14 w Fo(A)h Fu(its)e(orbit)g(under)g(the)h(action)g(of)g(the)g(spatial)f (isometries)f(whic)o(h)0 2250 y(lea)o(v)o(e)j(the)h(in)o(teger)f(lattice)h Fm(Z)563 2232 y Fl(3)599 2250 y Fu(in)o(v)m(arian)o(t.)j(By)p 906 2210 V 18 w Fo(A)943 2219 y Fl(123)1022 2250 y Fu(w)o(e)c(denote)h(its)f (orbit)g(under)g(the)g(action)h(of)g(the)0 2310 y(three)f(symmetries)g(with)g (resp)q(ect)h(to)g(the)f(planes)g(\()p Fo(e)1015 2317 y Fl(1)1038 2310 y Fo(;)8 b(e)1083 2317 y Fl(2)1105 2310 y Fu(\))p Fo(;)g Fu(\()p Fo(e)1188 2317 y Fl(2)1212 2310 y Fo(;)g(e)1257 2317 y Fl(3)1280 2310 y Fu(\))p Fo(;)g Fu(\()p Fo(e)1363 2317 y Fl(1)1386 2310 y Fo(;)g(e)1431 2317 y Fl(3)1454 2310 y Fu(\).)0 2400 y Fw(Prop)r(osition)17 b(7.1.)28 b Fh(F)l(or)14 b(eac)o(h)h(in)o(teger)f Fo(n)h Fh(there)g(exists)h(a)f(unique)f Fu(6)p Fh({uple)h Fu(\()p Fo(j;)8 b(l)q(;)g(k)r(;)g(\016)o(;)g(\022)q(;)g(\017)p Fu(\))17 b Fh(suc)o(h)d(that)0 2460 y Fo(\016)o(;)8 b(\022)q(;)g(\017)15 b Fp(2)f(f)p Fu(0)p Fo(;)8 b Fu(1)p Fp(g)p Fh(,)13 b Fo(\016)j Fp(\024)e Fo(\022)q Fh(,)g Fo(k)h(<)f(l)t Fu(+)s Fo(\017;)8 b(l)q Fu(\()p Fo(l)t Fu(+)s Fo(\017)p Fu(\))s(+)s Fo(k)15 b(<)f Fu(\()p Fo(j)6 b Fu(+)s Fo(\016)r Fu(\)\()p Fo(j)g Fu(+)s Fo(\022)q Fu(\))13 b Fh(and)f Fo(n)h Fu(=)h Fo(j)s Fu(\()p Fo(j)6 b Fu(+)s Fo(\016)r Fu(\)\()p Fo(j)g Fu(+)s Fo(\022)q Fu(\))s(+)s Fo(l)q Fu(\()p Fo(l)t Fu(+)s Fo(\017)p Fu(\))s(+)s Fo(k)r Fh(.)0 2550 y Fu(The)17 b(set)g(of)g(the)g(p)q(oly)o(omino)q(es)e(of)i(v)o(olume)f Fo(n)h Fu(is)f Fp(C)961 2557 y Fn(n)989 2550 y Fu(;)g(the)h(set)g Fp(M)1243 2557 y Fn(n)1287 2550 y Fu(of)g(the)g(minimal)e(p)q(oly)o(omino)q (es)h(of)954 2670 y Fs(29)p eop %%Page: 30 30 30 29 bop 0 200 a Fu(v)o(olume)13 b Fo(n)g Fu(is)g(the)h(set)f(of)h(the)g(p)q (oly)o(omino)q(es)e(of)i Fp(C)915 207 y Fn(n)956 200 y Fu(ha)o(ving)e (minimal)g(area.)20 b(Let)14 b Fo(n)g Fu(=)g Fo(j)s Fu(\()p Fo(j)8 b Fu(+)d Fo(\016)r Fu(\)\()p Fo(j)j Fu(+)d Fo(\022)q Fu(\))g(+)0 260 y Fo(l)q Fu(\()p Fo(l)h Fu(+)f Fo(\017)p Fu(\))g(+)g Fo(k)15 b Fu(b)q(e)e(the)h(decomp)q(osition)e(of)i Fo(n)p Fu(.)21 b(W)l(e)13 b(put)g Fo(r)j Fu(=)e Fo(l)q Fu(\()p Fo(l)6 b Fu(+)f Fo(\017)p Fu(\))g(+)g Fo(k)r Fu(.)20 b(The)13 b(canonical)g(p)q(oly)o(omino)f Fb(m)1923 267 y Fn(n)0 319 y Fu(is)17 b(obtained)g(b)o(y)g(adding)g(the)h(t)o (w)o(o)f(dimensional)e(canonical)i(p)q(oly)o(omino)f Fo(m)1456 326 y Fn(r)1496 319 y Fu(to)i(the)g(righ)o(t)e(side)h(of)h(a)0 379 y(quasicub)q(e)e(of)h(v)o(olume)e Fo(j)s Fu(\()p Fo(j)e Fu(+)e Fo(\016)r Fu(\)\()p Fo(j)j Fu(+)d Fo(\022)q Fu(\).)23 b(The)17 b(general)f(form)o(ula)e(is)382 497 y Fb(m)420 504 y Fn(n)462 497 y Fu(=)f(\()p Fo(j)h Fu(+)d Fo(\022)q Fu(\))h Fp(\002)f Fu(\()p Fo(j)j Fu(+)c Fo(\016)r Fu(\))i Fp(\002)f Fo(j)i Fu(+)1004 504 y Fl(1+)p Fn(\016)q Fl(+)p Fn(\022)1138 457 y Fk(\000)1161 497 y Fu(\()p Fo(l)f Fu(+)f Fo(\017)p Fu(\))g Fp(\002)g Fo(l)h Fu(+)1423 504 y Fl(1+)p Fn(\017)1503 497 y Fo(k)1531 457 y Fk(\001)1554 497 y Fo(:)0 615 y Fu(The)k(p)q(oly)o(omino)g Fb(m)383 622 y Fn(n)426 615 y Fu(is)h(called)f(the)g(canonical)g(p)q(oly)o (omino)f(of)i(v)o(olume)f Fo(n)p Fu(.)0 704 y Fw(Theorem)h(7.2.)28 b Fh(The)16 b(p)q(oly)o(omino)g Fb(m)744 711 y Fn(n)787 704 y Fh(is)h(minimal.)0 794 y Fu(W)l(e)g(de\014ne)f(sev)o(eral)f(imp)q(ortan)o (t)h(subsets)f(of)i Fp(M)918 801 y Fn(n)945 794 y Fu(.)22 b(The)16 b(set)h Fp(S)1191 801 y Fn(n)1235 794 y Fu(of)g(the)f(standard)f(p)q(oly)o (omino)q(es)h(is)408 924 y Fp(S)438 931 y Fn(n)487 924 y Fu(=)p 548 871 995 2 v 22 w(\()p Fo(j)e Fu(+)d Fo(\022)q Fu(\))h Fp(\002)f Fu(\()p Fo(j)j Fu(+)c Fo(\016)r Fu(\))i Fp(\002)f Fo(j)i Fp(\010)1038 931 y Fl(1+)p Fn(\016)q Fl(+)p Fn(\022)p 1172 880 371 2 v 1172 924 a Fu(\()p Fo(l)f Fu(+)f Fo(\017)p Fu(\))h Fp(\002)e Fo(l)i Fp(\010)1434 931 y Fl(1+)p Fn(\017)1515 924 y Fo(k)0 1049 y Fu(and)k(the)h(set)272 1037 y Fk(f)260 1049 y Fp(M)320 1056 y Fn(n)364 1049 y Fu(of)g(the)f(principal)f(p)q(oly)o(omino)q(es)g(is)413 1167 y Fk(f)401 1180 y Fp(M)461 1187 y Fn(n)510 1180 y Fu(=)606 1132 y Fk([)577 1231 y Fg(t)p Fd(=1)p Fg(;)p Fd(2)p Fg(;)p Fd(3)587 1260 y Fg(u)p Fd(=1)p Fg(;)p Fd(2)p 704 1126 846 2 v 704 1180 a Fu(\()p Fo(j)f Fu(+)d Fo(\022)q Fu(\))h Fp(\002)f Fu(\()p Fo(j)j Fu(+)c Fo(\016)r Fu(\))i Fp(\002)f Fo(j)i Fp(\010)1194 1187 y Fn(t)p 1223 1136 327 2 v 1223 1180 a Fu(\()p Fo(l)f Fu(+)f Fo(\017)p Fu(\))g Fp(\002)g Fo(l)h Fp(\010)1485 1187 y Fn(u)1522 1180 y Fo(k)0 1373 y Fw(Theorem)17 b(7.3.)50 b Fh(The)17 b(set)f Fp(M)623 1380 y Fn(n)667 1373 y Fh(is)g(reduced)f(to)i Fp(f)p Fb(m)1026 1380 y Fn(n)1054 1373 y Fp(g)f Fh(if)h(and)f(only)g(if)h Fo(n)f Fh(is)g(of)h(the)g(form)e Fo(j)1773 1355 y Fl(3)1795 1373 y Fh(.)0 1433 y(The)24 b(set)h Fp(M)256 1440 y Fn(n)308 1433 y Fh(is)f(equal)g(to)h Fp(S)604 1440 y Fn(n)656 1433 y Fh(if)g(and)f(only)g(if)h Fo(n)f Fh(is)g(of)h(the)g(form)e Fo(j)1406 1415 y Fl(3)1428 1433 y Fo(;)8 b(j)1474 1415 y Fl(2)1496 1433 y Fu(\()p Fo(j)19 b Fu(+)d(1\))p Fo(;)8 b(j)s Fu(\()p Fo(j)19 b Fu(+)d(1\))1858 1415 y Fl(2)1906 1433 y Fh(or)0 1493 y Fo(j)24 1475 y Fl(3)57 1493 y Fp(\000)11 b Fu(1)p Fo(;)d(j)178 1475 y Fl(2)199 1493 y Fu(\()p Fo(j)14 b Fu(+)d(1\))g Fp(\000)g Fu(1)p Fo(;)d(j)s Fu(\()p Fo(j)13 b Fu(+)e(1\))626 1475 y Fl(2)660 1493 y Fp(\000)g Fu(1)16 b Fh(\(in)h(whic)o(h)e(case)i Fo(S)1108 1500 y Fn(n)1148 1493 y Fu(=)p 1201 1463 39 2 v 14 w Fb(m)1239 1500 y Fn(n)1266 1493 y Fh(\),)g(or)f Fo(j)1401 1475 y Fl(3)1434 1493 y Fu(+)10 b Fo(l)1499 1475 y Fl(2)1522 1493 y Fo(;)e(j)1568 1475 y Fl(3)1601 1493 y Fu(+)i Fo(l)q Fu(\()p Fo(l)i Fu(+)f(1\))p Fh(.)0 1560 y(The)16 b(set)h Fp(M)240 1567 y Fn(n)284 1560 y Fh(coincides)e(with)620 1548 y Fk(f)608 1560 y Fp(M)668 1567 y Fn(n)711 1560 y Fh(if)i(and)f(only)g(if)h Fo(n)f Fh(is)g(of)h(the)g(form)f (quasicub)q(e+quasisquare)551 1670 y Fo(j)575 1652 y Fl(3)794 1670 y Fo(j)818 1652 y Fl(3)851 1670 y Fu(+)11 b Fo(l)917 1652 y Fl(2)1137 1670 y Fo(j)1161 1652 y Fl(3)1193 1670 y Fu(+)g Fo(l)q Fu(\()p Fo(l)h Fu(+)f(1\))477 1730 y Fo(j)501 1712 y Fl(2)522 1730 y Fu(\()p Fo(j)j Fu(+)d(1\))50 b Fo(j)744 1712 y Fl(2)766 1730 y Fu(\()p Fo(j)14 b Fu(+)c(1\))i(+)f Fo(l)991 1712 y Fl(2)1063 1730 y Fo(j)1087 1712 y Fl(2)1108 1730 y Fu(\()p Fo(j)j Fu(+)d(1\))g(+)g Fo(l)q Fu(\()p Fo(l)h Fu(+)f(1\))477 1790 y Fo(j)s Fu(\()p Fo(j)i Fu(+)e(1\))648 1772 y Fl(2)720 1790 y Fo(j)s Fu(\()p Fo(j)i Fu(+)e(1\))891 1772 y Fl(2)925 1790 y Fu(+)g Fo(l)991 1772 y Fl(2)1063 1790 y Fo(j)s Fu(\()p Fo(j)i Fu(+)e(1\))1234 1772 y Fl(2)1267 1790 y Fu(+)g Fo(l)q Fu(\()p Fo(l)h Fu(+)f(1\))0 1906 y Fh(\(where)17 b(the)h(quasisquare)d Fo(l)536 1888 y Fl(2)576 1906 y Fh(or)h Fo(l)q Fu(\()p Fo(l)d Fu(+)e(1\))18 b Fh(is)f(strictly)g(smaller)f(than)g(the)i(greatest)f(face)h Fo(j)1725 1888 y Fl(2)1747 1906 y Fh(,)f Fo(j)s Fu(\()p Fo(j)c Fu(+)f(1\))0 1966 y Fh(or)k Fu(\()p Fo(j)e Fu(+)d(1\))209 1948 y Fl(2)248 1966 y Fh(of)17 b(the)f(quasicub)q(e\))h(or)f(quasicub)q(e)g(min)o (us)e(one)j Fo(j)1192 1948 y Fl(3)1224 1966 y Fp(\000)11 b Fu(1)p Fo(;)d(j)1345 1948 y Fl(2)1367 1966 y Fu(\()p Fo(j)14 b Fu(+)d(1\))g Fp(\000)g Fu(1)p Fo(;)d(j)s Fu(\()p Fo(j)13 b Fu(+)e(1\))1794 1948 y Fl(2)1827 1966 y Fp(\000)g Fu(1)p Fh(.)0 2055 y Fw(Theorem)16 b(7.4.)50 b Fh(The)15 b(principal)e(p)q(oly)o (omino)q(es)i(can)g(b)q(e)h(completely)f(shrink)o(ed)e(through)i(the)g(min-)0 2120 y(imal)f(p)q(oly)o(omino)q(es:)20 b(for)14 b(an)o(y)g(in)o(teger)g Fo(n)h Fh(and)f(for)g(an)o(y)h(principal)e(p)q(oly)o(omino)g Fo(c)h Fh(in)1599 2107 y Fk(f)1588 2120 y Fp(M)1648 2127 y Fn(n)1674 2120 y Fh(,)h(there)g(exists)0 2180 y(an)f(increasing)f(sequence)h Fo(c)524 2187 y Fl(0)546 2180 y Fo(;)8 b Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(c)687 2187 y Fn(n)728 2180 y Fh(of)15 b(minimal)e(p)q(oly)o(omino) q(es)g(suc)o(h)g(that)i Fo(c)1496 2187 y Fl(0)1532 2180 y Fu(=)e Fp(;)i Fh(and)f Fo(c)1741 2187 y Fn(n)1781 2180 y Fu(=)g Fo(c)g Fh(and)0 2239 y Fo(q)r Fu(\()p Fo(c)65 2246 y Fn(i)p Fj(\000)p Fl(1)133 2239 y Fo(;)8 b(c)177 2246 y Fn(i)193 2239 y Fu(\))14 b Fo(>)g Fu(0)i Fh(for)g Fo(i)h Fh(in)f Fp(f)p Fu(1)8 b Fp(\001)g(\001)g (\001)h Fo(n)p Fp(g)p Fh(.)0 2329 y Fw(Theorem)26 b(7.5.)74 b Fh(The)24 b(standard)f(p)q(oly)o(omino)q(es)g(can)h(b)q(e)h(gro)o(wn)e(or)h (shrink)o(ed)f(arbitrarily)f(far)0 2388 y(through)12 b(minimal)f(p)q(oly)o (omino)q(es:)19 b(for)12 b(an)o(y)h(in)o(tegers)f Fo(m)h Fp(\024)h Fo(n)f Fh(and)f(for)h(an)o(y)f(standard)f(p)q(oly)o(omino)h Fo(c)g Fh(in)0 2448 y Fp(S)30 2455 y Fn(m)68 2448 y Fh(,)h(there)e(exists)h (an)g(increasing)e(sequence)i Fo(c)867 2455 y Fl(0)889 2448 y Fo(;)c Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(c)1030 2455 y Fn(n)1068 2448 y Fh(of)k(minimal)e(p)q(oly)o(omino)q(es)h(suc)o(h)g(that)h Fo(c)1823 2455 y Fl(0)1859 2448 y Fu(=)h Fp(;)p Fh(,)0 2508 y Fo(c)22 2515 y Fn(m)73 2508 y Fu(=)h Fo(c)i Fh(and)g Fo(q)r Fu(\()p Fo(c)326 2515 y Fn(i)p Fj(\000)p Fl(1)393 2508 y Fo(;)8 b(c)437 2515 y Fn(i)454 2508 y Fu(\))14 b Fo(>)f Fu(0)k Fh(for)f Fo(i)h Fh(in)f Fp(f)p Fu(1)8 b Fp(\001)g(\001)g(\001)h Fo(n)p Fp(g)p Fh(.)954 2670 y Fs(30)p eop %%Page: 31 31 31 30 bop 0 200 a Fw(Lemma)13 b(7.6.)49 b Fh(F)l(or)12 b Fo(n)h Fh(a)g(quasicub)q(e)g(plus)f(a)h(quasisquare)f(i.e.)21 b(of)13 b(the)h(form)e Fo(j)s Fu(\()p Fo(j)7 b Fu(+)t Fo(\016)r Fu(\)\()p Fo(j)g Fu(+)t Fo(\022)q Fu(\))t(+)t Fo(l)q Fu(\()p Fo(l)e Fu(+)t Fo(\017)p Fu(\))0 260 y Fh(\(where)16 b Fu(0)e Fo(<)g(l)q Fu(\()p Fo(l)e Fu(+)f Fo(\017)p Fu(\))j Fo(<)g Fu(\()p Fo(j)f Fu(+)e Fo(\016)r Fu(\)\()p Fo(j)k Fu(+)10 b Fo(\022)q Fu(\))p Fh(\))18 b(w)o(e)f(ha)o(v)o(e)421 383 y Fp(f)8 b Fo(c)14 b Fp(2)g(M)597 390 y Fn(n)p Fj(\000)p Fl(1)688 383 y Fu(:)g Fo(q)r Fu(\()771 371 y Fk(f)759 383 y Fp(M)819 390 y Fn(n)857 383 y Fp(n)d Fo(S)924 390 y Fn(n)951 383 y Fo(;)d(c)p Fu(\))14 b(=)g(1)8 b Fp(g)22 b(\033)1234 371 y Fk(f)1222 383 y Fp(M)1282 390 y Fn(n)p Fj(\000)p Fl(1)1371 383 y Fp(n)11 b Fo(S)1438 390 y Fn(n)p Fj(\000)p Fl(1)1515 383 y Fo(;)555 460 y Fp(f)d Fo(c)14 b Fp(2)g(M)731 467 y Fn(n)p Fj(\000)p Fl(1)823 460 y Fu(:)f Fo(q)r Fu(\()p Fp(S)923 467 y Fn(n)951 460 y Fo(;)8 b(c)p Fu(\))14 b Fo(>)g Fu(0)8 b Fp(g)22 b(\033)g(S)1252 467 y Fn(n)p Fj(\000)p Fl(1)1330 460 y Fo(;)526 550 y Fp(f)8 b Fo(c)14 b Fp(2)g(M)702 557 y Fn(n)p Fl(+1)793 550 y Fu(:)g Fo(q)r Fu(\()p Fp(M)924 557 y Fn(n)951 550 y Fo(;)8 b(c)p Fu(\))14 b Fo(>)g Fu(0)8 b Fp(g)22 b Fu(=)1234 538 y Fk(f)1222 550 y Fp(M)1282 557 y Fn(n)p Fl(+1)1359 550 y Fo(;)0 669 y Fh(where)16 b Fo(q)r Fu(\()p Fo(A;)8 b(c)p Fu(\))15 b(=)e(min)o Fp(f)p Fo(q)r Fu(\()p Fo(d;)8 b(c)p Fu(\))14 b(:)g Fo(q)i Fp(2)e Fo(A)p Fp(g)p Fh(.)0 758 y Fw(Notation)k(7.7.)28 b Fu(W)l(e)16 b(ha)o(v)o(e)f(also)f(some)h(results)g(concerning)f(the)i(b)q (est)g(w)o(a)o(y)f(to)g(shrink)g(or)g(to)h(gro)o(w)e(a)0 817 y(parallelepip)q(ed)h(plus)g(a)h(rectangle,)g(that)h(is)f(on)g(the)g(sets)g (\(where)h Fo(\033)h Fu(is)e(either)g(a)h(parallelepip)q(ed)d(or)i(a)0 877 y(parallelepip)q(ed)f(plus)g(a)i(rectangle\):)422 993 y Fp(M)p Fu(\()p Fo(\033)o(;)8 b Fp(\000)p Fo(k)r Fu(\))22 b(=)g Fp(f)8 b Fo(c)14 b Fp(2)g Fo(C)871 1002 y Fn(v)q Fl(\()p Fn(\033)q Fl(\))p Fj(\000)p Fn(k)1017 993 y Fu(:)f Fo(c)h Fp(\032)f Fo(\033)o(;)26 b(a)p Fu(\()p Fo(c)p Fu(\))17 b(minimal)6 b Fp(g)p Fo(;)422 1071 y Fp(M)p Fu(\()p Fo(\033)o(;)i Fu(+)p Fo(k)r Fu(\))22 b(=)g Fp(f)8 b Fo(c)14 b Fp(2)g Fo(C)871 1080 y Fn(v)q Fl(\()p Fn(\033)q Fl(\)+)p Fn(k)1016 1071 y Fu(:)g Fo(\033)i Fp(\032)d Fo(c;)25 b(a)p Fu(\()p Fo(c)p Fu(\))17 b(minimal)6 b Fp(g)p Fo(:)0 1189 y Fu(W)l(e)17 b(denote)f(b)o(y)g Fo(E)s Fu(\()p Fo(\033)o(;)8 b(k)r Fu(\))18 b(the)e(energy)h(of)f(an)h(elemen)o(t)f(of)g Fp(M)p Fu(\()p Fo(\033)o(;)8 b(k)r Fu(\))18 b(for)e Fo(k)i Fu(in)e Fm(Z)-11 b Fu(.)60 1278 y(The)16 b(next)g(t)o(w)o(o)f(results)g(are)h (restatemen)o(ts)f(of)h(Theorem)f(3)p Fo(:)p Fu(26,)g(Corollary)g(3)p Fo(:)p Fu(27)p Fo(;)h Fu(3)p Fo(:)p Fu(28)f(and)h(Theo-)0 1338 y(rem)g(3)p Fo(:)p Fu(30)g(of)21 b([1].)0 1427 y Fw(Theorem)h(7.8.)62 b Fh(Let)21 b Fo(j)514 1434 y Fl(1)536 1427 y Fo(;)8 b(j)579 1434 y Fl(2)601 1427 y Fo(;)g(j)644 1434 y Fl(3)666 1427 y Fo(;)g(r)22 b Fh(b)q(e)f(p)q(ositiv)o(e)g(in)o(tegers)e(suc)o(h)g(that)i Fo(r)i(<)d Fu(min)o(\()p Fo(j)1636 1434 y Fl(1)1658 1427 y Fo(j)1679 1434 y Fl(2)1701 1427 y Fo(;)8 b(j)1744 1434 y Fl(2)1766 1427 y Fo(j)1787 1434 y Fl(3)1809 1427 y Fo(;)g(j)1852 1434 y Fl(3)1874 1427 y Fo(j)1895 1434 y Fl(1)1917 1427 y Fu(\))p Fh(.)0 1486 y(The)15 b(set)h Fp(M)p Fu(\()p Fo(j)278 1493 y Fl(1)309 1486 y Fp(\002)9 b Fo(j)378 1493 y Fl(2)409 1486 y Fp(\002)g Fo(j)478 1493 y Fl(3)500 1486 y Fo(;)f Fp(\000)p Fo(r)q Fu(\))17 b Fh(is)e(the)h(set)f(of)h(the)g(p)q(oly)o(omino)q(es)e (obtained)h(b)o(y)g(remo)o(ving)f(from)h Fo(j)1880 1493 y Fl(1)1911 1486 y Fp(\002)0 1546 y Fo(j)21 1553 y Fl(2)57 1546 y Fp(\002)f Fo(j)131 1553 y Fl(3)173 1546 y Fh(as)21 b(man)o(y)f(bars)g(as)g(p)q (ossible,)h(and)f(then)h(remo)o(ving)e(a)i(succession)e(of)j(corner)d(cub)q (es)i(un)o(til)0 1606 y(reac)o(hing)16 b(the)h(v)o(olume)g Fo(j)479 1613 y Fl(1)501 1606 y Fo(j)522 1613 y Fl(2)543 1606 y Fo(j)564 1613 y Fl(3)598 1606 y Fp(\000)11 b Fo(r)q Fh(.)25 b(In)17 b(particular,)f(a)i(p)q(oly)o(omino)d(obtained)i(from)g Fo(j)1649 1613 y Fl(1)1682 1606 y Fp(\002)11 b Fo(j)1753 1613 y Fl(2)1787 1606 y Fp(\002)g Fo(j)1858 1613 y Fl(3)1897 1606 y Fh(b)o(y)0 1666 y(the)i(successiv)o(e)g(remo)o(v)m(al)f(of)h Fo(r)i Fh(cub)q(es)e(in)g(suc)o(h)f(a)h(w)o(a)o(y)g(that)g(eac)o(h)g(cub)q(e) g(remo)o(v)m(al)g(tak)o(es)g(place)g(on)g(a)g(bar)0 1725 y(of)f(minimal)e (length)h(is)h(in)f Fp(M)p Fu(\()p Fo(j)587 1732 y Fl(1)610 1725 y Fp(\002)q Fo(j)671 1732 y Fl(2)695 1725 y Fp(\002)q Fo(j)756 1732 y Fl(3)778 1725 y Fo(;)d Fp(\000)p Fo(r)q Fu(\))p Fh(.)22 b(Th)o(us)10 b(for)h(eac)o(h)h(p)q(oly)o(omino)e Fo(c)i Fh(in)f Fp(M)p Fu(\()p Fo(j)1641 1732 y Fl(1)1664 1725 y Fp(\002)q Fo(j)1725 1732 y Fl(2)1749 1725 y Fp(\002)q Fo(j)1810 1732 y Fl(3)1832 1725 y Fo(;)d Fp(\000)p Fo(r)q Fu(\))p Fh(,)0 1785 y(there)17 b(exists)h(a)f(p)q(oly)o(omino)f Fo(d)i Fh(in)f Fp(M)p Fu(\()p Fo(j)755 1792 y Fl(1)789 1785 y Fp(\002)11 b Fo(j)860 1792 y Fl(2)894 1785 y Fp(\002)g Fo(j)965 1792 y Fl(3)987 1785 y Fo(;)d Fp(\000)p Fo(r)13 b Fu(+)f(1\))18 b Fh(suc)o(h)e(that)i Fo(q)r Fu(\()p Fo(d;)8 b(c)p Fu(\))16 b Fo(>)f Fu(0)i Fh(i.e.)25 b Fo(c)17 b Fh(ma)o(y)g(b)q(e)0 1845 y(obtained)f(b)o(y)g(remo)o(ving)f(a)h (unit)h(cub)q(e)f(from)g Fo(d)p Fh(.)0 1934 y Fw(Theorem)d(7.9.)49 b Fh(Let)13 b Fo(j)484 1941 y Fl(1)506 1934 y Fo(;)8 b(j)549 1941 y Fl(2)571 1934 y Fo(;)g(j)614 1941 y Fl(3)636 1934 y Fo(;)g(r)15 b Fh(b)q(e)e(p)q(ositiv)o(e)f(in)o(tegers)g(suc)o(h)g(that)h Fo(r)j(<)d Fu(min)o(\()p Fo(j)1549 1916 y Fl(2)1546 1946 y(1)1571 1934 y Fo(;)8 b(j)1617 1916 y Fl(2)1614 1946 y(2)1639 1934 y Fo(;)g(j)1685 1916 y Fl(2)1682 1946 y(3)1707 1934 y Fu(\))p Fh(.)21 b(The)13 b(b)q(est)0 1994 y(w)o(a)o(y)i(to)i(add)e Fo(r)j Fh(cub)q(es)d(to)i(the)f(parallelepip)q(ed)e Fo(j)919 2001 y Fl(1)950 1994 y Fp(\002)c Fo(j)1020 2001 y Fl(2)1052 1994 y Fp(\002)f Fo(j)1121 2001 y Fl(3)1159 1994 y Fh(is)15 b(to)i(add)e(a)h(minimal)e(t)o(w)o(o)h(dimensional)0 2053 y(p)q(oly)o(omino)g (of)i Fo(M)347 2060 y Fn(r)386 2053 y Fh(on)f(one)g(side)g(of)h(the)g (parallelepip)q(ed.)j(Equiv)m(alen)o(tly)l(,)c(w)o(e)g(ha)o(v)o(e)256 2182 y Fp(M)p Fu(\()p Fo(j)356 2189 y Fl(1)389 2182 y Fp(\002)10 b Fo(j)459 2189 y Fl(2)492 2182 y Fp(\002)h Fo(j)563 2189 y Fl(3)585 2182 y Fo(;)d Fu(+)p Fo(r)q Fu(\))23 b(=)p 772 2139 847 2 v 22 w Fp(f)8 b Fo(j)826 2189 y Fl(1)859 2182 y Fp(\002)j Fo(j)930 2189 y Fl(2)963 2182 y Fp(\002)g Fo(j)1034 2189 y Fl(3)1067 2182 y Fp(\010)1106 2189 y Fn(i)1133 2182 y Fo(d;)25 b Fu(1)14 b Fp(\024)f Fo(i)h Fp(\024)g Fu(3)p Fo(;)i(d)e Fp(2)g Fo(M)1563 2189 y Fn(r)1594 2182 y Fp(g)1619 2147 y Fl(123)1681 2182 y Fo(:)0 2298 y Fh(In)i(particular,)f Fo(j)327 2305 y Fl(1)360 2298 y Fp(\002)c Fo(j)431 2305 y Fl(2)464 2298 y Fp(\002)g Fo(j)535 2305 y Fl(3)567 2298 y Fp(\010)g Fo(m)661 2305 y Fn(r)697 2298 y Fp(\032)j(M)p Fu(\()p Fo(j)850 2305 y Fl(1)883 2298 y Fp(\002)c Fo(j)953 2305 y Fl(2)986 2298 y Fp(\002)h Fo(j)1057 2305 y Fl(3)1079 2298 y Fo(;)d Fu(+)p Fo(r)q Fu(\))p Fh(.)0 2416 y Fw(Application)22 b(to)d(the)g(three)h(dimensional)f(Ising)g(mo)r (del.)60 2490 y Fu(W)l(e)c(\014rst)g(express)f(the)h(energy)g(of)g(the)h (Ising)e(mo)q(del)h(with)g(the)g(help)g(of)g(the)g(area)g(and)g(the)g(v)o (olume)0 2550 y(of)i(the)g(p)q(oly)o(omino)e(asso)q(ciated)h(to)h(the)g (con\014guration.)954 2670 y Fs(31)p eop %%Page: 32 32 32 31 bop 0 200 a Fw(Lemma)17 b(7.10.)28 b Fh(F)l(or)15 b(an)o(y)h (con\014guration)f Fo(\033)k Fh(in)d Fo(X)t Fh(,)g(w)o(e)h(ha)o(v)o(e)99 345 y Fo(E)s Fu(\()p Fo(\033)r Fu(\))31 b(=)f Fp(\000)352 311 y Fu(1)p 352 334 25 2 v 352 379 a(2)447 298 y Fk(X)391 407 y Fj(f)p Fn(x;y)q Fj(g)p Fl(:)p Fn(x)p Fj(\030)p Fn(y)583 345 y Fo(\033)r Fu(\()p Fo(x)p Fu(\))p Fo(\033)r Fu(\()p Fo(y)r Fu(\))22 b Fp(\000)859 311 y Fo(h)p 859 334 29 2 v 861 379 a Fu(2)904 298 y Fk(X)902 405 y Fn(x)p Fj(2)p Fl(\003)987 345 y Fo(\033)r Fu(\()p Fo(x)p Fu(\))32 b(=)e Fo(a)p Fu(\()p Fo(\033)r Fu(\))21 b Fp(\000)e Fo(hv)r Fu(\()p Fo(\033)r Fu(\))h(+)f Fo(N)1603 324 y Fl(3)1626 345 y Fu(\()p Fo(h)11 b Fp(\000)g Fu(3\))p Fo(=)p Fu(2)d Fo(:)60 523 y Fu(W)l(e)19 b(do)f(not)h(c)o(hange)f (the)h(dynamics)e(nor)h(the)h(cycle)h(decomp)q(osition)d(b)o(y)h(adding)g(a)h (constan)o(t)f(to)0 583 y(the)j(energy)f Fo(E)s Fu(.)33 b(In)20 b(dimension)f(three,)i(w)o(e)f(will)g(w)o(ork)g(with)g(the)h(energy)f Fo(E)s Fu(\()p Fo(\033)r Fu(\))h(=)f Fo(a)p Fu(\()p Fo(\033)r Fu(\))15 b Fp(\000)f Fo(hv)r Fu(\()p Fo(\033)r Fu(\).)0 643 y(W)l(e)k(denote)f(b)o(y)h Fo(E)s Fu(\()p Fo(n)p Fu(\))g(the)f(minimal)f (energy)h(of)h(a)g(con\014guration)e(of)h Fp(C)1373 650 y Fn(n)1418 643 y Fu(\(so)h(that)g Fo(E)s Fu(\()p Fo(n)p Fu(\))e(=)f Fo(E)s Fu(\()p Fo(\033)r Fu(\))h(=)0 702 y Fo(a)p Fu(\()p Fo(n)p Fu(\))c Fp(\000)f Fo(hn)16 b Fu(for)g(an)o(y)g Fo(\033)j Fu(in)d Fp(M)566 709 y Fn(n)593 702 y Fu(\).)60 795 y Fe(Hyp)m(othesis)g(on)f(the)f(magnetic)g (\014eld)g Fo(h)g Fe(and)h(the)f(size)h Fo(N)5 b Fe(.)26 b Fu(W)l(e)13 b(supp)q(ose)e(that)i Fo(h)f 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5 1 4 0 4 1 4 1 4 1 -20 1 4 1 4 1 4 1 4 1 -20 0 4 1 4 1 5 1 4 1 -20 1 4 1 4 1 4 1 -20 0 4 1 4 1 4 1 -20 1 4 1 4 1 -20 1 4 1 5 1 -20 0 4 1 -20 1 4 1 -20 1 -20 1 -45 1 5 1 4 1 4 0 4 1 4 1 4 1 4 1 4 1 4 1 4 1 5 1 -20 0 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 5 0 -20 1 4 1 4 1 4 1 4 1 4 1 4 1 4 2543 610 100 MP stroke 1 4 0 4 1 5 1 -20 1 4 1 4 1 4 1 4 1 4 1 4 1 4 0 4 1 5 1 -21 1 5 1 4 1 4 1 4 1 4 1 4 0 4 1 4 1 4 1 -20 1 4 1 5 1 4 1 4 1 4 0 4 1 4 1 4 1 -20 1 4 1 4 1 4 1 5 1 4 1 4 0 4 1 4 1 -20 1 4 1 4 1 4 1 4 1 4 1 5 0 4 1 -20 1 4 1 4 1 4 1 4 1 4 1 4 1 4 0 -20 1 4 1 5 1 4 1 4 1 4 1 4 1 -20 1 4 1 4 0 4 1 4 1 4 1 5 1 -20 1 4 1 4 1 4 1 4 0 4 1 -20 1 4 1 4 1 4 1 4 1 5 1 -21 1 5 0 4 1 4 1 4 1 -20 1 4 1 4 1 4 1 4 1 -20 1 4 0 4 1 5 1 -20 1 4 2455 541 100 MP stroke 1 4 1 4 1 -20 1 4 1 4 0 -20 1 4 1 4 1 -20 1 4 1 -20 1 4 1 -20 1 -20 1 -44 0 4 1 4 1 4 1 4 1 5 1 4 1 4 1 4 1 4 0 4 1 4 1 -20 1 4 1 4 1 4 1 5 1 4 1 4 0 4 1 4 1 4 1 4 1 -20 1 4 1 4 1 4 1 5 1 4 0 4 1 4 1 4 1 4 1 4 1 -20 1 4 1 4 1 4 0 5 1 4 1 4 1 4 1 4 1 4 1 -20 1 4 1 4 0 4 1 4 1 5 1 4 1 4 1 4 1 4 1 -20 1 4 1 4 0 4 1 4 1 4 1 5 1 4 1 4 1 -20 1 4 1 4 0 4 1 4 1 4 1 4 1 4 1 5 1 -20 1 4 1 4 0 4 1 4 1 4 1 4 1 4 1 -20 1 4 1 4 1 5 1 4 2366 521 100 MP stroke 0 4 1 4 1 4 1 -20 1 4 1 4 1 4 1 4 1 4 0 5 1 -20 1 4 1 4 1 4 1 4 1 4 1 4 1 -20 0 4 1 4 1 4 1 5 1 4 1 -20 1 4 1 4 1 4 1 4 0 4 1 -20 1 4 1 4 1 4 1 5 1 -21 1 5 1 4 0 4 1 4 1 -20 1 4 1 4 1 4 1 -20 1 4 1 4 1 4 0 -20 1 4 1 5 1 -20 1 4 1 4 1 -20 1 4 1 -20 0 4 1 -20 1 -20 1 -44 1 4 1 4 1 4 1 4 1 4 0 4 1 4 1 4 1 5 1 4 1 -20 1 4 1 4 1 4 1 4 0 4 1 4 1 4 1 4 1 5 1 4 1 -20 1 4 1 4 0 4 1 4 1 4 1 4 1 4 1 4 1 5 1 -20 1 4 0 4 1 4 1 4 1 4 1 4 1 4 2278 574 100 MP stroke 1 4 1 4 1 -20 1 5 0 4 1 4 1 4 1 4 1 4 1 4 1 4 1 -20 1 4 0 4 1 5 1 4 1 4 1 4 1 4 1 4 1 -20 1 4 0 4 1 4 1 4 1 5 1 4 1 4 1 -20 1 4 1 4 1 4 0 4 1 4 1 4 1 4 1 -20 1 5 1 4 1 4 1 4 0 4 1 4 1 -20 1 4 1 4 1 4 1 4 1 5 1 4 1 -20 0 4 1 4 1 4 1 4 1 4 1 -20 1 4 1 4 1 4 0 4 1 5 1 -20 1 4 1 4 1 4 1 4 1 -20 1 4 0 4 1 4 1 4 1 -20 1 4 1 5 1 4 1 -20 1 4 1 4 0 4 1 -20 1 4 1 4 1 -20 1 4 1 4 1 -20 1 4 0 -20 1 4 1 -20 1 -20 1 -44 1 4 1 4 1 4 1 5 0 4 1 4 2190 650 100 MP stroke 1 4 1 4 1 4 1 4 1 -20 1 4 1 4 1 4 0 5 1 4 1 4 1 4 1 4 1 4 1 -20 1 4 1 4 0 4 1 4 1 5 1 4 1 4 1 4 1 4 1 -20 1 4 0 4 1 4 1 4 1 4 1 5 1 4 1 4 1 -20 1 4 1 4 0 4 1 4 1 4 1 4 1 4 1 5 1 -21 1 5 1 4 0 4 1 4 1 4 1 4 1 4 1 -20 1 4 1 4 1 5 0 4 1 4 1 4 1 4 1 -20 1 4 1 4 1 4 1 4 1 4 0 5 1 -21 1 5 1 4 1 4 1 4 1 4 1 4 1 -20 0 4 1 4 1 4 1 4 1 5 1 -20 1 4 1 4 1 4 1 4 0 4 1 -20 1 4 1 4 1 4 1 4 1 -20 1 5 1 4 0 4 1 4 1 -20 1 4 1 4 1 4 1 -20 2101 582 100 MP stroke 1 4 1 4 0 4 1 -20 1 4 1 5 1 -21 1 5 1 4 1 -20 1 4 1 -20 0 4 1 -20 1 -20 1 -44 1 4 1 4 1 4 1 4 1 4 0 4 1 4 1 4 1 4 1 5 1 -20 1 4 1 4 1 4 0 4 1 4 1 4 1 4 1 4 1 5 1 -21 1 5 1 4 1 4 0 4 1 4 1 4 1 4 1 4 1 4 1 -20 1 4 1 5 0 4 1 4 1 4 1 4 1 4 1 4 1 -20 1 4 1 4 0 4 1 5 1 4 1 4 1 4 1 4 1 -20 1 4 1 4 1 4 0 4 1 4 1 5 1 4 1 -20 1 4 1 4 1 4 1 4 0 4 1 4 1 4 1 -20 1 4 1 5 1 4 1 4 1 4 1 4 0 -20 1 4 1 4 1 4 1 4 1 4 1 5 1 -20 1 4 0 4 1 4 1 4 2013 586 100 MP stroke 1 4 1 -20 1 4 1 4 1 4 1 4 0 5 1 -21 1 5 1 4 1 4 1 4 1 -20 1 4 1 4 1 4 0 4 1 -20 1 4 1 4 1 5 1 -20 1 4 1 4 1 4 0 -20 1 4 1 4 1 -20 1 4 1 4 1 -20 1 4 1 -20 0 4 1 -20 1 -20 1 -44 1 4 1 4 1 4 1 4 1 5 1 4 0 4 1 4 1 4 1 -20 1 4 1 4 1 4 1 4 1 4 0 5 1 4 1 4 1 4 1 -20 1 4 1 4 1 4 1 4 0 4 1 4 1 5 1 4 1 -20 1 4 1 4 1 4 1 4 1 4 0 4 1 4 1 4 1 -20 1 5 1 4 1 4 1 4 1 4 0 4 1 4 1 -20 1 4 1 4 1 4 1 5 1 4 1 4 0 4 1 -20 1 4 1 4 1 4 1 4 1 4 1 4 1 -20 1924 663 100 MP stroke 1 5 0 4 1 4 1 4 1 4 1 4 1 -20 1 4 1 4 1 4 0 4 1 5 1 -21 1 5 1 4 1 4 1 4 1 4 1 -20 1 4 0 4 1 4 1 4 1 -20 1 4 1 5 1 4 1 4 1 -20 0 4 1 4 1 4 1 -20 1 4 1 4 1 4 1 -20 1 4 0 4 1 -20 1 5 1 4 1 -20 1 4 1 -20 1 4 1 -20 1 -20 0 -45 1 5 1 4 1 4 1 4 1 4 1 4 1 4 1 4 0 4 1 -20 1 4 1 5 1 4 1 4 1 4 1 4 1 4 0 4 1 -20 1 4 1 4 1 4 1 5 1 4 1 4 1 4 1 4 0 -20 1 4 1 4 1 4 1 4 1 5 1 4 1 4 1 -20 0 4 1 4 1 4 1 4 1 4 1 4 1 4 1 -20 1 4 0 5 1 4 1 4 1 4 1 4 1836 715 100 MP stroke 1 -20 1 4 1 4 1 4 1 4 0 4 1 5 1 -20 1 4 1 4 1 4 1 4 1 4 1 -20 0 4 1 4 1 4 1 4 1 5 1 -20 1 4 1 4 1 4 1 4 0 -20 1 4 1 4 1 4 1 4 1 -20 1 4 1 4 1 5 0 -20 1 4 1 4 1 4 1 -20 1 4 1 4 1 -20 1 4 0 4 1 -20 1 4 1 -20 1 4 1 -20 1 -20 1 -44 1 4 1 4 0 4 1 4 1 5 1 4 1 4 1 4 1 4 1 -20 1 4 0 4 1 4 1 4 1 4 1 5 1 4 1 4 1 -20 1 4 0 4 1 4 1 4 1 4 1 4 1 4 1 5 1 -20 1 4 1 4 0 4 1 4 1 4 1 4 1 4 1 -20 1 4 1 4 1 5 0 4 1 4 1 4 1 4 1 -20 1 4 1 4 1 4 1 4 0 4 1748 792 100 MP stroke 1 5 1 -20 1 4 1 4 1 4 1 4 1 4 1 4 1 -20 0 4 1 4 1 4 1 5 1 4 1 -20 1 4 1 4 1 4 0 4 1 4 1 -20 1 4 1 4 1 4 1 4 1 -20 1 5 1 4 0 4 1 4 1 -20 1 4 1 4 1 4 1 -20 1 4 1 4 0 4 1 -20 1 4 1 5 1 -20 1 4 1 4 1 -20 1 4 0 -20 1 4 1 -20 1 -20 1 -44 1 4 1 4 1 4 1 4 1 4 0 4 1 4 1 4 1 -20 1 4 1 5 1 4 1 4 1 4 0 4 1 4 1 4 1 -20 1 4 1 4 1 4 1 5 1 4 0 4 1 4 1 -20 1 4 1 4 1 4 1 4 1 4 1 5 1 4 0 -20 1 4 1 4 1 4 1 4 1 4 1 4 1 -20 1 4 0 4 1 4 1 5 1 4 1 4 1 -20 1659 892 100 MP stroke 1 4 1 4 1 4 0 4 1 4 1 -20 1 4 1 4 1 5 1 4 1 4 1 -20 1 4 0 4 1 4 1 4 1 -20 1 4 1 4 1 4 1 5 1 -21 0 5 1 4 1 4 1 -20 1 4 1 4 1 4 1 -20 1 4 0 4 1 -20 1 4 1 4 1 -20 1 4 1 -20 1 5 1 -21 1 -20 0 -44 1 4 1 4 1 5 1 4 1 4 1 4 1 4 1 4 0 -20 1 4 1 4 1 4 1 4 1 5 1 4 1 4 1 -20 1 4 0 4 1 4 1 4 1 4 1 4 1 4 1 -20 1 5 1 4 0 4 1 4 1 4 1 4 1 -20 1 4 1 4 1 4 1 4 0 5 1 4 1 -20 1 4 1 4 1 4 1 4 1 4 1 -20 1 4 0 4 1 4 1 4 1 5 1 -20 1 4 1 4 1 4 1 4 0 -20 1 4 1571 993 100 MP stroke 1 4 1 4 1 4 1 -20 1 4 1 5 1 4 0 -20 1 4 1 4 1 4 1 -20 1 4 1 4 1 -20 1 4 1 4 0 -20 1 4 1 -20 1 4 1 -20 1 -20 1 -44 1 4 1 4 0 4 1 5 1 4 1 4 1 4 1 4 1 -20 1 4 1 4 0 4 1 4 1 4 1 5 1 4 1 -20 1 4 1 4 1 4 1 4 0 4 1 4 1 4 1 -20 1 4 1 4 1 5 1 4 1 4 0 4 1 -20 1 4 1 4 1 4 1 4 1 4 1 4 1 -20 1 5 0 4 1 4 1 4 1 4 1 -20 1 4 1 4 1 4 1 4 0 4 1 -20 1 5 1 4 1 4 1 4 1 -20 1 4 1 4 0 4 1 4 1 -20 1 4 1 4 1 4 1 -20 1 5 1 4 1 4 0 -20 1 4 1 4 1 -20 1 4 1 4 1 -20 1482 1142 100 MP stroke 1 4 1 -20 0 4 1 -20 1 -20 1 -44 1 4 1 4 1 4 1 4 1 4 0 5 1 4 1 -20 1 4 1 4 1 4 1 4 1 4 1 4 1 4 0 -20 1 4 1 5 1 4 1 4 1 4 1 4 1 -20 1 4 0 4 1 4 1 4 1 4 1 5 1 -21 1 5 1 4 1 4 0 4 1 4 1 -20 1 4 1 4 1 4 1 4 1 5 1 -21 1 5 0 4 1 4 1 4 1 -20 1 4 1 4 1 4 1 4 1 -20 0 4 1 4 1 5 1 -21 1 5 1 4 1 4 1 -20 1 4 0 4 1 -20 1 4 1 4 1 -20 1 4 1 -20 1 4 1 -20 1 -20 0 -44 1 4 1 4 1 4 1 4 1 5 1 4 1 4 1 -20 0 4 1 4 1 4 1 4 1 4 1 4 1 -20 1 4 1 4 1 5 0 4 1 4 1 4 1394 1315 100 MP stroke 1 -20 1 4 1 4 1 4 1 4 1 4 0 -20 1 4 1 5 1 4 1 4 1 4 1 -20 1 4 1 4 0 4 1 4 1 -20 1 4 1 4 1 5 1 4 1 -20 1 4 1 4 0 4 1 -20 1 4 1 4 1 4 1 -20 1 4 1 4 1 -20 0 4 1 5 1 -21 1 5 1 -20 1 4 1 -20 1 -20 1 -45 0 4 1 5 1 4 1 4 1 4 1 4 1 4 1 -20 1 4 1 4 0 4 1 4 1 5 1 4 1 -20 1 4 1 4 1 4 1 4 0 4 1 4 1 -20 1 4 1 4 1 5 1 4 1 4 1 -20 0 4 1 4 1 4 1 4 1 4 1 -20 1 4 1 4 1 4 1 5 0 -20 1 4 1 4 1 4 1 4 1 -20 1 4 1 4 1 4 0 -20 1 4 1 4 1 5 1 -21 1 5 1 4 1 -20 1 4 1305 1488 100 MP stroke 1 4 0 -20 1 4 1 -20 1 4 1 -20 1 -20 1 -44 1 4 1 4 0 4 1 4 1 4 1 4 1 -20 1 4 1 5 1 4 1 4 0 4 1 4 1 -20 1 4 1 4 1 4 1 4 1 4 1 -20 1 4 0 5 1 4 1 4 1 4 1 -20 1 4 1 4 1 4 1 4 0 -20 1 4 1 4 1 5 1 4 1 -20 1 4 1 4 1 4 0 -20 1 4 1 4 1 4 1 -20 1 4 1 4 1 -20 1 4 1 5 0 -21 1 5 1 -20 1 4 1 -20 1 -20 1 -45 1 4 1 5 0 4 1 4 1 4 1 4 1 -20 1 4 1 4 1 4 1 4 0 4 1 -20 1 5 1 4 1 4 1 4 1 4 1 -20 1 4 1 4 0 4 1 4 1 -20 1 4 1 5 1 4 1 4 1 -20 1 4 0 4 1 4 1 -20 1 4 1 4 1217 1734 100 MP stroke 1 4 1 -20 1 4 1 4 0 -20 1 5 1 4 1 -20 1 4 1 -20 1 4 1 -20 1 -20 1 -45 0 5 1 4 1 4 1 4 1 4 1 4 1 -20 1 4 1 4 0 4 1 4 1 5 1 -20 1 4 1 4 1 4 1 4 1 4 1 -20 0 4 1 4 1 4 1 4 1 -20 1 4 1 5 1 4 1 4 0 -20 1 4 1 4 1 4 1 -20 1 4 1 4 1 4 1 -20 0 4 1 5 1 -21 1 5 1 4 1 -20 1 4 1 -20 1 4 1 -20 0 -20 1 -44 1 4 1 4 1 4 1 4 1 4 1 -20 1 4 0 4 1 4 1 4 1 5 1 -21 1 5 1 4 1 4 1 4 0 -20 1 4 1 4 1 4 1 4 1 -20 1 4 1 4 1 5 1 -20 0 4 1 4 1 4 1 -20 1 4 1 4 1 -20 1 4 1 4 0 -20 1129 2052 100 MP stroke 1 4 1 -20 1 4 1 -20 1 -20 1 -44 1 4 1 4 0 4 1 4 1 5 1 -20 1 4 1 4 1 4 1 4 1 -20 1 4 0 4 1 4 1 4 1 -20 1 4 1 4 1 5 1 -20 1 4 0 4 1 4 1 -20 1 4 1 4 1 -20 1 4 1 4 1 -20 1 4 0 -20 1 4 1 -20 1 -20 1 -44 1 4 1 4 1 4 1 4 0 5 1 -20 1 4 1 4 1 4 1 4 1 -20 1 4 1 4 0 4 1 4 1 -20 1 4 1 4 1 5 1 -20 1 4 1 4 1 4 0 -20 1 4 1 4 1 -20 1 4 1 4 1 -20 1 4 1 -20 0 4 1 -20 1 -20 1 -44 1 4 1 4 1 4 1 4 1 -20 0 5 1 4 1 4 1 4 1 -20 1 4 1 4 1 4 1 -20 1 4 0 4 1 4 1 -20 1 4 1 5 1 -21 1040 2467 100 MP stroke 1 5 1 4 1 -20 0 4 1 -20 1 4 1 -20 1 -20 1 -44 1 4 1 4 1 4 0 4 1 -20 1 4 1 4 1 4 1 -20 1 4 1 4 1 5 1 -21 0 5 1 4 1 -20 1 4 1 4 1 -20 1 4 1 -20 1 4 0 -20 1 -20 1 -44 1 4 1 4 1 4 1 4 1 -20 1 4 0 4 1 4 1 -20 1 4 1 5 1 4 1 -20 1 4 1 4 1 -20 0 4 1 4 1 -20 1 4 1 -20 1 4 1 -20 1 -20 1 -44 0 4 1 4 1 4 1 -20 1 4 1 4 1 4 1 -20 1 4 1 5 0 -20 1 4 1 4 1 -20 1 4 1 -20 1 4 1 -20 1 -20 0 -44 1 4 1 4 1 4 1 -20 1 4 1 4 1 -20 1 4 0 4 1 -20 1 4 1 -20 1 4 1 -20 1 -20 1 -44 1 4 1 4 0 4 1 -20 952 3123 100 MP stroke 1 4 1 5 1 -20 1 4 1 4 1 -20 1 4 0 -20 1 4 1 -20 1 -20 1 -44 1 4 1 4 1 -20 1 4 0 4 1 -20 1 4 1 -20 1 4 1 -20 1 -20 1 -44 1 4 1 4 0 -20 1 4 1 -20 1 4 1 -20 1 -20 1 -44 1 4 1 4 0 -20 1 4 1 -20 1 4 1 -20 1 -20 1 -44 1 4 1 -20 0 4 1 -20 1 -20 1 -44 1 4 1 -20 1 -20 1 -44 1 4 1 -20 0 -20 1 -45 1 -20 1 -44 1 -44 899 3943 60 MP stroke gr 2991 274 mt (3D energy, h=0.34) s 3349 4944 mt (volume) s 566 2990 mt -90 rotate (minimal energy) s 90 rotate end eplot %%EndObject graph 1 epage end showpage %%Trailer %%EOF %%EndDocument endTexFig 60 1140 a Fu(Let)j(us)f(p)q(oin)o(t)g(out)h(that)g(eac)o(h)f(arc)o(h)g (corresp)q(onds)f(to)i(a)f(t)o(w)o(o)h(dimensional)d(energy)i(curv)o(e.)22 b(Belo)o(w)0 1200 y(is)16 b(a)h(narro)o(w)d(view)j(of)g(the)g(graph)e(around) g(its)h(maxim)o(um.)266 1228 y 18646798 13052758 3552215 12959006 36574658 39732183 startTexFig 266 1228 a %%BeginDocument: e3d2.eps %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The Mathworks, Inc. %%Title: e3d2.eps %%CreationDate: 07/06/95 09:09:58 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Pages: 1 %%BoundingBox: 54 197 556 604 %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 150 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /reencode { exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } bdef /isroman { findfont /CharStrings get /Agrave known } bdef /FMSR { 3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS } bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath fill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { 2 copy moveto lineto stroke } bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 54 197 556 604 MathWorks begin bpage %%EndPageSetup %%BeginObject: graph1 1 bplot /dpi2point 12 def portraitMode 0216 7344 csm 436 92 6031 4884 MR c np 76 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef %%IncludeResource: font Helvetica /Helvetica /ISOLatin1Encoding 144 FMSR 1 j c1 0 0 6917 5187 PR 6 w DO 4 w SO 6 w c0 899 4615 mt 6258 4615 L 899 389 mt 6258 389 L 899 4615 mt 899 389 L 6258 4615 mt 6258 389 L 899 4615 mt 899 4615 L 6258 4615 mt 6258 4615 L 899 4615 mt 6258 4615 L 899 4615 mt 899 389 L 899 4615 mt 899 4615 L 899 4615 mt 899 4561 L 899 389 mt 899 443 L 739 4784 mt (1300) s 1665 4615 mt 1665 4561 L 1665 389 mt 1665 443 L 1505 4784 mt (1400) s 2430 4615 mt 2430 4561 L 2430 389 mt 2430 443 L 2270 4784 mt (1500) s 3196 4615 mt 3196 4561 L 3196 389 mt 3196 443 L 3036 4784 mt (1600) s 3961 4615 mt 3961 4561 L 3961 389 mt 3961 443 L 3801 4784 mt (1700) s 4727 4615 mt 4727 4561 L 4727 389 mt 4727 443 L 4567 4784 mt (1800) s 5492 4615 mt 5492 4561 L 5492 389 mt 5492 443 L 5332 4784 mt (1900) s 6258 4615 mt 6258 4561 L 6258 389 mt 6258 443 L 6098 4784 mt (2000) s 899 4615 mt 953 4615 L 6258 4615 mt 6204 4615 L 624 4668 mt (272) s 899 4192 mt 953 4192 L 6258 4192 mt 6204 4192 L 624 4245 mt (274) s 899 3770 mt 953 3770 L 6258 3770 mt 6204 3770 L 624 3823 mt (276) s 899 3347 mt 953 3347 L 6258 3347 mt 6204 3347 L 624 3400 mt (278) s 899 2925 mt 953 2925 L 6258 2925 mt 6204 2925 L 624 2978 mt (280) s 899 2502 mt 953 2502 L 6258 2502 mt 6204 2502 L 624 2555 mt (282) s 899 2079 mt 953 2079 L 6258 2079 mt 6204 2079 L 624 2132 mt (284) s 899 1657 mt 953 1657 L 6258 1657 mt 6204 1657 L 624 1710 mt (286) s 899 1234 mt 953 1234 L 6258 1234 mt 6204 1234 L 624 1287 mt (288) s 899 812 mt 953 812 L 6258 812 mt 6204 812 L 624 865 mt (290) s 899 389 mt 953 389 L 6258 389 mt 6204 389 L 624 442 mt (292) s 899 4615 mt 6258 4615 L 899 389 mt 6258 389 L 899 4615 mt 899 389 L 6258 4615 mt 6258 389 L 899 389 mt 899 389 L 6258 389 mt 6258 389 L gs 899 389 5360 4227 MR c np 7 72 8 71 8 72 7 72 8 72 8 72 7 -351 8 72 8 72 7 72 8 71 8 72 7 72 8 72 7 72 8 72 8 72 7 -351 8 72 8 71 7 72 8 72 8 72 7 72 8 72 8 71 7 72 8 -350 8 71 7 72 8 72 8 72 7 72 8 72 8 71 7 72 8 72 8 -351 7 72 8 72 8 72 7 72 8 72 8 71 7 72 8 72 7 -351 8 72 8 72 7 72 8 72 8 72 7 71 8 72 8 72 7 -351 8 72 8 72 7 72 8 72 8 72 7 71 8 72 8 -351 7 72 8 72 8 72 7 72 8 72 8 72 7 71 8 -350 8 71 7 72 8 72 7 72 8 72 8 72 7 -351 8 72 8 72 7 71 8 72 8 72 7 72 8 -351 8 72 7 72 8 72 8 72 7 71 8 -350 8 72 7 71 8 72 8 72 7 72 8 -351 8 72 5492 1306 100 MP stroke 7 72 8 72 8 71 7 -350 8 72 8 71 7 72 8 72 7 -351 8 72 8 72 7 72 8 -351 8 72 7 72 8 72 8 -351 7 72 8 72 8 -351 7 72 8 71 8 -350 7 72 8 -351 8 72 7 -351 8 -351 8 -773 7 72 8 71 8 72 7 72 8 72 8 72 7 72 8 72 7 71 8 72 8 72 7 -351 8 72 8 72 7 72 8 72 8 71 7 72 8 72 8 72 7 72 8 72 8 -351 7 72 8 72 8 71 7 72 8 72 8 72 7 72 8 72 8 72 7 71 8 -350 8 71 7 72 8 72 8 72 7 72 8 72 7 71 8 72 8 72 7 -351 8 72 8 72 7 72 8 72 8 72 7 71 8 72 8 72 7 72 8 -351 8 72 7 72 8 72 8 72 7 71 8 72 8 72 7 72 8 -351 8 72 7 72 8 72 8 72 7 71 8 72 7 72 4735 1378 100 MP stroke 8 72 8 -351 7 72 8 72 8 72 7 72 8 71 8 72 7 72 8 -351 8 72 7 72 8 72 8 72 7 72 8 71 8 72 7 -351 8 72 8 72 7 72 8 72 8 72 7 72 8 -351 8 72 7 71 8 72 8 72 7 72 8 72 7 -351 8 72 8 72 7 72 8 71 8 72 7 -350 8 71 8 72 7 72 8 72 8 72 7 -351 8 72 8 72 7 71 8 72 8 -350 7 71 8 72 8 72 7 72 8 -351 8 72 7 72 8 72 8 -351 7 72 8 72 7 71 8 -350 8 72 7 71 8 -350 8 71 7 72 8 -350 8 71 7 -350 8 71 8 -350 7 -351 8 -773 8 71 7 72 8 72 8 72 7 72 8 72 8 72 7 71 8 72 8 72 7 72 8 -351 8 72 7 72 8 72 8 71 7 72 8 72 7 72 8 72 8 72 7 72 8 -351 8 72 7 71 3977 2718 100 MP stroke 8 72 8 72 7 72 8 72 8 72 7 72 8 71 8 72 7 -351 8 72 8 72 7 72 8 72 8 72 7 72 8 71 8 72 7 72 8 -351 8 72 7 72 8 72 7 72 8 71 8 72 7 72 8 72 8 72 7 -351 8 72 8 72 7 72 8 71 8 72 7 72 8 72 8 72 7 -351 8 72 8 72 7 72 8 71 8 72 7 72 8 72 8 72 7 -351 8 72 8 72 7 72 8 71 7 72 8 72 8 72 7 -351 8 72 8 72 7 72 8 72 8 71 7 72 8 72 8 -351 7 72 8 72 8 72 7 72 8 72 8 71 7 -350 8 71 8 72 7 72 8 72 8 72 7 72 8 -351 8 72 7 72 8 71 8 72 7 72 8 -351 7 72 8 72 8 72 7 72 8 72 8 -351 7 72 8 71 8 72 7 72 8 -351 8 72 7 72 8 72 8 72 7 -351 3219 1099 100 MP stroke 8 72 8 72 7 72 8 -351 8 72 7 71 8 72 8 -350 7 71 8 72 8 -351 7 72 8 72 7 -351 8 72 8 -350 7 71 8 -350 8 -351 7 -774 8 72 8 72 7 72 8 72 8 72 7 72 8 71 8 72 7 72 8 72 8 -351 7 72 8 72 8 72 7 72 8 71 8 72 7 72 8 72 8 72 7 72 8 -351 8 72 7 71 8 72 7 72 8 72 8 72 7 72 8 72 8 71 7 -350 8 71 8 72 7 72 8 72 8 72 7 72 8 72 8 71 7 72 8 -351 8 72 7 72 8 72 8 72 7 72 8 71 8 72 7 72 8 -351 8 72 7 72 8 72 7 72 8 72 8 71 7 72 8 72 8 -351 7 72 8 72 8 72 7 72 8 72 8 71 7 72 8 -350 8 71 7 72 8 72 8 72 7 72 8 72 8 71 7 -350 8 72 8 71 7 72 2461 1171 100 MP stroke 8 72 8 72 7 72 8 -351 8 72 7 72 8 71 7 72 8 72 8 72 7 -351 8 72 8 72 7 72 8 72 8 71 7 -350 8 72 8 71 7 72 8 72 8 72 7 -351 8 72 8 72 7 72 8 72 8 -351 7 72 8 71 8 72 7 72 8 -351 8 72 7 72 8 72 7 -351 8 72 8 72 7 72 8 -351 8 72 7 72 8 -351 8 72 7 72 8 -351 8 72 7 -351 8 72 8 -351 7 -351 8 -773 8 72 7 71 8 72 8 72 7 72 8 72 8 72 7 72 8 71 8 72 7 -351 8 72 8 72 7 72 8 72 7 72 8 72 8 71 7 72 8 72 8 -351 7 72 8 72 8 72 7 72 8 72 8 71 7 72 8 72 8 72 7 -351 8 72 8 72 7 72 8 71 8 72 7 72 8 72 8 72 7 -351 8 72 8 72 7 72 8 71 7 72 8 72 1703 2088 100 MP stroke 8 72 7 72 8 -351 8 72 7 72 8 72 8 71 7 72 8 72 8 72 7 -351 8 72 8 72 7 72 8 72 8 71 7 72 8 72 8 -351 7 72 8 72 8 72 7 72 8 72 8 71 7 -350 8 71 8 72 7 72 8 72 7 72 8 72 8 -351 7 72 8 72 8 71 7 72 8 72 8 -351 7 72 8 72 8 72 7 72 8 72 8 -351 7 72 8 72 8 71 7 72 8 -351 8 72 7 72 8 72 8 72 7 -351 8 72 8 72 7 72 8 -351 7 72 8 72 8 71 7 -350 8 71 8 72 7 -350 8 71 8 72 7 -351 8 72 8 -350 7 71 8 -350 8 -351 7 -774 8 72 8 72 7 72 8 72 8 72 7 72 8 71 8 72 7 72 8 72 8 -351 7 72 8 72 8 72 7 72 8 71 7 72 8 72 8 72 7 72 8 -351 8 72 7 72 8 72 945 3427 100 MP stroke 8 71 7 72 8 72 8 72 7 72 8 72 899 2996 7 MP stroke gr 2289 274 mt (3D energy around the maximum, h=0.34) s 3349 4944 mt (volume) s 570 2990 mt -90 rotate (minimal energy) s 90 rotate end eplot %%EndObject graph 1 epage end showpage %%Trailer %%EOF %%EndDocument endTexFig 60 2138 a Fu(W)l(e)22 b(denote)f(b)o(y)g Fo(E)427 2145 y Fl(2)471 2138 y Fu(the)h(t)o(w)o(o)f(dimensional)f(energy)l(,)i(so)f(that)h Fo(E)s Fu(\()p Fo(n)p Fu(\))h(=)f Fo(E)s Fu(\()p Fo(j)s Fu(\()p Fo(j)16 b Fu(+)f Fo(\016)r Fu(\)\()p Fo(j)i Fu(+)d Fo(\022)q Fu(\)\))i(+)0 2198 y Fo(E)37 2205 y Fl(2)59 2198 y Fu(\()p Fo(l)q Fu(\()p Fo(l)10 b Fu(+)f Fo(\017)p Fu(\))g(+)g Fo(k)r Fu(\).)21 b(W)l(e)16 b(\014rst)f(compute)g(the)g(energy)g(barrier)f(b)q(et)o (w)o(een)h(t)o(w)o(o)g(consecutiv)o(e)g(quasicub)q(es:)520 2299 y(max)435 2334 y Fn(j)453 2324 y Fd(3)473 2334 y Fj(\024)p Fn(n)p Fj(\024)p Fn(j)578 2324 y Fd(2)598 2334 y Fl(\()p Fn(j)r Fl(+1\))706 2299 y Fo(E)s Fu(\()p Fo(n)p Fu(\))23 b(=)f Fo(E)s Fu(\()p Fo(j)981 2279 y Fl(3)1002 2299 y Fu(\))12 b(+)37 b(max)1083 2334 y Fl(0)p Fj(\024)p Fn(n)p Fj(\024)p Fn(j)1208 2324 y Fd(2)1236 2299 y Fo(E)1273 2306 y Fl(2)1295 2299 y Fu(\()p Fo(n)p Fu(\))8 b Fo(;)470 2411 y Fu(max)335 2445 y Fn(j)353 2435 y Fd(2)373 2445 y Fl(\()p Fn(j)r Fl(+1\))p Fj(\024)p Fn(n)p Fj(\024)p Fn(j)r Fl(\()p Fn(j)r Fl(+1\))680 2435 y Fd(2)706 2411 y Fo(E)s Fu(\()p Fo(n)p Fu(\))23 b(=)f Fo(E)s Fu(\()p Fo(j)981 2390 y Fl(2)1002 2411 y Fu(\()p Fo(j)14 b Fu(+)d(1\)\))h(+)77 b(max)1230 2445 y Fl(0)p Fj(\024)p Fn(n)p Fj(\024)p Fn(j)r Fl(\()p Fn(j)r Fl(+1\))1464 2411 y Fo(E)1501 2418 y Fl(2)1523 2411 y Fu(\()p Fo(n)p Fu(\))8 b Fo(;)479 2522 y Fu(max)354 2556 y Fn(j)r Fl(\()p Fn(j)r Fl(+1\))473 2546 y Fd(2)492 2556 y Fj(\024)p Fn(n)p Fj(\024)p Fl(\()p Fn(j)r Fl(+1\))680 2546 y Fd(3)706 2522 y Fo(E)s Fu(\()p Fo(n)p Fu(\))23 b(=)f Fo(E)s Fu(\()p Fo(j)s Fu(\()p Fo(j)13 b Fu(+)e(1\))1128 2501 y Fl(2)1150 2522 y Fu(\))h(+)77 b(max)1230 2556 y Fl(0)p Fj(\024)p Fn(n)p Fj(\024)p Fl(\()p Fn(j)r Fl(+1\))1438 2546 y Fd(2)1465 2522 y Fo(E)1502 2529 y Fl(2)1524 2522 y Fu(\()p Fo(n)p Fu(\))8 b Fo(:)954 2670 y Fs(33)p eop %%Page: 34 34 34 33 bop 0 200 a Fu(W)l(e)17 b(next)g(compare)e(these)i(quan)o(tities.)k (Let)15 276 y(\001)57 283 y Fl(1)101 276 y Fu(=)74 b(max)129 311 y Fn(j)147 301 y Fd(3)167 311 y Fj(\024)p Fn(n)p Fj(\024)p Fn(j)272 301 y Fd(2)292 311 y Fl(\()p Fn(j)r Fl(+1\))350 276 y Fo(E)s Fu(\()p Fo(n)p Fu(\))20 b Fp(\000)38 b Fu(max)470 311 y Fl(\()p Fn(j)r Fj(\000)p Fl(1\))p Fn(j)589 301 y Fd(2)608 311 y Fj(\024)p Fn(n)p Fj(\024)p Fn(j)713 301 y Fd(3)658 276 y Fo(E)s Fu(\()p Fo(n)p Fu(\))23 b(=)e Fo(j)s Fu(\(4)11 b Fp(\000)g Fo(hj)s Fu(\))d Fo(;)15 382 y Fu(\001)57 389 y Fl(2)101 382 y Fu(=)124 b(max)129 416 y Fn(j)147 406 y Fd(2)167 416 y Fl(\()p Fn(j)r Fl(+1\))p 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Fu(+)e Fo(\016)r Fu(\)\()p Fo(j)k Fu(+)c Fo(\022)q Fu(\).)37 b(The)0 1248 y(inequalit)o(y)15 b Fo(n)f Fp(\024)f Fu(\()p Fo(j;)8 b(\016)o(;)g(\022)q Fu(\))17 b(then)e(means)f Fo(n)g Fp(\024)g Fo(j)s Fu(\()p Fo(j)c Fu(+)f Fo(\016)r Fu(\)\()p Fo(j)i Fu(+)e Fo(\022)q Fu(\).)22 b(W)l(e)16 b(consider)e(the)h(function)g Fo(\036)g Fu(de\014ned)0 1307 y(on)h(the)h(set)g Fo(D)h Fu(b)o(y)0 1384 y Fo(\036)p Fu(\()p Fo(j;)8 b(\016)o(;)g(\022)q Fu(\))23 b(=)f(max)p Fp(f)8 b Fo(E)s Fu(\()p Fo(n)p Fu(\))14 b(:)g(\()p Fo(j;)8 b(\016)o(;)g(\022)q Fu(\))16 b Fp(\024)d Fo(n)h Fp(\024)f Fu(succ)8 b(\()p Fo(j;)g(\016)o(;)g (\022)q Fu(\))g Fp(g)1005 1458 y(\000)16 b Fu(max)p Fp(f)8 b Fo(E)s Fu(\()p Fo(n)p Fu(\))14 b(:)g(pred)7 b(\()p Fo(j;)h(\016)o(;)g(\022) q Fu(\))15 b Fp(\024)f Fo(n)g Fp(\024)f Fu(\()p Fo(j;)8 b(\016)o(;)g(\022)q Fu(\))g Fp(g)p Fo(:)0 1535 y Fu(W)l(e)17 b(ha)o(v)o(e)f(sho)o(wn)f(that)i Fo(\036)f Fu(is)g(p)q(ositiv)o(e)g(for)60 1594 y Fp(\017)49 b Fo(\016)16 b Fu(=)e(0,)i Fo(\022)g 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1953 y Fu(\(case)j(of)f(\001)802 1960 y Fl(3)824 1953 y Fu(\))0 2013 y(W)l(e)h(no)o(w)f(de\014ne)f(the)i(critical)f(quasicub)q (e.)60 2073 y Fp(\017)49 b Fu(If)17 b Fp(b)p Fo(j)230 2055 y Fl(1)227 2085 y Fn(c)252 2073 y Fp(c)d Fu(=)g Fp(b)p Fo(j)387 2055 y Fl(2)384 2085 y Fn(c)409 2073 y Fp(c)i Fu(w)o(e)h(put)f Fo(j)633 2080 y Fn(c)667 2073 y Fu(=)d Fp(b)p Fo(j)765 2055 y Fl(1)762 2085 y Fn(c)787 2073 y Fp(c)k Fu(and)f Fo(\016)945 2080 y Fn(c)979 2073 y Fu(=)d(1.)60 2132 y Fp(\017)49 b Fu(If)17 b Fp(b)p Fo(j)230 2114 y Fl(1)227 2145 y Fn(c)252 2132 y Fp(c)d Fo(<)g Fp(b)p Fo(j)387 2114 y Fl(2)384 2145 y Fn(c)409 2132 y Fp(c)i Fu(w)o(e)h(put)f Fo(j)633 2139 y Fn(c)667 2132 y Fu(=)d Fp(b)p Fo(j)765 2114 y Fl(2)762 2145 y Fn(c)787 2132 y Fp(c)k Fu(and)f Fo(\016)945 2139 y Fn(c)979 2132 y Fu(=)d(0.)0 2192 y(The)23 b(critical)g(quasicub)q(e)f(is)h Fo(j)596 2199 y Fn(c)615 2192 y Fu(\()p Fo(j)655 2199 y Fn(c)691 2192 y Fu(+)15 b Fo(\016)767 2199 y Fn(c)787 2192 y Fu(\)\()p Fo(j)846 2199 y Fn(c)882 2192 y Fu(+)g(1\).)42 b(W)l(e)23 b(can)g(no)o(w)f(compute)h(the)g(energy)g (barrier)0 2252 y(b)q(et)o(w)o(een)16 b(t)o(w)o(o)g(remote)h(quasicub)q(es.)0 2328 y(max)p Fp(f)8 b Fo(E)s Fu(\()p Fo(n)p Fu(\))14 b(:)g(\()p Fo(j)316 2335 y Fl(1)338 2328 y Fo(;)8 b(\016)382 2335 y Fl(1)405 2328 y Fo(;)g(\022)450 2335 y Fl(1)472 2328 y Fu(\))15 b Fp(\024)e Fo(n)h Fp(\024)g Fu(\()p Fo(j)695 2335 y Fl(2)717 2328 y Fo(;)8 b(\016)761 2335 y Fl(2)783 2328 y Fo(;)g(\022)828 2335 y Fl(2)851 2328 y Fu(\))g Fp(g)31 b Fu(=)184 2361 y Fk(8)184 2406 y(>)184 2420 y(<)184 2510 y(>)184 2525 y(:)245 2403 y Fo(E)s Fu(\(\()p Fo(j)344 2410 y Fn(c)364 2403 y Fo(;)8 b(\016)408 2410 y Fn(c)429 2403 y Fo(;)g Fu(1\)\))k(+)e Fo(E)612 2410 y Fl(2)634 2403 y Fu(\()p Fo(l)668 2410 y Fn(c)689 2403 y Fu(\()p Fo(l)723 2410 y Fn(c)755 2403 y Fu(+)g(1\))i(+)f(1\))180 b(if)17 b(\()p Fo(j)1220 2410 y Fl(1)1242 2403 y Fo(;)8 b(\016)1286 2410 y Fl(1)1308 2403 y Fo(;)g(\022)1353 2410 y Fl(1)1376 2403 y Fu(\))14 b Fp(\024)g Fu(\()p Fo(j)1502 2410 y Fn(c)1522 2403 y Fo(;)8 b(\016)1566 2410 y Fn(c)1587 2403 y Fo(;)g Fu(1\))14 b Fo(<)g Fu(\()p Fo(j)1760 2410 y Fl(2)1782 2403 y Fo(;)8 b(\016)1826 2410 y Fl(2)1848 2403 y Fo(;)g(\022)1893 2410 y Fl(2)1916 2403 y Fu(\))245 2478 y Fo(E)s Fu(\(pred)f(\()p Fo(j)449 2485 y Fl(2)471 2478 y Fo(;)h(\016)515 2485 y Fl(2)538 2478 y Fo(;)g(\022)583 2485 y Fl(2)606 2478 y Fu(\)\))k(+)f Fo(E)743 2485 y Fl(2)765 2478 y Fu(\()p Fo(l)799 2485 y Fn(c)819 2478 y Fu(\()p Fo(l)853 2485 y Fn(c)885 2478 y Fu(+)g(1\))g(+)g(1\))50 b(if)17 b(\()p Fo(j)1220 2485 y Fl(2)1242 2478 y Fo(;)8 b(\016)1286 2485 y Fl(2)1308 2478 y Fo(;)g(\022)1353 2485 y Fl(2)1376 2478 y Fu(\))14 b Fp(\024)g Fu(\()p Fo(j)1502 2485 y Fn(c)1522 2478 y Fo(;)8 b(\016)1566 2485 y Fn(c)1587 2478 y Fo(;)g Fu(1\))p Fo(;)17 b(j)1705 2485 y Fl(2)1740 2478 y Fo(>)d(l)1808 2485 y Fn(c)245 2553 y Fo(E)s Fu(\(\()p Fo(j)344 2560 y Fl(1)366 2553 y Fo(;)8 b(\016)410 2560 y Fl(1)433 2553 y Fo(;)g(\022)478 2560 y Fl(1)501 2553 y Fu(\)\))j(+)g Fo(E)637 2560 y Fl(2)659 2553 y Fu(\()p Fo(l)693 2560 y Fn(c)714 2553 y Fu(\()p Fo(l)748 2560 y Fn(c)779 2553 y Fu(+)g(1\))h(+)e(1\))156 b(if)17 b(\()p Fo(j)1220 2560 y Fn(c)1240 2553 y Fo(;)8 b(\016)1284 2560 y Fn(c)1304 2553 y Fo(;)g Fu(1\))15 b Fo(<)e Fu(\()p Fo(j)1477 2560 y Fl(1)1499 2553 y Fo(;)8 b(\016)1543 2560 y Fl(1)1566 2553 y Fo(;)g(\022)1611 2560 y Fl(1)1634 2553 y Fu(\))954 2670 y Fs(34)p eop %%Page: 35 35 35 34 bop 0 260 a Fu(W)l(e)17 b(\014nally)f(obtain)g(the)g(global)g(energy)g (barrier.)0 348 y Fw(Prop)r(osition)h(7.13.)28 b Fh(W)l(e)15 b(set)g Fo(\016)637 355 y Fn(c)672 348 y Fu(=)e(0)i Fh(if)h Fu(4)8 b(+)889 307 y Fp(p)p 931 307 162 2 v 41 x Fu(16)j(+)f Fo(h)1070 334 y Fl(2)1115 348 y Fo(<)22 b(h)p Fu(\(2)p Fp(b)p Fu(4)p Fo(=h)p Fp(c)8 b Fu(+)g(1\))15 b Fh(and)g Fo(\016)1604 355 y Fn(c)1638 348 y Fu(=)f(1)h Fh(otherwise.)0 408 y(The)h(comm)o (unication)f(altitude)h(b)q(et)o(w)o(een)g Fp(\000)p Fu(1)p 827 420 64 2 v 17 w Fh(and)g Fu(+1)p 1005 420 V 16 w Fh(is)351 522 y Fo(E)s Fu(\()p Fp(\000)p Fu(1)p 410 534 V Fo(;)8 b Fu(+1)p 496 534 V -1 w(\))31 b(=)f Fo(E)s Fu(\()p Fo(n)767 529 y Fn(c)788 522 y Fu(\))g(=)14 b Fo(E)s Fu(\(\()p Fo(j)989 529 y Fn(c)1009 522 y Fo(;)8 b(\016)1053 529 y Fn(c)1074 522 y Fo(;)g Fu(1\)\))k(+)e Fo(E)1257 529 y Fl(2)1280 522 y Fu(\()p Fo(l)1314 529 y Fn(c)1334 522 y Fu(\()p Fo(l)1368 529 y Fn(c)1400 522 y Fu(+)h(1\))g(+)g(1\))0 636 y Fh(where)i Fo(l)156 643 y Fn(c)190 636 y Fu(=)g Fp(b)p Fu(2)p Fo(=h)p Fp(c)g Fh(is)g(the)h(t)o(w)o(o)e(dimensional)f(critical)i (length,)h Fo(j)1217 643 y Fn(c)1250 636 y Fu(=)g Fp(b)p Fu(4)p Fo(=h)p Fp(c)f Fh(is)g(the)g(three)g(dimensional)0 696 y(critical)j(length,)g Fo(n)362 703 y Fn(c)396 696 y Fu(=)e Fo(j)470 703 y Fn(c)490 696 y Fu(\()p Fo(j)530 703 y Fn(c)561 696 y Fu(+)d Fo(\016)633 703 y Fn(c)653 696 y Fu(\)\()p Fo(j)712 703 y Fn(c)744 696 y Fu(+)g(1\))g(+)g Fo(l)914 703 y Fn(c)934 696 y Fu(\()p Fo(l)968 703 y Fn(c)999 696 y Fu(+)g(1\))h(+)e(1)17 b Fh(is)f(the)h(critical)f(v)o (olume.)0 784 y Fe(R)m(emark.)27 b Fu(W)l(e)16 b(ha)o(v)o(e)f Fo(E)s Fu(\()p Fp(\000)p Fu(1)p 466 796 V Fo(;)8 b Fu(+1)p 552 796 V(\))14 b Fp(\030)g Fu(32)p Fo(=h)806 766 y Fl(2)844 784 y Fu(as)h Fo(h)f Fp(!)g Fu(0:)21 b(the)c(global)e(energy)h(barrier)e(go)q (es)j(to)f(in\014nit)o(y)0 844 y(lik)o(e)g Fo(h)120 826 y Fj(\000)p Fl(2)190 844 y Fu(as)g Fo(h)h Fu(go)q(es)f(to)h(zero.)60 918 y(F)l(rom)e(this)h(w)o(e)g(deduce)g(the)h(lev)o(el)f(of)h(the)g(greatest)g (cycle)g(con)o(taining)e Fp(\000)p Fu(1)p 1423 930 V 16 w(and)h(not)h(+1)p 1689 930 V -1 w(.)0 1007 y Fw(Corollary)j(7.14.)27 b Fh(The)17 b(lev)o(el)f(of)h(the)g(cycle)g Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 932 1019 V Fo(;)8 b Fu(+1)p 1018 1019 V -20 x Fn(c)1102 1007 y Fu(\))17 b Fh(is)f(pred)7 b Fo(E)s Fu(\()p Fo(n)1382 1014 y Fn(c)1403 1007 y Fu(\))p Fh(.)0 1095 y Fw(Notation)19 b(7.15.)28 b Fu(If)17 b Fo(Y)27 b Fu(is)17 b(a)f(subset)g(of)h Fo(X)t Fu(,)g(its)f(minimal)f(and)h(maximal)g(v)o(olumes)f Fo(v)p 1627 1103 26 2 v 2 w Fu(\()p Fo(Y)c Fu(\))17 b(and)p 1845 1068 V 16 w Fo(v)r Fu(\()p Fo(Y)12 b Fu(\))0 1155 y(are)314 1215 y Fo(v)p 314 1223 V 2 w Fu(\()p Fo(Y)g Fu(\))22 b(=)g(min)o Fp(f)8 b Fo(v)r Fu(\()p Fo(\033)r Fu(\))15 b(:)f Fo(\033)i Fp(2)e Fo(Y)19 b Fp(g)8 b Fo(;)p 999 1187 V 58 w(v)r Fu(\()p Fo(Y)k Fu(\))22 b(=)g(max)p Fp(f)8 b Fo(v)r Fu(\()p Fo(\033)r Fu(\))15 b(:)e Fo(\033)j Fp(2)e Fo(Y)20 b Fp(g)8 b Fo(:)60 1335 y Fu(Since)17 b Fo(E)s Fu(\()p Fp(\000)p Fu(1)p 250 1348 64 2 v Fo(;)8 b Fp(C)362 1342 y Fn(n)389 1335 y Fu(\))17 b Fp(\025)f Fo(E)s Fu(\()p Fo(n)569 1342 y Fn(c)589 1335 y Fu(\))i(for)g(an)o (y)f Fo(n)f Fp(\025)g Fo(n)930 1342 y Fn(c)968 1335 y Fu(\(b)o(y)i(prop)q (osition)e(7)p Fo(:)p Fu(11\),)i(then)g(all)g(the)g(con\014gura-)0 1395 y(tions)13 b(of)g(the)h(cycle)g Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 424 1407 V Fo(;)8 b Fu(+1)p 510 1407 V -20 x Fn(c)594 1395 y Fu(\))14 b(ha)o(v)o(e)f(a)g(v)o(olume)f(less)h(then)g Fo(n)1174 1402 y Fn(c)1199 1395 y Fp(\000)5 b Fu(1)13 b(i.e.)p 1365 1368 26 2 v 20 w Fo(v)r Fu(\()p Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 1459 1407 64 2 v 1 w Fo(;)8 b Fu(+1)p 1546 1407 V -20 x Fn(c)1630 1395 y Fu(\)\))15 b Fp(\024)e Fo(n)1765 1402 y Fn(c)1790 1395 y Fp(\000)5 b Fu(1.)20 b(T)l(o)0 1455 y(complete)e(step)f Fo(i)p Fu(\))i(of)f(the)g(general)f(strategy)l(,)h(w)o(e)g(determine)f(the)h (con\014gurations)e(of)i(the)g(principal)0 1515 y(b)q(oundary)d(of)i(the)g (cycle)g Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 539 1527 V Fo(;)8 b Fu(+1)p 625 1527 V -20 x Fn(c)709 1515 y Fu(\).)0 1611 y Fw(Theorem)23 b(7.16.)k Fh(The)22 b(principal)e(b)q(oundary)956 1598 y Fk(e)945 1611 y Fo(B)r Fu(\()p Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 1053 1623 V 1 w Fo(;)8 b Fu(+1)p 1140 1623 V -20 x Fn(c)1224 1611 y Fu(\)\))22 b Fh(of)g(the)g(cycle)g Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 1614 1623 V 1 w Fo(;)8 b Fu(+1)p 1701 1623 V 1764 1591 a Fn(c)1785 1611 y Fu(\))22 b Fh(is)f(the)0 1671 y(set)89 1658 y Fk(f)77 1671 y Fp(M)137 1678 y Fn(n)162 1683 y Fg(c)199 1671 y Fh(of)c(the)f(principal)f(con\014gurations)f(of)i(v)o (olume)g Fo(n)1128 1678 y Fn(c)1148 1671 y Fh(.)22 b(F)l(or)15 b Fo(\033)j Fh(in)1389 1658 y Fk(f)1377 1671 y Fp(M)1437 1678 y Fn(n)1462 1683 y Fg(c)1499 1671 y Fh(there)e(exists)g(a)h(unique)0 1731 y(con\014guration)j(in)h Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 419 1743 V Fo(;)8 b Fu(+1)p 505 1743 V -20 x Fn(c)589 1731 y Fu(\))22 b Fh(comm)o(unicating)e(with)h Fo(\033)r Fh(,)j(whic)o(h)c(is)h (the)h(quasicub)q(e)f(plus)g(a)h(qua-)0 1790 y(sisquare)15 b Fo(j)211 1797 y Fn(c)242 1790 y Fp(\002)c Fu(\()p Fo(j)332 1797 y Fn(c)363 1790 y Fu(+)g Fo(\016)435 1797 y Fn(c)455 1790 y Fu(\))h Fp(\002)f Fu(\()p Fo(j)576 1797 y Fn(c)607 1790 y Fu(+)g(1\))g Fp(\010)g Fo(l)777 1797 y Fn(c)808 1790 y Fp(\002)g Fu(\()p Fo(l)892 1797 y Fn(c)924 1790 y Fu(+)f(1\))17 b Fh(or)f Fu(\()p Fo(l)1129 1797 y Fn(c)1161 1790 y Fu(+)11 b(1\))g Fp(\002)g Fo(l)1331 1797 y Fn(c)1368 1790 y Fh(included)k(in)h Fo(\033)r Fh(.)0 1886 y Fe(Pr)m(o)m(of.)28 b Fu(Let)17 b Fo(\033)h Fu(b)q(elong)e(to) 524 1874 y Fk(f)512 1886 y Fp(M)572 1893 y Fn(n)597 1898 y Fg(c)617 1886 y Fu(.)22 b(Then)16 b Fo(E)s Fu(\()p Fo(\033)r Fu(\))e(=)g Fo(E)s Fu(\()p Fo(n)1047 1893 y Fn(c)1067 1886 y Fu(\))j(and)e(theorem)h(7)p Fo(:)p Fu(4)g(yields)g(the)g(existence)h(of)0 1946 y(a)g(sequence)f Fo(\033)276 1953 y Fl(0)299 1946 y Fo(;)8 b Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(\033)446 1953 y Fn(n)471 1958 y Fg(c)509 1946 y Fu(suc)o(h)15 b(that)187 2060 y Fo(\033)215 2067 y Fl(0)252 2060 y Fu(=)e Fp(\000)p Fu(1)p 304 2072 V Fo(;)58 b(\033)468 2067 y Fn(n)493 2072 y Fg(c)528 2060 y Fu(=)14 b Fo(\033)o(;)58 b Fp(8)p Fo(j)15 b Fp(2)f(f)p Fu(0)8 b Fp(\001)g(\001)g(\001)h Fo(n)947 2067 y Fn(c)978 2060 y Fp(\000)i Fu(1)p Fp(g)49 b Fo(\033)1155 2067 y Fn(j)1191 2060 y Fp(2)14 b(M)1298 2067 y Fn(j)1318 2060 y Fo(;)58 b(q)1414 2040 y Fl(+)1447 2060 y Fu(\()p Fo(\033)1494 2067 y Fn(j)1516 2060 y Fo(;)8 b(\033)1566 2067 y Fn(j)r Fl(+1)1638 2060 y Fu(\))15 b(=)e(1)p Fo(:)0 2174 y Fu(In)25 b(particular,)i(w)o(e)e(ha)o(v)o(e)g(max)p Fp(f)8 b Fo(E)s Fu(\()p Fo(\033)746 2181 y Fn(n)774 2174 y Fu(\))29 b(:)g(0)h Fp(\024)e Fo(n)i(<)e(n)1144 2181 y Fn(c)1173 2174 y Fp(g)h Fo(<)g(E)s Fu(\()p Fo(n)1384 2181 y Fn(c)1404 2174 y Fu(\))d(so)g(that)g Fo(\033)1665 2181 y Fn(n)1690 2186 y Fg(c)1709 2181 y Fj(\000)p Fl(1)1788 2174 y Fu(b)q(elongs)0 2234 y(to)16 b Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 109 2246 V Fo(;)8 b Fu(+1)p 195 2246 V -20 x Fn(c)279 2234 y Fu(\).)22 b(Since)15 b Fo(q)r Fu(\()p Fo(\033)534 2241 y Fn(n)559 2246 y Fg(c)579 2241 y Fj(\000)p Fl(1)632 2234 y Fo(;)8 b(\033)682 2241 y Fn(n)707 2246 y Fg(c)728 2234 y Fu(\))15 b Fo(>)e Fu(0,)j(the)g (con\014guration)d Fo(\033)1282 2241 y Fn(n)1307 2246 y Fg(c)1344 2234 y Fu(is)j(in)f(the)h(principal)d(b)q(oundary)0 2303 y(of)k Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 106 2315 V Fo(;)8 b Fu(+1)p 192 2315 V -20 x Fn(c)276 2303 y Fu(\).)22 b(Th)o(us)469 2290 y Fk(f)458 2303 y Fp(M)518 2310 y Fn(n)543 2315 y Fg(c)577 2303 y Fp(\032)640 2290 y Fk(e)630 2303 y Fo(B)r Fu(\()p Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 738 2315 V Fo(;)8 b Fu(+1)p 824 2315 V -20 x Fn(c)909 2303 y Fu(\)\).)0 2371 y(Con)o(v)o(ersely)l(,)16 b(let)i Fo(\033)h Fu(b)q(elong)e(to)615 2358 y Fk(e)605 2371 y Fo(B)r Fu(\()p Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 713 2383 V 1 w Fo(;)8 b Fu(+1)p 800 2383 V 863 2351 a Fn(c)884 2371 y Fu(\)\).)25 b(Necessarily)17 b Fo(E)s Fu(\()p Fo(\033)r Fu(\))e(=)g Fo(E)s Fu(\()p Fo(n)1486 2378 y Fn(c)1506 2371 y Fu(\))j(so)f(that)h Fo(\033)h Fu(is)e(of)h(v)o(ol-)0 2430 y(ume)12 b Fo(n)134 2437 y Fn(c)166 2430 y Fu(and)f(it)i(is)f(a)g(minimal)e(con\014guration.)19 b(In)12 b(addition)f(there)h(m)o(ust)f(exist)h Fo(\021)i Fu(in)e Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 1643 2443 V Fo(;)c Fu(+1)p 1729 2443 V -19 x Fn(c)1813 2430 y Fu(\))13 b(com-)0 2490 y(m)o(unicating)h (with)i Fo(\033)r Fu(.)22 b(This)15 b Fo(\021)i Fu(is)e(in)h Fp(C)725 2497 y Fn(n)750 2502 y Fg(c)768 2497 y Fj(\000)p Fl(1)838 2490 y Fu(and)f(satis\014es)f Fo(E)s Fu(\()p Fo(\021)r Fu(\))g Fo(<)g(E)s Fu(\()p Fo(\033)r Fu(\).)22 b(Th)o(us)15 b Fo(a)p Fu(\()p Fo(\021)r Fu(\))f Fp(\024)g Fo(a)p Fu(\()p Fo(\033)r Fu(\))c Fp(\000)g Fo(h)j(<)0 2550 y(a)p Fu(\()p Fo(n)75 2557 y Fn(c)96 2550 y Fu(\))h(=)g Fo(a)p Fu(\()p Fo(n)257 2557 y Fn(c)286 2550 y Fp(\000)8 b Fu(1\))g(+)g(2)14 b(and)g Fo(a)p Fu(\()p Fo(\021)r Fu(\))h Fp(\025)e Fo(a)p Fu(\()p Fo(n)799 2557 y Fn(c)828 2550 y Fp(\000)8 b Fu(1\).)22 b(Ho)o(w)o(ev)o(er)14 b(the)h(area)g(is)f(an)h(ev)o(en)g(in)o(teger.)20 b(The)15 b(only)954 2670 y Fs(35)p eop %%Page: 36 36 36 35 bop 0 200 a Fu(p)q(ossibilit)o(y)11 b(is)h Fo(a)p Fu(\()p Fo(\021)r Fu(\))i(=)g Fo(a)p Fu(\()p Fo(n)512 207 y Fn(c)535 200 y Fp(\000)s Fu(1\))e(whence)h Fo(\021)g Fu(is)f(minimal.)19 b(Y)l(et)13 b Fo(n)1217 207 y Fn(c)1240 200 y Fp(\000)s Fu(1)g(=)h Fo(j)1394 207 y Fn(c)1413 200 y Fu(\()p Fo(j)1453 207 y Fn(c)1476 200 y Fu(+)s Fo(\016)1540 207 y Fn(c)1560 200 y Fu(\)\()p Fo(j)1619 207 y Fn(c)1642 200 y Fu(+)s(1\))s(+)s Fo(l)1788 207 y Fn(c)1807 200 y Fu(\()p Fo(l)1841 207 y Fn(c)1864 200 y Fu(+)s(1\))0 267 y(is)20 b(a)h(quasicub)q(e)f(plus)g(a)g(quasisquare;)h(b)o(y)g(theorem)f (7)p Fo(:)p Fu(3,)h Fp(M)1209 274 y Fn(n)1234 279 y Fg(c)1252 274 y Fj(\000)p Fl(1)1326 267 y Fu(=)1398 255 y Fk(f)1386 267 y Fp(M)1446 274 y Fn(n)1471 279 y Fg(c)1489 274 y Fj(\000)p Fl(1)1563 267 y Fu(so)f(that)i Fo(\021)g Fu(b)q(elongs)0 335 y(to)68 322 y Fk(f)56 335 y Fp(M)116 342 y Fn(n)141 347 y Fg(c)159 342 y Fj(\000)p Fl(1)224 335 y Fu(and)11 b(it)g(is)g(a)h(quasicub)q(e)e(plus) h(a)g(quasisquare)f Fo(j)t Fp(\002)q Fu(\()p Fo(j)t Fu(+)q Fo(\016)1238 342 y Fn(c)1256 335 y Fu(\))q Fp(\002)q Fo(j)t Fu(+)q(1\))q Fp(\010)q Fo(l)1481 342 y Fn(c)1501 335 y Fp(\002)q Fu(\()p Fo(l)1575 342 y Fn(c)1596 335 y Fu(+)q(1\).)20 b(Lemma)10 b(7)p Fo(:)p Fu(6)0 394 y(sho)o(ws)j(that)h(the)h(only)f(p)q(oin)o(ts)f(of)h Fp(M)698 401 y Fn(n)723 406 y Fg(c)758 394 y Fu(whic)o(h)f(comm)o(unicate)g (with)h Fp(M)1370 401 y Fn(n)1395 406 y Fg(c)1413 401 y Fj(\000)p Fl(1)1481 394 y Fu(are)g(the)g(con\014gurations)0 462 y(of)68 449 y Fk(f)56 462 y Fp(M)116 469 y Fn(n)141 474 y Fg(c)162 462 y Fu(.)22 b(Th)o(us)14 b Fo(\033)k Fu(is)e(a)g(principal)e (con\014guration)g(of)i(v)o(olume)f Fo(n)1225 469 y Fn(c)1262 462 y Fu(and)1368 449 y Fk(e)1358 462 y Fo(B)r Fu(\()p Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 1466 474 64 2 v 1 w Fo(;)8 b Fu(+1)p 1553 474 V -20 x Fn(c)1637 462 y Fu(\)\))14 b Fp(\032)1754 449 y Fk(f)1742 462 y Fp(M)1802 469 y Fn(n)1827 474 y Fg(c)1848 462 y Fu(.)49 b Fc(\003)60 552 y Fu(W)l(e)19 b(no)o(w)e(pro)q(ceed)i(to)f (steps)h Fo(ii)p Fu(\))g(and)f Fo(iii)p Fu(\))h(of)g(the)g(general)f (strategy)l(.)28 b(On)18 b(the)g(set)h(of)g(the)g(cycles)0 612 y(included)c(in)h Fp(f\000)p Fu(1)p 282 624 V Fo(;)8 b Fu(+1)p 368 624 V Fp(g)457 594 y Fn(c)477 612 y Fu(,)16 b(w)o(e)h(consider)e (the)i(graph)e Fp(G)k Fu(de\014ned)d(b)o(y)490 740 y(\()p Fo(\031)537 747 y Fl(1)574 740 y Fp(!)d Fo(\031)665 747 y Fl(2)688 740 y Fu(\))h Fp(2)g(G)116 b(\()-8 b(\))1130 727 y Fk(e)1119 740 y Fo(B)r Fu(\()p Fo(\031)1206 747 y Fl(1)1230 740 y Fu(\))11 b Fp(\\)g Fo(\031)1332 747 y Fl(2)1369 740 y Fp(6)p Fu(=)i Fp(;)p Fo(:)0 860 y Fu(W)l(e)k(denote)g(b)o(y)g Fp(G)350 842 y Fl(+)399 860 y Fu(the)h(restriction)d(of)j Fp(G)h Fu(to)e Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 941 872 V 1 w Fo(;)8 b Fu(+1)p 1028 872 V 1091 840 a Fn(c)1112 860 y Fu(\))1131 842 y Fn(c)1168 860 y Fu(\(obtained)17 b(b)o(y)g(deleting)f(all)h(the)g(arro)o(ws)0 920 y(whose)23 b(starting)g(cycle)h(is)f(included)f(in)h Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 854 932 V Fo(;)8 b Fu(+1)p 940 932 V -20 x Fn(c)1024 920 y Fu(\)\).)43 b(According)23 b(to)h(the)g(general)e (metho)q(d)h(for)0 979 y(determining)16 b(the)h(saddle)g(paths)g(of)g (minimal)f(cost)i(exiting)f(from)g Fp(\000)p Fu(1)p 1295 991 V 17 w(and)g(reac)o(hing)f(+1)p 1673 991 V -1 w(,)i(the)g(next)0 1039 y(step)j(is)g(to)h(\014nd,)g(for)f(eac)o(h)g(p)q(oin)o(t)g Fo(\033)714 1021 y Fj(0)750 1039 y Fu(in)g(the)g(principal)f(b)q(oundary)g (of)i Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 1456 1051 V Fo(;)8 b Fu(+1)p 1542 1051 V -20 x Fn(c)1626 1039 y Fu(\),)23 b(the)f(minimal)0 1099 y(stable)14 b(subgraph)f(of)i Fp(G)445 1081 y Fl(+)492 1099 y Fu(con)o(taining)f Fp(f)p Fo(\033)788 1081 y Fj(0)802 1099 y Fp(g)p Fu(.)21 b(If)15 b(there)g(do)q(es)g(not)f(exist)i(a)e(path)h (in)f(this)h(graph)e(leading)0 1159 y(from)e Fp(f)p Fo(\033)168 1141 y Fj(0)182 1159 y Fp(g)h Fu(to)g(+1)p 275 1171 V(,)g(then)g Fo(\033)504 1141 y Fj(0)530 1159 y Fu(is)f(a)h(dead{end)f(and)g(the)h(last)f (exit)i(from)e Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 1347 1171 V Fo(;)d Fu(+1)p 1433 1171 V -20 x Fn(c)1517 1159 y Fu(\))k(b)q(efore)g(reac) o(hing)e(+1)p 1886 1171 V 0 1218 a(w)o(on't)15 b(tak)o(e)h(place)g(through)e Fo(\033)587 1200 y Fj(0)601 1218 y Fu(.)22 b(Otherwise,)15 b Fo(\033)916 1200 y Fj(0)946 1218 y Fu(is)g(a)h(p)q(ossible)e(global)h (saddle)g(b)q(et)o(w)o(een)g Fp(\000)p Fu(1)p 1711 1231 V 15 w(and)g(+1)p 1886 1231 V 0 1278 a(and)21 b(w)o(e)h(ha)o(v)o(e)f(to)h (determine)f(all)g(the)h(paths)f(in)h Fp(G)1008 1260 y Fl(+)1062 1278 y Fu(leading)f(from)g Fp(f)p Fo(\033)1417 1260 y Fj(0)1431 1278 y Fp(g)h Fu(to)g(+1)p 1544 1290 V(;)i(then)e(w)o(e)f(m)o(ust)0 1338 y(searc)o(h)f(for)i(all)f(the)h(cycle)h(paths)e(in)g Fp(f\000)p Fu(1)p 761 1350 V -1 w Fo(;)8 b Fu(+1)p 846 1350 V Fp(g)935 1320 y Fn(c)977 1338 y Fu(of)22 b(n)o(ull)f(cost)g(realizing)g(the)h(exit)g (of)g Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 1760 1350 V 1 w Fo(;)8 b Fu(+1)p 1847 1350 V 1910 1318 a Fn(c)1931 1338 y Fu(\))0 1398 y(at)20 b Fo(\033)94 1380 y Fj(0)109 1398 y Fu(.)32 b(More)20 b(precisely)l(,)f(if)i(w)o(e)e(denote)i(b)o(y)e Fp(G)901 1380 y Fj(\000)955 1398 y Fu(the)h(restriction)f(of)h Fp(G)j Fu(to)d Fo(\031)r Fu(\(+1)p 1512 1410 V 1 w Fo(;)8 b Fp(\000)p Fu(1)p 1599 1410 V 1662 1378 a Fn(c)1683 1398 y Fu(\))1702 1380 y Fn(c)1742 1398 y Fu(\(obtained)0 1457 y(b)o(y)20 b(deleting)h(all)f(the)h (arro)o(ws)e(whose)h(starting)g(cycle)i(is)e(in)g Fo(\031)r Fu(\(+1)p 1227 1470 V 1 w Fo(;)8 b Fp(\000)p Fu(1)p 1314 1470 V 1377 1438 a Fn(c)1398 1457 y Fu(\)\),)22 b(w)o(e)f(ha)o(v)o(e)f(to)h (determine)0 1517 y(the)d(minimal)d(stable)i(subgraph)f(of)h Fp(G)734 1499 y Fj(\000)785 1517 y Fu(con)o(taining)f Fo(\033)1058 1499 y Fj(0)1090 1517 y Fu(and)h(all)g(the)h(paths)e(in)h(this)h(graph)e (starting)0 1577 y(at)i Fp(f)p Fo(\033)117 1559 y Fj(0)131 1577 y Fp(g)g Fu(and)f(ending)g(at)h Fp(\000)p Fu(1)p 496 1589 V -1 w(.)26 b(T)l(o)17 b(ac)o(hiev)o(e)g(these)h(goals,)f(w)o(e)h(will)f (next)h(describ)q(e)f(a)h(list)f(of)h(relev)m(an)o(t)0 1637 y(cycles,)24 b(whic)o(h)d(are)g(included)g(in)h Fp(f\000)p Fu(1)p 694 1649 V Fo(;)8 b Fu(+1)p 780 1649 V -1 w Fp(g)868 1619 y Fn(c)911 1637 y Fu(and)21 b(w)o(e)h(will)g(precise)f(their)h (principal)e(b)q(oundaries.)0 1697 y(Notice)f(that)g(w)o(e)f(don't)g(kno)o(w) g Fe(a)i(priori)g Fu(that)e(these)h(cycles)f(are)g(in)g Fp(M)p Fu(\()p Fp(f\000)p Fu(1)p 1426 1709 V Fo(;)8 b Fu(+1)p 1512 1709 V Fp(g)1601 1679 y Fn(c)1621 1697 y Fu(\).)28 b(Lemma)17 b(6)p Fo(:)p Fu(19)0 1756 y(will)h(yield)g("instan)o(taneously")e(the)j(list) f(of)g(the)g(relev)m(an)o(t)g(cycles)h(and)e(will)h(designate)g(those)g (cycles)0 1816 y(of)h(the)h(list)e(whic)o(h)h(are)f(in)h Fp(M)p Fu(\()p Fp(f\000)p Fu(1)p 629 1828 V Fo(;)8 b Fu(+1)p 715 1828 V -1 w Fp(g)803 1798 y Fn(c)824 1816 y Fu(\).)30 b(Our)18 b(list)h(of)g (cycles)g(will)g(b)q(e)g(large)g(enough)f(to)h(ensure)0 1884 y(that)d(for)g(eac)o(h)f(con\014guration)f Fo(\033)625 1866 y Fj(0)655 1884 y Fu(in)723 1871 y Fk(e)712 1884 y Fo(B)r Fu(\()p Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 820 1896 V 1 w Fo(;)8 b Fu(+1)p 907 1896 V -20 x Fn(c)991 1884 y Fu(\)\),)17 b(it)f(includes)e(all)h (the)h(v)o(ertices)g(of)g(the)g(minimal)0 1944 y(stable)22 b(subgraph)d(of)j Fp(G)j Fu(con)o(taining)c Fp(f)p Fo(\033)791 1926 y Fj(0)805 1944 y Fp(g)p Fu(.)38 b(Moreo)o(v)o(er,)21 b(w)o(e)h(will)f(handle)g(separately)h(the)g(case)g(of)0 2011 y(the)f(con\014gurations)e(in)h Fp(S)508 2018 y Fn(n)533 2023 y Fg(c)575 2011 y Fu(and)g(in)751 1998 y Fk(f)739 2011 y Fp(M)799 2018 y Fn(n)824 2023 y Fg(c)858 2011 y Fp(n)14 b(S)927 2018 y Fn(n)952 2023 y Fg(c)973 2011 y Fu(:)31 b(it)21 b(turns)f(out)h(that)g (step)g Fo(ii)p Fu(\))g(succeeds)f(for)h(the)0 2071 y(standard)13 b(con\014gurations)g(and)h(fails)h(for)f(the)h(principal)e(non)i(standard)e (con\014gurations.)19 b(The)c(latter)0 2131 y(are)h(dead{ends.)0 2221 y Fw(The)26 b(standard)f(con\014gurations.)j Fu(W)l(e)23 b(start)g(b)o(y)f(describing)f(the)j(relev)m(an)o(t)e(list)h(of)g(cycles)g (for)0 2280 y(determining)18 b(the)h(minimal)f(stable)g(subgraph)f(of)j Fp(G)i Fu(con)o(taining)c(the)h(standard)f(con\014gurations)f(of)0 2340 y(v)o(olume)d Fo(n)200 2347 y Fn(c)220 2340 y Fu(.)22 b(These)15 b(are)g(the)g(cycles)h(around)d(a)j(parallelepip)q(ed)d Fo(j)1254 2347 y Fl(1)1284 2340 y Fp(\002)c Fo(j)1353 2347 y Fl(2)1383 2340 y Fp(\002)f Fo(j)1451 2347 y Fl(3)1473 2340 y Fu(,)15 b(a)h(parallelepip)q(ed)d(plus)0 2400 y(a)19 b(rectangle)g Fo(j)281 2407 y Fl(1)316 2400 y Fp(\002)13 b Fo(j)389 2407 y Fl(2)423 2400 y Fp(\002)g Fo(j)496 2407 y Fl(3)531 2400 y Fp(\010)f Fo(l)597 2407 y Fl(1)631 2400 y Fp(\002)f Fo(l)696 2407 y Fl(2)737 2400 y Fu(and)19 b(\014nally)f(a)i(parallelepip)q(ed)d(plus)h (a)h(rectangle)g(plus)g(a)g(bar)0 2460 y Fo(j)21 2467 y Fl(1)50 2460 y Fp(\002)7 b Fo(j)117 2467 y Fl(2)146 2460 y Fp(\002)g Fo(j)213 2467 y Fl(3)242 2460 y Fp(\010)g Fo(l)303 2467 y Fl(1)336 2460 y Fp(\002)k Fo(l)401 2467 y Fl(2)431 2460 y Fp(\010)c Fo(k)r Fu(,)14 b(where)g(eac)o(h)g(con\014guration)f(con)o(tains)g(strictly)i (a)g(critical)f(con\014guration.)0 2550 y Fw(Theorem)k(7.17.)50 b Fh(Let)17 b Fo(j)523 2557 y Fl(1)545 2550 y Fo(;)8 b(j)588 2557 y Fl(2)610 2550 y Fo(;)g(j)653 2557 y Fl(3)692 2550 y Fh(b)q(e)17 b(three)g(in)o(tegers)e(suc)o(h)h(that)h(the)g(parallelepip)q(ed) e Fo(j)1720 2557 y Fl(1)1753 2550 y Fp(\002)c Fo(j)1824 2557 y Fl(2)1857 2550 y Fp(\002)g Fo(j)1928 2557 y Fl(3)954 2670 y Fs(36)p eop %%Page: 37 37 37 36 bop 0 200 a Fh(con)o(tains)15 b(a)i(con\014guration)e(of)h Fp(S)625 207 y Fn(n)650 212 y Fg(c)671 200 y Fh(.)22 b(The)17 b(cycle)338 307 y Fo(\031)f Fu(=)d Fo(\031)r Fu(\()p Fo(j)504 314 y Fl(1)538 307 y Fp(\002)e Fo(j)609 314 y Fl(2)642 307 y Fp(\002)f Fo(j)712 314 y Fl(3)734 307 y Fo(;)e Fh(pred)g Fo(E)s Fu(\()p Fo(j)942 314 y Fl(1)975 307 y Fp(\002)i Fo(j)1045 314 y Fl(2)1078 307 y Fp(\002)h Fo(j)1149 314 y Fl(3)1182 307 y Fp(\010)g Fo(l)1247 314 y Fn(c)1278 307 y Fp(\002)g Fu(\()p Fo(l)1362 314 y Fn(c)1394 307 y Fu(+)f(1\))i Fp(\010)f Fu(1\)\))0 415 y Fh(do)q(es)16 b(not)h(con)o(tain)f Fp(\000)p Fu(1)p 377 427 64 2 v 16 w Fh(and)g Fu(+1)p 554 427 V -1 w Fh(.)22 b(Moreo)o(v)o(er)398 522 y Fo(v)p 398 530 26 2 v 2 w Fu(\()p Fo(\031)r Fu(\))h Fo(>)f(j)597 529 y Fl(1)619 522 y Fo(j)640 529 y Fl(2)661 522 y Fo(j)682 529 y Fl(3)715 522 y Fp(\000)11 b Fu(min)o(\()p Fo(j)888 529 y Fl(1)910 522 y Fo(j)931 529 y Fl(2)953 522 y Fo(;)d(j)996 529 y Fl(1)1018 522 y Fo(j)1039 529 y Fl(3)1061 522 y Fo(;)g(j)1104 529 y Fl(2)1126 522 y Fo(j)1147 529 y Fl(3)1169 522 y Fu(\))k(+)e Fo(l)1264 529 y Fn(c)1284 522 y Fu(\()p Fo(l)1318 529 y Fn(c)1350 522 y Fu(+)h(1\))g(+)g(1)d Fo(;)p 683 570 V 683 597 a(v)r Fu(\()p Fo(\031)r Fu(\))23 b(=)f Fo(j)882 604 y Fl(1)904 597 y Fo(j)925 604 y Fl(2)946 597 y Fo(j)967 604 y Fl(3)1000 597 y Fu(+)11 b Fo(l)1065 604 y Fn(c)1085 597 y Fu(\()p Fo(l)1119 604 y Fn(c)1151 597 y Fu(+)g(1\))d Fo(:)0 705 y Fh(The)16 b(b)q(ottom)h(of)g(this)f(cycle)h (is)f Fp(f)p Fo(j)651 712 y Fl(1)684 705 y Fp(\002)11 b Fo(j)755 712 y Fl(2)788 705 y Fp(\002)g Fo(j)859 712 y Fl(3)880 705 y Fp(g)17 b Fh(and)f(its)g(principal)f(b)q(oundary)g(is)525 806 y Fk(e)515 818 y Fo(B)r Fu(\()p Fo(\031)r Fu(\))23 b(=)p 707 765 653 2 v 22 w Fo(j)728 825 y Fl(1)761 818 y Fp(\002)11 b Fo(j)832 825 y Fl(2)864 818 y Fp(\002)g Fo(j)935 825 y Fl(3)968 818 y Fp(\010)p 1018 775 342 2 v 11 w Fo(l)1033 825 y Fn(c)1064 818 y Fp(\002)g Fu(\()p Fo(l)1148 825 y Fn(c)1180 818 y Fu(+)f(1\))i Fp(\010)f Fu(1)1360 774 y Fl(123)1422 818 y Fo(:)0 953 y Fe(Pr)m(o)m(of.)28 b Fu(W)l(e)15 b(c)o(hec)o(k)g(that)g(the)g(parallelepip)q(ed)f Fo(j)904 960 y Fl(1)934 953 y Fp(\002)8 b Fo(j)1002 960 y Fl(2)1031 953 y Fp(\002)g Fo(j)1099 960 y Fl(3)1136 953 y Fu(and)15 b(the)g(cycle)h Fo(\031)h Fu(satisfy)e(the)g(h)o(yp)q(othesis)0 1013 y(of)h(theorem)f(5)p Fo(:)p Fu(5.)21 b(Let)c Fo(x)465 1020 y Fl(1)487 1013 y Fo(;)8 b Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)634 1020 y Fn(r)673 1013 y Fu(b)q(e)16 b(a)f(sequence)h(of)g(sites)f(suc)o(h)g(that)h Fo(F)7 b Fu(\()p Fo(x)1457 1020 y Fl(1)1480 1013 y Fo(;)h Fp(\001)g(\001)g (\001)17 b Fo(;)8 b(x)1627 1020 y Fn(s)1648 1013 y Fu(\)\()p Fo(j)1707 1020 y Fl(1)1739 1013 y Fp(\002)h Fo(j)1808 1020 y Fl(2)1840 1013 y Fp(\002)g Fo(j)1909 1020 y Fl(3)1931 1013 y Fu(\))0 1073 y(is)24 b(in)h Fo(\031)i Fu(for)d Fo(s)i Fu(in)e Fp(f)p Fu(1)8 b Fp(\001)g(\001)g(\001)h Fo(r)q Fp(g)25 b Fu(\(i.e.)47 b(these)25 b(con\014gurations)e(ha)o(v)o(e)h(an)h(energy)f(less)g(or)h(equal) f(than)0 1133 y(pred)7 b Fo(E)s Fu(\()p Fo(j)185 1140 y Fl(1)218 1133 y Fp(\002)12 b Fo(j)290 1140 y Fl(2)323 1133 y Fp(\002)f Fo(j)394 1140 y Fl(3)427 1133 y Fp(\010)g Fo(l)492 1140 y Fn(c)523 1133 y Fp(\002)h Fu(\()p Fo(l)608 1140 y Fn(c)639 1133 y Fu(+)f(1\))h Fp(\010)f Fu(1\)\).)24 b(W)l(e)17 b(put)g Fo(\021)1099 1140 y Fn(s)1134 1133 y Fu(=)d Fo(F)7 b Fu(\()p Fo(x)1273 1140 y Fl(1)1296 1133 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1443 1140 y Fn(s)1465 1133 y Fu(\)\()p Fo(j)1524 1140 y Fl(1)1558 1133 y Fp(\002)j Fo(j)1629 1140 y Fl(2)1662 1133 y Fp(\002)g Fo(j)1733 1140 y Fl(3)1755 1133 y Fu(\))18 b(for)e Fo(s)h Fu(in)0 1192 y Fp(f)p Fu(0)8 b Fp(\001)g(\001)g(\001)h Fo(r)q Fp(g)p Fu(.)0 1252 y Fp(\017)50 b Fu(First)14 b(case:)22 b(all)15 b(the)h(sites)f(are)g(outside)g Fo(j)857 1259 y Fl(1)888 1252 y Fp(\002)9 b Fo(j)957 1259 y Fl(2)989 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1613 y Fo(j)1131 1620 y Fl(3)1170 1613 y Fu(and)f Fo(v)r Fu(\()p Fo(\021)1337 1620 y Fn(r)1359 1613 y Fu(\).)22 b(Henceforth)17 1721 y(max)3 1753 y Fl(0)p Fj(\024)p Fn(s)p Fj(\024)p Fn(r)131 1721 y Fo(E)s Fu(\()p Fo(j)211 1728 y Fl(1)244 1721 y Fp(\002)11 b Fo(j)315 1728 y Fl(2)348 1721 y Fp(\002)g Fo(j)419 1728 y Fl(3)441 1721 y Fo(;)d(v)r Fu(\()p Fo(\021)533 1728 y Fn(s)554 1721 y Fu(\))k Fp(\000)f Fo(j)656 1728 y Fl(1)677 1721 y Fo(j)698 1728 y Fl(2)720 1721 y Fo(j)741 1728 y Fl(3)763 1721 y Fu(\))31 b Fp(\025)f Fu(max)p Fp(f)8 b Fo(E)s Fu(\()p Fo(j)1088 1728 y Fl(1)1121 1721 y Fp(\002)j Fo(j)1192 1728 y Fl(2)1225 1721 y Fp(\002)g Fo(j)1296 1728 y Fl(3)1317 1721 y Fo(;)d(k)r Fu(\))14 b(:)g(0)g Fp(\024)f Fo(k)j Fp(\024)d Fo(v)r Fu(\()p Fo(\021)1683 1728 y Fn(r)1705 1721 y Fu(\))f Fp(\000)f Fo(j)1807 1728 y Fl(1)1828 1721 y Fo(j)1849 1728 y Fl(2)1871 1721 y Fo(j)1892 1728 y Fl(3)1922 1721 y Fp(g)0 1852 y Fu(and)16 b(the)h(v)o(olume)e(of)i Fo(\021)436 1859 y Fn(r)474 1852 y Fu(m)o(ust)f(satisfy)0 1959 y(max)p Fp(f)8 b Fo(E)s Fu(\()p Fo(j)206 1966 y Fl(1)234 1959 y Fp(\002)e Fo(j)300 1966 y Fl(2)329 1959 y Fp(\002)g Fo(j)395 1966 y Fl(3)417 1959 y Fo(;)i(k)r Fu(\))13 b(:)h(0)g Fp(\024)f Fo(k)j Fp(\024)d Fo(v)r Fu(\()p Fo(\021)782 1966 y Fn(r)804 1959 y Fu(\))6 b Fp(\000)g Fo(j)895 1966 y Fl(1)918 1959 y Fo(j)939 1966 y Fl(2)961 1959 y Fo(j)982 1966 y Fl(3)1012 1959 y Fp(g)30 b(\024)h Fu(pred)7 b Fo(E)s Fu(\()p Fo(j)1322 1966 y Fl(1)1350 1959 y Fp(\002)f Fo(j)1416 1966 y Fl(2)1445 1959 y Fp(\002)g Fo(j)1511 1966 y Fl(3)1539 1959 y Fp(\010)g Fo(l)1599 1966 y Fn(c)1626 1959 y Fp(\002)g Fu(\()p Fo(l)1705 1966 y Fn(c)1732 1959 y Fu(+)g(1\))g Fp(\010)g Fu(1\)\))p Fo(:)0 2067 y Fu(By)17 b(theorem)f(7)p Fo(:)p Fu(9,)g(w)o(e)g(ha)o(v)o(e)251 2174 y Fo(E)s Fu(\()p Fo(j)331 2181 y Fl(1)364 2174 y Fp(\002)11 b Fo(j)435 2181 y Fl(2)467 2174 y Fp(\002)g Fo(j)538 2181 y Fl(3)560 2174 y Fo(;)d(l)597 2181 y Fn(c)617 2174 y Fu(\()p Fo(l)651 2181 y Fn(c)683 2174 y Fu(+)j(1\))g(+)g(1\))j(=)g Fo(E)s Fu(\()p Fo(j)1029 2181 y Fl(1)1062 2174 y Fp(\002)c Fo(j)1132 2181 y Fl(2)1165 2174 y Fp(\002)h Fo(j)1236 2181 y Fl(3)1269 2174 y Fp(\010)g Fo(l)1334 2181 y Fn(c)1365 2174 y Fp(\002)g Fu(\()p Fo(l)1449 2181 y Fn(c)1481 2174 y Fu(+)f(1\))i Fp(\010)f Fu(1\)\))0 2282 y(so)16 b(that)h Fo(v)r Fu(\()p Fo(\021)239 2289 y Fn(r)261 2282 y Fu(\))d Fo(<)g(j)368 2289 y Fl(1)390 2282 y Fo(j)411 2289 y Fl(2)433 2282 y Fo(j)454 2289 y Fl(3)487 2282 y Fu(+)c Fo(l)551 2289 y Fn(c)572 2282 y Fu(\()p Fo(l)606 2289 y Fn(c)637 2282 y Fu(+)h(1\))g(+)g(1.)0 2341 y Fp(\017)50 b Fu(Second)16 b(case:)21 b(all)c(the)f(sites)h(are)f(inside)f Fo(j)880 2348 y Fl(1)913 2341 y Fp(\002)c Fo(j)984 2348 y Fl(2)1017 2341 y Fp(\002)g Fo(j)1088 2348 y Fl(3)1109 2341 y Fu(.)22 b(No)o(w)17 b Fo(\021)1284 2348 y Fn(s)1319 2341 y Fp(\032)c Fo(j)1392 2348 y Fl(1)1425 2341 y Fp(\002)e Fo(j)1496 2348 y Fl(2)1529 2341 y Fp(\002)g Fo(j)1600 2348 y Fl(3)1638 2341 y Fu(so)16 b(that)0 2449 y(pred)7 b Fo(E)s Fu(\()p Fo(j)185 2456 y Fl(1)218 2449 y Fp(\002)k Fo(j)289 2456 y Fl(2)322 2449 y Fp(\002)g Fo(j)393 2456 y Fl(3)426 2449 y Fp(\010)g Fo(l)491 2456 y Fn(c)522 2449 y Fp(\002)g Fu(\()p Fo(l)606 2456 y Fn(c)637 2449 y Fu(+)g(1\))g Fp(\010)g Fu(1\)\))31 b Fp(\025)44 b Fu(max)955 2481 y Fl(0)p Fj(\024)p Fn(s)p Fj(\024)p Fn(r)1084 2449 y Fo(E)s Fu(\()p Fo(j)1164 2456 y Fl(1)1197 2449 y Fp(\002)11 b Fo(j)1268 2456 y Fl(2)1301 2449 y Fp(\002)g Fo(j)1372 2456 y Fl(3)1394 2449 y Fo(;)d(v)r Fu(\()p Fo(\021)1486 2456 y Fn(s)1507 2449 y Fu(\))j Fp(\000)g Fo(j)1608 2456 y Fl(1)1630 2449 y Fo(j)1651 2456 y Fl(2)1673 2449 y Fo(j)1694 2456 y Fl(3)1716 2449 y Fu(\))802 2550 y Fp(\025)30 b Fu(max)p Fp(f)8 b Fo(E)s Fu(\()p Fo(j)1077 2557 y Fl(1)1110 2550 y Fp(\002)j Fo(j)1181 2557 y Fl(2)1214 2550 y Fp(\002)f Fo(j)1284 2557 y Fl(3)1306 2550 y Fo(;)e(k)r Fu(\))14 b(:)g Fo(v)r Fu(\()p Fo(\021)1487 2557 y Fn(r)1509 2550 y Fu(\))d Fp(\000)g Fo(j)1610 2557 y Fl(1)1632 2550 y Fo(j)1653 2557 y Fl(2)1675 2550 y Fo(j)1696 2557 y Fl(3)1732 2550 y Fp(\024)i Fo(k)i Fp(\024)f Fu(0)8 b Fp(g)p Fo(:)954 2670 y Fs(37)p eop %%Page: 38 38 38 37 bop 0 200 a Fu(Supp)q(ose)11 b(for)i(instance)f(that)h Fo(j)579 207 y Fl(1)601 200 y Fo(j)622 207 y Fl(2)656 200 y Fu(is)g(the)f(smallest)g(side)g(of)h(the)g(parallelepip)q(ed.)19 b(Theorem)11 b(7)p Fo(:)p Fu(8)i(sho)o(ws)0 260 y(that)k(for)f Fo(j)205 267 y Fl(1)227 260 y Fu(\()p Fo(j)267 267 y Fl(2)300 260 y Fp(\000)11 b Fo(j)371 267 y Fl(1)393 260 y Fu(\))j Fp(\024)g Fo(k)h(<)f(j)594 267 y Fl(1)616 260 y Fo(j)637 267 y Fl(2)658 260 y Fu(,)382 340 y Fo(E)s Fu(\()p Fo(j)462 347 y Fl(1)495 340 y Fp(\002)d Fo(j)566 347 y Fl(2)599 340 y Fp(\002)g Fo(j)670 347 y Fl(3)692 340 y Fo(;)d Fp(\000)p Fo(k)r Fu(\))21 b(=)h Fo(E)s Fu(\()p Fo(j)962 347 y Fl(1)995 340 y Fp(\002)11 b Fo(j)1066 347 y Fl(2)1099 340 y Fp(\002)g Fu(\()p Fo(j)1189 347 y Fl(3)1222 340 y Fp(\000)g Fu(1\))g Fp(\010)g Fo(m)1421 347 y Fn(j)1437 352 y Fd(1)1457 347 y Fn(j)1473 352 y Fd(2)1493 347 y Fj(\000)p Fn(k)1548 340 y Fu(\))0 420 y(Ho)o(w)o(ev)o(er,)16 b Fo(E)s Fu(\()p Fo(j)295 427 y Fl(1)328 420 y Fp(\002)11 b Fo(j)399 427 y Fl(2)432 420 y Fp(\002)g Fu(\()p Fo(j)522 427 y Fl(3)556 420 y Fp(\000)g Fu(1\))h Fp(\010)f Fo(l)727 427 y Fn(c)758 420 y Fp(\002)g Fu(\()p Fo(l)842 427 y Fn(c)874 420 y Fu(+)g(1\))g Fp(\010)1018 427 y Fl(1)1052 420 y Fu(1\))31 b Fo(>)g(E)s Fu(\()p Fo(j)1277 427 y Fl(1)1310 420 y Fp(\002)11 b Fo(j)1381 427 y Fl(2)1414 420 y Fp(\002)g Fo(j)1485 427 y Fl(3)1518 420 y Fp(\010)g Fo(l)1583 427 y Fn(c)1615 420 y Fp(\002)g Fu(\()p Fo(l)1699 427 y Fn(c)1731 420 y Fu(+)g(1\))g Fp(\010)g Fu(1\)\))0 480 y(whence)16 b(necessarily)g Fo(j)444 487 y Fl(1)466 480 y Fo(j)487 487 y Fl(2)520 480 y Fp(\000)10 b Fo(k)16 b(>)d(l)678 487 y Fn(c)698 480 y Fu(\()p Fo(l)732 487 y Fn(c)764 480 y Fu(+)e(1\))g(+)g(1)16 b(and)g Fo(v)r Fu(\()p Fo(\021)1127 487 y Fn(r)1149 480 y Fu(\))f Fo(>)e(j)1256 487 y Fl(1)1278 480 y Fo(j)1299 487 y Fl(2)1321 480 y Fo(j)1342 487 y Fl(3)1375 480 y Fp(\000)e Fo(j)1446 487 y Fl(1)1468 480 y Fo(j)1489 487 y Fl(2)1521 480 y Fu(+)g Fo(l)1586 487 y Fn(c)1606 480 y Fu(\()p Fo(l)1640 487 y Fn(c)1672 480 y Fu(+)g(1\))g(+)g(1.)0 539 y(In)16 b(addition,)g(w)o(e)g(ha)o(v)o(e)g(that)h Fo(E)s Fu(\()p Fo(j)649 546 y Fl(1)682 539 y Fp(\002)10 b Fo(j)752 546 y Fl(2)785 539 y Fp(\002)h Fo(j)856 546 y Fl(3)878 539 y Fo(;)d(k)r Fu(\))14 b Fo(>)f(E)s Fu(\()p Fo(j)1093 546 y Fl(1)1126 539 y Fp(\002)e Fo(j)1197 546 y Fl(2)1230 539 y Fp(\002)g Fo(j)1301 546 y Fl(3)1323 539 y Fu(\))17 b(for)f(all)g Fo(k)i Fu(suc)o(h)e(that)483 620 y Fp(\000)p Fo(j)543 627 y Fl(1)564 620 y Fo(j)585 627 y Fl(2)618 620 y Fu(+)11 b Fo(l)683 627 y Fn(c)703 620 y Fu(\()p Fo(l)737 627 y Fn(c)769 620 y Fu(+)g(1\))g(+)g(1)j Fo(<)f(k)j Fp(\024)d Fo(l)1124 627 y Fn(c)1144 620 y Fu(\()p Fo(l)1178 627 y Fn(c)1210 620 y Fu(+)e(1\))p Fo(;)16 b(k)g Fp(6)p Fu(=)d(0)p Fo(:)0 700 y Fu(W)l(e)h(ha)o(v)o(e)f(th)o(us)g(pro)o(v)o(ed)f(that)i Fo(j)587 707 y Fl(1)614 700 y Fp(\002)5 b Fo(j)679 707 y Fl(2)707 700 y Fp(\002)g Fo(j)772 707 y Fl(3)808 700 y Fu(and)13 b(the)h(cycle)g Fo(\031)i Fu(satisfy)e(the)g(h)o(yp)q(othesis)f(of)g(theorem)h(5)p Fo(:)p Fu(5.)0 759 y(Th)o(us)22 b(the)h(b)q(ottom)h(of)f(the)g(cycle)h(is)f Fp(f)p Fo(j)796 766 y Fl(1)833 759 y Fp(\002)15 b Fo(j)908 766 y Fl(2)946 759 y Fp(\002)g Fo(j)1021 766 y Fl(3)1043 759 y Fp(g)p Fu(.)41 b(W)l(e)24 b(ha)o(v)o(e)e(also)h(pro)o(v)o(ed)e(the)j (inequalities)0 819 y(on)p 74 792 26 2 v 21 w Fo(v)f Fu(and)d Fo(v)p 222 827 V 2 w Fu(.)36 b(That)21 b Fp(M)p Fu(\()p Fo(j)527 826 y Fl(1)563 819 y Fp(\002)14 b Fo(j)637 826 y Fl(2)673 819 y Fp(\002)g Fo(j)747 826 y Fl(3)769 819 y Fo(;)8 b(k)r Fu(\))21 b(is)f(included)g(in)h(the)g(cycle)h(for)f(0)g Fp(\024)g Fo(k)i Fp(\024)e Fo(l)1709 826 y Fn(c)1730 819 y Fu(\()p Fo(l)1764 826 y Fn(c)1798 819 y Fu(+)14 b(1\))22 b(is)0 879 y(ob)o(vious:)31 b(eac)o(h)22 b(con\014guration)e(of)i(this)f(set)h(comm)o(unicates)e(with)i Fo(j)1350 886 y Fl(1)1386 879 y Fp(\002)15 b Fo(j)1461 886 y Fl(2)1497 879 y Fp(\002)g Fo(j)1572 886 y Fl(3)1615 879 y Fu(under)21 b(the)h(lev)o(el)0 939 y(pred)7 b Fo(E)s Fu(\()p Fo(j)185 946 y Fl(1)219 939 y Fp(\002)k Fo(j)290 946 y Fl(2)323 939 y Fp(\002)g Fo(j)394 946 y Fl(3)428 939 y Fp(\010)g Fo(l)493 946 y Fn(c)525 939 y Fp(\002)g Fu(\()p Fo(l)609 946 y Fn(c)641 939 y Fu(+)g(1\))h Fp(\010)f Fu(1\)\))18 b(\(cf)g(theorem)f(7)p Fo(:)p Fu(9\).)23 b(Finally)17 b(a)g(con\014guration)e Fo(\030)k Fu(of)f(the)0 999 y(principal)11 b(b)q(oundary)h(of)i(this)e(cycle)i(is)f(of) g(energy)g Fo(E)s Fu(\()p Fo(\030)r Fu(\))i(=)e Fo(E)s Fu(\()p Fo(j)1197 1006 y Fl(1)1224 999 y Fp(\002)t Fo(j)1288 1006 y Fl(2)1314 999 y Fp(\002)t Fo(j)1378 1006 y Fl(3)1404 999 y Fp(\010)t Fo(l)1462 1006 y Fn(c)1487 999 y Fp(\002)t Fu(\()p Fo(l)1564 1006 y Fn(c)1589 999 y Fu(+)t(1\))t Fp(\010)t Fu(1\)\))i(so)e(that) 0 1058 y(its)f(v)o(olume)g(is)g(equal)g(to)h Fo(j)484 1065 y Fl(1)506 1058 y Fo(j)527 1065 y Fl(2)549 1058 y Fo(j)570 1065 y Fl(3)594 1058 y Fu(+)s Fo(l)651 1065 y Fn(c)671 1058 y Fu(\()p Fo(l)705 1065 y Fn(c)729 1058 y Fu(+)s(1\))s(+)s(1,)f(and)g(its)g (area)g(is)g(2\()p Fo(j)1284 1065 y Fl(1)1306 1058 y Fo(j)1327 1065 y Fl(2)1352 1058 y Fu(+)s Fo(j)1415 1065 y Fl(2)1437 1058 y Fo(j)1458 1065 y Fl(3)1482 1058 y Fu(+)s Fo(j)1545 1065 y Fl(1)1567 1058 y Fo(j)1588 1065 y Fl(3)1610 1058 y Fu(\))s(+)s(2\(2)p Fo(l)1758 1065 y Fn(c)1781 1058 y Fu(+)s(1\))s(+)s(2.)0 1118 y(Let)20 b Fo(\021)g Fu(b)q(e)f(a)g(con\014guration)e(of)j(the)f(cycle)g(suc) o(h)f(that)h Fo(q)r Fu(\()p Fo(\021)r(;)8 b(\030)r Fu(\))19 b Fo(>)e Fu(0.)29 b(Necessarily)l(,)19 b(the)g(v)o(olume)f(of)h Fo(\021)0 1178 y Fu(is)f Fo(j)73 1185 y Fl(1)95 1178 y Fo(j)116 1185 y Fl(2)137 1178 y Fo(j)158 1185 y Fl(3)192 1178 y Fu(+)12 b Fo(l)258 1185 y Fn(c)278 1178 y Fu(\()p Fo(l)312 1185 y Fn(c)345 1178 y Fu(+)g(1\).)27 b(Th)o(us)17 b Fo(\021)i Fu(is)f(a)g(con\014guration)f (of)h(maximal)f(v)o(olume)g(of)h(the)h(cycle)g Fo(\031)h Fu(and)d(as)0 1238 y(suc)o(h)f(it)i(is)f(a)h(maximal)e(con\014guration)g(of)i(the)g(cycle)g (for)f(the)h(inclusion)e(relation.)24 b(By)18 b(theorem)f(5)p Fo(:)p Fu(3)0 1297 y(the)22 b(parallelepip)q(ed)e Fo(j)437 1304 y Fl(1)473 1297 y Fp(\002)14 b Fo(j)547 1304 y Fl(2)584 1297 y Fp(\002)g Fo(j)658 1304 y Fl(3)702 1297 y Fu(is)21 b(included)f(in)i Fo(\021)r Fu(.)37 b(In)21 b(addition)g Fo(E)s Fu(\()p Fo(\021)r Fu(\))h Fo(<)g(E)s Fu(\()p Fo(\030)r Fu(\))g(implies)f(that)0 1357 y Fo(a)p Fu(\()p Fo(\021)r Fu(\))14 b Fp(\024)g Fu(2\()p Fo(j)223 1364 y Fl(1)245 1357 y Fo(j)266 1364 y Fl(2)292 1357 y Fu(+)t Fo(j)356 1364 y Fl(2)377 1357 y Fo(j)398 1364 y Fl(3)423 1357 y Fu(+)t Fo(j)487 1364 y Fl(1)508 1357 y Fo(j)529 1364 y Fl(3)551 1357 y Fu(\))t(+)t(2\(2)p Fo(l)701 1364 y Fn(c)725 1357 y Fu(+)t(1\))e(whence)h(in)f(fact)i Fo(\021)g Fu(b)q(elongs)e(to)h Fp(M)p Fu(\()p Fo(j)1515 1364 y Fl(1)1541 1357 y Fp(\002)t Fo(j)1605 1364 y Fl(2)1630 1357 y Fp(\002)t Fo(j)1694 1364 y Fl(3)1715 1357 y Fo(;)8 b(l)1752 1364 y Fn(c)1772 1357 y Fu(\()p Fo(l)1806 1364 y Fn(c)1830 1357 y Fu(+)t(1\)\).)0 1417 y(Th)o(us)k Fo(\021)i Fu(is)e(the)h(sum)f(of)h Fo(j)466 1424 y Fl(1)492 1417 y Fp(\002)t Fo(j)556 1424 y Fl(2)581 1417 y Fp(\002)t Fo(j)645 1424 y Fl(3)679 1417 y Fu(and)f(a)h(quasisquare)f Fo(l)1091 1424 y Fn(c)1114 1417 y Fp(\002)t Fu(\()p Fo(l)1191 1424 y Fn(c)1215 1417 y Fu(+)t(1\))h(\(theorem)f(7)p Fo(:)p Fu(9\).)21 b(Also)13 b Fo(\030)i Fu(b)q(elongs)0 1477 y(to)i Fp(M)p Fu(\()p Fo(j)161 1484 y Fl(1)195 1477 y Fp(\002)11 b Fo(j)266 1484 y Fl(2)299 1477 y Fp(\002)g Fo(j)370 1484 y Fl(3)392 1477 y Fo(;)d(l)429 1484 y Fn(c)449 1477 y Fu(\()p Fo(l)483 1484 y Fn(c)515 1477 y Fu(+)j(1\))h(+)f(1\).)24 b(The)17 b(only)g (con\014gurations)e(of)i(this)f(set)i(whic)o(h)e(comm)o(unicate)0 1537 y(with)g Fp(M)p Fu(\()p Fo(j)213 1544 y Fl(1)247 1537 y Fp(\002)10 b Fo(j)317 1544 y Fl(2)350 1537 y Fp(\002)h Fo(j)421 1544 y Fl(3)443 1537 y Fo(;)d(l)480 1544 y Fn(c)500 1537 y Fu(\()p Fo(l)534 1544 y Fn(c)566 1537 y Fu(+)j(1\)\))17 b(are)f(the)h (con\014gurations)d(of)p 611 1588 653 2 v 611 1642 a Fo(j)632 1649 y Fl(1)665 1642 y Fp(\002)c Fo(j)735 1649 y Fl(2)768 1642 y Fp(\002)h Fo(j)839 1649 y Fl(3)872 1642 y Fp(\010)p 922 1598 342 2 v 11 w Fo(l)937 1649 y Fn(c)968 1642 y Fp(\002)g Fu(\()p Fo(l)1052 1649 y Fn(c)1084 1642 y Fu(+)f(1\))i Fp(\010)f Fu(1)1263 1597 y Fl(123)1325 1642 y Fo(:)0 1722 y Fu(Con)o(v)o(ersely)l(,)j(it)h(is)g (clear)f(that)i(all)e(these)h(con\014gurations)e(b)q(elong)i(to)g(the)h (principal)d(b)q(oundary)h(of)h(the)0 1782 y(cycle.)50 b Fc(\003)0 1863 y Fw(Theorem)18 b(7.18.)52 b Fh(Let)18 b Fo(j)526 1870 y Fl(1)548 1863 y Fo(;)8 b(j)591 1870 y Fl(2)613 1863 y Fo(;)g(j)656 1870 y Fl(3)678 1863 y Fo(;)g(l)715 1870 y Fl(1)738 1863 y Fo(;)g(l)775 1870 y Fl(2)814 1863 y Fh(b)q(e)18 b(in)o(tegers)e(suc)o(h)g (that)i(the)g(parallelepip)q(ed)d Fo(j)1718 1870 y Fl(1)1752 1863 y Fp(\002)c Fo(j)1823 1870 y Fl(2)1857 1863 y Fp(\002)g Fo(j)1928 1870 y Fl(3)0 1922 y Fh(con)o(tains)16 b(a)i(critical)f(quasicub)q (e)g(and)g(the)h(rectangle)f Fo(l)1051 1929 y Fl(1)1085 1922 y Fp(\002)12 b Fo(l)1151 1929 y Fl(2)1191 1922 y Fh(con)o(tains)k(a)i (critical)f(t)o(w)o(o)g(dimensional)0 1982 y(con\014guration.)j(The)d(cycle) 278 2062 y Fo(\031)f Fu(=)d Fo(\031)r Fu(\()p Fo(j)444 2069 y Fl(1)478 2062 y Fp(\002)d Fo(j)548 2069 y Fl(2)581 2062 y Fp(\002)h Fo(j)652 2069 y Fl(3)685 2062 y Fu(+)724 2069 y Fl(1)757 2062 y Fo(l)772 2069 y Fl(1)806 2062 y Fp(\002)f Fo(l)870 2069 y Fl(2)893 2062 y Fo(;)e Fh(pred)f Fo(E)s Fu(\()p Fo(j)1100 2069 y Fl(1)1133 2062 y Fp(\002)k Fo(j)1204 2069 y Fl(2)1237 2062 y Fp(\002)g Fo(j)1308 2069 y Fl(3)1340 2062 y Fu(+)1379 2069 y Fl(1)1413 2062 y Fo(l)1428 2069 y Fl(1)1461 2062 y Fp(\002)g Fo(l)1526 2069 y Fl(2)1559 2062 y Fp(\010)g Fu(1\)\))0 2142 y Fh(do)q(es)16 b(not)h(con)o(tain)f Fp(\000)p Fu(1)p 377 2154 64 2 v 16 w Fh(and)g Fu(+1)p 554 2154 V -1 w Fh(.)22 b(Moreo)o(v)o(er)391 2222 y Fo(v)p 391 2230 26 2 v 2 w Fu(\()p Fo(\031)r Fu(\))h Fo(>)f(j)590 2229 y Fl(1)612 2222 y Fo(j)633 2229 y Fl(2)655 2222 y Fo(j)676 2229 y Fl(3)709 2222 y Fu(+)10 b Fo(l)773 2229 y Fl(1)796 2222 y Fo(l)811 2229 y Fl(2)844 2222 y Fp(\000)h Fo(l)909 2229 y Fn(c)940 2222 y Fu(+)g(1)d Fo(;)p 1095 2195 V 58 w(v)r Fu(\()p Fo(\031)r Fu(\))23 b(=)f Fo(j)1294 2229 y Fl(1)1316 2222 y Fo(j)1337 2229 y Fl(2)1359 2222 y Fo(j)1380 2229 y Fl(3)1412 2222 y Fu(+)11 b Fo(l)1477 2229 y Fl(1)1500 2222 y Fo(l)1515 2229 y Fl(2)1545 2222 y Fo(:)0 2303 y Fh(The)16 b(b)q(ottom)h(of)g(this)f(cycle)h(is)f Fp(f)p Fo(j)651 2310 y Fl(1)684 2303 y Fp(\002)11 b Fo(j)755 2310 y Fl(2)788 2303 y Fp(\002)g Fo(j)859 2310 y Fl(3)892 2303 y Fu(+)931 2310 y Fl(1)964 2303 y Fo(l)979 2310 y Fl(1)1012 2303 y Fp(\002)g Fo(l)1077 2310 y Fl(2)1099 2303 y Fp(g)17 b Fh(and)e(its)i(principal)e(b)q (oundary)g(is)605 2376 y Fk(e)595 2389 y Fo(B)r Fu(\()p Fo(\031)r Fu(\))23 b(=)f Fo(j)808 2396 y Fl(1)841 2389 y Fp(\002)11 b Fo(j)912 2396 y Fl(2)945 2389 y Fp(\002)f Fo(j)1015 2396 y Fl(3)1048 2389 y Fu(+)1087 2396 y Fl(1)1120 2389 y Fo(l)1135 2396 y Fl(1)1169 2389 y Fp(\002)h Fo(l)1234 2396 y Fl(2)1267 2389 y Fp(\010)g Fu(1)p Fo(:)0 2490 y Fe(R)m(emark.)27 b Fu(Ob)o(viously)l(,) 13 b(similar)f(results)h(hold)g(for)h(an)o(y)g(parallelepip)q(ed)f(plus)g (rectangle)h Fo(j)1694 2497 y Fl(1)1722 2490 y Fp(\002)6 b Fo(j)1788 2497 y Fl(2)1817 2490 y Fp(\002)g Fo(j)1883 2497 y Fl(3)1911 2490 y Fp(\010)0 2550 y Fo(l)15 2557 y Fl(1)48 2550 y Fp(\002)11 b Fo(l)113 2557 y Fl(2)152 2550 y Fu(satisfying)16 b(the)h(requiremen)o(ts)d(of)j(the)g(theorem.)954 2670 y Fs(38)p eop %%Page: 39 39 39 38 bop 0 200 a Fw(Theorem)12 b(7.19.)48 b Fh(Let)12 b Fo(j)510 207 y Fl(1)534 200 y Fp(\002)r Fo(j)596 207 y Fl(2)619 200 y Fp(\002)r Fo(j)681 207 y Fl(3)714 200 y Fh(b)q(e)g(a)g(parallelepip)q(ed)e (con)o(taining)h(strictly)h(a)g(three)f(dimensional)0 260 y(critical)16 b(con\014guration.)k(The)d(cycle)168 399 y Fo(\031)f Fu(=)e Fo(\031)r Fu(\()p Fo(j)335 406 y Fl(1)368 399 y Fp(\002)d Fo(j)439 406 y Fl(2)472 399 y Fp(\002)g Fo(j)543 406 y Fl(3)576 399 y Fu(+)g Fo(l)641 406 y Fn(c)672 399 y Fp(\002)g Fu(\()p Fo(l)756 406 y Fn(c)787 399 y Fu(+)g(2\))p Fo(;)d Fh(pred)g Fo(E)s Fu(\()p Fo(j)1089 406 y Fl(1)1122 399 y Fp(\002)j Fo(j)1193 406 y Fl(2)1226 399 y Fp(\002)f Fo(j)1296 406 y Fl(3)1329 399 y Fu(+)h Fo(l)1394 406 y Fn(c)1425 399 y Fp(\002)g Fu(\()p Fo(l)1509 406 y Fn(c)1541 399 y Fu(+)g(1\))g(+)1685 406 y Fl(2)1718 399 y Fu(1\)\))0 539 y Fh(is)16 b(included)f(in)h Fp(f\000)p Fu(1)p 332 551 64 2 v Fo(;)8 b Fu(+1)p 418 551 V Fp(g)507 521 y Fn(c)527 539 y Fh(.)22 b(Moreo)o(v)o(er,)319 678 y Fo(v)p 319 686 26 2 v 2 w Fu(\()p Fo(\031)r Fu(\))h(=)f Fo(j)518 685 y Fl(1)539 678 y Fo(j)560 685 y Fl(2)582 678 y Fo(j)603 685 y Fl(3)636 678 y Fu(+)11 b Fo(l)701 685 y Fn(c)721 678 y Fu(\()p Fo(l)755 685 y Fn(c)787 678 y Fu(+)g(1\))g(+)g(2)d Fo(;)p 1047 651 V 58 w(v)r Fu(\()p Fo(\031)r Fu(\))23 b(=)f Fo(j)1246 685 y Fl(1)1268 678 y Fo(j)1289 685 y Fl(2)1311 678 y Fo(j)1332 685 y Fl(3)1364 678 y Fu(+)11 b Fo(l)1429 685 y Fn(c)1449 678 y Fu(\()p Fo(l)1483 685 y Fn(c)1515 678 y Fu(+)g(2\))d Fo(:)287 753 y Fp(8)p Fo(k)14 b Fp(2)g(f)p Fu(1)8 b Fp(\001)g(\001)g(\001)h Fo(l)543 760 y Fn(c)574 753 y Fp(\000)i Fu(2)p Fp(g)99 b(M)p Fu(\()p Fo(j)873 760 y Fl(1)907 753 y Fp(\002)10 b Fo(j)977 760 y Fl(2)1010 753 y Fp(\002)h Fo(j)1081 760 y Fl(3)1114 753 y Fu(+)g Fo(l)1179 760 y Fn(c)1210 753 y Fp(\002)g Fu(\()p Fo(l)1294 760 y Fn(c)1325 753 y Fu(+)g(2\))p Fo(;)d Fp(\000)p Fo(k)r Fu(\))22 b Fp(\032)g Fo(\031)11 b(;)0 893 y Fh(The)16 b(b)q(ottom)h(of)g(this)f(cycle)h(is)f Fp(f)p Fo(j)651 900 y Fl(1)684 893 y Fp(\002)11 b Fo(j)755 900 y Fl(2)788 893 y Fp(\002)g Fo(j)859 900 y Fl(3)892 893 y Fu(+)f Fo(l)956 900 y Fn(c)988 893 y Fp(\002)g Fu(\()p Fo(l)1071 900 y Fn(c)1103 893 y Fu(+)h(2\))p Fp(g)p Fh(;)16 b(its)h(principal)e(b)q (oundary)g(is)446 1026 y Fk(e)436 1039 y Fo(B)r Fu(\()p Fo(\031)r Fu(\))23 b(=)f Fp(M)p Fu(\()p Fo(j)728 1046 y Fl(1)761 1039 y Fp(\002)11 b Fo(j)832 1046 y Fl(2)865 1039 y Fp(\002)g Fo(j)936 1046 y Fl(3)969 1039 y Fu(+)f Fo(l)1033 1046 y Fn(c)1065 1039 y Fp(\002)h Fu(\()p Fo(l)1149 1046 y Fn(c)1180 1039 y Fu(+)g(2\))p Fo(;)d Fp(\000)p Fu(\()p Fo(l)1369 1046 y Fn(c)1401 1039 y Fp(\000)j Fu(1\)\))0 1178 y Fh(and)16 b(th)o(us)f(con)o(tains)h Fo(j)423 1185 y Fl(1)456 1178 y Fp(\002)11 b Fo(j)527 1185 y Fl(2)559 1178 y Fp(\002)g Fo(j)630 1185 y Fl(3)663 1178 y Fu(+)p 713 1135 364 2 v 11 w Fo(l)728 1185 y Fn(c)759 1178 y Fp(\002)g Fu(\()p Fo(l)843 1185 y Fn(c)875 1178 y Fu(+)g(1\))g Fp(\010)1019 1185 y Fl(2)1052 1178 y Fu(1)1077 1143 y Fl(12)1119 1178 y Fh(.)0 1288 y Fe(Pr)m(o)m(of.)28 b Fu(Using)16 b(theorem)g(7)p Fo(:)p Fu(8,)g(the)g(pro)q(of)h(is)f(similar)e(to)j(the)g(pro)q(of)f(of)h (theorem)f(6)p Fo(:)p Fu(25.)49 b Fc(\003)0 1398 y Fe(R)m(emark.)27 b Fu(Ob)o(viously)l(,)12 b(similar)g(statemen)o(ts)h(are)h(true)g(for)f(the)h (con\014gurations)e(in)h Fo(j)1593 1405 y Fl(1)1621 1398 y Fp(\002)6 b Fo(j)1687 1405 y Fl(2)1714 1398 y Fp(\002)g Fo(j)1780 1405 y Fl(3)1807 1398 y Fp(\010)g Fu(\()p Fo(l)1886 1405 y Fn(c)1911 1398 y Fu(+)0 1458 y(2\))11 b Fp(\002)g Fo(l)120 1465 y Fn(c)157 1458 y Fu(and)16 b Fo(j)275 1465 y Fl(1)308 1458 y Fp(\002)11 b Fo(j)379 1465 y Fl(2)411 1458 y Fp(\002)g Fo(j)482 1465 y Fl(3)515 1458 y Fp(\010)g Fo(l)580 1465 y Fn(c)611 1458 y Fp(\002)g Fu(\()p Fo(l)695 1465 y Fn(c)727 1458 y Fu(+)f(2\).)0 1567 y Fw(Lemma)20 b(7.20.)56 b Fh(Let)19 b Fo(j)493 1574 y Fl(1)515 1567 y Fo(;)8 b(j)558 1574 y Fl(2)580 1567 y Fo(;)g(j)623 1574 y Fl(3)645 1567 y Fo(;)g(l)682 1574 y Fl(1)704 1567 y Fo(;)g(l)741 1574 y Fl(2)782 1567 y Fh(b)q(e)19 b(p)q(ositiv)o(e)g(in)o (tegers.)27 b(Let)19 b Fo(k)r(;)8 b(i)19 b Fh(b)q(e)g(suc)o(h)e(that)i Fo(k)g(>)e Fu(1)p Fo(;)g(i)g Fp(\025)0 1627 y Fu(0)p Fo(;)f(l)70 1634 y Fl(1)107 1627 y Fo(>)e(k)f Fu(+)e Fo(i)p Fh(.)24 b(The)17 b(cycle)g Fp(f)p Fo(j)576 1634 y Fl(1)609 1627 y Fp(\002)12 b Fo(j)681 1634 y Fl(2)714 1627 y Fp(\002)f Fo(j)785 1634 y Fl(3)818 1627 y Fu(+)857 1634 y Fl(1)891 1627 y Fo(l)906 1634 y Fl(1)939 1627 y Fp(\002)g Fo(l)1004 1634 y Fl(2)1038 1627 y Fu(+)1077 1609 y Fn(i)1077 1639 y Fl(2)1110 1627 y Fo(k)r Fp(g)17 b Fh(has)f(one)h(or)f(t)o(w)o(o)h(con\014gurations)e(in)h(its)0 1693 y(principal)f(b)q(oundary)442 1680 y Fk(e)432 1693 y Fo(B)r Fh(:)60 1756 y Fp(\017)49 b Fh(if)17 b Fo(i)d Fu(=)g(0)i Fh(then)429 1744 y Fk(e)419 1756 y Fo(B)g Fu(=)d Fp(f)p Fo(j)571 1763 y Fl(1)604 1756 y Fp(\002)e Fo(j)675 1763 y Fl(2)708 1756 y Fp(\002)g Fo(j)779 1763 y Fl(3)812 1756 y Fu(+)851 1763 y Fl(1)884 1756 y Fo(l)899 1763 y Fl(1)932 1756 y Fp(\002)g Fo(l)997 1763 y Fl(2)1030 1756 y Fu(+)1069 1738 y Fn(i)1069 1769 y Fl(2)1102 1756 y Fu(\()p Fo(k)i Fu(+)e(1\))p Fp(g)p Fh(,)60 1822 y Fp(\017)49 b Fh(if)17 b Fo(i)d Fu(=)g Fo(l)279 1829 y Fl(1)312 1822 y Fp(\000)d Fu(1)g Fp(\000)g Fo(k)18 b Fh(then)616 1810 y Fk(e)605 1822 y Fo(B)e Fu(=)e Fp(f)p Fo(j)758 1829 y Fl(1)791 1822 y Fp(\002)d Fo(j)862 1829 y Fl(2)894 1822 y Fp(\002)g Fo(j)965 1829 y Fl(3)998 1822 y Fu(+)1037 1829 y Fl(1)1070 1822 y Fo(l)1085 1829 y Fl(1)1119 1822 y Fp(\002)f Fo(l)1183 1829 y Fl(2)1217 1822 y Fu(+)1256 1801 y Fn(i)p Fj(\000)p Fl(1)1256 1836 y(2)1334 1822 y Fu(\()p Fo(k)j Fu(+)e(1\))p Fp(g)p Fh(,)60 1889 y Fp(\017)49 b Fh(if)16 b Fu(0)d Fo(<)h(i)g(<)g(l)369 1896 y Fl(1)399 1889 y Fp(\000)8 b Fu(1)g Fp(\000)g Fo(k)17 b Fh(then)691 1877 y Fk(e)681 1889 y Fo(B)f Fu(=)d Fp(f)p Fo(j)833 1896 y Fl(1)863 1889 y Fp(\002)8 b Fo(j)931 1896 y Fl(2)961 1889 y Fp(\002)g Fo(j)1029 1896 y Fl(3)1059 1889 y Fu(+)1098 1896 y Fl(1)1129 1889 y Fo(l)1144 1896 y Fl(1)1174 1889 y Fp(\002)g Fo(l)1236 1896 y Fl(2)1267 1889 y Fu(+)1306 1868 y Fn(i)p Fj(\000)p Fl(1)1306 1902 y(2)1381 1889 y Fu(\()p Fo(k)i Fu(+)e(1\))p Fo(;)17 b(l)1573 1896 y Fl(1)1603 1889 y Fp(\002)8 b Fo(l)1665 1896 y Fl(2)1696 1889 y Fu(+)1735 1871 y Fn(i)1735 1901 y Fl(2)1765 1889 y Fu(\()p Fo(k)i Fu(+)e(1\))p Fp(g)p Fh(.)0 1999 y Fw(Lemma)22 b(7.21.)62 b Fh(Let)21 b Fo(j)503 2006 y Fl(1)525 1999 y Fo(;)8 b(j)568 2006 y Fl(2)590 1999 y Fo(;)g(j)633 2006 y Fl(3)655 1999 y Fo(;)g(l)692 2006 y Fl(1)715 1999 y Fo(;)g(l)752 2006 y Fl(2)795 1999 y Fh(b)q(e)21 b(p)q(ositiv)o(e)f(in)o(tegers)g(with)g Fo(l)1376 2006 y Fl(1)1413 1999 y Fp(\002)13 b Fo(l)1480 2006 y Fl(2)1523 1999 y Fp(\032)21 b Fo(j)1604 2006 y Fl(1)1639 1999 y Fp(\002)14 b Fo(j)1713 2006 y Fl(2)1756 1999 y Fh(and)20 b(let)h Fo(i)0 2059 y Fh(b)q(e)e(suc)o(h)e(that)i Fo(i)f Fp(\025)f Fu(0)p Fo(;)f(l)455 2066 y Fl(1)495 2059 y Fo(>)h(i)c Fu(+)f(1)p Fh(.)28 b(The)19 b(cycle)g Fp(f)p Fo(j)975 2066 y Fl(1)1009 2059 y Fp(\002)13 b Fo(j)1082 2066 y Fl(2)1116 2059 y Fp(\002)f Fo(j)1188 2066 y Fl(3)1223 2059 y Fu(+)1262 2066 y Fl(1)1296 2059 y Fo(l)1311 2066 y Fl(1)1346 2059 y Fp(\002)g Fo(l)1412 2066 y Fl(2)1447 2059 y Fu(+)1486 2041 y Fn(i)1486 2071 y Fl(2)1520 2059 y Fu(1)p Fp(g)19 b Fh(has)f(t)o(w)o(o)g(or)g(three)0 2125 y(con\014gurations)c(in)j(its)f(principal)f(b)q(oundary)891 2112 y Fk(e)881 2125 y Fo(B)r Fh(:)60 2188 y Fp(\017)49 b Fh(if)17 b Fo(i)d Fu(=)g(0)i Fh(then)429 2175 y Fk(e)419 2188 y Fo(B)g Fu(=)d Fp(f)p Fo(j)571 2195 y Fl(1)604 2188 y Fp(\002)e Fo(j)675 2195 y Fl(2)708 2188 y Fp(\002)g Fo(j)779 2195 y Fl(3)812 2188 y Fu(+)851 2195 y Fl(1)884 2188 y Fo(l)899 2195 y Fl(1)932 2188 y Fp(\002)g Fo(l)997 2195 y Fl(2)1019 2188 y Fo(;)17 b(j)1071 2195 y Fl(1)1104 2188 y Fp(\002)10 b Fo(j)1174 2195 y Fl(2)1207 2188 y Fp(\002)h Fo(j)1278 2195 y Fl(3)1311 2188 y Fu(+)1350 2195 y Fl(1)1383 2188 y Fo(l)1398 2195 y Fl(1)1431 2188 y Fp(\002)g Fo(l)1496 2195 y Fl(2)1530 2188 y Fu(+)1569 2170 y Fn(i)1569 2200 y Fl(2)1602 2188 y Fu(2)p Fp(g)p Fh(,)60 2254 y Fp(\017)49 b Fh(if)17 b Fo(i)d Fu(=)g Fo(l)279 2261 y Fl(1)312 2254 y Fp(\000)d Fu(1)g Fp(\000)g Fo(k)18 b Fh(then)616 2241 y Fk(e)605 2254 y Fo(B)e Fu(=)e Fp(f)p Fo(j)758 2261 y Fl(1)791 2254 y Fp(\002)d Fo(j)862 2261 y Fl(2)894 2254 y Fp(\002)g Fo(j)965 2261 y Fl(3)998 2254 y Fu(+)1037 2261 y Fl(1)1070 2254 y Fo(l)1085 2261 y Fl(1)1119 2254 y Fp(\002)f Fo(l)1183 2261 y Fl(2)1206 2254 y Fo(;)16 b(j)1257 2261 y Fl(1)1290 2254 y Fp(\002)11 b Fo(j)1361 2261 y Fl(2)1394 2254 y Fp(\002)g Fo(j)1465 2261 y Fl(3)1498 2254 y Fu(+)1537 2261 y Fl(1)1570 2254 y Fo(l)1585 2261 y Fl(1)1618 2254 y Fp(\002)g Fo(l)1683 2261 y Fl(2)1716 2254 y Fu(+)1755 2233 y Fn(i)p Fj(\000)p Fl(1)1755 2267 y(2)1833 2254 y Fu(2)p Fp(g)p Fh(,)60 2321 y Fp(\017)49 b Fh(if)17 b Fu(0)d Fo(<)f(i)h(<)g(l)370 2328 y Fl(1)402 2321 y Fp(\000)d Fu(1)f Fp(\000)g Fo(k)18 b Fh(then)704 2308 y Fk(e)693 2321 y Fo(B)e Fu(=)e Fp(f)p Fo(j)846 2328 y Fl(1)878 2321 y Fp(\002)d Fo(j)949 2328 y Fl(2)981 2321 y Fp(\002)f Fo(j)1051 2328 y Fl(3)1083 2321 y Fu(+)1122 2328 y Fl(1)1155 2321 y Fo(l)1170 2328 y Fl(1)1203 2321 y Fp(\002)g Fo(l)1267 2328 y Fl(2)1289 2321 y Fo(;)17 b(j)1341 2328 y Fl(1)1373 2321 y Fp(\002)10 b Fo(j)1443 2328 y Fl(2)1476 2321 y Fp(\002)g Fo(j)1546 2328 y Fl(3)1578 2321 y Fu(+)1617 2328 y Fl(1)1650 2321 y Fo(l)1665 2328 y Fl(1)1697 2321 y Fp(\002)h Fo(l)1762 2328 y Fl(2)1794 2321 y Fu(+)1833 2299 y Fn(i)p Fj(\000)p Fl(1)1833 2334 y(2)1911 2321 y Fu(2)p Fo(;)120 2380 y(j)141 2387 y Fl(1)173 2380 y Fp(\002)g Fo(j)244 2387 y Fl(2)277 2380 y Fp(\002)g Fo(j)348 2387 y Fl(3)381 2380 y Fu(+)420 2387 y Fl(1)453 2380 y Fo(l)468 2387 y Fl(1)501 2380 y Fp(\002)g Fo(l)566 2387 y Fl(2)599 2380 y Fu(+)638 2362 y Fn(i)638 2393 y Fl(2)672 2380 y Fu(2)p Fp(g)p Fh(.)0 2490 y Fe(R)m(emark.)27 b Fu(Results)10 b(similar)g(to)j(those)e (stated)h(in)g(lemmas)e(7)p Fo(:)p Fu(20)i(and)f(7)p Fo(:)p Fu(21)g(hold)g(for)h(the)g(con\014gurations)0 2550 y(in)p 58 2509 533 2 v 16 w Fo(j)79 2557 y Fl(1)112 2550 y Fp(\002)f Fo(j)183 2557 y Fl(2)216 2550 y Fp(\002)g Fo(j)287 2557 y Fl(3)320 2550 y Fp(\010)f Fo(l)384 2557 y Fl(1)418 2550 y Fp(\002)g Fo(l)482 2557 y Fl(2)516 2550 y Fp(\010)h Fu(1)o(.)954 2670 y Fs(39)p eop %%Page: 40 40 40 39 bop 0 200 a Fw(Corollary)16 b(7.22.)49 b Fh(The)13 b(follo)o(wing)f (cycles)h(are)g(maximal)f(cycles)i(of)f Fp(f\000)p Fu(1)p 1362 212 64 2 v Fo(;)8 b Fu(+1)p 1448 212 V Fp(g)1537 182 y Fn(c)1557 200 y Fh(,)14 b(where)f Fo(j)1747 207 y Fl(1)1773 200 y Fp(\002)5 b Fo(j)1838 207 y Fl(2)1864 200 y Fp(\002)g Fo(j)1929 207 y Fl(3)0 260 y Fh(con)o(tains)15 b(a)i(con\014guration)e(of)h Fp(S)625 267 y Fn(n)650 272 y Fg(c)671 260 y Fh(,)h Fo(l)717 267 y Fl(1)750 260 y Fp(\002)11 b Fo(l)815 267 y Fl(2)854 260 y Fh(con)o(tains)k(a)i(critical)f(t)o(w)o(o)g(dimensional)e(con\014guration,) 18 380 y Fu(1\))51 b Fo(\031)r Fu(\()p Fo(j)183 387 y Fl(1)216 380 y Fp(\002)11 b Fo(j)287 387 y Fl(2)320 380 y Fp(\002)g Fo(j)391 387 y Fl(3)412 380 y Fo(;)d Fh(pred)g Fo(E)s Fu(\()p Fo(j)620 387 y Fl(1)653 380 y Fp(\002)j Fo(j)724 387 y Fl(2)757 380 y Fp(\002)f Fo(j)827 387 y Fl(3)860 380 y Fu(+)899 387 y Fl(1)933 380 y Fo(l)948 387 y Fn(c)979 380 y Fp(\002)h Fu(\()p Fo(l)1063 387 y Fn(c)1094 380 y Fu(+)g(1\))g(+)1238 387 y Fl(2)1271 380 y Fu(1\)\))d Fo(;)18 457 y Fu(2\))51 b Fo(\031)r Fu(\()p Fo(j)183 464 y Fl(1)216 457 y Fp(\002)11 b Fo(j)287 464 y Fl(2)320 457 y Fp(\002)g Fo(j)391 464 y Fl(3)424 457 y Fu(+)463 464 y Fl(1)496 457 y Fo(l)511 464 y Fn(c)542 457 y Fp(\002)g Fu(\()p Fo(l)626 464 y Fn(c)657 457 y Fu(+)g(2\))p Fo(;)d Fh(pred)g Fo(E)s Fu(\()p Fo(j)959 464 y Fl(1)992 457 y Fp(\002)j Fo(j)1063 464 y Fl(2)1096 457 y Fp(\002)f Fo(j)1166 464 y Fl(3)1199 457 y Fu(+)1238 464 y Fl(1)1271 457 y Fo(l)1286 464 y Fn(c)1318 457 y Fp(\002)g Fu(\()p Fo(l)1401 464 y Fn(c)1433 457 y Fu(+)h(1\))g(+)1577 464 y Fl(2)1610 457 y Fu(1\)\))d Fo(;)18 535 y Fu(3\))51 b Fo(\031)r Fu(\()p Fo(j)183 542 y Fl(1)216 535 y Fp(\002)11 b Fo(j)287 542 y Fl(2)320 535 y Fp(\002)g Fo(j)391 542 y Fl(3)424 535 y Fu(+)463 542 y Fl(1)496 535 y Fo(l)511 542 y Fl(1)544 535 y Fp(\002)g Fo(l)609 542 y Fl(2)631 535 y Fo(;)d Fh(pred)f Fo(E)s Fu(\()p Fo(j)838 542 y Fl(1)871 535 y Fp(\002)k Fo(j)942 542 y Fl(2)975 535 y Fp(\002)g Fo(j)1046 542 y Fl(3)1079 535 y Fu(+)1118 542 y Fl(1)1151 535 y Fo(l)1166 542 y Fl(1)1199 535 y Fp(\002)g Fo(l)1264 542 y Fl(2)1297 535 y Fp(\010)g Fu(1\)\))d Fo(;)113 612 y Fh(and)15 b(this)i(cycle)g(is)f(not)g(included)g(in)g(a)g (cycle)h(of)g(t)o(yp)q(e)g(1\),)18 689 y Fu(4\))51 b Fo(\031)r Fu(\()p Fo(j)183 696 y Fn(c)214 689 y Fp(\002)11 b Fu(\()p Fo(j)304 696 y Fn(c)335 689 y Fu(+)g Fo(\016)407 696 y Fn(c)427 689 y Fu(\))h Fp(\002)f Fu(\()p Fo(j)548 696 y Fn(c)579 689 y Fu(+)g(1\))g(+)723 696 y Fl(1)756 689 y Fo(l)771 696 y Fl(1)804 689 y Fp(\002)g Fo(l)869 696 y Fl(2)891 689 y Fo(;)d Fh(pred)g Fo(E)s Fu(\()p Fo(j)1099 696 y Fn(c)1130 689 y Fp(\002)j Fu(\()p Fo(j)1220 696 y Fn(c)1251 689 y Fu(+)g Fo(\016)1323 696 y Fn(c)1343 689 y Fu(\))h Fp(\002)e Fu(\()p Fo(j)1463 696 y Fn(c)1495 689 y Fu(+)g(1\))i(+)f Fo(l)1665 696 y Fl(1)1698 689 y Fp(\002)g Fo(l)1763 696 y Fl(2)1796 689 y Fp(\010)g Fu(1\)\))d Fo(;)113 766 y Fh(and)15 b(this)i(cycle)g(is)f(not)g(included)g(in)g(a)g(cycle)h(of)g (t)o(yp)q(e)g Fu(1\))p Fh(,)18 844 y Fu(5\))51 b Fp(f)p Fo(\021)r Fp(g)8 b Fo(;)57 b(\021)16 b Fp(2)e(M)p Fu(\()p Fo(j)457 851 y Fl(1)490 844 y Fp(\002)c Fo(j)560 851 y Fl(2)593 844 y Fp(\002)h Fo(j)664 851 y Fl(3)697 844 y Fu(+)736 851 y Fl(1)769 844 y Fo(l)784 851 y Fn(c)816 844 y Fp(\002)f Fu(\()p Fo(l)899 851 y Fn(c)931 844 y Fu(+)h(2\))p Fo(;)d Fp(\000)p Fu(\()p Fo(l)1120 851 y Fn(c)1152 844 y Fp(\000)i Fu(1\)\))e Fo(;)18 940 y Fu(6\))51 b Fp(f)p Fo(\021)r Fp(g)8 b Fo(;)57 b(\021)16 b Fp(2)p 357 887 535 2 v 14 w Fo(j)381 923 y Fj(0)378 953 y Fl(1)410 940 y Fp(\002)11 b Fo(j)484 923 y Fj(0)481 953 y Fl(2)514 940 y Fp(\002)g Fo(j)588 923 y Fj(0)585 953 y Fl(3)618 940 y Fp(\010)p 668 897 224 2 v 11 w Fo(l)684 923 y Fj(0)683 953 y Fl(1)716 940 y Fp(\002)g Fo(l)782 923 y Fj(0)781 953 y Fl(2)814 940 y Fp(\010)g Fo(k)18 b Fh(where)e Fo(j)1076 919 y Fj(0)1073 952 y Fl(1)1106 940 y Fp(\002)11 b Fo(j)1180 919 y Fj(0)1177 952 y Fl(2)1210 940 y Fp(\002)g Fo(j)1284 919 y Fj(0)1281 952 y Fl(3)1319 940 y Fh(con)o(tains)k(a)i(quasicub)q(e)113 1020 y Fo(j)134 1027 y Fn(c)164 1020 y Fp(\002)11 b Fu(\()p Fo(j)254 1027 y Fn(c)286 1020 y Fu(+)f Fo(\016)357 1027 y Fn(c)378 1020 y Fu(\))h Fp(\002)g Fu(\()p Fo(j)498 1027 y Fn(c)529 1020 y Fu(+)g(1\))p Fh(,)17 b Fo(l)670 999 y Fj(0)669 1032 y Fl(1)702 1020 y Fp(\002)11 b Fo(l)768 999 y Fj(0)767 1032 y Fl(2)806 1020 y Fh(con)o(tains)k(a)i(quasisquare)e Fo(l)1327 1027 y Fn(c)1358 1020 y Fp(\002)c Fu(\()p Fo(l)1442 1027 y Fn(c)1473 1020 y Fu(+)g(1\))p Fh(,)17 b Fo(k)e(>)f Fu(0)p Fh(,)113 1097 y(and)h(this)i(cycle)g(is)f(not)g(included)g(in)g(a)g(cycle)h(of)g(t)o(yp)q (e)g Fu(1\))p Fo(;)8 b Fu(2\))p Fo(;)g Fu(3\))p Fo(;)g Fu(4\))p Fo(;)g Fu(5\))p Fh(.)0 1220 y(The)16 b(list)g(should)e(b)q(e)j(completed)e (with)h(all)g(the)g(isometric)f(cycles)h(\(obtained)g(b)o(y)g(applying)f(an)g (isom-)0 1280 y(etry\))h(as)f(w)o(ell)f(as)h(all)g(the)g(cycles)g(whose)g(b)q (ottom)g(is)g(obtained)f(b)o(y)h(translating)e(the)j(rectangle)f Fo(l)1836 1287 y Fl(1)1866 1280 y Fp(\002)8 b Fo(l)1928 1287 y Fl(2)0 1340 y Fh(along)16 b(the)h(side)f(of)g(the)h(parallelepip)q(ed)e (\(i.e.)22 b(the)17 b(con\014gurations)d(in)i Fo(j)1372 1347 y Fl(1)1405 1340 y Fp(\002)11 b Fo(j)1476 1347 y Fl(2)1509 1340 y Fp(\002)g Fo(j)1580 1347 y Fl(3)1613 1340 y Fp(\010)g Fo(l)1678 1347 y Fl(1)1711 1340 y Fp(\002)g Fo(l)1776 1347 y Fl(2)1798 1340 y Fh(\).)0 1430 y Fe(Pr)m(o)m(of.)28 b Fu(This)18 b(corollary)g(is)h(a)g(consequence)g(of)g(lemma)g(6)p Fo(:)p Fu(19)g(together)g(with)g(theorems)f(7)p Fo(:)p Fu(17,)i(7)p Fo(:)p Fu(18,)0 1490 y(7)p Fo(:)p Fu(19)13 b(and)g(lemmas)f(7)p Fo(:)p Fu(20)h(and)g(7)p Fo(:)p Fu(21.)21 b(Notice)14 b(that)g(w)o(e)f(ha)o (v)o(e)g(to)h(put)f(together)h(the)f(descriptions)f(of)i(the)0 1550 y(cycles)g(of)f(theorems)f(7)p Fo(:)p Fu(17,)i(7)p Fo(:)p Fu(18,)f(7)p Fo(:)p Fu(19,)h(lemmas)e(7)p Fo(:)p Fu(20)p Fo(;)c Fu(7)p Fo(:)p Fu(21)k(in)h(order)f(to)i(c)o(hec)o(k)f(that)h(for)f(eac)o(h)g (cycle)h Fo(\031)0 1613 y Fu(in)e(the)h(ab)q(o)o(v)o(e)f(list,)h(there)f(is)g (a)g(sequence)h(of)f(cycles)h Fo(\031)995 1620 y Fl(0)1018 1613 y Fo(;)8 b Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(\031)1165 1620 y Fn(r)1200 1613 y Fu(suc)o(h)j(that)i Fo(\031)1440 1620 y Fl(0)1476 1613 y Fu(=)h Fo(\031)r Fu(,)1597 1601 y Fk(e)1586 1613 y Fo(B)s Fu(\()p Fo(\031)1674 1620 y Fn(i)1691 1613 y Fu(\))s Fp(\\)s Fo(\031)1777 1620 y Fn(i)p Fl(+1)1859 1613 y Fp(6)p Fu(=)f Fp(;)p Fu(,)0 1679 y(0)h Fp(\024)f Fo(i)h(<)g(r)k Fu(and)e(+1)p 312 1691 64 2 v 14 w Fp(2)447 1667 y Fk(e)437 1679 y Fo(B)r Fu(\()p Fo(\031)524 1686 y Fn(r)547 1679 y Fu(\).)50 b Fc(\003)0 1770 y Fw(Corollary)25 b(7.23.)63 b Fh(Let)22 b Fo(\033)h Fh(b)q(elong)e(to)h Fp(S)847 1777 y Fn(n)872 1782 y Fg(c)890 1777 y Fl(+1)943 1770 y Fh(.)36 b(The)20 b(minimal)g(stable)g (subgraph)f Fp(G)1700 1752 y Fl(+)1733 1770 y Fu(\()p Fo(\033)r Fu(\))j Fh(of)f Fp(G)1917 1752 y Fl(+)0 1830 y Fh(con)o(taining)f Fo(\033)k Fh(is)d(the)h(restriction)f(of)h Fp(G)i Fh(to)e(the)g(v)o(ertices)f (listed)g(in)g(corollary)g Fu(7)p Fo(:)p Fu(22)p Fh(.)36 b(The)22 b(arro)o(ws)0 1890 y(of)h Fp(G)96 1871 y Fl(+)129 1890 y Fu(\()p Fo(\033)r Fu(\))h Fh(are)f(\(where)g Fo(\021)i Fh(is)d(the)i(parallelepip)q (ed)d Fo(j)1027 1897 y Fl(1)1064 1890 y Fp(\002)15 b Fo(j)1139 1897 y Fl(2)1177 1890 y Fp(\002)g Fo(j)1252 1897 y Fl(3)1297 1890 y Fh(and)22 b(the)i(cycles)f(whic)o(h)f(are)h(the)0 1949 y(starting)16 b(p)q(oin)o(t)g(of)h(the)f(arro)o(ws)f(b)q(elong)h(to)h(the)g (list)f(of)h(cycles)g(of)f(corollary)g Fu(7)p Fo(:)p Fu(22)p Fh(\):)60 2009 y(1\))h(around)e(the)h(parallelepip)q(eds:)0 2085 y Fo(\031)r Fu(\()p Fo(\021)r(;)8 b Fh(pred)g Fo(E)s Fu(\()p Fo(\021)k Fp(\010)f Fo(l)365 2092 y Fn(c)396 2085 y Fp(\002)g Fu(\()p Fo(l)480 2092 y Fn(c)512 2085 y Fu(+)f(1\))i Fp(\010)f Fu(1\)\))j Fp($)g(f)p Fo(\030)r Fp(g)p Fo(;)i(\030)g Fp(2)p 997 2032 429 2 v 14 w Fo(\021)d Fp(\010)p 1084 2042 342 2 v 10 w Fo(l)1099 2092 y Fn(c)1131 2085 y Fp(\002)d Fu(\()p Fo(l)1214 2092 y Fn(c)1246 2085 y Fu(+)h(1\))g Fp(\010)g Fu(1)1426 2040 y Fl(123)1488 2085 y Fh(,)60 2176 y(2\))17 b(gro)o(wing)e(in)h(the)h (direction)e(of)i Fo(e)739 2183 y Fl(1)761 2176 y Fh(:)0 2251 y Fo(\031)r Fu(\()p Fo(\021)c Fu(+)126 2258 y Fl(1)159 2251 y Fo(l)174 2258 y Fn(c)206 2251 y Fp(\002)d Fu(\()p Fo(l)289 2258 y Fn(c)321 2251 y Fu(+)h(2\))p Fo(;)d Fh(pred)f Fo(E)s Fu(\()p Fo(\021)13 b Fu(+)678 2258 y Fl(1)711 2251 y Fo(l)726 2258 y Fn(c)757 2251 y Fp(\002)e Fu(\()p Fo(l)841 2258 y Fn(c)873 2251 y Fu(+)g(1\))g Fp(\010)g Fu(1\)\))j Fp($)g(f)p Fo(\030)r Fp(g)p Fo(;)120 2311 y(\030)h Fp(2)g(M)p Fu(\()p Fo(\021)d Fu(+)360 2318 y Fl(1)393 2311 y Fo(l)408 2318 y Fn(c)440 2311 y Fp(\002)e Fu(\()p Fo(l)523 2318 y Fn(c)555 2311 y Fu(+)h(2\))p Fo(;)d Fp(\000)p Fu(\()p Fo(l)744 2318 y Fn(c)776 2311 y Fp(\000)i Fu(1\)\))e Fo(;)0 2371 y(\031)r Fu(\()p Fo(\021)13 b Fu(+)126 2378 y Fl(1)159 2371 y Fo(l)174 2378 y Fl(1)208 2371 y Fp(\002)d Fo(l)272 2378 y Fl(2)295 2371 y Fo(;)e Fh(pred)f Fo(E)s Fu(\()p Fo(\021)13 b Fu(+)558 2378 y Fl(1)591 2371 y Fo(l)606 2378 y Fl(1)639 2371 y Fp(\002)e Fo(l)704 2378 y Fl(2)737 2371 y Fp(\010)g Fu(1\)\))j Fp($)g(f)p Fo(\030)r Fp(g)p Fo(;)i(\030)g Fp(2)e Fo(\021)f Fu(+)1194 2378 y Fl(1)1227 2371 y Fo(l)1242 2378 y Fl(1)1275 2371 y Fp(\002)e Fo(l)1340 2378 y Fl(2)1374 2371 y Fp(\010)f Fu(1)0 2430 y Fp(f)p Fo(\021)i Fu(+)101 2437 y Fl(1)135 2430 y Fo(l)150 2437 y Fl(1)183 2430 y Fp(\002)f Fo(l)248 2437 y Fl(2)281 2430 y Fu(+)320 2437 y Fl(1)353 2430 y Fo(k)r Fp(g)i(!)h(f)p Fo(\021)f Fu(+)585 2437 y Fl(1)618 2430 y Fo(l)633 2437 y Fl(1)666 2430 y Fp(\002)e Fo(l)731 2437 y Fl(2)764 2430 y Fu(+)803 2437 y Fl(1)836 2430 y Fu(\()p Fo(k)i Fu(+)e(1\))p Fp(g)p Fo(;)17 b Fu(0)c Fo(<)h(k)h(<)f(l)1244 2437 y Fl(2)1277 2430 y Fp(\000)d Fo(l)1342 2437 y Fn(c)1370 2430 y Fo(;)0 2490 y Fp(f)p Fo(\021)h Fu(+)101 2497 y Fl(1)135 2490 y Fo(l)150 2497 y Fl(1)183 2490 y Fp(\002)f Fo(l)248 2497 y Fl(2)281 2490 y Fu(+)320 2497 y Fl(2)353 2490 y Fo(k)r Fp(g)i(!)h(f)p Fo(\021)f Fu(+)585 2497 y Fl(1)618 2490 y Fo(l)633 2497 y Fl(1)666 2490 y Fp(\002)e Fo(l)731 2497 y Fl(2)764 2490 y Fu(+)803 2497 y Fl(2)836 2490 y Fu(\()p Fo(k)i Fu(+)e(1\))p Fp(g)p Fo(;)17 b Fu(0)c Fo(<)h(k)h(<)f(l)1244 2497 y Fl(1)1277 2490 y Fp(\000)d Fo(l)1342 2497 y Fn(c)1370 2490 y Fo(;)0 2550 y Fp(f)p Fo(\021)h Fu(+)101 2557 y Fl(1)135 2550 y Fo(l)150 2557 y Fl(1)183 2550 y Fp(\002)f Fo(l)248 2557 y Fl(2)281 2550 y Fu(+)320 2557 y Fl(2)353 2550 y Fo(l)368 2557 y Fl(1)401 2550 y Fp(\000)g Fo(l)466 2557 y Fn(c)486 2550 y Fp(g)j(!)g Fo(\031)r Fu(\()p Fo(\021)f Fu(+)715 2557 y Fl(1)748 2550 y Fo(l)763 2557 y Fl(1)796 2550 y Fp(\002)e Fu(\()p Fo(l)880 2557 y Fl(2)914 2550 y Fu(+)g(1\))p Fo(;)d Fh(pred)f Fo(E)s Fu(\()p Fo(\021)13 b Fu(+)1271 2557 y Fl(1)1304 2550 y Fo(l)1319 2557 y Fl(1)1352 2550 y Fp(\002)e Fu(\()p Fo(l)1436 2557 y Fl(2)1470 2550 y Fu(+)g(1\))g(+)g(1\)\))d Fo(;)954 2670 y Fs(40)p eop %%Page: 41 41 41 40 bop 0 200 a Fp(f)p Fo(\021)12 b Fu(+)101 207 y Fl(1)135 200 y Fo(l)150 207 y Fl(1)183 200 y Fp(\002)f Fo(l)248 207 y Fl(2)281 200 y Fu(+)320 207 y Fl(1)353 200 y Fo(l)368 207 y Fl(2)401 200 y Fp(\000)g Fo(l)466 207 y Fn(c)486 200 y Fp(g)j(!)g Fo(\031)r Fu(\()p Fo(\021)f Fu(+)715 207 y Fl(1)748 200 y Fu(\()p Fo(l)782 207 y Fl(1)816 200 y Fu(+)e(1\))g Fp(\002)g Fo(l)986 207 y Fl(2)1008 200 y Fo(;)d Fh(pred)f Fo(E)s Fu(\()p Fo(\021)13 b Fu(+)1271 207 y Fl(1)1304 200 y Fu(\()p Fo(l)1338 207 y Fl(1)1372 200 y Fu(+)e(1\))g Fp(\002)g Fo(l)1542 207 y Fl(2)1575 200 y Fu(+)g(1\)\))d Fo(;)0 260 y Fp(f)p Fo(\021)k Fu(+)101 267 y Fl(1)135 260 y Fo(l)150 267 y Fl(1)183 260 y Fp(\002)f Fo(l)248 267 y Fl(2)281 260 y Fu(+)320 267 y Fl(2)353 260 y Fo(l)368 267 y Fl(1)401 260 y Fp(\000)g Fo(l)466 267 y Fn(c)486 260 y Fp(g)j(!)g Fo(\031)r Fu(\()p Fo(\021)f Fu(+)715 267 y Fl(1)748 260 y Fo(j)769 267 y Fl(2)802 260 y Fp(\002)e Fo(j)873 267 y Fl(3)895 260 y Fo(;)d Fh(pred)f Fo(E)s Fu(\()p Fo(\021)13 b Fu(+)1158 267 y Fl(1)1191 260 y Fo(j)1212 267 y Fl(2)1245 260 y Fp(\002)e Fo(j)1316 267 y Fl(3)1349 260 y Fu(+)1388 267 y Fl(1)1421 260 y Fo(l)1436 267 y Fn(c)1467 260 y Fp(\002)g Fu(\()p Fo(l)1551 267 y Fn(c)1582 260 y Fu(+)g(1\))h(+)1727 267 y Fl(1)1760 260 y Fu(1\)\))c Fo(;)0 319 y Fp(f)p Fo(\021)k Fu(+)101 326 y Fl(1)135 319 y Fo(l)150 326 y Fl(1)183 319 y Fp(\002)f Fo(l)248 326 y Fl(2)281 319 y Fu(+)320 326 y Fl(1)353 319 y Fo(l)368 326 y Fl(2)401 319 y Fp(\000)g Fo(l)466 326 y Fn(c)486 319 y Fp(g)j(!)g Fo(\031)r Fu(\()p Fo(\021)f Fu(+)715 326 y Fl(1)748 319 y Fo(j)769 326 y Fl(2)802 319 y Fp(\002)e Fo(j)873 326 y Fl(3)895 319 y Fo(;)d Fh(pred)f Fo(E)s Fu(\()p Fo(\021)13 b Fu(+)1158 326 y Fl(1)1191 319 y Fo(j)1212 326 y Fl(2)1245 319 y Fp(\002)e Fo(j)1316 326 y Fl(3)1349 319 y Fu(+)1388 326 y Fl(1)1421 319 y Fo(l)1436 326 y Fn(c)1467 319 y Fp(\002)g Fu(\()p Fo(l)1551 326 y Fn(c)1582 319 y Fu(+)g(1\))h(+)1727 326 y Fl(1)1760 319 y Fu(1\)\))c Fo(;)60 402 y Fh(3\))17 b(gro)o(wing)e(in)h (the)h(direction)e(of)i Fo(e)739 409 y Fl(2)761 402 y Fh(:)0 473 y Fo(\031)r Fu(\()p Fo(\021)c Fu(+)126 480 y Fl(2)159 473 y Fo(l)174 480 y Fn(c)206 473 y Fp(\002)d Fu(\()p Fo(l)289 480 y Fn(c)321 473 y Fu(+)h(2\))p Fo(;)d Fh(pred)f Fo(E)s Fu(\()p Fo(\021)13 b Fu(+)678 480 y Fl(2)711 473 y Fo(l)726 480 y Fn(c)757 473 y Fp(\002)e Fu(\()p Fo(l)841 480 y Fn(c)873 473 y Fu(+)g(1\))g Fp(\010)g Fu(1\)\))j Fp($)g(f)p Fo(\030)r Fp(g)p Fo(;)120 533 y(\030)h Fp(2)g(M)p Fu(\()p Fo(\021)d Fu(+)360 540 y Fl(2)393 533 y Fo(l)408 540 y Fn(c)440 533 y Fp(\002)e Fu(\()p Fo(l)523 540 y Fn(c)555 533 y Fu(+)h(2\))p Fo(;)d Fp(\000)p Fu(\()p Fo(l)744 540 y Fn(c)776 533 y Fp(\000)i Fu(1\)\))e Fo(;)0 593 y(\031)r Fu(\()p Fo(\021)13 b Fu(+)126 600 y Fl(2)159 593 y Fo(l)174 600 y Fl(1)208 593 y Fp(\002)d Fo(l)272 600 y Fl(2)295 593 y Fo(;)e Fh(pred)f Fo(E)s Fu(\()p Fo(\021)13 b Fu(+)558 600 y Fl(2)591 593 y Fo(l)606 600 y Fl(1)639 593 y Fp(\002)e Fo(l)704 600 y Fl(2)737 593 y Fp(\010)g Fu(1\)\))j Fp($)g(f)p Fo(\030)r Fp(g)p Fo(;)i(\030)g Fp(2)e Fo(\021)f Fu(+)1194 600 y Fl(2)1227 593 y Fo(l)1242 600 y Fl(1)1275 593 y Fp(\002)e Fo(l)1340 600 y Fl(2)1374 593 y Fp(\010)f Fu(1)0 653 y Fp(f)p Fo(\021)i Fu(+)101 660 y Fl(2)135 653 y Fo(l)150 660 y Fl(1)183 653 y Fp(\002)f Fo(l)248 660 y Fl(2)281 653 y Fu(+)320 660 y Fl(1)353 653 y Fo(k)r Fp(g)i(!)h(f)p Fo(\021)f Fu(+)585 660 y Fl(2)618 653 y Fo(l)633 660 y Fl(1)666 653 y Fp(\002)e Fo(l)731 660 y Fl(2)764 653 y Fu(+)803 660 y Fl(1)836 653 y Fu(\()p Fo(k)i Fu(+)e(1\))p Fp(g)p Fo(;)17 b Fu(0)c Fo(<)h(k)h(<)f(l)1244 660 y Fl(2)1277 653 y Fp(\000)d Fo(l)1342 660 y Fn(c)1370 653 y Fo(;)0 712 y Fp(f)p Fo(\021)h Fu(+)101 719 y Fl(2)135 712 y Fo(l)150 719 y Fl(1)183 712 y Fp(\002)f Fo(l)248 719 y Fl(2)281 712 y Fu(+)320 719 y Fl(2)353 712 y Fo(k)r Fp(g)i(!)h(f)p Fo(\021)f Fu(+)585 719 y Fl(2)618 712 y Fo(l)633 719 y Fl(1)666 712 y Fp(\002)e Fo(l)731 719 y Fl(2)764 712 y Fu(+)803 719 y Fl(2)836 712 y Fu(\()p Fo(k)i Fu(+)e(1\))p Fp(g)p Fo(;)17 b Fu(0)c Fo(<)h(k)h(<)f(l) 1244 719 y Fl(1)1277 712 y Fp(\000)d Fo(l)1342 719 y Fn(c)1370 712 y Fo(;)0 772 y Fp(f)p Fo(\021)h Fu(+)101 779 y Fl(2)135 772 y Fo(l)150 779 y Fl(1)183 772 y Fp(\002)f Fo(l)248 779 y Fl(2)281 772 y Fu(+)320 779 y Fl(2)353 772 y Fo(l)368 779 y Fl(1)401 772 y Fp(\000)g Fo(l)466 779 y Fn(c)486 772 y Fp(g)j(!)g Fo(\031)r Fu(\()p Fo(\021)f Fu(+)715 779 y Fl(2)748 772 y Fo(l)763 779 y Fl(1)796 772 y Fp(\002)e Fu(\()p Fo(l)880 779 y Fl(2)914 772 y Fu(+)g(1\))p Fo(;)d Fh(pred)f Fo(E)s Fu(\()p Fo(\021)13 b Fu(+)1271 779 y Fl(2)1304 772 y Fo(l)1319 779 y Fl(1)1352 772 y Fp(\002)e Fu(\()p Fo(l)1436 779 y Fl(2)1470 772 y Fu(+)g(1\))g(+)g (1\)\))d Fo(;)0 832 y Fp(f)p Fo(\021)k Fu(+)101 839 y Fl(2)135 832 y Fo(l)150 839 y Fl(1)183 832 y Fp(\002)f Fo(l)248 839 y Fl(2)281 832 y Fu(+)320 839 y Fl(1)353 832 y Fo(l)368 839 y Fl(2)401 832 y Fp(\000)g Fo(l)466 839 y Fn(c)486 832 y Fp(g)j(!)g Fo(\031)r Fu(\()p Fo(\021)f Fu(+)715 839 y Fl(2)748 832 y Fu(\()p Fo(l)782 839 y Fl(1)816 832 y Fu(+)e(1\))g Fp(\002)g Fo(l)986 839 y Fl(2)1008 832 y Fo(;)d Fh(pred)f Fo(E)s Fu(\()p Fo(\021)13 b Fu(+)1271 839 y Fl(2)1304 832 y Fu(\()p Fo(l)1338 839 y Fl(1)1372 832 y Fu(+)e(1\))g Fp(\002)g Fo(l)1542 839 y Fl(2)1575 832 y Fu(+)g(1\)\))d Fo(;)0 892 y Fp(f)p Fo(\021)k Fu(+)101 899 y Fl(2)135 892 y Fo(l)150 899 y Fl(1)183 892 y Fp(\002)f Fo(l)248 899 y Fl(2)281 892 y Fu(+)320 899 y Fl(2)353 892 y Fo(l)368 899 y Fl(1)401 892 y Fp(\000)g Fo(l)466 899 y Fn(c)486 892 y Fp(g)j(!)g Fo(\031)r Fu(\()p Fo(\021)f Fu(+)715 899 y Fl(2)748 892 y Fo(j)769 899 y Fl(1)802 892 y Fp(\002)e Fo(j)873 899 y Fl(3)895 892 y Fo(;)d Fh(pred)f Fo(E)s Fu(\()p Fo(\021)13 b Fu(+)1158 899 y Fl(2)1191 892 y Fo(j)1212 899 y Fl(1)1245 892 y Fp(\002)e Fo(j)1316 899 y Fl(3)1349 892 y Fu(+)1388 899 y Fl(2)1421 892 y Fo(l)1436 899 y Fn(c)1467 892 y Fp(\002)g Fu(\()p Fo(l)1551 899 y Fn(c)1582 892 y Fu(+)g(1\))h(+)1727 899 y Fl(1)1760 892 y Fu(1\)\))c Fo(;)0 952 y Fp(f)p Fo(\021)k Fu(+)101 959 y Fl(2)135 952 y Fo(l)150 959 y Fl(1)183 952 y Fp(\002)f Fo(l)248 959 y Fl(2)281 952 y Fu(+)320 959 y Fl(1)353 952 y Fo(l)368 959 y Fl(2)401 952 y Fp(\000)g Fo(l)466 959 y Fn(c)486 952 y Fp(g)j(!)g Fo(\031)r Fu(\()p Fo(\021)f Fu(+)715 959 y Fl(2)748 952 y Fo(j)769 959 y Fl(1)802 952 y Fp(\002)e Fo(j)873 959 y Fl(3)895 952 y Fo(;)d Fh(pred)f Fo(E)s Fu(\()p Fo(\021)13 b Fu(+)1158 959 y Fl(2)1191 952 y Fo(j)1212 959 y Fl(1)1245 952 y Fp(\002)e Fo(j)1316 959 y Fl(3)1349 952 y Fu(+)1388 959 y Fl(2)1421 952 y Fo(l)1436 959 y Fn(c)1467 952 y Fp(\002)g Fu(\()p Fo(l)1551 959 y Fn(c)1582 952 y Fu(+)g(1\))h(+)1727 959 y Fl(1)1760 952 y Fu(1\)\))c Fo(;)60 1034 y Fh(4\))17 b(gro)o(wing)e(in)h (the)h(direction)e(of)i Fo(e)739 1041 y Fl(3)761 1034 y Fh(:)0 1105 y Fo(\031)r Fu(\()p Fo(\021)c Fu(+)126 1112 y Fl(3)159 1105 y Fo(l)174 1112 y Fn(c)206 1105 y Fp(\002)d Fu(\()p Fo(l)289 1112 y Fn(c)321 1105 y Fu(+)h(2\))p Fo(;)d Fh(pred)f Fo(E)s Fu(\()p Fo(\021)13 b Fu(+)678 1112 y Fl(3)711 1105 y Fo(l)726 1112 y Fn(c)757 1105 y Fp(\002)e Fu(\()p Fo(l)841 1112 y Fn(c)873 1105 y Fu(+)g(1\))g Fp(\010)g Fu(1\)\))j Fp($)g(f)p Fo(\030)r Fp(g)p Fo(;)120 1165 y(\030)h Fp(2)g(M)p Fu(\()p Fo(\021)d Fu(+)360 1172 y Fl(3)393 1165 y Fo(l)408 1172 y Fn(c)440 1165 y Fp(\002)e Fu(\()p Fo(l)523 1172 y Fn(c)555 1165 y Fu(+)h(2\))p Fo(;)d Fp(\000)p Fu(\()p Fo(l)744 1172 y Fn(c)776 1165 y Fp(\000)i Fu(1\)\))e Fo(;)0 1225 y(\031)r Fu(\()p Fo(\021)13 b Fu(+)126 1232 y Fl(3)159 1225 y Fo(l)174 1232 y Fl(1)208 1225 y Fp(\002)d Fo(l)272 1232 y Fl(2)295 1225 y Fo(;)e Fh(pred)f Fo(E)s Fu(\()p Fo(\021)13 b Fu(+)558 1232 y Fl(3)591 1225 y Fo(l)606 1232 y Fl(1)639 1225 y Fp(\002)e Fo(l)704 1232 y Fl(2)737 1225 y Fp(\010)g Fu(1\)\))j Fp($)g(f)p Fo(\030)r Fp(g)p Fo(;)i(\030)g Fp(2)e Fo(\021)f Fu(+)1194 1232 y Fl(3)1227 1225 y Fo(l)1242 1232 y Fl(1)1275 1225 y Fp(\002)e Fo(l)1340 1232 y Fl(2)1374 1225 y Fp(\010)f Fu(1)0 1285 y Fp(f)p Fo(\021)i Fu(+)101 1292 y Fl(3)135 1285 y Fo(l)150 1292 y Fl(1)183 1285 y Fp(\002)f Fo(l)248 1292 y Fl(2)281 1285 y Fu(+)320 1292 y Fl(1)353 1285 y Fo(k)r Fp(g)i(!)h(f)p Fo(\021)f Fu(+)585 1292 y Fl(3)618 1285 y Fo(l)633 1292 y Fl(1)666 1285 y Fp(\002)e Fo(l)731 1292 y Fl(2)764 1285 y Fu(+)803 1292 y Fl(1)836 1285 y Fu(\()p Fo(k)i Fu(+)e(1\))p Fp(g)p Fo(;)17 b Fu(0)c Fo(<)h(k)h(<)f(l)1244 1292 y Fl(2)1277 1285 y Fp(\000)d Fo(l)1342 1292 y Fn(c)1370 1285 y Fo(;)0 1345 y Fp(f)p Fo(\021)h Fu(+)101 1352 y Fl(3)135 1345 y Fo(l)150 1352 y Fl(1)183 1345 y Fp(\002)f Fo(l)248 1352 y Fl(2)281 1345 y Fu(+)320 1352 y Fl(2)353 1345 y Fo(k)r Fp(g)i(!)h(f)p Fo(\021)f Fu(+)585 1352 y Fl(3)618 1345 y Fo(l)633 1352 y Fl(1)666 1345 y Fp(\002)e Fo(l)731 1352 y Fl(2)764 1345 y Fu(+)803 1352 y Fl(2)836 1345 y Fu(\()p Fo(k)i Fu(+)e(1\))p Fp(g)p Fo(;)17 b Fu(0)c Fo(<)h(k)h(<)f(l)1244 1352 y Fl(1)1277 1345 y Fp(\000)d Fo(l)1342 1352 y Fn(c)1370 1345 y Fo(;)0 1404 y Fp(f)p Fo(\021)h Fu(+)101 1411 y Fl(3)135 1404 y Fo(l)150 1411 y Fl(1)183 1404 y Fp(\002)f Fo(l)248 1411 y Fl(2)281 1404 y Fu(+)320 1411 y Fl(2)353 1404 y Fo(l)368 1411 y Fl(1)401 1404 y Fp(\000)g Fo(l)466 1411 y Fn(c)486 1404 y Fp(g)j(!)g Fo(\031)r Fu(\()p Fo(\021)f Fu(+)715 1411 y Fl(3)748 1404 y Fo(l)763 1411 y Fl(1)796 1404 y Fp(\002)e Fu(\()p Fo(l)880 1411 y Fl(2)914 1404 y Fu(+)g(1\))p Fo(;)d Fh(pred)f Fo(E)s Fu(\()p Fo(\021)13 b Fu(+)1271 1411 y Fl(3)1304 1404 y Fo(l)1319 1411 y Fl(1)1352 1404 y Fp(\002)e Fu(\()p Fo(l)1436 1411 y Fl(2)1470 1404 y Fu(+)g(1\))g(+)g(1\)\))d Fo(;)0 1464 y Fp(f)p Fo(\021)k Fu(+)101 1471 y Fl(3)135 1464 y Fo(l)150 1471 y Fl(1)183 1464 y Fp(\002)f Fo(l)248 1471 y Fl(2)281 1464 y Fu(+)320 1471 y Fl(1)353 1464 y Fo(l)368 1471 y Fl(2)401 1464 y Fp(\000)g Fo(l)466 1471 y Fn(c)486 1464 y Fp(g)j(!)g Fo(\031)r Fu(\()p Fo(\021)f Fu(+)715 1471 y Fl(3)748 1464 y Fu(\()p Fo(l)782 1471 y Fl(1)816 1464 y Fu(+)e(1\))g Fp(\002)g Fo(l)986 1471 y Fl(2)1008 1464 y Fo(;)d Fh(pred)f Fo(E)s Fu(\()p Fo(\021)13 b Fu(+)1271 1471 y Fl(3)1304 1464 y Fu(\()p Fo(l)1338 1471 y Fl(1)1372 1464 y Fu(+)e(1\))g Fp(\002)g Fo(l)1542 1471 y Fl(2)1575 1464 y Fu(+)g(1\)\))d Fo(;)0 1524 y Fp(f)p Fo(\021)k Fu(+)101 1531 y Fl(3)135 1524 y Fo(l)150 1531 y Fl(1)183 1524 y Fp(\002)f Fo(l)248 1531 y Fl(2)281 1524 y Fu(+)320 1531 y Fl(2)353 1524 y Fo(l)368 1531 y Fl(1)401 1524 y Fp(\000)g Fo(l)466 1531 y Fn(c)486 1524 y Fp(g)j(!)g Fo(\031)r Fu(\()p Fo(\021)f Fu(+)715 1531 y Fl(3)748 1524 y Fo(j)769 1531 y Fl(1)802 1524 y Fp(\002)e Fo(j)873 1531 y Fl(2)895 1524 y Fo(;)d Fh(pred)f Fo(E)s Fu(\()p Fo(\021)13 b Fu(+)1158 1531 y Fl(3)1191 1524 y Fo(j)1212 1531 y Fl(1)1245 1524 y Fp(\002)e Fo(j)1316 1531 y Fl(2)1349 1524 y Fu(+)1388 1531 y Fl(3)1421 1524 y Fo(l)1436 1531 y Fn(c)1467 1524 y Fp(\002)g Fu(\()p Fo(l)1551 1531 y Fn(c)1582 1524 y Fu(+)g(1\))h(+)1727 1531 y Fl(1)1760 1524 y Fu(1\)\))c Fo(;)0 1584 y Fp(f)p Fo(\021)k Fu(+)101 1591 y Fl(3)135 1584 y Fo(l)150 1591 y Fl(1)183 1584 y Fp(\002)f Fo(l)248 1591 y Fl(2)281 1584 y Fu(+)320 1591 y Fl(1)353 1584 y Fo(l)368 1591 y Fl(2)401 1584 y Fp(\000)g Fo(l)466 1591 y Fn(c)486 1584 y Fp(g)j(!)g Fo(\031)r Fu(\()p Fo(\021)f Fu(+)715 1591 y Fl(3)748 1584 y Fo(j)769 1591 y Fl(1)802 1584 y Fp(\002)e Fo(j)873 1591 y Fl(2)895 1584 y Fo(;)d Fh(pred)f Fo(E)s Fu(\()p Fo(\021)13 b Fu(+)1158 1591 y Fl(3)1191 1584 y Fo(j)1212 1591 y Fl(1)1245 1584 y Fp(\002)e Fo(j)1316 1591 y Fl(2)1349 1584 y Fu(+)1388 1591 y Fl(3)1421 1584 y Fo(l)1436 1591 y Fn(c)1467 1584 y Fp(\002)g Fu(\()p Fo(l)1551 1591 y Fn(c)1582 1584 y Fu(+)g(1\))h(+)1727 1591 y Fl(1)1760 1584 y Fu(1\)\))c Fo(:)60 1666 y Fh(The)18 b(sym)o(b)q(ol)g Fp($)h Fh(means)f(that)h(b)q(oth)g(arro)o(ws)e Fp(!)i Fh(and)f Fp( )h Fh(are)f(presen)o(t.)28 b(The)19 b(ab)q(o)o(v)o(e)f(list)h(should)0 1726 y(b)q(e)h(completed)g(with)g(all)g(the)g(isometric)f(arro)o(ws)g (\(obtained)h(b)o(y)f(applying)g(the)i(same)e(isometry)h(to)0 1786 y(b)q(oth)e(ends)e(of)i(an)f(arro)o(w\),)g(as)g(w)o(ell)g(as)g(all)g (the)h(arro)o(ws)e(obtained)h(b)o(y)g(sliding)f(either)h(the)h(bar)f(along)0 1846 y(the)j(rectangle)f(\(i.e.)32 b(the)20 b(con\014gurations)e Fo(l)849 1853 y Fl(1)884 1846 y Fp(\002)13 b Fo(l)951 1853 y Fl(2)987 1846 y Fu(+)1026 1828 y Fn(i)1026 1858 y Fl(1)1061 1846 y Fo(k)r Fh(,)20 b Fu(0)f Fp(\024)g Fo(i)h(<)f(l)1335 1853 y Fl(2)1370 1846 y Fp(\000)13 b Fo(k)r Fh(\))20 b(or)f(b)o(y)g (translating)g(the)0 1905 y(rectangle)d Fo(l)228 1912 y Fl(1)261 1905 y Fp(\002)11 b Fo(l)326 1912 y Fl(2)365 1905 y Fh(on)16 b(the)h(side)e(of)i(the)g(parallelepip)q(ed)d Fo(\021)k Fh(\(i.e.)23 b(the)16 b(con\014gurations)f(in)h Fo(\021)c Fp(\010)f Fo(l)1797 1912 y Fl(1)1830 1905 y Fp(\002)g Fo(l)1895 1912 y Fl(2)1917 1905 y Fh(\).)60 1988 y Fu(The)16 b(lo)q(ops)g(in)g(the)h(graph)e Fp(G)609 1970 y Fl(+)642 1988 y Fu(\()p Fo(\033)r Fu(\))j(are)220 2102 y Fo(\031)r Fu(\()p Fo(\021)r(;)8 b Fu(pred)f Fo(E)s Fu(\()p Fo(\021)13 b Fp(\010)e Fo(l)585 2109 y Fn(c)616 2102 y Fp(\002)g Fu(\()p Fo(l)700 2109 y Fn(c)732 2102 y Fu(+)f(1\))i Fp(\010)e Fu(1\)\))15 b Fp($)f(f)p Fo(\030)r Fp(g)p Fo(;)i(\030)g Fp(2)p 1217 2048 429 2 v 14 w Fo(\021)d Fp(\010)p 1304 2058 342 2 v 10 w Fo(l)1319 2109 y Fn(c)1351 2102 y Fp(\002)d Fu(\()p Fo(l)1434 2109 y Fn(c)1466 2102 y Fu(+)h(1\))g Fp(\010)g Fu(1)1646 2057 y Fl(123)1716 2102 y Fo(;)0 2176 y(\031)r Fu(\()p Fo(\021)c Fu(+)120 2183 y Fl(1)148 2176 y Fo(l)163 2183 y Fn(c)189 2176 y Fp(\002)e Fu(\()p Fo(l)267 2183 y Fn(c)293 2176 y Fu(+)g(2\))p Fo(;)j Fu(pred)g Fo(E)s Fu(\()p Fo(\021)f Fu(+)639 2183 y Fl(1)666 2176 y Fo(l)681 2183 y Fn(c)707 2176 y Fp(\002)e Fu(\()p Fo(l)785 2183 y Fn(c)811 2176 y Fu(+)g(1\))g Fp(\010)g Fu(1\)\))16 b Fp($)d(f)p Fo(\030)r Fp(g)8 b Fo(;)17 b(\030)f Fp(2)e(M)p Fu(\()p Fo(\021)7 b Fu(+)1438 2183 y Fl(1)1466 2176 y Fo(l)1481 2183 y Fn(c)1506 2176 y Fp(\002)e Fu(\()p Fo(l)1584 2183 y Fn(c)1611 2176 y Fu(+)g(2\))p Fo(;)j Fp(\000)p Fu(\()p Fo(l)1794 2183 y Fn(c)1820 2176 y Fp(\000)d Fu(1\)\))j Fo(;)244 2251 y(\031)r Fu(\()p Fo(\021)13 b Fu(+)370 2258 y Fl(1)403 2251 y Fo(l)418 2258 y Fl(1)452 2251 y Fp(\002)d Fo(l)516 2258 y Fl(2)539 2251 y Fo(;)e Fu(pred)f Fo(E)s Fu(\()p Fo(\021)13 b Fu(+)802 2258 y Fl(1)835 2251 y Fo(l)850 2258 y Fl(1)883 2251 y Fp(\002)e Fo(l)948 2258 y Fl(2)981 2251 y Fp(\010)g Fu(1\)\))j Fp($)g(f)p Fo(\030)r Fp(g)p Fo(;)i(\030)g Fp(2)e Fo(\021)f Fu(+)1438 2258 y Fl(1)1471 2251 y Fo(l)1486 2258 y Fl(1)1519 2251 y Fp(\002)e Fo(l)1584 2258 y Fl(2)1618 2251 y Fp(\010)f Fu(1)p Fo(;)0 2326 y(\031)r Fu(\()p Fo(\021)d Fu(+)120 2333 y Fl(2)148 2326 y Fo(l)163 2333 y Fn(c)189 2326 y Fp(\002)e Fu(\()p Fo(l)267 2333 y Fn(c)293 2326 y Fu(+)g(2\))p Fo(;)j Fu(pred)g Fo(E)s Fu(\()p Fo(\021)f Fu(+)639 2333 y Fl(2)666 2326 y Fo(l)681 2333 y Fn(c)707 2326 y Fp(\002)e Fu(\()p Fo(l)785 2333 y Fn(c)811 2326 y Fu(+)g(1\))g Fp(\010)g Fu(1\)\))16 b Fp($)d(f)p Fo(\030)r Fp(g)8 b Fo(;)17 b(\030)f Fp(2)e(M)p Fu(\()p Fo(\021)7 b Fu(+)1438 2333 y Fl(2)1466 2326 y Fo(l)1481 2333 y Fn(c)1506 2326 y Fp(\002)e Fu(\()p Fo(l)1584 2333 y Fn(c)1611 2326 y Fu(+)g(2\))p Fo(;)j Fp(\000)p Fu(\()p Fo(l)1794 2333 y Fn(c)1820 2326 y Fp(\000)d Fu(1\)\))j Fo(;)244 2401 y(\031)r Fu(\()p Fo(\021)13 b Fu(+)370 2408 y Fl(2)403 2401 y Fo(l)418 2408 y Fl(1)452 2401 y Fp(\002)d Fo(l)516 2408 y Fl(2)539 2401 y Fo(;)e Fu(pred)f Fo(E)s Fu(\()p Fo(\021)13 b Fu(+)802 2408 y Fl(2)835 2401 y Fo(l)850 2408 y Fl(1)883 2401 y Fp(\002)e Fo(l)948 2408 y Fl(2)981 2401 y Fp(\010)g Fu(1\)\))j Fp($)g(f)p Fo(\030)r Fp(g)p Fo(;)i(\030)g Fp(2)e Fo(\021)f Fu(+)1438 2408 y Fl(2)1471 2401 y Fo(l)1486 2408 y Fl(1)1519 2401 y Fp(\002)e Fo(l)1584 2408 y Fl(2)1618 2401 y Fp(\010)f Fu(1)p Fo(;)0 2475 y(\031)r Fu(\()p Fo(\021)d Fu(+)120 2482 y Fl(3)148 2475 y Fo(l)163 2482 y Fn(c)189 2475 y Fp(\002)e Fu(\()p Fo(l)267 2482 y Fn(c)293 2475 y Fu(+)g(2\))p Fo(;)j Fu(pred)g Fo(E)s Fu(\()p Fo(\021)f Fu(+)639 2482 y Fl(3)666 2475 y Fo(l)681 2482 y Fn(c)707 2475 y Fp(\002)e Fu(\()p Fo(l)785 2482 y Fn(c)811 2475 y Fu(+)g(1\))g Fp(\010)g Fu(1\)\))16 b Fp($)d(f)p Fo(\030)r Fp(g)8 b Fo(;)17 b(\030)f Fp(2)e(M)p Fu(\()p Fo(\021)7 b Fu(+)1438 2482 y Fl(3)1466 2475 y Fo(l)1481 2482 y Fn(c)1506 2475 y Fp(\002)e Fu(\()p Fo(l)1584 2482 y Fn(c)1611 2475 y Fu(+)g(2\))p Fo(;)j Fp(\000)p Fu(\()p Fo(l)1794 2482 y Fn(c)1820 2475 y Fp(\000)d Fu(1\)\))j Fo(;)244 2550 y(\031)r Fu(\()p Fo(\021)13 b Fu(+)370 2557 y Fl(3)403 2550 y Fo(l)418 2557 y Fl(1)452 2550 y Fp(\002)d Fo(l)516 2557 y Fl(2)539 2550 y Fo(;)e Fu(pred)f Fo(E)s Fu(\()p Fo(\021)13 b Fu(+)802 2557 y Fl(3)835 2550 y Fo(l)850 2557 y Fl(1)883 2550 y Fp(\002)e Fo(l)948 2557 y Fl(2)981 2550 y Fp(\010)g Fu(1\)\))j Fp($)g(f)p Fo(\030)r Fp(g)p Fo(;)i(\030)g Fp(2)e Fo(\021)f Fu(+)1438 2557 y Fl(3)1471 2550 y Fo(l)1486 2557 y Fl(1)1519 2550 y Fp(\002)e Fo(l)1584 2557 y Fl(2)1618 2550 y Fp(\010)f Fu(1)p Fo(:)954 2670 y Fs(41)p eop %%Page: 42 42 42 41 bop 0 200 a Fu(All)11 b(these)h(lo)q(ops)f(are)g(of)g(length)g(t)o(w)o (o.)20 b(An)o(y)12 b(other)f(arro)o(w)e Fo(\031)1100 207 y Fl(1)1137 200 y Fp(!)14 b Fo(\031)1229 207 y Fl(2)1263 200 y Fu(of)d Fp(G)1347 182 y Fl(+)1380 200 y Fu(\()p Fo(\033)r Fu(\))i(satis\014es)p 1639 172 26 2 v 10 w Fo(v)r Fu(\()p Fo(\031)1712 207 y Fl(1)1735 200 y Fu(\))h Fo(<)g(v)p 1821 208 V 2 w Fu(\()p Fo(\031)1894 207 y Fl(2)1917 200 y Fu(\).)0 260 y(As)j(a)f(consequence)g(a)h (path)f(in)g Fp(G)645 242 y Fl(+)678 260 y Fu(\()p Fo(\033)r Fu(\))h(starting)f(at)h Fp(f)p Fo(\033)r Fp(g)g Fu(with)f(no)g(lo)q(op)h (ends)e(in)i Fp(f)p Fu(+1)p 1596 272 64 2 v -1 w Fp(g)p Fu(.)60 351 y(W)l(e)23 b(are)g(done)g(with)h(the)f(standard)f(con\014gurations.)40 b(W)l(e)24 b(ha)o(v)o(e)f(to)g(examine)g(the)h(remaining)0 411 y(con\014gurations)c(of)i(the)g(principal)e(b)q(oundary)g(of)i(the)g (cycle)h Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 1253 423 V Fo(;)8 b Fu(+1)p 1339 423 V -20 x Fn(c)1423 411 y Fu(\))22 b(i.e.)38 b(the)22 b(principal)e(non)0 471 y(standard)15 b(con\014gurations.)0 562 y Fw(The)k(principal)j(non)d(standard)f(con\014gurations.)27 b Fu(W)l(e)17 b(no)o(w)f(do)h(the)g(same)f(w)o(ork)g(for)h(the)g(con-)0 627 y(\014gurations)i(in)325 614 y Fk(f)313 627 y Fp(M)373 634 y Fn(n)398 639 y Fg(c)432 627 y Fp(n)14 b(S)501 634 y Fn(n)526 639 y Fg(c)547 627 y Fu(.)34 b(The)20 b(relev)m(an)o(t)h(cycles)g(are)f(the)h (cycles)f(around)f(the)i(parallelepip)q(eds)0 694 y Fo(j)21 701 y Fn(c)52 694 y Fp(\002)11 b Fu(\()p Fo(j)142 701 y Fn(c)173 694 y Fu(+)g Fo(\016)245 701 y Fn(c)265 694 y Fu(\))h Fp(\002)e Fu(\()p Fo(j)385 701 y Fn(c)417 694 y Fu(+)g(2\))17 b(and)f(around)f(the)i (con\014gurations)d(of)1269 681 y Fk(f)1257 694 y Fp(M)1317 701 y Fn(n)1342 706 y Fg(c)1360 701 y Fl(+1)1424 694 y Fp(n)d(S)1490 701 y Fn(n)1515 706 y Fg(c)1534 701 y Fl(+1)1587 694 y Fu(.)0 785 y Fw(Theorem)21 b(7.24.)59 b Fh(The)19 b(cycle)i Fo(\031)g Fu(=)e Fo(\031)r Fu(\()p Fo(j)834 792 y Fn(c)867 785 y Fp(\002)13 b Fu(\()p Fo(j)959 792 y Fn(c)992 785 y Fu(+)g Fo(\016)1066 792 y Fn(c)1087 785 y Fu(\))g Fp(\002)g Fu(\()p Fo(j)1211 792 y Fn(c)1245 785 y Fu(+)f(2\))p Fo(;)c Fh(pred)g Fo(E)s Fu(\()p Fo(n)1557 792 y Fn(c)1577 785 y Fu(\)\))21 b Fh(is)e(the)h(greatest)0 845 y(cycle)d(con)o(taining)e Fo(j)386 852 y Fn(c)417 845 y Fp(\002)c Fu(\()p Fo(j)507 852 y Fn(c)538 845 y Fu(+)g Fo(\016)610 852 y Fn(c)630 845 y Fu(\))h Fp(\002)f Fu(\()p Fo(j)751 852 y Fn(c)782 845 y Fu(+)g(2\))17 b Fh(included)e(in)h Fp(f\000)p Fu(1)p 1175 857 V Fo(;)8 b Fu(+1)p 1261 857 V -1 w Fp(g)1349 827 y Fn(c)1370 845 y Fh(.)22 b(Moreo)o(v)o(er,)313 967 y Fo(v)p 313 974 26 2 v 2 w Fu(\()p Fo(\031)r Fu(\))h(=)f Fo(n)521 974 y Fn(c)552 967 y Fu(+)11 b(1)d Fo(;)p 707 939 V 58 w(v)r Fu(\()p Fo(\031)r Fu(\))23 b Fo(<)f(j)906 974 y Fn(c)925 967 y Fu(\()p Fo(j)965 974 y Fn(c)997 967 y Fu(+)11 b Fo(\016)1069 974 y Fn(c)1089 967 y Fu(\)\()p Fo(j)1148 974 y Fn(c)1180 967 y Fu(+)f(2\))i(+)f Fo(l)1350 974 y Fn(c)1370 967 y Fu(\()p Fo(l)1404 974 y Fn(c)1435 967 y Fu(+)g(1\))g(+)g(1)d Fo(:)0 1088 y Fh(The)16 b(b)q(ottom)h(of)g(this)f (cycle)h(is)f Fp(f)p Fo(j)651 1095 y Fn(c)682 1088 y Fp(\002)11 b Fu(\()p Fo(j)772 1095 y Fn(c)803 1088 y Fu(+)g Fo(\016)875 1095 y Fn(c)895 1088 y Fu(\))h Fp(\002)f Fu(\()p Fo(j)1016 1095 y Fn(c)1047 1088 y Fu(+)g(2\))p Fp(g)p Fh(;)16 b(its)h(principal)d(b)q (oundary)i(con)o(tains)498 1217 y Fp(f)8 b Fo(\030)16 b Fu(:)e Fo(\030)i Fp(\032)d Fo(j)708 1224 y Fn(c)739 1217 y Fp(\002)e Fu(\()p Fo(j)829 1224 y Fn(c)860 1217 y Fu(+)g Fo(\016)932 1224 y Fn(c)952 1217 y Fu(\))h Fp(\002)f Fu(\()p Fo(j)1073 1224 y Fn(c)1104 1217 y Fu(+)g(2\))p Fo(;)16 b(\030)g Fp(2)1325 1204 y Fk(f)1313 1217 y Fp(M)1373 1224 y Fn(n)1398 1229 y Fg(c)1427 1217 y Fp(g)0 1338 y Fh(and)g(is)g(included)f(in)h Fp(M)p Fu(\()p Fo(j)504 1345 y Fn(c)535 1338 y Fp(\002)11 b Fu(\()p Fo(j)625 1345 y Fn(c)656 1338 y Fu(+)g Fo(\016)728 1345 y Fn(c)749 1338 y Fu(\))g Fp(\002)g Fu(\()p Fo(j)869 1345 y Fn(c)900 1338 y Fu(+)g(2\))p Fo(;)d(n)1046 1345 y Fn(c)1078 1338 y Fp(\000)i Fo(j)1148 1345 y Fn(c)1168 1338 y Fu(\()p Fo(j)1208 1345 y Fn(c)1240 1338 y Fu(+)g Fo(\016)1311 1345 y Fn(c)1332 1338 y Fu(\)\()p Fo(j)1391 1345 y Fn(c)1422 1338 y Fu(+)h(1\)\))p Fh(.)0 1429 y Fe(R)m(emark.)27 b Fu(Ob)o(viously)l(,)18 b(similar)f(statemen) o(ts)i(are)g(true)f(for)h(the)h(parallelepip)q(eds)d(isometric)h(to)i Fo(j)1879 1436 y Fn(c)1911 1429 y Fp(\002)0 1489 y Fu(\()p Fo(j)40 1496 y Fn(c)71 1489 y Fu(+)11 b Fo(\016)143 1496 y Fn(c)163 1489 y Fu(\))h Fp(\002)f Fu(\()p Fo(j)284 1496 y Fn(c)315 1489 y Fu(+)g(2\).)0 1581 y Fe(Pr)m(o)m(of.)28 b Fu(W)l(e)23 b(c)o(hec)o(k)g(that)g(the)g(parallelepip)q(ed)f Fo(j)944 1588 y Fn(c)979 1581 y Fp(\002)15 b Fu(\()p Fo(j)1073 1588 y Fn(c)1108 1581 y Fu(+)h Fo(\016)1185 1588 y Fn(c)1205 1581 y Fu(\))g Fp(\002)f Fu(\()p Fo(j)1334 1588 y Fn(c)1369 1581 y Fu(+)h(2\))23 b(and)f(the)i(cycle)f Fo(\031)i Fu(sat-)0 1640 y(isfy)i(the)g(h)o(yp)q (othesis)e(of)i(theorem)f(5)p Fo(:)p Fu(5.)53 b(Let)27 b Fo(x)982 1647 y Fl(1)1005 1640 y Fo(;)8 b Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1152 1647 y Fn(r)1201 1640 y Fu(b)q(e)27 b(a)g(sequence)g(of)g (sites)f(suc)o(h)g(that)0 1700 y Fo(F)7 b Fu(\()p Fo(x)86 1707 y Fl(1)109 1700 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)256 1707 y Fn(s)278 1700 y Fu(\)\()p Fo(j)337 1707 y Fn(c)364 1700 y Fp(\002)e Fu(\()p Fo(j)449 1707 y Fn(c)475 1700 y Fu(+)g Fo(\016)542 1707 y Fn(c)562 1700 y Fu(\))g Fp(\002)g Fu(\()p Fo(j)672 1707 y Fn(c)698 1700 y Fu(+)g(2\))15 b(is)e(in)h Fo(\031)i Fu(for)e Fo(s)g Fu(in)g Fp(f)p Fu(1)8 b Fp(\001)g(\001)g(\001)h Fo(r)q Fp(g)14 b Fu(\(i.e.)22 b(these)14 b(con\014gurations)e(ha)o(v)o(e)0 1760 y(an)g(energy)g(less)f(or)h(equal)g(than)g(pred)c Fo(E)s Fu(\()p Fo(n)798 1767 y Fn(c)818 1760 y Fu(\)\).)21 b(W)l(e)13 b(put)f Fo(\021)1085 1767 y Fn(s)1119 1760 y Fu(=)i Fo(F)7 b Fu(\()p Fo(x)1258 1767 y Fl(1)1281 1760 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1428 1767 y Fn(s)1450 1760 y Fu(\)\()p Fo(j)1509 1767 y Fn(c)1532 1760 y Fp(\002)r Fu(\()p Fo(j)1613 1767 y Fn(c)1636 1760 y Fu(+)r Fo(\016)1699 1767 y Fn(c)1719 1760 y Fu(\))r Fp(\002)r Fu(\()p Fo(j)1821 1767 y Fn(c)1845 1760 y Fu(+)r(2\)\))0 1820 y(for)16 b Fo(s)h Fu(in)f Fp(f)p Fu(0)8 b Fp(\001)g(\001)g(\001)h Fo(r)q Fp(g)p Fu(.)0 1880 y Fp(\017)50 b Fu(First)17 b(case:)26 b(all)18 b(the)h(sites)f(are)g(outside)g Fo(j)879 1887 y Fn(c)911 1880 y Fp(\002)12 b Fu(\()p Fo(j)1002 1887 y Fn(c)1034 1880 y Fu(+)g Fo(\016)1107 1887 y Fn(c)1128 1880 y Fu(\))h Fp(\002)f Fu(\()p Fo(j)1251 1887 y Fn(c)1283 1880 y Fu(+)g(2\).)28 b(W)l(e)19 b(ha)o(v)o(e)f(that)g Fo(j)1755 1887 y Fn(c)1788 1880 y Fp(\002)12 b Fu(\()p Fo(j)1879 1887 y Fn(c)1911 1880 y Fu(+)0 1939 y Fo(\016)22 1946 y Fn(c)42 1939 y Fu(\))f Fp(\002)f Fu(\()p Fo(j)161 1946 y Fn(c)191 1939 y Fu(+)f(2\))14 b Fp(\032)g Fo(\021)375 1946 y Fn(s)412 1939 y Fu(whence)i Fo(E)s Fu(\()p Fo(\021)670 1946 y Fn(s)691 1939 y Fu(\))23 b Fp(\025)e Fo(E)s Fu(\()p Fo(j)873 1946 y Fn(c)903 1939 y Fp(\002)10 b Fu(\()p Fo(j)992 1946 y Fn(c)1022 1939 y Fu(+)g Fo(\016)1093 1946 y Fn(c)1113 1939 y Fu(\))h Fp(\002)e Fu(\()p Fo(j)1231 1946 y Fn(c)1261 1939 y Fu(+)h(2\))p Fo(;)e(v)r Fu(\()p Fo(\021)1446 1946 y Fn(s)1468 1939 y Fu(\))i Fp(\000)g Fo(j)1567 1946 y Fn(c)1587 1939 y Fu(\()p Fo(j)1627 1946 y Fn(c)1657 1939 y Fu(+)g Fo(\016)1728 1946 y Fn(c)1748 1939 y Fu(\)\()p Fo(j)1807 1946 y Fn(c)1838 1939 y Fu(+)f(2\)\))0 1999 y(and)151 2120 y(pred)e Fo(E)s Fu(\()p Fo(n)345 2127 y Fn(c)365 2120 y Fu(\))31 b Fp(\025)44 b Fu(max)484 2153 y Fl(0)p Fj(\024)p Fn(s)p Fj(\024)p Fn(r)613 2120 y Fo(E)s Fu(\()p Fo(j)693 2127 y Fn(c)724 2120 y Fp(\002)11 b Fu(\()p Fo(j)814 2127 y Fn(c)845 2120 y Fu(+)g Fo(\016)917 2127 y Fn(c)937 2120 y Fu(\))h Fp(\002)f Fu(\()p Fo(j)1058 2127 y Fn(c)1089 2120 y Fu(+)g(2\))p Fo(;)d(v)r Fu(\()p Fo(\021)1275 2127 y Fn(s)1297 2120 y Fu(\))j Fp(\000)g Fo(j)1398 2127 y Fn(c)1418 2120 y Fu(\()p Fo(j)1458 2127 y Fn(c)1489 2120 y Fu(+)g Fo(\016)1561 2127 y Fn(c)1581 2120 y Fu(\)\()p Fo(j)1640 2127 y Fn(c)1672 2120 y Fu(+)g(2\)\))p Fo(:)0 2268 y Fu(Since)16 b(the)g(sequence)h Fo(\021)447 2275 y Fl(0)469 2268 y Fo(;)8 b Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(\021)613 2275 y Fn(r)651 2268 y Fu(is)16 b(a)g(sequence)g(of)h(spin)e (\015ips)g(w)o(e)h(ha)o(v)o(e)g Fp(j)p Fo(v)r Fu(\()p Fo(\021)1487 2275 y Fn(s)p Fl(+1)1558 2268 y Fu(\))11 b Fp(\000)g Fo(v)r Fu(\()p Fo(\021)1708 2275 y Fn(s)1729 2268 y Fu(\))p Fp(j)j(\024)f Fu(1)k(and)0 2328 y(\()p Fo(v)r Fu(\()p Fo(\021)89 2335 y Fn(s)111 2328 y Fu(\))p Fo(;)g Fu(0)c Fp(\024)h Fo(s)g Fp(\024)g Fo(r)q Fu(\))j(tak)o(es)g(all)f(the)h(v)m(alues)f(b)q(et)o(w)o(een)h Fo(j)1046 2335 y Fn(c)1065 2328 y Fu(\()p Fo(j)1105 2335 y Fn(c)1137 2328 y Fu(+)11 b Fo(\016)1209 2335 y Fn(c)1229 2328 y Fu(\)\()p Fo(j)1288 2335 y Fn(c)1320 2328 y Fu(+)f(2\))17 b(and)f Fo(v)r Fu(\()p Fo(\021)1597 2335 y Fn(r)1619 2328 y Fu(\).)23 b(Henceforth)14 2449 y(max)0 2481 y Fl(0)p Fj(\024)p Fn(s)p Fj(\024)p Fn(r)129 2449 y Fo(E)s Fu(\()p Fo(j)209 2456 y Fn(c)240 2449 y Fp(\002)10 b Fu(\()p Fo(j)329 2456 y Fn(c)361 2449 y Fu(+)h Fo(\016)433 2456 y Fn(c)453 2449 y Fu(\))g Fp(\002)g Fu(\()p Fo(j)573 2456 y Fn(c)604 2449 y Fu(+)g(2\))p Fo(;)d(v)r Fu(\()p Fo(\021)790 2456 y Fn(s)812 2449 y Fu(\))j Fp(\000)g Fo(j)913 2456 y Fn(c)933 2449 y Fu(\()p Fo(j)973 2456 y Fn(c)1004 2449 y Fu(+)g Fo(\016)1076 2456 y Fn(c)1097 2449 y Fu(\)\()p Fo(j)1156 2456 y Fn(c)1187 2449 y Fu(+)g(2\)\))31 b Fp(\025)364 2550 y Fu(max)p Fp(f)8 b Fo(E)s Fu(\()p Fo(j)570 2557 y Fn(c)601 2550 y Fp(\002)j Fu(\()p Fo(j)691 2557 y Fn(c)722 2550 y Fu(+)g Fo(\016)794 2557 y Fn(c)814 2550 y Fu(\))h Fp(\002)e Fu(\()p Fo(j)934 2557 y Fn(c)966 2550 y Fu(+)g(2\))p Fo(;)e(k)r Fu(\))15 b(:)e(0)h Fp(\024)g Fo(k)h Fp(\024)e Fo(v)r Fu(\()p Fo(\021)1425 2557 y Fn(r)1448 2550 y Fu(\))e Fp(\000)g Fo(j)1549 2557 y Fn(c)1569 2550 y Fu(\()p Fo(j)1609 2557 y Fn(c)1640 2550 y Fu(+)g Fo(\016)1712 2557 y Fn(c)1732 2550 y Fu(\)\()p Fo(j)1791 2557 y Fn(c)1823 2550 y Fu(+)g(2\))d Fp(g)954 2670 y Fs(42)p eop %%Page: 43 43 43 42 bop 0 200 a Fu(and)16 b(the)h(v)o(olume)e(of)i Fo(\021)436 207 y Fn(r)474 200 y Fu(m)o(ust)f(satisfy)8 322 y(max)p Fp(f)8 b Fo(E)s Fu(\()p Fo(j)214 329 y Fn(c)245 322 y Fp(\002)j Fu(\()p Fo(j)335 329 y Fn(c)366 322 y Fu(+)g Fo(\016)438 329 y Fn(c)458 322 y Fu(\))h Fp(\002)f Fu(\()p Fo(j)579 329 y Fn(c)610 322 y Fu(+)g(2\))p Fo(;)d(k)r Fu(\))14 b(:)g(0)f Fp(\024)h Fo(k)h Fp(\024)f Fo(v)r Fu(\()p Fo(\021)1070 329 y Fn(r)1092 322 y Fu(\))d Fp(\000)g Fo(j)1193 329 y Fn(c)1213 322 y Fu(\()p Fo(j)1253 329 y Fn(c)1284 322 y Fu(+)g Fo(\016)1356 329 y Fn(c)1376 322 y Fu(\)\()p Fo(j)1435 329 y Fn(c)1467 322 y Fu(+)g(2\))d Fp(g)31 b(\024)f Fu(pred)7 b Fo(E)s Fu(\()p Fo(n)1888 329 y Fn(c)1908 322 y Fu(\))p Fo(:)0 444 y Fu(By)15 b(theorem)f(7)p Fo(:)p Fu(9,)h(w)o(e)g(ha)o(v)o(e)f Fo(E)s Fu(\()p Fo(j)626 451 y Fn(c)653 444 y Fp(\002)7 b Fu(\()p Fo(j)739 451 y Fn(c)767 444 y Fu(+)g Fo(\016)835 451 y Fn(c)855 444 y Fu(\))g Fp(\002)g Fu(\()p Fo(j)967 451 y Fn(c)996 444 y Fu(+)g(2\))p Fo(;)h(l)1123 451 y Fn(c)1144 444 y Fu(\()p Fo(l)1178 451 y Fn(c)1206 444 y Fu(+)f(1\))g(+)g(1\))15 b(=)f Fo(E)s Fu(\()p Fo(j)1541 451 y Fn(c)1568 444 y Fp(\002)7 b Fu(\()p Fo(j)1654 451 y Fn(c)1682 444 y Fu(+)g Fo(\016)1750 451 y Fn(c)1771 444 y Fu(\))g Fp(\002)g Fu(\()p Fo(j)1883 451 y Fn(c)1911 444 y Fu(+)0 504 y(2\))13 b Fp(\010)f Fo(l)123 511 y Fn(c)155 504 y Fp(\002)g Fu(\()p Fo(l)240 511 y Fn(c)273 504 y Fu(+)g(1\))h(+)f(1\))19 b(whic)o(h)e(is)h (strictly)g(greater)g(than)h(the)f(lev)o(el)h(of)f Fo(\031)r Fu(.)28 b(Th)o(us)17 b Fo(v)r Fu(\()p Fo(\021)1684 511 y Fn(r)1706 504 y Fu(\))i(is)f(strictly)0 564 y(less)e(than)g Fo(j)229 571 y Fn(c)260 564 y Fp(\002)11 b Fu(\()p Fo(j)350 571 y Fn(c)381 564 y Fu(+)g Fo(\016)453 571 y Fn(c)473 564 y Fu(\))h Fp(\002)f Fu(\()p Fo(j)594 571 y Fn(c)625 564 y Fu(+)g(2\))g(+)g Fo(l)795 571 y Fn(c)815 564 y Fu(\()p Fo(l)849 571 y Fn(c)881 564 y Fu(+)f(1\))i(+)e(1.)0 623 y Fp(\017)50 b Fu(Second)11 b(case:)19 b(all)11 b(the)h(sites)g(are)f(inside)g Fo(j)849 630 y Fn(c)870 623 y Fp(\002)q Fu(\()p Fo(j)950 630 y Fn(c)972 623 y Fu(+)q Fo(\016)1034 630 y Fn(c)1054 623 y Fu(\))q Fp(\002)q Fu(\()p Fo(j)1154 630 y Fn(c)1177 623 y Fu(+)q(2\).)21 b(No)o(w)11 b Fo(\021)1429 630 y Fn(s)1464 623 y Fp(\032)j Fo(j)1538 630 y Fn(c)1559 623 y Fp(\002)q Fu(\()p Fo(j)1639 630 y Fn(c)1661 623 y Fu(+)q Fo(\016)1723 630 y Fn(c)1743 623 y Fu(\))q Fp(\002)q Fu(\()p Fo(j)1843 630 y Fn(c)1866 623 y Fu(+)q(2\))0 683 y(so)i(that)0 805 y(pred)7 b Fo(E)s Fu(\()p Fo(n)194 812 y Fn(c)215 805 y Fu(\))30 b Fp(\025)44 b Fu(max)334 837 y Fl(0)p Fj(\024)p Fn(s)p Fj(\024)p Fn(r)462 805 y Fo(E)s Fu(\()p Fo(j)542 812 y Fn(c)573 805 y Fp(\002)11 b Fu(\()p Fo(j)663 812 y Fn(c)694 805 y Fu(+)g Fo(\016)766 812 y Fn(c)786 805 y Fu(\))h Fp(\002)f Fu(\()p Fo(j)907 812 y Fn(c)938 805 y Fu(+)g(2\))p Fo(;)d(v)r Fu(\()p Fo(\021)1124 812 y Fn(s)1146 805 y Fu(\))j Fp(\000)g Fo(j)1247 812 y Fn(c)1267 805 y Fu(\()p Fo(j)1307 812 y Fn(c)1338 805 y Fu(+)g Fo(\016)1410 812 y Fn(c)1430 805 y Fu(\)\()p Fo(j)1489 812 y Fn(c)1521 805 y Fu(+)g(2\)\))262 906 y Fp(\025)30 b Fu(max)o Fp(f)8 b Fo(E)s Fu(\()p Fo(j)536 913 y Fn(c)568 906 y Fp(\002)i Fu(\()p Fo(j)657 913 y Fn(c)689 906 y Fu(+)g Fo(\016)760 913 y Fn(c)781 906 y Fu(\))h Fp(\002)g Fu(\()p Fo(j)901 913 y Fn(c)932 906 y Fu(+)g(2\))p Fo(;)d(k)r Fu(\))14 b(:)g Fo(v)r Fu(\()p Fo(\021)1207 913 y Fn(r)1229 906 y Fu(\))e Fp(\000)e Fo(j)1330 913 y Fn(c)1350 906 y Fu(\()p Fo(j)1390 913 y Fn(c)1421 906 y Fu(+)h Fo(\016)1493 913 y Fn(c)1514 906 y Fu(\)\()p Fo(j)1573 913 y Fn(c)1604 906 y Fu(+)g(2\)\))k Fp(\024)e Fo(k)i Fp(\024)f Fu(0)8 b Fp(g)p Fo(:)0 1028 y Fu(Theorem)16 b(7)p Fo(:)p Fu(8)h(sho)o(ws)f (that)h Fo(E)s Fu(\()p Fo(j)624 1035 y Fn(c)656 1028 y Fp(\002)11 b Fu(\()p Fo(j)746 1035 y Fn(c)778 1028 y Fu(+)g Fo(\016)850 1035 y Fn(c)871 1028 y Fu(\))g Fp(\002)h Fu(\()p Fo(j)992 1035 y Fn(c)1023 1028 y Fu(+)g(2\))p Fo(;)c Fp(\000)p Fo(j)1200 1035 y Fn(c)1220 1028 y Fu(\()p Fo(j)1260 1035 y Fn(c)1291 1028 y Fu(+)k Fo(\016)1364 1035 y Fn(c)1384 1028 y Fu(\))g(+)f Fo(l)1480 1035 y Fn(c)1500 1028 y Fu(\()p Fo(l)1534 1035 y Fn(c)1566 1028 y Fu(+)h(1\))g(+)f(1\))17 b(is)g(equal)0 1088 y(to)g Fo(E)s Fu(\()p Fo(n)150 1095 y Fn(c)170 1088 y Fu(\).)22 b(Necessarily)l(,)16 b Fo(v)r Fu(\()p Fo(\021)563 1095 y Fn(r)585 1088 y Fu(\))h(is)f(strictly)h(greater)f(than)g Fo(n)1157 1095 y Fn(c)1177 1088 y Fu(.)0 1148 y(In)h(addition,)f(w)o(e)h(ha)o(v)o(e)f(that)h Fo(E)s Fu(\()p Fo(j)651 1155 y Fn(c)683 1148 y Fp(\002)11 b Fu(\()p Fo(j)773 1155 y Fn(c)804 1148 y Fu(+)h Fo(\016)877 1155 y Fn(c)897 1148 y Fu(\))g Fp(\002)f Fu(\()p Fo(j)1018 1155 y Fn(c)1050 1148 y Fu(+)g(2\))p Fo(;)d(k)r Fu(\))15 b Fo(>)f(E)s Fu(\()p Fo(j)1361 1155 y Fn(c)1392 1148 y Fp(\002)e Fu(\()p Fo(j)1483 1155 y Fn(c)1514 1148 y Fu(+)f Fo(\016)1586 1155 y Fn(c)1607 1148 y Fu(\))h Fp(\002)f Fu(\()p Fo(j)1728 1155 y Fn(c)1759 1148 y Fu(+)h(2\)\))17 b(for)0 1208 y(all)f Fo(k)i Fu(suc)o(h)e(that)269 1330 y Fo(v)p 269 1338 26 2 v 1 w Fu(\()p Fo(\031)r Fu(\))c Fp(\000)f Fo(j)445 1337 y Fn(c)465 1330 y Fu(\()p Fo(j)505 1337 y Fn(c)536 1330 y Fu(+)g Fo(\016)608 1337 y Fn(c)629 1330 y Fu(\)\()p Fo(j)688 1337 y Fn(c)719 1330 y Fu(+)g(2\))j Fp(\024)g Fo(k)h Fp(\024)p 974 1302 V 14 w Fo(v)r Fu(\()p Fo(\031)r Fu(\))d Fp(\000)e Fo(j)1150 1337 y Fn(c)1170 1330 y Fu(\()p Fo(j)1210 1337 y Fn(c)1241 1330 y Fu(+)h Fo(\016)1313 1337 y Fn(c)1334 1330 y Fu(\)\()p Fo(j)1393 1337 y Fn(c)1424 1330 y Fu(+)g(2\))p Fo(;)17 b(k)e Fp(6)p Fu(=)f(0)p Fo(:)0 1452 y Fu(W)l(e)k(ha)o(v)o(e)f(th)o(us)g(pro)o(v)o(ed)f(that)i Fo(j)607 1459 y Fn(c)638 1452 y Fp(\002)12 b Fu(\()p Fo(j)729 1459 y Fn(c)761 1452 y Fu(+)g Fo(\016)834 1459 y Fn(c)854 1452 y Fu(\))g Fp(\002)g Fu(\()p Fo(j)976 1459 y Fn(c)1008 1452 y Fu(+)f(2\))18 b(and)f(the)h(cycle)h Fo(\031)h Fu(satisfy)d(the)h(h)o(yp)q (othesis)0 1512 y(of)f(theorem)g(5)p Fo(:)p Fu(5.)24 b(It)17 b(follo)o(ws)f(that)i(the)f(b)q(ottom)h(of)f(the)h(cycle)f(is)g Fp(f)p Fo(j)1308 1519 y Fn(c)1339 1512 y Fp(\002)12 b Fu(\()p Fo(j)1430 1519 y Fn(c)1461 1512 y Fu(+)g Fo(\016)1534 1519 y Fn(c)1554 1512 y Fu(\))g Fp(\002)f Fu(\()p Fo(j)1675 1519 y Fn(c)1707 1512 y Fu(+)g(2\))p Fp(g)p Fu(.)24 b(The)0 1571 y(inequalities)12 b(on)p 323 1544 V 12 w Fo(v)i Fu(and)e Fo(v)p 454 1579 V 14 w Fu(ha)o(v)o(e)g(also)g(b)q(een)g(pro)o(v)o(ed.)20 b(Moreo)o(v)o(er)10 b(the)j(set)g Fp(f)8 b Fo(\030)16 b Fu(:)d Fo(\030)j Fp(\032)e Fo(j)1573 1578 y Fn(c)1596 1571 y Fp(\002)s Fu(\()p Fo(j)1678 1578 y Fn(c)1700 1571 y Fu(+)s Fo(\016)1764 1578 y Fn(c)1784 1571 y Fu(\))s Fp(\002)s Fu(\()p Fo(j)1888 1578 y Fn(c)1911 1571 y Fu(+)0 1639 y(2\))p Fo(;)j(\030)f Fp(2)172 1626 y Fk(f)160 1639 y Fp(M)220 1646 y Fn(n)245 1651 y Fg(c)263 1646 y Fl(+1)324 1639 y Fp(g)f Fu(is)g(included)f(in)g(the)i(cycle:)21 b(eac)o(h)15 b(con\014guration)f(of)h(this)g(set)g(comm)o(unicates)e(with)0 1698 y Fo(j)21 1705 y Fn(c)51 1698 y Fp(\002)e Fu(\()p Fo(j)141 1705 y Fn(c)172 1698 y Fu(+)f Fo(\016)243 1705 y Fn(c)264 1698 y Fu(\))h Fp(\002)f Fu(\()p Fo(j)383 1705 y Fn(c)414 1698 y Fu(+)g(2\))17 b(under)e(the)i(lev)o(el)f(pred)7 b Fo(E)s Fu(\()p Fo(n)1058 1705 y Fn(c)1079 1698 y Fu(\))16 b(\(this)h(is)f(a)g(consequence)g (of)g(theorem)g(7)p Fo(:)p Fu(8\).)0 1766 y(Th)o(us)c Fp(f)c Fo(\030)16 b Fu(:)e Fo(\030)i Fp(\032)e Fo(j)334 1773 y Fn(c)359 1766 y Fp(\002)5 b Fu(\()p Fo(j)443 1773 y Fn(c)468 1766 y Fu(+)g Fo(\016)534 1773 y Fn(c)554 1766 y Fu(\))g Fp(\002)g Fu(\()p Fo(j)662 1773 y Fn(c)688 1766 y Fu(+)g(2\))p Fo(;)17 b(\030)f Fp(2)904 1753 y Fk(f)892 1766 y Fp(M)952 1773 y Fn(n)977 1778 y Fg(c)1006 1766 y Fp(g)d Fu(is)h(included)e(in)h(the)h(principal)e(b)q (oundary)g(of)i Fo(\031)r Fu(.)0 1826 y(Finally)g(a)h(con\014guration)f Fo(\030)j Fu(of)f(the)f(principal)e(b)q(oundary)h(of)i(this)f(cycle)g(is)g (of)h(energy)f Fo(E)s Fu(\()p Fo(\030)r Fu(\))f(=)f Fo(E)s Fu(\()p Fo(n)1910 1833 y Fn(c)1931 1826 y Fu(\))0 1885 y(so)g(that)h(its)f(v) o(olume)f(is)h(equal)g(to)h Fo(n)660 1892 y Fn(c)680 1885 y Fu(,)g(and)e(its)i(area)e(to)i Fo(a)p Fu(\()p Fo(n)1105 1892 y Fn(c)1126 1885 y Fu(\).)21 b(Let)14 b Fo(\021)h Fu(b)q(e)f(a)f (con\014guration)e(of)j(the)f(cycle)0 1945 y(suc)o(h)j(that)i Fo(q)r Fu(\()p Fo(\021)r(;)8 b(\030)r Fu(\))15 b Fo(>)g Fu(0.)24 b(Necessarily)l(,)17 b(the)g(v)o(olume)f(of)i Fo(\021)h Fu(is)e Fo(n)1199 1952 y Fn(c)1230 1945 y Fu(+)12 b(1.)24 b(Th)o(us)16 b Fo(\021)i Fu(is)f(a)g(con\014guration)f(of)0 2005 y(minimal)f(v)o(olume)g (of)i(the)g(cycle)g(and)f(as)g(suc)o(h)g(it)g(is)h(a)f(minimal)f (con\014guration)g(of)i(the)f(cycle)i(for)e(the)0 2065 y(inclusion)11 b(relation.)20 b(By)13 b(theorem)f(5)p Fo(:)p Fu(3)h(it)g(is)f(included)g(in) g(the)h(parallelepip)q(ed)e Fo(j)1507 2072 y Fn(c)1530 2065 y Fp(\002)s Fu(\()p Fo(j)1612 2072 y Fn(c)1636 2065 y Fu(+)s Fo(\016)1700 2072 y Fn(c)1721 2065 y Fu(\))s Fp(\002)s Fu(\()p Fo(j)1825 2072 y Fn(c)1850 2065 y Fu(+)s(2\).)0 2125 y(Th)o(us)k Fo(\030)k Fu(b)q(elongs)d(to)h Fp(M)p Fu(\()p Fo(j)507 2132 y Fn(c)538 2125 y Fp(\002)10 b Fu(\()p Fo(j)627 2132 y Fn(c)659 2125 y Fu(+)g Fo(\016)730 2132 y Fn(c)751 2125 y Fu(\))h Fp(\002)g Fu(\()p Fo(j)871 2132 y Fn(c)902 2125 y Fu(+)g(2\))p Fo(;)d(n)1048 2132 y Fn(c)1080 2125 y Fp(\000)j Fo(j)1151 2132 y Fn(c)1171 2125 y Fu(\()p Fo(j)1211 2132 y Fn(c)1242 2125 y Fu(+)g Fo(\016)1314 2132 y Fn(c)1334 2125 y Fu(\)\()p Fo(j)1393 2132 y Fn(c)1425 2125 y Fu(+)f(2\)\).)51 b Fc(\003)0 2224 y Fw(Corollary)18 b(7.25.)28 b Fh(Let)15 b Fo(\033)j Fh(b)q(elong)c(to)762 2212 y Fk(f)750 2224 y Fp(M)810 2231 y Fn(n)835 2236 y Fg(c)853 2231 y Fl(+1)914 2224 y Fp(n)8 b(S)977 2231 y Fn(n)1002 2236 y Fg(c)1021 2231 y Fl(+1)1074 2224 y Fh(.)21 b(Supp)q(ose)14 b Fo(j)1324 2231 y Fn(c)1352 2224 y Fp(\002)8 b Fu(\()p Fo(j)1439 2231 y Fn(c)1467 2224 y Fu(+)g Fo(\016)1536 2231 y Fn(c)1557 2224 y Fu(\))g Fp(\002)g Fu(\()p Fo(j)1671 2231 y Fn(c)1699 2224 y Fu(+)g(1\))g(+)1837 2231 y Fl(3)1868 2224 y Fo(l)1883 2231 y Fn(c)1911 2224 y Fp(\002)0 2284 y Fu(\()p Fo(l)34 2291 y Fn(c)62 2284 y Fu(+)g(1\))14 b Fp(\032)f Fo(\033)r Fh(.)22 b(Let)16 b Fp(G)406 2266 y Fl(+)438 2284 y Fu(\()p Fo(\033)r Fu(\))g Fh(b)q(e)g(the)f(minimal)e(stable)h(subgraph)f(of)i Fp(G)1310 2266 y Fl(+)1358 2284 y Fh(con)o(taining)e Fo(\031)r Fu(\()p Fo(\033)o(;)8 b Fp(f\000)p Fu(1)p 1721 2296 64 2 v 1 w Fo(;)g Fu(+1)p 1808 2296 V Fp(g)1897 2266 y Fn(c)1917 2284 y Fu(\))p Fh(.)0 2344 y(The)16 b(only)h(arro)o(ws)d(of)j Fp(G)461 2326 y Fl(+)494 2344 y Fu(\()p Fo(\033)r Fu(\))g Fh(en)o(tering)f Fo(\031)r Fu(\()p Fp(\000)p Fu(1)p 821 2356 V Fo(;)8 b Fu(+1)p 907 2356 V -20 x Fn(c)991 2344 y Fu(\))17 b Fh(are)287 2466 y Fp(f)p Fo(\030)r Fp(g)d(!)g Fo(\031)r Fu(\()p Fo(j)509 2473 y Fn(c)540 2466 y Fp(\002)d Fu(\()p Fo(j)630 2473 y Fn(c)661 2466 y Fu(+)g Fo(\016)733 2473 y Fn(c)753 2466 y Fu(\))h Fp(\002)f Fu(\()p Fo(j)874 2473 y Fn(c)905 2466 y Fu(+)g(1\))g(+)1049 2473 y Fl(3)1082 2466 y Fo(l)1097 2473 y Fn(c)1128 2466 y Fp(\002)g Fu(\()p Fo(l)1212 2473 y Fn(c)1244 2466 y Fu(+)g(1\))p Fo(;)d Fp(f\000)p Fu(1)p 1385 2478 V Fo(;)g Fu(+1)p 1471 2478 V -1 w Fp(g)1559 2445 y Fn(c)1580 2466 y Fu(\))p Fo(;)305 2550 y(\030)16 b Fp(2)402 2537 y Fk(f)390 2550 y Fp(M)450 2557 y Fn(n)475 2562 y Fg(c)507 2550 y Fp(n)11 b(S)573 2557 y Fn(n)598 2562 y Fg(c)619 2550 y Fo(;)58 b(j)712 2557 y Fn(c)743 2550 y Fp(\002)10 b Fu(\()p Fo(j)832 2557 y Fn(c)864 2550 y Fu(+)g Fo(\016)935 2557 y Fn(c)956 2550 y Fu(\))h Fp(\002)g Fu(\()p Fo(j)1076 2557 y Fn(c)1107 2550 y Fu(+)g(1\))h(+)1252 2557 y Fl(3)1285 2550 y Fo(l)1300 2557 y Fn(c)1331 2550 y Fp(\002)f Fu(\()p Fo(l)1415 2557 y Fn(c)1446 2550 y Fu(+)g(1\))j Fp(\032)g Fo(\030)r(:)954 2670 y Fs(43)p eop %%Page: 44 44 44 43 bop 0 200 a Fh(The)16 b(remaining)f(arro)o(ws)g(of)h Fp(G)584 182 y Fl(+)617 200 y Fu(\()p Fo(\033)r Fu(\))i Fh(are)434 315 y Fo(\031)r Fu(\()p Fo(\033)o(;)8 b Fp(f\000)p Fu(1)p 557 327 64 2 v 1 w Fo(;)g Fu(+1)p 644 327 V -1 w Fp(g)732 295 y Fn(c)753 315 y Fu(\))14 b Fp($)f(f)p Fo(\021)r Fp(g)p Fo(;)58 b(\021)15 b Fp(2)1096 303 y Fk(e)1085 315 y Fo(B)r Fu(\()p Fo(\031)r Fu(\()p Fo(\033)o(;)8 b Fp(f\000)p Fu(1)p 1267 327 V 2 w Fo(;)g Fu(+1)p 1355 327 V -1 w Fp(g)1443 295 y Fn(c)1464 315 y Fu(\)\))p Fo(:)0 425 y Fh(There)16 b(is)g(no)g(arro)o(w)f(in)h Fp(G)494 407 y Fl(+)527 425 y Fu(\()p Fo(\033)r Fu(\))i Fh(ending)d(at)i Fp(f)p Fu(+1)p 859 437 V Fp(g)p Fh(.)0 512 y Fe(R)m(emark.)27 b Fu(W)l(e)16 b(ha)o(v)o(e)g(pro)o(v)o(ed)f(in)h(theorem)g(7)p Fo(:)p Fu(24)g(that)171 612 y Fk(e)161 625 y Fo(B)r Fu(\()p Fo(\031)r Fu(\()p Fo(\033)o(;)8 b Fp(f\000)p Fu(1)p 343 637 V 1 w Fo(;)g Fu(+1)p 430 637 V Fp(g)519 604 y Fn(c)539 625 y Fu(\)\))14 b Fp(\032)g(M)p Fu(\()p Fo(j)744 632 y Fn(c)775 625 y Fp(\002)d Fu(\()p Fo(j)865 632 y Fn(c)896 625 y Fu(+)g Fo(\016)968 632 y Fn(c)988 625 y Fu(\))h Fp(\002)f Fu(\()p Fo(j)1109 632 y Fn(c)1140 625 y Fu(+)g(2\))p Fo(;)d(n)1286 632 y Fn(c)1317 625 y Fp(\000)j Fo(j)1388 632 y Fn(c)1408 625 y Fu(\()p Fo(j)1448 632 y Fn(c)1479 625 y Fu(+)g Fo(\016)1551 632 y Fn(c)1572 625 y Fu(\)\()p Fo(j)1631 632 y Fn(c)1662 625 y Fu(+)g(2\)\))p Fo(:)0 734 y Fe(Pr)m(o)m(of.)28 b Fu(That)12 b(these)g(arro)o(ws)e(b)q(elong)i(to)h Fp(G)800 716 y Fl(+)832 734 y Fu(\()p Fo(\033)r Fu(\))h(is)d(a)h(straigh)o(tforw)o(ard)e(consequence) i(of)g(theorem)f(7)p Fo(:)p Fu(24)0 794 y(whic)o(h)k(implies)f(in)i (particular)e(that)i Fo(\031)r Fu(\()p Fo(\033)o(;)8 b Fp(f\000)p Fu(1)p 827 806 V 1 w Fo(;)g Fu(+1)p 914 806 V Fp(g)1003 776 y Fn(c)1023 794 y Fu(\))23 b(=)e Fo(\031)r Fu(\()p Fo(j)1195 801 y Fn(c)1226 794 y Fp(\002)9 b Fu(\()p Fo(j)1314 801 y Fn(c)1344 794 y Fu(+)g Fo(\016)1414 801 y Fn(c)1435 794 y Fu(\))h Fp(\002)f Fu(\()p Fo(j)1552 801 y Fn(c)1582 794 y Fu(+)h(2\))p Fo(;)e Fp(f\000)p Fu(1)p 1722 806 V Fo(;)g Fu(+1)p 1808 806 V Fp(g)1897 776 y Fn(c)1917 794 y Fu(\).)0 854 y(W)l(e)20 b(ha)o(v)o(e)f(to)h(c)o(hec)o (k)f(that)h(there)g(is)f(no)g(other)h(arro)o(w.)30 b(Let)20 b Fo(\021)h Fu(b)q(elong)f(to)g(the)f(principal)f(b)q(oundary)0 918 y(of)f Fo(\031)r Fu(\()p Fo(j)127 925 y Fn(c)158 918 y Fp(\002)11 b Fu(\()p Fo(j)248 925 y Fn(c)279 918 y Fu(+)g Fo(\016)351 925 y Fn(c)371 918 y Fu(\))h Fp(\002)e Fu(\()p Fo(j)491 925 y Fn(c)523 918 y Fu(+)g(2\))p Fo(;)e Fu(pred)g Fo(E)s Fu(\()p Fo(n)833 925 y Fn(c)853 918 y Fu(\)\).)23 b(If)17 b Fo(\021)h Fu(is)e(in)1141 906 y Fk(f)1129 918 y Fp(M)1189 925 y Fn(n)1214 930 y Fg(c)1235 918 y Fu(,)g(then)h(all)f(arro)o(ws)e(of)j Fo(G)g Fu(starting)f(at)0 986 y(the)f(cycle)h Fp(f)p Fo(\021)r Fp(g)f Fu(are)g(presen)o(t)f(in)h(the)g(ab)q(o)o(v)o(e)g(list)g(\(lemma)g(7)p Fo(:)p Fu(21\).)21 b(If)15 b Fo(\021)i Fu(is)e(not)g(in)1532 973 y Fk(f)1520 986 y Fp(M)1580 993 y Fn(n)1605 998 y Fg(c)1626 986 y Fu(,)g(w)o(e)g(claim)g(that)480 1095 y Fp(f)p Fo(\021)r Fp(g)e(!)h Fo(\031)r Fu(\()p Fo(j)704 1102 y Fn(c)735 1095 y Fp(\002)d Fu(\()p Fo(j)825 1102 y Fn(c)857 1095 y Fu(+)f Fo(\016)928 1102 y Fn(c)949 1095 y Fu(\))h Fp(\002)g Fu(\()p Fo(j)1069 1102 y Fn(c)1100 1095 y Fu(+)g(2\))p Fo(;)d Fu(pred)g Fo(E)s Fu(\()p Fo(n)1411 1102 y Fn(c)1431 1095 y Fu(\)\))0 1204 y(is)14 b(the)h(unique)e(arro)o(w)g(of)i Fp(G)i Fu(starting)c(at)i Fp(f)p Fo(\021)r Fp(g)p Fu(.)21 b(Let)15 b Fo(\030)h Fu(b)q(e)f(a)f(p)q(oin)o (t)g(suc)o(h)f(that)i Fo(q)r Fu(\()p Fo(\021)r(;)8 b(\030)r Fu(\))14 b Fo(>)g Fu(0)g(and)g Fo(E)s Fu(\()p Fo(\030)r Fu(\))g Fp(\024)0 1264 y Fo(E)s Fu(\()p Fo(\021)r Fu(\).)21 b(Since)11 b Fo(v)r Fu(\()p Fo(\021)r Fu(\))j(=)g Fo(n)453 1271 y Fn(c)473 1264 y Fu(,)f(then)g Fo(v)r Fu(\()p Fo(\030)r Fu(\))g(is)f(equal)g(to)h Fo(n)971 1271 y Fn(c)994 1264 y Fp(\000)s Fu(1)f(or)g Fo(n)1160 1271 y Fn(c)1183 1264 y Fu(+)s(1.)20 b(Moreo)o(v)o(er)11 b Fo(\021)j Fu(is)e(minimal,)f(and)h(the)0 1324 y(inequalit)o(y)17 b Fo(E)s Fu(\()p Fo(\030)r Fu(\))e Fp(\024)f Fo(E)s Fu(\()p Fo(\021)r Fu(\))j(implies)e(in)i(b)q(oth)g(cases)f(that)i Fo(\030)h Fu(is)d(also)h(minimal.)k(Th)o(us)16 b Fo(\030)j Fu(cannot)e(b)q(e)g(of)0 1384 y(v)o(olume)d Fo(n)200 1391 y Fn(c)227 1384 y Fp(\000)7 b Fu(1)15 b(\(b)o(y)f(lemma)g(7)p Fo(:)p Fu(6,)h(the)f(only)h (con\014gurations)d(of)j Fp(M)1276 1391 y Fn(n)1301 1396 y Fg(c)1336 1384 y Fu(comm)o(unicating)e(with)h Fp(M)1853 1391 y Fn(n)1878 1396 y Fg(c)1897 1391 y Fj(\000)p Fl(1)0 1443 y Fu(are)i(the)h(principal)e(con\014gurations)711 1431 y Fk(f)699 1443 y Fp(M)759 1450 y Fn(n)784 1455 y Fg(c)805 1443 y Fu(\).)23 b(Th)o(us)15 b Fo(v)r Fu(\()p Fo(\030)r Fu(\))g(=)f Fo(n)1173 1450 y Fn(c)1205 1443 y Fu(+)d(1.)23 b(W)l(e)17 b(next)g(sho)o(w)f(that)h Fo(\030)i Fu(b)q(elongs)0 1503 y(to)g Fo(\031)r Fu(\()p Fo(j)133 1510 y Fn(c)166 1503 y Fp(\002)12 b Fu(\()p Fo(j)257 1510 y Fn(c)290 1503 y Fu(+)g Fo(\016)363 1510 y Fn(c)383 1503 y Fu(\))h Fp(\002)f Fu(\()p Fo(j)506 1510 y Fn(c)539 1503 y Fu(+)g(2\))p Fo(;)c Fu(pred)g Fo(E)s Fu(\()p Fo(n)851 1510 y Fn(c)871 1503 y Fu(\)\).)29 b(Let)19 b Fo(x)1071 1510 y Fl(1)1094 1503 y Fo(;)8 b Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)1241 1510 y Fn(r)1282 1503 y Fu(b)q(e)19 b(a)g(sequence)f(of)h(sites)f(inside)g Fo(\033)0 1563 y Fu(suc)o(h)13 b(that)i Fo(F)7 b Fu(\()p Fo(x)302 1570 y Fl(1)325 1563 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)472 1570 y Fn(s)494 1563 y Fu(\)\()p Fo(j)553 1570 y Fn(c)580 1563 y Fp(\002)f Fu(\()p Fo(j)666 1570 y Fn(c)693 1563 y Fu(+)g Fo(\016)761 1570 y Fn(c)781 1563 y Fu(\))g Fp(\002)g Fu(\()p Fo(j)893 1570 y Fn(c)920 1563 y Fu(+)g(2\)\))15 b(is)f(in)g Fo(\031)r Fu(\()p Fo(j)1218 1570 y Fn(c)1245 1563 y Fp(\002)7 b Fu(\()p Fo(j)1331 1570 y Fn(c)1358 1563 y Fu(+)g Fo(\016)1426 1570 y Fn(c)1446 1563 y Fu(\))g Fp(\002)g Fu(\()p Fo(j)1558 1570 y Fn(c)1585 1563 y Fu(+)g(2\))p Fo(;)h Fu(pred)f Fo(E)s Fu(\()p Fo(n)1891 1570 y Fn(c)1911 1563 y Fu(\)\))0 1623 y(for)23 b Fo(s)h Fu(in)f Fp(f)p Fu(1)8 b Fp(\001)g(\001)g(\001)h Fo(r)17 b Fp(\000)f Fu(1)p Fp(g)23 b Fu(and)g Fo(\021)k Fu(=)e Fo(F)7 b Fu(\()p Fo(x)793 1630 y Fl(1)816 1623 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)963 1630 y Fn(r)986 1623 y Fu(\)\()p Fo(j)1045 1630 y Fn(c)1081 1623 y Fp(\002)16 b Fu(\()p Fo(j)1176 1630 y Fn(c)1211 1623 y Fu(+)g Fo(\016)1288 1630 y Fn(c)1308 1623 y Fu(\))g Fp(\002)g Fu(\()p Fo(j)1438 1630 y Fn(c)1474 1623 y Fu(+)f(2\)\).)44 b(W)l(e)23 b(put)h Fo(\021)1865 1630 y Fn(s)1911 1623 y Fu(=)0 1682 y Fo(F)7 b Fu(\()p Fo(x)86 1689 y Fl(1)109 1682 y Fo(;)h Fp(\001)g(\001)g(\001)17 b Fo(;)8 b(x)256 1689 y Fn(j)278 1682 y Fu(\)\()p Fo(j)337 1689 y Fn(c)372 1682 y Fp(\002)14 b Fu(\()p Fo(j)465 1689 y Fn(c)499 1682 y Fu(+)g Fo(\016)574 1689 y Fn(c)595 1682 y Fu(\))g Fp(\002)h Fu(\()p Fo(j)722 1689 y Fn(c)756 1682 y Fu(+)f(2\)\),)23 b(0)f Fp(\024)g Fo(s)h Fp(\024)e Fo(r)q Fu(.)38 b(Since)21 b Fo(\021)i Fu(is)e(in)g(the)h(b)q (oundary)e(of)i(the)0 1742 y(cycle,)16 b(w)o(e)e(ha)o(v)o(e)h Fo(E)s Fu(\()p Fo(\021)405 1749 y Fn(r)q Fj(\000)p Fl(1)478 1742 y Fu(\))f Fo(<)f(E)s Fu(\()p Fo(\021)647 1749 y Fn(r)669 1742 y Fu(\).)22 b(Moreo)o(v)o(er)13 b Fo(\021)966 1749 y Fn(r)q Fj(\000)p Fl(1)1054 1742 y Fu(is)i(a)f(minimal)g(con\014guration)f(of)i(the)g (cycle,)h(of)0 1802 y(v)o(olume)c Fo(n)198 1809 y Fn(c)221 1802 y Fu(+)s(1,)h(so)g(that)g(the)g(last)f(spin)g(\015ip)g(at)h(site)f Fo(x)1005 1809 y Fn(r)1041 1802 y Fu(has)g(decreased)f(the)i(v)o(olume.)20 b(Let)13 b Fo(x)1731 1809 y Fn(r)q Fl(+1)1817 1802 y Fu(b)q(e)g(the)0 1862 y(unique)18 b(site)h(suc)o(h)e(that)i Fo(\030)h Fu(=)d Fo(F)7 b Fu(\()p Fo(x)666 1869 y Fn(r)q Fl(+1)739 1862 y Fu(\)\()p Fo(\021)r Fu(\).)29 b(W)l(e)19 b(ha)o(v)o(e)f Fo(\030)h Fu(=)e Fo(F)7 b Fu(\()p Fo(x)1254 1869 y Fn(r)q Fl(+1)1328 1862 y Fu(\)\()p Fo(\021)1391 1869 y Fn(r)1413 1862 y Fu(\))18 b(=)f Fo(F)7 b Fu(\()p Fo(x)1592 1869 y Fn(r)q Fl(+1)1665 1862 y Fo(;)h(x)1715 1869 y Fn(r)1738 1862 y Fu(\)\()p Fo(\021)1801 1869 y Fn(r)q Fj(\000)p Fl(1)1875 1862 y Fu(\))17 b(=)0 1922 y Fo(F)7 b Fu(\()p Fo(x)86 1929 y Fn(r)109 1922 y Fo(;)h(x)159 1929 y Fn(r)q Fl(+1)232 1922 y Fu(\)\()p Fo(\021)295 1929 y Fn(r)q Fj(\000)p Fl(1)368 1922 y Fu(\))26 b(=)f Fo(F)7 b Fu(\()p Fo(x)563 1929 y Fn(r)586 1922 y Fu(\)\()p Fo(\021)651 1903 y Fj(0)649 1934 y Fn(r)672 1922 y Fu(\))23 b(where)g Fo(\021)892 1903 y Fj(0)890 1934 y Fn(r)937 1922 y Fu(=)i Fo(F)7 b Fu(\()p Fo(x)1087 1929 y Fn(r)q Fl(+1)1161 1922 y Fu(\)\()p Fo(\021)1224 1929 y Fn(r)q Fj(\000)p Fl(1)1297 1922 y Fu(\).)43 b(The)23 b(energy)g(inequalit)o(y)g(5)p Fo(:)p Fu(1)0 1981 y(yields)13 b Fo(E)s Fu(\()p Fo(\021)223 1963 y Fj(0)221 1994 y Fn(r)243 1981 y Fu(\))6 b Fp(\000)g Fo(E)s Fu(\()p Fo(\021)397 1988 y Fn(r)q Fj(\000)p Fl(1)469 1981 y Fu(\))14 b Fp(\024)g Fo(E)s Fu(\()p Fo(\030)r Fu(\))6 b Fp(\000)g Fo(E)s Fu(\()p Fo(\021)r Fu(\))13 b(whence)h Fo(E)s Fu(\()p Fo(\021)1084 1963 y Fj(0)1082 1994 y Fn(r)1103 1981 y Fu(\))h Fp(\024)e Fo(E)s Fu(\()p Fo(\021)1273 1988 y Fn(r)q Fj(\000)p Fl(1)1346 1981 y Fu(\).)21 b(It)15 b(follo)o(ws)d(that)i Fo(\021)1744 1963 y Fj(0)1742 1994 y Fn(r)1778 1981 y Fu(is)f(in)h(the)0 2041 y(cycle,)i(as)f(w)o(ell)g(as)g Fo(\030)i Fu(\(their)e(energies)f(are)h(less)g(or)g(equal)g(than)g Fo(E)s Fu(\()p Fo(n)1286 2048 y Fn(c)1306 2041 y Fu(\))h(and)e(they)i(b)q (oth)g(comm)o(unicate)0 2101 y(with)g(a)h(con\014guration)e(of)h(the)h (cycle\).)51 b Fc(\003)0 2188 y Fw(Corollary)26 b(7.26.)h Fh(The)21 b(principal)f(non)h(standard)f(con\014gurations)f(of)j(v)o(olume)e Fo(n)1636 2195 y Fn(c)1678 2188 y Fh(\(i.e.)37 b(the)22 b(set)12 2243 y Fk(f)0 2256 y Fp(M)60 2263 y Fn(n)85 2268 y Fg(c)112 2256 y Fp(n)7 b(S)174 2263 y Fn(n)199 2268 y Fg(c)220 2256 y Fh(\))15 b(are)f(dead{ends:)19 b(there)c(is)f(no)g(saddle)f(path)h(of)h(n)o (ull)e(cost)h(b)q(et)o(w)o(een)h Fp(\000)p Fu(1)p 1541 2268 V 14 w Fh(and)f Fu(+1)p 1714 2268 V 14 w Fh(passing)0 2315 y(through)h(them.)0 2403 y Fw(Corollary)21 b(7.27.)28 b Fh(The)17 b(set)h(of)g(the)g(global)f(saddle)f(p)q(oin)o(ts)h(b)q(et)o(w)o(een)h Fp(\000)p Fu(1)p 1376 2415 V 17 w Fh(and)f Fu(+1)p 1555 2415 V 17 w Fh(is)h(exactly)g Fp(S)1890 2410 y Fn(n)1915 2415 y Fg(c)1936 2403 y Fh(.)0 2463 y(These)e(con\014gurations)f(are)h(the)h (critical)f(three)g(dimensional)e(con\014gurations.)60 2550 y Fu(Steps)i Fo(ii)p Fu(\))h(and)f Fo(iii)p Fu(\))i(are)e(no)o(w)f(completed) h(and)g(w)o(e)h(pro)q(ceed)f(to)h(steps)f Fo(iv)r Fu(\))h(and)f Fo(v)r Fu(\).)954 2670 y Fs(44)p eop %%Page: 45 45 45 44 bop 0 200 a Fw(The)17 b(ascending)h(part.)28 b Fu(W)l(e)15 b(no)o(w)f(pro)q(ceed)g(to)i(the)f(last)g(part)f(of)h(the)g(program:)k(for)c (eac)o(h)f(con\014gu-)0 260 y(ration)h Fo(\033)176 242 y Fj(0)206 260 y Fu(in)g Fp(S)293 267 y Fn(n)318 272 y Fg(c)339 260 y Fu(,)g(w)o(e)h(m)o(ust)e(determine)h(the)h(minimal)e(stable)h(subgraph)e(of)j Fp(G)1520 242 y Fj(\000)1569 260 y Fu(con)o(taining)e Fo(\033)1840 242 y Fj(0)1870 260 y Fu(and)0 319 y(all)19 b(the)g(paths)g(in)g(this)g (graph)f(starting)g(at)i Fp(f)p Fo(\033)911 301 y Fj(0)925 319 y Fp(g)f Fu(and)g(ending)f(at)i Fp(\000)p Fu(1)p 1296 331 64 2 v(.)30 b(Our)18 b(exp)q(osition)h(is)g(similar)0 379 y(as)f(b)q(efore:) 24 b(w)o(e)18 b(\014rst)f(list)h(the)g(set)h(of)f(the)g(relev)m(an)o(t)g (cycles)g(and)f(w)o(e)h(use)f(lemma)g(6)p Fo(:)p Fu(19)h(to)g(\014nd)f(those) 0 439 y(b)q(elonging)e(to)h Fp(M)p Fu(\()p Fp(f\000)p Fu(1)p 389 451 V -1 w Fo(;)8 b Fu(+1)p 474 451 V Fp(g)563 421 y Fn(c)583 439 y Fu(\).)23 b(W)l(e)15 b(\014nally)g(c)o(hec)o(k)h(that)g(w)o(e)f(ha)o(v) o(e)g(in)h(hand)e(all)i(the)g(v)o(ertices)f(of)h Fp(G)1903 421 y Fj(\000)1936 439 y Fu(.)0 521 y Fw(Theorem)h(7.28.)50 b Fh(Let)17 b Fo(j)i Fh(b)q(e)e(an)f(in)o(teger)g(less)g(or)g(equal)g(than)g Fo(j)1264 528 y Fn(c)1301 521 y Fh(and)g(greater)g(than)g Fo(l)1698 528 y Fn(c)1718 521 y Fh(.)0 581 y(The)g(cycle)h Fo(\031)f Fu(=)e Fo(\031)r Fu(\()p Fo(j)f Fp(\002)d Fu(\()p Fo(j)j Fu(+)d Fo(\016)r Fu(\))h Fp(\002)f Fu(\()p Fo(j)k Fu(+)c Fo(\022)q Fu(\))p Fo(;)e Fh(pred)g Fo(E)s Fu(\()p Fo(j)s Fu(\()p Fo(j)13 b Fu(+)d Fo(\016)r Fu(\)\()p Fo(j)k Fu(+)c Fo(\022)i Fp(\000)e Fu(1\))h(+)f Fo(l)1467 588 y Fn(c)1487 581 y Fu(\()p Fo(l)1521 588 y Fn(c)1552 581 y Fu(+)h(1\))f(+)g(1\))17 b Fh(do)q(es)f(not)0 640 y(con)o(tain)g Fp(\000)p Fu(1)p 176 652 V 16 w Fh(and)g Fu(+1)p 353 652 V -1 w Fh(.)22 b(Moreo)o(v)o(er)43 725 y Fo(v)p 43 733 26 2 v 2 w Fu(\()p Fo(\031)r Fu(\))h(=)f Fo(j)s Fu(\()p Fo(j)13 b Fu(+)e Fo(\016)r Fu(\)\()p Fo(j)j Fu(+)d Fo(\022)i Fp(\000)e Fu(1\))g(+)g Fo(l)701 732 y Fn(c)721 725 y Fu(\()p Fo(l)755 732 y Fn(c)787 725 y Fu(+)g(1\))g(+)g(2)d Fo(;)p 1047 697 V 58 w(v)r Fu(\()p Fo(\031)r Fu(\))23 b Fo(<)f(j)s Fu(\()p Fo(j)13 b Fu(+)e Fo(\016)r Fu(\)\()p Fo(j)j Fu(+)d Fo(\022)q Fu(\))h(+)f Fo(l)1619 732 y Fn(c)1639 725 y Fu(\()p Fo(l)1673 732 y Fn(c)1705 725 y Fu(+)f(1\))i(+)f(1)d Fo(:)0 809 y Fh(The)16 b(b)q(ottom)g(of)g(this)f(cycle)i(is)e Fp(f)p Fo(j)d Fp(\002)d Fu(\()p Fo(j)k Fu(+)c Fo(\016)r Fu(\))h Fp(\002)g Fu(\()p Fo(j)i Fu(+)e Fo(\022)q Fu(\))p Fp(g)p Fh(;)17 b(its)e(principal)f(b)q(oundary)h (con)o(tains)g(the)h(set)323 894 y Fp(f)8 b Fo(\030)16 b Fu(:)d Fo(\030)j Fp(\032)e Fo(j)f Fp(\002)e Fu(\()p Fo(j)j Fu(+)d Fo(\016)r Fu(\))g Fp(\002)g Fu(\()p Fo(j)j Fu(+)d Fo(\022)q Fu(\))p Fo(;)17 b(\030)f Fp(2)e(S)1097 903 y Fn(j)r Fl(\()p Fn(j)r Fl(+)p Fn(\016)q Fl(\)\()p Fn(j)r Fl(+)p Fn(\022)q Fj(\000)p Fl(1\)+)p Fn(l)1411 908 y Fg(c)1428 903 y Fl(\()p Fn(l)1457 908 y Fg(c)1475 903 y Fl(+1\)+1)1603 894 y Fp(g)0 979 y Fh(and)i(is)g (included)f(in)h Fp(M)p Fu(\()p Fo(j)e Fp(\002)d Fu(\()p Fo(j)j Fu(+)c Fo(\016)r Fu(\))i Fp(\002)f Fu(\()p Fo(j)i Fu(+)e Fo(\022)q Fu(\))p Fo(;)d(l)959 986 y Fn(c)981 979 y Fu(\()p Fo(l)1015 986 y Fn(c)1046 979 y Fu(+)j(1\))g(+)g(1)g Fp(\000)g Fo(j)s Fu(\()p Fo(j)i Fu(+)e Fo(\016)r Fu(\)\))p Fh(.)0 1061 y Fw(Theorem)20 b(7.29.)57 b Fh(Let)20 b Fo(j)h Fh(b)q(e)f(an)e(in)o(teger)h(less)f(or)h (equal)g(than)f Fo(l)1291 1068 y Fn(c)1311 1061 y Fh(.)30 b(Let)20 b Fo(\016)g Fp(\024)d Fo(\022)k Fh(b)q(elong)e(to)g Fp(f)p Fu(0)p Fo(;)8 b Fu(1)p Fp(g)p Fh(.)0 1120 y(The)16 b(cycle)327 1180 y Fo(\031)24 b Fu(=)e Fo(\031)r Fu(\()p Fo(j)14 b Fp(\002)d Fu(\()p Fo(j)i Fu(+)e Fo(\016)r Fu(\))h Fp(\002)f Fu(\()p Fo(j)i Fu(+)e Fo(\022)q Fu(\))p Fo(;)d Fh(pred)h Fo(E)s Fu(\()p Fo(j)s Fu(\()p Fo(j)k Fu(+)e Fo(\016)r Fu(\)\()p Fo(j)j Fu(+)d Fo(\022)q Fu(\))h Fp(\000)f Fo(j)i Fu(+)e(1\))0 1259 y Fh(do)q(es)16 b(not)h(con)o(tain)f Fp(\000)p Fu(1)p 377 1271 64 2 v 16 w Fh(and)g Fu(+1)p 554 1271 V -1 w Fh(.)22 b(Moreo)o(v)o(er)343 1344 y Fo(v)p 343 1352 26 2 v 2 w Fu(\()p Fo(\031)r Fu(\))h(=)f Fo(j)s Fu(\()p Fo(j)13 b Fu(+)e Fo(\016)r Fu(\)\()p Fo(j)j Fu(+)d Fo(\022)q Fu(\))h Fp(\000)f Fo(j)i Fu(+)e(2)d Fo(;)p 1089 1316 V 58 w(v)r Fu(\()p Fo(\031)r Fu(\))23 b(=)f Fo(j)s Fu(\()p Fo(j)13 b Fu(+)e Fo(\016)r Fu(\)\()p Fo(j)j Fu(+)d Fo(\022)q Fu(\))d Fo(;)357 1418 y Fp(8)p Fo(k)14 b Fp(2)g(f)p Fu(1)8 b Fp(\001)g(\001)g(\001)g Fo(j)14 b Fp(\000)d Fu(2)p Fp(g)99 b(M)p Fu(\()p Fo(j)14 b Fp(\002)c Fu(\()p Fo(j)k Fu(+)d Fo(\016)r Fu(\))h Fp(\002)e Fu(\()p Fo(j)k Fu(+)d Fo(\022)q Fu(\))p Fo(;)d Fp(\000)p Fo(k)r Fu(\))23 b Fp(\032)f Fo(\031)10 b(:)0 1503 y Fh(The)16 b(b)q(ottom)h(of)g(this)f(cycle)h(is)f Fp(f)p Fo(j)e Fp(\002)c Fu(\()p Fo(j)k Fu(+)d Fo(\016)r Fu(\))g Fp(\002)g Fu(\()p Fo(j)j Fu(+)d Fo(\022)q Fu(\))p Fp(g)p Fh(;)17 b(its)g(principal)e(b)q(oundary)g(is)535 1581 y Fk(e)525 1594 y Fo(B)r Fu(\()p Fo(\031)r Fu(\))23 b(=)f Fp(M)p Fu(\()p Fo(j)13 b Fp(\002)e Fu(\()p Fo(j)j Fu(+)d Fo(\016)r Fu(\))g Fp(\002)g Fu(\()p Fo(j)j Fu(+)d Fo(\022)q Fu(\))p Fo(;)d Fp(\000)p Fo(j)14 b Fu(+)d(1\))0 1678 y Fh(and)16 b(th)o(us)f(con)o(tains)h Fp(f)8 b Fo(\030)16 b Fu(:)e Fo(\030)i Fp(\032)d Fo(j)h Fp(\002)c Fu(\()p Fo(j)k Fu(+)d Fo(\016)r Fu(\))g Fp(\002)g Fu(\()p Fo(j)j Fu(+)d Fo(\022)q Fu(\))p Fo(;)17 b(\030)f Fp(2)e(S)1176 1687 y Fn(j)r Fl(\()p Fn(j)r Fl(+)p Fn(\016)q Fl(\)\()p Fn(j)r Fl(+)p Fn(\022)q Fl(\))p Fj(\000)p Fn(j)r Fl(+1)1506 1678 y Fp(g)p Fh(.)0 1760 y Fw(Theorem)20 b(7.30.)56 b Fh(Let)20 b Fo(j)h Fh(b)q(e)e(an)f(in)o(teger)h(less)f(or)g(equal)h(than)f Fo(j)1294 1767 y Fn(c)1314 1760 y Fh(.)29 b(Let)19 b Fo(\016)h Fp(\024)d Fo(\022)k Fh(b)q(elong)d(to)h Fp(f)p Fu(0)p Fo(;)8 b Fu(1)p Fp(g)p Fh(.)0 1820 y(Let)17 b Fo(l)12 b Fp(\002)f Fu(\()p Fo(l)h Fu(+)f Fo(\017)p Fu(\))17 b Fh(b)q(e)g(a)g(quasisquare)e(strictly)h(included) g(in)g Fo(j)d Fp(\002)e Fu(\()p Fo(j)j Fu(+)d Fo(\016)r Fu(\))p Fh(.)22 b(The)16 b(cycle)76 1905 y Fo(\031)25 b Fu(=)d Fo(\031)r Fu(\()p Fo(j)14 b Fp(\002)c Fu(\()p Fo(j)k Fu(+)d Fo(\016)r Fu(\))g Fp(\002)g Fu(\()p Fo(j)j Fu(+)d Fo(\022)q Fu(\))h(+)729 1912 y Fl(3)762 1905 y Fo(l)g Fp(\002)f Fu(\()p Fo(l)h Fu(+)f Fo(\017)p Fu(\))p Fo(;)d Fh(pred)g Fo(E)s Fu(\()p Fo(j)s Fu(\()p Fo(j)13 b Fu(+)e Fo(\016)r Fu(\)\()p Fo(j)j Fu(+)d Fo(\022)q Fu(\))h(+)f Fo(l)q Fu(\()p Fo(l)h Fu(+)f Fo(\017)p Fu(\))h Fp(\000)f Fo(l)h Fu(+)e(1\))0 1989 y Fh(do)q(es)16 b(not)h(con)o(tain)f Fp(\000)p Fu(1)p 377 2001 64 2 v 16 w Fh(and)g Fu(+1)p 554 2001 V -1 w Fh(.)22 b(Moreo)o(v)o(er)134 2074 y Fo(v)p 134 2082 26 2 v 2 w Fu(\()p Fo(\031)r Fu(\))h(=)f Fo(j)s Fu(\()p Fo(j)13 b Fu(+)e Fo(\016)r Fu(\)\()p Fo(j)j Fu(+)d Fo(\022)q Fu(\))h(+)f Fo(l)q Fu(\()p Fo(l)h Fu(+)f Fo(\017)p Fu(\))h Fp(\000)e Fo(l)i Fu(+)f(2)d Fo(;)p 1085 2046 V 58 w(v)r Fu(\()p Fo(\031)r Fu(\))23 b(=)f Fo(j)s Fu(\()p Fo(j)13 b Fu(+)e Fo(\016)r Fu(\)\()p Fo(j)j Fu(+)d Fo(\022)q Fu(\))h(+)f Fo(l)q Fu(\()p Fo(l)h Fu(+)f Fo(\017)p Fu(\))d Fo(;)213 2149 y Fp(8)p Fo(k)14 b Fp(2)g(f)p Fu(1)8 b Fp(\001)g(\001)g(\001)g Fo(l)k Fp(\000)f Fu(2)p Fp(g)99 b(M)p Fu(\()p Fo(j)14 b Fp(\002)d Fu(\()p Fo(j)j Fu(+)d Fo(\016)r Fu(\))g Fp(\002)g Fu(\()p Fo(j)j Fu(+)c Fo(\022)q Fu(\))j(+)1249 2156 y Fl(3)1282 2149 y Fo(l)f Fp(\002)f Fu(\()p Fo(l)h Fu(+)f Fo(\017)p Fu(\))p Fo(;)d Fp(\000)p Fo(k)r Fu(\))22 b Fp(\032)g Fo(\031)10 b(:)0 2233 y Fh(The)16 b(b)q(ottom)h(of)f(this)g (cycle)h(is)f Fp(f)p Fo(j)d Fp(\002)e Fu(\()p Fo(j)i Fu(+)e Fo(\016)r Fu(\))g Fp(\002)f Fu(\()p Fo(j)k Fu(+)c Fo(\022)q Fu(\))i(+)1116 2240 y Fl(3)1149 2233 y Fo(l)g Fp(\002)e Fu(\()p Fo(l)i Fu(+)f Fo(\017)p Fu(\))p Fp(g)p Fh(;)17 b(its)f(principal)e(b)q (oundary)i(is)391 2311 y Fk(e)381 2324 y Fo(B)r Fu(\()p Fo(\031)r Fu(\))23 b(=)f Fp(M)p Fu(\()p Fo(j)13 b Fp(\002)e Fu(\()p Fo(j)j Fu(+)d Fo(\016)r Fu(\))g Fp(\002)g Fu(\()p Fo(j)j Fu(+)d Fo(\022)q Fu(\))h(+)1142 2331 y Fl(3)1175 2324 y Fo(l)g Fp(\002)f Fu(\()p Fo(l)h Fu(+)f Fo(\017)p Fu(\))p Fo(;)d Fp(\000)p Fo(l)k Fu(+)f(1\))0 2408 y Fh(and)16 b(th)o(us)f(con)o(tains)h Fp(f)8 b Fo(\030)16 b Fu(:)e Fo(\030)i Fp(\032)d Fo(j)h Fp(\002)c Fu(\()p Fo(j)k Fu(+)d Fo(\016)r Fu(\))g Fp(\002)g Fu(\()p Fo(j)j Fu(+)d Fo(\022)q Fu(\))p Fo(;)17 b(\030)f Fp(2)e(S)1176 2417 y Fn(j)r Fl(\()p Fn(j)r Fl(+)p Fn(\016)q Fl(\)\()p Fn(j)r Fl(+)p Fn(\022)q Fl(\)+)p Fn(l)p Fl(\()p Fn(l)p Fl(+)p Fn(\017)p Fl(\))p Fj(\000)p Fn(l)p Fl(+1)1635 2408 y Fp(g)p Fh(.)0 2490 y Fe(R)m(emark.)27 b Fu(This)17 b(theorem)g(co)o(v)o(er)h(the)g(cases)g Fo(j)884 2497 y Fn(c)920 2490 y Fp(\024)e Fo(l)990 2497 y Fn(c)1028 2490 y Fu(and)i Fo(j)1148 2497 y Fn(c)1184 2490 y Fo(>)e(l)1254 2497 y Fn(c)1274 2490 y Fu(,)j(whic)o(h)e(are)h(of)g(sligh)o(tly)f(di\013eren)o(t)0 2550 y(natures.)954 2670 y Fs(45)p eop %%Page: 46 46 46 45 bop 0 200 a Fw(Corollary)20 b(7.31.)49 b Fh(The)17 b(follo)o(wing)e (cycles)i(are)f(maximal)f(cycles)i(of)g Fp(f\000)p Fu(1)p 1390 212 64 2 v Fo(;)8 b Fu(+1)p 1476 212 V -1 w Fp(g)1564 182 y Fn(c)1601 200 y Fh(\(where)17 b Fo(j)f Fp(\024)d Fo(j)1875 207 y Fn(c)1895 200 y Fh(\):)52 287 y Fo(\031)r Fu(\()p Fo(j)h Fp(\002)d Fu(\()p Fo(j)j Fu(+)d Fo(\016)r Fu(\))g Fp(\002)g Fu(\()p Fo(j)j Fu(+)c Fo(\022)q Fu(\))p Fo(;)e Fh(pred)h Fo(E)s Fu(\()p Fo(j)s Fu(\()p Fo(j)k Fu(+)e Fo(\016)r Fu(\)\()p Fo(j)j Fu(+)d Fo(\022)i Fp(\000)e Fu(1\))g(+)g Fo(l)1208 294 y Fn(c)1228 287 y Fu(\()p Fo(l)1262 294 y Fn(c)1294 287 y Fu(+)f(1\))i(+)f(1\)\))p Fo(;)17 b(l)1558 294 y Fn(c)1592 287 y Fo(<)c(j)e(;)52 364 y Fp(f)p Fo(\021)r Fp(g)p Fo(;)57 b(\021)16 b Fp(2)e(M)p Fu(\()p Fo(j)f Fp(\002)e Fu(\()p Fo(j)j Fu(+)d Fo(\016)r Fu(\))g Fp(\002)g Fu(\()p Fo(j)j Fu(+)d Fo(\022)q Fu(\))p Fo(;)d(l)843 371 y Fn(c)864 364 y Fu(\()p Fo(l)898 371 y Fn(c)930 364 y Fu(+)j(1\))g(+)g(1)g Fp(\000)g Fo(j)s Fu(\()p Fo(j)i Fu(+)e Fo(\016)r Fu(\)\))p Fo(;)17 b(l)1406 371 y Fn(c)1440 364 y Fo(<)c(j;)52 441 y(\031)r Fu(\()p Fo(j)h Fp(\002)d Fu(\()p Fo(j)j Fu(+)d Fo(\016)r Fu(\))g Fp(\002)g Fu(\()p Fo(j)j Fu(+)c Fo(\022)q Fu(\))p Fo(;)e Fh(pred)h Fo(E)s Fu(\()p Fo(j)s Fu(\()p Fo(j)k Fu(+)e Fo(\016)r Fu(\)\()p Fo(j)j Fu(+)d Fo(\022)q Fu(\))h Fp(\000)f Fo(j)j Fu(+)c(1\))p Fo(;)17 b(j)f Fp(\024)e Fo(l)1371 448 y Fn(c)1399 441 y Fo(;)52 518 y Fp(f)p Fo(\021)r Fp(g)p Fo(;)57 b(\021)16 b Fp(2)e(M)p Fu(\()p Fo(j)f Fp(\002)e Fu(\()p Fo(j)j Fu(+)d Fo(\016)r Fu(\))g Fp(\002)g Fu(\()p Fo(j)j Fu(+)d Fo(\022)q Fu(\))p Fo(;)d Fp(\000)p Fo(j)14 b Fu(+)d(1\))p Fo(;)17 b(j)f Fp(\024)e Fo(l)1132 525 y Fn(c)1160 518 y Fo(;)52 596 y(\031)r Fu(\()p Fo(j)g Fp(\002)d Fu(\()p Fo(j)j Fu(+)d Fo(\016)r Fu(\))g Fp(\002)g Fu(\()p Fo(j)j Fu(+)c Fo(\022)q Fu(\))j(+)592 603 y Fl(3)625 596 y Fo(l)f Fp(\002)f Fu(\()p Fo(l)h Fu(+)f Fo(\017)p Fu(\))p Fo(;)d Fh(pred)g Fo(E)s Fu(\()p Fo(j)s Fu(\()p Fo(j)13 b Fu(+)e Fo(\016)r Fu(\)\()p Fo(j)j Fu(+)d Fo(\022)q Fu(\))h(+)f Fo(l)q Fu(\()p Fo(l)h Fu(+)f Fo(\017)p Fu(\))g Fp(\000)g Fo(l)h Fu(+)f(1\))p Fo(;)17 b(l)d Fp(\024)g Fo(l)1864 603 y Fn(c)1884 596 y Fo(;)52 673 y Fp(f)p Fo(\021)r Fp(g)p Fo(;)57 b(\021)16 b Fp(2)e(M)p Fu(\()p Fo(j)f Fp(\002)e Fu(\()p Fo(j)j Fu(+)d Fo(\016)r Fu(\))g Fp(\002)g Fu(\()p Fo(j)j Fu(+)d Fo(\022)q Fu(\))h(+)857 680 y Fl(3)890 673 y Fo(l)g Fp(\002)f Fu(\()p Fo(l)h Fu(+)f Fo(\017)p Fu(\))p Fo(;)d Fp(\000)p Fo(l)k Fu(+)f(1\))p Fo(;)17 b(l)e Fp(\024)e Fo(l)1412 680 y Fn(c)1441 673 y Fo(:)0 782 y Fe(Pr)m(o)m(of.)28 b Fu(This)18 b(corollary)g(is)h(a)g(consequence)g(of)g(lemma)g(6)p Fo(:)p Fu(19)g(together)g(with)g(theorems)f(7)p Fo(:)p Fu(28,)i(7)p Fo(:)p Fu(29,)0 842 y(7)p Fo(:)p Fu(30.)g(Notice)14 b(that)f(w)o(e)f(ha)o(v)o (e)g(to)h(put)f(together)h(the)g(descriptions)d(of)j(the)g(cycles)g(of)f (these)h(theorems)e(in)0 902 y(order)g(to)i(c)o(hec)o(k)g(that)g(for)f(eac)o (h)g(cycle)i Fo(\031)h Fu(in)d(the)h(ab)q(o)o(v)o(e)f(list,)h(there)g(is)f(a) g(sequence)h(of)g(cycles)g Fo(\031)1758 909 y Fl(0)1781 902 y Fo(;)8 b Fp(\001)g(\001)g(\001)16 b Fo(;)8 b(\031)1927 909 y Fn(r)0 965 y Fu(suc)o(h)15 b(that)i Fo(\031)248 972 y Fl(0)285 965 y Fu(=)c Fo(\031)r Fu(,)409 952 y Fk(e)398 965 y Fo(B)r Fu(\()p Fo(\031)485 972 y Fn(i)503 965 y Fu(\))e Fp(\\)g Fo(\031)605 972 y Fn(i)p Fl(+1)687 965 y Fp(6)p Fu(=)i Fp(;)p Fu(,)k(0)c Fp(\024)h Fo(i)g(<)g(r)k Fu(and)e Fp(\000)p Fu(1)p 1107 977 V 13 w Fp(2)1242 952 y Fk(e)1231 965 y Fo(B)s Fu(\()p Fo(\031)1319 972 y Fn(r)1341 965 y Fu(\).)50 b Fc(\003)0 1047 y Fw(Corollary)25 b(7.32.)63 b Fh(Let)22 b Fo(\033)h Fh(b)q(elong)d(to)i Fp(S)846 1054 y Fn(n)871 1059 y Fg(c)889 1054 y Fj(\000)p Fl(1)943 1047 y Fh(.)35 b(The)21 b(minimal)e(stable)i(subgraph)d Fp(G)1699 1029 y Fj(\000)1733 1047 y Fu(\()p Fo(\033)r Fu(\))k Fh(of)f Fp(G)1917 1029 y Fj(\000)0 1107 y Fh(con)o(taining)f Fo(\033)k Fh(is)d(the)h(restriction)f(of)h Fp(G)i Fh(to)e(the)g(v)o(ertices)f(listed)g (in)g(corollary)g Fu(7)p Fo(:)p Fu(31)p Fh(.)36 b(The)22 b(arro)o(ws)0 1167 y(of)17 b Fp(G)90 1149 y Fj(\000)123 1167 y Fu(\()p Fo(\033)r Fu(\))h Fh(are)e(\(where)g(the)h(starting)f(cycles)h(b)q(elong)f(to)h(the)f (list)h(giv)o(en)f(in)g(corollary)f Fu(7)p Fo(:)p Fu(31)p Fh(\):)0 1254 y Fo(\031)r Fu(\()p Fo(j)f Fp(\002)d Fu(\()p Fo(j)j Fu(+)c Fo(\016)r Fu(\))i Fp(\002)f Fu(\()p Fo(j)j Fu(+)c Fo(\022)q Fu(\))p Fo(;)e Fh(pred)h Fo(E)s Fu(\()p Fo(j)s Fu(\()p Fo(j)k Fu(+)e Fo(\016)r Fu(\)\()p Fo(j)j Fu(+)d Fo(\022)i Fp(\000)e Fu(1\))g(+)g Fo(l)1156 1261 y Fn(c)1176 1254 y Fu(\()p Fo(l)1210 1261 y Fn(c)1241 1254 y Fu(+)g(1\))h(+)e(1\)\))15 b Fp($)f(f)p Fo(\021)r Fp(g)8 b Fo(;)1419 1335 y(\021)15 b Fp(2)1517 1322 y Fk(e)1506 1335 y Fo(B)s Fu(\()p Fo(\031)r Fu(\))8 b Fo(;)59 b(l)1711 1342 y Fn(c)1745 1335 y Fo(<)13 b(j)j Fp(\024)e Fo(j)1908 1342 y Fn(c)1936 1335 y Fo(;)0 1409 y Fp(f)p Fo(j)f Fp(\002)e Fu(\()p Fo(j)j Fu(+)d Fo(\016)r Fu(\))g Fp(\002)g Fu(\()p Fo(j)j Fu(+)d Fo(\022)i Fp(\000)d Fu(1\))i(+)601 1416 y Fl(3)634 1409 y Fo(l)649 1416 y Fn(c)680 1409 y Fp(\002)f Fu(\()p Fo(l)764 1416 y Fn(c)795 1409 y Fu(+)g(1\))h(+)940 1416 y Fl(1)973 1409 y Fu(1)p Fp(g)h(!)490 1484 y Fo(\031)r Fu(\()p Fo(j)h Fp(\002)d Fu(\()p Fo(j)j Fu(+)c Fo(\016)r Fu(\))i Fp(\002)f Fu(\()p Fo(j)j Fu(+)c Fo(\022)j Fp(\000)e Fu(1\))g(+)1115 1491 y Fl(3)1148 1484 y Fo(l)1163 1491 y Fn(c)1195 1484 y Fp(\002)g Fu(\()p Fo(l)1279 1491 y Fn(c)1310 1484 y Fu(+)g(1\))p Fo(;)d Fp(f\000)p Fu(1)p 1451 1496 V Fo(;)g Fu(+1)p 1537 1496 V Fp(g)1626 1463 y Fn(c)1646 1484 y Fu(\))g Fo(;)g(l)1710 1491 y Fn(c)1745 1484 y Fo(<)13 b(j)j Fp(\024)e Fo(j)1908 1491 y Fn(c)1936 1484 y Fo(;)0 1559 y(\031)r Fu(\()p Fo(j)g Fp(\002)d Fu(\()p Fo(j)j Fu(+)c Fo(\016)r Fu(\))i Fp(\002)f Fu(\()p Fo(j)j Fu(+)c Fo(\022)q Fu(\))p Fo(;)e Fh(pred)h Fo(E)s Fu(\()p Fo(j)s Fu(\()p Fo(j)k Fu(+)e Fo(\016)r Fu(\)\()p Fo(j)j Fu(+)d Fo(\022)q Fu(\))h Fp(\000)f Fo(j)i Fu(+)e(1\)\))k Fp($)e(f)p Fo(\021)r Fp(g)8 b Fo(;)927 1634 y(\021)15 b Fp(2)f(M)p Fu(\()p Fo(j)g Fp(\002)d Fu(\()p Fo(j)i Fu(+)e Fo(\016)r Fu(\))h Fp(\002)e Fu(\()p Fo(j)k Fu(+)d Fo(\022)q Fu(\))p Fo(;)d Fp(\000)p Fo(j)15 b Fu(+)10 b(1\))e Fo(;)59 b(j)16 b Fp(\024)e Fo(l)1908 1641 y Fn(c)1936 1634 y Fo(;)0 1708 y Fp(f)p Fo(j)f Fp(\002)e Fu(\()p Fo(j)j Fu(+)d Fo(\016)r Fu(\))g Fp(\002)g Fu(\()p Fo(j)j Fu(+)d Fo(\022)i Fp(\000)d Fu(1\))i(+)601 1715 y Fl(3)634 1708 y Fo(j)h Fp(\002)e Fu(\()p Fo(j)j Fu(+)d Fo(\016)i Fp(\000)e Fu(1\))g(+)1001 1715 y Fl(1)1034 1708 y Fu(1)p Fp(g)j(!)536 1783 y Fo(\031)r Fu(\()p Fo(j)g Fp(\002)d Fu(\()p Fo(j)j Fu(+)c Fo(\016)r Fu(\))i Fp(\002)f Fu(\()p Fo(j)i Fu(+)e Fo(\022)i Fp(\000)e Fu(1\))g(+)1161 1790 y Fl(3)1194 1783 y Fo(j)j Fp(\002)d Fu(\()p Fo(j)i Fu(+)e Fo(\016)i Fp(\000)e Fu(1\))p Fo(;)d Fp(f\000)p Fu(1)p 1558 1795 V Fo(;)g Fu(+1)p 1644 1795 V Fp(g)1733 1762 y Fn(c)1753 1783 y Fu(\))g Fo(;)g(j)17 b Fp(\024)d Fo(l)1908 1790 y Fn(c)1936 1783 y Fo(;)0 1858 y(\031)r Fu(\()p Fo(j)g Fp(\002)d Fu(\()p Fo(j)j Fu(+)c Fo(\016)r Fu(\))i Fp(\002)f Fu(\()p Fo(j)j Fu(+)c Fo(\022)q Fu(\))j(+)540 1865 y Fl(3)573 1858 y Fo(l)f Fp(\002)e Fu(\()p Fo(l)j Fu(+)e Fo(\017)p Fu(\))p Fo(;)d Fh(pred)g Fo(E)s Fu(\()p Fo(j)s Fu(\()p Fo(j)13 b Fu(+)e Fo(\016)r Fu(\)\()p Fo(j)j Fu(+)d Fo(\022)q Fu(\))h(+)f Fo(l)q Fu(\()p Fo(l)h Fu(+)f Fo(\017)p Fu(\))g Fp(\000)g Fo(l)h Fu(+)f(1\)\))j Fp($)g(f)p Fo(\021)r Fp(g)8 b Fo(;)646 1932 y(\021)16 b Fp(2)e(M)p Fu(\()p Fo(j)f Fp(\002)e Fu(\()p Fo(j)j Fu(+)d Fo(\016)r Fu(\))g Fp(\002)g Fu(\()p Fo(j)j Fu(+)d Fo(\022)q Fu(\))h(+)1303 1939 y Fl(3)1336 1932 y Fo(l)g Fp(\002)f Fu(\()p Fo(l)h Fu(+)f Fo(\017)p Fu(\))p Fo(;)d Fp(\000)p Fo(l)k Fu(+)f(1\))d Fo(;)58 b(l)15 b Fp(\024)f Fo(l)1908 1939 y Fn(c)1928 1932 y Fo(;)0 2007 y Fp(f)p Fo(j)f Fp(\002)e Fu(\()p Fo(j)j Fu(+)d Fo(\016)r Fu(\))g Fp(\002)g Fu(\()p Fo(j)j Fu(+)d Fo(\022)q Fu(\))h(+)515 2014 y Fl(3)548 2007 y Fo(l)g Fp(\002)f Fu(\()p Fo(l)h Fu(+)f Fo(\017)g Fp(\000)g Fu(1\))g(+)896 2014 y Fl(1)929 2007 y Fu(1)p Fp(g)j(!)648 2082 y Fo(\031)r Fu(\()p Fo(j)g Fp(\002)d Fu(\()p Fo(j)j Fu(+)c Fo(\016)r Fu(\))i Fp(\002)f Fu(\()p Fo(j)j Fu(+)c Fo(\022)q Fu(\))j(+)1188 2089 y Fl(3)1221 2082 y Fo(l)f Fp(\002)e Fu(\()p Fo(l)j Fu(+)e Fo(\017)g Fp(\000)g Fu(1\))p Fo(;)d Fp(f\000)p Fu(1)p 1566 2094 V Fo(;)g Fu(+1)p 1652 2094 V -1 w Fp(g)1740 2061 y Fn(c)1761 2082 y Fu(\))g Fo(;)g(l)15 b Fp(\024)f Fo(l)1908 2089 y Fn(c)1936 2082 y Fo(:)0 2169 y Fh(As)19 b(usual,)f(this)h(list)g (should)e(b)q(e)j(completed)e(with)h(all)g(the)g(isometric)f(arro)o(ws)f(as)i (w)o(ell)f(as)h(with)g(the)0 2229 y(arro)o(ws)d(b)q(et)o(w)o(een)h (con\014gurations)e(where)i(the)h(quasisquares)e Fo(l)c Fp(\002)g Fu(\()p Fo(l)h Fu(+)e Fo(\017)p Fu(\))18 b Fh(are)g(slid)e(against)h(the)h (side)0 2288 y(of)f(the)h(quasicub)q(es,)e(and)g(the)i(unit)f(cub)q(e)g Fu(1)g Fh(is)g(slid)f(against)g(the)i(side)e(of)i(the)f(quasisquare)f(\(i.e.) 24 b(the)0 2348 y(con\014gurations)14 b(in)j Fo(j)c Fp(\002)e Fu(\()p Fo(j)j Fu(+)c Fo(\016)r Fu(\))i Fp(\002)f Fu(\()p Fo(j)j Fu(+)c Fo(\022)q Fu(\))j Fp(\010)870 2355 y Fl(3)903 2348 y Fo(l)f Fp(\002)e Fu(\()p Fo(l)j Fu(+)e Fo(\017)p Fu(\))g Fp(\010)g Fu(1)p Fh(\).)60 2430 y Fu(The)23 b(only)h(lo)q(ops)f(in)g(the)h(graph)f Fp(G)762 2412 y Fj(\000)795 2430 y Fu(\()p Fo(\033)r Fu(\))i(are)e(lo)q(ops)g (around)f(t)o(w)o(o)i(cycles.)43 b(An)o(y)24 b(other)g(arro)o(w)0 2490 y Fo(\031)28 2497 y Fl(1)72 2490 y Fp(!)e Fo(\031)172 2497 y Fl(2)216 2490 y Fu(of)f Fp(G)310 2472 y Fj(\000)343 2490 y Fu(\()p Fo(\033)r Fu(\))i(satis\014es)d Fo(v)p 622 2498 26 2 v 2 w Fu(\()p Fo(\031)695 2497 y Fl(1)718 2490 y Fu(\))i Fo(>)p 819 2463 V 21 w(v)r Fu(\()p Fo(\031)892 2497 y Fl(2)915 2490 y Fu(\).)36 b(As)22 b(a)f(consequence)g(a)g(path)g(in)f Fp(G)1657 2472 y Fj(\000)1691 2490 y Fu(\()p Fo(\033)r Fu(\))i(starting)0 2550 y(at)17 b Fp(f)p Fo(\033)r Fp(g)g Fu(with)f(no)g(lo)q(op)h(ends)e(in)h Fp(f\000)p Fu(1)p 646 2562 64 2 v Fp(g)p Fu(.)954 2670 y Fs(46)p eop %%Page: 47 47 47 46 bop 0 200 a Fw(The)25 b(exit)i(path.)h Fu(W)l(e)22 b(ha)o(v)o(e)f (\014nally)g(reac)o(hed)g(the)h(last)f(step)h Fo(v)r(i)p Fu(\))g(whic)o(h)f (consists)g(in)g(doing)g(the)0 260 y(syn)o(thesis)16 b(of)i(all)f(the)h (results)e(gathered)h(in)g(the)h(previous)e(steps)i Fo(i)p Fu(\))12 b Fp(\000)f Fo(v)r Fu(\).)26 b(The)17 b(set)h(of)f(all)h(the)f (cycle)0 319 y(paths)c(of)i(n)o(ull)e(cost)h(is)g(obtained)f(b)o(y)h(glueing) 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(cycle)i(of)f Fp(M)p Fu(\()p Fp(f\000)p Fu(1)p 1678 1467 V -1 w Fo(;)8 b Fu(+1)p 1763 1467 V Fp(g)1852 1437 y Fn(c)1872 1455 y Fu(\))26 b Fh(it)0 1515 y(crosses:)31 b(the)21 b(distribution)f(of)i (the)f(pro)q(cess)g(b)q(efore)h(the)f(exit)i(of)e(the)h(cycle)g(is)f(v)o(ery) h(close)f(to)h(the)0 1575 y(Gibbs)16 b(distribution)e(at)j(in)o(v)o(erse)e (temp)q(erature)h Fo(\014)j Fh(restricted)d(to)h(the)g(cycle.)830 1720 y Fv(References)33 1805 y Fs(1.)k(L.)14 b(Alonso{R.)g(Cerf,)e Fr(The)j(thr)n(e)n(e)f(dimensional)h(p)n(olyomino)n(es)g(of)g(minimal)g(ar)n (e)n(a)p Fs(,)e(preprin)o(t)f(\(1995\).)33 1860 y(2.)21 b(M.)e(Cassandro{A.)f (Galv)o(es{E.)h(Olivieri{M.E.)e(V)l(ares,)k Fr(Metastable)f(b)n(ehaviour)f (of)h(sto)n(chastic)h(dynamics:)30 b(a)87 1910 y(p)n(athwise)16 b(appr)n(o)n(ach)p Fs(,)c(Jour.)h(Stat.)h(Ph)o(ys.)f Fa(35)18 b(nos.)e(5/6)f Fs(\(1984\),)h(603{634.)33 1965 y(3.)21 b(O.)14 b(Catoni,)f Fr(L)n(ar)n(ge)h(deviations)i(for)f(anne)n(aling)p 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