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/Subtype /Link /Dest (cite.hurwitz) cvn H.B /ANN pdfmark end 10352 11782 a 0 TeXcolorgray Fl(])h(and)e(B.)i(Minnegero)36 b(de)332 b([)p 0 0.5 0 TeXcolorrgb 23079 11782 a SDict begin H.S end 23079 11782 a 0 0.5 0 TeXcolorrgb Fl(6)p 0 0.5 0 TeXcolorrgb 23730 10926 a SDict begin H.R end 23730 10926 a 23730 11782 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.minnegerode) cvn H.B /ANN pdfmark end 23730 11782 a 0 TeXcolorgray Fl(])h(de\014ned)e(a)h(related)h(nearest)f(in)-36 b(teger)332 b(con)-36 b(tin)g(ued)331 b(fraction)800 13387 y(\(NICF-H\))433 b(b)-36 b(y)8693 11782 y SDict begin H.S end 8693 11782 a 8693 11782 a SDict begin 14.5 H.A end 8693 11782 a 8693 11782 a SDict begin [ /View [/XYZ H.V] /Dest (equation.4) cvn H.B /DEST pdfmark end 8693 11782 a 17164 15584 a Fj(\030)17796 15036 y Ff(0)17732 15913 y Fi(n)18727 15584 y Fl(=)368 b Fj(q)20732 15036 y Ff(0)20684 15913 y Fi(n)21606 15584 y Fh(\000)23940 14685 y Fl(1)p 23067 15279 2397 54 v 23067 16495 a Fj(\030)23699 16038 y Ff(0)23635 16821 y Fi(n)p Fo(+1)25596 15584 y Fj(;)221 b Fh(\000)27344 15222 y Fg(1)p 27345 15383 406 40 v 27345 15931 a(2)28253 15584 y Fj(<)368 b(\030)30265 15036 y Ff(0)30201 15913 y Fi(n)31123 15584 y Fh(\000)295 b Fj(q)33076 15036 y Ff(0)33028 15913 y Fi(n)34023 15584 y Fj(<)35537 15222 y Fg(1)p 35537 15383 V 35537 15931 a(2)36075 15584 y Fj(;)14702 b Fl(\(4\))800 18717 y(where)434 b Fj(q)5183 18235 y Ff(0)5135 19046 y Fi(n)6194 18717 y Fl(is)g(an)f(in)-36 b(teger.)579 b(Then)17510 17112 y SDict begin H.S end 17510 17112 a 17510 17112 a SDict begin 14.5 H.A end 17510 17112 a 17510 17112 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5) cvn H.B /DEST pdfmark end 17510 17112 a 18357 22444 a Fj(\030)18989 21896 y Ff(0)18925 22773 y Fo(0)19820 22444 y Fl(=)369 b Fj(q)21826 21896 y Ff(0)21778 22773 y Fo(0)22598 22444 y Fh(\000)24413 21608 y Fl(1)p 25527 22090 45 1606 v 23927 22134 1623 45 v 23905 23739 45 1606 v 24187 23258 a Fj(q)24812 22776 y Ff(0)24764 23586 y Fo(1)25845 22444 y Fh(\000)295 b(\001)221 b(\001)g(\001)296 b(\000)30883 21608 y Fl(1)p 32047 22090 V 30346 22134 1724 45 v 30324 23739 45 1606 v 30606 23258 a Fj(q)31231 22776 y Ff(0)31183 23586 y Fi(n)32365 22444 y Fh(\000)f(\001)221 b(\001)g(\001)15896 b Fl(\(5\))800 26175 y(and)496 b(w)-36 b(e)497 b(ha)-36 b(v)g(e)497 b Fh(j)p Fj(q)9426 25693 y Ff(0)9378 26503 y Fi(n)10004 26175 y Fh(j)476 b(\025)g Fl(2)434 b(for)g Fj(n)477 b Fh(\025)f Fl(1)p Fj(:)497 b Fl(Also)g(if)g(one)g(of)g Fj(q)28567 25693 y Ff(0)28519 26503 y Fo(1)29045 26175 y Fj(;)221 b(q)30252 25693 y Ff(0)30204 26503 y Fo(2)30730 26175 y Fj(;)g(:)g(:)g(:)r Fl(,)513 b(sa)-36 b(y)497 b Fj(q)36648 25693 y Ff(0)36600 26503 y Fi(n)37226 26175 y Fl(,)512 b(equals)498 b(2)f(\(resp.)766 b Fh(\000)p Fl(2\),)513 b(then)800 27780 y Fj(q)1425 27298 y Ff(0)1377 28108 y Fi(n)p Fo(+1)3574 27780 y Fj(<)369 b Fl(0)402 b(\(resp.)567 b Fj(q)10386 27298 y Ff(0)10338 28108 y Fi(n)p Fo(+1)12536 27780 y Fj(>)368 b Fl(0\))402 b(\(Hurwitz)g([)p 0 0.5 0 TeXcolorrgb 21331 27780 a SDict begin H.S end 21331 27780 a 0 0.5 0 TeXcolorrgb Fl(1)p 0 0.5 0 TeXcolorrgb 21982 26924 a SDict begin H.R end 21982 26924 a 21982 27780 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.hurwitz) cvn H.B /ANN pdfmark end 21982 27780 a 0 TeXcolorgray Fl(,)408 b(p.)568 b(372]\).)h(Section)p 0 0.5 0 TeXcolorrgb 32670 27780 a SDict begin H.S end 32670 27780 a 0 0.5 0 TeXcolorrgb Fl(2)p 0 0.5 0 TeXcolorrgb 33321 26924 a SDict begin H.R end 33321 26924 a 33321 27780 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (section.2) cvn H.B /ANN pdfmark end 33321 27780 a 0 TeXcolorgray 402 w Fl(relates)402 b(the)f(t)-36 b(w)g(o)403 b(t)-36 b(yp)36 b(es)402 b(of)g(con)-36 b(tin)g(ued)800 29385 y(fraction.)2751 30990 y(In)396 b(1980,)405 b(H.)397 b(C.)g(Williams)h(ga)-36 b(v)g(e)397 b(six)g(midp)36 b(oin)-36 b(t)395 b(criteria)i(for)g(solving)g(P)-36 b(ell's)397 b(equation)g Fj(x)47099 30508 y Fo(2)47844 30990 y Fh(\000)219 b Fj(D)36 b(y)50894 30508 y Fo(2)51788 30990 y Fl(=)800 32595 y Fh(\006)p Fl(1)481 b(in)f(terms)f(of)i(the)f (NICF-H)g(expansion)g(of)24745 31470 y Fh(p)p 25852 31470 1117 54 v 1125 x Fj(D)516 b Fl(\(see)480 b(Theorems)h(6)f(and)g(7,)492 b([)p 0 0.5 0 TeXcolorrgb 41725 32595 a SDict begin H.S end 41725 32595 a 0 0.5 0 TeXcolorrgb Fl(9)p 0 0.5 0 TeXcolorrgb 42376 31739 a SDict begin H.R end 42376 31739 a 42376 32595 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.hcw) cvn H.B /ANN pdfmark end 42376 32595 a 0 TeXcolorgray Fl(,)g(pp.)717 b(12{13]\).)j(His)800 34200 y(pro)36 b(of)589 b(made)f(extensiv)-36 b(e)589 b(use)f(of)h(the)f(singular)g(con)-36 b(tin)g(ued)587 b(fraction)i(expansion)g(of)44115 33075 y Fh(p)p 45222 33075 1117 54 v 1125 x Fj(D)37 b Fl(.)1042 b(Theorem)p 0 0.5 0 TeXcolorrgb 800 35805 a SDict begin H.S end 800 35805 a 0 0.5 0 TeXcolorrgb Fl(6)p 0 0.5 0 TeXcolorrgb 1450 34949 a SDict begin H.R end 1450 34949 a 1450 35805 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.6) cvn H.B /ANN pdfmark end 1450 35805 a 0 TeXcolorgray 568 w Fl(of)568 b(section)p 0 0.5 0 TeXcolorrgb 8109 35805 a SDict begin H.S end 8109 35805 a 0 0.5 0 TeXcolorrgb Fl(3)p 0 0.5 0 TeXcolorrgb 8759 34949 a SDict begin H.R end 8759 34949 a 8759 35805 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (section.3) cvn H.B /ANN pdfmark end 8759 35805 a 0 TeXcolorgray 568 w Fl(of)g(our)f(pap)36 b(er)566 b(giv)-36 b(es)569 b(the)e(corresp)36 b(onding)566 b(criteria)i(for)g (the)f(NICF-P)g(expansion)g(of)800 36286 y Fh(p)p 1907 36286 1117 54 v 1124 x Fj(D)36 b Fl(.)900 b(In)540 b(an)h(attempt)f(to) g(giv)-36 b(e)542 b(a)f(deriv)-72 b(ation)541 b(of)g(the)g(latter)f (criteria)h(without)g(the)f(use)g(of)i(singular)800 39015 y(con)-36 b(tin)g(ued)646 b(fractions,)703 b(the)647 b(author)g(studied)f(the)i(con)-36 b(v)g(ersion)648 b(of)g(the)f (regular)h(con)-36 b(tin)g(ued)646 b(fraction)800 40621 y(\(R)-36 b(CF\))547 b(expansion)g(of)12817 39496 y Fh(p)p 13924 39496 V 1125 x Fj(D)583 b Fl(to)548 b(the)e(NICF)i(giv)-36 b(en)547 b(b)-36 b(y)547 b(Lemma)p 0 0.5 0 TeXcolorrgb 33609 40621 a SDict begin H.S end 33609 40621 a 0 0.5 0 TeXcolorrgb Fl(9)p 0 0.5 0 TeXcolorrgb 34260 39764 a SDict begin H.R end 34260 39764 a 34260 40621 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.9) cvn H.B /ANN pdfmark end 34260 40621 a 0 TeXcolorgray 547 w Fl(of)g(section)h(5,)576 b(where)547 b(the)f(R)-36 b(CF)548 b(is)800 42226 y(de\014ned)432 b(recursiv)-36 b(ely)435 b(b)-36 b(y)22361 44349 y Fj(\030)22929 44548 y Fi(n)23924 44349 y Fl(=)369 b Fj(a)25988 44548 y Fi(n)26909 44349 y Fl(+)29222 43451 y(1)p 28349 44044 2397 54 v 28349 45260 a Fj(\030)28917 45459 y Fi(n)p Fo(+1)30878 44349 y Fj(;)800 47378 y Fl(with)396 b Fj(a)4408 47577 y Fi(n)5403 47378 y Fl(=)369 b Fh(b)p Fj(\030)7942 47577 y Fi(n)8568 47378 y Fh(c)q Fl(,)404 b(the)395 b(in)-36 b(teger)396 b(part)f(of)i Fj(\030)21180 47577 y Fi(n)21806 47378 y Fl(.)566 b(Theorem)p 0 0.5 0 TeXcolorrgb 28187 47378 a SDict begin H.S end 28187 47378 a 0 0.5 0 TeXcolorrgb Fl(8)p 0 0.5 0 TeXcolorrgb 28838 46522 a SDict begin H.R end 28838 46522 a 28838 47378 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.8) cvn H.B /ANN pdfmark end 28838 47378 a 0 TeXcolorgray 396 w Fl(of)396 b(section)p 0 0.5 0 TeXcolorrgb 34983 47378 a SDict begin H.S end 34983 47378 a 0 0.5 0 TeXcolorrgb Fl(4)p 0 0.5 0 TeXcolorrgb 35633 46522 a SDict begin H.R end 35633 46522 a 35633 47378 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (section.4) cvn H.B /ANN pdfmark end 35633 47378 a 0 TeXcolorgray 396 w Fl(sho)-36 b(ws)396 b(that)g(the)f(cen)-36 b(tral)396 b(part)f(of)800 48984 y(a)337 b(least)g(p)36 b(erio)g(d)336 b(determines)g(whic)-36 b(h)336 b(of)h(the)f(criteria)h(hold.)546 b(Finally)-108 b(,)357 b(Theorem)p 0 0.5 0 TeXcolorrgb 40060 48984 a SDict begin H.S end 40060 48984 a 0 0.5 0 TeXcolorrgb Fl(18)p 0 0.5 0 TeXcolorrgb 41361 48128 a SDict begin H.R end 41361 48128 a 41361 48984 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.18) cvn H.B /ANN pdfmark end 41361 48984 a 0 TeXcolorgray Fl(,)f(section)p 0 0.5 0 TeXcolorrgb 46324 48984 a SDict begin H.S end 46324 48984 a 0 0.5 0 TeXcolorrgb Fl(7)p 0 0.5 0 TeXcolorrgb 46974 48128 a SDict begin H.R end 46974 48128 a 46974 48984 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (section.7) cvn H.B /ANN pdfmark end 46974 48984 a 0 TeXcolorgray Fl(,)g(describ)36 b(es)800 50589 y(the)427 b(case)g(where)g(there)g (are)g(only)h(o)36 b(dd-length)426 b Fk(unise)-66 b(quenc)g(es)p Fl(,)427 b(i.e.,)i(consecutiv)-36 b(e)428 b(sequences)f(of)h(partial) 800 52194 y(quotien)-36 b(ts)513 b(equal)h(to)f(1,)533 b(in)513 b(the)f(R)-36 b(CF)513 b(expansion)g(of)28067 51069 y Fh(p)p 29174 51069 1117 54 v 1125 x Fj(D)36 b Fl(;)553 b(in)513 b(this)f(case)i(the)e(NICF-P)h(expansion)g(of)800 52674 y Fh(p)p 1907 52674 V 1125 x Fj(D)470 b Fl(exhibits)433 b(the)g(usual)h(symmetry)g(prop)36 b(erties)433 b(of)h(the)f(R)-36 b(CF)434 b(expansion.)800 56059 y SDict begin H.S end 800 56059 a 800 56059 a SDict begin 14.5 H.A end 800 56059 a 800 56059 a SDict begin [ /View [/XYZ H.V] /Dest (section.2) cvn H.B /DEST pdfmark end 800 56059 a 2177 x Fm(2)2152 b(Connections)821 b(b)60 b(et)-60 b(w)g(een)821 b(the)h(NICF-H)g(and)g (NICF-P)g(ex-)4028 60671 y(pansions)715 b(of)i(an)g(irrational)f(n)-60 b(um)g(b)60 b(er.)800 62359 y SDict begin H.S end 800 62359 a 800 62359 a SDict begin 14.5 H.A end 800 62359 a 800 62359 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.1) cvn H.B /DEST pdfmark end 800 62359 a 0 TeXcolorgray 1233 x Fe(Lemma)635 b(1.)p 0 TeXcolorgray 614 w Fk(L)-66 b(et)572 b Fj(q)10983 63110 y Ff(0)10935 63920 y Fi(n)11561 63592 y Fj(;)221 b(\030)12775 63110 y Ff(0)12711 63920 y Fi(n)13338 63592 y Fj(;)g(A)14895 63110 y Ff(0)14895 63920 y Fi(n)15522 63592 y Fj(=B)17227 63110 y Ff(0)17160 63920 y Fi(n)18360 63592 y Fk(denote)573 b(the)g Fj(n)p Fk(-th)g(p)-66 b(artial)573 b(denominator,)599 b(c)-66 b(omplete)572 b(quotient)g(and)800 65197 y(c)-66 b(onver)g(gent)429 b(of)j(the)h Fl(NICF-H)f Fk(exp)-66 b(ansion)431 b(of)h(an)g(irr)-66 b(ational)431 b(numb)-66 b(er)431 b Fj(\030)36069 65396 y Fo(0)37027 65197 y Fk(and)i Fj(q)40096 65396 y Fi(n)40722 65197 y Fj(;)221 b(\017)41829 65396 y Fi(n)42456 65197 y Fj(;)g(\030)43606 65396 y Fi(n)44233 65197 y Fj(;)g(A)45790 65396 y Fi(n)46417 65197 y Fj(=B)48055 65396 y Fi(n)49114 65197 y Fk(denote)p 0 TeXcolorgray 26475 70521 a Fl(2)p 0 TeXcolorgray eop end end %%Page: 3 3 TeXDict begin HPSdict begin 3 2 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show 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Fh(\000)295 b Fj(A)24107 12997 y Ff(0)24107 13874 y Fi(n)p Ff(\000)p Fo(1)25936 13545 y Fj(;)1328 b(B)28680 12997 y Ff(0)28613 13874 y Fi(n)p Fo(+1)30811 13545 y Fl(=)369 b Fj(q)32817 12997 y Ff(0)32769 13874 y Fi(n)p Fo(+1)34597 13545 y Fj(B)35652 12997 y Ff(0)35585 13874 y Fi(n)36506 13545 y Fh(\000)296 b Fj(B)38890 12997 y Ff(0)38823 13874 y Fi(n)p Ff(\000)p Fo(1)40651 13545 y Fj(;)800 18416 y Fk(wher)-66 b(e)465 b Fj(\017)4995 18615 y Fo(0)5890 18416 y Fl(=)368 b(1)p Fk(.)598 b(Then)12304 16811 y SDict begin H.S end 12304 16811 a 12304 16811 a SDict begin 14.5 H.A end 12304 16811 a 12304 16811 a SDict begin [ /View [/XYZ H.V] /Dest (equation.6) cvn H.B /DEST pdfmark end 12304 16811 a 17867 20021 a Fj(q)18492 19473 y Ff(0)18444 20349 y Fi(n)19439 20021 y Fl(=)368 b Fj(t)21289 20220 y Fi(n)21915 20021 y Fj(q)22492 20220 y Fi(n)23118 20021 y Fj(;)1550 b(\030)25661 19473 y Ff(0)25597 20349 y Fi(n)26592 20021 y Fl(=)369 b Fj(t)28443 20220 y Fi(n)29069 20021 y Fj(\030)29637 20220 y Fi(n)30263 20021 y Fj(;)1550 b(n)370 b Fh(\025)f Fl(0)p Fj(;)15405 b Fl(\(6\))800 22346 y Fk(wher)-66 b(e)465 b Fj(t)4940 22545 y Fo(0)5834 22346 y Fl(=)369 b(1)465 b Fk(and)g Fj(t)11324 22545 y Fi(n)12319 22346 y Fl(=)368 b(\()p Fh(\000)p Fl(1\))16394 21864 y Fi(n)17021 22346 y Fj(\017)17546 22545 y Fo(1)18293 22346 y Fh(\001)221 b(\001)g(\001)h Fj(\017)20589 22545 y Fi(n)21216 22346 y Fk(,)464 b(if)g Fj(n)370 b Fh(\025)f Fl(1)p Fk(.)27535 20741 y SDict begin H.S end 27535 20741 a 27535 20741 a SDict begin 14.5 H.A end 27535 20741 a 27535 20741 a SDict begin [ /View [/XYZ H.V] /Dest (equation.7) cvn H.B /DEST pdfmark end 27535 20741 a 16389 25279 a Fj(A)17364 24731 y Ff(0)17364 25607 y Fi(n)18359 25279 y Fl(=)g Fj(s)20353 25478 y Fi(n)20978 25279 y Fj(A)21953 25478 y Fi(n)22580 25279 y Fj(;)1550 b(B)25546 24731 y Ff(0)25479 25607 y Fi(n)26474 25279 y Fl(=)369 b Fj(s)28468 25478 y Fi(n)29094 25279 y Fj(B)30082 25478 y Fi(n)30708 25279 y Fj(;)1550 b(n)369 b Fh(\025)h(\000)p Fl(2)p Fj(;)13927 b Fl(\(7\))800 28213 y Fk(wher)-66 b(e)465 b Fj(s)5083 28412 y Ff(\000)p Fo(2)6709 28213 y Fl(=)369 b Fh(\000)p Fl(1)p Fj(;)221 b(s)10968 28412 y Ff(\000)p Fo(1)12595 28213 y Fl(=)369 b(1)465 b Fk(and)g Fj(s)18228 28412 y Fi(n)p Fo(+1)20425 28213 y Fl(=)369 b Fh(\000)p Fj(s)23452 28412 y Fi(n)p Ff(\000)p Fo(1)25280 28213 y Fj(\017)25805 28412 y Fi(n)p Fo(+1)28099 28213 y Fk(for)464 b Fj(n)370 b Fh(\025)f(\000)p Fl(1)p Fk(.)800 29611 y SDict begin H.S end 800 29611 a 800 29611 a SDict begin 14.5 H.A end 800 29611 a 800 29611 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.2) cvn H.B /DEST pdfmark end 800 29611 a 0 TeXcolorgray 1314 x Fe(Remark)499 b(2.)p 0 TeXcolorgray 2288 w Fl(It)433 b(follo)-36 b(ws)436 b(that)d Fj(s)19026 31124 y Fo(0)19920 30925 y Fl(=)369 b(1)434 b(and)16707 33858 y Fj(s)17320 34057 y Fo(2)p Fi(i)18535 33858 y Fl(=)368 b(\()p Fh(\000)p Fl(1\))22610 33310 y Fi(i)22986 33858 y Fj(\017)23511 34057 y Fo(2)p Fi(i)24357 33858 y Fj(\017)24882 34057 y Fo(2)p Fi(i)p Ff(\000)p Fo(2)27152 33858 y Fh(\001)221 b(\001)g(\001)h Fj(\017)29448 34057 y Fo(2)29974 33858 y Fj(;)1956 b Fl(if)435 b Fj(i)368 b Fh(\025)h Fl(1)p Fj(;)51138 31921 y SDict begin H.S end 51138 31921 a 51138 31921 a SDict begin 17.5 H.A end 51138 31921 a 51138 31921 a SDict begin [ /View [/XYZ H.V] /Dest (equation.8) cvn H.B /DEST pdfmark end 51138 31921 a 1937 x Fl(\(8\))15504 35916 y Fj(s)16117 36115 y Fo(2)p Fi(i)p Fo(+1)18535 35916 y Fl(=)f(\()p Fh(\000)p Fl(1\))22610 35368 y Fi(i)p Fo(+1)24188 35916 y Fj(\017)24713 36115 y Fo(2)p Fi(i)p Fo(+1)26762 35916 y Fj(\017)27287 36115 y Fo(2)p Fi(i)p Ff(\000)p Fo(1)29557 35916 y Fh(\001)221 b(\001)g(\001)h Fj(\017)31853 36115 y Fo(1)32379 35916 y Fj(;)1956 b Fl(if)434 b Fj(i)369 b Fh(\025)g Fl(0)p Fj(;)51138 33979 y SDict begin H.S end 51138 33979 a 51138 33979 a SDict begin 17.5 H.A end 51138 33979 a 51138 33979 a SDict begin [ /View [/XYZ H.V] /Dest (equation.9) cvn H.B /DEST pdfmark end 51138 33979 a 1937 x Fl(\(9\))14485 37853 y Fj(s)15098 38052 y Fi(n)p Fo(+1)16927 37853 y Fj(s)17540 38052 y Fi(n)18535 37853 y Fl(=)f Fj(t)20385 38052 y Fi(n)p Fo(+1)22213 37853 y Fj(;)1956 b Fl(if)435 b Fj(n)369 b Fh(\025)g(\000)p Fl(1)p Fj(:)50488 35916 y SDict begin H.S end 50488 35916 a 50488 35916 a SDict begin 17.5 H.A end 50488 35916 a 50488 35916 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10) cvn H.B /DEST pdfmark end 50488 35916 a 1937 x Fl(\(10\))p 0 TeXcolorgray 800 40787 a Fk(Pr)-66 b(o)g(of.)p 0 TeXcolorgray 649 w Fl(W)-108 b(e)417 b(pro)-36 b(v)g(e)416 b(\()p 0 0.5 0 TeXcolorrgb 11088 40787 a SDict begin H.S end 11088 40787 a 0 0.5 0 TeXcolorrgb Fl(6)p 0 0.5 0 TeXcolorrgb 11738 39931 a SDict begin H.R end 11738 39931 a 11738 40787 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.6) cvn H.B /ANN pdfmark end 11738 40787 a 0 TeXcolorgray Fl(\))h(b)-36 b(y)416 b(induction)g(on)g Fj(n)369 b Fh(\025)h Fl(0.)573 b(These)416 b(are)h(true)f(when)g Fj(n)369 b Fl(=)g(0.)573 b(So)416 b(w)-36 b(e)417 b(assume)f(that)800 42392 y Fj(n)369 b Fh(\025)h Fl(0)433 b(and)h(\()p 0 0.5 0 TeXcolorrgb 7467 42392 a SDict begin H.S end 7467 42392 a 0 0.5 0 TeXcolorrgb Fl(6)p 0 0.5 0 TeXcolorrgb 8117 41536 a SDict begin H.R end 8117 41536 a 8117 42392 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.6) cvn H.B /ANN pdfmark end 8117 42392 a 0 TeXcolorgray Fl(\))f(hold.)578 b(Then)17356 45918 y Fj(\030)17988 45369 y Ff(0)17924 46246 y Fi(n)p Fo(+1)20121 45918 y Fl(=)23320 45019 y(1)p 21635 45612 4021 54 v 21635 46829 a Fj(q)22260 46445 y Ff(0)22212 47157 y Fi(n)23133 46829 y Fh(\000)295 b Fj(\030)25093 46445 y Ff(0)25029 47157 y Fi(n)25788 45918 y Fj(;)1301 b(\030)28018 46117 y Fi(n)p Fo(+1)30216 45918 y Fl(=)32563 45019 y Fj(\017)33088 45218 y Fi(n)p Fo(+1)p 31729 45612 V 31729 46829 a Fj(q)32306 47028 y Fi(n)33228 46829 y Fh(\000)295 b Fj(\030)35124 47028 y Fi(n)35883 45918 y Fj(;)50488 43981 y SDict begin H.S end 50488 43981 a 50488 43981 a SDict begin 17.5 H.A end 50488 43981 a 50488 43981 a SDict begin [ /View [/XYZ H.V] /Dest (equation.11) cvn H.B /DEST pdfmark end 50488 43981 a 1937 x Fl(\(11\))20919 48724 y Fj(q)21544 48175 y Ff(0)21496 49052 y Fi(n)22491 48724 y Fl(=)368 b([)p Fj(\030)24864 48175 y Ff(0)24800 49052 y Fi(n)25427 48724 y Fl(])p Fj(;)1301 b(q)28027 48923 y Fi(n)29022 48724 y Fl(=)369 b([)p Fj(\030)31332 48923 y Fi(n)31959 48724 y Fl(])p Fj(;)50488 46787 y SDict begin H.S end 50488 46787 a 50488 46787 a SDict begin 17.5 H.A end 50488 46787 a 50488 46787 a SDict begin [ /View [/XYZ H.V] /Dest (equation.12) cvn H.B /DEST pdfmark end 50488 46787 a 1937 x Fl(\(12\))800 51657 y(where)434 b([)p Fj(x)p Fl(])g(denotes)f(the)g(nearest)g(in)-36 b(teger)434 b(to)f Fj(x)p Fl(.)579 b(Then)17763 55183 y Fj(\030)18395 54635 y Ff(0)18331 55511 y Fi(n)p Fo(+1)20529 55183 y Fl(=)24824 54284 y(1)p 22042 54877 6213 54 v 22042 56094 a Fj(t)22512 56293 y Fi(n)23138 56094 y Fj(q)23715 56293 y Fi(n)24636 56094 y Fh(\000)296 b Fj(t)26435 56293 y Fi(n)27061 56094 y Fj(\030)27629 56293 y Fi(n)20529 58511 y Fl(=)23505 57612 y Fj(t)23975 57811 y Fi(n)p 22042 58205 4021 54 v 22042 59422 a Fj(q)22619 59621 y Fi(n)23541 59422 y Fh(\000)f Fj(\030)25437 59621 y Fi(n)26565 58511 y Fl(=)369 b Fj(t)28416 58710 y Fi(n)29042 58511 y Fl(\()p Fh(\000)p Fj(\017)31106 58710 y Fi(n)p Fo(+1)32934 58511 y Fj(\030)33502 58710 y Fi(n)p Fo(+1)35331 58511 y Fl(\))20529 61247 y(=)g Fj(t)22380 61446 y Fi(n)p Fo(+1)24208 61247 y Fj(\030)24776 61446 y Fi(n)p Fo(+1)26604 61247 y Fj(:)800 64180 y Fl(Next,)12589 65785 y Fj(q)13214 65237 y Ff(0)13166 66114 y Fi(n)p Fo(+1)15363 65785 y Fl(=)g([)p Fj(\030)17737 65237 y Ff(0)17673 66114 y Fi(n)p Fo(+1)19502 65785 y Fl(])g(=)f([)p Fj(t)22443 65984 y Fi(n)p Fo(+1)24272 65785 y Fj(\030)24840 65984 y Fi(n)p Fo(+1)26669 65785 y Fl(])h(=)f Fj(t)29249 65984 y Fi(n)p Fo(+1)31078 65785 y Fl([)p Fj(\030)32007 65984 y Fi(n)p Fo(+1)33836 65785 y Fl(])h(=)g Fj(t)36417 65984 y Fi(n)p Fo(+1)38245 65785 y Fj(q)38822 65984 y Fi(n)p Fo(+1)40650 65785 y Fj(:)p 0 TeXcolorgray 26475 70521 a Fl(3)p 0 TeXcolorgray eop end end %%Page: 4 4 TeXDict begin HPSdict begin 4 3 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.4) cvn H.B /DEST pdfmark end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 1328 x Fl(Finally)-108 b(,)381 b(w)-36 b(e)366 b(pro)-36 b(v)g(e)366 b(\()p 0 0.5 0 TeXcolorrgb 11215 -2672 a SDict begin H.S end 11215 -2672 a 0 0.5 0 TeXcolorrgb Fl(7)p 0 0.5 0 TeXcolorrgb 11865 -3528 a SDict begin H.R end 11865 -3528 a 11865 -2672 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.7) cvn H.B /ANN pdfmark end 11865 -2672 a 0 TeXcolorgray Fl(\))g(b)-36 b(y)366 b(induction)f(on)h Fj(n)j Fh(\025)g(\000)p Fl(2.)557 b(These)366 b(hold)g(for)g Fj(n)k Fl(=)e Fh(\000)p Fl(2)f(and)e Fh(\000)p Fl(1.)557 b(So)366 b(w)-36 b(e)366 b(assume)800 -1067 y Fj(n)j Fh(\025)h(\000)p Fl(1)434 b(and)7779 1867 y Fj(A)8754 1318 y Ff(0)8754 2195 y Fi(n)p Ff(\000)p Fo(1)10952 1867 y Fl(=)368 b Fj(s)12945 2066 y Fi(n)p Ff(\000)p Fo(1)14774 1867 y Fj(A)15749 2066 y Fi(n)p Ff(\000)p Fo(1)17577 1867 y Fj(;)1522 b(B)20515 1318 y Ff(0)20448 2195 y Fi(n)p Ff(\000)p Fo(1)22646 1867 y Fl(=)369 b Fj(s)24640 2066 y Fi(n)p Ff(\000)p Fo(1)26468 1867 y Fj(B)27456 2066 y Fi(n)p Ff(\000)p Fo(1)29285 1867 y Fj(;)1522 b(A)32143 1318 y Ff(0)32143 2195 y Fi(n)33138 1867 y Fl(=)369 b Fj(s)35132 2066 y Fi(n)35758 1867 y Fj(A)36733 2066 y Fi(n)37359 1867 y Fj(;)1522 b(B)40297 1318 y Ff(0)40230 2195 y Fi(n)41226 1867 y Fl(=)368 b Fj(s)43219 2066 y Fi(n)43845 1867 y Fj(B)44833 2066 y Fi(n)45460 1867 y Fj(:)800 4800 y Fl(Then)15666 7734 y Fj(A)16641 7185 y Ff(0)16641 8062 y Fi(n)p Fo(+1)18839 7734 y Fl(=)g Fj(q)20844 7185 y Ff(0)20796 8062 y Fi(n)p Fo(+1)22625 7734 y Fj(A)23600 7185 y Ff(0)23600 8062 y Fi(n)24521 7734 y Fh(\000)295 b Fj(A)26824 7185 y Ff(0)26824 8062 y Fi(n)p Ff(\000)p Fo(1)18839 9671 y Fl(=)368 b(\()p Fj(t)21195 9870 y Fi(n)p Fo(+1)23023 9671 y Fj(q)23600 9870 y Fi(n)p Fo(+1)25429 9671 y Fl(\)\()p Fj(s)27054 9870 y Fi(n)27679 9671 y Fj(A)28654 9870 y Fi(n)29280 9671 y Fl(\))295 b Fh(\000)h Fj(s)32023 9870 y Fi(n)p Ff(\000)p Fo(1)33851 9671 y Fj(A)34826 9870 y Fi(n)p Ff(\000)p Fo(1)18839 11608 y Fl(=)368 b Fj(q)20796 11807 y Fi(n)p Fo(+1)22625 11608 y Fj(s)23238 11807 y Fi(n)p Fo(+1)25066 11608 y Fj(A)26041 11807 y Fi(n)26962 11608 y Fh(\000)296 b Fl(\()p Fh(\000)p Fj(s)30443 11807 y Fi(n)p Fo(+1)32271 11608 y Fj(\017)32796 11807 y Fi(n)p Fo(+1)34625 11608 y Fl(\))p Fj(A)36106 11807 y Fi(n)p Ff(\000)p Fo(1)18839 13545 y Fl(=)368 b Fj(s)20832 13744 y Fi(n)p Fo(+1)22661 13545 y Fl(\()p Fj(q)23744 13744 y Fi(n)p Fo(+1)25572 13545 y Fj(A)26547 13744 y Fi(n)27468 13545 y Fl(+)295 b Fj(\017)29300 13744 y Fi(n)p Fo(+1)31129 13545 y Fj(A)32104 13744 y Fi(n)p Ff(\000)p Fo(1)33932 13545 y Fl(\))18839 15482 y(=)368 b Fj(s)20832 15681 y Fi(n)p Fo(+1)22661 15482 y Fj(A)23636 15681 y Fi(n)p Fo(+1)25464 15482 y Fj(:)800 18416 y Fl(Similarly)435 b Fj(B)7383 17934 y Ff(0)7316 18744 y Fi(n)p Fo(+1)9514 18416 y Fl(=)368 b Fj(s)11507 18615 y Fi(n)p Fo(+1)13335 18416 y Fj(B)14323 18615 y Fi(n)p Fo(+1)16152 18416 y Fl(.)p 51860 18416 45 878 v 51905 17582 781 45 v 51905 18416 V 52684 18416 45 878 v 800 19833 a SDict begin H.S end 800 19833 a 800 19833 a SDict begin 14.5 H.A end 800 19833 a 800 19833 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.3) cvn H.B /DEST pdfmark end 800 19833 a 0 TeXcolorgray 1295 x Fe(Corollary)623 b(3.)p 0 TeXcolorgray 600 w Fk(Supp)-66 b(ose)547 b Fj(\030)14921 21327 y Fi(n)15547 21128 y Fj(;)221 b(:)g(:)g(:)j(;)d(\030)19028 21327 y Fi(n)p Fo(+)p Fi(k)24 b Ff(\000)p Fo(1)22650 21128 y Fk(is)547 b(a)g(le)-66 b(ast)547 b(p)-66 b(erio)g(d)546 b(of)h Fl(NICF-P)433 b Fk(c)-66 b(omplete)546 b(quotients)g(for)h(a)800 22733 y(quadr)-66 b(atic)464 b(irr)-66 b(ational)464 b Fj(\030)12809 22932 y Fo(0)13334 22733 y Fk(.)p 0 TeXcolorgray 1675 25445 a(\(a\))p 0 TeXcolorgray 651 w(If)g Fj(\017)5941 25644 y Fi(n)p Fo(+1)7991 25445 y Fh(\001)221 b(\001)g(\001)h Fj(\017)10287 25644 y Fi(n)p Fo(+)p Fi(k)12528 25445 y Fl(=)368 b(\()p Fh(\000)p Fl(1\))16603 24963 y Fi(k)17172 25445 y Fk(,)465 b(then)24551 27050 y Fj(\030)25183 26502 y Ff(0)25119 27379 y Fi(n)25745 27050 y Fj(;)221 b(:)g(:)g(:)j(;)d (\030)29290 26502 y Ff(0)29226 27379 y Fi(n)p Fo(+)p Fi(k)24 b Ff(\000)p Fo(1)4052 29375 y Fk(is)464 b(a)i(le)-66 b(ast)464 b(p)-66 b(erio)g(d)464 b(of)h Fl(NICF-H)433 b Fk(c)-66 b(omplete)464 b(quotients)g(for)g Fj(\030)33538 29574 y Fo(0)34064 29375 y Fk(.)p 0 TeXcolorgray 1741 32087 a(\(b\))p 0 TeXcolorgray 651 w(If)g Fj(\017)5941 32286 y Fi(n)p Fo(+1)7991 32087 y Fh(\001)221 b(\001)g(\001)h Fj(\017)10287 32286 y Fi(n)p Fo(+)p Fi(k)12528 32087 y Fl(=)368 b(\()p Fh(\000)p Fl(1\))16603 31605 y Fi(k)24 b Fo(+1)18375 32087 y Fk(,)464 b(then)22127 30482 y SDict begin H.S end 22127 30482 a 22127 30482 a SDict begin 14.5 H.A end 22127 30482 a 22127 30482 a SDict begin [ /View [/XYZ H.V] /Dest (Item.1) cvn H.B /DEST pdfmark end 22127 30482 a 19352 35020 a Fj(\030)19984 34472 y Ff(0)19920 35349 y Fi(n)20546 35020 y Fj(;)221 b(:)g(:)g(:)j(;)d(\030)24091 34472 y Ff(0)24027 35349 y Fi(n)p Fo(+)p Fi(k)24 b Ff(\000)p Fo(1)27101 35020 y Fj(;)221 b Fh(\000)p Fj(\030)29348 34472 y Ff(0)29284 35349 y Fi(n)29912 35020 y Fj(;)g(:)g(:)g(:)i(;)e Fh(\000)p Fj(\030)34489 34472 y Ff(0)34425 35349 y Fi(n)p Fo(+)p Fi(k)24 b Ff(\000)p Fo(1)50488 35020 y Fl(\(13\))4052 37954 y Fk(is)464 b(a)i(le)-66 b(ast)464 b(p)-66 b(erio)g(d)464 b(of)h Fl(NICF-H)433 b Fk(c)-66 b(omplete)464 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Fh(\000)295 b(\001)221 b(\001)g(\001)296 b(\000)43871 40775 y Fl(1)p 46781 41256 V 41588 41300 5216 45 v 41566 42906 45 1606 v 41854 42424 a Fh(\000)p Fj(q)43512 41942 y Ff(0)43464 42770 y Fi(n)p Fo(+)p Fi(k)24 b Ff(\000)p Fo(1)44477 43659 y Ff(\003)46803 41610 y Fj(;)4052 45341 y Fk(wher)-66 b(e)464 b(the)h(asterisks)g(c)-66 b(orr)g(esp)g(ond)464 b(to)g(the)h(le)-66 b(ast)465 b(p)-66 b(erio)g(d)601 b Fl(\()p 0 0.5 0 TeXcolorrgb 32840 45341 a SDict begin H.S end 32840 45341 a 0 0.5 0 TeXcolorrgb Fl(13)p 0 0.5 0 TeXcolorrgb 34141 44485 a SDict begin H.R end 34141 44485 a 34141 45341 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (Item.1) cvn H.B /ANN pdfmark end 34141 45341 a 0 TeXcolorgray Fl(\))o Fk(.)p 0 TeXcolorgray 800 48053 a(Pr)-66 b(o)g(of.)p 0 TeXcolorgray 649 w Fl(Supp)36 b(ose)477 b Fj(\030)10586 48252 y Fi(n)11213 48053 y Fj(;)221 b(:)g(:)g(:)i(;)e(\030)14693 48252 y Fi(n)p Fo(+)p Fi(k)24 b Ff(\000)p Fo(1)18246 48053 y Fl(is)478 b(a)h(least)f(p)36 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Fl(1\))37265 46385 y Fi(k)24 b Fo(+1)39395 46867 y Fl(and)359 b(Corollary)p 0 0.5 0 TeXcolorrgb 47520 46867 a SDict begin H.S end 47520 46867 a 0 0.5 0 TeXcolorrgb Fl(3)p 0 0.5 0 TeXcolorrgb 48170 46011 a SDict begin H.R end 48170 46011 a 48170 46867 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.3) cvn H.B /ANN pdfmark end 48170 46867 a 0 TeXcolorgray 360 w Fl(applies.)p 51860 48472 45 878 v 51905 47638 781 45 v 51905 48472 V 52684 48472 45 878 v 800 49428 a SDict begin H.S end 800 49428 a 800 49428 a SDict begin 14.5 H.A end 800 49428 a 800 49428 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.5) cvn H.B /DEST pdfmark end 800 49428 a 0 TeXcolorgray 1635 x Fe(Remark)547 b(5.)p 0 TeXcolorgray 2352 w Fl(This)476 b(result)g(w)-36 b(as)476 b(obtained)g(b)-36 b(y)475 b(Hurwitz)h(and)g(Minnegero)36 b(de)475 b(for)i(the)e(sp)36 b(ecial)477 b(case)800 52668 y Fj(\030)1368 52867 y Fo(0)2263 52668 y Fl(=)3644 51543 y Fh(p)p 4750 51543 1117 54 v 4750 52668 a Fj(D)37 b Fl(.)2751 55259 y(W)-108 b(e)434 b(giv)-36 b(e)434 b(some)g(examples.)p 0 TeXcolorgray 1739 57850 a(\(1\))p 0 TeXcolorgray 651 w Fj(\030)4620 58049 y Fo(0)5762 57850 y Fl(=)615 b(\(12)395 b(+)10995 56751 y Fh(p)p 12102 56751 2602 54 v 1099 x Fl(1792)r(\))p Fj(=)p Fl(16,)616 b(\(T)-108 b(ables)579 b(1)g(and)f(2\).)2027 b(Here)579 b Fj(k)661 b Fl(=)616 b(4)p Fj(;)221 b(r)653 b Fl(=)616 b(2,)f Fj(\030)42799 58049 y Fo(1)43325 57850 y Fj(;)221 b(\030)44475 58049 y Fo(2)45002 57850 y Fj(;)g(\030)46152 58049 y Fo(3)46678 57850 y Fj(;)g(\030)47828 58049 y Fo(4)48934 57850 y Fl(form)579 b(a)4052 59455 y(p)36 b(erio)g(d)417 b(of)h(NICF-P)f (complete)h(quotien)-36 b(ts,)420 b Fj(\017)26507 59654 y Fo(2)27033 59455 y Fj(\017)27558 59654 y Fo(3)28084 59455 y Fj(\017)28609 59654 y Fo(4)29135 59455 y Fj(\017)29660 59654 y Fo(5)30555 59455 y Fl(=)369 b(\()p Fh(\000)p Fl(1\)\(1\)\()p Fh(\000)p Fl(1\)\(1\))g(=)g(1)g(=)g(\()p Fh(\000)p Fl(1\))47495 58973 y Fi(k)48481 59455 y Fl(and)417 b(the)4052 61060 y(NICF-P)433 b(and)g(NICF-H)h(expansions)g(ha)-36 b(v)g(e)434 b(the)f(same)g(p)36 b(erio)g(d-length.)578 b(Also)434 b Fj(p)369 b Fl(=)g(6.)p 0 TeXcolorgray 1739 63731 a(\(2\))p 0 TeXcolorgray 651 w Fj(\030)4620 63930 y Fo(0)5659 63731 y Fl(=)513 b(\(5)353 b(+)10058 62633 y Fh(p)p 11165 62633 1301 54 v 1098 x Fl(13\))p Fj(=)p Fl(4,)541 b(\(T)-108 b(able)518 b(3)h(and)f(4\).)1666 b(Here)518 b Fj(k)559 b Fl(=)513 b(3)p Fj(;)221 b(r)550 b Fl(=)513 b(2)519 b(and)f Fj(\030)40465 63930 y Fo(0)40991 63731 y Fj(;)221 b(\030)42141 63930 y Fo(1)42667 63731 y Fj(;)g(\030)43817 63930 y Fo(2)44862 63731 y Fl(form)519 b(a)g(p)36 b(erio)g(d)4052 65337 y(of)455 b(NICF-P)g(complete)g (quotien)-36 b(ts,)460 b Fj(\017)22629 65536 y Fo(1)23155 65337 y Fj(\017)23680 65536 y Fo(2)24206 65337 y Fj(\017)24731 65536 y Fo(3)25663 65337 y Fl(=)404 b(\(1\)\()p Fh(\000)p Fl(1\)\()p Fh(\000)p Fl(1\))i(=)f(1)g(=)g(\()p Fh(\000)p Fl(1\))41121 64855 y Fi(k)24 b Fo(+1)43347 65337 y Fl(and)455 b(the)f(NICF-H)4052 66942 y(p)36 b(erio)g(d-length)432 b(is)i(t)-36 b(wice)434 b(the)f(NICF-P)h(p)36 b(erio)g(d-length.)578 b(Also)434 b Fj(p)369 b Fl(=)f(5.)p 0 TeXcolorgray 26475 70521 a(5)p 0 TeXcolorgray eop end end %%Page: 6 6 TeXDict begin HPSdict begin 6 5 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.6) cvn H.B /DEST pdfmark end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 800 -4000 a SDict begin 14.5 H.A end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (Item.2) cvn H.B /DEST pdfmark end 800 -4000 a 0 TeXcolorgray 12924 -1688 a Fl(T)-108 b(able)434 b(1:)p 0 TeXcolorgray 18090 -3293 a SDict begin H.S end 18090 -3293 a 0 TeXcolorgray 0 TeXcolorgray 18090 -3293 a SDict begin H.R end 18090 -3293 a 18090 -3293 a SDict begin [ /View [/XYZ H.V] /Dest (table.1) cvn H.B /DEST pdfmark end 18090 -3293 a 0 TeXcolorgray 1605 x Fl(NICF-P)g(expansion)g(for)g(\(12)295 b(+)34511 -2787 y Fh(p)p 35618 -2787 2602 54 v 1099 x Fl(1792)r(\))p Fj(=)p Fl(16)p 0 TeXcolorgray 800 -1201 19712 45 v 778 404 45 1606 v 1671 -77 a Fj(i)p 2757 404 V 3158 w(\030)5841 122 y Fi(i)p 8689 404 V 10158 -77 a Fj(\017)10683 122 y Fi(i)p 11701 404 V 12388 -77 a Fj(b)12941 122 y Fi(i)p 13958 404 V 15563 -77 a Fj(A)16538 122 y Fi(i)16914 -77 y Fj(=B)18552 122 y Fi(i)p 20489 404 V 800 448 19712 45 v 778 2267 45 1819 v 1464 1786 a Fl(0)p 2757 2267 V 3576 1261 a Fo(12+)5248 528 y Ff(p)p 6033 528 1882 40 v 6033 1261 a Fo(1792)p 3576 1480 4339 54 v 5275 2244 a(16)p 8689 2267 45 1819 v 10409 1786 a Fl(1)p 11701 2267 V 1468 w(3)p 13958 2267 V 3093 w(3)p Fj(=)p Fl(1)p 20489 2267 V 778 4086 V 1464 3604 a(1)p 2757 4086 V 3576 3080 a Fo(36+)5248 2347 y Ff(p)p 6033 2347 1882 40 v 6033 3080 a Fo(1792)p 3576 3299 4339 54 v 5275 4063 a(31)p 8689 4086 45 1819 v 10409 3604 a Fl(1)p 11701 4086 V 1468 w(3)p 13958 4086 V 2768 w(10)p Fj(=)p Fl(3)p 20489 4086 V 778 5905 V 1464 5423 a(2)p 2757 5905 V 3576 4899 a Fo(57+)5248 4166 y Ff(p)p 6033 4166 1882 40 v 6033 4899 a Fo(1792)p 3576 5118 4339 54 v 5275 5881 a(47)p 8689 5905 45 1819 v 9376 5423 a Fh(\000)p Fl(1)p 11701 5905 V 1468 w(2)p 13958 5905 V 2768 w(17)p Fj(=)p Fl(5)p 20489 5905 V 778 7723 V 1464 7242 a(3)p 2757 7723 V 3576 6718 a Fo(37+)5248 5984 y Ff(p)p 6033 5984 1882 40 v 6033 6718 a Fo(1792)p 3576 6936 4339 54 v 5510 7700 a(9)p 8689 7723 45 1819 v 10409 7242 a Fl(1)p 11701 7723 V 1468 w(9)p 13958 7723 V 2118 w(163)p Fj(=)p Fl(48)p 20489 7723 V 778 9542 V 1464 9061 a(4)p 2757 9542 V 3576 8536 a Fo(44+)5248 7803 y Ff(p)p 6033 7803 1882 40 v 6033 8536 a Fo(1792)p 3576 8755 4339 54 v 5275 9519 a(16)p 8689 9542 45 1819 v 9376 9061 a Fh(\000)p Fl(1)p 11701 9542 V 1468 w(5)p 13958 9542 V 1793 w(798)p Fj(=)p Fl(235)p 20489 9542 V 778 11361 V 1464 10879 a(5)p 2757 11361 V 3576 10355 a Fo(36+)5248 9622 y Ff(p)p 6033 9622 1882 40 v 6033 10355 a Fo(1792)p 3576 10574 4339 54 v 5275 11337 a(31)p 8689 11361 45 1819 v 10409 10879 a Fl(1)p 11701 11361 V 1468 w(3)p 13958 11361 V 1468 w(2557)p Fj(=)p Fl(753)p 20489 11361 V 800 11405 19712 45 v 0 TeXcolorgray 0 TeXcolorgray 800 12622 a SDict begin H.S end 800 12622 a 800 12622 a SDict begin 14.5 H.A end 800 12622 a 800 12622 a SDict begin [ /View [/XYZ H.V] /Dest (Item.3) cvn H.B /DEST pdfmark end 800 12622 a 0 TeXcolorgray 12879 14934 a Fl(T)-108 b(able)434 b(2:)p 0 TeXcolorgray 18045 13329 a SDict begin H.S end 18045 13329 a 0 TeXcolorgray 0 TeXcolorgray 18045 13329 a SDict begin H.R end 18045 13329 a 18045 13329 a SDict begin [ /View [/XYZ H.V] /Dest (table.2) cvn H.B /DEST pdfmark end 18045 13329 a 0 TeXcolorgray 1605 x Fl(NICF-H)g(expansion)g(for)g(\(12)295 b(+)34556 13835 y Fh(p)p 35663 13835 2602 54 v 1099 x Fl(1792)r(\))p Fj(=)p Fl(16)p 0 TeXcolorgray 800 15421 22533 45 v 778 17026 45 1606 v 1671 16545 a Fj(i)p 2757 17026 V 3158 w(\030)5841 16744 y Fi(i)p 8689 17026 V 10158 16545 a Fj(\017)10683 16744 y Fi(i)p 11701 17026 V 12765 16545 a Fj(b)13318 16744 y Fi(i)p 14713 17026 V 17351 16545 a Fj(A)18326 16744 y Fi(i)18702 16545 y Fj(=B)20340 16744 y Fi(i)p 23310 17026 V 800 17070 22533 45 v 778 18889 45 1819 v 1464 18408 a Fl(0)p 2757 18889 V 3576 17883 a Fo(12+)5248 17150 y Ff(p)p 6033 17150 1882 40 v 6033 17883 a Fo(1792)p 3576 18102 4339 54 v 5275 18866 a(16)p 8689 18889 45 1819 v 9376 18408 a Fh(\000)p Fl(1)p 11701 18889 V 1845 w(3)p 14713 18889 V 4504 w(3)p Fj(=)p Fl(1)p 23310 18889 V 778 20777 45 1888 v 1464 20226 a(1)p 2757 20777 V 3576 19702 a Fo(36+)5248 18969 y Ff(p)p 6033 18969 1882 40 v 6033 19702 a Fo(1792)p 3576 19921 4339 54 v 4909 20684 a Ff(\000)p Fo(31)p 8689 20777 45 1888 v 9376 20226 a Fh(\000)p Fl(1)p 11701 20777 V 1329 w Fh(\000)p Fl(3)p 14713 20777 V 2629 w Fh(\000)p Fl(10)q Fj(=)p Fh(\000)p Fl(3)p 23310 20777 V 778 22664 V 1464 22114 a(2)p 2757 22664 V 3576 21589 a Fo(57+)5248 20856 y Ff(p)p 6033 20856 1882 40 v 6033 21589 a Fo(1792)p 3576 21808 4339 54 v 4909 22572 a Ff(\000)p Fo(47)p 8689 22664 45 1888 v 9376 22114 a Fh(\000)p Fl(1)p 11701 22664 V 1329 w Fh(\000)p Fl(2)p 14713 22664 V 3662 w(17)p Fj(=)p Fl(5)p 23310 22664 V 778 24483 45 1819 v 1464 24001 a(3)p 2757 24483 V 3576 23477 a Fo(37+)5248 22744 y Ff(p)p 6033 22744 1882 40 v 6033 23477 a Fo(1792)p 3576 23696 4339 54 v 5510 24459 a(9)p 8689 24483 45 1819 v 9376 24001 a Fh(\000)p Fl(1)p 11701 24483 V 1845 w(9)p 14713 24483 V 3529 w(163)p Fj(=)p Fl(48)p 23310 24483 V 778 26302 V 1464 25820 a(4)p 2757 26302 V 3576 25296 a Fo(44+)5248 24563 y Ff(p)p 6033 24563 1882 40 v 6033 25296 a Fo(1792)p 3576 25515 4339 54 v 5275 26278 a(16)p 8689 26302 45 1819 v 9376 25820 a Fh(\000)p Fl(1)p 11701 26302 V 1845 w(5)p 14713 26302 V 3204 w(798)p Fj(=)p Fl(235)p 23310 26302 V 778 28189 45 1888 v 1464 27639 a(5)p 2757 28189 V 3576 27114 a Fo(36+)5248 26381 y Ff(p)p 6033 26381 1882 40 v 6033 27114 a Fo(1792)p 3576 27333 4339 54 v 4909 28097 a Ff(\000)p Fo(31)p 8689 28189 45 1888 v 9376 27639 a Fh(\000)p Fl(1)p 11701 28189 V 1329 w Fh(\000)p Fl(3)p 14713 28189 V 1328 w Fh(\000)p Fl(2557)r Fj(=)p Fh(\000)p Fl(753)p 23310 28189 V 800 28233 22533 45 v 0 TeXcolorgray 800 30336 a SDict begin H.S end 800 30336 a 800 30336 a SDict begin 14.5 H.A end 800 30336 a 800 30336 a SDict begin [ /View [/XYZ H.V] /Dest (section.3) cvn H.B /DEST pdfmark end 800 30336 a 1328 x Fm(3)2152 b(NICF-P)718 b(midp)60 b(oin)-60 b(t)715 b(criteria)i(for)g(P)-60 b(ell's)716 b(equation)800 33351 y SDict begin H.S end 800 33351 a 800 33351 a SDict begin 14.5 H.A end 800 33351 a 800 33351 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.6) cvn H.B /DEST pdfmark end 800 33351 a 0 TeXcolorgray 1233 x Fe(Theorem)569 b(6.)p 0 TeXcolorgray 585 w Fk(L)-66 b(et)520 b Fj(k)566 b Fk(and)520 b Fj(p)h Fk(b)-66 b(e)520 b(the)g(r)-66 b(esp)g(e)g(ctive)518 b(p)-66 b(erio)g(d-lengths)520 b(of)g(NICF-P)g(and)h(R)-33 b(CF)520 b(exp)-66 b(ansions)800 36190 y(of)2328 35065 y Fh(p)p 3435 35065 1117 54 v 1125 x Fj(D)36 b Fk(.)597 b(Then)464 b(pr)-66 b(e)g(cisely)464 b(one)g(of)h(the)f(fol)66 b(lowing)465 b(must)g(hold)h(for)e(the)h (NICF-P)f(exp)-66 b(ansion)464 b(of)48593 35065 y Fh(p)p 49700 35065 V 1125 x Fj(D)36 b Fk(:)p 0 TeXcolorgray 2205 38734 a(1\))p 0 TeXcolorgray 652 w Fj(P)4890 38933 y Fi(\032)5796 38734 y Fl(=)369 b Fj(P)8015 38933 y Fi(\032)p Fo(+1)9755 38734 y Fk(,)464 b Fj(k)414 b Fl(=)369 b(2)p Fj(\032;)221 b(p)369 b Fl(=)g(2)p Fj(h)p Fk(.)597 b(Then)19831 41472 y Fj(A)20806 41671 y Fi(k)24 b Ff(\000)p Fo(1)22946 41472 y Fl(=)369 b Fj(B)25315 41671 y Fi(\032)p Ff(\000)p Fo(1)27054 41472 y Fj(A)28029 41671 y Fi(\032)28862 41472 y Fl(+)294 b Fj(\017)30693 41671 y Fi(\032)31231 41472 y Fj(A)32206 41671 y Fi(\032)p Ff(\000)p Fo(1)33945 41472 y Fj(B)34933 41671 y Fi(\032)p Ff(\000)p Fo(2)36673 41472 y Fj(;)19818 43409 y(B)20806 43608 y Fi(k)24 b Ff(\000)p Fo(1)22946 43409 y Fl(=)369 b Fj(B)25315 43608 y Fi(\032)p Ff(\000)p Fo(1)27054 43409 y Fl(\()p Fj(B)28548 43608 y Fi(\032)29380 43409 y Fl(+)295 b Fj(\017)31212 43608 y Fi(\032)31750 43409 y Fj(B)32738 43608 y Fi(\032)p Ff(\000)p Fo(2)34477 43409 y Fl(\))p Fj(:)p 0 TeXcolorgray 2205 46673 a Fk(2\))p 0 TeXcolorgray 652 w Fj(P)4890 46872 y Fi(\032)p Fo(+1)6999 46673 y Fl(=)368 b Fj(P)9217 46872 y Fi(\032)10050 46673 y Fl(+)295 b Fj(Q)12387 46872 y Fi(\032)12924 46673 y Fk(,)464 b Fj(k)414 b Fl(=)369 b(2)p Fj(\032;)221 b(p)370 b Fl(=)e(2)p Fj(h)p Fk(.)598 b(Then)16829 49411 y Fj(A)17804 49610 y Fi(k)24 b Ff(\000)p Fo(1)19944 49411 y Fl(=)369 b Fj(B)22313 49610 y Fi(\032)p Ff(\000)p Fo(1)24052 49411 y Fj(A)25027 49610 y Fi(\032)25860 49411 y Fl(+)295 b Fj(A)28142 49610 y Fi(\032)p Ff(\000)p Fo(1)29881 49411 y Fj(B)30869 49610 y Fi(\032)p Ff(\000)p Fo(2)32904 49411 y Fh(\000)g Fj(A)35207 49610 y Fi(\032)p Ff(\000)p Fo(1)36947 49411 y Fj(B)37935 49610 y Fi(\032)p Ff(\000)p Fo(1)39674 49411 y Fj(;)16816 51348 y(B)17804 51547 y Fi(k)24 b Ff(\000)p Fo(1)19944 51348 y Fl(=)369 b Fj(B)22313 51547 y Fi(\032)p Ff(\000)p Fo(1)24052 51348 y Fl(\()p Fj(B)25546 51547 y Fi(\032)26379 51348 y Fl(+)295 b Fj(B)28674 51547 y Fi(\032)p Ff(\000)p Fo(2)30708 51348 y Fh(\000)h Fj(B)33025 51547 y Fi(\032)p Ff(\000)p Fo(1)34764 51348 y Fl(\))p Fj(:)p 0 TeXcolorgray 2205 54611 a Fk(3\))p 0 TeXcolorgray 652 w Fj(Q)5082 54810 y Fi(\032)5988 54611 y Fl(=)369 b Fj(Q)8399 54810 y Fi(\032)p Fo(+1)10603 54611 y Fk(and)17483 57349 y Fl(\(a\))1328 b Fj(\017)20998 57548 y Fi(\032)p Fo(+1)23107 57349 y Fl(=)368 b Fh(\000)p Fl(1)p Fj(;)221 b(k)415 b Fl(=)369 b(2)p Fj(\032)295 b Fl(+)g(1)p Fj(;)221 b(p)370 b Fl(=)f(2)p Fj(h;)686 b Fk(or)17410 59286 y Fl(\(b\))1328 b Fj(\017)20998 59485 y Fi(\032)p Fo(+1)23107 59286 y Fl(=)368 b(1)p Fj(;)221 b(k)415 b Fl(=)369 b(2)p Fj(\032)295 b Fl(+)g(1)p Fj(;)221 b(p)370 b Fl(=)e(2)p Fj(h)296 b Fh(\000)f Fl(1)p Fj(:)4052 62024 y Fk(Then)19831 64762 y Fj(A)20806 64961 y Fi(k)24 b Ff(\000)p Fo(1)22946 64762 y Fl(=)369 b Fj(A)25302 64961 y Fi(\032)25839 64762 y Fj(B)26827 64961 y Fi(\032)27659 64762 y Fl(+)295 b Fj(\017)29491 64961 y Fi(\032)p Fo(+1)31231 64762 y Fj(A)32206 64961 y Fi(\032)p Ff(\000)p Fo(1)33945 64762 y Fj(B)34933 64961 y Fi(\032)p Ff(\000)p Fo(1)36673 64762 y Fj(;)19818 66699 y(B)20806 66898 y Fi(k)24 b Ff(\000)p Fo(1)22946 66699 y Fl(=)369 b Fj(B)25382 66151 y Fo(2)25315 67028 y Fi(\032)26203 66699 y Fl(+)294 b Fj(\017)28034 66898 y Fi(\032)p Fo(+1)29774 66699 y Fj(B)30829 66151 y Fo(2)30762 67028 y Fi(\032)p Ff(\000)p Fo(1)32502 66699 y Fj(:)p 0 TeXcolorgray 26475 70521 a Fl(6)p 0 TeXcolorgray eop end end %%Page: 7 7 TeXDict begin HPSdict begin 7 6 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.7) cvn H.B /DEST pdfmark end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 800 -4000 a SDict begin 14.5 H.A end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (Item.4) cvn H.B /DEST pdfmark end 800 -4000 a 0 TeXcolorgray 14224 -1688 a Fl(T)-108 b(able)434 b(3:)p 0 TeXcolorgray 19391 -3293 a SDict begin H.S end 19391 -3293 a 0 TeXcolorgray 0 TeXcolorgray 19391 -3293 a SDict begin H.R end 19391 -3293 a 19391 -3293 a SDict begin [ /View [/XYZ H.V] /Dest (table.3) cvn H.B /DEST pdfmark end 19391 -3293 a 0 TeXcolorgray 1605 x Fl(NICF-P)f(expansion)h(for)g(\(5)296 b(+)35162 -2787 y Fh(p)p 36269 -2787 1301 54 v 1099 x Fl(13\))p Fj(=)p Fl(4)p 0 TeXcolorgray 800 -1201 16463 45 v 778 404 45 1606 v 1671 -77 a Fj(i)p 2757 404 V 2453 w(\030)5136 122 y Fi(i)p 7278 404 V 8747 -77 a Fj(\017)9272 122 y Fi(i)p 10290 404 V 10976 -77 a Fj(b)11529 122 y Fi(i)p 12547 404 V 13233 -77 a Fj(A)14208 122 y Fi(i)14584 -77 y Fj(=B)16222 122 y Fi(i)p 17240 404 V 800 448 16463 45 v 778 2267 45 1819 v 1464 1786 a Fl(0)p 2757 2267 V 3576 1261 a Fo(5+)4778 528 y Ff(p)p 5562 528 941 40 v 733 x Fo(13)p 3576 1480 2928 54 v 4804 2244 a(4)p 7278 2267 45 1819 v 8997 1786 a Fl(1)p 10290 2267 V 1468 w(2)p 12547 2267 V 2175 w(2)p Fj(=)p Fl(1)p 17240 2267 V 778 4086 V 1464 3604 a(1)p 2757 4086 V 3576 3080 a Fo(3+)4778 2347 y Ff(p)p 5562 2347 941 40 v 733 x Fo(13)p 3576 3299 2928 54 v 4804 4063 a(1)p 7278 4086 45 1819 v 8997 3604 a Fl(1)p 10290 4086 V 1468 w(7)p 12547 4086 V 1850 w(15)p Fj(=)p Fl(7)p 17240 4086 V 778 5905 V 1464 5423 a(2)p 2757 5905 V 3576 4899 a Fo(4+)4778 4166 y Ff(p)p 5562 4166 941 40 v 733 x Fo(13)p 3576 5118 2928 54 v 4804 5881 a(3)p 7278 5905 45 1819 v 7964 5423 a Fh(\000)p Fl(1)p 10290 5905 V 1468 w(3)p 12547 5905 V 1525 w(43)p Fj(=)p Fl(20)p 17240 5905 V 778 7723 V 1464 7242 a(3)p 2757 7723 V 3576 6718 a Fo(5+)4778 5984 y Ff(p)p 5562 5984 941 40 v 734 x Fo(13)p 3576 6936 2928 54 v 4804 7700 a(4)p 7278 7723 45 1819 v 7964 7242 a Fh(\000)p Fl(1)p 10290 7723 V 1468 w(2)p 12547 7723 V 1525 w(71)p Fj(=)p Fl(33)p 17240 7723 V 800 7768 16463 45 v 0 TeXcolorgray 0 TeXcolorgray 800 9090 a SDict begin H.S end 800 9090 a 800 9090 a SDict begin 14.5 H.A end 800 9090 a 800 9090 a SDict begin [ /View [/XYZ H.V] /Dest (Item.5) cvn H.B /DEST pdfmark end 800 9090 a 0 TeXcolorgray 14179 11402 a Fl(T)-108 b(able)434 b(4:)p 0 TeXcolorgray 19346 9797 a SDict begin H.S end 19346 9797 a 0 TeXcolorgray 0 TeXcolorgray 19346 9797 a SDict begin H.R end 19346 9797 a 19346 9797 a SDict begin [ /View [/XYZ H.V] /Dest (table.4) cvn H.B /DEST pdfmark end 19346 9797 a 0 TeXcolorgray 1605 x Fl(NICF-H)f(expansion)h(for)g(\(5)296 b(+)35207 10304 y Fh(p)p 36314 10304 1301 54 v 1098 x Fl(13\))p Fj(=)p Fl(4)p 0 TeXcolorgray 800 11889 21772 45 v 778 13495 45 1606 v 1671 13013 a Fj(i)p 2757 13495 V 2453 w(\030)5136 13212 y Fi(i)p 7278 13495 V 8747 13013 a Fj(\017)9272 13212 y Fi(i)p 10290 13495 V 11354 13013 a Fj(b)11907 13212 y Fi(i)p 13302 13495 V 16265 13013 a Fj(A)17240 13212 y Fi(i)17616 13013 y Fj(=B)19254 13212 y Fi(i)p 22549 13495 V 800 13539 21772 45 v 778 15358 45 1819 v 1464 14876 a Fl(0)p 2757 15358 V 3576 14352 a Fo(5+)4778 13619 y Ff(p)p 5562 13619 941 40 v 733 x Fo(13)p 3576 14571 2928 54 v 4804 15334 a(4)p 7278 15358 45 1819 v 8997 14876 a Fl(1)p 10290 15358 V 1846 w(2)p 13302 15358 V 4829 w(2)p Fj(=)p Fl(1)p 22549 15358 V 778 17245 45 1888 v 1464 16695 a(1)p 2757 17245 V 3576 16170 a Fo(3+)4778 15437 y Ff(p)p 5562 15437 941 40 v 733 x Fo(13)p 3576 16389 2928 54 v 4438 17153 a Ff(\000)p Fo(1)p 7278 17245 45 1888 v 7964 16695 a Fh(\000)p Fl(1)p 10290 17245 V 1329 w Fh(\000)p Fl(7)p 13302 17245 V 2955 w Fh(\000)p Fl(15)p Fj(=)p Fh(\000)p Fl(7)p 22549 17245 V 778 19133 V 1464 18582 a(2)p 2757 19133 V 3576 18058 a Fo(4+)4778 17325 y Ff(p)p 5562 17325 941 40 v 733 x Fo(13)p 3576 18277 2928 54 v 4438 19040 a Ff(\000)p Fo(3)p 7278 19133 45 1888 v 7964 18582 a Fh(\000)p Fl(1)p 10290 19133 V 1329 w Fh(\000)p Fl(3)p 13302 19133 V 3663 w(43)p Fj(=)p Fl(20)p 22549 19133 V 778 21020 V 1464 20470 a(3)p 2757 21020 V 3576 19945 a Fo(5+)4778 19212 y Ff(p)p 5562 19212 941 40 v 733 x Fo(13)p 3576 20164 2928 54 v 4438 20928 a Ff(\000)p Fo(4)p 7278 21020 45 1888 v 7964 20470 a Fh(\000)p Fl(1)p 10290 21020 V 1329 w Fh(\000)p Fl(2)p 13302 21020 V 2630 w Fh(\000)p Fl(71)p Fj(=)p Fh(\000)p Fl(33)p 22549 21020 V 778 22839 45 1819 v 1464 22357 a(4)p 2757 22839 V 3576 21833 a Fo(3+)4778 21100 y Ff(p)p 5562 21100 941 40 v 733 x Fo(13)p 3576 22052 2928 54 v 4804 22815 a(1)p 7278 22839 45 1819 v 7964 22357 a Fh(\000)p 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y Fj(Q)13250 30256 y Fi(\032)p Fo(+1)14990 30057 y Fj(;)221 b(\017)16097 30256 y Fi(\032)p Fo(+1)18206 30057 y Fl(=)369 b Fh(\000)p Fl(1)p Fj(;)221 b(k)415 b Fl(=)368 b(2)p Fj(\032)296 b Fl(+)f(1)p Fj(;)221 b(p)369 b Fl(=)g(2)p Fj(h)295 b Fh(\000)h Fl(1)p Fk(.)597 b(Then)14279 32981 y Fj(A)15254 33180 y Fi(k)24 b Ff(\000)p Fo(1)17395 32981 y Fl(=)368 b Fj(A)19750 33180 y Fi(\032)20287 32981 y Fj(B)21275 33180 y Fi(\032)22108 32981 y Fl(+)295 b(2)p Fj(A)25040 33180 y Fi(\032)p Ff(\000)p Fo(1)26780 32981 y Fj(B)27768 33180 y Fi(\032)p Ff(\000)p Fo(1)29802 32981 y Fh(\000)h Fl(\()p Fj(A)32612 33180 y Fi(\032)33149 32981 y Fj(B)34137 33180 y Fi(\032)p Ff(\000)p Fo(1)36172 32981 y Fl(+)e Fj(B)38466 33180 y Fi(\032)39004 32981 y Fj(A)39979 33180 y Fi(\032)p Ff(\000)p Fo(1)41718 32981 y Fl(\))p Fj(;)14266 34918 y(B)15254 35117 y Fi(k)24 b Ff(\000)p Fo(1)17395 34918 y Fl(=)368 b Fj(B)19830 34369 y Fo(2)19763 35246 y Fi(\032)20651 34918 y Fl(+)295 b(2)p Fj(B)23663 34369 y Fo(2)23596 35246 y Fi(\032)p Ff(\000)p Fo(1)25631 34918 y Fh(\000)h Fl(2)p Fj(B)28598 35117 y Fi(\032)29135 34918 y Fj(B)30123 35117 y Fi(\032)p Ff(\000)p Fo(1)31863 34918 y Fj(:)p 0 TeXcolorgray 2205 38393 a Fk(5\))p 0 TeXcolorgray 652 w Fj(P)4890 38592 y Fi(\032)5796 38393 y Fl(=)369 b Fj(Q)8207 38592 y Fi(\032)9039 38393 y Fl(+)10479 38031 y Fg(1)p 10479 38192 V 10479 38741 a(2)11018 38393 y Fj(Q)12048 38592 y Fi(\032)p Ff(\000)p Fo(1)13787 38393 y Fj(;)221 b(\017)14894 38592 y Fi(\032)15801 38393 y Fl(=)369 b Fh(\000)p Fl(1)p Fj(;)221 b(k)415 b Fl(=)369 b(2)p Fj(\032;)221 b(p)369 b Fl(=)g(2)p Fj(h)295 b Fh(\000)h Fl(1)p Fk(.)597 b(Then)12056 41317 y Fj(A)13031 41516 y Fi(k)24 b Ff(\000)p Fo(1)15171 41317 y Fl(=)368 b(2)p Fj(A)18176 41516 y Fi(\032)p Ff(\000)p Fo(1)19916 41317 y Fj(B)20904 41516 y Fi(\032)p Ff(\000)p Fo(1)22939 41317 y Fl(+)295 b Fj(A)25221 41516 y Fi(\032)p Ff(\000)p Fo(2)26960 41317 y Fj(B)27948 41516 y Fi(\032)p Ff(\000)p Fo(2)29983 41317 y Fh(\000)g Fl(\()p Fj(A)32792 41516 y Fi(\032)p Ff(\000)p Fo(1)34532 41317 y Fj(B)35520 41516 y Fi(\032)p Ff(\000)p Fo(2)37555 41317 y Fl(+)f Fj(B)39849 41516 y Fi(\032)p Ff(\000)p Fo(1)41589 41317 y Fj(A)42564 41516 y Fi(\032)p Ff(\000)p Fo(2)44303 41317 y Fl(\))12042 43254 y Fj(B)13030 43453 y Fi(k)24 b Ff(\000)p Fo(1)15171 43254 y Fl(=)368 b(2)p Fj(B)18256 42706 y Fo(2)18189 43582 y Fi(\032)p Ff(\000)p Fo(1)20224 43254 y Fl(+)295 b Fj(B)22586 42706 y Fo(2)22519 43582 y Fi(\032)p Ff(\000)p Fo(2)24554 43254 y Fh(\000)g Fl(2)p Fj(B)27520 43453 y Fi(\032)p Ff(\000)p Fo(1)29260 43254 y Fj(B)30248 43453 y Fi(\032)p Ff(\000)p Fo(2)31988 43254 y Fj(:)p 0 TeXcolorgray 800 46730 a Fk(Pr)-66 b(o)g(of.)p 0 TeXcolorgray 649 w Fl(W)-108 b(e)469 b(only)h(exhibit)f(the)g(calculations)h(for)g (criterion)f(1\))g(of)h(Theorem)p 0 0.5 0 TeXcolorrgb 39775 46730 a SDict begin H.S end 39775 46730 a 0 0.5 0 TeXcolorrgb Fl(6)p 0 0.5 0 TeXcolorrgb 40425 45874 a SDict begin H.R end 40425 45874 a 40425 46730 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.6) cvn H.B /ANN pdfmark end 40425 46730 a 0 TeXcolorgray Fl(.)685 b(This)469 b(corresp)36 b(onds)469 b(to)800 48335 y(criterion)444 b(1\))g(of)g(Theorem)g(6,)j(Williams)e([)p 0 0.5 0 TeXcolorrgb 21897 48335 a SDict begin H.S end 21897 48335 a 0 0.5 0 TeXcolorrgb Fl(9)p 0 0.5 0 TeXcolorrgb 22548 47479 a SDict begin H.R end 22548 47479 a 22548 48335 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.hcw) cvn H.B /ANN pdfmark end 22548 48335 a 0 TeXcolorgray Fl(,)h(p.)608 b(12],)448 b(whic)-36 b(h)443 b(states)h(that)f Fj(P)38805 47853 y Ff(0)38624 48663 y Fi(\032)39548 48335 y Fl(=)385 b Fj(P)41964 47853 y Ff(0)41783 48663 y Fi(\032)p Fo(+1)43523 48335 y Fl(,)447 b Fj(k)431 b Fl(=)386 b(2)p Fj(\032)p Fl(,)446 b Fj(p)386 b Fl(=)g(2)p Fj(h)800 49940 y Fl(and)17987 52863 y Fh(j)p Fj(A)19331 52315 y Ff(0)19331 53192 y Fi(k)24 b Ff(\000)p Fo(1)21103 52863 y Fh(j)368 b Fl(=)h Fh(j)p Fj(B)24645 52315 y Ff(0)24578 53192 y Fi(\032)p Ff(\000)p Fo(1)26318 52863 y Fj(A)27293 52315 y Ff(0)27293 53192 y Fi(\032)28125 52863 y Fh(\000)295 b Fj(A)30428 52315 y Ff(0)30428 53192 y Fi(\032)p Ff(\000)p Fo(1)32168 52863 y Fj(B)33223 52315 y Ff(0)33156 53192 y Fi(\032)p Ff(\000)p Fo(2)34896 52863 y Fh(j)p Fj(;)50488 50926 y SDict begin H.S end 50488 50926 a 50488 50926 a SDict begin 17.5 H.A end 50488 50926 a 50488 50926 a SDict begin [ /View [/XYZ H.V] /Dest (equation.16) cvn H.B /DEST pdfmark end 50488 50926 a 1937 x Fl(\(16\))17974 54930 y Fh(j)p Fj(B)19398 54382 y Ff(0)19331 55259 y Fi(k)24 b Ff(\000)p Fo(1)21103 54930 y Fh(j)368 b Fl(=)h Fh(j)p Fj(B)24645 54382 y Ff(0)24578 55259 y Fi(\032)p Ff(\000)p Fo(1)26318 54930 y Fl(\()p Fj(B)27879 54382 y Ff(0)27812 55259 y Fi(\032)28644 54930 y Fh(\000)295 b Fj(B)31027 54382 y Ff(0)30960 55259 y Fi(\032)p Ff(\000)p Fo(2)32700 54930 y Fl(\))p Fh(j)p Fj(:)50488 52993 y SDict begin H.S end 50488 52993 a 50488 52993 a SDict begin 17.5 H.A end 50488 52993 a 50488 52993 a SDict begin [ /View [/XYZ H.V] /Dest (equation.17) cvn H.B /DEST pdfmark end 50488 52993 a 1937 x Fl(\(17\))800 57854 y(Then)433 b(from)h(equations)g(\()p 0 0.5 0 TeXcolorrgb 13597 57854 a SDict begin H.S end 13597 57854 a 0 0.5 0 TeXcolorrgb Fl(7)p 0 0.5 0 TeXcolorrgb 14247 56998 a SDict begin H.R end 14247 56998 a 14247 57854 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.7) cvn H.B /ANN pdfmark end 14247 57854 a 0 TeXcolorgray Fl(\),)10517 60777 y Fj(B)11572 60229 y Ff(0)11505 61106 y Fi(\032)p Ff(\000)p Fo(1)13244 60777 y Fj(A)14219 60229 y Ff(0)14219 61106 y Fi(\032)15052 60777 y Fh(\000)295 b Fj(A)17355 60229 y Ff(0)17355 61106 y Fi(\032)p Ff(\000)p Fo(1)19094 60777 y Fj(B)20149 60229 y Ff(0)20082 61106 y Fi(\032)p Ff(\000)p Fo(2)22191 60777 y Fl(=)369 b Fj(s)24185 60976 y Fi(\032)p Ff(\000)p Fo(1)25924 60777 y Fj(s)26537 60976 y Fi(\032)27074 60777 y Fj(B)28062 60976 y Fi(\032)p Ff(\000)p Fo(1)29801 60777 y Fj(A)30776 60976 y Fi(\032)31609 60777 y Fh(\000)295 b Fj(s)33550 60976 y Fi(\032)p Ff(\000)p Fo(1)35289 60777 y Fj(s)35902 60976 y Fi(\032)p Ff(\000)p Fo(2)37641 60777 y Fj(A)38616 60976 y Fi(\032)p Ff(\000)p Fo(1)40356 60777 y Fj(B)41344 60976 y Fi(\032)p Ff(\000)p Fo(2)22191 62844 y Fl(=)369 b Fj(t)24042 63043 y Fi(\032)24579 62844 y Fj(B)25567 63043 y Fi(\032)p Ff(\000)p Fo(1)27306 62844 y Fj(A)28281 63043 y Fi(\032)29114 62844 y Fh(\000)295 b Fj(t)30912 63043 y Fi(\032)p Ff(\000)p Fo(1)32651 62844 y Fj(A)33626 63043 y Fi(\032)p Ff(\000)p Fo(1)35366 62844 y Fj(B)36354 63043 y Fi(\032)p Ff(\000)p Fo(2)22191 64781 y Fl(=)369 b Fh(\000)p Fj(t)25075 64980 y Fi(\032)p Ff(\000)p Fo(1)26814 64781 y Fj(\017)27339 64980 y Fi(\032)27876 64781 y Fj(B)28864 64980 y Fi(\032)p Ff(\000)p Fo(1)30604 64781 y Fj(A)31579 64980 y Fi(\032)32411 64781 y Fh(\000)296 b Fj(t)34210 64980 y Fi(\032)p Ff(\000)p Fo(1)35949 64781 y Fj(A)36924 64980 y Fi(\032)p Ff(\000)p Fo(1)38663 64781 y Fj(B)39651 64980 y Fi(\032)p Ff(\000)p Fo(2)22191 66718 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23493 6214 a SDict begin H.R end 23493 6214 a 23493 7070 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.20) cvn H.B /ANN pdfmark end 23493 7070 a 0 TeXcolorgray Fl(\))o Fk(,)465 b(in)f(r)-66 b(elation)463 b(to)i(The)-66 b(or)g(em)p 0 0.5 0 TeXcolorrgb 38179 7070 a SDict begin H.S end 38179 7070 a 0 0.5 0 TeXcolorrgb Fk(6)p 0 0.5 0 TeXcolorrgb 38843 6208 a SDict begin H.R end 38843 6208 a 38843 7070 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.6) cvn H.B /ANN pdfmark end 38843 7070 a 0 TeXcolorgray Fk(,)464 b(we)h(have)p 0 TeXcolorgray 1675 9782 a(\(1\))p 0 TeXcolorgray 651 w(If)f Fj(p)369 b Fl(=)f(2)p Fj(h)295 b Fh(\000)h Fl(1)p Fj(;)221 b(h)369 b(>)g Fl(1)465 b Fk(and)g Fj(a)18894 9981 y Fi(h)p Ff(\000)p Fo(1)21063 9782 y Fj(>)368 b Fl(1)p Fk(,)465 b(or)g Fj(p)369 b Fl(=)g(1)p Fk(,)464 b(we)i(get)e Fl(criterion)433 b(3\))p Fk(.)p 0 TeXcolorgray 1675 12494 a(\(2\))p 0 TeXcolorgray 651 w(If)464 b Fj(p)369 b 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Fj(M)604 b Fk(is)465 b(o)-66 b(dd,)464 b(we)i(get)e Fl(criterion)433 b(1\))p Fk(.)p 0 TeXcolorgray 1675 24394 a(\(4\))p 0 TeXcolorgray 651 w(Supp)-66 b(ose)465 b(the)f(c)-66 b(entr)g(e)463 b(of)i(a)g(p)-66 b(erio)g(d)464 b(c)-66 b(ontains)463 b(an)i Fj(M)139 b Fk(-unise)-66 b(quenc)g(e,)462 b Fj(M)508 b Fh(\025)369 b Fl(1)p Fk(.)p 0 TeXcolorgray 4536 27106 a(\(a\))p 0 TeXcolorgray 651 w(If)533 b Fj(M)673 b Fk(is)533 b(o)-66 b(dd,)551 b(then)533 b Fj(p)496 b Fl(=)g(2)p Fj(h)534 b Fk(and)g(we)f(get)g Fl(criterion)509 b(1\))534 b Fk(if)f Fj(M)635 b Fl(=)496 b(4)p Fj(t)346 b Fl(+)g(3)p Fk(,)551 b Fl(criterion)508 b(3\))534 b Fk(if)6913 28711 y Fj(M)508 b Fl(=)369 b(4)p Fj(t)295 b Fl(+)g(1)p Fk(.)p 0 TeXcolorgray 4602 30869 a(\(b\))p 0 TeXcolorgray 651 w(If)508 b Fj(M)647 b Fk(is)509 b(even,)518 b(then)507 b Fj(p)450 b Fl(=)f(2)p Fj(h)328 b Fh(\000)f Fl(1)509 b Fk(and)g(we)f(get)g Fl(criterion)481 b(4\))509 b Fk(if)e Fj(M)589 b Fl(=)449 b(4)p Fj(t)p Fk(,)519 b Fl(criterion)481 b(5\))509 b Fk(if)6913 32474 y Fj(M)f Fl(=)369 b(4)p Fj(t)295 b Fl(+)g(2)p Fk(.)2751 35186 y Fl(Before)359 b(w)-36 b(e)358 b(can)g(pro)-36 b(v)g(e)358 b(Theorem)p 0 0.5 0 TeXcolorrgb 19733 35186 a SDict begin H.S end 19733 35186 a 0 0.5 0 TeXcolorrgb Fl(8)p 0 0.5 0 TeXcolorrgb 20383 34330 a SDict begin H.R end 20383 34330 a 20383 35186 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.8) cvn H.B /ANN pdfmark end 20383 35186 a 0 TeXcolorgray Fl(,)374 b(w)-36 b(e)358 b(need)f(some)h(results)g(on)g(the)f(R)-36 b(CF)359 b(to)f(NICF-P)f(con)-36 b(v)g(ersion.)800 37446 y SDict begin H.S end 800 37446 a 800 37446 a SDict begin 14.5 H.A end 800 37446 a 800 37446 a SDict begin [ /View [/XYZ H.V] /Dest (section.5) cvn H.B /DEST pdfmark end 800 37446 a 2177 x Fm(5)2152 b(The)716 b(R)-60 b(CF)718 b(to)f(NICF-P)g (con)-60 b(v)g(ersion)717 b(and)f(its)h(prop)60 b(erties)800 41311 y SDict begin H.S end 800 41311 a 800 41311 a SDict begin 14.5 H.A end 800 41311 a 800 41311 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.9) cvn H.B /DEST pdfmark end 800 41311 a 0 TeXcolorgray 1475 x Fe(Lemma)499 b(9.)p 0 TeXcolorgray 554 w Fk(L)-66 b(et)464 b Fj(\030)10622 42985 y Fo(0)11517 42786 y Fl(=)13030 42252 y Fi(P)13627 42375 y Fg(0)14088 42252 y Fo(+)14820 41501 y Ff(p)p 15604 41501 799 40 v 751 x Fi(D)p 13030 42481 3372 54 v 14115 43244 a(Q)14855 43367 y Fg(0)16999 42786 y Fk(have)h(NICF-P)f (and)h(R)-33 b(CF)465 b(exp)-66 b(ansions:)16296 46751 y Fj(\030)16864 46950 y Fo(0)17759 46751 y Fl(=)368 b Fj(a)19822 46202 y Ff(0)19822 47079 y Fo(0)20643 46751 y Fl(+)22162 45852 y Fj(\017)22687 46051 y Fo(1)23213 45852 y Fh(j)p 22083 46445 1578 54 v 22083 47662 a(j)p Fj(a)23135 47204 y Ff(0)23135 47987 y Fo(1)24088 46751 y Fl(+)295 b Fh(\001)221 b(\001)g(\001)370 b Fl(=)f Fj(a)29378 46950 y Fo(0)30198 46751 y Fl(+)31917 45852 y(1)p Fh(j)p 31638 46445 V 31638 47662 a(j)p Fj(a)32690 47861 y Fo(1)33644 46751 y Fl(+)294 b Fh(\001)221 b(\001)g(\001)444 b Fj(;)800 50429 y Fk(with)468 b(c)-66 b(omplete)466 b(quotients)h Fj(\030)15156 49947 y Ff(0)15092 50758 y Fi(m)15979 50429 y Fj(;)221 b(\030)17129 50628 y Fi(m)18018 50429 y Fk(,)468 b(r)-66 b(esp)g(e)g(ctively.)603 b(De\014ne)466 b Fj(f)142 b Fl(\()p Fj(m)p Fl(\))467 b Fk(r)-66 b(e)g(cursively)466 b(for)h Fj(m)374 b Fh(\025)h Fl(0)468 b Fk(by)f Fj(f)142 b Fl(\(0\))374 b(=)g(0)800 52034 y Fk(and)3324 50429 y SDict begin H.S end 3324 50429 a 3324 50429 a SDict begin 14.5 H.A end 3324 50429 a 3324 50429 a SDict begin [ /View [/XYZ H.V] /Dest (equation.24) cvn H.B /DEST pdfmark end 3324 50429 a 15440 54470 a Fj(f)142 b Fl(\()p Fj(m)295 b Fl(+)f(1\))370 b(=)22374 52198 y Fd(\()23444 53560 y Fj(f)142 b Fl(\()p Fj(m)p Fl(\))295 b(+)g(1)p Fj(;)1329 b Fk(if)463 b Fj(\017)32105 53759 y Fi(m)p Fo(+1)34565 53560 y Fl(=)368 b(1)p Fk(;)23444 55486 y Fj(f)142 b Fl(\()p Fj(m)p Fl(\))295 b(+)g(2)p Fj(;)1329 b Fk(if)463 b Fj(\017)32105 55685 y Fi(m)p Fo(+1)34565 55486 y Fl(=)368 b Fh(\000)p Fl(1)p Fk(.)50488 54470 y Fl(\(24\))800 57883 y Fk(Then)464 b(for)h Fj(m)369 b Fh(\025)g Fl(0)p Fk(,)10686 56278 y SDict begin H.S end 10686 56278 a 10686 56278 a SDict begin 14.5 H.A end 10686 56278 a 10686 56278 a SDict begin [ /View [/XYZ H.V] /Dest (equation.25) cvn H.B /DEST pdfmark end 10686 56278 a 18272 60576 a Fj(\017)18797 60775 y Fi(m)p Fo(+1)21256 60576 y Fl(=)22637 58305 y Fd(\()23707 59666 y Fl(1)p Fj(;)2362 b Fk(if)464 b Fj(a)29025 59873 y Fi(f)98 b Fo(\()p Fi(m)p Fo(\)+1)32765 59666 y Fj(>)369 b Fl(1)p Fk(;)23707 61593 y Fh(\000)p Fl(1)p Fj(;)1329 b Fk(if)464 b Fj(a)29025 61799 y Fi(f)98 b Fo(\()p Fi(m)p Fo(\)+1)32765 61593 y Fl(=)369 b(1)p Fk(,)50488 60576 y Fl(\(25\))800 61462 y SDict begin H.S end 800 61462 a 800 61462 a SDict begin 14.5 H.A end 800 61462 a 800 61462 a SDict begin [ /View [/XYZ H.V] /Dest (equation.26) cvn H.B /DEST pdfmark end 800 61462 a 18003 65502 a Fj(\030)18635 64954 y Ff(0)18571 65831 y Fi(m)19828 65502 y Fl(=)21208 63231 y Fd(\()22279 64592 y Fj(\030)22847 64799 y Fi(f)98 b Fo(\()p Fi(m)p Fo(\))25016 64592 y Fj(;)3581 b Fk(if)464 b Fj(\017)30745 64791 y Fi(m)32001 64592 y Fl(=)369 b(1)p Fk(;)22279 66519 y Fj(\030)22847 66725 y Fi(f)98 b Fo(\()p Fi(m)p Fo(\))25311 66519 y Fl(+)295 b(1)p Fj(;)1329 b Fk(if)464 b Fj(\017)30745 66718 y Fi(m)32001 66519 y Fl(=)369 b Fh(\000)p Fl(1)p Fk(,)50488 65502 y Fl(\(26\))p 0 TeXcolorgray 26475 70521 a(9)p 0 TeXcolorgray eop end end %%Page: 10 10 TeXDict begin HPSdict begin 10 9 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.10) cvn H.B /DEST pdfmark end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4277 a SDict begin H.S end 800 -4277 a 800 -4277 a SDict begin 14.5 H.A end 800 -4277 a 800 -4277 a SDict begin [ /View [/XYZ H.V] /Dest (equation.27) cvn H.B /DEST pdfmark end 800 -4277 a 13315 660 a Fj(a)13998 112 y Ff(0)13998 989 y Fi(m)15254 660 y Fl(=)16635 -2461 y Fd(8)16635 -1266 y(>)16635 -867 y(<)16635 1524 y(>)16635 1922 y(:)17815 -1213 y Fj(a)18498 -1006 y Fi(f)98 b Fo(\()p Fi(m)p Fo(\))20667 -1213 y Fj(;)3581 b Fk(if)464 b Fj(\017)26396 -1014 y Fi(m)27653 -1213 y Fl(=)369 b(1)465 b Fk(and)g Fj(\017)33198 -1014 y Fi(m)p Fo(+1)35657 -1213 y Fl(=)368 b(1)p Fk(;)17815 713 y Fj(a)18498 920 y Fi(f)98 b Fo(\()p Fi(m)p Fo(\))20962 713 y Fl(+)295 b(1)p Fj(;)1329 b Fk(if)464 b Fj(\017)26396 912 y Fi(m)27284 713 y Fj(\017)27809 912 y Fi(m)p Fo(+1)30268 713 y Fl(=)369 b Fh(\000)p Fl(1)p Fk(;)17815 2640 y Fj(a)18498 2846 y Fi(f)98 b Fo(\()p Fi(m)p Fo(\))20962 2640 y Fl(+)295 b(2)p Fj(;)1329 b Fk(if)464 b Fj(\017)26396 2839 y Fi(m)27653 2640 y Fl(=)369 b Fh(\000)p Fl(1)465 b Fk(and)g Fj(\017)34231 2839 y Fi(m)p Fo(+1)36690 2640 y Fl(=)369 b Fh(\000)p Fl(1)p Fk(.)50488 660 y Fl(\(27\))p 0 TeXcolorgray 800 5431 a Fk(Pr)-66 b(o)g(of.)p 0 TeXcolorgray 649 w Fl(See)433 b(Theorem)h(2,)g(Matthews)g(and)f(Rob)36 b(ertson)433 b([)p 0 0.5 0 TeXcolorrgb 29327 5431 a SDict begin H.S end 29327 5431 a 0 0.5 0 TeXcolorrgb Fl(5)p 0 0.5 0 TeXcolorrgb 29978 4575 a SDict begin H.R end 29978 4575 a 29978 5431 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.matthews-robertson) cvn H.B /ANN pdfmark end 29978 5431 a 0 TeXcolorgray Fl(].)p 51860 5431 45 878 v 51905 4598 781 45 v 51905 5431 V 52684 5431 45 878 v 800 6760 a SDict begin H.S end 800 6760 a 800 6760 a SDict begin 14.5 H.A end 800 6760 a 800 6760 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.10) cvn H.B /DEST pdfmark end 800 6760 a 0 TeXcolorgray 1383 x Fe(Remark)480 b(10.)p 0 TeXcolorgray 2260 w Fl(By)418 b(virtue)f(of)511 b(\()p 0 0.5 0 TeXcolorrgb 18482 8143 a SDict begin H.S end 18482 8143 a 0 0.5 0 TeXcolorrgb Fl(24)p 0 0.5 0 TeXcolorrgb 19783 7287 a SDict begin H.R end 19783 7287 a 19783 8143 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.24) cvn H.B /ANN pdfmark end 19783 8143 a 0 TeXcolorgray Fl(\))417 b(and)g(\()p 0 0.5 0 TeXcolorrgb 23725 8143 a SDict begin H.S end 23725 8143 a 0 0.5 0 TeXcolorrgb Fl(26)p 0 0.5 0 TeXcolorrgb 25026 7287 a SDict begin H.R end 25026 7287 a 25026 8143 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.26) cvn H.B /ANN pdfmark end 25026 8143 a 0 TeXcolorgray Fl(\))o(,)k(w)-36 b(e)418 b(sa)-36 b(y)418 b(that)f(the)g Fj(\030)36101 7661 y Ff(0)36037 8472 y Fi(m)37343 8143 y Fl(are)g(obtained)g(from)h(the)f Fj(\030)50672 8342 y Fi(n)51716 8143 y Fl(in)800 9748 y Fk(jumps)434 b Fl(of)g(1)g(or)g(2.)800 11114 y SDict begin H.S end 800 11114 a 800 11114 a SDict begin 14.5 H.A end 800 11114 a 800 11114 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.11) cvn H.B /DEST pdfmark end 800 11114 a 0 TeXcolorgray 1346 x Fe(Lemma)611 b(11.)p 0 TeXcolorgray 604 w Fl(Let)531 b Fj(\030)11716 12659 y Fo(0)12777 12460 y Fl(=)536 b(\()p Fj(a)15514 12659 y Fo(0)16039 12460 y Fj(;)221 b(a)17304 12659 y Fo(1)17830 12460 y Fj(;)g(:)g(:)g(:)s Fl(\))531 b(b)36 b(e)531 b(an)g(R)-36 b(CF)532 b(expansion.)872 b(Then)531 b(if)h([)p Fj(x)p Fl(])g(denotes)e(the)h(nearest)800 14066 y(in)-36 b(teger)433 b(to)h Fj(x)p Fl(,)g(w)-36 b(e)434 b(ha)-36 b(v)g(e)18521 16759 y([)p Fj(\030)19450 16958 y Fi(n)20077 16759 y Fl(])369 b(=)22188 14488 y Fd(\()23258 15849 y Fj(a)23941 16048 y Fi(n)24567 15849 y Fj(;)3553 b Fl(if)434 b Fj(a)30356 16048 y Fi(n)p Fo(+1)32554 15849 y Fj(>)368 b Fl(1;)23258 17775 y Fj(a)23941 17974 y Fi(n)24862 17775 y Fl(+)295 b(1)p Fj(;)1301 b Fl(if)434 b Fj(a)30356 17974 y Fi(n)p Fo(+1)32554 17775 y Fl(=)368 b(1.)p 0 TeXcolorgray 800 20731 a Fk(Pr)-66 b(o)g(of.)p 0 TeXcolorgray 649 w Fl(If)521 b([)p Fj(\030)7188 20930 y Fi(n)7815 20731 y Fl(])c(=)g Fj(a)10905 20930 y Fi(n)11886 20731 y Fl(+)354 b(1,)542 b(then)520 b Fj(\030)18423 20930 y Fi(n)19567 20731 y Fj(>)c(a)21778 20930 y Fi(n)22759 20731 y Fl(+)24258 20208 y Fo(1)p 24258 20425 471 54 v 24258 21189 a(2)25381 20731 y Fl(and)521 b(hence)e Fj(a)32380 20930 y Fi(n)p Fo(+1)34726 20731 y Fl(=)e(1,)543 b(whereas)520 b(if)i([)p Fj(\030)45025 20930 y Fi(n)45651 20731 y Fl(])c(=)e Fj(a)48741 20930 y Fi(n)49367 20731 y Fl(,)543 b(then)800 22336 y Fj(\030)1368 22535 y Fi(n)2363 22336 y Fj(<)369 b(a)4427 22535 y Fi(n)5348 22336 y Fl(+)6788 21813 y Fo(1)p 6788 22031 V 6788 22794 a(2)7825 22336 y Fl(and)433 b(hence)g Fj(a)14650 22535 y Fi(n)p Fo(+1)16847 22336 y Fj(>)369 b Fl(1.)p 51860 22336 45 878 v 51905 21502 781 45 v 51905 22336 V 52684 22336 45 878 v 800 23790 a SDict begin H.S end 800 23790 a 800 23790 a SDict begin 14.5 H.A end 800 23790 a 800 23790 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.12) cvn H.B /DEST pdfmark end 800 23790 a 0 TeXcolorgray 1587 x Fe(Lemma)499 b(12.)p 0 TeXcolorgray 554 w Fk(L)-66 b(et)464 b Fj(\030)11369 25576 y Fo(0)12264 25377 y Fl(=)13777 24842 y Fi(P)14374 24965 y Fg(0)14835 24842 y Fo(+)15567 24092 y Ff(p)p 16351 24092 799 40 v 750 x Fi(D)p 13777 25071 3372 54 v 14862 25835 a(Q)15602 25958 y Fg(0)17747 25377 y Fk(have)h(NICF-P)f (exp)-66 b(ansion)21073 29341 y Fj(\030)21641 29540 y Fo(0)22536 29341 y Fl(=)369 b Fj(a)24600 28792 y Ff(0)24600 29669 y Fo(0)25421 29341 y Fl(+)26939 28442 y Fj(\017)27464 28641 y Fo(1)27990 28442 y Fh(j)p 26860 29035 1578 54 v 26860 30252 a(j)p Fj(a)27912 29795 y Ff(0)27912 30578 y Fo(1)28866 29341 y Fl(+)295 b Fh(\001)221 b(\001)g(\001)443 b Fj(:)800 32946 y Fk(Then)4187 31341 y SDict begin H.S end 4187 31341 a 4187 31341 a SDict begin 14.5 H.A end 4187 31341 a 4187 31341 a SDict begin [ /View [/XYZ H.V] /Dest (equation.28) cvn H.B /DEST pdfmark end 4187 31341 a 17701 35381 a Fj(A)18676 34833 y Ff(0)18676 35709 y Fi(m)19932 35381 y Fl(=)21313 33109 y Fd(\()22383 34471 y Fj(A)23358 34678 y Fi(f)98 b Fo(\()p Fi(m)p Fo(\))28058 34471 y Fk(if)464 b Fj(\017)29845 34670 y Fi(m)p Fo(+1)32304 34471 y Fl(=)369 b(1)p Fk(;)22383 36397 y Fj(A)23358 36604 y Fi(f)98 b Fo(\()p Fi(m)p Fo(\)+1)28058 36397 y Fk(if)464 b Fj(\017)29845 36596 y Fi(m)p Fo(+1)32304 36397 y Fl(=)369 b Fh(\000)p Fl(1)p Fk(,)50488 35381 y Fl(\(28\))800 38868 y Fk(wher)-66 b(e)661 b Fj(f)142 b Fl(\()p Fj(m)p Fl(\))661 b Fk(is)g(de\014ne)-66 b(d)660 b(by)778 b Fl(\()p 0 0.5 0 TeXcolorrgb 16944 38868 a SDict begin H.S end 16944 38868 a 0 0.5 0 TeXcolorrgb Fl(24)p 0 0.5 0 TeXcolorrgb 18244 38012 a SDict begin H.R end 18244 38012 a 18244 38868 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.24) cvn H.B /ANN pdfmark end 18244 38868 a 0 TeXcolorgray Fl(\))p Fk(.)1187 b(Equivalently,)709 b(in)660 b(the)h(notation)f(of)h(Bosma)i([)p 0 0.5 0 TeXcolorrgb 44481 38868 a SDict begin H.S end 44481 38868 a 0 0.5 0 TeXcolorrgb Fk(2)p 0 0.5 0 TeXcolorrgb 45144 38007 a SDict begin H.R end 45144 38007 a 45144 38868 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.bosma) cvn H.B /ANN pdfmark end 45144 38868 a 0 TeXcolorgray Fk(,)710 b(p.)1187 b(372],)710 b(if)800 40473 y Fj(n)p Fl(\()p Fj(k)45 b Fl(\))369 b(=)g Fj(f)142 b Fl(\()p Fj(k)340 b Fl(+)295 b(1\))g Fh(\000)g Fl(1)465 b Fk(for)g Fj(k)414 b Fh(\025)369 b(\000)p Fl(1)p Fk(,)465 b(then)22569 38868 y SDict begin H.S end 22569 38868 a 22569 38868 a SDict begin 14.5 H.A end 22569 38868 a 22569 38868 a SDict begin [ /View [/XYZ H.V] /Dest (equation.29) cvn H.B /DEST pdfmark end 22569 38868 a 16211 44569 a Fj(n)p Fl(\()p Fj(k)45 b Fl(\))369 b(=)20470 42297 y Fd(\()21540 43659 y Fj(n)p Fl(\()p Fj(k)340 b Fh(\000)296 b Fl(1\))f(+)g(1)p Fj(;)1329 b Fk(if)464 b Fj(\017)32052 43858 y Fi(k)24 b Fo(+1)34192 43659 y Fl(=)369 b(1)p Fk(;)21540 45585 y Fj(n)p Fl(\()p Fj(k)340 b Fh(\000)296 b Fl(1\))f(+)g(2)p Fj(;)1329 b Fk(if)464 b Fj(\017)32052 45784 y Fi(k)24 b Fo(+1)34192 45585 y Fl(=)369 b Fh(\000)p Fl(1)50488 44569 y(\(29\))800 48665 y Fk(and)603 b Fl(\()p 0 0.5 0 TeXcolorrgb 3968 48665 a SDict begin H.S end 3968 48665 a 0 0.5 0 TeXcolorrgb Fl(28)p 0 0.5 0 TeXcolorrgb 5268 47809 a SDict begin H.R end 5268 47809 a 5268 48665 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.28) cvn H.B /ANN pdfmark end 5268 48665 a 0 TeXcolorgray Fl(\))465 b Fk(has)h(the)e(simpler)h(form)18460 47059 y SDict begin H.S end 18460 47059 a 18460 47059 a SDict begin 14.5 H.A end 18460 47059 a 18460 47059 a SDict begin [ /View [/XYZ H.V] /Dest (equation.30) cvn H.B /DEST pdfmark end 18460 47059 a 20708 51598 a Fj(A)21683 51050 y Ff(0)21683 51926 y Fi(k)22621 51598 y Fl(=)368 b Fj(A)24976 51805 y Fi(n)p Fo(\()p Fi(k)24 b Fo(\))27313 51598 y Fk(for)464 b Fj(k)414 b Fh(\025)370 b Fl(0)p Fj(:)17596 b Fl(\(30\))800 53354 y SDict begin H.S end 800 53354 a 800 53354 a SDict begin 14.5 H.A end 800 53354 a 800 53354 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.13) cvn H.B /DEST pdfmark 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b(and)f(use)g(the)g(equation)19050 11671 y Fj(A)20025 11122 y Ff(0)20025 11999 y Fi(k)24 b Fo(+2)22165 11671 y Fl(=)369 b Fj(a)24229 11122 y Ff(0)24229 11999 y Fi(k)24 b Fo(+2)26000 11671 y Fj(A)26975 11122 y Ff(0)26975 11999 y Fi(k)g Fo(+1)29041 11671 y Fl(+)295 b Fj(\017)30873 11870 y Fi(k)24 b Fo(+2)32645 11671 y Fj(A)33620 11122 y Ff(0)33620 11999 y Fi(k)34188 11671 y Fj(:)800 14515 y Fl(Then)433 b(from)h(\()p 0 0.5 0 TeXcolorrgb 7773 14515 a SDict begin H.S end 7773 14515 a 0 0.5 0 TeXcolorrgb Fl(27)p 0 0.5 0 TeXcolorrgb 9074 13659 a SDict begin H.R end 9074 13659 a 9074 14515 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.27) cvn H.B /ANN pdfmark end 9074 14515 a 0 TeXcolorgray Fl(\),)f(with)h Fj(j)444 b Fl(=)368 b Fj(n)p Fl(\()p Fj(k)341 b Fl(+)295 b(1\))g(+)g(1,)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)28422 12909 y SDict begin H.S end 28422 12909 a 28422 12909 a SDict begin 14.5 H.A end 28422 12909 a 28422 12909 a SDict begin [ /View [/XYZ H.V] 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b(1)g(=)f Fj(\017)17576 24445 y Fi(k)24 b Fo(+3)19717 24246 y Fl(=)368 b(1.)579 b(Then)9960 27090 y Fj(n)p Fl(\()p Fj(k)341 b Fl(+)295 b(1\))369 b(=)f Fj(n)p Fl(\()p Fj(k)45 b Fl(\))296 b(+)f(1)369 b(=)f Fj(j)i Fh(\000)296 b Fl(1)p Fj(;)1522 b(n)p Fl(\()p Fj(k)341 b Fl(+)295 b(2\))369 b(=)f Fj(n)p Fl(\()p Fj(k)341 b Fl(+)295 b(1\))g(+)g(1)369 b(=)g Fj(j)800 29933 y Fl(and)15816 31538 y Fj(A)16791 30990 y Ff(0)16791 31867 y Fi(k)24 b Fo(+2)18932 31538 y Fl(=)368 b Fj(a)20995 31737 y Fi(j)21482 31538 y Fj(A)22457 31737 y Fi(j)51 b Ff(\000)p Fo(1)24441 31538 y Fl(+)295 b Fj(A)26723 31737 y Fi(j)51 b Ff(\000)p Fo(2)28781 31538 y Fl(=)369 b Fj(A)31137 31737 y Fi(j)31993 31538 y Fl(=)f Fj(A)34348 31745 y Fi(n)p Fo(\()p Fi(k)24 b Fo(+2\))37422 31538 y Fj(:)800 33825 y Fl(Case)434 b(2.)579 b(Supp)36 b(ose)433 b Fj(\017)11131 34024 y Fi(k)24 b Fo(+2)13271 33825 y Fl(=)369 b(1)p Fj(;)221 b(\017)16409 34024 y Fi(k)24 b Fo(+3)18550 33825 y Fl(=)369 b Fh(\000)p Fl(1.)578 b(Then)8834 36668 y Fj(n)p Fl(\()p Fj(k)341 b Fl(+)295 b(1\))369 b(=)f Fj(n)p Fl(\()p Fj(k)45 b Fl(\))296 b(+)e(1)370 b(=)e Fj(j)i Fh(\000)296 b Fl(1)p Fj(;)1522 b(n)p Fl(\()p Fj(k)341 b Fl(+)295 b(2\))369 b(=)f Fj(n)p Fl(\()p Fj(k)341 b Fl(+)295 b(1\))g(+)g(2)369 b(=)g Fj(j)h Fl(+)294 b(1)800 39512 y(and)18689 41117 y Fj(A)19664 40568 y Ff(0)19664 41445 y Fi(k)24 b Fo(+2)21805 41117 y Fl(=)368 b(\()p Fj(a)24374 41316 y Fi(j)25156 41117 y Fl(+)295 b(1\))p Fj(A)28594 41316 y Fi(j)51 b Ff(\000)p Fo(1)30578 41117 y Fl(+)295 b Fj(A)32860 41316 y Fi(j)51 b Ff(\000)p Fo(2)34549 41117 y Fj(:)800 43403 y Fl(No)-36 b(w)434 b Fj(\017)4287 43602 y Fi(k)24 b Fo(+3)6428 43403 y Fl(=)368 b Fh(\000)p Fl(1)434 b(implies)g(1)370 b(=)e Fj(a)17423 43610 y Fi(n)p Fo(\()p Fi(k)24 b Fo(+2\))20866 43403 y Fl(=)368 b Fj(a)22929 43602 y Fi(j)51 b Fo(+1)24618 43403 y Fl(,)434 b(so)g Fj(A)27985 43602 y Fi(j)51 b Fo(+1)30043 43403 y Fl(=)369 b Fj(A)32399 43602 y Fi(j)33181 43403 y Fl(+)295 b Fj(A)35463 43602 y Fi(j)51 b Ff(\000)p Fo(1)37152 43403 y Fl(.)579 b(Hence)20140 46247 y Fj(A)21115 45698 y Ff(0)21115 46575 y Fi(k)24 b Fo(+2)23255 46247 y Fl(=)369 b Fj(A)25611 46446 y Fi(j)51 b Fo(+1)27669 46247 y Fl(=)369 b Fj(A)30025 46453 y Fi(n)p Fo(\()p Fi(k)24 b Fo(+2\))33099 46247 y Fj(:)800 49090 y Fl(Case)434 b(3.)579 b(Supp)36 b(ose)433 b Fj(\017)11131 49289 y Fi(k)24 b Fo(+2)13271 49090 y Fl(=)369 b Fh(\000)p Fl(1)p Fj(;)221 b(\017)17442 49289 y Fi(k)24 b Fo(+3)19583 49090 y Fl(=)369 b(1.)578 b(Then)9960 51934 y Fj(n)p Fl(\()p Fj(k)341 b Fl(+)295 b(1\))369 b(=)f Fj(n)p Fl(\()p Fj(k)45 b Fl(\))296 b(+)f(2)369 b(=)f Fj(j)i Fh(\000)296 b Fl(1)p Fj(;)1522 b(n)p Fl(\()p Fj(k)341 b Fl(+)295 b(2\))369 b(=)f Fj(n)p Fl(\()p Fj(k)341 b Fl(+)295 b(1\))g(+)g(1)369 b(=)g Fj(j)800 54778 y Fl(and)18679 56383 y Fj(A)19654 55834 y Ff(0)19654 56711 y Fi(k)24 b Fo(+2)21794 56383 y Fl(=)368 b(\()p Fj(a)24363 56582 y Fi(j)25145 56383 y Fl(+)295 b(1\))p Fj(A)28583 56582 y Fi(j)51 b Ff(\000)p Fo(1)30568 56383 y Fh(\000)295 b Fj(A)32871 56582 y Fi(j)51 b Ff(\000)p Fo(3)34560 56383 y Fj(:)800 58669 y Fl(No)-36 b(w)434 b Fj(\017)4287 58868 y Fi(k)24 b Fo(+2)6428 58669 y Fl(=)368 b Fh(\000)p Fl(1)434 b(implies)g(1)370 b(=)e Fj(a)17423 58876 y Fi(n)p Fo(\()p Fi(k)24 b Fo(+1\))20866 58669 y Fl(=)368 b Fj(a)22929 58868 y Fi(j)51 b Ff(\000)p Fo(1)24618 58669 y Fl(,)434 b(so)g Fj(A)27985 58868 y Fi(j)51 b Ff(\000)p Fo(1)30043 58669 y Fl(=)369 b Fj(A)32399 58868 y Fi(j)51 b Ff(\000)p Fo(2)34383 58669 y Fl(+)295 b Fj(A)36665 58868 y Fi(j)51 b Ff(\000)p Fo(3)38354 58669 y Fl(.)579 b(Hence)15816 61513 y Fj(A)16791 60964 y Ff(0)16791 61841 y Fi(k)24 b Fo(+2)18932 61513 y Fl(=)368 b Fj(a)20995 61712 y Fi(j)21482 61513 y Fj(A)22457 61712 y Fi(j)51 b Ff(\000)p Fo(1)24441 61513 y Fl(+)295 b Fj(A)26723 61712 y Fi(j)51 b Ff(\000)p Fo(2)28781 61513 y Fl(=)369 b Fj(A)31137 61712 y Fi(j)31993 61513 y Fl(=)f Fj(A)34348 61719 y Fi(n)p Fo(\()p Fi(k)24 b Fo(+2\))37422 61513 y Fj(:)800 64356 y Fl(Case)434 b(4.)579 b(Supp)36 b(ose)433 b Fj(\017)11131 64555 y Fi(k)24 b Fo(+2)13271 64356 y Fl(=)369 b Fh(\000)p Fl(1)g(=)g Fj(\017)18610 64555 y Fi(k)24 b Fo(+3)20381 64356 y Fl(.)578 b(Then)8834 67200 y Fj(n)p Fl(\()p Fj(k)341 b Fl(+)295 b(1\))369 b(=)f Fj(n)p Fl(\()p Fj(k)45 b Fl(\))296 b(+)e(2)370 b(=)e Fj(j)i Fh(\000)296 b Fl(1)p Fj(;)1522 b(n)p Fl(\()p Fj(k)341 b Fl(+)295 b(2\))369 b(=)f Fj(n)p Fl(\()p Fj(k)341 b Fl(+)295 b(1\))g(+)g(2)369 b(=)g Fj(j)h Fl(+)294 b(1)p 0 TeXcolorgray 26150 70521 a(12)p 0 TeXcolorgray eop end end %%Page: 13 13 TeXDict begin HPSdict begin 13 12 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.13) cvn H.B /DEST pdfmark end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 1328 x Fl(and)18679 -1067 y Fj(A)19654 -1615 y Ff(0)19654 -738 y Fi(k)24 b Fo(+2)21794 -1067 y Fl(=)368 b(\()p Fj(a)24363 -868 y Fi(j)25145 -1067 y Fl(+)295 b(2\))p Fj(A)28583 -868 y Fi(j)51 b Ff(\000)p Fo(1)30568 -1067 y Fh(\000)295 b Fj(A)32871 -868 y Fi(j)51 b Ff(\000)p Fo(3)34560 -1067 y Fj(:)800 1130 y Fl(No)-36 b(w)434 b Fj(\017)4287 1329 y Fi(k)24 b Fo(+3)6428 1130 y Fl(=)368 b Fh(\000)p Fl(1)739 b(=)-221 b Fh(\))737 b Fl(1)369 b(=)g Fj(a)16169 1336 y Fi(n)p Fo(\()p Fi(k)24 b Fo(+2\))19612 1130 y Fl(=)368 b Fj(a)21675 1329 y Fi(j)51 b Fo(+1)23798 1130 y Fl(and)433 b Fj(\017)26852 1329 y Fi(k)24 b Fo(+2)28992 1130 y Fl(=)369 b Fh(\000)p Fl(1)738 b(=)-221 b Fh(\))738 b Fl(1)369 b(=)g Fj(a)38734 1336 y Fi(n)p Fo(\()p Fi(k)24 b Fo(+1\))42176 1130 y Fl(=)369 b Fj(a)44240 1329 y Fi(j)51 b Ff(\000)p Fo(1)45929 1130 y Fl(,)434 b(so)14395 3763 y Fj(A)15370 3962 y Fi(j)51 b Fo(+1)17428 3763 y Fl(=)369 b Fj(A)19784 3962 y Fi(j)20566 3763 y Fl(+)295 b Fj(A)22848 3962 y Fi(j)51 b Ff(\000)p Fo(1)24970 3763 y Fl(and)434 b Fj(A)28475 3962 y Fi(j)51 b Ff(\000)p Fo(1)30533 3763 y Fl(=)368 b Fj(A)32888 3962 y Fi(j)51 b Ff(\000)p Fo(2)34873 3763 y Fl(+)295 b Fj(A)37155 3962 y Fi(j)51 b Ff(\000)p Fo(3)38844 3763 y Fj(:)800 6397 y Fl(Hence)16210 9031 y Fj(A)17185 8482 y Ff(0)17185 9359 y Fi(k)24 b Fo(+2)19325 9031 y Fl(=)368 b(\()p Fj(a)21894 9230 y Fi(j)22676 9031 y Fl(+)295 b(2\))p Fj(A)26114 9230 y Fi(j)51 b Ff(\000)p Fo(1)28098 9031 y Fh(\000)296 b Fl(\()p Fj(A)30908 9230 y Fi(j)51 b Ff(\000)p Fo(1)32892 9031 y Fh(\000)295 b Fj(A)35195 9230 y Fi(j)51 b Ff(\000)p Fo(2)36885 9031 y Fl(\))19325 10968 y(=)368 b Fj(A)21680 11167 y Fi(j)22463 10968 y Fl(+)294 b Fj(A)24744 11167 y Fi(j)51 b Ff(\000)p Fo(1)26803 10968 y Fl(=)368 b Fj(A)29158 11167 y Fi(j)51 b Fo(+1)31216 10968 y Fl(=)369 b Fj(A)33572 11175 y Fi(n)p Fo(\()p Fi(k)24 b Fo(+2\))36646 10968 y Fj(:)p 51860 13602 45 878 v 51905 12768 781 45 v 51905 13602 V 52684 13602 45 878 v 800 14512 a SDict begin H.S end 800 14512 a 800 14512 a SDict begin 14.5 H.A end 800 14512 a 800 14512 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.14) cvn H.B /DEST pdfmark end 800 14512 a 0 TeXcolorgray 1545 x Fe(Lemma)602 b(14.)p 0 TeXcolorgray 600 w Fk(Each)548 b Fj(a)12745 16256 y Fi(n)13893 16057 y Fj(>)522 b Fl(1)p Fk(,)568 b Fj(n)523 b Fh(\025)f Fl(1)548 b Fk(wil)66 b(l)548 b(b)-66 b(e)547 b(visite)-66 b(d)546 b(by)h(the)g(algorithm)g (of)h(L)-66 b(emma)p 0 0.5 0 TeXcolorrgb 45636 16057 a SDict begin H.S end 45636 16057 a 0 0.5 0 TeXcolorrgb Fk(9)p 0 0.5 0 TeXcolorrgb 46300 15196 a SDict begin H.R end 46300 15196 a 46300 16057 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.9) cvn H.B /ANN pdfmark end 46300 16057 a 0 TeXcolorgray Fk(,)568 b(i.e.,)e(ther)-66 b(e)800 17662 y(exists)464 b(an)h Fj(m)g Fk(such)g(that)g Fj(n)369 b Fl(=)g Fj(f)142 b Fl(\()p Fj(m)p Fl(\))p Fk(.)p 0 TeXcolorgray 800 20117 a(Pr)-66 b(o)g(of.)p 0 TeXcolorgray 649 w Fl(Let)446 b Fj(a)7898 20316 y Fi(n)8916 20117 y Fj(>)390 b Fl(1)447 b(and)f Fj(f)142 b Fl(\()p Fj(m)p Fl(\))391 b Fh(\024)g Fj(n)h(<)f(f)142 b Fl(\()p Fj(m)303 b Fl(+)h(1\).)617 b(If)448 b Fj(f)142 b Fl(\()p Fj(m)p Fl(\))390 b Fj(<)h(n)p Fl(,)451 b(then)445 b Fj(f)142 b Fl(\()p Fj(m)304 b Fl(+)f(1\))392 b(=)f Fj(f)142 b Fl(\()p Fj(m)p Fl(\))303 b(+)h(2)447 b(and)800 21722 y Fj(\017)1325 21921 y Fi(m)p Fo(+1)3784 21722 y Fl(=)369 b Fh(\000)p Fl(1;)434 b(also)h Fj(n)369 b Fl(=)g Fj(f)142 b Fl(\()p Fj(m)p Fl(\))294 b(+)h(1.)579 b(Then)433 b(from)h(\()p 0 0.5 0 TeXcolorrgb 25875 21722 a SDict begin H.S end 25875 21722 a 0 0.5 0 TeXcolorrgb Fl(25)p 0 0.5 0 TeXcolorrgb 27175 20866 a SDict begin H.R end 27175 20866 a 27175 21722 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.25) cvn H.B /ANN pdfmark end 27175 21722 a 0 TeXcolorgray Fl(\),)g Fj(a)29159 21921 y Fi(n)30154 21722 y Fl(=)369 b Fj(a)32218 21929 y Fi(f)98 b Fo(\()p Fi(m)p Fo(\)+1)35958 21722 y Fl(=)368 b(1.)p 51860 21722 45 878 v 51905 20889 781 45 v 51905 21722 V 52684 21722 45 878 v 2751 24238 a(Note)587 b(that)f(in)h(the)f(R)-36 b(CF)586 b(to)h(NICF-P)g(transformation,)625 b(w)-36 b(e)587 b(ha)-36 b(v)g(e)587 b Fj(f)142 b Fl(\()p Fj(k)45 b Fl(\))629 b(=)g Fj(p)p Fl(,)c(where)587 b Fj(k)632 b Fl(is)587 b(the)800 25843 y(NICF-P)434 b(p)36 b(erio)g(d-length.)800 26101 y SDict begin H.S end 800 26101 a 800 26101 a SDict begin 14.5 H.A end 800 26101 a 800 26101 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.15) cvn H.B /DEST pdfmark end 800 26101 a 0 TeXcolorgray 2197 x Fe(Lemma)628 b(15.)p 0 TeXcolorgray 611 w Fk(Supp)-66 b(ose)568 b(that)f(R)-33 b(CF)568 b(p)-66 b(artial)567 b(quotients)g Fj(a)30351 28497 y Fi(r)31424 28298 y Fk(and)h Fj(a)34734 28497 y Fi(s)35792 28298 y Fk(satisfy)f Fj(a)40629 28497 y Fi(r)41695 28298 y Fj(>)559 b Fl(1)p Fj(;)221 b(a)45181 28497 y Fi(s)46232 28298 y Fj(>)559 b Fl(1)p Fj(;)221 b(r)597 b(<)559 b(s)p Fk(.)800 29903 y(Then)491 b(the)g(numb)-66 b(er)490 b Fj(J)614 b Fk(of)491 b(jumps)g(in)f(the)h(R)-33 b(CF)491 b(to)g(NICF-P)g(tr)-66 b(ansformation)489 b(when)i(starting)f (fr)-66 b(om)490 b Fj(a)52294 30102 y Fi(r)800 31509 y Fk(and)447 b(\014nishing)d(at)j Fj(a)10760 31708 y Fi(s)11696 31509 y Fk(is)g Fj(J)491 b Fl(=)369 b(\()p Fj(s)254 b Fh(\000)g Fj(r)291 b Fl(+)254 b Fj(E)78 b Fl(\))p Fj(=)p Fl(2)p Fk(,)449 b(wher)-66 b(e)447 b Fj(E)524 b Fk(is)446 b(the)g(numb)-66 b(er)446 b(of)g(even)f(unise)-66 b(quenc)g(es)445 b(in)g(the)800 33114 y(interval)464 b Fl([)p Fj(a)6610 33313 y Fi(r)7116 33114 y Fj(;)221 b(a)8381 33313 y Fi(s)8872 33114 y Fl(])p Fk(.)597 b(Her)-66 b(e)464 b(we)h(include)f(zer)-66 b(o)465 b(unise)-66 b(quenc)g(es)463 b Fl([)p Fj(a)31168 33313 y Fi(i)31544 33114 y Fj(;)221 b(a)32809 33313 y Fi(i)p Fo(+1)34387 33114 y Fl(])p Fk(,)465 b(wher)-66 b(e)464 b Fj(a)39964 33313 y Fi(i)40709 33114 y Fj(>)369 b Fl(1)465 b Fk(and)g Fj(a)46412 33313 y Fi(i)p Fo(+1)48358 33114 y Fj(>)369 b Fl(1)p Fk(.)p 0 TeXcolorgray 800 35569 a(Pr)-66 b(o)g(of.)p 0 TeXcolorgray 649 w Fl(Supp)36 b(ose)330 b(the)g(unisequences)g(in)h ([)p Fj(a)22074 35768 y Fi(r)22580 35569 y Fj(;)221 b(a)23845 35768 y Fi(s)24336 35569 y Fl(])331 b(ha)-36 b(v)g(e)330 b(lengths)h Fj(m)33446 35768 y Fo(1)33971 35569 y Fj(;)221 b(:)g(:)g(:)j(;)d(m)38022 35768 y Fi(N)39250 35569 y Fl(and)330 b(let)h Fj(j)43990 35768 y Fi(i)44696 35569 y Fl(b)36 b(e)330 b(the)h(n)-36 b(um)g(b)36 b(er)800 37174 y(of)434 b(jumps)f(o)36 b(ccurring)434 b(in)f(a)h(unisequence)f (of)i(length)e Fj(m)28291 37373 y Fi(i)28666 37174 y Fl(.)579 b(Then)13166 40896 y Fj(j)13704 41095 y Fi(i)14449 40896 y Fl(=)15962 39998 y Fj(m)17100 40197 y Fi(i)17771 39998 y Fl(+)294 b(1)i(+)f Fj(e)21933 40197 y Fi(i)p 15962 40591 6347 54 v 18810 41807 a Fl(2)22441 40896 y Fj(;)655 b Fl(where)434 b Fj(e)27818 41095 y Fi(i)28562 40896 y Fl(=)29943 38625 y Fd(\()31013 39986 y Fl(0)1301 b(if)434 b Fj(m)35294 40185 y Fi(i)36103 39986 y Fl(is)g(o)36 b(dd;)31013 41912 y(1)1301 b(if)434 b Fj(m)35294 42111 y Fi(i)36103 41912 y Fl(is)g(ev)-36 b(en.)800 44618 y(Then)18549 48156 y Fj(J)492 b Fl(=)21679 46496 y Fi(N)21141 46894 y Fd(X)21339 49694 y Fi(i)p Fo(=1)23281 48156 y Fj(j)23819 48355 y Fi(i)24564 48156 y Fl(=)26483 46496 y Fi(N)25944 46894 y Fd(X)26142 49694 y Fi(i)p Fo(=1)28217 47258 y Fj(m)29355 47457 y Fi(i)30026 47258 y Fl(+)295 b(1)g(+)g Fj(e)34188 47457 y Fi(i)p 28217 47851 V 31065 49067 a Fl(2)19760 52668 y(=)21274 52306 y Fg(1)p 21274 52467 406 40 v 21274 53015 a(2)21812 52668 y Fl(\()p Fj(N)434 b Fl(+)25639 51007 y Fi(N)25100 51406 y Fd(X)25298 54205 y Fi(i)p Fo(=1)27240 52668 y Fj(m)28378 52867 y Fi(i)28754 52668 y Fl(\))295 b(+)30994 51769 y Fj(E)p 30994 52362 1041 54 v 31189 53579 a Fl(2)19760 56587 y(=)21274 55688 y Fj(s)f Fh(\000)i Fj(r)p 21274 56282 2859 54 v 22378 57498 a Fl(2)24560 56587 y(+)26000 55688 y Fj(E)p 26000 56282 1041 54 v 26195 57498 a Fl(2)27542 56587 y(=)29056 55688 y Fj(s)e Fh(\000)i Fj(r)331 b Fl(+)295 b Fj(E)p 29056 56282 5502 54 v 31481 57498 a Fl(2)34690 56587 y Fj(:)p 51860 59516 45 878 v 51905 58682 781 45 v 51905 59516 V 52684 59516 45 878 v 800 60426 a SDict begin H.S end 800 60426 a 800 60426 a SDict begin 14.5 H.A end 800 60426 a 800 60426 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.16) cvn H.B /DEST pdfmark end 800 60426 a 0 TeXcolorgray 1545 x Fe(Corollary)588 b(16.)p 0 TeXcolorgray 584 w Fk(The)518 b(numb)-66 b(er)517 b(of)i(jumps)f(in)g(the)g(interval)f Fl([)p Fj(a)32592 62170 y Fo(0)33118 61971 y Fj(;)221 b(a)34383 62170 y Fi(r)34890 61971 y Fl(])p Fk(,)532 b Fj(a)36865 62170 y Fi(r)37839 61971 y Fj(>)468 b Fl(1)p Fk(,)532 b(e)-66 b(quals)518 b(the)h(numb)-66 b(er)517 b(in)800 63576 y(the)465 b(interval)e Fl([)p Fj(a)8768 63775 y Fi(p)p Ff(\000)p Fi(r)10480 63576 y Fj(;)221 b(a)11745 63775 y Fi(p)12274 63576 y Fl(])p Fk(,)465 b(wher)-66 b(e)465 b Fj(p)f Fk(is)h(the)g(p)-66 b(erio)g(d-length)463 b(and)h Fj(r)406 b Fh(\024)369 b Fj(p=)p Fl(2)p Fk(.)p 0 TeXcolorgray 800 66031 a(Pr)-66 b(o)g(of.)p 0 TeXcolorgray 649 w Fl(This)434 b(follo)-36 b(ws)436 b(from)e(Lemma)p 0 0.5 0 TeXcolorrgb 19826 66031 a SDict begin H.S end 19826 66031 a 0 0.5 0 TeXcolorrgb Fl(15)p 0 0.5 0 TeXcolorrgb 21127 65175 a SDict begin H.R end 21127 65175 a 21127 66031 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.15) cvn H.B /ANN pdfmark end 21127 66031 a 0 TeXcolorgray 433 w Fl(and)f(the)g(symmetry)h(of)h(an)e(R)-36 b(CF)434 b(p)36 b(erio)g(d.)p 51860 66031 45 878 v 51905 65197 781 45 v 51905 66031 V 52684 66031 45 878 v 0 TeXcolorgray 26150 70521 a(13)p 0 TeXcolorgray eop end end %%Page: 14 14 TeXDict begin HPSdict begin 14 13 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.14) cvn H.B /DEST pdfmark end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 800 -4000 a SDict begin 14.5 H.A end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (section.6) cvn H.B /DEST pdfmark end 800 -4000 a 1328 x Fm(6)2152 b(Pro)60 b(of)716 b(of)h(Theorem)p 0 0.5 0 TeXcolorrgb 21536 -2672 a SDict begin H.S end 21536 -2672 a 0 0.5 0 TeXcolorrgb Fm(8)p 0 0.5 0 TeXcolorrgb 22612 -3904 a SDict begin H.R end 22612 -3904 a 22612 -2672 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.8) cvn H.B /ANN pdfmark end 22612 -2672 a 0 TeXcolorgray 800 249 a Fl(W)-108 b(e)434 b(use)f(the)g(notation)h (of)g(Lemmas)p 0 0.5 0 TeXcolorrgb 19366 249 a SDict begin H.S end 19366 249 a 0 0.5 0 TeXcolorrgb Fl(9)p 0 0.5 0 TeXcolorrgb 20017 -607 a SDict begin H.R end 20017 -607 a 20017 249 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.9) cvn H.B /ANN pdfmark end 20017 249 a 0 TeXcolorgray 433 w Fl(and)p 0 0.5 0 TeXcolorrgb 22979 249 a SDict begin H.S end 22979 249 a 0 0.5 0 TeXcolorrgb Fl(12)p 0 0.5 0 TeXcolorrgb 24280 -607 a SDict begin H.R end 24280 -607 a 24280 249 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.12) cvn H.B /ANN pdfmark end 24280 249 a 0 TeXcolorgray Fl(.)800 1854 y(Case)587 b(\(1\)\(i\).)1037 b(Assume)586 b Fj(p)629 b Fl(=)g(2)p Fj(h)399 b Fh(\000)h Fl(1)p Fj(;)221 b(h)629 b(>)g Fl(1)587 b(and)f(there)g(is)g(an)h(ev)-36 b(en)586 b(length)g(unisequence,)625 b(or)586 b(no)800 3459 y(unisequence,)567 b(on)540 b(eac)-36 b(h)540 b(side)g(of)h Fj(a)18424 3658 y Fi(h)p Ff(\000)p Fo(1)20774 3459 y Fj(>)550 b Fl(1)p Fj(;)221 b(a)24251 3658 y Fi(h)25400 3459 y Fj(>)550 b Fl(1,)567 b(e.g.,)31419 2360 y Fh(p)p 32526 2360 1301 54 v 1099 x Fl(73)551 b(=)g([)36301 3459 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n0 16 {InitRnode } NewNode end end 36301 3459 a Fl(8)p Fj(;)221 b Fl(1)37624 4256 y Ff(\003)38184 3459 y Fj(;)38766 3459 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n1 16 {InitRnode } NewNode end end 38766 3459 a Fl(1)q Fj(;)39999 3459 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n2 16 {InitRnode } NewNode end end 39999 3459 a Fl(5)q Fj(;)41232 3459 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n3 16 {InitRnode } NewNode end end 41232 3459 a Fl(5)q Fj(;)g Fl(1)p Fj(;)43697 3459 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n4 16 {InitRnode } NewNode end end 43697 3459 a Fl(1)r Fj(;)44931 3459 y tx@Dict begin tx@NodeDict begin {7.73332 8.4 11.74988 5.87494 7.73332 } false /N@n5 16 {InitRnode } NewNode end end 44931 3459 a Fl(16)45347 4256 y Ff(\003)46233 3459 y Fl(])46594 3459 y tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n0 /N@n1 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 46594 3459 a 46594 3459 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n1 /N@n2 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 46594 3459 a 46594 3459 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n2 /N@n3 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 46594 3459 a 46594 3459 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n3 /N@n4 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 46594 3459 a 46594 3459 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n4 /N@n5 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 46594 3459 a 540 w Fl(or)48830 2360 y Fh(p)p 49937 2360 1301 54 v 1099 x Fl(89)551 b(=)800 5496 y([)1161 5496 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n0 16 {InitRnode } NewNode end end 1161 5496 a Fl(9)q Fj(;)2394 5496 y tx@Dict begin tx@NodeDict begin {7.73332 8.4 5.87494 2.93747 7.73332 } false /N@n1 16 {InitRnode } NewNode end end 2394 5496 a Fl(2)2484 6293 y Ff(\003)3045 5496 y Fj(;)3627 5496 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n2 16 {InitRnode } NewNode end end 3627 5496 a Fl(3)q Fj(;)4860 5496 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n3 16 {InitRnode } NewNode end end 4860 5496 a Fl(3)q Fj(;)6093 5496 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n4 16 {InitRnode } NewNode end end 6093 5496 a Fl(2)q Fj(;)7326 5496 y tx@Dict begin tx@NodeDict begin {7.73332 8.4 11.74988 5.87494 7.73332 } false /N@n5 16 {InitRnode } NewNode end end 7326 5496 a Fl(18)7742 6293 y Ff(\003)8627 5496 y Fl(])8988 5496 y tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n0 /N@n1 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 8988 5496 a 8988 5496 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n1 /N@n2 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 8988 5496 a 8988 5496 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n2 /N@n3 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 8988 5496 a 8988 5496 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n3 /N@n4 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 8988 5496 a 8988 5496 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n4 /N@n5 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 8988 5496 a Fl(.)959 b(Let)560 b Fj(f)142 b Fl(\()p Fj(m)p Fl(\))584 b(=)g Fj(h)381 b Fh(\000)h Fl(1,)592 b(where)560 b Fj(m)g Fl(is)h(the)f(n)-36 b(um)g(b)36 b(er)558 b(of)j(jumps)f(in)g([)p Fj(a)44565 5695 y Fo(0)45091 5496 y Fj(;)221 b(a)46356 5695 y Fi(h)p Ff(\000)p Fo(1)48157 5496 y Fl(].)958 b(Then)800 7533 y Fj(f)142 b Fl(\()p Fj(m)295 b Fl(+)f(1\))370 b(=)e Fj(h)434 b Fl(and)22019 10466 y Fj(\030)22651 9918 y Ff(0)22587 10795 y Fi(m)23844 10466 y Fl(=)369 b Fj(\030)25793 10665 y Fi(h)p Ff(\000)p Fo(1)27593 10466 y Fj(;)655 b(\017)29134 10665 y Fi(m)30391 10466 y Fl(=)368 b(1)p Fj(;)20817 12403 y(\030)21449 11855 y Ff(0)21385 12732 y Fi(m)p Fo(+1)23844 12403 y Fl(=)h Fj(\030)25793 12602 y Fi(h)26390 12403 y Fj(;)656 b(\017)27932 12602 y Fi(m)p Fo(+1)30391 12403 y Fl(=)368 b(1)p Fj(:)800 15337 y Fl(By)618 b(Corollary)p 0 0.5 0 TeXcolorrgb 8954 15337 a SDict begin H.S end 8954 15337 a 0 0.5 0 TeXcolorrgb Fl(16)p 0 0.5 0 TeXcolorrgb 10255 14481 a SDict begin H.R end 10255 14481 a 10255 15337 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.16) cvn H.B /ANN pdfmark end 10255 15337 a 0 TeXcolorgray Fl(,)664 b Fj(m)618 b Fl(is)g(also)h(the)e(n)-36 b(um)g(b)36 b(er)616 b(of)j(jumps)e(in)h([)p Fj(a)33131 15536 y Fi(h)33728 15337 y Fj(;)221 b(a)34993 15536 y Fi(p)35523 15337 y Fl(].)1131 b(There)618 b(is)g(also)h(one)f(jump)f(in)800 16942 y([)p Fj(a)1844 17141 y Fi(h)p Ff(\000)p Fo(1)3644 16942 y Fj(;)221 b(a)4909 17141 y Fi(h)5507 16942 y Fl(].)620 b(Hence)447 b Fj(k)438 b Fl(=)392 b(2)p Fj(m)305 b Fl(+)f(1.)620 b(As)447 b Fj(Q)21251 17141 y Fi(h)p Ff(\000)p Fo(1)23444 16942 y Fl(=)392 b Fj(Q)25878 17141 y Fi(h)26476 16942 y Fl(,)451 b(w)-36 b(e)447 b(ha)-36 b(v)g(e)448 b Fj(Q)33259 16460 y Ff(0)33259 17270 y Fi(m)34539 16942 y Fl(=)392 b Fj(Q)36973 16460 y Ff(0)36973 17270 y Fi(m)p Fo(+1)39063 16942 y Fl(,)451 b(whic)-36 b(h)448 b(is)f(criterion)h(3\))f(of)800 18547 y(Theorem)p 0 0.5 0 TeXcolorrgb 6292 18547 a SDict begin H.S end 6292 18547 a 0 0.5 0 TeXcolorrgb Fl(6)p 0 0.5 0 TeXcolorrgb 6942 17691 a SDict begin H.R end 6942 17691 a 6942 18547 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.6) cvn H.B /ANN pdfmark end 6942 18547 a 0 TeXcolorgray Fl(.)578 b(Also)12358 21480 y Fj(\017)12883 21679 y Fi(m)14140 21480 y Fl(=)369 b(1)738 b(=)-221 b Fh(\))737 b Fj(f)142 b Fl(\()p Fj(m)p Fl(\))369 b(=)f Fj(f)142 b Fl(\()p Fj(m)295 b Fh(\000)g Fl(1\))h(+)e(1)p Fj(;)656 b(A)33897 20932 y Ff(0)33897 21809 y Fi(m)p Ff(\000)p Fo(1)36356 21480 y Fl(=)369 b Fj(A)38712 21687 y Fi(f)98 b Fo(\()p Fi(m)p Ff(\000)p Fo(1\))42083 21480 y Fj(;)11156 23418 y(\017)11681 23617 y Fi(m)p Fo(+1)14140 23418 y Fl(=)369 b(1)738 b(=)-221 b Fh(\))737 b Fj(f)142 b Fl(\()p Fj(m)295 b Fl(+)g(1\))369 b(=)g Fj(f)142 b Fl(\()p Fj(m)p Fl(\))294 b(+)h(1)p Fj(;)655 b(A)33875 22869 y Ff(0)33875 23746 y Fi(m)35132 23418 y Fl(=)369 b Fj(A)37488 23624 y Fi(f)98 b Fo(\()p Fi(m)p Fo(\))39657 23418 y Fj(:)800 26351 y Fl(Hence)433 b Fj(A)5640 25869 y Ff(0)5640 26679 y Fi(m)p Ff(\000)p Fo(1)8099 26351 y Fl(=)369 b Fj(A)10455 26550 y Fi(h)p Ff(\000)p Fo(2)12255 26351 y Fj(;)655 b(A)14246 25869 y Ff(0)14246 26679 y Fi(m)15503 26351 y Fl(=)368 b Fj(A)17858 26550 y Fi(h)p Ff(\000)p Fo(1)19658 26351 y Fj(;)656 b(B)21730 25869 y Ff(0)21663 26679 y Fi(m)p Ff(\000)p Fo(1)24122 26351 y Fl(=)368 b Fj(B)26490 26550 y Fi(h)p Ff(\000)p Fo(2)28290 26351 y Fj(;)655 b(B)30361 25869 y Ff(0)30294 26679 y Fi(m)31551 26351 y Fl(=)369 b Fj(B)33920 26550 y Fi(h)p Ff(\000)p Fo(1)36153 26351 y Fl(and)17342 29284 y Fj(A)18317 29483 y Fi(p)p Ff(\000)p Fo(1)20417 29284 y Fl(=)g Fj(A)22773 29483 y Fi(h)p Ff(\000)p Fo(1)24573 29284 y Fj(B)25561 29483 y Fi(h)p Ff(\000)p Fo(1)27656 29284 y Fl(+)295 b Fj(A)29938 29483 y Fi(h)p Ff(\000)p Fo(2)31738 29284 y Fj(B)32726 29483 y Fi(h)p Ff(\000)p Fo(2)20417 31222 y Fl(=)369 b Fj(A)22773 30673 y Ff(0)22773 31550 y Fi(m)23661 31222 y Fj(B)24716 30673 y Ff(0)24649 31550 y Fi(m)25832 31222 y Fl(+)295 b Fj(A)28114 30673 y Ff(0)28114 31550 y Fi(m)p Ff(\000)p Fo(1)30203 31222 y Fj(B)31258 30673 y Ff(0)31191 31550 y Fi(m)p Ff(\000)p Fo(1)20417 33159 y Fl(=)369 b Fj(A)22773 32610 y Ff(0)22773 33487 y Fi(m)23661 33159 y Fj(B)24716 32610 y Ff(0)24649 33487 y Fi(m)25832 33159 y Fl(+)295 b Fj(\017)27664 33358 y Fi(m)p Fo(+1)29754 33159 y Fj(A)30729 32610 y Ff(0)30729 33487 y Fi(m)p Ff(\000)p Fo(1)32819 33159 y Fj(B)33874 32610 y Ff(0)33807 33487 y Fi(m)p Ff(\000)p Fo(1)35897 33159 y Fj(:)800 36092 y Fl(Also)19422 39026 y Fj(B)20410 39225 y Fi(p)p Ff(\000)p Fo(1)22510 39026 y Fl(=)369 b Fj(B)24946 38477 y Fo(2)24879 39354 y Fi(h)p Ff(\000)p Fo(1)26974 39026 y Fl(+)295 b Fj(B)29336 38477 y Fo(2)29269 39354 y Fi(h)p Ff(\000)p Fo(2)22510 41211 y Fl(=)369 b Fj(B)24946 40662 y Ff(0)25257 40521 y Fo(2)25257 41539 y Fi(m)26439 41211 y Fl(+)295 b Fj(B)28801 40662 y Ff(0)29112 40521 y Fo(2)29112 41539 y Fi(m)p Ff(\000)p Fo(1)22510 43396 y Fl(=)369 b Fj(B)24946 42847 y Ff(0)25257 42706 y Fo(2)25257 43724 y Fi(m)26439 43396 y Fl(+)295 b Fj(\017)28271 43595 y Fi(m)p Fo(+1)30361 43396 y Fj(B)31416 42847 y Ff(0)31727 42706 y Fo(2)31727 43724 y Fi(m)p Ff(\000)p Fo(1)33817 43396 y Fj(:)800 46547 y Fl(If)487 b Fj(p)458 b Fl(=)g(1,)6245 45422 y Fh(p)p 7352 45422 1117 54 v 1125 x Fj(D)495 b Fl(=)458 b([)p Fj(a;)p 12023 45532 1334 54 v 221 w Fl(2)p Fj(a)q Fl(])486 b(and)g(the)g(NICF)g(and)g(R)-36 b(CF)486 b(expansions)h(are)f(iden)-36 b(tical.)736 b(Also)487 b Fj(Q)47319 46065 y Ff(0)47319 46875 y Fo(0)48304 46547 y Fl(=)458 b Fj(Q)50804 46065 y Ff(0)50804 46875 y Fo(1)51788 46547 y Fl(=)800 48152 y(1)p Fj(;)221 b(\017)2557 48351 y Fo(1)3453 48152 y Fl(=)369 b(1)434 b(and)f(w)-36 b(e)433 b(ha)-36 b(v)g(e)434 b(criterion)g(3\).)800 49757 y(Case)511 b(\(1\)\(ii\).)809 b(Assume)510 b Fj(p)500 b Fl(=)f(2)p Fj(h)348 b Fh(\000)g Fl(1,)530 b(with)510 b(an)h(o)36 b(dd)510 b(length)g(unisequence)g(on)g(eac)-36 b(h)510 b(side)h(of)g Fj(a)49489 49956 y Fi(h)p Ff(\000)p Fo(1)51788 49757 y Fj(>)800 51362 y Fl(1)p Fj(;)221 b(a)2715 51561 y Fi(h)3682 51362 y Fj(>)369 b Fl(1,)401 b(e.g.,)9188 50263 y Fh(p)p 10295 50263 1951 54 v 1099 x Fl(113)370 b(=)f([)14357 51362 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 11.74988 5.87494 7.73332 } false /N@n0 16 {InitRnode } NewNode end end 14357 51362 a Fl(10)q Fj(;)221 b Fl(1)16330 52159 y Ff(\003)16891 51362 y Fj(;)17473 51362 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n1 16 {InitRnode } NewNode end end 17473 51362 a Fl(1)q Fj(;)g Fl(1)p Fj(;)19938 51362 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n2 16 {InitRnode } NewNode end end 19938 51362 a Fl(2)r Fj(;)21172 51362 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n3 16 {InitRnode } NewNode end end 21172 51362 a Fl(2)q Fj(;)g Fl(1)p Fj(;)23637 51362 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n4 16 {InitRnode } NewNode end end 23637 51362 a Fl(1)r Fj(;)g Fl(1)p Fj(;)26103 51362 y tx@Dict begin tx@NodeDict begin {7.73332 8.4 11.74988 5.87494 7.73332 } false /N@n5 16 {InitRnode } NewNode end end 26103 51362 a Fl(20)26520 52159 y Ff(\003)27405 51362 y Fl(])27766 51362 y tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n0 /N@n1 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 27766 51362 a 27766 51362 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n1 /N@n2 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 27766 51362 a 27766 51362 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n2 /N@n3 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 27766 51362 a 27766 51362 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n3 /N@n4 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 27766 51362 a 27766 51362 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n4 /N@n5 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 27766 51362 a Fl(.)565 b(Let)392 b Fj(f)142 b Fl(\()p Fj(m)p Fl(\))368 b(=)h Fj(h)211 b Fh(\000)g Fl(1.)565 b(Then)392 b Fj(f)142 b Fl(\()p Fj(m)211 b Fl(+)g(1\))368 b(=)g Fj(h)393 b Fl(and)20377 54855 y Fj(\030)21009 54306 y Ff(0)20945 55183 y Fi(m)22201 54855 y Fl(=)369 b Fj(\030)24150 55054 y Fi(h)p Ff(\000)p Fo(1)26245 54855 y Fl(+)295 b(1)p Fj(;)656 b(\017)29744 55054 y Fi(m)31000 54855 y Fl(=)369 b Fh(\000)p Fl(1)p Fj(;)19174 56792 y(\030)19806 56244 y Ff(0)19742 57120 y Fi(m)p Fo(+1)22201 56792 y Fl(=)g Fj(\030)24150 56991 y Fi(h)24748 56792 y Fj(;)655 b(\017)26289 56991 y Fi(m)p Fo(+1)28748 56792 y Fl(=)369 b(1)p Fj(;)800 59725 y Fl(and)460 b(as)i(in)e(Case)i(\(1\)\(i\),)468 b Fj(k)460 b Fl(=)415 b(2)p Fj(m)314 b Fl(+)f(1.)660 b(As)461 b Fj(Q)24171 59924 y Fi(h)p Ff(\000)p Fo(1)26387 59725 y Fl(=)415 b Fj(Q)28844 59924 y Fi(h)29441 59725 y Fl(,)468 b(w)-36 b(e)461 b(ha)-36 b(v)g(e)461 b Fj(Q)36268 59243 y Ff(0)36268 60054 y Fi(m)37571 59725 y Fl(=)415 b Fj(Q)40028 59243 y Ff(0)40028 60054 y Fi(m)p Fo(+1)42118 59725 y Fl(,)468 b(whic)-36 b(h)461 b(is)g(criterion)800 61331 y(3\).)579 b(Also)10724 64264 y Fj(\017)11249 64463 y Fi(m)12506 64264 y Fl(=)368 b Fh(\000)p Fl(1)739 b(=)-221 b Fh(\))737 b Fj(f)142 b Fl(\()p Fj(m)p Fl(\))368 b(=)h Fj(f)142 b Fl(\()p Fj(m)295 b Fh(\000)g Fl(1\))g(+)g(2)p Fj(;)656 b(A)33296 63715 y Ff(0)33296 64592 y Fi(m)p Ff(\000)p Fo(1)35755 64264 y Fl(=)368 b Fj(A)38110 64471 y Fi(f)98 b Fo(\()p Fi(m)p Ff(\000)p Fo(1\)+1)42684 64264 y Fj(;)10555 66201 y(\017)11080 66400 y Fi(m)p Fo(+1)13539 66201 y Fl(=)368 b(1)739 b(=)-221 b Fh(\))737 b Fj(f)142 b Fl(\()p Fj(m)295 b Fl(+)g(1\))369 b(=)f Fj(f)142 b Fl(\()p Fj(m)p Fl(\))295 b(+)g(1)p Fj(;)655 b(A)33274 65653 y Ff(0)33274 66530 y Fi(m)34531 66201 y Fl(=)369 b Fj(A)36887 66408 y Fi(f)98 b Fo(\()p Fi(m)p Fo(\))39056 66201 y Fj(:)p 0 TeXcolorgray 26150 70521 a Fl(14)p 0 TeXcolorgray eop end end %%Page: 15 15 TeXDict begin HPSdict begin 15 14 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.15) cvn H.B /DEST pdfmark end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 1328 x Fl(Hence)479 b Fj(A)5686 -3154 y Ff(0)5686 -2343 y Fi(m)p Ff(\000)p Fo(1)8223 -2672 y Fl(=)447 b Fj(A)10657 -2473 y Fi(h)p Ff(\000)p Fo(2)12457 -2672 y Fj(;)701 b(A)14494 -3154 y Ff(0)14494 -2343 y Fi(m)15828 -2672 y Fl(=)447 b Fj(A)18262 -2473 y Fi(h)p Ff(\000)p Fo(1)20541 -2672 y Fl(and)479 b Fj(B)24171 -3154 y Ff(0)24104 -2343 y Fi(m)p Ff(\000)p Fo(1)26641 -2672 y Fl(=)447 b Fj(B)29088 -2473 y Fi(h)p Ff(\000)p Fo(2)30888 -2672 y Fj(;)701 b(B)33005 -3154 y Ff(0)32938 -2343 y Fi(m)34272 -2672 y Fl(=)447 b Fj(B)36719 -2473 y Fi(h)p Ff(\000)p Fo(1)38519 -2672 y Fl(.)716 b(Then)479 b(as)g(in)h(Case)g(\(1\)\(i\),)800 -1067 y(w)-36 b(e)434 b(get)8663 539 y Fj(A)9638 -10 y Ff(0)9638 867 y Fi(m)10526 539 y Fj(B)11581 -10 y Ff(0)11514 867 y Fi(m)12697 539 y Fl(+)295 b Fj(\017)14529 738 y Fi(m)p Fo(+1)16619 539 y Fj(A)17594 -10 y Ff(0)17594 867 y Fi(m)p Ff(\000)p Fo(1)19684 539 y Fj(B)20739 -10 y Ff(0)20672 867 y Fi(m)p Ff(\000)p Fo(1)23131 539 y Fl(=)368 b Fj(A)25486 738 y Fi(p)p Ff(\000)p Fo(1)27651 539 y Fl(and)433 b Fj(B)31235 -10 y Ff(0)31546 -151 y Fo(2)31546 867 y Fi(m)32729 539 y Fl(+)295 b Fj(\017)34561 738 y Fi(m)p Fo(+1)36651 539 y Fj(B)37706 -10 y Ff(0)38017 -151 y Fo(2)38017 867 y Fi(m)p Ff(\000)p Fo(1)40475 539 y Fl(=)369 b Fj(B)42844 738 y Fi(p)p Ff(\000)p Fo(1)44575 539 y Fj(:)800 2995 y Fl(Case)434 b(\(2\))g(Assume)f Fj(p)369 b Fl(=)g(2)p Fj(h;)221 b(h)369 b(>)g Fl(1)p Fj(;)221 b(a)19627 3194 y Fi(h)p Ff(\000)p Fo(1)21796 2995 y Fj(>)369 b Fl(1)p Fj(;)221 b(a)25092 3194 y Fi(h)26059 2995 y Fj(>)369 b Fl(1,)434 b(e.g.,)31631 1896 y Fh(p)p 32738 1896 1301 54 v 1099 x Fl(92)369 b(=)g([)36149 2995 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n0 16 {InitRnode } NewNode end end 36149 2995 a Fl(9)q Fj(;)221 b Fl(1)37472 3792 y Ff(\003)38033 2995 y Fj(;)38615 2995 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n1 16 {InitRnode } NewNode end end 38615 2995 a Fl(1)q Fj(;)39848 2995 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n2 16 {InitRnode } NewNode end end 39848 2995 a Fl(2)q Fj(;)41081 2995 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n3 16 {InitRnode } NewNode end end 41081 2995 a Fl(4)q Fj(;)42314 2995 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n4 16 {InitRnode } NewNode end end 42314 2995 a Fl(2)q Fj(;)g Fl(1)p Fj(;)44779 2995 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n5 16 {InitRnode } NewNode end end 44779 2995 a Fl(1)r Fj(;)46013 2995 y tx@Dict begin tx@NodeDict begin {7.73332 8.4 11.74988 5.87494 7.73332 } false /N@n6 16 {InitRnode } NewNode end end 46013 2995 a Fl(18)46429 3792 y Ff(\003)47314 2995 y Fl(])47675 2995 y tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n0 /N@n1 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 47675 2995 a 47675 2995 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n1 /N@n2 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 47675 2995 a 47675 2995 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n2 /N@n3 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 47675 2995 a 47675 2995 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n3 /N@n4 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 47675 2995 a 47675 2995 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n4 /N@n5 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 47675 2995 a 47675 2995 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n5 /N@n6 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 47675 2995 a Fl(.)800 5031 y(Let)433 b Fj(f)142 b Fl(\()p Fj(m)p Fl(\))368 b(=)h Fj(h)p Fl(.)578 b(Then)433 b Fj(f)142 b Fl(\()p Fj(m)295 b Fl(+)g(1\))369 b(=)f Fj(h)295 b Fl(+)g(1)434 b(and)21418 7965 y Fj(\030)22050 7416 y Ff(0)21986 8293 y Fi(m)23243 7965 y Fl(=)369 b Fj(\030)25192 8164 y Fi(h)25789 7965 y Fj(;)655 b(\017)27330 8164 y Fi(m)28587 7965 y Fl(=)369 b(1)p Fj(;)20216 9902 y(\030)20848 9353 y Ff(0)20784 10230 y Fi(m)p Fo(+1)23243 9902 y Fl(=)g Fj(\030)25192 10101 y Fi(h)p Fo(+1)26992 9902 y Fj(;)655 b(\017)28533 10101 y Fi(m)p Fo(+1)30992 9902 y Fl(=)368 b(1)p Fj(;)800 12835 y Fl(Also)410 b(b)-36 b(y)410 b(Corollary)p 0 0.5 0 TeXcolorrgb 11212 12835 a SDict begin H.S end 11212 12835 a 0 0.5 0 TeXcolorrgb Fl(16)p 0 0.5 0 TeXcolorrgb 12513 11979 a SDict begin H.R end 12513 11979 a 12513 12835 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.16) cvn H.B /ANN pdfmark end 12513 12835 a 0 TeXcolorgray Fl(,)k Fj(m)c Fl(is)g(the)f(n)-36 b(um)g(b)36 b(er)407 b(of)k(jumps)e(in)g([)p Fj(a)30888 13034 y Fi(h)31486 12835 y Fj(;)221 b(a)32751 13034 y Fi(p)33280 12835 y Fl(].)571 b(Hence)409 b Fj(k)414 b Fl(=)369 b(2)p Fj(m)p Fl(.)570 b(Then)409 b Fj(P)47814 13034 y Fi(h)48781 12835 y Fl(=)369 b Fj(P)51000 13034 y Fi(h)p Fo(+1)800 14440 y Fl(giv)-36 b(es)435 b Fj(P)5006 13958 y Ff(0)4825 14769 y Fi(m)6082 14440 y Fl(=)368 b Fj(P)8481 13958 y Ff(0)8300 14769 y Fi(m)p Fo(+1)10824 14440 y Fl(and)433 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b(criterion)f(1\))h(of)g(Theorem)p 0 0.5 0 TeXcolorrgb 32032 14440 a SDict begin H.S end 32032 14440 a 0 0.5 0 TeXcolorrgb Fl(6)p 0 0.5 0 TeXcolorrgb 32682 13584 a SDict begin H.R end 32682 13584 a 32682 14440 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.6) cvn H.B /ANN pdfmark end 32682 14440 a 0 TeXcolorgray Fl(.)579 b(Also)12358 17374 y Fj(\017)12883 17573 y Fi(m)14140 17374 y Fl(=)369 b(1)738 b(=)-221 b Fh(\))737 b Fj(f)142 b Fl(\()p Fj(m)p Fl(\))369 b(=)f Fj(f)142 b Fl(\()p Fj(m)295 b Fh(\000)g Fl(1\))h(+)e(1)p Fj(;)656 b(A)33897 16825 y Ff(0)33897 17702 y Fi(m)p Ff(\000)p Fo(1)36356 17374 y Fl(=)369 b Fj(A)38712 17581 y Fi(f)98 b Fo(\()p Fi(m)p Ff(\000)p Fo(1\))42083 17374 y Fj(;)11156 19311 y(\017)11681 19510 y Fi(m)p Fo(+1)14140 19311 y Fl(=)369 b(1)738 b(=)-221 b Fh(\))737 b Fj(f)142 b Fl(\()p Fj(m)295 b Fl(+)g(1\))369 b(=)g Fj(f)142 b Fl(\()p Fj(m)p Fl(\))294 b(+)h(1)p Fj(;)655 b(A)33875 18763 y Ff(0)33875 19639 y Fi(m)35132 19311 y Fl(=)369 b Fj(A)37488 19518 y Fi(f)98 b Fo(\()p Fi(m)p Fo(\))39657 19311 y Fj(:)800 22245 y Fl(Hence)433 b Fj(A)5640 21762 y Ff(0)5640 22573 y Fi(m)p Ff(\000)p Fo(1)8099 22245 y Fl(=)369 b Fj(A)10455 22444 y Fi(h)p Ff(\000)p Fo(1)12255 22245 y Fj(;)655 b(A)14246 21762 y Ff(0)14246 22573 y Fi(m)15503 22245 y Fl(=)368 b Fj(A)17858 22444 y Fi(h)18890 22245 y Fl(and)433 b Fj(B)22474 21762 y Ff(0)22407 22573 y Fi(m)p Ff(\000)p Fo(1)24866 22245 y Fl(=)368 b Fj(B)27234 22444 y Fi(h)p Ff(\000)p Fo(1)29034 22245 y Fj(;)655 b(B)31105 21762 y Ff(0)31038 22573 y Fi(m)32295 22245 y Fl(=)369 b Fj(B)34664 22444 y Fi(h)35261 22245 y Fl(.)579 b(Then)18194 25178 y Fj(A)19169 25377 y Fi(p)p Ff(\000)p Fo(1)21269 25178 y Fl(=)369 b Fj(A)23625 25377 y Fi(h)24222 25178 y Fj(B)25210 25377 y Fi(h)p Ff(\000)p Fo(1)27305 25178 y Fl(+)295 b Fj(A)29587 25377 y Fi(h)p Ff(\000)p Fo(1)31387 25178 y Fj(B)32375 25377 y Fi(h)p Ff(\000)p Fo(2)21269 27115 y Fl(=)369 b Fj(A)23625 26567 y Ff(0)23625 27444 y Fi(m)24512 27115 y Fj(B)25567 26567 y Ff(0)25500 27444 y Fi(m)p Ff(\000)p Fo(1)27885 27115 y Fl(+)295 b Fj(A)30167 26567 y Ff(0)30167 27444 y Fi(m)p Ff(\000)p Fo(1)32257 27115 y Fj(B)33245 27314 y Fi(h)p Ff(\000)p Fo(2)35045 27115 y Fj(:)50488 25178 y SDict begin H.S end 50488 25178 a 50488 25178 a SDict begin 17.5 H.A end 50488 25178 a 50488 25178 a SDict begin [ /View [/XYZ H.V] /Dest (equation.32) cvn H.B /DEST pdfmark end 50488 25178 a 1937 x Fl(\(32\))800 30049 y(But)433 b Fj(B)4371 30248 y Fi(h)p Ff(\000)p Fo(2)6540 30049 y Fl(=)369 b Fj(B)8909 30248 y Fi(h)9801 30049 y Fh(\000)296 b Fj(a)11813 30248 y Fi(h)12410 30049 y Fj(B)13398 30248 y Fi(h)p Ff(\000)p Fo(1)15567 30049 y Fl(=)369 b Fj(B)18003 29567 y Ff(0)17936 30377 y Fi(m)19119 30049 y Fh(\000)295 b Fj(a)21130 29567 y Ff(0)21130 30377 y Fi(m)22017 30049 y Fj(B)23005 30248 y Fi(m)p Ff(\000)p Fo(1)25464 30049 y Fl(=)369 b Fj(B)27900 29567 y Ff(0)27833 30377 y Fi(m)p Ff(\000)p Fo(2)29923 30049 y Fl(.)578 b(Hence)434 b(\()p 0 0.5 0 TeXcolorrgb 35234 30049 a SDict begin H.S end 35234 30049 a 0 0.5 0 TeXcolorrgb Fl(32)p 0 0.5 0 TeXcolorrgb 36534 29193 a SDict begin H.R end 36534 29193 a 36534 30049 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.32) cvn H.B /ANN pdfmark end 36534 30049 a 0 TeXcolorgray Fl(\))g(giv)-36 b(es)17342 32982 y Fj(A)18317 33181 y Fi(p)p Ff(\000)p Fo(1)20417 32982 y Fl(=)369 b Fj(A)22773 32434 y Ff(0)22773 33310 y Fi(m)23661 32982 y Fj(B)24716 32434 y Ff(0)24649 33310 y Fi(m)p Ff(\000)p Fo(1)27034 32982 y Fl(+)295 b Fj(A)29316 32434 y Ff(0)29316 33310 y Fi(m)p Ff(\000)p Fo(1)31406 32982 y Fj(B)32461 32434 y Ff(0)32394 33310 y Fi(m)p Ff(\000)p Fo(2)20417 34919 y Fl(=)369 b Fj(A)22773 34371 y Ff(0)22773 35248 y Fi(m)23661 34919 y Fj(B)24716 34371 y Ff(0)24649 35248 y Fi(m)p Ff(\000)p Fo(1)27034 34919 y Fl(+)295 b Fj(\017)28866 35118 y Fi(m)29754 34919 y Fj(A)30729 34371 y Ff(0)30729 35248 y Fi(m)p Ff(\000)p Fo(1)32819 34919 y Fj(B)33874 34371 y Ff(0)33807 35248 y Fi(m)p Ff(\000)p Fo(2)35897 34919 y Fj(:)800 37853 y Fl(Also)19062 40786 y Fj(B)20050 40985 y Fi(p)p Ff(\000)p Fo(1)22151 40786 y Fl(=)368 b Fj(B)24519 40985 y Fi(h)p Ff(\000)p Fo(1)26319 40786 y Fl(\()p Fj(B)27813 40985 y Fi(h)28706 40786 y Fl(+)295 b Fj(B)31001 40985 y Fi(h)p Ff(\000)p Fo(2)32801 40786 y Fl(\))22151 42723 y(=)368 b Fj(B)24586 42175 y Ff(0)24519 43052 y Fi(m)p Ff(\000)p Fo(1)26609 42723 y Fl(\()p Fj(B)28170 42175 y Ff(0)28103 43052 y Fi(m)29286 42723 y Fl(+)295 b Fj(B)31648 42175 y Ff(0)31581 43052 y Fi(m)p Ff(\000)p Fo(2)33671 42723 y Fl(\))p Fj(:)800 45657 y Fl(Case)546 b(\(3\)\(a\).)915 b(Assume)545 b Fj(p)560 b Fl(=)f(2)p Fj(h)p Fl(,)574 b(with)546 b(an)f(ev)-36 b(en)546 b(length)f (unisequence)g(eac)-36 b(h)546 b(side)g(of)g Fj(a)46174 45856 y Fi(h)47331 45657 y Fj(>)559 b Fl(1,)575 b(e.g.,)800 46163 y Fh(p)p 1907 46163 1301 54 v 1099 x Fl(21)479 b(=)f([)5537 47262 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n0 16 {InitRnode } NewNode end end 5537 47262 a Fl(4)q Fj(;)221 b Fl(1)6860 48059 y Ff(\003)7421 47262 y Fj(;)8003 47262 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n1 16 {InitRnode } NewNode end end 8003 47262 a Fl(1)q Fj(;)9236 47262 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n2 16 {InitRnode } NewNode end end 9236 47262 a Fl(2)q Fj(;)g Fl(1)p Fj(;)11701 47262 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n3 16 {InitRnode } NewNode end end 11701 47262 a Fl(1)r Fj(;)12935 47262 y tx@Dict begin tx@NodeDict begin {7.73332 8.4 5.87494 2.93747 7.73332 } false /N@n4 16 {InitRnode } NewNode end end 12935 47262 a Fl(8)13025 48059 y Ff(\003)13586 47262 y Fl(])13947 47262 y tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n0 /N@n1 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 13947 47262 a 13947 47262 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n1 /N@n2 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 13947 47262 a 13947 47262 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n2 /N@n3 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 13947 47262 a 13947 47262 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n3 /N@n4 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 13947 47262 a Fl(.)771 b(Let)497 b Fj(f)142 b Fl(\()p Fj(m)p Fl(\))478 b(=)g Fj(h)p Fl(.)771 b(Then)497 b Fj(f)142 b Fl(\()p Fj(m)338 b Fl(+)h(1\))478 b(=)g Fj(h)339 b Fl(+)g(2.)771 b(As)498 b(in)g(Case\(2\),)514 b Fj(k)524 b Fl(=)478 b(2)p Fj(m)p Fl(.)800 49225 y(Also)19776 52158 y Fj(\030)20408 51610 y Ff(0)20344 52487 y Fi(m)21600 52158 y Fl(=)369 b Fj(\030)23549 52357 y Fi(h)24146 52158 y Fj(;)656 b(\017)25688 52357 y Fi(m)26945 52158 y Fl(=)368 b(1)p Fj(;)18573 54095 y(\030)19205 53547 y Ff(0)19141 54424 y Fi(m)p Fo(+1)21600 54095 y Fl(=)h Fj(\030)23549 54294 y Fi(h)p Fo(+2)25644 54095 y Fl(+)295 b(1)p Fj(;)655 b(\017)29142 54294 y Fi(m)p Fo(+1)31601 54095 y Fl(=)369 b Fh(\000)p Fl(1)p Fj(:)800 57029 y Fl(Then)12875 59962 y Fj(\017)13400 60161 y Fi(m)14657 59962 y Fl(=)f(1)738 b(=)-221 b Fh(\))738 b Fj(f)142 b Fl(\()p Fj(m)p Fl(\))368 b(=)h Fj(f)142 b Fl(\()p Fj(m)294 b Fh(\000)i Fl(1\))f(+)g(1)p Fj(;)656 b(A)34414 59414 y Ff(0)34414 60291 y Fi(m)p Ff(\000)p Fo(1)36872 59962 y Fl(=)369 b Fj(A)39228 60169 y Fi(f)98 b Fo(\()p Fi(m)p Ff(\000)p Fo(1\))42600 59962 y Fj(;)10639 61900 y(\017)11164 62099 y Fi(m)p Fo(+1)13623 61900 y Fl(=)369 b Fh(\000)p Fl(1)738 b(=)-221 b Fh(\))738 b Fj(f)142 b Fl(\()p Fj(m)295 b Fl(+)f(1\))370 b(=)e Fj(f)142 b Fl(\()p Fj(m)p Fl(\))295 b(+)f(2)p Fj(;)656 b(A)34392 61351 y Ff(0)34392 62228 y Fi(m)35649 61900 y Fl(=)368 b Fj(A)38004 62106 y Fi(f)98 b Fo(\()p Fi(m)p Fo(\)+1)41376 61900 y Fj(:)800 64833 y Fl(Hence)433 b Fj(A)5640 64351 y Ff(0)5640 65161 y Fi(m)p Ff(\000)p Fo(1)8099 64833 y Fl(=)369 b Fj(A)10455 65032 y Fi(h)p Ff(\000)p Fo(1)12255 64833 y Fj(;)655 b(A)14246 64351 y Ff(0)14246 65161 y Fi(m)15503 64833 y Fl(=)368 b Fj(A)17858 65032 y Fi(h)p Fo(+1)20092 64833 y Fl(and)433 b Fj(B)23676 64351 y Ff(0)23609 65161 y Fi(m)p Ff(\000)p Fo(1)26068 64833 y Fl(=)369 b Fj(B)28437 65032 y Fi(h)p Ff(\000)p Fo(1)30237 64833 y Fj(;)655 b(B)32308 64351 y Ff(0)32241 65161 y Fi(m)33497 64833 y Fl(=)369 b Fj(B)35866 65032 y Fi(h)p Fo(+1)37666 64833 y Fl(.)800 66438 y(W)-108 b(e)434 b(pro)-36 b(v)g(e)433 b(criterion)h(2\))g(of)g(Theorem)p 0 0.5 0 TeXcolorrgb 20310 66438 a SDict begin H.S end 20310 66438 a 0 0.5 0 TeXcolorrgb Fl(6)p 0 0.5 0 TeXcolorrgb 20960 65582 a SDict begin H.R end 20960 65582 a 20960 66438 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.6) cvn H.B /ANN pdfmark end 20960 66438 a 0 TeXcolorgray Fl(,)g Fj(P)22774 65956 y Ff(0)22593 66766 y Fi(m)p Fo(+1)25052 66438 y Fl(=)369 b Fj(P)27452 65956 y Ff(0)27271 66766 y Fi(m)28454 66438 y Fl(+)295 b Fj(Q)30791 65956 y Ff(0)30791 66766 y Fi(m)31678 66438 y Fl(,)434 b(i.e.,)h Fj(P)35768 66637 y Fi(h)p Fo(+2)37863 66438 y Fl(+)295 b Fj(Q)40200 66637 y Fi(h)p Fo(+2)42369 66438 y Fl(=)369 b Fj(P)44588 66637 y Fi(h)45481 66438 y Fl(+)295 b Fj(Q)47818 66637 y Fi(h)48415 66438 y Fj(:)p 0 TeXcolorgray 26150 70521 a Fl(15)p 0 TeXcolorgray eop end end %%Page: 16 16 TeXDict begin HPSdict begin 16 15 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.16) cvn H.B /DEST pdfmark end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 1328 x Fl(W)-108 b(e)434 b(note)f(from)h(Theorem)g(10.19,) h(Rosen)f([)p 0 0.5 0 TeXcolorrgb 22466 -2672 a SDict begin H.S end 22466 -2672 a 0 0.5 0 TeXcolorrgb Fl(8)p 0 0.5 0 TeXcolorrgb 23117 -3528 a SDict begin H.R end 23117 -3528 a 23117 -2672 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.rosen) cvn H.B /ANN pdfmark end 23117 -2672 a 0 TeXcolorgray Fl(],)g(that)f Fj(a)27774 -2473 y Fi(h)p Fo(+1)29942 -2672 y Fl(=)369 b(1)434 b(implies)g Fj(P)37660 -2473 y Fi(h)p Fo(+2)39755 -2672 y Fl(+)295 b Fj(P)41900 -2473 y Fi(h)p Fo(+1)44069 -2672 y Fl(=)369 b Fj(Q)46480 -2473 y Fi(h)p Fo(+1)48280 -2672 y Fl(.)578 b(Also)20226 190 y Fj(P)21245 -358 y Fo(2)21064 519 y Fi(h)p Fo(+2)23233 190 y Fl(=)369 b Fj(D)331 b Fh(\000)295 b Fj(Q)28383 389 y Fi(h)p Fo(+1)30183 190 y Fj(Q)31213 389 y Fi(h)p Fo(+2)33013 190 y Fj(;)20226 2127 y(P)21245 1579 y Fo(2)21064 2456 y Fi(h)p Fo(+1)23233 2127 y Fl(=)369 b Fj(D)331 b Fh(\000)295 b Fj(Q)28383 2326 y Fi(h)28981 2127 y Fj(Q)30011 2326 y Fi(h)p Fo(+1)31811 2127 y Fj(:)800 4989 y Fl(Hence)14803 7851 y Fj(P)15822 7302 y Fo(2)15641 8179 y Fi(h)p Fo(+2)17737 7851 y Fh(\000)g Fj(P)20084 7302 y Fo(2)19903 8179 y Fi(h)p Fo(+1)22072 7851 y Fl(=)369 b Fj(Q)24483 8050 y Fi(h)p Fo(+1)26283 7851 y Fl(\()p Fj(Q)27819 8050 y Fi(h)28711 7851 y Fh(\000)296 b Fj(Q)31070 8050 y Fi(h)p Fo(+2)32870 7851 y Fl(\))22072 9788 y(=)369 b(\()p Fj(P)24797 9987 y Fi(h)p Fo(+2)26892 9788 y Fl(+)295 b Fj(P)29037 9987 y Fi(h)p Fo(+1)30837 9788 y Fl(\)\()p Fj(Q)32879 9987 y Fi(h)33771 9788 y Fh(\000)h Fj(Q)36130 9987 y Fi(h)p Fo(+2)37930 9788 y Fl(\))p Fj(:)800 12650 y Fl(Hence)19339 15512 y Fj(P)20177 15711 y Fi(h)p Fo(+2)22273 15512 y Fh(\000)f Fj(P)24439 15711 y Fi(h)p Fo(+1)26608 15512 y Fl(=)369 b Fj(Q)29019 15711 y Fi(h)29912 15512 y Fh(\000)295 b Fj(Q)32270 15711 y Fi(h)p Fo(+2)34070 15512 y Fj(;)19169 17449 y(P)20007 17648 y Fi(h)p Fo(+2)22102 17449 y Fl(+)g Fj(Q)24439 17648 y Fi(h)p Fo(+2)26608 17449 y Fl(=)369 b Fj(P)28827 17648 y Fi(h)p Fo(+1)30922 17449 y Fl(+)295 b Fj(Q)33259 17648 y Fi(h)26608 19386 y Fl(=)369 b Fj(P)28827 19585 y Fi(h)29720 19386 y Fl(+)295 b Fj(Q)32057 19585 y Fi(h)50488 17449 y SDict begin H.S end 50488 17449 a 50488 17449 a SDict begin 17.5 H.A end 50488 17449 a 50488 17449 a SDict begin [ /View [/XYZ H.V] /Dest (equation.33) cvn H.B /DEST pdfmark end 50488 17449 a 1937 x Fl(\(33\))800 22248 y(W)-108 b(e)434 b(next)f(pro)-36 b(v)g(e)15205 25110 y Fj(A)16180 25309 y Fi(p)p Ff(\000)p Fo(1)18280 25110 y Fl(=)369 b Fj(B)20716 24561 y Ff(0)20649 25438 y Fi(m)p Ff(\000)p Fo(1)22739 25110 y Fl(\()p Fj(A)24220 24561 y Ff(0)24220 25438 y Fi(m)25403 25110 y Fh(\000)295 b Fj(A)27706 24561 y Ff(0)27706 25438 y Fi(m)p Ff(\000)p Fo(1)29796 25110 y Fl(\))g(+)g Fj(A)32879 24561 y Ff(0)32879 25438 y Fi(m)p Ff(\000)p Fo(1)34969 25110 y Fj(B)36024 24561 y Ff(0)35957 25438 y Fi(m)p Ff(\000)p Fo(2)38047 25110 y Fj(;)50488 23172 y SDict begin H.S end 50488 23172 a 50488 23172 a SDict begin 17.5 H.A end 50488 23172 a 50488 23172 a SDict begin [ /View [/XYZ H.V] /Dest (equation.34) cvn H.B /DEST pdfmark end 50488 23172 a 1938 x Fl(\(34\))15192 27047 y Fj(B)16180 27246 y Fi(p)p Ff(\000)p Fo(1)18280 27047 y Fl(=)369 b Fj(B)20716 26498 y Ff(0)20649 27375 y Fi(m)p Ff(\000)p Fo(1)22739 27047 y Fl(\()p Fj(B)24300 26498 y Ff(0)24233 27375 y Fi(m)25416 27047 y Fh(\000)295 b Fj(B)27799 26498 y Ff(0)27732 27375 y Fi(m)p Ff(\000)p Fo(1)30117 27047 y Fl(+)g Fj(B)32479 26498 y Ff(0)32412 27375 y Fi(m)p Ff(\000)p Fo(2)34502 27047 y Fl(\))p Fj(:)50488 25110 y SDict begin H.S end 50488 25110 a 50488 25110 a SDict begin 17.5 H.A end 50488 25110 a 50488 25110 a SDict begin [ /View [/XYZ H.V] /Dest (equation.35) cvn H.B /DEST pdfmark end 50488 25110 a 1937 x Fl(\(35\))800 29909 y(First)433 b(note)h(that)f(b)-36 b(y)433 b(equations)h(\()p 0 0.5 0 TeXcolorrgb 17813 29909 a SDict begin H.S end 17813 29909 a 0 0.5 0 TeXcolorrgb Fl(27)p 0 0.5 0 TeXcolorrgb 19114 29053 a SDict begin H.R end 19114 29053 a 19114 29909 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.27) cvn H.B /ANN pdfmark end 19114 29909 a 0 TeXcolorgray Fl(\),)g Fj(\017)20940 30108 y Fi(m)22196 29909 y Fl(=)369 b(1)434 b(and)f Fj(\017)27715 30108 y Fi(m)p Fo(+1)30174 29909 y Fl(=)369 b(1)434 b(imply)19766 32770 y Fj(a)20449 32222 y Ff(0)20449 33099 y Fi(m)21706 32770 y Fl(=)368 b Fj(a)23769 32977 y Fi(f)98 b Fo(\()p Fi(m)p Fo(\))26233 32770 y Fl(+)295 b(1)369 b(=)g Fj(a)30623 32969 y Fi(h)31515 32770 y Fl(+)295 b(1)p Fj(:)800 35632 y Fl(Also)434 b Fj(a)4416 35831 y Fi(h)p Fo(+1)6585 35632 y Fl(=)369 b(1)433 b(implies)h Fj(B)14452 35831 y Fi(h)p Fo(+1)16621 35632 y Fl(=)369 b Fj(B)18990 35831 y Fi(h)19883 35632 y Fl(+)295 b Fj(B)22178 35831 y Fi(h)p Ff(\000)p Fo(1)23978 35632 y Fl(,)433 b(i.e.,)i Fj(B)28284 35150 y Ff(0)28217 35961 y Fi(m)29474 35632 y Fl(=)369 b Fj(B)31843 35831 y Fi(h)32735 35632 y Fl(+)295 b Fj(B)35097 35150 y Ff(0)35030 35961 y Fi(m)p Ff(\000)p Fo(1)37120 35632 y Fj(:)434 b Fl(Hence)15958 38494 y Fj(B)16946 38693 y Fi(h)p Ff(\000)p Fo(2)19115 38494 y Fl(=)369 b Fj(B)21484 38693 y Fi(h)22376 38494 y Fh(\000)296 b Fj(a)24388 38693 y Fi(h)24985 38494 y Fj(B)25973 38693 y Fi(h)p Ff(\000)p Fo(1)19115 40431 y Fl(=)369 b(\()p Fj(B)22057 39883 y Ff(0)21990 40760 y Fi(m)23172 40431 y Fh(\000)296 b Fj(B)25556 39883 y Ff(0)25489 40760 y Fi(m)p Ff(\000)p Fo(1)27579 40431 y Fl(\))f Fh(\000)g Fl(\()p Fj(a)30897 39883 y Ff(0)30897 40760 y Fi(m)32079 40431 y Fh(\000)h Fl(1\))p Fj(B)35619 39883 y Ff(0)35552 40760 y Fi(m)p Ff(\000)p Fo(1)19115 42368 y Fl(=)369 b Fj(B)21551 41820 y Ff(0)21484 42697 y Fi(m)22667 42368 y Fh(\000)295 b Fj(a)24678 41820 y Ff(0)24678 42697 y Fi(m)25565 42368 y Fj(B)26620 41820 y Ff(0)26553 42697 y Fi(m)p Ff(\000)p Fo(1)29012 42368 y Fl(=)369 b Fj(B)31448 41820 y Ff(0)31381 42697 y Fi(m)p Ff(\000)p Fo(2)33471 42368 y Fj(:)800 45230 y Fl(Then)15199 48092 y Fj(A)16174 48291 y Fi(p)p Ff(\000)p Fo(1)18274 48092 y Fl(=)f Fj(A)20629 48291 y Fi(h)21227 48092 y Fj(B)22215 48291 y Fi(h)p Ff(\000)p Fo(1)24310 48092 y Fl(+)295 b Fj(A)26592 48291 y Fi(h)p Ff(\000)p Fo(1)28392 48092 y Fj(B)29380 48291 y Fi(h)p Ff(\000)p Fo(2)18274 50029 y Fl(=)368 b(\()p Fj(A)21135 49481 y Ff(0)21135 50358 y Fi(m)22318 50029 y Fh(\000)295 b Fj(A)24621 49481 y Ff(0)24621 50358 y Fi(m)p Ff(\000)p Fo(1)26711 50029 y Fl(\))p Fj(B)28272 49481 y Ff(0)28205 50358 y Fi(m)p Ff(\000)p Fo(1)30590 50029 y Fl(+)g Fj(A)32872 49481 y Ff(0)32872 50358 y Fi(m)p Ff(\000)p Fo(1)34962 50029 y Fj(B)36017 49481 y Ff(0)35950 50358 y Fi(m)p Ff(\000)p Fo(2)38040 50029 y Fj(;)800 52891 y Fl(pro)-36 b(ving)452 b(\()p 0 0.5 0 TeXcolorrgb 6021 52891 a SDict begin H.S end 6021 52891 a 0 0.5 0 TeXcolorrgb Fl(34)p 0 0.5 0 TeXcolorrgb 7321 52035 a SDict begin H.R end 7321 52035 a 7321 52891 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.34) cvn H.B /ANN pdfmark end 7321 52891 a 0 TeXcolorgray Fl(\).)579 b(Also)16711 55753 y Fj(B)17699 55952 y Fi(p)p Ff(\000)p Fo(1)19800 55753 y Fl(=)368 b Fj(B)22168 55952 y Fi(h)p Ff(\000)p Fo(1)23968 55753 y Fl(\()p Fj(B)25462 55952 y Fi(h)26355 55753 y Fl(+)295 b Fj(B)28650 55952 y Fi(h)p Ff(\000)p Fo(2)30450 55753 y Fl(\))19800 57690 y(=)368 b Fj(B)22235 57141 y Ff(0)22168 58018 y Fi(m)p Ff(\000)p Fo(1)24258 57690 y Fl(\()p Fj(B)25819 57141 y Ff(0)25752 58018 y Fi(m)26935 57690 y Fh(\000)296 b Fj(B)29319 57141 y Ff(0)29252 58018 y Fi(m)p Ff(\000)p Fo(1)31637 57690 y Fl(+)f Fj(B)33999 57141 y Ff(0)33932 58018 y Fi(m)p Ff(\000)p Fo(2)36022 57690 y Fl(\))p Fj(;)800 60552 y Fl(pro)-36 b(ving)452 b(\()p 0 0.5 0 TeXcolorrgb 6021 60552 a SDict begin H.S end 6021 60552 a 0 0.5 0 TeXcolorrgb Fl(35)p 0 0.5 0 TeXcolorrgb 7321 59696 a SDict begin H.R end 7321 59696 a 7321 60552 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.35) cvn H.B /ANN pdfmark end 7321 60552 a 0 TeXcolorgray Fl(\).)800 62801 y(If)479 b Fj(p)445 b Fl(=)g(2)479 b(and)f Fj(a)9086 63000 y Fo(1)10057 62801 y Fj(>)444 b Fl(1,)491 b(then)477 b Fj(D)482 b Fl(=)444 b Fj(a)19723 62319 y Fo(2)20575 62801 y Fl(+)325 b Fj(b;)221 b Fl(1)446 b Fj(<)f(b)h(<)f Fl(2)p Fj(a;)221 b(b)479 b Fl(dividing)g(2)p Fj(a)f Fl(\(Rosen)g([)p 0 0.5 0 TeXcolorrgb 42647 62801 a SDict begin H.S end 42647 62801 a 0 0.5 0 TeXcolorrgb Fl(8)p 0 0.5 0 TeXcolorrgb 43298 61945 a SDict begin H.R end 43298 61945 a 43298 62801 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.rosen) cvn H.B /ANN pdfmark end 43298 62801 a 0 TeXcolorgray Fl(,)490 b(p.)712 b(389]\).)i(Then)800 63472 y Fh(p)p 1907 63472 1117 54 v 1124 x Fj(D)422 b Fl(=)386 b([)5168 64596 y tx@Dict begin tx@NodeDict begin {5.16667 0.0 6.16798 3.08398 5.16667 } false /N@n0 16 {InitRnode } NewNode end end 5168 64596 a Fj(a;)6433 64596 y tx@Dict begin tx@NodeDict begin {9.0 11.4 22.91362 11.4568 9.0 } false /N@n1 16 {InitRnode } NewNode end end 6433 64596 a Fl(2)p Fj(a=b)7466 65725 y Ff(\003)8970 64596 y Fj(;)9552 64596 y tx@Dict begin tx@NodeDict begin {7.73332 8.4 12.04292 6.02145 7.73332 } false /N@n2 16 {InitRnode } NewNode end end 9552 64596 a Fl(2)p Fj(a)9984 65393 y Ff(\003)10885 64596 y Fl(])11246 64596 y tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n0 /N@n1 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 11246 64596 a 11246 64596 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n1 /N@n2 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 11246 64596 a 444 w Fl(and)443 b(the)g(NICF)h(and)f(R)-36 b(CF)444 b(expansions)g(are)g (iden)-36 b(tical.)608 b(Then)443 b Fj(\030)44401 64114 y Ff(0)44337 64925 y Fo(1)45249 64596 y Fl(=)46779 64073 y Fi(a)p Fo(+)48011 63323 y Ff(p)p 48795 63323 799 40 v 750 x Fi(D)p 46779 64291 2814 54 v 47985 65054 a(b)49726 64596 y Fj(;)221 b(\030)50940 64114 y Ff(0)50876 64925 y Fo(2)51788 64596 y Fl(=)800 67200 y Fj(a)295 b Fl(+)3085 66075 y Fh(p)p 4192 66075 1117 54 v 1125 x Fj(D)36 b(;)221 b(P)6909 66718 y Ff(0)6728 67528 y Fo(1)7624 67200 y Fl(=)368 b Fj(P)10023 66718 y Ff(0)9842 67528 y Fo(2)10737 67200 y Fl(=)h Fj(a;)221 b(\017)13908 67399 y Fo(1)14803 67200 y Fl(=)369 b(1)434 b(and)f(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b(criterion)f(1\).)p 0 TeXcolorgray 26150 70521 a(16)p 0 TeXcolorgray eop end end %%Page: 17 17 TeXDict begin HPSdict begin 17 16 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.17) cvn H.B /DEST pdfmark end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 1328 x Fl(Case)561 b(\(3\)\(b\).)958 b(Assume)559 b Fj(p)585 b Fl(=)f(2)p Fj(h)p Fl(,)592 b(with)560 b(an)h(o)36 b(dd)559 b(length)h(unisequence)g(eac)-36 b(h)560 b(side)g(of)h Fj(a)46106 -2473 y Fi(h)47288 -2672 y Fj(>)584 b Fl(1,)593 b(e.g.,)800 -2165 y Fh(p)p 1907 -2165 1301 54 v 1098 x Fl(14)370 b(=)e([)5318 -1067 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n0 16 {InitRnode } NewNode end end 5318 -1067 a Fl(3)q Fj(;)221 b Fl(1)6642 -270 y Ff(\003)7202 -1067 y Fj(;)7784 -1067 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n1 16 {InitRnode } NewNode end end 7784 -1067 a Fl(2)q Fj(;)g Fl(1)p Fj(;)10249 -1067 y tx@Dict begin tx@NodeDict begin {7.73332 8.4 5.87494 2.93747 7.73332 } false /N@n2 16 {InitRnode } NewNode end end 10249 -1067 a Fl(6)10339 -270 y Ff(\003)10901 -1067 y tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n0 /N@n1 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 10901 -1067 a 10901 -1067 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n1 /N@n2 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 10901 -1067 a Fl(].)579 b(Let)433 b Fj(f)142 b Fl(\()p Fj(m)p Fl(\))368 b(=)h Fj(h)p Fl(.)578 b(Then)433 b Fj(f)142 b Fl(\()p Fj(m)295 b Fl(+)f(1\))369 b(=)g Fj(h)295 b Fl(+)g(2)434 b(and)f Fj(k)414 b Fl(=)368 b(2)p Fj(m)p Fl(.)579 b(Then)19776 2415 y Fj(\030)20408 1867 y Ff(0)20344 2744 y Fi(m)21600 2415 y Fl(=)369 b Fj(\030)23549 2614 y Fi(h)24442 2415 y Fl(+)294 b(1)p Fj(;)656 b(\017)27940 2614 y Fi(m)29197 2415 y Fl(=)368 b Fh(\000)p Fl(1)p Fj(;)18573 4353 y(\030)19205 3804 y Ff(0)19141 4681 y Fi(m)p Fo(+1)21600 4353 y Fl(=)h Fj(\030)23549 4552 y Fi(h)p Fo(+2)25644 4353 y Fl(+)295 b(1)p Fj(;)655 b(\017)29142 4552 y Fi(m)p Fo(+1)31601 4353 y Fl(=)369 b Fh(\000)p Fl(1)p Fj(:)800 7276 y Fl(Then)11240 10199 y Fj(\017)11765 10398 y Fi(m)13022 10199 y Fl(=)g Fh(\000)p Fl(1)738 b(=)-221 b Fh(\))738 b Fj(f)142 b Fl(\()p Fj(m)p Fl(\))368 b(=)h Fj(f)142 b Fl(\()p Fj(m)294 b Fh(\000)i Fl(1\))f(+)g(2)p Fj(;)655 b(A)33812 9650 y Ff(0)33812 10527 y Fi(m)p Ff(\000)p Fo(1)36271 10199 y Fl(=)369 b Fj(A)38627 10405 y Fi(f)98 b Fo(\()p Fi(m)p Ff(\000)p Fo(1\)+1)43201 10199 y Fj(;)10038 12136 y(\017)10563 12335 y Fi(m)p Fo(+1)13022 12136 y Fl(=)369 b Fh(\000)p Fl(1)738 b(=)-221 b Fh(\))738 b Fj(f)142 b Fl(\()p Fj(m)294 b Fl(+)h(1\))369 b(=)g Fj(f)142 b Fl(\()p Fj(m)p Fl(\))294 b(+)h(2)p Fj(;)656 b(A)33791 11587 y Ff(0)33791 12464 y Fi(m)35047 12136 y Fl(=)369 b Fj(A)37403 12342 y Fi(f)98 b Fo(\()p Fi(m)p Fo(\)+1)40775 12136 y Fj(:)800 15059 y Fl(W)-108 b(e)434 b(ha)-36 b(v)g(e)433 b Fj(A)7013 14577 y Ff(0)7013 15387 y Fi(m)p Ff(\000)p Fo(1)9472 15059 y Fl(=)369 b Fj(A)11828 15258 y Fi(h)p Ff(\000)p Fo(1)13628 15059 y Fj(;)655 b(A)15619 14577 y Ff(0)15619 15387 y Fi(m)16876 15059 y Fl(=)368 b Fj(A)19231 15258 y Fi(h)p Fo(+1)21031 15059 y Fl(,)434 b Fj(B)22881 14577 y Ff(0)22814 15387 y Fi(m)p Ff(\000)p Fo(1)25273 15059 y Fl(=)369 b Fj(B)27642 15258 y Fi(h)p Ff(\000)p Fo(1)29442 15059 y Fj(;)655 b(B)31513 14577 y Ff(0)31446 15387 y Fi(m)32703 15059 y Fl(=)368 b Fj(B)35071 15258 y Fi(h)p Fo(+1)36871 15059 y Fl(.)579 b(Then)433 b(using)451 b(\()p 0 0.5 0 TeXcolorrgb 45134 15059 a SDict begin H.S end 45134 15059 a 0 0.5 0 TeXcolorrgb Fl(33)p 0 0.5 0 TeXcolorrgb 46435 14203 a SDict begin H.R end 46435 14203 a 46435 15059 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.33) cvn H.B /ANN pdfmark end 46435 15059 a 0 TeXcolorgray Fl(\))o(,)434 b(w)-36 b(e)434 b(get)15800 17982 y Fj(P)16819 17433 y Ff(0)16638 18310 y Fi(m)p Fo(+1)19097 17982 y Fl(=)369 b Fj(P)21316 18181 y Fi(h)p Fo(+2)23411 17982 y Fl(+)295 b Fj(Q)25748 18181 y Fi(h)p Fo(+2)27917 17982 y Fl(=)369 b Fj(P)30136 18181 y Fi(h)31029 17982 y Fl(+)295 b Fj(Q)33366 18181 y Fi(h)34332 17982 y Fl(=)369 b Fj(P)36732 17433 y Ff(0)36551 18310 y Fi(m)37439 17982 y Fj(;)800 20905 y Fl(whic)-36 b(h)433 b(is)h(criterion)g(1\))g(of)g(Theorem)p 0 0.5 0 TeXcolorrgb 19595 20905 a SDict begin H.S end 19595 20905 a 0 0.5 0 TeXcolorrgb Fl(6)p 0 0.5 0 TeXcolorrgb 20245 20049 a SDict begin H.R end 20245 20049 a 20245 20905 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.6) cvn H.B /ANN pdfmark end 20245 20905 a 0 TeXcolorgray Fl(.)800 22510 y(As)g Fj(\017)3247 22709 y Fi(m)4503 22510 y Fl(=)369 b Fh(\000)p Fl(1,)434 b(it)g(remains)g(to)f(pro)-36 b(v)g(e)18225 25433 y Fj(A)19200 25632 y Fi(p)p Ff(\000)p Fo(1)21300 25433 y Fl(=)369 b Fj(B)23736 24884 y Ff(0)23669 25761 y Fi(m)p Ff(\000)p Fo(1)25759 25433 y Fj(A)26734 24884 y Ff(0)26734 25761 y Fi(m)27917 25433 y Fh(\000)295 b Fj(A)30220 24884 y Ff(0)30220 25761 y Fi(m)p Ff(\000)p Fo(1)32310 25433 y Fj(B)33365 24884 y Ff(0)33298 25761 y Fi(m)p Ff(\000)p Fo(2)18212 27370 y Fj(B)19200 27569 y Fi(p)p Ff(\000)p Fo(1)21300 27370 y Fl(=)369 b Fj(B)23736 26822 y Ff(0)23669 27699 y Fi(m)p Ff(\000)p Fo(1)25759 27370 y Fl(\()p Fj(B)27320 26822 y Ff(0)27253 27699 y Fi(m)28436 27370 y Fh(\000)295 b Fj(B)30819 26822 y Ff(0)30752 27699 y Fi(m)p Ff(\000)p Fo(2)32842 27370 y Fl(\))p Fj(:)800 30293 y Fl(First,)548 b Fj(a)5127 30492 y Fi(h)p Fo(+1)7451 30293 y Fl(=)524 b(1)i(implies)f Fj(A)15644 30492 y Fi(h)p Fo(+1)17969 30293 y Fl(=)f Fj(A)20480 30492 y Fi(h)21435 30293 y Fl(+)357 b Fj(A)23779 30492 y Fi(h)p Ff(\000)p Fo(1)25579 30293 y Fl(,)548 b(i.e.,)h Fj(A)30034 29811 y Ff(0)30034 30622 y Fi(m)31446 30293 y Fl(=)524 b Fj(A)33957 30492 y Fi(h)34912 30293 y Fl(+)357 b Fj(A)37256 30492 y Fi(h)p Ff(\000)p Fo(1)39056 30293 y Fj(:)525 b Fl(Also)h Fj(\017)43492 30492 y Fi(m)44904 30293 y Fl(=)f Fh(\000)p Fl(1)g(=)f Fj(\017)50710 30492 y Fi(m)p Fo(+1)800 31898 y Fl(implies)434 b Fj(a)5898 31416 y Ff(0)5898 32227 y Fi(m)7154 31898 y Fl(=)369 b Fj(a)9218 32105 y Fi(f)98 b Fo(\()p Fi(m)p Fo(\))11682 31898 y Fl(+)295 b(2)369 b(=)g Fj(a)16072 32097 y Fi(h)16964 31898 y Fl(+)295 b(2.)579 b(Hence)15898 34821 y Fh(\000)p Fj(B)17986 34273 y Ff(0)17919 35150 y Fi(m)p Ff(\000)p Fo(2)20378 34821 y Fl(=)369 b Fj(\017)22284 35020 y Fi(m)23172 34821 y Fj(B)24227 34273 y Ff(0)24160 35150 y Fi(m)p Ff(\000)p Fo(2)20378 36758 y Fl(=)g Fj(B)22814 36210 y Ff(0)22747 37087 y Fi(m)23930 36758 y Fh(\000)295 b Fj(a)25941 36210 y Ff(0)25941 37087 y Fi(m)26829 36758 y Fj(B)27884 36210 y Ff(0)27817 37087 y Fi(m)p Ff(\000)p Fo(1)20378 38696 y Fl(=)369 b(\()p Fj(B)23253 38895 y Fi(h)24146 38696 y Fl(+)294 b Fj(B)26440 38895 y Fi(h)p Ff(\000)p Fo(1)28240 38696 y Fl(\))h Fh(\000)h Fl(\()p Fj(a)31559 38895 y Fi(h)32451 38696 y Fl(+)f(2\))p Fj(B)35902 38895 y Fi(h)p Ff(\000)p Fo(1)20378 40633 y Fl(=)369 b Fj(B)22747 40832 y Fi(h)23640 40633 y Fh(\000)295 b Fj(B)25956 40832 y Fi(h)p Ff(\000)p Fo(1)28051 40633 y Fh(\000)h Fj(a)30063 40832 y Fi(h)30660 40633 y Fj(B)31648 40832 y Fi(h)p Ff(\000)p Fo(1)20378 42570 y Fl(=)369 b Fj(B)22747 42769 y Fi(h)p Ff(\000)p Fo(2)24842 42570 y Fh(\000)295 b Fj(B)27158 42769 y Fi(h)p Ff(\000)p Fo(1)28958 42570 y Fj(:)800 45493 y Fl(Hence)8392 48416 y Fj(B)9447 47867 y Ff(0)9380 48744 y Fi(m)p Ff(\000)p Fo(1)11470 48416 y Fj(A)12445 47867 y Ff(0)12445 48744 y Fi(m)13628 48416 y Fh(\000)h Fj(A)15932 47867 y Ff(0)15932 48744 y Fi(m)p Ff(\000)p Fo(1)18022 48416 y Fj(B)19077 47867 y Ff(0)19010 48744 y Fi(m)p Ff(\000)p Fo(2)21469 48416 y Fl(=)368 b Fj(B)23837 48615 y Fi(h)p Ff(\000)p Fo(1)25637 48416 y Fl(\()p Fj(A)27118 48615 y Fi(h)28011 48416 y Fl(+)295 b Fj(A)30293 48615 y Fi(h)p Ff(\000)p Fo(1)32092 48416 y Fl(\))g Fh(\000)h Fj(A)35197 48615 y Fi(h)p Ff(\000)p Fo(1)36997 48416 y Fl(\()p Fj(B)38491 48615 y Fi(h)p Ff(\000)p Fo(1)40586 48416 y Fh(\000)f Fj(B)42902 48615 y Fi(h)p Ff(\000)p Fo(2)44702 48416 y Fl(\))21469 50353 y(=)368 b Fj(B)23837 50552 y Fi(h)p Ff(\000)p Fo(1)25637 50353 y Fj(A)26612 50552 y Fi(h)27505 50353 y Fl(+)295 b Fj(A)29787 50552 y Fi(h)p Ff(\000)p Fo(1)31587 50353 y Fj(B)32575 50552 y Fi(h)p Ff(\000)p Fo(2)21469 52290 y Fl(=)368 b Fj(A)23824 52489 y Fi(p)p Ff(\000)p Fo(1)25555 52290 y Fj(:)800 55213 y Fl(Finally)-108 b(,)10292 58136 y Fj(B)11347 57588 y Ff(0)11280 58465 y Fi(m)p Ff(\000)p Fo(1)13370 58136 y Fl(\()p Fj(B)14931 57588 y Ff(0)14864 58465 y Fi(m)16047 58136 y Fh(\000)295 b Fj(B)18430 57588 y Ff(0)18363 58465 y Fi(m)p Ff(\000)p Fo(2)20453 58136 y Fl(\))369 b(=)f Fj(B)23696 58335 y Fi(h)p Ff(\000)p Fo(1)25496 58136 y Fl(\(\()p Fj(B)27496 58335 y Fi(h)28389 58136 y Fl(+)295 b Fj(B)30684 58335 y Fi(h)p Ff(\000)p Fo(1)32484 58136 y Fl(\))g(+)f(\()p Fj(B)36085 58335 y Fi(h)p Ff(\000)p Fo(2)38180 58136 y Fh(\000)i Fj(B)40497 58335 y Fi(h)p Ff(\000)p Fo(1)42297 58136 y Fl(\)\))21328 60073 y(=)368 b Fj(B)23696 60272 y Fi(h)p Ff(\000)p Fo(1)25496 60073 y Fl(\()p Fj(B)26990 60272 y Fi(h)27883 60073 y Fl(+)295 b Fj(B)30178 60272 y Fi(h)p Ff(\000)p Fo(2)31978 60073 y Fl(\))21328 62011 y(=)368 b Fj(B)23696 62210 y Fi(p)p Ff(\000)p Fo(1)25428 62011 y Fj(:)800 65595 y Fl(Case)556 b(\(4\)\(a\).)943 b(Assume)554 b Fj(p)576 b Fl(=)f(2)p Fj(h)555 b Fl(with)g(an)g Fj(M)139 b Fl(-unisequence,)585 b Fj(M)694 b Fl(o)36 b(dd,)586 b(at)555 b(the)f(cen)-36 b(tre)555 b(of)g(a)h(p)36 b(erio)g(d.)800 67200 y(There)434 b(are)f(t)-36 b(w)g(o)434 b(cases:)p 0 TeXcolorgray 26150 70521 a(17)p 0 TeXcolorgray eop end end %%Page: 18 18 TeXDict begin HPSdict begin 18 17 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.18) cvn H.B /DEST pdfmark end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 1328 x Fj(M)508 b Fl(=)369 b(4)p Fj(t)295 b Fl(+)g(3,)434 b(e.g.,)10860 -3770 y Fh(p)p 11967 -3770 1301 54 v 1098 x Fl(88)369 b(=)g([)15378 -2672 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n0 16 {InitRnode } NewNode end end 15378 -2672 a Fl(9)q Fj(;)16611 -2672 y tx@Dict begin tx@NodeDict begin {7.73332 8.4 5.87494 2.93747 7.73332 } false /N@n1 16 {InitRnode } NewNode end end 16611 -2672 a Fl(2)16701 -1875 y Ff(\003)17262 -2672 y Fj(;)221 b Fl(1)p Fj(;)19076 -2672 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n2 16 {InitRnode } NewNode end end 19076 -2672 a Fl(1)r Fj(;)g Fl(1)p Fj(;)21542 -2672 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n3 16 {InitRnode } NewNode end end 21542 -2672 a Fl(2)r Fj(;)22776 -2672 y tx@Dict begin tx@NodeDict begin {7.73332 8.4 11.74988 5.87494 7.73332 } false /N@n4 16 {InitRnode } NewNode end end 22776 -2672 a Fl(18)23191 -1875 y Ff(\003)24077 -2672 y Fl(])24438 -2672 y tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n0 /N@n1 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 24438 -2672 a 24438 -2672 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n1 /N@n2 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 24438 -2672 a 24438 -2672 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n2 /N@n3 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 24438 -2672 a 24438 -2672 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n3 /N@n4 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 24438 -2672 a Fl(.)579 b(Let)433 b Fj(f)142 b Fl(\()p Fj(m)p Fl(\))368 b(=)h Fj(h)p Fl(.)578 b(Then)433 b Fj(f)142 b Fl(\()p Fj(m)295 b Fl(+)f(1\))370 b(=)e Fj(h)295 b Fl(+)g(2)434 b(and)19776 613 y Fj(\030)20408 65 y Ff(0)20344 942 y Fi(m)21600 613 y Fl(=)369 b Fj(\030)23549 812 y Fi(h)24442 613 y Fl(+)294 b(1)p Fj(;)656 b(\017)27940 812 y Fi(m)29197 613 y Fl(=)368 b Fh(\000)p Fl(1;)18573 2551 y Fj(\030)19205 2002 y Ff(0)19141 2879 y Fi(m)p Fo(+1)21600 2551 y Fl(=)h Fj(\030)23549 2750 y Fi(h)p Fo(+2)25644 2551 y Fl(+)295 b(1)p Fj(;)655 b(\017)29142 2750 y Fi(m)p Fo(+1)31601 2551 y Fl(=)369 b Fh(\000)p Fl(1)p Fj(:)800 5277 y Fl(and)431 b(as)h(with)f(case)h(\(3\)\(b\),)g(w)-36 b(e)431 b(ha)-36 b(v)g(e)432 b(criterion)f(1\))h(of)g(Theorem)p 0 0.5 0 TeXcolorrgb 33489 5277 a SDict begin H.S end 33489 5277 a 0 0.5 0 TeXcolorrgb Fl(6)p 0 0.5 0 TeXcolorrgb 34139 4421 a SDict begin H.R end 34139 4421 a 34139 5277 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.6) cvn H.B /ANN pdfmark end 34139 5277 a 0 TeXcolorgray Fl(.)578 b(No)-36 b(w)432 b Fj(m)f Fl(is)h(the)e(n)-36 b(um)g(b)36 b(er)430 b(of)j(jumps)800 6882 y(in)488 b([)p Fj(a)3416 7081 y Fo(0)3942 6882 y Fj(;)221 b(a)5207 7081 y Fi(h)5805 6882 y Fl(].)741 b(Then)487 b(w)-36 b(e)488 b(ha)-36 b(v)g(e)488 b(a)g(cen)-36 b(tral)488 b(unisequence)f([)p Fj(a)29527 7081 y Fi(r)30033 6882 y Fj(;)221 b(a)31298 7081 y Fi(p)p Ff(\000)p Fi(r)33010 6882 y Fl(])488 b(of)g(length)g(4)p Fj(t)332 b Fl(+)g(3.)741 b(There)487 b(are)h Fj(t)332 b Fl(+)g(1)800 8487 y(jumps)492 b(of)h(2)g(in)f([)p Fj(a)10035 8686 y Fi(r)10541 8487 y Fj(;)221 b(a)11806 8686 y Fi(h)12404 8487 y Fl(],)507 b(so)493 b Fj(r)371 b Fl(+)335 b(2)p Fj(t)g Fl(+)g(2)470 b(=)e Fj(h)p Fl(.)754 b(Let)492 b Fj(J)615 b Fl(b)36 b(e)492 b(the)g(n)-36 b(um)g(b)36 b(er)491 b(of)i(jumps)f(in)g([)p Fj(a)45594 8686 y Fo(0)46120 8487 y Fj(;)221 b(a)47385 8686 y Fi(r)47891 8487 y Fl(].)755 b(Hence)800 10092 y Fj(m)369 b Fl(=)f Fj(J)418 b Fl(+)295 b(\()p Fj(t)g Fl(+)g(1\).)578 b(There)434 b(are)f Fj(t)295 b Fl(+)g(1)434 b(jumps)f(of)h(2)g(in)g([)p Fj(a)28885 10291 y Fi(h)29482 10092 y Fj(;)221 b(a)30747 10291 y Fi(p)p Ff(\000)p Fi(r)32459 10092 y Fl(])434 b(and)f Fj(J)557 b Fl(jumps)433 b(in)g([)p Fj(a)43493 10291 y Fi(p)p Ff(\000)p Fi(r)45205 10092 y Fj(;)221 b(a)46470 10291 y Fi(p)46999 10092 y Fl(].)579 b(Hence)12348 12818 y Fj(k)414 b Fl(=)369 b(\()p Fj(J)418 b Fl(+)294 b Fj(t)h Fl(+)g(1\))g(+)g(\()p Fj(t)g Fl(+)g(1\))g(+)g Fj(J)492 b Fl(=)368 b(2\()p Fj(J)419 b Fl(+)294 b Fj(t)h Fl(+)g(1\))369 b(=)g(2)p Fj(m:)800 15544 y(M)546 b Fl(=)406 b(4)p Fj(t)310 b Fl(+)g(1,)461 b(e.g.,)11020 14445 y Fh(p)p 12127 14445 1301 54 v 1099 x Fl(91)407 b(=)f([)15613 15544 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n0 16 {InitRnode } NewNode end end 15613 15544 a Fl(9)q Fj(;)221 b Fl(1)16936 16341 y Ff(\003)17497 15544 y Fj(;)18079 15544 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n1 16 {InitRnode } NewNode end end 18079 15544 a Fl(1)q Fj(;)19312 15544 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n2 16 {InitRnode } NewNode end end 19312 15544 a Fl(5)q Fj(;)g Fl(1)p Fj(;)21777 15544 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n3 16 {InitRnode } NewNode end end 21777 15544 a Fl(5)r Fj(;)g Fl(1)p Fj(;)24243 15544 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n4 16 {InitRnode } NewNode end end 24243 15544 a Fl(1)r Fj(;)25477 15544 y tx@Dict begin tx@NodeDict begin {7.73332 8.4 11.74988 5.87494 7.73332 } false /N@n5 16 {InitRnode } NewNode end end 25477 15544 a Fl(18)25892 16341 y Ff(\003)26778 15544 y Fl(])27139 15544 y tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n0 /N@n1 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 27139 15544 a 27139 15544 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n1 /N@n2 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 27139 15544 a 27139 15544 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n2 /N@n3 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 27139 15544 a 27139 15544 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n3 /N@n4 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 27139 15544 a 27139 15544 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n4 /N@n5 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 27139 15544 a Fl(.)645 b(Let)455 b Fj(f)142 b Fl(\()p Fj(m)p Fl(\))406 b(=)g Fj(h)310 b Fh(\000)g Fl(1.)645 b(Then)455 b Fj(f)142 b Fl(\()p Fj(m)310 b Fl(+)f(1\))407 b(=)f Fj(h)310 b Fl(+)g(1)800 17507 y(and)18573 19112 y Fj(\030)19205 18564 y Ff(0)19141 19441 y Fi(m)p Fo(+1)21600 19112 y Fl(=)369 b Fj(\030)23549 19311 y Fi(h)p Fo(+1)25644 19112 y Fl(+)295 b(1)p Fj(;)655 b(\017)29142 19311 y Fi(m)p Fo(+1)31601 19112 y Fl(=)369 b Fh(\000)p Fl(1)p Fj(:)800 21348 y Fl(Also)418 b Fj(\030)4349 20866 y Ff(0)4285 21676 y Fi(m)5542 21348 y Fl(=)368 b Fj(\030)7490 21547 y Fi(h)p Ff(\000)p Fo(1)9708 21348 y Fl(or)417 b Fj(\030)11849 21547 y Fi(h)p Ff(\000)p Fo(1)13911 21348 y Fl(+)262 b(1.)573 b(W)-108 b(e)418 b(ha)-36 b(v)g(e)417 b(a)h(cen)-36 b(tral)417 b(unisequence)g([)p Fj(a)35551 21547 y Fi(r)36057 21348 y Fj(;)221 b(a)37322 21547 y Fi(p)p Ff(\000)p Fi(r)39034 21348 y Fl(])418 b(of)g(length)f(4)p Fj(t)262 b Fl(+)g(1.)573 b(There)800 22953 y(are)484 b Fj(t)f Fl(jumps)g(of)h(2)f(in)h([)p Fj(a)13170 23152 y Fi(r)13676 22953 y Fj(;)221 b(a)14941 23152 y Fi(h)p Ff(\000)p Fo(1)16741 22953 y Fl(],)497 b(so)483 b Fj(r)366 b Fl(+)328 b(2)p Fj(t)454 b Fl(=)g Fj(h)328 b Fh(\000)i Fl(1.)728 b(Let)483 b Fj(J)606 b Fl(b)36 b(e)483 b(the)g(n)-36 b(um)g(b)36 b(er)482 b(of)i(jumps)f(in)g ([)p Fj(a)49780 23152 y Fo(0)50306 22953 y Fj(;)221 b(a)51571 23152 y Fi(r)52077 22953 y Fl(].)800 24558 y(Hence)373 b Fj(m)p Fl(,)386 b(b)36 b(eing)374 b(the)f(n)-36 b(um)g(b)36 b(er)371 b(of)k(jumps)e(in)g([)p Fj(a)24523 24757 y Fo(0)25049 24558 y Fj(;)221 b(a)26314 24757 y Fi(h)p Ff(\000)p Fo(1)28114 24558 y Fl(])374 b(satis\014es)g Fj(m)369 b Fl(=)f Fj(J)296 b Fl(+)173 b Fj(t)p Fl(.)557 b(There)374 b(is)f(also)i(one)f(jump)800 26163 y(in)433 b([)p Fj(a)3361 26362 y Fi(h)p Ff(\000)p Fo(1)5161 26163 y Fj(;)221 b(a)6426 26362 y Fi(h)p Fo(+1)8227 26163 y Fl(],)434 b Fj(t)f Fl(jumps)g(of)i(2)e(in)h([)p Fj(a)19286 26362 y Fi(h)p Fo(+1)21086 26163 y Fj(;)221 b(a)22351 26362 y Fi(p)p Ff(\000)p Fi(r)24063 26163 y Fl(])434 b(and)f Fj(J)556 b Fl(jumps)433 b(in)h([)p Fj(a)35097 26362 y Fi(p)p Ff(\000)p Fi(r)36808 26163 y Fj(;)221 b(a)38073 26362 y Fi(p)38603 26163 y Fl(].)578 b(Hence)12854 28889 y Fj(k)414 b Fl(=)368 b(\()p Fj(J)418 b Fl(+)295 b Fj(t)p Fl(\))g(+)g(1)g(+)g Fj(t)g Fl(+)g Fj(J)492 b Fl(=)368 b(2\()p Fj(J)418 b Fl(+)295 b Fj(t)p Fl(\))g(+)g(1)369 b(=)g(2)p Fj(m)295 b Fl(+)g(1)p Fj(:)800 31615 y Fl(Then)541 b Fj(Q)5334 31133 y Ff(0)5334 31944 y Fi(m)p Fo(+1)7976 31615 y Fl(=)551 b Fj(Q)10569 31814 y Fi(h)p Fo(+1)12910 31615 y Fl(and)541 b Fj(Q)16577 31133 y Ff(0)16577 31944 y Fi(m)18016 31615 y Fl(=)552 b Fj(Q)20610 31814 y Fi(h)p Ff(\000)p Fo(1)22410 31615 y Fl(,)568 b(so)541 b Fj(Q)26073 31814 y Fi(h)p Ff(\000)p Fo(1)28425 31615 y Fl(=)551 b Fj(Q)31018 31814 y Fi(h)p Fo(+1)33359 31615 y Fl(implies)542 b Fj(Q)38912 31133 y Ff(0)38912 31944 y Fi(m)40352 31615 y Fl(=)551 b Fj(Q)42945 31133 y Ff(0)42945 31944 y Fi(m)p Fo(+1)45576 31615 y Fl(and)541 b(w)-36 b(e)541 b(ha)-36 b(v)g(e)800 33220 y(criterion)443 b(3\))f(of)h(Theorem)p 0 0.5 0 TeXcolorrgb 14601 33220 a SDict begin H.S end 14601 33220 a 0 0.5 0 TeXcolorrgb Fl(6)p 0 0.5 0 TeXcolorrgb 15251 32364 a SDict begin H.R end 15251 32364 a 15251 33220 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.6) cvn H.B /ANN pdfmark end 15251 33220 a 0 TeXcolorgray Fl(.)605 b(W)-108 b(e)443 b(ha)-36 b(v)g(e)442 b Fj(A)22448 32738 y Ff(0)22448 33549 y Fi(m)23720 33220 y Fl(=)384 b Fj(A)26091 33427 y Fi(f)98 b Fo(\()p Fi(m)p Fo(\)+1)29846 33220 y Fl(=)384 b Fj(A)32217 33419 y Fi(h)32815 33220 y Fl(.)605 b(Also,)445 b(regardless)e(of)g(the)f(sign)h(of)g Fj(\017)51551 33419 y Fi(m)52439 33220 y Fl(,)800 34826 y(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b Fj(A)6689 34343 y Ff(0)6689 35154 y Fi(m)p Ff(\000)p Fo(1)9148 34826 y Fl(=)368 b Fj(A)11503 35025 y Fi(h)p Ff(\000)p Fo(2)13303 34826 y Fl(.)578 b(W)-108 b(e)434 b(no)-36 b(w)434 b(pro)-36 b(v)g(e)18626 37552 y Fj(A)19601 37751 y Fi(k)24 b Ff(\000)p Fo(1)21741 37552 y Fl(=)368 b Fj(A)24096 37003 y Ff(0)24096 37880 y Fi(m)24984 37552 y Fj(B)26039 37003 y Ff(0)25972 37880 y Fi(m)27155 37552 y Fh(\000)295 b Fj(A)29458 37003 y Ff(0)29458 37880 y Fi(m)p Ff(\000)p Fo(1)31548 37552 y Fj(B)32603 37003 y Ff(0)32536 37880 y Fi(m)p Ff(\000)p Fo(1)34626 37552 y Fj(;)50488 35614 y SDict begin H.S end 50488 35614 a 50488 35614 a SDict begin 17.5 H.A end 50488 35614 a 50488 35614 a SDict begin [ /View [/XYZ H.V] /Dest (equation.36) cvn H.B /DEST pdfmark end 50488 35614 a 1938 x Fl(\(36\))18613 39737 y Fj(B)19601 39936 y Fi(k)24 b Ff(\000)p Fo(1)21741 39737 y Fl(=)368 b Fj(B)24176 39188 y Ff(0)24487 39047 y Fo(2)24487 40065 y Fi(m)25670 39737 y Fh(\000)295 b Fj(B)28053 39188 y Ff(0)28364 39047 y Fo(2)28364 40065 y Fi(m)p Ff(\000)p Fo(1)30454 39737 y Fj(:)50488 37799 y SDict begin H.S end 50488 37799 a 50488 37799 a SDict begin 17.5 H.A end 50488 37799 a 50488 37799 a SDict begin [ /View [/XYZ H.V] /Dest (equation.37) cvn H.B /DEST pdfmark end 50488 37799 a 1938 x Fl(\(37\))800 42463 y(Noting)434 b(that)f Fj(a)8600 42662 y Fi(h)9566 42463 y Fl(=)369 b(1)434 b(giv)-36 b(es)435 b Fj(A)16193 42662 y Fi(h)17159 42463 y Fl(=)369 b Fj(A)19515 42662 y Fi(h)p Ff(\000)p Fo(1)21610 42463 y Fl(+)295 b Fj(A)23892 42662 y Fi(h)p Ff(\000)p Fo(2)26125 42463 y Fl(and)433 b Fj(B)29642 42662 y Fi(h)30609 42463 y Fl(=)368 b Fj(B)32977 42662 y Fi(h)p Ff(\000)p Fo(1)35073 42463 y Fl(+)294 b Fj(B)37367 42662 y Fi(h)p Ff(\000)p Fo(2)39167 42463 y Fl(,)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)9595 45189 y Fj(A)10570 44640 y Ff(0)10570 45517 y Fi(m)11457 45189 y Fj(B)12512 44640 y Ff(0)12445 45517 y Fi(m)13628 45189 y Fh(\000)296 b Fj(A)15932 44640 y Ff(0)15932 45517 y Fi(m)p Ff(\000)p Fo(1)18022 45189 y Fj(B)19077 44640 y Ff(0)19010 45517 y Fi(m)p Ff(\000)p Fo(1)21469 45189 y Fl(=)368 b Fj(A)23824 45388 y Fi(h)24422 45189 y Fj(B)25410 45388 y Fi(h)26303 45189 y Fh(\000)295 b Fj(A)28606 45388 y Fi(h)p Ff(\000)p Fo(2)30406 45189 y Fj(B)31394 45388 y Fi(h)p Ff(\000)p Fo(2)21469 47126 y Fl(=)368 b Fj(A)23824 47325 y Fi(h)24422 47126 y Fl(\()p Fj(B)25916 47325 y Fi(h)p Ff(\000)p Fo(1)28011 47126 y Fl(+)295 b Fj(B)30306 47325 y Fi(h)p Ff(\000)p Fo(2)32105 47126 y Fl(\))g Fh(\000)h Fl(\()p Fj(A)35716 47325 y Fi(h)36608 47126 y Fh(\000)g Fj(A)38912 47325 y Fi(h)p Ff(\000)p Fo(1)40712 47126 y Fl(\))p Fj(B)42206 47325 y Fi(h)p Ff(\000)p Fo(2)21469 49063 y Fl(=)368 b Fj(A)23824 49262 y Fi(h)24422 49063 y Fj(B)25410 49262 y Fi(h)p Ff(\000)p Fo(1)27505 49063 y Fl(+)295 b Fj(A)29787 49262 y Fi(h)p Ff(\000)p Fo(1)31587 49063 y Fj(B)32575 49262 y Fi(h)p Ff(\000)p Fo(2)21469 51000 y Fl(=)368 b Fj(A)23824 51199 y Fi(p)p Ff(\000)p Fo(1)25555 51000 y Fj(:)800 53726 y Fl(Also)15261 56452 y Fj(B)16316 55904 y Ff(0)16627 55762 y Fo(2)16627 56781 y Fi(m)17809 56452 y Fh(\000)296 b Fj(B)20193 55904 y Ff(0)20504 55762 y Fo(2)20504 56781 y Fi(m)p Ff(\000)p Fo(1)22962 56452 y Fl(=)369 b Fj(B)25398 55904 y Fo(2)25331 56781 y Fi(h)26224 56452 y Fh(\000)295 b Fj(B)28607 55904 y Fo(2)28540 56781 y Fi(h)p Ff(\000)p Fo(2)22962 58390 y Fl(=)369 b(\()p Fj(B)25837 58589 y Fi(h)26730 58390 y Fh(\000)295 b Fj(B)29046 58589 y Fi(h)p Ff(\000)p Fo(2)30846 58390 y Fl(\)\()p Fj(B)32846 58589 y Fi(h)33739 58390 y Fl(+)f Fj(B)36033 58589 y Fi(h)p Ff(\000)p Fo(2)37833 58390 y Fl(\))22962 60327 y(=)369 b Fj(B)25331 60526 y Fi(h)p Ff(\000)p Fo(1)27131 60327 y Fl(\()p Fj(B)28625 60526 y Fi(h)29518 60327 y Fl(+)295 b Fj(B)31813 60526 y Fi(h)p Ff(\000)p Fo(2)33612 60327 y Fl(\))22962 62264 y(=)369 b Fj(B)25331 62463 y Fi(p)p Ff(\000)p Fo(1)27062 62264 y Fj(:)800 65595 y Fl(Case)444 b(\(4\)\(b\))e(Assume)g Fj(p)385 b Fl(=)g(2)p Fj(h)301 b Fh(\000)h Fl(1)443 b(with)g(an)g Fj(M)139 b Fl(-unisequence,)445 b Fj(M)582 b Fl(ev)-36 b(en,)445 b(at)e(the)g(cen)-36 b(tre)442 b(of)i(a)f(p)36 b(erio)g(d.)800 67200 y(There)434 b(are)f(t)-36 b(w)g(o)434 b(cases:)p 0 TeXcolorgray 26150 70521 a(18)p 0 TeXcolorgray eop end end %%Page: 19 19 TeXDict begin HPSdict begin 19 18 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.19) cvn H.B /DEST pdfmark end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 1328 x Fj(M)508 b Fl(=)369 b(4)p Fj(t)p Fl(,)434 b(e.g.,)8607 -3770 y Fh(p)p 9714 -3770 1301 54 v 1098 x Fl(13)370 b(=)f([)13126 -2672 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n0 16 {InitRnode } NewNode end end 13126 -2672 a Fl(3)p Fj(;)221 b Fl(1)14449 -1875 y Ff(\003)15009 -2672 y Fj(;)15591 -2672 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n1 16 {InitRnode } NewNode end end 15591 -2672 a Fl(1)q Fj(;)g Fl(1)p Fj(;)18056 -2672 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n2 16 {InitRnode } NewNode end end 18056 -2672 a Fl(1)r Fj(;)19290 -2672 y tx@Dict begin tx@NodeDict begin {7.73332 8.4 5.87494 2.93747 7.73332 } false /N@n3 16 {InitRnode } NewNode end end 19290 -2672 a Fl(6)19380 -1875 y Ff(\003)19941 -2672 y Fl(])20302 -2672 y tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n0 /N@n1 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 20302 -2672 a 20302 -2672 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n1 /N@n2 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 20302 -2672 a 20302 -2672 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n2 /N@n3 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 20302 -2672 a Fl(.)579 b(Let)433 b Fj(f)142 b Fl(\()p Fj(m)p Fl(\))368 b(=)h Fj(h)295 b Fh(\000)g Fl(1.)579 b(Then)433 b Fj(f)142 b Fl(\()p Fj(m)295 b Fl(+)g(1\))369 b(=)f Fj(h)295 b Fl(+)g(1)434 b(and)19776 466 y Fj(\030)20408 -82 y Ff(0)20344 795 y Fi(m)21600 466 y Fl(=)369 b Fj(\030)23549 665 y Fi(h)p Ff(\000)p Fo(1)25644 466 y Fl(+)295 b(1)p Fj(;)655 b(\017)29142 665 y Fi(m)30399 466 y Fl(=)369 b Fh(\000)p Fl(1)p Fj(;)18573 2403 y(\030)19205 1855 y Ff(0)19141 2732 y Fi(m)p Fo(+1)21600 2403 y Fl(=)g Fj(\030)23549 2602 y Fi(h)p Fo(+1)25644 2403 y Fl(+)295 b(1)p Fj(;)655 b(\017)29142 2602 y Fi(m)p Fo(+1)31601 2403 y Fl(=)369 b Fh(\000)p Fl(1)p Fj(:)800 4982 y Fl(W)-108 b(e)553 b(ha)-36 b(v)g(e)554 b(a)f(cen)-36 b(tral)554 b(unisequence)e([)p Fj(a)20261 5181 y Fi(r)20767 4982 y Fj(;)221 b(a)22032 5181 y Fi(p)p Ff(\000)p Fi(r)23744 4982 y Fl(])554 b(of)g(length)f(4)p Fj(t)g Fl(with)h Fj(r)413 b Fl(+)376 b(2)p Fj(t)573 b Fl(=)f Fj(h)377 b Fh(\000)g Fl(1.)937 b(Let)553 b Fj(J)676 b Fl(b)36 b(e)554 b(the)800 6587 y(n)-36 b(um)g(b)36 b(er)381 b(of)i(jumps)f(in)h([)p Fj(a)13245 6786 y Fo(0)13771 6587 y Fj(;)221 b(a)15036 6786 y Fi(r)15542 6587 y Fl(].)562 b(There)382 b(are)h Fj(t)f Fl(jumps)g(of)i(2)e(in)h([)p Fj(a)32297 6786 y Fi(r)32803 6587 y Fj(;)221 b(a)34068 6786 y Fi(h)p Ff(\000)p Fo(1)35868 6587 y Fl(].)562 b(Hence)382 b Fj(m)p Fl(,)393 b(b)36 b(eing)383 b(the)f(n)-36 b(um)g(b)36 b(er)800 8192 y(of)464 b(jumps)e(in)h([)p Fj(a)8804 8391 y Fo(0)9330 8192 y Fj(;)221 b(a)10595 8391 y Fi(h)p Ff(\000)p Fo(1)12395 8192 y Fl(])463 b(satis\014es)g Fj(m)418 b Fl(=)h Fj(J)438 b Fl(+)315 b Fj(t)p Fl(.)665 b(There)463 b(are)g(also)h Fj(t)e Fl(jumps)h(of)g(2)g(in)g([)p Fj(a)43677 8391 y Fi(h)p Ff(\000)p Fo(1)45477 8192 y Fj(;)221 b(a)46742 8391 y Fi(p)p Ff(\000)p Fi(r)26 b Ff(\000)p Fo(1)49656 8192 y Fl(],)471 b(one)800 9797 y(jump)433 b(in)h([)p Fj(a)6722 9996 y Fi(p)p Ff(\000)p Fi(r)26 b Ff(\000)p Fo(1)9635 9797 y Fj(;)221 b(a)10900 9996 y Fi(p)p Ff(\000)p Fi(r)12612 9797 y Fl(])434 b(and)f Fj(J)557 b Fl(jumps)433 b(in)g([)p Fj(a)23646 9996 y Fi(p)p Ff(\000)p Fi(r)25358 9797 y Fj(;)221 b(a)26623 9996 y Fi(p)27152 9797 y Fl(].)579 b(Hence)12854 12376 y Fj(k)414 b Fl(=)368 b(\()p Fj(J)418 b Fl(+)295 b Fj(t)p Fl(\))g(+)g Fj(t)g Fl(+)f(1)i(+)f Fj(J)492 b Fl(=)368 b(2\()p Fj(J)418 b Fl(+)295 b Fj(t)p Fl(\))g(+)g(1)369 b(=)g(2)p Fj(m)295 b Fl(+)g(1)p Fj(:)800 14955 y Fl(Then)11240 17534 y Fj(\017)11765 17733 y Fi(m)13022 17534 y Fl(=)369 b Fh(\000)p Fl(1)738 b(=)-221 b Fh(\))738 b Fj(f)142 b Fl(\()p Fj(m)p Fl(\))368 b(=)h Fj(f)142 b Fl(\()p Fj(m)294 b Fh(\000)i Fl(1\))f(+)g(2)p Fj(;)655 b(A)33812 16985 y Ff(0)33812 17862 y Fi(m)p Ff(\000)p Fo(1)36271 17534 y Fl(=)369 b Fj(A)38627 17740 y Fi(f)98 b Fo(\()p Fi(m)p Ff(\000)p Fo(1\)+1)43201 17534 y Fj(;)10038 19471 y(\017)10563 19670 y Fi(m)p Fo(+1)13022 19471 y Fl(=)369 b Fh(\000)p Fl(1)738 b(=)-221 b Fh(\))738 b Fj(f)142 b Fl(\()p Fj(m)294 b Fl(+)h(1\))369 b(=)g Fj(f)142 b Fl(\()p Fj(m)p Fl(\))294 b(+)h(2)p Fj(;)656 b(A)33791 18922 y Ff(0)33791 19799 y Fi(m)35047 19471 y Fl(=)369 b Fj(A)37403 19677 y Fi(f)98 b Fo(\()p Fi(m)p Fo(\)+1)40775 19471 y Fj(:)800 22050 y Fl(W)-108 b(e)434 b(ha)-36 b(v)g(e)433 b Fj(A)7013 21567 y Ff(0)7013 22378 y Fi(m)p Ff(\000)p Fo(1)9472 22050 y Fl(=)369 b Fj(A)11828 22249 y Fi(h)p Ff(\000)p Fo(2)13628 22050 y Fj(;)655 b(A)15619 21567 y Ff(0)15619 22378 y Fi(m)16876 22050 y Fl(=)368 b Fj(A)19231 22249 y Fi(h)19829 22050 y Fl(.)578 b(W)-108 b(e)434 b(no)-36 b(w)434 b(v)-36 b(erify)435 b(criterion)e(4\))h(of)g(Theorem)p 0 0.5 0 TeXcolorrgb 43096 22050 a SDict begin H.S end 43096 22050 a 0 0.5 0 TeXcolorrgb Fl(6)p 0 0.5 0 TeXcolorrgb 43746 21193 a SDict begin H.R end 43746 21193 a 43746 22050 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.6) cvn H.B /ANN pdfmark end 43746 22050 a 0 TeXcolorgray Fl(.)44686 20444 y SDict begin H.S end 44686 20444 a 44686 20444 a SDict begin 14.5 H.A end 44686 20444 a 44686 20444 a SDict begin [ /View [/XYZ H.V] /Dest (equation.38) cvn H.B /DEST pdfmark end 44686 20444 a 20625 24628 a Fj(P)21644 24080 y Ff(0)21463 24957 y Fi(m)p Fo(+1)23922 24628 y Fl(=)369 b Fj(Q)26333 24080 y Ff(0)26333 24957 y Fi(m)27516 24628 y Fl(+)28955 24266 y Fg(1)p 28955 24427 406 40 v 28955 24976 a(2)29494 24628 y Fj(Q)30524 24080 y Ff(0)30524 24957 y Fi(m)p Fo(+1)32614 24628 y Fj(:)17513 b Fl(\(38\))800 27207 y(W)-108 b(e)434 b(note)f(that)g Fj(a)9431 27406 y Fi(h)10397 27207 y Fl(=)369 b(1)434 b(implies)g Fj(P)18115 27406 y Fi(h)p Fo(+1)20284 27207 y Fl(=)369 b Fj(Q)22695 27406 y Fi(h)23588 27207 y Fh(\000)295 b Fj(P)25754 27406 y Fi(h)26352 27207 y Fl(.)578 b(Then)19721 29786 y Fj(P)20740 29237 y Ff(0)20559 30114 y Fi(m)p Fo(+1)23018 29786 y Fl(=)369 b Fj(P)25237 29985 y Fi(h)p Fo(+1)27332 29786 y Fl(+)295 b Fj(Q)29669 29985 y Fi(h)p Fo(+1)23018 31723 y Fl(=)369 b Fj(Q)25429 31922 y Fi(h)26322 31723 y Fh(\000)295 b Fj(P)28488 31922 y Fi(h)29381 31723 y Fl(+)g Fj(Q)31718 31922 y Fi(h)p Fo(+1)33518 31723 y Fj(:)800 34302 y Fl(Also)18122 35907 y Fj(Q)19152 35358 y Ff(0)19152 36235 y Fi(m)20335 35907 y Fl(+)21775 35545 y Fg(1)p 21775 35706 V 21775 36254 a(2)22313 35907 y Fj(Q)23343 35358 y Ff(0)23343 36235 y Fi(m)p Fo(+1)25803 35907 y Fl(=)368 b Fj(Q)28213 36106 y Fi(h)p Ff(\000)p Fo(1)30308 35907 y Fl(+)31748 35545 y Fg(1)p 31748 35706 V 31748 36254 a(2)32287 35907 y Fj(Q)33317 36106 y Fi(h)p Fo(+1)35117 35907 y Fj(:)800 38079 y Fl(Hence)433 b(\()p 0 0.5 0 TeXcolorrgb 5171 38079 a SDict begin H.S end 5171 38079 a 0 0.5 0 TeXcolorrgb Fl(38)p 0 0.5 0 TeXcolorrgb 6472 37223 a SDict begin H.R end 6472 37223 a 6472 38079 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.38) cvn H.B /ANN pdfmark end 6472 38079 a 0 TeXcolorgray Fl(\))g(holds)h(if)g(and)f(only)h(if)19048 40658 y Fj(Q)20078 40857 y Fi(h)20971 40658 y Fh(\000)295 b Fj(P)23137 40857 y Fi(h)24030 40658 y Fl(+)g Fj(Q)26367 40857 y Fi(h)p Fo(+1)28536 40658 y Fl(=)369 b Fj(Q)30947 40857 y Fi(h)p Ff(\000)p Fo(1)33042 40658 y Fl(+)34481 40296 y Fg(1)p 34481 40457 406 40 v 34481 41005 a(2)35020 40658 y Fj(Q)36050 40857 y Fi(h)p Fo(+1)37850 40658 y Fj(;)15389 42595 y Fl(i.e.,)435 b Fj(Q)18876 42794 y Fi(h)p Ff(\000)p Fo(1)20971 42595 y Fh(\000)295 b Fj(P)23137 42794 y Fi(h)24030 42595 y Fl(+)g Fj(Q)26367 42794 y Fi(h)p Fo(+1)28536 42595 y Fl(=)369 b Fj(Q)30947 42794 y Fi(h)p Ff(\000)p Fo(1)33042 42595 y Fl(+)34481 42233 y Fg(1)p 34481 42394 V 34481 42943 a(2)35020 42595 y Fj(Q)36050 42794 y Fi(h)p Fo(+1)37850 42595 y Fj(;)24274 44533 y Fl(i.e.,)435 b Fj(P)27569 44732 y Fi(h)28536 44533 y Fl(=)30049 44171 y Fg(1)p 30049 44332 V 30049 44880 a(2)30588 44533 y Fj(Q)31618 44732 y Fi(h)p Fo(+1)33418 44533 y Fj(:)800 47111 y Fl(Ho)-36 b(w)g(ev)g(er)23617 49690 y Fj(P)24636 49142 y Fo(2)24455 50018 y Fi(h)25531 49690 y Fl(=)369 b Fj(D)331 b Fh(\000)295 b Fj(Q)30681 49889 y Fi(h)p Ff(\000)p Fo(1)32481 49690 y Fj(Q)33511 49889 y Fi(h)34109 49690 y Fj(;)22524 51627 y(P)23543 51079 y Fo(2)23362 51956 y Fi(h)p Fo(+1)25531 51627 y Fl(=)369 b Fj(D)331 b Fh(\000)295 b Fj(Q)30681 51826 y Fi(h)31279 51627 y Fj(Q)32309 51826 y Fi(h)p Fo(+1)34109 51627 y Fj(;)19356 53564 y(P)20375 53016 y Fo(2)20194 53893 y Fi(h)21196 53564 y Fh(\000)g Fj(P)23543 53016 y Fo(2)23362 53893 y Fi(h)p Fo(+1)25531 53564 y Fl(=)369 b Fj(Q)27942 53763 y Fi(h)28540 53564 y Fl(\()p Fj(Q)30076 53763 y Fi(h)p Fo(+1)32171 53564 y Fh(\000)295 b Fj(Q)34529 53763 y Fi(h)p Ff(\000)p Fo(1)36329 53564 y Fl(\))p Fj(;)19465 55502 y(P)20303 55701 y Fi(h)21196 55502 y Fh(\000)g Fj(P)23362 55701 y Fi(h)p Fo(+1)25531 55502 y Fl(=)369 b Fj(Q)27942 55701 y Fi(h)p Fo(+1)30037 55502 y Fh(\000)295 b Fj(Q)32395 55701 y Fi(h)p Ff(\000)p Fo(1)34195 55502 y Fj(;)16404 57439 y(P)17242 57638 y Fi(h)18135 57439 y Fh(\000)h Fl(\()p Fj(Q)21000 57638 y Fi(h)21892 57439 y Fh(\000)g Fj(P)24059 57638 y Fi(h)24656 57439 y Fl(\))369 b(=)g Fj(Q)27942 57638 y Fi(h)p Fo(+1)30037 57439 y Fh(\000)295 b Fj(Q)32395 57638 y Fi(h)32993 57439 y Fj(;)23076 59376 y Fl(2)p Fj(P)24564 59575 y Fi(h)25531 59376 y Fl(=)369 b Fj(Q)27942 59575 y Fi(h)p Fo(+1)29742 59376 y Fj(:)800 61955 y Fl(Next)434 b(w)-36 b(e)434 b(pro)-36 b(v)g(e)11272 64533 y Fj(A)12247 64732 y Fi(p)p Ff(\000)p Fo(1)14347 64533 y Fl(=)369 b Fj(A)16703 63985 y Ff(0)16703 64862 y Fi(m)17590 64533 y Fj(B)18645 63985 y Ff(0)18578 64862 y Fi(m)19761 64533 y Fl(+)295 b(2)p Fj(A)22693 63985 y Ff(0)22693 64862 y Fi(m)p Ff(\000)p Fo(1)24783 64533 y Fj(B)25838 63985 y Ff(0)25771 64862 y Fi(m)p Ff(\000)p Fo(1)28157 64533 y Fh(\000)g Fl(\()p Fj(A)30966 63985 y Ff(0)30966 64862 y Fi(m)31853 64533 y Fj(B)32908 63985 y Ff(0)32841 64862 y Fi(m)p Ff(\000)p Fo(1)35227 64533 y Fl(+)g Fj(B)37589 63985 y Ff(0)37522 64862 y Fi(m)38409 64533 y Fj(A)39384 63985 y Ff(0)39384 64862 y Fi(m)p Ff(\000)p Fo(1)41474 64533 y Fl(\))p Fj(;)50488 62596 y SDict begin H.S end 50488 62596 a 50488 62596 a SDict begin 17.5 H.A end 50488 62596 a 50488 62596 a SDict begin [ /View [/XYZ H.V] /Dest (equation.39) cvn H.B /DEST pdfmark end 50488 62596 a 1937 x Fl(\(39\))11259 66718 y Fj(B)12247 66917 y Fi(p)p Ff(\000)p Fo(1)14347 66718 y Fl(=)369 b Fj(B)16783 66170 y Ff(0)17093 66029 y Fo(2)17093 67047 y Fi(m)18276 66718 y Fl(+)295 b(2)p Fj(B)21288 66170 y Ff(0)21599 66029 y Fo(2)21599 67047 y Fi(m)p Ff(\000)p Fo(1)23984 66718 y Fh(\000)h Fl(2)p Fj(B)27018 66170 y Ff(0)26951 67047 y Fi(m)27839 66718 y Fj(B)28894 66170 y Ff(0)28827 67047 y Fi(m)p Ff(\000)p Fo(1)30917 66718 y Fj(:)50488 64781 y SDict begin H.S end 50488 64781 a 50488 64781 a SDict begin 17.5 H.A end 50488 64781 a 50488 64781 a SDict begin [ /View [/XYZ H.V] /Dest (equation.40) cvn H.B /DEST pdfmark end 50488 64781 a 1937 x Fl(\(40\))p 0 TeXcolorgray 26150 70521 a(19)p 0 TeXcolorgray eop end end %%Page: 20 20 TeXDict begin HPSdict begin 20 19 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.20) cvn H.B /DEST pdfmark end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 1328 x Fl(Let)12147 -1067 y Fj(T)550 b Fl(=)369 b Fj(A)15815 -1615 y Ff(0)15815 -738 y Fi(m)16702 -1067 y Fj(B)17757 -1615 y Ff(0)17690 -738 y Fi(m)18873 -1067 y Fl(+)295 b(2)p Fj(A)21805 -1615 y Ff(0)21805 -738 y Fi(m)p Ff(\000)p Fo(1)23895 -1067 y Fj(B)24950 -1615 y Ff(0)24883 -738 y Fi(m)p Ff(\000)p Fo(1)27268 -1067 y Fh(\000)h Fl(\()p Fj(A)30078 -1615 y Ff(0)30078 -738 y Fi(m)30965 -1067 y Fj(B)32020 -1615 y Ff(0)31953 -738 y Fi(m)p Ff(\000)p Fo(1)34339 -1067 y Fl(+)e Fj(B)36700 -1615 y Ff(0)36633 -738 y Fi(m)37521 -1067 y Fj(A)38496 -1615 y Ff(0)38496 -738 y Fi(m)p Ff(\000)p Fo(1)40586 -1067 y Fl(\))p Fj(:)800 998 y Fl(Then)11888 3324 y Fj(T)550 b Fl(=)368 b Fj(A)15555 3523 y Fi(h)16153 3324 y Fj(B)17141 3523 y Fi(h)18034 3324 y Fl(+)295 b(2)p Fj(A)20966 3523 y Fi(h)p Ff(\000)p Fo(2)22766 3324 y Fj(B)23754 3523 y Fi(h)p Ff(\000)p Fo(2)25849 3324 y Fh(\000)g Fl(\()p Fj(A)28658 3523 y Fi(h)29256 3324 y Fj(B)30244 3523 y Fi(h)p Ff(\000)p Fo(2)32339 3324 y Fl(+)g Fj(B)34634 3523 y Fi(h)35231 3324 y Fj(A)36206 3523 y Fi(h)p Ff(\000)p Fo(2)38006 3324 y Fl(\))13200 5261 y(=)368 b Fj(A)15555 5460 y Fi(h)16153 5261 y Fl(\()p Fj(B)17647 5460 y Fi(h)18540 5261 y Fh(\000)295 b Fj(B)20856 5460 y Fi(h)p Ff(\000)p Fo(2)22656 5261 y Fl(\))g(+)g Fj(A)25739 5460 y Fi(h)p Ff(\000)p Fo(2)27539 5261 y Fl(\()p Fj(B)29033 5460 y Fi(h)p Ff(\000)p Fo(2)31128 5261 y Fh(\000)g Fj(B)33444 5460 y Fi(h)34042 5261 y Fl(\))g(+)f Fj(A)37124 5460 y Fi(h)p Ff(\000)p Fo(2)38924 5261 y Fj(B)39912 5460 y Fi(h)p Ff(\000)p Fo(2)13200 7198 y Fl(=)368 b Fj(A)15555 7397 y Fi(h)16153 7198 y Fj(B)17141 7397 y Fi(h)p Ff(\000)p Fo(1)19236 7198 y Fh(\000)295 b Fj(A)21539 7397 y Fi(h)p Ff(\000)p Fo(2)23339 7198 y Fj(B)24327 7397 y Fi(h)p Ff(\000)p Fo(1)26423 7198 y Fl(+)f Fj(A)28704 7397 y Fi(h)p Ff(\000)p Fo(2)30504 7198 y Fj(B)31492 7397 y Fi(h)p Ff(\000)p Fo(2)13200 9135 y Fl(=)368 b Fj(A)15555 9334 y Fi(h)16153 9135 y Fj(B)17141 9334 y Fi(h)p Ff(\000)p Fo(1)19236 9135 y Fl(+)295 b Fj(A)21518 9334 y Fi(h)p Ff(\000)p Fo(2)23318 9135 y Fl(\()p Fj(B)24812 9334 y Fi(h)p Ff(\000)p Fo(2)26907 9135 y Fh(\000)g Fj(B)29223 9334 y Fi(h)p Ff(\000)p Fo(1)31023 9135 y Fl(\))13200 11072 y(=)368 b(\()p Fj(A)16061 11271 y Fi(h)p Ff(\000)p Fo(1)18156 11072 y Fl(+)295 b Fj(A)20438 11271 y Fi(h)p Ff(\000)p Fo(2)22238 11072 y Fl(\))p Fj(B)23732 11271 y Fi(h)p Ff(\000)p Fo(1)25827 11072 y Fl(+)g Fj(A)28109 11271 y Fi(h)p Ff(\000)p Fo(2)29909 11072 y Fl(\()p Fj(B)31403 11271 y Fi(h)p Ff(\000)p Fo(2)33498 11072 y Fh(\000)g Fj(B)35814 11271 y Fi(h)p Ff(\000)p Fo(1)37614 11072 y Fl(\))13200 13010 y(=)368 b Fj(A)15555 13209 y Fi(h)p Ff(\000)p Fo(1)17355 13010 y Fj(B)18343 13209 y Fi(h)p Ff(\000)p Fo(1)20438 13010 y Fl(+)295 b Fj(A)22720 13209 y Fi(h)p Ff(\000)p Fo(2)24520 13010 y Fj(B)25508 13209 y Fi(h)p Ff(\000)p Fo(2)13200 14947 y Fl(=)368 b Fj(A)15555 15146 y Fi(p)p Ff(\000)p Fo(1)17287 14947 y Fj(:)800 17273 y Fl(Also)7610 19599 y Fj(B)8665 19050 y Ff(0)8976 18909 y Fo(2)8976 19927 y Fi(m)10159 19599 y Fl(+)295 b(2)p Fj(B)13171 19050 y Ff(0)13482 18909 y Fo(2)13482 19927 y Fi(m)p Ff(\000)p Fo(1)15867 19599 y Fh(\000)g Fl(2)p Fj(B)18900 19050 y Ff(0)18833 19927 y Fi(m)19721 19599 y Fj(B)20776 19050 y Ff(0)20709 19927 y Fi(m)p Ff(\000)p Fo(1)23168 19599 y Fl(=)369 b Fj(B)25604 19050 y Fo(2)25537 19927 y Fi(h)26430 19599 y Fl(+)294 b(2)p Fj(B)29441 19050 y Fo(2)29374 19927 y Fi(h)p Ff(\000)p Fo(2)31470 19599 y Fh(\000)h Fl(2)p Fj(B)34436 19798 y Fi(h)35034 19599 y Fj(B)36022 19798 y Fi(h)p Ff(\000)p Fo(2)23168 21536 y Fl(=)369 b Fj(B)25537 21735 y Fi(h)26135 21536 y Fl(\()p Fj(B)27629 21735 y Fi(h)28521 21536 y Fh(\000)296 b Fl(2)p Fj(B)31488 21735 y Fi(h)p Ff(\000)p Fo(2)33288 21536 y Fl(\))f(+)g(2)p Fj(B)37101 20987 y Fo(2)37034 21864 y Fi(h)p Ff(\000)p Fo(2)23168 23473 y Fl(=)369 b(\()p Fj(B)26043 23672 y Fi(h)p Ff(\000)p Fo(1)28138 23473 y Fl(+)295 b Fj(B)30433 23672 y Fi(h)p Ff(\000)p Fo(2)32233 23473 y Fl(\)\()p Fj(B)34233 23672 y Fi(h)p Ff(\000)p Fo(1)36327 23473 y Fh(\000)h Fj(B)38644 23672 y Fi(h)p Ff(\000)p Fo(2)40444 23473 y Fl(\))f(+)f(2)p Fj(B)44256 22925 y Fo(2)44189 23801 y Fi(h)p Ff(\000)p Fo(2)23168 25410 y Fl(=)369 b Fj(B)25604 24862 y Fo(2)25537 25739 y Fi(h)p Ff(\000)p Fo(1)27632 25410 y Fh(\000)295 b Fj(B)30015 24862 y Fo(2)29948 25739 y Fi(h)p Ff(\000)p Fo(2)32043 25410 y Fl(+)g(2)p Fj(B)35055 24862 y Fo(2)34988 25739 y Fi(h)p Ff(\000)p Fo(2)23168 27347 y Fl(=)369 b Fj(B)25604 26799 y Fo(2)25537 27676 y Fi(h)p Ff(\000)p Fo(1)27632 27347 y Fl(+)295 b Fj(B)29994 26799 y Fo(2)29927 27676 y Fi(h)p Ff(\000)p Fo(2)23168 29285 y Fl(=)369 b Fj(B)25537 29484 y Fi(p)p Ff(\000)p Fo(1)27268 29285 y Fj(:)800 32293 y(M)508 b Fl(=)369 b(4)p Fj(t)114 b Fl(+)g(2,)362 b(e.g.,)10355 31195 y Fh(p)p 11462 31195 1301 54 v 1098 x Fl(29)369 b(=)g([)14873 32293 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n0 16 {InitRnode } NewNode end end 14873 32293 a Fl(5)q Fj(;)16106 32293 y tx@Dict begin tx@NodeDict begin {7.73332 8.4 5.87494 2.93747 7.73332 } false /N@n1 16 {InitRnode } NewNode end end 16106 32293 a Fl(2)16196 33090 y Ff(\003)16757 32293 y Fj(;)221 b Fl(1)p Fj(;)18571 32293 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n2 16 {InitRnode } NewNode end end 18571 32293 a Fl(1)r Fj(;)19805 32293 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n3 16 {InitRnode } NewNode end end 19805 32293 a Fl(2)q Fj(;)21038 32293 y tx@Dict begin tx@NodeDict begin {7.73332 8.4 11.74988 5.87494 7.73332 } false /N@n4 16 {InitRnode } NewNode end end 21038 32293 a Fl(10)21454 33090 y Ff(\003)22339 32293 y Fl(])22700 32293 y tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n0 /N@n1 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 22700 32293 a 22700 32293 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n1 /N@n2 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 22700 32293 a 22700 32293 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n2 /N@n3 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 22700 32293 a 22700 32293 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n3 /N@n4 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 22700 32293 a Fl(.)549 b(Let)344 b Fj(f)142 b Fl(\()p Fj(m)p Fl(\))369 b(=)f Fj(h)p Fl(.)549 b(Then)344 b Fj(\017)36024 32492 y Fi(m)37281 32293 y Fl(=)369 b Fh(\000)p Fl(1)345 b(and)f Fj(\030)43762 31811 y Ff(0)43698 32622 y Fi(m)44955 32293 y Fl(=)368 b Fj(\030)46903 32492 y Fi(h)47615 32293 y Fl(+)114 b(1.)548 b(Also)800 34330 y Fj(f)142 b Fl(\()p Fj(m)p Fl(\))455 b(=)h Fj(f)142 b Fl(\()p Fj(m)329 b Fh(\000)h Fl(1\))g(+)g(2,)498 b(so)485 b Fj(A)16735 33848 y Ff(0)16735 34659 y Fi(m)p Ff(\000)p Fo(1)19281 34330 y Fl(=)455 b Fj(A)21723 34537 y Fi(f)98 b Fo(\()p Fi(m)p Ff(\000)p Fo(1\)+1)26753 34330 y Fl(=)456 b Fj(A)29196 34529 y Fi(h)p Ff(\000)p Fo(1)30996 34330 y Fl(.)731 b(Then)484 b(w)-36 b(e)485 b(ha)-36 b(v)g(e)485 b(a)g(cen)-36 b(tral)485 b(unisequence)800 35935 y([)p Fj(a)1844 36134 y Fi(r)2350 35935 y Fj(;)221 b(a)3615 36134 y Fi(p)p Ff(\000)p Fi(r)5327 35935 y Fl(])519 b(of)g(length)f(4)p Fj(t)353 b Fl(+)f(2.)833 b(There)518 b(are)h Fj(t)352 b Fl(+)h(1)519 b(jumps)e(in)i([)p Fj(a)32568 36134 y Fi(r)33074 35935 y Fj(;)221 b(a)34339 36134 y Fi(h)34937 35935 y Fl(],)540 b(so)518 b Fj(r)390 b Fl(+)352 b(2)p Fj(t)h Fl(+)g(2)513 b(=)g Fj(h)p Fl(.)832 b(Let)518 b Fj(J)641 b Fl(b)36 b(e)800 37540 y(the)536 b(n)-36 b(um)g(b)36 b(er)536 b(of)h(jumps)f(in)h([)p Fj(a)16205 37739 y Fo(0)16731 37540 y Fj(;)221 b(a)17996 37739 y Fi(r)18502 37540 y Fl(].)889 b(Hence)536 b Fj(m)p Fl(,)563 b(b)36 b(eing)537 b(the)f(n)-36 b(um)g(b)36 b(er)535 b(of)j(jumps)e(in)h([)p Fj(a)45156 37739 y Fo(0)45682 37540 y Fj(;)221 b(a)46947 37739 y Fi(h)47545 37540 y Fl(])537 b(satis\014es)800 39146 y Fj(m)389 b Fl(=)f Fj(J)426 b Fl(+)303 b Fj(t)g Fl(+)g(1.)614 b(There)445 b(are)h(also)g Fj(t)f Fl(jumps)g(of)h(2)g(in) f([)p Fj(a)28431 39345 y Fi(h)29028 39146 y Fj(;)221 b(a)30293 39345 y Fi(p)p Ff(\000)p Fi(r)26 b Ff(\000)p Fo(1)33208 39146 y Fl(],)448 b(one)e(jump)f(in)g([)p Fj(a)42720 39345 y Fi(p)p Ff(\000)p Fi(r)26 b Ff(\000)p Fo(1)45634 39146 y Fj(;)221 b(a)46899 39345 y Fi(p)p Ff(\000)p Fi(r)48611 39146 y Fl(])445 b(and)g Fj(J)800 40751 y Fl(jumps)433 b(in)h([)p Fj(a)7235 40950 y Fi(p)p Ff(\000)p Fi(r)8946 40751 y Fj(;)221 b(a)10211 40950 y Fi(p)10741 40751 y Fl(].)578 b(Hence)12854 43077 y Fj(k)414 b Fl(=)368 b(\()p Fj(J)418 b Fl(+)295 b Fj(t)g Fl(+)g(1\))g(+)g Fj(t)g Fl(+)g(1)g(+)g Fj(J)492 b Fl(=)368 b(2\()p Fj(J)419 b Fl(+)295 b Fj(t)f Fl(+)h(1\))369 b(=)g(2)p Fj(m:)800 45403 y Fl(W)-108 b(e)434 b(no)-36 b(w)433 b(v)-36 b(erify)435 b(Case)g(5\))e(of)i(the)e(midp)36 b(oin)-36 b(t)433 b(criteria:)28338 43798 y SDict begin H.S end 28338 43798 a 28338 43798 a SDict begin 14.5 H.A end 28338 43798 a 28338 43798 a SDict begin [ /View [/XYZ H.V] /Dest (equation.41) cvn H.B /DEST pdfmark end 28338 43798 a 21226 47729 a Fj(P)22245 47180 y Ff(0)22064 48057 y Fi(m)23321 47729 y Fl(=)369 b Fj(Q)25732 47180 y Ff(0)25732 48057 y Fi(m)26914 47729 y Fl(+)28354 47367 y Fg(1)p 28354 47528 406 40 v 28354 48076 a(2)28893 47729 y Fj(Q)29923 47180 y Ff(0)29923 48057 y Fi(m)p Ff(\000)p Fo(1)32013 47729 y Fj(:)18114 b Fl(\(41\))800 50055 y(Then,)434 b(as)f Fj(\030)6786 49573 y Ff(0)6722 50383 y Fi(m)p Ff(\000)p Fo(1)9181 50055 y Fl(=)369 b Fj(\030)11130 50254 y Fi(h)p Ff(\000)p Fo(2)13364 50055 y Fl(or)433 b Fj(\030)15521 50254 y Fi(h)p Ff(\000)p Fo(2)17616 50055 y Fl(+)295 b(1,)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b Fj(Q)26312 49573 y Ff(0)26312 50383 y Fi(m)p Ff(\000)p Fo(1)28771 50055 y Fl(=)369 b Fj(Q)31182 50254 y Fi(h)p Ff(\000)p Fo(2)33415 50055 y Fl(and)12878 52381 y Fj(P)13897 51832 y Ff(0)13716 52709 y Fi(m)14973 52381 y Fl(=)f Fj(Q)17383 51832 y Ff(0)17383 52709 y Fi(m)18566 52381 y Fl(+)20006 52019 y Fg(1)p 20006 52180 V 20006 52728 a(2)20545 52381 y Fj(Q)21575 51832 y Ff(0)21575 52709 y Fi(m)p Ff(\000)p Fo(1)24403 52381 y Fh(\()-221 b(\))738 b Fj(P)28414 52580 y Fi(h)29307 52381 y Fl(+)295 b Fj(Q)31644 52580 y Fi(h)32611 52381 y Fl(=)368 b Fj(Q)35021 52580 y Fi(h)35914 52381 y Fl(+)37354 52019 y Fg(1)p 37354 52180 V 37354 52728 a(2)37892 52381 y Fj(Q)38922 52580 y Fi(h)p Ff(\000)p Fo(2)24403 54318 y Fh(\()-221 b(\))738 b Fj(P)28414 54517 y Fi(h)29381 54318 y Fl(=)30894 53956 y Fg(1)p 30894 54117 V 30894 54665 a(2)31433 54318 y Fj(Q)32463 54517 y Fi(h)p Ff(\000)p Fo(2)34263 54318 y Fj(:)800 56644 y Fl(But)433 b Fj(a)4066 56843 y Fi(h)p Ff(\000)p Fo(1)6235 56644 y Fl(=)368 b(1)434 b(implies)g Fj(P)13952 56843 y Fi(h)14919 56644 y Fl(=)369 b Fj(Q)17330 56843 y Fi(h)p Ff(\000)p Fo(1)19425 56644 y Fh(\000)295 b Fj(P)21591 56843 y Fi(h)p Ff(\000)p Fo(1)23391 56644 y Fl(.)579 b(Hence)15086 58970 y Fj(P)16105 58421 y Fo(2)15924 59298 y Fi(h)p Ff(\000)p Fo(1)18019 58970 y Fh(\000)296 b Fj(P)20367 58421 y Fo(2)20186 59298 y Fi(h)21261 58970 y Fl(=)369 b(\()p 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Fj(P)20294 64980 y Fi(h)21261 64781 y Fl(=)369 b Fj(Q)23672 64980 y Fi(h)p Ff(\000)p Fo(1)25767 64781 y Fh(\000)296 b Fj(Q)28126 64980 y Fi(h)p Ff(\000)p Fo(2)29926 64781 y Fj(;)18806 66718 y Fl(2)p Fj(P)20294 66917 y Fi(h)21261 66718 y Fl(=)369 b Fj(Q)23672 66917 y Fi(h)p Ff(\000)p Fo(2)25472 66718 y Fj(:)p 0 TeXcolorgray 26150 70521 a Fl(20)p 0 TeXcolorgray eop end end %%Page: 21 21 TeXDict begin HPSdict begin 21 20 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.21) cvn H.B /DEST pdfmark end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 1328 x Fl(Next)434 b(w)-36 b(e)434 b(pro)-36 b(v)g(e)8867 262 y Fj(A)9842 461 y Fi(p)p Ff(\000)p Fo(1)11943 262 y Fl(=)368 b(2)p Fj(A)14948 -287 y Ff(0)14948 590 y Fi(m)p Ff(\000)p Fo(1)17038 262 y Fj(B)18093 -287 y Ff(0)18026 590 y Fi(m)p Ff(\000)p Fo(1)20412 262 y Fl(+)294 b Fj(A)22693 -287 y Ff(0)22693 590 y Fi(m)p Ff(\000)p Fo(2)24783 262 y Fj(B)25838 -287 y Ff(0)25771 590 y Fi(m)p Ff(\000)p Fo(2)28157 262 y Fh(\000)h Fl(\()p Fj(A)30966 -287 y Ff(0)30966 590 y Fi(m)p Ff(\000)p Fo(1)33056 262 y Fj(B)34111 -287 y Ff(0)34044 590 y Fi(m)p Ff(\000)p Fo(2)36429 262 y Fl(+)g Fj(B)38791 -287 y Ff(0)38724 590 y Fi(m)p Ff(\000)p Fo(1)40814 262 y Fj(A)41789 -287 y Ff(0)41789 590 y Fi(m)p Ff(\000)p Fo(2)43879 262 y Fl(\))p Fj(;)50488 -1675 y SDict begin H.S end 50488 -1675 a 50488 -1675 a SDict begin 17.5 H.A end 50488 -1675 a 50488 -1675 a SDict begin [ /View [/XYZ H.V] /Dest (equation.42) cvn H.B /DEST pdfmark end 50488 -1675 a 1937 x Fl(\(42\))8854 2447 y Fj(B)9842 2646 y Fi(p)p Ff(\000)p Fo(1)11943 2447 y Fl(=)368 b(2)p Fj(B)15028 1898 y Ff(0)15339 1757 y Fo(2)15339 2775 y Fi(m)p Ff(\000)p Fo(1)17724 2447 y Fl(+)295 b Fj(B)20086 1898 y Ff(0)20397 1757 y Fo(2)20397 2775 y Fi(m)p Ff(\000)p Fo(2)22782 2447 y Fh(\000)g Fl(2)p Fj(B)25815 1898 y Ff(0)25748 2775 y Fi(m)p Ff(\000)p Fo(1)27839 2447 y Fj(B)28894 1898 y Ff(0)28827 2775 y Fi(m)p Ff(\000)p Fo(2)30917 2447 y Fj(:)50488 510 y SDict begin H.S end 50488 510 a 50488 510 a SDict begin 17.5 H.A end 50488 510 a 50488 510 a SDict begin [ /View [/XYZ H.V] /Dest (equation.43) cvn H.B /DEST pdfmark end 50488 510 a 1937 x Fl(\(43\))800 5380 y(Note)434 b(that)f(regardless)h(of)g(the)f (sign)h(of)g Fj(\017)21181 5579 y Fi(m)p Ff(\000)p Fo(1)23272 5380 y Fl(,)f(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b Fj(A)29955 4898 y Ff(0)29955 5709 y Fi(m)p Ff(\000)p Fo(2)32414 5380 y Fl(=)369 b Fj(A)34770 5579 y Fi(h)p Ff(\000)p Fo(3)36569 5380 y Fl(.)800 6985 y(T)-108 b(o)434 b(pro)-36 b(v)g(e)433 b(\()p 0 0.5 0 TeXcolorrgb 6725 6985 a SDict begin H.S end 6725 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15985 y Fi(h)p Ff(\000)p Fo(1)17939 15786 y Fj(B)18927 15985 y Fi(h)p Ff(\000)p Fo(1)21022 15786 y Fl(+)295 b Fj(A)23304 15985 y Fi(h)p Ff(\000)p Fo(3)25104 15786 y Fj(B)26092 15985 y Fi(h)p Ff(\000)p Fo(3)28187 15786 y Fh(\000)g Fl(\()p Fj(A)30996 15985 y Fi(h)p Ff(\000)p Fo(1)32796 15786 y Fj(B)33784 15985 y Fi(h)p Ff(\000)p Fo(3)35879 15786 y Fl(+)g Fj(B)38174 15985 y Fi(h)p Ff(\000)p Fo(1)39974 15786 y Fj(A)40949 15985 y Fi(h)p Ff(\000)p Fo(3)42749 15786 y Fl(\))12395 17723 y(=)1107 b(2)p Fj(A)16139 17922 y Fi(h)p Ff(\000)p Fo(1)17939 17723 y Fj(B)18927 17922 y Fi(h)p Ff(\000)p Fo(1)21022 17723 y Fl(+)295 b(\()p Fj(A)23810 17922 y Fi(h)p Ff(\000)p Fo(1)25905 17723 y Fh(\000)g Fj(A)28208 17922 y Fi(h)p Ff(\000)p Fo(2)30008 17723 y Fl(\)\()p Fj(B)32008 17922 y Fi(h)p Ff(\000)p Fo(1)34103 17723 y Fh(\000)g Fj(B)36419 17922 y Fi(h)p Ff(\000)p Fo(2)38219 17723 y Fl(\))14809 19660 y Fh(\000)g Fj(A)17112 19859 y Fi(h)p Ff(\000)p Fo(1)18912 19660 y Fl(\()p Fj(B)20406 19859 y Fi(h)p 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12705 26468 a 0 TeXcolorgray Fl(\))o(.)8551 29401 y(2)p Fj(B)10256 28853 y Ff(0)10568 28711 y Fo(2)10568 29730 y Fi(m)p Ff(\000)p Fo(1)12953 29401 y Fl(+)294 b Fj(B)15314 28853 y Ff(0)15625 28711 y Fo(2)15625 29730 y Fi(m)p Ff(\000)p Fo(2)18010 29401 y Fh(\000)i Fl(2)p Fj(B)21044 28853 y Ff(0)20977 29730 y Fi(m)p Ff(\000)p Fo(1)23067 29401 y Fj(B)24122 28853 y Ff(0)24055 29730 y Fi(m)p Ff(\000)p Fo(2)27252 29401 y Fl(=)1107 b(2)p Fj(B)31076 28853 y Fo(2)31009 29730 y Fi(h)p Ff(\000)p Fo(1)33104 29401 y Fl(+)295 b Fj(B)35466 28853 y Fo(2)35399 29730 y Fi(h)p Ff(\000)p Fo(3)37494 29401 y Fh(\000)g Fl(2)p Fj(B)40460 29600 y Fi(h)p Ff(\000)p Fo(1)42261 29401 y Fj(B)43249 29600 y Fi(h)p Ff(\000)p Fo(3)27252 31338 y Fl(=)1107 b(2)p Fj(B)31076 30790 y Fo(2)31009 31667 y Fi(h)p Ff(\000)p Fo(1)33104 31338 y Fl(+)295 b(\()p Fj(B)35905 31537 y Fi(h)p Ff(\000)p Fo(1)38000 31338 y Fh(\000)g Fj(B)40316 31537 y Fi(h)p Ff(\000)p Fo(2)42116 31338 y Fl(\))42622 30790 y Fo(2)29666 33276 y Fh(\000)g Fl(2)p Fj(B)32632 33475 y Fi(h)p Ff(\000)p Fo(1)34432 33276 y Fl(\()p Fj(B)35926 33475 y Fi(h)p Ff(\000)p Fo(1)38021 33276 y Fh(\000)h Fj(B)40338 33475 y Fi(h)p Ff(\000)p Fo(2)42138 33276 y Fl(\))27252 35213 y(=)1107 b Fj(B)30426 34664 y Fo(2)30359 35541 y Fi(h)p Ff(\000)p Fo(1)32454 35213 y Fl(+)295 b Fj(B)34816 34664 y Fo(2)34749 35541 y Fi(h)p Ff(\000)p Fo(2)27252 37150 y Fl(=)1107 b Fj(B)30359 37349 y Fi(p)p Ff(\000)p Fo(1)32090 37150 y Fj(:)800 40083 y Fl(This)434 b(completes)g(the)f(pro)36 b(of)434 b(of)g(Theorem)p 0 0.5 0 TeXcolorrgb 22384 40083 a SDict begin H.S end 22384 40083 a 0 0.5 0 TeXcolorrgb Fl(8)p 0 0.5 0 TeXcolorrgb 23034 39227 a SDict begin H.R end 23034 39227 a 23034 40083 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.8) cvn H.B /ANN pdfmark end 23034 40083 a 0 TeXcolorgray Fl(.)800 42343 y SDict begin H.S end 800 42343 a 800 42343 a SDict begin 14.5 H.A end 800 42343 a 800 42343 a SDict begin [ /View [/XYZ H.V] /Dest (section.7) cvn H.B /DEST pdfmark end 800 42343 a 2177 x Fm(7)2152 b(R)-60 b(CF)717 b(p)60 b(erio)g(ds)716 b(with)g(only)h(o)60 b(dd)716 b(length)f(unisequences)800 46208 y SDict begin H.S end 800 46208 a 800 46208 a SDict begin 14.5 H.A end 800 46208 a 800 46208 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.17) cvn H.B /DEST pdfmark end 800 46208 a 0 TeXcolorgray 1233 x Fe(Lemma)383 b(17.)p 0 TeXcolorgray 477 w Fk(Supp)-66 b(ose)372 b(ther)-66 b(e)371 b(ar)-66 b(e)372 b(no)g(even)f(length)g (\()p Fh(\025)e Fl(2)p Fk(\))k(unise)-66 b(quenc)g(es)370 b(in)i(an)g(R)-33 b(CF)371 b(p)-66 b(erio)g(d)371 b(of)h(length)800 49046 y Fj(p)465 b Fk(for)3993 47921 y Fh(p)p 5100 47921 1117 54 v 1125 x Fj(D)36 b Fk(.)598 b(If)464 b Fj(k)510 b Fk(is)464 b(the)h(NICF)f(p)-66 b(erio)g(d-length)463 b(and)i Fl(0)369 b Fh(\024)g Fj(t)g Fh(\024)g Fj(k)45 b(=)p Fl(2)p Fk(,)465 b(then)37673 47441 y SDict begin H.S end 37673 47441 a 37673 47441 a SDict begin 14.5 H.A end 37673 47441 a 37673 47441 a SDict begin [ /View [/XYZ H.V] /Dest (equation.44) cvn H.B /DEST pdfmark end 37673 47441 a 21170 51979 a Fj(f)142 b Fl(\()p Fj(k)340 b Fh(\000)295 b Fj(t)p Fl(\))369 b(=)f Fj(p)295 b Fh(\000)h Fj(f)142 b Fl(\()p Fj(t)p Fl(\))p Fj(;)18057 b Fl(\(44\))p 0 TeXcolorgray 800 54913 a Fk(Pr)-66 b(o)g(of.)p 0 TeXcolorgray 649 w Fl(Let)547 b Fj(f)142 b Fl(\()p Fj(t)p Fl(\))561 b(=)g Fj(r)36 b Fl(.)919 b(Then)547 b(w)-36 b(e)547 b(ha)-36 b(v)g(e)547 b Fj(t)g Fl(jumps)f(on)h([)p Fj(a)30234 55112 y Fo(0)30760 54913 y Fj(;)221 b(a)32025 55112 y Fi(r)32531 54913 y Fl(].)919 b(Because)547 b(of)h(the)f(symmetry)g(of)h(the)800 56518 y(partial)589 b(quotien)-36 b(ts)588 b(and)g(the)g(absence)g(of)h (ev)-36 b(en)589 b(length)f(unisequences,)627 b(w)-36 b(e)588 b(get)h(the)f(same)g Fj(t)h Fl(jumps)800 58123 y(but)492 b(in)h(rev)-36 b(erse)493 b(order)f(on)h([)p Fj(a)15588 58322 y Fi(p)p Ff(\000)p Fi(r)17299 58123 y Fj(;)221 b(a)18564 58322 y Fi(p)19094 58123 y Fl(].)756 b(There)493 b(are)g Fj(k)538 b Fl(jumps)492 b(on)h([)p Fj(a)34672 58322 y Fo(0)35198 58123 y Fj(;)221 b(a)36463 58322 y Fi(p)36992 58123 y Fl(])493 b(and)g(hence)f Fj(k)381 b Fh(\000)335 b Fj(t)493 b Fl(jumps)f(on)800 59728 y([)p Fj(a)1844 59927 y Fo(0)2370 59728 y Fj(;)221 b(a)3635 59927 y Fi(p)p Ff(\000)p Fi(r)5347 59728 y Fl(].)579 b(Hence)433 b Fj(f)142 b Fl(\()p Fj(k)340 b Fh(\000)295 b Fj(t)p Fl(\))369 b(=)f Fj(p)295 b Fh(\000)h Fj(r)405 b Fl(=)369 b Fj(p)295 b Fh(\000)g Fj(f)142 b Fl(\()p Fj(t)p Fl(\).)p 51860 59728 45 878 v 51905 58895 781 45 v 51905 59728 V 52684 59728 45 878 v 800 61096 a SDict begin H.S end 800 61096 a 800 61096 a SDict begin 14.5 H.A end 800 61096 a 800 61096 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.18) cvn H.B /DEST pdfmark end 800 61096 a 0 TeXcolorgray 1344 x Fe(Theorem)668 b(18.)p 0 TeXcolorgray 629 w Fk(Supp)-66 b(ose)600 b(ther)-66 b(e)598 b(ar)-66 b(e)600 b(no)f(even)g(length)g(\()p Fh(\025)619 b Fl(2)p Fk(\))600 b(unise)-66 b(quenc)g(es)598 b(in)h(a)h(R)-33 b(CF)599 b(p)-66 b(erio)g(d)598 b(of)800 64045 y(length)464 b Fj(p)h Fk(for)7812 62921 y Fh(p)p 8919 62921 1117 54 v 1124 x Fj(D)36 b Fk(.)598 b(Then)464 b(if)g Fj(k)510 b Fk(is)465 b(the)f(NICF)g(p)-66 b(erio)g(d-length,)463 b(for)h Fl(1)370 b Fh(\024)f Fj(t)f Fh(\024)i Fj(k)45 b(=)p Fl(2)p Fk(,)465 b(we)g(have)p 0 TeXcolorgray 1940 66757 a(\(i\))p 0 TeXcolorgray 651 w Fj(\017)4577 66956 y Fi(t)5341 66757 y Fl(=)369 b Fj(\017)7247 66956 y Fi(k)24 b Fo(+1)p Ff(\000)p Fi(t)10090 66757 y Fk(;)p 0 TeXcolorgray 26150 70521 a Fl(21)p 0 TeXcolorgray eop end end %%Page: 22 22 TeXDict begin HPSdict begin 22 21 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 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y Fl(+)295 b Fj(Q)23435 -49 y Fi(r)26 b Ff(\000)p Fo(1)25512 -248 y Fl(=)369 b Fj(P)27731 -49 y Fi(r)26 b Fo(+1)29735 -248 y Fl(+)295 b Fj(Q)32072 -49 y Fi(r)26 b Fo(+1)33780 -248 y Fj(:)16347 b Fl(\(45\))800 2175 y(W)-108 b(e)434 b(ha)-36 b(v)g(e)14230 4599 y Fj(P)15249 4050 y Fo(2)15068 4927 y Fi(r)26 b Fo(+1)17072 4599 y Fh(\000)295 b Fj(P)19419 4050 y Fo(2)19238 4927 y Fi(r)20314 4599 y Fl(=)369 b(\()p Fj(D)331 b Fh(\000)295 b Fj(Q)25970 4798 y Fi(r)26476 4599 y Fj(Q)27506 4798 y Fi(r)26 b Fo(+1)29215 4599 y Fl(\))295 b Fh(\000)g Fl(\()p Fj(D)331 b Fh(\000)296 b Fj(Q)35620 4798 y Fi(r)26 b Ff(\000)p Fo(1)37328 4599 y Fj(Q)38358 4798 y Fi(r)38864 4599 y Fl(\))20314 6536 y(=)369 b Fj(Q)22725 6735 y Fi(r)23231 6536 y Fl(\()p Fj(Q)24767 6735 y Fi(r)26 b Ff(\000)p Fo(1)26770 6536 y Fh(\000)296 b Fj(Q)29129 6735 y Fi(r)26 b Fo(+1)30837 6536 y Fl(\))p Fj(:)50488 4599 y SDict begin H.S end 50488 4599 a 50488 4599 a SDict begin 17.5 H.A end 50488 4599 a 50488 4599 a SDict begin [ /View [/XYZ H.V] /Dest (equation.46) cvn H.B /DEST pdfmark end 50488 4599 a 1937 x Fl(\(46\))800 8960 y(No)-36 b(w)457 b Fj(\017)4310 9159 y Fi(t)5112 8960 y Fl(=)407 b Fh(\000)p Fl(1)456 b(implies)h Fj(a)13791 9166 y Fi(f)98 b Fo(\()p Fi(t)p Ff(\000)p Fo(1\)+1)18279 8960 y Fl(=)407 b(1)g(=)g Fj(a)22857 9166 y Fi(f)98 b Fo(\()p Fi(t)p Fo(\))p Ff(\000)p Fo(1)26143 8960 y Fl(=)407 b Fj(a)28245 9159 y Fi(r)29207 8960 y Fl(and)455 b(hence)h Fj(P)36232 9159 y Fi(r)26 b Fo(+1)38251 8960 y Fl(+)310 b Fj(P)40411 9159 y Fi(r)41325 8960 y Fl(=)406 b Fj(a)43426 9159 y Fi(r)43932 8960 y Fj(Q)44962 9159 y Fi(r)45876 8960 y Fl(=)g Fj(Q)48324 9159 y Fi(r)48831 8960 y Fl(.)645 b(Then)800 10565 y(\()p 0 0.5 0 TeXcolorrgb 1306 10565 a SDict begin H.S end 1306 10565 a 0 0.5 0 TeXcolorrgb Fl(46)p 0 0.5 0 TeXcolorrgb 2607 9709 a SDict begin H.R end 2607 9709 a 2607 10565 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.46) cvn H.B /ANN pdfmark end 2607 10565 a 0 TeXcolorgray Fl(\))433 b(giv)-36 b(es)19618 12170 y Fj(P)20456 12369 y Fi(r)26 b Fo(+1)22460 12170 y Fh(\000)295 b Fj(P)24626 12369 y Fi(r)25501 12170 y Fl(=)369 b Fj(Q)27912 12369 y Fi(r)26 b Ff(\000)p Fo(1)29916 12170 y Fh(\000)295 b Fj(Q)32274 12369 y Fi(r)26 b Fo(+1)800 14276 y Fl(and)433 b(hence)g(\()p 0 0.5 0 TeXcolorrgb 7448 14276 a SDict begin H.S end 7448 14276 a 0 0.5 0 TeXcolorrgb Fl(45)p 0 0.5 0 TeXcolorrgb 8749 13420 a SDict begin H.R end 8749 13420 a 8749 14276 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.45) cvn H.B /ANN pdfmark end 8749 14276 a 0 TeXcolorgray Fl(\))o(.)800 15881 y(\(iii\))h(T)-108 b(o)434 b(pro)-36 b(v)g(e)434 b Fj(Q)9779 15399 y Ff(0)9779 16209 y Fi(t)10543 15881 y Fl(=)368 b Fj(Q)12953 15399 y Ff(0)12953 16226 y Fi(k)24 b Ff(\000)p Fi(t)14594 15881 y Fl(,)434 b(w)-36 b(e)434 b(observ)-36 b(e)434 b(that)20041 18304 y Fj(\030)20673 17756 y Ff(0)20609 18633 y Fi(t)21373 18304 y Fl(=)369 b Fj(\030)23322 18511 y Fi(f)98 b Fo(\()p 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Fd(8)14229 28933 y(>)14229 29331 y(<)14229 31722 y(>)14229 32121 y(:)15409 28986 y Fj(a)16092 29192 y Fi(f)98 b Fo(\()p Fi(t)p Fo(\))18202 28986 y Fl(if)435 b(\()p Fj(\017)20426 29185 y Fi(t)20821 28986 y Fj(;)221 b(\017)21928 29185 y Fi(t)p Fo(+1)23526 28986 y Fl(\))369 b(=)g(\(1)p Fj(;)221 b Fl(1\);)15409 30912 y Fj(a)16092 31118 y Fi(f)98 b Fo(\()p Fi(t)p Fo(\))18064 30912 y Fl(+)295 b(1)434 b(if)g(\()p Fj(\017)22678 31111 y Fi(t)23073 30912 y Fj(;)221 b(\017)24180 31111 y Fi(t)p Fo(+1)25778 30912 y Fl(\))369 b(=)g(\(1)p Fj(;)221 b Fh(\000)p Fl(1\))435 b(or)e(\()p Fh(\000)p Fl(1)p Fj(;)221 b Fl(1\);)15409 32838 y Fj(a)16092 33045 y Fi(f)98 b Fo(\()p Fi(t)p Fo(\))18064 32838 y Fl(+)295 b(2)434 b(if)g(\()p Fj(\017)22678 33037 y Fi(t)23073 32838 y Fj(;)221 b(\017)24180 33037 y Fi(t)p Fo(+1)25778 32838 y Fl(\))369 b(=)g(\()p Fh(\000)p Fl(1)p Fj(;)221 b Fh(\000)p Fl(1\).)10156 37080 y Fj(a)10839 36531 y Ff(0)10839 37408 y Fi(k)24 b Ff(\000)p Fi(t)12848 37080 y Fl(=)14229 33958 y 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Fl(\))369 b(=)g(\()p Fh(\000)p Fl(1)p Fj(;)221 b Fh(\000)p Fl(1\).)800 43627 y(Then)433 b(as)h(\()p Fj(\017)6824 43826 y Fi(t)7219 43627 y Fj(;)221 b(\017)8326 43826 y Fi(t)p Fo(+1)9925 43627 y Fl(\))368 b(=)h(\()p Fj(\017)13211 43826 y Fi(k)24 b Fo(+1)p Ff(\000)p Fi(t)16054 43627 y Fj(;)221 b(\017)17161 43826 y Fi(k)24 b Ff(\000)p Fi(t)18802 43627 y Fl(\))433 b(and)h Fj(a)22954 43833 y Fi(f)98 b Fo(\()p Fi(k)24 b Ff(\000)p Fi(t)p Fo(\))26244 43627 y Fl(=)369 b Fj(a)28308 43833 y Fi(p)p Ff(\000)p Fi(f)98 b Fo(\()p Fi(t)p Fo(\))31559 43627 y Fl(=)368 b Fj(a)33622 43833 y Fi(f)98 b Fo(\()p Fi(t)p Fo(\))35299 43627 y Fl(,)434 b(it)f(follo)-36 b(ws)436 b(that)d Fj(a)45166 43145 y Ff(0)45166 43955 y Fi(t)45930 43627 y Fl(=)368 b Fj(a)47993 43145 y Ff(0)47993 43972 y Fi(k)24 b Ff(\000)p Fi(t)49634 43627 y Fl(.)p 51860 43627 45 878 v 51905 42793 781 45 v 51905 43627 V 52684 43627 45 878 v 800 46082 a(W)-108 b(e)315 b(giv)-36 b(e)317 b(examples)f(of)g(ev)-36 b(en)316 b(and)f(o)36 b(dd)315 b(NICF)g(p)36 b(erio)g(d-length)315 b(in)g(whic)-36 b(h)315 b(only)h(o)36 b(dd)315 b(length)h(unisequences) 800 47687 y(o)36 b(ccur.)6531 50111 y(\(a\))9493 48947 y Fh(p)p 10600 48947 2602 54 v 1164 x Fl(1532)371 b(=)d([)15312 50111 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 11.74988 5.87494 7.73332 } false /N@n0 16 {InitRnode } NewNode end end 15312 50111 a Fl(39)q Fj(;)17195 50111 y tx@Dict begin tx@NodeDict begin {7.73332 8.4 5.87494 2.93747 7.73332 } false /N@n1 16 {InitRnode } NewNode end end 17195 50111 a Fl(7)17285 50908 y Ff(\003)17846 50111 y Fj(;)18428 50111 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n2 16 {InitRnode } NewNode end end 18428 50111 a Fl(9)q Fj(;)221 b Fl(1)p Fj(;)20893 50111 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n3 16 {InitRnode } NewNode end end 20893 50111 a Fl(1)r Fj(;)g Fl(1)p Fj(;)23359 50111 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n4 16 {InitRnode } NewNode end end 23359 50111 a Fl(3)r Fj(;)g Fl(1)p Fj(;)25825 50111 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 11.74988 5.87494 7.73332 } false /N@n5 16 {InitRnode } NewNode end end 25825 50111 a Fl(18)s Fj(;)g Fl(1)p Fj(;)28942 50111 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n6 16 {InitRnode } NewNode end end 28942 50111 a Fl(3)r Fj(;)g Fl(1)p Fj(;)31408 50111 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n7 16 {InitRnode } NewNode end end 31408 50111 a Fl(1)r Fj(;)g Fl(1)p Fj(;)33874 50111 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n8 16 {InitRnode } NewNode end end 33874 50111 a Fl(9)r Fj(;)35108 50111 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n9 16 {InitRnode } NewNode end end 35108 50111 a Fl(7)q Fj(;)36341 50111 y tx@Dict begin tx@NodeDict begin {7.73332 8.4 11.74988 5.87494 7.73332 } false /N@n10 16 {InitRnode } NewNode end end 36341 50111 a Fl(78)36756 50908 y Ff(\003)37642 50111 y Fl(])38003 50111 y tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n0 /N@n1 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 38003 50111 a 38003 50111 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n1 /N@n2 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 38003 50111 a 38003 50111 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n2 /N@n3 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 38003 50111 a 38003 50111 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n3 /N@n4 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 38003 50111 a 38003 50111 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n4 /N@n5 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 38003 50111 a 38003 50111 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n5 /N@n6 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 38003 50111 a 38003 50111 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n6 /N@n7 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 38003 50111 a 38003 50111 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n7 /N@n8 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 38003 50111 a 38003 50111 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n8 /N@n9 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 38003 50111 a 38003 50111 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n9 /N@n10 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 38003 50111 a 13571 53443 a Fl(=)368 b(39)296 b(+)18114 52607 y(1)p 19002 53089 45 1606 v 17854 53133 1171 45 v 17832 54738 45 1606 v 18114 54256 a(7)18204 55053 y Ff(\003)19320 53443 y Fl(+)21212 52607 y(1)p 22425 53089 V 20626 53133 1821 45 v 20604 54738 45 1606 v 20886 54256 a(10)22742 53443 y Fh(\000)24331 52607 y Fl(1)p 25219 53089 V 24071 53133 1171 45 v 24049 54738 45 1606 v 24331 54256 a(3)25537 53443 y Fh(\000)27125 52607 y Fl(1)p 28013 53089 V 26865 53133 1171 45 v 26843 54738 45 1606 v 27125 54256 a(5)28331 53443 y Fh(\000)30244 52607 y Fl(1)p 31458 53089 V 29659 53133 1821 45 v 29637 54738 45 1606 v 29919 54256 a(20)31775 53443 y Fh(\000)33364 52607 y Fl(1)p 34252 53089 V 33104 53133 1171 45 v 33082 54738 45 1606 v 33364 54256 a(5)34569 53443 y Fh(\000)36158 52607 y Fl(1)p 37046 53089 V 35898 53133 1171 45 v 35876 54738 45 1606 v 36158 54256 a(3)37363 53443 y Fh(\000)39277 52607 y Fl(1)p 40491 53089 V 38692 53133 1821 45 v 38670 54738 45 1606 v 38952 54256 a(10)40808 53443 y(+)42375 52607 y(1)p 43263 53089 V 42115 53133 1171 45 v 42093 54738 45 1606 v 42375 54256 a(7)43580 53443 y(+)45473 52607 y(1)p 46686 53089 V 44887 53133 1821 45 v 44865 54738 45 1606 v 45147 54256 a(78)45562 55053 y Ff(\003)46708 53443 y Fj(:)2751 56641 y Fl(Here)434 b Fj(p)368 b Fl(=)h(16)p Fj(;)221 b(k)415 b Fl(=)369 b(10)p Fj(;)221 b(P)15479 56159 y Ff(0)15298 56970 y Fo(5)16194 56641 y Fl(=)369 b Fj(P)18594 56159 y Ff(0)18413 56970 y Fo(6)18939 56641 y Fl(.)4931 60670 y(\(b\))7966 59506 y Fh(p)p 9072 59506 1951 54 v 9072 60670 a Fl(277)h(=)f([)13134 60670 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 11.74988 5.87494 7.73332 } false /N@n0 16 {InitRnode } NewNode end end 13134 60670 a Fl(16)q Fj(;)221 b Fl(1)15108 61467 y Ff(\003)15668 60670 y Fj(;)16250 60670 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n1 16 {InitRnode } NewNode end end 16250 60670 a Fl(1)q Fj(;)g Fl(1)p Fj(;)18715 60670 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n2 16 {InitRnode } NewNode end end 18715 60670 a Fl(4)r Fj(;)19949 60670 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 11.74988 5.87494 7.73332 } false /N@n3 16 {InitRnode } NewNode end end 19949 60670 a Fl(10)q Fj(;)g Fl(1)p Fj(;)23064 60670 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n4 16 {InitRnode } NewNode end end 23064 60670 a Fl(7)r Fj(;)24298 60670 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n5 16 {InitRnode } NewNode end end 24298 60670 a Fl(2)q Fj(;)25531 60670 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n6 16 {InitRnode } NewNode end end 25531 60670 a Fl(2)r Fj(;)26765 60670 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n7 16 {InitRnode } NewNode end end 26765 60670 a Fl(3)q Fj(;)27998 60670 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n8 16 {InitRnode } NewNode end end 27998 60670 a Fl(3)q Fj(;)29231 60670 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n9 16 {InitRnode } NewNode end end 29231 60670 a Fl(2)q Fj(;)30464 60670 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n10 16 {InitRnode } NewNode end end 30464 60670 a Fl(2)q Fj(;)31697 60670 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n11 16 {InitRnode } NewNode end end 31697 60670 a Fl(7)q Fj(;)g Fl(1)p Fj(;)34162 60670 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 11.74988 5.87494 7.73332 } false /N@n12 16 {InitRnode } NewNode end end 34162 60670 a Fl(10)r Fj(;)36046 60670 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n13 16 {InitRnode } NewNode end end 36046 60670 a Fl(4)q Fj(;)g Fl(1)p Fj(;)38511 60670 y tx@Dict begin tx@NodeDict begin {7.73332 0.0 5.87494 2.93747 7.73332 } false /N@n14 16 {InitRnode } NewNode end end 38511 60670 a Fl(1)r Fj(;)g Fl(1)p Fj(;)40977 60670 y tx@Dict begin tx@NodeDict begin {7.73332 8.4 11.74988 5.87494 7.73332 } false /N@n15 16 {InitRnode } NewNode end end 40977 60670 a Fl(32)41394 61467 y Ff(\003)42279 60670 y Fl(])370 b(=)44022 60670 y tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n0 /N@n1 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 44022 60670 a 44022 60670 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n1 /N@n2 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 44022 60670 a 44022 60670 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n2 /N@n3 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 44022 60670 a 44022 60670 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n3 /N@n4 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 44022 60670 a 44022 60670 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n4 /N@n5 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 44022 60670 a 44022 60670 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n5 /N@n6 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 44022 60670 a 44022 60670 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n6 /N@n7 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 44022 60670 a 44022 60670 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n7 /N@n8 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 44022 60670 a 44022 60670 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n8 /N@n9 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 44022 60670 a 44022 60670 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n9 /N@n10 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 44022 60670 a 44022 60670 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n10 /N@n11 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 44022 60670 a 44022 60670 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n11 /N@n12 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 44022 60670 a 44022 60670 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n12 /N@n13 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 44022 60670 a 44022 60670 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n13 /N@n14 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 44022 60670 a 44022 60670 a tx@Dict begin gsave STV newpath 0.6 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 0.8 0.6 scale false 0.4 1.4 1.5 2. Arrow EndArrow } def /NCLW CLW def tx@NodeDict begin 0.0 0.0 neg 1.17505 1.17505 0 0 /N@n14 /N@n15 InitNC { yB yA sub xB xA sub Atan dup 45. add /AngleA ED 45. sub 180 add /AngleB ED 0.67 0.67 NCCurve } if end gsave 0.6 SLW 0 setgray 0 setlinecap stroke grestore grestore end 44022 60670 a 4135 64002 a Fl(17)295 b Fh(\000)7319 63166 y Fl(1)p 8207 63647 45 1606 v 7059 63692 1171 45 v 7037 65297 45 1606 v 7319 64815 a(3)7409 65612 y Ff(\003)8525 64002 y Fh(\000)10113 63166 y Fl(1)p 11001 63647 V 9853 63692 1171 45 v 9831 65297 45 1606 v 10113 64815 a(5)11319 64002 y(+)13211 63166 y(1)p 14424 63647 V 12625 63692 1821 45 v 12603 65297 45 1606 v 12886 64815 a(11)14742 64002 y Fh(\000)16330 63166 y Fl(1)p 17218 63647 V 16070 63692 1171 45 v 16048 65297 45 1606 v 16330 64815 a(8)17536 64002 y(+)19103 63166 y(1)p 19991 63647 V 18843 63692 1171 45 v 18821 65297 45 1606 v 19103 64815 a(2)20308 64002 y(+)21875 63166 y(1)p 22764 63647 V 21615 63692 1171 45 v 21593 65297 45 1606 v 21875 64815 a(2)23081 64002 y(+)24648 63166 y(1)p 25536 63647 V 24388 63692 1171 45 v 24366 65297 45 1606 v 24648 64815 a(3)25853 64002 y(+)27420 63166 y(1)p 28309 63647 V 27160 63692 1171 45 v 27138 65297 45 1606 v 27420 64815 a(3)28626 64002 y(+)30193 63166 y(1)p 31081 63647 V 29933 63692 1171 45 v 29911 65297 45 1606 v 30193 64815 a(2)31399 64002 y(+)32966 63166 y(1)p 33854 63647 V 32705 63692 1171 45 v 32683 65297 45 1606 v 32966 64815 a(2)34171 64002 y(+)35738 63166 y(1)p 36626 63647 V 35478 63692 1171 45 v 35456 65297 45 1606 v 35738 64815 a(8)36944 64002 y Fh(\000)38857 63166 y Fl(1)p 40071 63647 V 38272 63692 1821 45 v 38250 65297 45 1606 v 38532 64815 a(11)40388 64002 y(+)41955 63166 y(1)p 42844 63647 V 41695 63692 1171 45 v 41673 65297 45 1606 v 41955 64815 a(5)43161 64002 y Fh(\000)44749 63166 y Fl(1)p 45638 63647 V 44489 63692 1171 45 v 44467 65297 45 1606 v 44749 64815 a(3)45955 64002 y Fh(\000)47869 63166 y Fl(1)p 49082 63647 V 47283 63692 1821 45 v 47261 65297 45 1606 v 47543 64815 a(34)47959 65612 y Ff(\003)49104 64002 y Fj(:)2751 67200 y Fl(Here)434 b Fj(p)368 b Fl(=)h(21)p Fj(;)221 b(k)415 b Fl(=)369 b(15)p Fj(;)221 b(Q)15490 66718 y Ff(0)15490 67528 y Fo(7)16386 67200 y Fl(=)369 b Fj(Q)18797 66718 y Ff(0)18797 67528 y Fo(8)19323 67200 y Fl(.)p 0 TeXcolorgray 26150 70521 a(23)p 0 TeXcolorgray eop end end %%Page: 24 24 TeXDict begin HPSdict begin 24 23 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.24) cvn H.B /DEST pdfmark end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 800 -4000 a SDict begin 14.5 H.A end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (section.8) cvn H.B /DEST pdfmark end 800 -4000 a 1328 x Fm(8)2152 b(Ac)-60 b(kno)g(wledgemen)g(ts)800 249 y Fl(The)477 b(author)g(thanks)h(John)f (Rob)36 b(ertson)477 b(and)g(Jim)h(White)f(for)h(helpful)g(remarks)f (on)h(the)f(pap)36 b(er)477 b(of)h(H.)800 1854 y(C.)595 b(Williams.)1063 b(The)595 b(author)f(also)h(thanks)g(Alan)g(O\013er)e (for)i(his)g(help)f(with)h(the)f(graphics)g(pac)-36 b(k)-72 b(age)800 3459 y Fa(pstricks)p Fl(,)430 b(and)j(the)g(referee)h(for)g (impro)-36 b(ving)434 b(the)f(presen)-36 b(tation)433 b(of)i(the)e(pap)36 b(er.)800 7896 y Fm(References)800 9212 y SDict begin H.S end 800 9212 a 800 9212 a SDict begin 14.5 H.A end 800 9212 a 800 9212 a SDict begin [ /View [/XYZ H.V] /Dest (section*.1) cvn H.B /DEST pdfmark end 800 9212 a 800 8658 a SDict begin H.S end 800 8658 a 800 8658 a SDict begin 14.5 H.A end 800 8658 a 800 8658 a SDict begin [ /View [/XYZ H.V] /Dest (cite.hurwitz) cvn H.B /DEST pdfmark end 800 8658 a 0 TeXcolorgray 1450 10817 a Fl([1])p 0 TeXcolorgray 652 w(A.)306 b(Hurwitz,)10560 10481 y(\177)10398 10817 y(Ub)36 b(er)306 b(eine)g(b)36 b(esondere)306 b(Art)g(der)g (Ketten)-36 b(bruc)g(h-En)g(t)g(wic)g(klung)304 b(reeller)j(Gr\177)-650 b(ossen,)331 b Fk(A)-66 b(cta)3474 12422 y(Math.)433 b Fe(12)h Fl(\(1889\),)g(367{405.)800 13307 y SDict begin H.S end 800 13307 a 800 13307 a SDict begin 14.5 H.A end 800 13307 a 800 13307 a SDict begin [ /View [/XYZ H.V] /Dest (cite.bosma) cvn H.B /DEST pdfmark end 800 13307 a 0 TeXcolorgray 1450 15134 a Fl([2])p 0 TeXcolorgray 652 w(W.)g(Bosma,)g(Optimal)g(con) -36 b(tin)g(ued)432 b(fractions,)j Fk(Indag.)463 b(Math.)433 b Fe(49)h Fl(\(1987\),)h(353|379.)800 16019 y SDict begin H.S end 800 16019 a 800 16019 a SDict begin 14.5 H.A end 800 16019 a 800 16019 a SDict begin [ /View [/XYZ H.V] /Dest (cite.dickson) cvn H.B /DEST pdfmark end 800 16019 a 0 TeXcolorgray 1450 17846 a Fl([3])p 0 TeXcolorgray 652 w(L.)e(E.)h(Dic)-36 b(kson,)435 b Fk(History)464 b(of)h(the)f(The)-66 b(ory)465 b(of)f(Numb)-66 b(ers)p Fl(,)433 b(V)-108 b(ol.)434 b(2,)g(Chelsea)h (AMS)e(reprin)-36 b(t,)432 b(1999.)800 18731 y SDict begin H.S end 800 18731 a 800 18731 a SDict begin 14.5 H.A end 800 18731 a 800 18731 a SDict begin [ /View [/XYZ H.V] /Dest (cite.matthews-robertson-white) cvn H.B /DEST pdfmark end 800 18731 a 0 TeXcolorgray 1450 20558 a Fl([4])p 0 TeXcolorgray 652 w(K.)598 b(R.)h(Matthews,)640 b(J.)599 b(P)-108 b(.)598 b(Rob)36 b(ertson)598 b(and)g(J.)g(White,)640 b(Midp)36 b(oin)-36 b(t)598 b(criteria)h(for)g(solving)g(P)-36 b(ell's)3474 22163 y(equation)434 b(using)f(the)g(nearest)g(square)h (con)-36 b(tin)g(ued)433 b(fraction,)h(to)g(app)36 b(ear)433 b(in)h Fk(Math.)464 b(Comp.)800 22975 y SDict begin H.S end 800 22975 a 800 22975 a SDict begin 14.5 H.A end 800 22975 a 800 22975 a SDict begin [ /View [/XYZ H.V] /Dest (cite.matthews-robertson) cvn H.B /DEST pdfmark end 800 22975 a 0 TeXcolorgray 1450 24875 a Fl([5])p 0 TeXcolorgray 652 w(K.)596 b(R.)g(Matthews)g (and)f(J.)i(P)-108 b(.)595 b(Rob)36 b(ertson,)637 b(Equalit)-36 b(y)597 b(of)f(quadratic)h(surd)d(p)36 b(erio)g(d-lengths)595 b(for)3474 26480 y(NICF)434 b(and)f(NSCF,)g(in)h(preparation.)800 27292 y SDict begin H.S end 800 27292 a 800 27292 a SDict begin 14.5 H.A end 800 27292 a 800 27292 a SDict begin [ /View [/XYZ H.V] /Dest (cite.minnegerode) cvn H.B /DEST pdfmark end 800 27292 a 0 TeXcolorgray 1450 29192 a Fl([6])p 0 TeXcolorgray 652 w(B.)590 b(Minnegero)36 b(de,)13796 28856 y(\177)13634 29192 y(Ub)g(er)590 b(eine)g(neue)f(Metho)36 b(de,)629 b(die)590 b(P)-36 b(ellisc)g(he)590 b(Gleic)-36 b(h)g(ung)589 b(aufzul\177)-650 b(osen,)630 b Fk(G\177)-664 b(ott.)3474 30797 y(Nachr.)432 b Fe(23)i Fl(\(1873\),)h(619{653.)800 31683 y SDict begin H.S end 800 31683 a 800 31683 a SDict begin 14.5 H.A end 800 31683 a 800 31683 a SDict begin [ /View [/XYZ H.V] /Dest (cite.perron) cvn H.B /DEST pdfmark end 800 31683 a 0 TeXcolorgray 1450 33509 a Fl([7])p 0 TeXcolorgray 652 w(O.)e(P)-36 b(erron,)433 b Fk(Die)465 b(L)-66 b(ehr)g(e)463 b(von)i(den)f(Kettenbr)17 b(\177)-681 b(uchen)p Fl(,)432 b(Band)h(1,)h(T)-108 b(eubner,)433 b(1954.)800 34395 y SDict begin H.S end 800 34395 a 800 34395 a SDict begin 14.5 H.A end 800 34395 a 800 34395 a SDict begin [ /View [/XYZ H.V] /Dest (cite.rosen) cvn H.B /DEST pdfmark end 800 34395 a 0 TeXcolorgray 1450 36221 a Fl([8])p 0 TeXcolorgray 652 w(K.)h(Rosen,)g Fk(Intr)-66 b(o)g(duction)462 b(to)j(Numb)-66 b(er)464 b(The)-66 b(ory)464 b(and)h(its)f(Applic)-66 b(ations)p Fl(,)433 b(Addison-W)-108 b(esley)g(,)433 b(1984.)800 37107 y SDict begin H.S end 800 37107 a 800 37107 a SDict begin 14.5 H.A end 800 37107 a 800 37107 a SDict begin [ /View [/XYZ H.V] /Dest (cite.hcw) cvn H.B /DEST pdfmark end 800 37107 a 0 TeXcolorgray 1450 38933 a Fl([9])p 0 TeXcolorgray 652 w(H.)297 b(C.)h(Williams,)327 b(Some)297 b(results)g(concerning)g (the)g(nearest)g(in)-36 b(teger)298 b(con)-36 b(tin)g(ued)296 b(fraction)i(expansion)3474 40539 y(of)4955 39414 y Fh(p)p 6062 39414 1117 54 v 1125 x Fj(D)36 b Fl(,)434 b Fk(J.)464 b(R)-66 b(eine)463 b(A)-33 b(ngew.)463 b(Math.)433 b Fe(315)h Fl(\(1980\),)h(1{15.)p 800 43682 52000 45 v 800 45933 a(2000)g Fk(Mathematics)464 b(Subje)-66 b(ct)463 b(Classi\014c)-66 b(ation)p Fl(:)577 b(Primary)434 b(11A55,)h(11Y65.) 800 47538 y Fk(Keywor)-66 b(ds:)638 b Fl(nearest)359 b(in)-36 b(teger)359 b(con)-36 b(tin)g(ued)358 b(fraction,)375 b(p)36 b(erio)g(d{length,)374 b(midp)36 b(oin)-36 b(t)358 b(criteria,)375 b(unisequence.)p 800 50516 V 800 53567 a(Receiv)-36 b(ed)453 b(Ma)-36 b(y)452 b(18)h(2009;)463 b(revised)452 b(v)-36 b(ersion)453 b(receiv)-36 b(ed)452 b(Septem)-36 b(b)36 b(er)451 b(3)h(2009.)635 b(Published)451 b(in)h Fk(Journal)482 b(of)800 55172 y(Inte)-66 b(ger)463 b(Se)-66 b(quenc)g(es)p Fl(,)431 b(Septem)-36 b(b)36 b(er)433 b(22)h(2009.)p 800 56803 V 800 59054 a(Return)f(to)p 0 1 0 0 TeXcolorcmyk 6815 59312 a SDict begin H.S end 6815 59312 a 0 1 0 0 TeXcolorcmyk -258 x Fl(Journal)h(of)g(In)-36 b(teger)434 b(Sequences)f(home)g(page)p 0 1 0 0 TeXcolorcmyk 29602 58131 a SDict begin H.R end 29602 58131 a 29602 59054 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://www.cs.uwaterloo.ca/journals/JIS/) >> /Subtype /Link H.B /ANN pdfmark end 29602 59054 a 0 TeXcolorgray Fl(.)p 0 TeXcolorgray 26150 70521 a(24)p 0 TeXcolorgray eop end end %%Trailer end