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%%Creator: dvips(k) 5.85 Copyright 1999 Radical Eye Software
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% @psencodingfile{
% author = "Y&Y, Inc.",
% version = "1.1",
% date = "1 December 1996",
% filename = "texnansi.enc",
% email = "help@YandY.com",
% address = "45 Walden Street // Concord, MA 01742, USA",
% codetable = "ISO/ASCII",
% checksum = "xx",
% docstring = "Encoding for fonts in Adobe Type 1 format for use with TeX."
% }
%
% The idea is to have all 228 characters normally included in Type 1 text
% fonts (plus a few more) available for typesetting. This is effectively
% the character set in Adobe Standard Encoding, ISO Latin 1, plus a few more.
%
% Character code assignments were made as follows:
%
% (1) The character layout largely matches `ASCII' in the 32 -- 126 range,
% except for `circumflex' in 94 and `tilde' in 126, to match `TeX text'
% (`asciicircumflex' and `asciitilde' appear in 158 and 142 instead).
%
% (2) The character layout matches `Windows ANSI' in almost all places,
% except for `quoteright' in 39 and `quoteleft' in 96 to match ASCII
% (`quotesingle' and `grave' appear in 129 and 18 instead).
%
% (3) The character layout matches `TeX typewriter' used by CM text fonts
% in most places (except for discordant positions such as hungarumlaut
% (instead of braceright), dotaccent (instead of underscore) etc.
%
% (4) Remaining characters are assigned arbitrarily to the `control character'
% range (0 -- 31), avoiding 0, 9, 10 and 13 in case we meet dumb software
% - similarly one should really avoid 127 and 128 if possible.
% In addition, the 8 open slots in Windows ANSI between 128 and 159 are used.
%
% (5) Y&Y Lucida Bright includes some extra ligatures and such; ff, ffi, ffl,
% and `dotlessj,' these are included 11 -- 15, and 17.
%
% (6) Hyphen appears both at 45 and 173 for compatibility with both ASCII
% and Windows ANSI.
%
% (7) It doesn't really matter where ligatures appear (both real, such as ffi,
% and pseudo such as ---) since these should not be accessed directly, only
% via ligature information in the TFM file.
%
% SAMPLE USAGE (in `psfonts.map' file for DVIPS):
%
% lbr LucidaBright "TeXnANSIEncoding ReEncodeFont" <texnansi.enc <lbr.pfb
%
% This tells DVIPS that the font called `lbr' in TeX has PostScript
% FontName `LucidaBright.' It also asks DVIPS to expand the file `lbr.pfb'
% into PFA form, to include the attached `texnansi.enc' encoding vector,
% and to then actually reencode the font based on that encoding vector.
%
% Revised 1996 June 1 by adding second position for `fl' to avoid Acrobat bug.
% Revised 1996 June 1 by adding second position for `fraction' for same reason.
%
/TeXnANSIEncoding [
/.notdef /uni20AC /.notdef /.notdef % 0, 1, 2, 3
/fraction % 4
/dotaccent % 5
/hungarumlaut % 6
/ogonek % 7
/fl % 8
/.notdef % /fraction % 9 not used (see 4), backward compatability only
/cwm % 10 not used, except boundary char internally maybe
/ff % 11
/fi % 12
/.notdef % /fl % 13 not used (see 8), backward compatability only
/ffi % 14
/ffl % 15
/dotlessi % 16
/dotlessj % 17
/grave % 18
/acute % 19
/caron % 20
/breve % 21
/macron % 22
/ring % 23
/cedilla % 24
/germandbls % 25
/ae % 26
/oe % 27
/oslash % 28
/AE % 29
/OE % 30
/Oslash % 31
/space % 32 % /suppress in TeX text
/exclam % 33
/quotedbl % 34 % /quotedblright in TeX text
/numbersign % 35
/dollar % 36
/percent % 37
/ampersand % 38
/quoteright % 39 % /quotesingle in ANSI
/parenleft % 40
/parenright % 41
/asterisk % 42
/plus % 43
/comma % 44
/hyphen % 45
/period % 46
/slash % 47
/zero % 48
/one % 49
/two % 50
/three % 51
/four % 52
/five % 53
/six % 54
/seven % 55
/eight % 56
/nine % 57
/colon % 58
/semicolon % 59
/less % 60 % /exclamdown in Tex text
/equal % 61
/greater % 62 % /questiondown in TeX text
/question % 63
/at % 64
/A % 65
/B % 66
/C % 67
/D % 68
/E % 69
/F % 70
/G % 71
/H % 72
/I % 73
/J % 74
/K % 75
/L % 76
/M % 77
/N % 78
/O % 79
/P % 80
/Q % 81
/R % 82
/S % 83
/T % 84
/U % 85
/V % 86
/W % 87
/X % 88
/Y % 89
/Z % 90
/bracketleft % 91
/backslash % 92 % /quotedblleft in TeX text
/bracketright % 93
/circumflex % 94 % /asciicircum in ASCII
/underscore % 95 % /dotaccent in TeX text
/quoteleft % 96 % /grave accent in ANSI
/a % 97
/b % 98
/c % 99
/d % 100
/e % 101
/f % 102
/g % 103
/h % 104
/i % 105
/j % 106
/k % 107
/l % 108
/m % 109
/n % 110
/o % 111
/p % 112
/q % 113
/r % 114
/s % 115
/t % 116
/u % 117
/v % 118
/w % 119
/x % 120
/y % 121
/z % 122
/braceleft % 123 % /endash in TeX text
/bar % 124 % /emdash in TeX test
/braceright % 125 % /hungarumlaut in TeX text
/tilde % 126 % /asciitilde in ASCII
/dieresis % 127 not used (see 168), use higher up instead
/Lslash % 128 this position is unfortunate, but now too late to fix
/quotesingle % 129
/quotesinglbase % 130
/florin % 131
/quotedblbase % 132
/ellipsis % 133
/dagger % 134
/daggerdbl % 135
/circumflex % 136
/perthousand % 137
/Scaron % 138
/guilsinglleft % 139
/OE % 140
/Zcaron % 141
/asciicircum % 142
/minus % 143
/lslash % 144
/quoteleft % 145
/quoteright % 146
/quotedblleft % 147
/quotedblright % 148
/bullet % 149
/endash % 150
/emdash % 151
/tilde % 152
/trademark % 153
/scaron % 154
/guilsinglright % 155
/oe % 156
/zcaron % 157
/asciitilde % 158
/Ydieresis % 159
/nbspace % 160 % /space (no break space)
/exclamdown % 161
/cent % 162
/sterling % 163
/currency % 164
/yen % 165
/brokenbar % 166
/section % 167
/dieresis % 168
/copyright % 169
/ordfeminine % 170
/guillemotleft % 171
/logicalnot % 172
/sfthyphen % 173 % /hyphen (hanging hyphen)
/registered % 174
/macron % 175
/degree % 176
/plusminus % 177
/twosuperior % 178
/threesuperior % 179
/acute % 180
/mu % 181
/paragraph % 182
/periodcentered % 183
/cedilla % 184
/onesuperior % 185
/ordmasculine % 186
/guillemotright % 187
/onequarter % 188
/onehalf % 189
/threequarters % 190
/questiondown % 191
/Agrave % 192
/Aacute % 193
/Acircumflex % 194
/Atilde % 195
/Adieresis % 196
/Aring % 197
/AE % 198
/Ccedilla % 199
/Egrave % 200
/Eacute % 201
/Ecircumflex % 202
/Edieresis % 203
/Igrave % 204
/Iacute % 205
/Icircumflex % 206
/Idieresis % 207
/Eth % 208
/Ntilde % 209
/Ograve % 210
/Oacute % 211
/Ocircumflex % 212
/Otilde % 213
/Odieresis % 214
/multiply % 215 % OE in T1
/Oslash % 216
/Ugrave % 217
/Uacute % 218
/Ucircumflex % 219
/Udieresis % 220
/Yacute % 221
/Thorn % 222
/germandbls % 223 % SS in T1
/agrave % 224
/aacute % 225
/acircumflex % 226
/atilde % 227
/adieresis % 228
/aring % 229
/ae % 230
/ccedilla % 231
/egrave % 232
/eacute % 233
/ecircumflex % 234
/edieresis % 235
/igrave % 236
/iacute % 237
/icircumflex % 238
/idieresis % 239
/eth % 240
/ntilde % 241
/ograve % 242
/oacute % 243
/ocircumflex % 244
/otilde % 245
/odieresis % 246
/divide % 247 % oe in T1
/oslash % 248
/ugrave % 249
/uacute % 250
/ucircumflex % 251
/udieresis % 252
/yacute % 253
/thorn % 254
/ydieresis % 255 % germandbls in T1
] def
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3 3 bop Black Black Black Black 1241 274 a Fu(The)33
b(Univ)m(ersit)m(y)h(of)e(Southern)h(Mississippi)p Black
Black 625 781 a Ft(P)-11 b(ARALLEL)44 b(MONTE)g(CARLO)h(COMPUT)-11
b(A)g(TION)1111 1051 y(OF)44 b(INV)-15 b(ARIANT)44 b(MEASURES)p
Black Black 2019 1505 a Fu(b)m(y)p Black Black 1644 1734
a(Zizhong)31 b(John)i(W)-8 b(ang)p Black Black 1760 2310
a(A)33 b(Dissertation)1333 2435 y(Submitted)f(to)g(the)h(Graduate)f(Sc)
m(ho)s(ol)1185 2559 y(of)g(The)i(Univ)m(ersit)m(y)f(of)f(Southern)i
(Mississippi)1175 2684 y(in)e(P)m(artial)e(F)-8 b(ul\014llmen)m(t)30
b(of)i(the)h(Requiremen)m(ts)1241 2809 y(for)f(the)h(Degree)g(of)f(Do)s
(ctor)g(of)g(Philosoph)m(y)1986 3273 y(Appro)m(v)m(ed:)2695
3394 y Fs(Jiu)38 b(Ding)p 1986 3514 1854 4 v 1986 3642
a Fu(Director)p 1986 3809 V 1986 4048 V 1986 4286 V 1986
4525 V 1986 4763 V 1986 4891 a(Dean)32 b(of)g(the)i(Graduate)e(Sc)m(ho)
s(ol)p Black Black 1803 5387 a(August)h(2000)p Black
Black eop
%%Page: 2 5
2 4 bop Black Black Black Black 1112 150 a Fn(A)l(CKNO)l(WLEDGEMENTS)
479 851 y Fu(F)-8 b(or)40 b(their)g(assistance)h(in)f(the)h(course)g
(of)f(m)m(y)g(do)s(ctoral)f(study)-8 b(,)44 b(man)m(y)c(p)s(eople)h
(deserv)m(e)300 972 y(thanks.)65 b(Sp)s(ecial)38 b(thank)h(m)m(ust)h
(go)e(to)h(m)m(y)h(advisor)f(Jiu)f(Ding)g(for)h(his)g(ongoing)e(helps)j
(and)300 1092 y(supp)s(orts)32 b(in)e(m)m(y)h(progress)h(of)e(course)j
(study)-8 b(,)32 b(researc)m(h,)h(m)m(y)e(job)g(h)m(un)m(ting,)g(and)g
(other)h(\014elds.)300 1212 y(His)26 b(con)m(tin)m(ual)g(encouragemen)m
(t,)j(advise,)g(and)e(understanding)g(are)g(all)e(greatly)h
(appreciated.)446 1333 y(I)36 b(will)c(b)s(e)k(alw)m(a)m(ys)f(grateful)
f(to)h(James)g(Ca)m(v)m(en)m(y)-8 b(,)38 b(Ras)d(P)m(andey)-8
b(,)38 b(Marcin)c(P)m(aprzyc)m(ki,)k(and)300 1453 y(Ra)m(y)43
b(Seyfarth.)75 b(They)45 b(are)e(all)d(m)m(y)j(PhD)g(committee)e(mem)m
(b)s(ers)i(and)g(m)m(y)g(course)h(profes-)300 1574 y(sors,)31
b(and)f(they)i(ga)m(v)m(e)f(me)e(v)m(ery)j(helpful)d(directions)g(and)h
(commen)m(ts)g(during)g(the)g(prosp)s(ectus)300 1694
y(preparation)42 b(and)h(the)h(writing)d(of)i(m)m(y)g(dissertation.)74
b(In)43 b(particular)e(I)i(really)f(appreciate)300 1814
y(that)35 b(Marcin)g(P)m(aprzyc)m(ki)i(ga)m(v)m(e)f(me)e(abundan)m(t)i
(ideas)f(in)g(m)m(y)g(n)m(umerical)f(exp)s(erimen)m(ts)h(with)300
1935 y(parallel)c(computers.)47 b(Besides,)35 b(he)f(help)s(ed)g(me)f
(a)g(lot)g(for)g(job)g(h)m(un)m(ting.)47 b(Ra)m(y)34
b(Seyfarth)g(also)300 2055 y(merits)c(particular)g(thanks)i(for)f(his)g
(v)-5 b(alued)31 b(advise)h(in)f(m)m(y)g(parallel)e(computing)h(w)m
(ork.)44 b(I)32 b(ad-)300 2175 y(mire)g(James)h(Ca)m(v)m(en)m(y)j(for)d
(his)g(graceful)f(teac)m(hing)h(st)m(yle)h(and)f(resp)s(ectable)h(p)s
(ersonalit)m(y)-8 b(.)45 b(Ras)300 2296 y(P)m(andey's)35
b(profound)d(kno)m(wledge)i(and)f(sc)m(holarly)f(c)m(haracter)h
(impress)f(me)g(greatly)-8 b(.)446 2416 y(I)29 b(m)m(ust)g(thank)h
(Joseph)g(Kolibal)25 b(for)k(his)f(time-consuming)f(w)m(ork)j(on)e(the)
i(uniform)d(disser-)300 2537 y(tation)33 b(\(Latex\))i(pac)m(k)-5
b(age.)50 b(Indeed)36 b(it)d(sa)m(v)m(ed)j(me)f(a)f(lot)f(of)h(time)f
(in)h(t)m(yping)g(m)m(y)h(dissertation)300 2657 y(with)d(this)g(pac)m
(k)-5 b(age.)446 2777 y(I)23 b(thank)h(Charles)f(W)-8
b(righ)m(t)22 b(for)h(his)f(e\016cien)m(t)i(managemen)m(t)e(of)h
(Wiglaf)d(parallel)g(computers.)300 2898 y(He)30 b(is)f(a)g(w)m
(arm-hearted,)h(helpful,)f(and)g(friendly)g(p)s(erson)h(and)f(he)h(ga)m
(v)m(e)g(me)f(a)g(lot)f(of)h(supp)s(orts)300 3018 y(in)j(m)m(y)g
(parallel)e(computing)i(exp)s(erimen)m(ts.)446 3139 y(I)41
b(am)e(also)h(grateful)f(to)h(W)-8 b(allace)39 b(Py)m(e,)44
b(Je\013)d(Stuart)g(and)f(man)m(y)h(other)f(professors)i(and)300
3259 y(friends)k(whose)h(en)m(th)m(usiastic)f(helps)g(to)f(me)g(during)
g(m)m(y)h(do)s(ctoral)e(study)j(will)c(alw)m(a)m(ys)k(b)s(e)300
3379 y(memorized.)446 3500 y(Finally)-8 b(,)22 b(I)h(w)m(ould)f(lik)m
(e)g(to)g(dedicate)h(this)f(dissertation)g(to)g(m)m(y)g(grandparen)m
(ts)i(who)f(brough)m(t)300 3620 y(me)46 b(up,)j(to)d(m)m(y)g(wife)g
(Huiqing)f(Y)-8 b(ang)45 b(who)i(is)e(also)g(m)m(y)h(fello)m(w)f
(classmate)h(of)f(the)i(Ph.D.)300 3740 y(program)37 b(in)h(scien)m
(ti\014c)h(computing,)g(and)g(to)f(our)h(lo)m(v)m(ely)g(daugh)m(ter)g
(Angela)f(W)-8 b(ang,)40 b(for)e(all)300 3861 y(the)24
b(happiness)h(of)e(our)h(family)d(life)h(here)j(at)f(The)g(Univ)m
(ersit)m(y)h(of)e(Southern)i(Mississippi)e(during)300
3981 y(the)33 b(past)g(y)m(ears.)p Black 2043 5764 a(ii)p
Black eop
%%Page: 3 6
3 5 bop Black Black Black Black 1181 150 a Fn(T)-13 b(ABLE)53
b(OF)h(CONTENTS)300 851 y Fm(A)m(CKNO)m(WLEDGEMENTS)72
b Fr(:)56 b(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f
(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)158 b Fu(ii)304 1096 y
Fm(LIST)38 b(OF)g(ILLUSTRA)-9 b(TIONS)69 b Fr(:)55 b(:)h(:)f(:)h(:)f(:)
h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)152
b Fu(iv)305 1341 y Fm(LIST)38 b(OF)g(T)-9 b(ABLES)74
b Fr(:)55 b(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f
(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)159
b Fu(v)305 1586 y Fm(LIST)38 b(OF)g(ABBREVIA)-9 b(TIONS)86
b Fr(:)56 b(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f
(:)h(:)f(:)h(:)f(:)h(:)172 b Fu(vi)300 1887 y Fm(1)90
b(INTR)m(ODUCTION)84 b Fr(:)55 b(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)
h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)166
b Fm(1)300 2105 y(2)90 b(FR)m(OBENIUS-PERR)m(ON)35 b(OPERA)-9
b(TORS)72 b Fr(:)56 b(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f
(:)h(:)154 b Fm(8)300 2323 y(3)90 b(ULAM'S)39 b(METHOD)88
b Fr(:)56 b(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f
(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)114 b Fm(12)300
2541 y(4)90 b(PIECEWISE)35 b(LINEAR)h(MARK)m(O)m(V)h(APPR)m(O)m(XIMA)-9
b(TION)90 b Fr(:)55 b(:)g(:)h(:)f(:)118 b Fm(20)300 2759
y(5)90 b(QUASI-MONTE)38 b(CARLO)f(ALGORITHM)67 b Fr(:)55
b(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)92
b Fm(25)300 2977 y(6)e(P)-9 b(ARALLEL)36 b(ALGORITHMS)82
b Fr(:)56 b(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g
(:)h(:)f(:)h(:)f(:)107 b Fm(29)300 3195 y(7)90 b(NUMERICAL)37
b(RESUL)-9 b(TS)81 b Fr(:)56 b(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f
(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)106 b
Fm(36)300 3315 y Fu(7.1)168 b(Numerical)30 b(Results)j(from)f(the)h
(Quasi-Mon)m(te)f(Carlo)g(Approac)m(h)681 b(36)300 3435
y(7.2)168 b(Numerical)30 b(Results)j(from)f(the)h(P)m(arallel)d
(Computation)1017 b(39)300 3556 y(7.3)168 b(An)32 b(Applied)g
(Computational)e(Problem)1592 b(45)300 3774 y Fm(8)90
b(CONCLUDING)37 b(REMARKS)85 b Fr(:)55 b(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)
h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)110 b
Fm(53)305 4060 y(APPENDIX)300 4278 y(A)61 b(BRIEF)37
b(MA)-9 b(TH)36 b(REVIEW)84 b Fr(:)56 b(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g
(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)111
b Fm(54)300 4496 y(B)66 b(A)37 b(SOUR)m(CE)g(CODE)g(IN)g(C)90
b Fr(:)55 b(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h
(:)f(:)h(:)f(:)h(:)f(:)h(:)114 b Fm(61)305 4741 y(BIBLIOGRAPHY)99
b Fr(:)55 b(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h
(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)183 b
Fu(89)p Black 2029 5764 a(iii)p Black eop
%%Page: 4 7
4 6 bop Black Black Black Black 1066 150 a Fn(LIST)54
b(OF)h(ILLUSTRA)-13 b(TIONS)446 851 y Fu(1.1)100 b(The)33
b(orbits)f(of)g Fr(S)6 b Fu(\()p Fr(x)p Fu(\))28 b(=)g(4)p
Fr(x)p Fu(\(1)22 b Fq(\000)g Fr(x)p Fu(\))33 b(with)g(\(ab)s(o)m(v)m
(e\))g Fr(x)2656 866 y Fp(0)2723 851 y Fu(=)28 b(0)p
Fr(:)p Fu(037)k(and)g(\(b)s(elo)m(w\))671 972 y Fr(x)726
987 y Fp(0)793 972 y Fu(=)c(0)p Fr(:)p Fu(038)51 b(.)f(.)g(.)g(.)g(.)f
(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)
g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p Black 132 w(1)p
Black 446 1092 a(1.2)100 b(The)37 b(frequency)i(of)d(the)h(orbits)f(of)
g Fr(S)6 b Fu(\()p Fr(x)p Fu(\))35 b(=)f(4)p Fr(x)p Fu(\(1)25
b Fq(\000)g Fr(x)p Fu(\))37 b(of)f(10000)g(iterations)671
1212 y(and)c(100)g(partitions)f(in)h([0)p Fr(;)17 b Fu(1])32
b(with)g(di\013eren)m(t)h(initial)c(states)135 b(.)50
b(.)f(.)h(.)g(.)g(.)g(.)g(.)p Black 132 w(3)p Black 446
1416 a(2.1)100 b(The)30 b(\014xed)h(densit)m(y)f Fr(f)1493
1380 y Fl(\003)1562 1416 y Fu(of)f Fr(P)43 b Fu(for)29
b Fr(S)6 b Fu(\()p Fr(x)p Fu(\))28 b(=)f(4)p Fr(x)p Fu(\(1)16
b Fq(\000)g Fr(x)p Fu(\))30 b(with)f(100)g(partitions)f(in)671
1536 y([0,1])90 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)
g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f
(.)h(.)g(.)g(.)g(.)g(.)p Black 83 w(10)p Black 446 1740
a(7.1)100 b Fr(L)737 1703 y Fp(1)809 1740 y Fu(error)32
b(with)h(c)m(hange)g(of)f Fr(n)h Fu(for)f Fr(S)2003 1755
y Fp(1)2094 1740 y Fu(.)50 b(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)
g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p Black 83 w(39)p
Black 446 1860 a(7.2)100 b(E\016ciency)34 b(of)e(UMP)-8
b(A)33 b(for)f Fr(S)1760 1875 y Fp(3)1832 1860 y Fu(on)h(Wiglaf)73
b(.)49 b(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)
g(.)p Black 83 w(42)p Black 446 1980 a(7.3)100 b(E\016ciency)34
b(of)e(CP)-8 b(A)33 b(for)f Fr(S)1668 1995 y Fp(3)1740
1980 y Fu(on)h(Wiglaf)88 b(.)50 b(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g
(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p Black 83 w(42)p
Black 446 2101 a(7.4)100 b(E\016ciency)34 b(of)e(UMP)-8
b(A)33 b(for)f Fr(S)1760 2116 y Fp(3)1832 2101 y Fu(on)h(Sw)m(eetgum)80
b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p
Black 83 w(44)p Black 446 2221 a(7.5)100 b(E\016ciency)34
b(of)e(CP)-8 b(A)33 b(for)f Fr(S)1668 2236 y Fp(3)1740
2221 y Fu(on)h(Sw)m(eetgum)95 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p Black 83 w(44)p Black
446 2342 a(7.6)100 b(E\016ciency)34 b(comparison)d(of)h(UMP)-8
b(A)34 b(for)e Fr(S)2276 2357 y Fp(3)2348 2342 y Fu(on)g(Sw)m(eetgum)h
(and)g(Wiglaf)58 b(.)50 b(.)g(.)p Black 83 w(45)p Black
446 2462 a(7.7)100 b(The)33 b(graph)g(of)f(DPLL)110 b(.)50
b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g
(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p Black 83 w(47)p
Black 446 2582 a(7.8)100 b(E\016ciency)34 b(of)e(UMP)-8
b(A)33 b(for)f(DPLL)h(on)f(Wiglaf)63 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g
(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p Black 83 w(49)p
Black 446 2703 a(7.9)100 b(E\016ciency)34 b(of)e(the)h(CP)-8
b(A)33 b(for)f(DPLL)g(on)h(Wiglaf)64 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g
(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p Black 83 w(49)p Black
446 2823 a(7.10)h(E\016ciency)34 b(of)e(UMP)-8 b(A)33
b(for)f(DPLL)h(on)f(Sw)m(eetgum)71 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g(.)f
(.)h(.)g(.)g(.)g(.)g(.)p Black 83 w(51)p Black 446 2943
a(7.11)h(E\016ciency)34 b(of)e(CP)-8 b(A)33 b(for)f(DPLL)g(on)h(Sw)m
(eetgum)86 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g
(.)p Black 83 w(51)p Black 446 3064 a(7.12)h(E\016ciency)34
b(Comparison)d(of)h(UMP)-8 b(A)34 b(for)e(DPLL)g(on)h(Sw)m(eetgum)g
(and)g(Wiglaf)p Black 131 w(52)p Black 446 3184 a(7.13)51
b(The)33 b(stationary)f(PDF)g(graph)h(of)f(DPLL)g(with)g
Fr(n)c Fu(=)g(4096)77 b(.)50 b(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p
Black 83 w(52)p Black Black 2031 5764 a(iv)p Black eop
%%Page: 5 8
5 7 bop Black Black Black Black 1411 150 a Fn(LIST)54
b(OF)h(T)-13 b(ABLES)446 851 y Fu(3.1)100 b(P)m(erformance)33
b(comparison)e(b)s(et)m(w)m(een)k(IA)e(and)f(GA)101 b(.)50
b(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p Black
83 w(19)p Black 446 1055 a(7.1)100 b Fr(L)737 1019 y
Fp(1)809 1055 y Fu(error/time)31 b(comparison)g(for)h(the)h(logistic)d
(mo)s(del)78 b(.)50 b(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p
Black 83 w(37)p Black 446 1175 a(7.2)100 b Fr(L)737 1139
y Fp(1)809 1175 y Fu(error/time)31 b(comparison)g(for)h
Fr(S)2010 1190 y Fp(2)2094 1175 y Fu(.)50 b(.)g(.)g(.)f(.)h(.)g(.)g(.)g
(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p
Black 83 w(37)p Black 446 1295 a(7.3)100 b Fr(L)737 1259
y Fp(1)809 1295 y Fu(error)32 b(comparisons)g(of)g(the)h(QMC)h(to)e
(the)h(MC)k(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p
Black 83 w(38)p Black 446 1416 a(7.4)100 b Fr(L)737 1380
y Fp(1)809 1416 y Fu(error/time)31 b(comparison)g(of)h(QMC,)i(MC,)f
(and)g(Exact)g(for)f(logistic)e(mo)s(del)p Black 106
w(40)p Black 446 1536 a(7.5)100 b Fr(L)737 1500 y Fp(1)809
1536 y Fu(error)32 b(comparison)g(of)g(QMC,)h(MC,)h(and)e(Exact)i(for)e
Fr(S)2868 1551 y Fp(2)2940 1536 y Fu(.)50 b(.)g(.)g(.)f(.)h(.)g(.)g(.)g
(.)g(.)p Black 83 w(40)p Black 446 1657 a(7.6)100 b(P)m(erformance)33
b(of)f(UMP)-8 b(A)33 b(and)g(CP)-8 b(A)33 b(for)f Fr(S)2306
1672 y Fp(3)2378 1657 y Fu(on)h(Wiglaf)65 b(.)50 b(.)g(.)g(.)g(.)f(.)h
(.)g(.)g(.)g(.)g(.)p Black 83 w(41)p Black 446 1777 a(7.7)100
b(P)m(erformance)33 b(of)f(UMP)-8 b(A)33 b(and)g(CP)-8
b(A)33 b(for)f Fr(S)2306 1792 y Fp(3)2378 1777 y Fu(on)h(Sw)m(eetgum)73
b(.)50 b(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p Black 83 w(43)p
Black 446 1897 a(7.8)100 b(P)m(erformance)33 b(of)f(UMP)-8
b(A)33 b(and)g(CP)-8 b(A)33 b(for)f(DPLL)g(on)h(Wiglaf)56
b(.)50 b(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p Black 83 w(48)p
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@endspecial 300 3245 a(Figure)30 b(1.2:)43 b(The)32
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@endspecial 300 2655 a(Figure)j(2.1:)51 b(The)37 b(\014xed)h(densit)m
(y)f Fr(f)1657 2619 y Fl(\003)1732 2655 y Fu(of)f Fr(P)50
b Fu(for)36 b Fr(S)6 b Fu(\()p Fr(x)p Fu(\))34 b(=)g(4)p
Fr(x)p Fu(\(1)25 b Fq(\000)g Fr(x)p Fu(\))36 b(with)g(100)g(partitions)
f(in)300 2776 y([0,1])p Black 446 3126 a(Comparing)30
b(Figure)h(1.2)g(with)g(Figure)g(2.1,)g(w)m(e)h(could)g(imagine)d(that)
i(when)i(the)f(n)m(um)m(b)s(er)300 3246 y(of)g(iterations)f(go)s(es)i
(to)f(in\014nit)m(y)-8 b(,)32 b(the)h(three)g(pictures)g(will)e(b)s(e)h
(exactly)h(the)g(same.)446 3408 y(F)-8 b(or)31 b(m)m(ulti-dimensional)
26 b(cases,)32 b(let)f Fr(X)39 b Fu(b)s(e)31 b(an)g Fr(d)p
Fu(-dimensional)c(rectangle)k([)p Fr(a)3320 3423 y Fp(1)3360
3408 y Fr(;)17 b(b)3445 3423 y Fp(1)3485 3408 y Fu(])i
Fq(\002)g(\001)e(\001)g(\001)h(\002)300 3528 y Fu([)p
Fr(a)378 3543 y Fo(d)419 3528 y Fr(;)f(b)504 3543 y Fo(d)544
3528 y Fu(].)44 b(Then)34 b(di\013eren)m(tiating)c(the)j(equalit)m(y)
1173 3664 y Fi(Z)1273 3690 y Fo(x)1313 3699 y Fg(1)1228
3889 y Fo(a)1265 3898 y Fg(1)1368 3799 y Fq(\001)17 b(\001)g(\001)1501
3664 y Fi(Z)1600 3690 y Fo(x)1640 3702 y Ff(d)1556 3889
y Fo(a)1593 3901 y Ff(d)1697 3799 y Fr(P)d(f)d Fu(\()p
Fr(s)1917 3814 y Fp(1)1955 3799 y Fr(;)17 b Fq(\001)g(\001)g(\001)31
b Fr(;)17 b(s)2238 3814 y Fo(d)2279 3799 y Fu(\))p Fr(dt)2403
3814 y Fp(1)2458 3799 y Fq(\001)g(\001)g(\001)e Fr(dt)2677
3814 y Fo(d)1014 4070 y Fu(=)1173 3934 y Fi(Z)1289 4070
y Fq(\001)i(\001)g(\001)1422 3934 y Fi(Z)1477 4160 y
Fo(S)1524 4141 y Fh(\000)p Fg(1)1607 4160 y Fp(\([)p
Fo(a)1691 4169 y Fg(1)1726 4160 y Fo(;x)1786 4169 y Fg(1)1820
4160 y Fp(])p Fl(\002\001\001\001)n(\002)p Fp([)p Fo(a)2065
4172 y Ff(d)2101 4160 y Fo(;x)2161 4172 y Ff(d)2197 4160
y Fp(]\))2264 4070 y Fr(f)11 b Fu(\()p Fr(s)2407 4085
y Fp(1)2446 4070 y Fr(;)17 b Fq(\001)g(\001)g(\001)32
b Fr(;)17 b(s)2730 4085 y Fo(d)2770 4070 y Fu(\))p Fr(dt)2894
4085 y Fp(1)2950 4070 y Fq(\001)g(\001)g(\001)d Fr(dt)3168
4085 y Fo(d)300 4353 y Fu(with)32 b(resp)s(ect)i(to)e
Fr(x)1030 4368 y Fp(1)1070 4353 y Fr(;)17 b Fq(\001)g(\001)g(\001)31
b Fr(;)17 b(x)1362 4368 y Fo(d)1435 4353 y Fu(successiv)m(ely)-8
b(,)35 b(w)m(e)f(ha)m(v)m(e)408 4630 y Fr(P)14 b(f)d
Fu(\()p Fr(x)637 4645 y Fp(1)676 4630 y Fr(;)17 b Fq(\001)g(\001)g
(\001)31 b Fr(;)17 b(x)968 4645 y Fo(d)1009 4630 y Fu(\))28
b(=)1367 4563 y Fr(@)1423 4527 y Fo(d)p 1188 4607 455
4 v 1188 4699 a Fr(@)5 b(x)1299 4714 y Fp(1)1357 4699
y Fq(\001)17 b(\001)g(\001)d Fr(@)5 b(x)1600 4714 y Fo(d)1669
4495 y Fi(Z)1785 4630 y Fq(\001)17 b(\001)g(\001)1918
4495 y Fi(Z)1973 4720 y Fo(S)2020 4701 y Fh(\000)p Fg(1)2102
4720 y Fp(\([)p Fo(a)2186 4729 y Fg(1)2222 4720 y Fo(;x)2282
4729 y Fg(1)2315 4720 y Fp(])p Fl(\002\001\001\001)o(\002)p
Fp([)p Fo(a)2561 4732 y Ff(d)2597 4720 y Fo(;x)2657 4732
y Ff(d)2692 4720 y Fp(]\))2760 4630 y Fr(f)11 b Fu(\()p
Fr(s)2903 4645 y Fp(1)2942 4630 y Fr(;)17 b Fq(\001)g(\001)g(\001)31
b Fr(;)17 b(s)3225 4645 y Fo(d)3265 4630 y Fu(\))p Fr(dt)3389
4645 y Fp(1)3445 4630 y Fq(\001)g(\001)g(\001)e Fr(dt)3664
4645 y Fo(d)3705 4630 y Fr(:)3591 4829 y Fu(\(2.14\))300
5048 y Fm(Example)26 b(2)d(The)k(bak)m(er)g(transformation)p
Fu(.)40 b(Let)23 b Fr(X)35 b Fu(=)28 b([0)p Fr(;)17 b
Fu(1])s Fq(\002)s Fu([0)p Fr(;)g Fu(1].)39 b(De\014ne)24
b Fr(S)34 b Fu(:)27 b Fr(X)36 b Fq(!)27 b Fr(X)300 5169
y Fu(b)m(y)843 5441 y Fr(S)6 b Fu(\()p Fr(x;)17 b(y)t
Fu(\))27 b(=)1266 5301 y Fi(\032)1382 5380 y Fu(\(2)p
Fr(x;)1578 5341 y Fp(1)p 1578 5357 36 4 v 1578 5414 a(2)1624
5380 y Fr(y)t Fu(\))428 b(0)27 b Fq(\024)h Fr(x)g(<)2520
5341 y Fp(1)p 2520 5357 V 2520 5414 a(2)2565 5380 y Fr(;)44
b Fu(0)28 b Fq(\024)g Fr(y)j Fq(\024)d Fu(1)1382 5500
y(\(2)p Fr(x)23 b Fq(\000)f Fu(1)p Fr(;)1749 5461 y Fp(1)p
1749 5478 V 1749 5535 a(2)1794 5500 y Fr(y)j Fu(+)1975
5461 y Fp(1)p 1975 5478 V 1975 5535 a(2)2021 5500 y Fu(\))2151
5461 y Fp(1)p 2151 5478 V 2151 5535 a(2)2224 5500 y Fq(\024)k
Fr(x)f Fq(\024)g Fu(1)p Fr(;)44 b Fu(0)27 b Fq(\024)i
Fr(y)h Fq(\024)f Fu(1)p Fr(:)p Black 3591 5441 a Fu(\(2.15\))p
Black Black Black eop
%%Page: 11 20
11 19 bop Black 300 10 a Fk(CHAPTER)34 b(2.)76 b(FR)m(OBENIUS-PERR)m
(ON)33 b(OPERA)-8 b(TORS)1082 b Fu(11)p Black 300 274
a(Since)33 b Fr(S)621 238 y Fl(\000)p Fp(1)715 274 y
Fu(\([0)p Fr(;)17 b(x)p Fu(])22 b Fq(\002)h Fu([0)p Fr(;)17
b(y)t Fu(]\))26 b(=)i([0)p Fr(;)1573 235 y Fp(1)p 1573
251 36 4 v 1573 309 a(2)1618 274 y Fr(x)p Fu(])23 b Fq(\002)g
Fu([0)p Fr(;)17 b Fu(2)p Fr(y)t Fu(])31 b(for)h(0)27
b Fq(\024)h Fr(y)j(<)2625 235 y Fp(1)p 2625 251 V 2625
309 a(2)2670 274 y Fu(,)i(w)m(e)g(ha)m(v)m(e)667 574
y Fr(P)14 b(f)d Fu(\()p Fr(x;)17 b(y)t Fu(\))26 b(=)1231
507 y Fr(@)1287 471 y Fp(2)p 1169 552 221 4 v 1169 643
a Fr(@)5 b(x@)g(y)1416 439 y Fi(Z)1526 438 y Ff(x)p 1526
450 35 3 v 1528 491 a Fg(2)1471 664 y Fp(0)1591 574 y
Fr(ds)1705 439 y Fi(Z)1804 465 y Fp(2)p Fo(y)1760 664
y Fp(0)1897 574 y Fr(f)11 b Fu(\()p Fr(s;)17 b(t)p Fu(\))p
Fr(dt)27 b Fu(=)h Fr(f)11 b Fu(\()2481 507 y(1)p 2481
552 49 4 v 2481 643 a(2)2539 574 y Fr(x;)17 b Fu(2)p
Fr(y)t Fu(\))p Fr(;)44 b Fu(0)27 b Fq(\024)h Fr(y)j(<)3222
507 y Fu(1)p 3222 552 V 3222 643 a(2)3280 574 y Fr(:)p
Black 284 w Fu(\(2.16\))p Black 300 854 a(F)-8 b(or)485
815 y Fp(1)p 485 831 36 4 v 485 889 a(2)558 854 y Fq(\024)28
b Fr(y)j Fq(\024)d Fu(1,)595 1128 y Fr(S)661 1087 y Fl(\000)p
Fp(1)755 1128 y Fu(\([0)p Fr(;)17 b(x)p Fu(])22 b Fq(\002)g
Fu([0)p Fr(;)17 b(y)t Fu(]\))27 b(=)g(\([0)p Fr(;)1651
1061 y Fu(1)p 1651 1105 49 4 v 1651 1197 a(2)1710 1128
y Fr(x)p Fu(])22 b Fq(\002)h Fu([0)p Fr(;)17 b Fu(1]\))k
Fq([)i Fu(\([)2333 1061 y(1)p 2333 1105 V 2333 1197 a(2)2392
1128 y Fr(;)2445 1061 y Fu(1)p 2445 1105 V 2445 1197
a(2)2526 1128 y(+)2634 1061 y(1)p 2634 1105 V 2634 1197
a(2)2693 1128 y Fr(x)p Fu(])g Fq(\002)f Fu([0)p Fr(;)17
b Fu(2)p Fr(y)25 b Fq(\000)e Fu(1]\))p Fr(;)p Black 211
w Fu(\(2.17\))p Black 300 1374 a(hence,)633 1670 y Fr(P)14
b(f)d Fu(\()p Fr(x;)17 b(y)t Fu(\))81 b(=)1308 1602 y
Fr(@)1364 1566 y Fp(2)p 1246 1647 221 4 v 1246 1738 a
Fr(@)5 b(x@)g(y)1476 1670 y Fq(f)1526 1534 y Fi(Z)1636
1533 y Ff(x)p 1636 1545 35 3 v 1638 1586 a Fg(2)1581
1759 y Fp(0)1701 1670 y Fr(ds)1815 1534 y Fi(Z)1914 1560
y Fp(1)1870 1759 y(0)1970 1670 y Fr(f)11 b Fu(\()p Fr(s;)17
b(t)p Fu(\))p Fr(dt)k Fu(+)2435 1534 y Fi(Z)2545 1533
y Fg(1)p 2545 1545 31 3 v 2545 1586 a(2)2585 1560 y Fp(+)2650
1533 y Ff(x)p 2650 1545 35 3 v 2652 1586 a Fg(2)2491
1759 y Fp(0)2716 1670 y Fr(ds)2830 1534 y Fi(Z)2929 1560
y Fp(2)p Fo(y)r Fl(\000)p Fp(1)2884 1759 y(0)3112 1670
y Fr(f)11 b Fu(\()p Fr(s;)17 b(t)p Fu(\))p Fr(dt)p Fq(g)1077
1924 y Fu(=)83 b Fr(f)11 b Fu(\()1343 1857 y(1)p 1343
1901 49 4 v 1343 1993 a(2)1423 1924 y(+)1531 1857 y(1)p
1531 1901 V 1531 1993 a(2)1590 1924 y Fr(x;)17 b Fu(2)p
Fr(y)25 b Fq(\000)e Fu(1\))p Fr(;)2107 1857 y Fu(1)p
2107 1901 V 2107 1993 a(2)2193 1924 y Fq(\024)28 b Fr(y)j
Fq(\024)d Fu(1)p Fr(:)300 2170 y Fu(In)33 b(summary)-8
b(,)32 b(w)m(e)h(ha)m(v)m(e)1010 2442 y Fr(P)14 b(f)d
Fu(\()p Fr(x;)17 b(y)t Fu(\))26 b(=)1503 2302 y Fi(\032)1619
2381 y Fr(f)11 b Fu(\()1726 2342 y Fp(1)p 1726 2358 36
4 v 1726 2416 a(2)1771 2381 y Fr(x;)17 b Fu(2)p Fr(y)t
Fu(\))p Fr(;)428 b Fu(0)28 b Fq(\024)g Fr(y)j(<)2838
2342 y Fp(1)p 2838 2358 V 2838 2416 a(2)1619 2502 y Fr(f)11
b Fu(\()1726 2462 y Fp(1)p 1726 2479 V 1726 2536 a(2)1793
2502 y Fu(+)1901 2462 y Fp(1)p 1901 2479 V 1901 2536
a(2)1946 2502 y Fr(x;)17 b Fu(2)p Fr(y)26 b Fq(\000)c
Fu(1\))p Fr(;)2474 2462 y Fp(1)p 2474 2479 V 2474 2536
a(2)2547 2502 y Fq(\024)28 b Fr(y)j Fq(\024)d Fu(1)p
Fr(:)p Black 3591 2442 a Fu(\(2.18\))p Black 300 2715
a(Since)33 b Fr(P)14 b Fu(1)27 b(=)g(1,)33 b(the)g(Leb)s(esgue)g
(measure)g Fr(m)g Fu(is)f(in)m(v)-5 b(arian)m(t)31 b(under)j
Fr(S)6 b Fu(.)446 2835 y(The)24 b(existence)g(of)f(an)g(absolutely)f
(con)m(tin)m(uous)h(\014nite)g(in)m(v)-5 b(arian)m(t)21
b(measure)i(is)g(equiv)-5 b(alen)m(t)22 b(to)300 2956
y(that)28 b(of)g(a)g(non)m(trivial)e(solution)h(to)h(the)g(\014xed)i(p)
s(oin)m(t)d(equation)h Fr(P)14 b(f)2786 2920 y Fl(\003)2853
2956 y Fu(=)27 b Fr(f)3015 2920 y Fl(\003)3083 2956 y
Fu(for)g(the)i(F)-8 b(rob)s(enius-)300 3076 y(P)m(erron)33
b(op)s(erator)f Fr(P)46 b Fu(asso)s(ciated)33 b(with)f
Fr(S)6 b Fu(.)p Black 300 3238 a Fj(The)-5 b(or)g(em)34
b(2.2.)p Black 48 w Fu([36])26 b(Let)g Fr(f)38 b Fq(2)28
b Fr(L)1505 3202 y Fp(1)1571 3238 y Fu(b)s(e)f(a)e(densit)m(y)i
(function.)41 b(If)26 b(the)h(Ces\023)-49 b(aro)27 b(a)m(v)m(erages)g
(sequence)1622 3548 y Fr(A)1695 3563 y Fo(n)1742 3548
y Fr(f)38 b Fq(\021)1948 3481 y Fu(1)p 1943 3525 59 4
v 1943 3616 a Fr(n)2034 3424 y Fo(n)p Fl(\000)p Fp(1)2028
3453 y Fi(X)2043 3663 y Fo(i)p Fp(=0)2189 3548 y Fr(P)2266
3507 y Fo(i)2293 3548 y Fr(f)p Black 1250 w Fu(\(2.19\))p
Black 300 3860 a(is)29 b(w)m(eakly)h(pre-compact,)g(then)g(it)e(con)m
(v)m(erges)k(strongly)d(to)g(some)g(in)m(v)-5 b(arian)m(t)28
b(densit)m(y)i Fr(f)3573 3824 y Fl(\003)3640 3860 y Fq(2)e
Fr(L)3800 3824 y Fp(1)300 3981 y Fu(of)k Fr(P)14 b Fu(,)32
b(i.e.,)g Fr(P)14 b(f)867 3944 y Fl(\003)934 3981 y Fu(=)27
b Fr(f)1096 3944 y Fl(\003)1168 3981 y Fu(and)1558 4201
y(lim)1534 4260 y Fo(n)p Fl(!1)1735 4201 y Fq(k)p Fr(A)1858
4216 y Fo(n)1904 4201 y Fr(f)33 b Fq(\000)23 b Fr(f)2144
4160 y Fl(\003)2183 4201 y Fq(k)28 b Fu(=)f(0)p Fr(:)p
Black 1151 w Fu(\(2.20\))p Black Black Black eop
%%Page: 12 21
12 20 bop Black Black Black Black 1714 122 a Fn(Chapter)53
b(3)p Black Black 1357 554 a(ULAM'S)g(METHOD)477 1255
y Fu(No)m(w)41 b(w)m(e)g(in)m(tro)s(duce)f(the)g(idea)f(b)s(ehind)h
(Ulam's)e(piecewise)j(constan)m(t)g(appro)m(ximations)300
1376 y(for)30 b(computing)e(the)j(\014xed)g(densit)m(y)g(of)f(the)h(F)
-8 b(rob)s(enius-P)m(erron)29 b(op)s(erator.)42 b(W)-8
b(e)31 b(\014rst)g(consider)300 1496 y(the)g(one)g(dimensional)d(case,)
k(whic)m(h)f(w)m(as)g(initially)26 b(prop)s(osed)31 b(b)m(y)h(Ulam)d
(in)g(his)i(famous)e(b)s(o)s(ok)300 1616 y([47])j(on)h(mathematical)c
(problems.)446 1737 y(Assume)44 b(that)e Fr(S)50 b Fu(:)45
b([0)p Fr(;)17 b Fu(1])44 b Fq(!)g Fu([0)p Fr(;)17 b
Fu(1])42 b(is)g(a)h(nonsingular)e(transformation)f(suc)m(h)k(that)f
(the)300 1857 y(corresp)s(onding)28 b(F)-8 b(rob)s(enius-P)m(erron)27
b(op)s(erator)g Fr(P)41 b Fu(:)28 b Fr(L)2297 1821 y
Fp(1)2337 1857 y Fu(\(0)p Fr(;)17 b Fu(1\))27 b Fq(!)g
Fr(L)2775 1821 y Fp(1)2815 1857 y Fu(\(0)p Fr(;)17 b
Fu(1\))27 b(has)h(a)f(\014xed)i(densit)m(y)300 1978 y
Fr(f)359 1941 y Fl(\003)398 1978 y Fu(.)53 b(T)-8 b(o)36
b(illustrate)e(the)i(basic)f(idea)g(b)s(ehind)h(the)g(probabilit)m(y)e
(argumen)m(t)h(of)h(Ulam's)e(for)i(the)300 2098 y(motiv)-5
b(ation)35 b(of)i(his)h(metho)s(d,)g(w)m(e)h(\014rst)f(use)h(a)f
(simple)e(partition)f(of)j(the)g(in)m(terv)-5 b(al)37
b([0)p Fr(;)17 b Fu(1])37 b(in)m(to)300 2218 y(three)c(subin)m(terv)-5
b(als,)33 b(and)g(then)g(generalize)f(it)f(to)h(a)h(partition)d(of)i
Fr(n)h Fu(subin)m(terv)-5 b(als.)446 2339 y(Supp)s(ose)40
b([0)p Fr(;)17 b Fu(1])37 b(is)h(divided)g(in)m(to)g(three)h(subin)m
(terv)-5 b(als)38 b Fr(I)2567 2354 y Fp(1)2606 2339 y
Fr(;)17 b(I)2693 2354 y Fp(2)2733 2339 y Fr(;)g(I)2820
2354 y Fp(3)2859 2339 y Fu(.)60 b(Let)39 b Fr(f)49 b
Fu(b)s(e)38 b(a)g(piecewise)300 2459 y(constan)m(t)44
b(densit)m(y)g(suc)m(h)h(that)e(the)h(p)s(ossibilit)m(y)d(of)i
Fr(I)2323 2474 y Fp(1)2406 2459 y Fu(is)f Fr(a)2565 2474
y Fp(1)2605 2459 y Fu(,)k(the)e(probabilit)m(y)d(of)i
Fr(I)3531 2474 y Fp(2)3613 2459 y Fu(is)g Fr(a)3773 2474
y Fp(2)3813 2459 y Fu(,)300 2580 y(and)37 b(the)g(probabilit)m(y)d(of)j
Fr(I)1327 2595 y Fp(3)1403 2580 y Fu(is)f Fr(a)1556 2595
y Fp(3)1596 2580 y Fu(.)55 b(Ulam's)36 b(idea)g(is)g(that)g
Fr(P)14 b(f)47 b Fu(can)37 b(b)s(e)g(appro)m(ximated)f(b)m(y)h(a)300
2700 y(piecewise)i(constan)m(t)h(densit)m(y)f(with)f(the)h
(probabilities)d(of)i Fr(I)2598 2715 y Fp(1)2638 2700
y Fr(;)17 b(I)2725 2715 y Fp(2)2764 2700 y Fr(;)39 b
Fu(and)f Fr(I)3068 2715 y Fp(3)3146 2700 y Fu(to)g(b)s(e)h
Fr(b)3451 2715 y Fp(1)3491 2700 y Fr(;)17 b(b)3576 2715
y Fp(2)3616 2700 y Fu(,)40 b(and)300 2820 y Fr(b)341
2835 y Fp(3)381 2820 y Fu(.)j(W)-8 b(e)33 b(w)m(an)m(t)h(to)e(\014nd)h
(ho)m(w)g Fr(b)1416 2835 y Fp(1)1456 2820 y Fr(;)17 b(b)1541
2835 y Fp(2)1581 2820 y Fu(,)32 b(and)h Fr(b)1871 2835
y Fp(3)1944 2820 y Fu(dep)s(end)g(on)g Fr(a)2469 2835
y Fp(1)2508 2820 y Fr(;)17 b(a)2603 2835 y Fp(2)2643
2820 y Fu(,)32 b(and)h Fr(a)2943 2835 y Fp(3)2983 2820
y Fu(.)446 2941 y(F)-8 b(rom)32 b(the)j(de\014nition)d(of)i(the)g(F)-8
b(rob)s(enius-P)m(erron)33 b(op)s(erator)g Fr(P)14 b
Fu(,)34 b(the)g(new)g(probabilit)m(y)e Fr(b)3800 2956
y Fp(1)300 3061 y Fu(of)k Fr(I)458 3076 y Fp(1)533 3061
y Fu(should)g(carry)g(the)h(old)e(probabilit)m(y)f(of)h
Fr(S)2121 3025 y Fl(\000)p Fp(1)2215 3061 y Fu(\()p Fr(I)2296
3076 y Fp(1)2336 3061 y Fu(\))g(whic)m(h)i(is)e(the)i(disjoin)m(t)e
(union)g(of)h(the)300 3181 y(three)d(parts)g(of)f Fr(S)976
3145 y Fl(\000)p Fp(1)1070 3181 y Fu(\()p Fr(I)1151 3196
y Fp(1)1191 3181 y Fu(\))g(in)g Fr(I)1418 3196 y Fp(1)1457
3181 y Fr(;)17 b(I)1544 3196 y Fp(2)1584 3181 y Fu(,)32
b(and)h Fr(I)1876 3196 y Fp(3)1915 3181 y Fu(:)1229 3377
y Fr(I)1272 3392 y Fp(1)1334 3377 y Fq(\\)22 b Fr(S)1488
3336 y Fl(\000)p Fp(1)1582 3377 y Fu(\()p Fr(I)1663 3392
y Fp(1)1703 3377 y Fu(\))p Fr(;)17 b(I)1828 3392 y Fp(2)1889
3377 y Fq(\\)23 b Fr(S)2044 3336 y Fl(\000)p Fp(1)2138
3377 y Fu(\()p Fr(I)2219 3392 y Fp(1)2258 3377 y Fu(\))p
Fr(;)17 b(I)2383 3392 y Fp(3)2444 3377 y Fq(\\)23 b Fr(S)2599
3336 y Fl(\000)p Fp(1)2693 3377 y Fu(\()p Fr(I)2774 3392
y Fp(1)2813 3377 y Fu(\))p Fr(:)300 3573 y Fu(So)32 b(w)m(e)i(can)f
(obtain)492 3792 y Fr(b)533 3807 y Fp(1)656 3792 y Fu(=)815
3657 y Fi(Z)870 3882 y Fo(I)901 3891 y Fg(1)956 3792
y Fr(P)14 b(f)d Fu(\()p Fr(x)p Fu(\))p Fr(dx)27 b Fu(=)1460
3657 y Fi(Z)1515 3882 y Fo(S)1562 3863 y Fh(\000)p Fg(1)1644
3882 y Fp(\()p Fo(I)1702 3891 y Fg(1)1737 3882 y Fp(\))1785
3792 y Fr(f)11 b Fu(\()p Fr(x)p Fu(\))p Fr(dx)656 4068
y Fu(=)815 3932 y Fi(Z)870 4158 y Fo(S)917 4139 y Fh(\000)p
Fg(1)999 4158 y Fp(\()p Fo(I)1057 4167 y Fg(1)1092 4158
y Fp(\))p Fl(\\)p Fo(I)1197 4167 y Fg(1)1252 4068 y Fr(f)g
Fu(\()p Fr(x)p Fu(\))p Fr(dx)23 b Fu(+)1669 3932 y Fi(Z)1724
4158 y Fo(S)1771 4139 y Fh(\000)p Fg(1)1853 4158 y Fp(\()p
Fo(I)1911 4167 y Fg(1)1946 4158 y Fp(\))p Fl(\\)p Fo(I)2051
4167 y Fg(2)2106 4068 y Fr(f)11 b Fu(\()p Fr(x)p Fu(\))p
Fr(dx)23 b Fu(+)2523 3932 y Fi(Z)2578 4158 y Fo(S)2625
4139 y Fh(\000)p Fg(1)2707 4158 y Fp(\()p Fo(I)2765 4167
y Fg(1)2800 4158 y Fp(\))p Fl(\\)p Fo(I)2905 4167 y Fg(3)2960
4068 y Fr(f)11 b Fu(\()p Fr(x)p Fu(\))p Fr(dx)656 4354
y Fu(=)825 4287 y Fr(m)p Fu(\()p Fr(I)991 4302 y Fp(1)1053
4287 y Fq(\\)22 b Fr(S)1207 4250 y Fl(\000)p Fp(1)1301
4287 y Fu(\()p Fr(I)1382 4302 y Fp(1)1422 4287 y Fu(\)\))p
825 4331 673 4 v 1039 4422 a Fr(m)p Fu(\()p Fr(I)1205
4437 y Fp(1)1245 4422 y Fu(\))1530 4354 y Fq(\001)f Fr(a)1630
4369 y Fp(1)1692 4354 y Fu(+)1800 4287 y Fr(m)p Fu(\()p
Fr(I)1966 4302 y Fp(2)2028 4287 y Fq(\\)i Fr(S)2183 4250
y Fl(\000)p Fp(1)2277 4287 y Fu(\()p Fr(I)2358 4302 y
Fp(1)2397 4287 y Fu(\)\))p 1800 4331 V 2015 4422 a Fr(m)p
Fu(\()p Fr(I)2181 4437 y Fp(2)2220 4422 y Fu(\))2505
4354 y Fq(\001)f Fr(a)2606 4369 y Fp(2)2668 4354 y Fu(+)2776
4287 y Fr(m)p Fu(\()p Fr(I)2942 4302 y Fp(3)3004 4287
y Fq(\\)g Fr(S)3158 4250 y Fl(\000)p Fp(1)3252 4287 y
Fu(\()p Fr(I)3333 4302 y Fp(1)3373 4287 y Fu(\)\))p 2776
4331 V 2990 4422 a Fr(m)p Fu(\()p Fr(I)3156 4437 y Fp(3)3196
4422 y Fu(\))3481 4354 y Fq(\001)f Fr(a)3581 4369 y Fp(3)3621
4354 y Fr(;)300 4600 y Fu(or)32 b(in)g(this)g(w)m(a)m(y)-8
b(,)1390 4881 y Fr(b)1431 4896 y Fp(1)1498 4881 y Fu(=)1656
4756 y Fp(3)1602 4786 y Fi(X)1617 4996 y Fo(i)p Fp(=1)1772
4813 y Fr(m)p Fu(\()p Fr(I)1938 4828 y Fo(i)1989 4813
y Fq(\\)22 b Fr(S)2143 4777 y Fl(\000)p Fp(1)2238 4813
y Fu(\()p Fr(I)2319 4828 y Fp(1)2358 4813 y Fu(\))p 1772
4858 624 4 v 1968 4949 a Fr(m)p Fu(\()p Fr(I)2134 4964
y Fo(i)2162 4949 y Fu(\))2428 4881 y Fq(\001)g Fr(a)2529
4896 y Fo(i)2557 4881 y Fr(:)p Black 1055 w Fu(\(3.1\))p
Black 300 5159 a(By)33 b(the)g(same)g(tok)m(en,)1390
5439 y Fr(b)1431 5454 y Fp(2)1498 5439 y Fu(=)1656 5315
y Fp(3)1602 5345 y Fi(X)1617 5554 y Fo(i)p Fp(=1)1772
5372 y Fr(m)p Fu(\()p Fr(I)1938 5387 y Fo(i)1989 5372
y Fq(\\)22 b Fr(S)2143 5336 y Fl(\000)p Fp(1)2238 5372
y Fu(\()p Fr(I)2319 5387 y Fp(2)2358 5372 y Fu(\))p 1772
5416 V 1968 5508 a Fr(m)p Fu(\()p Fr(I)2134 5523 y Fo(i)2162
5508 y Fu(\))2428 5439 y Fq(\001)g Fr(a)2529 5454 y Fo(i)2557
5439 y Fr(;)p Black 1055 w Fu(\(3.2\))p Black Black 2021
5764 a(12)p Black eop
%%Page: 13 22
13 21 bop Black 300 10 a Fk(CHAPTER)34 b(3.)76 b(ULAM'S)33
b(METHOD)1907 b Fu(13)p Black 1390 352 a Fr(b)1431 367
y Fp(3)1498 352 y Fu(=)1656 227 y Fp(3)1602 257 y Fi(X)1617
467 y Fo(i)p Fp(=1)1772 285 y Fr(m)p Fu(\()p Fr(I)1938
300 y Fo(i)1989 285 y Fq(\\)22 b Fr(S)2143 248 y Fl(\000)p
Fp(1)2238 285 y Fu(\()p Fr(I)2319 300 y Fp(3)2358 285
y Fu(\))p 1772 329 624 4 v 1968 420 a Fr(m)p Fu(\()p
Fr(I)2134 435 y Fo(i)2162 420 y Fu(\))2428 352 y Fq(\001)g
Fr(a)2529 367 y Fo(i)2557 352 y Fr(:)p Black 1055 w Fu(\(3.3\))p
Black 446 653 a(In)31 b(general,)f(if)e([0)p Fr(;)17
b Fu(1])30 b(is)f(divided)h(in)m(to)f Fr(n)h Fu(subin)m(terv)-5
b(als)30 b Fr(I)2537 668 y Fp(1)2577 653 y Fr(;)17 b(I)2664
668 y Fp(2)2703 653 y Fr(;)g Fq(\001)g(\001)g(\001)31
b Fr(;)17 b(I)2983 668 y Fo(n)3059 653 y Fu(and)31 b(if)d
Fr(f)41 b Fu(is)29 b(a)h(piece-)300 774 y(wise)i(constan)m(t)h(densit)m
(y)g(suc)m(h)g(that)f(the)h(probabilit)m(y)d(of)h Fr(I)2489
789 y Fo(i)2549 774 y Fu(is)h Fr(a)2698 789 y Fo(i)2726
774 y Fu(,)g(then)h(the)f(new)h(probabilit)m(y)300 894
y(of)f Fr(I)454 909 y Fo(j)523 894 y Fu(is)g Fr(b)662
909 y Fo(j)732 894 y Fu(with)1374 1186 y Fr(b)1415 1201
y Fo(j)1479 1186 y Fu(=)1633 1062 y Fo(n)1583 1091 y
Fi(X)1598 1301 y Fo(i)p Fp(=1)1753 1119 y Fr(m)p Fu(\()p
Fr(I)1919 1134 y Fo(i)1970 1119 y Fq(\\)22 b Fr(S)2124
1082 y Fl(\000)p Fp(1)2219 1119 y Fu(\()p Fr(I)2300 1134
y Fo(j)2336 1119 y Fu(\)\))p 1753 1163 659 4 v 1966 1254
a Fr(m)p Fu(\()p Fr(I)2132 1269 y Fo(i)2161 1254 y Fu(\))2444
1186 y Fq(\001)g Fr(a)2545 1201 y Fo(i)2573 1186 y Fr(:)p
Black 1039 w Fu(\(3.4\))p Black 300 1487 a(Let)35 b Fr(a)c
Fu(=)f([)p Fr(a)743 1502 y Fp(1)783 1487 y Fr(;)17 b(a)878
1502 y Fp(2)917 1487 y Fr(;)g Fq(\001)g(\001)g(\001)31
b Fr(;)17 b(a)1205 1502 y Fo(n)1252 1487 y Fu(])35 b(and)f
Fr(b)e Fu(=)e([)p Fr(b)1752 1502 y Fp(1)1792 1487 y Fr(;)17
b(b)1877 1502 y Fp(2)1917 1487 y Fr(;)g Fq(\001)g(\001)g(\001)31
b Fr(;)17 b(b)2195 1502 y Fo(n)2242 1487 y Fu(],)35 b(and)g(let)e(the)i
Fr(n)24 b Fq(\002)g Fr(n)34 b Fu(matrix)f Fr(P)3492 1502
y Fo(n)3570 1487 y Fu(=)d([)p Fr(p)3752 1502 y Fo(ij)3813
1487 y Fu(])300 1608 y(b)s(e)j(de\014ned)h(b)m(y)1514
1875 y Fr(p)1563 1890 y Fo(ij)1651 1875 y Fu(=)1764 1807
y Fr(m)p Fu(\()p Fr(I)1930 1822 y Fo(i)1981 1807 y Fq(\\)23
b Fr(S)2136 1771 y Fl(\000)p Fp(1)2230 1807 y Fu(\()p
Fr(I)2311 1822 y Fo(j)2347 1807 y Fu(\)\))p 1764 1852
V 1978 1943 a Fr(m)p Fu(\()p Fr(I)2144 1958 y Fo(i)2172
1943 y Fu(\))2433 1875 y Fr(:)p Black 1179 w Fu(\(3.5\))p
Black 300 2138 a(Then)34 b(w)m(e)f(ha)m(v)m(e)h Fr(b)28
b Fu(=)g Fr(aP)1210 2153 y Fo(n)1257 2138 y Fu(.)43 b(Since)33
b Fr(P)1645 2153 y Fo(n)1725 2138 y Fu(is)f(a)g(nonnegativ)m(e)h
(matrix)e(and)1205 2297 y Fo(n)1155 2327 y Fi(X)1165
2537 y Fo(j)t Fp(=1)1315 2422 y Fr(p)1364 2437 y Fo(ij)1452
2422 y Fu(=)1606 2297 y Fo(n)1556 2327 y Fi(X)1566 2537
y Fo(j)t Fp(=1)1726 2354 y Fr(m)p Fu(\()p Fr(I)1892 2369
y Fo(i)1943 2354 y Fq(\\)22 b Fr(S)2097 2318 y Fl(\000)p
Fp(1)2192 2354 y Fu(\()p Fr(I)2273 2369 y Fo(j)2309 2354
y Fu(\)\))p 1726 2399 V 1939 2490 a Fr(m)p Fu(\()p Fr(I)2105
2505 y Fo(i)2134 2490 y Fu(\))2423 2422 y(=)2536 2354
y Fr(m)p Fu(\()p Fr(I)2702 2369 y Fo(i)2730 2354 y Fu(\))p
2536 2399 233 4 v 2536 2490 a Fr(m)p Fu(\()p Fr(I)2702
2505 y Fo(i)2730 2490 y Fu(\))2806 2422 y(=)28 b(1)p
Fr(;)300 2730 y Fu(so)46 b Fr(P)496 2745 y Fo(n)588 2730
y Fu(is)f(a)g(sto)s(c)m(hastic)g(matrix.)81 b(By)46 b(the)f(F)-8
b(rob)s(enius-P)m(erron)46 b(theorem)f(for)g(nonnegativ)m(e)300
2851 y(matrices,)52 b(there)d(is)f(a)g(nonnegativ)m(e)h(v)m(ector)h
Fr(c)2123 2815 y Fo(T)2233 2851 y Fu(=)k(\()p Fr(c)2443
2866 y Fp(1)2483 2851 y Fr(;)17 b(c)2569 2866 y Fp(2)2608
2851 y Fr(;)g Fq(\001)g(\001)g(\001)31 b Fr(;)17 b(c)2887
2866 y Fo(n)2934 2851 y Fu(\),)52 b(the)d(sum)g(of)f(whose)300
2971 y(comp)s(onen)m(ts)33 b(is)f(1,)h(suc)m(h)h(that)1245
3185 y(\()p Fr(c)1325 3200 y Fp(1)1364 3185 y Fr(;)17
b(c)1450 3200 y Fp(2)1489 3185 y Fr(;)g Fq(\001)g(\001)g(\001)31
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%%Page: 14 23
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%%Page: 15 24
15 23 bop Black 300 10 a Fk(CHAPTER)34 b(3.)76 b(ULAM'S)33
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(er)i(Ulam's)e(sc)m(heme:)45 b(divide)300 2744 y(the)36
b(in)m(terv)-5 b(al)35 b([0)p Fr(;)17 b Fu(1])35 b(in)m(to)g
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2759 y Fo(i)2260 2744 y Fu(=)d([)p Fr(x)2451 2759 y Fo(i)p
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2744 y Fu(])36 b(with)g(the)g(length)f Fr(h)f Fu(=)f(1)p
Fr(=n)p Fu(.)300 2864 y(De\014ne)1283 3078 y Fr(\036)1341
3093 y Fo(i)1397 3078 y Fu(=)27 b Fr(\037)1561 3093 y
Fo(I)1592 3103 y Ff(i)1622 3078 y Fr(;)45 b( )1757 3093
y Fo(i)1813 3078 y Fu(=)27 b(1)1965 3093 y Fo(i)1993
3078 y Fr(;)45 b(i)28 b Fu(=)f(1)p Fr(;)17 b Fu(2)p Fr(;)g(:)g(:)g(:)32
b(;)17 b(n:)p Black 900 w Fu(\(3.11\))p Black 446 3292
a(Then,)40 b(a)e(simple)e(calculation)f(giv)m(es)j(that)g(the)g(\()p
Fr(i;)17 b(j)6 b Fu(\))37 b(elemen)m(t)h(of)f(the)h(matrix)e
Fr(A)i Fu(in)f(the)300 3413 y(general)32 b(algorithm)d(is)300
3686 y Fr(a)351 3701 y Fo(ij)495 3686 y Fu(=)654 3551
y Fi(Z)753 3577 y Fp(1)709 3776 y(0)793 3686 y Fu(\()p
Fr(\037)892 3701 y Fo(I)923 3711 y Ff(i)953 3686 y Fu(\()p
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3711 y Ff(i)1328 3686 y Fu(\()p Fr(S)6 b Fu(\()p Fr(x)p
Fu(\)\)\)1)1688 3701 y Fo(j)1724 3686 y Fu(\()p Fr(x)p
Fu(\))p Fr(dx)28 b Fu(=)2093 3551 y Fi(Z)2193 3577 y
Fp(1)2148 3776 y(0)2249 3686 y Fr(\037)2310 3701 y Fo(I)2341
3711 y Ff(i)2371 3686 y Fu(\()p Fr(x)p Fu(\)1)2551 3701
y Fo(j)2587 3686 y Fu(\()p Fr(x)p Fu(\))p Fr(dx)23 b
Fq(\000)2947 3551 y Fi(Z)3046 3577 y Fp(1)3002 3776 y(0)3102
3686 y Fr(\037)3163 3703 y Fo(S)3210 3684 y Fh(\000)p
Fg(1)3293 3703 y Fp(\()p Fo(I)3351 3713 y Ff(i)3377 3703
y Fp(\))3409 3686 y Fu(\()p Fr(x)p Fu(\)1)3589 3701 y
Fo(j)3625 3686 y Fu(\()p Fr(x)p Fu(\))p Fr(dx)495 3956
y Fu(=)83 b Fr(n)729 3820 y Fi(Z)784 4046 y Fo(I)815
4056 y Ff(i)841 4046 y Fl(\\)p Fo(I)919 4056 y Ff(j)972
3956 y Fr(dx)22 b Fq(\000)h Fr(n)1275 3820 y Fi(Z)1330
4046 y Fo(S)1377 4027 y Fh(\000)p Fg(1)1459 4046 y Fp(\()p
Fo(I)1517 4056 y Ff(i)1543 4046 y Fp(\))p Fl(\\)p Fo(I)1648
4056 y Ff(j)1702 3956 y Fr(dx)28 b Fu(=)f Fr(\016)1982
3971 y Fo(ij)2065 3956 y Fq(\000)2175 3888 y Fr(m)p Fu(\()p
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3903 y Fo(i)2567 3888 y Fu(\))22 b Fq(\\)h Fr(I)2759
3903 y Fo(j)2795 3888 y Fu(\))p 2175 3933 659 4 v 2383
4024 a Fr(m)p Fu(\()p Fr(I)2549 4039 y Fo(j)2586 4024
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4252 y Fo(ij)721 4237 y Fu(is)e(the)i(Kronec)m(k)m(er)h(sym)m(b)s(ol,)d
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Fu(=)c(0)300 4357 y(if)h(and)i(only)f(if)g Fr(v)934 4321
y Fo(T)1016 4357 y Fu(=)c Fr(v)1171 4321 y Fo(T)1225
4357 y Fr(P)1288 4372 y Fo(n)1335 4357 y Fu(,)33 b(where)868
4625 y Fr(P)931 4640 y Fo(n)1006 4625 y Fu(=)27 b([)p
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4640 y Fo(ij)1509 4625 y Fu(=)1622 4557 y Fr(m)p Fu(\()p
Fr(I)1788 4572 y Fo(i)1839 4557 y Fq(\\)23 b Fr(S)1994
4521 y Fl(\000)p Fp(1)2088 4557 y Fu(\()p Fr(I)2169 4572
y Fo(j)2205 4557 y Fu(\)\))p 1622 4602 V 1836 4693 a
Fr(m)p Fu(\()p Fr(I)2002 4708 y Fo(i)2030 4693 y Fu(\))2291
4625 y Fr(;)72 b(i;)17 b(j)34 b Fu(=)27 b(1)p Fr(;)17
b Fu(2)p Fr(;)g(:)g(:)g(:)32 b(;)17 b(n:)p Black 485
w Fu(\(3.12\))p Black 300 4895 a(\(3.12\))32 b(is)g(exactly)h(the)g
(matrix)e(\(3.5\))h(in)g(Ulam's)f(metho)s(d.)636 5015
y(F)-8 b(or)43 b(m)m(ulti-dimensional)d(cases,)48 b(let)c(\012)k
Fq(\032)g Fr(R)2386 4979 y Fo(d)2471 5015 y Fu(b)s(e)d(a)f(b)s(ounded)h
(op)s(en)f(set)h(and)g(let)300 5136 y Fr(S)33 b Fu(:)28
b(\012)g Fq(!)g Fu(\012)j(b)s(e)g(a)f(nonsingular)g(transformation)e
(suc)m(h)33 b(that)d(the)h(corresp)s(onding)g(F)-8 b(rob)s(enius-)300
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b(a)f(non)m(trivial)f(\014xed)j(p)s(oin)m(t.)85 b(Let)47
b Fr(T)2736 5271 y Fo(h)2832 5256 y Fu(=)52 b Fq(f)p
Fu(\012)3080 5271 y Fo(i)3109 5256 y Fq(g)3159 5220 y
Fo(n)3159 5281 y(i)p Fp(=1)3323 5256 y Fu(b)s(e)47 b(a)g(shap)s(e-)300
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5391 y Fo(i)3227 5376 y Fu(=)3422 5337 y Fp(1)p 3343
5353 195 4 v 3343 5411 a Fo(m)p Fp(\(\012)3483 5421 y
Ff(i)3510 5411 y Fp(\))3547 5376 y Fr(\037)3608 5391
y Fp(\012)3659 5401 y Ff(i)3723 5376 y Fu(for)300 5515
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b Fr(;)17 b(n)p Fu(.)44 b(Then)34 b(eac)m(h)f(1)1403
5530 y Fo(i)1459 5515 y Fq(2)c Fr(L)1620 5478 y Fp(1)1660
5515 y Fu(\(\012\))k(is)f(a)g(densit)m(y)i(function)e(with)h(supp1)3209
5530 y Fo(i)3266 5515 y Fu(=)27 b(\012)3439 5530 y Fo(i)3468
5515 y Fu(.)44 b(Let)33 b(\001)3795 5530 y Fo(h)p Black
Black eop
%%Page: 16 25
16 24 bop Black 300 10 a Fk(CHAPTER)34 b(3.)76 b(ULAM'S)33
b(METHOD)1907 b Fu(16)p Black 300 274 a(b)s(e)31 b(the)g
Fr(n)p Fu(-dimensional)d(subspace)33 b(of)d Fr(L)1804
238 y Fp(1)1844 274 y Fu(\(\012\))h(spanned)h(b)m(y)g(1)2582
289 y Fp(1)2621 274 y Fr(;)17 b(:)g(:)g(:)32 b(;)17 b
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289 y Fo(h)3349 274 y Fu(is)g(the)g(space)300 395 y(of)h(piecewise)h
(constan)m(t)g(functions)g(asso)s(ciated)f(with)g Fr(T)2391
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395 y Fu(\(\012\).)43 b(De\014ne)300 515 y Fr(P)363 530
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Fu(1)1795 802 y Fo(i)1850 787 y Fu(=)2004 662 y Fo(n)1954
692 y Fi(X)1964 902 y Fo(j)t Fp(=1)2114 787 y Fr(p)2163
802 y Fo(ij)2224 787 y Fu(1)2273 802 y Fo(j)2309 787
y Fr(;)p Black 1255 w Fu(\(3.13\))p Black 300 1078 a(where)1486
1307 y Fr(p)1535 1322 y Fo(ij)1624 1307 y Fu(=)1737 1240
y Fr(m)p Fu(\(\012)1930 1255 y Fo(i)1981 1240 y Fq(\\)c
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1376 a Fr(m)p Fu(\(\012)2157 1391 y Fo(i)2186 1376 y
Fu(\))2460 1307 y Fr(:)p Black 1104 w Fu(\(3.14\))p Black
446 1553 a(Ulam's)37 b(metho)s(d)g(is)h(to)f(compute)h(a)g(\014xed)h
(densit)m(y)g Fr(f)2487 1568 y Fo(h)2569 1553 y Fq(2)e
Fu(\001)2753 1568 y Fo(h)2836 1553 y Fu(of)g Fr(P)3015
1568 y Fo(h)3098 1553 y Fu(to)g(appro)m(ximate)g(a)300
1674 y(\014xed)j(densit)m(y)f Fr(f)943 1638 y Fl(\003)1020
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m(ergence)i(of)d(Ulam's)f(metho)s(d)h(as)g Fr(h)g Fq(!)f
Fu(0,)j(w)m(e)300 1794 y(need)34 b(to)e(de\014ne)i(a)e(discretized)h
(op)s(erator)f Fr(Q)1963 1809 y Fo(h)2035 1794 y Fu(:)c
Fr(L)2156 1758 y Fp(1)2196 1794 y Fu(\(\012\))g Fq(!)f
Fr(L)2563 1758 y Fp(1)2603 1794 y Fu(\(\012\))33 b(b)m(y)1628
2066 y Fr(Q)1705 2081 y Fo(h)1750 2066 y Fr(f)38 b Fu(=)1990
1941 y Fo(n)1940 1971 y Fi(X)1955 2181 y Fo(i)p Fp(=1)2100
2066 y Fr(f)2148 2081 y Fo(i)2177 2066 y Fr(\037)2238
2081 y Fp(\012)2289 2091 y Ff(i)2319 2066 y Fr(;)p Black
1245 w Fu(\(3.15\))p Black 300 2344 a(where)1561 2563
y Fr(f)1609 2578 y Fo(i)1665 2563 y Fu(=)1884 2496 y(1)p
1778 2540 260 4 v 1778 2631 a Fr(m)p Fu(\(\012)1971 2646
y Fo(i)2000 2631 y Fu(\))2065 2427 y Fi(Z)2120 2653 y
Fp(\012)2171 2663 y Ff(i)2218 2563 y Fr(f)11 b(dm)p Black
1178 w Fu(\(3.16\))p Black 300 2816 a(is)35 b(the)h(a)m(v)m(erage)g(v)
-5 b(alue)35 b(of)g Fr(f)46 b Fu(o)m(v)m(er)36 b(\012)1668
2831 y Fo(i)1697 2816 y Fu(.)52 b(It)35 b(is)g(ob)m(vious)g(that)h
Fr(Q)2632 2831 y Fo(h)2712 2816 y Fu(is)f(a)g(Mark)m(o)m(v)i(op)s
(erator)d(with)300 2936 y(dim)p Fr(R)q Fu(\()p Fr(Q)654
2951 y Fo(h)698 2936 y Fu(\))27 b(=)h Fr(n)p Fu(.)44
b(Moreo)m(v)m(er)34 b Fr(Q)1501 2951 y Fo(h)1578 2936
y Fu(is)e(a)h(Galerkin)e(pro)5 b(jection)32 b(on)m(to)g(\001)2920
2951 y Fo(h)2998 2936 y Fu(in)g(the)h(sense)h(that)1280
3132 y Fr(<)28 b(Q)1461 3147 y Fo(h)1506 3132 y Fr(f)k
Fq(\000)23 b Fr(f)5 b(;)17 b Fu(1)1832 3147 y Fo(i)1888
3132 y Fr(>)p Fu(=)27 b(0)p Fr(;)44 b(i)28 b Fu(=)g(1)p
Fr(;)17 b(:)g(:)g(:)32 b(;)17 b(n)p Black 897 w Fu(\(3.17\))p
Black 300 3328 a(for)32 b(all)f Fr(f)38 b Fq(2)28 b Fr(L)831
3291 y Fp(1)871 3328 y Fu(\(\012\).)44 b(Also)32 b(it)f(is)h(easy)i(to)
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b(k)p Fr(f)11 b Fq(k)2853 3343 y Fl(1)2960 3328 y Fu(for)32
b(an)m(y)h Fr(f)38 b Fq(2)28 b Fr(L)3539 3291 y Fl(1)3615
3328 y Fu(\(\012\).)446 3448 y(Some)h(basic)f(prop)s(erties)h(of)f
(Ulam's)g(metho)s(d)g(are)h(summarized)f(in)g(the)h(follo)m(wing)d
(prop)s(o-)300 3568 y(sition.)p Black 300 3720 a Fj(Pr)-5
b(op)g(osition)34 b(3.1.)p Black 48 w Fu(\(i\))k(Let)g(\001)1429
3684 y Fl(0)1429 3746 y Fo(h)1512 3720 y Fu(=)f Fq(f)1675
3645 y Fi(P)1780 3671 y Fo(n)1780 3749 y(i)p Fp(=1)1915
3720 y Fr(a)1966 3735 y Fo(i)1994 3720 y Fu(1)2043 3735
y Fo(i)2109 3720 y Fu(:)h Fr(a)2225 3735 y Fo(i)2291
3720 y Fq(\025)g Fu(0)p Fr(;)2499 3645 y Fi(P)2603 3671
y Fo(n)2603 3749 y(i)p Fp(=1)2738 3720 y Fr(a)2789 3735
y Fo(i)2855 3720 y Fu(=)f(1)p Fq(g)p Fu(.)61 b(Then)39
b Fr(P)3478 3735 y Fo(h)3523 3720 y Fu(\(\001)3642 3684
y Fl(0)3642 3746 y Fo(h)3687 3720 y Fu(\))f Fq(\032)300
3840 y Fu(\001)381 3804 y Fl(0)381 3866 y Fo(h)426 3840
y Fu(.)44 b(Hence,)34 b(there)f(is)f(a)g(\014xed)i(densit)m(y)g
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y Fo(h)2159 3840 y Fu(.)446 3961 y(\(ii\))f(lim)744 3976
y Fo(h)p Fl(!)p Fp(0)912 3961 y Fr(Q)989 3976 y Fo(h)1034
3961 y Fr(f)38 b Fu(=)28 b Fr(f)43 b Fu(for)32 b(an)m(y)h
Fr(f)39 b Fq(2)28 b Fr(L)1895 3924 y Fp(1)1935 3961 y
Fu(\(\012\).)446 4081 y(\(iii\))i Fr(P)699 4096 y Fo(h)771
4081 y Fu(=)e Fr(Q)952 4096 y Fo(h)997 4081 y Fr(P)46
b Fu(on)32 b(\001)1322 4096 y Fo(h)1368 4081 y Fu(.)300
4232 y Fm(Pro)s(of.)41 b Fu(\(i\))24 b Fr(P)835 4247
y Fo(h)904 4232 y Fu(is)h(a)f(Mark)m(o)m(v)j(op)s(erator)d(since)h(b)s
(oth)g Fr(Q)2332 4247 y Fo(h)2402 4232 y Fu(and)g Fr(P)39
b Fu(are,)26 b(so)f Fr(P)3044 4247 y Fo(h)3089 4232 y
Fu(\(\001)3208 4196 y Fl(0)3208 4258 y Fo(h)3253 4232
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y Fu(,)g(b)m(y)i(Brou)m(w)m(er's)g(\014xed)f(p)s(oin)m(t)f(theorem,)g
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b(some)g Fr(f)741 4488 y Fo(h)814 4473 y Fq(2)c Fu(\001)989
4437 y Fl(0)989 4499 y Fo(h)1034 4473 y Fu(.)446 4594
y(\(ii\))i(It)i(is)g(enough)g(to)f(assume)i(that)e Fr(f)39
b Fq(2)28 b Fr(C)7 b Fu(\()2118 4568 y(\026)2107 4594
y(\012\).)43 b(Then)33 b(from)e(the)h(uniform)e(con)m(tin)m(uit)m(y)i
(of)300 4714 y Fr(f)11 b Fu(,)32 b(for)g(an)m(y)i Fr(\017)28
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4927 y Fq(j)p Fr(f)1650 4942 y Fo(i)1700 4927 y Fq(\000)23
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Black 3591 4927 a(\(3.18\))p Black 300 5173 a(for)32
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Fu(\(\012)2858 5454 y Fo(i)2887 5439 y Fu(\))g(=)f Fr(\017:)p
Black 469 w Fu(\(3.19\))p Black Black Black eop
%%Page: 17 26
17 25 bop Black 300 10 a Fk(CHAPTER)34 b(3.)76 b(ULAM'S)33
b(METHOD)1907 b Fu(17)p Black 300 274 a(\(iii\))30 b(F)-8
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y Fi(#)3178 547 y Fu(1)3227 562 y Fo(j)3264 547 y Fr(m)p
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814 y Fr(m)p Fu(\(\012)1398 829 y Fo(i)1449 814 y Fq(\\)23
b Fr(S)1604 778 y Fl(\000)p Fp(1)1698 814 y Fu(\(\012)1806
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787 y Fi(X)2156 997 y Fo(j)t Fp(=1)2306 881 y Fr(p)2355
896 y Fo(ij)2415 881 y Fu(1)2464 896 y Fo(j)2501 881
y Fr(:)p 2611 856 57 4 v 2611 906 4 50 v 2664 906 V 2611
909 57 4 v 446 1162 a Fu(W)-8 b(e)38 b(also)f(use)i Fr(P)1057
1177 y Fo(h)1139 1162 y Fu(to)e(denote)i(the)f(matrix)e(represen)m
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(whic)m(h)f(is)e(also)h(called)f Fj(Ulam's)j(matrix)e
Fu(or)g(the)h(companion)d(matrix.)300 1403 y(Since)k
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%%Page: 18 27
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%%Page: 19 28
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%%Page: 24 33
24 32 bop Black 300 10 a Fk(CHAPTER)34 b(4.)76 b(PIECEWISE)35
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%%Page: 25 34
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(\014xed,)g(then)f(w)m(e)h(can)f(get)f(the)h(relation)e(b)s(et)m(w)m
(een)k Fr(E)j Fu(and)33 b Fr(p)f Fu(as)h(follo)m(ws.)758
3206 y Fr(E)6 b Fu(\()p Fr(p)p Fu(\))83 b(=)1937 3139
y Fr(n)1995 3102 y Fp(2)2035 3139 y Fu(\()p Fr(t)2108
3154 y Fo(mc)2228 3139 y Fu(+)22 b Fr(M)10 b(t)2465 3154
y Fo(pr)r(od)2609 3139 y Fu(\))p 1213 3183 2159 4 v 1213
3274 a Fr(pl)r(og)1387 3289 y Fp(2)1426 3274 y Fr(p)p
Fu(\()p Fr(t)1548 3289 y Fo(s)1607 3274 y Fu(+)23 b Fr(nt)1799
3289 y Fo(w)1856 3274 y Fu(\))f(+)g Fr(pn)2121 3246 y
Fp(2)2160 3274 y Fu(\()p Fr(t)2233 3289 y Fo(w)2312 3274
y Fu(+)g Fr(M)10 b(t)2549 3289 y Fo(pr)r(od)2694 3274
y Fu(\))22 b(+)g Fr(n)2910 3246 y Fp(2)2950 3274 y Fu(\()p
Fr(t)3023 3289 y Fo(mc)3142 3274 y Fq(\000)h Fr(t)3277
3289 y Fo(w)3334 3274 y Fu(\))1044 3471 y(=)1554 3403
y Fr(K)1637 3418 y Fo(n)p 1213 3448 811 4 v 1213 3539
a Fr(pl)r(og)1387 3554 y Fp(2)1426 3539 y Fr(p)g Fu(+)f
Fr(pK)1735 3505 y Fp(1)1728 3567 y Fo(d)1796 3539 y Fu(+)g
Fr(K)1984 3505 y Fp(2)1977 3567 y Fo(d)2034 3471 y Fr(;)300
3744 y Fu(where)31 b(a)e(series)h(of)g Fr(K)7 b Fu(s)30
b(are)f(di\013eren)m(t)h(constan)m(ts.)44 b(F)-8 b(rom)28
b(the)i(form)m(ula)e(w)m(e)j(can)f(de\014nitely)f(sa)m(y)300
3864 y(that)j(for)g(the)h(UMP)-8 b(A,)34 b(when)g Fr(p)e
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y(If)i Fr(p)f Fu(is)g(k)m(ept)i(\014xed,)g(w)m(e)g(can)e(get)h(the)g
(relation)e(b)s(et)m(w)m(een)j Fr(E)39 b Fu(and)33 b
Fr(n)f Fu(as)h(follo)m(ws.)706 4259 y Fr(E)6 b Fu(\()p
Fr(n)p Fu(\))84 b(=)1942 4191 y Fr(n)2000 4155 y Fp(2)2040
4191 y Fu(\()p Fr(t)2113 4206 y Fo(mc)2232 4191 y Fu(+)22
b Fr(M)10 b(t)2469 4206 y Fo(pr)r(od)2614 4191 y Fu(\))p
1171 4236 2254 4 v 1171 4327 a Fr(n)1229 4298 y Fp(2)1268
4327 y Fu(\()p Fr(t)1341 4342 y Fo(mc)1461 4327 y Fu(+)22
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b Fr(t)2047 4342 y Fo(w)2103 4327 y Fu(\()p Fr(p)g Fq(\000)h
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4342 y Fp(2)2920 4327 y Fr(p)g Fu(+)g Fr(t)3124 4342
y Fo(s)3161 4327 y Fr(pl)r(og)3335 4342 y Fp(2)3375 4327
y Fr(p)1002 4534 y Fu(=)1391 4467 y Fr(n)1449 4431 y
Fp(2)1488 4467 y Fr(E)1560 4482 y Fo(u)p 1171 4511 656
4 v 1171 4603 a Fr(n)1229 4574 y Fp(2)1290 4603 y Fu(+)g
Fr(nK)1536 4568 y Fp(1)1529 4630 y Fo(d)1598 4603 y Fu(+)g
Fr(K)1786 4568 y Fp(2)1779 4630 y Fo(d)1836 4534 y Fr(:)300
4807 y Fu(F)-8 b(rom)45 b(the)i(form)m(ula)e(w)m(e)i(kno)m(w)h(that)e
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e(complicated)f(b)s(eha)m(vior.)60 b(That)39 b(means)f(that)h
Fr(E)44 b Fu(ma)m(y)38 b(decrease,)k(and)300 5048 y(also)47
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g(the)g(parameters)g(of)g(the)300 5169 y(parallel)30
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5431 y Fr(E)1365 5446 y Fo(u)1438 5431 y Fu(=)1829 5364
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5500 a Fr(t)1586 5515 y Fo(mc)1706 5500 y Fu(+)22 b Fr(M)10
b(pt)1992 5515 y Fo(pr)r(od)2159 5500 y Fu(+)22 b Fr(t)2292
5515 y Fo(w)2349 5500 y Fu(\()p Fr(p)g Fq(\000)g Fu(1\))2654
5431 y Fr(:)p Black 910 w Fu(\(6.15\))p Black Black Black
eop
%%Page: 34 43
34 42 bop Black 300 10 a Fk(CHAPTER)34 b(6.)76 b(P)-8
b(ARALLEL)33 b(ALGORITHMS)1528 b Fu(34)p Black 446 274
a(No)m(w)42 b(w)m(e)g(tak)m(e)g(a)f(lo)s(ok)f(at)h(the)h(equation)e
(\(6.15\).)69 b(When)42 b(the)g(pro)s(cessor)g(n)m(um)m(b)s(er)f
Fr(p)g Fu(is)300 395 y(1,)i Fr(E)491 410 y Fo(u)577 395
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b Fr(p)g Fu(go)s(es)h(to)e(in\014nit)m(y)-8 b(,)43 b
Fr(E)3602 410 y Fo(u)3688 395 y Fu(will)300 515 y(approac)m(h)d(0.)62
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b(the)f(parallel)e(time)g Fr(T)3116 530 y Fo(par)r(a)3304
515 y Fu(of)i(\(6.4\))f(con-)300 635 y(tains)31 b(a)g(sequen)m(tial)g
(part)h Fr(M)10 b(n)1445 599 y Fp(2)1485 635 y Fr(t)1520
650 y Fo(pr)r(od)1696 635 y Fu(whic)m(h)31 b(will)e(de\014nitely)j
(degrade)g(the)f(e\016ciency)i(when)g Fr(n)300 756 y
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(c)m(hapter)g(will)d(illustrate)g(this)h(conclusion.)446
876 y(F)-8 b(rom)31 b Fr(T)759 891 y Fo(par)r(a)941 876
y Fu(of)h(equation)g(\(6.4\),)g(w)m(e)i(obtain)d(the)i(cost)g(of)f(the)
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1111 y Fo(par)r(a)1085 1096 y Fu(=)g Fr(n)1246 1055 y
Fp(2)1286 1096 y Fu(\()p Fr(t)1359 1111 y Fo(mc)1478
1096 y Fu(+)22 b Fr(t)1611 1111 y Fo(w)1668 1096 y Fu(\()p
Fr(p)g Fq(\000)h Fu(1\)\))f(+)g(\()p Fr(t)2195 1111 y
Fo(s)2254 1096 y Fu(+)g Fr(nt)2445 1111 y Fo(w)2502 1096
y Fu(\))p Fr(pl)r(og)2714 1111 y Fp(2)2753 1096 y Fr(p)g
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1615 y Fp(2)1514 1657 y Fr(t)1549 1672 y Fo(mc)1668 1657
y Fu(+)22 b Fr(M)10 b(t)1905 1672 y Fo(pr)r(od)2050 1657
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1657 y Fu(+)22 b Fr(M)10 b(t)2806 1672 y Fo(pr)r(od)2951
1657 y Fu(\))p Fr(:)575 b Fu(\(6.17\))300 1877 y(Considering)32
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2241 a Fr(n)1187 2213 y Fp(2)1227 2241 y Fr(t)1262 2256
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2241 y Fu(+)22 b Fr(t)2351 2256 y Fo(w)2408 2241 y Fu(\))p
Fr(pl)r(og)2620 2256 y Fp(2)2659 2241 y Fr(p)g Fu(+)g
Fr(t)2863 2256 y Fo(w)2920 2241 y Fr(M)10 b Fu(\()p Fr(p)23
b Fq(\000)f Fu(1\))p Fr(n)3388 2173 y(:)p Black 176 w
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2738 y Fr(E)698 2673 y Fh(0)752 2738 y Fu(=)866 2671
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2738 y Fu(=)1848 2671 y Fr(n)1906 2634 y Fp(2)1946 2671
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1089 2715 2228 4 v 1089 2806 a Fr(n)1147 2778 y Fp(2)1187
2806 y Fu(\()p Fr(t)1260 2821 y Fo(mc)1380 2806 y Fu(+)22
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Fr(pl)r(og)2558 2821 y Fp(2)2597 2806 y Fr(p)h Fu(+)f
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b Fq(\000)g Fu(1\))p Fr(n)3327 2738 y(:)p Black 237 w
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(\014xed,)g(w)m(e)g(can)e(get)h(the)g(relation)e(b)s(et)m(w)m(een)j
Fr(E)2531 2979 y Fl(0)2587 3015 y Fu(and)f Fr(p)f Fu(as)h(follo)m(ws.)
647 3289 y Fr(E)725 3248 y Fl(0)749 3289 y Fu(\()p Fr(p)p
Fu(\))82 b(=)1949 3221 y Fr(n)2007 3185 y Fp(2)2047 3221
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1125 3266 2358 4 v 1125 3357 a Fr(pl)r(og)1299 3372 y
Fp(2)1339 3357 y Fr(pM)g Fu(\()p Fr(t)1565 3372 y Fo(s)1624
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Fr(n)2454 3328 y Fp(2)2494 3357 y Fu(\()p Fr(t)2567 3372
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3501 y Fo(d)p 1125 3531 811 4 v 1125 3622 a Fr(pl)r(og)1299
3637 y Fp(2)1339 3622 y Fr(p)22 b Fu(+)g Fr(pK)1647 3588
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3588 y Fp(2)1889 3650 y Fo(d)1946 3554 y Fr(:)300 3831
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y Fl(0)2578 3951 y Fu(and)e Fr(n)f Fu(as)h(follo)m(ws.)708
4225 y Fr(E)786 4184 y Fl(0)809 4225 y Fu(\()p Fr(n)p
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4308 y Fo(s)2932 4293 y Fu(+)22 b Fr(t)3065 4308 y Fo(w)3122
4293 y Fu(\))p Fr(pl)r(og)3334 4308 y Fp(2)3373 4293
y Fr(p)1026 4501 y Fu(=)1474 4433 y Fr(n)1532 4397 y
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(that)e(with)g(the)i(increase)f(of)f(the)i(problem)d(size)i
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(of)f(the)300 5033 y(parallel)30 b(computers.)44 b(When)33
b Fr(n)g Fu(go)s(es)g(to)f(in\014nit)m(y)-8 b(,)32 b
Fr(E)2299 4973 y Fh(0)2358 5033 y Fu(will)e Fj(eventual)5
b(ly)35 b(incr)-5 b(e)g(ase)66 b Fu(to)33 b(1.)446 5153
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-8 b(A)34 b(is)550 5373 y Fr(C)627 5308 y Fh(0)681 5373
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y Fo(w)2939 5373 y Fr(M)10 b(n)p Fu(\()p Fr(p)23 b Fq(\000)g
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Black eop
%%Page: 35 44
35 43 bop Black 300 10 a Fk(CHAPTER)34 b(6.)76 b(P)-8
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b(and)f(the)g(CP)-8 b(A)38 b(is)f(decreasing.)57 b(When)38
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300 2279 y(of)32 b(the)h(algorithms)d(will)g(b)s(e)j(presen)m(ted,)i
(to)s(o.)p Black Black eop
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36 44 bop Black Black Black Black 1714 150 a Fn(Chapter)53
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(for)i(most)g(applied)f(problems)g(due)i(to)f(the)h(fact)f(in)g(suc)m
(h)i(cases)g(the)300 3624 y(the)d(in)m(v)m(erse)g(image)e(of)g(a)h
(subset)i(under)f(the)g(mapping)d Fr(S)35 b Fu(can)30
b(not)f(b)s(e)g(obtained)g(analytically)300 3744 y(\(e.g.,)38
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h(the)g(expression)h(of)e Fr(S)43 b Fu(is)36 b(to)s(o)g(complicated)300
3865 y(to)e(use\),)h(the)f(only)g(w)m(a)m(y)h(to)e(ev)-5
b(aluate)34 b(the)g(Ulam)e(matrix)h(appro)m(ximately)f(is)i(via)f(the)h
(Mon)m(te)300 3985 y(Carlo)g(approac)m(h.)52 b(Of)35
b(course,)i(for)e(one)g(dimensional)e(mappings)h(as)i(simple)d(as)j
(the)f(logistic)300 4106 y(mo)s(del,)j(the)h(exact)h(Ulam)c(metho)s(d)i
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(to)300 4226 y(adopt)26 b(the)h(Mon)m(te)g(Carlo)e(approac)m(h)i(since)
f(the)h(exact)g(ev)-5 b(aluation)25 b(of)g(the)i(matrix)e(is)h(p)s
(ossible.)p Black Black 713 4496 a Fm(7.2)112 b(Numerical)35
b(Results)i(from)g(the)g(P)m(arallel)e(Computation)456
4716 y Fu(No)m(w)c(w)m(e)h(presen)m(t)h(the)e(n)m(umerical)f(results)h
(from)f(the)h(parallel)d(quasi-Mon)m(te)j(Carlo)f(algo-)300
4836 y(rithms)d(to)i(the)g(mapping)e Fr(S)1342 4851 y
Fp(3)1381 4836 y Fu(.)42 b(The)30 b(parallel)c(algorithms)g(w)m(ere)k
(implemen)m(ted)d(on)h(the)h(Wiglaf)300 4956 y(parallel)34
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(\(see)h(App)s(endix)f(B\).)p Black Black eop
%%Page: 40 49
40 48 bop Black 300 10 a Fk(CHAPTER)34 b(7.)76 b(NUMERICAL)34
b(RESUL)-8 b(TS)1678 b Fu(40)p Black Black Black Black
640 178 2861 4 v 638 298 4 121 v 642 298 V 693 262 a
Fm(n)p 936 298 V 936 298 V 460 w(Exact)p 1772 298 V 1772
298 V 614 w(MC)p 2633 298 V 2633 298 V 633 w(QMC)p 3495
298 V 3499 298 V 938 302 2563 4 v 638 419 4 121 v 642
419 V 936 419 V 1102 383 a Fu(Err)p 1406 419 V 213 w(T\(ms\))p
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419 V 227 w(Err)p 3103 419 V 227 w(T\(ms\))p 3495 419
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V 693 506 a(16)p 936 542 V 197 w(2.420E-1)p 1406 542
V 207 w(0)p 1772 542 V 208 w(2.615E-1)p 2241 542 V 196
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V 171 w(220)p 3495 666 V 3499 666 V 640 669 2861 4 v
638 790 4 121 v 642 790 V 693 754 a(64)p 936 790 V 197
w(1.464E-1)p 1406 790 V 183 w(10)p 1772 790 V 183 w(1.486E-1)p
2241 790 V 147 w(1060)p 2633 790 V 148 w(1.478E-1)p 3103
790 V 171 w(860)p 3495 790 V 3499 790 V 640 793 2861
4 v 638 914 4 121 v 642 914 V 693 877 a(128)p 936 914
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2861 4 v 638 1037 4 121 v 642 1037 V 693 1001 a(256)p
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1037 V 3499 1037 V 640 1041 2861 4 v 638 1161 4 121 v
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2241 1161 V 123 w(66450)p 2633 1161 V 123 w(6.323E-2)p
3103 1161 V 123 w(55850)p 3495 1161 V 3499 1161 V 640
1164 2861 4 v 638 1285 4 121 v 642 1285 V 693 1249 a(1024)p
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1285 V 134 w(6.301E-2)p 2241 1285 V 99 w(265540)p 2633
1285 V 98 w(4.621E-2)p 3103 1285 V 98 w(223540)p 3495
1285 V 3499 1285 V 640 1288 2861 4 v Black 340 1458 a(T)-8
b(able)32 b(7.4:)43 b Fr(L)869 1422 y Fp(1)942 1458 y
Fu(error/time)31 b(comparison)g(of)h(QMC,)h(MC,)h(and)e(Exact)i(for)e
(logistic)e(mo)s(del)p Black Black Black Black Black
1239 1694 1662 4 v 1237 1814 4 121 v 1241 1814 V 1293
1778 a Fr(n)p 1487 1814 V 249 w Fu(Exact)p 1957 1814
V 266 w(MC)p 2426 1814 V 273 w(QMC)p 2896 1814 V 2900
1814 V 1239 1818 1662 4 v 1237 1938 4 121 v 1241 1938
V 1293 1902 a(16)p 1487 1938 V 148 w(1.027E-1)p 1957
1938 V 98 w(1.172E-1)p 2426 1938 V 99 w(1.025E-1)p 2896
1938 V 2900 1938 V 1239 1941 1662 4 v 1237 2062 4 121
v 1241 2062 V 1293 2026 a(32)p 1487 2062 V 148 w(5.257E-2)p
1957 2062 V 98 w(6.122E-2)p 2426 2062 V 99 w(5.264E-2)p
2896 2062 V 2900 2062 V 1239 2065 1662 4 v 1237 2186
4 121 v 1241 2186 V 1293 2149 a(64)p 1487 2186 V 148
w(2.564E-2)p 1957 2186 V 98 w(3.603E-2)p 2426 2186 V
99 w(2.581E-2)p 2896 2186 V 2900 2186 V 1239 2189 1662
4 v 1237 2309 4 121 v 1241 2309 V 1293 2273 a(128)p 1487
2309 V 99 w(1.337E-2)p 1957 2309 V 98 w(3.682E-2)p 2426
2309 V 99 w(1.344E-2)p 2896 2309 V 2900 2309 V 1239 2313
1662 4 v 1237 2433 4 121 v 1241 2433 V 1293 2397 a(256)p
1487 2433 V 99 w(6.450E-3)p 1957 2433 V 98 w(3.389E-2)p
2426 2433 V 99 w(6.644E-3)p 2896 2433 V 2900 2433 V 1239
2436 1662 4 v Black 720 2607 a(T)-8 b(able)32 b(7.5:)43
b Fr(L)1249 2570 y Fp(1)1322 2607 y Fu(error)32 b(comparison)g(of)g
(QMC,)h(MC,)g(and)g(Exact)h(for)e Fr(S)3381 2622 y Fp(2)p
Black Black Black Black 1170 2999 a Fm(7.2.1)150 b(Numerical)35
b(Results)h(on)i(Wiglaf)476 3184 y Fu(Wiglaf)f(is)i(the)g(USM)h(Beo)m
(wulf)f(Cluster)h(with)f(32)g(P)m(en)m(tium)g(I)s(I)g(pro)s(cessors)i
(whic)m(h)f(run)300 3305 y(LAM)g(6.3.1/MPI)g(2)f(C++/R)m(OMIO)h(of)f
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300 3425 y(used)43 b(is)e Fr(hcc)h Fu(with)f(the)h(optimizing)c
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300 3545 y(of)36 b(Wiglaf)e(is:)51 b(233)36 b(MHz)h(CPU)g(sp)s(eed,)i
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3666 y(3c905)32 b(10/100)f(PCI)j(Ethernet)f(with)g(Ba)m(y)g(Net)m(w)m
(orks)h(350T)f(F)-8 b(ast)32 b(Ethernet)i(Switc)m(h.)446
3786 y(In)25 b(the)g(exp)s(erimen)m(ts,)i(all)22 b(times)i(rep)s(orted)
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b(In)25 b(the)300 3906 y(follo)m(wing)i(tables)j(and)g(\014gures,)h
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(parallel)e(pro)s(cessors.)300 4027 y Fr(T)58 b Fu(means)44
b(the)h(computation)e(time)g(in)h(seconds.)80 b Fr(s)45
b Fu(is)e(the)i(sp)s(eedup,)k(and)c(the)g(e\016ciency)300
4147 y(is)f(denoted)h(b)m(y)g Fr(E)6 b Fu(.)79 b(All)43
b(the)h(data)g(are)g(rounded)i(o\013)e(to)f(the)i(second)h(decimal)c
(place.)79 b(In)300 4268 y(the)32 b(quasi-Mon)m(te)f(Carlo)f
(computation,)g(1000)g(sampling)f(n)m(um)m(b)s(ers)j(are)f(used,)i
(that)e(is,)g(eac)m(h)300 4388 y(partition)h([)p Fr(ih;)17
b Fu(\()p Fr(i)23 b Fu(+)g(1\))p Fr(h)p Fu(])34 b(\()p
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Fq(\001)g(\001)g(\001)30 b Fr(;)17 b(n)23 b Fq(\000)g
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e(in)m(to)g(1000)300 4508 y(sub-in)m(terv)-5 b(als,)32
b(where)i Fr(h)28 b Fu(=)f(1)p Fr(=n)p Fu(.)446 4629
y(W)-8 b(e)36 b(discuss)h(the)f(p)s(erformances)g(of)f(the)h(UMP)-8
b(A)36 b(and)g(the)g(CP)-8 b(A)36 b(on)g(the)g(Wiglaf.)50
b(T)-8 b(able)300 4749 y(7.6)31 b(sho)m(ws)i(the)f(computation)e(time)g
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(mapping)e Fr(S)1860 4885 y Fp(3)1900 4870 y Fu(.)446
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b(the)i(partition)d(of)h([0)p Fr(;)17 b Fu(1]\).)43 b(Th)m(us)32
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Black Black eop
%%Page: 41 50
41 49 bop Black 300 10 a Fk(CHAPTER)34 b(7.)76 b(NUMERICAL)34
b(RESUL)-8 b(TS)1678 b Fu(41)p Black Black Black Black
782 178 2576 4 v 780 298 4 121 v 784 298 V 835 262 a
Fm(n)p 1030 298 V 1030 298 V 592 w(UMP)-9 b(A)p 2290
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782 302 2576 4 v 780 422 4 121 v 784 422 V 1030 422 V
1105 386 a Fr(p)p 1227 422 V 164 w(T)14 b Fu(\()p Fr(s)p
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V 123 w(89.50)p 3353 1870 V 3357 1870 V 782 1873 2576
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1994 V 123 w(1.00)p 2982 1994 V 124 w(100.00)p 3353 1994
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V 147 w(7.69)p 2982 2355 V 148 w(96.13)p 3353 2355 V
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V 3357 2596 V 782 2599 2576 4 v Black 760 2759 a(T)-8
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b(A)34 b(and)e(CP)-8 b(A)34 b(for)e Fr(S)2859 2774 y
Fp(3)2931 2759 y Fu(on)g(Wiglaf)p Black Black 300 3152
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@endspecial Black 1001 4588 a(Figure)f(7.5:)43 b(E\016ciency)34
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%%Page: 51 60
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@endspecial Black 849 2268 a(Figure)31 b(7.10:)43 b(E\016ciency)34
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@endspecial Black 895 4588 a(Figure)f(7.11:)43 b(E\016ciency)34
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g(the)g(Sw)m(eetgum)g(if)f(the)h(problem)e(size)i Fr(n)g
Fu(is)f(v)m(ery)i(large.)p Black Black eop
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@endspecial Black 325 2268 a(Figure)32 b(7.12:)43 b(E\016ciency)34
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(measurable)f(if)f Fr(f)2652 1043 y Fl(\000)p Fp(1)2746
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b(A.7.)p Black 48 w Fu(A)23 b(measurable)e(function)h
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(ts.)42 b(In)30 b(other)300 1482 y(w)m(ords,)k Fr(f)43
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b Fu(=)1739 1648 y Fo(n)1688 1678 y Fi(X)1703 1888 y
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y Fo(i)3197 2081 y Fu(and)i Fr(\037)3445 2096 y Fo(A)3498
2106 y Ff(i)3529 2081 y Fu(\()p Fr(x)p Fu(\))e(=)f(0)300
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1322 w Fu(\(A.3\))p Black 300 3062 a(where)33 b Fq(f)p
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eop
%%Page: 56 65
56 64 bop Black 300 10 a Fk(APPENDIX)34 b(A.)65 b(BRIEF)33
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a(if)31 b(at)i(least)f(one)h(of)f(the)h(terms)1466 817
y Fi(R)1513 932 y Fo(A)1587 897 y Fr(f)1646 861 y Fp(+)1705
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b(in)m(tegrable)f(and)1568 1540 y Fq(j)1613 1405 y Fi(Z)1667
1630 y Fo(X)1751 1540 y Fr(f)11 b(d\026)p Fq(j)27 b(\024)2080
1405 y Fi(Z)2135 1630 y Fo(X)2219 1540 y Fr(g)t(d\026:)p
Black 1208 w Fu(\(A.8\))p Black 446 1799 a(\(ii\))609
1719 y Fi(R)656 1834 y Fo(X)740 1799 y Fq(j)p Fr(f)11
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2040 y(and)1136 2128 y Fi(Z)1191 2354 y Fo(X)1259 2264
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y Fl([)2177 2864 y Ff(n)2219 2856 y Fo(A)2272 2864 y
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3169 y(space,)c(as)f(the)g(follo)m(wing)c(theorem)k(indicates.)p
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y Fo(X)2185 4071 y Fr(g)t(f)11 b(d\026)p Black 1161 w
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Fr(\026)p Black 1445 w Fu(\(A.13\))p Black 300 5515 a(if)k
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57 65 bop Black 300 10 a Fk(APPENDIX)34 b(A.)65 b(BRIEF)33
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801 y(777,)32 b(1998.)p Black 300 988 a([19])p Black
50 w(J.)g(Ding,)h(M.)f(P)m(aprzyc)m(ki,)h(and)f(Z.)g(W)-8
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24 b(op)s(erators.)29 b(a)g(solution)490 3243 y(of)22
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b(transformations.)25 b Fb(Physic)-5 b(a)25 b(D)p Fc(,)c(92:61{68,)27
b(1996.)p Black 300 3431 a([26])p Black 50 w(J.)f(Ding)g(and)f(A.)i
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y(Appl.)p Fc(,)i(to)g(app)s(ear.)p Black 300 3731 a([27])p
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b(arian)m(t)38 b(measures)h(for)g(piecewise)490 3844
y(expanding)29 b Fa(c)966 3811 y Fp(2)1036 3844 y Fc(transformations)h
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b(dimensional)e(p)s(oin)m(t)i(transformations)g(and)h(asymptotic)490
4145 y(measures)h(for)g(cellular)f(automata.)42 b Fb(Comput.)34
b(Math.)f(Appl.)p Fc(,)e(19:13{31,)j(1990.)p Black 300
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Black 50 w(F.)j(Y.)g(Hun)m(t.)64 b(Erratum:)55 b(`a)39
b(mon)m(te)h(carlo)e(approac)m(h)h(to)g(the)f(appro)m(ximation)f(of)i
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5347 y(Wiley)30 b(&)g(Sons,)g(New)g(Y)-8 b(ork,)31 b(1976.)p
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%%Page: 91 100
91 99 bop Black 300 10 a Fk(BIBLIOGRAPHY)2663 b Fu(91)p
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b Fb(L)-5 b(e)g(ctur)g(e)27 b(Notes)f(in)g(Phys.)p Fc(,)490
688 y(179:192{193,)36 b(1983.)p Black 300 875 a([35])p
Black 50 w(V.)23 b(Kumar,)g(A.)g(Grama,)i(A.)d(Gupta,)j(and)d(G.)g
(Karypis.)k Fb(Intr)-5 b(o)g(duction)28 b(to)e(Par)-5
b(al)5 b(lel)27 b(Computing)p Fc(.)490 988 y(Benjamin/Cummings,)h(San)i
(F)-8 b(rancisco,)31 b(1994.)p Black 300 1176 a([36])p
Black 50 w(A.)i(Lasota)g(and)f(M.)h(Mac)m(k)m(ey)-8 b(.)49
b Fb(Chaos,)36 b(F)-7 b(r)i(actals,)37 b(and)e(Noise)p
Fc(.)46 b(Springer-V)-8 b(erlag,)32 b(New)g(Y)-8 b(ork,)490
1289 y(1994.)p Black 300 1476 a([37])p Black 50 w(A.)42
b(Lasota)g(and)f(J.)g(A.)g(Y)-8 b(ork)m(e.)75 b(On)40
b(the)i(existence)f(of)h(in)m(v)-5 b(arian)m(t)40 b(measures)h(for)g
(piecewise)490 1589 y(monotonic)31 b(transformations.)39
b Fb(T)-7 b(r)i(ans.)34 b(A)n(mer.)f(Math.)g(So)-5 b(c.)p
Fc(,)31 b(186:481{488,)k(1973.)p Black 300 1777 a([38])p
Black 50 w(T.)24 b(Y.)h(Li.)k(Finite)23 b(appro)m(ximation)g(for)h(the)
h(frob)s(enius-p)s(erron)20 b(op)s(erator,)26 b(a)e(solution)f(to)i
(ulam's)490 1890 y(conjecture.)42 b Fb(J.)32 b(Appr)-5
b(ox.)34 b(The)-5 b(ory)p Fc(,)32 b(17:177{186,)j(1976.)p
Black 300 2077 a([39])p Black 50 w(T.)45 b(Y.)g(Li.)82
b Fb(Er)-5 b(go)g(dic)47 b(the)-5 b(ory)47 b(on)f Fc([0)p
Fa(;)15 b Fc(1].)85 b(Lecture)45 b(Notes)h(at)f(the)g(Dept.)h(of)f
(Math,)k(Ky)m(oto)490 2190 y(Univ)m(ersit)m(y)-8 b(,)30
b(1988.)p Black 300 2378 a([40])p Black 50 w(T.)48 b(Y.)g(Li)f(and)g
(J.)g(A.)h(Y)-8 b(ork)m(e.)94 b(P)m(erio)s(d)47 b(three)h(implies)d(c)m
(haos.)93 b Fb(A)n(mer.)49 b(Math.)g(Monthly)p Fc(,)490
2491 y(82\(10\):985{992,)37 b(1975.)p Black 300 2679
a([41])p Black 50 w(G.)31 b(G.)g(Loren)m(tz.)42 b Fb(Appr)-5
b(oximations)35 b(of)e(F)-7 b(unctions)p Fc(.)42 b(Holt,)31
b(Rinehart)e(and)h(Winston,)g(1966.)p Black 300 2866
a([42])p Black 50 w(E.)42 b(N.)h(Lorenz.)75 b(Deterministic)41
b(nonp)s(erio)s(dic)e(\015o)m(ws.)75 b Fb(J.)43 b(A)n(tmospheric)i
(Sci.)p Fc(,)f(20:130{141,)490 2979 y(1963.)p Black 300
3167 a([43])p Black 50 w(R.)35 b(M.)h(Ma)m(y)-8 b(.)56
b(Biological)35 b(p)s(opulations)e(with)h(nono)m(v)m(erlapping)g
(generations:)50 b(stable)35 b(p)s(oin)m(ts,)490 3280
y(stable)30 b(cycles,)h(and)f(c)m(haos.)42 b Fb(Scienc)-5
b(e)p Fc(,)30 b(186:645{647,)35 b(1974.)p Black 300 3467
a([44])p Black 50 w(W.)44 b(J.)f(Morok)m(o\013.)82 b(Generating)43
b(quasi-random)f(paths)h(for)g(sto)s(c)m(hastic)i(pro)s(cesses.)79
b Fb(SIAM)490 3580 y(R)-5 b(eview)p Fc(,)31 b(40\(4\):765{788,)36
b(1998.)p Black 300 3768 a([45])p Black 50 w(R.)k(Murra)m(y)-8
b(.)67 b(Appro)m(ximation)38 b(error)h(for)g(in)m(v)-5
b(arian)m(t)38 b(densit)m(y)h(calculations.)66 b Fb(J.)41
b(Discr)-5 b(ete)40 b(&)490 3881 y(Continuous)34 b(Dyn.)e(Sys.)p
Fc(,)f(4:535{558,)j(1998.)p Black 300 4068 a([46])p Black
50 w(P)-8 b(.)41 b(P)m(ac)m(heco.)74 b Fb(Par)-5 b(al)5
b(lel)44 b(Pr)-5 b(o)g(gr)g(amming)45 b(with)e(MPI)p
Fc(.)70 b(Morgan)42 b(Kaufmann,)g(San)e(F)-8 b(rancisco,)490
4181 y(1996.)p Black 300 4369 a([47])p Black 50 w(S.)29
b(M.)h(Ulam.)39 b Fb(A)31 b(Col)5 b(le)-5 b(ction)33
b(of)f(Mathematic)-5 b(al)34 b(Pr)-5 b(oblems)p Fc(.)40
b(In)m(ter-science,)31 b(New)e(Y)-8 b(ork,)31 b(1960.)p
Black 300 4557 a([48])p Black 50 w(M.)37 b(A.)g(v)-5
b(an)36 b(Wyk)h(and)f(T.)g(S.)h(Durrani.)57 b(Classi\014cation)34
b(of)j(system)g(resp)s(onse)e(using)g(proba-)490 4669
y(bilistic)28 b(mo)s(delling.)38 b Fb(R)-5 b(ese)g(ar)g(ch)34
b(R)-5 b(ep)g(ort,)35 b(Kentr)-5 b(on,)34 b(South)f(Afric)-5
b(a)p Fc(,)31 b(1999.)p Black 300 4857 a([49])p Black
50 w(M.)h(A.)g(v)-5 b(an)31 b(Wyk)h(and)e(W.)i(H.)g(Steeb.)44
b(Chaos)31 b(in)f(electronics.)43 b(In)31 b Fb(Mathematic)-5
b(al)36 b(Mo)-5 b(del)5 b(ling:)490 4970 y(The)-5 b(ory)34
b(and)g(Applic)-5 b(ations,)34 b(No.2)p Fc(,)d(South)f(Africa,)g(1997.)
j(Klu)m(w)m(er)c(Academic)i(Publishers.)p Black 300 5158
a([50])p Black 50 w(P)-8 b(.)31 b(W)-8 b(alters.)41 b
Fb(A)n(n)32 b(Intr)-5 b(o)g(duction)35 b(to)f(Er)-5 b(go)g(dic)33
b(The)-5 b(ory)p Fc(.)42 b(Springer-V)-8 b(erlag,)30
b(New)g(Y)-8 b(ork,)32 b(1982.)p Black Black eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF