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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
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/Aacute % 193
/Acircumflex % 194
/Atilde % 195
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/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
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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
Black 446 2018 a(7.9)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(Sw)m(eetgum)64
b(.)49 b(.)h(.)g(.)g(.)g(.)g(.)p Black 83 w(50)p Black
Black 2044 5764 a(v)p Black eop
%%Page: 6 9
6 8 bop Black Black Black Black 1037 150 a Fn(LIST)53
b(OF)i(ABBREVIA)-13 b(TIONS)509 823 y Fq(k)22 b(\001)g(k)99
b Fu(-)h Fr(L)979 787 y Fp(1)1018 823 y Fu(-)32 b(norm)630
944 y Fr(d)99 b Fu(-)h(dimension)31 b(of)h(the)h(system)583
1064 y Fr(f)642 1028 y Fl(\003)780 1064 y Fu(-)100 b(\014xed)33
b(densit)m(y)h(of)e Fr(P)623 1185 y(n)99 b Fu(-)h(n)m(um)m(b)s(er)33
b(of)f(subin)m(terv)-5 b(als)32 b(from)g(the)h(partition)d(of)i([0)p
Fr(;)17 b Fu(1])32 b(in)m(terv)-5 b(al)454 1305 y Fr(C)7
b(P)14 b(A)99 b Fu(-)h(completely)31 b(parallel)f(algorithm)499
1425 y Fr(M)10 b(C)107 b Fu(-)100 b(Mon)m(te)33 b(Carlo)f(approac)m(h)
592 1546 y Fr(N)110 b Fu(-)100 b(n)m(um)m(b)s(er)33 b(of)f(random)f(n)m
(um)m(b)s(ers)j(generated)f(in)f(Mon)m(te)h(Carlo)f(metho)s(d)604
1666 y Fr(P)113 b Fu(-)100 b(F)-8 b(rob)s(enius-P)m(erron)32
b(op)s(erator)573 1786 y Fr(P)636 1801 y Fo(h)780 1786
y Fu(-)100 b(Ulam's)31 b(matrix)g(or)h(companion)f(matrix)571
1907 y Fr(P)634 1922 y Fo(n)780 1907 y Fu(-)100 b(Ulam's)31
b(matrix)g(or)h(companion)f(matrix)g(\()p Fr(d)d Fu(=)f(1\))422
2027 y Fr(QM)10 b(C)107 b Fu(-)100 b(quasi-Mon)m(te)33
b(Carlo)e(approac)m(h)350 2148 y Fr(U)10 b(M)g(P)k(A)100
b Fu(-)g(Ulam's)31 b(matrix)g(parallel)f(algorithm)p
Black 2031 5764 a(vi)p Black eop
%%Page: 1 10
1 9 bop Black Black Black Black 1714 150 a Fn(Chapter)53
b(1)p Black Black 1410 581 a(INTR)l(ODUCTION)300 1283
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@endspecial 300 4380 a(Figure)38 b(1.1:)55 b(The)40
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Fu(037)f(and)i(\(b)s(elo)m(w\))300 4501 y Fr(x)355 4516
y Fp(0)423 4501 y Fu(=)27 b(0)p Fr(:)p Fu(038)p Black
446 4841 a(Figure)34 b(1.1)g(sho)m(ws)i(the)e(\014rst)h(500)f
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@endspecial 300 3245 a(Figure)30 b(1.2:)43 b(The)32
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Fr(\026)p Fu(.)81 b(Here)46 b Fr(m)f Fu(is)g(the)h(Leb)s(esgue)300
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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
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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
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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
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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(\()p Fr(c)1325 3200 y Fp(1)1364 3185 y Fr(;)17
b(c)1450 3200 y Fp(2)1489 3185 y Fr(;)g Fq(\001)g(\001)g(\001)31
b Fr(;)17 b(c)1768 3200 y Fo(n)1815 3185 y Fu(\))p Fr(P)1916
3200 y Fo(n)1990 3185 y Fu(=)28 b(\()p Fr(c)2174 3200
y Fp(1)2213 3185 y Fr(;)17 b(c)2299 3200 y Fp(2)2338
3185 y Fr(;)g Fq(\001)g(\001)g(\001)32 b Fr(;)17 b(c)2618
3200 y Fo(n)2664 3185 y Fu(\))p Fr(:)p Black 910 w Fu(\(3.6\))p
Black 300 3398 a(The)34 b(corresp)s(onding)e(densit)m(y)1544
3681 y Fr(f)1592 3696 y Fo(n)1640 3681 y Fu(\()p Fr(x)p
Fu(\))c(=)1953 3557 y Fo(n)1902 3587 y Fi(X)1917 3797
y Fo(i)p Fp(=1)2154 3614 y Fr(c)2196 3629 y Fo(i)p 2073
3659 V 2073 3750 a Fr(m)p Fu(\()p Fr(I)2239 3765 y Fo(i)2267
3750 y Fu(\))2315 3681 y Fr(\037)2376 3696 y Fo(I)2407
3706 y Ff(i)2437 3681 y Fu(\()p Fr(x)p Fu(\))p Fr(:)300
3977 y Fu(This)c(famous)f(sc)m(heme)i(is)e(called)g(Ulam's)f(metho)s
(d.)40 b(The)25 b(matrix)d Fr(P)2793 3992 y Fo(n)2863
3977 y Fu(is)i(called)e(the)i(companion)300 4098 y(matrix)31
b(or)h(Ulam's)g(matrix.)446 4218 y(In)43 b(1960)e(Ulam)f(conjectured)j
(that)f(the)g(piecewise)h(constan)m(t)g(appro)m(ximations)d
Fr(f)3599 4233 y Fo(n)3688 4218 y Fu(will)300 4338 y(con)m(v)m(erge)32
b(to)e Fr(f)872 4302 y Fl(\003)942 4338 y Fu(as)g Fr(n)h
Fu(go)s(es)f(to)g(in\014nit)m(y)g([47)o(].)43 b(In)31
b(1976)e(T.)i(Y.)g(Li)e(pro)m(v)m(ed)j(this)e(conjecture)h(for)300
4459 y(a)h(class)h(of)f(piecewise)h Fr(C)1221 4423 y
Fp(2)1293 4459 y Fu(and)g(stretc)m(hing)g(mappings)f([38)o(].)446
4579 y(Ulam's)46 b(metho)s(d)g(can)g(b)s(e)h(in)m(tro)s(duced)g(in)e
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%%Page: 14 23
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Black eop
%%Page: 16 25
16 24 bop Black 300 10 a Fk(CHAPTER)34 b(3.)76 b(ULAM'S)33
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%%Page: 17 26
17 25 bop Black 300 10 a Fk(CHAPTER)34 b(3.)76 b(ULAM'S)33
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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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(d,)300 1117 y(based)43 b(on)f(the)g(theory)h(of)f(appro)m(ximations)e
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300 1237 y(op)s(erators)32 b(in)g([10].)446 1358 y(It)24
b(is)e(w)m(ell)h(kno)m(wn)h(that)f(the)h(error)f Fq(k)p
Fr(f)1807 1373 y Fo(n)1857 1358 y Fq(\000)s Fr(f)1996
1322 y Fl(\003)2035 1358 y Fq(k)g Fu(of)g(the)h(appro)m(ximate)e
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(cal)e(error")i Fq(k)p Fr(Q)2605 1493 y Fo(n)2652 1478
y Fr(f)2711 1442 y Fl(\003)2766 1478 y Fq(\000)16 b Fr(f)2918
1442 y Fl(\003)2957 1478 y Fq(k)29 b Fu(of)g(the)h(appro)m(ximate)300
1598 y(op)s(erator)c Fr(Q)764 1613 y Fo(n)838 1598 y
Fu(applied)g(to)h(the)g(densit)m(y)h Fr(f)1840 1562 y
Fl(\003)1906 1598 y Fu(\(see)g(equalit)m(y)e(\(4.15\))g(b)s(elo)m(w)h
(or)g([18)o(]\).)42 b(So)27 b(in)f(order)300 1719 y(to)h(obtain)g(a)g
(con)m(v)m(ergence)j(order)e(for)f Fq(k)p Fr(f)1815 1734
y Fo(n)1874 1719 y Fq(\000)12 b Fr(f)2022 1683 y Fl(\003)2062
1719 y Fq(k)p Fu(,)28 b(it)f(is)g(su\016cien)m(t)h(to)g(ha)m(v)m(e)h
(an)e(appro)m(ximation)300 1839 y(order)k(for)f Fq(k)p
Fr(Q)827 1854 y Fo(n)875 1839 y Fr(f)f Fq(\000)19 b Fr(f)11
b Fq(k)p Fu(.)43 b(Since)31 b(w)m(e)h(only)e(consider)h(con)m(tin)m
(uous)h(functions)f Fr(f)42 b Fu(and)31 b(use)h(results)300
1960 y(in)g([10)o(],)h(w)m(e)h(need)f(to)g(study)g(the)g(functions)g
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1960 y Fu(,)f(where)i Fr(e)2557 1975 y Fo(i)2585 1960
y Fu(\()p Fr(x)p Fu(\))28 b(=)g Fr(x)2903 1923 y Fo(i)2964
1960 y Fu(for)k Fr(i)c Fu(=)g(0)p Fr(;)17 b Fu(1)p Fr(;)g
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y Fo(n)1246 2080 y Fr(e)1291 2095 y Fp(0)1363 2080 y
Fu(=)e Fr(e)1517 2095 y Fp(0)1557 2080 y Fu(.)52 b(But)36
b(since)g Fr(Q)2152 2095 y Fo(n)2235 2080 y Fu(is)f Fj(not)h
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2095 y Fo(n)3813 2080 y Fu(,)300 2200 y(in)44 b(general)g
Fr(Q)851 2215 y Fo(n)942 2200 y Fu(do)s(es)h(not)f(map)g(a)g(piecewise)
i(linear)c(function)i(in)g(\001)3000 2215 y Fo(n)3092
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(follo)m(wing)300 2441 y(result.)p Black 300 2603 a Fj(L)-5
b(emma)34 b(4.1.)p Black 48 w Fu(Let)f Fr(f)11 b Fu(\()p
Fr(t)p Fu(\))27 b(=)h Fr(at)23 b Fu(+)f Fr(b)33 b Fu(on)f([0)p
Fr(;)17 b Fu(1].)43 b(Then)1074 2941 y Fr(Q)1151 2956
y Fo(n)1198 2941 y Fr(f)11 b Fu(\()p Fr(t)p Fu(\))28
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2781 y Fo(a)p 1639 2797 38 4 v 1640 2854 a Fp(2)1687
2820 y Fu(\()p Fr(t)22 b Fu(+)g Fr(h)p Fu(\))g(+)g Fr(b;)255
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3061 y Fu(\()p Fr(t)22 b Fu(+)g(1)g Fq(\000)h Fr(h)p
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Fo(n)2664 3061 y Fr(:)p Black 3639 2941 a Fu(\(4.4\))p
Black 446 3289 a Fm(Pro)s(of.)42 b Fu(F)-8 b(rom)27 b(the)i
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y Fu(it)g(is)g(ob)m(vious)g(that)h Fr(Q)2717 3304 y Fo(n)2764
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3734 y Fp(2)2194 3770 y Fr(;)17 b(x)p Fu(\))38 b(and)f
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3891 y(Com)m(bining)31 b(the)i(ab)s(o)m(v)m(e)g(with)f(\(A.22\))h(in)e
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300 4053 a Fj(L)-5 b(emma)34 b(4.2.)p Black 48 w Fu(If)f
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4670 y Fo(n)2727 4655 y Fr(e)2772 4670 y Fp(0)2839 4655
y Fu(=)27 b Fr(e)2987 4670 y Fp(0)3027 4655 y Fu(,)33
b(w)m(e)g(ha)m(v)m(e)p Black 300 4817 a Fj(L)-5 b(emma)34
b(4.3.)p Black 48 w Fu(If)f Fr(x)28 b Fq(2)g Fu([0)p
Fr(;)17 b Fu(1])32 b(is)g(\014xed)i(and)e(if)g Fr(f)38
b Fq(2)28 b Fr(C)2226 4780 y Fp(1)2266 4817 y Fu([0)p
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b Fq(\000)g Fr(f)11 b Fu(\()p Fr(x)p Fu(\))p Fq(j)83
b(\024)1884 5024 y(j)p Fr(f)1971 4988 y Fl(0)1993 5024
y Fu(\()p Fr(x)p Fu(\))p Fq(j)p 1884 5069 269 4 v 1993
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a Fq(j)p Fr(Q)1316 5466 y Fo(n)1363 5451 y Fr(f)11 b
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5451 y Fu(\()p Fr(x)p Fu(\))p Fr(;)45 b(x)28 b Fq(2)g([)2720
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5451 y Fr(I)2900 5466 y Fo(i)p Black 3639 5451 a Fu(\(4.7\))p
Black Black Black eop
%%Page: 23 32
23 31 bop Black 300 10 a Fk(CHAPTER)34 b(4.)76 b(PIECEWISE)35
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b Fu(23)p Black 861 317 a Fq(j)p Fr(Q)966 332 y Fo(n)1013
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Fr(\013)1068 573 y Fo(n)1115 558 y Fu(\()p Fr(x)p Fu(\))p
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b Fr(\013)1545 573 y Fo(n)1592 558 y Fu(\()p Fr(x)p Fu(\)\).)446
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b(emma)34 b(4.4.)p Black 48 w Fr(\013)915 889 y Fo(n)962
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b Fu(Since)32 b Fr(Q)1124 1046 y Fo(n)1204 1031 y Fu(is)g(linear,)f(b)m
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Fr(Q)1514 1257 y Fo(n)1561 1242 y Fu(\(\()p Fr(t)22 b
Fq(\000)h Fr(x)p Fu(\))1887 1200 y Fp(2)1927 1242 y Fr(;)17
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y Fu(\()p Fr(t)2392 1200 y Fp(2)2454 1242 y Fq(\000)22
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y Fr(;)17 b(x)p Fu(\))1278 1387 y(=)83 b Fr(Q)1514 1402
y Fo(n)1561 1387 y Fu(\()p Fr(t)1634 1346 y Fp(2)1673
1387 y Fr(;)17 b(x)p Fu(\))23 b(+)f Fr(Q)2008 1402 y
Fo(n)2055 1387 y Fu(\()p Fq(\000)p Fu(2)p Fr(xt)h Fu(+)f
Fr(x)2485 1346 y Fp(2)2525 1387 y Fr(;)17 b(x)p Fu(\))1278
1650 y(=)83 b Fr(Q)1514 1665 y Fo(n)1561 1650 y Fu(\()p
Fr(t)1634 1609 y Fp(2)1673 1650 y Fr(;)17 b(x)p Fu(\))23
b(+)1931 1446 y Fi(8)1931 1536 y(<)1931 1715 y(:)2061
1529 y Fq(\000)p Fr(hx)330 b Fu(if)27 b Fr(x)h Fq(2)g
Fr(I)2884 1544 y Fp(1)2061 1650 y Fq(\000)p Fr(x)2193
1613 y Fp(2)2579 1650 y Fu(if)f Fr(x)h Fq(2)g([)2907
1608 y Fo(n)p Fl(\000)p Fp(1)2907 1675 y Fo(i)p Fp(=2)3045
1650 y Fr(I)3088 1665 y Fo(i)2061 1770 y Fq(\000)p Fr(x)p
Fu(\(1)23 b Fq(\000)f Fr(h)p Fu(\))83 b(if)27 b Fr(x)h
Fq(2)g Fr(I)2884 1785 y Fo(n)1278 2017 y Fu(=)83 b Fr(Q)1514
2032 y Fo(n)1561 2017 y Fu(\()p Fr(t)1634 1976 y Fp(2)1673
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Fl(\000)p Fp(1)1932 1922 y Fi(X)1947 2132 y Fo(i)p Fp(=1)2076
2017 y Fu(\()p Fr(ih)p Fu(\))2241 1976 y Fp(2)2281 2017
y Fr(\036)2339 2032 y Fo(i)2367 2017 y Fu(\()p Fr(x)p
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2548 y Fu(3\(1)g Fq(\000)h Fr(h)p Fu(\))p 2578 2592 352
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y Fr(\036)1306 2931 y Fo(i)1334 2916 y Fu(\()p Fr(x)p
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b(=)792 3153 y Fr(h)848 3117 y Fp(2)p 792 3198 96 4 v
816 3289 a Fu(3)914 3050 y Fi(")972 3220 y Fr(\036)1030
3235 y Fp(0)1070 3220 y Fu(\()p Fr(x)p Fu(\))22 b(+)1327
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3179 y Fp(2)2037 3220 y Fu(\))p Fr(\036)2133 3235 y Fo(i)2160
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3153 y Fu(3\(1)f Fq(\000)g Fr(h)p Fu(\))p 2495 3198 352
4 v 2623 3289 a Fr(h)2679 3260 y Fp(2)2879 3220 y Fu(+)g(1)g
Fq(\000)3157 3153 y Fu(3\(1)g Fq(\000)g Fr(h)p Fu(\))p
3157 3198 V 3285 3289 a Fr(h)3341 3260 y Fp(2)3518 3080
y Fi(\023)3608 3220 y Fr(\036)3666 3235 y Fo(n)3713 3220
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1649 3517 89 4 v 1649 3567 4 50 v 1735 3567 V 1649 3570
89 4 v 446 3835 a Fu(The)35 b(follo)m(wing)30 b(t)m(w)m(o)k(theorems)g
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3919 y Fp(1)1384 3955 y Fu(-norm,)e(resp)s(ectiv)m(ely)-8
b(.)p Black 300 4113 a Fj(The)j(or)g(em)34 b(4.1.)p Black
48 w Fu(\(i\))d(Let)i Fr(f)39 b Fq(2)28 b Fr(C)7 b Fu([0)p
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4388 y Fo(n)1587 4373 y Fr(f)32 b Fq(\000)23 b Fr(f)11
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Fr(w)s Fu(\()p Fr(f)11 b Fu(;)2392 4305 y Fr(h)p 2355
4350 132 4 v 2355 4370 a Fq(p)p 2438 4370 49 4 v 82 x
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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
25 33 bop Black Black Black Black 1714 150 a Fn(Chapter)53
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%%Page: 32 41
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eop
%%Page: 34 43
34 42 bop Black 300 10 a Fk(CHAPTER)34 b(6.)76 b(P)-8
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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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%%Page: 37 46
37 45 bop Black 300 10 a Fk(CHAPTER)34 b(7.)76 b(NUMERICAL)34
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%%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
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(logistic)e(mo)s(del)p Black Black Black Black Black
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99 w(2.581E-2)p 2896 2186 V 2900 2186 V 1239 2189 1662
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b(where)i Fr(h)28 b Fu(=)f(1)p Fr(=n)p Fu(.)446 4629
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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
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@endspecial Black 948 2849 a(Figure)e(7.8:)43 b(E\016ciency)34
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@endspecial Black 910 5282 a(Figure)e(7.9:)43 b(E\016ciency)34
b(of)e(the)h(CP)-8 b(A)34 b(for)e(DPLL)g(on)h(Wiglaf)p
Black Black Black Black eop
%%Page: 50 59
50 58 bop Black 300 10 a Fk(CHAPTER)34 b(7.)76 b(NUMERICAL)34
b(RESUL)-8 b(TS)1678 b Fu(50)p Black Black Black 847
274 a Fm(7.3.2)113 b(Numerical)35 b(Results)h(for)i(DPLL)f(on)h(Sw)m
(eetgum)470 459 y Fu(T)-8 b(able)37 b(7.9)g(sho)m(ws)h(the)g
(computation)e(time)f Fr(T)14 b Fu(\()p Fr(s)p Fu(\),)38
b(the)g(sp)s(eedup)g Fr(s)p Fu(,)h(and)e(the)h(e\016ciency)300
579 y Fr(E)h Fu(of)32 b(the)h(UMP)-8 b(A)34 b(and)f(the)g(CP)-8
b(A)34 b(for)e(the)i(DPLL)e(on)h(Sw)m(eetgum.)45 b(Figure)32
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b(on)f(Sw)m(eetgum)h(with)e(the)i(pro)s(cessor)300 820
y(n)m(um)m(b)s(er)33 b Fr(p)f Fu(and)h(the)g(problem)e(size)i
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1079 4 121 v 857 1079 V 908 1043 a Fm(n)p 1103 1079 V
1103 1079 V 568 w(UMP)-9 b(A)p 2314 1079 V 2314 1079
V 799 w(CP)g(A)p 3280 1079 V 3284 1079 V 855 1083 2430
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a Fr(p)p 1300 1203 V 163 w(T)14 b Fu(\()p Fr(s)p Fu(\))p
1671 1203 V 202 w Fr(s)p 1944 1203 V 181 w(E)6 b Fu(\(\045\))p
2314 1203 V 133 w Fr(T)14 b Fu(\()p Fr(s)p Fu(\))p 2636
1203 V 177 w Fr(s)p 2909 1203 V 182 w(E)6 b Fu(\(\045\))p
3280 1203 V 3284 1203 V 1105 1206 2181 4 v 853 1324 4
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b(A)24 b(and)g(the)g(CP)-8 b(A)300 3644 y(will)37 b(decrease.)64
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(UMP)-8 b(A)36 b(do)s(es)g(not)f(tak)m(e)h(to)s(o)e(m)m(uc)m(h)300
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/LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def
/LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def
/Pnt { stroke [] 0 setdash
gsave 1 setlinecap M 0 0 V stroke grestore } def
/Dia { stroke [] 0 setdash 2 copy vpt add M
hpt neg vpt neg V hpt vpt neg V
hpt vpt V hpt neg vpt V closepath stroke
Pnt } def
/Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V
currentpoint stroke M
hpt neg vpt neg R hpt2 0 V stroke
} def
/Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M
0 vpt2 neg V hpt2 0 V 0 vpt2 V
hpt2 neg 0 V closepath stroke
Pnt } def
/Crs { stroke [] 0 setdash exch hpt sub exch vpt add M
hpt2 vpt2 neg V currentpoint stroke M
hpt2 neg 0 R hpt2 vpt2 V stroke } def
/TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M
hpt neg vpt -1.62 mul V
hpt 2 mul 0 V
hpt neg vpt 1.62 mul V closepath stroke
Pnt } def
/Star { 2 copy Pls Crs } def
/BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M
0 vpt2 neg V hpt2 0 V 0 vpt2 V
hpt2 neg 0 V closepath fill } def
/TriUF { stroke [] 0 setdash vpt 1.12 mul add M
hpt neg vpt -1.62 mul V
hpt 2 mul 0 V
hpt neg vpt 1.62 mul V closepath fill } def
/TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M
hpt neg vpt 1.62 mul V
hpt 2 mul 0 V
hpt neg vpt -1.62 mul V closepath stroke
Pnt } def
/TriDF { stroke [] 0 setdash vpt 1.12 mul sub M
hpt neg vpt 1.62 mul V
hpt 2 mul 0 V
hpt neg vpt -1.62 mul V closepath fill} def
/DiaF { stroke [] 0 setdash vpt add M
hpt neg vpt neg V hpt vpt neg V
hpt vpt V hpt neg vpt V closepath fill } def
/Pent { stroke [] 0 setdash 2 copy gsave
translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
closepath stroke grestore Pnt } def
/PentF { stroke [] 0 setdash gsave
translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
closepath fill grestore } def
/Circle { stroke [] 0 setdash 2 copy
hpt 0 360 arc stroke Pnt } def
/CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def
/C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def
/C1 { BL [] 0 setdash 2 copy moveto
2 copy vpt 0 90 arc closepath fill
vpt 0 360 arc closepath } bind def
/C2 { BL [] 0 setdash 2 copy moveto
2 copy vpt 90 180 arc closepath fill
vpt 0 360 arc closepath } bind def
/C3 { BL [] 0 setdash 2 copy moveto
2 copy vpt 0 180 arc closepath fill
vpt 0 360 arc closepath } bind def
/C4 { BL [] 0 setdash 2 copy moveto
2 copy vpt 180 270 arc closepath fill
vpt 0 360 arc closepath } bind def
/C5 { BL [] 0 setdash 2 copy moveto
2 copy vpt 0 90 arc
2 copy moveto
2 copy vpt 180 270 arc closepath fill
vpt 0 360 arc } bind def
/C6 { BL [] 0 setdash 2 copy moveto
2 copy vpt 90 270 arc closepath fill
vpt 0 360 arc closepath } bind def
/C7 { BL [] 0 setdash 2 copy moveto
2 copy vpt 0 270 arc closepath fill
vpt 0 360 arc closepath } bind def
/C8 { BL [] 0 setdash 2 copy moveto
2 copy vpt 270 360 arc closepath fill
vpt 0 360 arc closepath } bind def
/C9 { BL [] 0 setdash 2 copy moveto
2 copy vpt 270 450 arc closepath fill
vpt 0 360 arc closepath } bind def
/C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill
2 copy moveto
2 copy vpt 90 180 arc closepath fill
vpt 0 360 arc closepath } bind def
/C11 { BL [] 0 setdash 2 copy moveto
2 copy vpt 0 180 arc closepath fill
2 copy moveto
2 copy vpt 270 360 arc closepath fill
vpt 0 360 arc closepath } bind def
/C12 { BL [] 0 setdash 2 copy moveto
2 copy vpt 180 360 arc closepath fill
vpt 0 360 arc closepath } bind def
/C13 { BL [] 0 setdash 2 copy moveto
2 copy vpt 0 90 arc closepath fill
2 copy moveto
2 copy vpt 180 360 arc closepath fill
vpt 0 360 arc closepath } bind def
/C14 { BL [] 0 setdash 2 copy moveto
2 copy vpt 90 360 arc closepath fill
vpt 0 360 arc } bind def
/C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill
vpt 0 360 arc closepath } bind def
/Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto
neg 0 rlineto closepath } bind def
/Square { dup Rec } bind def
/Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def
/S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def
/S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def
/S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def
/S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def
/S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def
/S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill
exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def
/S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def
/S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill
2 copy vpt Square fill
Bsquare } bind def
/S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def
/S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def
/S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill
Bsquare } bind def
/S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill
Bsquare } bind def
/S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def
/S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill
2 copy vpt Square fill Bsquare } bind def
/S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill
2 copy exch vpt sub exch vpt Square fill Bsquare } bind def
/S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def
/D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def
/D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def
/D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def
/D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def
/D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def
/D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def
/D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def
/D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def
/D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def
/D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def
/D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def
/D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def
/D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def
/D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def
/D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def
/D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def
/DiaE { stroke [] 0 setdash vpt add M
hpt neg vpt neg V hpt vpt neg V
hpt vpt V hpt neg vpt V closepath stroke } def
/BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M
0 vpt2 neg V hpt2 0 V 0 vpt2 V
hpt2 neg 0 V closepath stroke } def
/TriUE { stroke [] 0 setdash vpt 1.12 mul add M
hpt neg vpt -1.62 mul V
hpt 2 mul 0 V
hpt neg vpt 1.62 mul V closepath stroke } def
/TriDE { stroke [] 0 setdash vpt 1.12 mul sub M
hpt neg vpt 1.62 mul V
hpt 2 mul 0 V
hpt neg vpt -1.62 mul V closepath stroke } def
/PentE { stroke [] 0 setdash gsave
translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
closepath stroke grestore } def
/CircE { stroke [] 0 setdash
hpt 0 360 arc stroke } def
/Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def
/DiaW { stroke [] 0 setdash vpt add M
hpt neg vpt neg V hpt vpt neg V
hpt vpt V hpt neg vpt V Opaque stroke } def
/BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M
0 vpt2 neg V hpt2 0 V 0 vpt2 V
hpt2 neg 0 V Opaque stroke } def
/TriUW { stroke [] 0 setdash vpt 1.12 mul add M
hpt neg vpt -1.62 mul V
hpt 2 mul 0 V
hpt neg vpt 1.62 mul V Opaque stroke } def
/TriDW { stroke [] 0 setdash vpt 1.12 mul sub M
hpt neg vpt 1.62 mul V
hpt 2 mul 0 V
hpt neg vpt -1.62 mul V Opaque stroke } def
/PentW { stroke [] 0 setdash gsave
translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
Opaque stroke grestore } def
/CircW { stroke [] 0 setdash
hpt 0 360 arc Opaque stroke } def
/BoxFill { gsave Rec 1 setgray fill grestore } def
end
%%EndProlog
gnudict begin
gsave
50 50 translate
0.050 0.050 scale
0 setgray
newpath
(Helvetica) findfont 120 scalefont setfont
1.000 UL
LTb
420 240 M
63 0 V
2913 0 R
-63 0 V
348 240 M
(0) Rshow
420 507 M
63 0 V
2913 0 R
-63 0 V
348 507 M
(0.5) Rshow
420 774 M
63 0 V
2913 0 R
-63 0 V
348 774 M
(1) Rshow
420 1041 M
63 0 V
2913 0 R
-63 0 V
-2985 0 R
(1.5) Rshow
420 1308 M
63 0 V
2913 0 R
-63 0 V
-2985 0 R
(2) Rshow
420 1575 M
63 0 V
2913 0 R
-63 0 V
-2985 0 R
(2.5) Rshow
420 1842 M
63 0 V
2913 0 R
-63 0 V
-2985 0 R
(3) Rshow
420 2109 M
63 0 V
2913 0 R
-63 0 V
-2985 0 R
(3.5) Rshow
420 2376 M
63 0 V
2913 0 R
-63 0 V
-2985 0 R
(4) Rshow
420 240 M
0 63 V
0 2073 R
0 -63 V
420 120 M
(0) Cshow
894 240 M
0 63 V
0 2073 R
0 -63 V
894 120 M
(1) Cshow
1367 240 M
0 63 V
0 2073 R
0 -63 V
0 -2193 R
(2) Cshow
1841 240 M
0 63 V
0 2073 R
0 -63 V
0 -2193 R
(3) Cshow
2315 240 M
0 63 V
0 2073 R
0 -63 V
0 -2193 R
(4) Cshow
2788 240 M
0 63 V
0 2073 R
0 -63 V
0 -2193 R
(5) Cshow
3262 240 M
0 63 V
0 2073 R
0 -63 V
0 -2193 R
(6) Cshow
1.000 UL
LTb
420 240 M
2976 0 V
0 2136 V
-2976 0 V
420 240 L
1.000 UL
LT0
2829 2253 M
(PDF for DPLL with n=4096) Rshow
3076 2253 Pnt
421 551 Pnt
421 551 Pnt
422 550 Pnt
423 551 Pnt
424 551 Pnt
424 550 Pnt
425 552 Pnt
426 551 Pnt
427 552 Pnt
427 551 Pnt
428 552 Pnt
429 551 Pnt
429 552 Pnt
430 552 Pnt
431 553 Pnt
432 552 Pnt
432 552 Pnt
433 553 Pnt
434 552 Pnt
435 553 Pnt
435 554 Pnt
436 553 Pnt
437 553 Pnt
437 554 Pnt
438 555 Pnt
439 554 Pnt
440 555 Pnt
440 555 Pnt
441 555 Pnt
442 555 Pnt
443 556 Pnt
443 555 Pnt
444 555 Pnt
445 556 Pnt
445 556 Pnt
446 556 Pnt
447 556 Pnt
448 556 Pnt
448 556 Pnt
449 555 Pnt
450 557 Pnt
451 557 Pnt
451 557 Pnt
452 558 Pnt
453 558 Pnt
453 558 Pnt
454 557 Pnt
455 559 Pnt
456 559 Pnt
456 559 Pnt
457 559 Pnt
458 560 Pnt
459 560 Pnt
459 559 Pnt
460 560 Pnt
461 560 Pnt
461 559 Pnt
462 561 Pnt
463 560 Pnt
464 561 Pnt
464 561 Pnt
465 562 Pnt
466 561 Pnt
467 562 Pnt
467 561 Pnt
468 562 Pnt
469 562 Pnt
469 563 Pnt
470 564 Pnt
471 563 Pnt
472 565 Pnt
472 564 Pnt
473 564 Pnt
474 564 Pnt
474 566 Pnt
475 565 Pnt
476 565 Pnt
477 565 Pnt
477 567 Pnt
478 565 Pnt
479 567 Pnt
480 567 Pnt
480 567 Pnt
481 568 Pnt
482 568 Pnt
482 568 Pnt
483 570 Pnt
484 570 Pnt
485 570 Pnt
485 570 Pnt
486 571 Pnt
487 571 Pnt
488 571 Pnt
488 571 Pnt
489 571 Pnt
490 572 Pnt
490 572 Pnt
491 574 Pnt
492 574 Pnt
493 574 Pnt
493 575 Pnt
494 574 Pnt
495 575 Pnt
496 576 Pnt
496 577 Pnt
497 577 Pnt
498 576 Pnt
498 579 Pnt
499 579 Pnt
500 577 Pnt
501 579 Pnt
501 580 Pnt
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503 580 Pnt
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510 585 Pnt
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512 588 Pnt
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514 594 Pnt
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516 595 Pnt
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517 595 Pnt
518 598 Pnt
519 600 Pnt
520 599 Pnt
520 599 Pnt
521 600 Pnt
522 602 Pnt
522 601 Pnt
523 603 Pnt
524 603 Pnt
525 605 Pnt
525 605 Pnt
526 606 Pnt
527 606 Pnt
528 607 Pnt
528 609 Pnt
529 609 Pnt
530 609 Pnt
530 613 Pnt
531 614 Pnt
532 614 Pnt
533 615 Pnt
533 615 Pnt
534 615 Pnt
535 614 Pnt
536 620 Pnt
536 620 Pnt
537 620 Pnt
538 621 Pnt
538 621 Pnt
539 621 Pnt
540 626 Pnt
541 634 Pnt
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550 668 Pnt
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557 690 Pnt
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562 699 Pnt
562 700 Pnt
563 702 Pnt
564 700 Pnt
565 705 Pnt
565 706 Pnt
566 708 Pnt
567 706 Pnt
567 711 Pnt
568 710 Pnt
569 711 Pnt
570 718 Pnt
570 723 Pnt
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575 737 Pnt
575 740 Pnt
576 738 Pnt
577 741 Pnt
578 746 Pnt
578 745 Pnt
579 746 Pnt
580 765 Pnt
581 766 Pnt
581 765 Pnt
582 800 Pnt
583 819 Pnt
583 821 Pnt
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585 825 Pnt
586 825 Pnt
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602 843 Pnt
602 691 Pnt
603 691 Pnt
604 689 Pnt
605 660 Pnt
605 661 Pnt
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607 626 Pnt
607 606 Pnt
608 606 Pnt
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610 602 Pnt
610 602 Pnt
611 602 Pnt
612 600 Pnt
613 601 Pnt
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614 601 Pnt
615 602 Pnt
615 602 Pnt
616 601 Pnt
617 602 Pnt
618 602 Pnt
618 602 Pnt
619 603 Pnt
620 602 Pnt
621 603 Pnt
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622 605 Pnt
623 604 Pnt
623 605 Pnt
624 605 Pnt
625 605 Pnt
626 603 Pnt
626 606 Pnt
627 607 Pnt
628 607 Pnt
629 606 Pnt
629 609 Pnt
630 609 Pnt
631 608 Pnt
631 610 Pnt
632 611 Pnt
633 610 Pnt
634 612 Pnt
634 614 Pnt
635 613 Pnt
636 614 Pnt
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640 619 Pnt
641 618 Pnt
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644 624 Pnt
645 624 Pnt
645 625 Pnt
646 625 Pnt
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649 630 Pnt
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652 633 Pnt
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655 638 Pnt
656 640 Pnt
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660 647 Pnt
660 645 Pnt
661 650 Pnt
662 650 Pnt
663 651 Pnt
663 651 Pnt
664 656 Pnt
665 654 Pnt
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666 661 Pnt
667 660 Pnt
668 661 Pnt
668 663 Pnt
669 667 Pnt
670 666 Pnt
671 667 Pnt
671 674 Pnt
672 673 Pnt
673 674 Pnt
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682 704 Pnt
682 706 Pnt
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684 713 Pnt
684 715 Pnt
685 713 Pnt
686 720 Pnt
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688 728 Pnt
689 735 Pnt
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690 737 Pnt
691 745 Pnt
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693 753 Pnt
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695 764 Pnt
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696 777 Pnt
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698 777 Pnt
698 789 Pnt
699 798 Pnt
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700 801 Pnt
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717 1191 Pnt
718 1267 Pnt
719 1426 Pnt
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720 1426 Pnt
721 2316 Pnt
723 1696 Pnt
724 1032 Pnt
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725 964 Pnt
726 368 Pnt
727 369 Pnt
727 368 Pnt
728 368 Pnt
729 369 Pnt
730 368 Pnt
730 368 Pnt
731 369 Pnt
732 368 Pnt
732 368 Pnt
733 369 Pnt
734 368 Pnt
735 368 Pnt
735 368 Pnt
736 368 Pnt
737 368 Pnt
737 368 Pnt
738 367 Pnt
739 368 Pnt
740 367 Pnt
740 367 Pnt
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760 365 Pnt
761 365 Pnt
761 366 Pnt
762 365 Pnt
763 366 Pnt
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764 366 Pnt
765 365 Pnt
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767 365 Pnt
767 365 Pnt
768 365 Pnt
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770 364 Pnt
771 364 Pnt
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781 364 Pnt
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785 363 Pnt
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790 362 Pnt
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800 361 Pnt
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812 360 Pnt
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838 358 Pnt
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866 356 Pnt
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887 355 Pnt
888 355 Pnt
889 355 Pnt
889 355 Pnt
890 355 Pnt
891 355 Pnt
892 355 Pnt
892 355 Pnt
893 355 Pnt
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894 355 Pnt
895 355 Pnt
896 355 Pnt
897 355 Pnt
897 355 Pnt
898 355 Pnt
899 354 Pnt
900 354 Pnt
900 355 Pnt
901 355 Pnt
902 355 Pnt
902 355 Pnt
903 355 Pnt
904 356 Pnt
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905 356 Pnt
906 356 Pnt
907 356 Pnt
908 356 Pnt
908 357 Pnt
909 357 Pnt
910 357 Pnt
910 357 Pnt
911 356 Pnt
912 357 Pnt
913 357 Pnt
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914 357 Pnt
915 357 Pnt
916 356 Pnt
916 357 Pnt
917 357 Pnt
918 357 Pnt
918 356 Pnt
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eop
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y Fi(Z)2092 4567 y Fo(X)2176 4477 y Fr(f)11 b(g)t(d\026)p
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59 67 bop Black 300 10 a Fk(APPENDIX)34 b(A.)65 b(BRIEF)33
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446 3251 a Fm(Pro)s(of.)44 b Fu(Let)32 b Fr(t)c Fq(2)g
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y(})928 2831 y(})928 2923 y(for\(j=s;j<=n;j++\))e({)1241
3014 y(t[j]=p[j][s];)1241 3105 y(p[j][s]=p[j][point];)1241
3197 y(p[j][point]=t[j];)928 3288 y(})928 3471 y
(if\(fabs\(p[s][s]\)<=5e-20\))h({)1241 3562 y(printf\("No)e
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3927 y(for\(j=s+1;j<=n;j++\))h({)1241 4019 y(for\(i=s+1;i<=n;i++\))1555
4110 y(p[j][i]-=p[j][s]*p[s][i]/p[s)q(][s])q(;)928 4201
y(})614 4293 y(})614 4384 y(if\(fabs\(p[n][n]\)>0.05\))614
4475 y({)275 b(printf\("No)41 b(solution!\\n"\);)928
4566 y(exit\(1\);)614 4658 y(})614 4840 y(z[n]=1;)614
4932 y(for\(j=1;j<=n-1;j++\))83 b(/*)39 b(backward)i(solving)g(*/)575
5023 y({)314 b(for\(i=1;i<=n-j;i++\))1241 5114 y
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5297 y(})614 5480 y(for\(i=1;i<=n;i++\))82 b(/*)40 b(Nomorlizing)i(*/)p
Black Black eop
%%Page: 88 97
88 96 bop Black 300 10 a Fk(APPENDIX)34 b(B.)65 b(A)33
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274 a Fd(sum+=z[i];)614 366 y(temp=\(float\)n/sum;)614
457 y(for\(i=1;i<=n;i++\))928 548 y(z[i]*=temp;)300 640
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914 y(void)g(density\(\))300 1005 y({)614 1096 y(double)g
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1279 y(long)g(start,end;)614 1370 y(double)g(Timei,)h(Time,)f(Tim;)300
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1644 y(start=clock\(\);)614 1735 y(matrix\(\);)614 1827
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614 2466 y(end=clock\(\);)614 2557 y
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2740 y(/*)g(Timing)g(Gaussian)h(Algorithm)g(*/)614 2831
y(start=clock\(\);)614 2923 y(gauss\(N,)g(x\);)614 3014
y(TotalErr=TE\(\);)81 b(/*)e(L^1)40 b(error)h(for)f(GA)f(*/)614
3105 y(end=clock\(\);)614 3197 y(Time=\(\(double\)end-start\)/\(1000)q
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3653 y(printf\("\0455.4f\\n",)k(x[i]\);)614 3836 y(*/)614
3927 y(printf\("L)e(\\t)f(N)f(\\t)h(L^1_Err_i)h(\\t)f(L^1_Err_g)h(\\t)f
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b(\045d)39 b(\\t)h(\0457.6f)g(\\t)g(\0457.6f)g(\\t)g(\0455.2f)h(\\t)e
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4384 y(})p Black Black eop
%%Page: 89 98
89 97 bop Black Black Black Black 1422 136 a Fn(BIBLIOGRAPHY)p
Black 345 831 a Fc([1])p Black 50 w(C.)27 b(Bec)m(k)i(and)e(F.)g(Sc)m
(hl\177)-45 b(ogl.)35 b Fb(Thermo)-5 b(dynamics)33 b(of)d(Chaotic)h
(Systems)p Fc(.)36 b(Cam)m(bridge)27 b(Univ)m(ersit)m(y)490
943 y(Press,)j(1993.)p Black 345 1128 a([2])p Black 50
w(C.)c(Bose)g(and)g(R.)f(Murra)m(y)-8 b(.)34 b(The)25
b(exact)i(rate)g(of)f(appro)m(ximation)e(in)h(ulam's)f(metho)s(d.)33
b Fb(pr)-5 b(eprint)p Fc(,)490 1241 y(1999.)p Black 345
1425 a([3])p Black 50 w(A.)23 b(Bo)m(y)m(arsky)g(and)f(P)-8
b(.)22 b(G\023)-45 b(ora.)28 b Fb(L)-5 b(aws)26 b(of)f(Chaos:)40
b(Invariant)26 b(Me)-5 b(asur)g(es)26 b(and)f(Chaotic)i(Dynamic)-5
b(al)490 1538 y(Systems)34 b(in)e(One)g(Dimension)p Fc(.)41
b(Birkh\177)-45 b(auser,)29 b(1997.)p Black 345 1722
a([4])p Black 50 w(A.)53 b(Bo)m(y)m(arsky)i(and)d(Y.)h(S.)g(Lou.)107
b(Appro)m(ximating)51 b(measures)i(in)m(v)-5 b(arian)m(t)52
b(under)f(higher-)490 1835 y(dimensional)28 b(c)m(haotic)j
(transformations.)40 b Fb(J.)32 b(Appr)-5 b(ox.)34 b(The)-5
b(ory)p Fc(,)33 b(65:231{244,)i(1991.)p Black 345 2019
a([5])p Black 50 w(A.)29 b(Bo)m(y)m(arsky)h(and)d(Y.)i(S.)f(Lou.)37
b(A)28 b(compactness)h(theorem)g(for)f(appro)m(ximating)f(the)i(in)m(v)
-5 b(arian)m(t)490 2132 y(densities)30 b(of)i(higher)f(dimensional)e
(transformations.)44 b Fb(J.)34 b(Math.)g(A)n(naly.)g(Appl.)p
Fc(,)f(65:231{244,)490 2245 y(1993.)p Black 345 2430
a([6])p Black 50 w(A.)i(Bo)m(y)m(arsky)-8 b(,)38 b(P)-8
b(.Gora,)37 b(and)d(Y.)h(S.)g(Lou.)53 b(Constructiv)m(e)34
b(appro)m(ximations)g(to)h(the)g(in)m(v)-5 b(arian)m(t)490
2542 y(densities)21 b(of)j(higher-dimensional)19 b(transformations.)28
b Fb(Constructive)e(Appr)-5 b(ox.)p Fc(,)26 b(10:1{13,)i(1994.)p
Black 345 2727 a([7])p Black 50 w(C.)g(Chiu,)f(Q.)i(Du,)g(and)e(T.)i
(Y.)f(Li.)37 b(Error)27 b(estimates)i(of)g(the)g(mark)m(o)m(v)g
(\014nite)e(appro)m(ximation)h(of)490 2840 y(the)j(frob)s(enius-p)s
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Fc(,)d(19\(4\):291{308,)36 b(1992.)p Black 345 3024 a([8])p
Black 50 w(I.)29 b(P)-8 b(.)29 b(Cornfeld,)e(S.)h(V.)h(F)-8
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b(go)g(dic)33 b(The)-5 b(ory)p Fc(.)39 b(Springer-V)-8
b(erlag,)28 b(New)490 3137 y(Y)-8 b(ork,)31 b(1982.)p
Black 345 3321 a([9])p Black 50 w(D.)k(Daems)g(and)e(G.)i(Nicolis.)50
b(Cluster)33 b(expansion)g(for)h(the)g(p)s(erron-frob)s(enius)c(op)s
(erator)35 b(in)e(a)490 3434 y(system)e(of)f(coupled)g(map)g(lattices.)
40 b Fb(R)-5 b(ese)g(ar)g(ch)35 b(R)-5 b(ep)g(ort,)35
b(1050)f(Bruxel)5 b(les,)33 b(Belgium)p Fc(,)d(1995.)p
Black 300 3618 a([10])p Black 50 w(R.)21 b(A.)g(DeV)-8
b(ore.)27 b(The)21 b(appro)m(ximation)f(of)h(con)m(tin)m(uous)f
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3731 y(In)30 b(Springer-V)-8 b(erlag,)29 b(editor,)h
Fb(L)-5 b(e)g(ctur)g(e)34 b(Notes)f(in)f(Math,)h(293)p
Fc(,)f(1972.)p Black 300 3916 a([11])p Black 50 w(J.)24
b(Ding.)30 b(Computing)22 b(in)m(v)-5 b(arian)m(t)23
b(measures)h(for)g(piecewise)f(con)m(v)m(ex)j(transformations.)j
Fb(J.)d(Stat.)490 4029 y(Phys.)p Fc(,)31 b(83\(3/4\):623{635,)37
b(1996.)p Black 300 4213 a([12])p Black 50 w(J.)22 b(Ding.)k(A)21
b(maxim)m(um)g(en)m(trop)m(y)h(metho)s(d)f(for)h(solving)e(frob)s
(enius-p)s(erron)e(op)s(erator)k(equations.)490 4326
y Fb(Applie)-5 b(d)34 b(Math.)f(Comp.)p Fc(,)f(93:155{168,)j(1998.)p
Black 300 4510 a([13])p Black 50 w(J.)42 b(Ding.)77 b(The)41
b(p)s(oin)m(t)h(sp)s(ectrum)f(of)i(frob)s(enius-p)s(erron)38
b(and)k(k)m(o)s(opman)h(op)s(erators.)76 b Fb(Pr)-5 b(o)g(c.)490
4623 y(A)n(mer.)32 b(Math.)h(So)-5 b(c.)p Fc(,)32 b
(126\(5\):1355{1361,)37 b(1998.)p Black 300 4807 a([14])p
Black 50 w(J.)21 b(Ding,)i(Q.)e(Du,)j(and)c(T.)i(Y.)f(Li.)k(High)c
(order)f(appro)m(ximation)h(of)g(frob)s(enius-p)s(erron)d(op)s
(erators.)490 4920 y Fb(Appl.)33 b(Math.)g(Comp.)p Fc(,)f(53:151{171,)j
(1993.)p Black 300 5104 a([15])p Black 50 w(J.)28 b(Ding,)f(Q.)h(Du,)g
(and)f(T.)g(Y.)h(Li.)35 b(The)27 b(sp)s(ectral)g(analysis)f(of)i(frob)s
(enius-p)s(erron)23 b(op)s(erators.)36 b Fb(J.)490 5217
y(Math.)d(A)n(nal.)g(Appl.)p Fc(,)e(184\(2\):285{301,)36
b(1994.)p Black 300 5402 a([16])p Black 50 w(J.)j(Ding)g(and)f(T.)h(Y.)
g(Li.)65 b(Mark)m(o)m(v)41 b(\014nite)d(appro)m(ximation)g(of)h(frob)s
(enius-p)s(erron)c(op)s(erator,.)490 5515 y Fb(Nonlin.)e(A)n(nal.)f
(Th.)h(Meth.)g(Appl.)p Fc(,)e(17\(8\):759{772,)36 b(1991.)p
Black 2021 5764 a Fu(89)p Black eop
%%Page: 90 99
90 98 bop Black 300 10 a Fk(BIBLIOGRAPHY)2663 b Fu(90)p
Black Black 300 274 a Fc([17])p Black 50 w(J.)32 b(Ding)g(and)f(T.)h
(Y.)g(Li.)44 b(Pro)5 b(jection)32 b(solutions)e(of)i(frob)s(enius-)e(p)
s(erron)g(op)s(erators.)46 b Fb(Internat.)490 387 y(J.)32
b(Math.)h(&)g(Math.)g(Sci.)p Fc(,)d(16\(3\):465{484,)36
b(1993.)p Black 300 575 a([18])p Black 50 w(J.)c(Ding)f(and)g(T.)h(Y.)g
(Li.)43 b(A)32 b(con)m(v)m(ergence)i(rate)f(analysis)d(for)i(mark)m(o)m
(v)g(\014nite)f(appro)m(ximations)490 688 y(to)26 b(a)g(class)f(of)h
(frob)s(enius-p)s(erron)21 b(op)s(erators.)33 b Fb(Nonlin.)28
b(A)n(nal.)g(Th.)g(Meth.)g(Appl.)p Fc(,)f(31\(5/6\):765{)490
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
b(ang.)47 b(E\016cien)m(t)32 b(computation)g(of)g(un)m(b)s(ounded)e(in)
m(v)-5 b(arian)m(t)490 1101 y(densities.)34 b(In)27 b
Fb(Pr)-5 b(o)g(c)g(e)g(e)g(dings)32 b(of)e(the)g(14th)i(Confer)-5
b(enc)g(e)30 b(on)h(Applie)-5 b(d)31 b(Mathematics)p
Fc(,)e(pages)f(126{)490 1214 y(131,)k(1998.)p Black 300
1402 a([20])p Black 50 w(J.)40 b(Ding)f(and)g(Z.)h(W)-8
b(ang.)69 b(A)40 b(mo)s(di\014ed)d(mon)m(te)k(carlo)f(approac)m(h)g(to)
g(the)g(appro)m(ximation)f(of)490 1515 y(in)m(v)-5 b(arian)m(t)39
b(measures.)66 b(In)39 b Fb(R)-5 b(ese)g(ar)g(ch)43 b(Notes)e(in)g
(Math,)i(418)p Fc(,)g(pages)d(125{130.)i(Chapman)c(&)490
1628 y(Hall/CPC,)30 b(2000.)p Black 300 1815 a([21])p
Black 50 w(J.)h(Ding)f(and)f(Z.)i(W)-8 b(ang.)42 b(Appro)m(ximation)29
b(order)h(analysis)f(for)h(the)g(piecewise)g(linear)f(mark)m(o)m(v)490
1928 y(metho)s(d.)40 b Fb(Sto)-5 b(chastic)35 b(A)n(nalysis)e(and)g
(Applic)-5 b(ations)p Fc(,)33 b(submitted.)p Black 300
2116 a([22])p Black 50 w(J.)d(Ding)f(and)g(Z.)g(W)-8
b(ang.)40 b(P)m(arallel)29 b(computation)g(of)h(in)m(v)-5
b(arian)m(t)28 b(measures.)39 b Fb(A)n(nnals)32 b(in)g(Op)-5
b(er)g(a-)490 2229 y(tions)34 b(R)-5 b(ese)g(ar)g(ch)p
Fc(,)32 b(to)f(app)s(ear,)f(2000.)p Black 300 2416 a([23])p
Black 50 w(J.)e(Ding)g(and)f(A.)h(Zhou.)36 b(The)27 b(pro)5
b(jection)28 b(metho)s(d)g(for)f(computing)g(m)m(ulti-dimensional)d
(abso-)490 2529 y(lutely)29 b(con)m(tin)m(uous)h(in)m(v)-5
b(arian)m(t)30 b(measures.)40 b Fb(J.)32 b(Stat.)i(Phys.)p
Fc(,)d(77\(3/4\):899{908,)37 b(1994.)p Black 300 2717
a([24])p Black 50 w(J.)g(Ding)f(and)g(A.)h(Zhou.)58 b(Piecewise)37
b(linear)e(mark)m(o)m(v)i(appro)m(ximations)f(of)h(frob)s(enius-p)s
(erron)490 2830 y(op)s(erators)23 b(asso)s(ciated)f(with)f(m)m
(ulti-dimensional)d(transformations.)27 b Fb(Nonline)-5
b(ar)26 b(A)n(nal.,)h(TMA)p Fc(,)490 2943 y(25\(4\):399{408,)36
b(1995.)p Black 300 3130 a([25])p Black 50 w(J.)29 b(Ding)f(and)g(A.)h
(Zhou.)37 b(Finite)27 b(appro)m(ximation)h(of)h(frob)s(enius-p)s(erron)
24 b(op)s(erators.)29 b(a)g(solution)490 3243 y(of)22
b(ulam's)e(conjecture)i(to)g(m)m(ulti-dimensional)17
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
(Zhou.)32 b(On)25 b(the)i(sp)s(ectrum)e(of)h(frob)s(enius-p)s(erron)c
(op)s(erators.)34 b Fb(J.)28 b(Math.)h(A)n(nal.)490 3544
y(Appl.)p Fc(,)i(to)g(app)s(ear.)p Black 300 3731 a([27])p
Black 50 w(P)-8 b(.)40 b(G\023)-45 b(ora)39 b(and)g(A.)g(Bo)m(y)m
(arsky)-8 b(.)68 b(Absolutely)38 b(con)m(tin)m(uous)g(in)m(v)-5
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
(in)f Fa(r)1842 3811 y Fo(n)1888 3844 y Fc(.)41 b Fb(Isr)-5
b(ael)34 b(J.)e(Math.)p Fc(,)f(67\(3\):272{286,)36 b(1989.)p
Black 300 4032 a([28])p Black 50 w(P)-8 b(.)30 b(G\023)-45
b(ora)29 b(and)g(A.)g(Bo)m(y)m(arsky)-8 b(.)40 b(Higher)28
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
4332 a([29])p Black 50 w(F.)c(Hun)m(t)f(and)f(W.)i(Miller.)37
b(On)28 b(the)h(appro)m(ximation)f(of)i(in)m(v)-5 b(arian)m(t)28
b(measures.)38 b Fb(J.)31 b(Stat.)h(Phys.)p Fc(,)490
4445 y(66:535{548,)j(1992.)p Black 300 4633 a([30])p
Black 50 w(F.)k(Y.)g(Hun)m(t.)64 b(A)39 b(mon)m(te)g(carlo)f(approac)m
(h)h(to)g(the)g(appro)m(ximation)e(of)h(in)m(v)-5 b(arian)m(t)38
b(measures.)490 4746 y Fb(R)-5 b(andom)35 b(&)d(Comput)i(Dynamics)p
Fc(,)d(2\(1\):111{133,)36 b(1994.)p Black 300 4933 a([31])p
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
(in)m(v)-5 b(arian)m(t)490 5046 y(measures'.)41 b Fb(R)-5
b(andom)35 b(&)d(Comput.)i(Dynam.)p Fc(,)d(5\(4\):361{362,)36
b(1998.)p Black 300 5234 a([32])p Black 50 w(D.)d(L.)g(Isaacson)g(and)f
(R.)g(W.)i(Madsen.)46 b Fb(Markov)35 b(Chains,)h(The)-5
b(ory)36 b(and)g(Applic)-5 b(ations)p Fc(.)49 b(John)490
5347 y(Wiley)30 b(&)g(Sons,)g(New)g(Y)-8 b(ork,)31 b(1976.)p
Black Black eop
%%Page: 91 100
91 99 bop Black 300 10 a Fk(BIBLIOGRAPHY)2663 b Fu(91)p
Black Black 300 274 a Fc([33])p Black 50 w(M.)48 b(Jablonski.)90
b(On)47 b(in)m(v)-5 b(arian)m(t)47 b(measures)g(for)g(piecewise)g
Fa(c)2718 241 y Fp(2)2758 274 y Fc(-transformations)g(of)g(the)h
Fa(n)p Fc(-)490 387 y(dimensional)28 b(cub)s(e.)40 b
Fb(A)n(nn.)31 b(Polon.)j(Math.)p Fc(,)d(XLI)s(I)s(I:185{195,)h(1983.)p
Black 300 575 a([34])p Black 50 w(G.)24 b(Keller.)k(Sto)s(c)m(hastic)c
(p)s(erturbation)e(of)h(some)h(strange)g(attractors.)32
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