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"(sorted by r elevant packages)" }}{PARA 0 "" 0 "" {TEXT 267 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEX T 268 10 "combstruct" }}{EXCHG {PARA 15 "" 0 "" {TEXT 206 64 "A calculus for the random generation of combinatorial structur es" }{TEXT 269 59 ", by P. Flajolet, P. Zimmermann, and B. V. Cutsem, \+ (1994), " }{TEXT 207 28 "Theoretical Computer Science" }{TEXT 269 78 " , vol. 135. Gives the theoretical background of the labelled structur es case." }}{PARA 15 "" 0 "" {TEXT 208 52 "Attribute grammars and auto matic complexity analysis" }{TEXT 269 24 ", by M. Mishna, (2003), " } {TEXT 209 31 "Advances in Applied Mathematics" }{TEXT 269 20 ", vol. 3 0, number 1." }}{PARA 15 "" 0 "" {TEXT 210 48 "Attribute grammars are \+ useful for combinatorics!" }{TEXT 269 42 ", by M. P. Delest and J. M F edou, (1992), " }{TEXT 211 28 "Theoretical Computer Science" }{TEXT 269 20 ", vol. 98, p. 65-76." }}{PARA 15 "" 0 "" {TEXT 212 45 "Automat ic average-case analysis of algorithms" }{TEXT 269 55 ", by P. Flajole t, B. Salvy, and P. Zimmermann, (1991), " }{TEXT 213 38 "Theoretical C omputer Science, Series A" }{TEXT 269 20 ", vol. 79, number 1." }} {PARA 15 "" 0 "" {TEXT 214 69 "Gaia: a package for the random generati on of combinatorial structures" }{TEXT 269 30 ", by Paul Zimmermann, ( 1994), " }{TEXT 215 26 "Maple Technical Newsletter" }{TEXT 269 44 ". \+ Gaia is an earlier version of combstruct." }}{PARA 15 "" 0 "" {TEXT 216 39 "Lambda-Upsilon-Omega, the 1989 cookbook" }{TEXT 269 110 ", by \+ P. Flajolet, B. Salvy and P. Zimmermann, (1989), INRIA Research Report , vol. 1073. Lamba-Upsilon-Omega, " }{TEXT 269 1 "-" }{TEXT 269 2 "- " }{TEXT 269 59 " or LUO for short, is a previous incarnation of combs truct." }}{PARA 15 "" 0 "" {TEXT 217 54 "Lambda-Upsilon-Omega: an assi stant algorithms analyzer" }{TEXT 269 59 ", by P. Flajolet, B. Salvy, \+ and P. Zimmermann, (1989). In: " }{TEXT 218 64 "Applied Algebra, Algeb raic Algorithms and Error-Correcting Codes" }{TEXT 269 64 ", T. Mora ( editor), Lecture Notes in Computer Science, vol. 357." }}{PARA 15 "" 0 "" {TEXT 219 37 "Object grammars and random generation" }{TEXT 269 39 ", by I. Dutour and J. M Fedou, (1998), " }{TEXT 220 53 "Discrete M athematics and Theoretical Computer Science" }{TEXT 269 19 ", vol. 2, \+ p. 49-63." }}{PARA 15 "" 0 "" {TEXT 221 56 "Random generation of unlab elled combinatorial structures" }{TEXT 269 79 ", by Eithne Murray, (19 93), summary of a seminar talk by Paul Zimmermann. In: " }{TEXT 222 29 "Algorithms Seminar, 1993-1994" }{TEXT 269 68 ", by Bruno Salvy (ed itor), (1994), INRIA Research Report, vol. 2381." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 268 4 "gdev" }}{EXCHG {PARA 15 "" 0 "" {TEXT 223 43 "Exa mples of automatic asymptotic expansions" }{TEXT 269 26 ", by Bruno Sa lvy, (1991), " }{TEXT 224 15 "SIGSAM Bulletin" }{TEXT 269 20 ", vol. 2 5, number 2." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 268 4 "gfun" }}{EXCHG {PARA 15 "" 0 "" {TEXT 225 96 "Gfun: a Maple package for the manipulat ion of generating and holonomic functions in one variable" }{TEXT 269 41 ", by B. Salvy and P. Zimmermann, (1994), " }{TEXT 226 41 "ACM Tran sactions on Mathematical Software" }{TEXT 269 16 ", vol. 20, n\342\210 \236 2." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 268 38 "Ore_algebra, Groebn er, Holonomy, Mgfun" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 270 12 "Publicati ons" }}{EXCHG {PARA 0 "" 0 "" {TEXT 267 97 "The following are publicat ions that describe the theoretical aspects implemented in the packages ." }}}{EXCHG {PARA 15 "" 0 "" {TEXT 269 0 "" }{TEXT 227 36 "Fonctions \+ holonomes en calcul formel" }{TEXT 269 108 ", by Fr\303\210d\303\210ri c Chyzak, (1998), \"Th\303\213se universitaire\" n\342\210\236\342\200 \240TU\342\200\2400531, INRIA. Defended on May\342\200\24027, 1998. 2 27\342\200\240pages." }}{PARA 15 "" 0 "" {TEXT 228 58 "Gr\313\206bner \+ Bases, Symbolic Summation and Symbolic Integration" }{TEXT 269 77 ", b y Fr\303\210d\303\210ric Chyzak, (1998). In: Buchberger, B. and Winkle r, F., (editors), " }{TEXT 229 86 "Gr\313\206bner Bases and Applicatio ns (Proc. of the Conference ``33\342\200\240Years of Gr\313\206bner Ba ses'')" }{TEXT 269 167 ", Cambridge University Press (London Mathemati cal Society Lecture Notes Series, vol.\342\200\240251), p.\342\200\240 32-60. Preliminary version available as: Research Report n\342\210\23 6 3297, INRIA." }}{PARA 15 "" 0 "" {TEXT 230 74 "An Extension of Zeilb erger's Fast Algorithm to General Holonomic Functions" }{TEXT 269 34 " , by Fr\303\210d\303\210ric Chyzak, (1997). In: " }{TEXT 231 63 "Forma l Power Series and Algebraic Combinatorics, 9th Conference" }{TEXT 269 122 ", Universit\342\200\260t Wien, p.\342\200\240172-183, Confere nce Proceedings. Preliminary version available as: Research Report n \342\210\236\342\200\2403195, INRIA." }}{PARA 15 "" 0 "" {TEXT 232 74 "Non-Commutative Elimination in Ore Algebras Proves Multivariate Ident ities" }{TEXT 269 46 ", by Fr\303\210d\303\210ric Chyzak and Bruno Sal vy, (1998), " }{TEXT 233 31 "Journal of Symbolic Computation" }{TEXT 269 28 ", vol.\342\200\24026, n\342\210\236\342\200\2402, p.\342\200\2 40187-227." }}{PARA 15 "" 0 "" {TEXT 234 52 "Holonomic Systems and Aut omatic Proofs of Identities" }{TEXT 269 72 ", by Fr\303\210d\303\210ri c Chyzak, (1994), Research Report n\342\210\236\342\200\2402371, INRIA . 61\342\200\240pages." }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 270 17 "Sem inar Summaries" }}{EXCHG {PARA 0 "" 0 "" {TEXT 267 120 "Holonomy is a \+ recurrent topic discussed at the Algorithm Seminar. Here are a few su mmaries of sessions dedicated to it." }}}{EXCHG {PARA 15 "" 0 "" {TEXT 235 52 "Introduction aux fonctions holonomes en une variable" }{TEXT 269 37 ", by Philippe Flajolet, (1992). In: " }{TEXT 236 29 "Algorith ms Seminar, 1991-1992" }{TEXT 269 95 ", Ph. Flajolet and P. Zimmermann , (editors), (1992), Research Report n\342\210\236 1779, INRIA. p. 41 -45." }}{PARA 15 "" 0 "" {TEXT 237 41 "Fonctions holonomes \342\200\24 1 plusieurs variables" }{TEXT 269 33 ", by Kevin Compton, (1992). In: " }{TEXT 238 29 "Algorithms Seminar, 1991-1992" }{TEXT 269 95 ", Ph. \+ Flajolet and P. Zimmermann, (editors), (1992), Research Report n\342\2 10\236 1779, INRIA. p. 47-49." }}{PARA 15 "" 0 "" {TEXT 239 29 "Holon omic Symmetric Functions" }{TEXT 269 47 ", by Dominique Gouyou-Beaucha mps, (1992). In: " }{TEXT 240 29 "Algorithms Seminar, 1991-1992" } {TEXT 269 95 ", Ph. Flajolet and P. Zimmermann, (editors), (1992), Res earch Report n\342\210\236 1779, INRIA. p. 51-55." }}{PARA 15 "" 0 "" {TEXT 241 52 "Holonomic Systems and Automatic Proofs of Identities" } {TEXT 269 35 ", by Fr\303\210d\303\210ric Chyzak, (1995). In: " } {TEXT 242 29 "Algorithms Seminar, 1994-1995" }{TEXT 269 22 ", B. Salvy , (editor),\n" }{TEXT 269 50 "(1995), Research Report n\342\210\236 26 69, INRIA. p. 39-42." }}{PARA 15 "" 0 "" {TEXT 243 37 "Creative Teles coping and Applications" }{TEXT 269 35 ", by Fr\303\210d\303\210ric Ch yzak, (1996). In: " }{TEXT 244 29 "Algorithms Seminar, 1995-1996" } {TEXT 269 72 ", B. Salvy, (editor), (1996), Research Report n\342\210 \236 2992, INRIA. p. 39-42." }}{PARA 15 "" 0 "" {TEXT 245 17 "-Finite Functions" }{TEXT 269 35 ", by Fr\303\210d\303\210ric Chyzak, (1996). In: " }{TEXT 246 29 "Algorithms Seminar, 1995-1996" }{TEXT 269 72 ", B. Salvy, (editor), (1996), Research Report n\342\210\236 2992, INRIA . p. 43-46." }}{PARA 15 "" 0 "" {TEXT 247 53 "New Algorithms for Defi nite Summation and Integration" }{TEXT 269 35 ", by Fr\303\210d\303\21 0ric Chyzak, (1997). In: " }{TEXT 248 29 "Algorithms Seminar, 1996-19 97" }{TEXT 269 72 ", B. Salvy, (editor), (1997), Research Report n\342 \210\236 3267, INRIA. p. 27-30." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 131 "Here are also seminar su mmaries of sessions about related topics, either applications or algor ithms to be used internally in Mgfun." }}}{EXCHG {PARA 15 "" 0 "" {TEXT 249 99 "Un algorithme efficace pour le calcul des solutions rati onnelles d'un syst\303\213me diff\303\210rentiel lin\303\210aire" } {TEXT 269 35 ", by Moulay Barkatou, (1997). In: " }{TEXT 250 29 "Algo rithms Seminar, 1996-1997" }{TEXT 269 72 ", B. Salvy, (editor), (1997) , Research Report n\342\210\236 3267, INRIA. p. 31-32." }}{PARA 15 "" 0 "" {TEXT 251 48 "Short and Easy Computer Proofs of Partition and " }{TEXT 252 11 "-Identities" }{TEXT 269 31 ", by Peter Paule, (1995). \+ In: " }{TEXT 253 29 "Algorithms Seminar, 1994-1995" }{TEXT 269 22 ", B . Salvy, (editor),\n" }{TEXT 269 50 "(1995), Research Report n\342\210 \236 2669, INRIA. p. 43-46." }}{PARA 15 "" 0 "" {TEXT 254 26 "Symboli c Computation with " }{TEXT 255 17 "-finite Sequences" }{TEXT 269 35 " , by Marko Petkovsek, (1993). In: " }{TEXT 256 29 "Algorithms Seminar , 1992-1993" }{TEXT 269 72 ", B. Salvy, (editor), (1992), Research Rep ort n\342\210\236 2130, INRIA. p. 51-54." }}{PARA 15 "" 0 "" {TEXT 257 49 "Polynomial Solutions of Linear Operator Equations" }{TEXT 269 35 ", by Marko Petkovsek, (1995). In: " }{TEXT 258 29 "Algorithms Sem inar, 1994-1995" }{TEXT 269 22 ", B. Salvy, (editor),\n" }{TEXT 269 50 "(1995), Research Report n\342\210\236 2669, INRIA. p. 31-34." }} {PARA 15 "" 0 "" {TEXT 259 70 "Linear Recurrences, Linear Differential Equations and Fast Computation" }{TEXT 269 31 ", by Bruno Salvy, (199 6). In: " }{TEXT 260 29 "Algorithms Seminar, 1995-1996" }{TEXT 269 72 ", B. Salvy, (editor), (1996), Research Report n\342\210\236 2992, \+ INRIA. p. 31-38." }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 268 11 "regexpco unt" }}{EXCHG {PARA 15 "" 0 "" {TEXT 261 40 "Regular expressions into \+ finite automata" }{TEXT 269 35 ", by A. Brueggemann-Klein, (1993), " } {TEXT 262 28 "Theoretical Computer Science" }{TEXT 269 21 ", vol. 120, number 2." }}{PARA 15 "" 0 "" {TEXT 263 16 "Motif Statistics" }{TEXT 269 53 ", by P. Nicodeme, B. Salvy, and P. Flajolet, (2002), " }{TEXT 264 30 "Theoretical Computer Science, " }{TEXT 269 19 "vol. 287, numbe r 2." }}}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } Sem inar, 1994-1995" }{TEXT 269 22 ", B. 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Our packages are primarily intended fo r the manipulation of combinatorial structures (specification, generat ion, enumeration, computation of generating functions), for their asym ptotic analysis, and2 the applications to the automatic complexity anal ysis of algorithms, but the library also contains packages for the man ipulation of linear differential and difference operators, Groebner ba sis calculations, and the symbolic summation and integration of specia l functions and combinatorial sequences." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 266 20 "The algolib Pa ckages" }}{EXCHG {PARA 0 "" 0 "" {TEXT 265 24 "Here is the list of the " }{TEXT 212 7 "algolib" }{TEXT 265 10 " packages:" }}}{EXCHG {PARA 15 "" 0 "" {TEXT 267 0 "" }{HYPERLNK 17 "combstruct." 2 "combstruct" " " }{TEXT 267 42 " Combinatorial structures package . The " }{TEXT 213 10 "combstruct" }{TEXT 267 228 " package is used to define and man ipulate a wide range of combinatorial structures. Structures can be c ounted and generated uniformly at random, and in some cases it is poss ible to generate all the structures of a given size. " }{TEXT 214 120 "[Latest contributions by Marni Mishna, Eithne Mur3ray, original ve rsion by Paul Zimmermann. Comments and bug reports to " }{TEXT 215 19 "combstruct@inria.fr" }{TEXT 216 60 ". This version is part of the commercial library of Maple.]" }}{PARA 15 "" 0 "" {TEXT 267 0 "" } {HYPERLNK 17 "encyclopedia." 2 "encyclopedia" "" }{TEXT 267 115 " An \+ encyclopedia of combinatorial structures (March 2002). An on-line ver sion of it is also available at the URL " }{TEXT 217 44 "http://algo.i nria.fr/encyclopedia/index.html" }{TEXT 267 3 ". " }{TEXT 218 111 "[W ritten by Stephanie Petit, with contributions by Bruno Salvy and Miche le Soria. Comments and bug reports to " }{TEXT 219 29 "encyclopedia@p ommard.inria.fr" }{TEXT 220 2 ".]" }{TEXT 267 0 "" }}{PARA 15 "" 0 "" {TEXT 267 0 "" }{HYPERLNK 17 "gdev." 2 "gdev" "" }{TEXT 267 127 " A f acility for more general series expansions and limits (April 2007). U ses a different model fo asymptotic expansions than " }{TEXT 221 5 "Ma ple" }{TEXT 267 3 "'s " }{TEXT 222 6 "asympt" }{TEXT 267 5 " and " } {TEXT 223 6 "s4eries" }{TEXT 267 346 " commands. Includes the equivale nt function mentioned in the survey article (see below) for the asympt otic expansions of coefficients of generating functions. It does asym ptotic expansion of Taylor coefficients, useful in the study of genera ting functions. Apart from the equivalent function, users should now t urn to MultiSeries (see below). " }{TEXT 224 25 "[Written by Bruno Sa lvy.]" }}{PARA 15 "" 0 "" {TEXT 267 0 "" }{HYPERLNK 17 "gfun." 2 "gfun " "" }{TEXT 267 64 " Generating functions package (version 3.21, Apri l 2007). The " }{TEXT 225 4 "gfun" }{TEXT 267 78 " package is used fo r the manipulation and discovery of generating functions. " }{TEXT 226 189 "[Maintained and extended by Bruno Salvy, contributions by Lud ovic Meunier, Marni Mishna and Eithne Murray, original version by Brun o Salvy and Paul Zimmermann. Comments and bug reports to " }{TEXT 227 13 "gfun@inria.fr" }{TEXT 228 2 ".]" }}{PARA 15 "" 0 "" {TEXT 267 0 "" }{HYPERLNK 17 "Groebner." 2 "Groebner" "5" }{TEXT 267 127 " Groeb ner bases package. Implements Groebner basis methods in commutative p olynomial algebras and skew polynomial algebras. " }{TEXT 229 58 "[Wr itten by Frederic Chyzak. Comments and bug reports to " }{TEXT 230 24 "frederic.chyzak@inria.fr" }{TEXT 231 60 ". This version is part o f the commercial library of Maple.]" }}{PARA 15 "" 0 "" {TEXT 267 0 "" }{HYPERLNK 17 "Holonomy." 2 "Holonomy" "" }{TEXT 267 48 " Package fo r the manipulation of holonomic and " }{XPPEDIT 18 0 "d;" "6#%\"dG" } {TEXT 267 187 "-finite functions (version 3.3, September 2006). Deals with functions and sequences that are implicitly defined as solutions of systems of linear differential and difference equations. " }{TEXT 232 58 "[Written by Frederic Chyzak. Comments and bug reports to " } {TEXT 233 24 "frederic.chyzak@inria.fr" }{TEXT 234 2 ".]" }}{PARA 15 " " 0 "" {TEXT 267 0 "" }{HYPERLNK 235 "M" 2 "MAD" "" }{TEXT 267 0 "" } {HYPERLNK 236 "AD" 2 "MAD" "" }{TEXT 205 74 ". Mathematical Abstract D ocum6ent package (version 1.444, September 2003). " }{TEXT 267 53 "A do cument preparation system integrated with Maple, " }{TEXT 237 3 "MAD" }{TEXT 267 11 " defines a " }{HYPERLNK 17 "syntax" 2 "DocumentGenerato r,text" "" }{TEXT 267 116 " for representing a collection of logically structured documents with mathematical content. Such a document is th en " }{HYPERLNK 17 "translated" 2 "DocumentGenerator,Export" "" }{TEXT 267 31 " into various formats, such as " }{HYPERLNK 17 "LaTeX, PostSc ript, PDF" 2 "LaTeX" "" }{TEXT 267 5 " and " }{HYPERLNK 17 "HTML" 2 "H TMX" "" }{TEXT 267 2 ". " }{TEXT 238 29 "[Written by Ludovic Meunier.] " }}{PARA 15 "" 0 "" {TEXT 267 0 "" }{HYPERLNK 17 "Mgfun." 2 "Mgfun" " " }{TEXT 267 80 " Multivariate generating functions package (version \+ 3.3, September 2006). The " }{TEXT 239 5 "Mgfun" }{TEXT 267 246 " pac kage is intended for calculations with multivariate generating functio ns, in particular for their symbolic summation and integration, and fo r the proof of special funct7ion and combinatorial identities. It is a user-oriented interface to the " }{HYPERLNK 17 "Holonomy" 2 "Holonomy " "" }{TEXT 267 34 " package. More information about " }{TEXT 240 5 " Mgfun" }{TEXT 267 24 " to be found at the URL " }{TEXT 241 38 "http:// algo.inria.fr/chyzak/mgfun.html" }{TEXT 267 3 ". " }{TEXT 242 93 "[Wr itten by Frederic Chyzak, with contributions by Cyril Germa. Comments and bug reports to " }{TEXT 243 24 "frederic.chyzak@inria.fr" }{TEXT 244 2 ".]" }}{PARA 15 "" 0 "" {TEXT 267 0 "" }{HYPERLNK 245 "M" 2 "Mul tiSeries" "" }{TEXT 205 0 "" }{HYPERLNK 246 "ultiSeries" 2 "MultiSerie s" "" }{TEXT 205 70 ". Asymptotic expansions in general asymptotic sca les (November 2006). " }{TEXT 247 99 "[Written by Bruno Salvy. This ve rsion is more recent than that in the commercial library of Maple.]" } }{PARA 15 "" 0 "" {TEXT 267 0 "" }{HYPERLNK 17 "Ore_algebra." 2 "Ore_a lgebra" "" }{TEXT 267 114 " Ore algebras package. A package for the m anipulation of linear operators and skew polynomials (8Ore operators). \+ " }{TEXT 248 58 "[Written by Frederic Chyzak. Comments and bug repor ts to " }{TEXT 249 24 "frederic.chyzak@inria.fr" }{TEXT 250 59 ". Thi s version is part of the commercial library of Maple." }{TEXT 251 1 "] " }}{PARA 15 "" 0 "" {TEXT 267 0 "" }{HYPERLNK 17 "regexpcount." 2 "re gexpcount" "" }{TEXT 267 461 " Counting matches of regular expression s (version 1.4, August 2001). A package for general manipulations of \+ regular expressions and (marked) automata, with application to computi ng the probability distributions of motifs occurrences in a random tex t and of waiting times for first matches. Available models of random \+ texts are: uniform and non-uniform Bernoulli models, and Markov model. Approximate matching with bounded number of errors is also handled. \+ " }{TEXT 252 58 "[Written by Pierre Nicodeme. Comments and bug repor ts to " }{TEXT 253 24 "pierre.nicodeme@inria.fr" }{TEXT 254 2 ".]" }}} }{SECT 1 {PARA 3 "" 0 "" {TEXT 266 50 "Documentation, Demos, and Relat e9d Research Papers." }}{EXCHG {PARA 0 "" 0 "" {TEXT 265 13 "Beside usu al " }{TEXT 255 5 "Maple" }{TEXT 265 21 " on-line help pages, " } {HYPERLNK 17 "demonstration pages" 2 "autocomb" "" }{TEXT 265 5 " for " }{TEXT 256 4 "gfun" }{TEXT 265 2 ", " }{TEXT 257 10 "combstruct" } {TEXT 265 91 ", and other combinatorial packages are available in the \+ form of combinatorial case studies." }}{PARA 0 "" 0 "" {TEXT 265 0 "" }}{PARA 0 "" 0 "" {TEXT 265 281 "This software is part of an overall f ramework to study decomposable combinatorial structures and their gene rating functions. A survey article by covers this aspect or our resea rch (Computer Algebra Libraries for Combinatorial Structures, Philippe Flajolet and Bruno Salvy (1995), " }{TEXT 258 31 "Journal of Symbolic Computation" }{TEXT 265 125 ", 20:5-6, p. 653-671). A list of more r esearch papers related to theoretical aspects relevant to this softwar e can be found " }{HYPERLNK 17 "here" 2 "algolib,references" "" }{TEXT 265 1 "." }}}}{SECT 1 {PARA 3 ":" 0 "" {TEXT 266 29 "Updating your alg olib Library" }}{EXCHG {PARA 0 "" 0 "" {TEXT 265 70 "Parts of the libr ary have been introduced in various past releases of " }{TEXT 259 5 "M aple" }{TEXT 265 20 ", specifically, the " }{TEXT 260 10 "combstruct" }{TEXT 265 2 ", " }{TEXT 261 11 "Ore_algebra" }{TEXT 265 6 ", and " } {TEXT 262 8 "Groebner" }{TEXT 265 139 " packages. The present distrib ution contains more recent releases of some of these packages, which f ix bugs and/or add new functionality. " }}{PARA 0 "" 0 "" {TEXT 265 0 "" }}{PARA 0 "" 0 "" {TEXT 265 76 "The most recent versions of the lib rary are available on the web at the URL " }{TEXT 263 31 "http://algo. inria.fr/libraries/" }{TEXT 265 1 "." }}}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } ", 20:5-6, p. 653-671). A list of more r esearch papers related to theoretical aspects relevant to this softwar e can be found " }{HYPERLNK 17 "here" 2 "algolib,references" "" }{TEXT 265 1 "." }}}}{SECT 1 {PARA 3 ":MAD,MADMADDocument,MADMath,DocumentGeneratorgMathAlias,DocumentGenerator~MathKeyword,DocumentGeneratorZMathKeyword,LaTeXMathStyle,DocumentGeneratortMathStyle,HTMXMathStyle,LaTeXkMathSymbol,LaTeXRMathType,DocumentGeneratorNth,DocumentGenerator,OreAlgebra,type1Ore_to_DESol,Ore_algebra0ROre_to_RESol,Ore_algebra0ROre_to_diff,Ore_algebra0ROre_to_shift,Ore_algebra0RParagraph,DocumentGeneratorPlot,DocumentGenerator@Pollard_algo,autocomb;Ref,DocumentGeneratorRelativePath,CommonLib$Section,DocumentGeneratorShortTermOrder,type/SingPlur,DocumentGeneratoro=tool+ ,8o=Mrtop#'XOmV|stopictopmosttord  toroidaltorontortot 4;total; $+4mVlPJ;rStouchtowertrXO dy trac@(track@(lrtrad  traditional trailktraingltran pH^;otransact transcendental transf transformS,4 B78[:pHclՓ)&'@(introductmgfunpackagconsistprocedurcomputwithspecialfunctioncombinatorialsequencimplicitdefinsolutionsystemlineardifferencdifferentialequationsyntaxveryclosgfunsamespiritlatttypicalusesumintegralfollowfirstobtainsummandintegrandcallcombinknownusingderivfinalsolvformexistweexemplifworkoutevaluatmehlformulahermitpolynomialinputsincoverneeddependenctermsessfoculeavparameterbutotherchoichavemadeyieldtracsysdfinitexprshiftdiffsummatitselfperformviewformalpowerserigoestendinfinitatboundarrecurrencprovnegatoptionthereforfulljustifinaturalonlyremainequatdsolvunionsimplifexpandsymbolicwordstudparametvalumakewellsayhereagainourvariablrespectwantcontainintegratnotakeplacjustificatmaythuswillconsidneighbourhoodexponentialsmallvaluatanycombinatderivatfunctcoefficientprovidbecomsufficientlarginttakayamaalgoabovcancelunfortunatedoknowpointlowerboundvalidfacenongfun,cauchyproduct=gfun,diffeq*diffeqtgfun,diffeq+diffeqtgfun,diffeqtohomdiffeqOgfun,diffeqtorec gfun,gftypesg gfun,guesseqnZv gfun,guessgfZvgfun,hadamardproducttgfun,holexprtodiffeq ; gfun,invborel'gfun,listtoalgeqgfun,listtodiffeqdgfun,listtohypergeomMgfun,listtolist[gfun,listtoratpoly gfun,listtorec·gfun,listtoseries[gfun,maxdegcoeff7pHGvariatsequencaprynumberfrricchyzakversjanuarearldoronzeilbergherbertwilfdeveloppnewmethodologsymbolicsummatintegratalgorithmicprooftheorhypergeometricordinarmultisumintegralidentitinventionmathematicamostfamousuccesscallwzmethodprovidcomputcombinatorialprovthessatisfsecondorderrecurrencequatcrucialstepirrationalitotherhanditselfstemnumbtheoreticquestraisschmidtasmugenerallegendrpolynomialreinangewmathseveralabovrelatfranelweregivenstrehlvolkbinomialaspectdiscretthemparticularbasedalgorithmyieldproductfollowsectionwefirstrecalhowledborrowvanderpoortenreportalfreulermissintelligencnextgivebothresultusingourpackagfinalexploitbeatmaplmanydigitsketchremarkdoubltenduniforminfinitalternatseriboundhowevconvergencstrongenoughshowusedacceleratnamedefinalsohereappearsatisfiwithinitialconditargumpositsufficiirrationalmeasuratleastsesschiefuseuserorientmgfunlefipl ; iplbt;iplcc;iplfg;iplt;iquo6 irrationalpH irrationalit pH  irregular 4  irregularito irrelevantisingiso XOisolat 4l;isolatinLisomera isomorphic(la isomorphism(issacissuvistingitalic XOitem [dyintegralproductfourbesselfunctionfrricchyzakversjanuarglassmontaldisomeinvolvmathanalapplcomputclosformsuggesttheirtreatmextendfollowexamplinterestbecauscontaineachtypenumeroumoreinstancprudnikovbrychkovyumarichevserivolumspecialgordonbreachsecsesswedealwithabovderivusingourpackagintimatinteractmgfungfunspecificalwillusedpreparsystempdeapplicatsolvodeoutputsearchsatisfiintegrandusefactornextackagidentitfuncttrivialsatisfdifferentialentrsetdenotequatfirstrespectvariablconsidersymmetrcompletholexprtodiffeqdeqopremovsyssamewaypreviousectencoddoesdependnowsimplnoteroutinperformsimilartaskbutrestrictunivariatcasedescribsinglthuscanonlycrosderivatturnoutinduclossinformatprecludcalculatbesidalsomixeddifferencequationintegratagainitselfintinfinittakayamaalgohowevjustificatalgorithmselectoptionunderconsideratrathtechnicalbeyondpresentatresolutfinalsolutco algebraic{? 0#$+467}gv;0e^)S algebraical gS algebraicsub4^)algeq }v algeqtodiffeq  algeqtoser 6 algfuntoalgeq6^algoC. "@(pHngl  o0valgolib   algorithm  Z*!+468pH^l;;0e rmialskewfredericchyzakholonomicfinitseptembdealimplicitsolutionsystemequationmathematicalabstractdocumpreparatmadrepresentcollectlogicaldocumentcontsuchintovariouformatmultivariatmgfunparticularproofidentitorientinterfacinformataboutfoundcyrilgermascalnovembrecentorematchregularexpressaugustmarkautomataprobabilitdistributmotifoccurrenctextwaittimefirstnonbernoullimarkovapproximatboundnumberrorhandlpierrnicodemdocumentatdemorelatpaperbesidusualhelppageotherformsoftwaroveralframeworkdecomposablcoveraspectphilippflajoletjournaltheoreticalrelevantupdatyourhaveintroducpastreleasspecificalpresthesfixbugsaddnewfunctionalitmostversionwebardfacilitmoregeneralseriexpanslimitaprilusesdiffermodelfoasymptcommandincludequivalfunctmentionsurvearticlbelowcoefficientdoestaylorusefulstudapartusernowturnmultisergfundiscovermaintainextendludovicmeunibaseimplementmethodcommutatpolynoC "Holonomy[algeq_to_dfinite] - Compute system of linear differentia l operators that cancel an algebraic function" }}{PARA 0 "" 0 "usage" {TEXT 26 17 "Calling Sequence:" }{TEXT -1 21 "\n algeq_to_dfinite(" }{XPPEDIT 18 0 "P;" "6#%\"PG" }{TEXT -1 1 "," }{XPPEDIT 18 0 "f;" "6#% \"fG" }{TEXT -1 1 "," }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 1 ")" }} {PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 4 "\n " } {XPPEDIT 18 0 "P;" "6#%\"PG" }{TEXT 23 3 " - " }{TEXT -1 54 "a polynom ial that determines an algebraic function\n " }{XPPEDIT 18 0 "f;" "6 #%\"fG" }{TEXT 23 3 " - " }{TEXT -1 38 "the name of the algebraic func tion\n " }{XPPEDIT 18 0 "T;" "6#%\"TG" }{TEXT 23 3 " - " }{TEXT -1 58 "a term order over an Ore algebra of differential operators" }}} {SECT 0 {PARA 0 "" 0 "synopsis" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 114 "The command algeq_to_dfinite computes a system o f linear differential operators that cancel an algebraic function " } {XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 1 G)TX{VERSION 2 3 "DEC ALPHA UNIX" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "2 D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0F 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Out put" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "T itle" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0G 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 16 "GFUN AND THE AGM" }} {PARA 19 "" 0 "" {TEXT 257 11 "Bruno Salvy" }}{PARA 261 "" 0 "" {TEXT -1 12 "January 1998" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 4 "Let " }{XPPEDIT 18 0 "a" "I\"aG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b" "I\"bG6\"" }{TEXT -1 36 " be two positive real numb ers, with " }{XPPEDIT 18 0 "a>b" "2%\"bG%\"aG" }{TEXT -1 6 ". The " } {TEXT 258 20 "arithmetic-geometric" }{TEXT -1 9 " mean of " }{XPPEDHIT 18 0 "a" "I\"aG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b" "I\"bG6\"" }{TEXT -1 61 " is classically defined as the common limit of the seque nces " }{XPPEDIT 18 0 "a[k]" "&%\"aG6#%\"kG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "b[k]" "&%\"bG6#%\"kG" }{TEXT -1 12 " defined by\n" } {XPPEDIT 18 0 "a[k+1]=(a[k]+b[k])/2, b[k+1]=sqrt(a[k]*b[k])" "6$/&%\"a G6#,&%\"kG\"\"\"\"\"\"F)*&,&&F%6#F(F)&%\"bG6#F(F)F)\"\"#!\"\"/&F06#,&F (F)\"\"\"F)-%%sqrtG6#*&&F%6#F(F)&F06#F(F)" }{TEXT -1 7 ", with " } {XPPEDIT 18 0 "a[0]=a" "/&%\"aG6#\"\"!F$" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "b[0]=b" "/&%\"bG6#\"\"!F$" }{TEXT -1 68 ".\nThat the se quences converge to the same limit can be inferred from" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[k+1]^2-b[k+1]^2=((a [k]-b[k])/2)^2" "/,&*$&%\"aG6#,&%\"kG\"\"\"\"\"\"F*\"\"#F**$&%\"bG6#,& F)F*\"\"\"F*\"\"#!\"\"*$*&,&&F&6#F)F*&F/6#F)F4F*\"\"#F4\"\"#" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "This common limit is k nown by Maple as " }{HYPERLNK 17 "GaussAGM" I2 "GaussAGM" "" }{XPPEDIT 18 0 "``(a,b)" "-%!G6$%\"aG%\"bG" }{TEXT -1 105 ". It was discovered b y Gauss that the arithmetic-geometric mean is related to hypergeometri c functions by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "GaussAGM( a,b)=a/hypergeom([1/2, 1/2],[1],1-b^2/a^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%)GaussAGMG6$%\"aG%\"bG*&F'\"\"\"-%*hypergeomG6%7$#F* \"\"#F/7#F*,&F*F**&F(F0F'!\"#!\"\"F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eval(subs(a=3.,b=2.,\"));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/$\"+O/ouC!\"*$\"+P/ouCF&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "This worksheet, largely inspired by [1], shows how " } {HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 123 " can be used to guess a nd then prove this result, as well as a generalization of it due to J. M. Borwein and P. B. Borwein." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 23 "The functional equation" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 143 "Fo llowing [1], we start by introducing a generalization of the arithmeti c-geometric mean obtaiJned by considering the following iteration where " }{XPPEDIT 18 0 "N>1" "2\"\"\"%\"NG" }{TEXT -1 16 " is an integer: \+ " }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[k+1]=(a [k]+(N-1)*b[k])/N,b[k+1]=(a[k+1]^N-((a[k]-b[k])/N)^N)^(1/N)" "6$/&%\"a G6#,&%\"kG\"\"\"\"\"\"F)*&,&&F%6#F(F)*&,&%\"NGF)\"\"\"!\"\"F)&%\"bG6#F (F)F)F)F1F3/&F56#,&F(F)\"\"\"F)),&)&F%6#,&F(F)\"\"\"F)F1F))*&,&&F%6#F( F)&F56#F(F3F)F1F3F1F3*&\"\"\"F)F1F3" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "where the second equality is motivated by" }}} {EXCHG {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[k+1]^N-b[k+1 ]^N=((a[k]-b[k])/N)^N" "/,&)&%\"aG6#,&%\"kG\"\"\"\"\"\"F*%\"NGF*)&%\"b G6#,&F)F*\"\"\"F*F,!\"\")*&,&&F&6#F)F*&F/6#F)F3F*F,F3F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "from which follows that both sequences co nverge to a common limit, which is denoted by " }{XPPEDIT 18 0 "M[N](a ,b)" "-&%\"MG6#%\"NG6$%\"aG%\"bG" }{TEXT -1 56 ". The arithmetic-geome tric mean corresponds to the case " }{XPPEDIT 18 0 "KN=2" "/%\"NG\"\"# " }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The function \+ " }{XPPEDIT 18 0 "M[N](a,b)" "-&%\"MG6#%\"NG6$%\"aG%\"bG" }{TEXT -1 35 " is easily seen to be homogeneous: " }{XPPEDIT 18 0 "M[N](lambda*a ,lambda*b)=lambda*M[N](a,b)" "/-&%\"MG6#%\"NG6$*&%'lambdaG\"\"\"%\"aGF +*&F*F+%\"bGF+*&F*F+-&F%6#F'6$F,F.F+" }{TEXT -1 6 ", for " }{XPPEDIT 18 0 "lambda>0" "2\"\"!%'lambdaG" }{TEXT -1 42 ". Together with the ob vious property that " }{XPPEDIT 18 0 "M[N](a[0],b[0])=M[N](a[1],b[1]) " "/-&%\"MG6#%\"NG6$&%\"aG6#\"\"!&%\"bG6#F,-&F%6#F'6$&F*6#\"\"\"&F.6# \"\"\"" }{TEXT -1 24 ", this implies that for " }{XPPEDIT 18 0 "x" "I \"xG6\"" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "` `(0,1)" "-%\"~G6$\"\"!\" \"\"" }{TEXT -1 1 "," }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "M[N](1,(1-x^N)^(1/N))=(1+(N-1)*x)*M[N](1,(1-x)/(1+(N-1) *x))" "/-&%\"MG6#%\"NG6$\"\"\"),&\"\"\"\"\"\")%\"xGF'!\"\"*&\"\"\"F-F' F0*&,&\"\"\"F-*&,&F'F-\"\"\"F0F-F/F-F-F--&F%6#F'6$\"\"\"*&,&\"\"\"F-F/ F0F-,&\"\"\"F-*&L,&F'F-\"\"\"F0F-F/F-F-F0F-" }{TEXT -1 2 ". " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Defining the function " }{XPPEDIT 18 0 "A[N](x)" "-&%\"AG6#%\"NG6#%\"xG" }{TEXT -1 3 " by" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[N](x)=1/M[N](1,(1-x )^(1/N))" "/-&%\"AG6#%\"NG6#%\"xG*&\"\"\"\"\"\"-&%\"MG6#F'6$\"\"\"),& \"\"\"F,F)!\"\"*&\"\"\"F,F'F6F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "the equation above translates into the following " }{TEXT 259 19 " functional equation" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "A[N](x)" "-&% \"AG6#%\"NG6#%\"xG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "funeq:=(1+(N-1)*x)*A[N](x^N)=A[N](1-((1-x)/(1+(N-1)*x)) ^N):" ">%&funeqG/*&,&\"\"\"\"\"\"*&,&%\"NGF(\"\"\"!\"\"F(%\"xGF(F(F(-& %\"AG6#F+6#)F.F+F(-&F16#F+6#,&\"\"\"F()*&,&\"\"\"F(F.F-F(,&\"\"\"F(*&, &F+F(\"\"\"F-F(F.F(F(F-F+F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "wh ich plays a central r\364le in this worksheet. It is not too difficult to show that " }{XPPEDIT 18 0 "A[N](x)" "-&%\"AG6#%\"NG6#%M\"xG" } {TEXT -1 139 " is analytic in the neighborhood of the origin and that \+ the functional equation above has a unique analytic solution in this n eighborhood. " }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 18 "The quadratic \+ case" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "This is the case " } {XPPEDIT 18 0 "N=2" "/%\"NG\"\"#" }{TEXT -1 51 " and Gauss's theorem i s equivalent to stating that " }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[2](x)=hypergeom([1/2,1/2],[1],x)" "/-&%\"AG6#\" \"#6#%\"xG-%*hypergeomG6%7$*&\"\"\"\"\"\"\"\"#!\"\"*&\"\"\"F0\"\"#F27# \"\"\"F)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "We no w use " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 126 " to first gues s and then prove this result. The first step is to use the functional \+ equation to compute a series expansion of " }{XPPEDIT 18 0 "A[2](x)" " -&%\"AG6#\"\"#6#%\"xG" }{TEXT -1 168 ", then we use this series to gue ss a possible closed form which turns out to be analytic, then we show that this analyticN function does satisfy the functional equation." }} }{SECT 0 {PARA 4 "" 0 "" {TEXT -1 16 "Series expansion" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Starting from the functional equation," } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "funeq2:=subs(N=2,A[2]=A,op (1,funeq)-op(2,funeq));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'funeq2G, &*&,&\"\"\"F(%\"xGF(F(-%\"AG6#*$F)\"\"#F(F(-F+6#,&F(F(*&,&F(F(F)!\"\"F .F'!\"#F4F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "a series solution \+ is easily obtained by a method of undeterminate coefficients:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "sol:=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "for i to 12 do \n sol:=sol+x^i*solve(op(1, series(eval(\n subs(A=unapply(sol+a*x^i,x),funeq2)),x,i+2)),a) od:so l;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,<\"\"\"F$%\"xG#F$\"\"%*$F%\"\"# #\"\"*\"#k*$F%\"\"$#\"#D\"$c#*$F%F'#\"%D7\"&%Q;*$F%\"\"&#\"%pR\"&Ob'*$ F%\"\"'#\"&hL&\"(w&[5*$F%\"\"(#\"'TS=\"(/V>%*$F%\"\")#\")D#49%\"+C=ut5 *$F%F+#\"*DSuZ\"\"+'Hn\\H%*$F%\"#5#\"+@PUL@\",OnZ>(oO*$F%\"#6#\"+Tg`vx \"-Wp!z([F*$F%\"#7#\"-@&H(GqX\"/;W/'=#f<" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 21 "Guessing the solution" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "From this series, " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 61 " guesses a differential equation which could be satisfied by " } {XPPEDIT 18 0 "A(x)" "-%\"AG6#%\"xG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "deq:=op(1,gfun[seriestodiffeq](series(sol ,x,13),y(x),[ogf]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$deqG<%/-%\" yG6#\"\"!\"\"\",(-F(6#%\"xGF+*&,&!\"%F+F/\"\")F+-%%diffG6$F-F/F+F+*&,& F/F2*$F/\"\"#\"\"%F+-F56$F4F/F+F+/--%\"DG6#F(F)#F+F;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "It turns out that Maple's dsolve function is un able to solve this differential equation:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "dsolve(deq,y(x));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "We then use " }{HYPERLNK 17 "gfun[diffeqtorec]" 2 "gfun[diffeqt orec]" "" }{TEXT -1 116 " which deduces from this differential equatio n the recurrencPe satisfied by the Taylor coefficients of its solutions :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "gfun[diffeqtorec](deq, y(x),u(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%,&*&,(\"\"\"F'%\"nG\" \"%*$F(\"\"#F)F'-%\"uG6#F(F'F'*&,(F(!\")!\"%F'F*F2F'-F-6#,&F(F'F'F'F'F '/-F-6#\"\"!F'/-F-6#F'#F'F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Fr om this first order linear recurrence, a solution is easily found:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "rsolve(\",u(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%&GAMMAG6#,&%\"nG\"\"\"#F)\"\"#F)F+-F%6#, &F(F)F)F)!\"#%#PiG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "hence \+ the sum:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "y(x)=sum(\"*x^n ,n=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%*h ypergeomG6%7$#\"\"\"\"\"#F,7#F-F'" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 31 "Proving the result of the guess" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "The proof consists in showing that the function " } {XPPEDIT 18 0 "y(x)" "-%\"yG6#%\"xG" }{TEXT -1 64 ",Q which is obviousl y analytic, satisfies the functional equation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs(A=y,funeq2)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,&\"\"\"F'%\"xGF'F'-%\"yG6#*$F(\"\"#F'F'-F*6#,&F'F '*&,&F'F'F(!\"\"F-F&!\"#F3F3\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "Our approach consists in using closure properties of solutions of linear differential equations that are implemented in " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 284 " to compute a linear differential equation satisfied by the left-hand side of this equation. The proof \+ then reduces to showing that 0 is the only solution of this differenti al equation that is compatible with the initial conditions, which are \+ 0 up to a large order by construction of " }{XPPEDIT 18 0 "y" "I\"yG6 \"" }{TEXT -1 144 ".\nIt turns out that this proof can be performed di rectly from the differential equation, and would apply even if no clos ed-form had been found. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Given a linear diffRerential equation satisfied by a series " }{XPPEDIT 18 0 "y(x)" "-%\"yG6#%\"xG" }{TEXT -1 15 ", the function " }{HYPERLNK 17 "gfun[algebraicsubs]" 2 "gfun[algebraicsubs]" "" }{TEXT -1 55 " comput es a linear differential equation satisfied by " }{XPPEDIT 18 0 "y(f( x))" "-%\"yG6#-%\"fG6#%\"xG" }{TEXT -1 51 " for any algebraic function , given by a polynomial " }{XPPEDIT 18 0 "P" "I\"PG6\"" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "P(x,f(x))=0" "/-%\"PG6$%\"xG-%\"fG6#F&\" \"!" }{TEXT -1 44 ". Thus a differential equation satisfied by " } {XPPEDIT 18 0 "y(1-(1-x)^2/(1+x)^2)" "-%\"yG6#,&\"\"\"\"\"\"*&,&\"\"\" F'%\"xG!\"\"\"\"#*$,&\"\"\"F'F+F'\"\"#F,F," }{TEXT -1 43 " is easily c omputed from that satisfied by " }{XPPEDIT 18 0 "y(x)" "-%\"yG6#%\"xG " }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "deq:=op (select(has,deq,x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "deq 1:=gfun[algebraicsubs](deq,numer(y-(1-(1-x)^2/(1+x)^2)),y(x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%deq1G,(*&,&!\"\"S\"\"\"%\"xGF)F)-%\" yG6#F*F)F)*&,**$F*\"\"$F(*$F*\"\"#!\"$F*F(F)F)F)-%%diffG6$F+F*F)F)*&,* *$F*\"\"%F(F0F(F2F)F*F)F)-F66$F5F*F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Similarly, " }{XPPEDIT 18 0 "y(x^2)" "-%\"yG6#*$%\"xG\"\" #" }{TEXT -1 10 " satisfies" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "gfun[algebraicsubs](deq,y-x^2,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&-%\"yG6#%\"xG\"\"\"F(F)F)*&,&!\"\"F)*$F(\"\"#\"\"$F)-%%diffG 6$F%F(F)F)*&,&F(F,*$F(F/F)F)-F16$F0F(F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "and its product by " }{XPPEDIT 18 0 "-(1+x)" ",$,&\"\"\" \"\"\"%\"xGF%!\"\"" }{TEXT -1 10 " satisfies" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "deq2:=gfun[`diffeq*diffeq`](\",y(x)+1+x,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%deq2G,(*&,&\"\"\"F(%\"xG!\"\"F(-% \"yG6#F)F(F(*&,*F)F(*$F)\"\"$F(*$F)\"\"#F1F*F(F(-%%diffG6$F+F)F(F(*&,* F)F*F2F*F0F(*$F)\"\"%F(F(-F56$F4F)F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 165 "From there, we deduce a differential equation satisfied \+ by the left-hanTd side of the functional equation when applied to the h ypergeometric function we have guessed:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "gfun[`diffeq+diffeq`](deq1,deq2,y(x));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,(*&,&\"\"\"F&%\"xG!\"\"F&-%\"yG6#F'F&F&*&,*F'F& *$F'\"\"$F&*$F'\"\"#F/F(F&F&-%%diffG6$F)F'F&F&*&,*F'F(F0F(F.F&*$F'\"\" %F&F&-F36$F2F'F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Analytic so lutions of this equation have a coefficient sequence which satisfies \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "gfun[diffeqtorec](\",y( x),u(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<&,**&%\"nG\"\"#-%\"uG6#F &\"\"\"F+*&,(F'F+*$F&F'F+F&\"\"%F+-F)6#,&F&F+F+F+F+F+*&,(F&!\"#F+F+F.! \"\"F+-F)6#,&F&F+F'F+F+F+*&,(F&!\"'!\"*F+F.F6F+-F)6#,&F&F+\"\"$F+F+F+/ -F)6#F+,$&%#_CG6#\"\"!F//-F)6#F'FF/-F)FHFE" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "and thus the first three zeroes of the Taylor expansion \+ of the left-hand side of the functional equation conclude the proof." }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1U4 "The cubic case" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "It has been discovered by J. M. Borwein \+ and P. B. Borwein that a hypergeometric expression also exists when " }{XPPEDIT 18 0 "N=3" "/%\"NG\"\"$" }{TEXT -1 91 ". Again, the same ste ps as above lead to guessing and then proving the following result by \+ " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 " " 0 "" {TEXT 262 7 "Theorem" }{TEXT -1 25 ". [Borwein & Borwein 90] " }{TEXT 263 13 "The function " }{XPPEDIT 264 0 "A[3](x)" "-&%\"AG6#\"\" $6#%\"xG" }{TEXT 265 77 " corresponding to the AGM iteration of order \+ 3 has the following closed form:" }}}{EXCHG {PARA 263 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[3](x)=hypergeom([1/3,2/3],[1],x)" "/-&%\"AG 6#\"\"$6#%\"xG-%*hypergeomG6%7$*&\"\"\"\"\"\"\"\"$!\"\"*&\"\"#F0\"\"$F 27#\"\"\"F)" }{TEXT -1 1 "." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 16 "S eries expansion" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "We start from t he functional equation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0V 49 "f uneq3:=subs(N=3,A[3]=A,op(1,funeq)-op(2,funeq));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'funeq3G,&*&,&\"\"\"F(%\"xG\"\"#F(-%\"AG6#*$F)\"\"$F( F(-F,6#,&F(F(*&,&F(F(F)!\"\"F/F'!\"$F5F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "and compute the first terms of the series expansion of th e solution:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "sol:=1:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "for i to 12 do \n sol:=so l+x^i*solve(op(1,series(eval(\n subs(A=unapply(sol+a*x^i,x),funeq3)) ,x,i+2)),a) od:sol;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,<\"\"\"F$%\"xG #\"\"#\"\"**$F%F'#\"#5\"#\")*$F%\"\"$#\"$g&\"%hl*$F%\"\"%#\"%]Q\"&\\!f *$F%\"\"&#\"&G!G\"'T9`*$F%\"\"'#\"(/f!>\")@n/V*$F%\"\"(#\")![!y9\"**[? uQ*$F%\"\")#\"*q\"eo6\"+,Wy'[$*$F%F(#\",+0\"y&f(\".H$Ge'=a#*$F%F+#\"-g 7Uxnh\"/h\\X#zwG#*$F%\"#6#\".?vQbl0&\"0\\Y4K6*e?*$F%\"#7#\"0+o%pI\"3w$ \"2plm*p\"=xm\"" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 21 "Guessing the solution" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Again, this is a luck y situation where a difWferential equation can be guessed:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "deq:=op(1,gfun[seriestodiffeq](seri es(sol,x,13),y(x),[ogf]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$deqG< %/--%\"DG6#%\"yG6#\"\"!#\"\"#\"\"*/-F+F,\"\"\",(-F+6#%\"xGF/*&,&!\"*F3 F7\"#=F3-%%diffG6$F5F7F3F3*&,&F7F:*$F7F/F0F3-F=6$F " 0 "" {MPLTEXT 1 0 33 "gfun[diffeqtorec](deq ,y(x),u(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/-%\"uG6#\"\"\"#\"\" #\"\"*/-F&6#\"\"!F(,&*&,(F*F(%\"nGF+*$F3F*F+F(-F&6#F3F(F(*&,(F3!#=!\"* F(F4F:F(-F&6#,&F3F(F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "rsolve(\",u(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,-%&GAMMAG6 #,&%\"nG\"\"\"#\"\"#\"\"$F*F*-F&6#,&F)F*#F*F-F*F*-F&6#,&F)F*F*F*!\"#%# PiG!\"\"F-#F*F,F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "y(x)=s um(\"*x^n,n=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6# %\"xG-%*hypergeomG6%7$#\"\"\"\"\"$#\"\"#F.7#F-F'" }X}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 31 "Proving the result of the guess" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "The same routine applies:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs(A=y,funeq3)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,&\"\"\"F'%\"xG\"\"#F'-%\"yG6#*$F(\"\"$F'F'-F+6#,& F'F'*&,&F'F'F(!\"\"F.F&!\"$F4F4\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "deq:=op(select(has,deq,x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "deq1:=gfun[algebraicsubs](deq,numer(y-(1-(1-x)^3 /(1+2*x)^3)),y(x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%deq1G,(*&,( \"\"#\"\"\"%\"xG!\"%*$F*F(F(F)-%\"yG6#F*F)F)*&,,!\"\"F)*$F*\"\"&\"\")* $F*\"\"%\"#7*$F*\"\"$F7F,F7F)-%%diffG6$F-F*F)F)*&,.*$F*\"\"'F7F3F7F6F) F9F+F,F+F*F2F)-F<6$F;F*F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "gfun[algebraicsubs](deq,y-x^3,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&-%\"yG6#%\"xG\"\"\"F(\"\"#F**&,&!\"\"F)*$F(\"\"$\"\"%F)-%%di ffG6$F%F(F)F)*&,&F(F-*$F(F0F)F)-F26$F1F(F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4Y7 "deq2:=gfun[`diffeq*diffeq`](\",y(x)+1+2*x,y(x)); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%deq2G,(*&,(\"\"#\"\"\"%\"xG!\"% *$F*F(F(F)-%\"yG6#F*F)F)*&,,!\"\"F)*$F*\"\"&\"\")*$F*\"\"%\"#7*$F*\"\" $F7F,F7F)-%%diffG6$F-F*F)F)*&,.*$F*\"\"'F7F3F7F6F)F9F+F,F+F*F2F)-F<6$F ;F*F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "gfun[`diffeq+dif feq`](deq1,deq2,y(x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,(*&,(\"\"# \"\"\"%\"xG!\"%*$F(F&F&F'-%\"yG6#F(F'F'*&,,!\"\"F'*$F(\"\"&\"\")*$F(\" \"%\"#7*$F(\"\"$F5F*F5F'-%%diffG6$F+F(F'F'*&,.*$F(\"\"'F5F1F5F4F'F7F)F *F)F(F0F'-F:6$F9F(F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "g fun[diffeqtorec](\",y(x),u(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<(, .*&,&*$%\"nG\"\"#\"\"%F(F*\"\"\"-%\"uG6#F(F+F+*&,(F(\"#;\"#7F+F'F*F+-F -6#,&F(F+F+F+F+F+*&,(F'F+F(\"\"(F2F+F+-F-6#,&F(F+F)F+F+F+*&,(!#;F+F(F> F'!\"%F+-F-6#,&F(F+\"\"$F+F+F+*&,(!#YF+F'F?F(!#GF+-F-6#,&F(F+F*F+F+F+* &,(F(!#5!#DF+F'!\"\"F+-F-6#,&F(F+\"\"&F+F+F+/-F-6#F)\"\"!/-F-6#F*&%#_C G6#FW/-F-6#FC,$Fen#F+F)/-F-Fgn,$Fen#\"\"*F*/-FZ-6#F+,$Fen#FaoF)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "and thus the first five zeroes of the Taylor expansion of the left-hand side of the functional equation conclude the proof." }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Conclu sion" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "These results are very goo d examples of the use of " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 384 ": experiments first lead to conjecture a general form for the sol ution to a problem and then a completely different process leads to a \+ proof. However, the apparent ease with which the problems treated here are solved using gfun hides the preliminary work which led to the for m under which this approach could work. For example this approach does not seem to work for higher values of " }{XPPEDIT 18 0 "N" "I\"NG6\" " }{TEXT -1 36 ", where similar results might exist." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "Bibliography" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "[1] Arithmetic-Goemetric Means Revisited. Jonathan M. Bor wein, Pet[r Lisonek and John A. Macdonald. " }{TEXT 260 9 "MapleTech" } {TEXT -1 2 ", " }{TEXT 261 3 "4-1" }{TEXT -1 19 ", pp. 20-27 (1997)." }}}}}{MARK "0 4 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 } f." }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Conclu sion" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "These results are very goo d examples of the use of " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 384 ": experiments first lead to conjecture a general form for the sol ution to a problem and then a completely different process leads to a \+ proof. However, the apparent ease with which the problems treated here are solved using gfun hides the preliminary work which led to the for m under which this approach could work. For example this approach does not seem to work for higher values of " }{XPPEDIT 18 0 "N" "I\"NG6\" " }{TEXT -1 36 ", where similar results might exist." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "Bibliography" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "[1] Arithmetic-Goemetric Means Revisited. Jonathan M. Bor wein, Pet[awenumeratalcoholotherclasschemicalmoleculexamplpolyatheorfredericchyzakversjanuaralkansimplclascompoundtheygenericaldescribformulafirstsmallmethanethanpropanbutangivenhowevexistseveraldifferisomerstructurbondbetweenatomchemistrmuchinterestknownumbbettyetlistsuchobtainreplachydrogengroupfollowisomorphiccarbonchainwithdistinguishnodeagainalkylradicalmisswedisregardgeometricalconstraintconsidstructuralonlyconformationalleadpuregraphtheoreticalproblemhowmanyroottreeinternaleachdegresessthuscountgenericcombinatoricalsocorrespondorganometalicanymonosubstitutnexttreatcasedisubstituttrisubstitutdevelopstudourmodelusingpackagcombstructpartreferbookreadcombinatorialspringverlagmoreextensresultsectwithoutaccordheightgeneraldefinitcanviewlinkatmosttakeintoaccountimplicitnolossinformatsincalwayrecoverskeletonyieldequatmapgrammargrammprodsetcardspecunlabellnoteconstg gfungftypdescribavailabltypegeneratfunctionfunctanalyticencodnumericaldataformalpowersericanmanipulatalgebraicalwaysparalleloftencombinatorialobjecttheyreprespackagrecognisseveraldifferinformatlistfollowknownogfegfrevogfrevegflgdogflgdegflaplacordinarcoefficientelementexamplcorrespondexponentialthcoefficiopreciprocallogarithmicderivatuserdefinhisowncreatprocedurlisttosermytypeofgftakevariablinputyieldmusttaylorparticularcannothavenegatexponentalsorballurnsetcphilippflajoletversdecembmodelbasiccombinatoricstatisticanalysialgorithmstatisticalphysicthesnicedecomposabltheirpropertcanexplorusingtooldevelopautomaticmanipulatcombinatoriallikewellknownfourtypedependwhethtakendistinguishablweconsidplacemintoallpossiblwaysdefinitenesexaminonlysituatnonemptnumbconfiguratfixedsizealwayfinitmayassumthemnumberconsecutiveintegercasedealwithlabellstructuratomindistinguishablsimpregardanonymouunlabellgenericalcallglobalconventcombstructviewarrangrowconstructotherwishavemeanmultisetsaysetrepetitalloworderwithinurnpriorigiverisedifferdbdusequencsetsuniversdbiuibduibiuexpressbutsimilarlookspecificatcardspecdodrawodcorrespondcountsatisfnaturaldominatconditsummarizinformalinequalitseqsequelconvenirepresobjectmoreconcisnotatthusintroducreductprocedurlreducprocevalsubsargsendureducsincmapldoeskeepmultiwillrepresentsu3plargconstructperformdirectapplevennohadgivenanyalgebraicpolynomialsuchthusselectalgebraicsubnumersimilarproductdiffeqapplihavecoefficizeroconcludcubicexpressalsoexistagainleadtermlucksituatfindbeforroutinfiveconclusthesverygoodexamplexperimentconjecturgeneralproblemcompletedifferproceshowevappareasetreatherehidepreliminarworkledunderseemhighvalumightbibliographgoemetricrevisitjonathanpetrlisonekjohnmacdonaldmapletechpptranslatintoplaycentralletoodifficultanalyticneighborhoodoriginuniqusolutquadratictheoremequivalstatnowusefirststepcomputseriexpanspossiblclosformturnoutdoessatisffuneqopmethodundeterminatcoefficientsoldosolvunappodguessdifferentialsatisfideqseriestodiffeqogfdsolvunabldeducrecurrenctaylorsolutiondiffeqtorecorderlinearfoundrsolvhencsuminfinitproofconsistobviousourapproachusingclosurequationimplementlefthandsidereduconlycompatiblinitialconditu_pHz|{VERSION 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1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 26c6 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 43 "Variations on the Sequenc e of Ap\351ry Numbers" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 19 "" 0 "" {TEXT -1 15 "Fr\351d\351ric Chyzak" }}{PARA 268 "" 0 "" {TEXT -1 28 "(Version of January 9, 1998)" }}{PARA 269 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 234 "In the early 1990's, Doron Zeilb erger and Herbert Wilf, developped a new methodology for symbolic summ ation and integration (Wilf, Herbert S. and Zeilberger, Doron (1992):A n algorithmic proof theory for hypergeometric (ordinary and ``" } {XPPEDIT 18 0 "q" "I\"qG6\"" }{TEXT -1 34 "'') multisdum/integral ident ities, " }{TEXT 269 24 "Inventiones Mathematicae" }{TEXT -1 2 ", " } {TEXT 270 3 "108" }{TEXT -1 143 ":575-633). One of the most famous su ccesses of this so-called ``WZ-method'' has been to provide a computer proof of the combinatorial identity" }}}{EXCHG {PARA 256 "" 0 "" {XPPEDIT 18 0 "sum(binomial(n,k)^2*binomial(n+k,k)^2,k=0..n)=sum(binom ial(n,k)*binomial(n+k,k)*sum(binomial(k,j)^3,j=0..k),k=0..n)" "/-%$sum G6$*&-%)binomialG6$%\"nG%\"kG\"\"#-F(6$,&F*\"\"\"F+F0F+\"\"#/F+;\"\"!F *-F$6$*(-F(6$F*F+F0-F(6$,&F*F0F+F0F+F0-F$6$*$-F(6$F+%\"jG\"\"$/FB;F4F+ F0/F+;F4F*" }{TEXT -1 1 "," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "and to prove that the sequence of these numbers " }{XPPEDIT 18 0 "a[n]" " &%\"aG6#%\"nG" }{TEXT -1 47 " satisfies the second order recurrence eq uation" }}}{EXCHG {PARA 257 "" 0 "" {XPPEDIT 18 0 "(n+2)^3*u[n+2]-((n+ 2)^3+(n+1)^3+4*(2*n+3)^3)*u[n+1]+(n+1)^3*u[n]=0" "/,(*&,&%\"nG\"\"\"\" \"#F'\"\"$&%\"uG6#,&F&F'\"\"#F'F'F'*&,(*$,&F&F'\"\"#F'\"\"$F'*$,&F&F' \"\"\"F'\"\"$F'e*&\"\"%F'*$,&*&\"\"#F'F&F'F'\"\"$F'\"\"$F'F'F'&F+6#,&F& F'\"\"\"F'F'!\"\"*&,&F&F'\"\"\"F'\"\"$&F+6#F&F'F'\"\"!" }{TEXT -1 1 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Proving this recurrence was a crucial step of Ap\351ry's proof for the irrationality of" }}}{EXCHG {PARA 259 "" 0 "" {XPPEDIT 18 0 "Zeta(3)=sum(1/k^3,k=1..infinity)" "/- %%ZetaG6#\"\"$-%$sumG6$*&\"\"\"\"\"\"*$%\"kG\"\"$!\"\"/F.;\"\"\"%)infi nityG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 160 "On the \+ other hand, the identity itself stems from a number-theoretic question raised by Schmidt in (Schmidt, Asmus L. (1990): Generalized Legendre \+ polynomials, " }{TEXT 256 21 "J. reine angew. Math." }{TEXT -1 2 ", " }{TEXT 272 3 "404" }{TEXT -1 10 ":192-202)." }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 69 "Several proofs of the identity above, that relates the \+ Ap\351ry numbers " }{XPPEDIT 18 0 "a[n]" "&%\"aG6#%\"nG" }{TEXT -1 22 " to the Franel numbers" }}}{EXCHG {PARA 258 "" 0 "" {XPPEDIT 18 0 "f[ n]=sum(binomial(n,k)^3,k=0..n)" "/&%f\"fG6#%\"nG-%$sumG6$*$-%)binomialG 6$F&%\"kG\"\"$/F.;\"\"!F&" }{TEXT -1 1 "," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "were given in (Strehl, Volker (1994): Binomial Identities , Combinatorial and Algorithmic Aspects, " }{TEXT 257 14 "Discrete Mat h." }{TEXT -1 2 ", " }{TEXT 271 3 "136" }{TEXT -1 380 ":309-346). One of them in particular is based on Zeilberger's algorithm for hypergeo metric summation, and yields the recurrence equation above as a by-pro duct. In the following sections, we first recall how Ap\351ry was led to the identity, borrowing from Van der Poorten's report (Van der Poo rten, Alfred (1979): A Proof that Euler missed... Ap\351ry's Proof of \+ the Irrationality of " }{XPPEDIT 18 0 "Zeta(3)" "-%%ZetaG6#\"\"$" } {TEXT -1 2 ", " }{TEXT 259 19 "Math. Intelligencer" }{TEXT -1 2 ", " } {TEXT 273 1 "1" }{TEXT -1 67 ":195-203); we next give a proof for both results using our package " }{HYPERLNK 17 "Mgfun" 2 "Mgfun" "" } {TEXT -1 54 ", and finally exploit the recurrence equation to beat " g} {TEXT 258 5 "Maple" }{TEXT -1 26 " computing many digits of " } {XPPEDIT 18 0 "Zeta(3)" "-%%ZetaG6#\"\"$" }{TEXT -1 1 "." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "Sketch of Ap\351ry's Proof" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Ap\351ry's first remark is that the doubl e sequence" }}}{EXCHG {PARA 262 "" 0 "" {XPPEDIT 18 0 "c[n,k]=sum(1/m^ 3,m=1..n)+sum((-1)^(m+1)/(2*m^3*binomial(n,m)*binomial(n+m,m)),m=1..k) " "/&%\"cG6$%\"nG%\"kG,&-%$sumG6$*&\"\"\"\"\"\"*$%\"mG\"\"$!\"\"/F0;\" \"\"F&F.-F*6$*&),$\"\"\"F2,&F0F.\"\"\"F.F.**\"\"#F.*$F0\"\"$F.-%)binom ialG6$F&F0F.-FC6$,&F&F.F0F.F0F.F2/F0;\"\"\"F'F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "tends to " }{XPPEDIT 18 0 "Zeta(3)" "-%%ZetaG6#\"\"$ " }{TEXT -1 14 " uniformly in " }{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 4 " in " } {XPPEDIT 18 0 "``(1..infinity)" "-%!G6#;\"\"\"%)infinityG" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 87 " tends to infin ity. This stems from the alternating series bheing uniformly bounded b y " }{XPPEDIT 18 0 "1/n" "*&\"\"\"\"\"\"%\"nG!\"\"" }{TEXT -1 99 ". H owever, the convergence of this series is not strong enough so as to s how the irrationality of " }{XPPEDIT 18 0 "Zeta(3)" "-%%ZetaG6#\"\"$" }{TEXT -1 78 ". Ap\351ry used summation methods to accelerate the con vergence. Namely, define" }}}{EXCHG {PARA 263 "" 0 "" {XPPEDIT 18 0 " a[n]=sum(binomial(n,k)^2*binomial(n+k,k)^2*c[n,k],k=0..n)" "/&%\"aG6#% \"nG-%$sumG6$*(-%)binomialG6$F&%\"kG\"\"#-F,6$,&F&\"\"\"F.F3F.\"\"#&% \"cG6$F&F.F3/F.;\"\"!F&" }{TEXT -1 11 " and " }{XPPEDIT 18 0 "b[ n]=sum(binomial(n,k)^2*binomial(n+k,k)^2,k=0..n" "/&%\"bG6#%\"nG-%$sum G6$*&-%)binomialG6$F&%\"kG\"\"#-F,6$,&F&\"\"\"F.F3F.\"\"#/F.;\"\"!F&" }{TEXT -1 1 "," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "then " } {XPPEDIT 18 0 "a[n]/b[n]" "*&&%\"aG6#%\"nG\"\"\"&%\"bG6#F&!\"\"" } {TEXT -1 15 " also tends to " }{XPPEDIT 18 0 "Zeta(3)" "-%%ZetaG6#\"\" $" }{TEXT -1 100 ". Here appears the crucial recurrence of Ap\351ry: \+ one remarks ithat it is satisfied by both sequences " }{XPPEDIT 18 0 "a " "I\"aG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b" "I\"bG6\"" }{TEXT -1 25 ", with initial conditions" }}}{EXCHG {PARA 264 "" 0 "" {XPPEDIT 18 0 "a[0]=0" "/&%\"aG6#\"\"!F&" }{TEXT -1 5 ", " } {XPPEDIT 18 0 "a[1]=6" "/&%\"aG6#\"\"\"\"\"'" }{TEXT -1 11 ", and \+ " }{XPPEDIT 18 0 "b[0]=1" "/&%\"bG6#\"\"!\"\"\"" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "b[1]=5" "/&%\"bG6#\"\"\"\"\"&" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "By a number-theoretic argument, it follows from this recurrence that" }}}{EXCHG {PARA 265 "" 0 "" {XPPEDIT 18 0 "Zeta(3)-a[n]/b[n]=O(q[n]^(-1+delta)" "/,&-%%ZetaG6#\"\" $\"\"\"*&&%\"aG6#%\"nGF(&%\"bG6#F-!\"\"F1-%\"OG6#)&%\"qG6#F-,&\"\"\"F1 %&deltaGF(" }{TEXT -1 1 "," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "with " }}}{EXCHG {PARA 266 "" 0 "" {XPPEDIT 18 0 "q[n]=2*lcm(1,2,``..``,n)^ 3*b[n]" "/&%\"qG6#%\"nG*(\"\"#\"\"\"*$-%$lcmG6&\"\"\"\"\"#;%!GF1F&\"\" $F)&%\"bG6#F&F)" }{TEXT -1 11 " and " }{XPPEDIT 18 0 "djelta=(4*l n(1+sqrt(2))-3)/(4*ln(1+sqrt(2))+3)" "/%&deltaG*&,&*&\"\"%\"\"\"-%#lnG 6#,&\"\"\"F(-%%sqrtG6#\"\"#F(F(F(\"\"$!\"\"F(,&*&\"\"%F(-F*6#,&\"\"\"F (-F/6#\"\"#F(F(F(\"\"$F(F3" }{TEXT -1 20 ", which is positive." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "This is sufficient to prove that \+ " }{XPPEDIT 18 0 "Zeta(3)" "-%%ZetaG6#\"\"$" }{TEXT -1 63 " is irratio nal, and yields an irrationality measure of at least" }}}{EXCHG {PARA 267 "" 0 "" {XPPEDIT 18 0 "1+1/delta=13.417820..``" "/,&\"\"\"\"\"\"*& \"\"\"F%%&deltaG!\"\"F%;$\")?yT8!\"'%!G" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 27 "Proof of Ap\351ry's Recurrence" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "In this session, we chiefly use the user-oriented package " }{HYPERLNK 17 "Mgfun" 2 "Mgfun" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(Mgfun);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(%,diag_of_sysG%+int_of_sysG%+pol_to_sysG%+sum_of_sysG %(sys*sysG%(sys+sysG" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 33 "Recurren ce for the Left-Hand Side" k}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "We fi rst prove that the Ap\351ry numbers, as defined by the left-hand side " }}}{EXCHG {PARA 260 "" 0 "" {XPPEDIT 18 0 "a[n]=sum(binomial(n,k)^2* binomial(n+k,k)^2,k=0..n)" "/&%\"aG6#%\"nG-%$sumG6$*&-%)binomialG6$F&% \"kG\"\"#-F,6$,&F&\"\"\"F.F3F.\"\"#/F.;\"\"!F&" }{TEXT -1 1 "," }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "satisfy the announced recurrence. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "The summand" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "f:=binomial(n,k)^2*binomial(n+k,k)^2:" ">% \"fG*&-%)binomialG6$%\"nG%\"kG\"\"#-F&6$,&F(\"\"\"F)F.F)\"\"#" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "satisfies both following equations :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "h(n+1,k)/h(n,k)=factor (normal(subs(n=n+1,f)/f,expanded));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/*&-%\"hG6$,&%\"nG\"\"\"F*F*%\"kGF*-F&6$F)F+!\"\"*&,(F)F*F*F*F+F*\"\" #,(F)F.F.F*F+F*!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "h(n, k+1)/h(n,k)=factor(normal(subs(k=k+1,f)/f,expanded));" }}{PARA l11 "" 1 "" {XPPMATH 20 "6#/*&-%\"hG6$%\"nG,&%\"kG\"\"\"F+F+F+-F&6$F(F*!\"\"* (,(F(F+F+F+F*F+\"\"#,&F(F.F*F+F1F)!\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "This yields the following system" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "sys:=collect(map(numer,map(eq->op(1,eq)-op(2,eq) ,\{\"\",\"\})),h);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$sysG<$,&*&,.* $%\"nG\"\"#\"\"\"F*F+*&F*F,%\"kGF,!\"#F,F,F.F/*$F.F+F,F,-%\"hG6$,&F*F, F,F,F.F,F,*&,.F-F/!\"\"F,F)F7F*F/F.F/F0F7F,-F26$F*F.F,F,,&*&,6*$F*\"\" %F7*$F*\"\"$F/F)F7*&F*F+F.F,F+*&F*F+F.F+F+F-F+*&F*F,F.F+F+F0F7*$F.F@F/ *$F.F>F7F,F8F,F,*&,,FEF,FDF>F0\"\"'F.F>F,F,F,-F26$F*,&F.F,F,F,F,F," }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "where each element " }{XPPEDIT 18 0 "expr" "I%exprG6\"" }{TEXT -1 33 " in the set denotes the equatio n " }{XPPEDIT 18 0 "expr=0" "/%%exprG\"\"!" }{TEXT -1 31 ". The defin ite summation over " }{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "``(0,n)" "-%!G6$\"\"!%\"nG" }{TEXT -1 39 " is performe d by the following call tmo " }{HYPERLNK 17 "Mgfun[sum_of_sys]" 2 "Mgfu n[sum_of_sys]" "" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "sum_of_sys(sys,k=-infinity..infinity,takayama_algo); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#,(*&,**$%\"nG\"\"$\"\"\"*$F(\"\" #F)F(F)F*F*F*-%\"hG6#F(F*F**&,*F'!#MF(!$J#F+!$`\"!$<\"F*F*-F.6#,&F(F*F *F*F*F**&,*F'F*F+\"\"'F(\"#7\"\")F*F*-F.6#,&F(F*F,F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "rec[left]:=op(collect(\",h,factor)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$recG6#%%leftG,(*&,&%\"nG\"\" \"F,F,\"\"$-%\"hG6#F+F,F,*&,&F+F,\"\"#F,F--F/6#F2F,F,*(,&F+F3F-F,F,,(* $F+F3\"# " 0 "" {XPPEDIT 19 1 "f:=binomial(n,k)^3:" ">%\"fG*$ -%)binomialG6$%\"nG%\"kG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "h(n+1,k)/h(n,k)-factor(normal(subs(n=n+1,f)/f,expanded)):" ",&*& -%\"hG6$,&%\"nG\"\"\"\"\"\"F)%\"kGF)-F%6$F(F+!\"\"F)-%'factorG6#-%'nor malG6$*&-%%subsG6$/F(,&F(F)\"\"\"F)%\"fGF)F " 0 "" {XPPEDIT 19 1 "h(n,k+1)/h(n,k)-factor(normal( subs(k=k+1,f)/f,expanded)):" ",&*&-%\"hG6$%\"nG,&%\"kG\"\"\"\"\"\"F*F* -F%6$F'F)!\"\"F*-%'factorG6#-%'normalG6$*&-%%subsG6$/F),&F)F*\"\"\"F*% \"fGF*F " 0 "" {MPLTEXT 1 0 34 " sys:=collect(map(numer,\{\"\",\"\}),h);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$sysG<$,&*&,**$%\"nG\"\"$!\"\"*&F*\"\"#%\"kG\"\"\"F+*&F*F0F/F. !\"$*$F/F+F0F0-%\"hG6$F*F/F0F0*&,*F3Fo0*$F/F.F+F/F+F0F0F0-F56$F*,&F/F0F 0F0F0F0,&*&,6F)F,*$F*F.F2F-F+F*F2*&F*F0F/F0\"\"'F1F2F,F0F/F+F9F2F3F0F0 -F56$,&F*F0F0F0F/F0F0*&,*F)F0F@F+F*F+F0F0F0F4F0F0" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 52 "Summing, we get a recurrence for the Franel number s " }{XPPEDIT 18 0 "f[n]" "&%\"fG6#%\"nG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "sys:=sum_of_sys(sys,k=-infinity..in finity,natural_boundaries);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$sysG <#,**&,*!#c\"\"\"%\"nG!$O\"*$F+\"\"#!$/\"*$F+\"\"$!#CF*-%\"hG6#F+F*F** &,*F0F1F-\"#AF+\"#^\"#OF*F*-F46#,&F+F*F1F*F*F**&,*F0!#X!$S#F*F-FAF+!$> %F*-F46#,&F+F*F*F*F*F**&,*F0!#=!$[\"F*F-!$9\"F+!$K#F*-F46#,&F+F*F.F*F* F*" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 33 "System for the Right-Hand Product" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Let us multiply the " }{XPPEDIT 18 0 "f[k]" "&%\"fG6#%\"kG" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "binomial(n,k)*binomial(n+k,k)" "*&-%)binomialG6$%\"nG%\"kG\"\"\"-F$ 6$,&F&F(F'F(F'F(" }{TEXT -1 81 ". To do so, we prepare two systepms of recurrence equations, one for each factor." }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 31 "The following system describes " }{XPPEDIT 18 0 "f[k]" "&%\"fG6#%\"kG" }{TEXT -1 28 ", which is independent from " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "sys1:=\{h(n+1,k)-h(n,k)\} union eval(subs(h=proc(k) h (n,k) end,subs(n=k,sys)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%sys1G <$,&-%\"hG6$,&%\"nG\"\"\"F,F,%\"kGF,-F(6$F+F-!\"\",**&,*!#cF,F-!$O\"*$ F-\"\"#!$/\"*$F-\"\"$!#CF,F.F,F,*&,*F9F:F6\"#AF-\"#^\"#OF,F,-F(6$F+,&F -F,F:F,F,F,*&,*F9!#X!$S#F,F6FGF-!$>%F,-F(6$F+,&F-F,F,F,F,F,*&,*F9!#=!$ [\"F,F6!$9\"F-!$K#F,-F(6$F+,&F-F,F7F,F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "We next obtain a system that describes " }{XPPEDIT 18 0 " binomial(n,k)*binomial(n+k,k)" "*&-%)binomialG6$%\"nG%\"kG\"\"\"-F$6$, &F&F(F'F(F'F(" }{TEXT -1 30 ", the weight to multiply with:" }}} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "f:=binomial(n,k)*binomial(n+k, k):" ">%\"fG*&-%)binomialG6$%\"nG%q\"kG\"\"\"-F&6$,&F(F*F)F*F)F*" }}} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "h(n+1,k)/h(n,k)-factor(normal( subs(n=n+1,f)/f,expanded)):" ",&*&-%\"hG6$,&%\"nG\"\"\"\"\"\"F)%\"kGF) -F%6$F(F+!\"\"F)-%'factorG6#-%'normalG6$*&-%%subsG6$/F(,&F(F)\"\"\"F)% \"fGF)F " 0 "" {XPPEDIT 19 1 "h( n,k+1)/h(n,k)-factor(normal(subs(k=k+1,f)/f,expanded)):" ",&*&-%\"hG6$ %\"nG,&%\"kG\"\"\"\"\"\"F*F*-F%6$F'F)!\"\"F*-%'factorG6#-%'normalG6$*& -%%subsG6$/F),&F)F*\"\"\"F*%\"fGF*F " 0 "" {MPLTEXT 1 0 35 "sys2:=collect(map(numer,\{\"\",\"\}),h); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sys2G<$,&*&,**$%\"nG\"\"#!\"\"F *F,%\"kG\"\"\"*$F-F+F.F.-%\"hG6$F*F-F.F.*&,(F/F.F-F+F.F.F.-F16$F*,&F-F .F.F.F.F.,&*&,(F*F,F,F.F-F.F.-F16$,&F*F.F.F.F-F.F.*&,(F*F.F.F.F-F.F.F0 F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "We finally perform the pr oduct by a call to " }{HYPERLNK 17 "Mgfun[`sys*sys`]" 2 "Mgfun[`sys*sy s`]" "" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0r 26 "s ys:=`sys*sys`(sys1,sys2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$sysG<$ ,&*&,(%\"nG!\"\"F*\"\"\"%\"kGF*F+-%\"hG6$F)F,F+F+*&,(F)F+F+F+F,F*F+-F. 6$,&F)F+F+F+F,F+F+,**&,jnF,!$s'F)\"$s'*$F,\"\"'!$s#*$F)\"\"#\"$C#*&F)F +F,F+\"%/B*$F,F>!%GD*&F)F>F,F+\"%3;*&F)F>F,F>\"%kE*&F)F+F,F>\"%[I*$F, \"\"$!%;Q*$F,\"\"%!%gH*&F,FNF)F>\"$+'*&F,\"\"&F)F+\"#s*$F)FN!$!G*$F)FK !$S)*&F,FKF)F+\"%W>*&F)FSF,F+FT*$F)F;\"#c*&F)FNF,F>!$%Q*&F)FKF,F+!%o8* $F)FS\"$o\"*&F)F;F,F+\"#C*&F)FKF,F>!$o(*&F)FNF,F+!$C'*&F,FKF)FN!#s*&F, FKF)FK!$W\"*&F,FNF)F+FQ*$F,\"\"(!#C*&F,FKF)F>\"%s=*$F,FS!%[7*&F,FSF)F> FTF+F-F+F+*&,Z\"%!)GF+F,\"&[<\"F)!%?>F:FQF=!%!o\"F@!%s_FB\"&K,#FD!%`[F F!%7bFH!%_dFJ\"&<)=FM\"&s.\"FPFXFR!#!*FU\"$S#FW\"$![FY!%=JFhnF_qFjn\"$ Q)F`oF`qFbo\"$>%Fdo\"#XFfo\"#!*FhoFXFioFdqF\\p!%tIF^p\"%uLF`pF^qF+-F.6 $F),&F,F+F+F+F+F+*&,JFB!&%QF+F+F+*&,2FB!%%=&F:!#_Fio!\"$FJ!%vOFM!%S:F ,!%'*R!%'H\"F+F^p!$#sQF+-F.6$F),&F,F+FKF+F+F+" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "Recurrence for the Right-Hand Side" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "We obtain a recurrence for the right-hand side \+ by applying " }{HYPERLNK 17 "Mgfun[sum_of_sys]" 2 "Mgfun[sum_of_sys]" "" }{TEXT -1 48 " on the system computed in the previous section." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "sum_of_sys(sys,k=-infinity.. infinity,takayama_algo);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<#,,*&,0!$ 7\"\"\"\"%\"nG!$m%*$F)\"\"#!$v(*$F)\"\"%!$&H*$F)\"\"$!$b'*$F)\"\"'!\"' *$F)\"\"&!#nF(-%\"hG6#F)F(F(*&,0F1\"&Uq$\"&!G9F(F4\"$/#F.\"&6O\"F7\"%7 EF+\"&&obF)\"&mR%F(-F;6#,&F)F(F(F(F(F(*&,0F4F5F7\"$8\"F.\"$q)F1\"%&\\$ F+\"%+xF)\"%%y)\"%KSF(F(-F;6#,&F)F(F/F(F(F(*&,0!&+#))F(F1!&y@*F+!'0#*= F4!$/#F7!%3NF)!'MF?F.!&6[#F(-F;6#,&F)F(F2F(F(F(*&,.F1!&wz&!&Sy(F(F+!'X /9F)!''3n\"F.!&v<\"F7!$U*F(-F;6#,&F)F(F,F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "This recurrence is different from the one obtained f or the left-hand side." }}}{EXCHG {PARA 0 "> " t0 "" {MPLTEXT 1 0 36 "r ec[right]:=op(collect(\",h,factor));" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#>&%$recG6#%&rightG,,*,,&%\"nG\"\"#\"\"(\"\"\"F.,&F+F,\"\"$F.F.,&F+F. F0F.F.,**$F+F0\"#^*$F+F,\"$p%F+\"%=9\"%+9F.F.-%\"hG6#F1F.!\"\"*,F/F.,& F+F0F-F.F.F1F.,&F+F.\"\"%F.F0-F:6#F?F.F.*,,&F+F0\"\")F.F.F*F.,&F+F.F,F .F.,&F+F.F.F.F0-F:6#F+F.F<*(,&F+F,\"\"&F.F.,,*$F+F@\"$r%F3\"%5ZF5\"&8s \"F+\"&!>F\"&ob\"F.F.-F:6#FFF.F<*,F*F.F/F.FFF.,*F3F4F5\"$'HF+\"$`&\"$S $F.F.-F:6#FGF.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "We thus need \+ more work to prove that both sides agree." }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 62 "Final Proof of the Identity and of the Second Order Rec urrence" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "h[ n]" "&%\"hG6#%\"nG" }{TEXT -1 95 " be any solution of the second order recurrence which has been obtained for the left-hand side:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "rec[left];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,&%\"nG\"\"\"F'F'\"\"$-%\"hG6#F&F'F'*&,&F&F'\"\" #F'F(-Fu*6#F-F'F'*(,&F&F.F(F'F',(*$F&F.\"# " 0 "" {MPLTEXT 1 0 41 "h(n+2)=collect(solve(\",h(n+2)),h,normal);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"hG6#,&%\"nG\"\"\"\"\"#F),&*(,**$F (\"\"$F)*$F(F*F/F(F/F)F)F),*F.F)F0\"\"'F(\"#7\"\")F)!\"\"-F%6#F(F)F5*( ,*F.\"#MF0\"$`\"F(\"$J#\"$<\"F)F)F1F5-F%6#,&F(F)F)F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(n=n+1,\");" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"hG6#,&%\"nG\"\"\"\"\"$F),&*(,**$,&F(F)F)F)F*F)*$ F/\"\"#F*F(F*\"\"%F)F),*F.F)F0\"\"'F(\"#7\"#?F)!\"\"-F%6#F/F)F7*(,*F. \"#MF0\"$`\"F(\"$J#\"$[$F)F)F3F7-F%6#,&F(F)F1F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "and" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(n=n+1,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"hG6#,&%\"n G\"\"\"\"\"%F),&*(,**$,&F(F)\"\"#F)\"\"$F)*$F/F0F1F(F1\"\"(F)F),*F.F)F 2\"\"'F(\"#7\"#KF)!\"\"-F%6#F/F)F8*(,*F.\"#MF2\"$`\"F(\"$J#\"$z&F)F)F4 F8-F%6#,&F(F)F1F)F)F)" }}}{EXCHG {PARA 0 "v" 0 "" {TEXT -1 6 "Then, " } {XPPEDIT 18 0 "h[n]" "&%\"hG6#%\"nG" }{TEXT -1 52 " also solves the re currence for the right-hand side:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "collect(subs(\",\"\",\"\"\",rec[right]),h,normal);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 241 "At this point, we have proved that both sides of the equ ation satisfy the same recurrence of order 4. To prove the announced \+ equality, we simply need to check 4 initial conditions, since the lead ing coefficient of the recurrence of order 4," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "coeff(rec[right],h(n+3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**,&%\"nG\"\"#\"\"(\"\"\"F),&F&F'\"\"$F)F),&F&F)F+F)F ),**$F&F+\"#^*$F&F'\"$p%F&\"%=9\"%+9F)F)!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "never vanishes for non-negative " }{XPPEDIT 18 0 "n" " I\"nG6\"" }{TEXT -1 32 ". Now the proof of the identity" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "eq:=Sum(binomial(n,k)^2*binomial(n+k, k)^2,k=0..wn)=Sum(binomial(n,k)*binomial(n+k,k)*Sum(binomial(k,j)^3,j=0 ..k),k=0..n):" ">%#eqG/-%$SumG6$*&-%)binomialG6$%\"nG%\"kG\"\"#-F*6$,& F,\"\"\"F-F2F-\"\"#/F-;\"\"!F,-F&6$*(-F*6$F,F-F2-F*6$,&F,F2F-F2F-F2-F& 6$*$-F*6$F-%\"jG\"\"$/FD;F6F-F2/F-;F6F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "simplify follows from" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eval(subs(n=0,Sum=add,eq));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"\"F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eval(subs(n=1,Sum=add,eq));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"& F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eval(subs(n=2,Sum=add ,eq));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"#tF$" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 27 "eval(subs(n=3,Sum=add,eq));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"%X9F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "T herefore, the Ap\351ry numbers also satisfy the announced second order recurrence." }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Computation of " }{XPPEDIT 18 0 "Zeta(3)" "-%%ZetaG6#\"\"$" }{TEXxT -1 48 " Using Sta ndard Maple and the Holonomic Approach" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "Standard Maple" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "Ma ple has numerical routines for almost all special functions it knows a bout. Here is the corresponding calculation for " }{XPPEDIT 18 0 "Zet a(3)" "-%%ZetaG6#\"\"$" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "ti[0]:=time():" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Z3[standard]:=evalf(Zeta(3),332);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%#Z3G6#%)standardG$\"g_l`yBvt*\\R'ydr9q06=mT\\)pLANg( ymD`F)R?5\"y4rV:!))fD:d50%=*z7l9b'4y&4f6H.Tm4RchRC%\\UA_v:>>n1g$\\HyTR Xh,!z?PoRwP@$Gt')*f*>\\=%>2E&zd\"*>Y\"e_L4O(eX'=!48jy0#Q=Mb:F#z\")))\\ SBH')\\w!**\\9^h\"Q(*R&G%ffJ!p0-7!$J$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "ti[standard]:=time()-ti[0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#tiG6#%)standardG$\"&:$Q!\"$" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 18 "Holonomic Approach" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "We compute an approxiymation of " }{XPPEDIT 18 0 "Zeta(3) " "-%%ZetaG6#\"\"$" }{TEXT -1 61 " using Ap\351ry's recurrence. More \+ precisely, we compute it as " }{XPPEDIT 18 0 "a[200]/b[200]" "*&&%\"aG 6#\"$+#\"\"\"&%\"bG6#\"$+#!\"\"" }{TEXT -1 18 ". (Remember that " } {XPPEDIT 18 0 "a[n]/b[n]" "*&&%\"aG6#%\"nG\"\"\"&%\"bG6#F&!\"\"" } {TEXT -1 10 " tends to " }{XPPEDIT 18 0 "Zeta(3)" "-%%ZetaG6#\"\"$" } {TEXT -1 2 ".)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "ti[0]:=ti me():" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "N:=200:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "To do so, we use the " }{HYPERLNK 17 "gfu n" 2 "gfun" "" }{TEXT -1 174 " package by Salvy and Zimmermann (Salvy, Bruno and Zimmermann, Paul (1994): Gfun: a Maple package for the mani pulation of generating and holonomic functions in one variable, " } {TEXT 261 25 "ACM Trans. Math. Software" }{TEXT -1 2 ", " }{TEXT 274 2 "20" }{TEXT -1 13 "(2):163-177)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "with(gfun);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7V%(La placezG%.algebraicsubsG%.algeqtodiffeqG%.algeqtoseriesG%.algfuntoalgeqG %&borelG%.cauchyproductG%.diffeq*diffeqG%.diffeq+diffeqG%2diffeqtohomd iffeqG%,diffeqtorecG%)guesseqnG%(guessgfG%0hadamardproductG%0holexprto diffeqG%)invborelG%,listtoalgeqG%-listtodiffeqG%0listtohypergeomG%+lis ttolistG%.listtoratpolyG%*listtorecG%-listtoseriesG%5listtoseries/Lapl aceG%1listtoseries/egfG%4listtoseries/lgdegfG%4listtoseries/lgdogfG%1l isttoseries/ogfG%4listtoseries/revegfG%4listtoseries/revogfG%,maxdegco effG%*maxdegeqnG%,maxordereqnG%,mindegcoeffG%*mindegeqnG%,minordereqnG %*optionsgfG%,poltodiffeqG%)poltorecG%/ratpolytocoeffG%(rec*recG%(rec+ recG%,rectodiffeqG%,rectohomrecG%*rectoprocG%.seriestoalgeqG%/seriesto diffeqG%2seriestohypergeomG%-seriestolistG%0seriestoratpolyG%,seriesto recG%/seriestoseriesG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 262 4 "gfun" }{TEXT -1 79 " package provides us with a routine f or transforming a recurrence equation like" }}}{EXCHG {PARA 0 "> " 0 " " {XPPEDIT 19 1 "eq:=n^3*u(n)-(34{*n^3-51*n^2+27*n-5)*u(n-1)+(n-1)^3*u( n-2):" ">%#eqG,(*&%\"nG\"\"$-%\"uG6#F&\"\"\"F+*&,**&\"#MF+*$F&\"\"$F+F +*&\"#^F+*$F&\"\"#F+!\"\"*&\"#FF+F&F+F+\"\"&F6F+-F)6#,&F&F+\"\"\"F6F+F 6*&,&F&F+\"\"\"F6\"\"$-F)6#,&F&F+\"\"#F6F+F+" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 90 "into a procedure. Each of the following procedures enc odes the calculation of a sequence " }{XPPEDIT 18 0 "u[n]" "&%\"uG6#% \"nG" }{TEXT -1 23 " given by the equation " }{XPPEDIT 18 0 "eq" "I#eq G6\"" }{TEXT -1 24 " and its initial values " }{XPPEDIT 18 0 "u[0]" "& %\"uG6#\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "u[1]" "&%\"uG6#\"\" \"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "A:=re ctoproc(\{eq,u(0)=0,u(1)=6\},u(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #>%\"AG:6#%\"nG6&%\"iG%#u0G%#u1G%#u2G6\"E\\s#\"\"\"\"\"'\"\"!F1C&>8%F1 >8&F0?(8$\"\"#F/,&9$F/!\"\"F/%%trueGC%>8',$*&,(F4FF4F6 >F6F@,$*&,(F4F " 0 "" {MPLTEXT 1 0 38 "B:=rectopr oc(\{eq,u(0)=1,u(1)=5\},u(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\" BG:6#%\"nG6&%\"iG%#u0G%#u1G%#u2G6\"E\\s#\"\"\"\"\"&\"\"!F/C&>8%F/>8&F0 ?(8$\"\"#F/,&9$F/!\"\"F/%%trueGC%>8',$*&,(F4FF4F6>F6F@,$* &,(F4F " 0 "" {MPLTEXT 1 0 8 "a:=A(N);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"aG#\"]^m*HbNsb*zn 4KF)QKPAj@Xex]&G%y`$fh&Q!y?!ez5!pY0N)\\i0hqbzyeMh(>\\o%o(fXN9LQ5C$*4!H CkZ0sh\\$))p_kRtK+'*G;OG&34:%)3K6!z+v?ca&Q#>@vw>X'*R\\_'=b>NuiNZSKa>Z0 BS1HDNfy]T(Gs0yRJ2'4d<]syf&Qqpb*p2fm@m8sC\")fb-ie4=t(*3'*=MvE-A&\\c'H4 %)>Dg)[pY8&oR5w*>^%pJeX:SefU;r\\QB'4T/Oy,:g#G#ps4)[z3'RGT*QShXAWRyPo%* oq#pt'e!HstpN(Q!}\\0\"=e['z/a_9-7F\"_[l++++#RfsaYgl=SYT\\5%>-![;W$RWZ0f a&pBU[2/!QK1XVx$4m'fyY@'\\/v!=F.qRy8][]x%*yYoN%*\\,;\\f>z@6%oo*3sr9$oh *fz5P&=>$))=4#>\"\\*))f*G*QLZuMT*=9Xh)Q[Es\"=Mo%)*)[<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "b:=B(N);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"bG\"j]lDu?l%eHmOf:Ow9LZb%f'p23Li]gMN1\\]$o>0QBk&*)3 !fPMA.M@fdRZ![0y8N9<#)fOuuBf/2-9X;Yt=;]\"Hb(Gw#*4XP7\"QM9-hXFIII$p[)\\ '*3zCyC*3>z![\"QA#>1aUt6?Am%=+VI(G]azsa'Qf%e?o,Ek#Rm\\4/!H\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 263 8 "A priori" }{TEXT -1 51 ", it is no t clear how many digits we can guarantee." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 34 "P:=round(evalf(ln(a)/ln(10))*1.1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG\"$K$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Z3[holonomy]:=evalf(a/b,P);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #>&%#Z3G6#%)holonomyG$\"g_l`yBvt*\\R'ydr9q06=mT\\)pLANg(ymD`F)R?5\"y4r V:!))fD:d50%=*z7l9b'4y&4f6H.Tm4RchRC%\\UA_v:>>n1g$\\HyTRXh,!z?PoRwP@$G t')*f*>\\=%>2E&zd\"*>Y\"e_L4O(eX'=!48jy0#Q=Mb:F#z\")))\\SBH')~\\w!**\\9 ^h\"Q(*R&G%ffJ!p0-7!$J$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "The ti me used is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "ti[holonomy]: =time()-ti[0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#tiG6#%)holonomyG $\"%'G(!\"$" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Comparison" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "The holonomic approach is several \+ time faster." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "ti[standard ]/ti[holonomy];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+W`re_!\"*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "Moreover, this ratio would increa se with the accuracy of the calculations. In this session, we have ob tained " }{XPPEDIT 18 0 "Zeta(3)" "-%%ZetaG6#\"\"$" }{TEXT -1 18 " up \+ to 332 digits." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Z3[standa rd]-Z3[holonomy];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "However, " }{TEXT 268 5 "Maple" }{TEXT -1 26 " would be able to compute " }{XPPEDIT 18 0 "Zeta(z)" "-%%ZetaG6#% \"zG" }{TEXT -1 5 " for " }{TEXT 267 3 "any" }{TEXT -1 15 " complex va lue " }{XPPEDIT 18 0 "z" "I\"zG6\"" }{TEXT -1 46 ", while the holonomi c approach only works for " }{XPPEDIT 18 0 "z=3" "/%\"zG\"\"$" }{TEXT -1 1 "." }}}}}}{MARK "0 1 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 } "" 0 "" {TEXT -1 10 "Comparison" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "The holonomic approach is several \+ time faster." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "ti[standard ]/ti[holonomy];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+W`re_!\"*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "Moreover, this ratio would increa se with the accuracy of the calculations. In this session, we have ob tained " }{XPPEDIT 18 0 "Zeta(3)" "-%%ZetaG6#\"\"$" }{TEXT -1 18 " up \+ to 332 digits." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Z3[standa rd]-Z3[holonomy];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "However, " }{TEXT 268 5 "Maple" }{TEXT -1 26 " would be able to compute " }{XPPEDIT 18 0 "Zeta(z)" "-%%ZetaG6#% \"zG" }{TEXT -1 Holonomy,hypergeom_to_dfinite Holonomy,takayama_algoLaTeX,DerivedFormatJLaTeX,DocStyleLLaTeX,FormatNumberingA LaTeX,LaTeXzLaTeX,MathKeywordLaTeX,MathStylekLaTeX,MathSymbolRMAD MAD,ExportMAD,MADMAD,MADDocumentMAD,cgingMadMgfunvMgfun,`sys*sys`ZMgfun,`sys+sys`ZMgfun,creative_telescopingOMgfun,dfinite_expr_to_diffeq Mgfun,dfinite_expr_to_rec 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0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Out put" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "T itle" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 49 "An Integral of a Product \+ of four Bessel Functions" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 19 "" 0 "" {TEXT -1 15 "Fr\351d\351ric Chyzak" }}{PARA 257 "" 0 "" {TEXT -1 28 "(Version of January 8, 1998)" }}{PARA 258 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "In (Glasser, M. L. and Montal di E. (1994): Some Integrals Involving Bessel Functions, " }{TEXT 259 20 "J. Math. Anal. Appl." }{TEXT -1 2 ", " }{TEXT 276 3 "183" }{TEXT -1 179 ":577-590), Glasser and Montaldi compute a closed form for an i ntegral of a product of two Bessel functions, and suggest that their t reatment should extend to the following example" }}}{EXCHG {PARA 256 " " 0 "" {XPPEDIT 18 0 "int(x*J[1](a*x)*I[1](a*x)*Y[0](x)*K[0](x),x=0..i nfinity)=-ln(1-a^4)/2/Pi/a^2" "/-%$intG6$*,%\"xG\"\"\"-&%\"JG6#\"\"\"6 #*&%\"aGF(F'F(F(-&%\"IG6#\"\"\"6#*&F0F(F'F(F(-&%\"YG6#\"\"!6#F'F(-&%\" KG6#F<6#F'F(/F';F<%)infinityG,$**-%#lnG6#,&\"\"\"F(*$F0\"\"%!\"\"F(\" \"#FO%#PiGFO*$F0\"\"#FOFO" }{TEXT -1 1 "," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 255 "which is of interest because it contains each of the fou r types of Bessel functions. This integral is one of numerous integra ls containing four (or more) Bessel functions. See for instance (Prud nikov, A. P., Brychkov, Yu. A. and Marichev, O. I. (1986): " }{TEXT 260 49 "Integrals and Series. Volume 2: Special functions" }{TEXT -1 35 ", Gordon and Breach; Sec. 2.16.47)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "In this session, we deal with the integral above and deri ve a closed form for it using our " }{HYPERLNK 17 "Mgfun" 2 "Mgfun" " " }{TEXT -1 1 " " }{TEXT -1 44 "package in an intimate interaction wit h the " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 1 " " }{TEXT -1 8 " package." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(Mgfun);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#7(%,diag_of_sysG%+int_of_sysG%+pol_to _sysG%+sum_of_sysG%(sys*sysG%(sys+sysG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "with(gfun);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7V%(La placeG%.algebraicsubsG%.algeqtodiffeqG%.algeqtoseriesG%.algfuntoalgeqG %&borelG%.cauchyproductG%.diffeq*diffeqG%.diffeq+diffeqG%2diffeqtohomd iffeqG%,diffeqtorecG%)guesseqnG%(guessgfG%0hadamardproductG%0holexprto diffeqG%)invborelG%,listtoalgeqG%-listtodiffeqG%0listtohypergeomG%+lis ttolistG%.listtoratpolyG%*listtorecG%-listtoseriesG%5listtoseries/Lapl aceG%1listtoseries/egfG%4listtoseries/lgdegfG%4listtoseries/lgdogfG%1l isttoseries/ogfG%4listtoseries/revegfG%4listtoseries/revogfG%,maxdegco effG%*maxdegeqnG%,maxordereqnG%,mindegcoeffG%*mindegeqnG%,minordereqnG %*optionsgfG%,poltodiffeqG%)poltorecG%/ratpolytocoeffG%(rec*recG%(rec+ recG%,rectodiffeqG%,rectohomrecG%*rectoprocG%.seriestoalgeqG%/seriesto diffeqG%2seriestohypergeomG%-seriestolistG%0seriestoratpolyG%,seriesto recG%/seriestoseriesG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "More spe cifically, the " }{TEXT 264 4 "gfun" }{TEXT -1 74 " package will be us ed to prepare a system of PDE's for the application of " }{TEXT 263 5 "Mgfun" }{TEXT -1 55 " functions, and to solve the ODE that is output \+ by the " }{TEXT 265 5 "Mgfun" }{TEXT -1 9 " package." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 56 "Search for a System of PDE's Satisfied by the Integrand " }{XPPEDIT 18 0 "x*J[1](a*x)*I[1](a*x)*Y[0](x)*K[0](x) " "*,%\"xG\"\"\"-&%\"JG6#\"\"\"6#*&%\"aGF$F#F$F$-&%\"IG6#\"\"\"6#*&F,F $F#F$F$-&%\"YG6#\"\"!6#F#F$-&%\"KG6#F86#F#F$" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 11 "We use the " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 1 " " }{TEXT -1 98 "package to compute a system of PDE's satisfied \+ by each factor of the integrand. Next, we use the " }{HYPERLNK 17 "Mg fun" 2 "Mgfun" "" }{TEXT -1 2 " p" }{TEXT -1 62 "ackage to derive a sy stem of PDE's satisfied by their product." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "System of PDE's Satisfied by " }{XPPEDIT 18 0 "x" "I\"xG6 \"" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "The identity function " } {XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 54 " trivially satisfies the fo llowing differential system" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "sys1:=\{x*diff(h(x,a),x)-1,diff(h(x,a),a)\}:" ">%%sys1G<$,&*&%\"xG \"\"\"-%%diffG6$-%\"hG6$F'%\"aGF'F(F(\"\"\"!\"\"-F*6$-F-6$F'F/F/" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "where each entry " }{XPPEDIT 18 0 "expr" "I%exprG6\"" }{TEXT -1 33 " in the set denotes the equation " } {XPPEDIT 18 0 "expr=0" "/%%exprG\"\"!" }{TEXT -1 1 "." }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "System of PDE's Satisfied by " }{XPPEDIT 18 0 "J[1](a*x)" "-&%\"JG6#\"\"\"6#*&%\"aG\"\"\"%\"xGF*" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "We compute a system of PDE's for" }}} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "f:=BesselJ(1,a*x):" ">%\"fG-%( BesselJG6$\"\"\"*&%\"aG\"\"\"%\"xGF*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "by first computing an ODE with respect to the variable " } {XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 7 " using " }{HYPERLNK 17 "gfun [holexprtodiffeq]" 2 "gfun[holexprtodiffeq]" "" }{TEXT -1 1 "," } {TEXT -1 36 " and next considering symmetries of " }{XPPEDIT 18 0 "f" "I\"fG6\"" }{TEXT -1 29 " to derive a complete system." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "holexprtodiffeq(f,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%,(*&,&!\"\"\"\"\"*&%\"aG\"\"#%\"xGF+F(F(-% \"yG6#F,F(F(*&F,F(-%%diffG6$F-F,F(F(*&F,F+-F26$F1F,F(F(/-F.6#\"\"!F:/- -%\"DG6#F.F9,$F*#F(F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "de q:=op(remove(type,\",equation)):" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "subs(y(x)=h(x,a),deq);" "-%%subsG6$/-%\"yG6#%\"xG-%\"hG 6$F)%\"aG%$deqG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,&!\"\"\"\"\"*& %\"aG\"\"#%\"xGF*F'F'-%\"hG6$F+F)F'F'*&F+F'-%%diffG6$F,F+F'F'*&F+F*-F1 6$F0F+F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "subs(\{x=a,a=x, y(x)=h(x,a)\},deq);" "-%%subsG6$<%/%\"xG%\"aG/F(F'/-%\"yG6#F'-%\"hG6$F 'F(%$deqG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,&!\"\"\"\"\"*&%\"aG \"\"#%\"xGF*F'F'-%\"hG6$F+F)F'F'*&F)F'-%%diffG6$F,F)F'F'*&F)F*-F16$F0F )F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "x*diff(h(x,a),x)-a*d iff(h(x,a),a):" ",&*&%\"xG\"\"\"-%%diffG6$-%\"hG6$F$%\"aGF$F%F%*&F,F%- F'6$-F*6$F$F,F,F%!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "s ys2:=\{\"\"\",\"\",\"\}:" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "Sys tem of PDE's Satisfied by " }{XPPEDIT 18 0 "I[1](a*x)" "-&%\"IG6#\"\" \"6#*&%\"aG\"\"\"%\"xGF*" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "We com pute a system of PDE's for" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "f:=BesselI(1,a*x):" ">%\"fG-%(BesselIG6$\"\"\"*&%\"aG\"\"\"%\"xGF*" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "in the same way as in the previo us section." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "holexprtodif feq(f,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%,(*&,&!\"\"\"\"\"*&% \"aG\"\"#%\"xGF+F'F(-%\"yG6#F,F(F(*&F,F(-%%diffG6$F-F,F(F(*&F,F+-F26$F 1F,F(F(/-F.6#\"\"!F:/--%\"DG6#F.F9,$F*#F(F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "deq:=op(remove(type,\",equation)):" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "subs(y(x)=h(x,a),deq);" "-%%subsG6$/- %\"yG6#%\"xG-%\"hG6$F)%\"aG%$deqG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, (*&,&!\"\"\"\"\"*&%\"aG\"\"#%\"xGF*F&F'-%\"hG6$F+F)F'F'*&F+F'-%%diffG6 $F,F+F'F'*&F+F*-F16$F0F+F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "subs(\{x=a,a=x,y(x)=h(x,a)\},deq);" "-%%subsG6$<%/%\"xG%\"aG/F(F'/- %\"yG6#F'-%\"hG6$F'F(%$deqG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,&! \"\"\"\"\"*&%\"aG\"\"#%\"xGF*F&F'-%\"hG6$F+F)F'F'*&F)F'-%%diffG6$F,F)F 'F'*&F)F*-F16$F0F)F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "x*d iff(h(x,a),x)-a*diff(h(x,a),a):" ",&*&%\"xG\"\"\"-%%diffG6$-%\"hG6$F$% \"aGF$F%F%*&F,F%-F'6$-F*6$F$F,F,F%!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sys3:=\{\"\"\",\"\",\"\}:" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 29 "System of PDE's Satisfied by " }{XPPEDIT 18 0 "Y[0](x) " "-&%\"YG6#\"\"!6#%\"xG" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "We com pute a system of PDE's for" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "f:=BesselY(0,x):" ">%\"fG-%(BesselYG6$\"\"!%\"xG" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 55 "by first computing an ODE with respect to the vari able " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 10 " by using " } {HYPERLNK 17 "gfun[holexprtodiffeq]" 2 "gfun[holexprtodiffeq]" "" } {TEXT -1 25 ", and next encoding that " }{XPPEDIT 18 0 "f" "I\"fG6\"" }{TEXT -1 20 " does not depend on " }{XPPEDIT 18 0 "a" "I\"aG6\"" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "holexprtod iffeq(f,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%\"xG\"\"\"-%%di ffG6$-F(6$-%\"yG6#F%F%F%F&F&F*F&*&F%F&F,F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "sys4:=\{subs(y(x)=h(x,a),\"),diff(h(x,a),a)\};" "> %%sys4G<$-%%subsG6$/-%\"yG6#%\"xG-%\"hG6$F,%\"aG%\"\"G-%%diffG6$-F.6$F ,F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sys4G<$-%%diffG6$-%\"hG6$% \"xG%\"aGF-,(*&F,\"\"\"-F'6$-F'6$F)F,F,F0F0F3F0*&F,F0F)F0F0" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "System of PDE's Satisfied by " } {XPPEDIT 18 0 "K[0](x)" "-&%\"KG6#\"\"!6#%\"xG" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "We compute a system of PDE's for" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "f:=BesselK(0,x):" ">%\"fG-%(BesselKG6$\"\"!%\"x G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "in the same way as in the pr evious section." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "holexprt odiffeq(f,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%\"xG\"\"\"-%% diffG6$-F(6$-%\"yG6#F%F%F%F&F&F*F&*&F%F&F,F&!\"\"" }}}{EXCHG {PARA 0 " > " 0 "" {XPPEDIT 19 1 "sys5:=\{subs(y(x)=h(x,a),\"),diff(h(x,a),a)\}; " ">%%sys5G<$-%%subsG6$/-%\"yG6#%\"xG-%\"hG6$F,%\"aG%\"\"G-%%diffG6$-F .6$F,F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sys5G<$-%%diffG6$-%\"h G6$%\"xG%\"aGF-,(*&F,\"\"\"-F'6$-F'6$F)F,F,F0F0F3F0*&F,F0F)F0!\"\"" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "System of PDE's Satisfied by th e Product" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Computing a system fo r the product is now a simple call to " }{HYPERLNK 17 "Mgfun[`sys*sys` ]" 2 "Mgfun[`sys*sys`]" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 41 "sys:=`sys*sys`(sys1,sys2,sys3,sys4,sys5);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$sysG<',D*&,(*&%\"aG\"\"%%\"xGF+!$_%\"%\"4 #\"\"\"*$F,F+\"$S(F/-%\"hG6$F,F*F/F/*(F*\"\"#F,F6-%%diffG6$-F86$-F86$- F86$F2F*F*F,F,F/\"$5#*&,(*$F,\"\"$!$3\"*&F*F+F,\"\"(\"$+(*$F,FG\"#?F/- F86$-F86$-F86$F2F,F,F,F/F/*&F,\"\"&-F86$-F86$FKF,F,F/!#)**(F*F/F,F+-F8 6$-F86$-F86$-F86$F>F,F,F,F,F/\"$S\"*&,(*$F,\"\"'F[o*$F,F6\"$0)*&F,F_oF *F+\"$?%F/FMF/F/*&F,F_o-F86$FSF,F/\"#N*&,(*&F*F/F,F/!%3>*&F*FRF,FR\"%+ G*&F*F/F,FR\"$+%F/FinF/F/*(F*F/F,FDFenF/!$q(*&F,F+FUF/\"$v\"*&,(*&F*F/ F,F+!%G:*&F*FRF,F+!%sKF*\"%77F/F>F/F/*&,(*$F,FR!$7%*&F*F+F,FRFHF,!%\"4 #F/FOF/F/*&F,FG-F86$FeoF,F/FR*&,(*$F*FD!#[*&F*FGF,F+!$+%*&F*FDF,F+F_pF /-F86$FF/!#E*&F,F/FOF/FD*&F*FDF[rF/!\")*(F*F/F,F6FgnF/!#7*&F *F6FF/F/*&,(F,!#@F]qFhtF_qF`uF/FOF/F/FetFftFgt\"#kFitFJF[u!#W*(F,FDF *F6-F86$F7F,F/!#5,B*&,(FDF/F)\"#7F0FJF/F2F/F/F5\"#;FcuFDFQFis*&,(F^oFi sF`o!\"#FboFJF/FMF/F/FdoF\\t*&,(F^pFftFjo!$!>F\\p!#CF/FinF/F/F`pF`uFbp F+*&,(Ffp!#;FhpFhvF*\"$Q#F/F>F/F/*&,(F,FhsF]qFbwF_qFhvF/FOF/F/FetF+Fgt \"#w*&,(Fir\"#)*F[sFiuFgrFftF/FFhs*&F*F6F[rF/Fis*&F*F/F " 0 "" {MPLTEXT 1 0 61 "ode:=op(int_of_sys(sys,x=-infinity..infinity,takayama_algo));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG,,*&%\"aG\"\"$-%\"hG6#F'\"\"\" \"#K*&,&*$F'\"\"(F,*$F'F(!\"\"F,-%%diffG6$-F56$-F56$-F56$F)F'F'F'F'F,F ,*&,&!\"$F,*$F'\"\"%\"$.\"F,F;F,F,*&,&*$F'\"\"#!\"%*$F'\"\"'\"#;F,F7F, F,*&,&*$F'\"\"&\"#tF'F(F,F9F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "However, the justification that the algorithm selected by the opti on " }{XPPEDIT 18 0 "takayama_algo" "I.takayama_algoG6\"" }{TEXT -1 0 "" }{TEXT -1 97 " applies to the integral under consideration is rathe r technical and is beyond this presentation." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 27 "Resolution of the Final ODE" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 11 "We use the " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 1 " " }{TEXT -1 34 "package to solve for the solution " }{XPPEDIT 18 0 "h(a)" "-%\"hG6#%\"aG" }{TEXT -1 199 " which corresponds to the i ntegral to be computed. Due to its integral representation, this func tion is analytic at 0, hence admits a Taylor expansion at 0. We proce ed to compute a closed form for " }{XPPEDIT 18 0 "h(a)" "-%\"hG6#%\"aG " }{TEXT -1 132 " by summation of this expansion. To this end, we det ermine a recurrence equation on the coefficients of the Taylor expansi on using " }{HYPERLNK 17 "gfun[diffeqtorec]" 2 "gfun[diffeqtorec]" "" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "ore:=diffeq torec(ode,h(a),u(n));" ">%$oreG-%,diffeqtorecG6%%$odeG-%\"hG6#%\"aG-% \"uG6#%\"nG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$oreG<%,&*&,&%\"nG\" \"\"\"\"#F*F*-%\"uG6#F)F*F**&,&F)!\"\"!\"'F*F*-F-6#,&F)F*\"\"%F*F*F*/- F-6#F*\"\"!/-F-6#\"\"$F:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 274 5 "Maple " }{TEXT -1 32 " readily solves this recurrence:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "rsol:=rsolve(ore,u(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%rsolG,**(,&-%\"uG6#\"\"##\"\"\"F+-F)6#\"\"!#F-\"\"%F --%&GAMMAG6#,&%\"nGF-F+F-F--F46#,&F7F-\"\"$F-!\"\"F+**,&F.F1F(#F F-),$FAF " 0 "" {MPLTEXT 1 0 44 "collect(map(normal,rsol,expanded),u,factor); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&**,&\"\"\"F&)-%'RootOfG6#,&*$%#_ ZG\"\"#F&F&F&%\"nGF&F&,&F&F&)!\"\"F/F&F&,&F/F&F.F&F2-%\"uG6#\"\"!F&#F& F.**,&F2F&F'F&F&F0F&F3F2-F56#F.F&F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "We perform the corresponding change of variable, " }{XPPEDIT 18 0 "n=2*p" "/%\"nG*&\"\"#\"\"\"%\"pGF&" }{TEXT -1 1 "," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "subs(\{n=2*p,(-1)^n=1,RootOf(_Z^2+1 )^n=(-1)^p\},\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(,&\"\"\"F&)! \"\"%\"pGF&F&,&F)\"\"#F+F&F(-%\"uG6#\"\"!F&F&*(,&F(F&F'F&F&F*F(-F-6#F+ F&!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "so that " }{TEXT 275 5 " Maple" }{TEXT -1 27 " can sum the Taylor series:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "sum(\"*a^(2*p),p=0..infinity);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$sumG6$*&,&*(,&\"\"\"F*)!\"\"%\"pGF*F*,&F-\"\" #F/F*F,-%\"uG6#\"\"!F*F**(,&F,F*F+F*F*F.F,-F16#F/F*!\"#F*)%\"aG,$F-F/F */F-;F3%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "h:=co llect(value(expand(\")),u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG, &*&,&*&%\"aG!\"#-%#lnG6#,&*$F)\"\"#!\"\"\"\"\"F2F2#F1F0*&F)F*-F,6#,&F/ F2F2F2F2#F2F0F2-%\"uG6#\"\"!F2F2*&,&F(F1F4F1F2-F:6#F0F2F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "It only remains to evaluate " }{XPPEDIT 18 0 "u[0]" "&%\"uG6#\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "u[2]" "&%\"uG6#\"\"#" }{TEXT -1 20 ". We first compute " }{XPPEDIT 18 0 "u[ 0]" "&%\"uG6#\"\"!" }{TEXT -1 46 " and find it is 0 by inversion of li mits. Let" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "f:=x*BesselJ(1, a*x)*BesselI(1,a*x)*BesselY(0,x)*BesselK(0,x):" ">%\"fG*,%\"xG\"\"\"-% (BesselJG6$\"\"\"*&%\"aGF&F%F&F&-%(BesselIG6$\"\"\"*&F,F&F%F&F&-%(Bess elYG6$\"\"!F%F&-%(BesselKG6$F5F%F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "be the integrand. We have:" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "int(limit(f,a=0),x=0..infinity);" "-%$intG6$-%&limitG6$ %\"fG/%\"aG\"\"!/%\"xG;F+%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "In the same way, each coefficient of the Taylor series for the integral is obtained by inve rsion of limits. In particular, " }{XPPEDIT 18 0 "kappa=u[2]" "/%&kap paG&%\"uG6#\"\"#" }{TEXT -1 6 ", but " }{TEXT 257 5 "Maple" }{TEXT -1 31 " is not capable of integrating:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "kappa=int(coeff(series(normal(diff(f,a,a)),a=0),a,0)/ 2,x=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&kappaG-%$intG6 $,$*(%\"xG\"\"$-%(BesselYG6$\"\"!F*\"\"\"-%(BesselKGF.F0#F0\"\"%/F*;F/ %)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "(This integral for " }{XPPEDIT 18 0 "kappa" "I&kappaG6\"" }{TEXT -1 33 " cannot be compu ted by a call to " }{TEXT 258 3 "int" }{TEXT -1 82 " using the Release 4, but the next release will probably be able to integrate it.)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "We obtain the following form for \+ " }{XPPEDIT 18 0 "h(a)" "-%\"hG6#%\"aG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "-combine(normal(-subs(\{u(0)=0,u(2) =kappa\},h)),ln,symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(-%#l nG6#*&,&*$%\"aG\"\"#!\"\"\"\"\"F.F.,&F*F.F.F.F.F.%&kappaGF.F+!\"#F-" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "It only remains to be proved tha t " }{XPPEDIT 18 0 "kappa=1/2/Pi" "/%&kappaG*(\"\"\"\"\"\"\"\"#!\"\"%# PiGF(" }{TEXT -1 300 ". We do not do it, since computing this last in tegral which is a constant lies outside the scope of the theory of hol onomy. With this example, we have reduced the problem of evaluating a parametrized integral to the evaluation of a non-parametrized integra l. In case there were no closed form for " }{XPPEDIT 18 0 "kappa" "I& kappaG6\"" }{TEXT -1 136 ", we could at least perform a simple numeric al evaluation and return a result in terms of this numerical value and the series above for " }{XPPEDIT 18 0 "kappa=1" "/%&kappaG\"\"\"" } {TEXT -1 1 "." }}}}}{MARK "0 1 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 } #%\"aG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "-combine(normal(-subs(\{u(0)=0,u(2) =kappa\},h)),ln,symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(-%#l nG6#*&,&*$%\"aG\"\"#!\"\"\"\"\"F.F.,&F*F.F.F.F.F.%&kappaGF.F+!\"#F-" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "It only remains to be proved tha t " }{XPPEDIT 18 0 "kappa=1!b({VERSION 3 0 "IBM INTEL LINUX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 } {CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 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"Norma l" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 } 0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT 256 18 "PATTERNS IN WORDS\n" }}{PARA 257 "" 0 "" {TEXT 260 11 "Bruno Salvy" }}{PARA 258 "" 0 "" {TEXT -1 29 "(Version of February 7, 1997)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 23 "This worksheet applies " }{HYPERLNK 17 "c ombstruct" 2 "combstruct" "" }{TEXT -1 5 " and " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 766 " to a simple combinatorial model of a probl em from computational biology and the study of DNA sequences. The DNA \+ can be viewed as a long text on an alphabet of four letters (A,C,G,T) . Large fragments of this text are tabulated. In particular, there are huge bases of genes, a gene being a few thousand letters long. Given \+ a short word, it is interesting to determine whether its number of occ urrences in a gene (or a virus) is far away from the most probable num ber of occurrences. If this number of occurrences is very improbable, \+ then this particular word may have a biological function. \n\nThe comb inatorial model is rational. The text is a word on the alphabet (a,b,c ,d) where all words of the same length are equiprobable. The probabili ty that a pattern occurs " }{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 72 " times in the text depends on the way the pattern overlaps with itsel f. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "libname:=`/net/blagny /algo/maple/5.4`,libname:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with(combstruct): with(gfun):" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 264 70 "Specification and univariate generating functions for the patt ern abab" }}{PARA 0 "" 0 "" {TEXT -1 190 "Working over the alphabet (a ,b,c,d), we first concentrate on a specific pattern (abab). To attack \+ problems related to occurrences of this pattern in words using combstr uct, we first write a " }{TEXT 261 7 "grammar" }{TEXT -1 293 " which d escribes a corresponding automaton.This grammar recognizes all the wor ds on (a,b,c,d). It is written in such a way that a mark (named Mark) \+ is present in a word everytime the pattern abab occurs. Then counting \+ the number of marks in a word gives the number of occurences of abab i n it." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 371 "G:=\{w=Union(Epsil on,Prod(a,wa),Prod(b,w),\n Prod(c,w),Prod(d,w)),\n wa=Un ion(Epsilon,Prod(a,wa),Prod(b,wab),\n Prod(c,w),Prod(d,w)), \n wab=Union(Epsilon,Prod(a,waba),Prod(b,w),\n Prod(c,w) ,Prod(d,w)),\n waba=Union(Epsilon,Prod(a,wa),Prod(b,Prod(Mark,w)), \n Prod(c,w),Prod(d,w)),\n Mark=Epsilon,a=Atom,b=Atom,c= Atom,d=Atom\}:" }}}{PARA 0 "" 0 "" {TEXT -1 20 "We use the function " }{HYPERLNK 17 "combstruct[count]" 2 "combstruct[count]" "" }{TEXT -1 45 " to check that the number of words of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "4^n" ")\"\"%%\"nG" } {TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "count([w,G, unlabelled],size=10),4^10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"(w&[5F #" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "count([w,G,unlabelled] ,size=20),4^20;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\".wxi6&*4\"F#" }}} {PARA 0 "" 0 "" {TEXT -1 23 "It is also possible to " }{TEXT 257 5 "pr ove" }{TEXT -1 64 " this by computing the generating function of the l anguage with " }{HYPERLNK 17 "combstruct[gfsolve]" 2 "combstruct[gfsol ve]" "" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "gf solve(G,unlabelled,z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<+/-%$wabG6# %\"zG,$*$,&F(\"\"%!\"\"\"\"\"F-F-/-%#waGF'F)/-%\"wGF'F)/-%%wabaGF'F)/- %\"cGF'F(/-%\"dGF'F(/-%\"aGF'F(/-%\"bGF'F(/-%%MarkGF'F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs(\",w(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$,&%\"zG\"\"%!\"\"\"\"\"F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 19 "The coefficient of " }{XPPEDIT 18 0 "z^n" ")%\"zG%\"nG" } {TEXT -1 67 " in the Taylor expansion of this generating function is t he number " }{XPPEDIT 18 0 "4^n" ")\"\"%%\"nG" }{TEXT -1 20 " of words of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 127 " in the lang uage, which confirms the correctness of our grammar.\n\nHere are a few typical words of the language obtained by the " }{TEXT 262 24 "unifor m random generator" }{TEXT -1 13 " provided by " }{HYPERLNK 17 "combst ruct[draw]" 2 "combstruct[draw]" "" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 76 "to 20 do eval(subs(Prod=proc() args end,draw ([w,G,unlabelled],size=30))) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A% \"cG%\"dG%\"aGF$F$F%%\"bGF#F&F$F#F&F%F$F&F%F#F&F#F$F#F%F%F#F%F%F$F&F%F #%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"bGF#%\"cGF$%\"dG%\"a GF#F#F#F#F$F&F#F$F&F#F%F$F&F&F$F%F$F%F&F$F$F&F&F&%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"dGF#%\"cG%\"aG%\"bGF&F%F$F$F$F&F&F$F%F#F& F$F&F%F&F#F&F#F$F#F%F&F%F$F%%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"bG%\"aG%\"cGF$%\"dGF%F$F$F$F&F#F#F#F%F%F&F$F&F#F%F#F$F#F#F$F% F#F%F&F#%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"dG%\"aG%\"cGF #F%F#%\"bGF#F&F$F%F#F#F&F#F&F%F#F$F%F#F#F%F$F#F$F#F$F#F#%(EpsilonG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6A%\"cG%\"bGF$F$F$F#%\"aGF#F%F$%\"dGF$F$ F$F%F$F$F%F$F#F%F#F$F#F%F%F$F$F$F%%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"bG%\"cGF#%\"dGF%%\"aGF$F$F&F$F$F&F%F#F$F$F&F%F$F#F%F &F&F#F$F&F&F&F&F%%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"cG% \"aG%\"bGF#%\"dGF#F%F#F$F$F%F%F$F#F$F&F#F&F&F&F%F&F&F&F#F&F%F$F&F%%(Ep silonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"cG%\"dGF$F$%\"bGF$F%%\"aG F$F#F&F$F$F%F%F&F#F$F#F&F%F%F#F&F$F&F&F%F%F%%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"aGF#%\"bGF$%\"dGF#F#F$F#F%F%F%F%F#F#F#F#%\"cGF% F$F&F$F$F$F#F#F&F#F#F%%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A% \"aG%\"bG%\"cG%\"dGF#F$F$F%F&F%F&F%F#F%F%F%F&F%F&F#F&F#F#F&F&F#F$F#F%F #%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"aG%\"dG%\"bGF%F$F#F$ F$F#F%F$F#F#F$%\"cGF#F$F&F$F$F$F%F$F&F#F$F&F%F%F$%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6B%\"dG%\"bGF$%\"aGF$F%F#F$F%F$F%F$%%MarkGF#F$ F%%\"cGF$F#F'F#F#F$F#F#F'F%F'F'F$F#%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"bG%\"dGF#%\"aGF%F#F$F$%\"cGF#F&F#F&F#F%F%F&F#F#F$F$F $F&F#F%F%F#F$F#F#%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"bGF# F#%\"cG%\"aGF$F$F%F#%\"dGF&F$F#F%F%F#F#F#F$F&F%F%F%F%F&F&F$F#F$F$%(Eps ilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"aG%\"bGF#%\"cGF$F%%\"dGF$F $F$F#F%F&F$F$F&F$F$F&F%F#F#F$F%F$F%F%F$F&F%%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"aG%\"cG%\"dG%\"bGF$F#F#F&F&F&F&F&F&F&F#F&F$F&F& F#F#F#F$F%F#F&F%F%F&F%%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6B% \"bG%\"dGF#%\"aGF%F#F$F#F#%\"cGF&F#F&F&F&F&F&F$F&F&F$F%F&F%F#F%F#%%Mar kGF&F%F#%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"dG%\"bGF$%\"a G%\"cGF&F#F#F%F$F&F&F&F&F&F&F%F$F&F&F%F$F$F%F&F&F&F$F$F&%(EpsilonG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6A%\"bG%\"cGF$F$F$F$F#%\"dGF$F%F#F#F$F%F %F%F$%\"aGF#F#F$F&F&F&F&F$F&F#F%F&%(EpsilonG" }}}{PARA 0 "" 0 "" {TEXT -1 109 "Some of these words countain the pattern abab, as indica ted by the letter `Mark' right after its occurrence. " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 265 30 "Bivariate generating functions" }}{PARA 0 "" 0 "" {TEXT -1 50 "From the grammar specification above, the comma nd " }{HYPERLNK 17 "combstruct[gfsolve]" 2 "combstruct[gfsolve]" "" } {TEXT -1 29 " can also be used to derive " }{TEXT 263 12 "multivariat e" }{TEXT -1 96 " generating functions. From this, it is easy to compu te the probability that the pattern occurs " }{XPPEDIT 18 0 "k" "I\"kG 6\"" }{TEXT -1 34 " times in a random word of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 109 ", the expectation of the number of occur rences of the pattern in such a word, and the corresponding variance. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "gfsolve(G,unlabelled,z,[ [u,Mark]]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<+/-%\"cG6$%\"zG%\"uGF( /-%\"dGF'F(/-%\"aGF'F(/-%\"bGF'F(/-%%MarkGF'F)/-%%wabaGF',$*&,.\"\"\"F F=F@FE*$F(FEF=*&F(FEF)FFF " 0 "" {MPLTEXT 1 0 20 "sol:=subs(\",w(z,u)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG,$*&,&\"\"\"F(*$%\"zG\"\"#F (F(,.!\"\"F(F*\"\"%F)F-*$F*\"\"$F.*$F*F.F-*&F*F.%\"uGF(F(F-F-" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "The coefficient of " }{XPPEDIT 18 0 "z^n*u^k" "*&)%\"zG%\"nG\"\"\")%\"uG%\"kGF&" }{TEXT -1 44 " in the T aylor series of this expression at " }{XPPEDIT 18 0 "z=0" "/%\"zG\"\"! " }{TEXT -1 34 " is the number of words of length " }{XPPEDIT 18 0 "n " "I\"nG6\"" }{TEXT -1 31 " where the pattern abab occurs " }{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 18 " times. Replacing " }{XPPEDIT 18 0 " z" "I\"zG6\"" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "z/4" "*&%\"zG\"\"\"\" \"%!\"\"" }{TEXT -1 21 " directly yields the " }{TEXT 258 31 "probabil ity generating function" }{TEXT -1 30 " under the uniform model (see \+ " }{HYPERLNK 17 "below" 1 "" "biased" }{TEXT -1 21 " for a biased mode l):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "GF:=normal(subs(z=z/4,sol)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#GFG,$*&,&\"#;\"\"\"*$%\"zG\"\"# F)F),.!$c#F)F+\"$c#F*!#;*$F+\"\"$F(*$F+\"\"%!\"\"*&F+F4%\"uGF)F)F5F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Here are the first coefficients :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "S:=map(normal,series(GF,u));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"SG+1%\"uG,$*&,&\"#;\"\"\"*$%\"zG \"\"#F+F+,,\"$c#F+F-!$c#F,F**$F-\"\"$!#;*$F-\"\"%F+!\"\"F*\"\"!,$*(F)F +F-F6F/!\"#F*\"\"\",$*(F)F+F-\"\")F/!\"$F*\"\"#,$*(F)F+F-\"#7F/!\"%F* \"\"$,$*(F)F+F-F*F/!\"&F*\"\"%,$*(F)F+F-\"#?F/!\"'F*\"\"&-%\"OG6#F+\" \"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "For instance, the coeffici ent of " }{XPPEDIT 18 0 "u^0" "*$%\"uG\"\"!" }{TEXT -1 77 " in this se ries gives the probabilities that the pattern abab does not occur:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "series(coeff(S,u,0),z,31); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+]o%\"zG\"\"\"\"\"!F%\"\"\"F%\"\"# F%\"\"$#\"$b#\"$c#\"\"%#\"$F\"\"$G\"\"\"&#\"%\\S\"%'4%\"\"'#\"%SF4\"\")#\"&lS'\"&Ob'\"\"*#\"(J7-\"\"(w&[5\"#5#\"'fVD\"'W@E \"#6#\")\"[=i\"\");sx;\"#7#\")6#eh\"FK\"#8#\")l\")4;FK\"#9#\"*PnIG\"\" *Gx@M\"\"#:#\"+Ryb!4%\"+'Hn\\H%\"#;#\"+(Ryw.#\"+[O[Z@\"#<#\",4eRj\\'\" ,OnZ>(o\"#=#\",Xe\\!=;\",%=p)zr\"\"#>#\",r!o9[kF[o\"#?#\".rZapy-\"\".w xi6&*4\"\"#@#\"/rac(z%Q;\"/;W/'=#f<\"#A#\".23p)[S?\"._bDB!*>#\"#B#\"0D ]3k>@g#\"0c1rw\\Z\"G\"#C#\"0p#zD)\\Cf#Fbp\"#D#\"0pD]%f\"Ge#Fbp\"#E#\"1 ,^+YrGH5\"1CE%o!***e7\"\"#F#\"2R'o%*ez&Hc'\"2Oz#z.%fd?(\"#G#\"2N]Xnd%G pK\"2oR'*=qzGg$\"#H#\"4TyP,oN$GU5\"4wp%og/:#H:\"\"#I-%\"OG6#F%\"#J" }} }{PARA 0 "" 0 "" {TEXT -1 144 "Thus the random draws of words of lengt h 30 that we did before are typical: the probability that the pattern \+ does not occur in such a word being" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "coeff(\",z,30)=evalf(coeff(\",z,30));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#\"4TyP,oN$GU5\"4wp%og/:#H:\"$\"+QqOS!*!#5" }}} {PARA 0 "" 0 "" {TEXT -1 135 "The expected number of occurrences is ob tained very directly from the bivariate generating function GF. Here i s its generating function" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "mom1:=factor(subs(u=1,diff(GF,u)));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%%mom1G,$*(%\"zG\"\"%,&F'\"\"\"!\"\"F*!\"#,&\"#;F**$F'\"\"#F*F+#F*F ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "smom1:=series(\",z,31) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&smom1G+en%\"zG#\"\"\"\"$c#\"\" %#F(\"$G\"\"\"&#\"#Z\"%'4%\"\"'#\"#J\"%[?\"\"(#\"%L7\"&Ob'\"\")#\"$P( \"&oF$\"\"*#\"&Ru#\"(w&[5\"#5#\"&Zc\"\"')GC&\"#6#\"'&Qi&\");sx;\"#7#\" 'L?J\"(3')Q)\"#8#\")^>(4\"\"*caVo#\"#9#\"(B%zf\"*Gx@M\"\"#:#\"*d=82#\" +'Hn\\H%\"#;#\"**3h96\"+[O[Z@\"#<#\"+$)**R>Q\",OnZ>(o\"#=#\"+fD-O?\",o $Q(fV$\"#>#\",\\S/&>p\".wxi6&*4\"\"#?#\",0Jo=m$\"-))Q\"ev\\&\"#@#\".NR ,vkB\"\"/;W/'=#f<\"#A#\"-bU3w0l\".3A-$4'z)\"#B#\"/h$p\"pK&=#\"0c1rw\\Z \"G\"#C#\"/\"G@=0W9\"\"0G`N)[P29\"#D#\"02A&)4qw#Q\"1'\\qti*f.X\"#E#\"0 6\"1[_o8)*z^A\"#F#\"1$>9zS2Tl'\"2Oz#z.%fd?(\"#G#\"1*3JD3V\\6\"4wp%og/:#H:\"\"#I-%\"OG6#F(\"#J" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(\");" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#+en%\"zG$\"+++D1R!#7\"\"%$\"+++]7yF'\"\"&$\"+Q4YZ6!#6 \"\"'$\"+v=n8:F.\"\"($\"+\"p39)=F.\"\")$\"+3b9\\AF.\"\"*$\"+dpy;EF.\"# 5$\"+1%GW)HF.\"#6$\"+:e2_LF.\"#7$\"+DKs>PF.\"#8$\"+i-P(3%F.\"#9$\"+)H< ]X%F.\"#:$\"+fVmA[F.\"#;$\"+>9J!>&F.\"#<$\"+y%ezb&F.\"#=$\"+ObgDfF.\"# >$\"+&f_KH'F.\"#?$\"+a'**3m'F.\"#@$\"+8naGqF.\"#A$\"+sP>'R(F.\"#B$\"+I 3%Qw(F.\"#C$\"+*)y[J\")F.\"#D$\"+[\\8*\\)F.\"#E$\"+2?ym))F.\"#F$\"+m!H WB*F.\"#G$\"+Dh2-'*F.\"#H$\"+$=B(p**F.\"#I-%\"OG6#\"\"\"\"#J" }}} {PARA 0 "" 0 "" {TEXT -1 106 "Thus in a sequence of 20 draws as above, we can expect the following number of occurrences of the pattern:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "20*coeff(\",z,30);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+PY%R*>!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "The variances are computed as easily as the expectations: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "mom2:=factor(subs(u=1,d iff(u*diff(GF,u),u)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%mom2G,$** %\"zG\"\"%,,\"$G\"\"\"\"F'!$G\"*$F'\"\"#\"\")*$F'\"\"$!\")*$F'F(F+F+,& F'F+!\"\"F+!\"$,&\"#;F+F-F+!\"##F5F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "evalf(series(mom2-add(coeff(smom1,z,i)^2*z^i,i=0..30) ,z,31));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+en%\"zG$\"+67*4*Q!#7\"\"% $\"+W[Y^xF'\"\"&$\"+rUHM6!#6\"\"'$\"+])f2\\\"F.\"\"($\"+jM1\\=F.\"\")$ \"+EUr2AF.\"\"*$\"+%HSic#F.\"#5$\"+YMtCHF.\"#6$\"+/]B$G$F.\"#7$\"+@$R< k$F.\"#8$\"+uIC+SF.\"#9$\"+6mueVF.\"#:$\"+'=]sr%F.\"#;$\"+xPvv]F.\"#<$ \"+ltDMaF.\"#=$\"+_4w#z&F.\"#>$\"+SXE^hF.\"#?$\"+F\"o(4lF.\"#@$\"+:!*F.\"#G$\"+Fozx$*F.\"#H$\"+9/IO(*F.\"#I-%\"O G6#\"\"\"\"#J" }}}{PARA 0 "" 0 "" {TEXT -1 19 "The coefficient of " } {XPPEDIT 18 0 "z^n" ")%\"zG%\"nG" }{TEXT -1 96 " in this series is the variance of the number of occurrences of the pattern in a word of len gth " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 1 "." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 266 27 "Fixed number of occurrences" }}{PARA 0 "" 0 " " {TEXT -1 51 "We now consider the probabilities that abab occurs " } {XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 12 " times, for " }{XPPEDIT 18 0 "k=0..5" "/%\"kG;\"\"!\"\"&" }{TEXT -1 8 ". Using " }{HYPERLNK 17 "g fun" 2 "gfun" "" }{TEXT -1 64 ", we can compute these probabilities fo r random words of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 7 " , with " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 23 " up to a few thou sands." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "maxnb:=5:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "for i from 0 to maxnb do pro ba[i]:=coeff(S,u,i) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&probaG6 #\"\"!,$*&,&\"#;\"\"\"*$%\"zG\"\"#F,F,,,\"$c#F,F.!$c#F-F+*$F.\"\"$!#;* $F.\"\"%F,!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&probaG6#\"\" \",$*(,&\"#;F'*$%\"zG\"\"#F'F'F-\"\"%,,\"$c#F'F-!$c#F,F+*$F-\"\"$!#;*$ F-F/F'!\"#F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&probaG6#\"\"#,$*(, &\"#;\"\"\"*$%\"zGF'F,F,F.\"\"),,\"$c#F,F.!$c#F-F+*$F.\"\"$!#;*$F.\"\" %F,!\"$F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&probaG6#\"\"$,$*(,&\" #;\"\"\"*$%\"zG\"\"#F,F,F.\"#7,,\"$c#F,F.!$c#F-F+*$F.F'!#;*$F.\"\"%F,! \"%F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&probaG6#\"\"%,$*(,&\"#;\" \"\"*$%\"zG\"\"#F,F,F.F+,,\"$c#F,F.!$c#F-F+*$F.\"\"$!#;*$F.F'F,!\"&F+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&probaG6#\"\"&,$*(,&\"#;\"\"\"* $%\"zG\"\"#F,F,F.\"#?,,\"$c#F,F.!$c#F-F+*$F.\"\"$!#;*$F.\"\"%F,!\"'F+ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 231 "Since we want to investigate these probabilities for texts of large size (a typical gene is a few \+ thousand letters long), we need the Taylor expansions of these rationa l functions for very large orders. These can be computed using " } {HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 31 ", which will first compu te the " }{TEXT 259 6 "linear" }{TEXT -1 52 " recurrence satisfied by \+ these Taylor coefficients (" }{HYPERLNK 17 "gfun[diffeqtorec]" 2 "gfun [diffeqtorec]" "" }{TEXT -1 73 "), and then exploit these recurrences \+ to compute the series efficiently (" }{HYPERLNK 17 "gfun[rectoproc]" 2 "gfun[rectoproc]" "" }{TEXT -1 2 "):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 216 "for i from 0 to maxnb do \n rec:=diffeqtorec(y(z )-proba[i],y(z),u(n));\n print(i,rec);\n rec:=select(has,rec,n) un ion \{seq(op(1,i)=evalf(op(2,i)),i=remove(has,rec,n))\};\n P[i]:=rec toproc(rec,u(n),remember) \nod:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\" \"!<',,-%\"uG6#%\"nG\"\"\"-F'6#,&F)F*F*F*!#;-F'6#,&F)F*\"\"#F*\"#;-F'6 #,&F)F*\"\"$F*!$c#-F'6#,&F)F*\"\"%F*\"$c#/-F'6#F2F*/-F'6#F7F*/-F'6#F#F */-F'6#F*F*" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$\"\"\"<+/-%\"uG6#F#\"\" !/-F'6#\"\"$F)/-F'6#\"\"#F),4-F'6#%\"nGF#-F'6#,&F5F#F#F#!#K-F'6#,&F5F# F1F#\"$)G-F'6#,&F5F#F-F#!%C5-F'6#,&F5F#\"\"%F#\"%g*)-F'6#,&F5F#\"\"&F# !&%Q;-F'6#,&F5F#\"\"'F#\"&GP(-F'6#,&F5F#\"\"(F#!'s58-F'6#,&F5F#\"\")F# \"&Ob'/-F'6#F)F)/-F'6#FE#F#\"$c#/-F'6#FJ#F#\"$G\"/-F'6#FO#\"#Z\"%'4%/- F'6#FT#\"#J\"%[?" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$\"\"#\"-F'6#,&FHF)\"\"'F)\"(obS(-F'6#,&FHF)\"\"(F)!)+!oF$-F' 6#,&FHF)FDF)\"*SI#[;-F'6#,&FHF)F8F)!*+!)GC&-F'6#,&FHF)F04\"-F'6#,&FHF)\"#8F)!,![O [Z@-F'6#,&FHF)\"#9F)\",+caVo#-F'6#,&FHF)\"#:F)!,%=p)zr\"-F'6#,&FHF)\"# ;F)\"+'Hn\\H%/-F'6#FXF*/-F'6#FgnF*/-F'6#F\\oF*/-F'6#FaoF*/-F'6#F`q#\"$ d\"\"*caVo#/-F'6#Feq#\"#x\")k)3r'/-F'6#Ffp#F)\");sx;/-F'6#F[q#F)\"(/V> %" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$\"\"%<7/-%\"uG6#\"\"\"\"\"!/-F'6# \"\"$F*/-F'6#\"\"#F*/-F'6#F*F*/-F'6#\"\"*F*/-F'6#\"#5F*/-F'6#\"#6F*/-F '6#\"\")F*/-F'6#\"#8F*/-F'6#\"#9F*/-F'6#\"#:F*/-F'6#\"#7F*,L-F'6#%\"nG F)-F'6#,&FYF)F)F)!#!)-F'6#,&FYF)F2F)\"%SE-F'6#,&FYF)F.F)!>%-F'6#,&FY F)F#F)\"'?j`-F'6#,&FYF)\"\"&F)!('47Y-F'6#,&FYF)\"\"'F)\")gp&[$-F'6#,&F YF)\"\"(F)!*!)[B=#-F'6#,&FYF)FEF)\"+?FQc6-F'6#,&FYF)F9F)!+SUv()e-F'6#, &FYF)F=F)\",ca.1K#-F'6#,&FYF)FAF)!,Sy1/U*-F'6#,&FYF)FUF)\"-?j(R.'H-F'6 #,&FYF)FIF)!-![o+*Q*)-F'6#,&FYF)FMF)\".g0t&Q%G#-F'6#,&FYF)FQF)!.'HvJ8O [-F'6#,&FYF)\"#;F)\".?^[cz**)-F'6#,&FYF)\"#.Jr7-F'6#,&FYF)\"#= F)\"/S9m8(Q8\"-F'6#,&FYF)\"#>F)!.!))Q\"ev\\&-F'6#,&FYF)\"#?F)\".wxi6&* 4\"/-F'6#F#F*/-F'6#FgoF*/-F'6#F\\pF*/-F'6#FapF*/-F'6#Ffr#F)\"+'Hn\\H%/ -F'6#F[s#FgoF\\u/-F'6#F`s#\"#f\",%=p)zr\"/-F'6#Fes#\"$N\"Ffu" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$\"\"&<;/-%\"uG6#\"\"\"\"\"!/-F'6#\"\"$F*/-F' 6#\"\"#F*/-F'6#F*F*/-F'6#\"\"*F*/-F'6#\"#5F*/-F'6#\"#6F*/-F'6#\"\")F*/ -F'6#\"#8F*/-F'6#\"#9F*/-F'6#\"#:F*/-F'6#\"#7F*,T-F'6#,&%\"nGF)FIF)!/; s$*3de9-F'6#,&FZF)FMF)\"/O6*o:!Q^-F'6#,&FZF)FQF)!0'*[E&[*eb\"-F'6#,&FZ F)\"#;F)\"0;wW9oaF%-F'6#,&FZF)F)F)!#'*-F'6#,&FZF)F2F)\"%OR-F'6#,&FZF)F .F)!&O6*-F'6#,&FZF)\"\"%F)\"(cqN\"-F'6#,&FZF)\"#=F)\"1c5Y3&o:=#-F'6#,& FZF)\"#>F)!1wp#4e`=$R-F'6#,&FZF)\"#?F)\"1w0%)Q6^Ge-F'6#,&FZF)\"# " 0 "" {MPLTEXT 1 0 52 "Digits:=30:for i from 0 to maxnb do i,P[i](1000) od; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"!$\"?H)f4f^R0U>$R8EUC!#J" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"\"$\"?gier+tR,%\\+Vs;=*!#J" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"#$\"?C=5L0q8+%e<-kTr\"!#I" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"$$\"?k@mY![0hd;Y0$))=@!#I" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"%$\"?]_Mx7(R*[%*oXK%3&>!#I" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"&$\"?CEykKv=B\">0`epU\"!#I" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "The following picture then shows h ow these probabilities evolve with the length of the word:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "plots[display](\{seq(plot([seq([10* i,P[j](10*i)],i=1..100)]),j=0..maxnb)\});" }}{PARA 13 "" 1 "" {INLPLOT "6(-%'CURVESG6$7`q7$$\"#5\"\"!$\"+u&QX'*)!#97$$\"#?F*$\"?%Re) zu!of-B1RtNC\"!#K7$$\"#IF*$\"?l3;/S#)3\">=fBD$)f$F37$$\"#SF*$\">/N&eJB H3Fn\"o#e7$$\"#gF*$\"?%\\?0.=^m:9' 4ZUb;F>7$$\"#qF*$\"?$Hd;mTp^[)[=HqWAF>7$$\"#!)F*$\"?6&fqK$fMl=$GJ$R&*G F>7$$\"#!*F*$\"?;1vjj@b/\\#o%3K)f$F>7$$\"$+\"F*$\"?vF(*f(eO*f%pldT]M%F >7$$\"$5\"F*$\">`.>e3t;F\"zoizF^!#I7$$\"$?\"F*$\">9P$yt:6P^B9mYRfF\\o7 $$\"$I\"F*$\">*pSG&RDPWv.\"R`tnF\\o7$$\"$S\"F*$\">`v>$3M1;_t%3SSi(F\\o 7$$\"$]\"F*$\">'[%Qm_5?9hk$4b&[)F\\o7$$\"$g\"F*$\">#Q.,0T9c[&)z97`$*F \\o7$$\"$q\"F*$\"?h3m!eP8]fKzKEA-\"F\\o7$$\"$!=F*$\"?$o+hCym0IsqO\"*)3 6F\\o7$$\"$!>F*$\"?L@\"G58*=1G#pnS\\>\"F\\o7$$\"$+#F*$\"?>qw['*zXUz*oh W+G\"F\\o7$$\"$5#F*$\"?.Wc)*4c89HVCz!RO\"F\\o7$$\"$?#F*$\"?yK$*4gk'Ge: nHniW\"F\\o7$$\"$I#F*$\"?)Hf!4!H'oO^*e6*)o_\"F\\o7$$\"$S#F*$\"?D1%p:$Q *G8\\+Jnbg\"F\\o7$$\"$]#F*$\"?I'*Q1I<3Z$zDM@@o\"F\\o7$$\"$g#F*$\"?/\\8 .*QMZ`xIK%RcO,?@7I()\\0L7DG=F\\o7$$\"$!GF*$\"?BLWD 9e*oZl-Exv*=F\\o7$$\"$!HF*$\"?$\\/Z_u'pH1nNiFk>F\\o7$$\"$+$F*$\"?!G@)3 K7X32Sn\"p#G?F\\o7$$\"$5$F*$\"?eSHxo=JXHNmG\\*3#F\\o7$$\"$?$F*$\"?A(yy l(**fSYMe')*y9#F\\o7$$\"$I$F*$\"?db_**))zT#*)**z<^M?#F\\o7$$\"$S$F*$\" ?<'HPO#*ovn(zRs7cAF\\o7$$\"$]$F*$\"?a%ohoj\"Qoc%>y9fI#F\\o7$$\"$g$F*$ \"?%*pVGj)Hw@p*y=\"GN#F\\o7$$\"$q$F*$\"?*RKOS(=HB*=\">e#oR#F\\o7$$\"$! 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Th)e@)[hTLFK#=iZ)F-7$F^jl$\"?wI(49Vm*Hanp')4D%)F-7$Fcjl$\"?rk9)GTj0V?& HACt$)F-Fgjl" 2 236 216 216 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 17 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "The complexity of these compu tations grows only linearly with the number " }{XPPEDIT 18 0 "k" "I\"k G6\"" }{TEXT -1 95 " of occurrences under study. Other kinds of constr aints like number of occurrences larger than " }{XPPEDIT 18 0 "k" "I\" kG6\"" }{TEXT -1 11 " for fixed " }{XPPEDIT 18 0 "k" "I\"kG6\"" } {TEXT -1 12 " or between " }{XPPEDIT 18 0 "k[1]" "&%\"kG6#\"\"\"" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "k[2]" "&%\"kG6#\"\"#" }{TEXT -1 141 " also give rise to rational generating functions that can be extr acted from the generating function GF and thus can be treated the same way. " }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 267 14 "Other patterns" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 241 "All the above computation was der ived from the grammar describing the language, a mark being appended t o every occurrence of the pattern. It is actually easy to write a Mapl e procedure taking as input a word, and producing the corresponding " }{HYPERLNK 17 "combstruct grammar" 2 "combstruct[specification]" "" } {TEXT -1 94 ". Then the whole computation above can be reproduced for \+ any pattern completely automatically." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 561 "gengram:=proc(pattern::list(\{identical(a),identical (b),identical(c),identical(d)\}))local i, eq, letter, state, j;\nfor i to nops(pattern) do for letter in [a,b,c,d] do for j from 0 to i-1 do if [op(1..i-j,pattern)]=[op(j+1..i-1,pattern),letter] then state[lett er]:=cat(w,op(1..i-j,pattern)); break fi od; if j=i then state[letter] :=w fi od;eq[i]:=cat(w,op(1..i-1,pattern))=Union(Epsilon,seq(Prod(lett er,state[letter]),letter=[a,b,c,d])) od; subs(cat(w,op(pattern))=Prod( Mark,w),\{seq(eq[i],i=1..nops(pattern)),seq(letter=Atom,letter=[a,b,c, d]),Mark=Epsilon\}) end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(gengram G:6#'%(patternG-%%listG6#<&-%*identicalG6#%\"aG-F.6#%\"bG-F.6#%\"cG-F. 6#%\"dG6'%\"iG%#eqG%'letterG%&stateG%\"jG6\"F@C$?(8$\"\"\"FD-%%nopsG6# 9$%%trueGC$?&8&7&F0F3F6F9FIC$?(8(\"\"!FD,&FCFD!\"\"FDFI@$/7#-%#opG6$;F D,&FCFDFPFSFH7$-FX6$;,&FPFDFDFDFRFHFLC$>&8'6#FL-%$catG6$%\"wGFW%&break G@$/FPFC>F]oFco>&8%6#FC/-Fao6$Fco-FX6$;FDFRFH-%&UnionG6$%(EpsilonG-%$s eqG6$-%%ProdG6$FLF]o/FLFM-%%subsG6$/-Fao6$Fco-FXFG-Fjp6$%%MarkGFco<%/F fqFep-Fgp6$/FL%%AtomGF\\q-Fgp6$Fio/FC;FDFEF@F@" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "Given a pattern, this procedure outputs the corresp onding combstruct grammar. Thus for instance, using the pattern abab, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "gengram([a,b,a,b]);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#<+/%\"wG-%&UnionG6'%(EpsilonG-%%ProdG6 $%\"aG%#waG-F+6$%\"bGF%-F+6$%\"cGF%-F+6$%\"dGF%/F.-F'6'F)F*-F+6$F1%$wa bGF2F5/F=-F'6'F)-F+6$F-%%wabaGF/F2F5/FC-F'6'F)F*-F+6$F1-F+6$%%MarkGF%F 2F5/FKF)/F-%%AtomG/F1FN/F4FN/F7FN" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 211 "we obtain the grammar we started with. It is now possible to stud y longer patterns easily. Here are the different states leading to the probability that the pattern abacab occurs twice in a word of length 5000:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "First the grammar is ge nerated" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "G:=gengram([a,b, a,c,a,b]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"GG<-/%\"wG-%&UnionG6 '%(EpsilonG-%%ProdG6$%\"aG%#waG-F-6$%\"bGF'-F-6$%\"cGF'-F-6$%\"dGF'/F0 -F)6'F+F,-F-6$F3%$wabGF4F7/F?-F)6'F+-F-6$F/%%wabaGF1F4F7/%%MarkGF+/F/% %AtomG/F3FI/F6FI/F9FI/FE-F)6'F+F,F=-F-6$F6%&wabacGF7/FR-F)6'F+-F-6$F/% 'wabacaGF1F4F7/FX-F)6'F+F,-F-6$F3-F-6$FGF'F4F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "then the bivariate generating functions are derived " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "gfsolve(G,unlabelled,z, [[u,Mark]]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<-/-%\"cG6$%\"zG%\"uGF (/-%\"dGF'F(/-%\"aGF'F(/-%\"bGF'F(/-%%MarkGF'F)/-%\"wGF',$*&,&\"\"\"F< *$F(\"\"%FF=F@*$F(\"\"&F>*$F(\"\"'F@*&F(FDF)FF)F " 0 "" {MPLTEXT 1 0 29 "normal(subs(\",z=z/4,w(z,u)) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&\"$c#\"\"\"*$%\"zG\"\"%F'F ',.!%'4%F'F)\"%'4%F(!#;*$F)\"\"&\"#;*$F)\"\"'!\"\"*&F)F3%\"uGF'F'F4F. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "extracting a coefficient we \+ get the probability generating function of words with 2 occurrences of the pattern" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "coeff(serie s(\",u,3),u,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**,&\"$c#\"\"\"*$ %\"zG\"\"%F'F'F)\"#7,,\"%'4%F'F)!%'4%F(\"#;*$F)\"\"&!#;*$F)\"\"'F'!\"# ,,F.F'F)F-F(F2F0F/F3!\"\"F7F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 " this gives rise to a linear recurrence satisfied by the Taylor coeffic ients" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "diffeqtorec(y(z)- \",y(z),u(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<5/-%\"uG6#\"\"\"\" \"!/-F&6#\"\"$F)/-F&6#\"\"#F)/-F&6#F)F)/-F&6#\"\"*F)/-F&6#\"#5F)/-F&6# \"#6F)/-F&6#\"\")F),H-F&6#%\"nGF(-F&6#,&FHF(F(F(!#[-F&6#,&FHF(F1F(\"$; )-F&6#,&FHF(F-F(!%Kc-F&6#,&FHF(\"\"%F(\"&cI\"-F&6#,&FHF(\"\"&F(!&wX#-F &6#,&FHF(\"\"'F(\"'+'4%-F&6#,&FHF(\"\"(F(!(g@$R-F&6#,&FHF(FDF(\"(+/$)* -F&6#,&FHF(F8F(!(%=P%*-F&6#,&FHF(FfT#-F&6#,&FHF(\"#9F(\"*oj I0)-F&6#,&FHF(\"#:F(!,OnZ>(o-F&6#,&FHF(\"#;F(\"-3-Veh?-F&6#,&FHF(\"#(o/-F&6#FXF)/-F&6#FgnF)/-F&6#F\\oF)/- F&6#FaoF)/-F&6#F_r#\"%&o#\"+[O[Z@/-F&6#Fjq#\"%>>Fgs/-F&6#Feq#Fgn\"(3') Q)/-F&6#F[q#F-\");sx;/-F&6#F`q#F-Fat/-F&6#Ffp#F(Fft" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "As before, in order to speed up the computatio n, we change the initial conditions into floating point numbers:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Digits:=50:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "select(has,\"\",n) union \{seq(op(1 ,i)=evalf(op(2,i)),i=remove(has,\"\",n))\}:" }}}{EXCHG {PARA 0 "" 0 " " {HYPERLNK 17 "gfun[rectoproc]" 2 "gfun[rectoproc]" "" }{TEXT -1 64 " then produces a procedure to evaluate this sequence efficiently" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "p:=rectoproc(\",u(n),remembe r):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "for i from 0 by 500 \+ to 10000 do i,p(i) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"!F#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"$+&$\"R#oj..Hv'*=N0T!zP<%p+!oVs6]J'! #^" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+5$\"S$e%!#^" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+?$\"R1/Lv5ZU> 29z5o8>)=/i&QSa#Rs!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+D$\"S*G.l yI5$onRiZhRsv*)o)3XnuN+\"!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+I$ \"S$[&*fl-_#eE#fZSF$)*)3hE?yxv7G\"!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+N$\"S-$z#\\*)o87w/ty/y#\\8R$GFAHeX:!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+S$\"SWxl&*H2Vo>x$\\l1B\\@3AO/9U')y\"!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+X$\"S(=A15$4Z$ziuB/N^SARc9-B5a+#!#]" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+]$\"S.ZskEY])4(fA0&HkiCO$*>9#z,$># !#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+b$\"S'[tBBA\"QI'4KfLfiLlP-s V2`-N#!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+g$\"SNIud*Q`*>#GqT\"o Dh$3+>AG]VrZ#!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+l$\"S/Z%)yRJ-v 6?q(=)z`Etf;3fAiuD!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+q$\"S<\\n pmo)R/Sz*yG/-wrgdOs4EWE!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+v$\" S)[N*GAL60[Lu7&)\\>k\"f!)o#!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$\"%+!)$\"Sr%H,g.9m0ffd^^BkA)*\\z!R%z#3F!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+&)$\"SM-I%f!>(Hcv%ft#RHq+@pkw6?tq#!#]" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$\"%+!*$\"S:48c+o=d#)=j()e#y#)\\o%oU)G=wo#!#]" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+&*$\"SAa(Hqm+*Q`\"RQm8f%eyo2tTud^E !#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"&++\"$\"SCEWCN4uk-#zc*RJ$\\Uv KA)\\[^,E!#]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "The following pic ture shows the evolution of this probability with the length of the wo rd" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot([seq([100*i,p(10 0*i)],i=0..100)]);" }}{PARA 13 "" 1 "" {INLPLOT "6#-%'CURVESG6$7aq7$\" \"!F(7$$\"$+\"F($\"SYX.5hLk[*z&)eKd_4G4(z.;D)yK#!#`7$$\"$+#F($\"SsRs4l 03(>v%ziq$H*\\7[:rJBF47 $$\"$+%F($\"SIFF3G/kNRuu)pN_T_FLF\\HAq4%F47$$\"$+&F($\"R#oj..Hv'*=N0T! 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)o#Ffp7$$\"%+wF($\"S[&R?iD,Mo8gL'z)=zFx+.r#Ffp7$$\"%+$)F($\" S^K!QrE\"\\grj!>M?/9hRw)eZ$*35FFfp7$$\"%+%)F($\"S=1wv/z>mF.;%yx\"3^>oR wz#*34FFfp7$$\"%+&)F($\"SM-I%f!>(Hcv%ft#RHq+@pkw6?tq#Ffp7$$\"%+')F($\" SE:AxvjF^$)*zU>GGY%eR10r8![q#Ffp7$$\"%+()F($\"S>9.^:tL'3z8p%[9o&[mP14 \\_:q#Ffp7$$\"%+))F($\"SoQ6nHC$)[@6Rt?T.#fJH<0&Gf(p#Ffp7$$\"%+*)F($\"S Y%>N>%fl\"\\Z'=x\"4_$R7j " 0 "" {MPLTEXT 1 0 2 "G;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<-/% \"wG-%&UnionG6'%(EpsilonG-%%ProdG6$%\"aG%#waG-F+6$%\"bGF%-F+6$%\"cGF%- F+6$%\"dGF%/F.-F'6'F)F*-F+6$F1%$wabGF2F5/F=-F'6'F)-F+6$F-%%wabaGF/F2F5 /%%MarkGF)/F-%%AtomG/F1FG/F4FG/F7FG/FC-F'6'F)F*F;-F+6$F4%&wabacGF5/FP- F'6'F)-F+6$F-%'wabacaGF/F2F5/FV-F'6'F)F*-F+6$F1-F+6$FEF%F2F5" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 327 "Marks are added by replacing a by Prod(Atom,Marka) everywhere it occurs, and similarly for the other le tters. There Marka, Markb, Markc and Markd have size 0 and do not modi fy the counting sequences and the related probabilities, but make it p ossible to extract multivariate generating functions which contain mor e information." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "Gprob:=s ubs(a=Prod(Atom,Marka),b=Prod(Atom,Markb),c=Prod(Atom,Markc),d=Prod(At om,Markd),G minus \{a=Atom,b=Atom,c=Atom,d=Atom\}) union \{Marka=Epsil on, Markb=Epsilon, Markc=Epsilon, Markd=Epsilon\};" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%&GprobG<-/%%MarkG%(EpsilonG/%&MarkaGF(/%&MarkbGF(/% &MarkcGF(/%&MarkdGF(/%'wabacaG-%&UnionG6'F(-%%ProdG6$-F76$%%AtomGF*%#w aG-F76$-F76$F;F,-F76$F'%\"wG-F76$-F76$F;F.FC-F76$-F76$F;F0FC/F<-F46'F( F6-F76$F?%$wabGFDFH/%&wabacG-F46'F(-F76$F9F2-F76$F?FCFDFH/FQ-F46'F(-F7 6$F9%%wabaGFXFDFH/FC-F46'F(F6FXFDFH/Fin-F46'F(F6FO-F76$FFFSFH" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Here is the multivariate generatin g function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "gfsolve(Gpro b,unlabelled,z,[[u,Mark],[a,Marka],[b,Markb],[c,Markc],[d,Markd]]);" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#<-/-%#waG6(%\"zG%\"uG%\"aG%\"bG%\"cG% \"dG,$*&,*\"\"\"F1*,F(\"\"&F*\"\"#F+F4F,F1F)F1F1**F(\"\"%F,F1F+F1F*F4F 1**F(F3F,F1F+F4F*F4!\"\"F1,:*&F(F1F*F1F1*,F(\"\"'F,F1F*\"\"$F+F4F)F1F1 *,F(F3F,F1F-F1F+F1F*F4F1**F(F3F,F4F+F1F*F4F1F7F1F5F8**F(F3F,F1F*F=F+F1 F1**F(F " 0 "" {MPLTEXT 1 0 27 "GF:=su bs(\",w(z,u,a,b,c,d));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#GFG,$*&,& \"\"\"F(**%\"zG\"\"%%\"cGF(%\"bGF(%\"aG\"\"#F(F(,:*&F*F(F.F(F(*,F*\"\" 'F,F(F.\"\"$F-F/%\"uGF(F(*,F*\"\"&F,F(%\"dGF(F-F(F.F/F(**F*F7F,F/F-F(F .F/F(**F*F7F,F(F-F/F.F/F(F)!\"\"**F*F7F,F(F.F4F-F(F(**F*F3F,F(F.F4F-F/ F;F;F(*&F*F(F-F(F(*&F*F(F,F(F(*&F*F(F8F(F(F;F;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "The coefficient of " }{XPPEDIT 18 0 "a^k*b^l*c^m*d^p *u^q*z^n" "*.)%\"aG%\"kG\"\"\")%\"bG%\"lGF&)%\"cG%\"mGF&)%\"dG%\"pGF&) %\"uG%\"qGF&)%\"zG%\"nGF&" }{TEXT -1 34 " in the Taylor expansion of G F in " }{XPPEDIT 18 0 "z" "I\"zG6\"" }{TEXT -1 34 " is the number of w ords of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 6 " with " } {XPPEDIT 18 0 "k " "I\"kG6\"" }{TEXT -1 30 " occurrences of the letter a, " }{XPPEDIT 18 0 "l" "I\"lG6\"" }{TEXT -1 37 " occurrences of the \+ letter b ... and " }{XPPEDIT 18 0 "q" "I\"qG6\"" }{TEXT -1 93 " occurr ences of the pattern. The probability generating function is obtained \+ by substituting " }{XPPEDIT 18 0 "a,b,c,d" "6&%\"aG%\"bG%\"cG%\"dG" } {TEXT -1 148 " by the corresponding probability. Thus GF now takes int o account both the specific pattern and the biased probabilities of th e letters in the text." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "We can \+ again compute the probability that the pattern occurs twice in words o f length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 41 " and compare th is with the uniform model:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "normal(coeff(series(GF,u,3),u,2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*.,&\"\"\"F%**%\"zG\"\"%%\"cGF%%\"bGF%%\"aG\"\"#F%F%F'\"#7F)F,F+ \"\"'F*F(,8*&F'F%F+F%!\"\"F&F%*,F'\"\"&F)F%%\"dGF%F*F%F+F,F1**F'F3F)F, F*F%F+F,F1**F'F3F)F%F*F,F+F,F1*&F'F%F*F%F1**F'F3F)F%F+\"\"$F*F%F1**F'F .F)F%F+F9F*F,F%F%F%*&F'F%F)F%F1*&F'F%F4F%F1!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 140 "Again we compute a linear recurrence satisfied by t he Taylor coefficients from which we produce an efficient procedure fo r their evaluation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "diff eqtorec(y(z)-\",y(z),u(n)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "nuprob:=rectoproc(subs([a=0.25,b=0.18,c=0.24,d=0.33],\"),u(n),reme mber);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'nuprobG:6#%\"nG6\"6#%)rem emberGE\\s3\"\"\"\"\"!\"#7$\"2+++++DiZ\"!#C\"\"&F-\"#9$\"6+++++++]t&)) !#G\"\"$F-\"\"(F-\"#6F-\"#:$\"9++++++++]Aw9!#I\"#;$\";+++++++]Q.a8A!#K \"#<$\"=++++++++]:5o(4$!#M\"\"*F-\"#8$\"4++++++v'GW!#E\"\"#F-\"\")F-\" \"'F-\"#5F-\"\"%F-F-F-,F-9!6#,&9$F,!#=F,$!:+++++++]P8Oz\"!#O-FR6#,&FUF ,!#\"FE-FR6#,&FUF,!#;F,$!9+++++++D#znx#FA-FR6#,&FUF ,!#:F,$\"8++++++++T(*\\#F=-FR6#,&FUF,!#9F,$!6++++++]?1<'F6-FR6#,&FUF,! #8F,$\"5++++++vNL5FJ-FR6#,&FUF,!#7F,$!4+++++vpA.#F1-FR6#,&FUF,!#6F,$\" 3++++++m!e#!#A-FR6#,&FUF,!#5F,$!1+++++$yv'!#?-FR6#,&FUF,!\"*F,$\"/++++ +hlFV-FR6#,&FUF,!\")F,$!.++++P'QF]o-FR6#,&FUF,!\"(F,$\"-++++H))Fio-FR6 #,&FUF,!\"'F,$!,++]kY#Fep-FR6#,&FUF,!\"&F,$\"*+++V#Fbq-FR6#,&FUF,!\"%F ,$!'++\")F_r-FR6#,&FUF,!\"$F,$\"(+++\"F[s-FR6#,&FUF,!\"#F,$!&++$Fgs-FR 6#,&FUF,!\"\"F,$\"$+$FctF(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 207 " The probabilities for this non-uniform model are rather different from those obtained in the uniform model: the pattern abacab is now less p robable, since b and c are less probable than in the uniform model." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "for i from 0 by 500 to 100 00 do i,nuprob(i) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"!F#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"$+&$\"S+\"[*))R%R^DT'*efa)Qa))oCVq*G 6m\"!#_" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+5$\"R@r=I`R]'G&p/loOnM* zKz)R0$)Q'!#^" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+:$\"SN&oyL$z1h-5- X#\\]1\"H#3[k=(Qi8!#^" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+?$\"Svj\" GW)>&H:u>'4#eivswgJfOuvG#!#^" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+D$ \"STHpgHM!*z/s$34`+FP27m$eH`d%!#^" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+N$\"Ru%R$Ra'*pP!zcD0**oRlAjni#oq'e!#]" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$\"%+S$\"RHn*3/Y2m48;t(\\YrA4)eO^7u5wx!>9^3z#f'4VT ++\"!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+b$\"S9X8\\;$pd'oGA5')[C e&>UB6T.$R6!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+g$\"Sb5u?GjkQ%*R wF&[+z7&Q;.N>^w7!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+l$\"S3)*))) Hp*=4O+$RmNb1lD'R)R4!Q59!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+q$ \"S6^]$p&f*eM:z(Q-v$[de;hw0T)R:!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ \"%+v$\"SV=IG+`YDJqeY!Qg!z@1dF8I-k;!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+!)$\"SEB+a.Z\\AARQO&Hh+p/bIY]EAy\"!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+&)$\"S,];'QAq'p(He.)RF%yr!)3HH'R!R*=!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+!*$\"S\">B:E0Cf\\D(oz1@xDP@e4Q7k)*>!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+&*$\"SDqt/[)y\"p\"pa0<-'p(yMew\\ &49'4#!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"&++\"$\"Ss!Hg*)RJ\\^.w5 L?GVYej!=Lv?'=#!#]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Here are th e curves corresponding to the non-uniform and to the uniform model:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "plots[display](\{plot([se q([100*i,p(100*i)],i=0..100)]),plot([seq([100*i,nuprob(100*i)],i=0..10 0)])\});" }}{PARA 13 "" 1 "" {INLPLOT 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fp7$Fb[l$\"SU1V%*=f8CUB6T.$R6Ffp7$F[]l$\"SaGJ)[1aBs?Z,B-%fvxJYtdR&p;\"Ffp7$F`]l$\"S4=bRb6 TEj<7')\\p<*p)\\^xT0^%>\"Ffp7$Fe]l$\"Sif[Mk()GzM(R8(Re-/5290A\"Ffp 7$Fj]l$\"Sop%)RiWMTRa+j^#[%eiDw$[h*H\\7Ffp7$F_^l$\"Sb5u?GjkQ%*RwF&[+z7 &Q;.N>^w7Ffp7$Fd^l$\"S2SF\"3Amd4)e\\m2?4s\\MY[P**e.8Ffp7$Fi^l$\"Sfn&GJ j$y#fRV'3F;x&HDt,NVC0L\"Ffp7$F^_l$\"S`%R)*=i6k7I%4oh6$4hBM5vd1tN\"Ffp7 $Fc_l$\"Sh.Q7AW0w]b4)4aG)[@=L*)ey#RQ\"Ffp7$Fh_l$\"S3)*)))Hp*=4O+$RmNb1 lD'R)R4!Q59Ffp7$F]`l$\"SWC\"zh))yUh=7_VKP.Hi)*>#=alO9Ffp7$Fb`l$\"S@m@- ^Ch:O@,>!f\"Ffp7$F`bl$\"S$\\EE?FmQ?!>`tDpF;J$[>)*3K]h\"Ffp 7$Febl$\"SmpH;K6^HWlsku+dt\"Ffp7$F^d l$\"S:PF\"*Q$>!>8BV2r5>$f;7E%*R#4fFfp7$Fffl$\"Sk85B[_Fq` ;$oSG`2VfSo_z]m$>Ffp7$F[gl$\"S^f$=9&f*fKdugn)zP(y%Q/!fh+w&>Ffp7$F`gl$ \"SIKFRa)*R29IoY1H$yc_*e!y#[Ey>Ffp7$Fegl$\"S\">B:E0Cf\\D(oz1@xDP@e4Q7k )*>Ffp7$Fjgl$\"S]=<2\"oYf7)G$zXEH(=kg/&4#ys=?Ffp7$F_hl$\"S:X\"3%\\HFc \"G/$=mn'>mP?MIrA&Q?Ffp7$Fdhl$\"SFR6#[c]`.W/kB@k%)=?N2!4U-e?Ffp7$Fihl$ \"Sd=S4,BW6%QiY)R)*pjY,H8d2Bx?Ffp7$F^il$\"SDqt/[)y\"p\"pa0<-'p(yMew\\& 49'4#Ffp7$Fcil$\"S$3gG)fU?^\"o$R-VDr8_F%)>XNv9@Ffp7$Fhil$\"S6okc'Q+s-C T+0bk2q>9M9TnI8#Ffp7$F]jl$\"Su)yVA-:J4nH[#oUUs&zjkUd\"3^@Ffp7$Fbjl$\"S AhmLNHq8r[d$Q-&e_/]?F(=&zo@Ffp7$Fgjl$\"Ss!Hg*)RJ\\^.w5L?GVYej!=Lv?'=#F fpF[[m" 2 294 214 214 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 7896 0 0 0 0 0 0 }}} {PARA 5 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 15 " Markovian model" }}{PARA 0 "" 0 "" {TEXT -1 444 "By slightly modifying the automaton, it is possible to consider a Markovian model, where in stead of giving the probabilities of occurrence of each letter, one gi ves the probabilities of transition from one letter to the next one. T he new automaton is almost the same as the previous one, except that t he transitions are marked and three more states are added at the begin ing. Here is the procedure generating the new automaton from the patte rn:" }}{EXCHG {PARA 4 "> " 0 "" {MPLTEXT 1 0 858 "gengram2 := proc (pa ttern::list(\{identical(a), identical(b),identical(c), identical(d)\}) )local i, eq, letter, state, j,alph;alph:=[a,b,c,d];for i to nops(patt ern) do for letter in alph do for j from 0 to i-1 do if [op(1 .. i-j,p attern)] =[op(j+1 .. i-1,pattern), letter] then state[letter] :=cat(w, op(1 .. i-j,pattern));break fi od;if j=i then state[letter] := cat(w,l etter) fi od;eq[i] := cat(w,op(1 .. i-1,pattern)) =Union(Epsilon,seq(P rod(letter,`if`(i>1,cat(Mark,pattern[i-1],letter),cat(Markini,letter)) ,state[letter]), letter = alph))od;subs(cat(w,op(pattern)) =Prod(Ma rk,w),\{Mark = Epsilon,seq(seq(cat(Mark,i,j)=Epsilon,j=alph),i=alph),s eq(eq[i],i = 1 .. nops(pattern)),seq(letter = Atom,letter=alph), \+ seq(cat(w,i)=Union(Epsilon,seq(Prod(j,cat(Mark,i,j),cat(w,j)),j=alph)) ,i=subs(pattern[1]=NULL,alph)),seq(cat(Markini,i)=Epsilon,i=alph)\})en d;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)gengram2G:6#'%(patternG-%%lis tG6#<&-%*identicalG6#%\"aG-F.6#%\"bG-F.6#%\"cG-F.6#%\"dG6(%\"iG%#eqG%' letterG%&stateG%\"jG%%alphG6\"FAC%>8)7&F0F3F6F9?(8$\"\"\"FH-%%nopsG6#9 $%%trueGC$?&8&FDFMC$?(8(\"\"!FH,&FGFH!\"\"FHFM@$/7#-%#opG6$;FH,&FGFHFS FVFL7$-Fen6$;,&FSFHFHFHFUFLFPC$>&8'6#FP-%$catG6$%\"wGFZ%&breakG@$/FSFG >F`o-Fdo6$FfoFP>&8%6#FG/-Fdo6$Ffo-Fen6$;FHFUFL-%&UnionG6$%(EpsilonG-%$ seqG6$-%%ProdG6%FP-%#ifG6%2FHFG-Fdo6%%%MarkG&FL6#FUFP-Fdo6$%(MarkiniGF PF`o/FPFD-%%subsG6$/-Fdo6$Ffo-FenFK-F_q6$FgqFfo<(/FgqFjp-F\\q6$/-Fdo6$ F\\rFGFjp/FGFD-F\\q6$/-Fdo6$FfoFG-Fhp6$Fjp-F\\q6$-F_q6%FS-Fdo6%FgqFGFS -Fdo6$FfoFS/FSFD/FG-F_r6$/&FL6#FH%%NULLGFD-F\\q6$/FP%%AtomGF]r-F\\q6$- F\\q6$/FjsFjpF^tF^s-F\\q6$F^p/FG;FHFIFAFA" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "For instance, the same pattern abacab as above yields th e automaton described by the following grammar" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 27 "G:=gengram2([a,b,a,c,a,b]);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%\"GG-F56'F(-F86%F*%'MarkbaGFY-F 86%F-%'MarkbbGFQ-F86%F/%'MarkbcGFB-F86%F1%'MarkbdGFF/FF-F56'F(-F86%F*% 'MarkdaGF3-F86%F-%'MarkdbGFQ-F86%F/%'MarkdcGFB-F86%F1%'MarkddGFF/FB-F5 6'F(-F86%F*%'MarkcaGF3-F86%F-%'MarkcbGFQ-F86%F/%'MarkccGFB-F86%F1%'Mar kcdGFF/FTF(/FWF(/FPF(/FMF(/Fhn-F56'F(-F86%F*F\\q%'wabacaGF]qF`qFcq/F:F (/FAF(/F=F(/F_r-F56'F(F7-F86%F-F=-F86$F'FHF?FC/FfpF(/FeqF(/F]pF(/F`pF( /FcpF(/FgoF(/F\\qF(/F_qF(/FbqF(/FEF(/FaoF(/FdoF(/F^oF(/FQ-F56'F(-F86%F *F^oF3F_oFboFeo" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 144 "From there, g enerating functions follow. For instance, we recover the generating fu nction obtained before when all transitions are equiprobable:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "subs(gfsolve(G,unlabelled,z ,[[u,Mark],seq([1/4,cat(Markini,i)],i=[a,b,c,d]),seq(seq([1/4,cat(Mark ,i,j)],j=[a,b,c,d]),i=[a,b,c,d])]),w(z,u));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&\"$c#\"\"\"*$%\"zG\"\"%F'F',.!%'4%F'F)\"%'4%*$F) \"\"'!\"\"*$F)\"\"&\"#;*&F)F/%\"uGF'F'F(!#;F0F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "The rest of the treatment is as before." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 17 "Multiple patterns" }}{PARA 0 "" 0 "" {TEXT -1 284 "It is also possible to writ e an automaton which will recognize not only a fixed pattern, but a se t of possible patterns. The procedure gengram3 below takes as input a \+ list of patterns, and produces the minimal grammar recognizing all wor ds on 4 letters, the patterns being ``marked''." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 913 "gengram3:=proc (patterns::list(list(\{identic al(a),identical(b),identical(c),identical(d)\}))) local alpha, states, state, eq, n, trans, letter, nbst, st, indst, i; alpha:=[a,b,c,d]; nb st:=1; states[1]:=[]; for indst while indst<=nbst do state:=states[ind st]; n:=nops(state); for letter in alpha do if member([op(state),lette r],patterns) then trans[letter]:=Prod(Mark,w) elif member([op(state),l etter],map( proc(x,n) if nops(x)>n then [op(1..n+1,x)] fi end, pattern s,nops(state))) then trans[letter]:=cat(w,op(state),letter); nbst:=nbs t+1; states[nbst]:=[op(state),letter] else for i from indst by -1 to 2 do st:=states[i]; if st=[op(n-nops(st)+2..n,state),letter] then trans [letter]:=cat(w,op(st)); break fi od; if i=1 then trans[letter]:=w fi \+ fi od; eq[indst]:=cat(w,op(state))= Union(Epsilon,seq(Prod(letter,tran s[letter]),letter=alpha)) od; \{seq(eq[i],i=1..nbst),Mark=Epsilon,seq( letter=Atom,letter=alpha)\} end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% )gengram3G:6#'%)patternsG-%%listG6#-F*6#<&-%*identicalG6#%\"aG-F06#%\" bG-F06#%\"cG-F06#%\"dG6-%&alphaG%'statesG%&stateG%#eqG%\"nG%&transG%'l etterG%%nbstG%#stG%&indstG%\"iG6\"FHC'>8$7&F2F5F8F;>8+\"\"\">&8%6#FO7 \"?(8-FOFOFH1FVFNC&>8&&FR6#FV>8(-%%nopsG6#FZ?&8*FK%%trueG@'-%'memberG6 $7$-%#opGF[oF]o9$>&8)6#F]o-%%ProdG6$%%MarkG%\"wG-Fao6$Fco-%$mapG6%:6$% \"xGFAFHFHFH@$29%-Fjn6#Ffo7#-Feo6$;FO,&FjpFOFOFOFfoFHFHFfoFinC%>Fho-%$ catG6%F_pFdoF]o>FN,&FNFOFOFO>&FR6#FNFcoC$?(8.FV!\"\"\"\"#F^oC$>8,&FR6# F^r@$/Fcr7$-Feo6$;,(FhnFO-Fjn6#FcrF_rF`rFOFhnFZF]oC$>Fho-Feq6$F_p-FeoF ^s%&breakG@$/F^rFO>FhoF_p>&8'Ffn/-Feq6$F_pFdo-%&UnionG6$%(EpsilonG-%$s eqG6$-F\\p6$F]oFho/F]oFK<%/F^pFat-Fct6$&FjsFer/F^r;FOFN-Fct6$/F]o%%Ato mGFgtFHFH" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "For instance, here i s the grammar recognizing the words abab and abacab:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "G:=gengram3([[a,b,a,b],[a,b,a,c,a,b]]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"GG<-/%\"wG-%&UnionG6'%(EpsilonG- %%ProdG6$%\"aG%#waG-F-6$%\"bGF'-F-6$%\"cGF'-F-6$%\"dGF'/F0-F)6'F+F,-F- 6$F3%$wabGF4F7/F?-F)6'F+-F-6$F/%%wabaGF1F4F7/%%MarkGF+/F/%%AtomG/F3FI/ F6FI/F9FI/%&wabacG-F)6'F+-F-6$F/%'wabacaGF1F4F7/FS-F)6'F+F,-F-6$F3-F-6 $FGF'F4F7/FE-F)6'F+F,FW-F-6$F6FNF7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Again, generating functions follow:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 48 "subs(gfsolve(G,unlabelled,z,[[u,Mark]]),w(z,u));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,(\"\"\"F&*$%\"zG\"\"#F&*$F(\"\"% F&F&,4!\"\"F&F(F+F'F-*$F(\"\"$F+F*!\"#*&F(F+%\"uGF&F&*$F(\"\"&F+*$F(\" \"'F-*&F(F6F2F&F&F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "The coef ficient of " }{XPPEDIT 18 0 "u^k*z^n" "*&)%\"uG%\"kG\"\"\")%\"zG%\"nGF &" }{TEXT -1 84 " in the Taylor expansion of this rational function is the number of words of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 75 " occurrences of the patterns abab and abacab. Here are the first few terms:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "map(expand,series(\",z,10)); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+9%\"zG\"\"\"\"\"!\"\"%\"\"\"\"#; \"\"#\"#k\"\"$,&\"$b#F%%\"uGF%\"\"%,&\"%;5F%F/\"\")\"\"&,&\"%[SF%F/\"# [\"\"',&\"&Gh\"F%F/\"$c#\"\"(,(\"&eU'F%F/\"%x7*$F/\"\"#F%\"\"),(\"'?gD F%F/\"%7hF@\"#7\"\"*-%\"OG6#F%\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "For instance the term " }{XPPEDIT 18 0 "48*u*z^6" "*(\"#[\"\"\" %\"uGF$%\"zG\"\"'" }{TEXT -1 364 " corresponds to the 47 words of leng th 6 containing abab (ababab is counted only once), plus the word abac ab itself. Of course, it would not be difficult to modify the grammar \+ so as to take into accounts overlapping words differently. From this g enerating function, the treatment proceeds as before. Again, non-unifo rm and Markovian extensions could be considered." }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 20 "Patterns with errors" }}{PARA 0 "" 0 "" {TEXT -1 612 "It is also possible to accomodate patterns whose occurrence is ex act except at one unspecified position. A direct way would be to apply the previous technique for multiple patterns after having generated a ll possible patterns obtained by introducing one error in the pattern \+ under study. However, the number of patterns obtained this way may be \+ much too large for this technique to be practical. A better way is to \+ produce directly the automaton corresponding to occurrences of the pat tern with at most one error, and this turns out not to be too difficul t. The following procedure generates all words of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 148 ", exact occurrences of the pattern \+ being tagged with a mark as before (Mark0err), while occurrences with \+ one mismatch are tagged by a Mark1err mark." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1479 "gengram4:=proc (pattern::list(\{identical(a),iden tical(b),identical(c),identical(d)\})) local alpha, state, eq, n, lett er, st, i, eq2, staterr, j; alpha:=[a,b,c,d]; for i to nops(pattern) d o staterr[i]:=\{\}; for letter in alpha do for j from 0 to i do if j=i or [op(1 .. i - j, pattern)] = [op(j + 1 .. i - 1, pattern), letter] \+ then if j=0 then state[letter] := cat(w, op(1 .. i , pattern)) else st aterr[i]:=staterr[i] union \{[[op(1..i,pattern)], [op(1..i-j,pattern)] ]\}; state[letter]:=cat(w,op(1..i,pattern),`|`, op(1..i-j,pattern)) fi ; break fi od; od; eq[i] := cat(w, op(1 .. i - 1, pattern)) = Union(Ep silon, seq(Prod(letter, state[letter]), letter = alpha)) od; for i to \+ nops(pattern)-1 do for st in staterr[i] do n:=nops(st[2]); for letter \+ in alpha do for j from 0 to n+1 do if j=n+1 or [op(1..n-j+1,pattern)]= [op(j+1..n,st[2]),letter] then if patt ern[i+1]=letter then state[lett er]:=cat(w,op(1..i+1,pattern),`|`, op(1..n-j+1,pattern)); staterr[i+1] :=staterr[i+1] union \{[[op(1..i+1,pattern)],[op(1..n-j+1,pattern)]]\} else state[letter]:=cat(w,op(1..n-j+1,pattern)) fi; break fi od od; e q2[st] := cat(w,op(st[1]),`|`,op(st[2])) = Union(Epsilon, seq(Prod(let ter, state[letter]), letter = alpha)) od od; \{seq(eq[i],i=1..nops(pat tern)), seq(seq(eq2[st],st=staterr[i]),i=1..nops(pattern)-1), cat(w,op (pattern))=Prod(Mark0err,w), seq(cat(w,op(st[1]),`|`,op(st[2]))=Prod(M ark1err,w), st=staterr[nops(pattern)]), Mark0err=Epsilon,Mark1err=Epsi lon,seq(letter=Atom,letter=alpha)\} end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)gengram4G:6#'%(patternG-%%listG6#<&-%*identicalG6#%\"aG-F.6#% \"bG-F.6#%\"cG-F.6#%\"dG6,%&alphaG%&stateG%#eqG%\"nG%'letterG%#stG%\"i G%$eq2G%(staterrG%\"jG6\"FEC&>8$7&F0F3F6F9?(8*\"\"\"FL-%%nopsG6#9$%%tr ueGC%>&8,6#FK<\"?&8(FHFQ?(8-\"\"!FLFKFQ@$5/FenFK/7#-%#opG6$;FL,&FKFLFe n!\"\"FP7$-F]o6$;,&FenFLFLFL,&FKFLFaoFLFPFYC$@%/FenFfn>&8%6#FY-%$ catG6 $%\"wG-F]o6$;FLFKFPC$>FT-%&unionG6$FT<#7$7#FcpF[o>F\\p-F`p6&FbpFcp%\"| grGF\\o%&breakG>&8&FV/-F`p6$Fbp-F]o6$;FLFgoFP-%&UnionG6$%(EpsilonG-%$s eqG6$-%%ProdG6$FYF\\p/FYFH?(FKFLFL,&FMFLFaoFLFQ?&8)FTFQC%>8'-FN6#&Fjr6 #\"\"#?&FYFHFQ?(FenFfnFL,&F]sFLFLFLFQ@$5/FenFes/7#-F]o6$;FL,(F]sFLFenF aoFLFLFP7$-F]o6$;FfoF]sF`sFYC$@%/&FP6#,&FKFLFLFLFYC$>F\\p-F`p6&Fbp-F]o 6$;FLFhtFPFaqF[t>&FUFgt-Fip6$Fau<#7$7#F]uFjs>F\\p-F`p6$FbpF[tFbq>&8+6# Fjr/-F`p6&Fbp-F]o6#&Fjr6#FLFaq-F]oF_sF\\r<)/%)Mark0errGF_r/%)Mark1errG F_r-Far6$/FY%%AtomGFfr-Far6$/F_v-Fdr6$FjvFbp/Fjr&FU6#FM/-F`p6$Fbp-F]oF O-Fdr6$FhvFbp-Far6$-Far6$F[v/FjrFT/FK;FLFhr-Far6$Fdq/FK;FLFMFEFE" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "Here is a simple example. The gen erated automaton is almost optimal (in general it has O(1) too many st ates):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "gengram4([a,b,b,c ]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<5/%\"aG%%AtomG/%\"bGF&/%\"cGF& /%\"dGF&/%#waG-%&UnionG6'%(EpsilonG-%%ProdG6$F%%&wab|graG-F46$F(%$wabG -F46$F*%%wab|grG-F46$ F,F " 0 "" {MPLTEXT 1 0 67 "subs(gfsolve(\" ,unlabelled,z,[[u,Ma rk0err],[v,Mark1err]]),w(z,u,v));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#, $*&,4!\"\"\"\"\"%\"zG!\"%*$F(\"\"#!\"'*$F(\"\"$F,*$F(\"\"%F)*$F(\"\"&! \"$*$F(\"\")F.*$F(\"\"'!\"#*$F(\"\"*F.F',L*&F(F2%\"uGF'F)*&F(\"\"(F=F' !\"&F'F'F1\"#@F-!#=*&F(F2%\"vGF'!#I*&F(F7F=F'F,*&F(F7FDF'!#@F4F3F9F3F6 \"#<*$F(F?F.*&F(F?FDF'F,*&F(F:FDF'F:*&F(\"#5FDF'F:*&F(FNF=F'F.*&F(F:F= F'F.*&F(F0F=F'F&F*!#5*&F(F0FDF'!#7F/!\"(F&F&" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 19 "The coefficient of " }{XPPEDIT 18 0 "u^k*v^l*z^n" "*()% \"uG%\"kG\"\"\")%\"vG%\"lGF&)%\"zG%\"nGF&" }{TEXT -1 54 " in the Taylo r expansion of this rational function at " }{XPPEDIT 18 0 "z=0" "/%\"z G\"\"!" }{TEXT -1 34 " is the number or words of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 31 " where the pattern abbc occurs " } {XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 25 " times without error and " }{XPPEDIT 18 0 "l" "I\"lG6\"" }{TEXT -1 149 " times with exactly one e rror. Again, the Markovian model could also be treated, and extensions to multi ple errors are likely to be possible as well." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Conclusion" }}{PARA 0 "" 0 "" {TEXT -1 575 "Various proba bilistic parameters related to the occurrence of specific patterns in \+ random words can be computed very easily using combstruct and gfun. He re, combstruct is used to model the combinatorics of the problem and g fun is very helpful to compute expansions to very large orders, thanks to the rational type of the corresponding generating functions. The m odel itself can be modified in various directions, to take into accoun t different probability models or different ways of counting occurrenc es of the pattern when they overlap, or several patterns simultaneousl y." }}}}{MARK "5 0 0" 46 }{VIEWOPTS 0 0 0 1 1 1803 } \"kG6\"" }{TEXT -1 25 " times without error and " }{XPPEDIT 18 0 "l" "I\"lG6\"" }{TEXT -1 149 " times with exactly one e rror. 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1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 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0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 273 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 274 1 {CSTYLE "" -1 -1 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 48 "Borel Resummation of Dive rgent Series Using Gfun" }}{PARA 19 "" 0 "" {TEXT 256 51 "Fr\351d\351r ic Chyzak, Marianne Durand, and Bruno Salvy\n\n" }{TEXT -1 10 "June, 2 001" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 262 8 "Abstract" }{TEXT -1 7 ": We ex" }{TEXT 265 0 "" }{TEXT -1 343 "pand on ideas of Balser, Lutz, and Sch\344fke by showing how coefficients and integrals involv ed in calculations related to the analytic continuation of Borel trans forms obey simple recurrences that lead to efficient numerical computa tions. This work is a follow-up to a talk by Donald Lutz at our Algor ithms seminar, and summarized in [Durand]." }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 33 "1. Borel-Laplace resummation and " }{TEXT 261 0 "" } {TEXT -1 9 "Euler acc" }{TEXT 260 0 "" }{TEXT -1 9 "eleration" }} {PARA 0 "" 0 "" {TEXT -1 108 "Starting with a linear differential equa tion with polynomial coefficients satisfied by a formal power series" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "x^~ = Sum(x[n]*z^n,n = 0 .. infinit y);" "6#/)%\"xG%\"|irG-%$SumG6$*&&F%6#%\"nG\"\"\")%\"zGF-F./F-;\"\"!%) infinityG" }{TEXT 257 0 "" }{TEXT 258 0 "" }{TEXT 259 0 "" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 87 "it is possible to compute a diffe rential equation satisfied by the Borel transform of " }{XPPEDIT 18 0 "x^~;" "6#)%\"xG%\"|irG" }{TEXT -1 17 ". We assume that " }{XPPEDIT 18 0 "x^~;" "6#)%\"xG%\"|irG" }{TEXT -1 31 " is Gevrey 1, which means \+ that " }{XPPEDIT 18 0 "x[n] <= A*c^n*n!;" "6#1&%\"xG6#%\"nG*(%\"AG\"\" \")%\"cGF'F*-%*factorialG6#F'F*" }{TEXT -1 21 " for some constants " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "c;" "6#%\"cG" }{TEXT -1 34 ", so that the Borel transform of " }{XPPEDIT 18 0 "x^~;" "6#)%\"xG%\"|irG" }{TEXT -1 11 " defined by" }}{PARA 257 " " 0 "" {XPPEDIT 18 0 "y(z) = Sum(x[n]*z^n/n!,n = 0 .. infinity);" "6#/ -%\"yG6#%\"zG-%$SumG6$*(&%\"xG6#%\"nG\"\"\")F'F/F0-%*factorialG6#F/!\" \"/F/;\"\"!%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 121 "is convergent on some neighbourhood of the origin. The Borel tran sform has an \"inverse\", the Laplace transform defined by" }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Laplace(y) = Int(exp(-t/z)* y(t),t = 0 .. infinity);" "6#/-%(LaplaceG6#%\"yG-%$IntG6$*&-%$expG6#,$ *&%\"tG\"\"\"%\"zG!\"\"F4F2-F'6#F1F2/F1;\"\"!%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 62 "Provided this integral converges, t he function it defines has " }{XPPEDIT 18 0 "z*x^~;" "6#*&%\"zG\"\"\") %\"xG%\"|irGF%" }{TEXT -1 18 " for expansion as " }{XPPEDIT 18 0 "proc (z) options operator, arrow; 0 end;" "6#R6#%\"zG7\"6$%)operatorG%&arr owG6\"\"\"!F*F*F*" }{TEXT -1 42 " +. Then applying the change of vari able " }{XPPEDIT 18 0 "t = phi(alpha);" "6#/%\"tG-%$phiG6#%&alphaG" } {TEXT -1 69 " to the integral computes the acceleration \"a la Euler\" [Lutz et al.]" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z*x ^~ = Int(exp(-t/z)*y(phi(psi(t))),t = 0 .. infinity);" "6#/*&%\"zG\"\" \")%\"xG%\"|irGF&-%$IntG6$*&-%$expG6#,$*&%\"tGF&F%!\"\"F4F&-%\"yG6#-%$ phiG6#-%$psiG6#F3F&/F3;\"\"!%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 7 "We note" }}{PARA 260 "" 0 "" {XPPEDIT 18 0 "y(phi(z)) = Sum(d[n]*z^n,n = 0 .. infinity);" "6#/-%\"yG6#-%$phiG6#%\"zG-%$SumG 6$*&&%\"dG6#%\"nG\"\"\")F*F2F3/F2;\"\"!%)infinityG" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "q[n](z) = Int(exp(-t/z)*psi(t)^n,t = 0 .. infinity); " "6#/-&%\"qG6#%\"nG6#%\"zG-%$IntG6$*&-%$expG6#,$*&%\"tG\"\"\"F*!\"\"F 6F5)-%$psiG6#F4F(F5/F4;\"\"!%)infinityG" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "psi;" "6#%$psiG" }{TEXT -1 53 " is the functional inverse of the rational function " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 47 ". In terms of formal power series , the product " }{XPPEDIT 18 0 "z*x^~;" "6#*&%\"zG\"\"\")%\"xG%\"|irGF %" }{TEXT -1 31 " equals the Taylor expansion of" }}{PARA 261 "" 0 "" {XPPEDIT 18 0 "Sum(q[n]*d[n],n = 0 .. infinity);" "6#-%$SumG6$*&&%\"qG 6#%\"nG\"\"\"&%\"dG6#F*F+/F*;\"\"!%)infinityG" }{TEXT -1 2 " ," }} {PARA 0 "" 0 "" {TEXT -1 20 "where the integrals " }{XPPEDIT 18 0 "q[n ];" "6#&%\"qG6#%\"nG" }{TEXT -1 21 " are independent of " }{XPPEDIT 18 0 "x^~;" "6#)%\"xG%\"|irG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "This process is illustrated in \+ the present worksheet using a simple mapping function " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 153 " on the double confluent Heun equat ion. The Heun equation is the generic differential equation with four \+ regular singular points located at 0, 1, c, and " }{XPPEDIT 18 0 "infi nity;" "6#%)infinityG" }{TEXT -1 134 ", see [DuLoRi92]. The double con fluent Heun equation is obtained by letting the singularity located a t c tend to the one located at " }{XPPEDIT 18 0 "infinity;" "6#%)infi nityG" }{TEXT -1 127 ", and the singularity located at 1 tend to 0. T he equation obtained then has two irregular \nsingular points located at 0 and " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 77 ". The example we study is the double confluent Heun equation in the f orm " }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "heun_infp := z^2*di ff(f(z),z,z)+(z+alpha*z^2+alpha)*diff(f(z),z)+(2*alpha*z^2*beta[1]+alp ha*z^2+alpha^2*z-2*gamma*z+2*alpha*beta[-1]-alpha)*f(z)/(2*z):" "6#>%* heun_infpG,(*&%\"zG\"\"#-%%diffG6%-%\"fG6#F'F'F'\"\"\"F/*&,(F'F/*&%&al phaGF/*$F'\"\"#F/F/F3F/F/-F*6$-F-6#F'F'F/F/*(,.**\"\"#F/F3F/F'\"\"#&%% betaG6#\"\"\"F/F/*&F3F/*$F'\"\"#F/F/*&F3\"\"#F'F/F/*(\"\"#F/%&gammaGF/ F'F/!\"\"*(\"\"#F/F3F/&F@6#,$\"\"\"FKF/F/F3FKF/-F-6#F'F/*&\"\"#F/F'F/F KF/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "readlib(gfun):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Since the point of interest is inf inity and the " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 66 " packag e works at the origin, we first change the variable (using " } {HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 2 "):" }{TEXT 267 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "heunp:=gfun[algebraicsubs](h eun_infp,z*f-1,f(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&heunpG,(*& ,.*()%\"zG\"\"#\"\"\"%&alphaG\"\"\"&%%betaG6#!\"\"F.!\"#*&F-F,F)F,F.*& )F-F+F,F*F.F2*&%&gammaGF.F*F,F+*&F-F,&F06#F.F.F3F-F2F.-%\"fG6#F*F.F.*& ,(*$F)F,F3*&)F*\"\"$F,F-F,F+*&F*F,F-F,F+F.-%%diffG6$F " 0 "" {XPPEDIT 19 1 "paramform:=[alpha=-1,beta[-1]= 1/2,beta[1]=1/2,gamma=1/3]:" "6#>%*paramformG7&/%&alphaG,$\"\"\"!\"\"/ &%%betaG6#,$\"\"\"F**&\"\"\"\"\"\"\"\"#F*/&F-6#\"\"\"*&\"\"\"F3\"\"#F* /%&gammaG*&\"\"\"F3\"\"$F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Sub stitution of these parameters in the differential equation gives" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1  0 28 "heun:=subs(paramform,heunp); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%heunG,(*&,&%\"zG#!\"\"\"\"$\"\" #\"\"\"F--%\"fG6#F(F-F-*&,(*$)F(F,\"\"\"!\"#*$)F(F+F5F6F(F6F--%%diffG6 $F.F(F-F-*&F8F5-F:6$F.-%\"$G6$F(F,F-F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "From this equation, we obtain a recurrence equation for t he Taylor series coefficients" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "recheunseries:=gfun[diffeqtorec](heun,f(z),u(n));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%.recheunseriesG<$,(*&%\"nG\"\"\"-%\"uG6#F(F)! \"'*&,(!\"(F)F(!#7*$)F(\"\"#\"\"\"F-F)-F+6#,&F(F)F)F)F)F)*&,&F-F)F(F-F )-F+6#,&F(F)F4F)F)F)/-F+6#\"\"!FA" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "This recurrence yields an efficient procedure to evaluate the coef fici" }{TEXT 266 0 "" }{TEXT -1 17 "ents recursively:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "heundiv:=gfun[rectoproc](recheunser ies union \{u(1)=1/2\},u(n),remember);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(heundivGR6#%\"nG6\"6#%)rememberGE\\s#\"\"\"#F,\"\"#\"\"!F/,$* &,(!-9!6#,&9$F,!\"#F,\"#7-F46#,&F7F,!\"\"F,!\"(*&,(F3!\"'F:F9*&F:F,F7F, FAF,F7\"\"\"F,FC,&\"\"'F,F7FA!\"\"F=F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "From this procedure, the divergence is clear from the gro wth of the first coefficients:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "seq(heundiv(i),i=1..15);" }}{PARA 12 "" 1 "" {XPPMATH 20 "61# \"\"\"\"\"##!\"(\"#7#\"$R\"\"$W\"#!%Pm\"%#f##\"'`()f\"&3A'#!)\"z)e')\" (Si'=#\",Z**)3D=\")SY=n#!.H7Y$zr_\"+!)[v@G#\"0^45;%\\`G\",?j<\\$>#!3V. Z`t%*41'*\".g*['))RJ(#\"6JV1$)z2E?]r&\"0+w$*=$R)Q%#!9PQ`%>@#[rxB?T\"2+ ;))\\]Rj*G#\" " 0 "" {XPPEDIT 19 1 "calculheundiv:=proc(heundiv,z)\nlocal total,previous ,last,n;\nprevious:=heundiv(1)*z;total:=previous;last:=heundiv(2)*z^2; \nfor n from 3 while abs(previous)>abs(last) do\ntotal:=total+last;\np revious:=last;\nlast:= heundiv(n)*z^n od;\nuserinfo(1,'heundiv',n,last );\nevalf(total)\nend:" "6#>%.calculheundivGR6$%(heundivG%\"zG7&%&tota lG%)previousG%%lastG%\"nG6\"F.C(>F+*&-F'6#\"\"\"\"\"\"F(F5>F*F+>F,*&-F '6#\"\"#F5*$F(\"\"#F5?(F-\"\"$F5F.2-%$absG6#F,-FB6#F+C%>F*,&F*F5F,F5>F +F,>F,*&-F'6#F-F5)F(F-F5-%)userinfoG6&\"\"\".F'F-F,-%&evalfG6#F*F.F.F. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot('calculheundiv'(he undiv,z),z=0..0.3);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7]p7$ \"\"!F(7$$\"+]i9Rl!#7$\"+w[*[C$F,7$$\"+WA)GA\"!#6$\"+Pw))Gg#F,7$$\"+Qeu i=F2$\"+FMF<\"*F,7$$\"+j3&o]#F2$\"+(3&>=7F27$$\"+pX*y9$F2$\"+#*)G*=:F2 7$$\"+WTAUPF2$\"+[t.%z\"F27$$\"+%*zhdVF2$\"+f0Bv?F27$$\"+%>fS*\\F2$\"+ 1d@iBF27$$\"+>$f%GcF2$\"+\"[0Xk#F27$$\"+Dy,\"G'F2$\"+:&R5$HF27$$\"+8LZF27$$\"+/QBE6Fcp$\"+X=k,]F27$$\"+:o?&=\" Fcp$\"+8G!\\B&F27$$\"+a&4*\\7Fcp$\"+sWo)[&F27$$\"+j=_68Fcp$\"+jwt6dF27 $$\"+Wy!eP\"Fcp$\"+[h&=&fF27$$\"+UC%[V\"Fcp$\"+BCPohF27$$\"+J#>&)\\\"F cp$\"+6InWkF27$$\"+>:mk:Fcp$\"+Fn/'p'F27$$\"+w&QAi\"Fcp$\"+t-r9pF27$$ \"+v4L`;Fcp$\"+JT!H.(F27$$\"+uLU%o\"Fcp$\"+&f#G^rF27$$\"+ZPX#p\"Fcp$\" +1w*==(F27$$\"+?T[+?sF27$$\"+2$* \\/luF27$$\"+M aKs=Fcp$\"+Z0Fcp$\"+h[LjyF27$$\"+:K^+?Fcp$\"+(eMI\"y?Fcp$\"+QgCL')F27$$\"+)[k*z?Fcp$\"+* GD-k)F27$$\"+v;I(3#Fcp$\"+as:o')F27$$\"+i)QY4#Fcp$\"+bO6'p)F27$$\"+OKJ 4@Fcp$\"+*o-@v)F27$$\"+4w)R7#Fcp$\"+0#)>3))F27$$\"+WN2c@Fcp$\"+lBDJ*)F 27$$\"+y%f\")=#Fcp$\"+rp!\\0*F27$$\"+/-a[AFcp$\"+:OY*G*F27$$\"+ial6BFc p$\"+vdlP&*F27$$\"+j@OtBFcp$\"+(*[!Qy*F27$$\"+fL'zV#Fcp$\"+t)zX+\"Fcp7 $$\"+!*>=+DFcp$\"+b#*GI5Fcp7$$\"+E&4Qc#Fcp$\"+g(Hr0\"Fcp7$$\"+g)f`f#Fc p$\"+&3m12\"Fcp7$$\"+%>5pi#Fcp$\"+!*[O%3\"Fcp7$$\"+NfSTEFcp$\"+fdr!4\" Fcp7$$\"+v;!fl#Fcp$\"+cP5(4\"Fcp7$$\"+WOrdEFcp$\"+7\\!z4\"Fcp7$$\"+6c_ fEFcp$\"+lmq)4\"Fcp7$$\"+zvLhEFcp$\"+nb45(*F27$$\"+Y&\\Jm#Fcp$\"+jYu8( *F27$$\"+\"[tnm#Fcp$\"+5^-@(*F27$$\"+;uRqEFcp$\"+4=GG(*F27$$\"+'GXwn#F cp$\"++QsU(*F27$$\"+bJ*[o#Fcp$\"+a-2d(*F27$$\"+k17=FFcp$\"+Jig@)*F27$$ \"+r\"[8v#Fcp$\"+Yd3%))*F27$$\"+Ijy5GFcp$\%"+ZKf!***F27$$\"+/)fT(GFcp$ \"+(Q\\'45Fcp7$$\"+1j\"[$HFcp$\"+j#=!>5Fcp7$$\"\"$!\"\"$\"+](=#G5Fcp-% 'COLOURG6&%$RGBG$\"#5FddlF(F(-%+AXESLABELSG6$Q\"z6\"%!G-%%VIEWG6$;F(Fb dl%(DEFAULTG" 2 376 376 376 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 -22808 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "As " }{XPPEDIT 18 0 "z;" " 6#%\"zG" }{TEXT -1 68 " tends to infinity, the imprecision of this sum mation grows quickly." }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 57 "2. Diffe rential equation satisfied by the Borel transform" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 116 "The class of solutions of linear differential equ ations enjoys many closure properties which are implemented in the " } {HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 225 " package for the case o f equations with polynomial coefficients. One of them is closure under Borel transform. Here is the differential equation satisfied by the B orel transform of the divergent solution of the Heun equation:" }}} {EXCHG &{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "bdeqp:=op(select(has,gfun[bo rel](heunp,f(z),'diffeq'),z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&b deqpG,**&,&*&%&alphaG\"\"\"&%%betaG6#!\"\"F*\"\"#F)F.F*-%\"fG6#%\"zGF* F**&,**&F3F*F)\"\"\"!\"#*$)F)F/F7F*F/F*%&gammaGF8F*-%%diffG6$F0F3F*F** &,(F3\"\"'*&F)F7&F,6#F*F*F/F)!\"$F*-F=6$F0-%\"$G6$F3F/F*F**&,&*$)F3F/F 7F/F6F8F*-F=6$F0-FI6$F3\"\"$F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "and the equation specialized at the parameters" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "bdeq:=subs(paramform,bdeqp):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "From this follows the recurrence satisfie d by the Taylor coefficients of the Borel transform:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "collect(gfun[diffeqtorec](bdeqp,f(z),a(n) ),a,factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(%&alphaG\"\"\",(&% %betaG6#!\"\"!\"#F&F&%\"nG\"\"#F&-%\"aG6#F-F&F+*(,&F-F&F&F&F&,,*$)F-F. \"\"\"F.F-\"\"%*$)F%F.F7F&F.F&%&gammaGF,F&-F06#F3F&F&*,F%F7F3F7,&F-F&F .F&F&,(F-F.&F)6#F&F,\"\"$F&F&-F06#F?F'&F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "l15:= gfun[rectoproc](\{subs(paramform,%),a(0)=0,a(1)=1/2\},a(n),list)(15); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$l15G72\"\"!#\"\"\"\"\"##!\"(\"# C#\"$R\"\"$k)#!%Pm\"&3A'#\"'`()f\"(g\\Y(#!)\"z)e')\"++GpV8#\",Z**)3D= \"-+ce5'Q$#!.H7Y$zr_\"0+;wcJx8\"#\"0^45;%\\`G\"1+;+.\"H9-(#!3V.Z`t%*41 '*\"5+![gM4?+Tl##\"6JV1$)z2E?]r&\"8++o\"RohKhq^<#!9PQ`%>@#[rxB?T\";++c 5iOD.d7N(Q\"#\"++;%3tFaCew2c)H\"#!?Fb!Htc\\\\$*HrRQ3 d$\"A++s-z?32S!G!o$G!=9#\"B$3lJrLh,5u(H=%[M\"=%\"D++?Fu&>M#*[I`rVdr'y \"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "This list corresponds to th e list above, the " }{XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT -1 27 "th ele ment being scaled by " }{XPPEDIT 18 0 "1/k!;" "6#*&\"\"\"\"\"\"-%*fact orialG6#%\"kG!\"\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 147 "This differential equation and this recurrence can be used to \+ compute (but not necessarily fast) the analytic continuation of the (Bo rel transform. " }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 93 "3. The coeffic ients of the composition with an algebraic function satisfy a linear r ecurrence" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "This is another clo sure property of solutions of linear differential equation that we exp loit here. The coefficients " }{XPPEDIT 18 0 "d[n];" "6#&%\"dG6#%\"nG " }{TEXT -1 15 " are defined by" }}{PARA 262 "" 0 "" {XPPEDIT 18 0 "y( phi(z)) = Sum(d[n]*z^n,n = 0 .. infinity);" "6#/-%\"yG6#-%$phiG6#%\"zG -%$SumG6$*&&%\"dG6#%\"nG\"\"\")F*F2F3/F2;\"\"!%)infinityG" }}{PARA 0 " " 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 28 " is a rational function and " }{XPPEDIT 18 0 "y;" "6#%\"yG" } {TEXT -1 15 " is defined by" }}{PARA 263 "" 0 "" {XPPEDIT 18 0 "y(z) \+ = Sum(x[n]*z^n/n!,n = 0 .. infinity);" "6#/-%\"yG6#%\"zG-%$SumG6$*(&% \"xG6#%\"nG\"\"\")F'F/F0-%*factorialG6#F/!\"\"/F/;\"\"!%)infinityG" } {TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 42 "On the example of the H eun equation, with " }{XPPEDI)T 18 0 "phi = 1/((1-t)^2)-1;" "6#/%$phiG, &*&\"\"\"\"\"\"*$,&\"\"\"F(%\"tG!\"\"\"\"#F-F(\"\"\"F-" }{TEXT -1 10 " we obtain" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "eq:=(1-z)^2*(f+ 1)-1:" "6#>%#eqG,&*&,&\"\"\"\"\"\"%\"zG!\"\"\"\"#,&%\"fGF)\"\"\"F)F)F) \"\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "dneqp:=gfun[alg ebraicsubs](bdeqp,eq,f(z));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&dneq pG,**&,&%&alphaG\"\"%*&F(\"\"\"&%%betaG6#!\"\"F+!\")F+-%\"fG6#%\"zGF+F +*&,hn!\"%F+%&gammaGF)*&F(\"\"\"&F-6#F+F+\"\"'F4\"#[*&)F(\"\"#F:)F4FAF :!\"'*&F(F:FBF:!$n#F(!\"**&F@F:F4F+F=*&F8F+F4F:!#7*$FBF:!$)>*(F(F:FBF: F;F:\"#g*&F8F:FBF:\"#7*&)F4\"\"$F:F(F:\"$p%*&F4F:F(F:\"#$)*$FQF:\"$1%* (F(F:F;F:F4F:!#I*&)F4F)F:F(F:!$&\\*&)F4\"\"&F:F(F:\"$4$*&)F4F=F:F(F:!$ 0\"*&F@F:FQF:FA*&)F4\"\"(F:F(F:\"#:*$FenF:!$l%*$FhnF:\"$.$*$F\\oF:F]o* $F`oF:Fbo*(F(F:FQF:F;F:!#g*&F8F:FQF:F7*(F(F:FhnF:F;F:FC*(FenF:F(F:F;F: \"#I*$F@F:!\"#F+-%%diffG6$F1F4F+F+*&,RF9F`pF4FIFD\"$i\"F(FRFJ\"#!*FLFY FP!$%QFT!#OFV!$w#FXFOFZ\"$S&Fgn!$o%F[o\"$Y#*&)F4\"\")F:*F(F:\"\"**$F_qF :FaqF_o!#sFco\"$`%Feo!$K%Fgo\"$S#FhoFcqFio\"#SF\\pFOF]pFY*(F(F:F\\oF:F ;F:F`pF+-Fbp6$F1-%\"$G6$F4FAF+F+*&,DFD!#:FJF7FP\"#\\FTFAFV\"#CFZ!#\"*F gn\"$0\"F[o!#x*$)F4FaqF:F+F^qFFFbqFFF_o\"#NFco!#hFeo\"#&)Fgo!#qFho\"#M *&FgrF:F(F:F+F+-Fbp6$F1-F\\r6$F4FRF+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "and the equation when the parameters are specialized:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "dneq:=gfun[algebraicsubs](b deq,eq,f(z));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%dneqG,(*&,,*$)%\"z G\"\"%\"\"\"\"\"**$)F*\"\"$F,!#O*$)F*\"\"#F,\"#hF*!#]F+\"\"\"F7-%%diff G6$-%\"fG6#F*F*F7F7*&,.F.\"$T\"F(!#v*$)F*\"\"&F,\"#:F2!$B\"F*\"#[!\"'F 7F7-F96$F;-%\"$G6$F*F4F7F7*&,.F(\"#UF.!#[FB!#=*$)F*\"\"'F,F0F2\"#FF*FH F7-F96$F;-FL6$F*F0F7F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "The lin ear recurrence satisfied by the " }{XPPEDIT 18 0 "d[n];" "6#&%\"dG6#% \"nG" }{TEXT -1 18 " follows directly" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "dnRec:=op(select(has,gfun[diffeqtorec](dneq,f(z),a(n) ),n));" }}{PARA 1+2 "" 1 "" {XPPMATH 20 "6#>%&dnRecG,.*&,&*$)%\"nG\"\"# \"\"\"\"\"'*$)F*\"\"$F,F0\"\"\"-%\"aG6#F*F1F1*&,*F*!#$*!#OF1F(!#vF.!#= F1-F36#,&F*F1F1F1F1F1*&,*F*\"$o&\"$/%F1F(\"$n#F.\"#UF1-F36#,&F*F1F+F1F 1F1*&,*F*!%$>\"!%w6F1F(!$6%F.!#[F1-F36#,&F*F1F0F1F1F1*&,*F*\"%U5\"%S7F 1F(\"$\"HF.\"#FF1-F36#,&F*F1\"\"%F1F1F1*&,*F(!#yF*!$O$!$![F1F.!\"'F1-F 36#,&F*F1\"\"&F1F1F1" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 17 "4. The in tegrals " }{XPPEDIT 18 0 "q[n];" "6#&%\"qG6#%\"nG" }{TEXT -1 28 " sati sfy a linear recurrence" }}{PARA 0 "" 0 "" {TEXT -1 68 "The property a bove does not depend on the specific divergent series " }{XPPEDIT 18 0 "x^~;" "6#)%\"xG%\"|irG" }{TEXT -1 80 " that one is resumming. This \+ allows one to precompute efficiently the integrals " }{XPPEDIT 18 0 "q [n](z);" "6#-&%\"qG6#%\"nG6#%\"zG" }{TEXT -1 28 " given the mapping fu nction " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 173 ". Indeed the \+ general theory of holonomic function has recently led to symbolic summ ation and integration algorithms that tu,rn out to apply to the integra l representation of " }{XPPEDIT 18 0 "q[n](z);" "6#-&%\"qG6#%\"nG6#%\" zG" }{TEXT -1 224 ". The goal of these algorithms is to derive (system s of) linear functional equations, differential or difference, satisfi ed by a sum or an integral. We now proceed to use a prototypical imple mentation of them in the package " }{HYPERLNK 17 "Mgfun" 2 "Mgfun" "" }{TEXT -1 31 " to obtain a recurrence on the " }{XPPEDIT 18 0 "q[n](z) ;" "6#-&%\"qG6#%\"nG6#%\"zG" }{TEXT -1 138 ". Then, we prove a theorem that by-passes the general theory of holonomic functions, and recover the same recurrence in a more direct way." }}}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 32 "(This section uses a version of " }{HYPERLNK 17 "Mgfun " 2 "Mgfun" "" }{TEXT -1 30 " that is not distributed yet.)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "readlib(Mgfun):" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "phi:=1/(1-t)^2-1:" "6#>%$phiG,&*&\"\"\"\"\"\" *$,&\"\"\"F(%\"tG!\"\"\"\"#F-F(\"\"\"F-" }}}{EXCHG {PARA 0 "> " 0 ""- {XPPEDIT 19 1 "F:=exp(-phi/z)*t^n*diff(phi,t):" "6#>%\"FG*(-%$expG6#,$ *&%$phiG\"\"\"%\"zG!\"\"F.F,)%\"tG%\"nGF,-%%diffG6$F+F0F," }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "ct:=Mgfun[creative_telescoping](F,n:: shift,t::diff);" "6#>%#ctG-&%&MgfunG6#%5creative_telescopingG6%%\"FG'% \"nG%&shiftG'%\"tG%%diffG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ctG7$, **&,&%\"zG\"\"\"*&%\"nGF*F)F*F*F*-%#_fG6$F,%\"tGF*F**&,&F)\"\"$F+F3F*- F.6$,&F,F*\"\"#F*F0F*F**&,&F+!\"\"F)F:F*-F.6$,&F,F*F3F*F0F*F**&,(F+!\" $F)F@!\"#F*F*-F.6$,&F,F*F*F*F0F*F***F)\"\"\"F0F*,*F:F*F0F3*$)F0F7FFF@* $)F0F3FFF*F*F-FF" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "h[n](t) :=%[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%\"hG6#%\"nG6#%\"tG,**&, &%\"zG\"\"\"*&F(F/F.F/F/F/-%#_fG6$F(F*F/F/*&,&F.\"\"$F0F6F/-F26$,&F(F/ \"\"#F/F*F/F/*&,&F0!\"\"F.F=F/-F26$,&F(F/F6F/F*F/F/*&,(F0!\"$F.FC!\"#F /F/-F26$,&F(F/F/F/F*F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "H[n](t):=%%[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%\"HG6#%\"nG6#% \"tG**%\"zG\"\"\"F*F-,*!\."\"F-F*\"\"$*$)F*\"\"#\"\"\"!\"$*$)F*F0F4F-F- -%#_fG6$F(F*F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "The meaning of \+ the previous computation is that the differential equation" }}{PARA 272 "" 0 "" {XPPEDIT 18 0 "h[n](t)+diff(H[n](t),t) = 0;" "6#/,&-&%\"hG 6#%\"nG6#%\"tG\"\"\"-%%diffG6$-&%\"HG6#F)6#F+F+F,\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 91 "holds. This can be viewed as a differential-differen ce relation satisfied by the integrand " }{XPPEDIT 18 0 "f[n](z,t);" " 6#-&%\"fG6#%\"nG6$%\"zG%\"tG" }{TEXT -1 27 ". Now, integrating between " }{XPPEDIT 18 0 "-1;" "6#,$\"\"\"!\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "1;" "6#\"\"\"" }{TEXT -1 64 " returns a non-homogeneous differential equation in the integral" }}{PARA 273 "" 0 "" {XPPEDIT 18 0 "q[n](z) = int(f[n](z,t),t = -1 .. 1);" "6#/-&%\"qG6#%\"nG6#%\"zG -%$intG6$-&%\"fG6#F(6$F*%\"tG/F3;,$\"\"\"!\"\"\"\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "namely" }}{PARA 271 "" 0 "" {XPPEDIT 18 0 "int(h[n](t),t = -1 .. 1)+H[n](1)-H[n](-1) = 0;" "6#/,(-%$/intG6$-&% \"hG6#%\"nG6#%\"tG/F.;,$\"\"\"!\"\"\"\"\"\"\"\"-&%\"HG6#F,6#\"\"\"F5-& F86#F,6#,$\"\"\"F3F3\"\"!" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 40 "where the integral rewrites in terms of " }{XPPEDIT 18 0 "q[n](z); " "6#-&%\"qG6#%\"nG6#%\"zG" }{TEXT -1 3 " as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "Int(h[n](t),t=-1..1)=eval(subs(_f=unapply(q[n](z ),n,t),ct[1]));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,**&,&%\" zG\"\"\"*&%\"nGF+F*F+F+F+-%#_fG6$F-%\"tGF+F+*&,&F*\"\"$F,F4F+-F/6$,&F- F+\"\"#F+F1F+F+*&,&F,!\"\"F*F;F+-F/6$,&F-F+F4F+F1F+F+*&,(F,!\"$F*FA!\" #F+F+-F/6$,&F-F+F+F+F1F+F+/F1;F;F+,**&F)\"\"\"-&%\"qG6#F-6#F*F+F+*&F3F J-&FM6#F7FOF+F+*&F:FJ-&FM6#F>FOF+F+*&F@FJ-&FM6#FEFOF+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "As to the non-homogeneous part " } {XPPEDIT 18 0 "H[n](1)-H[n](-1);" "6#,&-&%\"HG6#%\"nG6#\"\"\"\"\"\"-&F &6#F(6#,$\"\"\"!\"\"F2" }{TEXT -1 59 ", we readily evaluate it, verify ing that it is 0 by chance." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "H[n](t):=factor(eval(su0bs(_f=unapply(F,n,t),H[n](t))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%\"HG6#%\"nG6#%\"tG**%\"zG\"\"\"F*F-),&! \"\"F-F*F-\"\"$\"\"\"%\"fGF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "assume(z>0); assume(n,integer); factor(simplify(limit(op(2,%),t =1)-limit(op(2,%),t=-1))): non_hom:=subs([z='z',n='n'],%); z:='z': n:= 'n':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(non_homG\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Consequently, we have obtained the follow ing recurrence on the integrals " }{XPPEDIT 18 0 "q[n](z);" "6#-&%\"qG 6#%\"nG6#%\"zG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "collect(eval(subs(_f=unapply(q[n](z),n,t),ct[1])),q,factor)=0; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**&,(*&%\"nG\"\"\"%\"zGF)!\"$F*F +!\"#F)F)-&%\"qG6#,&F(F)F)F)6#F*F)F)*(F*\"\"\"F1F)-&F/6#,&F(F)\"\"$F)F 2F)!\"\"*(F*F4F1F4-&F/6#,&F(F)\"\"#F)F2F)F9*(F*F4F1F4-&F/6#F(F2F)F)\" \"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "collect(subs(n=n-4,% ),q,factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**(%\"zG\"1\"\",&%\"n GF'!\"$F'F'-&%\"qG6#,&F)F'!\"\"F'6#F&F'F0*(F&\"\"\"F(F3-&F-6#,&F)F'!\" #F'F1F'\"\"$*(F&F3F(F3-&F-6#,&F)F'!\"%F'F1F'F'*&,(*&F)F'F&F3F*F&\"\"*F 8F'F'-&F-6#F(F1F'F'\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "More \+ generally, a differential equation with respect to " }{XPPEDIT 18 0 "z ;" "6#%\"zG" }{TEXT -1 103 ", or even a system of mixed differential-d ifference equations could be obtained by the same algorithms." }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "The following result had not been noticed by Lutz et al., but might prove useful in numerical computati ons." }}{PARA 0 "" 0 "" {TEXT 263 7 "Theorem" }{TEXT -1 32 ". With the above notations, let " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 266 " " 0 "" {XPPEDIT 18 0 "sum(p[k](n)*a(n+k),k = 0 .. K) = 0;" "6#/-%$sumG 6$*&-&%\"pG6#%\"kG6#%\"nG\"\"\"-%\"aG6#,&F.F/F,F/F//F,;\"\"!%\"KGF6" } }{PARA 0 "" 0 "" {TEXT -1 150 "be the linear recurrence satisfied by t he Taylor coefficients at the origin of a power series solution of the first-order line2ar differential equation" }}{PARA 267 "" 0 "" {XPPEDIT 18 0 "diff(G(t),t) = (diff(phi(t),t,t)/diff(phi(t),t)-diff(ph i(t),t)/z)*G(t);" "6#/-%%diffG6$-%\"GG6#%\"tGF**&,&*&-F%6%-%$phiG6#F*F *F*\"\"\"-F%6$-F16#F*F*!\"\"F3*&-F%6$-F16#F*F*F3%\"zGF8F8F3-F(6#F*F3" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 19 "Then the integrals " } {XPPEDIT 18 0 "q[n];" "6#&%\"qG6#%\"nG" }{TEXT -1 23 " satisfy the rec urrence" }}{PARA 268 "" 0 "" {XPPEDIT 18 0 "sum(p[k](-n)*q[n-k-1](z),k = 0 .. K) = 0;" "6#/-%$sumG6$*&-&%\"pG6#%\"kG6#,$%\"nG!\"\"\"\"\"-&% \"qG6#,(F/F1F,F0\"\"\"F06#%\"zGF1/F,;\"\"!%\"KGF<" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT 264 6 "Proof." }{TEXT -1 78 " The differential e quation in the statement above is satisfied by the function" }}{PARA 269 "" 0 "" {XPPEDIT 18 0 "e^(-phi(u)/z)*diff(phi(u),u);" "6#*&)%\"eG, $*&-%$phiG6#%\"uG\"\"\"%\"zG!\"\"F.F,-%%diffG6$-F)6#F+F+F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "Since the integrals " }{XPPEDIT 18 0 "q[n](z);" "6#-&%\"qG6#%\"nG6#%\"zG" }{TEXT -1 11 3" rewrite as" } }{PARA 270 "" 0 "" {XPPEDIT 18 0 "int(e^(-phi(u)/z)*u^n*diff(phi(u),u) ,u = 0 .. 1);" "6#-%$intG6$*()%\"eG,$*&-%$phiG6#%\"uG\"\"\"%\"zG!\"\"F 1F/)F.%\"nGF/-%%diffG6$-F,6#F.F.F//F.;\"\"!\"\"\"" }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 114 "by integration by parts and differentiat ion under the integral sign, they satisfy the announced linear recurre nce." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "The following one-line pr ocedure computes a recurrence on the integral" }}{PARA 264 "" 0 "" {XPPEDIT 18 0 "q[n](z) = Int(exp(-t/z)*psi(t)^n,t = 0 .. infinity);" " 6#/-&%\"qG6#%\"nG6#%\"zG-%$IntG6$*&-%$expG6#,$*&%\"tG\"\"\"F*!\"\"F6F5 )-%$psiG6#F4F(F5/F4;\"\"!%)infinityG" }}{PARA 0 "" 0 "" {TEXT -1 6 "wh ere " }{XPPEDIT 18 0 "psi;" "6#%$psiG" }{TEXT -1 50 " is the functiona l inverse of a rational function " }{XPPEDIT 18 0 "phi;" "6#%$phiG" } {TEXT -1 20 ". It takes as input " }{XPPEDIT 18 0 "phi(t),t,g,n,z;" "6 '-%$phiG6#%\"tGF&%\"gG%\"nG%\"zG" }{TEXT -1 115 " where all the argume nts except the 4first one are symbols that appear in the output linear \+ recurrence relating the " }{XPPEDIT 18 0 "q[n](z);" "6#-&%\"qG6#%\"nG6 #%\"zG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "rec qnofz:=proc(phi::ratpoly,t::name,q::name,n::name,z::name)\nlocal gf,a; \n op(select(has,eval(subs(n=-n,a=subs(_A=q,proc(x) _A(-x-1) end), \n gfun[diffeqtorec](diff(gf(t),t)+(diff(phi,t)/z-diff(phi,t,t)/d iff(phi,t))*gf(t),gf(t),a(n)))),n))\nend:" "6#>%)recqnofzGR6''%$phiG%( ratpolyG'%\"tG%%nameG'%\"qGF,'%\"nGF,'%\"zGF,7$%#gfG%\"aG6\"F6-%#opG6# -%'selectG6%%$hasG-%%evalG6#-%%subsG6%/F0,$F0!\"\"/F5-FB6$/%#_AGF.R6#% \"xG7\"F6F6-FK6#,&FNFF\"\"\"FFF6F6F6-&%%gfunG6#%,diffeqtorecG6%,&-%%di ffG6$-F46#F+F+\"\"\"*&,&*&-Ffn6$F(F+FjnF2FFFjn*&-Ffn6%F(F+F+Fjn-Ffn6$F (F+FFFFFjn-F46#F+FjnFjn-F46#F+-F56#F0F0F6F6F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Example: " }{XPPEDIT 18 0 "phi = 1/((1-t)^2)-1;" "6#/%$ phiG,&*&\"\"\"\"\"\"*$,&\"\"\"F(%\"tG!\"\"\"\"#F-F(\"\"\"F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "qnR5ec:=recqnofz(1/(1-t)^2-1, t,q,n,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&qnRecG,**&,&*&%\"nG\" \"\"%\"zGF*!\"\"F+\"\"$F*-%\"qG6#,&F)F*F,F*F*F**&,&F+!\"*F(F-F*-F/6#,& F)F*!\"#F*F*F**&,(F(!\"$F+\"\"*F8F*F*-F/6#,&F)F*F;F*F*F**&,&F(F*F+F;F* -F/6#,&F)F*!\"%F*F*F*" }}}{EXCHG {PARA 274 "" 0 "" {TEXT -1 51 "We hav e obtained the same recurrence as when using " }{HYPERLNK 17 "Mgfun" 2 "Mgfun" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "qnRec:=applyop(factor,[2,2],applyop(collect,[2,1],readlib(isolate )(subs(n=n+1,qnRec),q(n)),q,normal));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&qnRecG/-%\"qG6#%\"nG,$*&,(*&,&%\"zG\"\"'*&F)\"\"\"F/F2!\"$F2-F'6 #,&F)F2!\"\"F2F2F2*&,(F/!\"'F1\"\"$\"\"#F2F2-F'6#,&F)F2!\"#F2F2F2*&,&F /F " 0 "" {MPLTEXT 1 0 27 "phi:=subs(z=u,solve(eq,f));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phiG,$*&*&%\"uG\"\"\",&!\"#F)F(F)F)\"\"\"*$),&F(F)! \"\"F)\"\"#F,!\"\"F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "ass ume(z>0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "q0:=int(exp(-p hi/z)*diff(phi,u),u =0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q0G% #z|irG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "q0:=subs(z='z',q0 ); z:='z':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q0G%\"zG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Note that this initia7l value is a posteri ori obvious." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Thus we now have \+ both recurrence and initial condition. The solution " }{XPPEDIT 18 0 " q[n];" "6#&%\"qG6#%\"nG" }{TEXT -1 276 " to the recurrence equation qn Rec is a dominated solution, which means that any numerical error grow s exponentially. To avoid this, we run the recurrence backwards from a ny non trivial initial conditions. The dominating solution disappears \+ quickly, and we obtain the solution " }{XPPEDIT 18 0 "q[n];" "6#&%\"qG 6#%\"nG" }{TEXT -1 109 " because when the recurrence is run backwards \+ it becomes a dominating solution. We therefore add a parameter " } {XPPEDIT 18 0 "NN;" "6#%#NNG" }{TEXT -1 50 " indicating from where we \+ start running backwards." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "eval(collect(op(2,isolate(subs(n=n+3,qnRec),q(n))),q,normal),q=proc(n ) q(n,z,NN) end);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%\"qG6%,&%\"nG \"\"\"\"\"$F)%\"zG%#NNGF)-F%6%,&F(F)\"\"#F)F+F,!\"$*&*&,(F+F**&F(F)F+F )8F*F0F)F)-F%6%,&F(F)F)F)F+F,F)\"\"\"*&F+\"\"\"F8\"\"\"!\"\"F)" }}} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "qnprocrev:=subs(_REC=%,q=qnpro crev,proc(n,z,NN)\noption remember;\nif n=NN then 0 \nelif n=NN-1 then 0\nelif n=NN-2 then 1\nelse _REC fi end);" "6#>%*qnprocrevG-%%subsG6% /%%_RECG%\"%G/%\"qGF$R6%%\"nG%\"zG%#NNG7\"6#%)rememberG6\"@)/F/F1\"\"! /F/,&F1\"\"\"\"\"\"!\"\"F8/F/,&F1F;\"\"#F=\"\"\"F)F5F5F5" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%*qnprocrevGR6%%\"nG%\"zG%#NNG6\"6#%)rememberGF *@)/9$9&\"\"!/F/,&F0\"\"\"!\"\"F4F1/F/,&F0F4!\"#F4F4,(-F$6%,&F/F4\"\"$ F49%F0F4-F$6%,&F/F4\"\"#F4F>F0!\"$*&*&,(F>F=*&F/F4F>F4F=FBF4F4-F$6%,&F /F4F4F4F>F0F4\"\"\"*&F>\"\"\"FJ\"\"\"!\"\"F4F*F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Here is a procedure to compute " }{XPPEDIT 18 0 " q[n];" "6#&%\"qG6#%\"nG" }{TEXT -1 13 " numerically." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "`evalf/q/digits`:=0:" }}}{EXCHG {PARA 0 " > " 0 "" {XPPEDIT 19 1 "`evalf/q`:=proc (n,z,NN)\noption remember; \ng lobal `evalf/q`,`evalf/q/digits9`;\nif Digits>`evalf/q/digits` \nthen ` evalf/q/digits`:=Digits;\n `evalf/q`:=subsop(4=NULL,op(`evalf/q`)) fi; \nif n = NN then 0 \n elif n = NN-1 then 0 \n elif n = NN-2 then \+ 1.0\n else \nprocname(n+3,z,NN)+(2+3*z+3*z*n)*procname(n+1,z,NN)/(z*(n +1))-3*procname(n+2,z,NN) fi end:" "6#>%(evalf/qGR6%%\"nG%\"zG%#NNG7\" 6#%)rememberG6\"C$@$2%/evalf/q/digitsG%'DigitsGC$>F1F2>F$-%'subsopG6$/ \"\"%%%NULLG-%#opG6#F$@)/F'F)\"\"!/F',&F)\"\"\"\"\"\"!\"\"FA/F',&F)FD \"\"#FF$\"#5!\"\",(-%)procnameG6%,&F'FD\"\"$FDF(F)FD*(,(\"\"#FD*&\"\"$ FDF(FDFD*(\"\"$FDF(FDF'FDFDFD-FO6%,&F'FD\"\"\"FDF(F)FD*&F(FD,&F'FD\"\" \"FDFDFFFD*&\"\"$FD-FO6%,&F'FD\"\"#FDF(F)FDFFF-6$F$F1F-" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "dnRec:=gfun[diffeqtorec](\{dneq,D(f)( 0)=1,f(0)=0\},f(z),a(n)):" "6#>%&dnRecG-&%%gfunG6#%,diffeqtorecG6%<%%% dneqG/--%\"DG6#%\"fG6#\"\"!\"\"\"/-F26#F4F4-F26#%\"zG-%\"aG6#%\"nG" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "dnproc:=gfun[rectoproc](dnR ec,a(n),remember):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 ":Here is a p rocedure to compute " }{XPPEDIT 18 0 "d[n];" "6#&%\"dG6#%\"nG" }{TEXT -1 13 " numerically." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "`ev alf/d/digits`:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "`evalf/d `:=proc (n) option remember; global `evalf/d`,`evalf/d/digits`; if n>5 then if Digits>`evalf/d/digits` then `evalf/d/digits`:=Digits;`evalf/ d`:=subsop(4=NULL,op(`evalf/d`)) fi; evalf(-(-225*procname(n-5)-70*pro cname(n-1)+804*procname(n-4)-1011*procname(n-3)+514*procname(n-2)+(165 *procname(n-5)-693*procname(n-4)+1048*procname(n-3)-683*procname(n-2)+ 157*procname(n-1)+(-39*procname(n-5)+195*procname(n-4)-363*procname(n- 3)+309*procname(n-2)-114*procname(n-1)+(-18*procname(n-4)+42*procname( n-3)-48*procname(n-2)+3*procname(n-5)+27*procname(n-1))*n)*n)*n)/((-6+ (12-6*n)*n)*n)) elif n=0 then 0 elif n=1 then 1 elif n=2 then evalf(1/ 3) elif n=3 then evalf(-23/108) else evalf(-2749/3888) fi end:" "6#>%( evalf/dGR6#%\"nG7\"6#%)rememberG6\"@-2\"\"&F'C$@$2%/evalf/d/digitsG%'D igitsGC$>F2F3>;F$-%'subsopG6$/\"\"%%%NULLG-%#opG6#F$-%&evalfG6#,$*&,.*& \"$D#\"\"\"-%)procnameG6#,&F'FH\"\"&!\"\"FHFN*&\"#qFH-FJ6#,&F'FH\"\"\" FNFHFN*&\"$/)FH-FJ6#,&F'FH\"\"%FNFHFH*&\"%65FH-FJ6#,&F'FH\"\"$FNFHFN*& \"$9&FH-FJ6#,&F'FH\"\"#FNFHFH*&,.*&\"$l\"FH-FJ6#,&F'FH\"\"&FNFHFH*&\"$ $pFH-FJ6#,&F'FH\"\"%FNFHFN*&\"%[5FH-FJ6#,&F'FH\"\"$FNFHFH*&\"$$oFH-FJ6 #,&F'FH\"\"#FNFHFN*&\"$d\"FH-FJ6#,&F'FH\"\"\"FNFHFH*&,.*&\"#RFH-FJ6#,& F'FH\"\"&FNFHFN*&\"$&>FH-FJ6#,&F'FH\"\"%FNFHFH*&\"$j$FH-FJ6#,&F'FH\"\" $FNFHFN*&\"$4$FH-FJ6#,&F'FH\"\"#FNFHFH*&\"$9\"FH-FJ6#,&F'FH\"\"\"FNFHF N*&,,*&\"#=FH-FJ6#,&F'FH\"\"%FNFHFN*&\"#UFH-FJ6#,&F'FH\"\"$FNFHFH*&\"# [FH-FJ6#,&F'FH\"\"#FNFHFN*&\"\"$FH-FJ6#,&F'FH\"\"&FNFHFH*&\"#FFH-FJ6#, &F'FH\"\"\"FNFHFHFHF'FHFHFHF'FHFHFHF'FHFHFH*&,&\"\"'FN*&,&\"#7FH*&\"\" 'FHF'FHFNFHF'FHFHFHF'FHFNFN/F'\"\"!Fju/F'\"\"\"\"\"\"/F'\"\"#-FA6#*&\" \"\"FH\"\"$FN/F'\"\"$-FA6#,$*&\"#BFH\"$3\"FNFN-FA6#,$*&\"%\\FFH\"%))QF NFNF+6$F$F2F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "Finally, the fo llowing procedure computes values of the< double confluent Heun functio n as follows. First, an upper value of " }{XPPEDIT 18 0 "NN;" "6#%#NNG " }{TEXT -1 17 " is selected and " }{XPPEDIT 18 0 "2*NN;" "6#*&\"\"#\" \"\"%#NNGF%" }{TEXT -1 70 " is used in the backward recurrence to comp ute the scaling to use for " }{XPPEDIT 18 0 "q[n](z);" "6#-&%\"qG6#%\" nG6#%\"zG" }{TEXT -1 84 " in view of the actual initial condition. The n the summation is performed up to the " }{XPPEDIT 18 0 "NN;" "6#%#NNG " }{TEXT -1 63 "th term. If the relative error of the last term is lar ger than " }{XPPEDIT 18 0 "10^(-Digits);" "6#)\"#5,$%'DigitsG!\"\"" } {TEXT -1 7 ", then " }{XPPEDIT 18 0 "NN;" "6#%#NNG" }{TEXT -1 89 " is \+ doubled and the computation starts again. Note that option remember ha s been used in " }{XPPEDIT 18 0 "qnprocrev;" "6#%*qnprocrevG" }{TEXT -1 45 " so as to avoid duplicating some of the work." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "time(evalf(q(0,10.2,3000),21));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$*R!\"$" }}}{EXCHG {PARA 0 "> " 0 "" = {MPLTEXT 1 0 450 "valheun:=subs(_q0=q0,proc(z)\n local tot,i,N,NN,l ambda,st,D;\nN:=10;\nst:=time();\ndo \n N:=floor(2*N);\n NN:=N+f loor(sqrt(N))+10;\n D:=Digits+3*ilog10(N) +floor(log(N));\n lamb da:=_q0/evalf(q(0,z,NN),D);\n \n tot:=add(evalf(d(i),D)*evalf(q(i ,z,NN),D),i=1..N)*lambda;\n if abs(evalf(d(N),D)*evalf(q(N,z,NN),D) *lambda) " 0 "" {XPPEDIT 19 1 "infolevel[ valheun]:=1:" "6#>&%*infolevelG6#%(valheunG\"\"\"" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 20 "plot(valheun,0..50);" }}{PARA 6 "" 1 "" {TEXT -1 75 "valheun: \"N=\" 80 \"z=\" 1.089857709 \"time:\" .141 \"digits:\" 17" }}{PARA 6 "" 1 "" {TEXT -1 76 "valheun: \+ \"N=\" 160 \"z=\" 2.0>38137074 \"time:\" .270 \"digits:\" \+ 21" }}{PARA 6 "" 1 "" {TEXT -1 76 "valheun: \"N=\" 320 \"z=\" \+ 3.104576397 \"time:\" .579 \"digits:\" 21" }}{PARA 6 "" 1 "" {TEXT -1 76 "valheun: \"N=\" 320 \"z=\" 4.178084772 \"time: \" .571 \"digits:\" 21" }}{PARA 6 "" 1 "" {TEXT -1 76 "valheun: \+ \"N=\" 320 \"z=\" 5.246490950 \"time:\" .599 \"digits:\" 21" }}{PARA 6 "" 1 "" {TEXT -1 77 "valheun: \"N=\" 640 \"z=\" 6.237040242 \"time:\" 1.221 \"digits:\" 22" }}{PARA 6 "" 1 "" {TEXT -1 77 "valheun: \"N=\" 640 \"z=\" 7.262696659 \"tim e:\" 1.080 \"digits:\" 22" }}{PARA 6 "" 1 "" {TEXT -1 77 "valheu n: \"N=\" 640 \"z=\" 8.323431992 \"time:\" 1.129 \"digit s:\" 22" }}{PARA 6 "" 1 "" {TEXT -1 77 "valheun: \"N=\" 640 \" z=\" 9.380765534 \"time:\" 1.240 \"digits:\" 22" }}{PARA 6 " " 1 "" {TEXT -1 77 "valheun: \"N=\" 640 \"z=\" 10.46836305 \+ \"time:\" 1.201 \"digits:\" 22" }}{PARA 6? "" 1 "" {TEXT -1 78 "v alheun: \"N=\" 1280 \"z=\" 11.42631952 \"time:\" 2.520 \+ \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 1 280 \"z=\" 12.50475163 \"time:\" 2.369 \"digits:\" 26" }} {PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 1280 \"z=\" 13.58 761188 \"time:\" 2.601 \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 1280 \"z=\" 14.63114838 \"time: \" 2.750 \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 1280 \"z=\" 15.57877949 \"time:\" 2.800 \"digits :\" 26" }}{PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 1280 \" z=\" 16.70560455 \"time:\" 3.110 \"digits:\" 26" }}{PARA 6 " " 1 "" {TEXT -1 78 "valheun: \"N=\" 1280 \"z=\" 17.66017317 \+ \"time:\" 3.219 \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 78 "v alheun: \"N=\" 1280 \"z=\" 18.77056341 \"time:\" 3.340 \+ \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 2 560 @ \"z=\" 19.75344692 \"time:\" 7.290 \"digits:\" 26" }} {PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 2560 \"z=\" 20.83 182591 \"time:\" 8.091 \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 2560 \"z=\" 21.85869773 \"time: \" 8.440 \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 2560 \"z=\" 22.93013074 \"time:\" 9.180 \"digits :\" 26" }}{PARA 6 "" 1 "" {TEXT -1 78 "valheun: \"N=\" 2560 \" z=\" 23.91404071 \"time:\" 9.710 \"digits:\" 26" }}{PARA 6 " " 1 "" {TEXT -1 79 "valheun: \"N=\" 2560 \"z=\" 24.97532053 \+ \"time:\" 10.209 \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 79 " valheun: \"N=\" 2560 \"z=\" 26.07769200 \"time:\" 10.941 \+ \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" \+ 2560 \"z=\" 27.03730960 \"time:\" 11.229 \"digits:\" 26" } }{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 2560 \"z=\" 28.0 7372290 \"time:\" 11.961 A \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 2560 \"z=\" 29.14443926 \"time: \" 12.589 \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun : \"N=\" 2560 \"z=\" 30.19192632 \"time:\" 12.810 \"digi ts:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 2560 \+ \"z=\" 31.20542392 \"time:\" 13.741 \"digits:\" 26" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" 32.33074020 \"time:\" 20.060 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" 33.34188693 \"time:\" 12.8 80 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N= \" 5120 \"z=\" 34.42150189 \"time:\" 15.039 \"digits:\" \+ 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" \+ 35.39979350 \"time:\" 17.241 \"digits:\" 27" }}{PARA 6 "" 1 " " {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" 36.46932464 \"tim e:\" 19.520 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEBXT -1 79 "valhe un: \"N=\" 5120 \"z=\" 37.47567008 \"time:\" 21.760 \"di gits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \+ \"z=\" 38.52759104 \"time:\" 24.110 \"digits:\" 27" }} {PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" 39.55 603605 \"time:\" 26.450 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" 40.63272267 \"time: \" 28.740 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun : \"N=\" 5120 \"z=\" 41.66969985 \"time:\" 31.479 \"digi ts:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \+ \"z=\" 42.73015878 \"time:\" 33.841 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" 43.78183659 \"time:\" 36.099 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" 44.74821926 \"time:\" 39.0 00 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N= \" 5120 \"z=C\" 45.85580287 \"time:\" 40.981 \"digits:\" \+ 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" \+ 46.84643885 \"time:\" 43.800 \"digits:\" 27" }}{PARA 6 "" 1 " " {TEXT -1 79 "valheun: \"N=\" 5120 \"z=\" 47.90266342 \"tim e:\" 45.850 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 79 "valhe un: \"N=\" 5120 \"z=\" 48.91360511 \"time:\" 48.260 \"di gits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 72 "valheun: \"N=\" 5120 \+ \"z=\" 50.0 \"time:\" 50.699 \"digits:\" 27" }}{PARA 6 "" 1 "" {TEXT -1 77 "valheun: \"N=\" 160 \"z=\" 1.563997392 \"t ime:\" 1.450 \"digits:\" 21" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'C URVESG6$7S7$$\"+4x&)*3\"!\"*$\"+7+&3\"H!#57$$\"+#R(*Rc\"F*$\"+mL$Q6&F- 7$$\"+uq8Q?F*$\"+&Q!)ps(F-7$$\"+(RwX5$F*$\"+'epd]\"F*7$$\"+sZ3yTF*$\"+ $Q+sV#F*7$$\"+]4\\Y_F*$\"+gn*z`$F*7$$\"+U-/PiF*$\"+w!)*[p%F*7$$\"+fmpi sF*$\"+nI,9gF*7$$\"+#*>VB$)F*$\"+\"=:D\\(F*7$$\"+Mbw!Q*F*$\"+**)\\'o!* F*7$$\"+0j$o/\"!\")$\"+CI\\y5Fhn7$$\D"+_>jU6Fhn$\"+N4'oB\"Fhn7$$\"+j^Z] 7Fhn$\"+4#pCU\"Fhn7$$\"+)=h(e8Fhn$\"+FQ(fh\"Fhn7$$\"+Q[6j9Fhn$\"+1_l3= Fhn7$$\"+\\z(yb\"Fhn$\"+ny]))>Fhn7$$\"+b/cq;Fhn$\"+Y8)z?#Fhn7$$\"+Fhn$\"+4W#*GGF hn7$$\"+\"f#=$3#Fhn$\"+-(*>dIFhn7$$\"+t(pe=#Fhn$\"+.OMyKFhn7$$\"+uI,$H #Fhn$\"+`+z7NFhn7$$\"+rSS\"R#Fhn$\"+enCJPFhn7$$\"+`?`(\\#Fhn$\"+\"30,( RFhn7$$\"++#pxg#Fhn$\"+J7e@UFhn7$$\"+g4t.FFhn$\"+&\\cJW%Fhn7$$\"+!Hst! GFhn$\"+RW7&o%Fhn7$$\"+ERW9HFhn$\"+.%yy$\\Fhn7$$\"+KE>>IFhn$\"+wKw(=&F hn7$$\"+#RU07$Fhn$\"+;#**=V&Fhn7$$\"+?S2LKFhn$\"+Q%zbq&Fhn7$$\"+$p)=ML Fhn$\"+(HVP&fFhn7$$\"+*=]@W$Fhn$\"+:3(4A'Fhn7$$\"+]$z*RNFhn$\"+j?1lkFh n7$$\"+kC$pk$Fhn$\"+df&Rt'Fhn7$$\"+3qcZPFhn$\"+uH%)))pFhn7$$\"+/\"fF&Q Fhn$\"+sQ:dsFhn7$$\"+0OgbRFhn$\"+DKF@vFhn7$$\"+nAFjSFhn$\"+Cdi*z(Fhn7$ $\"+&)*pp;%Fhn$\"+>yVp!)Fhn7$$\"+ye,tUFhn$\"+G40Z$)Fhn7$$\"+fO=yVFhn$ \"+Z!3Si)Fhn7$$\"+E>#[Z%Fhn$\"+$Q-*z))Fhn7$$\"+(G!e&e%Fhn$\"+Wnyu\"*Fh n7$$\"+&)Qk%o%Fhn$\"+`)Q*R%*Fhn7$$\"+UjE!z%Fhn$\"+:%ySs*Fhn7$$\E"+60O\" *[Fhn$\"+87Q(***Fhn7$\"#]$\"+U%[#H5!\"(-%'COLOURG6&%$RGBG$\"#5!\"\"\" \"!F`[l-%+AXESLABELSG6$%!GFd[l-%%VIEWG6$;F`[l$FezF`[l%(DEFAULTG" 2 762 762 762 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 2 0 0 0 0 0 0 }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 117 "This curve is to be contrasted with the irregular plot we got from the same series using summation to the least term." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 216 "Sections 1 and 2 of this works heet only depend on the Heun differential equation, and can easily be \+ adapted to any linear differential equation. Sections 3 and 4 compute \+ the recurrences satisfied by the coefficients " }{XPPEDIT 18 0 "q[n]; " "6#&%\"qG6#%\"nG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "d[n];" "6#&%\" dG6#%\"nG" }{TEXT -1 53 ", which depend on the choice of the mapping f unction " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 10 " only for " } {XPPEDIT 18 0 "q[n];" "6#&%\"qG6#%\"nG" }{TEXT -1 47 ", and on the dif ferential equation asF well for " }{XPPEDIT 18 0 "d[n];" "6#&%\"dG6#%\" nG" }{TEXT -1 62 ". Section 5 details the numerical computations. It d epends on " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 158 " and on th e recurrences found in the previous sections. This worksheet can be ad apted to another mapping function and to another linear differential e quation." }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 10 "References" }}{PARA 0 "" 0 "" {TEXT -1 15 "[Lutz et al.] " }{TEXT 268 40 "On the converge nce of Borel approximants" }{TEXT -1 62 ", by W. Balser, D. A. Lutz a nd R. Sch\344fke, (2000). Preprint." }}{PARA 0 "" 0 "" {TEXT -1 12 "[ DuLoRi92] " }{TEXT 269 77 "Kovacic's Algorithm and Its Application to Some Families of Special Functions" }{TEXT -1 43 ", by Anne Duval and Mich\350le Loday-Richaud, " }{TEXT 270 62 "Applicable Algebra in Engi neering, Communication and Computing" }{TEXT -1 29 ", (1992), vol. 3, \+ p. 211-246." }}{PARA 0 "" 0 "" {TEXT -1 10 "[Durand] " }{TEXT 272 75 "On the convergence of Borel approximants [sumGmary of a talk by Donald Lutz]" }{TEXT -1 45 ", by Marianne Durand, (2001). To appear in: " } {TEXT 271 29 "Algorithms Seminar, 2000-2001" }{TEXT -1 24 ", INRIA Res earch Report." }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 } ctions. This worksheet can be ad apted to another mapping function and to another linear differential e quation." }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 10 "References" }}{PARA 0 "" 0 "" {TEXT -1 15 "[Lutz et al.] " }{TEXT 268 40 "On the converge nce of Borel approximants" }{TEXT -1 62 ", by W. Balser, D. A. Lutz a nd R. Sch\344fke, (2000). Preprint." }}{PARA 0 "" 0 "" {TEXT -1 12 "[ DuLoRi92] " }{TEXT 269 77 "Kovacic's Algorithm and Its Application to Some Families of Special Functions" }{TEXT -1 43 ", by Anne Duval and Mich\350le Loday-Richaud, " }{TEXT 270 62 "Applicable Algebra in Engi neering, Communication and Computing" }{TEXT -1 29 ", (1992), vol. 3, \+ p. 211-246." }}{PARA 0 "" 0 "" {TEXT -1 10 "[Durand] " }{TEXT 272 75 "On the convergence of Borel approximants [sumGintroductsystemmadcallsequencfunctargumentdescriptmathematicalabstractdocumpreparatintegratwithmapldefinrepresentcollectlogicalstructurdocumentcontsuchintovariouformatkeypointallowbuildupotherwordrepresdatalanguagcanusedautomatproductreliexternalbinarthesinterfacwayporteverplatformdefaultconfigurworkunixlikeoperatnoteuserresponsabilitensuravailabldesignextendvirtualanyproducprovidcorrespondimplementfollowpackagmaintranslatorhtmllatexgeneralpurposroutinsimplifiabovforwardprocedurparametercrosreferenctranslatalsoincludspecificstylwhenevbringframeworkusingwebservremarkunderdevelopmsuggestcommentbugreportwelcompleassendemailludovicmeuniinriafrcommutation_rules,Ore_algebra*computeentry,encyclopediaTCcreative_telescoping,MgfunOcrossref,DocumentGenerator5P%crossref-translator,DocumentGeneratorcustommath,DocumentGeneratorMdeclaration_options,Ore_algebradfinite_add,Holonomy!dfinite_expr_to_diffeq,Mgfun dfinite_expr_to_rec,Mgfun dfinite_expr_to_sys,Mgfun 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"POLLARD'S RHO ALGORITHM "T }}{PARA 0 "" 0 "" {TEXT 280 0 "" }}{PARA 257 "" 0 "" {TEXT 256 11 "B runo Salvy" }}{PARA 258 "" 0 "" {TEXT -1 29 "(Version of January 27, 1 997)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "P ollard's " }{XPPEDIT 18 0 "rho" "I$rhoG6\"" }{TEXT -1 141 "-method is \+ an efficient technique used to find factors of integers. It is both ve ry efficient and very simple. We show in this worksheet how " } {HYPERLNK 17 "combstruct" 2 "combstruct" "" }{TEXT -1 5 " and " } {HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 162 " can be used to analyze a realistic combinatorial model of the algorithm and thus derive a pr obabilistic complexity analysis of this algorithm and variants of it. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with(combstruct): with( gfun):" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 33 "Algorithm and combinat orial model" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 13 "The algorithm" }} {PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "N" "I\"NG6\"" }{TEXT -1 94 " is the integer of which a factoUr is sought, the basic version \+ of the algorithm is as follows:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 29 "Pick up an arbitrary integer " }{XPPEDIT 18 0 "a" "I\"aG6\"" }{TEXT -1 5 " mod " }{XPPEDIT 18 0 "N" "I\"NG6\"" }{TEXT -1 6 ", set " }{XPPEDIT 18 0 "f(x)=x^2+a mod N" "/-%\"fG6#%\"xG -%$modG6$,&*$F&\"\"#\"\"\"%\"aGF-%\"NG" }}{PARA 0 "" 0 "" {TEXT -1 26 "Select at random a number " }{XPPEDIT 18 0 "x[0]" "&%\"xG6#\"\"!" } {TEXT -1 5 " mod " }{XPPEDIT 18 0 "N" "I\"NG6\"" }{TEXT -1 6 ", set " }{XPPEDIT 18 0 "y[0]=x[0]" "/&%\"yG6#\"\"!&%\"xG6#F&" }}{PARA 0 "" 0 " " {TEXT -1 8 "Iterate:" }}{PARA 0 "" 0 "" {TEXT -1 6 " " } {XPPEDIT 18 0 "i:=i+1; x[i]:=f(x[i-1]); y[i]:=f(f(y[i-1]))" "C%>%\"iG, &F$\"\"\"\"\"\"F&>&%\"xG6#F$-%\"fG6#&F*6#,&F$F&\"\"\"!\"\">&%\"yG6#F$- F-6#-F-6#&F66#,&F$F&\"\"\"F3" }}{PARA 0 "" 0 "" {TEXT -1 6 "until " } {XPPEDIT 18 0 "gcd(y[i]-x[i],N)<>1" "0-%$gcdG6$,&&%\"yG6#%\"iG\"\"\"&% \"xG6#F*!\"\"%\"NG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }V}{PARA 0 "" 0 "" {TEXT -1 151 "This is directly translated into \+ the following rough Maple procedure which returns a factor and the num ber of iterations performed to find this factor:" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 220 "pollard:=proc(N)\nlocal rnd, a, f, x, y, i, g ;\n rnd:=rand(N); a:=rnd(); x:=rnd(); y:=x;\n for i do\n \+ x:=x^2+a mod N; y:=(y^2+a mod N)^2+a mod N; g:=igcd(y-x,N);\n i f g<>1 then RETURN(g,i) fi\n od\nend: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Here are a few examples (Maple's " }{HYPERLNK 17 "nextpri me" 2 "nextprime" "" }{TEXT -1 67 " routine returns the smallest prime number following its argument)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "nextprime(900)*nextprime(20000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\")x*\\\"=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "pollard(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"$2*\"#J" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "pollard(\"\");" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$\"$2*\"#K" }}}{EXCWHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "pollard(\"\"\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"$2*\"#I " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "nextprime(10^5)*nextpri me(10^6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"-4+I.+5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "pollard(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"(.++\"\"$<\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "pollard(\"\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"'.+5\"$O&" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "pollard(\"\"\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"'.+5\"$L#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "nextprime(10^6)*nextprime(10^7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"/d++\\++5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "pollard(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"(.++\"\"$#z" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "pollard(\"\");" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$\"(.++\"\"$!R" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "pollard(\"\"\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ \")>++5\"%q8X" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 23 "The combinatori al model" }}{PARA 0 "" 0 "" {TEXT -1 78 "The algorithm relies on the s tructure of the functional graph of the function " }{XPPEDIT 18 0 "f(x )=x^2+a mod p" "/-%\"fG6#%\"xG-%$modG6$,&*$F&\"\"#\"\"\"%\"aGF-%\"pG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "p" "I\"pG6\"" }{TEXT -1 33 " is the smallest prime factor of " }{XPPEDIT 18 0 "N" "I\"NG6\"" }{TEXT -1 47 ". In this graph, the vertices are the integers " }{XPPEDIT 18 0 "0..p-1" ";\"\"!,&%\"pG\"\"\"\"\"\"!\"\"" }{TEXT -1 5 " mod " } {XPPEDIT 18 0 "p" "I\"pG6\"" }{TEXT -1 57 " and the directed edges lin k each vertex to its image by " }{XPPEDIT 18 0 "f" "I\"fG6\"" }{TEXT -1 236 ". Since the number of points is finite, it is not difficult to see that the graph has the structure of a union of connected componen ts, each of these components being constituted of a cycle to which con verge trees. Since the polynomial " }{XPPEDIT 18 0 "f(x)" "-%\"fG6#%\" xG" }{TEXT -1 18 " has degree 2 and " }{XPPEDIYT 18 0 "p" "I\"pG6\"" } {TEXT -1 36 " is prime, all the vertices (except " }{XPPEDIT 18 0 "a" "I\"aG6\"" }{TEXT -1 73 ") have in-degree 0 or 2, while they have out- degree 1. The prime factor " }{XPPEDIT 18 0 "p" "I\"pG6\"" }{TEXT -1 66 " is assumed to be large, therefore the special case of the vertex \+ " }{XPPEDIT 18 0 "a" "I\"aG6\"" }{TEXT -1 125 " which has in-degree 1 \+ can be discarded as a first approximation. The combinatorial model is \+ thus expressed by the following " }{HYPERLNK 17 "combstruct grammar" 2 "combstruct[specification]" "" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 137 "G:=\{fungraph=Set(connected_component),\n \+ connected_component=Cycle(Prod(Z,bintree)),\n bintree=Union(Z,Prod( Z,Set(bintree,card=2)))\}:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 " T he execution of the algorithm is interpreted on this graph as follows: a random point " }{XPPEDIT 18 0 "x[0]" "&%\"xG6#\"\"!" }{TEXT -1 46 " of the graph is selected. Then two sequences " }{XPPEDIT 18 0 "xZ[i]" "&%\"xG6#%\"iG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y[i]" "&%\"yG6#%\" iG" }{TEXT -1 32 " of iterates with initial value " }{XPPEDIT 18 0 "x[ 0]" "&%\"xG6#\"\"!" }{TEXT -1 374 " are computed, one of them proceedi ng one step at a time, while the other one proceeds by steps of length 2. Starting from a node of the graph, these two sequences eventually \+ reach a cycle, where they are bound to intersect. This is where the al gorithm stops.\n Under this model, the average number of steps requir ed by the algorithm is therefore related to two parameters: " }{TEXT 257 3 "(i)" }{TEXT -1 49 " the average distance from a point to its cy cle; " }{TEXT 258 4 "(ii)" }{TEXT -1 34 " the average length of the cy cles." }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 34 "Path length in planar binary trees" }}{PARA 0 "" 0 "" {TEXT -1 23 "We start with question \+ " }{TEXT 259 3 "(i)" }{TEXT -1 281 " above: the determination of the a verage distance from a point to its cycle. We first concentrate on a s imilar but simpler[ problem, the computation of the average distance fr om a node to the root in a planar binary tree. This is related to a cl assical combinatorial parameter: the " }{TEXT 260 20 "internal path le ngth" }{TEXT -1 136 " of the tree, which is the sum of the distances f rom each of the nodes to the root.\nBinary trees are described by the \+ following grammar:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "bin:= \{bintree=Union(Epsilon,Prod(Z,bintree,bintree))\}:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "The counting sequence given by " }{HYPERLNK 17 "combstruct[count]" 2 "combstruct[count]" "" }{TEXT -1 50 " is constit uted by the well-known Catalan numbers:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "seq(count([bintree,bin,unlabelled],size=i),i=0..15); " }}{PARA 11 "" 1 "" {XPPMATH 20 "62\"\"\"F#\"\"#\"\"&\"#9\"#U\"$K\"\" $H%\"%I9\"%i[\"&'z;\"&'ye\"'7!3#\"'+Hu\"(SWn#\"(X[p*" }}}{PARA 0 "" 0 "" {TEXT -1 152 "The following Maple procedure takes as input a binary tree as produced by combstruct using the spec\ification above and retu rns its internal path length:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 236 "ipl:=proc(t)\n if type(t,name) then 0 # the tree is of the fo rm Z or Epsilon\n else # the tree is of the form Prod(Z,t1,t2)\n \+ ipl(op(2,t))+ipl(op(3,t))+size(t)-1\n fi\nend:\nsize:=proc(t) eval (subs([Prod=`+`,Z=1,Epsilon=0],t)) end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 218 "The internal path length is computed using a simple bije ction: each of the nodes on a branch from the root to a leaf are count ed once for each of the nodes below it. Here is an example on a tree o f size 5 generated by " }{HYPERLNK 17 "combstruct[draw]" 2 "combstruct [draw]" "" }{TEXT -1 30 " from the specification above:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "T:=draw([bintree,bin,unlabelled],si ze=5); ipl(T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG-%%ProdG6%%\"Z G-F&6%F(%(EpsilonG-F&6%F(-F&6%F(-F&6%F(F+F+F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 175 "We shall \+ comp]ute the average internal path length by first computing the total \+ internal path length, i.e. the sum of the internal path lengths of all the binary trees of size " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 38 " and then dividing this number by the " }{XPPEDIT 18 0 "n" "I\"nG6 \"" }{TEXT -1 158 "th Catalan number. We first start by computing expe rimentally the first values of these numbers, generating all the trees of fixed sizes for small sizes with " }{HYPERLNK 17 "combstruct[allst ructs]" 2 "combstruct[allstructs]" "" }{TEXT -1 102 ", and computing t he sum of the internal path lengths of all these trees with the proced ure ipl above. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "for i to 5 do sum_ipl[i]:=`+`(op(map(ipl,allstructs([bintree,bin,unlabelled],s ize=i)))) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(sum_iplG6#\"\"\" \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(sum_iplG6#\"\"#F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(sum_iplG6#\"\"$\"#9" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%(sum_iplG6#^\"\"%\"#u" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%(sum_iplG6#\"\"&\"$_$" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 36 "Combinatorial model for path lengths" }}{PARA 0 "" 0 "" {TEXT -1 290 "The numbers computed above can actually be derived more \+ efficiently, together with many other results related to path lengths \+ using combstruct's ability to deal with marks (atoms of size 0). The i dea consists in writing a combinatorial specification such that the nu mber of objects of size " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 81 " is exactly the sum of the internal path lengths of all the binary tre es of size " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 136 ". The combin atorial technique used here extends to other combinatorial structures \+ and leads to a combstruct-based analysis of Pollard's " }{XPPEDIT 18 0 "rho" "I$rhoG6\"" }{TEXT -1 437 "-algorithm.\nThe grammar for binary trees is modified to take into account \"decorated\" binary trees. A \+ binary tree is decorated by putting two marks (An for Ancestor_ and De \+ for Descendant) on two nodes belonging to the same branch. The number \+ of ways of decorating a binary tree is then exactly its internal path \+ length. Couting the number of decorated trees therefore corresponds to summing the internal path lengths of all binary trees." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 390 "bin2:=\{bintree2=Prod(Z,\n \+ Union(leftright2,Prod(An,leftright1))),\n bintree1=Prod(Z,\n \+ Union(leftright1,Prod(De,bintree,bintree))),\n leftrig ht2=Union(Prod(bintree2,bintree),\n Prod(bintre e,bintree2)),\n leftright1=Union(Prod(bintree1,bintree),\n \+ Prod(bintree,bintree1)),\n An=Epsilon, De=Epsi lon\} union bin:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "The sequence \+ of cumulated internal path lengths is now derived in less than a secon d:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "seq(count([bintree2,b in2,unlabelled],size=i),i=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6, \"\"!\"\`"#\"#9\"#u\"$_$\"%)e\"\"%Yp\"&'yH\"'3g7\"'+z_" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Here are the 14 decorated trees of size 3 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "allstructs([bintree2,b in2,unlabelled],size=3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#70-%%ProdG 6$%\"ZG-F%6$%#AnG-F%6$%(EpsilonG-F%6$F'-F%6%%#DeGF--F%6%F'F-F--F%6$F'- F%6$F*-F%6$F.F--F%6$F'-F%6$F--F%6$F'-F%6$F*-F%6$F--F%6$F'-F%6%F2F-F--F %6$F'-F%6$F*-F%6$F3FE-F%6$F'-F%6$F*-F%6$F--F%6$F'-F%6$FEF--F%6$F'-F%6$ F--F%6$F'-F%6$F*FW-F%6$F'-F%6$F*-F%6$FEF3-F%6$F'-F%6$F*-F%6$-F%6$F'-F% 6%F2F3F-F--F%6$F'-F%6$F*-F%6$F--F%6$F'FC-F%6$F'-F%6$F*-F%6$F-Fgo-F%6$F '-F%6$FgnF--F%6$F'-F%6$F?F--F%6$F'-F%6$F*-F%6$FUF--F%6$F'-F%6$F*-F%6$F apF-" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 20 "Generating functions" } }{EXCHG {PARA 0 "" 0 "" {TEXT -1 151 "From the grammars describing bin ary trees and decorated binary trees, the average internal path length can be computed via generating functions. Using " }{HYPERLNK 17 "comb struct[gfsolve]" 2 "combstruct[gfsolve]a" "" }{TEXT -1 102 ", we first \+ derive the generating functions for the Catalan numbers and for the cu mulated path lengths:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "F: =subs(gfsolve(bin,unlabelled,z),bintree(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG,$*&%\"zG!\"\",&\"\"\"F**$,&F*F*F'!\"%#F*\"\"#F(F *F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "S:=series(F,z,30);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"SG+in%\"zG\"\"\"\"\"!F'\"\"\"\" \"#\"\"#\"\"&\"\"$\"#9\"\"%\"#U\"\"&\"$K\"\"\"'\"$H%\"\"(\"%I9\"\")\"% i[\"\"*\"&'z;\"#5\"&'ye\"#6\"'7!3#\"#7\"'+Hu\"#8\"(SWn#\"#9\"(X[p*\"#: \")qwNN\"#;\"*!zW'H\"\"#<\"*+(QwZ\"#=\"+!>jsw\"\"#>\"+?/7kl\"#?\",?qEm W#\"#@\",SOc#[\"*\"#A\"-]OhfIM\"#B\".Ct9/**G\"\"#C\"._9SY>'[\"#D\"/_@2 `tO=\"#E\"//g\"4bL&p\"#F\"0g.v^zuj#\"#G-%\"OG6#F'\"#H" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Fpl:=subs(gfsolve(bin2,unlabelled,z ),bintree2(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$FplG,$*&,(*&%\"z G!\"\",&\"\"\"F,*$,&F,F,F)!\"%#F,\"\"#F*F,#F*F1#\"\"$F1F,F-F2F,,&F*F,F )\"b\"%F*F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Spl:=series(F pl,z,30);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$SplG+en%\"zG\"\"#\"\"# \"#9\"\"$\"#u\"\"%\"$_$\"\"&\"%)e\"\"\"'\"%Yp\"\"(\"&'yH\"\")\"'3g7\" \"*\"'+z_\"#5\"(!e&>#\"#6\"(s23*\"#7\")kGRP\"#8\"*OXV`\"\"#9\"*a*yxi\" #:\"+m9WiD\"#;\",/,MQ/\"\"#<\",O#[$\\C%\"#=\"-C>kwB<\"#>\"-c@G+\"*p\"# ?\".C=U0A$G\"#@\"/O0GaGY6\"#A\"/k@ArXNY\"#B\"0/g&R(>J(=\"#C\"0sAL!)>Rc (\"#D\"1)[qYtyD0$\"#E\"2c'pVLZEJ7\"#F\"2Ot9R\"=!R'\\\"#G-%\"OG6#\"\"\" \"#H" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "The average path length i s simply the ratio of these coefficients:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 41 "seq(coeff(Spl,z,i)/coeff(S,z,i),i=0..28);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6?\"\"!F#\"\"\"#\"#9\"\"&#\"#P\"\"(#\"$w\"\"#@ #\"$(R\"#L#\"%Yp\"$H%#\"&$*[\"\"$:(#\"&/I'\"%JC#\"'v>8\"%*>%#\"(!z(4\" \"&$RH#\"($>qA\"&.?&#\"(;#[$*\"'Dd=#\")<$z\">\"'0VL#\"*a*yxi\"(X[p*#\" +L2A\"G\"\")N)yw\"#\"+_+<>_\")&RA['#\",fqL71\"\"*v'4%>\"#\",i4K)=')\"* &fJO))#\"-R02vZ<\"+0,.T;#\"-caNc^!3(\"+bnc;h#\".n]ycGV\"\",b/KN9\"#\"/# 36cGxJ#\"-Do!)H:<#\"/,!*[$*z#o%\"-Jo.wCK#\"0oI3&*z4*=\".j.g'[:7#\"06Q$ =Ms:Q\".>S8>fH##\"19CfL=;yI\"/,!HxQ$Q<#\"1 " 0 "" {MPLTEXT 1 0 11 "evalf([\"]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7?\"\"!F$$\"\"\"F$$\"+++++G!\"*$\"+'G9dG&F)$\"+\"Q _4Q)F)$\"+....7!\")$\"+>U6>;F0$\"+jq$H3#F0$\"+i1p\"f#F0$\"+T-,VJF0$\"+ Y)o[t$F0$\"+8R]lVF0$\"+$QkL.&F0$\"+O:2PdF0$\"+#G*QvkF0$\"+[6?ZsF0$\"+h @\\^!)F0$\"+YVL()))F0$\"+p\\(Qv*F0$\"+*oK]1\"!\"($\"+cgfd6FM$\"+N(4ID \"FM$\"+%Q57N\"FM$\"+1(Q@X\"FM$\"+N!Rdb\"FM$\"+y*e>m\"FM$\"+4\"\\2x\"F M$\"+(pi?)=FM" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 167 "Empirically, th ese numbers grow slightly faster than linearly with the size of the tr ee. A closed-form for the average path length can be established rigor ously using " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 112 ". The ge nerating functions being algebraic, they satisfy linear differential e quations. These can be derived by " }{HYPERLNK 17 "gfun[holedxprtodiffe q]" 2 "gfun[holexprtodiffeq]" "" }{TEXT -1 109 ". From these different ial equations, a linear recurrence satisfied by the Taylor coefficient s is obtained by " }{HYPERLNK 17 "gfun[diffeqtorec]" 2 "gfun[diffeqtor ec]" "" }{TEXT -1 74 ". It turns out that these recurrences fall into \+ a class for which Maple's " }{HYPERLNK 17 "rsolve" 2 "rsolve" "" } {TEXT -1 60 " can find a solution.\n Here are the differential equati ons:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "deqF:=holexprtodiff eq(F,y(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%deqFG,(\"\"\"F&*&,&! \"\"F&%\"zG\"\"#F&-%\"yG6#F*F&F&*&,&F*F)*$F*F+\"\"%F&-%%diffG6$F,F*F&F &" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "deqFpl:=holexprtodiffe q(Fpl,y(z));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'deqFplG<&/---%#@@G6 $%\"DG\"\"#6#%\"yG6#\"\"!\"\"%/-F/F0F1/--F,F.F0F1,(*$%\"zGF-\"\"'*&,** $F:\"\"$\"\")F9!#EF:\"#5!\"\"\"\"\"FD-F/6#F:FDFD*&,**$F:F2\"#;F>!#CF9 \"\"*F:FCFD-%%diffG6$FEF:FDFD" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 " From there, the reecurrences follow:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "recF:=diffeqtorec(deqF,y(z),u(n));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%recFG<$/-%\"uG6#\"\"!\"\"\",&*&,&\"\"#F+%\"nG\"\"% F+-F(6#F0F+F+*&,&!\"#F+F0!\"\"F+-F(6#,&F0F+F+F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "recFpl:=diffeqtorec(deqFpl,y(z),u(n));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'recFplG<&,**&,&\"\")\"\"\"%\"nG\"# ;F*-%\"uG6#F+F*F**&,&!#]F*F+!#CF*-F.6#,&F+F*F*F*F*F**&,&\"#GF*F+\"\"*F *-F.6#,&F+F*\"\"#F*F*F**&,&!\"%F*F+!\"\"F*-F.6#,&F+F*\"\"$F*F*F*/-F.6# F>F>/-F.6#\"\"!FM/-F.6#F*FM" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "T he recurrence satisfied by the Catalan numbers being linear of order 1 , it is obvious that Maple will be able to solve it:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Cat:=rsolve(recF,u(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$CatG**)\"\"%%\"nG\"\"\"-%&GAMMAG6#,&F(F)#F)\"\"#F )F)-F+6#,&F(F)F/F)!\"\"%#PiG#F3F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 261 "It is however more surprising that rsolve canf compute the solutio n of the 3rd order recurrence satisfied by the cumulated path lengths. This is due to the implementation of the recent algorithm by M. Petko vsek for hypergeometric solutions of linear recurrences." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Pl:=rsolve(recFpl,u(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#PlG,&*0)\"\"%%\"nG\"\"\",&F)F*#F*\"\"$F*F *)#F*\"\"#F)F*)F0F)F*-%&GAMMAG6#,&F)F*F/F*F*-F36#,&F)F*F0F*!\"\"%#PiG# F9F0!\"$F'F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "After some cleani ng up, we have thus proved the following." }}}{EXCHG {PARA 259 "" 0 " " {TEXT 264 11 "Proposition" }{TEXT -1 2 ". " }{TEXT 261 83 "The avera ge internal path length in a random planar unlabelled binary tree of s ize " }{XPPEDIT 262 0 "n" "I\"nG6\"" }{TEXT 263 29 " under the uniform model is \n" }{XPPEDIT 18 0 "4^n*(n+1)/binomial(2*n,n)-3*n-1" ",(*() \"\"%%\"nG\"\"\",&F&F'\"\"\"F'F'-%)binomialG6$*&\"\"#F'F&F'F&!\"\"F'*& \"\"$F'F&F'F/\"\"\"F/" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 g45 "seq(4^i*(i+1)/binomial(2*i,i)-3*i-1,i=1..10);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6,\"\"!\"\"\"#\"#9\"\"&#\"#P\"\"(#\"$w\" \"#@#\"$(R\"#L#\"%Yp\"$H%#\"&$*[\"\"$:(#\"&/I'\"%JC#\"'v>8\"%*>%" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "From there, the asymptotic behavio ur is well within the reach of Maple's asympt command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "map(combine,asympt(Pl/Cat,n),exp); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*&%#PiG#\"\"\"\"\"#*$%\"nG!\"\"# !\"$F(F'F*F-*&F%F&F)#F+F(#\"\"*\"\")F+F'*&F%F&F)F&#\"#<\"$G\"*&F%F&F)# \"\"$F(#F9\"%C5-%\"OG6#*$F)#\"\"&F(F'" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 40 "Average distance from a point to a cycle" }}{PARA 0 "" 0 "" {TEXT -1 46 "We now come back to the analysis of Pollard's " } {XPPEDIT 18 0 "rho" "I$rhoG6\"" }{TEXT -1 209 "-algorithm. The analysi s of the average distance from a node to its cycle proceeds exactly as in the case of binary trees by decorating the corresponding trees. Th e grammar is derived from the grammar G above:" }}{EXCHGh {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "G;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<%/%)fungra phG-%$SetG6#%4connected_componentG/F)-%&CycleG6#-%%ProdG6$%\"ZG%(bintr eeG/F2-%&UnionG6$F1-F/6$F1-F'6$F2/%%cardG\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 176 "The process of decoration consists in isolating the path between ancestor and descendant in the combinatorial structure. \+ For instance, non-planar binary trees are described by " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "npbin:=\{bintree=subs(G,bintree)\}; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&npbinG<#/%(bintreeG-%&UnionG6$% \"ZG-%%ProdG6$F+-%$SetG6$F'/%%cardG\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 335 "Here, Set indicates that the respective position of the \+ two subtrees does not count. In the labelled case, we can therefore de cide that the decorated branch will always be the leftmost one (instea d of considering all the paths leftright1 and leftright2 as in the cas e of planar binary trees). Thus the decorated grammar in this case is " }}}{EXCHG {PiARA 0 "> " 0 "" {MPLTEXT 1 0 277 "bin3:=\{bintree2=Prod( Z,Union(Prod(bintree2,bintree),\n Prod(An, bintree1,bintree))),\n bintree1=\n Prod(Z,Union(Prod(b intree1,bintree),\n Prod(De,Set(bintree,card=2) ),De)),\n An=Epsilon,De=Epsilon\} union npbin:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 149 "Again, we can check that the first values coin cide with the result produced by computing the internal path length on all the non-planar binary trees:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "ipl:=proc(t) if type(t,name) then 0 else size(t)-1+i pl(op([2,1],t))+ipl(op([2,2],t)) fi end:\nsize:=proc(t) local i;eval(s ubs([Prod=`+`,Set=`+`,seq(i=1,i=indets(t,name))],t)) end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "for i to 6 do i,`+`(op(map(ipl,alls tructs([bintree,npbin,labelled],size=i)))) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"#\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"$\"\"'" }j}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"&\"$g $" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"'\"\"!" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 52 "seq(count([bintree2,bin3,labelled],size=i),i=1 ..11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-\"\"!F#\"\"'F#\"$g$F#\"&Sl$F #\"(+o*eF#\"++w2-9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 159 "The same p rocess readily extends to the functional graphs involved in Pollard's \+ algorithm by a decomposition of sets and cycles which isolates the mar ked part." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 296 "G1:=\{fungrap h=Prod(connected_component1,\n Set(connected_compone nt)),\n connected_component1=Prod(Z,\n Union(Prod(An,bintr ee1),bintree2),\n Sequence(Prod(Z,bintree))),\n conn ected_component=Cycle(Prod(Z,bintree))\}\nunion bin3: \+ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 229 "We first write pr ocedures to compute the distance from nodes to their cycles in the non -kdecorated graphs so that we can check on the first few values that ou r grammar for decorated graphs is consistent with the non-decorated on e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 351 "iplfg:=proc(g) `+`(o p(map(iplcc,g))) end:\niplcc:=proc(cc) `+`(op(map(iplbt,cc))) end:\nip lbt:=proc(bt) if type(bt,name) then 0 else size(op(2,bt))+iplt(op(2,bt )) fi end:\niplt:=proc(t) if type(t,name) then 0 else size(t)-1+iplt(o p([2,1],t))+iplt(op([2,2],t)) fi end:\nsize:=proc(t) local i; eval(sub s([Set=`+`,Prod=`+`,seq(i=1,i=indets(t,name))],t)) end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Here are a few tests:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "to 4 do t:=draw([fungraph,G,labelled],size= 6); print(t,iplfg(t)) od:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%$SetG6$ -%&CycleG6#-%%ProdG6$&%\"ZG6#\"\"\"&F-6#\"\"&-F'6#-F*6$&F-6#\"\"$-F*6$ &F-6#\"\"%-F$6$&F-6#\"\"#&F-6#\"\"'FF" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%$SetG6#-%&CycleG6$-%%ProdG6$&%\"ZG6#\"\"%&F-6#\"\"\"-F*6$&F-6#\" \"'-F*6$&F-6#\"\"&-F$6$&F-6#\"\l"#&F-6#\"\"$F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%$SetG6$-%&CycleG6#-%%ProdG6$&%\"ZG6#\"\"&&F-6#\"\"\"- F'6#-F*6$&F-6#\"\"%-F*6$&F-6#\"\"'-F$6$&F-6#\"\"#&F-6#\"\"$F>" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$-%$SetG6#-%&CycleG6#-%%ProdG6$&%\"ZG6# \"\"&-F*6$&F-6#\"\"%-F$6$&F-6#\"\"\"-F*6$&F-6#\"\"'-F$6$&F-6#\"\"#&F-6 #\"\"$\"#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "The number of funct ional graphs of fixed size grows quite fast:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "seq(count([fungraph,G,labelled],size=i),i=1..10) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,\"\"!\"\"#F#\"#OF#\"%+=F#\"'+k " 0 "" {MPLTEXT 1 0 52 "map(iplfg,allstructs([fungra ph,G,labelled],size=2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"\"F$ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "sort(map(iplfg,allstruc ts([fungraph,G,labelled],size=4)));" }}{PARA 11 "" 1 "" {XPPMATH m20 "6 #7F\"\"#F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$\"\"&F%F%F%F%F%F %F%F%F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "convert(\",` +`);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$3\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "On the other hand, here is the counting sequence for decorated graphs:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "seq(c ount([fungraph,G1,labelled],size=i),i=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,\"\"!\"\"#F#\"$3\"F#\"&S/\"F#\"(+!Q;F#\"*+[Q#Q" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "It takes 11 minutes to check that 10440 is also the value we get by generating all the binary functiona l graphs of size 6." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "We now com pute the generating functions with " }{HYPERLNK 17 "combstruct[gfsolve ]" 2 "combstruct[gfsolve]" "" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "F:=subs(gfsolve(G,labelled,z),fungraph(z));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG*$,&*$%\"zG\"\"#!\"#\"\"\"F+#!\" \"F)" }}}{EXCHG {PARA 0 "> " 0 n"" {MPLTEXT 1 0 46 "Fpl:=subs(gfsolve(G 1,labelled,z),fungraph(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$FplG **%\"zG\"\"\",&*&F&!\"\",&\"\"#F'*$,&*$F&F,!\"#F'F'#F'F,F0F'#F*F,F&F,F ',(F'F'F/!\"%*$F&\"\"%F6F*F.F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Again, we obtain closed-forms from these generating functions by appl ying " }{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 110 " to find the di fferential equation they satisfy and from there the recurrence satisfi ed by their coefficients:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "deqF:=holexprtodiffeq(F,y(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%%deqFG<$,&*&%\"zG\"\"\"-%\"yG6#F(F)\"\"#*&,&!\"\"F)*$F(F-F-F)-%%diff G6$F*F(F)F)/-F+6#\"\"!F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "deqFpl:=holexprtodiffeq(Fpl,y(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%'deqFplG<%/-%\"yG6#\"\"!F*/--%\"DG6#F(F)F*,(*&,(*$%\"zG\"\"&\"#C*$F 4\"\"$!#CF4\"\"'\"\"\"-F(6#F4F;F;*&,**$F4\"\"#F:!\"\"F;*$F4F:\"\")*$F4 \"\"%!#7F;-%%diffG6$F " 0 o"" {MPLTEXT 1 0 34 "recF:=diffeqtorec(deqF,y(z),u(n));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%recFG<%/-%\"uG6#\"\"!\"\"\",&*&,&%\"nG\"\"#F0F+F+- F(6#F/F+F+*&,&!\"#F+F/!\"\"F+-F(6#,&F/F+F0F+F+F+/-F(6#F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "recFpl:=diffeqtorec(deqFpl,y(z),u(n ));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'recFplG<)/-%\"uG6#\"\"!F*/-F (6#\"\"\"F*/-F(6#\"\"$F*/-F(6#\"\"&F*/-F(6#\"\"#F./-F(6#\"\"%#\"\"*F:, **&,&\"#CF.%\"nG\"\")F.-F(6#FEF.F.*&,&!#[F.FE!#7F.-F(6#,&FEF.F:F.F.F.* &,&FE\"\"'\"#IF.F.-F(6#,&FEF.F>F.F.F.*&,&FE!\"\"!\"'F.F.-F(6#,&FEF.FRF .F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 119 "Unfortunately, due to a weakness in Maple's current version of rsolve, the solutions to these recurrences are not found" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "rsolve(recF,u(n)),rsolve(recFpl,u(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$-%'rsolveG6$<%/-%\"uG6#\"\"!\"\"\",&*&,&%\"nG\"\"#F1F,F ,-F)6#F0F,F,*&,&!\"#F,F0!\"\"F,-F)6#,&F0F,F1F,F,F,/-F)6#F,F+F2-F$6$<)/ F(F+F;/-F)6#\"\"$F+/-F)6#\"\"&pF+/-F)6#F1F,/-F)6#\"\"%#\"\"*F1,**&,&\"# CF,F0\"\")F,F2F,F,*&,&!#[F,F0!#7F,F8F,F,*&,&F0\"\"'\"#IF,F,-F)6#,&F0F, FPF,F,F,*&,&F0F7!\"'F,F,-F)6#,&F0F,FhnF,F,F,F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "However, Maple can find the solution if we help it \+ by taking into account the parity of the generating functions:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "deqF2:=holexprtodiffeq(subs( z=z^(1/2),F),y(z)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "deqF pl2:=holexprtodiffeq(subs(z=z^(1/2),normal(Fpl)),y(z)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "recF2:=diffeqtorec(deqF2,y(z),u(n)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&recF2G<$/-%\"uG6#\"\"!\"\"\",& *&,&F+F+%\"nG\"\"#F+-F(6#F/F+F+*&,&F/!\"\"F5F+F+-F(6#,&F/F+F+F+F+F+" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "recFpl2:=diffeqtorec(deqFp l2,y(z),u(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(recFpl2G<&/-%\"uG 6#\"\"!F*,**&,&\"#7\"\"\"%\"nG\"\")F/-F(6#F0F/F/*&,&F0!#7!#CF/F/-F(6#, &F0F/F/F/F/F/*&,&\"#:F/F0\"\"'F/-F(6#,&F0F/\"\"#F/F/F/*&,&F0!q\"\"!\"$F /F/-F(6#,&F0F/\"\"$F/F/F//-F(6#F/F//-F(6#FB#\"\"*FB" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "nb_fg:=subs(n=n/2,rsolve(recF2,u(n)));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&nb_fgG**)\"\"#,$%\"nG#\"\"\"F'F+-% &GAMMAG6#,&F*F+F)F*F+-F-6#,&F+F+F)F*!\"\"%#PiG#F3F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "tot_pl:=map(simplify,subs(n=n/2,rsolve(re cFpl2,u(n))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'tot_plG,()\"\"#,$ %\"nG#\"\"\"F'F+*&)F',&F)F*!\"\"F+F+F)F+F+**)F',&F+F+F)F*F+-%&GAMMAG6# ,&#\"\"$F'F+F)F*F+-F46#F2F/%#PiG#F/F'F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "This is summarized by:" }}}{EXCHG {PARA 260 "" 0 "" {TEXT 265 11 "Proposition" }{TEXT -1 2 ": " }{TEXT 266 100 "The averag e distance from a point to a cycle in a random binary non-planar funct ional graph of size " }{XPPEDIT 267 0 "n" "I\"nG6\"" }{TEXT 268 13 " i s given by " }{XPPEDIT 18 0 "sqrt(Pi)*GAMMA(n/2+2)/n/GAMMA(n/2+1/2)-1- 1/n" ",(**-%%sqrtG6#%#PiG\"\"\"-%&GAMMAG6#,&*&%\"nGF(\"\"#!\"\"F(\"\"# F(F(F.F0-F*6#,&*&F.F(\"\"#rF0F(*&\"\"\"F(\"\"#F0F(F0F(\"\"\"F0*&\"\"\"F (F.F0F0" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "From t here a direct asymptotic expansion follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "map(simplify,asympt(subs(tot_pl/nb_fg/n),n));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,2*(\"\"##\"\"\"F%%#PiGF&*$%\"nG!\"\"# F+F%#F'\"\"%F+F'*(F%F&F(F&F)F&#\"\"*\"#;F)F+**F%F&F(F&F*F+F)F&#\"#<\"$ G\"**F%F&F(F&F*!\"#F)F&#\"\"$\"$7&**F%F&F(F&F*!\"$F)F&#!$\"=\"%#>)-%\" OG6#*&F*!\"%F)F&F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "as_co st_1:=op(1,\"):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Thus under our probabilistic model, the first stage of Pollard's " }{XPPEDIT 18 0 "r ho" "I$rhoG6\"" }{TEXT -1 32 "-algorithm takes asymptotically " } {XPPEDIT 18 0 "C*sqrt(p)" "*&%\"CG\"\"\"-%%sqrtG6#%\"pGF$" }{TEXT -1 14 " steps, where " }{XPPEDIT 18 0 "p" "I\"pG6\"" }{TEXT -1 63 " is th e smallest prime factor of the number to be factored and " }{XPPEDIT 18 0 "C=sqrt(2*Pi)/4" "/%\"CG*&-%%sqrtG6#*&\"\"#\"\"\"%#PiGFs*F*\"\"%! \"\"" }{TEXT -1 1 "." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 28 "Average length of the cycles" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "After bot h sequences (" }{XPPEDIT 18 0 "x[i]" "&%\"xG6#%\"iG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y[i]" "&%\"yG6#%\"iG" }{TEXT -1 438 ") have reached the cycle, the number of steps before the end of Pollard's algorithm \+ is bounded by the length of this cycle. Since this length might be cor related to the number of steps before, it is not a priori sufficient t o compute the average length of the cycles in a random graph obeying o ur grammar. Instead, we modify the decorated graphs so that the Ancest or is now an element of the cycle. The number of decorated graphs of s ize " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 96 " is thus the sum for all the nodes of the length of their cycle, summed over all graphs of size " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 26 ". Here is the new \+ grammar:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 247 "G2:=remove(typ e,G1,identictal(connected_component1)=anything)\nunion \{connected_comp onent1=\n Prod(Z,\n Union(bintree1,Prod(De,bintree)),\n \+ Sequence(Prod(Z,bintree)),\n Prod(Z,An,bintree),\n \+ Sequence(Prod(Z,bintree)))\}:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "The corresponding generating function is again algebraic:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Fplcycle:=factor(normal(subs (gfsolve(G2,labelled,z),fungraph(z))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)FplcycleG,$*(%\"zG\"\"#,&!\"\"\"\"\"*$,&*$F'F(!\"#F+F+#F+F(F+ F+,&F*F+F.F(F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "series( Fplcycle,z,11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+-%\"zG\"\"#\"\"%\" \"*\"\"'\"#H\"\")#\"$D$\"\"%\"#5-%\"OG6#\"\"\"\"#7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Again, a closed form follows for the coefficients :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "holexprtodiffeq(subs(z =z^(1/2),Fplcycle)/z,y(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$,(\" \"#\"\"\"*&,(\"\"$F&*$%\"zGF%\"#7F+!#7F&-%u\"yG6#F+F&F&*&,*F+\"\"'!\"\" F&*$F+F)\"\")F*F-F&-%%diffG6$F.F+F&F&/-F/6#\"\"!F=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "diffeqtorec(\",y(z),u(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&/-%\"uG6#\"\"!F(/-F&6#\"\"\"\"\"#/-F&6#F-\"\"*,* *&,&\"#7F,%\"nG\"\")F,-F&6#F6F,F,*&,&F6!#7!#CF,F,-F&6#,&F6F,F,F,F,F,*& ,&\"#:F,F6\"\"'F,-F&6#,&F6F,F-F,F,F,*&,&F6!\"\"!\"$F,F,-F&6#,&F6F,\"\" $F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "rsolve(\",u(n)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,()\"\"#%\"nGF%*,F$F%)#\"\"\"F%F&F *-%&GAMMAG6#,&#\"\"$F%F*F&F*F*-F,6#,&F&F*F*F*!\"\"%#PiG#F4F%!\"%*&F$F* F&F*F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "T1:=map(simplify, subs(n=n/2-1,\")+tot_pl);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T1G,,) \"\"#,$%\"nG#\"\"\"F'F'**)F',&F+F+F)F*F+-%&GAMMAG6#,&F*F+F)F*F+-F06#F( !\"\"%#PiG#F5F'F5*&)F',&F)F*F5F+F+,&F)F+!\"#F+F+F+*&F9F+F)F+F+**F-F+-F 06#,&#\"\"$F'F+F)F*F+-F06#F.F5F6F7F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "We thus get the following conclusion:" }}}{EXCHG {PARA 261 "" v 0 "" {TEXT 272 7 "Theorem" }{TEXT -1 2 ". " }{TEXT 269 41 "The average number of steps of Pollard's " }{XPPEDIT 270 0 "rho" "I$rhoG6\"" } {TEXT 271 59 "-algorithm under the probabilistic model is bounded betw een" }{XPPEDIT 18 0 "sqrt(Pi)*GAMMA(n/2+2)/n/GAMMA(n/2+1/2)-1-1/n" ",( **-%%sqrtG6#%#PiG\"\"\"-%&GAMMAG6#,&*&%\"nGF(\"\"#!\"\"F(\"\"#F(F(F.F0 -F*6#,&*&F.F(\"\"#F0F(*&\"\"\"F(\"\"#F0F(F0F(\"\"\"F0*&\"\"\"F(F.F0F0 " }{TEXT -1 4 " " }{TEXT 273 3 "and" }{TEXT -1 3 " " }{XPPEDIT 18 0 "2*sqrt(Pi)/n*GAMMA(n/2+1)/GAMMA(n/2+1/2)-2-1/n" ",(*,\"\"#\"\"\" -%%sqrtG6#%#PiGF%%\"nG!\"\"-%&GAMMAG6#,&*&F*F%\"\"#F+F%\"\"\"F%F%-F-6# ,&*&F*F%\"\"#F+F%*&\"\"\"F%\"\"#F+F%F+F%\"\"#F+*&\"\"\"F%F*F+F+" } {TEXT -1 2 ",\n" }{TEXT 274 6 "where " }{XPPEDIT 275 0 "n" "I\"nG6\"" }{TEXT 276 137 " is the smallest prime factor of the integer whose fac torization is sought. \nAsymptotically, this number of steps therefore behaves like " }{XPPEDIT 18 0 "C*sqrt(n)" "*&%\"CG\"\"\"-%%sqrtG6#%\" nGF$" }{TEXT -1 2 ", " }{TEXT 277 5w "with " }{XPPEDIT 278 0 "C" "I\"CG 6\"" }{TEXT 279 3 " in" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(2*Pi)/4.. sqrt(2*Pi)/2" ";*&-%%sqrtG6#*&\"\"#\"\"\"%#PiGF)F)\"\"%!\"\"*&-F%6#*& \"\"#F)F*F)F)\"\"#F," }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Here is the verification of the second part:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "map(simplify,asympt(T1/nb_fg/n,n,3));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,,*(\"\"##\"\"\"F%%#PiGF&*$%\"nG!\"\"# F+F%F&!\"#F'*(F%F&F(F&F)F&#\"\"&\"\")F)F+-%\"OG6#*&F*F+F)F&F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "as_cost_2:=op(1,\"):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalf(\"\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*$*$%\"nG!\"\"#F'\"\"#$\"+PTJ`7!\"*$!\"#\"\"!\" \"\"*$F%#F0F)$\"+rEkm:F,F%$F'F/-%\"OG6#*&F&F'F%F2F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Since the smallest prime factor of a number " } {XPPEDIT 18 0 "N" "I\"NG6\"" }{TEXT -1 13 " is of order " }{XPPEDIT 18 0 "O(sqrt(N))" "-%\"OG6#-%%sqrtG6#%\"NG" }{TEXT -1 95 ", it follows thxat the (arithmetic) complexity of Pollard's algorithm under this mo del grows in " }{XPPEDIT 18 0 "O(N^(1/4))" "-%\"OG6#)%\"NG*&\"\"\"\"\" \"\"\"%!\"\"" }{TEXT -1 1 "." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 33 "Extension: polynomials of degree " }{XPPEDIT 18 0 "m" "I\"mG6\"" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "When the factor " } {XPPEDIT 18 0 "p" "I\"pG6\"" }{TEXT -1 42 " to be found is known a pri ori to satisfy " }{XPPEDIT 18 0 "p=1 mod m" "/%\"pG-%$modG6$\"\"\"%\"m G" }{TEXT -1 10 " for some " }{XPPEDIT 18 0 "m" "I\"mG6\"" }{TEXT -1 110 " known in advance, Pollard conjectured that the number of steps i n his algorithm could be reduced by a factor " }{XPPEDIT 18 0 "sqrt(m- 1)" "-%%sqrtG6#,&%\"mG\"\"\"\"\"\"!\"\"" }{TEXT -1 25 " by using the p olynomial " }{XPPEDIT 18 0 "f(x)=x^m+a mod N" "/-%\"fG6#%\"xG-%$modG6$ ,&)F&%\"mG\"\"\"%\"aGF-%\"NG" }{TEXT -1 12 " instead of " }{XPPEDIT 18 0 "f(x)=(x^2+a mod N" "/-%\"fG6#%\"xG-%$modG6$,&*$F&\"\"#\"\"\"%\"a GF-%\"NG" }{TEXT -1 98 ". Brent and Pollayrd used this idea to factor t he 8th Fermat number, whose factors are known to be " }{XPPEDIT 18 0 " 1 mod 2^(8+2)" "-%$modG6$\"\"\")\"\"#,&\"\")\"\"\"\"\"#F*" }{TEXT -1 63 ". We know consider Pollard's under the probabilistic model for " } {XPPEDIT 18 0 "m=6" "/%\"mG\"\"'" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Here is the grammar for functional graphs with " } {XPPEDIT 18 0 "m" "I\"mG6\"" }{TEXT -1 16 " as a parameter:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "Gm:=\{fg=Set(cc),cc=Cycle(Pr od(Z,Set(tree,card=m-1))),\n tree=Union(Z,Prod(Z,Set(tree,card=m))) \}:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Here is the corresponding \+ decorated grammar for the distances to the cycles:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 402 "Gm2:=\{fg1=Prod(cc1,Set(cc)),\n cc1=P rod(Z,Union(Prod(An,tree1),tree2),\n Set(tree,card=m-2), \n Sequence(Prod(Z,Set(tree,card=m-1)))),\n tree1=P rod(Z,Union(Prod(tree1,Set(tree,card=m-1)),\n \+ Pzrod(De,Set(tree,card=m)),De)),\n tree2=Prod(Z,Union(tree2,Prod(A n,tree1)),\n Set(tree,card=m-1)),\n An=Epsilon,De =Epsilon\} union Gm:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "And the g rammar for the distances to the cycles plus the length of the cycle:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 300 "Gm3:=\{fg2=Prod(cc2,Set( cc)),\n cc2=Prod(Z,Union(Prod(tree1,Set(tree,card=m-2)),\n \+ Prod(De,Set(tree,card=m-1))),\n Sequence (Prod(Z,Set(tree,card=m-1))),\n Prod(Z,An,Set(tree,card= m-1)),\n Sequence(Prod(Z,Set(tree,card=m-1))))\}\nunion \+ Gm2:" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m=2" " /%\"mG\"\"#" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "We first check on t he case " }{XPPEDIT 18 0 "m=2" "/%\"mG\"\"#" }{TEXT -1 56 " that we re cover the generating functions we had before." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "sol:=gfsolve(subs(m=2,Gm3),labelled,z);" }} {PARA 12 "" 1 "" {XPPMATH 20{ "6#>%$solG<./-%#AnG6#%\"zG\"\"\"/-%\"ZGF) F*/-%#DeGF)F+/-%#ccGF)-%#lnG6#*$,&*$F*\"\"#!\"#F+F+#!\"\"F;/-%%treeGF) ,$*&F*F>,&F;F+*$F9#F+F;FF9F=/-%&tree2GF),$*&,&FCF=F*F+F+,&F>F+F:F;F>F;/-%$fg2GF)** F*F;FDF+,(FEF+F:F<*&F*F;FDF+F+F>F9F=/-%&tree1GF),$*(F*F>FDF+F9F=FF/-%$ cc2GF)*(F*F;FDF+FenF>/-%$cc1GF)*(F*F+FKF+FLF>/-%#fgGF)F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "normal(subs(sol,fg(z))-F);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "normal(subs(sol,fg1(z))-Fpl);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "nor mal(subs(sol,fg2(z))-Fplcycle);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"!" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m=6" " /%\"mG\"\"'" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "The \+ grammar becomes" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "G6:=subs (m=6,Gm3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#G6G<-/%#fgG-%$Set|G6#% #ccG/%$fg1G-%%ProdG6$%$cc1GF(/%#AnG%(EpsilonG/%#DeGF4/%$fg2G-F/6$%$cc2 GF(/%%treeG-%&UnionG6$%\"ZG-F/6$FA-F)6$F=/%%cardG\"\"'/F1-F/6&FA-F?6$- F/6$F3%&tree1G%&tree2G-F)6$F=/FG\"\"%-%)SequenceG6#-F/6$FA-F)6$F=/FG\" \"&/FP-F/6$FA-F?6%-F/6$FPFen-F/6$F6FDF6/FQ-F/6%FA-F?6$FQFNFen/F;-F/6'F A-F?6$-F/6$FPFR-F/6$F6FenFV-F/6%FAF3FenFV/F+-%&CycleGFX" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "From there we compute the generating func tions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sol:=gfsolve(G6,l abelled,z):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Here is the genera ting function for the functional graphs:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "F:=subs(sol,fg(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"FG,$*$,&!$?\"\"\"\"*&%\"zGF)-%'RootOfG6#,(%#_ZG!$?(F+\"$?(*&F+F) F0\"\"'F)\"\"&F)!\"\"F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 232 "Again , the coefficients admit a closed-form which can be obtained using gfu n (it could also be derived by Lagrange's inversion formula). We rathe r compute directly the a}symptotic expansion of the number of functiona l graphs of size " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 269 " whose nodes have an in-degree of 0 or 6 by singularity analysis. Unfortunat ely, Maple's series command is not yet able to deal with algebraic fun ctions like the RootOf above. We therefore start from the equation in \+ the system which is at the origin of the singularity:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "T:=subs(sol,tree(z));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"TG-%'RootOfG6#,(%#_ZG!$?(%\"zG\"$?(*&F+\"\" \"F)\"\"'F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "tree(z)-subs (gfeqns(G6,labelled,z),tree(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,( -%%treeG6#%\"zG\"\"\"-%\"ZGF&!\"\"*&F)F(F$\"\"'#F+\"$?(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eq:=subs(tree(z)=y,Z(z)=z,\");" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG,(%\"yG\"\"\"%\"zG!\"\"*&F(F'F& \"\"'#F)\"$?(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "diff(eq,y) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$*&%\"zGF$%\"~yG\"\"&#!\" \"\"$?\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve(\{\"\",\" \},\{y,z\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<$/%\"zG,$-%'RootOfG6# ,&*$%#_ZG\"\"$\"\"\"!#7F.#\"\"&\"\"'/%\"yGF'<$/F4-F(6#,&F+F.\"#7F./F%, $F7F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "It follows that the domi nant singularity is at " }{XPPEDIT 18 0 "z=5/6*(12)^(1/3)" "/%\"zG*(\" \"&\"\"\"\"\"'!\"\")\"#7*&\"\"\"F&\"\"$F(F&" }{TEXT -1 8 ", where " } {XPPEDIT 18 0 "y=12^(1/3)" "/%\"yG)\"#7*&\"\"\"\"\"\"\"\"$!\"\"" } {TEXT -1 29 ". The singular expansion of " }{XPPEDIT 18 0 "y(z)" "-% \"yG6#%\"zG" }{TEXT -1 65 " can be computed at that point by power ser ies reversion (we set " }{XPPEDIT 18 0 "theta=sqrt(1-z/rho)" "/%&theta G-%%sqrtG6#,&\"\"\"\"\"\"*&%\"zGF)%$rhoG!\"\"F-" }{TEXT -1 7 " where \+ " }{XPPEDIT 18 0 "rho" "I$rhoG6\"" }{TEXT -1 49 " is the singularity, \+ which helps series a little)" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "sing:=5/6*12^(1/3):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 47 "series(subs(z=sing*(1-theta^2),eq),y=12^(1/3));" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#+1,&%\"yG\"\"\"*$\"#7#F&\"\"$!\"\",&F' F&*&F(F),&F&F&*$%&thetaG\"\"#F+F&F+\"\"!F/\"\"\",$*&F(#F1F*F.F&#!\"&\" #C\"\"#,$F-#F8\"#=\"\"$,&F7F&F/#\"\"&F9\"\"%,$F5#F+\"$W\"\"\"&-%\"OG6# F&\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "\{solve(\",y)\}; " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<$+/%&thetaG*$\"#7#\"\"\"\"\"$\"\" !,$*&\"#5#F)\"\"#F'F(#!\"\"\"\"&\"\"\",$F&#F2\"#:\"\"#,$F-#!#r\"$+*\" \"$,$F&#\"\"(\"%vL\"\"%-%\"OG6#F)\"\"&+/F%F&F+,$F-#F)F3\"\"\"F5\"\"#,$ F-#\"#rF<\"\"$F>\"\"%FC\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "sery:=op(select(proc(s,u) evalb(signum(coeff(s,u,1))=-1) end,\", theta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%seryG+/%&thetaG*$\"#7# \"\"\"\"\"$\"\"!,$*&\"#5#F*\"\"#F(F)#!\"\"\"\"&\"\"\",$F'#F3\"#:\"\"#, $F.#!#r\"$+*\"\"$,$F'#\"\"(\"%vL\"\"%-%\"OG6#F*\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "series(subs(T=sery,z=sing*(1-theta^2),F), theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+-%&thetaG,$*$\"#5#\"\"\"\" \"##F)F'!\"\"#\"\"%\"#:\"\"!,$F&#\"#Z\"$+'\"\"\"#\"$;'\"%vL\"\"#-%\"OG 6#F)\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "Hence the number of \+ functional graphs with in-degree in \{0,6\} is asymptotically:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "as_fg:=coeff(\",theta,-1)/sq rt(Pi)*n^(-1/2)*sing^(-n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&as_fg G,$**\"#5#\"\"\"\"\"#%#PiG#!\"\"F*%\"nGF,),$*$\"#7#F)\"\"$#\"\"&\"\"', $F.F-F)#F)F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "We next consider \+ the distance to the cycle" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Fpl:=subs(sol,fg1(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$FplG, $*,%\"zG\"\"\"-%'RootOfG6#,(%#_ZG!$?(F'\"$?(*&F'F(F-\"\"'F(\"\"%,&F)! \"&F'F1F(,,!#IF(*&F'F(F)\"\"&!#E*&F'\"\"#F)F2\"#***&F'\"\"$F)F>!$E\"*& F'F2F)F;\"#a!\"\",&!$?\"F(F7F(FB\"$]\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "series(subs(T=sery,z=sing*(1-theta^2),Fpl),theta);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#+'%&thetaG#\"\"\"\"#?!\"%-%\"OG6#F&! \"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "And the length of the cycl e:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Fpl2:=subs(sol,fg2(z) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Fpl2G,$*,%\"zG\"\"#-%'RootOfG 6#,(%#_ZG!$?(F'\"$?(*&F'\"\"\"F-\"\"'F1\"\"$,&F)!\"\"F'F1F1,*!#SF1*&F' F1F)\"\"&\"\"(*&F'F(F)\"\"%!#=*&F'F3F)F3\"#7F5,&!$?\"F1F8F1F5!%S9" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "series(subs(T=sery,z=sing*(1 -theta^2),Fpl2),theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+)%&thetaG# \"\"\"\"#?!\"%,$*$\"#5#F&\"\"##!#6\"$+$!\"$-%\"OG6#F&!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "Thus the distance to the cycle and the le ngth of this cycle are both asymptotic to:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 48 "asympt(coeff(\",theta,-4)*n*sing^(-n)/as_fg/n,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"#5#\"\"\"\"\"#%#PiGF&*$%\"nG! \"\"#F,F(#F'\"#?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 152 "Thus the rat io between this variant of Pollard's algorithm and the original one is in both cases (number of steps to the cycle and length of the cycle): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "ratio:=simplify(map(asy mpt,[as_cost_1/\",as_cost_2/\"/2],n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ratioG7$*$\"\"&#\"\"\"\"\"#F&" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Conclusion" }}{PARA 0 "" 0 "" {TEXT -1 374 "This workshee t shows that some algorithms whose complexity analysis does not reduce to counting the number of subcomponents in some kind of recursive com binatorial structure can sometimes be treated by combstruct using seve ral marks. This is in particular true of many recursive searching algo rithms whose complexity is related to the path length of an underlying structure." }}}}{MARK "1 0 0" 29 }{VIEWOPTS 1 1 0 1 1 1803 } \",theta,-4)*n*sing^(-n)/as_fg/n,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"#5#\"\"\"\"\"#%#PiGF&*$%\"nG! \"\"#F,F(#F'\"#?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 152 "Thus the rat io between this variant of Pollard's algorithm and the original one is in both cases (number of steps to t"=gfunpoltorecdeterminrecurrencsatisfipolynomialholonomicsequenccalllistreclistunknownparameterpossibshiftvariablcontaineacheithlinearequatsettogethwithinitialconditsameordernamegenericdescriptsolutionoutputverifiexamplreccassiniidentitfibalsoapex6apliappar8)appealappearW' 0#* +348pHlonaM rappendo_appendix6applK*46londyo )appli3  +lPJr) applicabl4applicat7 46l apply( +4pH;applyop 4applyopr+#!*0Rvvapproach48pHlr) appropriat +eapproxXO approximant  4 approximat' 8pHl;r approximate @(l approxmeanlapril  ar 6 arbitarlarbitraarbitrar'@l;r|shtmxdocstylsetreadoptionmadtextformatcallsequencoptvaludescriptfunctallowconversfirstreturnlistknownsecondrespthirdforminvokatequationnumberleftrightnonenumberequationbodyheadbooleantrueargumconsiderpartmaindocumbutbodydesignaddtopbannbodyfootbottomdocumentheadstringadditionalheaddocumenttitltitlcsstabltablassociatnamecssrepresentatplotwidthwidthpicturhtmlunitvoidprovidnosizespecifiplotnumberplotdisplayrelativposintinfinitchaptersectionhavenestlesshyperlinkothertoctitlcontenttocindwithindenttocmaxindmaximalexpresspercenclosblocktocnumbertoccssbasenamclasusefulhavingseveraldifferlookalso skewalgebra1skewparamalgebra1skewpolynomial 1 skippsl slic  slicingslight6o;sloa 6sloan 6larslop slowsm M smallO@(68XOldya;;rsmallest46g;smomosnSx !0#(*+*F0Ron }1M vsociet softwar pHsol%o; ) solb8solid XOsolutoA Z!@(+48pH}v;dt^r)SsolutionW' Z!$@(+468"=}; )Oreleas  relevant lreliXO;remain+ @(6lYremaind+e= remainder*IremarkpHc remarkabl8rematchrememb3+ + 468pHla oremember0# +areminiscrremovK00#@(+6IZkl~o; renamȁrencontr  rendXOrenderXOX0#functgroebngbasicomputreducbasipretendaddlistknownonescallsequencparameterpossibskewpolynomialtermordertablshortdescriptwithrespectresultcalculatrememberduringsessanothdoesrecomputmarkmakepossibluserletmaplknowoutsiddifferelementmusteithoperatororealgebrauseddefincommutatcasereturnleftidealrespmodulgeneratsidedmanytypeorderingcandetailparticulareliminatoftenleadcomputattimeordinarplexopposotherfunctionpackagalgebraicnumberdenotrootofmayappearinputincludcomplexsuchfunctionalitthoughviewlimitsyntacticalfacilitoutputcombinatthespartformonlyafterperformcommandalwayaccesslongexamplsimplwltdegherevivianicurvintersectspherparaboloidprojectontovarplangeronolemniscatparabolalexdegmoreinvolvnonzerocharacteristicpolyrationaltermordcoefficientsqrtgbusprovshiftsnskpolynommapfactorsubsremovassumfollowgxwehaverathnoinsteadnopsalsodiscretpH discus  discuss6disguispHdisjointldispla#[Z,9XOcodisplay Z,kon|sdy displaynumber 9 disregardadiss6dissect 6dissg6 dissociat6disssy6distancl;  distiguishl distiguishabldistinct aM distinguishladistinguishabl r distr8 distribut; !@(48ao rdistributional8'`{VERSION 5 0 "DEC ALPHA UNIX" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 } {CSTYLE "Help Normal" -1 30 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 35 "" 0 1 104 64 92 1 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Norm al" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 20 "CommonLib[FileClose]" }{TEXT 30 15 " - 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This worksheet details the computa tions involved in Section 3 of our article " }{TEXT 258 54 "Symbolic A symptotics: Multiseries of Inverse Functions" }{TEXT -1 185 ", availab le as INRIA Research Report #3264, 1997. The starting point for all th ese asymptotic expansions is the bivariate generating function of the \+ Stirling numbers of the second kind:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "F:=exp(u*(exp(x)-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG-%$expG6#*&%\"uG\"\"\",&-F&6#%\"xGF*!\"\"F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "series(F,x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+1%\"xG\"\"\"\"\"!%\"uG\"\"\",&F'#F%\"\"#*$F'F+F*\"\"#, (F'#F%\"\"'F,F**$F'\"\"$F/\"\"$,*F'#F%\"#CF,#\"\"(F6F1#F%\"\"%*$F'F:F5 \"\"%,,F'#F%\"$?\"F,#F%\"\")F1#\"\"&F6F;#F%\"#7*$F'FCF>\"\"&-%\"OG6#F% \"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "The coefficient of " } {XPPEDIT 18 0 "u^k*x^n" "*&)%\"uG%\"kG\"\"\")%\"xG%\"nGF&" }{TEXT -1 58 " in this series is the Stirling number of the second kind " } {XPPEDIT 18 0 "S[n,k]" "&%\"SG6$%\"nG%\"kG" }{TEXT -1 12 " divided by \+ " }{XPPEDIT 18 0 "n!" "-%*factorialG6#%\"nG" }{TEXT -1 65 ". These num bers count the number of partitions of the set \{1,...," }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 5 "\} in " }{XPPEDIT 18 0 "k" "I\"kG6\"" } {TEXT -1 7 " parts." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "For funct ions with a fast growth at infinity, like many functions related to " }{XPPEDIT 18 0 "F" "I\"FG6\"" }{TEXT -1 719 ", the asymptotic behaviou r of the Taylor coefficients can be computed by the saddle-point metho d. This method relies on using an integral along a closed contour in t he complex plane chosen in such a way as to pass through a special poi nt called the saddle-point. The integral is concentrated in the neighb orhood of the saddle-point and good asymptotic estimates are obtained \+ from a local expansion there. However, in these applications, the sadd le-point is defined implicitly by an equation and only an asymptotic e xpansion of it is available. This introduces technical difficulties in the manipulation of the expansions. This worksheet explores how our a lgorithm for asymptotic inversion applies in such circumstances." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "We first load our experimental imp lementation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "read `/expor t/salvy/Alcom/97/Lib/asympt_inv/load.mpl`;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 314 "Sufficient conditions for deriving an asymptotic expansi on by the saddle-point method have been given by Harris and Schoenfeld . These conditions are satisfied by all the generating functions we co nsider in this worksheet. The following procedure (which can be skippe d in a first reading) takes as input a function " }{XPPEDIT 18 0 "f(x) " "-%\"fG6#%\"xG" }{TEXT -1 117 ", the associated saddle-point equatio n, an asymptotic scale (see below) and computes the asymptotic expansi on of the " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 25 "th Taylor coef ficient of " }{XPPEDIT 18 0 "f(x)" "-%\"fG6#%\"xG" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 48 " tends to infinity in term s of the saddle-point." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 507 " harris_schoenfeld:=proc(f,x,R,nofR,nbterms,scale)local A, B, Bb, Cc, F , i, m, u, compo;A:=diff(log(f),x);for i to 2*nbterms do B[i]:=normal( x^i/i!*diff(A,[x$(i-1)])) od;Bb:=x*diff(B[1],x)/2;for i to 2*nbterms-2 do Cc[i]:=normal(-(B[i+2]+(-1)^i/(i+2)*B[1])/Bb) od;F[0]:=1;for i to \+ nbterms-1 do F[i]:=(-1)^i/sqrt(Pi)*add(GAMMA(m+i+1/2)/m!*add(mul(Cc[u] ,u=compo),compo=combinat[composition](2*i,m)),m=1..2*i)od;tower(subs([ x=R,n=nofR],f/2/sqrt(Pi)/sqrt(Bb)/x^n*add(normal(F[i]/Bb^i),i=0..nbter ms-1)),scale)end:" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 31 "Asymptotics of the Bell numbers" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "The Bell n umbers " }{XPPEDIT 18 0 "B[n]" "&%\"BG6#%\"nG" }{TEXT -1 18 " are the \+ sum over " }{TEXT 270 1 "k" }{TEXT -1 25 " of the Stirling numbers " } {XPPEDIT 18 0 "S[n,k]" "&%\"SG6$%\"nG%\"kG" }{TEXT -1 30 ". Their gene rating function is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "f:=su bs(u=1,F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG-%$expG6#,&-F&6#% \"xG\"\"\"!\"\"F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 "Alternative ly, since these numbers count the number of partitions of a set, this \+ generating function and the numbers can be obtained using " } {HYPERLNK 17 "combstruct" 2 "combstruct" "" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "G:=\{P=Set(Set(Atom,card>0))\}:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "combstruct[gfsolve](G,label led,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#/-%\"PG6#%\"xG-%$expG6#,& -F*F'\"\"\"!\"\"F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Here are th e first numbers:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "series( f,x,20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+M%\"xG\"\"\"\"\"!F%\"\"\" F%\"\"##\"\"&\"\"'\"\"$#F*\"\")\"\"%#\"#8\"#I\"\"&#\"$.#\"$?(\"\"'#\"$ x)\"%S]\"\"(#\"#B\"$C#\"\")#\"%25\"&!G<\"\"*#\"%RY\"'_^9\"#5#\"&>E#\"( g0L\"\"#6#\"((f8U\"*+;+z%\"#7#\")PWkF\"++3-Fi\"#8#\")h'\\a*\",+c9*eV\" #9#\"*4-%\"OG6#F% \"#?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "or, using " }{HYPERLNK 17 "combstruct" 2 "combstruct" "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "seq(combstruct[count]([P,G,labelled],size=i),i=0..19) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "66\"\"\"F#\"\"#\"\"&\"#:\"#_\"$.#\" $x)\"%ST\"&Z6#\"'vf6\"'q&y'\"((f8U\")PWkF\"*A$**3>\"+X&eHQ\"\",Z@9![5 \",/)p['G)\"-fh!o2#o\".d]?UF$e" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "the factor 1/" }{XPPEDIT 18 0 "n!" "-%*factorialG6#%\"nG" }{TEXT -1 25 " being due to the use of " }{TEXT 269 11 "exponential" }{TEXT -1 22 " generating functions." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "The saddle-point equation we use is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "speq:=subs(x=R,diff(log(f),x)*x)-1=n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%speqG/,&*&-%$expG6#%\"RG\"\"\"F+F,F,!\"\"F,%\"n G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "Then the asymptotic analysi s of the Bell numbers is classically based on the asymptotic behaviour of the saddle-point " }{XPPEDIT 18 0 "R" "I\"RG6\"" }{TEXT -1 25 " vi ewed as a function of " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 19 " tends to infinity." }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 178 "From the above equation our algo rithm produces the asymptotic expansion of the saddle-point in two sta ges: first a sequence of ``exact'' information is computed by the proc edure " }{TEXT 272 6 "Invert" }{TEXT -1 30 " which also creates the sc ale " }{TEXT 271 6 "scaleR" }{TEXT -1 46 " necessary for the expansion s of functions of " }{TEXT 259 1 "R" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "r1:=Invert(lhs(speq),R,'scaleR');" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r1G6&7%,*-%#lnG6#%\"RG\"\"\"F+F,-F) 6#,&F,F,*&-%$expGF*!\"\"F+F3F3F,*(,&*$F+F3F,*$F(F3F,F,F(F,F+F,F3F,F4F2 7%-F)6#*&,&*$F1F3F,F6F,F,F+F,\"\"#F+7%\"\"!F>F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "The last argument has been assigned the correspondin g asymptotic scale:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "scal eR[list];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%*$-%#lnG6#%\"RG!\"\"*$F (F)*$-%$expGF'F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "In a second s tage, asymptotic information is extracted from the exact result " } {TEXT 273 2 "r1" }{TEXT -1 60 " and the scale necessary for the expans ions of functions of " }{TEXT 260 1 "n" }{TEXT -1 11 " is created" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "exp1:=multiseriesinverse([r1 ],scaleR,0,2,n,3,'scalen');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exp1 G,*-%#lnG6#%\"nG\"\"\"-F'6#F&!\"\"*&F+F*F&F-F*-%\"OG6#*$F&!\"#F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Here again the variable " }{TEXT 274 6 "scalen" }{TEXT -1 52 " holds the asymptotic scale related to th e variable " }{TEXT 275 1 "n" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "scalen[list];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #7%*$-%#lnG6#-F&6#-F&6#%\"nG!\"\"*$F(F-*$F*F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 276 17 "harris_schoenfeld" } {TEXT -1 102 " written above is then able to compute the expansion of \+ the Bell numbers in terms of the saddle-point " }{TEXT 261 1 "R" } {TEXT -1 14 " in the scale " }{TEXT 262 6 "scaleR" }{TEXT -1 30 " whic h has been obtained above" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "d1:=harris_schoenfeld(f,x,R,lhs(speq),3,scaleR);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#d1G,$*4-%$expG6#-F(6#%\"RG\"\"\"-F(6#!\"\"F-%#PiG #F0\"\"#F3#F-F3*&F,F-,&F*F-*&F*F-F,F-F-F-F2-F(6#*(F,F--%#lnGF+F-F*F-F0 F,F--F(6#,(*&,&F7F-F0F-F-F;F-F-F:F0F;F-F0,(F-F-**,,*$F,\"\"$!\"$*$F,\" \"%F3F3F-F,!#=*$F,F3!#?F-,&F-F-F,F-FGF,F0F*F0#F0\"#C**,4F,!#s*$F,\"\"' !$&p*$F,\"\"&!$'pFH\"%#4\"FK\"%s>FE\"%;H*$F,\"\"(!$c\"*$F,\"\")FIFIF-F -FM!\"'F,!\"#F*F\\o#F-\"%_6F-F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "The result follows upon substitution of " }{TEXT 263 1 "R" }{TEXT -1 41 " by its asymptotic expansion in terms of " }{TEXT 264 1 "n" } {TEXT -1 2 " (" }{TEXT 265 4 "exp1" }{TEXT -1 88 " above). Since the l ogarithm of this expansion is easier to handle, we first compute it:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "d2:=`tower/ln`(d1,scaleR) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "multiseries(d2,scaleR, 0,3,nops(scaleR[list]),3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&,&\" \"\"F&*&-%#lnG6#%\"RGF&F+F&!\"\"F&-%$expGF*F&F&F+#F,\"\"#-F)6#,$*(-F.6 #F,F&F0#F&F0%#PiGF/F7F&*$F+F,F/-%\"OG6#*$F+!\"#F&*&,*#F,\"#7F&F9#\"\"$ \"\")F=#F,\"#C-F;6#*$F+!\"$F&F&F-F,F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Hence the result by substituting the expansion of " } {XPPEDIT 18 0 "R" "I\"RG6\"" }{TEXT -1 13 " in terms of " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 24 " in this last expansion:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "substitute(d2,scaleR,exp1,scalen,3) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,.*&,(-%#lnG6#-F'6#%\"nG!\"\"*(,& *$F&F,\"\"\"F0F0F0F&F0F)F,F0-%\"OG6#*$F)!\"#F0F0F+F0F0F)#F,\"\"#*&,&*& -F'6#,$*(-%$expG6#F,F0F7#F0F7%#PiGF6FBF0F&F,F7F0F0F0F&F0FBF-F6F1F0*&,( F)#F,\"#7*&,&F/\"\"*F7F0F0F&F0#F0\"#C-F26#*$F)F,F0F0F+F,F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "The first part of this expansion is the c lassical expansion of " }{XPPEDIT 18 0 "ln(B[n]*`/`*n!)" "-%#lnG6#*(&% \"BG6#%\"nG\"\"\"%\"/GF*-%*factorialG6#F)F*" }{TEXT -1 192 " which can be found in (de Bruijn 1981, p. 108). The remaining parts reflect the use of multiseries, a recent tool which makes it possible to deal wit h indefinite cancellation in a simple way." }}}}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 38 "Average number of parts in a partition" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 209 "The technique demonstrated above also applies \+ to the average number of parts in a partition, where an indefinite can cellation occurs and this exemplifies the use of multiple scales. The \+ generating function of " }{XPPEDIT 18 0 "Sum(k*S[n,k],k)" "-%$SumG6$*& %\"kG\"\"\"&%\"SG6$%\"nGF&F'F&" }{TEXT -1 4 " is" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 23 "g:=subs(u=1,diff(F,u));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG*&,&-%$expG6#%\"xG\"\"\"!\"\"F+F+-F(6#F&F+" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "The average number of parts in a p artition of a set of size " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 8 " is the " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 25 "th Taylor coeff icient of " }{XPPEDIT 18 0 "g" "I\"gG6\"" }{TEXT -1 16 " divided by th e " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 18 "th coefficient of " } {XPPEDIT 18 0 "f" "I\"fG6\"" }{TEXT -1 72 " whose asymptotics we have \+ just computed. The saddle-point equation for " }{XPPEDIT 18 0 "g" "I\" gG6\"" }{TEXT -1 4 " is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "speq2:=normal(subs(x=R,diff(log(g),x)*x)-1)=n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&speq2G/*&,(*&-%$expG6#%\"RG\"\"#F,\"\"\"F.F)!\"\"F.F .F.,&F)F.F/F.F/%\"nG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "From thi s equation our algorithm produces the asymptotic expansion of the sadd le-point in two stages as before:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "r2:=Invert(op(1,speq2),R):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "exp2:=multiseriesinverse([r2],scaleR,0,2,n,3,scale n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exp2G,*-%#lnG6#%\"nG\"\"\"-F '6#F&!\"\"*&F+F*F&F-F*-%\"OG6#*$F&!\"#F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 294 "This expansion of this new saddle-point is the same as t hat of the previous case. But this does not mean that both saddle-poin ts are equal. The difference between their asymptotic behaviours can b e obtained by working in a different scale. Here is a refined estimate for the first saddle-point:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "sp1:=multiseriesinverse([r1],scaleR,2,1,n,3,'scalezeta');" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sp1G,*-%'RootOfG6#,(-%#lnG6#%#_ZG\" \"\"F-F.-F+6#%\"nG!\"\"F.*(F&F2,&*$F&F2F.F.F.F2-%$expG6#F&F2F.**F&!\"# ,&F5\"\"#F.F.F.F4!\"$F6F:#F2F<-%\"OG6#*$F6F=F." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 15 "alias(zeta=%1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "sp1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%%zetaG\"\" \"*(F$!\"\",&*$F$F'F%F%F%F'-%$expG6#F$F'F%**F$!\"#,&F)\"\"#F%F%F%F(!\" $F*F.#F'F0-%\"OG6#*$F*F1F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "and here is an estimate for the second saddle-point refined similarly" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "sp2:=multiseriesinverse([r2 ],scaleR,2,1,n,3,scalezeta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sp2 G,*%%zetaG\"\"\"*(,&*$F&!\"\"F'F+F'F',&F*F'F'F'F+-%$expG6#F&F+F'*(,**$ F&!\"#F+F*\"\"#\"\"$F'*$F&!\"$\"\"%F'F,F7F-F3#F+F4-%\"OG6#*$F-F7F'" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Hence the difference can be compu ted at this level of the asymptotic scale" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 58 "`multiseries/expand`(x1-x2,[x1,x2],[sp1,sp2],scalez eta,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,&*$%%zetaG!\"\"\"\"\"F )F)F(-%$expG6#F'F(F)*(,**$F'!\"$\"\"#*$F'!\"#F3F&F1\"\"$F)F)F%F0F*F3#F )F1-%\"OG6#*$F*F0F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "or in term s of " }{XPPEDIT 18 0 "n" "I\"nG6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "substitute(SERIEStoseries(\",'polynom'),scalezeta,exp 1,scalen,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,(-%#lnG6#%\"nG\" \"\"*&,&*$-F'6#F&!\"\"F*F*F*F*F.F*F0-%\"OG6#*$F&F0F*F*F)F0F**&,(*$F&\" \"##\"\"$F8*(,&\"\"'F*F-\"\"(F*F.F*F&F*#F0F8-F26#F*F*F*F)!\"#F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Thus the saddle-points differ asym ptotically by " }{XPPEDIT 18 0 "ln(n)" "-%#lnG6#%\"nG" }{TEXT 266 1 "/ " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 82 ", which could not be det ected from the asymptotic expansion in the original scale." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "In order to compute the average number o f parts in a partition, we have to divide the estimate for the " } {XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 18 "th coefficient of " } {XPPEDIT 18 0 "g" "I\"gG6\"" }{TEXT -1 25 " by the estimate for the " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 18 "th coefficient of " } {XPPEDIT 18 0 "f" "I\"fG6\"" }{TEXT -1 169 ". Since these estimates ar e given in terms of different saddle-points, the computation is more e asily done by obtaining both estimates in the same scale, which involv es " }{XPPEDIT 18 0 "zeta" "I%zetaG6\"" }{TEXT -1 42 " , and then subs tituting the estimate for " }{XPPEDIT 18 0 "zeta" "I%zetaG6\"" }{TEXT -1 13 " in terms of " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 1 "." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "estden:=substitute(d1,scale R,sp1,scalezeta,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'estdenG*(,(* ,\"\"##\"\"\"F(-%$expG6#!\"\"F*%#PiG#F.F(*&%%zetaGF*,&F*F*F2F*F*F0*$-F ,6#F2F.F)F)*0,,*$F2!\"%F(*$F2!\"$\"\"'*$F2!\"#\"#;*$F2F.\"\"*F(F*F*F+F *F(F)F/F0F1F0,&FAF*F*F*F " 0 "" {MPLTEXT 1 0 50 "n1:=harris_schoenfeld(g,x,R,op(1,speq2),3,scaleR):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "estnum:=substitute(SERIES toseries(multiseries(n1,scaleR,2,1,nops(scaleR[list]),3),'polynom'),sc aleR,sp2,scalezeta,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'estnumG*( ,(*,\"\"##\"\"\"F(-%$expG6#!\"\"F*%#PiG#F.F(*&%%zetaGF*,&F*F*F2F*F*F0* $-F,6#F2F.F0F)*0,,*$F2!\"%F(*$F2!\"$\"#I*$F2!\"#\"#w*$F2F.\"#p\"#EF*F* F+F*F(F)F1F0,&FAF*F*F*F " 0 "" {MPLTEXT 1 0 71 "ratio:=`multiseries/e xpand`(x2/x1,[x1,x2],[estden,estnum],scalezeta,3);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%&ratioG,(-%$expG6#%%zetaG\"\"\"*&,(*$F)!\"#\"\"#*$F )!\"\"\"\"$F/F*F*,&F0F*F*F*F.#F1F/-%\"OG6#*$F&F1F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "And here is the final result in " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 48 ": the average number of parts in a partit ion of " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 10 " elements:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "substitute(SERIEStoseries(ra tio,'polynom'),scalezeta,exp1,scalen,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&,(*$-%#lnG6#%\"nG!\"\"\"\"\"*&-F(6#F'F,F'!\"#F,-%\"OG6#*$F'! \"$F,F,F*F,F,F+F,F&#F,\"\"#*(F.F,,&F+F,*$F.F+F7F,F'F0#F+F7F1F," }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 273 "This result is classical. However , it is worth noting that classical references give at best the first \+ term of the asymptotic behaviour (n/log n) and one of them is wrong by a factor of e=exp(1). This gives an idea of the difficulty of perform ing such computations by hand." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 8 "Variance" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "Computation of the variance leads to further cancellation. It is obtained from the gener ating function" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "h:=subs(u =1,diff(u*diff(F,u),u));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG,&*& ,&-%$expG6#%\"xG\"\"\"!\"\"F,F,-F)6#F'F,F,*&F'\"\"#F.F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "The corresponding saddle-point equation i s" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "speq3:=subs(x=R,normal (diff(log(h),x)*x))-1=n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&speq3G/ ,&*(,(!\"\"\"\"\"-%$expG6#%\"RGF**$F+\"\"#F*F*F.F*,&F+F*F)F*F)F*F)F*% \"nG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "The expansion of the sadd le-point in terms of " }{XPPEDIT 18 0 "zeta" "I%zetaG6\"" }{TEXT -1 22 " is obtained as before" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "r3:=Invert(op(1,speq3),R);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#r 3G6&7%,*-%#lnG6#%\"RG\"\"\"F+F,-F)6#,$*&,,*$-%$expGF*!\"#!\"\"*$F3F6F, F,F,*&F3F6F+F6F6*&F3F5F+F6F,F,,&F6F,F7F,F6F6F,*(,&*$F+F6F,*$F(F6F,F,F( F,F+F,F6F,F;F47%-F)6#*&,&F7F,F=F,F,F+F,\"\"#F+7%\"\"!FDF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "sp3:=multiseriesinverse([r3],scaleR ,2,1,n,3,scalezeta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sp3G,*%%zet aG\"\"\"*(,&*$F&!\"\"F'!\"#F'F',&F*F'F'F'F+-%$expG6#F&F+F'*(,(*$F&!\"$ \"\"#*$F&F,F4F5F'F'F-F4F.F,#F4F5-%\"OG6#*$F.F4F'" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 45 "This leads to the following estimate for the " } {XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 18 "th coefficient of " } {XPPEDIT 18 0 "h" "I\"hG6\"" }{TEXT -1 20 ", first in terms of " } {TEXT 267 1 "R" }{TEXT -1 22 " and then in terms of " }{XPPEDIT 18 0 " zeta" "I%zetaG6\"" }{TEXT -1 18 " by substituting " }{TEXT 268 1 "R" }{TEXT -1 41 " by its asymptotic expansion in terms of " }{XPPEDIT 18 0 "zeta" "I%zetaG6\"" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "n2:=harris_schoenfeld(h,x,R,op(1,speq3),3,scaleR):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "estnum2:=substitute(SERIE Stoseries(multiseries(n2,scaleR,2,1,nops(scaleR[list]),3),'polynom'),s caleR,sp3,scalezeta,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(estnum2G *(,(*,\"\"##\"\"\"F(-%$expG6#!\"\"F*%#PiG#F.F(*&%%zetaGF*,&F2F*F*F*F*F 0*$-F,6#F2F.#!\"$F(F)*0,,*$F2!\"%F(*$F2F8\"#I*$F2!\"#\"#))*$F2F.\"$0\" \"#]F*F*F+F*F(F),&FBF*F*F*F8F/F0F1F0F4F0#F.\"#[-%\"OG6#*$F4F)F*F*-F,6# F5F*-F,6#*(F2F*-%#lnGF6F*F5F*F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "From there we get the variance in terms of " }{XPPEDIT 18 0 "zeta " "I%zetaG6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "finalres:= `multiseries/expand`(x3/x1-(x2/x1)^2,[x1,x2,x3],[estden,estnum,estnum2 ],scalezeta,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)finalresG,&*(%%z etaG!\"\",&*$F'F(\"\"\"F+F+F(-%$expG6#F'F+F+-%\"OG6#F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "and in terms of " }{XPPEDIT 18 0 "n" "I\" nG6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "substitute(SERIESt oseries(\",'polynom'),scalezeta,exp1,scalen,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,(*$-%#lnG6#%\"nG!\"#\"\"\"*(,&F*F+*$-F'6#F&!\"\"F+F+ F/F+F&!\"$F1-%\"OG6#*$F&!\"%F+F+F)F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 148 "Again, only the first term is given in the literature. Our met hod can compute as many terms as desired, as well as the expansions of higher moments." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Conclusion " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 321 "Multiseries are specially use ful for saddle-point asymptotics that arise frequently in combinatoria l applications. Delicate expansions (that lead to errors in the techni cal literature) can be handled automatically. Such a process is specia lly useful for moment computations that involve minute saddle-point pe rturbations." }}}}}{MARK "12 0 0" 10 }{VIEWOPTS 1 1 0 1 1 1803 } pG6#F'F+F+-%\"OG6#F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "and in terms of " }{XPPEDIT 18 0 "n" "I\" nG6\"" 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2 3 "DEC ALPHA UNIX" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "2 D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 " " 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "H eading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 60 "Enumerating alcohols an d other classes of chemical moleculs," }}{PARA 257 "" 0 "" {TEXT 258 26 "an example of Polya theory" }}{PARA 258 "" 0 "" {TEXT -1 47 "\nFre deric Chyzak\n(Version of January 13, 1997)\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "Alkanes are a simple class of chemical compounds. Th ey are generically described by the chemical formula " }{XPPEDIT 18 0 "C[n]*H[2*n+2]" "*&&%\"CG6#%\"nG\"\"\"&%\"HG6#,&*&\"\"#F'F&F'F'\"\"#F' F'" }{TEXT -1 28 ". First examples for small " }{XPPEDIT 18 0 "n" "I \"nG6\"" }{TEXT -1 14 " are methane (" }{XPPEDIT 18 0 "n=1" "/%\"nG\" \"\"" }{TEXT -1 11 "), ethane (" }{XPPEDIT 18 0 "n=2" "/%\"nG\"\"#" } {TEXT -1 12 "), propane (" }{XPPEDIT 18 0 "n=3" "/%\"nG\"\"$" }{TEXT -1 11 "), butane (" }{XPPEDIT 18 0 "n=4" "/%\"nG\"\"%" }{TEXT -1 23 ") , a.s.o. For a given " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 40 " h owever, there exist several different " }{TEXT 271 7 "isomers" }{TEXT -1 226 ", i.e., different structures of bonds between atoms. In chemi stry, there is much interest in knowing the number, or better yet the \+ list, of such isomers. Alcohols are obtained from alkanes by replacin g a hydrogen atom by an " }{XPPEDIT 18 0 "OH" "I#OHG6\"" }{TEXT -1 116 " group. It follows that they are isomorphic to carbon chains wit h a distinguished node, or again to alkyl radicals " }{XPPEDIT 18 0 "C [n]*H[2*n+1]" "*&&%\"CG6#%\"nG\"\"\"&%\"HG6#,&*&\"\"#F'F&F'F'\"\"\"F'F '" }{TEXT -1 113 ", which are alkanes with a missing hydrogen atom. I f we disregard geometrical constraints (i.e., if we consider " }{TEXT 272 10 "structural" }{TEXT -1 23 " isomers only, and not " }{TEXT 273 14 "conformational" }{TEXT -1 152 " isomers), this leads to a pure gra ph-theoretical problem: how many rooted trees are there with n interna l nodes, where each internal node has degree 4?" }}{PARA 0 "" 0 "" {TEXT -1 35 "In this session, we thus consider " }{TEXT 257 6 "rooted " }{TEXT -1 68 " trees, so that we count and enumerate alkyls, with ge neric formula " }{XPPEDIT 18 0 "C[n]*H[2*n+1" "*&&%\"CG6#%\"nG\"\"\"&% \"HG6#,&*&\"\"#F'F&F'F'\"\"\"F'F'" }{TEXT -1 57 ". The combinatorics \+ also corresponds to simple alcohols " }{XPPEDIT 18 0 "C[n]*H[2*n+1]*O* H" "**&%\"CG6#%\"nG\"\"\"&%\"HG6#,&*&\"\"#F'F&F'F'\"\"\"F'F'%\"OGF'F)F '" }{TEXT -1 27 ", organo-metalic compounds " }{XPPEDIT 18 0 "C[n]*H[2 *n+1]*X" "*(&%\"CG6#%\"nG\"\"\"&%\"HG6#,&*&\"\"#F'F&F'F'\"\"\"F'F'%\"X GF'" }{TEXT -1 165 ", and any other monosubstituted alkanes. We next \+ treat the cases of disubstituted and trisubstituted alkanes. We devel op the study of our models using the package " }{HYPERLNK 17 "Combstru ct" 2 "combstruct" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "with(combstruct);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# 7.%+allstructsG%&countG%%drawG%)finishedG%'gfeqnsG%)gfseriesG%(gfsolve G%,iterstructsG%+nextstructG%,prog_gfeqnsG%.prog_gfseriesG%-prog_gfsol veG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "Enumerations of such clas ses of chemical compounds are part of Polya theory. We refer to the b ook by G. Polya and R. C. Read [" }{TEXT 266 67 "Combinatorial Enumera tion of Groups, Graphs, and Chemical Compounds" }{TEXT -1 54 ", (1987) , Springer-Verlag] for more extensive results." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 25 "Monosubstituted alkanes, " }{XPPEDIT 18 0 "C[n]*H[2* n+1]*X" "*(&%\"CG6#%\"nG\"\"\"&%\"HG6#,&*&\"\"#F'F&F'F'\"\"\"F'F'%\"XG F'" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "In this section, we study mo nosubstituted alkanes, i.e., " }{TEXT 275 6 "rooted" }{TEXT -1 60 " tr ees, first without any constraint, next according to the " }{TEXT 274 6 "height" }{TEXT -1 1 "." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 14 "Gen eral alkyls" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Definition" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 280 "An alkyl radical can be viewed as a carbon atom linked to at most 3 alkyl radicals. Thus, we only take into account hydrogen atoms implicitly. There is no loss of informati on, since hydrogen atoms can always be recovered from the carbon skele ton. This yields the class equation " }{XPPEDIT 18 0 "Alkyl=Carbon*(E psilon+Alkyl+Alkyl^2+Alkyl^3)" "/%&AlkylG*&%'CarbonG\"\"\",*%(EpsilonG F&F#F&*$F#\"\"#F&*$F#\"\"$F&F&" }{TEXT -1 42 ", which we map into the \+ following grammar:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "gram m_Alkyl:=Alkyl=Prod(Carbon,Set(Alkyl,card<=3)),Carbon=Atom:\nspecs_Alk yl:=[Alkyl,\{gramm_Alkyl\},unlabelled]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Note that since the " }{HYPERLNK 17 "Set" 2 "combstruct[s pecification]" "" }{TEXT -1 190 " construct denotes multisets, i.e., s ets with repetitions, a carbon atom of an alkyl is allowed to be bound to two copies of the same subtree (but the order of the subtrees does not matter)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "Define the size of an alkyl as the number of carbon atoms it contains. We compute th e number of alkyls of a given size using " }{HYPERLNK 17 "combstruct[c ount]" 2 "combstruct[count]" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 39 "seq(count(specs_Alkyl,size=i),i=0..50);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6U\"\"!\"\"\"F$\"\"#\"\"%\"\")\"#<\"#R\" #*)\"$6#\"$2&\"%Q7\"%dI\"%Rw\"&T#>\"&l)[\"'1\\7\"')>@$\"'>-$)\"(5g:#\" (4@i&\")8er9\")_\"\\'Q\"*F>#=5\"*&[5!p#\"*nlc7(\"+WL*>*=\"+G[qM]\",1y6 DM\"\",p3bme$\",)Gl8*f*\"-1X'GLd#\"-0TNG4p\".f:N@y&=\".`rg0B+&\"/pc2Sk [8\"/y-V#Q/k$\"/$Gq\"z2Q)*\"0x/+_&ehE\"0Z9$o(z!3s\"1>=m]?+a>\"1jq,#p]> I&\"2<#R6h\"**)R9\"2_dY^(ox8R\"3l9`eY&fY1\"\"38%Rk5*QT)*G\"3hd4\\v6U'* y\"4cblrf%\\\"G:#\"4a>Lh$3eAte\"5u!RqbZ_\\Lg\"\"5gxdP>%\\b(zV" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "This series appears as the entry \+ " }{TEXT 261 5 "M1146" }{TEXT -1 30 " (\"quartic planted trees with " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 56 " nodes\") in the book by N . J. A. Sloane and S. Plouffe [" }{TEXT 262 37 "The Encyclopedia of In teger Sequences" }{TEXT -1 26 ", (1995), Academic Press]." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Here is an example of an alkyl with 6 car bon atoms, obtained by the command " }{HYPERLNK 17 "combstruct[draw]" 2 "combstruct[draw]" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "alk:=draw(specs_Alkyl,size=6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$alkG-%%ProdG6$%'CarbonG-%$SetG6#-F&6$F(-F*6$-F&6$F(- F*6#-F&6$F(%(EpsilonGF0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "The fo llowing procedure rewrites an alkyl into a more readable way." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "nice:=proc(alk) eval(subs( \{Epsilon=NULL,Carbon=C,Prod=proc() global H; [args] end,Set=proc() ar gs end\},alk)) end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "nice (alk);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$%\"CG7%F$7$F$7#F$F&" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "The following procedure computes t he size of a given alkyl." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "size:=proc(alk) option remember; 1+convert(map(size,op(2,alk)),`+ `) end:\nsize(Prod(Carbon,Epsilon)):=1:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "The following procedure computes the height of a given al kyl." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "height:=proc(alk) \+ option remember; 1+max(op(map(height,op(2,alk)))) end:\nheight(Prod(Ca rbon,Epsilon)):=1:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Here is an \+ alkyl with 50 carbon atoms, its nice representation and height." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "alk:=draw(specs_Alkyl,size=5 0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "nice(alk);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7$%\"CG7$F$7%F$7#F$7$F$7%F$7%F$7&F$F'7$F$7%F $F'7%F$F'F'7&F$7$F$7$F$F'F07%F$7&F$F'F'7&F$F'F'F17%F$F'F17$F$7$F$7%F$F 1F.7%F$F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "height(alk); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#8" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "Empirical study" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 7 "Dr awing" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 171 "By drawing several rando m structures, we can study probabilistic properties of alkyls. For in stance, the following is a probabilistic estimate of their height on a verage:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "for i to 10 do h o[i]:=height(draw(specs_Alkyl,size=50)) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#hoG6#\"\"\"\"#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >&%#hoG6#\"\"#\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#hoG6#\"\"$\" #>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#hoG6#\"\"%\"#8" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%#hoG6#\"\"&\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#hoG6#\"\"'\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#hoG6# \"\"(\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#hoG6#\"\")\"#;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#hoG6#\"\"*\"#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>&%#hoG6#\"#5\"#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "add(ho[i],i=1..10)/10.;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++++I:!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "In the sam e way, we get a probabilistic estimate of their standard deviation:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sqrt(add((ho[i]-\")^2,i=1. .10)/10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+`%G=0#!\"*" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 22 "Exhaustive enumeration" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "The command " }{HYPERLNK 17 "combstruct[d raw]" 2 "combstruct[draw]" "" }{TEXT -1 20 " permits us to draw " } {TEXT 259 3 "one" }{TEXT -1 44 " structure at random. We can also gen erate " }{TEXT 260 3 "all" }{TEXT -1 31 " alkyls of a given size, usin g " }{HYPERLNK 17 "combstruct[allstructs]" 2 "combstruct[allstructs]" "" }{TEXT -1 222 ", so as to compute the mean of a particular paramete r exactly, or to count all those with a particular property. For inst ance, the height of trees cannot be represented in the class of combin atorial structures when using " }{HYPERLNK 17 "Combstruct" 2 "combstru ct" "" }{TEXT -1 134 ". For instance, by computing all alkyls of size 5, we get the distribution of height for these alkyls (in their nice \+ representation)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "allstru cts(specs_Alkyl,size=5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "map(nice,\");" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7*7$%\"CG7%F%7#F%7$F %F'7&F%F'F'F(7%F%F'7%F%F'F'7$F%7$F%7$F%F(7$F%7&F%F'F'F'7$F%7$F%F+7%F%F 'F.7%F%F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "sort(map(hei ght,\"\"));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*\"\"$F$F$F$\"\"%F%F% \"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "Here we count 4 alkyls \+ of size 5 and height 3, 3 alkyls of size 5 and height 4, and 1 alkyl o f size 5 and height 5." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "By the \+ same method, we get the exact mean and standard deviation of the heigh t for small sizes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "esd: =proc(n) local i,as,mean;\n as:=map(height,allstructs(specs_Alkyl,s ize=n));\n mean:=evalf(convert(as,`+`)/nops(as));\n nops(as),mea n,evalf(sqrt(add((i-mean)^2,i=as))/nops(as))\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "for i from 2 to 6 do i=esd(i) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"#6%\"\"\"$F$\"\"!F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/\"\"$6%\"\"#$\"+++++D!\"*$\"+1R`NN!#5" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/\"\"%6%F$$\"\"$\"\"!$\"+0R`NN!#5" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/\"\"&6%\"\")$\"++++DO!\"*$\"+huigC!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"'6%\"#<$\"+fqk " 0 "" {MPLTEXT 1 0 307 "gramm_ltd_height:=proc(n) option remembe r;\n Alkyl_height[n]=Prod(Carbon,Set(Alkyl_height[n-1],card<=3)),gr amm_ltd_height(n-1)\nend:\ngramm_ltd_height(1):=Alkyl_height[1]=Prod(C arbon,Epsilon),Carbon=Atom:\nspecs_ltd_height:=proc(n) option remember ;\n [Alkyl_height[n],\{gramm_ltd_height(n)\},unlabelled]\nend:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "The following procedure rewrites a n alkyl into a more readable way." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "nice:=proc(alk) eval(subs(\{Epsilon=NULL,Carbon=C,Pr od=proc() global H; [args] end,Set=proc() args end\},alk)) end:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "The following procedures compute t he size and height of a given alkyl." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "size:=proc(alk) option remember; 1+convert(map(size, op(2,alk)),`+`) end:\nsize(Prod(Carbon,Epsilon)):=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "height:=proc(alk) option remember; 1+max (op(map(height,op(2,alk)))) end:\nheight(Prod(Carbon,Epsilon)):=1:" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "For instance, we compute the heig ht of a random alkyl of size 10 and height at most 5." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "alk:=draw(specs_ltd_height(5),size= 10):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "nice(alk);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7&%\"CG7#F$7$F$F%7$F$7&F$F%F%F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "size(alk),height(alk);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"#5\"\"&" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 213 "In this section, we proceed to compute a table of the nu mber of alkyls according to their size and height. The first method i s by generating all structures. Next, we use generating functions to \+ extend the table." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "Generating \+ all structures" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "The following pr ocedure remembers all the alkyls of a given size and with bounded heig ht." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "list_all_st:=proc(d, s) option remember; allstructs(specs_ltd_height(d),size=s) end:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "An alkyl of height " }{XPPEDIT 18 0 "h" "I\"hG6\"" }{TEXT -1 20 " has a size at most " }{XPPEDIT 18 0 "( 3^h-1)/2" "*&,&)\"\"$%\"hG\"\"\"\"\"\"!\"\"F'\"\"#F)" }{TEXT -1 56 ". \+ Therefore, to produce all alkyls with height at most " }{XPPEDIT 18 0 "h[max]" "&%\"hG6#%$maxG" }{TEXT -1 48 ", we need to produce all alk yls with size up to " }{XPPEDIT 18 0 "s[max]=(3^h[max]-1)/2" "/&%\"sG6 #%$maxG*&,&)\"\"$&%\"hG6#F&\"\"\"\"\"\"!\"\"F.\"\"#F0" }{TEXT -1 1 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "s[max]:=(3^h[max]-1)/2; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"sG6#%$maxG,&)\"\"$&%\"hGF&#\" \"\"\"\"##!\"\"F/F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "To begin w ith, we enumerate all alkyls with height at most 3." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "h[max]:=3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"hG6#%$maxG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "for i from 1 to s[max] do i,nops(list_all_st(h[max],i)),map(nice,l ist_all_st(h[max],i)) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"\"F#7 #7#%\"CG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"#\"\"\"7#7$%\"CG7#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"$\"\"#7$7%%\"CG7#F'F(7$F'7$F'F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"%\"\"$7%7%%\"CG7#F'7$F'F(7$F'7 %F'F(F(7&F'F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"&\"\"%7&7&%\" CG7#F'F(7$F'F(7%F'F(7%F'F(F(7$F'7&F'F(F(F(7%F'F)F)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6%\"\"'\"\"%7&7&%\"CG7#F'7$F'F(F)7%F'F(7&F'F(F(F(7%F'F) 7%F'F(F(7&F'F(F(F-" }}{PARA 12 "" 1 "" {XPPMATH 20 "6%\"\"(\"\"&7'7&% \"CG7$F'7#F'F(F(7&F'F)F)7&F'F)F)F)7&F'F)F(7%F'F)F)7%F'F-F-7%F'F(F+" }} {PARA 12 "" 1 "" {XPPMATH 20 "6%\"\")\"\"%7&7&%\"CG7#F'7%F'F(F(F)7%F'F )7&F'F(F(F(7&F'F(7$F'F(F+7&F'F-F-F)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 %\"\"*\"\"%7&7&%\"CG7#F'7%F'F(F(7&F'F(F(F(7&F'7$F'F(F,F*7&F'F,F)F)7%F' F*F*" }}{PARA 12 "" 1 "" {XPPMATH 20 "6%\"#5\"\"$7%7&%\"CG7$F'7#F'7%F' F)F)7&F'F)F)F)7&F'F)F+F+7&F'F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6% \"#6\"\"#7$7&%\"CG7$F'7#F'7&F'F)F)F)F*7&F'7%F'F)F)F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#7\"\"\"7#7&%\"CG7%F'7#F'F)7&F'F)F)F)F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%\"#8\"\"\"7#7&%\"CG7&F'7#F'F)F)F(F(" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "In this way, we have obtained t he truncation of the bivariate generating function of alkyls with size marked by " }{XPPEDIT 18 0 "z" "I\"zG6\"" }{TEXT -1 15 " and height b y " }{XPPEDIT 18 0 "u" "I\"uG6\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "enum_BGF:=map(series,series(convert(map(proc (s,z,u) z^size(s)*u^height(s) end,map(op,[seq(list_all_st(h[max],i),i= 1..s[max])]),z,u),`+`),z,infinity),u,infinity);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)enum_BGFG+=%\"zG+%%\"uG\"\"\"\"\"\"\"\"\"+%F(F)\"\"# \"\"#+'F(F)\"\"#F)\"\"$\"\"$+'F(F)\"\"#\"\"#\"\"$\"\"%+%F(\"\"%\"\"$\" \"&F8\"\"'+%F(\"\"&\"\"$\"\"(F8\"\")F8\"\"*+%F(\"\"$\"\"$\"#5+%F(F5\" \"$\"#6+%F(F)\"\"$\"#7FJ\"#8" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Generating functions" }}{EXCHG {PARA 0 "" 0 "" {HYPERLNK 17 "combstru ct[gfeqns]" 2 "combstruct[gfeqns]" "" }{TEXT -1 210 " returns a system of functional equations satisfied by the generating functions of rela ted combinatorial structures. In the case of the alkyls with maximum \+ height above, we get the following triangular system." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "gfeqns(op(2..3,specs_ltd_height(4)) ,z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7'/-%'CarbonG6#%\"zGF(/-&%-Alk yl_heightG6#\"\"#F'*&F%\"\"\",0F0F0-&F,6#F0F'F0-F36#*$F(F.#F0F.*$F2F.F 8-F36#*$F(\"\"$#F0F=*&F2F0F5F0F8*$F2F=#F0\"\"'F0/-&F,6#F=F'*&F%F0,0F0F 0F*F0-F+F6F8*$F*F.F8-F+F;F>*&F*F0FIF0F8*$F*F=FAF0/F2F%/-&F,6#\"\"%F'*& F%F0,0F0F0FDF0-FEF6F8*$FDF.F8-FEF;F>*&FDF0FVF0F8*$FDF=FAF0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "gfsol:=gfsolve(op(2..3,specs_ltd_he ight(4)),z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&gfsolG<'/-&%-Alkyl_ heightG6#\"\"$6#%\"zG,\"$P\"F-F/F8\"$%=F6\" #qF<\"#**FFF1F0\"#?*$F-\"#:\"$+$*$F-\"#=\"$)\\*$F-FR\"$%f*$F-\"#E\"$`% *$F-\"#C\"$q&*$F-\"#;\"$p$*$F-\"#@\"$9'*$F-\"#F\"$y$*$F-\"#I\"$\"=*$F- \"#RF/*$F-\"#L\"#c*$F-\"#OF?*$F-\"#A\"$C'*$F-\"#9\"$R#*$F-\"#QF+*$F-\" #PFA*$F-\"#MFgp*$F-\"#NFR*$F-FL\"$G\"*$F-\"#G\"$7$*$F-\"#H\"$Q#*$F-\"# K\"#*)*$F-\"#B\"$,'*$F-\"#D\"$9&*$F-\"#<\"$K%*$F-\"#>\"$^&*$F-\"#SF/F: F/FDF;FEF5/-&F)6#F/F,F-/-%'CarbonGF,F-/-&F)6#F;F,,*F-F/F:F/FDF/FEF/" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 151 "In particular, we have obtained a truncation of the bivariate generating function of all alkyls (i.e. , with no constraint on height). In this series, " }{XPPEDIT 18 0 "u " "I\"uG6\"" }{TEXT -1 55 " marks the height. It extends the previous truncation " }{XPPEDIT 18 0 "enum_BGF" "I)enum_BGFG6\"" }{TEXT -1 1 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "BGF:=map(series,serie s(eval(subs(Alkyl_height[0]=0,gfsol,add(u^h*(Alkyl_height[h]-Alkyl_hei ght[h-1])(z),h=1..4))),z,infinity),u,infinity);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$BGFG+]p%\"zG+%%\"uG\"\"\"\"\"\"\"\"\"+%F(F)\"\"#\"\" #+'F(F)\"\"#F)\"\"$\"\"$+)F(F)\"\"#\"\"#\"\"$F)\"\"%\"\"%+'F(\"\"%\"\" $\"\"$\"\"%\"\"&+'F(F:\"\"$\"\")\"\"%\"\"'+'F(\"\"&\"\"$\"#:\"\"%\"\"( +'F(F:\"\"$\"#F\"\"%\"\")+'F(F:\"\"$\"#V\"\"%\"\"*+'F(F<\"\"$\"#n\"\"% \"#5+'F(F5\"\"$\"#(*\"\"%\"#6+'F(F)\"\"$\"$O\"\"\"%\"#7+'F(F)\"\"$\"$$ =\"\"%\"#8+%F(\"$R#\"\"%\"#9+%F(\"$+$\"\"%\"#:+%F(\"$p$\"\"%\"#;+%F(\" $K%\"\"%\"#<+%F(\"$)\\\"\"%\"#=+%F(\"$^&\"\"%\"#>+%F(\"$%f\"\"%\"#?+%F (\"$9'\"\"%\"#@+%F(\"$C'\"\"%\"#A+%F(\"$,'\"\"%\"#B+%F(\"$q&\"\"%\"#C+ %F(\"$9&\"\"%\"#D+%F(\"$`%\"\"%\"#E+%F(\"$y$\"\"%\"#F+%F(\"$7$\"\"%\"# G+%F(\"$Q#\"\"%\"#H+%F(\"$\"=\"\"%\"#I+%F(\"$G\"\"\"%\"#J+%F(\"#*)\"\" %\"#K+%F(\"#c\"\"%\"#L+%F(\"#P\"\"%\"#M+%F(\"#?\"\"%\"#N+%F(\"#7\"\"% \"#O+%F(\"\"'\"\"%\"#P+%F(F<\"\"%\"#Q+%F(F)\"\"%\"#RFeu\"#S" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "This is made explicit on the follo wing normalized difference: each entry starts with a term in " } {XPPEDIT 18 0 "u^4" "*$%\"uG\"\"%" }{TEXT -1 41 ", denoting alkyls wit h height at least 4." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "map (series,series(BGF-enum_BGF,z,infinity),u,infinity);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+go%\"zG+%%\"uG\"\"\"\"\"%\"\"%+%F&\"\"$\"\"%\"\"&+% F&\"\")\"\"%\"\"'+%F&\"#:\"\"%\"\"(+%F&\"#F\"\"%\"\")+%F&\"#V\"\"%\"\" *+%F&\"#n\"\"%\"#5+%F&\"#(*\"\"%\"#6+%F&\"$O\"\"\"%\"#7+%F&\"$$=\"\"% \"#8+%F&\"$R#\"\"%\"#9+%F&\"$+$\"\"%\"#:+%F&\"$p$\"\"%\"#;+%F&\"$K%\" \"%\"#<+%F&\"$)\\\"\"%\"#=+%F&\"$^&\"\"%\"#>+%F&\"$%f\"\"%\"#?+%F&\"$9 '\"\"%\"#@+%F&\"$C'\"\"%\"#A+%F&\"$,'\"\"%\"#B+%F&\"$q&\"\"%\"#C+%F&\" $9&\"\"%\"#D+%F&\"$`%\"\"%\"#E+%F&\"$y$\"\"%\"#F+%F&\"$7$\"\"%\"#G+%F& \"$Q#\"\"%\"#H+%F&\"$\"=\"\"%\"#I+%F&\"$G\"\"\"%\"#J+%F&\"#*)\"\"%\"#K +%F&\"#c\"\"%\"#L+%F&\"#P\"\"%\"#M+%F&\"#?\"\"%\"#N+%F&\"#7\"\"%\"#O+% F&\"\"'\"\"%\"#PF*\"#QF%\"#RF%\"#S" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 58 "Table of the number of alkyls according to size and height" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 235 "Calculations with respect to diff erent heights are much more efficient than the method of exhaustive en umeration. This makes it possible for us to set up the table of the n umber of alkyls according to size and height in a few minutes:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "h[max]:=5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"hG6#%$maxG\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "gfsol:=gfsolve(op(2..3,specs_ltd_height(h[max])),z); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&gfsolG<(/-&%-Alkyl_heightG6#\" \"&6#%\"zG,^z*$F-\"\"'\"#;*$F-\"\")\"#j*$F-\"\"*\"$@\"*$F-\"#7\"$\\(F- \"\"\"*$F-\"#8\"%W8*$F-\"#5\"$D#*$F-\"#6\"$:%*$F-F+F3*$F-\"\"(\"#L*$F- \"#$*\")yF#*R*$F-\"#%)\"*Nl/_&*$F-\"#()\"*3Mrh#*$F-\"#'*\")kzn7*$F-\"# :\"%HT*$F-\"#=\"&a.#*$F-\"#?\"&1d&*$F-\"#E\"'FF()*$F-\"#C\"'bXO*$F-F1 \"%1r*$F-\"#@\"&G1**$F-\"#F\"(X&G8*$F-\"#I\"((G$R%*$F-\"#R\")\\av*)*$F -FH\")EX?8*$F-\"#O\")-^4O*$F-\"#A\"'Hd9*$F-\"#9\"%lB*$F-\"#Q\")^9&p'*$ F-\"#P\"))**=%\\*$F-\"#M\")0zl=*$F-\"#N\")W#)3E*$F-\"#J\"($o2k*$F-\"#G \"(O0+#*$F-\"#H\"(a0)H*$F-\"#K\"(IoC**$F-\"#B\"',=B*$F-\"#D\"'1sc*$F- \"#<\"&/@\"*$F-\"#>\"&$)Q$*$F-\"#S\"*\\J1>\"*$F-\"\"#F;*$F-\"\"$F_s*$F -F7F;*$F-\"#p\"+'enS*R*$F-\"#v\"+YN@KD*$F-\"#^\"+QR_B8*$F-\"#d\"+rF*)[ G*$F-\"$?\"F;*$F-\"#i\"+XuV$*R*$F-\"#c\"+b:CwD*$F-\"#e\"+R[0;J*$F-\"#k \"+7L\"yA%*$F-\"#Y\"*\"Qf&>&*$F-\"#]\"+1q`@6*$F-\"#!)\"+*$ F-\"#*)\"*k_J[\"*$F-\"#\"*\")/!)Hz*$F-\"##*\")'>&pc*$F-\"#%*\")n`nF*$F -\"#&*\"):]))=*$F-\"#(*\"(+GP)*$F-\"#)*\"(EcV&*$F-\"$+\"\"(&Ru@*$F-\"$ ,\"\"(!zQ8*$F-\"$.\"\"'R$z%*$F-\"$/\"\"'!Gy#*$F-\"$1\"\"&/#))*$F-\"$2 \"\"&E\"[*$F-\"$4\"\"&[L\"*$F-\"$5\"\"%Wn*$F-\"$7\"\"%/;*$F-\"$8\"\"$e (*$F-\"$:\"\"$a\"*$F-\"$;\"Fbt*$F-\"$=\"F@*$F-\"$>\"F]z*$F-\"#f\"+T8+r L*$F-\"#h\"+7KQ;Q*$F-\"#l\"+R<(pF%*$F-\"#n\"+;rgHU*$F-\"#r\"+d_:*f$*$F -\"#t\"+b3u#4$*$F-\"#x\"+#HbP(>*$F-\"#z\"+wa@j9*$F-\"#T\"*IZGc\"*$F-\" #V\"*ee'3E*$F-\"#Z\"*<&R*R'*$F-\"#\\\"*_nBS**$F-\"#`\"+j!*e%y\"*$F-\"# b\"+N2S/B/-&F)6#FasF,, " 0 "" {MPLTEXT 1 0 147 "BGF:=map (series,series(eval(subs(Alkyl_height[0]=0,gfsol,add(u^hh*(Alkyl_heigh t[hh]-Alkyl_height[hh-1])(z),hh=1..h[max]))),z,infinity),u,infinity); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$BGFG+_z%\"zG+%%\"uG\"\"\"\"\"\" \"\"\"+%F(F)\"\"#\"\"#+'F(F)\"\"#F)\"\"$\"\"$+)F(F)\"\"#\"\"#\"\"$F)\" \"%\"\"%+)F(\"\"%\"\"$\"\"$\"\"%F)\"\"&\"\"&+)F(F:\"\"$\"\")\"\"%F:\" \"&\"\"'+)F(\"\"&\"\"$\"#:\"\"%\"#8\"\"&\"\"(+)F(F:\"\"$\"#F\"\"%\"#K \"\"&\"\")+)F(F:\"\"$\"#V\"\"%\"#u\"\"&\"\"*+)F(F<\"\"$\"#n\"\"%\"$b\" \"\"&\"#5+)F(F5\"\"$\"#(*\"\"%\"$;$\"\"&\"#6+)F(F)\"\"$\"$O\"\"\"%\"$7 '\"\"&\"#7+)F(F)\"\"$\"$$=\"\"%\"%g6\"\"&\"#8+'F(\"$R#\"\"%\"%E@\"\"& \"#9+'F(\"$+$\"\"%\"%HQ\"\"&\"#:+'F(\"$p$\"\"%\"%Pn\"\"&\"#;+'F(\"$K% \"\"%\"&s;\"\"\"&\"#<+'F(\"$)\\\"\"%\"&c)>\"\"&\"#=+'F(\"$^&\"\"%\"&KL $\"\"&\"#>+'F(\"$%f\"\"%\"&7^&\"\"&\"#?+'F(\"$9'\"\"%\"&9+*\"\"&\"#@+' F(\"$C'\"\"%\"'0^9\"\"&\"#A+'F(\"$,'\"\"%\"'+7B\"\"&\"#B+'F(\"$q&\"\"% \"'&)RO\"\"&\"#C+'F(\"$9&\"\"%\"'#pm&\"\"&\"#D+'F(\"$`%\"\"%\"'uA()\" \"&\"#E+'F(\"$y$\"\"%\"(n\"G8\"\"&\"#F+'F(\"$7$\"\"%\"(C-+#\"\"&\"#G+' F(\"$Q#\"\"%\"(;.)H\"\"&\"#H+'F(\"$\"=\"\"%\"(1JR%\"\"&\"#I+'F(\"$G\" \"\"%\"(bvS'\"\"&\"#J+'F(\"#*)\"\"%\"(TnC*\"\"&\"#K+'F(\"#c\"\"%\")qW? 8\"\"&\"#L+'F(\"#P\"\"%\")oyl=\"\"&\"#M+'F(\"#?\"\"%\")C#)3E\"\"&\"#N+ 'F(\"#7\"\"%\")!4&4O\"\"&\"#O+'F(\"\"'\"\"%\")#**=%\\\"\"&\"#P+'F(F<\" \"%\")[9&p'\"\"&\"#Q+'F(F)\"\"%\")[av*)\"\"&\"#R+'F(F)\"\"%\"*[J1>\"\" \"&\"#S+%F(\"*IZGc\"\"\"&\"#T+%F(\"*US)H?\"\"&\"#U+%F(\"*ee'3E\"\"&\"# V+%F(\"*VerJ$\"\"&\"#W+%F(\"*-)ftT\"\"&\"#X+%F(\"*\"Qf&>&\"\"&\"#Y+%F( \"*<&R*R'\"\"&\"#Z+%F(\"*HI%)z(\"\"&\"#[+%F(\"*_nBS*\"\"&\"#\\+%F(\"+1 q`@6\"\"&\"#]+%F(\"+QR_B8\"\"&\"#^+%F(\"+[z;X:\"\"&\"#_+%F(\"+j!*e%y\" \"\"&\"#`+%F(\"+z*G*Q?\"\"&\"#a+%F(\"+N2S/B\"\"&\"#b+%F(\"+b:CwD\"\"& \"#c+%F(\"+rF*)[G\"\"&\"#d+%F(\"+R[0;J\"\"&\"#e+%F(\"+T8+rL\"\"&\"#f+% F(\"+L/t1O\"\"&\"#g+%F(\"+7KQ;Q\"\"&\"#h+%F(\"+XuV$*R\"\"&\"#i+%F(\"+h 'p@8%\"\"&\"#j+%F(\"+7L\"yA%\"\"&\"#k+%F(\"+R<(pF%\"\"&\"#l+%F(\"+?Pmx U\"\"&\"#m+%F(\"+;rgHU\"\"&\"#n+%F(\"+.h3MT\"\"&\"#o+%F(\"+'enS*R\"\"& \"#p+%F(\"+[\"yQ\"Q\"\"&\"#q+%F(\"+d_:*f$\"\"&\"#r+%F(\"+7RQcL\"\"&\"# s+%F(\"+b3u#4$\"\"&\"#t+%F(\"+7,d:G\"\"&\"#u+%F(\"+YN@KD\"\"&\"#v+%F( \"+uI_\\A\"\"&\"#w+%F(\"+#HbP(>\"\"&\"#x+%F(\"+#)\\?5<\"\"&\"#y+%F(\"+ wa@j9\"\"&\"#z+%F(\"+\" \"&\"#))+%F(\"*k_J[\"\"\"&\"#*)+%F(\"*%R]#4\"\"\"&\"#!*+%F(\")/!)Hz\" \"&\"#\"*+%F(\")'>&pc\"\"&\"##*+%F(\")yF#*R\"\"&\"#$*+%F(\")n`nF\"\"& \"#%*+%F(\"):]))=\"\"&\"#&*+%F(\")kzn7\"\"&\"#'*+%F(\"(+GP)\"\"&\"#(*+ %F(\"(EcV&\"\"&\"#)*+%F(\"(N\"pM\"\"&\"#**+%F(\"(&Ru@\"\"&\"$+\"+%F(\" (!zQ8\"\"&\"$,\"+%F(\"'Z&3)\"\"&\"$-\"+%F(\"'R$z%\"\"&\"$.\"+%F(\"'!Gy #\"\"&\"$/\"+%F(\"'w%e\"\"\"&\"$0\"+%F(\"&/#))\"\"&\"$1\"+%F(\"&E\"[\" \"&\"$2\"+%F(\"&%eD\"\"&\"$3\"+%F(\"&[L\"\"\"&\"$4\"+%F(\"%Wn\"\"&\"$5 \"+%F(\"%`L\"\"&\"$6\"+%F(\"%/;\"\"&\"$7\"+%F(\"$e(\"\"&\"$8\"+%F(\"$Q $\"\"&\"$9\"+%F(\"$a\"\"\"&\"$:\"+%F(\"#i\"\"&\"$;\"+%F(FP\"\"&\"$<\"+ %F(\"#5\"\"&\"$=\"+%F(F:\"\"&\"$>\"+%F(F)\"\"&\"$?\"F[^m\"$@\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "In the following table, the entry \+ at row " }{XPPEDIT 18 0 "r" "I\"rG6\"" }{TEXT -1 12 " and column " } {XPPEDIT 18 0 "c" "I\"cG6\"" }{TEXT -1 33 " is the number of alkyls of size " }{XPPEDIT 18 0 "r" "I\"rG6\"" }{TEXT -1 0 "" }{TEXT -1 12 " an d height " }{XPPEDIT 18 0 "c" "I\"cG6\"" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "matrix([[` `,seq(`height = `.hh,hh =1..h[max])],seq([`size = `.ss,seq(coeff(coeff(BGF,z,ss),u,hh),hh=1..h [max])],ss=1..s[max])]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG 6#7fr7(%\"~G%+height~=~1G%+height~=~2G%+height~=~3G%+height~=~4G%+heig ht~=~5G7(%)size~=~1G\"\"\"\"\"!F1F1F17(%)size~=~2GF1F0F1F1F17(%)size~= ~3GF1F0F0F1F17(%)size~=~4GF1F0\"\"#F0F17(%)size~=~5GF1F1\"\"%\"\"$F07( %)size~=~6GF1F1F;\"\")F;7(%)size~=~7GF1F1\"\"&\"#:\"#87(%)size~=~8GF1F 1F;\"#F\"#K7(%)size~=~9GF1F1F;\"#V\"#u7(%*size~=~10GF1F1F<\"#n\"$b\"7( %*size~=~11GF1F1F8\"#(*\"$;$7(%*size~=~12GF1F1F0\"$O\"\"$7'7(%*size~=~ 13GF1F1F0\"$$=\"%g67(%*size~=~14GF1F1F1\"$R#\"%E@7(%*size~=~15GF1F1F1 \"$+$\"%HQ7(%*size~=~16GF1F1F1\"$p$\"%Pn7(%*size~=~17GF1F1F1\"$K%\"&s; \"7(%*size~=~18GF1F1F1\"$)\\\"&c)>7(%*size~=~19GF1F1F1\"$^&\"&KL$7(%*s ize~=~20GF1F1F1\"$%f\"&7^&7(%*size~=~21GF1F1F1\"$9'\"&9+*7(%*size~=~22 GF1F1F1\"$C'\"'0^97(%*size~=~23GF1F1F1\"$,'\"'+7B7(%*size~=~24GF1F1F1 \"$q&\"'&)RO7(%*size~=~25GF1F1F1\"$9&\"'#pm&7(%*size~=~26GF1F1F1\"$`% \"'uA()7(%*size~=~27GF1F1F1\"$y$\"(n\"G87(%*size~=~28GF1F1F1\"$7$\"(C- +#7(%*size~=~29GF1F1F1\"$Q#\"(;.)H7(%*size~=~30GF1F1F1\"$\"=\"(1JR%7(% *size~=~31GF1F1F1\"$G\"\"(bvS'7(%*size~=~32GF1F1F1\"#*)\"(TnC*7(%*size ~=~33GF1F1F1\"#c\")qW?87(%*size~=~34GF1F1F1\"#P\")oyl=7(%*size~=~35GF1 F1F1\"#?\")C#)3E7(%*size~=~36GF1F1F1\"#7\")!4&4O7(%*size~=~37GF1F1F1\" \"'\")#**=%\\7(%*size~=~38GF1F1F1F<\")[9&p'7(%*size~=~39GF1F1F1F0\")[a v*)7(%*size~=~40GF1F1F1F0\"*[J1>\"7(%*size~=~41GF1F1F1F1\"*IZGc\"7(%*s ize~=~42GF1F1F1F1\"*US)H?7(%*size~=~43GF1F1F1F1\"*ee'3E7(%*size~=~44GF 1F1F1F1\"*VerJ$7(%*size~=~45GF1F1F1F1\"*-)ftT7(%*size~=~46GF1F1F1F1\"* \"Qf&>&7(%*size~=~47GF1F1F1F1\"*<&R*R'7(%*size~=~48GF1F1F1F1\"*HI%)z(7 (%*size~=~49GF1F1F1F1\"*_nBS*7(%*size~=~50GF1F1F1F1\"+1q`@67(%*size~=~ 51GF1F1F1F1\"+QR_B87(%*size~=~52GF1F1F1F1\"+[z;X:7(%*size~=~53GF1F1F1F 1\"+j!*e%y\"7(%*size~=~54GF1F1F1F1\"+z*G*Q?7(%*size~=~55GF1F1F1F1\"+N2 S/B7(%*size~=~56GF1F1F1F1\"+b:CwD7(%*size~=~57GF1F1F1F1\"+rF*)[G7(%*si ze~=~58GF1F1F1F1\"+R[0;J7(%*size~=~59GF1F1F1F1\"+T8+rL7(%*size~=~60GF1 F1F1F1\"+L/t1O7(%*size~=~61GF1F1F1F1\"+7KQ;Q7(%*size~=~62GF1F1F1F1\"+X uV$*R7(%*size~=~63GF1F1F1F1\"+h'p@8%7(%*size~=~64GF1F1F1F1\"+7L\"yA%7( %*size~=~65GF1F1F1F1\"+R<(pF%7(%*size~=~66GF1F1F1F1\"+?PmxU7(%*size~=~ 67GF1F1F1F1\"+;rgHU7(%*size~=~68GF1F1F1F1\"+.h3MT7(%*size~=~69GF1F1F1F 1\"+'enS*R7(%*size~=~70GF1F1F1F1\"+[\"yQ\"Q7(%*size~=~71GF1F1F1F1\"+d_ :*f$7(%*size~=~72GF1F1F1F1\"+7RQcL7(%*size~=~73GF1F1F1F1\"+b3u#4$7(%*s ize~=~74GF1F1F1F1\"+7,d:G7(%*size~=~75GF1F1F1F1\"+YN@KD7(%*size~=~76GF 1F1F1F1\"+uI_\\A7(%*size~=~77GF1F1F1F1\"+#HbP(>7(%*size~=~78GF1F1F1F1 \"+#)\\?5<7(%*size~=~79GF1F1F1F1\"+wa@j97(%*size~=~80GF1F1F1F1\"+7(%*size~=~89GF1F1F1F1\"*k_J[\"7(%*s ize~=~90GF1F1F1F1\"*%R]#4\"7(%*size~=~91GF1F1F1F1\")/!)Hz7(%*size~=~92 GF1F1F1F1\")'>&pc7(%*size~=~93GF1F1F1F1\")yF#*R7(%*size~=~94GF1F1F1F1 \")n`nF7(%*size~=~95GF1F1F1F1\"):]))=7(%*size~=~96GF1F1F1F1\")kzn77(%* size~=~97GF1F1F1F1\"(+GP)7(%*size~=~98GF1F1F1F1\"(EcV&7(%*size~=~99GF1 F1F1F1\"(N\"pM7(%+size~=~100GF1F1F1F1\"(&Ru@7(%+size~=~101GF1F1F1F1\"( !zQ87(%+size~=~102GF1F1F1F1\"'Z&3)7(%+size~=~103GF1F1F1F1\"'R$z%7(%+si ze~=~104GF1F1F1F1\"'!Gy#7(%+size~=~105GF1F1F1F1\"'w%e\"7(%+size~=~106G F1F1F1F1\"&/#))7(%+size~=~107GF1F1F1F1\"&E\"[7(%+size~=~108GF1F1F1F1\" &%eD7(%+size~=~109GF1F1F1F1\"&[L\"7(%+size~=~110GF1F1F1F1\"%Wn7(%+size ~=~111GF1F1F1F1\"%`L7(%+size~=~112GF1F1F1F1\"%/;7(%+size~=~113GF1F1F1F 1\"$e(7(%+size~=~114GF1F1F1F1\"$Q$7(%+size~=~115GF1F1F1F1\"$a\"7(%+siz e~=~116GF1F1F1F1\"#i7(%+size~=~117GF1F1F1F1FG7(%+size~=~118GF1F1F1F1\" #57(%+size~=~119GF1F1F1F1F;7(%+size~=~120GF1F1F1F1F07(%+size~=~121GF1F 1F1F1F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "A (huge) table for " } {XPPEDIT 18 0 "h[max]=7" "/&%\"hG6#%$maxG\"\"(" }{TEXT -1 43 " could b e computed in less than 10 minutes." }}}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "Disubstituted alkanes, " }{XPPEDIT 18 0 "C[n]*H[2*n]*X*Y " "**&%\"CG6#%\"nG\"\"\"&%\"HG6#*&\"\"#F'F&F'F'%\"XGF'%\"YGF'" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Enumerating disubstituted alkanes \+ " }{XPPEDIT 18 0 "C[n]*H[2*n]*X*Y" "**&%\"CG6#%\"nG\"\"\"&%\"HG6#*&\" \"#F'F&F'F'%\"XGF'%\"YGF'" }{TEXT -1 53 " is equivalent to enumerating monosubstituted alkyls " }{XPPEDIT 18 0 "C[n]*H[2*n]*X" "*(&%\"CG6#% \"nG\"\"\"&%\"HG6#*&\"\"#F'F&F'F'%\"XGF'" }{TEXT -1 163 ". The latter can generically be viewed as a carbon atom linked to one monosubstitu ted alkyl and at least 2 nonsubstituted alkyls. This yields the class equation " }{XPPEDIT 18 0 "S1_Alkyl[X]=Carbon*S1_Alkyl[X]*(Epsilon+Al kyl)+Carbon*X*(Epsilon+Alkyl+Alkyl^2)" "/&%)S1_AlkylG6#%\"XG,&*(%'Carb onG\"\"\"&F$6#F&F*,&%(EpsilonGF*%&AlkylGF*F*F**(F)F*F&F*,(F.F*F/F**$F/ \"\"#F*F*F*" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "gramm_S1_Alkyl:=S1_Alkyl[X]=Union(Prod(Carbon,S1_Alkyl[X],Set(Alk yl,card<=2)),Prod(Prod(Carbon,X),Set(Alkyl,card<=2))),X=Epsilon:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "specs_S1_Alkyl:=[S1_Alkyl[X] ,\{gramm_S1_Alkyl,gramm_Alkyl\},unlabelled]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "seq(count(specs_S1_Alkyl,size=i),i=0..50);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6U\"\"!\"\"\"\"\"#\"\"&\"#7\"#J\"#!)\"$5 #\"$b&\"%z9\"%fR\"&_1\"\"&g(G\"&5z(\"'C;@\"'@id\"(5Ad\"\"(LxH%\")Gtw6 \"),oEK\")EYf))\"*>\\aV#\"*B'G-n\"+PRGY=\"+=^g!4&\",o!ow/9\",$HA\\zQ\" -d!Q_@2\"\"-/(y9]'H\"-&HhiZ?)\".j#eksrA\".c\"HIL$H'\"/;M;oJW<\"/amji+P [\"0yf)4p!>M\"\"0y:)Q$eVs$\"1%R3p')yS.\"\"1\\4P:<\"3/&4SfWxAx%\"4Za4)pc)[!G8\"4Ir?\\oJH np$\"5D\\EIwK[FH5\"5h^#)=Pi&*[mG\"5,<39N`J\"\\)z\"6Pfmzfs!yxCA\"6N[u% \\T_!e+?'\"7n'yl!f+)G*>G<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "This series appears as the entry " }{TEXT 264 5 "M1418" }{TEXT -1 18 " (\" paraffins with " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 63 " carbon a toms\") in the book by N. J. A. Sloane and S. Plouffe [" }{TEXT 265 37 "The Encyclopedia of Integer Sequences" }{TEXT -1 26 ", (1995), Aca demic Press]." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 161 "Of course, ther e are more monosubstituted alkyls than nonsubstituted ones. We give t he ratios number of monosubstituted alkyls/number of alkyls for small \+ sizes:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "seq([i=evalf(coun t(specs_S1_Alkyl,size=i)/count(specs_Alkyl,size=i))],i=1..50);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6T7#/\"\"\"$F%\"\"!7#/\"\"#$F*F'7#/\"\"$ $\"+++++D!\"*7#/\"\"%$F.F'7#/\"\"&$\"++++vQF17#/\"\"'$\"+`B)eq%F17#/\" \"($\"+&Q:YQ&F17#/\"\")$\"+c]&fB'F17#/\"\"*$\"+t'y%4qF17#/\"#5$\"+,&y' 3yF17#/\"#6$\"+B.?/')F17#/\"#7$\"+ei\"zS*F17#/\"#8$\"+#*y*)>5!\")7#/\" #9$\"+v'f)*4\"F]o7#/\"#:$\"+p+@z6F]o7#/\"#;$\"+`Xre7F]o7#/\"#<$\"+/B.Q 8F]o7#/\"#=$\"+zjP<9F]o7#/\"#>$\"+Hzf'\\\"F]o7#/\"#?$\"+ya#ed\"F]o7#/ \"#@$\"+iy)\\l\"F]o7#/\"#A$\"+\"RNTt\"F]o7#/\"#B$\"+%)yC8=F]o7#/\"#C$ \"++XM#*=F]o7#/\"#D$\"+^$=9(>F]o7#/\"#E$\"+$zy/0#F]o7#/\"#F$\"+5Q_H@F] o7#/\"#G$\"+Uxb3AF]o7#/\"#H$\"+01e(G#F]o7#/\"#I$\"+\"f%fmBF]o7#/\"#J$ \"+#>+cW#F]o7#/\"#K$\"+)o)fCDF]o7#/\"#L$\"+I1f.EF]o7#/\"#M$\"+Eod#o#F] o7#/\"#N$\"+rxbhFF]o7#/\"#O$\"+=S`SGF]o7#/\"#P$\"+vf]>HF]o7#/\"#Q$\"+[ SZ)*HF]o7#/\"#R$\"+l&Qu2$F]o7#/\"#S$\"+K)*RcJF]o7#/\"#T$\"+9\"e`B$F]o7 #/\"#U$\"+^OJ9LF]o7#/\"#V$\"+bmE$R$F]o7#/\"#W$\"+>t@sMF]o7#/\"#X$\"+:e ;^NF]o7#/\"#Y$\"+*H7,j$F]o7#/\"#Z$\"+6p04PF]o7#/\"#[$\"+!y**zy$F]o7#/ \"#\\$\"+C5%p'QF]o7#/\"#]$\"+Z2)e%RF]o" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Here is an example of a monosubstituted alkyl with 6 carb on atoms, obtained by the command " }{HYPERLNK 17 "combstruct[draw]" 2 "combstruct[draw]" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "alk:=draw(specs_S1_Alkyl,size=6);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$alkG-%%ProdG6%%'CarbonG-F&6%F(-F&6$-F&6$F(%\"XG-%$ SetG6#-F&6$F(-F16#-F&6$F(%(EpsilonGF5F9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "The following procedure rewrites a monosubstituted alkyl \+ into a more readable way." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "nice:=proc(alk) subs([C,X]=CX,eval(subs(\{Epsilon=NULL,Carbon=C,P rod=proc() [args] end,Set=proc() args end\},alk))) end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "nice(alk);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$%\"CG7%F$7$%#CXG7$F$7#F$F)" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 86 "The following procedures compute the size and height of a given monosubstituted alkyl." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "size:=proc(alk) option remember; 1+convert(map(op,map2(map,si ze,[op(2..-1,alk)])),`+`) end:\nsize(Carbon):=1:\nsize(X):=0:\nsize(Ep silon):=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "height:=proc (alk) option remember; `if`(nops(alk)=2,1+max(op(map(height,op(2,alk)) )),1+max(height(op(2,alk)),op(map(height,op(3,alk))))) end:\nheight(Ca rbon):=1:\nheight(X):=0:\nheight(Epsilon):=0:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Here is a monosubstituted alkyl with 50 carbon atoms, \+ its nice representation and height." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "alk:=draw(specs_S1_Alkyl,size=50):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "nice(alk);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7&%\"CG7%F$7$F$7$F$7&F$7#%#CXG7#F$7%F$F+7$F$7%F$7$F$7$F$F+7%F$F0 F0F+7%F$F+7&F$F+F+F07$F$7&F$F+F37%F$7$F$F/7$F$7&F$F07%F$F+F+F:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "height(alk);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#6" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Tri substituted alkanes, " }{XPPEDIT 18 0 "C[n]*H[2*n-1]*X*Y*Z" "*,&%\"CG6 #%\"nG\"\"\"&%\"HG6#,&*&\"\"#F'F&F'F'\"\"\"!\"\"F'%\"XGF'%\"YGF'%\"ZGF '" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Enumerating trisubstituted al kanes " }{XPPEDIT 18 0 "C[n]*H[2*n-1]*X*Y*Z" "*,&%\"CG6#%\"nG\"\"\"&% \"HG6#,&*&\"\"#F'F&F'F'\"\"\"!\"\"F'%\"XGF'%\"YGF'%\"ZGF'" }{TEXT -1 51 " is equivalent to enumerating disubstituted alkyls " }{XPPEDIT 18 0 "C[n]*H[2*n-1]*X*Y" "**&%\"CG6#%\"nG\"\"\"&%\"HG6#,&*&\"\"#F'F&F'F' \"\"\"!\"\"F'%\"XGF'%\"YGF'" }{TEXT -1 30 ". In this section, we assu me " }{XPPEDIT 18 0 "X" "I\"XG6\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Y " "I\"YG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Z" "I\"ZG6\"" }{TEXT -1 139 " to be distinct. The grammar is more involved than in the dis ubstituted case: we have to distinguish several cases, according to wh ich of " }{XPPEDIT 18 0 "X" "I\"XG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Y" "I\"YG6\"" }{TEXT -1 81 " go into subtrees, and into which su btrees. The corresponding class equation is " }{XPPEDIT 18 0 "S2_Alky l[X,Y]=Carbon*S2_Alkyl[X,Y]*(Epsilon+Alkyl+Alkyl^2)+Carbon*(X+S1_Alkyl [X])*(Y+S1_Alkyl[Y])*(Epsilon+Alkyl)" "/&%)S2_AlkylG6$%\"XG%\"YG,&*(%' CarbonG\"\"\"&F$6$F&F'F+,(%(EpsilonGF+%&AlkylGF+*$F0\"\"#F+F+F+**F*F+, &F&F+&%)S1_AlkylG6#F&F+F+,&F'F+&F66#F'F+F+,&F/F+F0F+F+F+" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "gramm_S2_Alkyl:=S2_A lkyl[X,Y]=Union(Prod(Carbon,S2_Alkyl[X,Y],Set(Alkyl,card<=2)),Prod(Car bon,Union(S1_Alkyl[X],X),Union(S1_Alkyl[Y],Y),Set(Alkyl,card<=1))):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "specs_S2_Alkyl:=[S2_Alkyl [X,Y],\{gramm_S2_Alkyl,gramm_S1_Alkyl,op(subs(X=Y,[gramm_S1_Alkyl])),g ramm_Alkyl\},unlabelled]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "seq(count(specs_S2_Alkyl,size=i),i=0..50);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6U\"\"!\"\"\"\"\"%\"#8\"#U\"$J\"\"$-%\"%=7\"%dO\"&**3\"\" &)HK\"&d_*\"'W)z#\"'!R>)\"(#R#R#\"(cz'p\")u4D?\")*3W(e\"*!)*=,<\"***R \">\\\"+iQ\\?9\"+I0%p4%\",_@<.=\"\",tuDqR$\",6t-yw*\"-J%GVi!G\"-iQs(e0 )\".#*****>4J#\".'>,,nCm\"/esD3*y*=\"/Ha/i0Ma\"0%>P7D.b:\"0$*H;kmwW%\" 1B\\,!)))\\r7\"1f>\"o`:Lj$\"2a'4pg\"zx.\"\"2$R'4-E,I'H\"2'=qi!p^lX)\"3 HPg)36rET#\"3(HOX&eX/\")o\"4\"R?ab;o'='>\"49\"*eR]\\&y\"f&\"5/#Q)yi(pA Lf\"\"5`gq(>D3R(QX\"6*[4QX#)[$fDH\"\"6j@SF0]&)e+o$\"7QaL6c:Ks\\Z5\"76y wJGxQO\"4)H\"7)>J.6q<0H5[)\"83$=`[. " 0 "" {MPLTEXT 1 0 33 "alk:=draw(specs_S2_Alkyl,size=6);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$alkG-%%ProdG6%%'CarbonG-F&6&F(%\"XG-F&6%F(-F&6$-F& 6$F(%\"YG-%$SetG6#-F&6$F(%(EpsilonGF3F8F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "The following procedure rewrites a disubstituted alkyl in to a more readable way." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "nice:=proc(alk) subs(\{[C,X]=CX,[C,Y]=CY,[C,X,Y]=CXY\},eval(subs(\{Ep silon=NULL,Carbon=C,Prod=proc() [args] end,Set=proc() args end\},alk)) ) end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "nice(alk);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$%\"CG7%F$%\"XG7%F$7$%#CYG7#F$F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "The following procedures compute t he size and height of a given disubstituted alkyl." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 224 "size:=proc(alk) option remember; `if`(nops (alk)=2,1+convert(map(size,op(2,alk)),`+`),1+convert(map(size,[op(2..- 2,alk)]),`+`)+convert(map(size,op(-1,alk)),`+`)) end:\nsize(Carbon):=1 :\nsize(X):=0:\nsize(Y):=0:\nsize(Epsilon):=0:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 223 "height:=proc(alk) option remember; `if`(nops( alk)=2,1+max(op(map(height,op(2,alk)))),1+max(op(map(height,[op(2..-2, alk)])),op(map(height,op(-1,alk))))) end:\nheight(Carbon):=1:\nheight( X):=0:\nheight(Y):=0:\nheight(Epsilon):=0:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "Here is a disubstituted alkyl with 50 carbon atoms, its n ice representation and height." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "alk:=draw(specs_S2_Alkyl,size=50):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 10 "nice(alk);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7$%\" CG7%F$7$F$7&F$7$F$7$F$7%F$7%F$7$F$7$F$7&F$7&F$7&F$7#%#CXG7$F$7&F$7#F$F 5F57$F$7$F$7$F$F5F87%F$F5F5F5F8F97%F$F8F9F5F57%F$7$%#CYGF6F8" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "height(alk);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Tri substituted alkanes, " }{XPPEDIT 18 0 "C[n]*H[2*n-1]*X[2]*Y" "**&%\"CG 6#%\"nG\"\"\"&%\"HG6#,&*&\"\"#F'F&F'F'\"\"\"!\"\"F'&%\"XG6#\"\"#F'%\"Y GF'" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Enumerating trisubstituted \+ alkanes " }{XPPEDIT 18 0 "C[n]*H[2*n-1]*X[2]*Y" "**&%\"CG6#%\"nG\"\"\" &%\"HG6#,&*&\"\"#F'F&F'F'\"\"\"!\"\"F'&%\"XG6#\"\"#F'%\"YGF'" }{TEXT -1 51 " is equivalent to enumerating disubstituted alkyls " }{XPPEDIT 18 0 "C[n]*H[2*n-1]*X[2]" "*(&%\"CG6#%\"nG\"\"\"&%\"HG6#,&*&\"\"#F'F&F 'F'\"\"\"!\"\"F'&%\"XG6#\"\"#F'" }{TEXT -1 30 ". In this section, we \+ assume " }{XPPEDIT 18 0 "X" "I\"XG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Y" "I\"YG6\"" }{TEXT -1 40 " to be distinct. The class equation is " }{XPPEDIT 18 0 "S2_Alkyl[X,X]=Carbon*S2_Alkyl[X,X]*(Epsilon+Alky l+Alkyl^2)+Carbon*(S1_Alkyl[X]^2+S1_Alkyl[X]*X+X^2)*(Epsilon+Alkyl)" " /&%)S2_AlkylG6$%\"XGF&,&*(%'CarbonG\"\"\"&F$6$F&F&F*,(%(EpsilonGF*%&Al kylGF**$F/\"\"#F*F*F**(F)F*,(*$&%)S1_AlkylG6#F&\"\"#F**&&F66#F&F*F&F*F **$F&\"\"#F*F*,&F.F*F/F*F*F*" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "gramm_S2b_Alkyl:=S2_Alkyl[X,X]=Union(Prod(Carbo n,S2_Alkyl[X,X],Set(Alkyl,card<=2)),Prod(Carbon,Union(Prod(S1_Alkyl[X] ,S1_Alkyl[X]),Prod(S1_Alkyl[X],X),Prod(X,X)),Set(Alkyl,card<=1))):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "specs_S2b_Alkyl:=[S2_Alkyl[ X,X],\{gramm_S2b_Alkyl,gramm_S1_Alkyl,gramm_Alkyl\},unlabelled]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "seq(count(specs_S2b_Alkyl,si ze=i),i=0..50);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6U\"\"!\"\"\"\"\"$\" \"*\"#H\"#\"*\"$#G\"$k)\"%CE\"%0z\"&lO#\"&e/(\"'8)3#\"'+kh\"(aL\"=\"(t '=`\")s!fb\"\")R-TX\"*_(eA8\"*4O\\%Q\"+&e2f6\"\"+$>#yLK\"+8RHe$*\",a)* =[q#\",;sV(3y\"-\\pt)>D#\"-VH5E)['\".Q&z'pw'=\".](RPnr`\"/a'Q_pPa\"\"/ (4DyYMV%\"0:!\\fCNs7\"0Y,zI6#\\O\"1(RJ(4Q+Y5\"1)G.!y)yl*H\"1!R#QcS1!e) \"2p4]()G/bX#\"26i<\"*peS-(\"308(>@Nq$3?\"3gr?G)Hy+u&\"4^C;qC83*R;\"4: 4NNM>gLo%\"5b'36&\\Lj-P8\"5eX+\\W$f*p:Q\"6B(Q#)zrAqf)3\"\"6INZ)3v\\1u/ J\"6()GI2WiVsA&))\"77j%*)\\6E'[DBD\"7ZJ9qP\\FQM!>(\"82MK768y*)\\%[?\"8 7!>Y" "0%\"XG%\"YG" }{TEXT -1 2 ")." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Here is an example of a disubstituted alkyl with \+ 6 carbon atoms, obtained by the command " }{HYPERLNK 17 "combstruct[dr aw]" 2 "combstruct[draw]" "" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "alk:=draw(specs_S2b_Alkyl,size=6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$alkG-%%ProdG6%%'CarbonG-F&6$-F&6%F(-F&6$-F&6$F( %\"XG-%$SetG6#-F&6$F(%(EpsilonG-F36$F5F5F1F7" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 80 "The following procedure rewrites a disubstituted alkyl \+ into a more readable way." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "nice:=proc(alk) subs(\{[C,X]=CX,[C,X,X]=CX[2]\},eval(subs(\{Epsil on=NULL,Carbon=C,Prod=proc() [args] end,Set=proc() args end\},alk))) e nd:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "nice(alk);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$%\"CG7$7&F$7$%#CXG7#F$F)F)%\"XG" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Here are the 9 disubstituted compo unds " }{XPPEDIT 18 0 "C[n]*H[2*n-1]*X[2]*Y" "**&%\"CG6#%\"nG\"\"\"&% \"HG6#,&*&\"\"#F'F&F'F'\"\"\"!\"\"F'&%\"XG6#\"\"#F'%\"YGF'" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "n=3" "/%\"nG\"\"$" }{TEXT -1 1 ":" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "map(nice,allstructs(specs_S2 b_Alkyl,size=3));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7+7%%\"CG7$7#%#CX G%\"XG7#F%7%F%7$F%7$F)F)F*7$F%7%F%F-F*7$F%7$7$F(F*F)7$F%7$7$F%F'F)7$F% 7$F%F,7$F%7$F%F&7%F%F-7$F%F*7$F%7$F'F'" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 39 "Conclusion: multiply substituted alkyls" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "In the previous sections, we have enumerated th e substituted compounds " }{XPPEDIT 18 0 "C[n]*H[2*n+1]*X" "*(&%\"CG6# %\"nG\"\"\"&%\"HG6#,&*&\"\"#F'F&F'F'\"\"\"F'F'%\"XGF'" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "C[n]*H[2*n]*X*Y" "**&%\"CG6#%\"nG\"\"\"&%\"HG6#*&\" \"#F'F&F'F'%\"XGF'%\"YGF'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "C[n]*H[2*n -1]*X*Y*Z" "*,&%\"CG6#%\"nG\"\"\"&%\"HG6#,&*&\"\"#F'F&F'F'\"\"\"!\"\"F '%\"XGF'%\"YGF'%\"ZGF'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "C[n]*H[2*n -1]*X[2]*Y" "**&%\"CG6#%\"nG\"\"\"&%\"HG6#,&*&\"\"#F'F&F'F'\"\"\"!\"\" F'&%\"XG6#\"\"#F'%\"YGF'" }{TEXT -1 45 ". We could in principle enume rate the class " }{XPPEDIT 18 0 "S[p[1],`...`,p[t]]" "&%\"SG6%&%\"pG6# \"\"\"%$...G&F&6#%\"tG" }{TEXT -1 42 " of compounds obtained after sub stituting " }{XPPEDIT 18 0 "p[1]" "&%\"pG6#\"\"\"" }{TEXT -1 19 " hydr ogen atoms by " }{XPPEDIT 18 0 "X^`(1)`" ")%\"XG%$(1)G" }{TEXT -1 8 " \+ atoms, " }{XPPEDIT 18 0 "p[2]" "&%\"pG6#\"\"#" }{TEXT -1 19 " hydrogen atoms by " }{XPPEDIT 18 0 "X^`(2)`" ")%\"XG%$(2)G" }{TEXT -1 13 " ato ms, ..., " }{XPPEDIT 18 0 "p[t]" "&%\"pG6#%\"tG" }{TEXT -1 19 " hydrog en atoms by " }{XPPEDIT 18 0 "X^`(t)`" ")%\"XG%$(t)G" }{TEXT -1 33 " a toms, and one hydrogen atom by " }{XPPEDIT 18 0 "Y" "I\"YG6\"" }{TEXT -1 73 " (so as to plant the trees). Doing so would require to define \+ the class " }{XPPEDIT 18 0 "S[q[1],`...`,q[t]]" "&%\"SG6%&%\"qG6#\"\" \"%$...G&F&6#%\"tG" }{TEXT -1 10 " for each " }{XPPEDIT 18 0 "q[1]<=p[ 1]" "1&%\"qG6#\"\"\"&%\"pG6#\"\"\"" }{TEXT -1 7 ", ..., " }{XPPEDIT 18 0 "q[t]<=p[t]" "1&%\"qG6#%\"tG&%\"pG6#F&" }{TEXT -1 15 ", and for e ach " }{XPPEDIT 18 0 "q[1]<=p[1]" "1&%\"qG6#\"\"\"&%\"pG6#\"\"\"" } {TEXT -1 7 ", ..., " }{XPPEDIT 18 0 "q[t]<=p[t]" "1&%\"qG6#%\"tG&%\"pG 6#F&" }{TEXT -1 75 ", to write a recursion involving partitions into \+ 4 parts of the multiset \{" }{XPPEDIT 18 0 "X^`(1)`" ")%\"XG%$(1)G" } {TEXT -1 2 " (" }{XPPEDIT 18 0 "q[1]" "&%\"qG6#\"\"\"" }{TEXT -1 14 " \+ times), ..., " }{XPPEDIT 18 0 "X^`(t)`" ")%\"XG%$(t)G" }{TEXT -1 2 " ( " }{XPPEDIT 18 0 "q[t]" "&%\"qG6#%\"tG" }{TEXT -1 20 " times)\}. When the " }{XPPEDIT 18 0 "q[i]" "&%\"qG6#%\"iG" }{TEXT -1 60 "'s are give n, those partitions can be computed by a call to " }{HYPERLNK 17 "comb struct[allstructs]" 2 "combstruct[allstructs]" "" }{TEXT -1 146 ". It follows that we would describe and generate the grammar for multiply \+ substituted alkyls in terms of the grammar for partitions into 4 parts !" }}}}}{MARK "0 2 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 } 6#\"\"\"" }{TEXT -1 7 ", ..., " }{XPPEDIT 18 0 "q[t]<=p[t]" "1&%\"qG6#%\"tG&%\"pG6#F&" }{TEXT -1 15 ", and for e ach " }{XPPEDIT 18 0 "q[1]<=p[1]" "1&%\"qG6#\"\"\"&%\"pG6#\"\"\"" } {TEXT -1 7 ", ..., " }{XPPEDIT 18 0 "q[t]<=p[t]" "1&%\"qG6#%\"tG&%\"pG 6#F&" }{TEXT -1 75 ", to write a recursion involving partitions into \+ 4 parts of the multiset \{" }{XPPEDIT 18 0 "X^`(1)`" ")%\"XG%$(1)G" } {TEXT -1 2 " (" }{XPPEDIT 18 0 "q[1]" "&%\"qG6#\"\"\"" }{TEXT -1 14 " \+ times), ..., " }{XPPEDIT 18 0 "X^`(t)`" ")%\"XG%$(t)G" }{TEXT -1 2 " ( " }{XPPEDIT 18 0 "q[t]" "&%\"qG6#%\"tG" }{TEXT -1 20 " times)\}. When the " }{XPPEDIT 18 0 "q[i]" "&%\"qG6#%\"iG" }{TEXT -1 60 "'s are give n, those partitions can be computed by a call to " }{HYPERLNK 17 "comb struct[allstructs]" 2 "combstruct[allstructs]" "" }{TEXT -1 146 ". It follows that we would describe and generate the grammar for multiply \+ substituted alkyls in terms of the grammar for partitions into 4 parts !" }}}}}{MARK "0 2 assign+ _#lonMrassignm 8 assistantassociat5 $46XO5PRcl|sdyzςPJ 'X&M_^rSassum70#$46ag;rSassumpt 8asylasympt+- 8l; r  asymptoticW "g%6 8TCll ; r asymptotical# 68I; asymptotiqug%at  "$g%@( * ++4 6 B78 pHMXOngkll dya Y1o;0 t^ vrSatablatomcv "6B7[:cȁaՓ)o;&ratomic fungraph ; furth# onȁ0M furthermorZfutur *O v0gagaia gaingammamV gap lgapsl gather +XOmVgaus )gaussagm)gaussian8gauto[:c)gb'E!0#8 I =gbasi+7!0#+IF =gcb gcd*(6egcdex(e gdev gdfa^;gebra gegenbauerg geneogeneralL * @(+46 8TCpHl aoeF/ vr) 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1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT 256 21 "BALLS AND URNS, ETC.\n" } {TEXT 268 1 "\n" }{TEXT 257 17 "Philippe Flajolet" }}{PARA 262 "" 0 " " {TEXT -1 30 "(Version of December 14, 1996)" }}{PARA 263 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 274 "Balls and urns models ar e basic in combinatorics, statistics, analysis of algorithms, and stat istical physics. These models are nicely decomposable and their basic \+ properties can be explored using tools developed for the automatic man ipulation of combinatorial models, like " }{HYPERLNK 17 "Combstruct" 2 "combstruct" "" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 125 "As i s well-known there are four types of models, depending on whether ball s and urns are taken to be distinguishable or not." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "The four basic models" }}{PARA 0 "" 0 "" {TEXT -1 350 "We consider the placement of balls into urns in all possible ways . For definiteness, we examine only the situation of nonempty urns, so that the number of possible configurations of a fixed size (i.e., a f ixed number of balls) is always finite. If the balls are distinguishab le, we may assume them to be numbered consecutively by integers 1, 2, \+ ..., " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 200 "; in this case, we are dealing with labelled structures, and balls are labelled atoms. I f the balls are indistinguishable, then we simply regard them as anony mous unlabelled atoms (generically called " }{HYPERLNK 17 "Z" 2 "combs truct[specification]" "" }{TEXT -1 143 ", by a global convention of Co mbstruct). If the urns are distinguishable, we may view them as arrang ed in a row, so that we are dealing with a " }{HYPERLNK 17 "Sequence" 2 "combstruct[specification]" "" }{TEXT -1 36 " construction; otherwis e, we have a " }{HYPERLNK 17 "Set" 2 "combstruct[specification]" "" } {TEXT -1 20 " construction. (The " }{HYPERLNK 17 "Set" 2 "combstruct[s pecification]" "" }{TEXT -1 98 " construction of Combstruct means a mu ltiset, that is to say a set where repetitions are allowed.)" }}{PARA 0 "" 0 "" {TEXT -1 66 "Balls are not ordered within an urn, so that an urn is a priori a " }{HYPERLNK 17 "Set" 2 "combstruct[specification] " "" }{TEXT -1 52 " of balls. This gives rise to four different models :" }}{PARA 16 "" 0 "" {TEXT -1 116 "DBDU: distinguishable balls and di stinguishable urns; we are dealing with Sequences of Sets, in a labell ed universe;" }}{PARA 16 "" 0 "" {TEXT -1 113 "DBIU: distinguishable b alls and indistinguishable urns; we are dealing with Sets of Sets, in \+ a labelled universe;" }}{PARA 16 "" 0 "" {TEXT -1 121 "IBDU: indisting uishable balls and distinguishable urns; we are dealing with Sequences of Sets, in an unlabelled universe;" }}{PARA 16 "" 0 "" {TEXT -1 118 "IBIU: indistinguishable balls and indistinguishable urns; we are deal ing with Sets of Sets, in an unlabelled universe." }}{PARA 0 "" 0 "" {TEXT -1 88 "In combstruct, this is expressed by four different, but s imilar looking, specifications:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "with(combstruct);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7.%+allst ructsG%&countG%%drawG%)finishedG%'gfeqnsG%)gfseriesG%(gfsolveG%,iterst ructsG%+nextstructG%,prog_gfeqnsG%.prog_gfseriesG%-prog_gfsolveG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "DBDU:=[S,\{S=Sequence(U),U=S et(Z,card>=1)\},labelled]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "DBIU:=[S,\{S=Set(U),U=Set(Z,card>=1)\},labelled]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "IBDU:=[S,\{S=Sequence(U),U=Set(Z,ca rd>=1)\},unlabelled]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "IB IU:=[S,\{S=Set(U),U=Set(Z,card>=1)\},unlabelled]:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 57 "for spec in DBDU,DBIU,IBDU,IBIU do draw(spec ,size=10) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%)SequenceG6*-%$SetG 6#&%\"ZG6#\"\"%-F'6#&F*6#\"\"$-F'6#&F*6#\"\"&-F'6$&F*6#\"\"*&F*6#\"\"( -F'6#&F*6#\"\"#-F'6#&F*6#\"\")-F'6#&F*6#\"\"\"-F'6$&F*6#\"#5&F*6#\"\"' " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SetG6&-F$6#&%\"ZG6#\"\"&-F$6%&F )6#\"\"#&F)6#\"\")&F)6#\"#5-F$6%&F)6#\"\"$&F)6#\"\"*&F)6#\"\"(-F$6%&F) 6#\"\"%&F)6#\"\"\"&F)6#\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%)Seq uenceG6%-%$SetG6$%\"ZGF)-F'6)F)F)F)F)F)F)F)-F'6#F)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$SetG6'-F$6#%\"ZGF&-F$6$F(F(F)-F$6&F(F(F(F(" }}} {PARA 0 "" 0 "" {TEXT -1 161 "The corresponding counting sequences sat isfy natural domination conditions that one can summarize by the infor mal inequality: \"Distinguishable>Indistinguishable\"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "for spec in DBDU,DBIU,IBDU,IBIU do seq(co unt(spec,size=j),j=1..12) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6.\"\" \"\"\"$\"#8\"#v\"$T&\"%$o%\"&$HZ\"'Nea\"(hs3(\"*jvC-\"\"+tDjA;\",&fn:4 G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6.\"\"\"\"\"#\"\"&\"#:\"#_\"$.#\"$x )\"%ST\"&Z6#\"'vf6\"'q&y'\"((f8U" }}{PARA 11 "" 1 "" {XPPMATH 20 "6.\" \"\"\"\"#\"\"%\"\")\"#;\"#K\"#k\"$G\"\"$c#\"$7&\"%C5\"%[?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6.\"\"\"\"\"#\"\"$\"\"&\"\"(\"#6\"#:\"#A\"#I\"#U \"#c\"#x" }}}{PARA 0 "" 0 "" {TEXT -1 158 "In the sequel, it is conven ient to represent objects by a more concise notation. We thus introduc e \"reduction\" procedures for labelled and unlabelled objects:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 177 "lreduce:=proc(e) eval(subs( \{Set=proc() \{args\} end, Sequence=proc() [args] end\},e)) end:\nured uce:=proc(e) eval(subs(\{Set=proc() \{[args]\} end, Sequence=proc() [a rgs] end\},e)) end:" }}}{PARA 0 "" 0 "" {TEXT -1 124 "Since the set co nstruction \"\{\}\" in Maple does not keep multisets, an unlabelled (m ulti)set will be represented as \"\{[...]\}\"." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "for spec in DBDU,DBIU do lreduce(draw(spec,size =25)) od;\nfor spec in IBDU,IBIU do ureduce(draw(spec,size=25)) od;" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#79<#&%\"ZG6#\"#C<#&F&6#\"#6<#&F&6#\"# 9<#&F&6#\"\"&<#&F&6#\"#D<$&F&6#\"\"(&F&6#\"#?<#&F&6#\"\"#<#&F&6#\"\"%< #&F&6#\"#8<#&F&6#\"#<<#&F&6#\"\"\"<#&F&6#\"\"$<#&F&6#\"#=<#&F&6#\"#5<# &F&6#\"#B<#&F&6#\"\"*<#&F&6#\"#@<#&F&6#\"#><#&F&6#\"#A<$&F&6#\"\")&F&6 #\"#:<#&F&6#\"#7<#&F&6#\"\"'<#&F&6#\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<-<#&%\"ZG6#\"\"#<#&F&6#\"#><$&F&6#\"\"*&F&6#\"#@<$&F&6#\"\"\"&F &6#\"#6<$&F&6#\"\"%&F&6#\"#?<$&F&6#\"#C&F&6#\"#7<&&F&6#\"\"$&F&6#\"#5& F&6#\"#A&F&6#\"#:<$&F&6#\"#8&F&6#\"#B<%&F&6#\"\")&F&6#\"#<&F&6#\"#=<&& F&6#\"\"&&F&6#\"\"(&F&6#\"#9&F&6#\"#;<$&F&6#\"\"'&F&6#\"#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7-<#7(%\"ZGF&F&F&F&F&<#7#F&F'F'<#7%F&F&F&F)< #7$F&F&F'F'F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#7&<#7%%\"ZGF'F'F% <#73F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'<#7$F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 105 "On such simulations, we see that there tends to be fewer urns in models of type IU, but more filled ones." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 258 43 "Distinguishable balls (labelled structures)" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Distinguishable urns" }}{PARA 0 " " 0 "" {TEXT -1 156 "In this model, we deal with distinguishable balls (labelled atoms) that go in all possible way into distinguishable urn s corresponding to the specification:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "DBDU:=[S,\{S=Sequence(U),U=Set(Z,card>=1)\},labelled] :" }}}{PARA 0 "" 0 "" {TEXT -1 63 "Combinatorially, this model is the \+ same as of Surjections from " }{XPPEDIT 18 0 "[1..n]" "7#;\"\"\"%\"nG " }{TEXT -1 94 " to an initial segment of the integers. It is the one \+ that leads to larger cardinality counts." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "for j to 3 do j=map(lreduce,allstructs(DBDU,size=j)) \+ od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"\"7#7#<#&%\"ZG6#F$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"#7%7$<#&%\"ZG6#F$<#&F)6#\"\"\"7$F +F'7#<$F(F," }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/\"\"$7/7#<%&%\"ZG6#F$& F)6#\"\"#&F)6#\"\"\"7%<#F.<#F(<#F+7%F4F2F37$<$F(F+F27$F3<$F+F.7$<$F(F. F47$F2F77$F4F;7%F3F2F47%F3F4F27%F4F3F27$F9F37%F2F4F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(count(DBDU,size=j),j=0..30);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6A\"\"\"F#\"\"$\"#8\"#v\"$T&\"%$o%\"&$HZ \"'Nea\"(hs3(\"*jvC-\"\"+tDjA;\",&fn:4G\"-\"Q[$eo_\"/V/(HMT1\"\"0`y(4> $GI#\"1b8)>oacJ&\"3,f8Hqwq.8\"4B`%oDjY`&Q$\"5L6TG$>te,G*\"7:J?%QWiz(ox E\"8@=)Q2/&)*\\#[7\")\":.Ua%Q!>.)>W%[d#\";8/e%H4duO8XQa)\"=v)[1lsa9u57 z#eH\"?T\")\\$eDexv%QaO(p1\"\"@$3Rv7'[#\\oT%)fdA-S\"B$p`iAw#\\Ki@'=Rw( *e:\"CNYLNRP=N.m+&\\1ivH'\"EhodW!HL*3-!pBIE&QyME\"Gj>!\\='>kCu$[!)=,%z oNS6" }}}{PARA 0 "" 0 "" {TEXT -1 133 "Such tables are quite useful fo r checking various combinatorial conjectures. Here, we may verify that these numbers are the sequence " }{TEXT 269 5 "M2952" }{TEXT -1 8 " o f the " }{TEXT 270 33 "Encyclopedia of Integer Sequences" }{TEXT -1 92 " by Sloane and Plouffe, where they are known as the numbers of pre ferential arrangements of " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 57 " things.\nThe counting problem is solved automatically by " } {HYPERLNK 17 "combstruct[gfeqns]" 2 "combstruct[gfeqns]" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "combstruct[gfseries]" 2 "combstruct[gfseries]" " " }{TEXT -1 26 " (a series alternative to " }{HYPERLNK 17 "combstruct[ count]" 2 "combstruct[count]" "" }{TEXT -1 6 ") and " }{HYPERLNK 17 "c ombstruct[gfsolve]" 2 "combstruct[gfsolve]" "" }{TEXT -1 1 ":" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "gfeqns(op(2..3,DBDU),z);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%/-%\"UG6#%\"zG,&-%$expG6#-%\"ZGF'\" \"\"!\"\"F//-%\"SGF'*$,&F/F/F%F0F0/F-F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Order:=12: gfseries(op(2..3,DBDU),z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7%/-%\"SG6#%\"zG+=F+\"\"\"\"\"!F-\"\"\" #\"\"$\"\"#\"\"##\"#8\"\"'\"\"$#\"#D\"\")\"\"%#\"$T&\"$?\"\"\"&#\"%h: \"$S#\"\"'#\"&$HZ\"%S]\"\"(#\"&*QO\"%)o#\"\")#\"(hs3(\"'!)GO\"\"*#\")@ D3M\"(+'47\"#5#\"+tDjA;\")+o\"*R\"#6-%\"OG6#F-\"#7/-%\"UGF*+;F+F-\"\" \"#F-F2\"\"##F-F6\"\"$#F-\"#C\"\"%#F-F>\"\"&#F-\"$?(\"\"'#F-FF\"\"(#F- \"&?.%\"\")#F-FN\"\"*#F-\"(+)GO\"#5#F-FV\"#6FX\"#7/-%\"ZGF*+%F+F-\"\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "gfsolve(op(2..3,DBDU) ,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/-%\"SG6#%\"zG,$*$,&!\"#\"\" \"-%$expGF'F-!\"\"F0/-%\"ZGF'F(/-%\"UGF',&F.F-F0F-" }}}{PARA 0 "" 0 " " {TEXT -1 82 "In particular, we have found the exponential generating function (EGF) explicitly:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "S_z:=subs(\",S(z)); series(S_z,z=0,7);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$S_zG,$*$,&!\"#\"\"\"-%$ex pG6#%\"zGF)!\"\"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"zG\"\"\"\" \"!F%\"\"\"#\"\"$\"\"#\"\"##\"#8\"\"'\"\"$#\"#D\"\")\"\"%#\"$T&\"$?\" \"\"&#\"%h:\"$S#\"\"'-%\"OG6#F%\"\"(" }}}{PARA 0 "" 0 "" {TEXT -1 35 " The EGF is singular with a pole at " }{XPPEDIT 18 0 "z=ln(2)" "/%\"zG- %#lnG6#\"\"#" }{TEXT -1 72 ". This immediately gives an approximate ex pression for the coefficients:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "series(S_z,z=log(2),3);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#+),&%\"zG\"\"\"-%#lnG6#\"\"#!\"\"#F+F* !\"\"#F&\"\"%\"\"!-%\"OG6#F&\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "S_n_asympt:=1/2*n!*log(2)^(-n-1);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%+S_n_asymptG,$*&-%*factorialG6#%\"nG\"\"\")-%#lnG6# \"\"#,&F*!\"\"F2F+F+#F+F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "As u sual with meromorphic functions, the approximation is extremely good: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "for j from 0 by 5 to 30 do j,ev alf(count(DBDU,size=j)/subs(n=j,S_n_asympt),30); od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"!$\"?#HCkW$)=1*)>6O%H'Q\"!#H" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$\"\"&$\"?W)ej0U#=TLQJ>(*****!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#5$\"?M>*)[b@Q#[%[**********!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#:$\"?y\"*QtZVP)***************!#I" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$\"#?$\"?X;%f-,+++++++++\"!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#D$\"?p5-+++++++++++5!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#I$\"?\"*****************************!#I" }}}{PARA 0 "" 0 "" {TEXT -1 259 "This type of analysis can be easily generalized \+ to determine for instance the expected number of urns in a random surj ection. Such analyses may then be used to validate an a priori statist ical model by comparing theoretical predictions against empirical data ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "Indistinguishable urns (set partitions)" }}{PARA 0 "" 0 "" {TEXT -1 82 "We are now dealing with indistinguishable urns. Equivalen tly, we consider the way " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 30 " elements (the labels 1, ..., " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 63 ") may be grouped into equivalence classes in all possible ways. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "DBIU:=[S,\{S=Set(U),U=Se t(Z,card>=1)\},labelled]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "for j to 4 do j=map(lreduce,allstructs(DBIU,size=j)) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"\"7#<#<#&%\"ZG6#F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"#7$<$<#&%\"ZG6#\"\"\"<#&F)6#F$<#<$F-F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"$7'<$<#&%\"ZG6#\"\"\"<$&F)6#F$&F)6#\"\" #<$<$F/F(<#F-<#<%F-F/F(<$<#F/<$F-F(<%F'F8F4" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/\"\"%71<$<%&%\"ZG6#\"\"$&F)6#\"\"#&F)6#\"\"\"<#&F)6#F$ <%<#F,<#F(<$F/F3<%<#F/F6<$F(F3<$F6<%F(F/F3<$F:<%F(F,F3<&F:F6F7F2<%<$F, F/F7F2<%F:F7<$F,F3<#<&F(F,F/F3<$FBF;<$F7<%F,F/F3<$<$F(F/FD<%F:<$F(F,F2 <$FMF8<%F6FKF2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(count (DBIU,size=j),j=0..30);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6A\"\"\"F#\" \"#\"\"&\"#:\"#_\"$.#\"$x)\"%ST\"&Z6#\"'vf6\"'q&y'\"((f8U\")PWkF\"*A$* *3>\"+X&eHQ\"\",Z@9![5\",/)p['G)\"-fh!o2#o\".d]?UF$e\"/s`BeTs^\"0^n:;) p[Z\"1BtWQdr1X\"2YV3be+_T%\"3*G0[Hp)efW\"4`$***HAL!fQY\"5uiv=O_Y7j\\\" 6*Q*)*fg$z/v-')Q>!)R8(\"9Z,XK$4=^9!\\n%)" }}} {PARA 0 "" 0 "" {TEXT -1 133 "Such tables are quite useful for checkin g various combinatorial conjectures. Here, we may verify that these nu mbers are the sequence " }{TEXT 271 5 "M1484" }{TEXT -1 8 " of the " } {TEXT 259 33 "Encyclopedia of Integer Sequences" }{TEXT -1 48 " by Slo ane and Plouffe. They are the well-known " }{HYPERLNK 17 "Bell numbers " 2 "combinat[bell]" "" }{TEXT -1 125 " of combinatorial theory that a lso appear as moments of the Poisson distribution, in the calculus of \+ finite differences, etc." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 62 "We automatically obtain the exponential generating function as" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "gfsolve(op(2 ..3,DBIU),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/-%\"ZG6#%\"zGF(/-% \"UGF',&-%$expGF'\"\"\"!\"\"F//-%\"SGF'-F.6#F," }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "P_z:=subs(\",S(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$P_zG-%$expG6#,&-F&6#%\"zG\"\"\"!\"\"F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "series(P_z,z=0,8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+5%\"zG\"\"\"\"\"!F%\"\"\"F%\"\"##\"\"&\"\"'\"\"$# F*\"\")\"\"%#\"#8\"#I\"\"&#\"$.#\"$?(\"\"'#\"$x)\"%S]\"\"(-%\"OG6#F%\" \")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Order:=8: gfseries(o p(2..3,DBIU),z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7%/-%\" SG6#%\"zG+5F+\"\"\"\"\"!F-\"\"\"F-\"\"##\"\"&\"\"'\"\"$#F2\"\")\"\"%# \"#8\"#I\"\"&#\"$.#\"$?(\"\"'#\"$x)\"%S]\"\"(-%\"OG6#F-\"\")/-%\"UGF*+ 3F+F-\"\"\"#F-\"\"#\"\"##F-F3\"\"$#F-\"#C\"\"%#F-\"$?\"\"\"&#F-F>\"\"' #F-FB\"\"(FD\"\")/-%\"ZGF*+%F+F-\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 131 "By expanding and truncating, we obtain excellent approximations ( this is in fact a version of a formula found by Dobinski in 1877):" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "P_n_asympt:=exp(-1)*Sum(k^n /k!,k=0..2*n);\nfor j by 3 to 20 do j,evalf(count(DBIU,size=j)/subs(n= j,P_n_asympt),30); od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+P_n_asymp tG*&-%$expG6#!\"\"\"\"\"-%$SumG6$*&)%\"kG%\"nGF*-%*factorialG6#F0F)/F0 ;\"\"!,$F1\"\"#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"\"$\"?oNP9!o< E_HU\"49f8!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"%$\"?=z)pS%o]A'[Y Y@0+\"!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"($\"?d8Vx^jn#H<21+++ \"!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#5$\"?fRk7d&yf6,+++++\"!#H " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#8$\"?'GC0GD0++++++++\"!#H" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"#;$\"?C$R3+++++++++++\"!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#>$\"?,+++++++++++++5!#H" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 47 "Indistinguishable balls (unlabelled struc tures)" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "Distinguishable urns (i nteger compositions)" }}{PARA 0 "" 0 "" {TEXT -1 31 "We start from the specification" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "IBDU:=[S, \{S=Sequence(U),U=Sequence(Z,card>=1)\},unlabelled]:" }}}{PARA 0 "" 0 "" {TEXT -1 89 "In this particular case, as balls are indistinguishabl e, we may as well consider urns as " }{HYPERLNK 17 "Sequence" 2 "combs truct[specification]" "" }{TEXT -1 250 " of atoms. The reason for doin g this is a simpler form of generating functions (as we do not have to go unnecessarily through Polya operators) as well as faster computati ons. We can check that this new version is equivalent to the earlier o ne, namely" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "IBDU_0:=[S,\{S =Sequence(U),U=Set(Z,card>=1)\},unlabelled]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "seq(count(IBDU,size=j),j=0..20); seq(count(IBDU_ 0,size=j),j=0..20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "67\"\"\"F#\"\"#\" \"%\"\")\"#;\"#K\"#k\"$G\"\"$c#\"$7&\"%C5\"%[?\"%'4%\"%#>)\"&%Q;\"&oF$ \"&Ob'\"'s58\"'W@E\"')GC&" }}{PARA 11 "" 1 "" {XPPMATH 20 "67\"\"\"F# \"\"#\"\"%\"\")\"#;\"#K\"#k\"$G\"\"$c#\"$7&\"%C5\"%[?\"%'4%\"%#>)\"&%Q ;\"&oF$\"&Ob'\"'s58\"'W@E\"')GC&" }}}{PARA 0 "" 0 "" {TEXT -1 91 "Of c ourse, here we recognize the powers of two: the result is combinatoria lly obvious since" }}{PARA 0 "" 0 "" {TEXT -1 88 "a partition can be o btained by inserting arbitrary cuts in the integer interval 1, ..., " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 49 ". We can also check this w ith combstruct[gfsolve]" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "g fsolve(op(2..3,IBDU),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/-%\"ZG6 #%\"zGF(/-%\"SGF'*&,&!\"\"\"\"\"F(F/F/,&F.F/F(\"\"#F./-%\"UGF',$*&F(F/ F-F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "SS_z:=subs(\",S(z ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%SS_zG*&,&!\"\"\"\"\"%\"zGF(F (,&F'F(F)\"\"#F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "series( SS_z,z=0,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+9%\"zG\"\"\"\"\"!F% \"\"\"\"\"#\"\"#\"\"%\"\"$\"\")\"\"%\"#;\"\"&\"#K\"\"'\"#k\"\"(\"$G\" \"\")\"$c#\"\"*-%\"OG6#F%\"#5" }}}}{SECT 1 {PARA 257 "" 0 "" {TEXT 260 43 "Indistinguishable urns (integer partitions)" }}{PARA 0 "" 0 " " {TEXT -1 31 "We start from the specification" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "IBIU:=[S,\{S=Set(U),U=Sequence(Z,card>=1)\},unla belled]:" }}}{PARA 0 "" 0 "" {TEXT -1 99 "Combinatorially, we are spec ifying integer partitions that describe the occupancy profile of urns ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "for j to 6 do j=map(ure duce,allstructs(IBIU,size=j)) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ \"\"\"7#<#7#7#%\"ZG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"#7$<#7$7#% \"ZGF(<#7#7$F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"$7%<#7#7%%\"Z GF)F)<#7%7#F)F,F,<#7$7$F)F)F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\" %7'<#7$7$%\"ZGF)F(<#7$7%F)F)F)7#F)<#7%F(F-F-<#7&F-F-F-F-<#7#7&F)F)F)F) " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/\"\"&7)<#7$7%%\"ZGF)F)7$F)F)<#7%F *F*7#F)<#7&F*F-F-F-<#7#7'F)F)F)F)F)<#7$F-7&F)F)F)F)<#7%F(F-F-<#7'F-F-F -F-F-" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/\"\"'7-<#7(7#%\"ZGF(F(F(F(F( <#7$7%F)F)F)F,<#7&F,F(F(F(<#7&7$F)F)F1F(F(<#7%F1F1F1<#7%F(F(7&F)F)F)F) <#7'F1F(F(F(F(<#7%F,F1F(<#7$F(7'F)F)F)F)F)<#7#7(F)F)F)F)F)F)<#7$F1F6" }}}{PARA 0 "" 0 "" {TEXT -1 178 "Naturally, since we are dealing with \+ sets of summands (the order does not count), we may as well regard the se objects as an increasing sequence of summands that sum to the size " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 79 ", or equivalently as \" staircases\" with size being the area below the staircase." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "preduce:=proc(e) sort(eval(subs(\{S et=proc() [args] end, Sequence=proc() nargs end\},e))) end:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "rand_part:=draw(IBIU,size=10 0): ureduce(rand_part); preduce(rand_part);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<#7C7$%\"ZGF&F%F%F%F%F%7#F&F'F'F'F'F'F'F'F'F'F'F'F'F'F' F'F'7'F&F&F&F&F&F(F(F(F(F(F(F(73F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&F&70F& F&F&F&F&F&F&F&F&F&F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7C\"\"\" F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$\"\"#F%F%F%F%F%\"\"&F&F&F&F&F&F&F&\"#9 \"#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 " There are much fewer structures than in previous models:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(count(IBIU,size=j),j=0..30);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6A\"\"\"F#\"\"#\"\"$\"\"&\"\"(\"#6\"#: \"#A\"#I\"#U\"#c\"#x\"$,\"\"$N\"\"$w\"\"$J#\"$(H\"$&Q\"$!\\\"$F'\"$#z \"%-5\"%b7\"%v:\"%e>\"%OC\"%5I\"%=P\"%lX\"%/c" }}}{PARA 0 "" 0 "" {TEXT -1 209 "The random generation process is nontrivial as one must \+ generate objects up to certain symmetries. The first time, counting ta bles are set up on the fly, so that random generation takes a few seco nds for size " }{XPPEDIT 18 0 "n<=100" "1%\"nG\"$+\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "for i from 0 by 20 to 100 \+ do i,preduce(draw(IBIU,size=i)); od;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$\"\"!%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#?7*\"\"\"F%F%F %F%\"\"$\"\"%\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#S77\"\"\"F%F% F%F%F%F%F%F%F%\"\"#F&F&F&F&F&F&F&\"\"%F'\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20