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Motif Statistics
, 1999
"... We present a complete analysis of the statistics of number of occurrences of a regular expression pattern in a random text. This covers "motifs" widely used in computational biology. Our approach is based on: (i) a constructive approach to classical results in theoretical computer science ..."
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We present a complete analysis of the statistics of number of occurrences of a regular expression pattern in a random text. This covers "motifs" widely used in computational biology. Our approach is based on: (i) a constructive approach to classical results in theoretical computer science (automata and formal language theory), in particular, the rationality of generating functions of regular languages; (ii) analytic combinatorics that is used for deriving asymptotic properties from generating functions; (iii) computer algebra for determining generating functions explicitly, analysing generating functions and extracting coefficients efficiently. We provide constructions for overlapping or nonoverlapping matches of a regular expression. A companion implementation produces multivariate generating functions for the statistics under study. A fast computation of Taylor coefficients of the generating functions then yields exact values of the moments with typical application to random t...
Convolutions of inverse linear functions via multivariate residues
, 2004
"... lj(z1,...,zd) n j be the quotient of an analytic function by a product of linear functions lj: = 1 − � bijzi. We compute asymptotic formulae for the Taylor coefficients of F via the multivariate residue approach begun by [BM93]. By means of stratified Morse theory, we are able to give a short and f ..."
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Cited by 10 (6 self)
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lj(z1,...,zd) n j be the quotient of an analytic function by a product of linear functions lj: = 1 − � bijzi. We compute asymptotic formulae for the Taylor coefficients of F via the multivariate residue approach begun by [BM93]. By means of stratified Morse theory, we are able to give a short and fully implementable algorithm for determining an asymptotic series expansion.
REGULAR LANGUAGES AND THEIR GENERATING FUNCTIONS: THE INVERSE PROBLEM
"... Abstract. The technique of determining a generating function for an unambiguous contextfree language is known as the Schützenberger methodology. For regular languages, Elena Barcucci et al. proposed an approach for inverting this methodology. This idea allows a combinatorial interpretation (by mean ..."
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Cited by 5 (0 self)
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Abstract. The technique of determining a generating function for an unambiguous contextfree language is known as the Schützenberger methodology. For regular languages, Elena Barcucci et al. proposed an approach for inverting this methodology. This idea allows a combinatorial interpretation (by means of a regular language) of certain positive integer sequences that are defined by Cfinite recurrences. In this paper we present a Maple implementation of this inverse methodology and describe various applications. We give a short introduction to the underlying theory, i.e., the question of deciding Nrationality. In addition, some aspects and problems concerning the implementation are discussed; some examples from combinatorics illustrate its applicability. 1.
Checking and bounding the solutions of some recurrence relations
, 2004
"... Recurrence relations play an important role in the field of complexity analysis since complexity measures can often be elegantly expressed by systems of such relations. This justifies the interest in automatic, precise, efficient and correct systems able to solve or to approximate the solution of s ..."
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Cited by 3 (1 self)
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Recurrence relations play an important role in the field of complexity analysis since complexity measures can often be elegantly expressed by systems of such relations. This justifies the interest in automatic, precise, efficient and correct systems able to solve or to approximate the solution of systems of recurrence relations. Assume such a system is built. Since closedform solutions for recurrences of even modest complexity can be so big and complex to be unmanageable, how can confidence on such a system be gained? How can we quickly validate or perhaps disprove its results? And, in those cases where the exact solution is too complex to be of practical use, how can we trade precision for efficiency by approximating them from below and from above? We also concern ourselves with a problem related to the handling of sets of solutions of recurrence relations: how can we confine by means of a lower bound and an upper bound a set of such solutions? We provide some solutions to these problems where we are careful to rely, whenever possible, on fast integer computations and/or conditions that are easy to check in a completely automatic way. The ongoing experimental evaluation of these ideas is giving very promising results, showing orderofmagnitude speedups over the more traditional methods.
MonomerDimer Tilings
, 1997
"... ,2*v[6]v[3]; , 1.934141616 1.372041543 , 1.939128039 1.797297765 Generating functions archive Conclusion We showed that various parameters related to dimermonomer tilings such as the average number of pieces or the relative numbers of horizontal dimers and monomers in a random tiling of height ..."
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,2*v[6]v[3]; , 1.934141616 1.372041543 , 1.939128039 1.797297765 Generating functions archive Conclusion We showed that various parameters related to dimermonomer tilings such as the average number of pieces or the relative numbers of horizontal dimers and monomers in a random tiling of height n in a strip of width m can be computed very easily using Combstruct and ratasympt. More precisely Combstruct is used to define the grammars the tilings are derived from, and ratasympt is used to perform asymptotic expansions on rational fractions with rational coeficients. About the number ( ) g n of different tilings of a nxn chessboard, altough the method presented here is limited due to the exponential growth of the grammar describing these tilings, the very first terms computed provide provably good upper and lower bounds for the connectivity constant ( ) g n
RIGOROUS UNIFORM APPROXIMATION OF DFINITE FUNCTIONS USING CHEBYSHEV EXPANSIONS
, 2013
"... A wide range of numerical methods exists for computing polynomial approximations of solutions of ordinary differential equations based on Chebyshev series expansions or Chebyshev interpolation polynomials. We consider the application of such methods in the context of rigorous computing (where we ne ..."
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A wide range of numerical methods exists for computing polynomial approximations of solutions of ordinary differential equations based on Chebyshev series expansions or Chebyshev interpolation polynomials. We consider the application of such methods in the context of rigorous computing (where we need guarantees on the accuracy of the result), and from the complexity point of view. It is wellknown that the ordern truncation of
The Positivity Set of a Recurrence Sequence
, 2008
"... We consider real sequences (fn) that satisfy a linear recurrence with constant coefficients. We show that the density of the positivity set of such a sequence always exists. In the special case where the sequence has no positive dominating characteristic root, we establish that the density is positi ..."
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We consider real sequences (fn) that satisfy a linear recurrence with constant coefficients. We show that the density of the positivity set of such a sequence always exists. In the special case where the sequence has no positive dominating characteristic root, we establish that the density is positive. Furthermore, we determine the values that can occur as density of such a positivity set, both for the special case just mentioned and in general. Keywords: Recurrence sequence, inequality, KroneckerWeyl theorem.
Das inverse Problem FriedrichAlexanderUniversität ErlangenNürnberg
"... anderer als der angegebenen Quellen angefertigt habe und dass die Arbeit in gleicher oder ähnlicher Form noch keiner anderen Prüfungsbehörde vorgelegen hat und von dieser als Teil einer Prüfungsleistung angenommen wurde. Alle Ausführungen, die wörtlich oder sinngemäß übernommen wurden, sind als solc ..."
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anderer als der angegebenen Quellen angefertigt habe und dass die Arbeit in gleicher oder ähnlicher Form noch keiner anderen Prüfungsbehörde vorgelegen hat und von dieser als Teil einer Prüfungsleistung angenommen wurde. Alle Ausführungen, die wörtlich oder sinngemäß übernommen wurden, sind als solche gekennzeichnet. Erlangen, den The technique of determining a generating function for an unambiguous contextfree language, is known as the Schützenberger methodology. For regular languages, Barcucci et al. proposed1 a technology for inverting this methodology, which allows to give a combinatorial interpretation (by means of a regular expression) of certain positive integer sequences that are defined by a linear recurrence. In this thesis, we provide an implementation of this inverse methodology in Maple. Therefore, a detailed introduction to the underlying theory, i.e., the theory of formal power series and especially the question of deciding Nrationality, is given. Further, various aspects and problems concerning the implementation are discussed, and some examples from combinatorics illustrate its applicability. Kurzzusammenfassung