on the closest string and substring problems the problem of finding a center string that is close to everygiven string arises in computational molecular biology and codingtheory this problem has two versions the closest string problemand the closest substring problem given a set of strings s s1 s2 sn each of length m the closest stringproblem is to find the smallest d and a string s of lengthm which is within hamming distance d to eachsi e s this problem comes fromcoding theory when we are looking for a code not too far away froma given set of codes closest substring problem with an additionalinput integer l asks for the smallest d and a strings of length l which is within hamming distance daway from a substring of length l of each si this problemis much more elusive than the closest string problem the closestsubstring problem is formulated from applications in findingconserved regions identifying genetic drug targets and generatinggenetic probes in molecular biology whether there are efficientapproximation algorithms for both problems are major open questionsin this area we present two polynomial time approximationalgorithms with approximation ratio 1 e for any smalle to settle both questions

a preparation theorem for weierstrass systems it is shown that lion and rolin s preparation theorem for globally subanalytic functions holds for the collection of definable functions in any expansion of the real ordered field by a weierstrass system

keeping track of limit cycles this article uses analytic geometry methods to bound the number of limit cycles of bounded period for a planar polynomial vector fields it introduces the notion of limit periodic sets and shows that the hilbert s 16th problem can be reduced to the existence of a uniform bound on its cyclicity it also shows the existence of a universal bound on the number of limit cycles in the perturbation of a linear focus as a consequence of the noetherian property of the bautin ideal

two non holonomic lattice walks in the quarter plane we present two classes of random walks restricted to the quarter plane with non holonomic generating functions the non holonomicity is established using the iterated kernel method a variant of the kernel method this adds evidence to a recent conjecture on combinatorial properties of walks with holonomic generating functions m mishna classifying lattice walks in the quarter plane j combin theory ser a 116 2009 460 477 the method also yields an asymptotic expression for the number of walks of length n

a holonomic systems approach to special functions identities abstract we observe that many special functions are solutions of so called holonomic systems bernstein s deep theory of holonomic systems is then invoked to show that any identity involving sums and integrals of products of these special functions can be verified in a finite number of steps this is partially substantiated by an algorithm that proves terminating hypergeometric series identities and that is given both in english and in maple

a polynomial time approximation scheme for embedding hypergraph in a weighted cycle the problem of minimum congestion hypergraph embedding in a weighted cycle mchewc is to embed the hyperedges of a hypergraph as paths in a weighted cycle such that the maximum congestion is minimized this problem is np hard in this paper we present a polynomial time approximation scheme ptas for this problem

algebraic families of analytic functions i the aim of this paper is to consider analytic functions arising in connection with algebraic differential equations from the following point of view taylor coefficients of such functions usually turn out to be polynomials or rational functions in the parameters of the problem following pioneering work of bautin 1 we relate analytic properties of the functions considered with algebraic properties of various ideals generated by their taylor coefficients bautin s approach has been successfully applied to the study of the poincare mapping of the planar polynomial systems in 4 9 27 and to the study of the period function of polynomial hamiltonian systems in 5 6 12 13 however the study of the poincare mapping of the polynomial differential equations closely related to the second part of hilbert 16th problem is probably the most difficult among a wide variety of problems to which bautin s approach can be applied in particular counting zeroes of solutions of algebraic differential equations is an important and mostly open problem to our best knowledge the first application of bautin s method to this problem was given in 26 in the present paper we systematically apply this method in the following two situations to algebraic functions and to solutions of the first order polynomial differential equations higher order linear equations are also considered in section 5 article no de963250

estimating species richness by a poisson compound gamma model we propose a poisson compound gamma approach for species richness estimation based on the denseness and nesting properties of the gamma mixture we fix the shape parameter of each gamma component at a unified value and estimate the mixture using nonparametric maximum likelihood a least squares crossvalidation procedure is proposed for the choice of the common shape parameter the performance of the resulting estimator of n is assessed using numerical studies and genomic data

an elementary solution of gessel s walks in the quadrant abstract around 2000 ira gessel conjectured that the number of lattice walks in the quadrant n 2 starting and ending at the origin 0 0 and taking their steps in had a simple hypergeometric form in the following decade this problem was recast in the systematic study of walks with small steps that is steps in 1 0 1 2 confined to the quadrant the generating functions of such walks are archetypal solutions of partial discrete differential equations a complete classification of quadrant walks according to the nature of their generating function algebraic d finite or not is now available but gessel s walks remained mysterious because they were the only model among the 23 d finite ones that had not been given an elementary solution instead gessel s conjecture was first proved using an inventive computer algebra approach in 2008 a year later the associated three variate generating function was proved to be algebraic by a computer algebra tour de force this was re proved recently using elaborate complex analysis machinery we give here an elementary and constructive proof our approach also solves other quadrant models with multiple steps recently proved to be algebraic via computer algebra

counting real zeros of analytic functions satisfying linear ordinary differential equations abstract we suggest an explicit procedure to establish upper bounds for the number of real zeros of analytic functions satisfying linear ordinary differential equations with meromorphic coefficients if the equation formula has no singular points in a small neighborhooduof a real segmentk all the coefficientsaj t have absolute value aonuanda0 t 1 then any solution of this equation may have no more than a zeros onk where u k is a geometric constant depending only onkandu if the principal coefficienta0 t is nonconstant but its modulus is at leasta 0 somewhere onk then the number of real zeros onkof any solution analytic inu does not exceed a a with some u k

small amplitude limit cycles and the distribution of zeros of families of analytic functions we estimate the expected number of limit cycles of a planar polynomial vector field situated in a neighbourhood of the origin provided that the field in a larger neighbourhood is close enough to a linear center our main tool is a distributional inequality for the number of zeros of some families of univariate holomorphic functions depending analytically on a parameter we obtain this inequality by methods of pluripotential theory this inequality also implies versions of a strong law of large numbers and the central limit theorem for a probabilistic scheme associated with the distribution of zeros

counting elements and geodesics in thompson s group f we present two quite different algorithms to compute the number of elements in the sphere of radius n of thompson s group f with standard generating set the first of these requires exponential time and polynomial space but additionally computes the number of geodesics and is generalisable to many other groups the second algorithm requires polynomial time and space and allows us to compute the size of the spheres of radius n with n 1500 using the resulting series data we find that the growth rate of the group is bounded above by 2 62167 this is very close to guba s lower bound of 3 5 2 indeed numerical analysis of the series data strongly suggests that the growth rate of the group is exactly 3 5 2

differential algebra on lattice green functions and calabi yau operators we revisit miscellaneous linear differential operators mostly associated with lattice green functions in arbitrary dimensions but also calabi yau operators and order 7 operators corresponding to exceptional differential galois groups we show that these irreducible operators are not only globally nilpotent but are such that they are homomorphic to their formal adjoints considering these operators or sometimes equivalent operators we show that they are also such that either their symmetric square or their exterior square have a rational solution this is a general result an irreducible linear differential operator homomorphic to its formal adjoint is necessarily such that either its symmetric square or its exterior square has a rational solution and this situation corresponds to the occurrence of a special differential galois group we thus define the notion of being special geometry for a linear differential operator if it is irreducible globally nilpotent and such that it is homomorphic to its formal adjoint since many derived from geometry n fold integrals periods occurring in physics are seen to be diagonals of rational functions we address several examples of minimal order operators annihilating diagonals of rational functions and remark that they also seem to be systematically associated with irreducible factors homomorphic to their adjoint

analytic models and ambiguity of context free languages we establish that several classical context free languages are inherently ambiguous by proving that their counting generating functions when considered as analytic functions exhibit some characteristic form of transcendental behaviour to that purpose we survey some general results on elementary analytic properties and enumerative uses of algebraic functions in relation to formal languages in particular the paper contains a general density theorem for unambiguous context free languages

complements of subanalytic sets and existential formulas for analytic functions we show that the complement of a subanalytic set defined by real analytic functions from any subalgebra closed under differentiation is a subanalytic set defined by the functions from the same subalgebra this result has an equivalent formulation in logic consider an expression built from functions as above using equalities and inequalities as well as existential and universal quantifiers such an expression is equivalent to an existential expression involving functions from the same class provided that the variables approach neither infinity nor the boundary of the domain

bernstein inequalities and applications to analytic geometry and differential equations we introduce new methods of complex analysis inequalities of bernstein type to study projections of analytic sets these techniques are applied to problems of bifurcations of periodic orbits of differential equations such as the local hilbert s 16th problem

an algorithmic proof theory for hypergeometric ordinary and q multisum integral identities it is shown that every proper hypergeometric multisum integral identity orq identity with a fixed number of summations and or integration signs possesses a short computer constructible proof we give a fast algorithm for finding such proofs most of the identities that involve the classical special functions of mathematical physics are readily reducible to the kind of identities treated here we give many examples of the method including computer generated proofs of identities of mehta dyson selberg hille hardy q saalschutz and others the prospect of using the method for proving multivariate identities that involve an arbitrary number of summations integrations is discussed

walks confined in a quadrant are not always d finite we consider planar lattice walks that start from a prescribed position take their steps in a given finite subset of z2 and always stay in the quadrant x 0 y 0 we first give a criterion which guarantees that the length generating function of these walks is d finite that is satisfies a linear differential equation with polynomial coefficients this criterion applies among others to the ordinary square lattice walks then we prove that walks that start from 1 1 take their steps in 2 1 1 2 and stay in the first quadrant have a non d finite generating function our proof relies on a functional equation satisfied by this generating function and on elementary complex analysis

directed paths in a wedge models of directed paths have been used extensively in the scientific literature to model linear polymers in this paper we examine a directed path model of a linear polymer in a confining geometry a wedge we are particularly interested in cn the number of directed lattice paths of length n steps which take steps in the north east and south east directions and are confined in the wedge y x p where p is an integer we examine the case p 2 in detail and show that the generating function satisfies a functional equation with quartic kernel we give a formal solution by using the kernel method show that this model is not d finite and determine asymptotics to leading order for cn in particular the number of paths of length n in the wedge y x 2 is given by where the constant 0 67874 can be determined to arbitrary accuracy with little computational effort

sister celine s technique and its generalizations abstract sister celine fasenmyer s technique for obtaining pure recurrence relations for hypergeometric polynomials is formalized and generalized in various directions applications include algorithms for verifying any given binomial coefficients identity and any identities involving sums and integrals of products of special functions this is shown to lead to a new approach to the theory of special functions which allows a natural definition of special functions of several variables

bayesian shrinkage estimation of the relative abundance of mrna transcripts using sage sage est une technique permettant de quantifier l expression d un gene dans un tissu biologique cette technique aboutit a l obtention de donnees de comptage modelisables par une loi multinomiale possedant les deux caracteristiques suivantes asymetrie de la distribution des frequences relatives et echantillon de petite taille par rapport a la dimension de la loi multinomiale en raison de ces caracteristiques un echantillon sage peut manquer un grand nombre de types d arnm pourtant presents dans le tissu des lors qu ils ignorent ce probleme des types manquants des estimateurs empiriques de la frequence relative des types d arnm tendent naturellement a surestimer la frequence des types rares trouves au moins une fois lesquels forment la grande majorite des types observes nous avons donc developpe une nouvelle procedure d estimation bayesienne comportant une information a priori tenant compte des caracteristiques precitees cette procedure aboutit a des estimateurs non lineaires a retrecisseur shrinkage plus efficaces que les estimateurs du maximum de vraisemblance la distribution a priori est constituee d un melange de lois de dirichlet les types arnm etant stochastiquement classes en types rares et abondants une distribution multivariee a priori est calculee separement pour chacune des deux classes differentes simulations montrent que sur les scenarios sage envisages cet estimateur a un meilleur lmse somme des ecarts quadratiques moyens des erreurs sur la frequence cumulee sur l ensemble des types que l estimateur du maximum de vraisemblance c est a dire qu il propose pour tous les echantillons simules des profils de frequences relatives plus proches de la realite au sens de la distance euclidienne nous appliquons cette methode a une bibliotheque sage de tissus issus de colons normaux et discutons de ses implications quant a l evaluation de l expression differentielle

elimination of variables in linear solvable polynomial algebras and holonomicity abstract let k be a field of characteristic 0 based on the gelfand kirillov dimension computation of modules over solvable polynomial k algebras where solvable polynomial algebras are in the sense of a kandri rody and v weispfenning 1990 j symbolic comput 9 1 26 we prove that the elimination lemma obtained from d zeilberger 1990 j comput appl math 32 321 368 by using holonomic modules over the weyl algebra a n k and used in the automatic proving of special function identities holds for a class of solvable polynomial k algebras without any holonomicity restriction this opens a way to the solution of the extension contraction problem stemming from the automatic proving of multivariate identities with respect to the finiteness in the sense of f chyzak and b salvy 1998 j symbolic comput 26 187 227 it also yields a holonomicity so that automatic proving of multivariate identities may be dealt with by manipulating polynomial function coefficients instead of rational functions

walks on the slit plane in the first part of this paper we enumerate exactly walks on the square lattice that start from the origin but otherwise avoid the half line we call them walks on the slit plane we count them by their length and by the coordinates of their endpoint the corresponding three variable is algebraic of degree 8 moreover for any point i j the length for walks of this type ending at i j is also algebraic of degree 2 or 4 and involves the famous catalan numbers

canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant this article contains an elementary constructive proof of resolution of singularities in characteristic zero our proof applies in particular to schemes of finite type and to analytic spaces so we recover the great theorems of hironaka we introduce a discrete local invariant rm inv x a whose maximum locus determines a smooth centre of blowing up leading to desingularization to define rm inv x we need only to work with a category of local ringed spaces x x cal o x satisfying certain natural conditions if a in x then rm inv x a depends only on widehat cal o x a more generally rm inv x is defined inductively after any sequence of blowings up whose centres have only normal crossings with respect to the exceptional divisors and lie in the constant loci of rm inv x cdot the paper is self contained and includes detailed examples one of our goals is that the reader understand the desingularization theorem rather than simply know it is true

bifurcation of critical periods for plane vector fields on etudie un probleme de bifurcation dans des familles de champs de vecteurs analytiques plans qui ont un centre non degenere a l origine pour toutes les valeurs d un parametre r n on definit la fonction periode p on determine les points critiques de cette fonction pres du centre avec comme parametre de bifurcation

ising n fold integrals as diagonals of rational functions and integrality of series expansions we show that the n fold integrals n of the magnetic susceptibility of the ising model as well as various other n fold integrals of the ising class or n fold integrals from enumerative combinatorics like lattice green functions correspond to a distinguished class of functions generalizing algebraic functions they are actually diagonals of rational functions as a consequence the power series expansions of the analytic at x 0 solutions of these linear differential equations derived from geometry are globally bounded which means that after just one rescaling of the expansion variable they can be cast into series expansions with integer coefficients we also give several results showing that the unique analytical solution of calabi yau odes and more generally picard fuchs linear odes with solutions of maximal weights are always diagonals of rational functions besides in a more enumerative combinatorics context generating functions whose coefficients are expressed in terms of nested sums of products of binomial terms can also be shown to be diagonals of rational functions we finally address the question of the relations between the notion of integrality series with integer coefficients or more generally globally bounded series and the modularity of odes

a human proof of gessel s lattice path conjecture gessel walks are lattice paths confined to the quarter plane that start at the origin and consist of unit steps going either west east south west or north east in 2001 ira gessel conjectured a nice closed form expression for the number of gessel walks ending at the origin in 2008 kauers koutschan and zeilberger gave a computer aided proof of this conjecture the same year bostan and kauers showed again using computer algebra tools that the complete generating function of gessel walks is algebraic in this article we propose the first human proofs of these results they are derived from a new expression for the generating function of gessel walks in terms of weierstrass zeta functions

on computing the coefficients of bivariate holonomic formal series in this work we study the problem of computing the coefficients of holonomic formal series in two commuting variables given a formal series x y n k 0 cnkxnyk specified by a holonomic system j 0d1 pj x y xj 0 and j 0d2qj x y yj 0 with a suitable finite set of intial conditions xayb x y we show that the coefficient xiyj x y can be computed in time o i j under the uniform cost criterion

twenty combinatorial examples of asymptotics derived from multivariate generating functions let a bf r bf r in mathbb n d be a d dimensional array of numbers for which the generating function f bf z sum bf r a bf r bf z bf r is meromorphic in a neighborhood of the origin for example f may be a rational multivariate generating function we discuss recent results that allow the effective computation of asymptotic expansions for the coefficients of f our purpose is to illustrate the use of these techniques on a variety of problems of combinatorial interest the survey begins by summarizing previous work on the asymptotics of univariate and multivariate generating functions next we describe the morse theoretic underpinnings of some new asymptotic techniques we then quote and summarize these results in such a way that only elementary analyses are needed to check hypotheses and carry out computations the remainder of the survey focuses on combinatorial applications such as enumeration of words with forbidden substrings edges and cycles in graphs polyominoes and descents in permutations after the individual examples we discuss three broad classes of examples namely functions derived via the transfer matrix method those derived via the kernel method and those derived via the method of lagrange inversion these methods have the property that generating functions derived from them are amenable to our asymptotic analyses and we describe further machinery that facilitates computations for these classes of examples

stability under integration of sums of products of real globally subanalytic functions and their logarithms we study lebesgue integration of sums of products of globally subanalytic functions and their logarithms called constructible functions our first theorem states that the class of constructible functions is stable under integration the second theorem treats integrability conditions in fubini type settings and the third result gives decay rates at infinity for constructible functions further we give preparation results for constructible functions related to integrability conditions

lebesgue classes and preparation of real constructible functions abstract we call a function constructible if it has a globally subanalytic domain and can be expressed as a sum of products of globally subanalytic functions and logarithms of positively valued globally subanalytic functions for any q 0 and constructible functions f and on e r n we prove a theorem describing the structure of the set x p e 0 f x l p x q where x q is the positive measure on r n whose radon nikodym derivative with respect to the lebesgue measure is x q y x y q we also prove a closely related preparation theorem for f and these results relate analysis the study of l p spaces with geometry the study of zero loci

minimax multiple shrinkage estimation on presente des nouveaux estimateurs qui incorporent l information partielle a priori en utilisant les donnees comme emulation du comportement et du risque des estimateurs consideres les plus efficaces

the elementary theory of restricted analytic fields with exponentiation numbers with exponentiation is model complete when we combine this with hovanskii s finiteness theorem 9 it follows that the real exponential field is o minimal in o minimal expansions of the real field the definable subsets of r share many of the nice structural properties of semialgebraic sets for example definable subsets have only finitely many connected components definable sets can be stratified and triangulated and continuous definable maps are piecewise trivial see 5 in this paper we will prove a quantifier elimination result for the real field augmented by exponentiation and all restricted analytic functions and use this result to obtain o minimality we were led to this while studying work of ressayre 13 and several of his ideas emerge here in simplified form however our treatment is formally independent of the results of 16 17 9 and 13

minimum congestion hypergraph embedding in a cycle the minimum congestion hypergraph embedding in a cycle mchec problem is to embed the n edges in an m vertex hypergraph as paths in a cycle on the same number of vertices such that congestion the maximum number of paths that use any single edge in the cycle is minimized the mchec problem has applications in electronic design automation and parallel computing in this paper it is proven that the mchec problem is np complete an o nm sup k 1 algorithm is described that computes an embedding with congestion k or determines that such an embedding does not exist finally a linear time approximation algorithm for arbitrary instances is presented that computes an embedding whose congestion is at most three times optimal

local zero estimates and effective division in rings of algebraic power series we give a necessary condition for algebraicity of finite modules over the ring of formal power series this condition is given in terms of local zero estimates in fact we show that this condition is also sufficient when the module is a ring with some additional properties to prove this result we show an effective weierstrass division theorem and an effective solution to the ideal membership problem in rings of algebraic power series finally we apply these results to prove a gap theorem for power series which are remainders of the grauert hironaka galligo division theorem

estimating the number of species a review how many kinds are there suppose that a population is partitioned into c classes in many situations interest focuses not on estimation of the relative sizes of the classes but on estimation of c itself for example biologists and ecologists may be interested in estimating the number of species in a population of plants or animals numismatists may be concemed with estimating the number of dies used to produce an ancient coin issue and linguists may be interested in estimating the size of an author s vocabulary in this article we review the problem of statistical estimation of c many approaches have been proposed some purely data analytic and others based in sampling theory in the latter case numerous variations have been considered the population may be finite or infinite if finite samples may be taken with replacement multinomial sampling or without replacement hypergeometric sampling or by bernoulli sampling if infinite sampling may be multinomial or bernoulli or the sample may be th

basic analytic combinatorics of directed lattice paths this paper develops a unified enumerative and asymptotic theory of directed two dimensional lattice paths in half planes and quarter planes the lattice paths are specified by a finite set of rules that are both time and space homogeneous and have a privileged direction of increase they are then essentially one dimensional objects the theory relies on a specific kernel method that provides an important decomposition of the algebraic generating functions involved as well as on a generic study of singularities of an associated algebraic curve consequences are precise computable estimates for the number of lattice paths of a given length under various constraints bridges excursions meanders as well as a characterization of the limit laws associated to several basic parameters of paths

integration of semialgebraic functions and integrated nash functions when integrating semialgebraic functions one has to leave the semialgebraic setting for example one gets the global logarithm and iterated antiderivatives of algebraic power series such as the arctangent we show that it is enough to enlarge the semialgebraic functions by these functions to completely describe parameterized integrals of semialgebraic functions to realize this we close the rings of algebraic power series in arbitrary dimension under taking antiderivatives we analyze these rings profoundly in particular we show that the weierstrass division theorem and the weierstrass preparation theorem hold this allows us to apply model theoretic results to obtain an explicit description of parameterized integrals of semialgebraic functions finally we investigate the structure generated by the integrated algebraic power series

e algebraic functions over fields of positive characteristic an analogue of differentially algebraic functions abstract a function or a power series f is called differentially algebraic if it satisfies a differential equation of the form p x y y y n 0 where p is a nontrivial polynomial this notion is usually defined only over fields of characteristic zero and is not so significant over fields of characteristic p 0 as f p 0 for a formal power series over a perfect field k of positive characteristic we shall define an analogue of the concept of a differentially algebraic power series we shall show that these series together with ordinary addition and multiplication of series form a field k with some natural properties we also show that k is not closed under the hadamard product operation

quasianalytic denjoy carleman classes and o minimality we show that the expansion of the real field generated by the functions of a quasianalytic denjoy carleman class is model complete and o minimal provided that the class satisfies certain closure conditions some of these structures do not admit analytic cell decomposition and they show that there is no largest o minimal expansion of the real field

generating functions fibonacci numbers and rational knots abstract we describe rational knots with any of the possible combinations of the properties a chirality non positivity non fiberedness and unknotting number one or higher and determine exactly their number for a given number of crossings in terms of their generating functions we show in particular how fibonacci numbers occur in the enumeration of fibered achiral and unknotting number one rational knots then we show how to enumerate rational knots of given crossing number depending on genus and or signature this allows to determine the asymptotical average value of these invariants among rational knots we give also an application to the enumeration of lens spaces