deviations from uniformity in random strings we show that almost all binary strings of length n contain all blocks of size 1 e log2n a close to uniform number of times from this we derive tight bounds on the discrepancy of random infinite strings our results are obtained through explicit generating function expressions and contour integration estimates

counting bps states on the enriques calabi yau we study topological string amplitudes for the fhsv model using various techniques this model has a type ii realization involving a calabi yau threefold with enriques fibres which we call the enriques calabi yau by applying heterotic type iia duality we compute the topological amplitudes in the fibre to all genera it turns out that there are two different ways to do the computation that lead to topological couplings with different bps content one of them gives the standard d0 d2 counting amplitudes and from the other one we obtain information about bound states of d0 d4 d2 branes on the enriques fibre we also study the model using mirror symmetry and the holomorphic anomaly equations we verify in this way the heterotic results for the d0 d2 generating functional for low genera and find closed expressions for the topological amplitudes on the total space in terms of modular forms and up to genus three this model turns out to be much simpler than the generic b model and might be exactly solvable

row polynomial matrices of riordan arrays abstract let r r n k n k 0 be a riordan array define the row polynomials r n q k 0 n r n k q k and the row polynomial matrix r q r n k q n k 0 by r n k q j k n r n j q j k then r q is also a riordan array with the r n q located on the leftmost column of r q in this paper we investigate combinatorial properties of the matrix r q and the sequence r n q n 0 including their characterizations the q total positivity of r q and the q log convexity of r n q n 0

row sums and alternating sums of riordan arrays abstract here we use row sum generating functions and alternating sum generating functions to characterize riordan arrays and subgroups of the riordan group numerous applications and examples are presented which include the construction of girard waring type identities we also show the extensions to weighted sum generating functions called the expected value generating functions of riordan arrays

universality classes for the fisher metric derived from relative group entropy abstract we consider the fisher metric which results from the hessian of the relative group entropy that we call group fisher metric in particular the metrics corresponding to the boltzmann gibbs tsallis kaniadakis and abe universality classes are obtained we prove that the scalar curvature derived from the group fisher metric results in a multiple of the boltzmann gibbs one with the factor of proportionality given by the local properties of the group entropy we analyze for the tsallis universality class the 2 d correlated model that presents a softening and strengthening of the scalar curvature and we illustrate with the canonical ensemble of a pair of interacting harmonic oscillators as well as a quartic harmonic oscillator

an application of gauge theory to four dimensional topology soit x une u variete compacte reguliere simplement connexe orientee avec la propriete que la forme associee q est definie positive alors cette forme est equivalente sur les entiers a la forme diagonale standard soit dans une base q u 1 u 2 u 2 u 2 1 u 2 2 u 2 2

the elements of finite order in the riordan group over the complex field abstract an element of finite order in the riordan group over the real field must have order 1 or 2 if we extend all the entries to be complex numbers then it may have any finite order in the present paper we investigate the elements of finite order of the riordan group over the complex field this notion leads us to define an element of pseudo order k 2 and k pseudo involution respectively it turns out that the inverse of a k pseudo involution only differs from it in signs we clarify some relationship between the elements of pseudo order k and k pseudo involutions in particular k pseudo involutions for k 0 mod 4 are characterized by a single sequence the subgroups of the riordan group formed by the elements having pseudo order of a prime power p r are also introduced

matching a set of strings with variable length don t cares given an alphabet a a pattern p is a word 1 m where i a and a is a distinguished symbol called a variable length don t care symbol pattern p matches a text t a if t u0 1u1 um 1 mum for some u0 um a we address the following problem given a set p of patterns and a text t test whether one of the patterns of p matches t we describe an algorithm that solves the problem in time o t p log p in contrast to most of the existing string matching algorithms such as that of aho corasick our algorithm is not composed of two successive stages preprocessing the pattern resp the text and reading through the text resp the pattern but has these two stages essentially interleaved our approach is based on using the dawg directed acyclic word graph a data structure studied by a blumer j blumer haussler ehrenfeucht crochemore chen seiferas

umbral presentations for polynomial sequences abstract using random variables as motivation this paper presents an exposition of formalisms developed in 1 2 for the classical umbral calculus a new construction and a variety of examples are presented culminating in several descriptions of sequences of binomial type in terms of umbral polynomials

donaldson invariants of product ruled surfaces and two dimensional gauge theories using the u plane integral of moore and witten we derive a simple expression for the donaldson invariants of product ruled surfaces g s 2 where g is a riemann surface of genus g this expression generalizes a theorem of morgan and szabo for g 1 to any genus g we give two applications of our results 1 we derive thaddeus formulae for the intersection pairings on the moduli space of rank two stable bundles over a riemann surface 2 we derive the eigenvalue spectrum of the fukaya floer cohomology of g s 1

the donaldson witten function for gauge groups of rank larger than one we study correlation functions in topologically twisted d 4 supersymmetric yang mills theory for gauge groups of rank larger than one on compact four manifolds x we find that the topological invariance of the generator of correlation functions of brst invariant observables is not spoiled by noncompactness of field space we show how to express the correlators on simply connected manifolds of b 2 x 0 in terms of seiberg witten invariants and the classical cohomology ring of x for manifolds x of simple type and gauge group su n we give explicit expressions of the correlators as a sum over 1 vacua we describe two applications of our expressions one to superconformal field theory and one to large n expansions of su n d 4 supersymmetric yang mills theory

superconformal simple type and witten s conjecture abstract let x be a smooth closed connected orientable four manifold with b 1 x 0 and b x 3 and odd we show that if x has seiberg witten simple type then the so 3 monopole cobordism formula of 14 implies witten s conjecture relating the donaldson and seiberg witten invariants

extremal bundles on calabi yau threefolds we study constructions of stable holomorphic vector bundles on calabi yau threefolds especially those with exact anomaly cancellation which we call extremal by going through the known databases we find that such examples are rare in general and can be ruled out for the spectral cover construction for all elliptic threefolds we then introduce a general hartshorne serre construction and use it to find extremal bundles of general ranks and study their stability as well as computing their chern numbers based on both existing and our new constructions we revisit the dry conjecture for the existence of stable sheaves on calabi threefolds and provide theoretical and numerical evidence for its correctness our construction can be easily generalized to bundles with no extremal conditions imposed

compact recognizers of episode sequences given two strings x a1 an and p b1 bm over an alphabet the problem of testing whether p occurs as a subsequence of x is trivially solved in linear time it is also known that a simple o n log time preprocessing of x makes it easy to decide subsequently for any p and in at most p log character comparisons whether p is a subsequence of x these problems become more complicated if one asks instead whether p occurs as a subsequence of some substring y of x of bounded length this paper presents an automaton built on the textstring x and capable of identifying all distinct minimal substrings y of x having p as a subsequence by a substring y being minimal with respect to p it is meant that p is not a subsequence of any proper substring of y for every minimal substring y the automaton recognizes the occurrence of p having the lexicographically smallest sequence of symbol positions in y it is not difficult to realize such an automaton in time and space o n2 for a text of n characters one result of this paper consists of bringing those bounds down to linear or o n log n respectively depending on whether the alphabet is bounded or of arbitrary size thereby matching the corresponding complexities of automata constructions for offline exact string searching having built the automaton the search for all lexicographically earliest occurrences of p in x is carried out in time o i 1mrocci i or o n i 1mrocci i log n depending on whether the alphabet is fixed or arbitrary where rocci is the number of distinct minimal substrings of x having b1 bi as a subsequence note that each such substring may occur many times in x but is counted only once in the bound all log factors appearing in the above bounds can be further reduced to log log by resorting to known integer handling data structures

a formula to construct all involutions in riordan matrix groups abstract we get a formula for all finite riordan involutions using the natural approximation induced by the inverse limit construction we also describe all infinite riordan matrices which are involutions in the riordan group finally we get some consequences

double parameter recurrences for polynomials in bi infinite riordan matrices and some derived identities abstract we present some double parameter recurrence relations for families of polynomials in bi infinite riordan matrices from this point of view we treat some riordan families of generalized fibonacci and lucas polynomials we use these to obtain some identities related mainly with binomial coefficients and fibonacci numbers

matrix characterizations of riordan arrays abstract here we discuss two matrix characterizations of riordan arrays p matrix characterization and a matrix characterization p matrix is an extension of the stieltjes matrix defined in 28 and the production matrix defined in 8 by modifying the marked succession rule introduced in 21 a combinatorial interpretation of the p matrix is given the p matrix characterizations of some subgroups of riordan group are presented which are used to find some algebraic structures of the subgroups we also give the p matrix characterizations of the inverse of a riordan array and the product of two riordan arrays a matrix characterization is defined in 20 and it is proved to be a useful tool for a riordan array while on the other side the a sequence characterization is very complex sometimes by using the fundamental theorem of riordan arrays a method of construction of a matrix characterizations from riordan arrays is given the converse process is also discussed several examples and applications of two matrix characterizations are presented

riordan group involutions we study involutions in the riordan group especially those with combinatorial meaning we give a new determinantal criterion for a matrix to be a riordan involution and examine several classes of examples a complete characterization of involutions in the appell subgroup is developed in another direction we find several examples that generalize the rna matrix but are of independent interest

algebraic aspects of some riordan arrays related to binary words avoiding a pattern we consider some riordan arrays related to binary words avoiding a pattern p which can be easily studied by means of an a matrix rather than their a sequence both concepts allow us to define every element as a linear combination of other elements in the array the a sequence is unique and corresponds to a linear dependence from the previous row the a matrix is not unique and corresponds to a linear dependence from several previous rows however for the problems considered in the present paper we show that the a matrix approach is more convenient we provide explicit algebraic generating functions for these riordan arrays and obtain many statistics on the corresponding languages we thus obtain a deeper insight of the languages l p of binary words avoiding p having a number of 0s less or equal to the number of 1s

riordan graphs ii spectral properties abstract the authors of this paper have used the theory of riordan matrices to introduce the notion of a riordan graph in 3 riordan graphs are proved to have a number of interesting fractal properties and they are a far reaching generalization of the well known and well studied pascal graphs and toeplitz graphs and also some other families of graphs the main focus in 3 is the study of structural properties of families of riordan graphs obtained from certain infinite riordan graphs in this paper we use a number of results in 3 to study spectral properties of riordan graphs our studies include but are not limited to the spectral graph invariants for riordan graphs such as the adjacency eigenvalues signless laplacian eigenvalues nullity positive and negative inertia indices and rank we also study determinants of riordan graphs in particular giving results about determinants of catalan graphs

an application of riordan arrays to the transient analysis of m m 1 queues abstract riordan arrays are used extensively in many contexts as a combinatorial tool for solving enumeration problems such as lattice path counting problems in this paper we first show how riordan array method can be applied to the transient analysis of m m 1 queue with zero customer at the initial point second we extend this method to the generalized riordan array with multiple support functions in order to deal with the transient analysis of m m 1 queue with non zero customers at the initial point numerical examples are also given to show how easy and quick the transient probability obtained from the riordan method can be computed

riordan graphs i structural properties abstract in this paper we use the theory of riordan matrices to introduce the notion of a riordan graph the riordan graphs are a far reaching generalization of the well known and well studied pascal graphs and toeplitz graphs and also some other families of graphs the riordan graphs are proved to have a number of interesting fractal properties which can be useful in creating computer networks with certain desirable features or in obtaining useful information when designing algorithms to compute values of graph invariants the main focus in this paper is the study of structural properties of families of riordan graphs obtained from infinite riordan graphs which includes a fundamental decomposition theorem and certain conditions on riordan graphs to have an eulerian trail cycle or a hamiltonian cycle we will study spectral properties of the riordan graphs in a follow up paper

simple proofs of open problems about the structure of involutions in the riordan group abstract we prove that if d g x f x is an element of order 2 in the riordan group then g x exp x xf x for some antisymmetric function x z also we prove that every element of order 2 in the riordan group can be written as bmb 1 for some element b and m 1 1 in the riordan group these proofs provide solutions to two open problems presented by l shapiro l w shapiro some open questions about random walks involutions limiting distributions and generating functions adv appl math 27 2001 585 596

a symbolic treatment of riordan arrays abstract we approach riordan arrays and their generalizations via umbral symbolic methods this new approach allows us to derive fundamental aspects of the theory of riordan arrays as immediate consequences of the umbral version of the classical abel s identity for polynomials in particular we obtain a novel non recursive formula for riordan arrays and derive from this new formula some known recurrences and a new recurrence relation for riordan arrays

superconformal invariance and the geography of four manifolds the correlation functions of supersymmetric gauge theories on a four manifold x can sometimes be expressed in terms of topological invariants of x we show how the existence of superconformal fixed points in the gauge theory can provide nontrivial information about four manifold topology in particular in the example of gauge group su 2 with one doublet hypermultiplet we derive a theorem relating classical topological invariants such as the euler character and signature to sum rules for seiberg witten invariants a short account of this paper can be found in 1

donaldson invariants for non simply connected manifolds we study coulomb branch u plane integrals for supersymmetric su 2 so 3 yang mills theory on 4 manifolds x of b 1 x 0 b 2 x 1 using wall crossing arguments we derive expressions for the donaldson invariants for manifolds with b 1 x 0 b 2 x 0 explicit expressions for where f g is a riemann surface of genus g are obtained using kronecker s double series identity the result might be useful in future studies of quantum cohomology

half of a riordan array and restricted lattice paths abstract for an infinite lower triangular matrix g g n k n k 0 we define the half of g to be the infinite lower triangular matrix h h n k n k 0 such that h n k g 2 n k n for all n k 0 in this paper we will show that if g g n k n k 0 is a riordan array then its half h h n k n k 0 is also a riordan array and we obtain new combinatorial interpretations for some riordan arrays in terms of weighted lattice paths

some inverse limit approaches to the riordan group abstract using the inverse limit tool we obtain the riordan group in its infinite and bi infinite representations from groups of finite riordan matrices we define two different reflections for bi infinite riordan matrices employing these definitions we give answers to the problems d d and d d in the riordan group so we describe the self complementary and self dual riordan matrices

elements of finite order in the riordan group and their eigenvectors abstract we consider elements of finite order in the riordan group r over a field f of characteristic 0 we solve for all n 2 two foundational questions posed by l shapiro 2001 for the case n 2 involutions given f x of finite compositional order n theorem 1 gives a formula for g x showing that those g x which make g x f x r a riordan element of order n are precisely those for which the matrix g x f x has an eigenvector h 0 h 1 t with h 0 0 theorem 2 classifies finite order riordan group elements up to conjugation in r and gives a normal form for finite order riordan arrays under similarity this leads to theorem 3 a formula for all eigenvectors of finite order riordan arrays and results in interesting combinatorial identities we relate our work to 2008 and 2013 papers of cheon and kim which motivated this paper and we solve their open question this circle of ideas closes with a new proof of c marshall s theorem which finds the unique f x given bi invertible g x such that g x f x is an involution

supersymmetric yang mills theory on a four manifold by exploiting standard facts about n 1 and n 2 supersymmetric yang mills theory the donaldson invariants of four manifolds that admit a kahler metric can be computed the results are in agreement with available mathematical computations and provide a powerful check on the standard claims about supersymmetric yang mills theory

recurrence relations for polynomial sequences via riordan matrices we give recurrence relations for any family of generalized appell polynomials unifying so some known recurrences for many classical sequences of polynomials our main tool to get our goal is the riordan group we use the product of riordan matrices to interpret some relationships between different polynomial families moreover using the hadamard product of series we get a general recurrence relation for the polynomial sequences associated to the so called generalized umbral calculus

hidden word statistics we consider the sequence comparison problem also known as hidden pattern problem where one searches for a given subsequence in a text rather than a string understood as a sequence of consecutive symbols a characteristic parameter is the number of occurrences of a given pattern w of length m as a subsequence in a random text of length n generated by a memoryless source spacings between letters of the pattern may either be constrained or not in order to define valid occurrences we determine the mean and the variance of the number of occurrences and establish a gaussian limit law and large deviations these results are obtained via combinatorics on words formal language techniques and methods of analytic combinatorics based on generating functions the motivations to study this problem come from an attempt at finding a reliable threshold for intrusion detections from textual data processing applications and from molecular biology

generalized humbert polynomials via generalized fibonacci polynomials in this paper we define the generalized p q fibonacci polynomials un m x and generalized p q lucas polynomials vn m x and further introduce the generalized humbert polynomials un m r x as the convolutions of un m x we give many expressions expansions recurrence relations and differential recurrence relations of un m r x and study the matrices and determinants related to the polynomials un m x vn m x and un m r x finally we present an algebraic interpretation for the generalized humbert polynomials un m r x it can be found that various well known polynomials are special cases of un m x vn m x or un m r x therefore by introducing the general polynomials un m x vn m x and un m r x we have a unified approach to dealing with many special polynomials in the literature

quantum cohomology of the moduli space of stable bundles over a riemann surface we determine the quantum cohomology of the moduli space m of odd degree rank two stable vector bundles over a riemann surface of genus g 1 this work together with 10 complete the proof of the existence of an isomorphism qh m hf s1

the blowup formula for donaldson invariants k k which is calculated in 4 below this formula has been the target of much recent work the abstract fact that there exists such a formula which is independent of x was first proved by c taubes using techniques of t j bryan b and p ozsvath o have independently calculated the coefficients throughb10 x quite recently j morgan and ozsvath have announced a scheme which can recursively compute all of the bk x the special case of the blowup formula for manifolds of simple type see 5 below was first given by p kronheimer and t mrowka however none of the techniques in these cases approach the simplicity of that offered here before presenting the formula we shall first establish notation for the donaldson in variants of a simply connected 4 manifold x with b 1 and odd the hypothesis of simple connectivity is not necessary but makes the exposition easier an orientation of x together with an orientation of h 2 x r is called a homology orientation of x such a homology orientation determines the su 2 donaldson invariant a linear function d dx a x sym h0 x h2 x r

a unified approach for the catalan matrices by using riordan arrays abstract we study generalized catalan matrices based on the riordan array and fuss catalan numbers a unified combinatorial interpretation for the entries of the generalized catalan matrices is presented by means of m dyck paths some properties and examples of the generalized catalan matrices are given including a new convolution formula for the generalized catalan numbers finally we present applications of generalized catalan matrices to the problems in counting the hill free and lower peak free m dyck paths

euler number of instanton moduli space and seiberg witten invariants we show that a partition function of topological twisted n 4 yang mills theory is given by seiberg witten invariants on a riemannian four manifolds under the condition that the sum of the euler number and the signature of the four manifolds vanishes the partition function is the sum of the euler number of instanton moduli space when it is possible to apply the vanishing theorem also we obtain a relation of the euler number labeled by the instanton number k with seiberg witten invariants all calculations in this article are done without assuming duality

fuss catalan matrices their weighted sums and stabilizer subgroups of the riordan group abstract in this paper we present the riordan arrays called fuss catalan matrices which are constructed by the convolutions of the generating functions of the fuss catalan numbers we also discuss weighted sums of the fuss catalan matrices using such matrices as transformations of recursive sequences and their connection with stabilizer subgroups of the riordan group

a combinatorial equivalence relation for formal power series abstract in this paper we examine an equivalence relation on the set of formal power series with nonzero constant term this is done both in terms of functional equations and also by interlacing two concepts from riordan group theory the a sequence and the bell subgroup the best known example gives an equivalence class 1 z 1 1 z c z t z q z where c z t z and q z are generating functions of the catalan numbers ternary numbers and quaternary numbers respectively a power series for one member of an equivalence class can be transformed into power series for the rest of members in the equivalence class and interpretations in terms of weighted lattice paths can also be given

groups of generalized pascal matrices abstract we present a new approach to the study of generalized pascal matrices that yields general results about group structures and explicit formulas for powers and inverses of the pascal matrices shows how the groups of pascal matrices are related to groups of 2 2 matrices and clarifies the way in which symmetries and duality relate to the algebraic aspects of the theory we consider bilaterally infinite generalized pascal matrices obtained as the matrix representations of certain linear operators on spaces of formal laurent series their column generating functions are simple rational functions closely related to the linear fractional maps of the complex plane we obtain explicit expressions for the inverses and the powers of the pascal matrices that generalize most of the analogous results in the literature

on the chromatic number of integral circulant graphs integral circulant graphs are a generalization of unitary cayley graphs recently studied by klotz and sander the integral circulant graph x n d has vertices 0 1 n 1 and two vertices a and b are adjacent iff gcd x y n d where d d d n 1 d 3 we construct a family of counterexamples using exhaustive search algorithm for graph coloring and disprove the conjecture in this case

identities induced by riordan arrays historically there exist two versions of the riordan array concept the older one better known as recursive matrix consists of bi infinite matrices d n k n k is an element of z k n implies d n k 0 deals with formal laurent series and has been mainly used to study algebraic properties of such matrices the more recent version consists of infinite lower triangular arrays d n k n k is an element of n k n implies d n k 0 deals with formal power series and has been used to study combinatorial problems here we show that every riordan array induces two characteristic combinatorial sums in three parameters n k m is an element of z these parameters can he specialized and generate an indefinite number of other combinatorial identities which are valid in the hi infinite realm of recursive matrices

some algebraic structure of the riordan group abstract in this paper we highlight some of the algebraic properties of the riordan group we present some new properties and provide proofs for some known properties for instance proofs of properties of the power bell and derivative subgroups are presented we also give a new property for generating stochastic riordan matrices which are elements of the stochastic riordan subgroup centralizers stabilizers and certain riordan group isomorphisms are also given and proved we conclude by proving some properties connecting similar riordan matrices and pseudo involutions in the riordan group

riordan arrays and related polynomial sequences abstract in this paper we generalize the results due to luzon and moron and present a characterization of the generalized riordan arrays using this characterization we establish a recurrence for the generalized sheffer sequences and study some special types of polynomial sequences including the generalized lucas u sequence and v sequence and the generalized humbert polynomial sequence in particular by appealing to the factorizations of riordan arrays the umbral composition of polynomial sequences and the hadamard product of series we establish various relations among these polynomial sequences thus we give a unified approach to some well known polynomial sequences appeared in combinatorics and the theory of special functions

combinatorics of a generalized narayana identity abstract the narayana identity is a well known formula that expresses the classical catalan numbers as sums of the ordinary narayana numbers in this paper we generalize the narayana identity to a family of riordan arrays including the array of ballot numbers the classical catalan triangle and several generalized catalan triangles recently studied a combinatorial description based on non crossing partitions is given for this identity for the column recursive rule and for the sheffer sequence associated with any array of the family

recursive matrices and umbral calculus two major objectives can be seen to guide much recent work in enumeration 1 to single out a limited variety of recurrences for numerical sequences which will encompass counting problems of wide enough type 2 to recover from empirical data an underlying set theoretic structure which would reveal the source of the given recursion we are here concerned with the first of these objectives though the eventual understanding of the second is tacitly present if only as a goal we noticed the coincidence of several computations which similar as they are in retrospect had failed to realize their kinship on leafing through the unique assembly of recursively solvable combinatorial problems in comtet s and sloane s invaluable collections one is struck by the repeated occurrence of one and the same kind of double recursion more strikingly the same recursion is seen to occur in the polynomial sequences of the umbra1 calculus of roman and rota see 25 everywhere the lagrange inversion formula for power series plays a pivotal role much work is nowadays going into the unraveling of the everdeeper layers of combinatorial significance of this formula both in the ordinary case and in its as yet partially worked out noncommutative and qanalogs andrews foata garsia gessel joni raney reiner schtitzenberger to name but a few whatever their origins the identities abutting lagrange inversion are expressed by integers alone this suggests not only a hidden set theoretic layer but a characteristic free generalization as well this generalization is the central theme of our work we define a monoid of infinite matrices recursive matrices for short the entries of these matrices give the sought out recursion for example that for coefficients of binomial and sheffer polynomials and factor sequences as well as that of the special sequences recently introduced by roman in 26 546 002 l 8693 82 040546 28 02 00 0

convolutions of the generalized morgan voyce polynomials we consider the following classes of polynomials b n m s x c n m s x b n m s x and c n m s x which are convolutions of the generalized morgan voyce polynomials where s is a nonnegative integer these convolutions are related to the classical morgan voyce chebyshev and jacobsthal polynomials

an algebraic structure for faber polynomials abstract let be the set of functions convergent for all z 1 with a laurent series of the form f z z n 0 a n z n in this paper we prove that the set of faber polynomial sequences over and the set of their normalized kth derivative sequences form groups which are isomorphic to the hitting time subgroup and the bell k subgroup of the riordan group respectively further a relationship between such faber polynomial sequences and lucas and sheffer polynomial sequences is derived

notes on the total positivity of riordan arrays abstract let r d t h t be a riordan array we show that if both d t and h t are polya frequency formal power series then r is totally positive

riordan arrays and harmonic number identities let the numbers p r n k be defined by p r n k p r h n 1 h k 1 h n r h k r where p r x 1 x r 1 ry r 0 x 1 1 x 2 r 1 x r and y r are the exponential complete bell polynomials by observing that the numbers p r n k generate two riordan arrays we establish several general summation formulas from which series of harmonic number identities are obtained in particular some of these harmonic number identities also involve other special combinatorial sequences such as the stirling numbers of both kinds the lah numbers the bernoulli numbers and polynomials and the cauchy numbers of both kinds

q riordan representation abstract the aim of this study is to establish further theory of q riordan representation we emphasize that any q matrices can be obtained by q riordan representation then we analyse the structure of the entries in the matrix product of general q riordan matrices

dual operators and lagrange inversion in several variables the original lagrange inversion formula which gives explicitly the inverse under composition of a formal series was obtained by lagrange in 1770 through formal computations involving logarithms of inlmite products see lagrange 121 there exists an extensive literature on various versions of the inversion formula obtained by algebraic analytic and combinatorial methods comtet 3 gives a survey and an ample bibliography during the last ten years there have appeared several papers on generalizations and new proofs of the inversion formula in 1974 abhyankar obtained an inversion formula using algebraic methods see bass connell and wright 11 several authors have related lagrange inversion with rota s finite operator calculus 151 for example joni 8 garsia and joni 4 hofhauer 7 and roman and rota 14 joyal and labelle used a combinatorial theory of formal series to obtain inversion formulas see 9 l i and viskov 161 used ideas from lie algebra theory to prove formulas similar to abhyankar s in the present paper we use operator methods to obtain inversion formulas in several variables that generalize the formulas of abhyankar joni and viskov we get our results studying an algebra with involution of linear operators on the algebra of formal laurent series in several indeterminates we also show that by means of an anti isomorphism of operator algebras which we call the operator bore1 transform the study of finite operator calculus can be reduced to the study of certain groups of operators on the algebra of formal laurent series our approach may be considered as an algebraic analogue of the complex variable methods used by good 5 to obtain the lagrange inversion formula as a consequence of the change of variables theorem for the

the smallest automaton recognizing the subwords of a text let a partial deterministic finite automaton be a dfa in which each state need not have a transition edge for each letter of the alphabet we demonstrate that the smallest partial dfa for the set of all subwords of a given word w iwl 2 has at most 21w 2 states and 3 wl 4 transition edges independently of the alphabet size we give an algorithm to build this smallest partial dfa from the input w on line in linear time

finite and infinite dimensional lie group structures on riordan groups abstract we introduce a frechet lie group structure on the riordan group we give a description of the corresponding lie algebra as a vector space of infinite lower triangular matrices we describe a natural linear action induced on the frechet space k n by any element in the lie algebra we relate this to a certain family of bivariate linear partial differential equations we obtain the solutions of such equations using one parameter groups in the riordan group we show how a particular semidirect product decomposition in the riordan group is reflected in the lie algebra we study the stabilizer of a formal power series under the action induced by riordan matrices we get stabilizers in the fraction field k x using bi infinite representations we provide some examples the main tool to get our results is the paper 18 where the riordan group was described using inverse sequences of groups of finite matrices

a general umbral calculus in infinitely many variables on etudie des suites de fonctions factorielles satisfaisant des analogues des recurrences binomiales et de sheffer et on montre comment un calcul operatoriel ressemblant au calcul umbral peut etre construit dessus

a determinantal approach to sheffer sequences abstract in this paper by using the theory of riordan arrays and the relations between sheffer sequences and riordan arrays we give a determinantal definition for sheffer sequences based on this new definition some general properties of sheffer sequences are reproved and the determinantal representations of some well known sheffer sequences are presented

twisted n 2 supersymmetry with central charge and equivariant cohomology we present an equivariant extension of the thom form with respect to a vector field action in the framework of the mathai quillen formalism the associated topological quantum field theories correspond to twisted n 2 supersymmetric theories with a central charge we analyze in detail two different cases topological sigma models and non abelian monopoles on four manifolds

bijections and the riordan group one of the cornerstone ideas in mathematics is to take a problem and to look at it in a bigger space in this paper we examine combinatorial sequences in the context of the riordan group various subgroups of the riordan group each give us a different view of the original sequence in many cases this leads to both a combinatorial interpretation and to eco rewriting rules in this paper we will concentrate on just four of the subgroups of the riordan group to demonstrate some of the possibilities of this approach

structural properties of riordan matrices and extending the matrices abstract we consider an infinite lower triangular matrix l l n k n k n 0 and a sequence n n n 0 called the a b sequence such that every element l n 1 k 1 except lying in column 0 can be expressed as l n 1 k 1 i 0 n k m i l n ai k bi 0 0 where a and b are integers with a b m 0 and b 0 this concept generalizes the a sequence of a riordan matrix as a result we explore several structural properties of riordan matrices by means of a b sequences in particular if b

the so 3 monopole cobordism and superconformal simple type abstract we show that the so 3 monopole cobordism formula from 15 implies that all closed oriented smooth four manifolds with b 1 0 and b 3 and odd with seiberg witten simple type satisfy the superconformal simple type condition defined by marino moore and peradze 28 27 this implies the lower bound conjectured by fintushel and stern 18 on the number of seiberg witten basic classes in terms of topological data

large scale typicality of markov sample paths and consistency of mdl order estimators for markov chains of arbitrary order with finite alphabet a almost sure sense limit theorems are proved on relative frequencies of k blocks and of symbols preceded by a given k block when k is permitted to grow as the sample size n grows as an application the consistency of two kinds of minimum description length mdl markov order estimators is proved with upper bound o log n respectively spl alpha log n with spl alpha 1 log a on the permissible value of the estimated order it was shown by csiszar and shields see ann statist vol 28 p 1601 1619 2000 that in the absence of any bound or with bound spl alpha log n with large spl alpha consistency fails

multimode uncertainty relations and separability of continuous variable states a multimode uncertainty relation generalizing the robertson schrodinger relation is derived as a necessary constraint on the second moments of n pairs of canonical operators in turn necessary conditions for the separability of multimode continuous variable states under m n mode bipartitions are derived from the uncertainty relation these conditions are proven to be necessary and sufficient for 1 n mode gaussian states and for m n mode bisymmetric gaussian states

log convexity of aigner catalan riordan numbers abstract let t t n k n k 0 be an infinite lower triangular matrix defined by t 0 0 1 t n 1 0 j 0 n z j t n j t n 1 k 1 j k n a j k t n j for n k 0 where all z j a j k are nonnegative and a j k 0 unless j k 0 we show that the sequence t n 0 n 0 is log convex if the coefficient matrix a is tp 2 where z 0 z 1 z 2 and a a i j i j 0 this gives a unified proof of the log convexity of many well known combinatorial sequences including the catalan numbers the motzkin numbers the central binomial coefficients the schroder numbers the bell numbers and so on

riordan matrices in the reciprocation of quadratic polynomials we iterate contractive one degree polynomials with coefficients in the ring k x of formal power series to calculate the reciprocal in k x of a quadratic polynomial doing this we meet the structure of riordan array we interpret certain changes of variable as a riordan array we finish the paper by using our techniques to find new ways to get known formulas for the sum of powers of natural numbers involving stirling and eulerian numbers

recurrence relations for the sheffer sequences abstract in this paper using the production matrix of an exponential riordan array g t f t we give a recurrence relation for the sheffer sequence for the ordered pair g t f t we also develop a new determinant representation for the general term of the sheffer sequence as applications determinant expressions for some classical sheffer polynomial sequences are derived

on effective computation of expectations in large or infinite dimension abstract this study is an analysis of the natural difficulties of integration by monte carlo or quasi monte carlo methods in spite of what is sometimes written these methods work only in some precise cases for the important problem of the computation of expectations of functionals of stochastic processes we present the advantages of a method based on the implementation of the bernoulli shift operator by pointers

the relevant prefixes of coloured motzkin walks an average case analysis in this paper we study some relevant prefixes of coloured motzkin walks otherwise called coloured motzkin words in these walks the three kinds of step can have a b and c colours respectively in particular when a b c 1 we have the classical motzkin walks while for a c 1 and b 0 we find the well known dyck walks by using the concept of riordan arrays and probability generating functions we find the average length of the relevant prefix in a walk of length n and the corresponding variance in terms of a b and c this result is interesting from a combinatorial point of view and also provides an average case analysis of algorithms related to the problem of ranking and generating uniformly at random the coloured motzkin words

pu 2 monopoles ii top level seiberg witten moduli spaces and witten s conjecture in low degrees in this article a continuation of 10 we complete the proof for a broad class of four manifolds of witten s conjecture that the donaldson and seiberg witten series coincide at least through terms of degree less than or equal to c 2 where c 1 4 7 11 andandare the euler characteristic and signature of the four manifold we use our computations of chern classes for the virtual normal bundles for the seiberg witten strata from the companion article 10 a comparison of all the orientations and the pu 2 monopole cobordism to compute pairings with the links of level zero seiberg witten moduli subspaces of the moduli space of pu 2 monopoles these calculations then allow us to compute low degree donaldson invariants in terms of seiberg witten invariants and provide a partial verification of witten s conjecture

standard forms and entanglement engineering of multimode gaussian states under local operations we investigate the action of local unitary operations on multimode pure or mixed gaussian states and single out the minimal number of locally invariant parameters which completely characterize the covariance matrix of such states for pure gaussian states central resources for continuous variable quantum information we investigate separately the parameter reduction due to the additional constraint of global purity and the one following by the local unitary freedom counting arguments and insights from the phase space schmidt decomposition and in general from the framework of symplectic analysis accompany our description of the standard form of pure n mode gaussian states in particular we clarify why only in pure states with n 3 modes all the direct correlations between position and momentum operators can be set to zero by local unitary operations for any n the emerging minimal set of parameters contains complete information about all forms of entanglement in the corresponding states an efficient state engineering scheme able to encode direct correlations between position and momentum operators as well is proposed to produce entangled multimode gaussian resources its number of optical elements matching the minimal number of locally invariant degrees of freedom of general pure n mode gaussian states finally we demonstrate that so called block diagonal gaussian states without direct correlations between position and momentum are systematically less entangled on average than arbitrary pure gaussian states