constructible differentially finite algebraic series in several variables we extend the concept of cdf series to the context of several variables and show that the series solution of first order differential equations y x t y and functional equation y x t y with x cdf in two variables are cdf series we also give many effective closure properties for cdf series in several variables

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

polya number and first return of bursty random walk rigorous solutions abstract the recurrence properties of random walks can be characterized by polya number i e the probability that the walker has returned to the origin at least once in this paper we investigate polya number and first return for bursty random walk on a line in which the walk has different step size and moving probabilities using the concept of the catalan number we obtain exact results for first return probability the average first return time and polya number for the first time we show that polya number displays two different functional behavior when the walk deviates from the recurrent point by utilizing the lagrange inversion formula we interpret our findings by transferring polya number to the closed form solutions of an inverse function we also calculate polya number using another approach which corroborates our results and conclusions finally we consider the recurrence properties and polya number of two variations of the bursty random walk model

directed compact percolation near a damp wall mean length and mean number of wall contacts key aspects of the cluster distribution in the case of directed compact percolation near a damp wall are derived as functions of the bulk occupation probability pand the wall occupation probability pw the mean length of finite clusters and mean number of contacts with the wall are derived exactly and we find that both results involve elliptic integrals and further multiple sum functions of two variables despite the added complication of these multiple sumfunctions ouranalysisshowsthatthecriticalbehaviourissimilartothedry wallcasewherethesefunctionsdonotappear wederivethecriticalamplitudes as a function of pw

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

anomalous diffusion of random walk on random planar maps we prove that the simple random walk on the uniform infinite planar triangulation uipt typically travels graph distance at most n 1 4 o n 1 in n units of time together with the complementary lower bound proven by gwynne and miller 2017 this shows that the typical graph distance displacement of the walk after n steps is n 1 4 o n 1 as conjectured by benjamini and curien geom funct anal 2 2 501 531 2013 arxiv 1202 5454 more generally we show that the simple random walks on a certain family of random planar maps in the gamma liouville quantum gravity lqg universality class for gamma in 0 2 including spanning tree weighted maps bipolar oriented maps and mated crt maps typically travels graph distance n 1 d gamma o n 1 in n units of time where d gamma is the growth exponent for the volume of a metric ball on the map which was shown to exist and depend only on gamma by ding and gwynne commun math phys 374 1877 1934 2018 arxiv 1807 01072 since d gamma 2 this shows that the simple random walk on each of these maps is subdiffusive our proofs are based on an embedding of the random planar maps under consideration into mathbb c wherein graph distance balls can be compared to euclidean balls modulo subpolynomial errors this embedding arises from a coupling of the given random planar map with a mated crt map together with the relationship of the latter map to sle decorated lqg

on the robustness of interconnections in random graphs a symbolic approach graphs are models of communication networks this paper applies symbolic combinatorial techniques in order to characterize the interplay between two parameters of a random graph namely its density the number of edges in the graph and its robustness to link failures here robustness means multiple connectivity by short disjoint paths a triple g s t where g is a graph and s t are designated vertices is called l robust if s and t are connected via at least two edge disjoint paths of length at most l we determine the expected number of ways to get from s to t via two edge disjoint paths of length l in the classical random graph model gn p by means of symbolic combinatorial methods we then derive bounds on related threshold probabilities as functions of l and n

rational function certification of multisum integral q identities the method of rational function certification for proving terminating hypergeometric identities is extended from single sums or integrals to multi integral sums and q integral sums

third order integrability conditions for homogeneous potentials of degree 1 we prove an integrability criterion of order 3 for a homogeneous potential of degree 1 in the plane still this criterion depends on some integer and it is impossible to apply it directly except for families of potentials whose eigenvalues are bounded to address this issue we use holonomic and asymptotic computations with error control of this criterion and apply it to the potential of the form v r r 1h exp i with h c z degh 3 we then find all meromorphically integrable potentials of this form

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

asymptotics of quantum spin networks at a fixed root of unity a classical spin network consists of a ribbon graph i e an abstract graph with a cyclic ordering of the vertices around each edge and an admissible coloring of its edges by natural numbers the standard evaluation of a spin network is an integer number in a previous paper we proved an existence theorem for the asymptotics of the standard evaluation of an arbitrary classical spin network when the coloring of its edges are scaled by a large natural number in the present paper we extend the results to the case of an evaluation of quantum spin networks of arbitrary valency at a fixed root of unity as in the classical case our proofs use the theory of g functions of andre together with some new results concerning holonomic and q holonomic sequences of wilf zeilberger

g functions and multisum versus holonomic sequences abstract the purpose of the paper is three fold a we prove that every sequence which is a multidimensional sum of a balanced hypergeometric term has an asymptotic expansion of gevrey type 1 with rational exponents b we construct a class of g functions that come from enumerative combinatorics and c we give a counterexample to a question of zeilberger that asks whether holonomic sequences can be written as multisums of balanced hypergeometric terms the proofs utilize the notion of a g function introduced by siegel and its analytic arithmetic properties shown recently by andre

current trends in asymptotics some problems and some solutions abstract i discuss five topics of current interest in asymptotic analysis the use of probabilistic methods to estimate the growth of combinatorial sequences asymptotic methods in the theory of random walks the asymptotic estimation of solutions of difference equations asymptotic expansions in generalized scales and the computation by asymptotic methods of distributions whose moments are known i give a number of examples and mention some outstanding unsolved problems

summation transformation and expansion formulas for bibasic series etude des extensions a deux bases de la formule de transformation d euler et de la formule de developpement de fields wimp etablissement d une formule de transformation a quatre bases independantes d une formule d inversion de lagrange et de quelques formules de sommation de degres 2 3 et 4

cocharacter sequences are holonomic abstract let c n n 0 be a codimension sequence of a characteristic zero p i algebra then the c n satisfy a linear homogenous recurrence relation whose coefficients are rational functions of n

on polynomially bounded operators acting on a banach space abstract by the von neumann inequality every contraction on a hilbert space is polynomially bounded a simple example shows that this result does not extend to banach space contractions in this paper we give general conditions under which an arbitrary banach space contraction is polynomially bounded these conditions concern the thinness of the spectrum and the behaviour of the resolvent or the sequence of negative powers to do this we use techniques from harmonic analysis in particular results concerning thin sets such as helson sets kronecker sets and sets that satisfy spectral synthesis

random walks in cones we study the asymptotic behavior of a multidimensional random walk in a general cone we find the tail asymptotics for the exit time and prove integral and local limit theorems for a random walk conditioned to stay in a cone the main step in the proof consists in constructing a positive harmonic function for our random walk under minimal moment restrictions on the increments for the proof of tail asymptotics and integral limit theorems we use a strong approximation of random walks by brownian motion for the proof of local limit theorems we suggest a rather simple approach which combines integral theorems for random walks in cones with classical local theorems for unrestricted random walks we also discuss some possible applications of our results to ordered random walks and lattice path enumeration

a mating of trees approach for graph distances in random planar maps we introduce a general technique for proving estimates for certain random planar maps which belong to the gamma liouville quantum gravity lqg universality class for gamma in 0 2 the family of random planar maps we consider are those which can be encoded by a two dimensional random walk with i i d increments via a mating of trees bijection and includes the uniform infinite planar triangulation uipt gamma sqrt 8 3 and planar maps weighted by the number of different spanning trees gamma sqrt 2 bipolar orientations gamma sqrt 4 3 or schnyder woods gamma 1 that can be put on the map using our technique we prove estimates for graph distances in the above family of random planar maps in particular we obtain non trivial upper and lower bounds for the cardinality of a graph distance ball consistent with the watabiki prog theor phys suppl 114 1 17 1993 prediction for the hausdorff dimension of gamma lqg and we establish the existence of an exponent for certain distances in the map the basic idea of our approach is to compare a given random planar map m to a mated crt map a random planar map constructed from a correlated two dimensional brownian motion using a strong coupling zaitsev in esaim probab stat 2 41 108 1998 of the encoding walk for m and the brownian motion used to construct the mated crt map this allows us to deduce estimates for graph distances in m from the estimates for graph distances in the mated crt map which we proved using continuum theory in a previous work in the special case when gamma sqrt 8 3 we instead deduce estimates for the sqrt 8 3 mated crt map from known results for the uipt the arguments of this paper do not directly use sle lqg and can be read without any knowledge of these objects

uniform infinite planar triangulations the existence of the weak limit as n of the uniform measure on rooted triangulations of the sphere with n vertices is proved some properties of the limit are studied in particular the limit is a probability measure on random triangulations of the plane

rigorous uniform approximation of d finite functions using chebyshev expansions 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 well known that the order n truncation of the chebyshev expansion of a function over a given interval is a near best uniform polynomial approximation of the function on that interval in the case of solutions of linear differential equations with polynomial coefficients the coefficients of the expansions obey linear recurrence relations with polynomial coefficients unfortunately these recurrences do not lend themselves to a direct recursive computation of the coefficients owing among other things to a lack of initial conditions we show how they can nevertheless be used as part of a validated process to compute good uniform approximations of d finite functions together with rigorous error bounds and we study the complexity of the resulting algorithms our approach is based on a new view of a classical numerical method going back to clenshaw combined with a functional enclosure method

the susceptibility of the square lattice ising model new developments we have made substantial advances in elucidating the properties of the susceptibility of the square lattice ising model we discuss its analyticity properties certain closed form expressions for subsets of the coefficients and give an algorithm of complexity o n6 to determine its first n coefficients as a result we have generated and analyzed series with more than 300 terms in both the high and low temperature regime we quantify the effect of irrelevant variables to the scaling amplitude functions in particular we find and quantify the breakdown of simple scaling in the absence of irrelevant scaling fields arising first at order t tc 9 4 though high low temperature symmetry is still preserved at terms of order t tc 17 4 and beyond this symmetry is no longer present the short distance terms are shown to have the form t tc p log t tc q with p q2 conjectured exact expressions for some correlation functions and series coefficients in terms of elliptic theta functions also foreshadow future developments

lattice green s functions of the higher dimensional face centered cubic lattices we study the face centered cubic fcc lattice in up to six dimensions in particular we are concerned with lattice green functions lgfs and return probabilities computer algebra techniques such as the method of creative telescoping are used for deriving an ode for a given lgf for the four and five dimensional fcc lattices we give rigorous proofs of the odes that were conjectured by guttmann and broadhurst additionally we find the ode of the lgf of the six dimensional fcc lattice a result that was not believed to be achievable with current computer hardware

rational approximations for values of derivatives of the gamma function the arithmetic nature of euler s constant is still unknown and even getting good rational approximations to it is difficult recently aptekarev managed to find a third order linear recurrence with polynomial coefficients which admits two rational solutions a n and b n such that a n b n converges sub exponentially to viewed as 1 where r is the usual gamma function although this is not yet enough to prove that it is a major step in this direction in this paper we present a different but related approach based on simultaneous pade approximants to euler s functions from which we construct and study a new third order recurrence that produces a sequence in z whose height grows like the factorial and that converges sub exponentially to log z for any complex number z 0 where log is defined by its principal branch we also show how our approach yields in theory rational approximations of numbers related to s 1 for any integer s 1 in particular we construct a sixth order recurrence which provides simultaneous rational approximations of factorial height converging sub exponentially to the numbers and 1 2 1 2 2 2

recursive evaluation of space time lattice green s functions up to a multiplicative constant the lattice green s function lgf as defined in condensed matter physics and lattice statistical mechanics is equivalent to the z domain counterpart of the finite difference time domain green s function gf on a lattice expansion of a well known integral representation for the lgf on a dimensional hyper cubic lattice in powers of z 1 and application of the chu vandermonde identity results in 1 nested finite sum representations for discrete space time gfs due to severe numerical cancellations these nested finite sums are of little practical use for 2 the finite sum may be evaluated in closed form in terms of a generalized hypergeometric function for special lattice points that representation simplifies considerably while on the other hand the finite difference stencil may be used to derive single lattice point second order recurrence schemes for generating 2d discrete space time gf time sequences on the fly for arbitrary symbolic lattice points zeilberger s algorithm produces a third order recurrence operator with polynomial coefficients of the sixth degree the corresponding recurrence scheme constitutes the most efficient numerical method for the majority of lattice points in spite of the fact that for explicit numeric lattice points the associated third order recurrence operator is not the minimum recurrence operator as regards the asymptotic bounds for the possible solutions to the recurrence scheme perron s theorem precludes factorial or exponential growth along horizontal lattices directions rapid initial growth does occur but poses no problems in augmented dynamic range fixed precision arithmetic by analysing long distance wave propagation along a horizontal lattice direction we have concluded that the chirp up oscillations of the discrete space time gf are the root cause of grid dispersion anisotropy with each factor of ten increase in the lattice distance one would have to roughly double the pulse width of the source signature to keep pulse distortion at bay the gf time sequences can also be used for an efficient computation of discrete space frequency lgfs especially if one employs aitken s 2 process for the acceleration of the convergence of the consecutive partial sums

lattice green functions the seven dimensional face centred cubic lattice we present a recursive method to generate the expansion of the lattice green function of the d dimensional face centred cubic fcc lattice we produce a long series for d 7 then we show and recall that in order to obtain the linear differential equation annihilating such a long power series the most economic way amounts to producing the non minimal order differential equations we use the method to obtain the minimal order linear differential equation of the lattice green function of the seven dimensional fcc lattice we give some properties of this irreducible order eleven differential equation we show that the differential galois group of the corresponding operator is included in this order eleven operator is non trivially homomorphic to its adjoint and we give a decomposition of this order eleven operator in terms of four order one self adjoint operators and one order seven self adjoint operator furthermore using the landau conditions on the integral we forward the regular singularities of the differential equation of the d dimensional lattice and show that they are all rational numbers we evaluate the return probability in random walks in the seven dimensional fcc lattice we show that the return probability in the d dimensional fcc lattice decreases as d 2 as the dimension d goes to infinity

subsequential scaling limits for liouville graph distance for 0 gamma 2 and delta 0 we consider the liouville graph distance which is the minimal number of euclidean balls of gamma liouville quantum gravity measure at most delta whose union contains a continuous path between two endpoints in this paper we show that the renormalized distance is tight and thus has subsequential scaling limits at delta rightarrow 0 in particular we show that for all delta 0 the diameter with respect to the liouville graph distance has the same order as the typical distance between two endpoints

directed compact percolation near a wall iii exact results for the mean length and number of contacts existing exact results for the percolation probability and mean cluster size for compact percolation near a dry wall are extended to the mean cluster length and the mean number of contacts with the wall the results are derived from our previous work on vesicles near an attractive wall and involve elliptic integrals as opposed to the simple rational forms found for the percolation probability and cluster size below the results for the cluster length satisfy previously conjectured differential equations a closed expression is conjectured for the mean size above in terms of a hypergeometric function

critical exponents on fortuin kasteleyn weighted planar maps in this paper we consider random planar maps weighted by the self dual fortuin kastelyn model with parameter q2 0 4 using a bijection due to sheeld and a connection to planar brownian motion in a cone we obtain rigorously the value of critical exponents associated with the length of cluster interfaces which is shown to be 4 arccos p 2 p q 2

gevrey series in quantum topology our aim is to prove that two formal power series of importance to quantum topology are gevrey these series are the kashaev invariant of a knot reformulated by huynh and the second author and the gromov norm of the lmo of an integral homology 3 sphere it follows that the power series associated to a simple lie algebra and a homology sphere is gevrey contrary to the case of analysis our formal power series are not solutions to dieren tial equations with polynomial coecien ts the rst author has conjectured and in some cases proved in joint work with costin that our formal power series have resurgent borel transform with geometrically interesting set of singularities

solving difference equations whose coefficients are not transcendental we consider a large class of sequences which are defined by systems of possibly nonlinear difference equations a procedure for recursively enumerating the algebraic dependencies of such sequences is presented also a procedure for solving linear difference equations with such sequences as coefficients is proposed the methods are illustrated on some problems arising in the literature on special functions and combinatorial sequences

edge grouping combining boundary and region information this paper introduces a new edge grouping method to detect perceptually salient structures in noisy images specifically we define a new grouping cost function in a ratio form where the numerator measures the boundary proximity of the resulting structure and the denominator measures the area of the resulting structure this area term introduces a preference towards detecting larger size structures and therefore makes the resulting edge grouping more robust to image noise to find the optimal edge grouping with the minimum grouping cost we develop a special graph model with two different kinds of edges and then reduce the grouping problem to finding a special kind of cycle in this graph with a minimum cost in ratio form this optimal cycle finding problem can be solved in polynomial time by a previously developed graph algorithm we implement this edge grouping method test it on both synthetic data and real images and compare its performance against several available edge grouping and edge linking methods furthermore we discuss several extensions of the proposed method including the incorporation of the well known grouping cues of continuity and intensity homogeneity introducing a factor to balance the contributions from the boundary and region information and the prevention of detecting self intersecting boundaries

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

proof of ira gessel s lattice path conjecture abstract we present a computer aided yet fully rigorous proof of ira gessel s tantalizingly simply stated conjecture that the number of ways of walking 2n steps in the region x y 0 y 0 of the square lattice with unit steps in the east west north and south directions that start and end at the origin equals

a class of constrained clustering algorithms for object boundary extraction boundary extraction is a key task in many image analysis operations this paper describes a class of constrained clustering algorithms for object boundary extraction that includes several well known algorithms proposed in different fields deformable models constrained clustering data ordering and traveling salesman problems the algorithms belonging to this class are obtained by the minimization of a cost function with two terms a quadratic regularization term and an image dependent term defined by a set of weighting functions the minimization of the cost function is achieved by lowpass filtering the previous model shape and by attracting the model units toward the centroids of their attraction regions to define a new algorithm belonging to this class the user has to specify a regularization matrix and a set of weighting functions that control the attraction of the model units toward the data the usefulness of this approach is twofold it provides a unified framework for many existing algorithms in pattern recognition and deformable models and allows the design of new recursive schemes

square lattice ising model susceptibility series expansion method and differential equation for χ 3 in a previous paper 2004 j phys a math gen 37 9651 68 we have given the fuchsian linear differential equation satisfied by 3 the three particle contribution to the susceptibility of the isotropic square lattice ising model this paper gives the details of the calculations with some useful tricks and tools allowing one to obtain long series in polynomial time the method is based on series expansion in the variables that appear in the n 1 dimensional integrals representing the n particle contribution to the isotropic square lattice ising model susceptibility the integration rules are straightforward due to remarkable formulae we derived for these variables we obtain without any numerical approximation 3 as a fully integrated series in the variable w s 2 1 s2 where s sh 2k with k j kt the conventional ising model coupling constant we also give some perspectives and comments on these results

the product of the eigenvalues of a symmetric tensor abstract we study e eigenvalues of a symmetric tensor f of degree d on a finite dimensional euclidean vector space v and their relation with the e characteristic polynomial of f we show that the leading coefficient of the e characteristic polynomial of f when it has maximum degree is the d 2 th power respectively the d 2 2 th power when d is odd respectively when d is even of the q discriminant where q is the d th veronese embedding of the isotropic quadric q p v this fact together with a known formula for the constant term of the e characteristic polynomial of f leads to a closed formula for the product of the e eigenvalues of f which generalizes the fact that the determinant of a symmetric matrix is equal to the product of its eigenvalues

some results of katznelson tzafriri type abstract for semigroups s r z and z k we show that if t is a representation of s by contractions on a hilbert space then inf t s t t f t sup f sp u t s where f l 1 s and sp u t s is the unitary spectrum of t with respect to s for t a representation of a suitable semigroup s by contractions on a banach space we give sharp conditions on sp u t s which guarantee that the equality above holds these conditions concern the thinness of sp u t s in the harmonic analysis sense these results are related to theorems of katznelson tzafriri type which give conditions guaranteeing that inf t s t t f t vanishes

a distance exponent for liouville quantum gravity let gamma in 0 2 and let h be the random distribution on mathbb c which describes a gamma liouville quantum gravity lqg cone also let kappa 16 gamma 2 4 and let eta be a whole plane space filling sle kappa curve sampled independent from h and parametrized by gamma quantum mass with respect to h we study a family mathcal g epsilon epsilon 0 of planar maps associated with h eta called the lqg structure graphs a k a mated crt maps which we conjecture converge in probability in the scaling limit with respect to the gromov hausdorff topology to a random metric space associated with gamma lqg in particular mathcal g epsilon is the graph whose vertex set is epsilon mathbb z with two such vertices x 1 x 2 in epsilon mathbb z connected by an edge if and only if the corresponding curve segments eta x 1 epsilon x 1 and eta x 2 epsilon x 2 share a non trivial boundary arc due to the peanosphere description of sle decorated lqg due to duplantier et al liouville quantum gravity as a mating of trees 2014 the graph mathcal g epsilon can equivalently be expressed as an explicit functional of a correlated two dimensional brownian motion so can be studied without any reference to sle or lqg we prove non trivial upper and lower bounds for the cardinality of a graph distance ball of radius n in mathcal g epsilon which are consistent with the prediction of watabiki prog theor phys suppl 114 1 17 1993 for the hausdorff dimension of lqg using subadditivity arguments we also prove that there is an exponent chi 0 for which the expected graph distance between generic points in the subgraph of mathcal g epsilon corresponding to the segment eta 0 1 is of order epsilon chi o epsilon 1 and this distance is extremely unlikely to be larger than epsilon chi o epsilon 1

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

a diagrammatic approach to the aj conjecture the aj conjecture relates a quantum invariant a minimal order recursion for the colored jones polynomial of a knot known as the hat a polynomial with a classical invariant namely the defining polynomial a of the mathrm psl 2 mathbb c character variety of a knot more precisely the aj conjecture asserts that the set of irreducible factors of the hat a polynomial after we set q 1 and excluding those of l degree zero coincides with those of the a polynomial in this paper we introduce a version of the hat a polynomial that depends on a planar diagram of a knot that conjecturally agrees with the hat a polynomial and we prove that it satisfies one direction of the aj conjecture our proof uses the octahedral decomposition of a knot complement obtained from a planar projection of a knot the r matrix state sum formula for the colored jones polynomial and its certificate

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

weak lqg metrics and liouville first passage percolation for gamma in 0 2 we define a weak gamma liouville quantum gravity lqg metric to be a function h mapsto d h which takes in an instance of the planar gaussian free field and outputs a metric on the plane satisfying a certain list of natural axioms we show that these axioms are satisfied for any subsequential limits of liouville first passage percolation such subsequential limits were proven to exist by ding et al tightness of liouville first passage percolation for gamma in 0 2 2019 arxiv e prints arxiv 1904 08021 it is also known that these axioms are satisfied for the sqrt 8 3 lqg metric constructed by miller and sheffield 2013 2016 for any weak gamma lqg metric we obtain moment bounds for diameters of sets as well as point to point set to set and point to set distances we also show that any such metric is locally bi holder continuous with respect to the euclidean metric and compute the optimal holder exponents in both directions finally we show that lqg geodesics cannot spend a long time near a straight line or the boundary of a metric ball these results are used in subsequent work by gwynne and miller which proves that the weak gamma lqg metric is unique for each gamma in 0 2 which in turn gives the uniqueness of the subsequential limit of liouville first passage percolation however most of our results are new even in the special case when gamma sqrt 8 3

fast evaluation of holonomic functions abstract a holonomic function is an analytic function which satisfies a linear differential equation with polynomial coefficients in particular the elementary functions exp log sin etc and many special functions like erf si bessel functions etc are holonomic functions given a holonomic function f determined by the linear differential equation it satisfies and initial conditions in a non singular point z we show how to perform arbitrary precision evaluations of f at a non singular point z on the riemann surface of f while estimating the error moreover if the coefficients of the polynomials in the equation for f are algebraic numbers then our algorithm is asymptotically very fast if m n is the time needed to multiply two n digit numbers then we need a time o m n log 2 n log log n to compute n digits of f z

an adaptive contour closure algorithm and its experimental evaluation the potential of edge based complete image segmentation into regions has not gained the due attention in the literature thus far the present paper attempts to explore this potential by proposing an adaptive grouping algorithm to solve the contour closure problem that is the key to a successful edge based complete image segmentation the effectiveness of the proposed algorithm is extensively tested in the range image domain and compared to several region based segmentation methods within a rigorous comparison framework on three range image databases of varying quality acquired by different range scanners it is shown that the proposed approach is able to achieve very appealing performance with respect to both segmentation quality and computation time

how likely is polya s drunkard to stay in x y z in his celebrated paper polya has considered the random walk in the three dimensional cubic lattice and showed that the probability of return to the origin is less than 1 subsequent authors have shown that the probability is 34 053 here we consider the same random walk with the restriction that the drunkard is only allowed to stay inx y z it is shown that his probability of returning to the originand staying in the allowed region is 6 4844

uniqueness and universality of the brownian map we consider a random planar map mn which is uniformly distributed over the class of all rooted q angulations with n faces we let mn be the vertex set of mn which is equipped with the graph distance dgr both when q 4 is an even integer and when q 3 there exists a positive constant cq such that the rescaled metric spaces mn cqn 1 4dgr converge in distribution in the gromov hausdorff sense toward a universal limit called the brownian map the particular case of triangulations solves a question of schramm

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

rationality of the mobius function of a composition poset we consider the zeta and mobius functions of a partial order on integer compositions first studied by bergeron bousquet melou and dulucq the mobius function of this poset was determined by sagan and vatter we prove rationality of various formal power series in noncommuting variables whose coefficients are evaluations of the zeta function and the mobius function the proofs are either directly from the definitions or by constructing finite state automata we also obtain explicit expressions for generating functions obtained by specializing the variables to commutative ones we reprove sagan and vatter s formula for using this machinery these results are closely related to those of bjorner and reutenauer about subword order and we discuss a common generalization

bose and fermi walk configurations on planar graphs the number fcn h of n walk configurations of type c is investigated on certain two rooted directed planar graphs h which will be always realized as plane graphs in r2 c may be bose or fermi as defined by inui and katori both types of configuration are collections of non crossing walks which follow the directed paths between the roots of the plane graph h in the case of configurations of fermi type each walk may be included only once the number fbosen h is shown to be a polynomial in n of degree nmax 1 where nmax is the maximum number of walks in a fermi configuration the coefficient of the highest power of n in this polynomial is simply related to the number of maximal fermi walk configurations it is also shown that nmax c h 1 where c h is the number of finite faces on h extension of these results to multi rooted graphs is also discussed when h is the union of paths between two sites of the directed square lattice subject to various boundary conditions kreweras showed that the number of bose configurations is equal to the number of n element multi chains on segments of young s lattice he expressed this number as a determinant the elements of which are polynomials in n we evaluate this determinant by the method of lu decomposition in the case of watermelon configurations above a wall in this case the polynomial is a product of linear factors but on introducing a second wall the polynomial does not completely factorize but has a factor which is the number of watermelon configurations on the largest rectangular subgraph the number of two rooted star configurations is found to be the product of the numbers of watermelon configurations on the three rectangular subgraphs into which it may be partitioned

the colored homflypt function is q holonomic we prove that the homflypt polynomial of a link colored by partitions with a fixed number of rows is a q holonomic function by specializing to the case of knots colored by a partition with a single row it proves the existence of an a q superpolynomial of knots in 3 space as was conjectured by string theorists our proof uses skew howe duality that reduces the evaluation of web diagrams and their ladders to a poincare birkhoff witt computation of an auxiliary quantum group of rank the number of strings of the ladder diagram the result is a concrete and algorithmic web evaluation algorithm that is manifestly q holonomic

the brownian map is the scaling limit of uniform random plane quadrangulations we prove that uniform random quadrangulations of the sphere with n faces endowed with the usual graph distance and renormalized by n 1 4 converge as n in distribution for the gromov hausdorff topology to a limiting metric space we validate a conjecture by le gall by showing that the limit is up to a scale constant the so called brownian map which was introduced by marckert mokkadem and le gall as the most natural candidate for the scaling limit of many models of random plane maps the proof relies strongly on the concept of geodesic stars in the map which are configurations made of several geodesics that only share a common endpoint and do not meet elsewhere

continuity and estimates of the liouville heat kernel with applications to spectral dimensions the liouville brownian motion lbm recently introduced by garban rhodes and vargas and in a weaker form also by berestycki is a diffusion process evolving in a planar random geometry induced by the liouville measure m gamma formally written as m gamma dz e gamma x z gamma 2 mathbb e x z 2 2 dz gamma in 0 2 for a massive gaussian free field x it is an m gamma symmetric diffusion defined as the time change of the two dimensional brownian motion by the positive continuous additive functional with revuz measure m gamma in this paper we provide a detailed analysis of the heat kernel p t x y of the lbm specifically we prove its joint continuity a locally uniform sub gaussian upper bound of the form p t x y le c 1 t 1 log t 1 exp bigl c 2 x y beta wedge 1 t frac 1 beta 1 bigr for t in bigl 0 frac 1 2 bigr for each beta frac 1 2 gamma 2 2 and an on diagonal lower bound of the form p t x x ge c 3 t 1 bigl log t 1 bigr eta for t in 0 t eta x with t eta x in bigl 0 frac 1 2 bigr heavily dependent on x for each eta 18 for m gamma almost every x as applications we deduce that the pointwise spectral dimension equals 2 m gamma a e and that the global spectral dimension is also 2

asymptotic values for degrees associated with strips of young diagrams abstract the sums s l n occur in the representations of the symmetric and the general linear groups in combinatorics and in pi algebras we give asymptotic values for these sums

on zeilberger s algorithm and its q analogue abstract gosper s and zeilberger s algorithms for summation of terminating hypergeometric series as well as the q versions of these algorithms are described in a very rigorous way the paper is a companion to maple v procedures implementing these algorithms it concludes with the help information for these procedures

the fuchsian differential equation of the square lattice ising model χ 3 susceptibility using an expansion method in the variables xi that appear in n 1 dimensional integrals representing the n particle contribution to the ising square lattice model susceptibility we generate a long series of coefficients for the three particle contribution 3 using an n4 polynomial time algorithm we give the fuchsian differential equation of order 7 for 3 that reproduces all the terms of our long series an analysis of the properties of this fuchsian differential equation is performed

random k noncrossing rna structures in this paper we introduce a combinatorial framework that provides an interpretation of rna pseudoknot structures as sampling paths of a markov process our results facilitate a variety of applications ranging from the energy based sampling of pseudoknot structures as well as the ab initio folding via hidden markov models our main result is an algorithm that generates rna pseudoknot structures with uniform probability this algorithm serves as a steppingstone to sequence specific as well as energy based transition probabilities the approach employs a correspondence between pseudoknot structures parametrized in terms of the maximal number of mutually crossing arcs and certain tableau sequences the latter can be viewed as lattice paths the main idea of this paper is to view each such lattice path as a sampling path of a stochastic process and to make use of d finiteness for the efficient computation of the corresponding transition probabilities

proof of george andrews s and david robbins s q tspp conjecture the conjecture that the orbit counting generating function for totally symmetric plane partitions can be written as an explicit product formula has been stated independently by george andrews and david robbins around 1983 we present a proof of this long standing conjecture

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

graph based representations and techniques for image processing and image analysis in this paper we will discuss the use of some graph based representations and techniques for image processing and analysis instead of making an extensive review of the graph techniques in this field we will explain how we are using these techniques in an active vision system for an autonomous mobile robot developed in the institut de robotica i informatica industrial within the project active vision system with automatic learning capacity for industrial applications cicyt tap98 0473 specifically we will discuss the use of graph based representations and techniques for image segmentation image perceptual grouping and object recognition we first present a generalisation of a graph partitioning greedy algorithm for colour image segmentation next we describe a novel fusion of colour based segmentation and depth from stereo that yields a graph representing every object in the scene finally we describe a new representation of a set of attributed graphs ags denominated function described graphs fdgs a distance measure for matching ags with fdgs and some applications for robot vision

on a class of languages with holonomic generating functions we define a class of languages rcm obtained by considering regular languages linear constraints on the number of occurrences of symbols and morphisms the class rcm presents some interesting closure properties and contains languages with holonomic generating functions as a matter of fact rcm is related to one way 1 reversal bounded k counter machines and also to parikh automata on letters indeed rcm is contained in l nfcm but not in l dfcm and strictly includes l lpa we conjecture that l dfcm rcm

densities of short uniform random walks in higher dimensions we study arithmetic properties of short uniform random walks in arbitrary dimensions with a focus on explicit hypergeometric evaluations of the moment functions and probability densities in the case of up to five steps somewhat to our surprise we are able to provide complete extensions to arbitrary dimensions for most of the central results known in the two dimensional case

segmentation of multiple salient closed contours from real images using a saliency measure based on the global property of contour closure we have developed a segmentation method which identifies smooth closed contours bounding objects of unknown shape in real images the saliency measure incorporates the gestalt principles of proximity and good continuity that previous methods have also exploited unlike previous methods we incorporate contour closure by finding the eigenvector with the largest positive real eigenvalue of a transition matrix for a markov process where edges from the image serve as states element i j of the transition matrix is the conditional probability that a contour which contains edge j will also contain edge i we show how the saliency measure defined for individual edges can be used to derive a saliency relation defined for pairs of edges and further show that strongly connected components of the graph representing the saliency relation correspond to smooth closed contours in the image finally we report for the first time results on large real images for which segmentation takes an average of about 10 seconds per object on a general purpose workstation

the continuum random tree iii let mathscr r k k geq 1 be random trees with k leaves satisfying a consistency condition removing a random leaf from mathscr r k gives mathscr r k 1 then under an extra condition this family determines a random continuum tree mathscr l which it is convenient to represent as a random subset of l 1 this leads to an abstract notion of convergence in distribution as n rightarrow infty of rescaled random trees mathscr j n on n vertices to a limit continuum random tree mathscr l the notion is based upon the assumption that for fixed k the subtrees of mathscr j n determined by k randomly chosen vertices converge to mathscr r k as our main example under mild conditions on the offspring distribution the family tree of a galton watson branching process conditioned on total population size equal to n can be rescaled to converge to a limit continuum random tree which can be constructed from brownian excursion

unsupervised contour closure algorithm for range image edge based segmentation this paper presents an efficient technique for extracting closed contours from range images edge points edge points are assumed to be given as input to the algorithm i e previously computed by an edge based range image segmentation technique the proposed approach consists of three steps initially a partially connected graph is generated from those input points then the minimum spanning tree of that graph is computed finally a postprocessing technique generates a single path through the regions boundaries by removing noisy links and closing open contours the novelty of the proposed approach lies in the fact that by representing edge points as nodes of a partially connected graph it reduces the contour closure problem to a minimum spanning tree partitioning problem plus a cost function minimization stage to generate closed contours experimental results with synthetic and real range images together with comparisons with a previous technique are presented