series representations for some mathematical constants abstract the authors apply a classical series identity involving the psi or digamma function with a view to deriving series representations for a number of known mathematical constants several closely related consequences and results are also considered

some classes of infinite series associated with the riemann zeta and polygamma functions and generalized harmonic numbers the authors aim at presenting a systematic investigation of several families of infinite series which are associated with the riemann zeta function the digamma and polygamma functions the harmonic and generalized harmonic numbers and the stirling numbers of the first kind relevant connections of the results derived here with those considered in many earlier works on this subject are also indicated

on some series containing p x p y and 9 p x p y 2 for certain values of x and y abstract in this article we discuss series containing the factor k 1 k 1 2 1 2 k 1 2 and 1 2 k 1 case1 2 k with k n here x d dxlog x and x is the gammafunction

an efficient algorithm for maximum entropy extension of block circulant covariance matrices article history this paper deals with maximum entropy completion of partially specified block circulant matrices since positive definite symmet ric circulants happen to be covariance matrices of stationary peri odic processes in particular of stationary reciprocal processes this problem has applications in signal processing in particular to im age modeling in fact it is strictly related to maximum likelihood estimation of bilateral ar type representations of acausal signals subject to certain conditional independence constraints the maxi mum entropy completion problem for block circulant matrices has recently been solved by the authors although leaving open the problem of an efficient computation of the solution in this pa per we provide an efficient algorithm for computing its solution which compares very favorably with existing algorithms designed for positive definite matrix extension problems the proposed al gorithm benefits from the analysis of the relationship between our problem and the band extension problem for block toeplitz matri ces also developed in this paper

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

mesures et equidistribution sur les espaces de berkovich the proof by ullmo and zhang of bogomolov s conjecture about points of small height in abelian varieties made a crucial use of an equidistribution property for small points in the associated complex abelian variety we study the analogous equidistribution property at p adic places our results can be conveniently stated within the framework of the analytic spaces defined by berkovich the first one is valid in any dimension but is restricted to algebraic metrics the second one is valid for curves but allows for more general metrics in particular to the normalized heights with respect to dynamical systems

reduction formulae for kampe de feriet functions fq 1 0p 2 1 abstract several reduction formulae for kampe de feriet functions f q 1 0 p 2 1 are given in terms of a finite number of generalized hypergeometric functions

spanning tree extensions of the hadamard fischer inequalities abstract all possible graph theoretic generalizations of a certain sort for the hadamard fischer determinantal inequalities are determined these involve ratios of products of principal minors which dominate the determinant furthermore the cases of equality in these inequalities are characterized and equality is possible for every set of values which can occur for the relevant minors this relates recent work of the authors on positive definite completions and determinantal identities when applied to the same collections of principal minors earlier generalizations give poorer more difficult to compute bounds than the present inequalities thus this work extends and in a certain sense completes a series of generalizations of hadamard fischer begun in the 1960s

a generalized levinson algorithm for covariance extension with application to multiscale autoregressive modeling efficient computation of extensions of banded partially known covariance matrices is provided by the classical levinson algorithm one contribution of this paper is the introduction of a generalization of this algorithm that is applicable to a substantially broader class of extension problems this generalized algorithm can compute unknown covariance elements in any order that satisfies certain graph theoretic properties which we describe this flexibility which is not provided by the classical levinson algorithm is then harnessed in a second contribution of this paper the identification of a multiscale autoregressive mar model for the maximum entropy me extension of a banded partially known covariance matrix the computational complexity of mar model identification is an order of magnitude below that of explicitly computing a full covariance extension and is comparable to that required to build a standard autoregressive ar model using the classical levinson algorithm

positive semidefinite matrix completions on chordal graphs and constraint nondegeneracy in semidefinite programming let g v e be a graph in matrix completion theory it is known that the following two conditions are equivalent i g is a chordal graph ii every g partial positive semidefinite matrix has a positive semidefinite matrix completion in this paper we relate these two conditions to constraint nondegeneracy condition in semidefinite programming and prove that they are each equivalent to iii for any g partial positive definite matrix that has a positive semidefinite completion constraint nondegeneracy is satisfied at each of its positive semidefinite matrix completions

further reducible cases of certain kampe de fe riet functions associated with incomplete integrals of cylindrical functions abstract new reduction formulas are derived for the kampe de feriet functions f q 1 0 p 2 1 and f 1 2 1 1 1 0 which are closely connected with representations for incomplete lipschitz hankel type integrals of cylindrical functions some general series identities and related reduction formulas relevant to the present investigation are also considered

on one dimensional digamma and polygamma series related to the evaluation of feynman diagrams we consider summations over digamma and polygamma functions often with summands of the form 1 n j n p q n r and 1 n j m n p q n r where m p q and r are positive integers we develop novel general integral representations and present explicit examples special cases of the sums reduce to known linear euler sums the sums of interest find application in quantum field theory including evaluation of feynman amplitudes

values of the polygamma functions at rational arguments gauss in 1812 in his celebrated memoir on the hypergeometric series presented a remarkable formula for the psi or digamma function z at rational arguments z which can be expressed in terms of elementary functions davis in 1935 extended gauss s result to the polygamma functions by using a known series representation of n z in an elementary yet technical way k lbig in 1996 in his cern technical report also gave two extensions to n z by using the series definition of polylogarithm function and the above known series representation here we aim at deriving general formulae expressing n z as rational arguments in terms of other functions which will be obtained in two ways in addition several special cases are also considered and as a by product of our main results we derive in a simple and unified manner all formulae given by gauss davis and k lbig finally it should be noted that all our results in view of the relationship between n z and the hurwitz zeta function s a could be rewritten in the representation of s a

closed form summations of certain hypergeometric type series containing the digamma function recently interesting novel summation formulae for hypergeometric type series containing a digamma function as a factor have been established by miller 2006 j phys a math gen 39 3011 20 mainly by exploiting already known results and using certain transformation and reduction formulae in the theory of the kampe de feriet double hypergeometric function it is shown here that these and several other series of this type can be closed form summed by simpler and more direct arguments based only on a derivative formula for the pochhammer symbol and the theory of the digamma or function and generalized hypergeometric function in addition a new reduction formula for the kampe de feriet function is obtained

positive definite completions and determinant maximization abstract a method is described for determining whether a positive definite completion of a given partial hermitian matrix exists and if so for finding the determinant maximizing positive definite completion no assumption is made about the arrangement of the specified entries the method employs iterative application to a modified problem of an explicit formula for the maximum determinant in case there is only one symmetrically placed pair of unspecified entries a robust and fast algorithm based upon this approach is shown to have global convergence to the necessarily unique solution

how many rational points does a random curve have a large part of modern arithmetic geometry is dedicated to or motivated by the study of rational points on varieties for an elliptic curve over q the set of rational points forms a finitely generated abelian group the ranks of these groups when ranging over all elliptic curves are conjectured to be evenly distributed between rank 0 and rank 1 with higher ranks being negligible we will describe these conjectures and discuss some results on bounds for average rank highlighting the recent work of bhargava and shankar

determinantal formulae for matrix completions associated with chordal graphs abstract let a be a partial positive definite hermitian matrix and let g a be the undirected graph of the specified off diagonal entries of a when g a is a chordal graph we present an explicit determinantal formula for the unique determinant maximizing positive definite completion maximum entropy completion of a in order to do this we first give a characterization of a chordal graph in terms of the existence of a proper spanning tree t of the intersection graph of its set of maximal cliques the formula for the maximum determinant is then a quotient of products of specified principal minors of a in which each minor in the numerator corresponds to a maximal clique of g a while each minor in the denominator corresponds to the intersection of two maximal cliques associated with an edge of t we next show that the denominator terms can be identified graph theoretically with the minimal vertex separators of g a and that these terms are independent of the particular proper spanning tree t then an alternative formula is given for the maximum determinant as an inclusion exclusion like ratio of principal minors associated with the maximal cliques of g a finally we show that the entries of the determinant maximizing completion may be calculated by solving a sequence of simple one variable maximization problems

gaussian markov distributions over finite graphs on s interesse aux modeles de selection de covariance introduites par dempster 1972 et etudies par wermuth 1976

summations for certain series containing the digamma function apparently new summations in terms of well known special functions are deduced for hypergeometric type series containing a digamma function as a factor as a by product of this investigation new reduction formulae for the kampe de feriet function f0 2 12 1 0 x x are obtained