spectral methods for weakly singular volterra integral equations with smooth solutions we propose and analyze a spectral jacobi collocation approximation for the linear volterra integral equations vies of the second kind with weakly singular kernels in this work we consider the case when the underlying solutions of the vies are sufficiently smooth in this case we provide a rigorous error analysis for the proposed method which shows that the numerical errors decay exponentially in the infinity norm and weighted sobolev space norms numerical results are presented to confirm the theoretical prediction of the exponential rate of convergence

a network algorithm for performing fisher s exact test in r c contingency tables abstract an exact test of significance of the hypothesis that the row and column effects are independent in an r c contingency table can be executed in principle by generalizing fisher s exact treatment of the 2 2 contingency table each table in a conditional reference set of r c tables with fixed marginal sums is assigned a generalized hypergeometric probability the significance level is then computed by summing the probabilities of all tables that are no larger on the probability scale than the observed table however the computational effort required to generate all r c contingency tables with fixed marginal sums severely limits the use of fisher s exact test a novel technique that considerably extends the bounds of computational feasibility of the exact test is proposed here the problem is transformed into one of identifying all paths through a directed acyclic network that equal or exceed a fixed length some interesting new optimization theorems are developed in the process the numer

barycentric interpolation collocation methods for solving linear and nonlinear high dimensional fredholm integral equations abstract in this article two barycentric interpolation collocation methods are proposed for solving linear and nonlinear high dimensional f r e d h o l m integral equations of the second kind the approaches respectively utilize the modified weighted l a g r a n g e functions and the novel rational functions as the interpolation basis functions they are effective schemes for evaluating the multidimensional undetermined function through the numerical strategies and some composite quadrature formulas the linear and nonlinear f r e d h o l m integral equations are transformed into the corresponding linear and nonlinear algebraic equations further we prove that the discrete collocation methods are equivalent to the n y s t r o m quadrature methods then the convergence analysis is established by the collectively compact theory moreover the error estimation of the approximate solution and the exact solution are also provided numerical examples are presented to illustrate the capability and efficiency of the techniques by compared with the classic l a g r a n g e interpolation collocation method and other methodologies

markov complexity of monomial curves abstract let a a 1 a n n m we give an algebraic characterization of the universal markov basis of the toric ideal i a we show that the markov complexity of a n 1 n 2 n 3 is equal to 2 if i a is complete intersection and equal to 3 otherwise answering a question posed by santos and sturmfels we prove that for any r 2 there is a unique minimal markov basis of a r moreover we prove that for any integer l there exist integers n 1 n 2 n 3 such that the graver complexity of a is greater than l

generalized monte carlo significance tests summary simple monte carlo significance testing has many applications particularly in the preliminary analysis of spatial data the method requires the value of the test statistic to be ranked among a random sample of values generated according to the null hypothesis however there are situations in which a sample of values can only be conveniently generated using a markov chain initiated by the observed data so that independence is violated this paper describes two methods that overcome the problem of dependence and allow exact tests to be carried out the methods are applied to the rasch model to the finite lattice ising model and to the testing of association between spatial processes power is discussed in a simple case

gomory integer programs the set of all group relaxations of an integer program contains certain special members called gomory relaxations a family of integer programs with a fixed coefficient matrix and cost vector but varying right hand sides is a gomory family if every program in the family can be solved by one of its gomory relaxations in this paper we characterize gomory families every tdi system gives a gomory family and we construct gomory families from matrices whose columns form a hilbert basis for the cone they generate the existence of gomory families is related to the hilbert covering problems that arose from the conjectures of sebo connections to commutative algebra are outlined at the end

methods for exact goodness of fit tests abstract numerous goodness of fit tests with asymptotic chi squared distributions have been proposed for discrete multivariate data and there has been much discussion about using asymptotic results for computing critical values when there are small expected cell values although exact methods would be preferred in these situations it generally is believed that such methods are computationally intractable we propose methods for calculating exact distributions and significance levels for goodness of fit statistics that are computationally feasible over a wide range of models in particular the distribution for a simple multinomial model can be evaluated in polynomial time for composite null hypotheses we calculate the distribution conditional on the sufficient statistics for the nuisance parameters we calculate the characteristic function of a distribution and invert the characteristic function using the fast fourier transform fft our approach emphasizes the relationship between exact methods and

unbiased testing in exponential family regression let x ij z i i 1 2 k j 1 2 n i be independent observations such that z i is a fixed r x 1 vector r can be 0 no z s observed or 1 2 k 1 and x ij is distributed according to a one parameter exponential family which is log concave with natural parameter i we test the hypothesis that z where 1 k z is the matrix whose ith row is z i and 1 r is a vector of parameters we focus on r 2 and z i 1 z i i 1 2 k z i z i 1 the null hypothesis on hand is thus of the form i 1 2 z i in such a case the model under the null hypothesis becomes logistic regression in the binomial case poisson regression in the poisson case and linear regression in the normal case we consider mostly the one sided alternative that the second order differences of the natural parameters are nonnegative such testing problems test goodness of fit vs alternatives in which the natural parameters behave in a convex way we find classes of tests that are unbiased and that lie in a complete class we also note that every admissible test of constant size is unbiased in some discrete situations we find the minimal complete class of unbiased admissible tests generalizations and examples are given

fixed width output analysis for markov chain monte carlo markov chain monte carlo is a method of producing a correlated sample to estimate features of a target distribution through ergodic averages a fundamental question is when sampling should stop that is at what point the ergodic averages are good estimates of the desired quantities we consider a method that stops the simulation when the width of a confidence interval based on an ergodic average is less than a user specified value hence calculating a monte carlo standard error is a critical step in assessing the simulation output we consider the regenerative simulation and batch means methods of estimating the variance of the asymptotic normal distribution we give sufficient conditions for the strong consistency of both methods and investigate their finite sample properties in various examples

constructions and complexity of secondary polytopes abstract the secondary polytope a of a configuration a of n points in affine d 1 space is an n d polytope whose vertices correspond to regular triangulations of conv a in this article we present three constructions of a and apply them to study various geometric combinatorial and computational properties of secondary polytopes the first construction is due to gel fand kapranov and zelevinsky who used it to describe the face lattice of a we introduce the universal polytope u a d r n a combinatorial object depending only on the oriented matroid of a the secondary a can be obtained as the image of u a under a canonical linear map onto r n the third construction is based upon gale transforms or oriented matroid duality it is used to analyze the complexity of computing a and to give bounds in terms of n and d for the number of faces of a

sampling for conditional inference on network data random graphs with given vertex degrees have been widely used as a model for many real world complex networks however both statistical inference and analytic study of such networks present great challenges in this article we propose a new sequential importance sampling method for sampling networks with a given degree sequence these samples can be used to approximate closely the null distributions of a number of test statistics involved in such networks and provide an accurate estimate of the total number of networks with given vertex degrees we study the asymptotic behavior of the proposed algorithm and prove that the importance weight remains bounded as the size of the graph grows this property guarantees that the proposed sampling algorithm can still work efficiently even for large sparse graphs we apply our method to a range of examples to demonstrate its efficiency in real problems

algebraic algorithms for sampling from conditional distributions we construct markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic examples include contingency tables logistic regression and spectral analysis of permutation data the algorithms involve computations in polynomial rings using grobner bases

a meshless discrete collocation method for the numerical solution of singular logarithmic boundary integral equations utilizing radial basis functions abstract the main intention of the current paper is to describe a scheme for the numerical solution of boundary integral equations of the second kind with logarithmic singular kernels these types of integral equations result from boundary value problems of laplace s equations with linear robin boundary conditions the method approximates the solution using the radial basis function rbf expansion with polynomial precision in the discrete collocation method the collocation method for solving logarithmic boundary integral equations encounters more difficulties for computing the singular integrals which cannot be approximated by the classical quadrature formulae to overcome this problem we utilize the non uniform composite gauss legendre integration rule and employ it to estimate the singular logarithm integrals appeared in the method since the scheme is based on the use of scattered points spread on the analyzed domain and does not need any domain elements we can call it as the meshless discrete collocation method the new algorithm is successful and easy to solve various types of boundary integral equations with singular kernels we also provide the error estimate of the proposed method the efficiency and accuracy of the new approach are illustrated by some numerical examples

generalized derivations of lie algebras abstract suppose l is a finite dimensional lie algebra with multiplication l l l let l denote the set of triples f f f with f f f hom l l such that f i l i l f f we consider the lie algebra gender l f hom l l f f f f f l well researched subalgebras of gender l include the derivation algebra der l f hom l l f f f l and the centroid c l f hom l l f 0 f l we now study the subalgebra qder l f hom l l f f f f l and the subspace qc l f hom l l f f 0 l in characteristic 2 gender l qder l qc l and we are concerned with the inclusions der l qder l and c l qc l qder l if z l 0 then c l qc l qder l and under reasonable conditions on lie algebras with toral cartan subalgebras we show qder l der l c l if l is a parabolic subalgebra of a simple lie algebra of rank 1 in characteristic 0 then we even have gender l ad l i l in general qc l is not closed under composition or lie bracket however if z l 0 then qc l is a commutative associative algebra and we describe conditions that force qc l c l or equivalently gender l qder l we show that in characteristic 0 gender l preserves the radical of l thus generalizing the classical result for der l we also discuss some applications of the main results to the study of functions f hom l l such that f or f i l defines a lie multiplication

binomial generation of the radical of a lattice ideal abstract let i l be a lattice ideal we provide a necessary and sufficient criterion under which a set of binomials in i l generates the radical of i l up to radical we apply our results to the problem of determining the minimal number of generators of i l or of the rad i l up to radical

the circuit ideal of a vector configuration abstract given a configuration a a 1 a n z d a basis ideal of a is an ideal j b x u x u u b k x 1 x n where b spans the lattice l a u z n a i u i 0 our main interest is to understand when the toric ideal i a of a equals a basis ideal j b with radical i a the circuit ideal j c a of a is an example of such a basis ideal we study such a j b in relation to i a from various algebraic and combinatorial perspectives with a special focus on j c a we prove that the obstruction to equality of the ideals is the existence of certain polytopes this result is based on a complete characterization of the standard pairs associated primes of a monomial initial ideal of j b and their differences from those for the corresponding toric initial ideal eisenbud and sturmfels proved that the embedded primes of j b are indexed by certain faces of the cone spanned by a we provide a necessary condition for a particular face to index an embedded prime and a partial converse finally we compare various polyhedral fans associated to i a and j c a the grobner fan of j c a is shown to refine that of i a when the codimension of the ideals is at most two

bernstein series solutions of multidimensional linear and nonlinear volterra integral equations with fractional order weakly singular kernels abstract this paper proposes a quadrature method based on multi variate bernstein polynomials the method is used to solve multidimensional volterra integral equations with weakly singular kernels firstly we use multi variate bernstein polynomials to approximate the unknown function of an equation then a discrete function equation can be obtained by substituting the approximate solution into the equation secondly the discrete function system is transformed into an algebra equation system by using some discrete points we can perform the integral operations without discrete kernel function and the weakly singular integrals can be calculated directly by using quadrature method so the method is easy to implement thirdly we prove the existence and uniqueness of the solution of the approximate equation as well as the error analysis of the proposed method six numerical examples are given to illustrate the efficiency of this method

space time spectral method for a weakly singular parabolic partial integro differential equation on irregular domains abstract the spectral method is proposed for the partial integro differential equations with a weakly singular kernel on irregular domains the space discretization is based on the nodal spectral element method using the lagrange polynomials basis associated with the gauss lobatto legendre quadrature nodes also the model is discretized in time with the legendre spectral galerkin method the discretization leads to conversion of the problem to a sylvester matrix equation which can be solved efficiently by the qz algorithm gardiner et al 1992 the convergence of the method is proven by providing a priori l 2 error estimate numerical results illustrate the efficiency and spectral accuracy of the proposed method

a geometric buchberger algorithm for integer programming let ip a c denote the family of integer programs of the form min cx ax b x nn obtained by varying the right hand side vector b but keeping a and c fixed a test set for ipa c is a set of vectors in zn such that for each nonoptimal solution to a program in this family there is at least one element g in this set such that g has an improved cost value as compared to we describe a unique minimal test set for this family called the reduced grobner basis of ip a c an algorithm for its construction is presented which we call a geometric buchberger algorithm for integer programming and we show how an integer program may be solved using this test set the reduced grobner basis is then compared with some other known test sets from the literature we also indicate an easy procedure to construct test sets with respect to all cost functions for a matrix a zn 2 n of full row rank

applying the modified block pulse functions to solve the three dimensional volterra fredholm integral equations the main aim of this work is to give further studies for the multi dimensional integral equations in this work we solve special types of the three dimensional volterra fredholm linear integral equations of the second kind via the modified block pulse functions some theorems are included to show convergence and advantage of the method we solve some examples to investigate the applicability and simplicity of the method the numerical results confirm that the method is efficient and simple

on the rosenberg zelinsky sequence in abelian monoidal categories we consider frobenius algebras and their bimodules in certain abelian monoidal categories in particular we study the picard group of the category of bimodules over a frobenius algebra i e the group of isomorphism classes of invertible bimodules the rosenberg zelinsky sequence describes a homomorphism from the group of algebra automorphisms to the picard group which however is typically not surjective we investigate under which conditions there exists a morita equivalent frobenius algebra for which the corresponding homomorphism is surjective one motivation for our considerations is the orbifold construction in conformal field theory

testing for independence in a two way table new interpretations of the chi square statistic the classical chi square test for independence in a two way contingency table often rejects the independence hypothesis at an extremely small significance level particularly when the sample size is large this paper proposes some alternative distributions to independence to help interpret the chi 2 statistic in such situations the uniform alternative in which every possible contingency table of the given dimension and sample size receives equal probability leads to the volume test as originally suggested in a regression context by h hotelling exponential family theory is used to generate a class of intermediate alternatives between independence and uniformity leading to a random effects model for contingency tables

approximate conditional inference in exponential families via the gibbs sampler abstract this article presents the gibbs skovgaard algorithm for approximate frequentist inference the method makes use of the double saddlepoint approximation of skovgaard to the conditional cumulative distribution function of a sufficient statistic given the remaining sufficient statistics this approximation is then used in the gibbs sampler to generate a markov chain the equilibrium distribution of this chain approximates the joint distribution of the sufficient statistics associated with the parameters of interest conditional on the observed values of the sufficient statistics associated with the nuisance parameters this gibbs skovgaard algorithm is applied to the cases of logistic and poisson regression

iterated collocation methods and their discretizations for volterra integral equations it is known that the numerical solution of volterra integral equations of the second kind by polynomial spline collocation at the gauss legendre points does not lead to local superconvergence at the knots of the approximating function in the present paper we show that iterated collocation approximation restores optimal local superconvergence at the knots but does not yield global superconvergence on the entire interval of integration in contrast to fredholm integral equations with smooth kernels we also analyze the discretized versions obtained by suitable numerical quadrature of the collocation and iterated collocation methods

combinatorics of binomial primary decomposition an explicit lattice point realization is provided for the primary components of an arbitrary binomial ideal in characteristic zero this decomposition is derived from a characteristic free combinatorial description of certain primary components of binomial ideals in affine semigroup rings namely those that are associated to faces of the semigroup these results are intimately connected to hypergeometric differential equations in several variables

exact sequences of inner automorphisms of tensors abstract we produce a long exact sequence whose terms are unit groups of associative algebras that behave as inner automorphisms of a given tensor our sequence generalizes known sequences for associative and non associative algebras in a manner similar to those our sequence facilitates inductive reasoning about and calculation of the groups of symmetries of a tensor the new insights these methods afford can be applied to problems ranging from understanding algebraic structures to distinguishing entangled states in particle physics

methods for the analysis of contingency tables with large and small cell counts abstract the traditional practice has been to analyze contingency tables using asymptotic 2 approximations for the tail probability of certain test statistics even when the approximation is known to be poor for two dimensional tables pearson s x 2 statistic is most commonly used whereas for multidimensional tables the likelihood ratio test is preferred because it makes it easy to test series of nested hypotheses when asymptotic approximations are not adequate exact tests are preferred recent advances in algorithms have broadened the range of two dimensional tables for which satisfactory answers can be obtained baglivo olivier and pagano 1985 mehta and patel 1983 1986 pagano and taylor halvorsen 1981 in this article we propose algorithms that use a mixture of exact and asymptotic techniques to calculate tail probabilities our methods can be used with several different test statistics for two dimensional and multidimensional contingency table models and hence further broaden the set of tables

performing the exact test of hardy weinberg proportion for multiple alleles summary the hardy weinberg law plays an important role in the field of population genetics and often serves as a basis for genetic inference because of its importance much attention has been devoted to tests of hardy weinberg proportions hwp over the decades it has long been recognized that largesample goodness of fit tests can sometimes lead to spurious results when the sample size and or some genotypic frequencies are small although a complete enumeration algorithm for the exact test has been proposed it is not of practical use for loci with more than a few alleles due to the amount of computation required we propose two algorithms to estimate the significance level for a test of hwp the algorithms are easily applicable to loci with multiple alleles both are remarkably simple and computationally fast relative efficiency and merits of the two algorithms are compared guidelines regarding their usage are given numerical examples are given to illustrate the practicality of the algorithms

low cost numerical solution for three dimensional linear and nonlinear integral equations via three dimensional jacobi polynomials in recent years numerical methods have been introduced to solve two dimensional volterra and fredholm integral equations in this study a numerical scheme is constructed to solve classes of linear and nonlinear three dimensional integral equations volterra fredholm and mixed volterrafredholm this operational approach is proposed to easily and directly solve these equations at low computational costs the scheme is based on the jacobi polynomials on the interval 0 1 where three variable jacobi polynomials are introduced and their operational matrices of integration and product are derived compared to other existing methods for multidimensional problems the jacobi operational method eliminates the time consuming computations and solely employs the one dimensional operational matrix to construct corresponding multidimensional operational matrices the absolute error of the proposed method is almost constant on the studied interval even at higher dimensions confirming the stability of the proposed operational jacobi method required theorems on the convergence of the method are proved in jacobi weighted sobolev space it is established that the error function vanishes as n increases the method is evaluated using several illustrative examples which indicate the proposed method with lesser computational size compared to the blockpulse functions differential transform and degenerate kernel methods

numerical solution for the weakly singular fredholm integro differential equations using legendre multiwavelets an effective method based upon legendre multiwavelets is proposed for the solution of fredholm weakly singular integro differential equations the properties of legendre multiwavelets are first given and their operational matrices of integral are constructed these wavelets are utilized to reduce the solution of the given integro differential equation to the solution of a sparse linear system of algebraic equations in order to save memory requirement and computational time a threshold procedure is applied to obtain the solution to this system of algebraic equations through numerical examples performance of the present method is investigated concerning the convergence and the sparseness of the resulted matrix equation

hit and run algorithms for generating multivariate distributions we introduce a general class of hit and run algorithms for generating essentially arbitrary absolutely continuous distributions on rd they include the hypersphere directions algorithm and the coordinate directions algorithm that have been proposed for identifying nonredundant linear constraints and for generating uniform distributions over subsets of rd given a bounded open set s in rd an absolutely continuous probability distribution on s the target distribution and an arbitrary probability distribution i on the boundary of the d dimensional unit sphere centered at the origin the direction distribution the i hit and run algorithm produces a sequence of iteration points as follows given the nth iteration point x choose a direction i according to the distribution i and then choose the n 1st iteration point according to the conditionalization of the distribution along the line defined by x and x i under mild conditions we show that this sequence of points is a harris recurrent reversible markov chain converging in total variation to the target distribution

picard groups and infinite matrix rings we describe a connection between the picard group of a ring with local units t and the picard group of the unital overring end tt using this connection we show that the three groups pic r pic fm r and pic rfm r are isomorphic for any unital ring r furthermore eaclh ele ment of pic rfm r arises from an automorphism of rfm r which yields an isomorphsm between pic rfm r and out rfm r as one applica tion we extend a classical result of rosenberg and zelinsky by showing that the group outr rfm r is abelian for any commutative unital ring r

on invertible bimodules and automorphisms of noncommutative rings in this article we attempt to generalize the result that for a commu tative ring r the outer automorphism group of i automorphisms of m r is abelian of exponent n it is shown that a slightly weaker stable version of the result is still valid for affine semiprime noetherian pi rings we also show that the automorphism group of an affine commutative domain of positive di mension acts faithfully on the spectrum of the domain we investigate other questions involving bimodules and automorphisms and extend a result of smith on the first weyl algebra as a fixed ring let r be a ring with 1 in this article we are interested in comparing the automorphism group of mn r the ring of x matrices over r and the automorphism group of r in particular if r is commutative it is a classical result that the outer automorphism group of r automorphisms of mn r is an abelian group of exponent dividing this can be restated in terms of invertible bimodules the result is equivalent to the easy fact that any projective rank one module x with nx free over a commutative ring has order dividing in the picent group of r see rz and ko in the noncommutative case there is still a well known relationship between automorphisms and the picard

using operational matrix of two dimensional bernstein polynomials for solving two dimensional integral equations of fractional order in this paper we first preset two dimensional bernstein polynomials operational matrix 2dbpom for two dimensional fractional integral equations 2dfie then we introduce a computational method based on 2dbpom to reduce the 2dfie to those of solving a system of algebraic equations also we give the error analysis finally this method is applied for solving three examples to illustrate the applicability and accuracy of presented method

monomial ideals and the scarf complex for coherent systems in reliability theory a certain type of integer grid called here an echelon grid is an object found both in coherent systems whose components have a finite or countable number of levels and in algebraic geometry if 1 d is an integer vector representing the state of a system then the corresponding algebraic object is a monomial x 1 1 x d d in the indeterminates x 1 x d the idea is to relate a coherent system to monomial ideals so that the so called scarf complex of the monomial ideal yields an inclusion exclusion identity for the probability of failure which uses many fewer terms than the classical identity moreover in the general position case we obtain via the scarf complex the tube bounds given by naiman and wynn j inequal pure appl math 2001 2 1 16 examples are given for the binary case but the full utility is for general multistate coherent systems and a comprehensive example is given

a meshless method for solving nonlinear two dimensional integral equations of the second kind on non rectangular domains using radial basis functions with error analysis in this paper we present a numerical method for solving two dimensional nonlinear fredholm integral equations of the second kind on a non rectangular domain the method utilizes radial basis functions rbfs constructed on scattered points as a basis in the discrete collocation method the proposed scheme is meshless since it does not need any domain element and so it is independent of the geometry of the domain the method reduces the solution of the two dimensional nonlinear integral equation to the solution of a nonlinear system of algebraic equations error analysis is presented for this method finally numerical examples are included to show the validity and efficiency of the new technique

markov chains for monte carlo tests of genetic equilibrium in multidimensional contingency tables hardy weinberg equilibrium and linkage equilibrium are fundamental concepts in population genetics in practice testing linkage equilibrium in haplotype data is equivalent to testing independence in a large sparse multidimensional contingency table testing hardy weinberg and linkage equilibrium simultaneously on multilocus genotype data introduces the additional complications of missing information and symmetry constraints on marginal probabilities to avoid unreliable large sample approximations for sparse contingency tables one can use exact tests like fisher s classical test that condition on observed marginal totals unfortunately computing p values for exact tests is often infeasible because of the large number of tables consistent with the marginal totals of an observed table we develop here markov chains for sampling from the appropriate conditional distributions for testing genetic equilibrium these chains compare favorably with a parallel independent sampling method that we present for n haplotype observations on j loci the markov chains converge to their stationary distributions in j 1 n ln n 2 0 n steps and can be an efficient tool for estimating p values our theoretical treatment of these results involves strong stationary stopping times order statistics large deviations and the embedding of poisson processes we include some general results on the application of strong stationary times to bounding the precision and bias of sample average estimators

on uniform generation of two way tables with fixed margins and the conditional volume test of diaconis and efron two efficient monte carlo algorithms are proposed for uniformly generating two way contingency tables with fixed margins these permit some improvements on recent work of diaconis efron and gangolli especially concerning estimates of the total number of such tables

accurate directional inference for vector parameters in linear exponential families we consider inference on a vector valued parameter of interest in a linear exponential family in the presence of a finite dimensional nuisance parameter based on higher order asymptotic theory for likelihood we propose a directional test whose p value is computed using one dimensional integration the work simplifies and develops earlier research on directional tests for continuous models and on higher order inference for discrete models and the examples include contingency tables and logistic regression examples and simulations illustrate the high accuracy of the method which we compare with the usual likelihood ratio test and with an adjusted version due to skovgaard in high dimensional settings such as covariance selection the approach works essentially perfectly whereas its competitors can fail catastrophically