fekete points and convergence towards equilibrium measures on complex manifolds building on bb1 we prove a general criterion for convergence of possibly singular bergman measures towards pluripotential theoretic equilibrium measures on complex manifolds the criterion may be formulated in terms of the growth properties of the unit balls of certain norms on holomorphic sections or equivalently as an asymptotic minimization property for generalized donaldson l functionals our result settles in particular a well known conjecture in pluripotential theory concerning the equidistribution of fekete points and it gives the convergence of bergman measures towards the equilibrium measure for bernstein markov measures applications to interpolation of holomorphic sections are also discussed

on the tricomi problem the tricomi equation is solved by the decomposition method 6 8 on the half space x y 0 and also for suitable boundary conditions given on the boundary of the hyperbolic elliptic domain

the auxiliary equation method for solving the zakharov kuznetsov zk equation in this paper the nonlinear dispersive zakharov kuznetsov equation is solved by using the auxiliary equation method as a result new solitary pattern solitary wave and singular solitary wave solutions are found

using the theory of cyclotomy to factor cyclotomic polynomials over finite fields we examine the problem of factoring the rth cyclotomic polynomial r x over fp r and p distinct primes given the traces of the roots of r x we construct the coefficients of r x in time o r 4 we demonstrate a deterministic algorithm for factoring r x in time o r 1 2 log p 9 when r x has precisely two irreducible factors finally we present a deterministic algorithm for computing the sum of the irreducible factors of r x in time o r 6

2 1 dimensional rossby waves with complete coriolis force and its solution by homotopy perturbation method abstract in this paper the effects of both complete coriolis force and dissipation on equatorial nonlinear rossby wave are investigated analytically from the quasi geostrophic potential vorticity equation by using methods of multiple scales and perturbation expansions a 2 1 dimensional nonlinear zakharov kuznetsov burgers equation is derived in describing the evolution of rossby wave amplitude the effects of generalized beta the horizontal component of coriolis parameter and the dissipation are presented from the zakharov kuznetsov burgers equation we also obtain the classical solitary solution of the zakharov kuznetsov equation when the dissipation is absent by elliptic function expansion method and the complete coriolis force effect can be seen by the solution but the method is failed to zakharov kuznetsov burgers equation therefore we use the efficient homotopy perturbation method to solve the zakharov kuznetsov burgers equation

bergman kernels and equilibrium measures for line bundles over projective manifolds let l be a holomorphic line bundle over a compact complex projective hermitian manifold x any fixed smooth hermitian metric h on l induces a hilbert space structure on the space of global holomorphic sections with values in the k th tensor power of l in this paper various convergence results are obtained for the corresponding bergman kernels i e orthogonal projection kernels the convergence is studied in the large k limit and is expressed in terms of the equilibrium metric h e associated to h as well as in terms of the monge ampere measure of h on a certain support set it is also shown that the equilibrium metric h e is in the class c 1 1 on the complement of the augmented base locus of l for l ample these results give generalizations of well known results concerning the case when the curvature of h is globally positive then h e h in general the results can be seen as local metrized versions of fujita s approximation theorem for the volume of l

a new zk ilw equation for algebraic gravity solitary waves in finite depth stratified atmosphere and the research of squall lines formation mechanism abstract the research of algebraic gravity solitary waves is an advanced field which has important practical and theoretical value in physical oceanography aerology and etc by calculation condition and theoretical method limit previous researches mainly focused on the 1 1 dimensional models and 2 1 dimensional models were few considered in this paper from the non static equilibrium equation a new zk ilw equation is derived by using multi scale analysis and perturbation method in finite depth stratified atmosphere which is the first time obtained the model can reduce to zk bo model h and zk model h 0 and is the generalization of the above two models in order to further understand the nature of algebraic gravity solitary waves we get the analytical solution of zk ilw equation by using the trial function method and discuss the conservation laws furthermore the fission process of algebraic gravity solitary waves is studied and we can judge that one of the possible formation mechanism of squall lines is the fission of algebraic gravity solitary waves

homotopy perturbation technique abstract the homotopy perturbation technique does not depend upon a small parameter in the equation by the homotopy technique in topology a homotopy is constructed with an imbedding parameter p 0 1 which is considered as a small parameter some examples are given the approximations obtained by the proposed method are uniformly valid not only for small parameters but also for very large parameters

algebraic equations with exponential terms abstract the decomposition method is applied to algebraic equations containing exponential terms the n term approximation n is rapidly damped as n increases yielding an oscillating convergence of superior accuracy n 5 and n 9 for k 2

analysis of lie symmetries with conservation laws and solutions for the generalized 3 1 dimensional time fractional camassa holm kadomtsev petviashvili equation abstract in this paper under investigated is a generalized 3 1 dimensional camassa holm kadomtsev petviashvili gch kp equation which describes the role of dispersion in the formation of patterns in liquid drops with the help of the semi inverse method the euler lagrange equation and agrawal s method the time fractional gch kp equation is derived in the sense of riemann liouville fractional derivatives further the symmetry of the 3 1 dimensional time fractional gch kp equation is studied by fractional order symmetry meanwhile based on the new conservation theorem the conservation laws of 3 1 dimensional time fractional gch kp equation are constructed finally the solutions to the equation are given via a bilinear method and the radial basis functions rbfs meshless approach

a new homotopy perturbation method for solving systems of partial differential equations abstract in this paper a new homotopy perturbation method nhpm is introduced for obtaining solutions of systems of non linear partial differential equations theoretical considerations are discussed to illustrate the capability and reliability of the method three examples are provided comparison of the results of applying nhpm with those of applying hpm reveal the effectiveness and convenience of the new technique

factoring polynomials over finite fields using differential equations and normal bases the deterministic factorization algorithm for polynomials over finite fields that was recently introduced by the author is based on a new type of linearization of the factorization problem the main ingredients are differential equations in rational function fields and normal bases of field extensions for finite fields of characteristic 2 it is known that this algorithm has several advantages over the classical berlekamp algorithm we develop the algorithm in a general framework and we show that it is feasible for arbitrary finite fields in the sense that the linearization can be achieved in polynomial time

approximation of quantities of interest in stochastic pdes by the random discrete l 2 projection on polynomial spaces in this work we consider the random discrete l 2 projection on polynomial spaces hereafter rdp for the approximation of scalar quantities of interest qois related to the solution of a partial differential equation model with random input parameters in the rdp technique the qoi is first computed for independent samples of the random input parameters as in a standard monte carlo approach and then the qoi is approximated by a multivariate polynomial function of the input parameters using a discrete least squares approach we consider several examples including the darcy equations with random permeability the linear elasticity equations with random elastic coefficient and the navier stokes equations in random geometries and with random fluid viscosity we show that the rdp technique is well suited to qois that depend smoothly on a moderate number of random parameters our numerical tests confirm the theoretical findings in g migliorati f nobile e von schwerin and r tempone analysis of the

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

univariate polynomial factorization over finite fields abstract this paper is a tutorial introduction to univariate polynomial factorization over finite fields we recall the classical methods that induced most factorization algorithms berlekamp s and the cantor zassenhaus ones and some refinements which can be applied to these methods explicit algorithms are presented in a form suitable for almost immediate implementation we give a detailed description of an efficient implementation of the cantor zassenhaus algorithm used in the release 2 of the axiom computer algebra system

on a new factorization algorithm for polynomials over finite fields a new deterministic factorization algorithm for polynomials over finite fields was recently developed by the first author the bottleneck in this algorithm is the last stage in which the irreducible factors of the polynomial are derived from the solutions of a system of linear equations an efficient approach to the last stage was designed by the second author for the case of finite fields of characteristic 2 in this paper we describe a different approach to the last stage which works for arbitrary fields of positive characteristic in particular we obtain in this way an acceleration of the factorization algorithm of the first author which makes this algorithm polynomial time for fixed characteristic

on a near optimal sampling strategy for least squares polynomial regression abstract we present a sampling strategy of least squares polynomial regression the strategy combines two recently developed methods for least squares method christoffel least squares algorithm and quasi optimal sampling more specifically our new strategy first choose samples from the pluripotential equilibrium measure and then re order the samples by the quasi optimal algorithm a weighted least squares problem is solved on a much smaller sample set to obtain the regression result it is then demonstrated that the new strategy results in a polynomial least squares method with high accuracy and robust stability at almost minimal number of samples

a review of the decomposition method and some recent results for nonlinear equations abstract the decomposition method can be an effective procedure for solution of nonlinear and or stochastic continous time dynamical systems without usual restrictive assumptions this paper is intended as a convenient tutorial review of the method 1

nonadaptive quasi optimal points selection for least squares linear regression in this paper we present a quasi optimal sample set for ordinary least squares ols regression the quasi optimal set is designed in such a way that for a given number of samples it can deliver the regression result as close as possible to the result obtained by a much larger set of candidate samples the quasi optimal set is determined by maximizing a quantity measuring the mutual column orthogonality and the determinant of the model matrix this procedure is nonadaptive in the sense that it does not depend on the sample data this is useful in practice as it allows one to determine prior to the potentially expensive data collection procedure where to sample the underlying system in addition to presenting the theoretical motivation of the quasi optimal set we also present its efficient implementation via a greedy algorithm along with several numerical examples to demonstrate its efficacy since the quasi optimal set allows one to obtain a near optimal regression result under any affordable nu

basis adaptive sample efficient polynomial chaos base pc abstract for a large class of orthogonal basis functions there has been a recent identification of expansion methods for computing accurate stable approximations of a quantity of interest this paper presents within the context of uncertainty quantification a practical implementation using basis adaptation and coherence motivated sampling which under assumptions has satisfying guarantees this implementation is referred to as basis adaptive sample efficient polynomial chaos base pc a key component of this is the use of anisotropic polynomial order which admits evolving global bases for approximation in an efficient manner leading to consistently stable approximation for a practical class of smooth functionals this fully adaptive non intrusive method requires no a priori information of the solution and has satisfying theoretical guarantees of recovery a key contribution to stability is the use of a presented correction sampling for coherence optimal sampling in order to improve stability and accuracy within the adaptive basis scheme theoretically the method may dramatically reduce the impact of dimensionality in function approximation and numerically the method is demonstrated to perform well on problems with dimension up to 1000

a convenient computational form for the adomian polynomials abstract recent important generalizations by g adomian stochastic systems academic press 1983 have extended the scope of his decomposition method for nonlinear stochastic operator equations see also iterative method inverse operator method symmetrized method or stochastic green s function method very considerably so that they are now applicable to differential partial differential delay and coupled equations which may be strongly nonlinear and or strongly stochastic or linear or deterministic as subcases thus for equations modeling physical problems solutions are obtained rapidly easily and accurately the methodology involves an analytic parametrization in which certain polynomials an dependent on the nonlinearity are derived this paper establishes simple symmetry rules which yield adomian s polynomials quickly to high orders

semilinear stochastic systems analysis with the method of the stochastic green s function and application in mechanics abstract this work deals with the analysis of a class of semilinear vector stochastic operator equations the method of the stochastic green s function due to adomian is extended to the above mentioned class of equations some perturbation techniques are also developed and a pertinent application in nonlinear mechanics is considered

2 1 dimensional zk burgers equation with the generalized beta effect and its exact solitary solution abstract the 1 1 dimensional mathematical model had been extensively derived to describe rossby solitary waves in a line in the past few decades but as is well known the 1 1 dimensional model cannot reflect the generation and evolution of rossby solitary waves in a plane in this paper a 2 1 dimensional nonlinear zakharov kuznetsov burgers equation is derived to describe the evolution of rossby wave amplitude by using methods of multiple scales and perturbation expansions from the quasi geostrophic potential vorticity equations with the generalized beta effect the effects of the generalized beta and dissipation are presented by the zakharov kuznetsov burgers equation we also obtain the new solitary solution of the zakharov kuznetsov equation when the dissipation is absent with the help of the bernoulli equation which is different from the common classical solitary solution based on the solution the features of the variable coefficient are discussed by geometric figures meanwhile the approximate solitary solution of zakharov kuznetsov burgers equation is given by using the homotopy perturbation method and the amplitude of solitary waves changing with time is depicted by figures undoubtedly these solitary solutions will extend previous results and better help to explain the feature of rossby solitary waves

factorization of polynomials and some linear algebra problems over finite fields abstract the recently developed algorithm of niederreiter for the factorization of polynomials over finite fields provides a linearization of the factorization problem via differential equations in rational function fields we show that the range of applicability of this algorithm can be extended by various techniques such as the application of normal bases and the use of hasse teichmuller derivatives we also discuss topics from pseudorandom vector generation and combinatorial linear algebra that lead to interesting linear algebra problems over finite fields

connections between the algorithms of berlekamp and niederreiter for factoring polynomials over fq abstract in this paper we determine an explicit isomorphism between the solution spaces of berlekamp s and niederreiter s algorithm for factoring polynomials over finite fields this can be used for the implementation of a procedure combining the benefits of both methods

coupled nonlinear stochastic differential equations abstract systems of n coupled linear or nonlinear differential equations which may be deterministic or stochastic are solved by methods of the first author and his co workers examples include two coupled riccati equations coupled linear equations stochastic coupled equations with product terms and n coupled stochastic differential equations

an acceleration of the niederreiter factorization algorithm in characteristic 2 a new deterministic factorization algorithm for polynomials over finite fields was recently developed by niederreiter the bottleneck in this algorithm is the last stage in which the irreducible factors of the polynomial are derived from the solutions of a system of linear equations in this paper we consider finite fields of characteristic 2 and we show that in this case there is a more efficient approach to the last stage of the niederreiter algorithm which speeds up the algorithm considerably

hirota bilinear equations with linear subspaces of solutions we explore when hirota bilinear equations possess linear subspaces of solutions first we establish a sufficient and necessary criterion for the existence of linear subspaces of exponential traveling wave solutions to hirota bilinear equations second we show that multivariate polynomials whose zeros form a vector space can generate the desired hirota bilinear equations with given linear subspaces of solutions and formulate such multivariate polynomials by using multivariate polynomials which have one and only one zero third applying an algorithm using weights we present parameterizations of wave numbers and frequencies achieved by using one parameter to compute the desired hirota bilinear equations

computing multivariate fekete and leja points by numerical linear algebra we discuss and compare two greedy algorithms that compute discrete versions of fekete like points for multivariate compact sets by basic tools of numerical linear algebra the first gives the so called approximate fekete points by qr factorization with column pivoting of vandermonde like matrices the second computes discrete leja points by lu factorization with row pivoting moreover we study the asymptotic distribution of such points when they are extracted from weakly admissible meshes