an iterative method for nonsymmetric systems with multiple right hand sides we propose a method for the solution of linear systems ax b where a is a large possibly sparse nonsymmetric matrix of order n and b is an arbitrary rectangular matrix of order n times s with s of moderate size the method uses a single krylov subspace per step as a generator of approximations a projection process and a richardson acceleration technique it thus combines the advantages of recent hybrid methods with those for solving symmetric systems with multiple right hand sides numerical experiments indicate that in several cases the method has better practical performance and significantly lower memory requirements than block versions of nonsymmetric solvers and other proposed methods for the solution of systems with multiple right hand sides

iterative solution of multiple radiation and scattering problems in structural acoustics using a block quasi minimal residual algorithm abstract finite element discretizations of time harmonic acoustic wave problems in exterior domains result in large sparse systems of linear equations with complex symmetric coefficient matrices in many situations these matrix problems need to be solved repeatedly for different right hand sides but with the same coefficient matrix recently freund and malhotra have proposed a block quasi minimal residual bl qmr algorithm 13 for the iterative solution of non hermitian linear systems with multiple right hand sides the bl qmr algorithm is a block krylov subspace iterative method that incorporates deflation to delete linearly and almost linearly dependent vectors in the underlying block krylov sequences in this paper we describe a j symmetric variant of the bl qmr algorithm that introduces important simplifications for the case when the coefficient matrix is symmetric with respect to a bilinear form induced by a certain matrix j in particular the j symmetric variant includes the complex symmetric form of bl qmr as a special case we identify suitable preconditioners for the bl qmr algorithm applied to multiple radiation and scattering problems our numerical tests with the preconditioned bl qmr algorithm for such multiple linear systems show that instead of solving each of the linear systems individually it is significantly more efficient to employ the block version of the iterative method moreover the numerical results clearly illustrate the importance of deflation and its effect on iterative convergence

gromov witten classes quantum cohomology and enumerative geometry the paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry in particular it contains an axiomatic treatment of gromov witten classes and a discussion of their properties for fano varieties cohomological field theories are defined and it is proved that tree level theories are determined by their correlation functions application to counting rational curves on del pezzo surfaces and projective spaces are given

mutations of puzzles and equivariant cohomology of two step flag varieties we introduce a mutation algorithm for puzzles that is a three direction analogue of the classical jeu de taquin algorithm for semistandard tableaux we apply this algorithm to prove our conjectured puzzle formula for the equivariant schubert structure constants of two step flag varieties this formula gives an expression for the structure constants that is positive in the sense of graham thanks to the equivariant version of the quantum equals classical result our formula specializes to a littlewood richardson rule for the equivariant quantum cohomology of grassmannians

infinitely many mountain pass solutions on a kind of fourth order neumann boundary value problem in this paper the existence of infinitely many mountain pass solutions are obtained for the fourth order boundary value problem bvp u 4 t 2u t u t f u t 0r is continuous the study of the problem is based on the variational methods and critical point theory we prove the conclusion by using sub sup solution method mountain pass theorem in order intervals leray schauder degree theory and morse theory

double step scale splitting real valued iteration method for a class of complex symmetric linear systems abstract a double step scale splitting dss real valued iteration method is constructed to solve an block two by two real linear system which is deduced equivalently from a complex symmetric linear system the convergence analysis for this dss real valued iteration method is presented under suitable conditions and the upper bound of its spectral radius is proved to be smaller than that of the pmhss real valued iteration method furthermore an improved dss idss real valued iteration method is derived by respectively adding two matrices to the coefficient matrices of the dss iterative scheme and the corresponding convergence analysis is also discussed finally some numerical examples are given to illustrate the effectiveness and robustness of the proposed methods

a splitting iterative method for the discrete dynamic linear systems this paper is concerned with a splitting iterative method for a class of complex symmetric linear systems from an n degree of freedom n dof discrete system this splitting iterative method is established by the complex symmetric and skew hermitian splitting of the coefficient matrix and is called the css method which is from the classical state space formulation of frequency analysis of the discrete dynamic linear systems the convergence properties of the css method are obtained the corresponding css preconditioner is proposed and some useful properties of the preconditioned matrix are established the presented numerical examples are to illustrate the efficiency of both the css method and the css preconditioner

a generalized shift splitting preconditioner for complex symmetric linear systems abstract in this paper the generalized shift splitting gss preconditioner is implemented for solving a class of generalized saddle point problems which stem from the solution of complex symmetric linear systems the gss preconditioner is induced by the generalized shift splitting iterative method theoretical analysis shows that the generalized shift splitting iterative method is unconditionally convergent some numerical experiments are provided to show the effectiveness of the proposed preconditioner

solving shifted linear systems with restarted gmres augmented with error approximations abstract in this paper we investigate a variant of the restarted gmres method for solving a series of large sparse linear systems restarting is carried out by augmenting krylov subspaces with some recently generated error approximations from the seed system the method can preserve a nice property that allows solving the seed and the added linear systems at the cost of only one matrix vector multiplication per iteration compared with solving each added linear system separately the advantage of the new scheme is to lower down the overall cost of solving all added linear systems numerical experiments illustrate the efficiency of the acceleration strategy

efficient parameterized hss iteration methods for complex symmetric linear systems abstract based on the new hss nhss iteration method proposed by pour and goughery 2015 and the efficient pshss iteration method by zeng and ma 2016 we introduce an efficient parameterized hss pnhss and a parameterized single step hss ps hss iteration methods for solving a class of complex symmetric linear systems convergence properties of the pnhss and the ps hss iteration methods are studied which show that the iterative sequences are convergent to the unique solution of the linear system for any initial guess under a loose restriction on the parameter furthermore we derive an upper bound for the spectral radius of the pnhss iteration matrix and the quasi optimal parameters and which minimize the above upper bound are also considered both theoretical and numerical results show that the pnhss and the ps hss iteration methods outperform the nhss and the shss iteration methods little difference about the computational efficiency from the point of view of the cpu times between the ps hss the pnhss and the pshss iteration methods is justified by using the experimental optimal parameters however sometimes the ps hss and the pnhss iteration methods are more efficient than the pshss iteration method when the experimental optimal parameters are not used

existence of infinitely many mountain pass solutions for some fourth order boundary value problems with a parameter abstract in this paper for the fourth order boundary value problem bvp u 4 t u t 2 4 u t f t u t 0 t 1 u 0 u 1 u 0 u 1 0 where f 0 1 r r is continuous 0 is a parameter the existence of infinitely many mountain pass solutions are obtained with the variational methods and critical point theory we prove the conclusion by combining sub sup solution method mountain pass theorem in order intervals leray schauder degree theory and morse theory

a class of accelerated parameterized inexact uzawa algorithms for complex symmetric linear systems abstract we establish a class of accelerated parameterized inexact uzawa apiu algorithms for solving the complex symmetric linear systems our main contribution is accelerating the convergence of the piu algorithm by making use of the extrapolation technique which is based on the eigenvalues of the iterative matrix these accelerated parameterized inexact uzawa algorithms involve two iteration parameters whose special choices can recover the parameterized inexact uzawa algorithm and some other methods first the accelerated model for the piu algorithm is established and the accelerated piu algorithm is presented then we study the convergence of the corrected piu algorithm moreover we present the optimal iteration parameter and the corresponding optimal convergence factor for the piu method we also consider acceleration of the piu iteraton by krylov subspace methods numerical experiments are presented to illustrate the theoretical results and examine the numerical effectiveness of the new method

sign changing solutions on a kind of fourth order neumann boundary value problem abstract in this paper we consider the fourth order neumann boundary value problem u 4 t 2 u t u t f t u t for all t 0 1 and subject to u 0 u 1 u 0 u 1 0 using the fixed point index and the critical group we establish the existence theorem of solutions that guarantees the problem has at least one positive solution and two sign changing solutions under certain conditions

existence of solutions to nonlinear hammerstein integral equations and applications abstract in this paper we study the existence and multiplicity of solutions of the operator equation k f u u in the real hilbert space l 2 g under certain conditions on the linear operator k we establish the conditions on f which are able to guarantee that the operator equation has at least one solution a unique solution and infinitely many solutions respectively the monotone operator principle and the critical point theory are employed to discuss this problem respectively in argument quadratic root operator k 1 2 and its properties play an important role as an application we investigate the existence and multiplicity of solutions to fourth order boundary value problems for ordinary differential equations with two parameters and give some new existence results of solutions

deflated gmres for systems with multiple shifts and multiple right hand sides abstract we consider solution of multiply shifted systems of nonsymmetric linear equations possibly also with multiple right hand sides first for a single right hand side the matrix is shifted by several multiples of the identity such problems arise in a number of applications including lattice quantum chromodynamics where the matrices are complex and non hermitian some krylov iterative methods such as gmres and bicgstab have been used to solve multiply shifted systems for about the cost of solving just one system restarted gmres can be improved by deflating eigenvalues for matrices that have a few small eigenvalues we show that a particular deflated method gmres dr can be applied to multiply shifted systems in quantum chromodynamics it is common to have multiple right hand sides with multiple shifts for each right hand side we develop a method that efficiently solves the multiple right hand sides by using a deflated version of gmres and yet keeps costs for all of the multiply shifted systems close to those for one shift an example is given showing this can be extremely effective with a quantum chromodynamics matrix

applications of linking to hammerstein integral equations in this paper we establish some results for asymptotic linear hammerstein integral equations using the notion of linking both homotopical and co homological

aor uzawa iterative method for a class of complex symmetric linear system of equations abstract in this paper based on accelerated overrelaxation aor method and uzawa method we present aor uzawa iterative method for solving a broad class of complex symmetric linear systems we investigate its convergence properties in addition the resulting aor uzawa preconditioner is used to precondition krylov subspace methods such as the restarted generalized minimal residual gmres method for solving the real equivalent formulation of the complex symmetric system finally two numerical examples are given to show the effectiveness of the proposed method

existence and multiplicity of solutions of a kind of fourth order boundary value problem abstract in this paper the existence and multiplicity results of the solutions are obtained for the fourth order boundary value problem u 4 t f t u t for all t 0 1 subject to u 0 u 1 u 0 u 1 0 where f is continuous the monotone operator theory and critical point theory are employed to discuss this problem respectively

quantum and affine schubert calculus and macdonald polynomials abstract we definitively establish that the theory of symmetric macdonald polynomials aligns with quantum and affine schubert calculus using a discovery that distinguished weak chains can be identified by chains in the strong bruhat order poset on the type a affine weyl group we construct two one parameter families of functions that respectively transition positively with hall littlewood and macdonald s p functions and specialize to the representatives for schubert classes of homology and cohomology of the affine grassmannian our approach leads us to conjecture that all elements in a defining set of 3 point genus 0 gromov witten invariants for flag manifolds can be formulated as strong covers

existence and multiplicity of solutions to even order ordinary differential equations via variational methods abstract in this paper we study the existence and multiplicity of solutions for two point boundary value problems for even order ordinary differential equation by using the critical point theory and the property of projective operator we establish some conditions on the nonlinearity which guarantee that the nonlinear hammerstein integral equation with quasi positive definite kernel has at least one solution two nontrivial solutions and finite pairs of solutions respectively as an application we get the corresponding results for two point boundary value problems for even order ordinary differential equation moreover an example is presented to illustrate our main results

a preconditioner for optimal control problems constrained by stokes equation with a time harmonic control in this article we construct an efficient preconditioner for solving the algebraic systems arising from discretized optimal control problems with time periodic stokes equations based on a preconditioning technique for stationary stokes constrained optimal control problems considered in an earlier paper by the authors a simplified analysis of the derivation of the preconditioner and its properties is presented the preconditioner is fully parameter independent and the condition number of the corresponding preconditioned matrix is bounded by 2 the so constructed preconditioner is favourably compared with another robust preconditioner for the same problem

gromov witten invariants on grassmannians we prove that any three point genus zero gromov witten invariant on a type a grassmannian is equal to a classical intersection number on a two step flag variety we also give symplectic and orthogonal analogues of this result in these cases the two step flag variety is replaced by a sub maximal isotropic grassmannian our theorems are applied in type a to formulate a conjectural quantum littlewood richardson rule and in the other classical lie types to obtain new proofs of the main structure theorems for the quantum cohomology of lagrangian and orthogonal grassmannians

existence and multiplicity of solutions to 2mth order ordinary differential equations abstract in this paper the existence and multiplicity of solutions are obtained for the 2mth order ordinary differential equation two point boundary value problems 1 m u 2 m t i 1 m 1 m i a i u 2 m i t f t u t for all t 0 1 subject to dirichlet neumann mixed and periodic boundary value conditions respectively where f is continuous a i r for all i 1 2 m since these four boundary value problems have some common properties and they can be transformed into the integral equation of form u i 1 m a i t i u t m f u we firstly deal with this nonlinear integral equation by using the strongly monotone operator principle and the critical point theory we establish some conditions on f which are able to guarantee that the integral equation has a unique solution at least one nonzero solution and infinitely many solutions furthermore we apply the abstract results on the integral equation to the above four 2mth order two point boundary problems and successfully resolve the existence and multiplicity of their solutions

efficient single step preconditioned hss iteration methods for complex symmetric linear systems abstract we propose a single step preconditioned variant of hss sphss and an efficient parameterized sphss psphss iteration method for solving a class of complex symmetric linear systems under suitable conditions we analyze the convergence properties of the sphss and psphss iteration methods theoretical analysis shows that the minimal upper bounds for the spectral radius of the sphss and psphss iteration matrices are less than those of the shss and pshss iteration matrices when using the optimal parameters respectively numerical results show that the psphss iteration method has comparable advantage over several other iteration methods whether the experimental optimal parameters are used or not

positive solutions of fourth order boundary value problems with two parameters abstract in this paper the existence results of positive solutions are obtained for fourth order boundary value problem u 4 u u f t u 0 where f 0 1 r r is continuous r and satisfy

on a nonlinear hammerstein integral equation with a parameter abstract this paper following theories for asymptotically linear operators the schaefer fixed point theorem decomposition of operators and critical point theory is mainly concerned with the existence and multiplicity of solutions to a nonlinear hammerstein integral equation with a parameter the results show that when the nonlinearity satisfies certain conditions different parametric intervals lead to different existence results however in some cases only the sign of the parameter makes a contribution to the existence of solutions for the problem our results can be applied to some well known boundary value problems and some examples are given

a new iterative method for solving complex symmetric linear systems based on implementation of the quasi minimal residual qmr and biconjugate a orthogonal residual bicor method a new krylov subspace method is presented for solving complex symmetric linear systems the new method can be combined with arbitrary symmetric preconditioners the preconditioned modified hermitian and skew hermitian splitting pmhss preconditioner is used to accelerate the convergence rate of this method numerical experiments indicate that the proposed method and its preconditioned version are efficient and robust in comparison with other krylov subspace methods

equivariant pieri rules for isotropic grassmannians we give a pieri rule for the torus equivariant cohomology of submaximal grassmannians of lie types b c and d to the authors best knowledge our rule is the first manifestly positive formula beyond the equivariant chevalley formula we also give a simple proof of the equivariant pieri rule for the ordinary type a grassmannian

preconditioning of complex linear systems from the helmholtz equation abstract in this paper for solving a class of complex linear systems from the helmholtz equation efficiently a new splitting preconditioner is established and a real valued preconditioned iterative method is presented spectral properties of the preconditioned matrix are discussed and bound on the eigenvalues of the preconditioned matrix is given theorem which provides the dimension of the krylov subspace methods for the preconditioned iteration method is obtained the implementation of the preconditioned method is given and the optimal choice of the accelerated parameter is derived in particular a more practical way to choose the accelerated parameter is also proposed numerical experiments arising from the helmholtz equation are used to illustrate the performance of the preconditioner which show the effectiveness and robustness of the new preconditioned gmres method and demonstrate meshsize independent and wave number insensitive convergence behavior

restarted weighted full orthogonalization method for shifted linear systems it is known that the restarted full orthogonalization method fom outperforms the restarted generalized minimum residual method gmres in several circumstances for solving shifted linear systems when the shifts are handled simultaneously on the basis of the weighted arnoldi process a weighted version of the restarted shifted fom is proposed which can provide accelerating convergence rate with respect to the number of restarts in the cases where our hybrid algorithm needs less enough number of restarts to converge than the restarted shifted fom the associated cpu consuming time is also reduced as shown by the numerical experiments moreover our algorithm is able to solve certain shifted systems which the restarted shifted fom cannot handle sometimes

equivariant littlewood richardson skew tableaux we give a positive equivariant littlewood richardson rule also discovered independently by molev our proof generalizes a proof by stembridge of the classical littlewood richardson rule we describe a weight preserving bijection between our indexing tableaux and trapezoid puzzles which restricts to a bijection between positive indexing tableaux and knutson tao puzzles

restarted gmres for shifted linear systems shifted matrices which differ by a multiple of the identity only generate the same krylov subspaces with respect to any fixed vector this fact has been exploited in lanczos based methods like cg qmr and bicg to simultaneously solve several shifted linear systems at the expense of only one matrix vector multiplication per iteration here we develop a variant of the restarted gmres method exhibiting the same advantage and we investigate its convergence for positive real matrices in some detail we apply our method to speed up multiple masses calculations arising in lattice gauge computations in quantum chromodynamics one of the most time consuming supercomputer applications

factorizing complex symmetric matrices with positive definite real and imaginary parts complex symmetric matrices whose real and imaginary parts are positive definite are shown to have a growth factor bounded by 2 for lu factorization this result adds to the classes of matrix for which it is known to be safe not to pivot in lu factorization block mathrm ldl t factorization with the pivoting strategy of bunch and kaufman is also considered and it is shown that for such matrices only 1 times 1 pivots are used and the same growth factor bound of 2 holds but that interchanges that destroy band structure may be made the latter results hold whether the pivoting strategy uses the usual absolute value or the modification employed in linpack and lapack

existence of solutions for some fourth order boundary value problems with parameters abstract in this paper existence and multiplicity results for solutions are obtained for the fourth order boundary value problem bvp u 4 t u t u t f t u t 0 t 1 u 0 u 1 u 0 u 1 0 where f 0 1 r r is continuous r and r are parameters by using the critical point theory and morse theory we obtain that if satisfy 4 2 1 then the above bvp has solutions where is in some different intervals

quantum k theory of grassmannians we show that equivariant k theoretic 3 point gromov witten invariants of genus zero on a grassmann variety are equal to triple intersec tions computed in the equivariant k theory of a two step flag manifold thus generalizing an earlier result of buch kresch and tamvakis in the process we show that the gromov witten variety of curves passing through 3 general points is irreducible and rational our applications include pieri and giambelli formulas for the quantum k theory ring of a grassmannian which determine the multiplication in this ring we also compute the dual schubert basis for this ring and show that its structure constants satisfy s3 symmetry our for mula for gromov witten invariants can be partially generalized to cominuscule homogeneous spaces by using a construction of chaput manivel and perrin

bounds for eigenvalues of matrix polynomials with applications to scalar polynomials abstract we first generalize to complex matrix polynomials an improvement of an upper bound by cauchy on the zeros of complex scalar polynomials the bound requires the unique positive root of a real scalar polynomial of the same degree as the given complex scalar or matrix polynomial we then create a recursive procedure to represent a matrix polynomial by another matrix polynomial of larger size but of lower degree we apply this procedure to scalar polynomials and then apply the generalized improved cauchy bound to their matrix polynomial representation often obtaining a better bound by solving a real scalar polynomial equation of significantly reduced degree

positive solutions of semi positone hammerstein integral equations and applications abstract existence of solutions for semi positone integral equations of hammerstein type is obtained by using the fixed point index method many boundary value problems of differential equations can be transformed into these integral equations a specific example is given

existence of solutions to a class of nonlinear second order two point boundary value problems in this paper the existence and multiplicity results of solutions are obtained for the second order two point boundary value problem u t f t u t for all t 0 1 subject to u 0 u 1 0 where f is continuous the monotone operator theory and critical point theory are employed to discuss this problem respectively in argument quadratic root operator and its properties play an important role

littlewood richardson polynomials abstract we introduce a family of rings of symmetric functions depending on an infinite sequence of parameters a distinguished basis of such a ring is comprised by analogues of the schur functions the corresponding structure coefficients are polynomials in the parameters which we call the littlewood richardson polynomials we give a combinatorial rule for their calculation by modifying an earlier result of b sagan and the author the new rule provides a formula for these polynomials which is positive in the sense of w graham we apply this formula for the calculation of the product of equivariant schubert classes on grassmannians which implies a stability property of the structure coefficients the first manifestly positive formula for such an expansion was given by a knutson and t tao by using combinatorics of puzzles while the stability property was not apparent from that formula we also use the littlewood richardson polynomials to describe the multiplication rule in the algebra of the casimir elements for the general linear lie algebra in the basis of the quantum immanants constructed by a okounkov and g olshanski

a littlewood richardson rule for factorial schur functions we give a combinatorial rule for calculating the coefficients in the expansion of a product of two factorial schur functions it is a special case of a more general rule which also gives the coefficients in the expansion of a skew factorial schur function applications to capelli operators and quantum immanants are also given

on equivariant quantum schubert calculus for g p we show a z2 filtered algebraic structure and a quantum to classical principle on the torus equivariant quantum cohomology of a complete flag variety of general lie type generalizing earlier works of leung and the second author we also provide various applications on equivariant quantum schubert calculus including an equivariant quantum pieri rule for partial flag variety fln1 nk n 1 of lie type a

automated adaptive multilevel solver abstract this paper presents an automated adaptive multilevel solver for linear or linearized system of equations the multilevel aspect of the solver is aimed at securing an optimal rate of convergence while keeping the size of the coarsest problem sufficiently small to ensure that the direct portion of the solution does not dominate the total computational cost adaptivity in terms of a priori selection of the number of levels one or more and construction of the optimal multilevel preconditioner is the key to the robustness of the method the number of levels is selected on the basis of estimated conditioning sparsity of the factor and available memory the auxiliary coarse models if required are automatically constructed on the basis of spectral characteristics of individual aggregates groups of neighboring elements an obstacle test consisting of twenty industry and model problems was designed to i determine the optimal values of computational parameters and to ii compare the adaptive multilevel scheme with existing state of the art equation solvers including the multifrontal solver 17 with the mmd reordering scheme and the pcg solver with the nearly optimal modified incomplete cholesky factorization preconditioner

nontrivial solutions for some fourth order boundary value problems with parameters abstract in this paper we prove some new existence results concerning nontrivial solutions to some fourth order boundary value problems u 4 t u t u t f t u t 0 t 1 u 0 u 1 u 0 u 1 0 where f c 1 0 1 r r r are parameters satisfying 4 2 1 the methods used here are based on combining the minimax methods and the morse theory

nontrivial solutions on a kind of fourth order neumann boundary value problems abstract nontrivial solutions for the fourth order neumann boundary value problem u 4 t 2 u t u t f t u t for all t 0 1 and subject to u 0 u 1 u 0 u 1 0 are obtained via morse theory to compute the critical groups at infinity of the relevant functional we propose an approach by combining the homotopy and reduction methods

iterative system solvers for the frequency analysis of linear mechanical systems abstract the paper deals with the numerical treatment of the direct frequency domain dfd analysis of linear mechanical systems attention is mainly focused on the solution of the complex system of equations needed for each analyzed frequency strategies for transforming the system into the shifted form t i z d are proposed and discussed where is related to the frequency for two formulations of the shifted system the performance of some krylov sub space iterative solvers is tested and compared to that of a multi frontal direct method advantage is taken of the shifted form in solving simultaneously a large number of systems resulting from different values of the shift frequency numerical experiments on some prototype structural dynamics problems are reported the results shown demonstrate how the devised strategies for the iterative solution can outperform in many cases the direct solver

numerical solution of nonlinear volterra fredholm hammerstein integral equations via collocation method based on radial basis functions abstract a numerical technique based on the spectral method is presented for the solution of nonlinear volterra fredholm hammerstein integral equations this method is a combination of collocation method and radial basis functions rbfs with the differentiation process drbf using zeros of the shifted legendre polynomial as the collocation points different applications of rbfs are used for this purpose the integral involved in the formulation of the problems are approximated based on legendre gauss lobatto integration rule the results of numerical experiments are compared with the analytical solution in illustrative examples to confirm the accuracy and efficiency of the presented scheme

a variant of the hss preconditioner for complex symmetric indefinite linear systems abstract using the equivalent block two by two real linear systems we establish a new variant of the hermitian and skew hermitian splitting hss preconditioner for a class of complex symmetric indefinite linear systems the new preconditioner is not only a better approximation to the block two by two real coefficient matrix than the well known hss preconditioner but also resulting in an unconditional convergent fixed point iteration the quasi optimal parameter which minimizes an upper bound of the spectral radius of the iteration matrix is analyzed eigen properties and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are discussed finally two numerical examples are provided to show the efficiency of the new preconditioner

equivariant quantum schubert calculus abstract we study the t equivariant quantum cohomology of the grassmannian we prove the vanishing of a certain class of equivariant quantum littlewood richardson coefficients which implies an equivariant quantum pieri rule as in the equivariant case this implies an algorithm to compute the equivariant quantum littlewood richardson coefficients

multiple solutions of some boundary value problems with parameters abstract in this paper we study the existence and multiplicity of nontrivial solutions for the following second order dirichlet nonlinear boundary value problem with odd order derivative u t a u t b u t f t u t for all t 0 1 with u 0 u 1 0 where a b r 1 f c 1 0 1 r 1 r 1 by using the morse theory we impose certain conditions on f which are able to guarantee that the problem has at least one nontrivial solution two nontrivial solutions and infinitely many solutions separately

optimality properties of a square block matrix preconditioner with applications abstract a preconditioned square block matrix called presb has previously been applied successfully and for more standard type of problems have been shown to have eigenvalue bounds in the interval 1 2 1 which holds uniformly with respect to all parameters involved having such fixed bounds enables the use of an inner product free acceleration method such as the chebyshev iterative method here it is shown that the method can be applied also for some more general problems where the spectrum contains outlier eigenvalues larger than unity and that one can apply a polynomially modified version of the chebyshev method for such problems

kostant polynomials and the cohomology ring for g b the schubert calculus for g b can be completely determined by a certain matrix related to the kostant polynomials introduced in section 5 of bernstein gelfand and gelfand bernstein i gelfand i gelfand s 1973 russ math surv 28 1 26 the polynomials are defined by vanishing properties on the orbit of a regular point under the action of the weyl group for each element w in the weyl group the polynomials also have nonzero values on the orbit points corresponding to elements which are larger in the bruhat order than w the main theorem given here is an explicit formula for these values the matrix of orbit values can be used to determine the cup product for the cohomology ring for g b using only linear algebra or as described by lascoux and schutzenberger lascoux a schutzenberger m p 1982 c r seances acad sci ser a 294 447 450 complete proofs of all the theorems will appear in a forthcoming paper

multiple solutions to fourth order boundary value problems in this paper we study the existence and multiplicity of solutions to the fourth order boundary value problem u 4 t email protected t email protected t f t u t for all email protected 0 1 subject to dirichlet boundary value condition where email protected c 1 0 1 xr 1 r 1 a email protected r 1 by using the critical point theory and the infinite dimensional morse theory we establish some conditions on f which are able to guarantee that this boundary value problem has at least one nontrivial two nontrivial m distinct pairs of solutions and infinitely many solutions respectively our results improve some recent works

positivity in equivariant quantum schubert calculus a conjecture of d peterson proved by w graham states that the structure constants of the t equivariant cohomology of a homogeneous space g p satisfy a certain positivity property in this paper we show that this positivity property holds in the more general situation of equivariant quantum cohomology 1 introduction it is well known that the integral cohomology of the homogeneous space x g p for g a connected semisimple complex lie group and p a parabolic subgroup satisfies a positivity property its structure constants are nonnegative integers equal to the number of intersection points of three schubert varieties in general position whose codimensions add up to the dimension of x recently graham gr has proved a conjecture of peterson p asserting that h x the t equivariant cohomology of x where t c r is a maximal torus in g enjoys a more general positivity property

fast multilevel augmentation methods for solving hammerstein equations fast multilevel augmentation methods based on projections for solving hammerstein equations are developed each of these methods requires availability of a multilevel decomposition of the solution space and a projection from the solution space onto a finite dimensional subspace a subspace at a level is obtained from the subspace at the previous level by adding a difference subspace accordingly the projection equation at the present level is obtained by augmenting the projection equation at the previous level using this idea recursively we solve the nonlinear equation by an iteration method only at an initial lower resolution level while obtaining its solution at a higher resolution level with the help of the multiscale analysis we separate the procedure of solving the nonlinear operator equation at a high level into two major components 1 solving the nonlinear equation only at an initial lower level and 2 compensating the error by solving a linear system at the high level we prove that the proposed methods require only linear up to a logarithmic factor computational complexity and have the optimal convergence order a relationship between the proposed method and the multigrid method is discussed two specific fast methods based on the galerkin projection and the collocation projection are developed numerical results are presented to confirm the theoretical estimates with comparisons to the two grid method

a new iteration method for a class of complex symmetric linear systems abstract in this paper a new iteration method is proposed for solving the complex symmetric linear systems in theory we show that the convergence factor or the upper bound of the spectral radius of the iteration matrix of the new method are smaller than that of the pmhss method proposed by bai et al 2011 moreover we analyze and compare the parameter free versions and the spectrum distributions of the preconditioned matrix of the new method and the pmhss method finally we present some numerical experiments on a few model problems to illustrate the theoretical results and show the effectiveness of our new method

arnoldi methods for large sylvester like observer matrix equations and an associated algorithm for partial spectrum assignment abstract we propose an arnoldi based numerical method for solving a sylvester type equation arising in the construction of the luenberger observer given an n n matrix a and an n m matrix g the method simultaneously constructs an m m hessenberg matrix h with a preassigned spectrum and an n m orthonormal matrix x such that ax xh g we consider the case when a is large and sparse so that the standard techniques such as the well known hessenberg schur method for solving a sylvester equation cannot be easily applied as a byproduct we propose an algorithm for the partial pole assignment problem for large matrices

a fast multiscale solver for modified hammerstein equations abstract this paper presents a fast solver called the multilevel augmentation method for modified nonlinear hammerstein equations when we utilize the method to solve a large scale problem most of components of the solution can be computed directly and lower frequency components can be obtained by solving a fixed low order algebraic nonlinear system the advantage of using the algorithm to modified equations is that it leads to reduce the cost of numerical integrations greatly the optimal error estimate of the method is established and the nearly linear computational complexity is proved finally numerical examples are presented to confirm the theoretical results and illustrate the efficiency of the method

nontrivial solutions to a class of systems of second order differential equations abstract in this paper we investigate the nontrivial solutions to a class of systems of second order ordinary differential equations by using the critical groups and establish the existence results under the resonant case

on positive solutions of some nonlinear fourth order beam equations abstract the existence uniqueness and multiplicity of positive solutions of the following boundary value problem is considered u 4 t f t u t 0 for 0 where 0 is a constant f 0 1 0 0 is continuous

a littlewood richardson rule for two step flag varieties this paper studies the geometry of one parameter specializations of subvarieties of grassmannians and two step flag varieties as a consequence we obtain a positive geometric rule for expressing the structure constants of the cohomology of two step flag varieties in terms of their schubert basis a corollary is a positive geometric rule for computing the structure constants of the small quantum cohomology of grassmannians we also obtain a positive geometric rule for computing the classes of subvarieties of grassmannians that arise as the projection of the intersection of two schubert varieties in a partial flag variety these rules have numerous applications to geometry representation theory and the theory of symmetric functions