the structured gerstenhaber problem iii abstract let b be a symmetric or alternating bilinear form on a finite dimensional vector space v when the characteristic of the underlying field is not 2 we determine the greatest dimension for a linear subspace of nilpotent b symmetric or b alternating endomorphisms of v expressing it as a function of the dimension the rank and the witt index of b similar results are obtained for subspaces of nilpotent b hermitian endomorphisms when b is a hermitian form with respect to a non identity involution in three situations b symmetric endomorphisms when b is symmetric b alternating endomorphisms when b is alternating and b hermitian endomorphisms when b is hermitian and the underlying field has more than 2 elements we also characterize the linear subspaces with the maximal dimension our results are wide generalizations of results of meshulam and radwan 7 who tackled the case of a non degenerate symmetric bilinear form over the field of complex numbers and recent results of kokol bukovsek and omladic 4 in which the spaces with maximal dimension were determined when the underlying field is the one of complex numbers the bilinear form b is symmetric and non degenerate and one considers b symmetric endomorphisms

large spaces of bounded rank matrices revisited let n p r be positive integers with n p r a rank r subset of n by p matrices with entries in a field is a subset in which every matrix has rank less than or equal to r a classical theorem of flanders states that the dimension of a rank r linear subspace must be less than or equal to nr and it characterizes the spaces with the critical dimension nr linear subspaces with dimension close to the critical one were later studied by atkinson lloyd and beasley over fields with large cardinality their results were recently extended to all fields 18 using a new method we obtain a classification of rank r affine subspaces with large dimension over all fields this classification is then used to double the range of large dimensions for which the structure of rank r linear subspaces is known for all fields

burton miller type singular boundary method for acoustic radiation and scattering abstract this paper proposes the singular boundary method sbm in conjunction with burton and miller s formulation for acoustic radiation and scattering the sbm is a strong form collocation boundary discretization technique using the singular fundamental solutions which is mathematically simple easy to program meshless and introduces the concept of source intensity factors sifs to eliminate the singularities of the fundamental solutions therefore it avoids singular numerical integrals in the boundary element method bem and circumvents the troublesome placement of the fictitious boundary in the method of fundamental solutions mfs in the present method we derive the sifs of exterior helmholtz equation by means of the sifs of exterior laplace equation owing to the same order of singularities between the laplace and helmholtz fundamental solutions in conjunction with the burton miller formulation the sbm enhances the quality of the solution particularly in the vicinity of the corresponding interior eigenfrequencies numerical illustrations demonstrate efficiency and accuracy of the present scheme on some benchmark examples under 2d and 3d unbounded domains in comparison with the analytical solutions the boundary element solutions and dirichlet to neumann finite element solutions

a geometric isomorphism with applications to closed loop controls feedback equivalence of n state n 1 control systems satisfying certain regularity conditions divides such systems into two invariant classes we show that class one corresponds via a geometric isomorphism to classical lagrangian variational problems we prove the existence of time critical closed loop controls for systems that satisfy the nondegeneracy condition that the analogue of the hessian for the lagrangian problem have full rank we show that the vanishing of this hessian characterizes the control linear systems in class one and identify the rank condition for local controllability for such systems as the nonvanishing of a differential invariant the control linear systems in class two are also characterized by the vanishing of an invariant and the rank condition is identified

fast computation of the spectral differentiation by the fast multipole method abstract this paper is concerned with the application of the fast multipole method fmm to the pseudospectral methods the spectral differentiation in pseudospectral methods involves dense matrix vector products which if evaluated directly entail o n 2 operations the core of this work lies in the fact that the fmm leads to a significant speed up in cpu time with a major reduction in the amount of computer memory needed when performing matrix vector products the fmm is exploited to estimate the barycentric weights and the pseudospectral differentiation which reduces the complexity to o n furthermore we show the convergence analysis and present the applications to the second order boundary value problems numerical examples confirm the advantages of the method

a fast singular boundary method for 3d helmholtz equation abstract this paper presents a fast singular boundary method sbm for three dimensional 3d helmholtz equation the sbm is a boundary type meshless method which incorporates the advantages of the boundary element method bem and the method of fundamental solutions mfs it is easy to program and attractive to the problems with complex geometries however the sbm is usually limited to small scale problems because of the operation count of o n 3 with direct solvers or o n 2 with iterative solvers as well as the memory requirement of o n 2 to overcome this drawback this study makes the first attempt to employ the precorrected fft pfft to accelerate the sbm matrix vector multiplication at each iteration step of the gmres for 3d helmholtz equation consequently the computational complexity can be reduced from o n 2 to o n l o g n or o n three numerical examples are successfully tested on a desktop computer the results clearly demonstrate the accuracy and efficiency of the developed fast pfft sbm strategy

spaces of matrices with few eigenvalues abstract let k be a commutative field with characteristic not 2 and v be a linear subspace of matrices of m n k that have at most two eigenvalues in k respectively at most one non zero eigenvalue in k we prove that dim v n 2 2 provided that n 3 respectively dim v n 2 1 we also classify up to similarity the linear subspaces of m n k in which every matrix has at most two eigenvalues respectively at most one non zero eigenvalue in an algebraic closure of k and which have the maximal dimension among such spaces

linear spaces of nilpotent matrices abstract we consider several questions on spaces of nilpotent matrices we present sufficient conditions for triangularizability and give examples of irreducible spaces we give a necessary and sufficient condition in terms of the trace for all linear combinations of a given set of operators to be nilpotent we also consider the question of the dimension of a space l of nilpotents on f n in particular we give a simple new proof of a theorem due to m gerstenhaber concerning the maximal dimension of such spaces

on two extremal matrix problems abstract let w be a linear subspace of symmetric n n matrices whose rank is at most t it is shown that if the underlying field has more than n elements then let l be an affine space of n n matrices all having rank at least k it is shown that if the underlying field is algebraically closed then dim l n 2 k 1 2 the last result is applied to a problem of valiant concerning permanents and determinants

diagonal form fast multipole singular boundary method applied to the solution of high frequency acoustic radiation and scattering summary the singular boundary method sbm is a recent strong form boundary collocation method that uses a linear combination of the fundamental solution of the governing equation to approximate the field variables because of its full interpolation matrix the sbm solution encounters the high computational complexity and storage requirement that limit its applications to large scale engineering problems this paper presents a way to overcome this drawback by introducing the diagonal form fast multipole method a diagonal form fast multipole singular boundary method is then developed to reduce the computational operations of the sbm with direct solvers from o n3 to o nlogn where n is the number of unknowns the proposed method works well for acoustic radiation and scattering with nondimensional wave number kd 110 k is the wave number and d the maximum diameter of the computational domain several numerical experiments are provided to show the efficiency and accuracy of the present strategy including the radiation from a nuclear submarine and the scattering from an airplane the numerical results clearly demonstrate that the present diagonal form fast multipole singular boundary method breaks the limitations of the sbm and successfully simulates the large scale acoustic problems with more than one million unknowns on a desktop computer copyright 2016 john wiley sons ltd

analysis of three dimensional transient acoustic wave propagation using the boundary integral equation method this paper describes a boundary integral equation boundary element method for the solution of a variety of transient acoustic problems the spatial and temporal discretization employs quadratic isoparametric elements with high order gauss quadrature and the ensuing equations are implicit the implicit formulation both eliminates the instabilities reported in explicit treatments and permits a freedom of choice of timestep which can reduce costs dramatically the accuracy of the approach is demonstrated by comparison with the analytical solution for a sphere results for more demanding sphere cone sphere geometries extending to seven wavelengths long are presented and compared to those obtained from a related frequency domain approach

a new family of continuous discontinuous three dimensional boundary elements with application to acoustic wave propagation this paper presents a new family of continuous discontinuous three dimensional boundary elements which has been implemented in a computer system for acoustic wave propagation in the frequency domain the system includes elements of different geometrical shapes order of interpolation and degree of continuity results of several tests are included to show the efficiency of the family of elements

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

the boundary element free method for 2d interior and exterior helmholtz problems abstract the boundary element free method befm is developed in this paper for numerical solutions of 2d interior and exterior helmholtz problems with mixed boundary conditions of dirichlet and neumann types a unified boundary integral equation is established for both interior and exterior problems by using the improved interpolating moving least squares method to form meshless shape functions mixed boundary conditions in the befm can be satisfied directly and easily detailed computational formulas are derived to compute weakly and strongly singular integrals over linear and higher order integration cells three numerical integration procedures are developed for the computation of strongly singular integrals numerical examples involving acoustic scattering and radiation problems are presented to show the accuracy and efficiency of the meshless method

fast multipole accelerated singular boundary method for the 3d helmholtz equation in low frequency regime this paper presents a fast multipole accelerated singular boundary method sbm to the solution of the large scale three dimensional helmholtz equation at low frequency by using a desingularization strategy to directly compute singular kernels in the strong form collocation discretization the sbm formulations are derived for the dirichlet and neumann problems a fast multipole method fmm is then introduced to expedite the solution process the cpu time and the memory requirement of the fmmsbm scheme are both reduced to o n where n is the number of boundary nodes numerical examples with up to one million unknowns have been tested on a desktop computer the results clearly illustrate that the proposed strategy appears very efficient and promising in solving large scale helmholtz problems

a meshless singular boundary method for transient heat conduction problems in layered materials abstract the transient heat conduction problems in layered materials are common encountered in practical application but bring great challenge to most numerical algorithms in this study the singular boundary method sbm a meshless boundary collocation technique is first employed to numerically simulate such problems based on the time dependent fundamental solution of diffusion equation taking the boundary conditions and interface conditions into account the computing system of the sbm for layered materials is established and then is solved the proposed method fully inherits the merits of conventional boundary type methods while possessing its distinctive advantages furthermore it is simple straightforward computationally efficient and stable since it does not need to discretize temporal derivative term three numerical experiments are performed to verify the efficiency and accuracy of the proposed scheme numerical results clearly indicate the efficiency accuracy and stability of the presented sbm for solving transient heat conduction problems in layered materials

three dimensional discontinuous galerkin elements with plane waves and lagrange multipliers for the solution of mid frequency helmholtz problems recently a discontinuous galerkin finite element method with plane wave basis functions and lagrange multiplier degrees of freedom was proposed for the efficient solution in two dimensions of helmholtz problems in the mid frequency regime in this paper this method is extended to three dimensions and several new elements are proposed computational results obtained for several wave guide and acoustic scattering model problems demonstrate one to two orders of magnitude solution time improvement over the higher order galerkin method

a semi analytical approach for radiation and scattering problems with circular boundaries in this paper the radiation and scattering problems with circular boundaries are studied by using the null field integral equations in conjunction with degenerate kernels and fourier series to avoid calculating the cauchy and hadamard principal values in implementation the null field point can be located on the real boundary owing to the introduction of degenerate kernels for fundamental solution an adaptive observer system of polar coordinate is considered to fully employ the property of degenerate kernels for the hypersingular equation vector decomposition for the radial and tangential gradient is carefully considered this method can be seen as a semi analytical approach since errors attribute from the truncation of fourier series neither hypersingularity in burton and miller approach nor the chief concepts were required to deal with the problem of irregular frequencies four gains well posed model singularity free boundary layer effect free and exponential convergence are achieved by using the present approach a fast convergence rate in exponential order than algebraic one in bem stems from the series expansions three examples were demonstrated to see the validity of the present formulation and show the better accuracy than bem

on the matrices of given rank in a large subspace abstract let v be a linear subspace of m n p k with codimension lesser than n where k is an arbitrary field and n p in a recent work of the author it was proven that v is always spanned by its rank p matrices unless n p 2 and k f 2 here we give a sufficient condition on codim v for v to be spanned by its rank r matrices for a given r 1 p 1 this involves a generalization of the gerstenhaber theorem on linear subspaces of nilpotent matrices

nonsingular aci matrices over integral domains abstract an aci matrix is a matrix whose entries are polynomials of degree at most one in a number of indeterminates where no indeterminate appears in two different columns consider the next two problems a characterize the aci matrices of order n all of whose completions have the same nonzero constant determinant b characterize the aci matrices of order n all of whose completions are nonsingular in 2010 brualdi huang and zhan solved both problems for fields of at least n 1 elements we extend their result on problem a to integral domains and extend their result on problem b to arbitrary fields

singular boundary method for solving plane strain elastostatic problems this study documents the first attempt to apply the singular boundary method sbm a novel boundary only collocation method to two dimensional 2d elasticity problems unlike the method of fundamental solutions mfs the source points coincide with the collocation points on the physical boundary by using an inverse interpolation technique to regularize the singularity of the fundamental solution of the equation governing the problems of interest three benchmark elasticity problems are tested to demonstrate the feasibility and accuracy of the proposed method through detailed comparisons with the mfs boundary element method bem and finite element method fem

the method of fundamental solutions and quasi monte carlo method for diffusion equations the laplace transform is applied to remove the time dependent variable in the diffusion equation for non harmonic initial conditions this gives rise to a non homogeneous modified helmholtz equation which we solve by the method of fundamental solutions to do this a particular solution must be obtained which we find through a method suggested by atkinson to avoid costly gaussian quadratures we approximate the particular solution using quasi monte carlo integration which has the advantage of ignoring the singularity in the integrand the approximate transformed solution is then inverted numerically using stehfest s algorithm two numerical examples are given to illustrate the simplicity and effectiveness of our approach to solving diffusion equations in 2 d and 3 d 1998 john wiley sons ltd