on the covering number of a matroid element abstract let s be a nonempty finite set with cardinality m let m s i m be a matroid on s with no loops we present results connecting the structure of the bases of m k the k th power of m and the covering number of elements of s in m

fast and explicit wentzel kramers brillouin mode sum for the bottom interacting field including leaky modes a fast and explicit method is presented for calculating the bottom interacting field by means of a mode sum using the wentzel kramers brillouin approximation for the eigenfunctions and the tindle weston formula for eigenvalue spacing both trapped and leaky modes are automatically included requiring no numerical inversions for either eigenfunctions or eigenvalues examples are provided which demonstrate the speed and accuracy of the approach by comparison with reference solutions the importance of leaky modes especially in range dependent environments with variable water depth is demonstrated a simple criterion for neglecting leaky modes is developed

estimation of the dispersion error in the numerical wave number of standard and stabilized finite element approximations of the helmholtz equation an estimator for the error in the wave number is presented in the context of finite element approximations of the helmholtz equation the proposed estimate is an extension of the ideas introduced in steffens and d iez comput methods appl mech engng 2009 198 1389 1400 in the previous work the error assessment technique was developed for standard galerkin approximations here the methodology is extended to deal also with stabilized approximations of the helmholtz equation thus the accuracy of the stabilized solutions is analyzed including also their sensitivity to the stabilization parameters depending on the mesh topology the procedure builds up an inexpensive approximation of the exact solution using post processing techniques standard in error estimation analysis from which the estimate of the error in the wave number is computed using a simple closed expression the recovery technique used in steffens and diez comput methods appl mech engng 2009 198 1389 1400 is based in a polynomial least squares fitting here a new recovery strategy is introduced using exponential in a complex setup trigonometric local approximations respecting the nature of the solution of the wave problem copyright 2011 john wiley sons ltd

conditions for a symmetrized decomposable tensor to be zero let x1ast xm be a decomposable symmetrized tensor corresponding to the symmetric group sm and an irreducible character a necessary and sufficient condition on the vectors x1 xm is given for x1 xm to be zero

boundary conditions for the numerical solution of elliptic equations in exterior regions elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the weill posedness of the problem examples of practical applications include the helmholtz equation and laplace s equation computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition at infinity by a boundary condition on a finite artificial surface direct imposition of the condition at infinity along the finite boundary results in large errors a sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain estimates of the error due to the finite boundary are obtained for several cases computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions the examples are based on a finite element formulation but finite difference methods can also be used

technical communique feedback nash equilibria in the scalar infinite horizon lq game in this note we consider linear stationary feedback nash equilibria of the scalar linear quadratic differential game the planning horizon considered is assumed to be infinite we present both necessary and sufficient conditions on the system parameters for the existence of a unique solution of the associated algebraic riccati equations are that stabilizes the closed loop system for the case of more solutions singleton valued refinements of the equilibrium concept are studied

bifurcation analysis of a non cooperative differential game with one weak player we study a bifurcation problem for a system of two differential equations in implicit form for each value of the parameter the solution yields a pair of nash equilibrium strategies in feedback form for a non cooperative differential game when 0 the second player has no power to influence the dynamics of the system and his optimal strategy is myopic the game thus reduces to an optimal control problem for the first player by studying the bifurcation in the solutions to the corresponding system of hamilton jacobi equations one can establish existence and multiplicity of solutions to the differential game as becomes strictly positive

flags and equality of tensors abstract we state a necessary and sufficient condition for equality of decomposable symmetrized tensors when the symmetrizer is associated to an irreducible character of s p corresponding to the partition of p and the tensors are the images by this symmetrizer of the tensor product of a family of vectors of rank partition we make an application of this result to the representation of flags by projective points

patchwork rejection algorithms abstract new methods for implementing rejection algorithms are suggested the objectives are to avoid the generation of unwanted envelope variates to use fewer pseudorandom uniform variates and to bypass as far as possible the need to decide whether to accept or to reject considerably fewer than two uniforms on average per attempted output of a target random variable can be achieved together with an increased acceptance rate the generation of tail variates from a normal distribution is used as illustration

on the powers of a vectorial matroid an equivalence relation abstract let v be a nonzero vector space over the field f of characteristic zero let x be a finite family of nonzero vectors of v for each a v we denote by ma the vectorial matroid of x a for each k n we denote by ma k the kth power of ma we define an equivalence relation k on v such that a kb if and only if the matroids ma k and mb k are on the same set and have the same set of independent sets we present results on this equivalence relation

a numerical method based on the complementarity and optimal control formulations for solving a family of zero sum pursuit evasion differential games abstract this paper presents an efficient numerical method for a class of zero sum pursuit evasion differential games zspedgs the aim of the presented method is to resolve the drawbacks of the indirect methods in solving zspedgs in the indirect methods the solution of zspedg is found by solving a two point boundary value problem tpbvp derived from the necessary conditions the indirect methods are accurate and fast however they are very sensitive to initial guess and when the control is bounded some non smooth equations appear in the resulting tpbvp these drawbacks restrict the use of indirect methods for solving zspedgs to overcome these drawbacks at first we reformulate the discontinuous equations of tpbvp by complementarity conditions and as a result a differential complementarity system dcs is obtained then the resulting dcs is reformulated to an optimal control problem which can be solved by a well developed direct or indirect method the efficiency and robustness of the method are reported by means of two benchmarks and two real life zspedg problems

computing acoustic waves in an inhomogeneous medium of the plane by a coupling of spectral and finite elements in this paper we analyze a galerkin procedure based on a combination of finite and spectral elements for approximating a time harmonic acoustic wave scattered by a bounded inhomogeneity the finite element method used to approximate the near field in the region of inhomogeneity is coupled with a nonlocal boundary condition which consists in a linear integral equation this integral equation is discretized by a spectral galerkin approximation method we provide error estimates for the galerkin method propose fully discrete schemes based on elementary quadrature formulas and show that the perturbation due to this numerical integration gives rise to a quasi optimal rate of convergence we also suggest a method for implementing the algorithm using the preconditioned gmres method and provide some numerical results

high frequency numerical analytic approximations for the separable acoustic wave equation abstract an algorithm which employs a combination of asymptotic and numerical methods to solve a sturm liouville problem arising from a separable acoustic wave equation is presented error criteria for switching between methods are derived implementation requires a decomposition of the index of refraction profile into separable subprofiles computing times and accuracies are compared with conventional methods

new conditions for equality of decomposable symmetrized tensors we determine conditions for equality of decomposable symmetrized tensors in arbitrary symmetry classes based on these conditions we derive results for equality of decomposable symmetrized tensors associated to certain special group characters

a numerical method for ocean acoustic normal modes the method of normal modes is frequently used to solve acoustic propagation problems in stratified oceans the propagation numbers for the modes are the eigenvalues of the boundary value problem to determine the depth dependent normal modes errors in the numerical determination of these eigenvalues appear as phase shifts in the range dependence of the acoustic field such errors can severely degrade the accuracy of the normal mode representation particularly at long ranges in this paper we present a fast finite difference method to accurately determine these propagation numbers and the corresponding normal modes it consists of a combination of well known numerical procedures such as sturm sequences the bisection method newton s and brent s methods richardson extrapolation and inverse iteration we also introduce a modified richardson extrapolation procedure that substantially increases the speed and accuracy of the computation

equality of decomposable symmetrized tensors and matrix groups let g be a subgroup of the full symmetric group sn and a character of g a matrix can be defined as an n n matrix b which satisfies dgx bx dgx x for every n n matrix x they form a multiplicative group denoted s g x which plays a fundamental role in the study of equality of two decomposable symmetrized tensors the main result of this paper theorems 2 2 2 3 and 2 4 is a complete description of the matrices in s g x this description has many consequences that we present there are also results on related questions

a numerical solution of open loop nash equilibrium in nonlinear differential games based on chebyshev pseudospectral method in general the applications of differential games for solving practical problems have been limited because all calculations had to be done analytically in this investigation a simple and efficient numerical method for solving nonlinear nonzero sum differential games with finite and infinite time horizon is presented in both cases derivation of open loop nash equilibria solutions usually leads to solving nonlinear boundary value problems for a system of odes the proposed numerical method is based on a combination of minimum principle of pontryagin and expanding the required approximate solutions as the elements of chebyshev polynomials applying chebyshev pseudospectral method two point boundary value problems in differential games are reduced to the solution of a system of algebraic equations finally several examples are given to demonstrate the accuracy and efficiency of the proposed method and a comparison is made with the results obtained by fourth order runge kutta method

the uniform wkb modal approach to pulsed and broadband propagation a uniform asymptotic approach is utilized to consider pulsed and broadband propagation for a given frequency the wkb approach with boundary layer corrections is used to approximate the normal modes of the sound channel this approach is demonstrated to yield significant reductions in computer processing time and high accuracy when compared to a conventional code it is shown that a substantial amount of the calculations in the uniform wkb approach required at each frequency is frequency independent and need be done only once in multiple frequency runs hence further efficiencies in broadband or pulsed propagation result without any sacrifice in accuracy comparison of waveform predictions between uniform wkb and conventional techniques show excellent agreement in a complex propagation environment with use of the uniform wkb approach resulting in reductions of computer time by a factor of over 100

on the kernel of the derivation operator abstract we state a necessary and sufficient condition for a symmetrized tensor to belong to the kernel of the derivation operator the derivation operator a generalization of the associated transformation introduced by m marcus in o shisha ed inequalities ii academic press new york 1970 p 223 is a linear operator on the range of the symmetrizer when the symmetrizer is associated to an absolutely irreducible character of s m corresponding to the partition of m and the tensor is the image by this symmetrizer of the tensor product of a family of vectors of rank partition

the covering number of elements of a matroid and associated transformations abstract let s be a nonempty finite set with cardinality m let m s i m be a matroid on s let x be an element of s which is not a loop of m the covering number of x in m is the smallest positive integer s such that x is a coloop of the union of s copies of m we investigate relations between the covering number of the elements in m and in its minors applications to the study of the rank partition and associated transformation are presented

the covering number of the elements of a matroid and generalized matrix functions abstract let s be a nonempty finite set with cardinality m let m be a matroid on s with no loops the covering number of an element x in s is the smallest positive integer k such that x is a coloop of the union of k copies of m we investigate connections between the structure of m and the values of the covering numbers of elements of s applications to the study of the rank partition and generalized matrix functions are presented

min max optimal control of linear systems with uncertainty and terminal state constraints in this paper a class of min max optimal control problems with continuous dynamical systems and quadratic terminal constraints is studied the main contribution is that the original terminal state constraint in which the disturbance is involved is transformed into an equivalent linear matrix inequality without disturbance under certain conditions then the original min max optimal control problem is solved via solving a sequence of semi definite programming problems an example is presented to illustrate the proposed method

the multilinear algebra of jose dias da silva and the portuguese school of mathematics abstract in portugal multilinear algebra has a strong group of researchers that achieved in the last years some impressive results in this area this paper surveys the results obtained by this group and the current state of knowledge on the different problems

edged based smoothed point interpolation method for acoustic radiation with perfectly matched layer abstract it is known that for exterior acoustics the domain based finite element method always encounters the accuracy loss for high wavenumbers and the difficulty in dealing with the sommerfeld radiation condition in this work by using the polynomial and radial bases with specific node selection schemes the point interpolation method with the generalized gradient smoothing technique is employed to improve the accuracy for acoustic computations the appropriately softened acoustical stiffness of the discrete model is constructed based on the edge based smoothing domains via the gradient smoothing technique in addition the perfectly matched layer technique enables the smoothed point interpolation method to investigate exterior acoustics the complex coordinate transformation is utilized to determine the proper parameters for the perfectly matched layer through theoretical and numerical analyses several typical acoustic problems are analyzed revealing that better accuracy is obtained by the present method compared with the finite element method especially for high wavenumbers

nash equilibrium approximation of some class of stochastic differential games a combined chebyshev spectral collocation method with policy iteration abstract this study relates to nonzero sum stochastic games with perfect information it proposes an efficient combined chebyshev spectral collocation method cscm with the policy iteration pi algorithm for solving nonlinear coupled hamilton jacobi hj equations the proposed approach is comprised of two steps first the pi algorithm is used to reduce the nonlinear coupled hj equations to a sequence of linear uncoupled pdes then these equations are approximated by the cscm the cscm pi is especially useful when the cscm fails due to the increasing number of collocation points for solving the associated system of nonlinear algebraic equations the main advantage of the resulting method is that it converts nonlinear coupled hj equations to the systems of linear algebraic equations which can be solved readily convergence analysis of this method is also provided in detail to confirm the accuracy and validity of the proposed computational algorithm several illustrative examples are presented

three dimensional approximate local dtn boundary conditions for prolate spheroid boundaries we propose a new class of approximate local dtn boundary conditions to be applied on prolate spheroidal shaped exterior boundaries when solving problems of acoustic scattering by elongated obstacles these conditions are a exact for the first modes b easy to implement and to parallelize c compatible with the local structure of the computational finite element scheme and d applicable to exterior ellipsoidal shaped boundaries that are more suitable in terms of cost effectiveness for surrounding elongated scatterers we investigate analytically and numerically the effect of the frequency regime and the slenderness of the boundary on the accuracy of these conditions we also compare their performance to the second order absorbing boundary condition bgt2 designed by bayliss gunzburger and turkel when expressed in prolate spheroid coordinates the analysis reveals that in the low frequency regime the new second order dtn condition dtn2 retains a good level of accuracy regardless of the slenderness of the boundary in addition the dtn2 boundary condition outperforms the bgt2 condition such superiority is clearly noticeable for large eccentricity values

open loop nash equilibrium in polynomial differential games via state dependent riccati equation this paper studies finite as well as infinite time horizon nonzero sum polynomial differential games in both cases we explore the so called state dependent riccati equations to find a set of strategies that guarantee an open loop nash equilibrium for this particular class of nonlinear games such a method presents advantages in simplicity of the design of equilibrium strategies and yields computationally effective solution algorithms we demonstrate that this solution leads the game to an e or quasi equilibrium and provide an upper bound for this e quantity the proposed solution is given as a set of n coupled polynomial riccati like state dependent differential equations where each equation includes a p linear form tensor representation for its polynomial part we provide an algorithm for finding the solution of the state dependent algebraic equation in the infinite time case based on a hamiltonian approach and give conditions on the existence of such stabilizing solutions for a third order polynomial a numerical example is presented to illustrate effectiveness of the approach

an iterative adaptive dynamic programming method for solving a class of nonlinear zero sum differential games in this paper a new iterative adaptive dynamic programming adp method is proposed to solve a class of continuous time nonlinear two person zero sum differential games the idea is to use the adp technique to obtain the optimal control pair iteratively which makes the performance index function reach the saddle point of the zero sum differential games if the saddle point does not exist the mixed optimal control pair is obtained to make the performance index function reach the mixed optimum stability analysis of the nonlinear systems is presented and the convergence property of the performance index function is also proved two simulation examples are given to illustrate the performance of the proposed method