a simplex based algorithm to solve separated continuous linear programs we consider the separated continuous linear programming problem with linear data we characterize the form of its optimal solution and present an algorithm which solves it in a finite number of steps using an analog of the simplex method in the space of bounded measurable functions

convergence of a general class of algorithms for separated continuous linear programs separated continuous linear programs sclp are a type of infinite dimensional linear program which can serve as a useful model for a variety of dynamic network problems where storage is permitted at the nodes this paper proves the convergence of a general class of algorithms for solving sclp under certain restrictions on the problem data this is the first such proof for any nondiscretization algorithm for solving any form of continuous linear programs

a moving finite element method with error estimation and refinement for one dimensional time dependent partial differential equations we discuss a moving finite element method for solving vector systems of time dependent partial differential equations in one space dimension the mesh is moved so as to equidistribute the spatial component of the discretization error in h 1 we present a method of estimating this error by using p hierarchic finite elements the error estimate is also used in an adaptive mesh refinement procedure to give an algorithm that combines mesh movement and refinement we discretize the partial differential equations in space using a galerkin procedure with piecewise linear elements to approximate the solution and quadratic elements to estimate the error a system of ordinary differential equations for mesh velocities are used to control element motions we use existing software for stiff ordinary differential equations for the temporal integration of the solution the error estimate and the mesh motion computational results using a code based on our method are presented for several examples

separated continuous conic programming strong duality and an approximation algorithm motivated by recent applications in robust optimization and in sign constrained linear quadratic control we study in this paper a new class of optimization problems called separated continuous conic programming sccp focusing on a symmetric primal dual pair we develop a strong duality theory for the sccp our idea is to use discretization to connect the sccp and its dual to two ordinary conic programs we show if the latter are strongly feasible and with finite optimal values a condition that is readily verifiable then the strong duality holds for the sccp this approach also leads to a polynomial time approximation algorithm that solves the sccp to any required accuracy

random positive operator valued measures we introduce several notions of random positive operator valued measures povms and we prove that some of them are equivalent we then study statistical properties of the effect operators for the canonical examples starting from the limiting eigenvalue distribution we derive the large system limit for several quantities of interest in quantum information theory such as the sharpness the noise content and the probability range finally we study different compatibility criteria and we compare them for generic povms

a new algorithm for state constrained separated continuous linear programs during the last few decades significant progress has been made in solving large scale finite dimensional and semi infinite linear programming problems in contrast little progress has been made in solving linear programs in infinite dimensional spaces despite their importance as models in manufacturing and communication systems inspired by the research on separated continuous linear programs we propose a new class of continuous linear programming problems that has a variety of important applications in communications manufacturing and urban traffic control this class of continuous linear programs contains the separated continuous linear programs as a subclass using ideas from quadratic programming we propose an efficient algorithm for solving large scale problems in this new class under mild assumptions on the form of the problem data we prove algorithmically the absence of a duality gap for this class of problems without any boundedness assumptions on the solution set we show this class of problems admits piecewise constant optimal control when the optimal solution exists we give conditions for the existence of an optimal solution we also report computational results which illustrate that the new algorithm is effective in solving large scale realistic problems with several hundred continuous variables arising in manufacturing systems

limit laws for random matrices and free products in earlier articles we studied a kind of probability theory in the framework of operator algebras with the tensor product replaced by the free product we prove here that free random variables naturally arise as limits of random matrices and that wigner s semicircle law is a consequence of the central limit theorem for free random variables in this way we obtain a non commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra wigner s law being given by the trace of the spectral measure of the selfadjoint component of this operator

adaptive zoning for singular problems in two dimensions abstract winslow s method for the automatic generation of computation meshes is extended to adaptively vary the zone sizes and orthogonality of grid lines in the resulting mesh through simple analysis and numerical examples the adaptive mesh is shown to give significant increases in accuracy in the computation of singular problems

adaptive methods for elliptic grid generation on developpe une methode generale pour l application de techniques de grille adaptative precises et efficaces a la generation de grilles elliptiques

an algorithm for a class of continuous linear programs this paper discusses a class of continuous linear programs posed in a function space called separated continuous linear programs sclp a dual linear program and a corresponding discrete approximation are introduced followed by a discussion of their properties the discrete approximation gives rise to an improvement step which is constructed from any given feasible nonoptimal solution for sclp a strong duality result follows from this there are a variety of possible implementations of an algorithm for solving sclp problems using this improvement step finally some computational results are given from one possible implementation

simple adaptive grids for 1 d initial value problems abstract in solving 1 d initial value problems by implicit finite difference methods it is advantageous to use an adaptive grid to provide local regions of high resolution and allow larger time steps to be taken a simple method of constructing such grids is presented and the benefits are demonstrated by calculations of sod s shock tube problem g a sod j comput phys 27 1 1978 and of a supernova explosion

grid adaption for problems in fluid dynamics analyse et application d une technique de grille adaptative aux problemes fortement non lineaires en mecanique des fluides et transfert de chaleur resolution de problemes d ecoulement externe avec decollement de propagation de flamme et d interaction choc couche limite

an extended algorithm for separated continuous linear programs separated continuous linear programs sclp are a class of continuous linear programs which among other things can serve as a useful model for dynamic network problems where storage is permitted at the nodes recent work on sclp has produced a detailed duality theory conditions under which an optimal solution exists with a finite number of breakpoints a purification algorithm as well as a convergent algorithm for solving sclp under certain assumptions on the problem data this paper combines much of this work to develop a possible approach for solving a wider range of sclp problems namely those with fairly general costs the techniques required to implement the algorithm are no more than standard finite dimensional linear programming and line searching and the resulting algorithm is simplex like in nature we conclude the paper with the numerical results obtained by using a simple implementation of the algorithm to solve a small problem

commutators of free random variables let a be a unital c algebra given together with a specified state a c consider two selfadjoint elements a b of a which are free with respect to in the sense of the free probability theory of voiculescu let us denote c i ab ba where the i in front of the commutator is introduced to make c selfadjoint in this paper we show how the spectral distribution of c can be calculated from the spectral distributions of a and b some properties of the corresponding operation on probability measures are also discussed the methods we use are combinatorial based on the description of freeness in terms of non crossing partitions an important ingredient is the notion of r diagonal pair introduced and studied in our previous paper 12

on the law of addition of random matrices normalized eigenvalue counting measure of the sum of two hermitian or real symmetric matrices a n and b n rotated independently with respect to each other by the random unitary or orthogonal haar distributed matrix u n i e a n u n b n u n is studied in the limit of large matrix order n convergence in probability to a limiting nonrandom measure is established a functional equation for the stieltjes transform of the limiting measure in terms of limiting eigenvalue measures of a n and b n is obtained and studied

a duality theory for separated continuous linear programs this paper presents a detailed duality theory for a class of continuous linear programs called separated continuous linear programs sclp based on a particular dual problem sclp using weak duality a notion of complementary slackness is introduced and several sufficient conditions for optimality of sclp are derived along with the existence of complementary slack variables for basic feasible solutions for sclp following this a fairly general condition for the absence of a duality gap between sclp and sclp is given as are several conditions for the existence of an optimal solution for sclp finally using all these ingredients a strong duality result between sclp and sclp is proven when the problem data are piecewise analytic a simple counterexample is presented to show that strong duality may not follow if the assumptions of piecewise analyticity do not hold

a new theory for one dimensional adaptive grid generation and its applications the theory of a new approach to adaptive grid generation in one dimension is developed the approach is based on approximating either the resolution or the grid spacing ratio on discrete lattice points by continuous variables the order of accuracy of these approximations in a suitable reference frame characterizes the various methods approximations that are first or second order accurate in a suitable reference coordinate are derived in this paper the free parameters associated with these methods provide flexibility in generating a large family of adaptive grids with smooth grid spacing ratio and high resolution a selected group of this family of adaptive grids may prove very useful in adaptive computations of partial differential equations the adaptive grids are numerically generated using these approximations numerical examples are given that exemplify the usefulness of these adaptive grids

symmetric strong duality for a class of continuous linear programs with constant coefficients we consider continuous linear programs over a continuous finite time horizon t with linear cost coefficient functions and linear right hand side functions and a constant coefficient matrix where we search for optimal solutions in the space of measures or of functions of bounded variation these models generalize the separated continuous linear programming models and their various duals as formulated in the past by anderson by pullan and by weiss we present simple necessary and sufficient conditions for feasibility we formulate a symmetric dual and investigate strong duality by considering discrete time approximations we prove that under a slater type condition there is no duality gap and there exist optimal solutions which have impulse controls at 0 and t and have piecewise constant densities in 0 t moreover we show that under nondegeneracy assumptions all optimal solutions are of this form and are uniquely determined over 0 t

on the asymptotic distribution of the commutator and anticommutator of random matrices we study in the limit of infinite matrix order the normalized eigenvalue counting measures of the commutator and anticommutator of two hermitian or real symmetric matrices rotated independently one respect to another by the random unitary or orthogonal haar distributed matrix we establish the convergence with probability 1 to a limiting nonrandom measure we obtain and analyze the functional equations for the stieltjes transforms of the limiting measures

an adaptive method of lines with error control for parabolic equations of the reaction diffusion type abstract a piecewise linear finite element based method of lines is presented for the numerical solution of coupled parabolic partial differential equations which model biological and physicochemical reaction diffusion processes in one space dimension the vertial lines emanating from the space nodes in this method change at automatically selected times when in order to control a norm of the space discretization error adaptive spatial regridding occurs the regridding algorithm is an extension of one described previously by the authors 7 and is implemented in the program femol 1 which uses the lsodi package 14 15 of hindmarsh and painter to integrate the ordinary differential equations in time along the vertical lines computational results show that the method is efficient that a posteriori estimates of the space discretization error are accurate and that the adaptive procedure reliably controls the space discretization error

some properties of a class of continuous linear programs this paper discusses a class of linear programs posed in a function space a member of this class is called a separated continuous linear program sclp such problems occur for example in the planning of production and inventory we characterize the l infty extreme point solutions of sclp in a manner analogous to the basic solutions of finite dimensional linear programming and give a sufficient condition for there to exist optimal extreme point solutions with finitely many constant basis intervals sclp is to date the most general continuous linear program for which such strong characterizations have been found

iterative schemes and algorithms for adaptive grid generation in one dimension some iterative adaptive grid generators developed by the author are numerically explored in detail to assess their relative merits against conventional grid generators based on a direct method of integration and interpolation we find that some of these iterative adaptive grid generators are preferable to a direct method of integration and interpolation in contrast with a direct method appropriate use of these iterative adaptive grid generators produces adaptive grids with a smooth variation of the grid spacing ratio and resolution all adaptive grid generators are a subclass of a more general iterative map general features of this iterated map which are related to the function to be resolved are briefly discussed the results obtained here supplements recent investigations on these adaptive grid generators by the author

conservative rezoning algorithm for generalized two dimensional meshes a method is presented for transferring a conserved quantity q from one generalized mesh to another when the volumetric density of q is uniform within each cell of the original mesh

simplified second order rezoning algorithm for generalized two dimensional meshes a second order conservative rezoning algorithm for generalized two dimensional meshes has recently been described by dukowicz and kodis siam j sci statist comput in press here it is shown that this algorithm can be simplified by a different and more symmetrical choice of the flux vector f whose divergence is the cell density the resulting formulation is the natural second order generalizaton of the simplified first order algorithm described earlier j comput phys 59 193 1985

forms of optimal solutions for separated continuous linear programs this paper discusses the nature of optimal solutions for a class of continuous linear programs called separated continuous linear programs it is shown that under various different assumptions on the problem data there exist optimal solutions that are piecewise constant piecewise polynomial or more generally piecewise analytic these results are reminiscent of bang bang results in linear optimal control

the moving finite element method applications to general partial differential equations with multiple large gradients abstract the moving finite element mfe method has been reduced to practice in the automatic solution program dyla for general systems of transient partial differential equations pdes in 1 d several test examples are presented which illustrate the unique node movement and systematic control features which are intrinsic in the mfe method computational dilemmas of numerical diffusion gibbs overshooting and undershooting zone tangling and grid remap or re connection aliasing which occur frequently in conventional pde methods are essentially eliminated in the mfe method arbitrarily large gradients or shocks can be solved with extremely high resolution and accuracy for non coincident or even counterdirected propagating wavefronts boundary layers of arbitrarily small dimensions are solved with high accuracy simultaneously with the large scale structures in reactive and non reactive fluid calculations the mfe method requires a small fraction of the grid nodes which are used in conventional pde solution methods because the nodes migrate continuously and systematically to those positions where they are most needed in order to yield accurate pde solutions on entire problem domains courant friedrichs lewy time step limits are exceeded by wide margins by factors of two to several thousand finally the extension of the mfe method to 2 d is briefly discussed