strategy iteration is strongly polynomial for 2 player turn based stochastic games with a constant discount factor ye 2011 showed recently that the simplex method with dantzig s pivoting rule as well as howard s policy iteration algorithm solve discounted markov decision processes mdps with a constant discount factor in strongly polynomial time more precisely ye showed that both algorithms terminate after at most o mn1 log n1 iterations where n is the number of states m is the total number of actions in the mdp and 0

las vegas algorithms for linear and integer programming when the dimension is small this paper gives an algorithm for solving linear programming problems for a problem with n constraints and d variables the algorithm requires an expected o d2n logn o d d 2 o 1 o d4 n logn arithmetic operations as n the constant factors do not depend on d also an algorithm is given for integer linear programming let 4 bound the number of bits required to specify the rational numbers defining an input constraint or the objective function vector let n and d be as before then the algorithm requires expected o 2ddn 8dd n l n n ln n do d 4 ln n operations on numbers with do 1 4 bits as n where the constant factors do not depend on d or 4 to other convex programming problems for example an algorithm for finding the smallest sphere enclosing a set of n points in ed has the same time bound

a note on the second order non autonomous neutral stochastic evolution equations with infinite delay under caratheodory conditions abstract in this note we aim to study a class of second order non autonomous neutral stochastic evolution equations with infinite delay driven by a standard cylindrical wiener process and an independent cylindrical fractional brownian motion with hurst parameter h 1 2 1 in which the initial value belongs to the abstract space b we establish the existence and uniqueness of mild solutions for this kind of equations under some caratheodory conditions by means of the successive approximation the obtained result extends some well known results an example is proposed to illustrate the theory

mild solutions of non autonomous second order problems with nonlocal initial conditions abstract in this paper we establish the existence of mild solutions for a non autonomous abstract semi linear second order differential equation submitted to nonlocal initial conditions our approach relies on the existence of an evolution operator for the corresponding linear equation and the properties of the hausdorff measure of non compactness

differentiability of solutions of second order functional differential equations with unbounded delay abstract in this paper we study the differentiability of solutions of the second order semilinear abstract retarded functional differential equation with unbounded delay specially when the underlying space is reflexive or at least has the radon nikodym property we apply our results to characterize the infinitesimal generators of several strongly continuous semigroups of linear operators that arise in the theory of linear abstract retarded functional differential equations with unbounded delay on a phase space defined axiomatically

linear programming in linear time when the dimension is fixed it is demonstrated that the linear programming problem in d variables and n constraints can be solved in o n time when d is fixed this bound follows from a multidimensional search technique which is applicable for quadratic programming as well there is also developed an algorithm that is polynomial in both n and d provided d is bounded by a certain slowly growing function of n

a quasi polynomial bound for the diameter of graphs of polyhedra the diameter of the graph of a d dimensional polyhedron with n facets is at most nlog d 2 let p be a convex polyhedron the graph of p denoted by g p is an abstract graph whose vertices are the extreme points of p and two vertices u and v are adjacent if the interval v u is an extreme edge 1 dimensional face of p the diameter of the graph of p is denoted by p let d n be the maximal diameter of the graphs of d dimensional polyhedra p with n facets a facet is a d 1 dimensional face thus p is the set of solutions of n linear inequalities in d variables it is an old standing problem to determine the behavior of the function d n the value of d n is a lower bound for the number of iterations needed for dantzig s simplex algorithm for linear programming with any pivot rule in 1957 hirsch conjectured 2 that d n n d klee and walkup 6 showed that the hirsch conjecture is false for unbounded polyhedra they proved that for n 2d d n n d d 5 this is the best known lower bound for d n the statement of the hirsch conjecture for bounded polyhedra is still open for a recent survey on the hirsch conjecture and its relatives see 5 in 1967 barnette proved 1 3 that d n n3 an improved upper bound d n n2 was proved in 1970 by larman 7 barnette s and larman s bounds are linear in n but exponential in the dimension d in 1990 the first author 4 proved a subexponential bound d n 2 n d the purpose of this paper is to announce and to give a complete proof of a quasipolynomial upper bound for d n such a bound was proved by the first author in march 1991 the proof presented here is a substantial simplification that was subsequently found by the second author see 4 for the original proof and related results the existence of a polynomial or even linear upper bound for d n is still open recently the first author found a randomized pivot rule for linear programming which requires an expected n d or less arithmetic operations for every linear programming problem with d variables and n constraints 1991 mathematics subject classification primary 52a25 90c05 received by the editors july 1 1991 the first author was supported in part by a bsf grant by a gif grant the second author was supported by an afosr grant c 1992 american mathematical society 0273 0979 92 1 00 25 per page

a sinh transformation for evaluating two dimensional nearly singular boundary element integrals a new transformation technique is introduced for evaluating the two dimensional nearly singular integrals which arise in the solution of laplace s equation in three dimensions using the boundary element method when the source point is very close to the element of integration the integrals are evaluated using in a product fashion a transformation which has recently been used to evaluate one dimensional near singular integrals this sinh transformation method automatically takes into account the position of the projection of the source point onto the element and also the distance b between the source point and the element the method is straightforward to implement and when it is compared with a number of existing techniques for evaluating two dimensional near singular integrals it is found that the sinh method is superior to the existing methods considered both for potential integrals across the full range of b values considered 0 0 01 for smaller values of b the use of the lmethod is recommended for flux integrals copyright 2006 john wiley sons ltd

evaluation of hypersingular and nearly singular integrals in the isogeometric boundary element method for acoustics abstract special integration routines for hypersingular and nearly singular integrals that arise in an isogeometric boundary element method igabem are presented the igabem is applied to an acoustic problem in the frequency domain that corresponds to the helmholtz equation in three dimensions a major topic of such an acoustical bem is the non uniqueness of the conventional boundary integral equation which can be corrected by the burton miller formulation at the cost of a hypersingular integral that requires the presented routines the combination of the bem with an isogeometric analysis iga is advantageous since the exact cad description is directly incorporated to the numerical simulation as the geometry is represented exactly the integration routines become even more relevant and cannot be masked by the geometrical approximation for the conventional approximation with plane lagrange elements a variety of different integration routines exist but are not investigated in the scope of an iga therefore the hypersingular integration scheme of guiggiani is adapted to be applicable to this new methodology a direct evaluation of the hypersingular integral is possible instead of an integration over the complete surface as in the often used regularized boundary integral equation additionally the sinh transformation for nearly singular integrals is introduced to achieve more accurate results with the same number of integration points as the conventional gaussian integration the new formulation is analyzed under the special case of the oscillatory kernels in acoustics

a randomized algorithm for fixed dimensional linear programming we give a las vegas randomized algorithm for linear programming in a fixed dimensiond for which the expected computation time is o d 3 varepsilon d d n where limd e d 0 this improves the corresponding worst case complexity o 3 d 2 n the method is based on a recent idea of clarkson two variations on the algorithm are examined briefly

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

adaptive h p and hp versions for boundary integral element methods recently very promising results in a so called hp version of the finite element method have been obtained the basic idea is a balanced combination of mesh refinement and increase of the polynomial degree of the shape functions this idea is applied to a boundary collocation method in this paper the new method is compared with adaptive h and p versions and it is shown in numerical examples that even in the presence of singularities in the exact solution exponential convergence is obtained

on simple polytopes letp be a simpled polytope ind dimensional euclidean space mathbb e d and let p be the subalgebra of the polytope algebra generated by the classes of summands ofp it is shown that the dimensions of the weight spaces r p of p are theh numbers ofp which describe the dehn sommerville equations between the numbers of faces ofp and reflect the duality between r p and d r p moreover p admits a lefschetz decomposition under multiplication by the element of 1 p corresponding top itself which yields a proof of the necessity of mcmullen s conditions in theg theorem on thef vectors of simple polytopes the lefschetz decomposition is closely connected with the new hodge riemann minkowski quadratic inequalities between mixed volumes which generalize minkowski s second inequality also proved are analogous generalizations of the aleksandrov fenchel inequalities a striking feature is that these are obtained without using brunn minkowski theory indeed the brunn minkowski theorem without characterization of the cases of equality can be deduced from them the connexion found between p and the face ring of the dual simplicial polytopep enables this ring to be looked at in two ways and a conjectured formulation of theg theorem in terms of a gale diagram ofp is also established

combinatorial structure and randomized subexponential algorithms for infinite games the complexity of solving infinite games including parity mean payoff and simple stochastic is an important open problem in verification automata and complexity theory in this paper we develop an abstract setting for studying and solving such games based on function optimization over certain discrete structures we introduce new classes of recursively local global rlg and partial recursively local global prlg functions and show that strategy evaluation functions for simple stochastic mean payoff and parity games belong to these classes in this setting we suggest randomized subexponential algorithms appropriate for rlg and prlg function optimization we show that the subexponential algorithms for combinatorial linear programming due to kalai and matousek sharir welzl can be adapted for optimizing the rlg and prlg functions

a strongly polynomial algorithm to solve combinatorial linear programs khachiyan and recently karmarkar gave polynomial algorithms to solve the linear programming problem these algorithms have a small theoretical drawback namely the number of arithmetic steps depends on the size of the input numbers we present a polynomial linear programming algorithm whose number of arithmetic steps depends only on the size of the numbers in the constraint matrix but is independent of the size of the numbers in the right hand side and objective vectors in particular it gives a polynomial algorithm for the minimum cost flow and multicommodity flow problems in which the number of arithmetic steps is independent of the size of the costs and capacities the algorithm makes use of an existing polynomial linear programming algorithm the problem of whether any algorithm has a running time that is independent even of the size of the numbers in the constraint matrix remains open

hp fast multipole boundary element method for 3d acoustics summary a fast multipole boundary element method fmbem extended by an adaptive mesh refinement algorithm for solving acoustic problems in three dimensional space is presented in this paper the collocation method is used and the burton miller formulation is employed to overcome the fictitious eigenfrequencies arising for exterior domain problems due to the application of the combined integral equation chbie the developed fmbem is feasible for all positive wave numbers even up to high frequencies in order to evaluate the hypersingular integral resulting from the burton miller formulation of the bie an integration technique for arbitrary element order is applied the fast multipole method combined with an arbitrary order h p mesh refinement strategy enables accurate computation of large scale systems numerical examples substantiate the high accuracy attainable by the developed fmbem while requiring only moderate computational effort at the same time this article is protected by copyright all rights reserved

necessary and sufficient conditions for the regularity and stability for some partial functional differential equations with infinite delay abstract in this work we give necessary and sufficient conditions for the regularity and stability of solutions for some partial functional differential equations with infinite delay we establish also a new characterization of the infinitesimal generator of the solution semigroup

parallel linear programming in fixed dimension almost surely in constant time for any fixed dimension d thelinear programming problem with n inequality constraints can be solved on a probabilistic crcw pram with o n processors almost surely in constant time the algorithm always findsthe correct solution with nd log 2 d processors the probability that the algorithm will not finish within o d 2 log 2 d time tends to zero exponentially with n authors abstract

frechet differentiability of solution mappings for semilinear second order evolution equations in this paper we establish the strong frechet differentiability of maps from the set of initial values and forcing terms into the set of solutions for semilinear second order evolution equations also under the weaker conditions of nonlinear terms we establish the strong g teaux differentiability of the solution maps an application of results to semilinear strongly damped wave equations is given

a linear algorithm for integer programming in the plane we show that a 2 variable integer program defined by m constraints involving coefficients with at most bits can be solved with o m arithmetic operations on rational numbers of size o

solutions to non autonomous integrodifferential equations with infinite delay abstract the conclusions of da prato and sinestrari concerning the non autonomous evolution operator of hyperbolic type for the equation u t a t u t f t t 0 t u 0 z are applied to find the uniqueness and existence of solution to u t a t u t 0 t k t u d l t u t f t t 0 t u 0 where a maybe nondensely defined moreover some relative problems are concerned

faedo galerkin approximation of second order nonlinear differential equation with deviated argument abstract in this manuscript we consider a second order nonlinear differential equation with deviated argument in a separable hilbert space x we used the strongly continuous cosine family of linear operators and fixed point method to study the existence of an approximate solution of the second order differential equation we define the fractional power of the closed linear operator and used it to prove the convergence of the approximate solution also we prove the existence and convergence of the faedo galerkin approximate solution finally we give an example to illustrate the application of these abstract results

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

voronoi cells probabilistic bounds and hypothesis testing in mixed integer linear models although real valued linear models whether or not of full rank have been thoroughly investigated and are well documented very little is known about statistical and probabilistic aspects of a mixed integer linear model which arose from space geodesy and serves as the standard starting model for precise positioning using the global positioning system gps voronoi cells play a fundamental role in the least squares estimation of the integer unknowns of the model in this paper we first develop a method to construct voronoi cells and study how to fit figures of simple shape to a voronoi cell both from inside and outside we then derive a number of new lower and upper bounds on the probability that the integers of the model are correctly estimated finally we discuss the tests of two hypotheses on the integer mean

stability and asymptotic behavior of solutions for some linear partial functional differential equations in critical cases abstract in this work we study the stability of the solution semigroup for some linear partial functional differential equations with infinite delay in a banach space when the exponential stability fails we use the so called characteristic equation to compute the order of each pole of the resolvent operator associated with the infinitesimal generator of the solution semigroup this result allows us to give sufficient conditions for having stability of the solution semigroup

integer programming with a fixed number of variables it is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable the proof depends on methods from geometry of numbers

the cauchy problem for time dependent abstract parabolic equations of higher order after the pioneering works by chazarain 2 and dubinskii 3 in the early seventies several papers were devoted to higher order differential equations in a banach space especially for either constant coefficient or second order equations detailed references can be find in 4 sect 25 c very recent results in the constant coefficient case are due to sandefur s for second order semilinear equations and to neubrander s for linear arbitrary order equations in a recent paper 7 we considered an equation such as

a fast algorithm for the two variable integer programming problem an algorithm that solves any two variable integer programming problem is presented a constant word length model for the data is assumed the complexity for a problem with m constraints and word length of l digits is bounded by the maximum of two values the first which is o mlogm steps is a bound on the complexity of finding the convex region bounded by the constraints each step being an arithmetic orperation or a compare the second which is o ml steps is the complexity of solving m greatest common divisor problems the algorithm finds a minimal binding set of constraints for any given problem in addition to finding the soluuon set a new method of solving three constraint problems is introduced

existence of solutions of non autonomous second order functional differential equations with infinite delay in this paper the existence of solutions of a non autonomous abstract retarded functional differential equation of second order with infinite delay is considered assuming the existence of an evolution operator corresponding to the associate abstract cauchy problem of second order we establish the existence of mild solutions of the functional equation furthermore we study the existence of classical solutions of the abstract cauchy problem of second order and we apply these results to establish the existence of classical solutions of the functional equation finally we apply our results to study the existence of solutions of the non autonomous wave equation with delay

grid optimization for the boundary element method the quality of solution obtained using the boundary element method bem is dependent on how the boundary is discretized this is particularly true in domains of complex geometry a rule for grid optimization for the bem is derived on the bases of an asymptotic measure of the boundary element error that preserves the number of elements degrees of freedom three example problems are provided to show the advantages of grid optimization in terms of accuracy and cost

behavior near hyperbolic stationary solutions for partial functional differential equations with infinite delay abstract the aim of this work is to investigate the asymptotic behavior of solutions near hyperbolic stationary solutions for partial functional differential equations with infinite delay we suppose that the linear part satisfies the hille yosida condition on a banach space and it is not necessarily densely defined firstly we establish a new variation of constants formula for the nonhomogeneous linear equations secondly we use this formula and the spectral decomposition of the phase space to show the existence of stable and unstable manifolds the estimations of solutions on these manifolds are obtained for illustration we propose to study the stability of stationary solutions for the lotka volterra model with diffusion

random walks totally unimodular matrices and a randomised dual simplex algorithm we discuss the application of random walks to generating a random basis of a totally unimodular matrix and to solving a linear program with such a constraint matrix we also derive polynomial upper bounds on the combinatorial diameter of an associated polyhedron

pseudo s asymptotically periodic solutions of second order abstract cauchy problems in this work we discuss the existence of pseudo s asymptotically periodic mild solutions for a second order abstract cauchy problem in particular we show that pseudo s asymptotically periodic cosine functions of operators are periodic the paper is completed with an application to the nonautonomous wave equation

non autonomous nonlinear integro differential equations with infinite delay abstract in this paper we consider a non autonomous abstract nonlinear volterra integro differential equation with infinite delay in a banach space we establish the existence and uniqueness of a strict solution under a certain local lipschitz condition on the nonlinear map and an integrability condition on the kernel

linear programming the simplex algorithm and simple polytopes in the first part of the paper we survey some far reaching applications of the basic facts of linear programming to the combinatorial theory of simple polytopes in the second part we discuss some recent developments concerning the simplex algorithm we describe subexponential randomized pivot rules and upper bounds on the diameter of graphs of polytopes 1997 the mathematical programming society inc published by elsevier science b v

second order nonlinear impulsive integro differential equations of mixed type with time varying generating operators and optimal controls on banach spaces in this paper a class of second order nonlinear impulsive integro differential equations of mixed type whose principle is time varying generating operators with unbounded perturbation on banach spaces is considered discussing the perturbation of time varying operator matrix and constructing corresponding the evolution system generated by operator matrix we introduce the reasonable mild solution of second order nonlinear impulsive integro differential equations of mixed type and prove the existence of mild solutions the existence of optimal controls for a lagrange problem of systems governed by the second order nonlinear impulsive integro equations of mixed type is also presented an example is given for demonstration

existence and controllability results for second order impulsive stochastic evolution systems with state dependent delay in this paper we discuss a kind of impulsive second order stochastic evolution systems with state dependent delay in a real separable hilbert space the results concerning the existence and controllability of mild solutions have been addressed by means of the fixed point techniques some sufficient conditions are formulated as well as an application involving partial differential equation with impulses is presented

minkowski s convex body theorem and integer programming the paper presents an algorithm for solving integer programming problems whose running time depends on the number n of variables as non this is done by reducing an n variable problem to 2n5i 2 problems in n i variables for some i greater than zero chosen by the algorithm the factor of on5 2 per variable improves the best previously known factor which is exponential in n minkowski s convex body theorem and other results from geometry of numbers play a crucial role in the algorithm several related algorithms for lattice problems are presented the complexity of these problems with respect to polynomial time reducibilities is studied

error indicators for adaptive mesh refinement in the boundary element method a new approach in this paper a new approach to the development of a posteriori error indicators for the direct boundary element method bem is presented it takes advantage of a unique opportunity offered by the collocation bem that is the possibility of obtaining more than one numerical solution from a given boundary element mesh unlike other approaches the error indicator introduced in the present paper does not require any explicit evaluation of the residual of the approximate solution in order to test the capability of the proposed error indicators to actually control a selective mesh refinement process an h adaptive refinement strategy is also described together with the criteria for refinement and for the termination of the adaptive process full details on the whole numerical implementation for two dimensional problems are provided in particular there is described a direct method for the numerical evaluation of cauchy principal value integrals numerical examples show that pointwise convergence is always obtained in a few steps and the final meshes look nearly as if they were designed by an expert analyst

adaptive boundary element methods for some first kind integral equations in this paper we present an adaptive boundary element method for the boundary integral equations of the first kind concerning the dirichlet problem and the neumann problem for the laplacian in a two dimensional lipschitz domain for the h version of the finite element galerkin discretization of the single layer potential and the hypersingular operator we derive a posteriori error estimates which guarantee a given bound for the error in the energy norm up to a multiplicative constant following eriksson and johnson this yields adaptive algorithms steering the mesh refinement numerical examples confirm that our adaptive algorithms yield automatically the expected convergence rate

a subexponential algorithm for abstract optimization problems an abstract optimization problem aop is a triple h phi where h is a finite set a total order on 2 h and phi an oracle that for given f subseteq g subseteq h either reports that f min f mid f subseteq g or returns a set f subseteq g with f e 2 sqrt n o sqrt 4 n ln n oracle calls n h in contrast any deterministic algorithm needs to make 2 n 1 oracle calls in the worst case the algorithm is applied to the problem of finding the distance between two n vertex or n facet convex polyhedra in d space and to the computation of the smallest ball containing n points in d space for both problems we give the first subexponential bounds in the arithmetic model of computation

the many facets of linear programming we examine the history of linear programming from computational geometric and complexity points of view looking at simplex ellipsoid interior point and other methods

variation of constants formula and almost periodic solutions for some partial functional differential equations with infinite delay abstract in this work we give a variation of constants formula for partial functional differential equations with infinite delay we assume that the linear part is not necessarily densely defined and its resolvent operator satisfies the hille yosida condition we establish a reduction of the problem to a finite dimensional space which allows us to prove the existence of almost periodic solutions

on a multidimensional search technique and its application to the euclidean one centre problem the paper is divided into two main sections the first deals with a multidimensional search technique of megiddo j assoc comput mach 31 1984 pp 114 127 and suggests an improvement the second gives an application of the technique to the euclidean one centre problem in mathbb r d an algorithm of time complexity o 3 d 2 2 n is derived for this problem this improves the best previous bound even in the case d 2