a canonical form for pencils of matrices with applications to asymptotic linear programs abstract this paper studies asymptotic pencils of linear programs in which the constraint matrix right hand sides and objective function coefficients depend linearly on a parameter and we seek a basis that is optimal for all large or small enough values of the parameter the asymptotic linear programming formulation of discounted markov decision chains has this form as do many other problems for this problem we introduce a new canonical form for bases that facilitates computation of laurent expansions of the solutions of the resulting asymptotic linear programs and show how to implement the simplex method efficiently in this setting we show that the computational work to solve the asymptotic linear program is the product of the number of rows of the problem and the work required to solve an ordinary linear program of the same size this improves computational efficiency of the asymptotic simplex method of jeroslow 1973 and a related algorithm considered by lamond 1989 in essence this work generalizes that of miller and veinott 1969 and rothblum 1974 1975 for markov ordinary and markov branching decision chains to arbitrary linear programs

on the matrix equation x a x na i the nonlinear matrix equation x a x na i is discussed n is a positive integer convergence of the iterative methods proposed by el sayed comput math appl 41 2001 579 588 is proved under weaker restrictions for the matrix a and an arbitrary integer n

multiplicative markov decision chains previous treatments of multiplicative markov decision chains eg bellman bellman r 1957 dynamic programming princeton university press princeton new jersey mandl mandl p 1967 an iterative method for maximizing the characteristic root of positive matrices rev roumaine math pures appl xii 1317 1322 and howard and matheson howard r a matheson j e 1972 risk sensitive markov decision processes management sci 8 356 369 restricted attention to stationary policies and assumed that all transition matrices are irreducible and aperiodic they also used a first term optimality criterion namely maximizing the spectral radius of the associated transition matrix we give a constructive proof of the existence of optimal policies among all policies under new cumulative average optimality criteria which are more sensitive than the maximization of the spectral radius the algorithm for finding an optimal policy first searches for a stationary policy with a nonnilpotent transition matrix provided such a rule exists otherwise the method still finds an optimal policy though in this case the set of optimal policies usually does not contain a stationary policy if a stationary policy with a nonnilpotent transition matrix exists then we develop a policy improvement algorithm which finds a stationary optimal policy

boundary layers on sobolev besov spaces and poisson s equation for the laplacian in lipschitz domains we study inhomogeneous boundary value problems for the laplacian in arbitrary lipschitz domains with data in sobolev besov spaces as such this is a natural continuation of work in jerison and kenig j funct anal 1995 16 219 where the inhomogeneous dirichlet problem is treated via harmonic measure techniques the novelty of our approach resides in the systematic use of boundary integral methods in this regard the key results are establishing the invertibility of the classical layer potential operators on scales of sobolev besov spaces on lipschitz boundaries for optimal ranges of indices applications tolp based helmholtz type decompositions of vector fields in lipschitz domains are also presented

an improved algorithm for the computation of kronecker s canonical form of a singular pencil we give an 0 t algorithm for computing the kronecker structure of an arbitrary m x n pencil xe a the algorithm is shown to he numerically stable because only unitary transformations are used the improved speed over earlier unitary methods is due to the efficient use of condensed forms which are maintained throughout the recursions of the algorithm

complex interpolation of some quasi banach spaces abstract we interpolate in the complex method some real intermediate quasi banach spaces this enables us for example to get in a unified way complex interpolation of h p spaces 0 p 0 p 1 h p spaces could be the standard ones as well as weighted h p spaces h p spaces on product domains etc

on the existence of a positive definite solution of the matrix equation x atx 1a i the question is raised under which conditions on the real square matrix a the matrix equation x atx 1a i has a real symmetric positive definite solution x both necessary and sufficient solvability conditions on a are derived moreover we give an algorithm to calculate the solution for a number of special cases we also present an analytic solution

a theory of complex interpolation for families of banach spaces abstract a detailed development is given of a theory of complex interpolation for families of banach spaces which extends the well known theory for pairs of spaces

a new inversion free iteration for solving the equation x a x 1 a q in this paper we introduce a new inversion free variant of the basic fixed point iteration method for obtaining a maximal positive definite solution of the nonlinear matrix equation x a x 1a q it is more accurate than zhan s algorithm j sci comput 17 1996 1167 and has less number of operations than the algorithm of guo and lancaster math comput 68 1999 1589 we derive convergence conditions of the iteration and existence conditions of a solution to the problem finally we give some numerical results to illustrate the behavior of the considered algorithm

on positive definite solutions of the nonlinear matrix equations x a x q a q in this paper we investigate the nonlinear matrix equations x a x q a q 0 q 1 and x a x q a q q 1 necessary and sufficient conditions for the unique existence of hermitian positive definite solutions of these equations are derived effective iterative algorithms are also provided to obtain the unique solution of x a x q a q 0 q 1 and the minimal and maximal hermitian positive definite solutions of x a x q a q q 1 numerical examples are included to illustrate the efficiency of the presented iteration algorithms

spectral theory and complex interpolation abstract some relations between the spectral theory for linear operators and the complex theory of interpolation for families of banach spaces are studied an invertibility theorem is proven and size estimates for the spectra of operators on the interpolation spaces are given

notes on limits of sobolev spaces and the continuity of interpolation scales we extend lemmas by bourgain brezis mironescu 2001 and maz ya shaposhnikova 2002 on limits of sobolev spaces to the setting of interpolation scales this is achieved by means of establishing the continuity of real and complex interpolation scales at the end points a connection to extrapolation theory is developed and a new application to limits of sobolev scales is obtained we also give a new approach to the problem of how to recognize constant functions via sobolev conditions

the laurent expansion of pencils that are singular at the origin abstract this paper gives two new proofs of a theorem of langenhop on the laurent expansion of a matrix pencil that is singular at the origin the second based on a decomposition of van dooren leads to a computational algorithm

a discrete transform and decompositions of distribution spaces abstract we study a representation formula of the form q q q for a distribution on r n this formula is obtained by discretizing and localizing a standard littlewood paley decomposition the map taking to the sequence q q with q running over the dyadic cubes in r n is called the transform the functions q and q have a particularly simple form moreover most of the familiar distribution spaces lp spaces 1 r n we also consider pointwise multipliers for the characteristic function of a domain we obtain boundedness results for a general class of domains which properly includes lipschitz domains several interpolation methods are easily analyzed via the sequence spaces for real interpolation we obtain among other things an extension to the case p 0 this in turn gives a new approach to the traditional atomic decomposition of hardy spaces

perturbation analysis of the hermitian positive definite solution of the matrix equation x a x 2a i abstract consider the nonlinear matrix equation x a x 2a i where a is an n n complex matrix i the identity matrix and a the conjugate transpose of a matrix a in this paper it is proved that this matrix equation has a unique hermitian positive definite solution provided a 2

hermitian solutions of the equation x q nx 1n we consider the matrix equation x q nx 1n its hermitian solutions are parametrized in terms of the generalized lagrangian eigenspaces of a certain matrix pencil we show that the equation admits both a largest and a smallest solution the largest solution corresponds to the unique positive definite solution the smallest solution is the unique negative definite solution if and only if n is nonsingular if n is singular no negative definite solution exists an interesting relation between the given equation and a standard algebraic riccati equation of kalman filtering theory is also obtained finally we present an algorithm which converges to the positive definite solution for a wide range of initial conditions

interpolation between hp spaces the complex method the corollary follows immediately from theorem 1 and fefferman s theorem see 111 that bmo is the dual space of h before proceeding to further results we make some comments on the history of theorem 1 unfortunately the theorem h lm e lp l p 1 0 has been folklore for many years and is stated e g in 2 221 this is due to a small but inopportune typographical error in 111 it is stated on page 157 of that paper that h lp lq l q 1 8 e p 1 p co

abstract capacitary estimates and the completeness and separability of certain classes of non locally convex topological vector spaces abstract we are concerned with establishing completeness and separability criteria for large classes of topological vector spaces which are typically non locally convex including lebesgue like spaces lorentz spaces orlicz spaces mixed normed spaces tent spaces and discrete triebel lizorkin and besov spaces for vector spaces of measurable functions we also derive pointwise convergence results our approach relies on abstract capacitary estimates and works in certain cases of interest even in the absence of a background measure space and or of a vector space structure

iterative solution of two matrix equations we study iterative methods for finding the maximal hermitian positive definite solutions of the matrix equations x a x 1 a q and x a x 1 a q where q is hermitian positive definite general convergence results are given for the basic fixed point iteration for both equations newton s method and inversion free variants of the basic fixed point iteration are discussed in some detail for the first equation numerical results are reported to illustrate the convergence behaviour of various algorithms

stability results on interpolation scales of quasi banach spaces and applications we investigate the stability of fredholm properties on interpolation scales of quasi banach spaces this analysis is motivated by problems arising in pde s and several applications are presented

perturbation and stability theory for markov control problems a unified approach to the asymptotic analysis of a markov decision process disturbed by an epsilon additive perturbation is proposed irrespective of whether the perturbation is regular or singular the underlying control problem that needs to be understood is the limit markov control problem the properties of this problem are studied

condition numbers for the nonlinear matrix equation and their statistical estimation abstract in this paper we consider four different types of condition numbers i e two normwise ones mixed and componentwise ones for the nonlinear matrix equation x a f x a q and derive their explicit expressions using the probabilistic spectral norm estimator and the statistical condition estimation method these condition numbers can be estimated with high reliability two numerical examples are provided to illustrate the difference between the normwise condition numbers and the mixed and componentwise condition numbers and the efficiency of the estimators respectively

contractive maps on normed linear spaces and their applications to nonlinear matrix equations in this paper we will give necessary and sufficient conditions under which a map is a contraction on a certain subset of a normed linear space these conditions are already well known for maps on intervals in r using the conditions and banach s fixed point theorem we can prove a fixed point theorem for operators on a normed linear space the fixed point theorem will be applied to the matrix equation x in a f x a where f is a map on the set of positive definite matrices induced by a real valued map on 0 this will give conditions on a and f under which the equation has a unique solution in a certain set we will consider two examples of f in detail in one example the application of the fixed point theorem is the first step in proving that the equation has a unique positive definite solution under the conditions on a

two iteration processes for computing positive definite solutions of the equation x a x na q abstract in the present paper we suggest two iteration methods for obtaining positive definite solutions of nonlinear matrix equation x a x n a q for the integer n 1 we obtain sufficient conditions for existence of the solutions for the matrix equation finally some numerical examples to illustrate the effectiveness of the algorithms and some remarks

a generalized inverse method for asymptotic linear programming consider a linear program in which the entries of the coefficient matrix vary linearly with time to study the behavior of optimal solutions as time goes to infinity it is convenient to express the inverse of the basis matrix as a series expansion of powers of the time parameter we show that an algorithm of wilkinson 1982 for solving singular differential equations can be used to obtain such an expansion efficiently the resolvent expansions of dynamic programming are a special case of this method

the asymptotic optimal partition and extensions of the nonsubstitution theorem the data describing an asymptotic linear program relies on a single parameter usually referred to as time and unlike parametric linear programming asymptotic linear programming is concerned with the steady state behavior as time increases to infinity the fundamental result of this work shows that the optimal partition of an asymptotic linear program attains a steady state for a large class of functions consequently this allows us to define an asymptotic center solution we show that this solution inherits the analytic properties of the functions used to describe the feasible region moreover our results allow significant extensions of an economics result known as the nonsubstitution theorem

an efficient basis update for asymptotic linear programming abstract for a linear program in which the constraint coefficients vary linearly with the time parameter we showed in a previous paper that a basic feasible solution can be evaluated using o k 1 m 3 arithmetic operations where m is the number of constraints and k is the index of the basis matrix pair here we show in the special case when k 1 for all basis matrix pairs and when one of the matrices in each pair has nearly full rank how the possibly singular matrix factorization can be updated with only o m 2 operations using rank one update techniques this makes the arithmetic complexity of updating the basis in asymptotic linear programming comparable to that of updating the inverse in ordinary linear programming in this case moreover we show that the result holds in particular when computing a blackwell optimal policy for markov decision chains in the unichain case or when all policies have only a small number of recurrent subchains

on the matrix equation x atx 1a i simple necessary and sufficient conditions for the matrix equation x atx 1a i to have a positive definite solution are given these conditions are used to explain some properties of the equation and to obtain new information finally we prove a conjecture on the extremal solutions

two structure preserving doubling like algorithms for obtaining the positive definite solution to a class of nonlinear matrix equation in this paper we present two structure preserving doubling like algorithms for obtaining the positive definite solution of the nonlinear matrix equation x a h x 1 a q where x c n i n is an unknown matrix and q c n i n is a hermitian positive definite matrix we prove that the sequences generated by the algorithms converge to the positive definite solution of the considered matrix equation r quadratically in addition we also present some numerical results to illustrate the behavior of the considered algorithm

hardy spaces and the neumann problem in l p for laplace s equation in lipschitz domains on donne des resultats optimaux pour la resolubilite du probleme de neumann dans des domaines de lipschitz a donnees dans l p on obtient des resultats du point extreme correspondant pour des espaces de hardy

properties of positive definite solutions of the equation x a x 2a i abstract in this paper we discuss some properties of a positive definite solution of the matrix equation x a x 2 a i two effective iterative methods for computing a positive definite solution of this equation are proposed necessary and sufficient conditions for existence of a positive definite solution are derived numerical experiments are executed with these methods