on newton s method and halley s method for the principal pth root of a matrix if a is a matrix with no negative real eigenvalues and all zero eigenvalues of a are semisimple the principal pth root of a can be computed by newton s method or halley s method with a preprocessing procedure if necessary we prove a new convergence result for newton s method and discover an interesting property of newton s method and halley s method in terms of series expansions we explain how the convergence of newton s method and halley s method can be improved when the eigenvalues of a are known or when a is a singular matrix we also prove new results on pth roots of m matrices and h matrices and consider the application of newton s method and halley s method to find the principal pth roots of these special matrices ams classification 65f30 41a58 15a48

reductions of minimal lagrangian submanifolds with symmetries let m be a fano manifold equipped with a kahler form omega in 2 pi c 1 m and k a connected compact lie group acting on m as holomorphic isometries in this paper we show the minimality of a k invariant lagrangian submanifold l in m with respect to a globally conformal kahler metric is equivalent to the minimality of the reduced lagrangian submanifold l 0 l k in a kahler quotient m 0 with respect to the hsiang lawson metric furthermore we give some examples of kahler reductions by using a circle action obtained from a cohomogenenity one action on a kahler einstein manifold of positive ricci curvature applying these results we obtain several examples of minimal lagrangian submanifolds via reductions

computing the frechet derivative of the matrix logarithm and estimating the condition number the most popular method for computing the matrix logarithm is the inverse scaling and squaring method which is the basis of the recent algorithm of al mohy and higham siam j sci comput 34 2012 pp c152 c169 for real matrices we develop a version of the latter algorithm that works entirely in real arithmetic and is twice as fast as and more accurate than the original algorithm we show that by differentiating the algorithms we obtain backward stable algorithms for computing the frechet derivative we demonstrate experimentally that our two algorithms are more accurate and efficient than existing algorithms for computing the frechet derivative and we also show how the algorithms can be used to produce reliable estimates of the condition number of the matrix logarithm

contraction conditions for average and α discount optimality in countable state markov games with unbounded rewards the goal of this paper is to provide a theory of n person markov games with unbounded cost for a countable state space and compact action spaces we investigate both the finite and infinite horizon problems for the latter we consider the discounted cost as well as the expected average cost we present conditions for the infinite horizon problems for which equilibrium policies exist for all players within the stationary policies and show that the costs in equilibrium policies exist for all players within the stationary policies and show that the costs in equilibrium satisfy the optimality equations similar results are obtained for the finite horizon costs for which equilibrium policies are shown to exist for all players within the markov policies as special case of n person games we investigate the zero sum 2 players game for which we establish the convergence of the value iteration algorithm we conclude by studying an application of a zero sum markov game in a queueing model

the complexity of divisibility we address two sets of long standing open questions in linear algebra and probability theory from a computational complexity perspective stochastic matrix divisibility and divisibility and decomposability of probability distributions we prove that finite divisibility of stochastic matrices is an np complete problem and extend this result to nonnegative matrices and completely positive trace preserving maps i e the quantum analogue of stochastic matrices we further prove a complexity hierarchy for the divisibility and decomposability of probability distributions showing that finite distribution divisibility is in p but decomposability is np hard for the former we give an explicit polynomial time algorithm all results on distributions extend to weak membership formulations proving that the complexity of these problems is robust to perturbations

on lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres the n dimensional complex hyperquadric is a compact complex algebraic hypersurface defined by the quadratic equation in the n 1 dimensional complex projective space which is isometric to the real grassmann manifold of oriented 2 planes and is a compact hermitian symmetric space of rank 2 in this paper we study geometry of compact lagrangian submanifolds in complex hyperquadrics from the viewpoint of the theory of isoparametric hypersurfaces in spheres from this viewpoint we provide a classification theorem of compact homogeneous lagrangian submanifolds in complex hyperquadrics by using the moment map technique moreover we determine the hamiltonian stability of compact minimal lagrangian submanifolds embedded in complex hyperquadrics which are obtained as gauss images of isoparametric hypersurfaces in spheres with g 1 2 3 distinct principal curvatures

third degree anomalies of traub s method traub s method is a tough competitor of newton s scheme for solving nonlinear equations as well as nonlinear systems due to its third order convergence and its low computational cost it is a good procedure to be applied on complicated multidimensional problems in order to better understand its behavior the stability of the method is analyzed on cubic polynomials showing the existence of very small regions with unstable behavior finally the performance of the method on cubic matrix equations arising in control theory is presented showing a good performance

the matrix sign function a survey of the matrix sign function is presented including some historical background definitions and properties approximation theory and computational methods and condition theory and estimation procedures applications to areas such as control theory eigendecompositions and roots of matrices are outlined and some new theoretical results are also given

second variation and stabilities of minimal lagrangian submanifolds in kahler manifolds first we derive a new second variation formula which holds for any minimal lagrangian submanifold in k ihler manifolds using this we prove that any minimal lagrangian submanifold in kfihler manifolds with nonpositive chern class is stable we also prove that in kfihler manifolds with positive chern class any stable minimal lagrangian submanifold l satisfies that h l 0 next we introduce the notion of hamiltonian stability either local or global of minimal lagrangian submanifolds in einstein kfihler manifolds using our second variation formula we find a general criterion for the local hamiltonian stability for minimal lagrangian submanifolds in einstein k ihler manifolds this criterion is satisfied for all real forms which are einstein of irreducible hermitian symmetric spaces e g the cannonical rp c cp it is also satisfied by the clifford torus t c cp

a taylor expansion of the square root matrix functional abstract this short note provides an explicit description of the frechet derivatives of the principal square root matrix function at any order we present an original formulation that allows to compute sequentially the frechet derivatives of the matrix square root at any order starting from the first order derivative a taylor expansion at any order with an integral remainder term is also provided yielding the first result of this type for this class of matrix function

the computation of the principal square roots of centrosymmetric h matrices in this paper two algorithms for computing the principal square root of a centrosymmetric h matrix with positive diagonal entries are proposed it is showed that our algorithms ensure significant savings in computational costs as compared to the case of an arbitrary h matrix a with positive diagonal entries

a nonzero sum stochastic differential game in the orthant we study a nonzero sum stochastic differential game where the state is a controlled reflecting diffusion in the nonnegative orthant under certain conditions we establish the existence of nash equilibria in stationary strategies for both discounted and average payoff criteria

roots of real matrices abstract an algorithm for computing the roots of a matrix with real elements and the real part of the eigenvalues not zero will be described the algorithm is based on the sign matrix given in an earlier paper and the projectors of the matrix

existence of correlated equilibria an elementary proof based on linear duality is provided for the existence of correlated equilibria in finite games the existence result is then extended to infinite games including some that possess no nash equilibria

considerations on computing real logarithms of matrices hamiltonian logarithms and skew symmetric logarithms abstract the issue of computing a real logarithm of a real matrix is addressed after a brief review of some known methods more attention is paid to three 1 pade approximation techniques 2 newton s method and 3 a series expansion method newton s method has not been previously treated in the literature we address commutativity issues and simplify the algorithmic formulation we also address general structure preserving issues for two applications in which we are interested finding the real hamiltonian logarithm of a symplectic matrix and finding the skew symmetric logarithm of an orthogonal matrix the diagonal pade approximants and the proposed series expansion technique are proven to be structure preserving some algorithmic issues are discussed

perturbation bounds for matrix square roots and pythagorean sums abstract perturbation estimates for the square root r a 1 2 and pythagorean sum p i a 2 1 2 of complex matrices are proved we present bounds in the spectral norm for the cases that r and p are accretive i e have positive definite hermitian parts with special attention to the case of a large condition of a the results are based on bounds for the solution of the matrix sylvester equation and for the separation of two matrices

j orthogonal matrices properties and generation a real square matrix q is j orthogonal if qtjq j where the signature matrix j diag pm 1 j orthogonal matrices arise in the analysis and numerical solution of various matrix problems involving indefinite inner products including in particular the downdating of cholesky factorizations we present techniques and tools useful in the analysis application and construction of these matrices giving a self contained treatment that provides new insights first we define and explore the properties of the exchange operator which maps j orthogonal matrices to orthogonal matrices and vice versa then we show how the exchange operator can be used to obtain a hyperbolic cs decomposition of a j orthogonal matrix directly from the usual cs decomposition of an orthogonal matrix we employ the decomposition to derive an algorithm for constructing random j orthogonal matrices with specified norm and condition number we also give a short proof of the fact that j orthogonal matrices are optimally scaled und

existence of stationary correlated equilibria with symmetric information for discounted stochastic games the following theorem is proved every nonzero sum discounted stochastic game in countably generated measurable state space with compact metric action spaces admits a stationary correlated equilibrium point with symmetric public information whenever the immediate rewards and transition densities are measurable with respect to the state variable and continuous with respect to joint actions

computing the square roots of matrices with central symmetry for computing square roots of a nonsingular matrix a which are functions of a two well known fast and stable algorithms which are based on the schur decomposition of a were proposed by bjork and hammarling a bjork s hammarling a schur method for the square root of a matrix linear alg appl 52 53 1983 127 140 for square roots of general complex matrices and by higham n j higham computing real square roots of a real matrix linear alg appl 88 89 1987 405 430 for real square roots of real matrices in this paper we further consider the computation of the square roots of matrices with central symmetry we first investigate the structure of the square roots of these matrices and then develop several algorithms for computing the square roots we show that our algorithms ensure significant savings in computational costs as compared to the use of standard algorithms for arbitrary matrices

on the positive definite solutions of the matrix equations xs atx ta in abstract the two matrix equations x s a t x t a i n and x s a t x t a i n are studied based on the fixed point theory the existence of the symmetric positive definite solutions are proved sensitivity analysis of the maximal solution is presented some elegant estimates of the positive definite solutions are obtained three iterative methods for computing the positive solutions are proposed

hamiltonian stability of the gauss images of homogeneous isoparametric hypersurfaces i the image of the gauss map of any oriented isopara metric hypersurface of the unit standard sphere s n 1 1 is a mini mal lagrangian submanifold in the complex hyperquadric qn c in this paper we determine the hamiltonian stability of all com pact minimal lagrangian submanifolds embedded in complex hy perquadrics which are obtained as the images of the gauss map of homogeneous isoparametric hypersurfaces in the unit spheres

efficient implementation of second order stochastic approximation algorithms in high dimensional problems stochastic approximation sa algorithms have been widely applied in minimization problems when the loss functions and or the gradient information are only accessible through noisy evaluations stochastic gradient sg descent a first order algorithm and a workhorse of much machine learning is perhaps the most famous form of sa among all sa algorithms the second order simultaneous perturbation stochastic approximation 2spsa and the second order stochastic gradient 2sg are particularly efficient in handling high dimensional problems covering both gradient free and gradient based scenarios however due to the necessary matrix operations the per iteration floating point operations flops cost of the standard 2spsa 2sg is o p 3 where p is the dimension of the underlying parameter note that the o p 3 flops cost is distinct from the classical spsa based per iteration o 1 cost in terms of the number of noisy function evaluations in this work we propose a technique to efficiently implement the 2spsa 2sg algorithms via the symmetric indefinite matrix factorization and show that the flops cost is reduced from o p 3 to o p 2 the formal almost sure convergence and rate of convergence for the newly proposed approach are directly inherited from the standard 2spsa 2sg the improvement in efficiency and numerical stability is demonstrated in two numerical studies

a survey of condition number estimation for triangular matrices we survey and compare a wide variety of techniques for estimating the condition number of a triangular matrix and make recommendations concerning the use of the estimates in applications each of the methods is shown to bound the condition number the bounds can broadly be categorised as upper bounds from matrix theory and lower bounds from heuristic or probabilistic algorithms for each bound we examine by how much at worst it can overestimate or underestimate the condition number numerical experiments are presented in order to illustrate and compare the practical performance of the condition estimators

hamiltonian square roots of skew hamiltonian matrices abstract we present a constructive existence proof that every real skew hamiltonian matrix w has a real hamiltonian square root the key step in this construction shows how one may bring any such w into a real quasi jordan canonical form via symplectic similarity we show further that every w has infinitely many real hamiltonian square roots and give a lower bound on the dimension of the set of all such square roots some extensions to complex matrices are also presented

a weak local linearization scheme for stochastic differential equations with multiplicative noise abstract in this paper a weak local linearization scheme for stochastic differential equations sdes with multiplicative noise is introduced first for a time discretization the solution of the sde is locally approximated by the solution of the piecewise linear sde that results from the local linearization strategy the weak numerical scheme is then defined as a sequence of random vectors whose first moments coincide with those of the piecewise linear sde on the time discretization the scheme is explicit preserves the first two moments of the solution of sdes with linear drift and diffusion coefficients in state and time and inherits the mean square stability or instability that such solution may have the rate of convergence is derived and numerical simulations are presented for illustrating the performance of the scheme

the computation of the square roots of circulant matrices abstract in the first part of this paper we investigate the reduced forms of circulant matrices and quasi skew circulant matrices by using their properties we present two efficient algorithms to compute the square roots of circulant matrices and quasi skew circulant matrices respectively those methods are faster than the traditional algorithm which is based on the schur decomposition in the second part we further consider circulant h matrices with positive diagonal entries and develop two algorithms for computing their principal square roots those two algorithms have the common advantage that is they only need matrix matrix multiplications in their iterative sequences an operation which can be done very efficiently on modern high performance computers

the existence of subgame perfect equilibrium in continuous games with almost perfect information a case for public randomization the starting point of this paper is a simple regular dynamic game in which subgame perfect equilibrium fails to exist examination of this example shows that existence would be restored if players were allowed to observe the output of a public randomization device the main result of the paper shows that the introduction of public randomization yields existence not only in the example but also in a large class of dynamic games it is also argued that the introduction of public randomization is the minimal robust extension of subgame perfect equilibrium in this class of games

selection of a correlated equilibrium in markov stopping games abstract this paper deals with an extension of the concept of correlated strategies to markov stopping games the nash equilibrium approach to solving nonzero sum stopping games may give multiple solutions an arbitrator can suggest to each player the decision to be applied at each stage based on a joint distribution over the players decisions according to some optimality criterion this is a form of equilibrium selection examples of correlated equilibria in nonzero sum games related to the best choice problem are given several concepts of criteria for selecting a correlated equilibrium are used

fast random field generation with h matrices we use the h matrix technology to compute the approximate square root of a covariance matrix in linear cost this allows us to generate normal and log normal random fields on general point sets with optimal cost we derive rigorous error estimates which show convergence of the method our approach requires only mild assumptions on the covariance function and on the point set therefore it might be also a nice alternative to the circulant embedding approach which applies only to regular grids and stationary covariance functions

the method of normalized correlations a fast parameter estimation method for random processes and isotropic random fields that focuses on short range dependence we focus on the inference of covariance parameters from spatial or temporal data using the method of normalized correlations monc we compare the monc with maximum likelihood estimation and weighted least squares using simulated data as well as gridded and scattered real data for the synthetic experiments none of the methods considered performed uniformly better than the others in terms of estimation accuracy and precision however for regularly spaced data the monc was significantly faster and practically insensitive to sample size for scattered data the monc s speed was reduced but still competitive potential applications of the monc include those involving large data sets and fast preconditioning

properties of a quadratic matrix equation and the solution of the continuous time algebraic riccati equation abstract we discuss some properties of a quadratic matrix equation with some restrictions then use these results on the algebraic riccati equation to get a new algorithm the algorithm sufficiently takes account of the structure of the associated matrix hence it is very effective

newton s method for the matrix square root one approach to computing a square root of a matrix a is to apply newton s method to the quadratic matrix equation f x x2 a 0 two widely quoted matrix square root iterations obtained by rewriting this newton iteration are shown to have excellent mathematical convergence properties however by means of a perturbation analysis and supportive numerical examples it is shown that these simplified iterations are numerically unstable a further variant of newton s method for the matrix square root recently proposed in the literature is shown to be for practical purposes numerically stable

a structured condition number for self adjoint polynomial matrix equations with applications in linear control abstract based on the classic definition of condition number a structured condition number is proposed for a class of self adjoint polynomial matrix equations the explicit formula of the structured condition number is derived with applying newly defined linear operators the structured condition number can be applied to some important polynomial matrix equations including the continuous time algebraic riccati equation care the discrete time algebraic lyapunov equation dale etc compared with the state of the art condition numbers for care and dale the newly proposed structured condition number can measure the sensitivity of the solution better which is validated by numerical examples

computing real square roots of a real matrix bjorck and hammarling 1 describe a fast stable schur method for computing a square root x of a matrix a x2 a we present an extension of their method which enables real arithmetic to be used throughout when computing a real square root of a real matrix for a nonsingular real matrix a conditions are given for the existence of a real square root and for the existence of a real square root which is a polynomial in a the number of square roots of the latter type is determined the conditioning of matrix square roots is investigated and an algorithm is given for the computation of a well conditioned square root

a schur method for the square root of a matrix abstract a fast and stable method for computing the square root x of a given matrix a x 2 a is developed the method is based on the schur factorization a qsq h and uses a fast recursion to compute the upper triangular square root of s it is shown that if x 2 a is not large then the computed square root is the exact square root of a matrix close to a the method is extended for computing the cube root of a matrices exist for which the square root computed by the schur method is ill conditioned but which nonetheless have well conditioned square roots an optimization approach is suggested for computing the well conditioned square roots in these cases

a parallel algorithm for principal n th roots of matrices an iterative algorithm for computing the principal nth root of a positive definite matrix is presented the algorithm is based on the gauss legendre approximation of a definite integral we present a parallelization in which we use as many processors as the order of the approximation an analysis of the error introduced at each step of the iteration indicates that the algorithm converges more rapidly as the order of the approximation and thus the number of processors increases we describe the results of our implementation of an eight processor meiko cs 2 comparing the parallel algorithm to the fastest sequential algorithm which is the hoskins walton method

the matrix sign decomposition and its relation to the polar decomposition the sign function of a square matrix was introduced by roberts in 1971 we show that it is useful to regard s sign a as being part of a matrix sign decomposition a sn where n a2 1 2 this decomposition leads to the new representation sign a a a2 1 2 most results for the matrix sign decomposition have a counterpart for the polar decomposition a uh and vice versa to illustrate this we derive best approximation properties of the factors u h and s determine bounds for a s and a u and describe integral formulas for s and u we also derive explicit expressions for the condition numbers of the factors s and n an important equation expresses the sign of a block 2 2 matrix involving a in terms of the polar factor u of a we apply this equation to a family of iterations for computing s by pandey kenney and laub to obtain a new family of iterations for computing u the iterations have some attractive properties including suitability for parallel computation

the first eigenvalue of the laplacian of an isoparametric minimal hypersurface in a unit sphere soit m n 1 une variete de riemann connexe compacte a n 1 dimensions sans bord et son laplacien agissant sur des fonctions lisses sur m a un spectre discret 0 0 m 1 m 2 m k m x soit f une immersion minimale de m n 1 dans la sphere unite a n dimensions s n 1 de l espace euclidien on etudie la premiere valeur propre du laplacien d une hypersurface isoparametrique minimale fermee dans s n 1

an efficient algorithm for the discrete time algebraic riccati equation the authors develop an algorithm to solve the standard discrete time algebraic riccati equation by using a skew hamiltonian transformation and the square root method the algorithm is structure preserving and efficient because the hamiltonian structure is fully exploited and only orthogonal transformations are used the efficiency and stability of the algorithm are analyzed numerical examples are included

on square roots of m matrices abstract the question of the existence and uniqueness of an m matrix which is a square root of an m matrix is discussed the results are then used to derive some new necessary and sufficient conditions for a real matrix with nonpositive off diagonal elements to be an m matrix

optimal finite difference grids and rational approximations of the square root i elliptic problems the main objective of this paper is optimization of second order finite difference schemes for elliptic equations in particular for equations with singular solutions and exterior problems a model problem corresponding to the laplace equation on a semi infinite strip is considered the boundary impedance neumann to dirichlet map is computed as the square root of an operator using the standard three point finite difference scheme with optimally chosen variable steps the finite difference approximation of the boundary impedance for data of given smoothness is the problem of rational approximation of the square root on the operator s spectrum we have implemented zolotarev s optimal rational approx imant obtained in terms of elliptic functions we have also found that a geometrical progression of the grid steps with optimally chosen parameters is almost as good as the optimal approximant for bounded operators it increases from second to exponential the convergence order of the finite difference impedance with the convergence rate proportional to the inverse of the logarithm of the condition number for the case of unbounded operators in sobolev spaces associated with elliptic equations the error decays as the exponential of the square root of the mesh dimension as an example we numerically compute the green function on the boundary for the laplace equation some features of the optimal grid obtained for the laplace equation remain valid for more general elliptic problems with variable coefficients 2000 john wiley sons inc

stochastic games of resource extraction we study stochastic games of resource extraction in which the transition probability is a convex combination of stochastic kernels with coefficients depending on the joint investments of the players our approach covers the unbounded utility case which was not examined in this class of games beforehand we give two theorems on the existence of pure stationary markov perfect equilibria for the models of games under consideration a detailed discussion with illustrative examples explains the meaning of our assumptions and their relation to the conditions used earlier in the literature

isoparametric hypersurfaces with four principal curvatures let m be an isoparametric hypersurface in the sphere s n with four distinct principal curvatures munzner showed that the four principal curvatures can have at most two distinct multiplicities m1 m 2 and stolz showed that the pair m1 m 2 must either be 2 2 4 5 or be equal to the multiplicities of an isoparametric hypersurface of fkm type constructed by ferus karcher and munzner from orthogonal representations of clifford algebras in this paper we prove that if the multiplicities satisfy m2 2m1 1 then the isoparametric hypersurface m must be of fkm type together with known results of takagi for the case m1 1 and ozeki and takeuchi for m1 2 this handles all possible pairs of multiplicities except for four cases for which the classification problem remains open

stationary almost markov perfect equilibria in discounted stochastic games in this paper we study discounted stochastic games with borel state and compact action spaces depending on the state variable the primitives of our model satisfy standard continuity and measurability conditions the transition probability is a convex combination of finitely many probability measures depending on states and it is dominated by some finite measure on the state space the coefficients of the combination depend on both states and action profiles this class of models contains stochastic games with borel state spaces and finite state dependent action sets our main result establishes the existence of subgame perfect equilibria which are stationary in the sense that the equilibrium strategy for each player is determined by a single function of the current and previous states of the game this dependence is called almost markov our result enhances both the theorem of mertens and parthasarathy established in 1991 for games with finite state independent action sets where the equilibrium str

a power method for computing square roots of complex matrices abstract in this paper higher order convergent methods for computing square roots of nonsingular complex matrices are derived these methods are globally convergent and are based on eigenvalue shifting and powering specifically it is shown for each positive integerr 2 a convergent method of orderrcan be developed these algorithms can be used to compute square roots of general nonsingular complex matrices such as computing square roots of matrices with negative eigenvalues

discounted stochastic games we give an alternative proof to a result of mertens and parthasarathy stating that every n player discounted stochastic game with general setup and with a norm continuous transition has a subgame perfect equilibrium

numerical approximation of the product of the square root of a matrix with a vector abstract given an n n symmetric positive definite matrix a and a vector c two numerical methods for approximating a 1 2 c are developed analyzed and computationally tested the first method applies a newton iteration to a specific nonlinear system to approximate a 1 2 c while the second method applies a step control method to numerically solve a specific initial value problem to approximate a 1 2 c assuming that a is first reduced to tridiagonal form the first method requires o n 2 operations per iteration while the second method requires o n operations per iteration in contrast numerical methods that first approximate a 1 2 and then compute a 1 2 c generally require o n 3 operations per iteration

some properties of generalized k centrosymmetric h matrices every nxn generalized k centrosymmetric matrix a can be reduced into a 2x2 block diagonal matrix see z liu h cao h chen a note on computing matrix vector products with generalized centrosymmetric centrohermitian matrices appl math comput 169 2 2005 1332 1345 this block diagonal matrix is called the reduced form of the matrix a in this paper we further investigate some properties of the reduced form of these matrices and discuss the square roots of these matrices finally exploiting these properties the development of structure preserving algorithms for certain computations for generalized k centrosymmetric h matrices is discussed

analytic properties and covariance functions for a new class of generalized gibbs random fields spartan spatial random fields ssrfs are generalized gibbs random fields equipped with a coarse graining kernel that acts as a low pass filter for the fluctuations ssrfs are defined by means of physically motivated spatial interactions and a small set of free parameters interaction couplings this paper focuses on the fluctuation gradient curvature fgc ssrf model henceforth referred to as fgc ssrf this model is defined on the euclidean space r by means of interactions proportional to the squares of the field realizations as well as their gradient and curvature the permissibility criteria of fgc ssrfs are extended by considering the impact of a finite bandwidth kernel it is proved that the fgc ssrfs are almost surely differentiable in the case of finite bandwidth asymptotic explicit expressions for the spartan covariance function are derived for d 1 and d 3 both known and new covariance functions are obtained depending on the value of the fgc ssrf shape parameter nonlinear dependence of the covariance integral scale on the fgc ssrf characteristic length is established and it is shown that the relation becomes linear asymptotically the results presented in this paper are useful in random field parameter inference and in spatial interpolation of irregularly spaced samples

computation of approximate null vectors of sylvester and lyapunov operators this paper describes an effective algorithm for computing approximate null vectors of certain matrix operators associated with sylvester or lyapunov equations the singular value decomposition and rank revealing qr methods are two widely used stable algorithms for numerical determination of the rank and nullity of a matrix a these methods however are not readily applicable to sylvester and lyapunov operators since they require on the order of n sup 6 arithmetic operations on order n sup 2 data for these problems a variant of inverse power iteration is employed to compute orthonormal bases for singular subspaces associated with the small singular values the method is practical since it relies only on the ability to solve a sylvester or lyapunov equation certain practical aspects are considered and a direct refinement technique is proposed to enhance the convergence of the algorithm

palindromic matrix polynomials matrix functions and integral representations abstract we study the properties of palindromic quadratic matrix polynomials z p qz pz 2 i e quadratic polynomials where the coefficients p and q are square matrices and where the constant and the leading coefficients are equal we show that for suitable choices of the matrix coefficients p and q it is possible to characterize by means of z well known matrix functions namely the matrix square root the matrix polar factor the matrix sign and the geometric mean of two matrices finally we provide some integral representations of these matrix functions

exact rational solution of the matrix equation a p x by linearization abstract for any given complex n n matrix a and any polynomial p with complex coefficients methods to obtain all complex n n matrix solutions x of a p x have been discussed from as early as 1906 however in practice the solutions obtained are only approximations i e 2 n 2 truncated decimal expansions for the real and imaginary parts of the n 2 entries of x the present article treats the corresponding diophantine problem where both a and p are defined over the rational field q and where if rational solutions x exist they are to be found exactly a complete solution is given when a has no repeated eigenvalue in which case all rational solutions x are obtained using only linear procedures and integer arithmetic the method generalizes at once from q to any finite algebraic extension of q or of any z p

extraction of nth roots of 2 2 matrices abstract this paper is concerned with the determination of algebraic formulae giving all the solutions of the matrix equation x n a where n is a positive integer greater than 2 and a is a 2 2 matrix with real or complex elements if a is a 2 2 scalar matrix the equation x n a has infinitely many solutions and we obtain explicit formulae giving all the solutions if a is a non scalar 2 2 matrix the equation x n a has a finite number of solutions and we give a formula expressing all solutions in terms of a and the roots of a suitably defined n th degree polynomial in a single variable this leads to very simple formulae for all the solutions when a is either a singular matrix or a non singular matrix with two coincident eigenvalues similarly when n 3 or 4 we get explicit algebraic formulae for all the solutions we also determine the precise number of solutions in various cases

subgame perfect equilibria for stochastic games for an n person stochastic game with borel state space s and compact metric action sets a1 a2 an sufficient conditions are given for the existence of subgame perfect equilibria one result is that such equilibria exist if the law of motion q s a is for fixed s continuous in a a1 a2 an for the total variation norm and the payoff functions f1 f2 fn are bounded borel measurable functions of the sequence of states s1 s2 sn and in addition are continuous when sn is given the product of discrete topologies on s

computing primary solutions of equations involving primary matrix functions abstract the matrix equation f x a where f is an analytic function and a is a square matrix is considered some results on the classification of solutions are provided when f is rational a numerical algorithm is proposed to compute all solutions that can be written as a polynomial of a for real data the algorithm yields the real solutions using only real arithmetic numerical experiments show that the algorithm performs in a stable fashion when run in finite precision arithmetic

iterative methods for finding commuting solutions of the yang baxter like matrix equation abstract the main goal of this paper is the numerical computation of solutions of the so called yang baxter like matrix equation a x a x a x where a is a given complex square matrix two novel matrix iterations are proposed both having second order convergence a sign modification in one of the iterations gives rise to a third matrix iteration strategies for finding starting approximations are discussed as well as a technique for estimating the relative error one of the methods involves a very small cost per iteration and is shown to be stable numerical experiments are carried out to illustrate the effectiveness of the new methods

reflection transmission quantum yang baxter equations we explore the reflection transmission quantum yang baxter equations arising in factorized scattering theory of integrable models with impurities the physical origin of these equations is clarified and three general families of solutions are described in detail explicit representatives of each family are also displayed these results allow us to establish for the first time a direct relationship with the different previous works on the subject and make evident the advantages of the reflection transmission algebra as a universal approach to integrable systems with impurities

existence of perfect equilibria in a class of multigenerational stochastic games of capital accumulation in this paper we introduce a model of multigenerational stochastic games of capital accumulation where each generation consists of m different players the main objective is to prove the existence of a perfect stationary equilibrium in an infinite horizon game a suitable change in the terminology used in this paper provides in the case of perfect altruism between generations a new nash equilibrium theorem for standard stochastic games with uncountable state space

varepsilon equilibria for stochastic games with uncountable state space and unbounded costs we study a class of noncooperative stochastic games with unbounded cost functions and an uncountable state space it is assumed that the transition law is absolutely continuous with respect to some probability measure on the state space undiscounted stochastic games with expected average costs are considered first it is shown under a uniform geometric ergodicity assumption that there exists a stationary varepsilon equilibrium for each varepsilon 0 the proof is based on recent results on uniform bounds for convergence rates of markov chains s p meyn and r l tweedie ann appl probab 4 1994 pp 981 1011 and on an approximation method similar to that used in a s nowak j optim theory appl 45 1985 pp 591 602 where an varepsilon equilibrium in stationary policies was shown to exist for the bounded discounted costs the stochastic game is approximated by one with a countable state space for which a stationary nash equilibrium exists see e altman a hordijk and f m spieksma math oper res 22 1997 pp 588 618 this equilibrium determines an epsilon equilibrium for the original game finally new results for the existence of stationary varepsilon equilibrium for discounted stochastic games are given