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

a new approach to the cramer rao type bound of the pure state model this paper sheds light on non commutativity in quantum theory as regards theoretical estimation in it we calculate the quantum cramer rao type bound for many cases by use of a newly proposed powerful technique we also discuss the use of collective measurement in statistical estimation

geometrically non linear enhanced strain mixed methods and the method of incompatible modes a class of assumed strain mixed finite element methods for fully non linear problems in solid mechanics is presented which when restricted to geometrically linear problems encompasses the classical method of incompatible modes as a particular case the method relies crucially on a local multiplicative decomposition of the deformation gradient into a conforming and an enhanced part formulated in the context of a three field variational formulation the resulting class of mixed methods provides a possible extension to the non linear regime of well known incompatible mode formulations in addition this class of methods includes non linear generalizations of recently proposed enhanced strain interpolations for axisymmetric problems which cannot be interpreted as incompatible modes elements the good performance of the proposed methodology is illustrated in a number of simulations including 2 d 3 d and axisymmetric finite deformation problems in elasticity and elastoplasticity remarkably these methods appear to be specially well suited for problems involving localization of the deformation as illustrated in several numerical examples

improved enhanced strain four node element with taylor expansion of the shape functions a class of enhanced strain four node elements with taylor expansion of the shape function derivatives is presented a new concept of enhancement using besides the standard enhanced strain fields also two other enhanced fields is developed on the basis of the hu washizu principle for first order taylor expansion enhanced modes become uncoupled thus only a negligible amount of computing effort for the static condensation of enhanced modes is needed furthermore the formulation permits a symbolic integration which leads to a closed form solution for the element tangent matrix several numerical examples show that the element is stable invariant passes the patch test and yields good results especially in the highly distorted regime 1997 by john wiley sons ltd

optimal steady states excessive functions and deterministic dynamic programs this paper studies the relationship between optimal steady states and excessive functions in undiscounted deterministic dynamic programming models optimal steady states arise in the turnpike theorems of optimal economic growth and the steady state policies of 3 4 81 excessive functions which play a critical role in constructing bounds on the performance of steady state policies in 3 81 are generalizations of lagrangian saddlepoints section 2 presents the dynamic programming model section 3 defines optimal steady states and discusses their importance section 4 characterizes the relationship between them and lagrangian saddle points section 5 introduces excessive functions and uses them to derive bounds on the optimal finite stage costs section 6 shows that the set of excessive functions associated with an oss contains a greatest element and a least element and proves that for s to be an optimal steady state it is necessary and sufficient that s be associated with an excessive function that is bounded above note that our model is quite general and does not assume convexity

monotone riemannian metrics and relative entropy on noncommutative probability spaces we use the relative modular operator to define a generalized relative entropy for any convex operator function g on 0 satisfying g 1 0 we show that these convex operator functions can be partitioned into convex subsets each of which defines a unique symmetrized relative entropy a unique family parametrized by density matrices of continuous monotone riemannian metrics a unique geodesic distance on the space of density matrices and a unique monotone operator function satisfying certain symmetry and normalization conditions we describe these objects explicitly in several important special cases including g w log w which yields the familiar logarithmic relative entropy the relative entropies riemannian metrics and geodesic distances obtained by our procedure all contract under completely positive trace preserving maps we then define and study the maximal contraction associated with these quantities

the simplex and policy iteration methods are strongly polynomial for the markov decision problem with a fixed discount rate we prove that the classic policy iteration method howard r a 1960 dynamic programming and markov processes mit cambridge and the original simplex method with the most negative reduced cost pivoting rule of dantzig are strongly polynomial time algorithms for solving the markov decision problem mdp with a fixed discount rate furthermore the computational complexity of the policy iteration and simplex methods is superior to that of the only known strongly polynomial time interior point algorithm ye y 2005 a new complexity result on solving the markov decision problem math oper res 30 3 733 749 for solving this problem the result is surprising because the simplex method with the same pivoting rule was shown to be exponential for solving a general linear programming problem klee v g j minty 1972 how good is the simplex method technical report o shisha ed inequalities iii academic press new york the simplex method with the smallest index pivoting rule was shown to be exponential for solving an mdp regardless of discount rates melekopoglou m a condon 1994 on the complexity of the policy improvement algorithm for markov decision processes informs j comput 6 2 188 192 and the policy iteration method was recently shown to be exponential for solving undiscounted mdps under the average cost criterion we also extend the result to solving mdps with transient substochastic transition matrices whose spectral radii are uniformly below one

mixed enhanced formulation for geometrically linear axisymmetric problems a four noded quadrilateral axisymmetric formulation in the context of a mixed enhanced method is presented the strain field is represented by two sets of element parameters which results in enhanced performance and coarse mesh accuracy in bending dominated problems and locking free response in the near incompressible limit the mixed fields presented are such that variational stress recovery is permissible in addition the formulation is cast such that the mixed parameters are obtained explicitly yielding finite element arrays with the proper rank using standard order quadrature in this paper our attention is restricted to the area of geometrically linear problems in solid mechanics representative simulations show favourable performance of the formulation copyright 2002 john wiley sons ltd

an algorithm for a dynamic programming model of fractional flows in an earlier paper the author introduced an n dimensional dynamic programming model of fractional flows characterized its solutions under general conditions and developed methods for their computation the latter which resemble algorithms for n state markovian decision processes break down when the costs are undiscounted and the flow matrices are nontransient here an algorithm is presented that computes many of the quantities of interest more efficiently under milder assumptions in addition some of our earlier results for the undiscounted case are extended to models with nontransient flow matrices

a mixed enhanced formulation tetrahedral finite elements this paper considers the solution of problems in three dimensional solid mechanics using tetrahedral finite elements a formulation based on a mixed enhanced treatment involving displacement pressure and volume effects is presented the displacement and pressure are used as nodal quantities while volume effects and enhanced modes belong to individual elements both small and finite deformation problems are addressed and sample solutions are given to illustrate the performance of the formulation copyright 2000 john wiley sons ltd

on fractional flow models and equivalent finite state processes this paper studies the relationships between a dynamic programming model of fractional flows in an n sector system and certain finite state processes the former which is described in section 2 is a nonstationary version of 4 under conditions given in section 3 the flow model is equivalent to an n state markovian decision process allowing for simpler computation in certain cases where that equivalence fails a partial solution to the flow model follows from the solution to an n l state semi markovian decision process this result which appears in section 6 helps to explain why the algorithms of 4 51 resemble ones for finite state processes it does not substantially reduce computation

mixed enhanced formulation for finite deformation axisymmetric torsion the development and implementation of a four noded quadrilateral element for the analysis of the axisymmetric torsion problem is presented in the context of the mixed enhanced method the deformation gradient is developed from two sets of element interpolants of which one may be explicitly obtained resulting in only two undetermined enhanced parameters needed for the satisfaction of the near incompressible limit the resulting element arrays yield proper rank with standard 2 2 quadrature the introduction of a three field variation statement for the problem along with the interpolations for the stress and deformation gradient render a variational stress recovery permissible several numerical examples are explored in the setting of finite kinematics with physically non linear material properties to further induce coupling between the radial and axial displacement and the longitudinal rotation copyright 2004 john wiley sons ltd

monotone metrics on matrix spaces abstract the study of monotone inner products under stochastic mappings on the space of matrices was initiated by morozova and chentsov motivated by information geometry they did not show a monotone metric but proposed several candidates the main result of the present paper is to provide an abundance of monotone metrics by means of operator monotone functions and to characterize them it turns out that there is a correspondence between monotone metrics and operator means in the sense of kubo and ando it follows that all proposals of morozova and chentsov are indeed monotone metrics

growth optimality for branching markov decision chains this paper considers a multiplicative process called branching markov decision chains in which the output at the end of the nth period equals the product of n nonnegative matrices chosen at the beginning of periods 1 n respectively times a positive fixed terminal reward vector it is assumed that the above transition matrices are drawn out of a finite set of matrices given in product form i e the rows of the matrices can be selected independently out of finite sets of nonnegative row vectors for each coordinate s we define the geometric and algebraic growth rates respectively of the sth coordinate of the stream of output these growth rates are defined so that the magnitude of the corresponding sequence is of the order nnk where is the geometric growth rate and k is the algebraic growth rate the main result of this paper is the constructive establishment of the existence of a transition matrix whose repeated use will guarantee for each coordinate the achievement of the best geometri

an incompatible element for axisymmetric structure and its modification by hybrid method this paper introduces three four node quadrilateral axisymmetric elements the first one is an incompatible element while the second one is a modification of the first element by an extended hellinger reissner principle the second element is then simplified by incorporating the conceptual idea of free formulation and thus forms the third element all of the three elements are found to be invariant insensitive to nodal point numbering free of locking for nearly incompressible material able to pass the patch test and do not exhibit the false shear phenomenon numerical examples are given and satisfactory predictions are achieved

improved versions of assumed enhanced strain tri linear elements for 3d finite deformation problems abstract improved three dimensional tri linear elements for finite deformation problems are developed based on an assumed enhanced strain methodology which in the linear regime incorporates the classical method of incompatible modes as a particular case three crucial modifications of a recently proposed element which reduces to wilson s brick in the linear regime are introduced to prevent locking response in distorted configurations and to maintain proper rank while preserving excellent performance in bending dominated and localization problems i a modified quadrature rule ii an additional enhancement of the divergence term and iii a modification of the isoparametric shape function derivatives for the three dimensional problem moreover these modified shape function derivatives are shown to improve the performance of the standard tri linear brick in distorted configurations in addition a strategy is described to circumvent the memory storage requirements in the static condensation procedure of the enhanced strain parameters the excellent performance of the improved methodology is illustrated in representative numerical simulations

a systematic construction of b bar functions for linear and non linear mixed enhanced finite elements for plane elasticity problems in a previous paper a modified hu washizu variational formulation has been used to derive an accurate four node plane strain stress finite element denoted qe2 for the mixed element qe2 two enhanced strain terms are used and the assumed stresses satisfy the equilibrium equations a priori for the linear elastic case in this paper an alternative approach is discussed the new formulation leads to the same accuracy for linear elastic problems as the qe2 element however it turns out to be more efficient in numerical simulations especially for large deformation problems using orthogonal stress and strain functions we derive b functions which avoid numerical inversion of matrices the b strain matrix is sparse and has the same structure as the strain matrix b obtained from a compatible displacement field the implementation of the derived mixed element is basically the same as the one for a compatible displacement element the only difference is that we have to compute a b strain matrix instead of the standard b matrix accordingly existing subroutines for a compatible displacement element can be easily changed to obtain the mixed enhanced finite element which yields a higher accuracy than the q4 and qm6 elements copyright 1999 john wiley sons ltd

connections and metrics respecting purification of quantum states standard purification interlaces hermitian and riemannian metrics on the space of density operators with metrics and connections on the purifying hilbert schmidt space we discuss connections and metrics which are well adopted to purification and present a selected set of relations between them a connection as well as a metric on state space can be obtained from a metric on the purification space we include a condition with which this correspondence becomes one to one our methods are borrowed from elementary representation and fiber space theory we lift as an example solutions of a von neumann equation write down holonomy invariants for cyclic ones and add noise to a curve of pure states

an estimation theoretical characterization of coherent states we introduce a class of quantum pure state models called the coherent models a coherent model is an even dimensional manifold of pure states whose tangent space is characterized by a symplectic structure in a rigorous framework of noncommutative statistics it is shown that a coherent model inherits and expands the original spirit of the minimum uncertainty property of coherent states

hamilton jacobi approach to potential functions in information geometry the search for a potential function s allowing us to reconstruct a given metric tensor g and a given symmetric covariant tensor t on a manifold m is formulated as the hamilton jacobi problem associated with a canonically defined lagrangian on tm the connection between this problem the geometric structure of the space of pure states of quantum mechanics and the theory of contrast functions of classical information geometry are outlined

duality versus dual flatness in quantum information geometry we investigate questions in quantum informationgeometry which concern the existence and nonexistence of dual and dually flat structures on stratified sets of density operators on finite dimensional hilbert spaces we show that the set of density operators of a given rank admits dually flat connections for which one connection is complete if and only if this rank is maximal we prove moreover that there is never a dually flat structure on the set of pure states thus any general theory of quantum informationgeometry that involves duality concepts must inevitably be based on dual structures which are nonflat

rational approach for assumed stress finite elements a new method for the formulation of hybrid elements by the hellinger reissner principle is established by expanding the essential terms of the assumed stresses as complete polynomials in the natural coordinates of the element the equilibrium conditions are imposed in a variational sense through the internal displacements which are also expanded in the natural co ordinates the resulting element possesses all the ideal qualities i e it is invariant it is less sensitive to geometric distortion it contains a minimum number of stress parameters and it provides accurate stress calculations for the formulation of a 4 node plane stress element a small perturbation method is used to determine the equilibrium constraint equations the element has been proved to be always rank sufficient

a class of mixed assumed strain methods and the method of incompatible modes a three field mixed formulation in terms of displacements stresses and an enhanced strain field is presented which encompasses as a particular case the classical method of incompatible modes within this frame work incompatible elements arise as particular compatible mixed approximations of the enhanced strain field the conditions that the stress interpolation contain piece wise constant functions and be l2 ortho gonal to the enhanced strain interpolation ensure satisfaction of the patch test and allow the elimination of the stress field from the formulation the preceding conditions are formulated in a form particularly convenient for element design as an illustration of the methodology three new elements are developed and shown to exhibit good performance a plane 3d elastic plastic quad an axisymmetric element and a thick plate bending quad the formulation described herein is suitable for non linear analysis

high accuracy low order three dimensional brick elements this paper presents a family of low order high accuracy three dimensional brick elements the elements are formulated via a three field variational principle the assumed independent strain field is constructed from two disjointed distributions the first contains the lower order distribution and its dimension is the minimum required to satisfy stability requirements an energy constraint which is enforced weakly at the element level is used to relate the second distribution to the first the stress field is chosen to a priori satisfy a similar energy constraint as a result internal constraints e g incompressibility are automatically satisfied by these fields and locking behaviour is avoided a j2 plasticity model illustrates the proposed elements performance in nonlinear solids the excellent performance of the proposed elements is demonstrated with numerous challenging examples including many that are usually modelled by shell elements

geometry of canonical correlation on the state space of a quantum system a riemannian metric is defined on the state space of a finite quantum system by the canonical correlation or kubo mori bogoliubov scalar product this metric is infinitesimally induced by the nonsymmetric relative entropy functional or the von neumann entropy of density matrices hence its geometry expresses maximal uncertainty it is proven that the metric is monotone under stochastic mappings however an example shows that it is not the only such riemannian metric this fact is remarkable because in the probabilistic case the markovian monotonicity property characterizes the fisher information metric the essential difference appears in the curvatures of a classical state space and a quantum one a conjecture is made that the scalar curvature is monotone with respect to the more mixed statistical partial order of density matrices furthermore an information inequality resembling the cramer rao inequality of classical statistics is established the inequality provides a lower bound for the c