a construction of a nonparametric quantum information manifold abstract we present a construction of a banach manifold structure on the set of faithful normal states of a von neumann algebra where the underlying banach space is a quantum analogue of an orlicz space on the manifold we introduce the exponential and mixture connections as dual pair of affine connections

on constructing matrices with prescribed singular values and diagonal elements abstract similar to the well known schur horn theorem that characterizes the relationship between the diagonal entries and the eigenvalues of a hermitian matrix the sing thompson theorem characterizes the relationship between the diagonal entries and the singular values of an arbitrary matrix it is noted in this paper that based on the induction principle such a matrix can be constructed numerically by a fast recursive algorithm provided that the given singular values and diagonal elements satisfy the sing thompson conditions

qualitative behavior of non abelian toda like flows in this paper we analyze and compare the long term behavior of certain non abelian generalizations of the integrable toda lattice flow in particular we show that these flows exhibit generalized sorting behavior analogous to the sorting behavior of the original tridiagonal toda lattice we show that one of these generalizations the block double bracket equation is gradient with respect to a normal metric and exactly solvable this provides a new solvable case of the double bracket system

differential equations for roaming pseudospectra paths to extremal points and boundary tracking when studying the varepsilon pseudospectrum of a matrix one is often interested in computing the extremal points having maximum real part or modulus this is a crucial step for example when computing the distance to instability of a stable system using the property that the pseudospectrum is determined via perturbations by rank 1 matrices we derive differential equations on the manifold of normalized rank 1 matrices whose solutions tend to the critical rank 1 perturbations associated with the extremal points of locally maximum real part and modulus this approach also allows us to track the boundary contour of the pseudospectrum in a neighborhood of the extremal points the technique we propose is related to an idea recently developed by guglielmi and overton who derived discrete dynamical systems instead of the continuous ones we present the method turns out to be fast in comparison with those previously proposed in the literature and appears to be promising in dealing with large sparse problems

a fast recursive algorithm for constructing matrices with prescribed eigenvalues and singular values the weyl horn theorem characterizes a relationship between the eigenvalues and the singular values of an arbitrary matrix based on that characterization a fast recursive algorithm is developed to construct numerically a matrix with prescribed eigenvalues and singular values besides being of theoretical interest the technique could be employed to create test matrices with desired spectral features numerical experiment shows this algorithm to be quite efficient and robust

fuzzy set valued gaussian processes and brownian motions gaussian processes and brownian motion are concepts and tools in modelling important uncertain systems in many areas in view of uncertainty complexity in many real world problems we extend these tools to the case where stochastic processes can take on fuzzy sets as values in this paper we discuss fuzzy set valued gaussian processes based on the results of s li y ogura v kreinovich limit theorems and applications of set valued and fuzzy set valued random variables kluwer academic publishers dordrecht 2002 s li y ogura h t nguyen gaussian processes and martingales for fuzzy valued variables with continuous parameter inform sci 133 2001 7 21 s li y ogura f n proske m l puri central limit theorems for generalized set valued random variables j math anal appl 285 2003 250 263 n n lyashenko on limit theorems for sums of independent compact random subsets in the euclidean space j soviet math 20 1982 2187 2196 and m l puri d a ralescu the concept of normality for fuzzy random variables ann probab 13 1985 1373 1379 we also introduce the concept of fuzzy set valued brownian motion and then prove several properties of such processes

schur majorization inequalities for symmetrized sums with applications to tensor products we show that if w y and x z are four vectors in rn then a number of schur majorizations hold between symmetrized vector functions of w x y and z e g wi xj i j yi zj i j where the left hand expression means the vector of dimension n2 consisting of all sums wi xj of the co ordinates of w and x arranged in lexicographic order among other things we get vector and matrix versions of muirhead s theorem for scalar inequalities from the vector inequalities follow many scalar inequalities for symmetrized sums some of which are scattered through the inequality literature in section 2 applications are given to matrix inequalities for tensor products e g if a b and c are hermitian and a b then a c b c

orthogonal eigensubspace estimation using neural networks we present a neural network nn approach for simultaneously estimating all or some of the orthogonal eigenvectors of a symmetric nonindefinite matrix corresponding to its repeated minimum in magnitude eigenvalue this problem has its origin in the constrained minimization framework and has extensive applications in signal processing we recast this problem into the nn framework by constructing an appropriate energy function which the nn minimizes the nn is of feedback type with the neurons having sigmoidal activation function the proposed approach is analyzed to characterize the nature of the minimizers of the energy function the main result is that the matrix w is a minimizer of the energy function if and only if the columns of w are the orthogonal eigenvectors with a given norm corresponding to the smallest eigenvalue of the given matrix further all minimizers are global minimizers bounds on the integration time step that is required to numerically solve the system of differential equations which define the dynamics of the nn have also been derived results of computer simulations are presented to support our analysis

simulation of fuzzy random variables this work deals with the simulation of fuzzy random variables which can be used to model various realistic situations where uncertainty is not only present in form of randomness but also in form of imprecision described by means of fuzzy sets utilizing the common arithmetics in the space of all fuzzy sets only induces a conical structure as a consequence it is difficult to directly apply the usual simulation techniques for functional data in order to overcome this difficulty two different approaches based on the concept of support functions are presented the first one makes use of techniques for simulating hilbert space valued random elements and afterwards projects on the cone of all fuzzy sets it is shown by empirical results that the practicability of this approach is limited the second approach imitates the representation of every element of a separable hilbert space in terms of an orthonormal basis directly on the space of fuzzy sets in this way a new approximation of fuzzy sets useful to approximate and simulate fuzzy random variables is developed this second approach is adequate to model various realistic situations

singular value decomposition via gradient and self equivalent flows the task of finding the singular value decomposition svd of a finite dimensional complex linear operator is here addressed via gradient flows evolving on groups of complex unitary matrices and associated self equivalent flows the work constitutes a generalization of that of brockett on the diagonalization of real symmetric matrices via gradient flows on orthogonal matrices and associated isospectral flows it complements results of symes chu and other authors on continuous analogs of the classical qr algorithm as well as earlier work by the authors on svd via gradient flows on positive definite matrices

recurrent neural networks for computing pseudoinverses of rank deficient matrices three recurrent neural networks are presented for computing the pseudoinverses of rank deficient matrices the first recurrent neural network has the dynamical equation similar to the one proposed earlier for matrix inversion and is capable of moore penrose inversion under the condition of zero initial states the second recurrent neural network consists of an array of neurons corresponding to a pseudoinverse matrix with decaying self connections and constant connections in each row or column the third recurrent neural network consists of two layers of neuron arrays corresponding respectively to a pseudoinverse matrix and a lagrangian matrix with constant connections all three recurrent neural networks are also composed of a number of independent subnetworks corresponding to the rows or columns of a pseudoinverse the proposed recurrent neural networks are shown to be capable of computing the pseudoinverses of rank deficient matrices

convergence theorem for fuzzy martingales abstract we study fuzzy set valued measures in a banach space and their relationships to fuzzy random variables our main result is a convergence theorem for fuzzy martingales our tool is a radon nikodym type theorem for fuzzy measures which are absolutely continuous with respect to probability measures

a trace inequality with a subtracted term abstract for fixed real or complex matrices a and b the well known von neumann trace inequality identifies the maximum of tr uavb z sfnc as u and v range over the unitary group the maximum being a bilinear expression in the singular values of a and b this paper establishes the analogue of this inequality for real matrices a and b when u and v range over the proper real orthogonal group the maximum is again a bilinear expression in the singular values but there is a subtracted term when a and b have determinants of opposite sign

positive and doubly stochastic maps and majorization in euclidean jordan algebras abstract a positive map between euclidean jordan algebras is a symmetric cone order preserving linear map we show that the norm of such a map is attained at the unit element thus obtaining an analog of the operator matrix theoretic russo dye theorem a doubly stochastic map between euclidean jordan algebras is a positive unital and trace preserving map we relate such maps to jordan algebra automorphisms and majorization

nonholonomic double bracket equations and the gauss thermostat in this rapid communication we consider certain equations that arise from imposing a constant kinetic energy constraint on a one dimensional set of oscillators this is a nonlinear nonholonomic constraint on these oscillators and the dynamics are consistent with gauss s law of least constraint dynamics of this sort are of interest in nonequilibrium molecular dynamics we show that under certain choices of external potential these equations give rise to a generalization of the so called double bracket equations which are of interest in studying gradient flows and integrable systems such as the toda lattice in the case of harmonic potentials the flow is described by a symmetric bracket and periodic solutions are obtained

cone orderings group majorizations and similarly separable vectors abstract we generalize some results on majorization in papers by wu and debnath s wu l debnath inequalities for convex sequences and their applications comput math appl 54 2007 525 534 and marshall et al a w marshall i olkin f proschan monotonicity of ratios of means and other applications of majorization in o shisha ed inequalities academic press new york 1967 pp 177 190 we present sufficient conditions for some vector inequalities to hold in the case of cone orderings and group induced cone orderings the framework used is based on results for similarly separable vectors given in paper m niezgoda bifractional inequalities and convex cones discrete math 306 2006 231 243

separability for graph convergence of sequences of fuzzy valued random variables abstract we prove that the graph convergence in hausdorff metric or kuratowski mosco topology of a sequence of fuzzy sets follows from the convergence of the sequences of the level sets for countable dense levels as an application we give a strong law of large numbers for fuzzy valued random variables including the case when the level sets may not be bounded

existence and uniqueness for solutions to fuzzy stochastic differential equations driven by local martingales under the non lipschitzian condition abstract this paper makes a research into a class of fuzzy stochastic differential equations fsdes driven by a continuous local martingale under the non lipschitzian condition such equations can be useful in modelling of hybrid systems where the phenomena are subjected to two kinds of uncertainties randomness and fuzziness simultaneously the solutions of fsdes are the fuzzy stochastic processes and their uniqueness is considered to be in a strong sense thus the existence and uniqueness of solutions to fsdes under the non lipschitzian condition is first proven and the continuity of solutions to fsdes with respect to the initial data or the coefficients of the equations is investigated

dynamical methods for polar decomposition and inversion of matrices we show how one may obtain polar decomposition as well as inversion of fixed and time varying matrices using a class of nonlinear continuous time dynamical systems first we construct a dynamic system that causes an initial approximation of the inverse of a time varying matrix to flow exponentially toward the true time varying inverse using a time parameterized homotopy from the identity matrix to a fixed matrix with unknown inverse and applying our result on the inversion of time varying matrices we show how any positive definite fixed matrix may be dynamically inverted by a prescribed time without an initial guess at the inverse we then construct a dynamical system that solves for the polar decomposition factors of a time varying matrix given an initial approximation for the inverse of the positive definite symmetric part of the polar decomposition as a byproduct this method gives another method of inverting time varying matrices finally using homotopy again we show how dynamic polar decomposition may be applied to fixed matrices with the added benefit that this allows us to dynamically invert any fixed matrix by a prescribed time

existence and uniqueness of solution for fuzzy random differential equations with non lipschitz coefficients in this paper a class of fuzzy random differential equations with non lipschitz coefficients is studied the existence and uniqueness of solutions for fuzzy random differential equations with non lipschitz coefficients is first proved then the dependence of fuzzy random differential equations on initial values is discussed finally the non confluence property of the solution for fuzzy random differential equation is investigated our main tool is the gronwall lemma

fuzzy set valued stochastic lebesgue integral this paper studies lebesgue integral of a fuzzy closed set valued stochastic process with respect to the time t firstly a progressively measurable fuzzy closed set valued stochastic process is discussed and an almost everywhere problem in the former aumann type lebesgue integral of the level set process is pointed out secondly a new definition of the lebesgue integral by decomposable closure is given focusing on aumann representation theorem representation theorem and property of convexity it is proved that the fuzzy closed set valued stochastic lebesgue integral is a fuzzy closed set valued stochastic process which is widely used in the fuzzy world with randomness finally the fuzzy closed set valued stochastic lebesgue integral in l p space is studied especially on an almost everywhere problem

group majorization the convex hulls of sets of matrices and the diagonal element singular value inequalities abstract a group induced preorder on matrix space permits the description of the convex hulls of sets of proper and improper real matrices with given singular values and also permits a new proof of the necessity of the known relations between the diagonal elements and singular values of a real matrix

topology of the cone of positive maps on qubit systems an alternative geometrical proof of a known theorem concerning the decomposition of positive maps of the matrix algebra has been presented the premise of the proof is the identification of positive maps with operators preserving the lorentz cone in four dimensions and it allows to decompose the positive maps with respect to those preserving the boundary of the cone in addition useful conditions implying complete positivity of a map of have been given together with a sufficient condition for complete positivity of an extremal schwarz map of lastly following the same geometrical approach a description in topological terms of maps that are simultaneously completely positive and completely copositive has been presented

numerical solution of isospectral flows in this paper we are concerned with the problem of solving numerically isospectral flows these flows are characterized by the differential equation l b l l l 0 l 0 where l 0 is a d x d symmetric matrix b l is a skew symmetric matrix function of l and b l is the lie bracket operator we show that standard runge kutta schemes fail in recovering the main qualitative feature of these flows that is isospectrality since they cannot recover arbitrary cubic conservation laws this failure motivates us to introduce an alternative approach and establish a framework for generation of isospectral methods of arbitrarily high order

dynamical systems that sort lists diagonalize matrices and solve linear programming problems abstract we establish a number of properties associated with the dynamical system h h h n where h and n are symmetric n by n matrices and a b ab ba the most important of these come from the fact that this equation is equivalent to a certain gradient flow on the space of orthogonal matrices we are especially interested in the role of this equation as an analog computer for example we show how to map the data associated with a linear programming problem into h 0 and n in such a way as to have h h h n evolve to a solution of the linear programming problem this result can be applied to find systems which solve a variety of genetic combinatorial optimization problems and it even provides an algorithm for diagonalizing symmetric matrices

g majorization with applications to matrix orderings abstract a vector x is said to g majorize a vector y if y lies in the convex hull of the orbit of x under a group g the present paper contains a straightforward account with two important statements equivalent to g majorization in certain cases for example when g is a finite reflection group one equivalent condition reduces to a finite set of linear inequalities representing a cone ordering in the fundamental region of the group the other condition is that every convex g invariant function of y is less than the same function of x upper and lower weak majorizations gw majorizations are introduced by combining g with a second ordering compatible with it the results are applied where possible to matrix orderings where a g b is g majorization when g is a subgroup of the orthogonal group o n acting by congruence i e g b qbq t with q the matrix representation for an element of o n when g is the symmetric group of permutation matrices this defines a new ordering and generalizes a proposition of kiefer in the design of experiments

on the continuous realization of iterative processes many important mathematical problems are solved by iterative methods many of these iterative schemes may be regarded as the discrete realization of certain continuous dynamical systems this paper summarizes some of the recent developments in the continuous realization of several popular basic iterative methods

a theory for learning by weight flow on stiefel grassman manifold recently we introduced the concept of neural network learning on stiefel grassman manifold for multilayer perceptron like networks contributions of other authors have also appeared in the scientific literature about this topic this article presents a general theory for it and illustrates how existing theories may be explained within the general framework proposed here

global convergence of nonlinear cascade flows with morse bott zero dynamics abstract we derive a sufficient condition that a flow captures the dynamics on an invariant submanifold this leads to a refinement of the lasalle invariance principle as a consequence we generalize a well known global asymptotic stability result of nonlinear cascade systems to show global convergence to a compact invariant set this includes the case where a globally asymptotically stable system is coupled to a morse bott flow

group majorization and schur type inequalities abstract the problem of the reduction of a group majorization induced by an infinite group to a finite group majorization is discussed a necessary and sufficient condition for a possibility of the reduction is given the condition generalizes some known inequalities in the matrix theory involving eigenvalues or singular values of a matrix a corresponding result for a skew symmetric matrix is derived

gaussian fuzzy random variables in this paper we define the concepts of a gaussian fuzzy random variable and a gaussian fuzzy random vector and discuss their properties the covariance operator the characteristic functional and the linear transformation of a gaussian fuzzy random variable are investigated

singular values diagonal elements and extreme matrices abstract for complex matrices a and b there are inequalities related to the diagonal elements of ab and the singular values of a and b we study the conditions on the matrices for which those inequalities become equalities in all cases the conditions are both necessary and sufficient

efficient recovery of low rank matrix via double nonconvex nonsmooth rank minimization recently there is a rapidly increasing attraction for the efficient recovery of low rank matrix in computer vision and machine learning the popular convex solution of rank minimization is nuclear norm based minimization nnm which usually leads to a biased solution since nnm tends to overshrink the rank components and treats each rank component equally to address this issue some nonconvex nonsmooth rank nnr relaxations have been exploited widely different from these convex and nonconvex rank substitutes this paper first introduces a general and flexible rank relaxation function named weighted nnr relaxation function which is actually derived from the initial double nnr dnnr relaxations i e dnnr relaxation function acts on the nonconvex singular values function svf an iteratively reweighted svf optimization algorithm with continuation technology through computing the supergradient values to define the weighting vector is devised to solve the dnnr minimization problem and the closed form solution of the subproblem can be efficiently obtained by a general proximal operator in which each element of the desired weighting vector usually satisfies the nondecreasing order we next prove that the objective function values decrease monotonically and any limit point of the generated subsequence is a critical point combining the kurdyka ojasiewicz property with some milder assumptions we further give its global convergence guarantee as an application in the matrix completion problem experimental results on both synthetic data and real world data can show that our methods are competitive with several state of the art convex and nonconvex matrix completion methods

central limit theorems for generalized set valued random variables abstract we give central limit theorems for generalized set valued random variables whose level sets are compact both in r d or in a banach space under milder conditions than those obtained recently by the latter two authors

an infinite dimensional geometric structure on the space of all the probability measures equivalent to a given one let m be the set of all probability densities equivalent to a given reference probability measure this set is thought of as the maximal regular i e with strictly positive densities dominated statistical model for each f m we define 1 a banach space l f with unit ball v f and 2 a mapping s f from a subset u f of m onto v f in such a way that the system s f u f f m is an affine atlas on m moreover each parametric exponential model dominated by is a finite dimensional affine submanifold and each parametric statistical model dominated by with a suitable regularity is a submanifold the global geometric framework given by the manifold structure adds some insight to the so called geometric theory of statistical models in particular the present paper gives some of the developments connected with the fisher information metrics rao and the hilbert bundle introduced by amari

on lei miranda and thompson s result on singular values and diagonal elements abstract thomspon and sing s result on the singular values and the diagonal elements of a complex n n matrix has been recently extended to arbitrary m matrices by lei and by miranda and thompson independently the real case in which the determinant of the product of the m real matrices is nonnegative or nonpositive was raised by miranda and thompson in this note we provide an answer to the question and we also extend some other results of lei

polynomial sized semidefinite representations of derivative relaxations of spectrahedral cones we give explicit polynomial sized in n n and k k semidefinite representations of the hyperbolicity cones associated with the elementary symmetric polynomials of degree k k in n n variables these convex cones form a family of non polyhedral outer approximations of the non negative orthant that preserve low dimensional faces while successively discarding high dimensional faces more generally we construct explicit semidefinite representations polynomial sized in k m k m and n n of the hyperbolicity cones associated with k kth directional derivatives of polynomials of the form p x det sum i 1 n a i x i p x det i 1naixi where the a i ai are m times m m m symmetric matrices these convex cones form an analogous family of outer approximations to any spectrahedral cone our representations allow us to use semidefinite programming to solve the linear cone programs associated with these convex cones as well as their less well understood dual cones

double bracket equations and geodesic flows on symmetric spaces in this paper we consider the geometry of hamiltonian flows on the cotangent bundle of coadjoint orbits of compact lie groups and on symmetric spaces a key idea here is the use of the normal metric to define the kinetic energy this leads to hamiltonian flows of the double bracket type we analyze the integrability of geodesic flows according to the method of thimm we obtain via the double bracket formalism a quite explicit form of the relevant commuting flows and a correspondingly transparent proof of involutivity we demonstrate for example integrability of the geodesic flow on the real and complex grassmannians we also consider right invariant systems and the generalized rigid body equations in this setting

lie algebraic aspects of the finite nonperiodic toda flows in this paper we review various aspects of nonperiodic full and tridiagonal toda flow theory in particular we consider gradient structures in the dynamics and geometry of these systems and we compare and contrast a number of formulations of the nonperiodic toda equations we describe the structure of these systems on general semisimple lie algebras in the case of the full kostant asymmetric toda flow we describe some of the background behind its integrability and explain the role of noncommutative integrability in its qualitative behavior we describe the relationship between the asymmetric toda flows and the symmetric and indefinite toda flows and show how one may conjugate from the full kostant toda flows to the full symmetric toda flows via a poisson map we describe briefly related toda systems such as nonabelian toda and the peakon flows

moyal implementation of flow equations a non perturbative approach to quantum many body systems we show how wegner s flow equations can be reformulated as ordinary differential equations through the use of the moyal bracket in finite dimensional hilbert spaces the introduction of the moyal bracket leads naturally to the identification of a small expansion parameter namely the inverse of the dimensionality of the space this expansion corresponds to a non perturbative treatment of the coupling constant in the case of infinite dimensional spaces plays the role of the small parameter and the moyal formulation then allows for a semi classical treatment of the flow equation we demonstrate these statements for the lipkin and dicke models as well as for the symmetric x4 and double well potentials

an extension of schur ostrowski s condition weak eaton triples and generalized ai functions abstract in this paper for a given eaton triple v g d we present a characterization of g increasing real functions by using the maximality property of their directional derivatives it generalizes schur ostrowski s condition related to schur convexity of a function it is also useful in establishing theorems of von neumann davis type regarding the form of group invariant real functions with certain kinds of convexity we introduce and investigate weak eaton triples and their chains we show a method for constructing new eaton triples we apply it to define generalized arrangement increasing ai functions we prove that the mentioned derivatives of g increasing maps are generalized ai functions

cases of equality for certain inequalities involving a g doubly stochastic map abstract we study inner product inequalities for a g doubly stochastic map with respect to a group induced cone ordering we give necessary and sufficient conditions for the inequalities to become equalities a special condition is obtained if a birkhoff type theorem holds for a finite reflection group related to the ordering the conditions are applied to derive some classical and new results on equality cases for majorization inequalities connected with certain group actions

on lattice property of group induced cone orderings abstract the problem of existence and uniqueness of greatest lower bound glb and least upper bound lub of a set in group induced cone gic ordering is discussed explicit formula for the glb is given the results are applied to obtain the best possible upper bound ingic ordering for values of linear and nonnegative linear maps

miranda and thompson s trace inequality and a log convexity result abstract a geometric proof of miranda and thompson s trace inequality is given via thompson s singular value diagonal element inequalities miranda and thompson s trace inequality is associated with the unitary group we then deal with the cases associated with the orthogonal group and the special unitary group we also discuss the convexity of the set of the diagonal elements of complex matrices with fixed singular values and determinant some question are asked a log convexity result related to gram schmidt decomposition is obtained

a convergence theorem of fuzzy valued martingales in the extended hausdorff metric h in this paper we shall give a new embedding method to prove a convergence theorem for fuzzy valued random variables in the sense of the extended hausdorff metric h without the restriction of fuzzy sets satisfying the lipschitz condition

convergence of set valued sub and supermartingales in the kuratowski mosco sense the purpose of this paper is to prove some convergence theorems of closed and convex set valued sub and supermartingales in the kuratowski mosco sense to get submartingale convergence theorems we give sufficient conditions for the kudo aumann integral and hiai umegaki conditional expectation to be closed both for compact convex set valued random variables and for closed convex set valued random variables we also give an example of a bounded closed convex set valued random variable whose kudo aumann integral is not closed

a q exponential statistical banach manifold abstract let be a given probability measure and m the set of equivalent strictly positive probability densities in this paper we construct a banach manifold on m modeled on the space l p where p is a reference density for the non parametric q exponential statistical models tsallis s deformed exponential where 0 q 1 is any real number this family is characterized by the fact that when q 1 then the non parametric exponential models are obtained and the manifold constructed by pistone and sempi is recovered up to continuous embeddings on the modeling space the coordinate mappings of the manifold are given in terms of csiszar s divergences the tangent vectors are identified with the one dimensional q exponential models and q deformations of the score function

on the dual of a result of miranda and thompson abstract for a and b real matrices with prescribed singular values miranda and thompson characterized max tr aubv when u and v range over so n the real proper orthogonal group motivated by this result we investigate the location of det a ubv for a b u v as previously the corresponding problem for a and b complex matrices and u and v ranging over the unitary group is also discussed analogies are drawn between the determinantal problems and their tracial versions

g majorization group induced cone orderings and reflection groups a vector y is g majorized by a vector x if y is an element of the convex hull of the orbit of x under the action of a group g it is known that if g is a finite reflection group then g majorization is equivalent to a group induced cone ordering in this paper it is established that if for a finite subgroup g of the orthogonal group g majorization is equivalent to a group induced cone ordering then g must be a reflection group

least squares matching problems abstract a large class of matching problems can be thought of in the following way one is given a group g which acts on a riemannian manifold and one is given two points x 1 and x 2 in the manifold the problem is to find g g such that d g x 1 x 2 is as small as possible if the group has finite cardinality this is a combinatorial optimization problem whereas if g is a lie group one can use the calculus to obtain necessary conditions our main results establish circumstances under which one can relax the combinatorial problem to an appropriate lie group problem and then solve it by the method of steepest descent an interesting class of applications involves embedding the permutation group on n letters in the orthogonal group o n with the latter acting on r n n 1 2 via a symmetricized tensor product the convergence properties of the descent equation which arises in this way are analyzed in some detail

statistical physics algorithms that converge in recent years there has been significant interest in adapting techniques from statistical physics in particular mean field theory to provide deterministic heuristic algorithms for obtaining app

universal computation and other capabilities of hybrid and continuous dynamical systems we explore the simulation and computational capabilities of hybrid and continuous dynamical systems the continuous dynamical systems considered are ordinary differential equations odes for hybrid systems we concentrate on models that combine odes and discrete dynamics e g finite automata we review and compare four such models from the literature notions of simulation of a discrete dynamical system by a continuous one are developed we show that hybrid systems whose equations can describe a precise binary timing pulse exact clock can simulate arbitrary reversible discrete dynamical systems defined on closed subsets of r n the simulations require continuous odes in ir2n with the exact clock as input all four hybrid systems models studied here can implement exact clocks we also prove that any discrete dynamical system in rn can be simulated by continuous odes in rt2n 1 we use this to show that smooth odes in ri3 can simulate arbitrary turing machines and hence possess the power of universal computation we use the famous asynchronous arbiter problem to distinguish between hybrid and continuous dynamical systems we prove that one cannot build an arbiter with devices described by a system of lipschitz odes on the other hand all four hybrid systems models considered can implement arbiters even if their odes are lipschitz

hamiltonian structure of dynamical systems which solve linear programming problems abstract a new class of completely integrable hamiltonian systems which solve linear programming problems is introduced the relations with reduced toda flows double bracket equations interior point methods are established an interpretation in terms of total actions on kahler manifolds is given a complete description of the phase portrait including the explicit description of action angle variables is provided

gaussian processes and martingales for fuzzy valued random variables with continuous parameter abstract the purpose of this paper is to discuss fuzzy valued gaussian processes and martingales with continuous parameter we mainly obtain a representation theorem for fuzzy valued gaussian processes after proving a new embedding theorem based on previous results about set valued and fuzzy valued martingales cf s li y ogura j fuzzy math 4 1996 905 s li y ogura fuzzy sets and systems 101 1999 453 s li y ogura ann probab 26 1998 1384 1402 s li y ogura in proc ifsa 97 vol 4 1997 pp 9 13 s li y ogura in c bertoluzza m a gil d a ralescu eds statistical modeling analysis and management of fuzzy data physica in press y ogura s li fuzzy sets and systems in press we prove optional sampling theorems for set valued and fuzzy valued martingales with continuous parameter

on solving an isospectral flow in this paper we expand the solution of the matrix ordinary differential system originally due to bloch and iserles of the form x n x 2 t 0 x 0 x 0 sym n n s o n where sym n denotes the space of real n n symmetric matrices and s o n denotes the lie algebra of real n n skew symmetric matrices the flow is solved using explicit magnus expansion which respects the isospectrality of the system we represent the terms of expansion as binary rooted trees and deduce an explicit formalism to construct the trees recursively

on computation of the stability radius for nonlinearly structured perturbations this paper considers the problem of finding a perturbation matrix with the least spectral norm such that a matrix valued function becomes singular where the dependence of the function on the perturbation is allowed to be nonlinear it is proved that such a problem can be approximated by a smooth unconstrained minimization problem with compact sublevel sets a computational procedure proposed based on this result is demonstrated to be effective in both linear and nonlinear cases

on the norm property of g c radii and eaton triples abstract let v be a euclidean space and o v be the orthogonal group the norm criterion of the g c radius is studied in the framework of an eaton triple v g d

reaching an optimal consensus dynamical systems that compute intersections of convex sets in this paper multi agent systems minimizing a sum of objective functions where each component is only known to a particular node is considered for continuous time dynamics with time varying interconnection topologies assuming that each node can observe a convex solution set of its optimization component and the intersection of all such sets is nonempty the considered optimization problem is converted to an intersection computation problem by a simple distributed control rule the considered multi agent system with continuous time dynamics achieves not only a consensus but also an optimal agreement within the optimal solution set of the overall optimization objective directed and bidirectional communications are studied respectively and connectivity conditions are given to ensure a global optimal consensus in this way the corresponding intersection computation problem is solved by the proposed decentralized continuous time algorithm we establish several important properties of the distance functions with respect to the global optimal solution set and a class of invariant sets with the help of convex and non smooth analysis

analog computation with dynamical systems a b s t r a c t physical systems exhibit various levels of complexity their long term dynamics may converge to fixed points or exhibit complex chaotic behavior this paper presents a theory that enables to interpret natural processes as special purpose analog computers since physical systems are naturally described in continuous time a definition of computational complexity for continuous time systems is required in analogy with the classical discrete theory we develop fundamentals of computational complexity for dynamical systems discrete or continuous in time on the basis of an intrinsic time scale of the system dissipative dynamical systems are classified into the computational complexity classes pd co rpd npd and exp t corresponding to their standard counterparts according to the complexity of their long term behavior the complexity of chaotic attractors relative to regular ones leads to the conjecture pa fi npj continuous time flows have been proven useful in solving various practical problems our theory provides the tools for an algorithmic analysis of such flows as an example we analyze the continuous hopfield network 1998 elsevier science b v all rights reserved

optimal control and geodesic flows in this paper we analyze and generalize from the point of view of the maximum principle a class of nonlinear optimal control problems originally introduced in brockett 1994 the optimal control problems discussed here lie on the adjoint orbits of compact lie groups or on grassmann manifolds and the structure of the equations that arise depends on the metric used the so called normal metric we consider some special cases of the general problem and their meaning from the point of view of riemannian geometry and hamiltonian mechanics

inverse eigenvalue problems a collection of inverse eigenvalue problems are identified and classified according to their characteristics current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper this exposition also reveals many open questions that deserve further study an extensive bibliography of pertinent literature is attached

functional dissipation microarrays for classification in this article we describe a new method of extracting information from signals called functional dissipation that proves to be very effective for enhancing classification of high resolution texture rich data our algorithm bypasses to some extent the need to have very specialized feature extraction techniques and can potentially be used as an intermediate feature enhancement step in any classification scheme functional dissipation is based on signal transforms but uses the transforms recursively to uncover new features we generate a variety of masking functions and extract features with several generalized matching pursuit iterations in each iteration the recursive process modifies several coefficients of the transformed signal with the largest absolute values according to the specific masking function in this way the greedy pursuit is turned into a slow controlled dissipation of the structure of the signal that for some masking functions enhances separation among classes our case study in this paper is the classification of crystallization patterns of amino acids solutions affected by the addition of small quantities of proteins

a supplement to the von neumann trace inequality for singular values abstract the von neumann trace inequality for singular values is reexamined by determining the possible values for the diagonal elements contributing to the trace the possible values for the eigenvalues contributing to the trace are also found

ordinary differential equations and the symmetric eigenvalue problem in this paper the authors develop a general framework for calculating the eigenvalues of a symmetric matrix using ordinary differential equations new algorithms are suggested and old algorithms including qr are interpreted

flow equations and normal ordering a survey first we give an introduction to the method of diagonalizing or block diagonalizing continuously a hamiltonian and explain how this procedure can be used to analyse the two dimensional hubbard model then we give a short survey on applications of this flow equation on other models finally we outline how symmetry breaking can be introduced by means of a symmetry breaking of the normal ordering not of the hamiltonian

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

convergence of set valued and fuzzy valued martingales the purpose of this paper is to prove the convergence theorems of set valued and fuzzy valued martingales in kuratowski mosco sense without assuming that their values are compact or of compact level sets

limit theorems for fuzzy random variables a strong law of large numbers and a central limit theorem are proved for independent and identically distributed fuzzy random variables whose values are fuzzy sets with compact levels the proofs are based on embedding theorems as well as on probability techniques in banach space

a statistical manifold modeled on orlicz spaces using kaniadakis κ exponential models abstract given a probability measure on the space of strictly positive densities m we construct a topological manifold on which the elements are connected by exponential models in the form q exp u k p u p where exp x 1 2 x 2 x 1 x y x 1 2 y 2 y 1 2 x 2 p q m and their local representations are elements of an orlicz space i e the manifold is modeled on orlicz spaces parameter k is the g kaniadakis parameter for deformed exponentials which is strongly relevant to relativity and statistical complex models in statistical mechanics functional k p is the deformed counterpart of the cumulant mapping and satisfies that if 0 we obtain the usual cumulant functional of the exponential manifold moreover in this limit case the exponential manifold constructed by pistone and sempi is recovered in the context of deformed exponentials we prove that the function cosh 1 where cosh x is the deformed hyperbolic cosine is a young function and generates the orlicz space on which the exponential manifold is modeled namely l p this construction differs from the one made by pistone on the paper exponential models from the geometrical viewpoint since this last one is based on divergence functionals and modeled on lebesgue spaces l 1 p the use of deformed models is interesting since they generalize the exponential models and extend them to non additive systems