dynamical system for computing the eigenvectors associated with the largest eigenvalue of a positive definite matrix a dynamical system for computing the largest eigenpair of a given positive matrix is introduced certain qualitative properties of the proposed system are analyzed in detail the authors results suggest that the weight bounding term in pca algorithms for extracting the first principal component could assume a class of forms

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

an algorithm for exact eigenvalue calculations for rotationally periodic structures an existing algorithm ensures that no eigenvalues are missed when using the stiffness matrix method of structural analysis where the eigenvalues are the natural frequencies of undamped free vibration analyses or the critical load factors of buckling problems the algorithm permits efficient multi level substructuring and gives exact results when the member equations used are those obtained by solving appropriate differential equations the present paper extends this algorithm to cover rotationally periodic i e cyclically symmetric three dimensional structures which are analysed by using complex arithmetic to obtain a stiffness matrix which involves only one of the rotationally repeating portions of the structure nodes and members are allowed to coincide with the axis of rotational periodicity and the resulting modes are classified rigid body freedoms are accounted for empirically and the exact member equations and efficient multi level substructuring of the earlier algorithm can be used when assembling the stiffness matrix of the repeating portion

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 proportional derivative control strategy for restarting the gmres m algorithm abstract restarted gmres or gmres m is normally used for solving large linear systems a x b with a general possibly nonsymmetric matrix a although the restarted gmres consumes less computational time than its counterpart full gmres if the restarting parameter is not correctly chosen its convergence cannot be guaranteed and the method may converge slowly unfortunately it is difficult to know how to choose this parameter a priori in this article we regard the gmres m method as a control problem in which the parameter m is the controlled variable and propose a new control inspired strategy for choosing the parameter m adaptively at each iteration the advantage of this control strategy method is that only a few additional vectors need to be stored and the controller has the capacity to modify the dimension of the krylov subspace whenever any convergence problem is detected numerical experiments based on benchmark problems show that the proposed control strategy accelerates the convergence of gmres m

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

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

the constrained newton method on a lie group and the symmetric eigenvalue problem the newton method for estimating a critical point of a real function is formulated in a coordinate free manner on an arbitrary lie group convergence proofs for the numerical method are given an application of the general approach to computing the eigenvalues of a symmetric matrix is given and the resultant algorithm is compared with the classical shifted qr algorithm properties of the method described suggest that it is of interest for certain computations in online and adaptive environments

on the decomposition of generalized eigenproblems for the free vibration analysis of cyclically symmetric finite element models free vibration analysis is a major part of any dynamic analysis natural frequencies and related mode shapes may be obtained from free vibration analysis as the solutions of generalized eigenproblems although the eigensolutions of large scale structures require large computational efforts these solutions may be achieved simply for symmetric structures we present an efficient method for the decomposition of generalized eigenproblems related to finite element models with cyclic symmetry having nodes at the axis of symmetry into eigensubproblems with significantly smaller dimensions this decomposition is obtained by block diagonalization of a matrix with a special pattern known as a block circulant using the concept of the kronecker product and similarity transformations the proposed method is applied to three finite element models discretized by triangular and four node quadrilateral plate and shell elements and its efficiency accuracy and simplicity are evaluated copyright 2009 john wiley sons ltd

vector fields global flows and the calculus of pulses the point of this paper is to show a remarkable property of functional expansions in the form of iterated integrals in capturing the dynamics of digital computation when realized in the form of pulse spike driven controlled smooth dynamical systems the key features of the structure thus captured by the formalism are a the combinatorics of spike trains in various input channels and their effect on the discrete time dynamics of the realized automaton and b the relative timing precedence of the spike trains in various input channels we do this by providing globally valid functional expansions for a certain kind of hybrid dynamical system when subject to inputs which resemble pulse trains we give these functional expansions in the form of a product of a nonlinear holonomy factor and a local fliess functional expansion we also obtain a computational scheme to determine the response of finite automata to input sequences which is meant as a substitute for the naive iteration of the next state maps we obtain this by embedding finite automata in suitable controlled dynamical systems on the special orthogonal group incidentally we show that our realization of automata can also be modified to realize hybrid machines in which a finite automaton is interacting with an external system some of whose analog variables are converted into interspike time intervals which in turn are simultaneously tokenized and acted upon by the modified realization of the finite automaton the latter is achieved intrinsically by exploiting the dynamics of the realization

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

model predictive control based on linear programming the explicit solution we study model predictive control mpc schemes for discrete time linear time invariant systems with constraints on inputs and states that can be formulated using a linear program lp in particular we focus our attention on performance criteria based on a mixed 1 norm namely 1 norm with respect to time and norm with respect to space first we provide a method to compute the terminal weight so that closed loop stability is achieved we then show that the optimal control profile is a piecewise affine and continuous function of the initial state and briefly describe the algorithm to compute it the piecewise affine form allows to eliminate online lp as the computation associated with mpc becomes a simple function evaluation besides practical advantages the availability of the explicit structure of the mpc controller provides an insight into the type of control action in different regions of the state space and highlights possible conditions of degeneracies of the lp such as multiple optima

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

a rasch model for partial credit scoring a unidimensional latent trait model for responses scored in two or more ordered categories is developed this partial credit model is a member of the family of latent trait models which share the property of parameter separability and so permit specifically objective comparisons of persons and items the model can be viewed as an extension of andrich s rating scale model to situations in which ordered response alternatives are free to vary in number and structure from item to item the difference between the parameters in this model and the category boundaries in samejima s graded response model is demonstrated an unconditional maximum likelihood procedure for estimating the model parameters is developed

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

on the universality of the probability distribution of the product b 1x of random matrices consider random matrices a of dimension m x m n drawn from an ensemble with probability density f tr aa with f x a given appropriate function break a b x into an m x m block b and the complementary m x n block x and define the random matrix z b 1 x we calculate the probability density function p z of the random matrix z and find that it is a universal function independent of f x the universal probability distribution p z is a spherically symmetric matrix variate t distribution universality of p z is essentially a consequence of rotational invariance of the probability ensembles we study as an application we study the distribution of solutions of systems of linear equations with random coefficients and extend a classic result due to girko

hessian riemannian gradient flows in convex programming in view of solving theoretically constrained minimization problems we investigate the properties of the gradient flows with respect to hessian riemannian metrics induced by legendre functions the first result characterizes hessian riemannian structures on convex sets as metrics that have a specific integration property with respect to variational inequalities giving a new motivation for the introduction of bregman type distances then the general evolution problem is introduced and global convergence is established under quasi convexity conditions with interesting refinements in the case of convex minimization some explicit examples of these gradient flows are discussed dual trajectories are identified and sufficient conditions for dual convergence are examined for a convex program with positivity and equality constraints some convergence rate results are established in the case of a linear objective function several optimality characterizations of the orbits are given optimal path of viscosi

a sort jacobi algorithm for semisimple lie algebras the classical jacobi algorithm is extended to an unified lie algebraic approach the conventional jacobi algorithm minimize the distance to diagonality they reduce the off norm i e the sum of squares of off diagonal entries sorting the diagonal elements after each step would accelerate the convergence but there are difficulties to apply this sorting to the off norm that has to minimize using the gradient flow of a trace function it r w brockett dynamical systems that sort lists diagonalize matrices and solve linear programming problems linear algebra appl 146 79 91 1991 zbl 0719 90045 as a more appropriate distance measure matrices can be diagonalized and the eigenvalues can be simultaneously sorted in this paper the sort jacobi algorithm is extended to a large class of structured matrices besides the semisimple lie algebra cases such as e g the symmetric hermitian and skew symmetric eigenvalue problem and the real and complex singular value decomposition par the local quadratic convergence of the sort jacobi method is proved for the regular case but not for the case in which the eigenvalues or singular values occur in clusters for these more complicated irregular problem a future publication is announced par the results are illustrated considering the lie algebra of derivations of the complex octonions

brief paper a hybrid steepest descent method for constrained convex optimization this paper describes a hybrid steepest descent method to decrease over time any given convex cost function while keeping the optimization variables in any given convex set the method takes advantage of the properties of hybrid systems to avoid the computation of projections or of a dual optimum the convergence to a global optimum is analyzed using lyapunov stability arguments a discretized implementation and simulation results are presented and analyzed this method is of practical interest to integrate real time convex optimization into embedded controllers thanks to its implementation as a dynamical system its simplicity and its low computation cost

non perturbative flow equations from continuous unitary transformations we use a novel parametrization of the flowing hamiltonian to show that the flow equations based on continuous unitary transformations as proposed by wegner can be implemented through a nonlinear partial differential equation involving one flow parameter and two system specific auxiliary variables the implementation is non perturbative as the partial differential equation involves a systematic expansion in fluctuations controlled by the size of the system rather than the coupling constant the method is applied to the lipkin model to construct a mapping which maps the non interacting spectrum onto the interacting spectrum to a very high accuracy this function is universal in the sense that the full spectrum for any large number of particles can be obtained from it in a similar way expectation values for a large class of operators can be obtained which also makes it possible to probe the structure of the eigenstates

a control theoretic study on iterative solutions to nonlinear equations for applications in embedded systems in this paper the fixed point iteration and newton s methods for iteratively solving nonlinear equations are studied in the control theoretical framework this work is motivated by the ever increasing demands for integrating iterative solutions of nonlinear functions into embedded control systems the use of the well established control theoretical methods for our application purpose is inspired by the recent control theoretical study on numerical analysis our study consists of two parts in the first part the existing fixed point iteration and newton s methods are analysed using the stability theory for the sector bounded lure s systems the second part is devoted to the modified iteration methods and the integration of sensor signals into the iterative computations the major results achieved in our study are besides some academic examples applied to the iterative computation of the air path model embedded in the engine control systems

an introduction to lie group integrators basics new developments and applications we give a short and elementary introduction to lie group methods a selection of applications of lie group integrators are discussed finally a family of symplectic integrators on cotangent bundles of lie groups is presented and the notion of discrete gradient methods is generalised to lie groups

an efficient analysis of repetitive structures generated by graph products in this paper an efficient method is developed for the analysis of regular structures a structure is called regular if its model can be formed by a graph product here instead of direct solution of the equations corresponding to a regular structure or finding the inverse of the stiffness matrix directly modal analysis is used and eigenvectors are employed for calculating the displacements and then internal forces of the structures for this purpose first an efficient method is developed for calculating the eigenvectors of the product graphs and then a method is presented for using these eigenvectors for evaluating the displacements of a structure

system theory for numerical analysis many numerical schemes can be suitably studied from a system theoretic point of view this paper studies the relationship between the two disciplines that is numerical analysis and system theory we first see that various iterative solution schemes for linear and nonlinear equations can be suitably transformed into the form of a closed loop feedback system and show the crucial role of the internal model principle in such a context this leads to new stability criteria for newton s method we then study runge kutta type methods for solving differential equations and also derive new stability criteria based on recent results on lmi a numerical example is given to illustrate the advantage of the present theory

dynamic stiffness matrix development and free vibration analysis of a moving beam abstract the dynamic stiffness matrix of a moving bernoulli euler beam is developed and used to investigate its free flexural vibration characteristics in order to develop the dynamic stiffness matrix it is necessary to derive and solve the governing differential equation of motion of the moving beam in closed analytical form the solution is then used to obtain the general expressions for both responses and loads boundary conditions are applied to determine the constants in the general solution leading to the formation of the frequency dependent dynamic stiffness matrix of the moving beam relating the amplitudes of the harmonically varying loads to those of the corresponding responses the application of the resulting dynamic stiffness matrix using the wittrick williams algorithm is demonstrated by some illustrative examples numerical results for both simply supported and fixed fixed end conditions of the beam are discussed and wherever possible some are compared with those available in the literature

spectral analysis of fokker planck and related operators arising from linear stochastic differential equations we study spectral properties of certain families of linear second order differential operators arising from linear stochastic differential equations we construct a basis in the hilbert space of square integrable functions using modified hermite polynomials and obtain a representation for these operators from which their eigenvalues and eigenfunctions can be computed in particular we completely describe the spectrum of the fokker planck operator on an appropriate invariant subspace of rapidly decaying functions the eigenvalues of the fokker planck operator provide information about the speed of convergence of the corresponding probability distribution to steady state which is important for stochastic estimation and control applications we show that the operator families under consideration can be realized as solutions of differential equations in the double bracket form on an operator lie algebra which leads to a simple expression for the flow of their eigenfunctions

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

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

on quadratic stability of systems with structured uncertainty this paper considers the problem of stability robustness with respect to a class of nonlinear time varying perturbations which are bounded in a component wise rather than aggregated manner a family of robustness bounds is parameterized in terms of a non singular symmetric matrix it is shown that the problem of computing the largest robustness bound over the set of non singular symmetric matrices can be approximated by a smooth minimization problem over a compact set a convergent algorithm for computing an optimal robustness bound is proposed in the form of a gradient flow

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

explicit magnus expansions for nonlinear equations in this paper we develop and analyse new explicit magnus expansions for the nonlinear equation y a t y y defined in a matrix lie group in particular integration methods up to order four are presented in terms of integrals which can be either evaluated exactly or replaced by conveniently adapted quadrature rules the structure of the algorithm allows us to change the step size and even the order during the integration process thus improving its efficiency several examples are considered including isospectral flows and highly oscillatory nonlinear differential equations

in memory of kenneth g wilson kenneth wilson had an enormous impact on the renormalization group and field theories in general i had the great pleasure to work in three fields to which he contributed essentially critical phenomena gauge invariance in duality and confinement and flow equations and similarity renormalization

exact eigenvalue calculations for structures with rotationally periodic substructures on considere 2 classes de sous structures a symetrie cyclique autour d un axe longitudinal l une n est reliee a sa structure mere qu aux extremites de son axe tandis que l autre est reliee par un nombre quelconque de n uds a sa structure mere qui partage alors sa periodicite rotationnelle

isospectral flows on a class of finite dimensional jacobi matrices abstract we present a new matrix valued isospectral ordinary differential equation that asymptotically block diagonalizes n n zero diagonal jacobi matrices employed as its initial condition this o d e features a right hand side with a nested commutator of matrices and structurally resembles the double bracket o d e studied by r w brockett in 1991 we prove that its solutions converge asymptotically that the limit is block diagonal and above all that the limit matrix is defined uniquely as follows for n even a block diagonal matrix containing 2 2 blocks such that the super diagonal entries are sorted by strictly increasing absolute value furthermore the off diagonal entries in these 2 2 blocks have the same sign as the respective entries in the matrix employed as the initial condition for n odd there is one additional 1 1 block containing a zero that is the top left entry of the limit matrix the results presented here extend some early work of kac and van moerbeke

on the finiteness of the weighted likelihood estimator of ability the purpose of this note is to focus on the finiteness of the weighted likelihood estimator wle of ability in the context of dichotomous and polytomous item response theory irt models it is established that the wle always returns finite ability estimates this general result is valid for dichotomous one two three and four parameter logistic irt models the class of polytomous difference models and divide by total models independently of the number of items the item parameters and the response patterns further implications of this result are outlined

weighted likelihood estimation of ability in item response theory applications of item response theory which depend upon its parameter invariance property require that parameter estimates be unbiased a new method weighted likelihood estimation wle is derived and proved to be less biased than maximum likelihood estimation mle with the same asymptotic variance and normal distribution wle removes the first order bias term from mle two monte carlo studies compare wle with mle and bayesian modal estimation bme of ability in conventional tests and tailored tests assuming the item parameters are known constants the monte carlo studies favor wle over mle and bme on several criteria over a wide range of the ability scale

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

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

on the relationships between jeffreys modal and weighted likelihood estimation of ability under logistic irt models this paper focuses on two estimators of ability with logistic item response theory models the bayesian modal bm estimator and the weighted likelihood wl estimator for the bm estimator jeffreys prior distribution is considered and the corresponding estimator is referred to as the jeffreys modal jm estimator it is established that under the three parameter logistic model the jm estimator returns larger estimates than the wl estimator several implications of this result are outlined

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

asymptotic hamiltonian dynamics the toda lattice the three wave interaction and the non holonomic chaplygin sleigh abstract in this paper we discuss asymptotic stability in energy preserving systems which have an almost poisson structure in particular we consider a class of poisson systems which includes the toda lattice in standard hamiltonian systems one of course does not expect asymptotic stability the key here is the structure of the phase space of the poisson or almost poisson systems and the nature of the equilibria our systems are in some sense generalizations of the integrable toda lattice system but are not integrable in general as a particular example we point out an interesting connection between three mechanical two degree of freedom systems that exhibit asymptotic stability two of them are classical hamiltonian systems while the third is a non holonomic system non holonomic systems are generally energy preserving but not hamiltonian but the system analyzed here turns out to have a phase space which is the union of hamiltonian ones we also discuss various higher dimensional examples

a control theoretic approach to the design of zero finding numerical methods in this paper it is shown how standard iterative methods for solving linear and nonlinear equations can be designed from the point of view of control appropriate choices of control lyapunov functions clfs lead to both continuous and discrete time versions of the newton raphson and conjugate gradient algorithms as well as new variants

on explicitly solvable gradient systems of moser karmarkar type abstract a generalization and a classification of explicitly solvable gradient systems which appear in linear programming and neurodynamics are studied it is shown that nonlinear dynamical systems of moser karmarkar type generically take both a lax representation and a gradient form a related optimization problem on a simplex is discussed an integration procedure and a complete description of the invariant manifolds and the phase portrait of the dynamical systems of moser karmarkar type are also investigated

asymptotically correct standardization of person fit statistics beyond dichotomous items the l z statistic drasgow et al in br j math stat psychol 38 67 86 1985 is one of the most popular person fit statistics armstrong et al in pract assess res eval 12 16 1 10 2007 snijders psychometrika 66 331 342 2001 derived the asymptotic null distribution of l z when the examinee ability parameter is estimated he also suggested the l z statistic which is the asymptotically correct standardized version of l z however snijders psychometrika 66 331 342 2001 only considered tests with dichotomous items in this paper the asymptotic null distribution of l z is derived for mixed format tests those that include both dichotomous and polytomous items the asymptotically correct standardized version of l z which can be considered as the extension of l z to such tests is suggested the type i error rate and power of the suggested statistic are examined from several simulated datasets the suggested statistic is computed using a real dataset the suggested statistic appears to be a satisfactory tool for assessing person fit for mixed format tests

the use of cyclic symmetry in efg analysis for heat transfer problems rotationally periodic or cyclic symmetry is exploited in the element free galerkin efg analysis for heat transfer problems of two dimensional systems it is proved that the coefficient matrices of the global efg equations are block circulant furthermore a partitioning algorithm is presented and the computational convenience and efficiency are demonstrated a technique dealing with asymmetric boundary conditions is developed to extend the application of the proposed approach numerical examples are given to illustrate the advantages of such exploitation of symmetry in the context copyright 2004 john wiley sons ltd

a note on weighted likelihood and jeffreys modal estimation of proficiency levels in polytomous item response models warm in psychometrika 54 427 450 1989 established the equivalence between the so called jeffreys modal and the weighted likelihood estimators of proficiency level with some dichotomous item response models the purpose of this note is to extend this result to polytomous item response models first a general condition is derived to ensure the perfect equivalence between these two estimators second it is shown that this condition is fulfilled by two broad classes of polytomous models including among others the partial credit rating scale graded response and nominal response models

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

steepest descent algorithms for optimization under unitary matrix constraint in many engineering applications we deal with constrained optimization problems with respect to complex valued matrices this paper proposes a riemannian geometry approach for optimization of a real valued cost function t of complex valued matrix argument w under the constraint that w is an n times n unitary matrix we derive steepest descent sd algorithms on the lie group of unitary matrices u n the proposed algorithms move towards the optimum along the geodesics but other alternatives are also considered we also address the computational complexity and the numerical stability issues considering both the geodesic and the nongeodesic sd algorithms armijo step size 1 adaptation rule is used similarly to 2 but with reduced complexity the theoretical results are validated by computer simulations the proposed algorithms are applied to blind source separation in mimo systems by using the joint diagonalization approach 3 we show that the proposed algorithms outperform other widely used algorithms

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

stabilization of nonholonomic systems using isospectral flows in this paper we derive and analyze a discontinuous stabilizing feedback for a lie algebraic generalization of a class of kinematic nonholonomic systems introduced by brockett new directions in applied mathematics p hilton and g young eds springer verlag new york 1982 pp 11 27 the algorithm involves discrete switching between isospectral and norm decreasing flows we include a rigorous analysis of the convergence

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

generalized kyp lemma unified frequency domain inequalities with design applications the celebrated kalman yakubovic spl caron popov kyp lemma establishes the equivalence between a frequency domain inequality fdi and a linear matrix inequality and has played one of the most fundamental roles in systems and control theory this paper first develops a necessary and sufficient condition for an s procedure to be lossless and uses the result to generalize the kyp lemma in two aspects the frequency range and the class of systems and to unify various existing versions by a single theorem in particular our result covers fdis in finite frequency intervals for both continuous discrete time settings as opposed to the standard infinite frequency range the class of systems for which fdis are considered is no longer constrained to be proper and nonproper transfer functions including polynomials can also be treated we study implications of this generalization and develop a proper interface between the basic result and various engineering applications specifically it is shown that our result allows us to solve a certain class of system design problems with multiple specifications on the gain phase properties in several frequency ranges the method is illustrated by numerical design examples of digital filters and proportional integral derivative controllers

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

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

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

experimental computation of real numbers by newtonian machines following a methodology we have proposed for analysing the nature of experimental computation we prove that there is a 3 dimensional newtonian machine which given any point x 2 0 1 can generate an infinite sequence pn qn for n 1 2 of rational number interval approximations that converges to x as n 1 the machine is a system for scattering and collecting particles the theorem implies that every point x 2 0 1 is computable by a simple newtonian kinematic system that is bounded in space and mass and for which the calculation of the n th approximation of x takes place in o n time with o n energy we describe a set of variants of the scatter machine which explain why our machine is non deterministic

lyapunov stability theory of nonsmooth systems this paper develops nonsmooth lyapunov stability theory and lasalle s invariance principle for a class of nonsmooth lipschitz continuous lyapunov functions and absolutely continuous state trajectories computable tests based on filipov s differential inclusion and clarke s generalized gradient are derived the primary use of these results is in analyzing the stability of equilibria of differential equations with discontinuous right hand side such as in nonsmooth dynamic systems or variable structure control

numerical integration methods for the double bracket flow in this paper new methods up to order four based on the magnus expansion are proposed for the numerical integration of the double bracket equation the magnus series is constructed term by term by means of recurrences and a bound on the convergence domain is also provided the new integrators preserve the most salient qualitative features of the flow and are computationally more efficient than other standard lie group solvers such as the runge kutta munthe kaas class of algorithms