optimal cycles in doubly weighted graphs and approximation of bivariate functions by univariate ones the problem of finding optimal cycles in a doubly weighted directed graph problem a is closely related to the problem of approximating bivariate functions by the sum of two univariate functions with respect to the supremum norm problem b the close relationship between problem a and problem b is detected by the characterization 7 4 of the distance dist f t of problem b in part 1 we construct an algorithm for problem a where the essential role is played by the minimal lengthsy j k defined by 2 3 if weight functiont 1 then the minimum of problem a is computed by equality 2 4 ift 1 then the minimum is obtained by a binary search procedure algorithm 3 in part 2 we construct our algorithms for solving problem b by following exactly the ideas of part 1 by algorithm 4 we compute the minimal pseudolengthsh k y m defined by 7 5 if weight functiont 1 then the infimum dist f t of problem b is obtained by equality 7 12 which is closely related to 2 4 ift 1 we compute the infimum dist f t by the binary search procedure algorithm 5 additionally algorithm 4 leads to a constructive proof of the existence of continuous optimal solutions of problem b see theorem 7 1e which is already known in caset 1 but unknown in caset 1 interesting applications to the steady state behaviour of industrial processes with interference sect 3 and the solution of integral equations problem c are included

a relative error bound for balanced stochastic truncation recently several methods have appeared for the approximation of power spectra notably balanced stochastic truncation bst it is shown that bst satisfies a relative error bound approximately twice the bound for the relative error method rem proper this offers a quantitative basis for the observation that bst and rem produce similar reduced order models balanced stochastic truncation can therefore be interpreted as providing a computationally simple algorithm for relative error approximation

boundary value problems for nth order lipschitz equations on considere l unicite des solutions de problemes aux valeurs limites pour une equation scalaire d ordre n y n f t y y y n 1 ou f est continue sur a b r n et satisfait une condition de lipschitz

iterative methods for solving right focal point boundary value problems abstract we provide necessary and sufficient conditions for the existence of a solution of n th order differential equation satisfying right focal point boundary conditions for the linear problems we propose forward forward and backward backward one pass practical shooting methods which convert the boundary value problems to its equivalent initial value problems for the nonlinear problems iterative methods are discussed and a priori conditions for the convergence are obtained several examples are also illustrated

internal positivity preserved model reduction this article studies model reduction of continuous time stable positive linear systems under the hankel norm h norm and h 2 norm performance the reduced order systems preserve the stability as well as the positivity of the original systems this is achieved by developing new necessary and sufficient conditions of the model reduction performances in which the lyapunov matrices are decoupled with the system matrices in this way the positivity constraints in the reduced order model can be imposed in a natural way as the model reduction performances are expressed in linear matrix inequalities with equality constraints the desired reduced order positive models can be obtained by using the cone complementarity linearisation iterative algorithm a numerical example is presented to illustrate the effectiveness of the given methods

optimality for boundary value problems for lipschitz equations on considere les solutions de problemes aux valeurs limites pour une equation scalaire d ordre n x n f t x x x n 1 ou f est continue sur a b r n et satisfait une condition de lipschitz on determine les sous intervalles de longueur optimale de a b en fonctions des coefficients de lipschitz sur lesquels des solutions sont uniques

multiple solutions and eigenvalues for third order right focal boundary value problems abstract we are concerned with the right focal boundary value problem x t f t x t t 1 t t 3 x t 1 x t 2 x t 3 0 and the associated eigenvalue problem x t a t f x t with the same boundary conditions under various assumptions on f a and we establish intervals of the parameter which yield the existence of a positive solution of the eigenvalue problem by placing certain restrictions on the nonlinearity we prove the existence of at least one at least two at least three and infinitely many positive solutions of the boundary value problem by applying some known fixed point theorems as well as some recent generalizations of these fixed point theorems

attractors via random perturbations we discuss conditions which ensure that weak limits of invariant measures of small random perturbations of dynamical systems have their supports on attractors

large deviation principle with generated pseudo measures contemporary large deviation theory uses various approaches one of possible recent approaches is based on the idempotent sup measure and related integrals based on operations sup and the usual product in this paper the large deviation convergence of a sequence of generated pseudo measures to sup decomposable measure is introduced the main result of the paper gives a necessary condition for introduced convergence

hankel norm approximation of linear systems with time varying delay continuous and discrete cases this paper investigates the problem of hankel norm model reduction for linear systems with time varying delay in the state for a given stable system our attention is focused on the construction of reduced order model which guarantees the corresponding error system to be asymptotically stable and has a specified hankel norm error performance two different approaches are proposed to solve this problem one casts the model reduction into a convex optimization problem by using a linearization procedure and the other is based on the cone complementarity linearization idea which casts the model reduction into a sequential minimization problem subject to linear matrix inequality constraints both continuous and discrete time cases are considered a numerical example is provided to show the effectiveness of the proposed theory

a new reproducing kernel method for linear nonlocal boundary value problems in the previous works the authors developed the reproducing kernel method rkm for nonlocal boundary value problems a key of the method is the construction of the reproducing kernel rk satisfying the homogenous boundary conditions bcs of the considered problems however it is very difficult to obtain the rk of a reproducing kernel space satisfying nonlocal bcs or nonlinear bcs even if the rk is found its representation is also very complicated compared with the rk without any bcs in this paper we will present a new rkm for linear nonlocal boundary value problems the method can avoid reducing the inhomogeneous bcs to homogeneous bcs and constructing rk satisfying homogeneous nonlocal linear bcs numerical examples are provided to show the effectiveness of the new method

l sub 2 optimal filter reduction a closed loop approach this paper is concerned with the problem of order reduction of a full order kalman filter for a stable linear signal model so that the steady state filtering error variance associated with the reduced order filter is minimized by an orthogonal parameterization the above problem is formulated to minimize the filtering error variance over a set of orthogonal matrices both continuous and iterative algorithms are derived to compute an optimal reduced order filter the algorithms are shown to possess good properties including the desirable convergence property the proposed algorithms are simple and effective numerical examples are presented to demonstrate the effectiveness and the significant advantages of the proposed algorithms over the existing open loop methods such as the well known balanced truncation method

spectral analysis in linear process identification a new look at an old problem the problem of estimating the transfer function frequency response of a linear process by means of a modified spectral analysis method is treated a method of determining the maximum time lags of the correlation functions which makes use of the available a priori physical knowledge about the process and the input signal is proposed this leads in a natural way to the use of an asymmetrical window technique some simulation results are used to illustrate the method based on the asymptotic properties of the estimate the input design is studied finally it will be shown that this work can be used for the estimation of an upper bound of the model uncertainty in the frequency domain

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

uniqueness and existence for bounded boundary value problems the existence and uniqueness of solutions for the boundary value problems with general linear point evaluation boundary conditions is established we assume that f is bounded and that there is uniqueness on a homogeneous problem and on the linear variational problems

dwell time and average dwell time methods based on the cycle ratio of the switching graph abstract a new method to find an upper bound on dwell time and average dwell time for switched linear systems is proposed the method is based on computing the maximum cycle ratio and the maximum cycle mean of the directed graph that governs switchings for planar switched systems an upper bound for dwell time and average dwell time can be estimated by considering only the cycles of length two

a conforming decomposition theorem a piecewise linear theorem of the alternative and scalings of matrices satisfying lower and upper bounds a scaling of a nonnegative matrixa is a matrixxay 1 wherex andy are nonsingular nonnegative diagonal matrices some condition may be imposed on the scaling for example whena is square x y or detx dety we characterize matrices for which there exists a scaling that satisfies predetermined upper and lower bound our principal tools are a piecewise linear theorem of the alternative and a theorem decomposing a solution of a system of equations as a sum of minimal support solutions which conform with the given solutions

probability one homotopy algorithms for solving the coupled lyapunov equations arising in reduced order h2 h modeling estimation and control optimal reduced order modeling estimation and control with respect to combined h 2 h criteria give rise to coupled lyapunov and riccati equations to develop reliable numerical algorithms for these problems this paper focuses on the coupled lyapunov equations which appear as a subset of the synthesis equations in particular this paper systematically examines the requirements of probability one homotopy algorithms to guarantee global convergence homotopy algorithms for nonlinear systems of equations construct a continuous family of systems and solve the given system by tracking the continuous curve of solutions to the family the main emphasis is on guaranteeing transversality for several homotopy maps based upon the pseudogramian formulation of the coupled lyapunov equations and variations based upon canonical forms these results are essential to the probability one homotopy approach by guaranteeing good numerical properties in the computational implementation of the homotopy algorithms

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

existence and uniqueness of solutions for nth order nonlinear two point boundary value problems in this paper we present some general results of existence and uniqueness of solutions for nth order nonlinear two point boundary value problems by using shooting method together with maximum principle and kneser hukahara continuum theorem as applications we give certain concrete sufficient conditions for the existence and uniqueness of solutions of nonlinear two point boundary value problems of nth order lipschitz equations

brief paper positivity preserving h model reduction for positive systems this paper is concerned with the model reduction of positive systems for a given stable positive system our attention is focused on the construction of a reduced order model in such a way that the positivity of the original system is preserved and the error system is stable with a prescribed h performance based upon a system augmentation approach a novel characterization on the stability with h performance of the error system is first obtained in terms of linear matrix inequality lmi then a necessary and sufficient condition for the existence of a desired reduced order model is derived accordingly furthermore iterative lmi approaches with primal and dual forms are developed to solve the positivity preserving h model reduction problem finally a compartmental network is provided to show the effectiveness of the proposed techniques

multiobjective h sub 2 h sub spl infin control for a linear time invariant system with several disturbance inputs and controlled outputs we show how to minimize the nominal h sub 2 norm performance in one channel while keeping bounds on the h sub 2 norm or h sub spl infin norm performance implying robust stability in the other channels this multiobjective h sub 2 h sub spl infin problem in an infinite dimensional space is reduced to sequences of finite dimensional convex optimization problems we show how to compute the optimal value and how to numerically detect the existence of a rational optimal controller if it exists we reveal how the novel trick of optimizing the trace norm of the youla parameter over certain convex constraints allows one to design a nearly optimal controller whose youla parameter is of the same order as the optimal one

uncertain dynamical systems defined by pseudomeasures this paper deals with uncertain dynamical systems in which predictions about the future state of a system are assessed by so called pseudomeasures two special cases are stochastic dynamical systems where the pseudomeasure is the conventional probability measure and fuzzy dynamical systems in which the pseudomeasure is a so called possibility measure new results about possibilistic systems and their relation to deterministic and to stochastic systems are derived by using idempotent pseudolinear algebra by expressing large deviation estimates for stochastic perturbations in terms of possibility measures we obtain a new interpretation of the freidlin wentzell quasipotentials for stochastic perturbations of dynamical systems as invariant possibility densities

optimal hankel model reduction for nonminimal systems a basis free descriptor system representation is shown to facilitate the computation of all minimum degree and optimal kth order all pass extensions and hankel norm approximants the descriptor representation has the same simple form for both the optimal and suboptimal cases the method makes hankel model reduction practical for nonminimal and nearly nonminimal systems by eliminating the ill conditioned calculation of a minimal balanced realization a simple numerically sound method based on singular value decomposition enables the results to be expressed in state space form

model simplification for switched hybrid systems abstract this paper is concerned with the problem of h model reduction for switched system which is an important class of hybrid systems frequently encountered in practical situations two sharply different approaches are proposed to solve this problem the first approach casts the model reduction into a convex optimization problem which is the first attempt to solve the model reduction problem by using linearization procedure the second one based on the cone complementarity linearization idea casts the model reduction problem into a sequential minimization problem subject to linear matrix inequality constraints both approaches have their own advantages and disadvantages concerning conservatism and computational complexity a numerical example illustrates the effectiveness of the proposed theories

multiplicative hankel norm approximation of linear multivariable systems abstract in control system design specifications are commonly given in terms of log magnitude quantities i e decibels this description induces a multiplicative or relative error criterion for plant model reduction techniques from the theory of additive hankel norm approximation and spectral factorization are synthesized to produce a multiplicative approximant of a possibly unstable or non square plant it is shown that the reduced order model preserves the unstable poles and right half plane zeros of the original system explicit state space formulae are provided for the construction of the reduced order model and its error properties are discussed the method is illustrated by a numerical example

existence of solutions of right focal point boundary value problems for ordinary differential equations y f x y y y 1 1 with respect to l two conditions which we will assume are a f x y 7 yz y is continuous on a b x r b solutions of initial value problems for 1 are unique motivated by the idea of right disfocality of a linear nth order differential operator given by muldowney 11 we shall be concerned with uniqueness of solutions of 1 which satisfy boundary conditions of the form y l x yi 1 i n where x1 x q a x for related results in the linear case where x1 x x x see for example nehari 2 peterson 3 and elias 4 5 1 to be more definite we define what is meant by a right m 1 m focal point boundary value problem for 1 on an interval i

existence and uniqueness of solutions of k 2 point nonlocal boundary value problems for ordinary differential equations abstract sufficient conditions are given under which the existence of solutions of 4 point nonlocal boundary value problems for n th order nonlinear ordinary differential equations yields the existence of unique solutions of k 2 point boundary value problems for 1 k n 1 the results are motivated by a henderson and jackson paper involving relationships between 2 point and n point conjugate boundary value problems

quasipotentials for simple noisy maps with complicated dynamics the theory of nonequilibrium potentials or quasipotentials is a physically motivated approach to small random perturbations of dynamical systems leading to exponential estimates of invariant probabilities and mean first exit times in the present article we develop the mathematical foundation of this theory for discrete time systems following and extending the work of freidlin and wentzell and kifer we discuss strategies for calculating and estimating quasipotentials and show their application to one dimensionals unimodal maps the method proves to be especially suited for describing the noise scaling behavior of invariant probabilities e g for the map occurring as the limit of the feigenbaum period doubling sequence we show that the method allows statements about the scaling behavior in the case of localized noise too which does not originally lie within the scope of the quasipotential formalism

optimal interval lengths for nonlocal boundary value problems associated with third order lipschitz equations abstract for the third order differential equation y f x y y y where f x y 1 y 2 y 3 is lipschitz continuous in terms of y i i 1 2 3 we obtain optimal bounds on the length of intervals on which there exist unique solutions of certain nonlocal three and four point boundary value problems these bounds are obtained through an application of the pontryagin maximum principle from the theory of optimal control

frequency weighted optimal hankel norm approximation of stable transfer functions abstract this paper shows how to introduce arbitrary frequency weighting into the optimal hankel norm approximation problem for scalar finite dimensional linear time invariant systems the procedure has applications to the closed form design of reduced order controllers

applications of shortest path algorithms to matrix scalings a symmetric scaling of a nonnegative square matrixa is a matrixxax 1 wherex is a nonsingular nonnegative diagonal matrix by associating a family of weighted directed graphs with the matrixa we are able to adapt the shortest path algorithms to compute an optimal scaling ofa where we call a symmetric scalinga ofa optimal if it minimizes the maximum of the ratio of non zero elements

formulation and solution of structured total least norm problems for parameter estimation the total least squares tls method is a generalization of the least squares ls method for solving overdetermined sets of linear equations ax spl ap b the tls method minimizes spl par e r spl par sub f where r b a e x so that b r spl isin range a e given a spl isin c sup m spl times n with m spl ges n and b spl isin c sup m spl times 1 the most common tls algorithm is based on the singular value decomposition svd of a b however the svd based methods may not be appropriate when the matrix a has a special structure since they do not preserve the structure previously a new problem formulation known as structured total least norm stln and the algorithm for computing the stln solution have been developed the stln method preserves the special structure of a or a b and can minimize the error in the discrete l sub p norm where p 1 2 or spl infin in this paper the stln problem formulation is generalized for computing the solution of stln problems with multiple right hand sides ax spl ap b it is shown that these problems can be converted to ordinary stln problems with one right hand side in addition the method is shown to converge to the optimal solution in certain model reduction problems furthermore the application of the stln method to various parameter estimation problems is studied in which the computed correction matrix applied to a or a b keeps the same toeplitz structure as the data matrix a of a b respectively in particular the l sub 2 norm stln method is compared with the ls and tls methods in deconvolution transfer function modeling and linear prediction problems

uniqueness implies existence on time scales abstract the uniqueness of solutions of initial value boundary value problems along with some uniqueness conditions on two point boundary value problems imply the existence of solutions for the same boundary value problems for the second order nonlinear dynamic equation y f t y y on a time scale this existence is established using shooting methods

model reduction of large scale systems by least squares abstract in this paper we introduce an approximation method for model reduction of large scale dynamical systems this is a projection which combines aspects of the svd and krylov based reduction methods this projection can be efficiently computed using tools from numerical analysis namely the rational krylov method for the krylov side of the projection and a low rank smith type iteration to solve a lyapunov equation for the svd side of the projection for discrete time systems the proposed approach is based on the least squares fit of the r 1 th column of a hankel matrix to the preceding r columns where r is the order of the reduced system the reduced system is asymptotically stable matches the first r markov parameters of the full order model and minimizes a weighted h 2 error the method is also generalized for moment matching at arbitrary interpolation points application to continuous time systems is achieved via the bilinear transformation numerical examples prove the effectiveness of the approach the proposed method is significant because it combines guaranteed stability and moment matching together with an optimization criterion

pseudo additive measures and integrals abstract we suggest pseudo additive measures based on a pseudo addition and discuss integrals with respect to pseudo additive measures a pseudo additive measure is a special type of fuzzy measures to define an integral a multiplication corresponding to a pseudo addition is introduced the resulting integral is an extension of the lebesgue integral in this context radon nikodym like theorems are shown

the chebyshev solution of the linear matrix equationax yb c in this paper we investigate the properties of the chebyshev solutions of the linear matrix equationax yb c wherea b andc are given matrices of dimensionsm r s n andm n respectively wherer m ands n we separately consider two particular cases in the first case we assumem r 1 andn s 1 in the second caser s 1 andm n are arbitrary for these two cases under the assumption that the matricesa andb are full rank we formulate necessary and sufficient conditions characterizing the chebyshev solution ofax yb c and we give the formulas for the chebyshev error then we propose an algorithm which may be applied to compute the chebyshev solution ofax yb c for some particular cases some numerical examples are also given

small random perturbations of dynamical systems and the definition of attractors the strange attractors plotted by computers and seen in physical experiments do not necessarily have an open basin of attraction in view of this we study a new definition of attractors based on ideas of conley we argue that the attractors observed in the presence of small random perturbations correspond to this new definition

on the right focal point boundary value problems for integro differential equations abstract for the nth order nonlinear integro differential equations subject to right focal point boundary conditions we provide necessary and sufficient conditions for the existence uniqueness and convergence of an approximate iterative method

existence and uniqueness of solutions of k point boundary value problems for ordinary differential equations for boundary value problems for linear differential equations in which the boundary conditions are linear the uniqueness of solutions implies the existence of solutions this is not the case in general for nonlinear differential equations but in the past several years a number of results of the uniqueness implies existence type for boundary value problems for nonlinear equations have been proven for example 1 2 4 5 9 111 in this paper we prove some additional results of this type we will be concerned with boundary value problems

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

large deviations from the thermodynamic limit in globally coupled maps abstract systems of a large number n of globally coupled maps have become popular as a relatively simple prototype of high dimensional dynamics showing many interesting and typical phenomena like synchronisation cluster formation and multistability and having potential applications in systems like josephson junction arrays or in biophysical models while much progress has been made in the explanation of the macroscopic behaviour of such systems in the limit n there is still need for a sound theory about the asymptotic behaviour of finite n systems as n approaches infinity this article introduces a method by which it is possible to obtain asymptotic estimates for finite size effects that show up as long term deviations from the thermodynamic limit behaviour this method works for systems of globally coupled noisy maps and is based upon the concept of quasipotentials originally developed by freidlin wentzell and others for describing the influence of small random perturbations on the long term behaviour of dynamical systems the problems of explicitly computing quasipotentials in the present context and potential approximation schemes are discussed all the concepts described in this article are illustrated with a simple example

uniqueness of solutions of right focal problems for third order differential equations on considere l unicite des solutions quand elles existent de certains problemes aux valeurs limites pour l equation differentielle d ordre 3 y f x y y y on suppose que f a b r 3 r est continue

paper relative multiplicative model reduction for unstable and non minimum phase systems this paper considers the relative multiplicative model reduction for unstable and non minimum phase systems using generalized frequency weighted normal representations the proposed method only requires solving two lyapunov equations with possibly indefinite solutions and preserves the number of unstable poles and the number of zeros in the square case furthermore a priori l norm error bounds are also derived for the relative approximation error and the multiplicative approximation error

conjugation interpolation and model matching in h abstract the notion of conjugation is introduced to give a unified framework of algebraic operations connected to the analytic aspects of transfer functions two classes of conjugations are shown to be important one is the lossless conjugation that performs inner outer factorization and the other is the j lossless conjugation that solves the model matching problem related to h 8 control the solution of the model matching problem obtained by j lossless conjugation is regarded as the state space version of the directional interpolation theory and is much simpler than those obtained via the nehari approximation

pseudo lp space and convergence in the framework of the pseudo analysis the classical l p space is generalized and there are proved important properties of introduced space three types of convergence of sequences of measurable functions are considered in this space the inequalities for integrals based on pseudo integral have been recently proposed as the holder minkowski and markov inequalities which are applied in observing relationship among introduced convergences

brief paper on the stability of a reduced order filter based on dominant singular value decomposition of the system dynamics the stability of a reduced order filter rof the gain of which is constructed on the basis of a subspace of dominant singular vectors of the system dynamics is examined a definition of s detectability is introduced it is found that the observability of all unstable and neutral singular vectors s detectability is a sufficient condition for the existence of a stable filter

computing optimal scalings by parametric network algorithms asymmetric scaling of a square matrixa 0 is a matrix of the formxax 1 wherex is a nonnegative nonsingular diagonal matrix having the same dimension ofa anasymmetric scaling of a rectangular matrixb 0 is a matrix of the formxby 1 wherex andy are nonnegative nonsingular diagonal matrices having appropriate dimensions we consider two objectives in selecting a symmetric scaling of a given matrix the first is to select a scalinga of a given matrixa such that the maximal absolute value of the elements ofa is lesser or equal that of any other corresponding scaling ofa the second is to select a scalingb of a given matrixb such that the maximal absolute value of ratios of nonzero elements ofb is lesser or equal that of any other corresponding scaling ofb we also consider the problem of finding an optimal asymmetric scaling under the maximal ratio criterion the maximal element criterion is of course trivial in this case we show that these problems can be converted to parametric network problems which can be solved by corresponding algorithms

hankel norm approximation for well posed linear systems the sub optimal hankel norm approximation problem is solved for a well posed linear system with generating operators a b c and transfer function g satisfying some mild assumptions in the special case of the sub optimal nehari problem an explicit parameterization of all solutions is obtained in terms of the system parameters a b c and g 0

the chebyshev solution of certain matrix equations the problem of approximating ann x m matrixc by matrices of the formx a b y wherea b x andy are of appropriate size is considered the measure of error is the supremum of the absolute values of the individual entries in the error matrix the problem is closely related to that of approximating a bivariate functionf by sums of functions of the formxh gy wherex andg are functions of the first variable alone andh andy are functions of the second variable an old algorithm for constructing best approximations is described and some of the properties of its convergence are discussed

possibility theory an integral theoretic approach this paper introduces possibility integrals as fuzzy integrals w r t possibility measures it is shown that this concept can be used to unify a lot of known results in possibility theory that up to now have been introduced on a more ad hoc basis the integral theoretic approach allows us amongst other things to introduce product and conditional possibility measures in very much the same way as product lebesgue measures and conditional probability measures are introduced furthermore to stress the fact that the concept of order relations is basic for possibility theory all the results are proved for a complete lattice as the evaluation set instead of the more familiar unit interval 0 1

model reduction for state space symmetric systems in this paper the model reduction problem for state space symmetric systems is investigated first it is shown that several model reduction methods such as balanced truncation balanced truncation which preserves the dc gain optimal and suboptimal hankel norm approximations inherit the state space symmetric property furthermore for single input and single output siso state space symmetric systems we prove that the h norm of its transfer functions can be calculated via two simple formulas moreover the siso state space symmetric systems are equivalent to systems with zeros interlacing the poles zip under mild conditions

a stable state space realization in the formulation of h norm computation in the two block hinftyoptimization problem usually we are given the state space realizations of the proper rational matrices r 1 s and r 2 s whose poles are all the open right half plane two problems are studied in the note the first is the evaluation of phi s r 1 s at s s k k 1 2 n where phi s is an inner function whose zeros s k k 1 2 n are the poles of r 1 s this evaluation is essential if chang and pearson s method is used for computing the optimal hinftynorm the problem is solved in state space via the solutions of lyapunov equations neither polynomial matrix manipulations nor numerical pole zero cancellations are involved in the evaluation the second problem is to find a stable state space realization of s s u s r 2 s where u s is an inner matrix this problem arises in the spectral factorization of gamma 2 r 2 ast r 2 doyle and chu had a method for constructing stable s s based on a minimal realization of r 2 s an alternate method is proposed the alternate method does not require a minimal realization of r 2 s and only a lyapunov equation is involved

numerical procedure for solving a minimization eigenvalue problem a numerical procedure for solving the left eigenvalue problem lambda l k min j l j k j k and its right counterpart is described k is a known realn n matrix the first stage uses a modification of the karp and von golitschek procedure for finding the minimum cyclic average ofk along with a minimizing cycle the second stage employs this information to generatel by a very similar algorithm

uniqueness existence and optimality for fourth order lipschitz equations for each t y y y y4 t zi z2 z3 z e a 6 x r4 in particular we will characterize optimal length subintervals of a b in terms of the lipschitz coefficients k i 1 2 3 4 on which certain two three and four point boundary value problems for 1 have unique solutions the techniques we employ here involve applications of the pontryagin maximum principle 14 p 3141 in conjunction with uniqueness implies existence results for solutions of boundary value problems for 1 these techniques are motivated by works of melentsova 15 and melentsova and mil shtein 16 171 and most notably by the two papers by jackson ll 121 furthermore the results contained herein can be considered as extensions of a recent paper by henderson 6 dealing with boundary value problems for third order equations in relating the results of this paper to previous works we will formulate the boundary value problems in terms of the nth order differential equation

max balancing weighted directed graphs and matrix scaling a weighted directed graph g is a triple v a g where v a is a directed graph and g is an arbitrary real valued function defined on the arc set a let g be a strongly connected simple weighted directed graph we say that g is max balanced if for every nontrivial subset of the vertices w the maximum weight over arcs leaving w equals the maximum weight over arcs entering w we show that there exists a up to an additive constant unique potential pv for v v such that v a gp is max balanced where gap pu ga pv for a u v a we describe an o v 2 a algorithm for computing p using an algorithm for computing the maximum cycle mean of g finally we apply our principal result to the similarity scaling of nonnegative matrices

a transformation approach to stochastic model reduction a new and direct approach to stochastic model reduction is developed the order reduction algorithm is obtained by establishing an equivalence between canonical correlation analysis and solutions to algebraic riccati equations also the concept of balanced stochastic realization bsr plays a fundamental role asymptotic stability of the reduced order realization is established and spectral domain interpretations for the bsr are given

focal boundary value problems for nonlinear difference equations ii on considere l unicite des solutions impliquant l existence des solutions de problemes aux valeurs limites de type focal pour l equation aux differences non lineaire d ordre n u t n f t u t u t n 1 ou f a r n r est continue et l equation u n 1 f t u 1 u n peut etre resolue pour u 1 comme fonction continue des variables t u 2 u u 1

performance evaluation of max automata automata with multiplicities over the max semiring can be used to represent the behavior of timed discrete event systems this formalism which extends both conventional automata and max linear representations covers a class of systems with synchronization phenomena and variable schedules performance evaluation is considered in the worst mean and optimal cases a simple algebraic reduction is provided for the worst case the last two cases are solved for the subclass of deterministic series recognized by deterministic automata deterministic series frequently arise due to the finiteness properties of max linear projective semigroups the mean performance is given by the kolmogorov equation of a markov chain the optimal performance is given by a hamilton jacobi bellman equation

idempotent version of the frechet contingency array problem in this paper we study the idempotent version of the so called frechet correlation array problem the problem is studied using an algebraic approach the major result is that there exists a unique upper bound and several lower bounds the formula for the upper bound is given an algorithm is proposed to compute one lower bound another algorithm is provided to compute all lower bounds but the number of lower bounds may be a very large number note that all these results are only based on the distributive lattice property of the idempotent algebraic structure

brief paper balanced truncation model reduction for systems with inhomogeneous initial conditions we present a rigorous approach to extend balanced truncation model reduction btmr to systems with inhomogeneous initial conditions we provide an estimate for the error between the input output maps of the original and of the reduced initial value system and we illustrate numerically the superiority of our approach over the naive application of btmr when btmr is applied to linear time invariant systems with inhomogeneous initial conditions it is crucial that the initial data are well represented by the subspaces generated by btmr this requirement is often ignored or it is avoided by making the restrictive assumption that the initial data are zero to ensure that the initial data are well represented by the btmr subspaces we add auxiliary inputs determined by the initial data

uniqueness implies existence and uniqueness conditions for nonlocal boundary value problems for nth order differential equations abstract for the nth order differential equation y n f x y y y n 1 we consider uniqueness implies existence results for solutions satisfying certain nonlocal k 2 point boundary conditions 1 k n 1 uniqueness of solutions when k n 1 is intimately related to uniqueness of solutions when 1 k n 2 these relationships are investigated as well

interface interactions in modulated phases and upsilon points certain features in frenkel kontorova and other models of phases with a one dimensional modulation can be analyzed by assuming parallel interfaces separating sets of lattice planes belonging to two different phases and treating the free energy to create interfaces as well as the interaction of two three or more interfaces as phenomenological parameters a strategy employed by fisher and szpilka for interacting defects can be extended to the case of interfaces allowing a systematic study of the phase diagram by ignoring all interface interactions and then successively taking into account pair triple and higher order terms the possible phase diagrams which can occur near the point where 0 include various sorts of endpoints analogous to critical endpoints an accumulation point of first order transitions and triple points and a self similar structure which we call an upsilon point which turns out to be an accumulation point of an infinite number of segments of first order transition lines each of which terminates in two upsilon points

locking to incommensurate structures a model with three competing lengths a classical pseudo spin model in one dimension is considered representing a variation on the frenkel kontorova model to include non convex interactions resulting in three competing length scales an exact algorithm is used numerically to determine the classical ground state as a function of a chemical potential approximate arguments suggest that only a first order transition should occur however when the lengths are not rationally related it is found that the mean lattice spacing appears to vary as a devil s staircase the plateaux of the staircase correspond to locking to incommensurate structures while there is no locking to the commensurate ones most of the locking values observed numerically belong to a series which can be analytically calculated on the basis of simple topological hypotheses which are also consistent with numerical observations

convolution and hankel operator norms for linear systems some norms are derived for convolution and hankel operators associated with linear time invariant systems in certain cases these norms are shown to be identical the tightest possible bound has been obtained for the absolute magnitude of the euclidean 2 or infinity norm of the time domain response of a multioutput system to certain classes of input disturbance

directional interpolation approach to h optimization and robust stabilization it is shown that under a mild condition h infty optimization and robust stabilization are equivalent to a directional interpolation problem which is a matrix extension of the classical pick nevanlinna interpolation problem a classical iterative method which is an extension of the schur nevanlinna algorithm is given for solving the problem this method does not require the inner outer factorization nor the balanced realization of the original plant a circuit theoretical parameterization of all solutions is derived that is expected to enhance the physical insight to the h infty optimal control and robust stabilization this parameterization has the degree much less than the one obtained previously

multiplicative approximation of transfer functions with frequency weighting abstract an algorithm for transfer function order reduction is presented which generalizes the balanced stochastic truncation algorithm to allow for input weighting an example illustrates use of the algorithm to secure smaller db error in selected frequency bands through the introduction of the weighting

phase diagram of a one dimensional model with nonconvex interactions using the method of effective potentials the ground states of a one dimensional system of the frenkel kontorova type but involving piecewise parabolic potentials including a nonconvex interatomic interaction have been studied numerically using the method of effective potentials part of the phase diagram is identical to one studied earlier for a convex interaction and part of it exhibits some new phases first order phase transitions multicritical points and an accumulation point of multicritical points all associated with the nonconvex interaction

right focal point boundary value problems for ordinary differential equations and variational equations where x g x such problems are called right focal point boundary value problems in the last few years a number of papers have appeared dealing with such boundary value problems most notable in the case when 1 is linear are papers by elias 11 muldowney 7 8 and peterson 19 io and for nonlinear considerations the papers by henderson 3 4 jackson 5 and peterson 11 121 in those papers several analogues of known results for conjugate type boundary value problems are considered for eq l we will have need of the hypotheses