positive linear operators with equidistant nodes abstract in the present paper the approximation power of positive linear operators with equidistant nodes is investigated new pointwise estimates are given in terms of first and second order moduli of continuity showing that for positive operators having uniform jackson orders of approximation one may expect interpolation at the endopoints in particular the first solution to butzer s problem with equidistant nodes is given a negative result yields an explanation why bernstein operators and some of their modifications are optimal in a certain sense

convexity and differentiability properties of spectral functions and spectral mappings on euclidean jordan algebras abstract we study in this paper several properties of the eigenvalues function of a euclidean jordan algebra extending several known results in the framework of symmetric matrices in particular we give a concise form for the directional differential of a single eigenvalue we especially focus on spectral functions f on euclidean jordan algebras which are the composition of a symmetric real valued function f with the eigenvalues function we explore several properties of f that are transferred to f in particular convexity strong convexity and differentiability spectral mappings are also considered a special case of which is the gradient mapping of a spectral function answering a problem proposed by h sendov we give a formula for the jacobian of these functions

identification of multivariable bilinear state space systems based on subspace techniques and separable least squares optimization we discuss identification of discrete time bilinear state space systems with multiple inputs and multiple outputs subspace identification methods for bilinear systems suffer from the curse of dimensionality already for relatively low order systems the matrices involved become so large that the method cannot be used in practice we have modified the subspace algorithm such that it reduces the dimension of the matrices involved only the rows that have the largest influence on the model are selected the remaining rows are discarded this obviously leads to an approximation error the initial model that we get from the subspace method is optimized using the principle of separable least squares according to this principle we can first solve for the matrices that enter non linearly in the output error criterion and then obtain the others by solving a linear least squares problem

extended generalized total least squares method for the identification of bilinear systems the extended generalized total least squares e gtls method that consider the special structure of the data matrix is proposed as one of the bilinear system parameters considering that the input is noise free and that bilinear system equation is linear with respect to the output we extend the gtls problem the extended gtls problem is reduced to an unconstrained minimization problem and is then solved by the newton raphson method we compare the gtls method and the extended gtls method as far as the accuracy of the estimated system parameters is concerned

an extended lie trotter formula and its applications abstract in this paper we present a class of lie trotter formulae for hermitian operators including the formulae derived by hiai petz and furuta a lie trotter formula for weighted log euclidean geometric means of several positive definite operators is given in terms of sagae tanabe geometric and spectral geometric means

a jordan algebraic approach to potential reduction algorithms we consider the linear monotone complementarity problem for domains obtained as the intersection of an affine subspace and the cartesian product of symmetric cones a primal dual potential reduction algorithm is described and its complexity estimates are established with the help of the jordan algebraic technique

geometry of homogeneous convex cones duality mapping and optimal self concordant barriers we study homogeneous convex cones we first characterize the extreme rays of such cones in the context of their primal construction due to vinberg and also in the context of their dual construction due to rothaus then using these results we prove that every homogeneous cone is facially exposed we provide an alternative proof of a result of guler and tunc el that the siegel rank of a symmetric cone is equal to its caratheodory number our proof does not use the jordan von neumann wigner characterization of the symmetric cones but it easily follows from the primal construction of the homogeneous cones and our results on the geometry of homogeneous cones in primal and dual forms we study optimal self concordant barriers in this context we briefly discuss the duality mapping in the context of automorphisms of convex cones and prove using numerical integration that the duality mapping is not an involution on certain self dual cones

nonlinearity tests for bilinear systems abstract the purpose of this paper is to develop nonlinearity tests for open loop bilinear systems lagrange multiplier tests of linear systems against a bilinear alternative are proposed a simulation study is performed to check the validity of the asymptotic null distributions of the test statistics and to investigate the power characteristics of the tests two recent nonlinearity tests in the time series context are adapted to linear systems and compared with lagrange multiplier tests simulation results show that the proposed lagrange multiplier tests are more powerful than the other tests

bilinear state space realization for polynomial stochastic systems the wiener approach to the non linear stochastic systems permits the representation of single valued systems with memory for which a small perturbation of the input produces a small perturbation of the output the wiener functional series representation contains infinitely many transfer functions to describe entirely the input output connections one of the most important class of stochastic systems especially in the statistical point of view is the case when all the transfer functions are determined by finitely many parameters the question of realizability i e to construct a state space model for a system given in terms of a set of transfer functions is a central problem in many cases for example in engineering there are several results in this direction for systems with deterministic inputs see 1 2 the realization of linear systems with stochastic gaussian input was treated by akaike 3 and studied intensively afterwards see 4 5 the bilinear system seems to be the next step leading from the linear towards non linear ones the realization problem of bilinear systems appeared in the papers 6 8 mainly by the time series side in this work we follow the early akaike work and the results for the deterministic systems by rugh 2 and also brillinger 9 who investigated the identification of finite degree functional series by spectral analysis first the bilinear realization problem is considered together with a simplest non linear system i e the second hermite degree bilinear process in the second section the transfer function system with its main properties is given by a recursive formula for the bilinear states next we show that these properties are necessary and sufficient for the transfer functions to be bilinear realizable in section four the stationarity of the bilinear state space model is investigated we are constructing the abstract bilinear minimal realization for a time invariant stationary degree n non linear system driven by gaussian white noise concerning to the identification of the bilinear model with hermite degree 2 explicit formula of the bispectrum is given for this family of models

maximum likelihood parameter estimation of bilinear systems this paper addresses the problem of estimating the parameters in a multivariable bilinear model on the basis of observed input output data the main contribution is to develop analyze and empirically study new techniques for computing a maximum likelihood based solution in particular the emphasis here is on developing practical methods that are illustrated to be numerically reliable robust to choice of initialization point and numerically efficient in terms of how computation and memory requirements scale relative to problem size this results in new methods that can be reliably deployed on systems of nontrivial state input and output dimension underlying these developments is a new approach in this context of employing the expectation maximization method as a means for robust and gradient free computation of the maximum likelihood solution

self scaled barriers for irreducible symmetric cones self scaled barrier functions are fundamental objects in the theory of interior point methods for linear optimization over symmetric cones of which linear and semidefinite programming are special cases in this article we classify the special class of self scaled barriers which are defined on irreducible symmetric cones together with a decomposition theorem for general self scaled barriers this concludes the algebraic classification theory of these functions

barrier functions in interior point methods we show that the universal barrier function of a convex cone introduced by nesterov and nemirovskii is the logarithm of the characteristic function of the cone this interpretation demonstrates the invariance of the universal barrier under the automorphism group of the underlying cone this provides a simple method for calculating the universal barrier for homogeneous cones we identify some known barriers as the universal barrier scaled by an appropriate constant we also calculate some new universal barrier functions our results connect the field of interior point methods to several branches of mathematics such as lie groups jordan algebras siegel domains differential geometry complex analysis of several variables etc

estimation of time delay and lag relationship of quadratic systems abstract this paper is concerned with the identification of a quadratic non linear system the identification includes the estimation of the lag delay relationship of the input process the lead lag time delay of the input output system and the unknown parameters the estimation procedures are based on the second order covariances cross covariances and spectral densities the estimation methods described here are illustrated with numerical examples

a recursive on line identification of bilinear systems a recursive identification of a discrete dynamical system is presented in this paper two methods of recursive identification namely the recursive extended least squares method and the recursive maximum likelihood method are described and analysed the techniques are illustrated numerically with simulations

approximation theory and functional analysis part 1 research and survey articles on bernstein durrineyer polynomials with jacob weights h berens and y xu an overview of wavelets c chui a note on weak inequalities in orlicz and lorentz spaces g a edgar and l sucheston bivariate birkhoff interpolation a survey r a lorentz linear approximations of functions with several restricted derivatives y makovoz some characterizations theorems for measures associated with orthogonal polynomials on the unit circle k pan and e b saff box splines cardinal series and wavelets s d reimenschneider and z shen some aspects of the subspace structure of infinite dimensional banach space h rosenthal fairness and monotone curvature j roulier et al projections on 2 dimensional spaces n tomczak jacgermann real versus complex best rational approximation r s varga and a ruttan

potential reduction methods in mathematical programming we provide a survey of interior point methods for linear programming and its extensions that are based on reducing a suitable potential function at each iteration we give a fairly complete overview of potential reduction methods for linear programming focusing on the possibility of taking long steps and the properties of the barrier function that are necessary for the analysis we then describe briefly how the methods and results can be extended to certain convex programming problems following the approach of nesterov and todd we conclude with some open problems

recursive identification of bilinear systems methods of identifying bilinear systems from recorded input output data are discussed in this article a short survey of the existing literature on the topic is given standard methods from linear systems identification such as least squares extended least squares recursive prediction error and instrumental variable methods are transferred to bilinear input output model structures and tested in simulation special attention is paid to problems of stabilizing the model predictor and it is shown how a time varying kalman filter and associated parameter estimation algorithm can deal with this problem

a long step primal dual algorithm for the symmetric programming problem abstract based on the techniques of euclidean jordan algebras we prove complexity estimates for a long step primal dual interior point algorithm for the optimization problem of the minimization of a linear function on a feasible set obtained as the intersection of an affine subspace and a symmetric cone this result provides a meaningful illustration of a power of the technique of euclidean jordan algebras applied to problems under consideration

primal dual algorithms and infinite dimensional jordan algebras of finite rank we consider primal dual algorithms for certain types of infinite dimensional optimization problems our approach is based on the generalization of the technique of finite dimensional euclidean jordan algebras to the case of infinite dimensional jb algebras of finite rank this generalization enables us to develop polynomial time primal dual algorithms for infinite dimensional second order cone programs we consider as an example a long step primal dual algorithm based on the nesterov todd direction it is shown that this algorithm can be generalized along with complexity estimates to the infinite dimensional situation under consideration an application is given to an important problem of control theory multi criteria analytic design of the linear regulator the calculation of the nesterov todd direction requires in this case solving one matrix differential riccati equation plus solving a finite dimensional system of linear algebraic equations on each iteration the number of equations and unknown variables of this algebraic system is m 1 where m is a number of quadratic performance criteria

similarity and other spectral relations for symmetric cones abstract the similarity relations that are derived in this paper reduce to well known results in the special case of symmetric matrices in particular for two positive definite matrices x and y the square of the spectral geometric mean is known to be similar to the matrix product xy it is shown in this paper that this property carries over to symmetric cones more elementary similarity relations such as xy 2 x yx 2 y are generalized as well we also extend the result that the eigenvalues of a matrix product xy are less dispersed than the eigenvalues of the jordan product xy yx 2 the paper further contains a number of inequalities on norms and spectral values this type of inequality is often used in the analysis of interior point methods in optimization we also derive an extension of stein s theorem to symmetric cones

variation on butzer s problem characterization of the solutions abstract the so called butzer problem had been a challenge to the approximation community for decades until its first constructive solution was given by cao and gonska at the end of the 1980s butzer s original problem can be turned into a strong form namely a pointwise one in the present note we characterize among other things and in an easy way certain operators which solve butzer s problem in this strong and most demanding version

on the identification of bilinear systems from operating records in this paper the identification of a non linear system when the input output record is available is considered the system discussed is an open loop system it is assumed that the unobservable output x t at time t and the observable input w s s t are related by a bilinear model it is further assumed that only a contaminated version of the output is observed i e y t whore for each t y t x t z t and x t and z t are mutually independent the identification of the system is considered in the following cases i where the external noise z t is a gaussian white noise and ii where z t is a gaussian stationary time series satisfying an autoregressive moving average arma model of order g j the estimation procedures are illustrated with examples

solving symmetric matrix word equations via symmetric space machinery abstract it has recently been proved by hillar and johnson c j hillar c r johnson symmetric word equations in two positive definite letters proc amer math soc 132 2004 945 953 that every symmetric word equation in positive definite matrices has a positive definite solution existence theorem in this paper we partially solve the uniqueness conjecture via the symmetric space and non positive curvature machinery existing in the open convex cone of positive definite matrices unique positive definite solutions are obtained in terms of geometric and weighted means and as fixed points of explicit strict contractions on the cone of positive definite matrices

on the optimal parameter of a self concordant barrier over a symmetric cone abstract the properties of the barrier f x log det x defined over the cone of squares of a euclidean jordan algebra are analyzed using pure algebraic techniques furthermore relating the caratheodory number of a symmetric cone with the rank of an underlying euclidean jordan algebra conclusions about the optimal parameter of f are suitably obtained namely in a more direct and suitable way than the one presented by guler and tuncel characterization of the barrier parameter of homogeneous convex cones mathematical programming 81 1998 55 76 it is proved that the caratheodory number of the cone of squares of a euclidean jordan algebra is equal to the rank of the algebra then taking into account the result obtained in the same paper where it is stated that the caratheodory number of a symmetric cone q is the optimal parameter of a self concordant barrier defined over q we may conclude that the rank of every underlying euclidean jordan algebra is also the self concordant barrier optimal parameter

linear systems in jordan algebras and primal dual interior point algorithms we discuss a possibility of the extension of a primal dual interior point algorithm suggested recently by alizadeh et al 1994 we consider optimization problems defined on the intersection of a symmetric cone and an affine subspace the question of solvability of a linear system arising in the implementation of the primal dual algorithm is analyzed a nondegeneracy theory for the considered class of problems is developed the jordan algebra technique suggested by faybusovich 1995 plays major role in the present paper

a new full nesterov todd step feasible interior point method for convex quadratic symmetric cone optimization in this paper we generalize the classical primal dual logarithmic barrier method for linear optimization to convex quadratic optimization over symmetric cone by using euclidean jordan algebras the symmetrization of the search directions used in this paper is based on the nesterov todd scaling scheme and only full nesterov todd step is used at each iteration we derive the iteration bound that matches the currently best known iteration bound for small update methods namely o r log r e some preliminary numerical results are provided to demonstrate the computational performance of the proposed algorithm