further relaxations of the semidefinite programming approach to sensor network localization recently a semidefinite programming sdp relaxation approach has been proposed to solve the sensor network localization problem although it achieves high accuracy in estimating the sensor locations the speed of the sdp approach is not satisfactory for practical applications in this paper we propose methods to further relax the sdp relaxation more precisely to relax the single semidefinite matrix cone into a set of small size semidefinite submatrix cones which we call a sub sdp ssdp approach we present two such relaxations although they are weaker than the original sdp relaxation they retain the key theoretical property and numerical experiments show that they are both efficient and accurate the speed of the ssdp is even faster than that of other approaches based on weaker relaxations the ssdp approach may also pave a way to efficiently solving general sdp problems without sacrificing the solution quality

weak majorization inequalities and convex functions abstract let f be a convex function defined on an interval i 0 1 and a b n n complex hermitian matrices with spectrum in i we prove that the eigenvalues of f a 1 b are weakly majorized by the eigenvalues of f a 1 f b further if f is log convex we prove that the eigenvalues of f a 1 b are weakly majorized by the eigenvalues of f a f b 1 as applications we obtain generalizations of the famous golden thomson trace inequality a representation theorem and a harmonic geometric mean inequality some related inequalities are discussed

norm inequalities for weighted power means of operators some norm inequalities for weighted power means of hilbert space operators are proved for the general class of unitarily invariant norms these inequalities generalize a recent inequality of jocic equality conditions of some of these inequalities for the schatten p norms are also obtained

estimation of positive definite m matrices and structure learning for attractive gaussian markov random fields abstract consider a random vector with finite second moments if its precision matrix is an m matrix then all partial correlations are non negative if that random vector is additionally gaussian the corresponding markov random field gmrf is called attractive we study estimation of m matrices taking the role of inverse second moment or precision matrices using sign constrained log determinant divergence minimization we also treat the high dimensional case with the number of variables exceeding the sample size the additional sign constraints turn out to greatly simplify the estimation problem we provide evidence that explicit regularization is no longer required to solve the resulting convex optimization problem we propose an algorithm based on block coordinate descent in which each sub problem can be recast as non negative least squares problem illustrations on both simulated and real world data are provided

equivalent relaxations of optimal power flow several convex relaxations of the optimal power flow opf problem have recently been developed using both bus injection models and branch flow models in this paper we prove relations among three convex relaxations a semidefinite relaxation that computes a full matrix a chordal relaxation based on a chordal extension of the network graph and a second order cone relaxation that computes the smallest partial matrix we prove a bijection between the feasible sets of the opf in the bus injection model and the branch flow model establishing the equivalence of these two models and their second order cone relaxations our results imply that for radial networks all these relaxations are equivalent and one should always solve the second order cone relaxation for mesh networks the semidefinite relaxation and the chordal relaxation are equally tight and both are strictly tighter than the second order cone relaxation therefore for mesh networks one should either solve the chordal relaxation or the socp relaxation trading off tightness and the required computational effort simulations are used to illustrate these results

connections between semidefinite relaxations of the max cut and stable set problems we describe links between a recently introduced semidefinite relaxation for the max cut problem and the well known semidefinite relaxation for the stable set problem underlying the lovasz s theta function it turns out that the connection between the convex bodies defining the semidefinite relaxations mimics the connection existing between the corresponding polyhedra we also show how the semidefinite relaxations can be combined with the classical linear relaxations in order to obtain tighter relaxations

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

shift invariant subspaces factorization and interpolation for matrices i the canonical case abstract this article begins a program of classifying subspaces of block matrices that are invariant under multiplication on the right by the algebra of block upper or lower triangular matrices this gives a unifying point of view to several general kinds of factorization problems including lu qr and cholesky factorization a second application is to some matrix completion problems such as that of filling in the upper triangular entries of a partially defined matrix to make the result of norm less than or equal to 1 included is a linear fractional parametrization of the set of all solutions of this problem

the real positive definite completion problem for a simple cycle abstract we consider the question of whether a real partial positive definite matrix in which the specified off diagonal entries consist of a full n cycle has a positive definite completion this lies in contrast to the previously studied chordal case we give two solutions in one we describe about n 2 independent conditions on angles associated with a normalization of the data that are necessary and sufficient the second is more computational and allows presentation of all positive definite completions as well as answering the existence question

extremal bipartite matrices abstract let g be an undirected graph on vertices 1 n let m g be the convex cone of all positive semidefinite hermitian matrices a satisfying aij 0 if i j is not an edge of g in the case that g is a complete bipartite graph we characterize all extreme rays of m g in doing so we shall also find the maximum rank that an extreme matrix in m g can have

typical and generic ranks in matrix completion abstract we consider the problem of exact low rank matrix completion from a geometric viewpoint given a partially filled matrix m we keep the positions of specified and unspecified entries fixed and study the minimal completion rank if the entries of the matrix are complex and the known entries are chosen randomly according to a continuous distribution then for a fixed pattern of locations of specified and unspecified entries there is a unique minimum completion rank which occurs with probability one we call this rank the generic completion rank over the real numbers there can be multiple ranks that occur with positive probability we call them typical completion ranks we introduce these notions formally and provide a number of inequalities and exact results on typical and generic ranks for different families of patterns of known and unknown entries

positive semidefinite completions of partial hermitian matrices abstract we classify the ranks of positive semidefinite completions of hermitian band matrices and other partially specified hermitian matrices with chordal graphs and specified main diagonals completing a partially specified matrix means filling in the unspecified entries

maximum likelihood estimation for linear gaussian covariance models summary we study parameter estimation in linear gaussian covariance models which are p dimensional gaussian models with linear constraints on the covariance matrix maximum likelihood estimation for this class of models leads to a non convex optimization problem which typically has many local maxima using recent results on the asymptotic distribution of extreme eigenvalues of the wishart distribution we provide sufficient conditions for any hill climbing method to converge to the global maximum although we are primarily interested in the case in which n p the proofs of our results utilize large sample asymptotic theory under the scheme n p 1 remarkably our numerical simulations indicate that our results remain valid for p as small as 2 an important consequence of this analysis is that for sample sizes n 14p maximum likelihood estimation for linear gaussian covariance models behaves as if it were a convex optimization problem

a classical perturbation technique which is valid for large parameters there are various perturbation techniques for constructing analytical approximations to the oscillatory solutions of second order non linear differential equations 1 2 but many of them apply to weakly non linear cases only to overcome the limitations many novel techniques have been proposed in recent years for example cheung et al 3 proposed a modified lindstedt poincare method and lim et al 4 presented a modified mickens procedure for certain non linear oscillators recently he 5 proposed a perturbation technique which is valid for large parameters but unfortunately o1 and b in ref 5 are the functions of z which is unknown the main purpose of this paper is to point out that there already exists a classical perturbation technique which is valid for large parameters in this paper the following well known duffing equation is considered

large scale semidefinite programming via a saddle point mirror prox algorithm in this paper we first demonstrate that positive semidefiniteness of a large well structured sparse symmetric matrix can be represented via positive semidefiniteness of a bunch of smaller matrices linked in a linear fashion to the matrix we derive also the dual counterpart of the outlined representation which expresses the possibility of positive semidefinite completion of a well structured partially defined symmetric matrix in terms of positive semidefiniteness of a specific bunch of fully defined submatrices of the matrix using the representations we then reformulate well structured large scale semidefinite problems into smooth convex concave saddle point problems which can be solved by a prox method developed in 6 with efficiency mathcal o epsilon 1 implementations and some numerical results for large scale lovasz capacity and maxcut problems are finally presented

on linstedt poincare techniques for the quintic duffing equation abstract an artificial parameter method for the quintic duffing equation is presented the method is based on the introduction of a linear stiffness term and a new dependent variable both of which are proportional to the unknown frequency of oscillation the introduction of an artificial parameter and the expansion of both the solution and the unknown frequency of oscillation in series of the artificial parameter and results in linear ordinary differential equations at each order in the parameter by imposing the nonsecularity condition at each order in the expansion the method provides different approximations to both the solution and the frequency of oscillation the method does not require that a small parameter be present in the governing equation and its results are compared with those of linstedt poincare modified linstedt poincare harmonic balance and galerkin techniques it is shown that these four techniques predict the same first order approximation to the frequency of oscillation as the artificial parameter method presented in this paper and the latter introduces higher order corrections at second order whereas similar corrections are introduced by the linstedt poincare and modified linstedt poincare methods at orders equal to and higher than three

the real positive semidefinite completion problem for series parallel graphs abstract we consider the partial real symmetric matrices x whose diagonal entries are equal to 1 and whose off diagonal entries are specified only on a subset of the positions the question is to determine whether x can be completed to a positive semidefinite matrix extending a result of barrett et al 3 we give a set of necessary conditions for x to be completable and show that these conditions are also sufficient if and only if the graph corresponding to the positions of the specified entries is series parallel i e has no k 4 minor

laguerre samuelson type inequalities abstract we give a generalization of laguerre samuelson s inequality in the framework of group theory this extends a recent result of trenkler and puntanen g trenkler s puntanen a multivariate version of samuelson s inequality linear algebra appl 410 2005 143 149 some examples are given

on the solvability of derived matrix problems including completions and duals abstract a class of m by n real complex matrices is semi algebraic sa if membership in it may be checked via a finite list of polynomial inequalities in the entries the real and imaginary parts of the entries real algebraic geometry is used to show that 1 very many familiar classes of matrices some perhaps surprising are sa and 2 that two important derived problems matrix completions and hadamard duals lead to an sa solution when they come from an sa class this means that at least there exists a finite solution to these problems though it may be difficult to find the methodology likely extends to other problems of interest

eigenvalue inequalities for differences of means of hilbert space operators abstract we prove several eigenvalue inequalities for the differences of various means of two positive invertible operators a and b on a separable hilbert space under the assumption that a b is compact equality conditions of these inequalities are also obtained

inheritance principles for chordal graphs abstract we prove two results concerning positive completions of partial positive matrices first we give a parametrization of all positive invertible completions of a partial positive matrix having as support a chordal graph along a fixed increasing chordal sequence then a formula for the determinant of each completion is obtained in terms of the associated parameters as an application we obtained a proof of a conjecture of w w barrett c r johnson and l rodman regarding an inheritance principle for chordal graphs

closed loop optimal experiment design solution via moment extension we consider optimal experiment design for parametric prediction error system identification of linear time invariant multiple input multiple output systems in closed loop when the true system is in the model set the optimization is performed jointly over the controller and the spectrum of the external excitation which can be reparametrized as a joint spectral density matrix the optimal solution consists of first computing a finite set of generalized moments of this spectrum as the solution of a semi definite program a second step then consists of constructing a spectrum that matches this finite set of optimal moments and satisfies some constraints due to the particular closed loop nature of the optimization problem this problem can be seen as a moment extension problem under constraints here we first show that the so called central extension always satisfies these constraints leading to a constructive procedure for the optimal controller and excitation spectrum we then show that one can construct a broader set of parametrized optimal solutions that also satisfy the constraints the additional degrees of freedom can then be used to achieve additional objectives

exploiting sparsity in linear and nonlinear matrix inequalities via positive semidefinite matrix completion a basic framework for exploiting sparsity via positive semidefinite matrix completion is presented for an optimization problem with linear and nonlinear matrix inequalities the sparsity characterized with a chordal graph structure can be detected in the variable matrix or in a linear or nonlinear matrix inequality constraint of the problem we classify the sparsity in two types the domain space sparsity d space sparsity for the symmetric matrix variable in the objective and or constraint functions of the problem which is required to be positive semidefinite and the range space sparsity r space sparsity for a linear or nonlinear matrix inequality constraint of the problem four conversion methods are proposed in this framework two for exploiting the d space sparsity and the other two for exploiting the r space sparsity when applied to a polynomial semidefinite program sdp these conversion methods enhance the structured sparsity of the problem called the correlative sparsity as a result the resulting polynomial sdp can be solved more effectively by applying the sparse sdp relaxation preliminary numerical results on the conversion methods indicate their potential for improving the efficiency of solving various problems

geometry of maximum likelihood estimation in gaussian graphical models author s uhler caroline advisor s sturmfels bernd abstract algebraic statistics exploits the use of algebraic techniques to develop new paradigms and algorithms for data analysis the development of computational algebra software provides a powerful tool to analyze statistical models in part i of this thesis we use methods from computational algebra and algebraic geometry to study gaussian graphical models algebraic methods have proven to be useful for statistical theory and applications alike we describe a particular application to computational biology in part ii part i of this thesis investigates geometric aspects of maximum likelihood estimation in gaussian graphical models more generally we study multivariate normal models that are described by linear constraints on the inverse of the covariance matrix maximum likelihood estimation for such models leads to the problem of maximizing the determinant function over a spectrahedron and to the problem of characterizing the image of the positive definite cone under an arbitrary linear projection in chapter 2 we examine these problems at the interface of statistics and optimization from the perspective of convex algebraic geometry and characterize the cone of all sufficient statistics for which the maximum likelihood estimator mle exists in chapter 3 we develop an algebraic elimination criterion which allows us to find exact lower bounds on the number of observations needed to ensure that the mle exists with probability one this is applied to bipartite graphs grids and colored graphs we also present the first instance of a graph for which the mle exists with probability one even when the number of observations equals the treewidth computational algebra software can be used to study graphs with a limited number of vertices and edges in chapter 4 we study the problem of existence of the mle from an asymptotic point of view by fixing a class of graphs and letting the number of vertices grow to infinity we prove that for very large cycles already two observations are sufficient for the existence of the mle with probability one part ii of this thesis describes an application of algebraic statistics to association studies rapid research progress in genotyping techniques have allowed large genome wide association studies existing methods often focus on determining associations between single loci and a specific phenotype however a particular phenotype is usually the result of complex relationships between multiple loci and the environment we develop a method for finding interacting genes i e epistasis using markov bases we test our method on simulated data and compare it to a two stage logistic regression method and to a fully bayesian method showing that we are able to detect the interacting loci when other methods fail to do so finally we apply our method to a genome wide dog data set and identify epistasis associated with canine hair length

completing a block diagonal matrix with a partially prescribed inverse abstract a complete solution of the matrix completion problem a b 1 c d is obtained in terms of solutions of a riccati type matrix equation several special cases are described in detail

inequalities for sums and direct sums of hilbert space operators abstract we prove several singular value inequalities and norm inequalities involving sums and direct sums of hilbert space operators it is shown among other inequalities that if x and y are compact operators then the singular values of x y 2 are dominated by those of x y applications of these inequalities are also given

exploiting sparsity in semidefinite programming via matrix completion i general framework a critical disadvantage of primal dual interior point methods compared to dual interior point methods for large scale semidefinite programs sdps has been that the primal positive semidefinite matrix variable becomes fully dense in general even when all data matrices are sparse based on some fundamental results about positive semidefinite matrix completion this article proposes a general method of exploiting the aggregate sparsity pattern over all data matrices to overcome this disadvantage our method is used in two ways one is a conversion of a sparse sdp having a large scale positive semidefinite matrix variable into an sdp having multiple but smaller positive semidefinite matrix variables to which we can effectively apply any interior point method for sdps employing a standard block diagonal matrix data structure the other way is an incorporation of our method into primal dual interior point methods which we can apply directly to a given sdp in part ii of this article we will investigate an implementation of such a primal dual interior point method based on positive definite matrix completion and report some numerical results

robust estimators for nondecomposable elliptical graphical models robust estimators of the restricted covariance matrices associated with elliptical graphical models are studied general asymptotic results which apply to both decomposable and nondecomposable graphical models are presented for robust plug in type estimators these extend results previously established only for the decomposable case furthermore a class of graphical m estimators for the restricted covariance matrices is introduced and compared with the corresponding plug in m estimators the two approaches are shown to be asymptotically equivalent under random sampling from an elliptical distribution a simulation study demonstrates the superiority of the graphical m estimators for small samples

invariants and approximate solutions for certain non linear oscillators by means of the field method certain strongly non linear conservative oscillators are approached with the field method which is combined with the convolution integral method a complete set of their adiabatic invariants are derived on the basis of which approximate solutions for motion can be obtained

the symmetric m matrix and symmetric inverse m matrix completion problems abstract the symmetric m matrix and symmetric m 0 matrix completion problems are solved and results of johnson and smith linear algebra appl 290 1999 193 are extended to solve the symmetric inverse m matrix completion problem 1 a pattern i e a list of positions in an n n matrix has symmetric m completion i e every partial symmetric m matrix specifying the pattern can be completed to a symmetric m matrix if and only if the principal subpattern r determined by its diagonal is permutation similar to a pattern that is block diagonal with each diagonal block complete or in graph theoretic terms if and only if each component of the graph of r is a clique 2 a pattern has symmetric m 0 completion if and only if the pattern is permutation similar to a pattern that is block diagonal with each diagonal block either complete or omitting all diagonal positions or in graph theoretic terms if and only if every principal subpattern corresponding to a component of the graph of the pattern either omits all diagonal positions or includes all positions 3 a pattern has symmetric inverse m completion if and only if its graph is block clique and no diagonal position is omitted that corresponds to a vertex in a graph block of order 2 the techniques used are also applied to matrix completion problems for other classes of symmetric matrices

linear estimating equations for exponential families with application to gaussian linear concentration models abstract in many families of distributions maximum likelihood estimation is intractable because the normalization constant for the density which enters into the likelihood function is not easily available the score matching estimator 35 provides an alternative where this normalization constant is not required for an exponential family e g a gaussian linear concentration model the corresponding estimating equations become linear 2 36 and the score matching estimator is shown to be consistent and asymptotically normally distributed as the number of observations increase to infinity although not necessarily efficient for linear concentration models that are also linear in the covariance 37 we show that the score matching estimator is identical to the maximum likelihood estimator hence in such cases it is also efficient gaussian graphical models and graphical models with symmetries 32 form particularly interesting subclasses of linear concentration models and we investigate the potential use of the score matching estimator for this case

a convolution integral method for certain strongly nonlinear oscillators a modification of the convolution integral method for linear oscillators is presented for the analysis of certain strongly nonlinear oscillators the modification provides an iteration scheme two examples are given to illustrate the effectiveness of the proposed method

limit cycles of non smooth oscillators abstract an artificial parameter linstedt poincare method based on the introduction of a linear stiffness term a change of independent variable and the expansion of both the solution and the frequency of oscillation in power series of the artificial parameter and the theory of generalized functions are used to determine the frequency of limit cycles of non smooth oscillators the method is applied to three non smooth oscillators and it is shown that the first order approximation coincides with those obtained by means of harmonic balance and parameter expansion techniques whereas higher order ones are given in terms of convergent series

dualistic differential geometry of positive definite matrices and its applications to related problems abstract analysis on the set of positive definite matrices is involved in many engineering problems we develop a differential geometric theory of the set of positive definite matrices by means of a specific class of connections introduced into it consequently various dualistic aspects of the set are elucidated next using the theory we derive new results and interesting geometrical interpretations for matrix approximation positive definite matrix completion and linear matrix inequality problems

an artificial parameter decomposition method for nonlinear oscillators applications to oscillators with odd nonlinearities abstract a method for obtaining series solutions of nonlinear second order ordinary differential equations based on the introduction of an artificial parameter is presented and shown to be identical to the well known adomian s decomposition technique the method is formulated in both integral and differential forms for the determination of the limit cycle of oscillators with odd nonlinearities two differential forms and one integral form of the artificial parameter method are presented these versions are based on introducing a linear stiffness term with an unknown frequency and the use of either the original independent variable or a new independent variable that depends linearly on the unknown frequency of the oscillator the three formulations provide identical results and their application to eight oscillators with odd nonlinearities shows that the artificial parameter technique presented in this paper predicts the same frequency of oscillation as the harmonic balance and iterative techniques as well as modified linstedt poincare methods however the method presented here is based on the introduction of an artificial parameter and does not require the presence of small perturbation parameters in the ordinary differential equation it is also shown that two and three level iterative methods yield the same frequency of oscillation as the artificial parameter technique presented in this paper provided that the initial iterate of the former coincides with the leading order solution of the latter and only one iteration of iterative techniques and only the second approximation of the artificial parameter method are determined

positive p matrices completion problem with zeros in the inverse completion in this paper based on the completion problem of partial p matrix for 1 chordal graph we obtain the partial positive p matrix completions of k chordal graphs and k chordal block graphs under some conditions and solve partial positive p matrix completion problems with zeros in the inverse completion for k chordal graphs and k chordal block graphs k 2

extensions of band matrices with band inverses abstract let r ij be a given set of i j matrices for i j 1 n and i j m where 0 m n 1 necessary and sufficient conditions are established for the existence and uniqueness of an invertible block matrix f ij i j 1 n such that f ij r ij for i j m f admits either a left or right block triangular factorization and f 1 ij 0 for i j m the well known conditions for an invertible block matrix to admit a block triangular factorization emerge for the particular choice m n 1 the special case in which the given r ij are positive definite in an appropriate sense is explored in detail and an inequality which corresponds to burg s maximal entropy inequality in the theory of covariance extension is deduced the block toeplitz case is also studied

on representing the positive semidefinite cone using the second order cone the second order cone plays an important role in convex optimization and has strong expressive abilities despite its apparent simplicity second order cone formulations can also be solved more efficiently than semidefinite programming problems in general we consider the following question posed by lewis and glineur parrilo saunderson is it possible to express the general positive semidefinite cone using second order cones we provide a negative answer to this question and show that the 3 times 3 positive semidefinite cone does not admit any second order cone representation in fact we show that the slice consisting of 3 times 3 positive semidefinite hankel matrices does not admit a second order cone representation our proof relies on exhibiting a sequence of submatrices of the slack matrix of the 3 times 3 positive semidefinite cone whose second order cone rank grows to infinity

distributing the kalman filter for large scale systems this paper presents a distributed kalman filter to estimate the state of a sparsely connected large scale n dimensional dynamical system monitored by a network of n sensors local kalman filters are implemented on nl dimensional subsystems nl lt n obtained by spatially decomposing the large scale system the distributed kalman filter is optimal under an lth order gauss markov approximation to the centralized filter we quantify the information loss due to this lth order approximation by the divergence which decreases as l increases the order of the approximation l leads to a bound on the dimension of the subsystems hence providing a criterion for subsystem selection the approximated centralized riccati and lyapunov equations are computed iteratively with only local communication and low order computation by a distributed iterate collapse inversion dici algorithm we fuse the observations that are common among the local kalman filters using bipartite fusion graphs and consensus averaging algorithms the proposed algorithm achieves full distribution of the kalman filter nowhere in the network storage communication or computation of n dimensional vectors and matrices is required only nl lt n dimensional vectors and matrices are communicated or used in the local computations at the sensors in other words knowledge of the state is itself distributed

on the variational iteration method and other iterative techniques for nonlinear differential equations abstract a variety of iterative methods for the solution of initial and or boundary value problems in ordinary and partial differential equations is presented these iterative procedures provide the solution or an approximation to it as a sequence of iterates for initial value problems it is shown that these iterative procedures can be written in either an integral or differential form the integral form is governed by a volterra integral equation whereas the differential one can be obtained from the volterra representation by simply differentiation it is also shown that integration by parts variation of parameters adjoint operators green s functions and the method of weighted residuals provide the same volterra integral equation and that this equation in turn can be written as that of the variational iteration method it is therefore shown that the variational iteration method is nothing else by the picard lindelof theory for initial value problems in ordinary differential equations and banach s fixed point theory for initial value problems in partial differential equations and the convergence of these iterative procedures is ensured provided that the resulting mapping is lipschitz continuous and contractive it is also shown that some of the iterative methods for initial value problems presented here are special cases of the bellman kalaba quasilinearization technique provided that the nonlinearities are differentiable with respect to the dependent variable and its derivatives but such a condition is not required by the techniques presented in this paper for boundary value problems it is shown that one may use the iterative procedures developed for initial value problems but the resulting iterates may not satisfy the boundary conditions and two new iterative methods governed by fredholm integral equations are proposed it is shown that the resulting iterates satisfy the boundary conditions if the first one does so the iterative integral formulation presented here is applied to ten nonlinear oscillators with odd nonlinearities and it is shown that its results coincide with those of differential two and three level iterative techniques harmonic balance procedures and standard and modified linstedt poincare techniques the method is also applied to two boundary value problems

maximum entropy kernels for system identification bayesian nonparametric approaches have been recently introduced in system identification scenario where the impulse response is modeled as the realization of a zero mean gaussian process whose covariance kernel has to be estimated from data in this scheme quality of the estimates crucially depends on the parametrization of the covariance of the gaussian process a family of kernels that have been shown to be particularly effective in the system identification framework is the family of diagonal correlated dc kernels maximum entropy properties of a related family of kernels the tuned correlated tc kernels have been recently pointed out in the literature in this technical note we show that maximum entropy properties indeed extend to the whole family of dc kernels the maximum entropy interpretation can be exploited in conjunction with results on matrix completion problems in the graphical models literature to shed light on the structure of the dc kernel in particular we prove that the dc kernel admits a closed form factorization inverse and determinant these results can be exploited both to improve the numerical stability and to reduce the computational complexity associated with the computation of the dc estimator

solutions of nonlinear oscillators with fractional powers by an iteration procedure a modified iteration procedure is applied to nonlinear oscillations with fractional powers with the procedure the excellent approximate frequencies and the corresponding periodic solutions can be easily obtained

a classical iteration procedure valid for certain strongly nonlinear oscillators a classical iteration procedure is applied to nonlinear oscillations of conservative single degree of freedom systems with odd nonlinearity with the procedure the analytical approximate frequency and the corresponding periodic solution valid for small as well as large amplitudes of oscillation can be obtained two examples are given to illustrate the accuracy and effectiveness of the method another advantage of this classical iteration approach is that it also works if the linear part of restoring force is zero

graphical gaussian models with edge and vertex symmetries summary we introduce new types of graphical gaussian models by placing symmetry restrictions on the concentration or correlation matrix the models can be represented by coloured graphs where parameters that are associated with edges or vertices of the same colour are restricted to being identical we study the properties of such models and derive the necessary algorithms for calculating maximum likelihood estimates we identify conditions for restrictions on the concentration and correlation matrices being equivalent this is for example the case when symmetries are generated by permutation of variable labels for such models a particularly simple maximization of the likelihood function is available

solutions of the duffing harmonic oscillator by an iteration procedure a modified iteration procedure is applied to the duffing harmonic oscillator with the procedure the excellent approximate frequencies and the corresponding periodic solutions can be easily obtained

operator ky fan type inequalities abstract in this paper we extend some significant ky fan type inequalities in a large setting to operators on hilbert spaces and derive their equality conditions among other things we prove that if f 0 0 is an operator monotone function with f 1 1 f 1 and associated mean then for all operators a and b on a complex hilbert space h such that 0 a b 1 2 i we have a b a b a b a b where i is the identity operator on h a i a b i b and is the weighted arithmetic mean

a non simultaneous variational approach for the oscillators with fractional order power nonlinearities abstract this paper is concerned with a class of conservative oscillators the restitution force of which is of a power form which includes positive non integer exponents it is shown how an approximate lagrangian and hamilton s variational principle can be used to obtain a second order approximate solution for their free vibrations due to the fact that in a general case when the restoring force is multi term the period cannot be obtained from the energy conservation law in a closed form the problem is formulated as a one point boundary value problem and a non simultaneous variation is introduced the explicit expressions for the amplitudes and frequency of oscillations are derived in which there are no restrictions on the values of the non integer powers the analytically obtained results are compared with numerical results as well as with some approximate analytical results from the literature

extensions of positive definite functions on free groups abstract an analogue of krein s extension theorem is proved for operator valued positive definite functions on free groups the proof gives also the parametrization of all extensions by means of a generalized type of szego parameters one singles out a distinguished completion called central which is related to quasi multiplicative positive definite functions an application is given to factorization of noncommutative polynomials

norm inequalities for self adjoint derivations for bounded hilbert space operators and for all unitarily invariant norms there hold a b p 2p 1 a a p 1 b b p 1 for all realp 2 and ifaandbare additionally self adjoint then ax xb p 2p 1 x p 1 a p 1ax xb b p 1 for all realp 3