the elliptic matrix completion problem abstract completions of partial elliptic matrices are studied given an undirected graph g it is shown that every partial elliptic matrix with graph g can be completed to an elliptic matrix if and only if the maximal cliques of g are pairwise disjoint further given a partial elliptic matrix a with undirected graph g it is proved that if g is chordal and each specified principal submatrix defined by a pair of intersecting maximal cliques is nonsingular then a can be completed to an elliptic matrix conversely if g is nonchordal or if the regularity condition is relaxed it is shown that there exist partial elliptic matrices which are not completable to an elliptic matrix in the process we obtain several results concerning chordal graphs that may be of independent interest

an autoregressive growth model for longitudinal item analysis a first order autoregressive growth model is proposed for longitudinal binary item analysis where responses to the same items are conditionally dependent across time given the latent traits specifically the item response probability for a given item at a given time depends on the latent trait as well as the response to the same item at the previous time or the lagged response an initial conditions problem arises because there is no lagged response at the initial time period we handle this problem by adapting solutions proposed for dynamic models in panel data econometrics asymptotic and finite sample power for the autoregressive parameters are investigated the consequences of ignoring local dependence and the initial conditions problem are also examined for data simulated from a first order autoregressive growth model the proposed methods are applied to longitudinal data on korean students self esteem

treediagonal matrices and their inverses abstract a generalization of tridiagonal matrices is considered namely treediagonal matrices which have nonzero off diagonal elements only in positions where the adjacency matrix of a tree has nonzero elements some properties of treediagonal matrices are given and their inverses are characterized and shown to have an interesting structure

matrix sandwich problems abstract the matrix sandwich problem msp is introduced here as follows given a 0 1 valued matrix a where is interpreted as do not care does there exist a fill in of the asterisks with 0s and 1s such that the completed 0 1 valued matrix m satisfies property we study the computational complexity of this problem for several matrix properties including the ferrers property block decompositions and certain forbidden submatrices matrix sandwich problems are an important special case of matrix completion problems the latter being generally defined over the real numbers rather than simply 0 1

an efficient algorithm for maximum entropy extension of block circulant covariance matrices article history this paper deals with maximum entropy completion of partially specified block circulant matrices since positive definite symmet ric circulants happen to be covariance matrices of stationary peri odic processes in particular of stationary reciprocal processes this problem has applications in signal processing in particular to im age modeling in fact it is strictly related to maximum likelihood estimation of bilateral ar type representations of acausal signals subject to certain conditional independence constraints the maxi mum entropy completion problem for block circulant matrices has recently been solved by the authors although leaving open the problem of an efficient computation of the solution in this pa per we provide an efficient algorithm for computing its solution which compares very favorably with existing algorithms designed for positive definite matrix extension problems the proposed al gorithm benefits from the analysis of the relationship between our problem and the band extension problem for block toeplitz matri ces also developed in this paper

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

determinantal formulae for matrices with sparse inverses ii asymmetric zero patterns abstract in an earlier paper formulae for det a as a ratio of products of principal minors of a were exhibited for any given symmetric zero pattern of a 1 these formulae may be presented in terms of a spanning tree of the intersection graph of certain index sets associated with the zero pattern of a 1 however just as the determinant of a diagonal and of a triangular matrix are both the product of the diagonal entries the symmetry of the zero pattern is not essential for these formulae we describe here how analogous formulae for det a may be obtained in the asymmetric zero pattern case by introducing a directed spanning tree we also examine the converse question of determining all possible zero patterns of a 1 which guarantee that a certain determinantal formula holds

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

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

logistic latent trait models with linear constraints two linearly constrained logistic models which are based on the well known dichotomous rasch model the linear logistic test model lltm and the linear logistic model with relaxed assumptions llra are discussed necessary and sufficient conditions for the existence of unique conditional maximum likelihood estimates of the structural model parameters are derived methods for testing composite hypotheses within the framework of these models and a number of typical applications to real data are mentioned

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

spanning tree extensions of the hadamard fischer inequalities abstract all possible graph theoretic generalizations of a certain sort for the hadamard fischer determinantal inequalities are determined these involve ratios of products of principal minors which dominate the determinant furthermore the cases of equality in these inequalities are characterized and equality is possible for every set of values which can occur for the relevant minors this relates recent work of the authors on positive definite completions and determinantal identities when applied to the same collections of principal minors earlier generalizations give poorer more difficult to compute bounds than the present inequalities thus this work extends and in a certain sense completes a series of generalizations of hadamard fischer begun in the 1960s

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

estimating latent correlations between repeated testings a model for longitudinal latent structure analysis is proposed we assume that test scores for a given mental or attitudinal test are observed for the same individuals at two different points in time the purpose of the analysis is to fit a model that combines the values of the latent variable at the two time points in a two dimensional latent density the correlation coefficient between the two values of the latent variable can then be estimated the theory and methods are illustrated by a danish dataset concerning psychic vulnerability

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

determinantal formulae for matrices with sparse inverses abstract the determinant of a matrix is expressed in terms of certain of its principal minors by a formula which can be read off from the graph of the inverse of the matrix the only information used is the zero pattern of the inverse and each zero pattern yields one or more corresponding formulae for the determinant

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

inverses of banded matrices abstract we establish a correspondence between the vanishing of a certain set of minors of a matrix a and the vanishing of a related set of minors of a 1 in particular inverses of banded matrices are characterized we then use our results to find patterns for toeplitz matrices with banded inverses finally we give an interesting determinant formula for inverses of banded matrices and show that in general a banded partial matrix may be completed in a unique way to give a banded inverse of the same bandwidth

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

the positive definite completion problem revisited abstract in r grone c r johnson e sa h wolkowicz positive definite completions of partial hermitian matrices linear algebra appl 58 1984 109 124 the positive definite semi completion problem in which the underlying graph is chordal was solved for the positive definite case the process was constructive and the completion was obtained by completing the partial matrix an entry at a time for the positive semidefinite case they obtained completions of a particular sequence of partial positive definite matrices with the same underlying graph and noted that there is a convergent subsequence of these completions that converges to the desired completion here in the chordal case we provide a constructive solution based entirely on matrix graph theoretic methods to the positive semi definite completion problem our solution associates a specific tree called the clique tree c r johnson m lundquist matrices with chordal inverse zero patterns linear and multilinear algebra 36 1993 1 17 with the chordal graph of the given partial positive semi definite matrix this tree structure allows us to complete the matrix a block at a time as opposed to an entry at a time as in grone et al 1984 for the positive definite case in grone et al 1984 using complex analytic techniques the completion for the positive definite case was shown to be the unique determinant maximizing completion and was shown to be the unique completion that has zeros in its inverse in the positions corresponding to the unspecified entries of the partial matrix here we show the same using only matrix graph theoretic tools

a generalized inverse eigenvalue problem in structural dynamic model updating this paper is concerned with the problem of the best approximation for a given matrix pencil under a given spectral constraint and a submatrix pencil constraint such a problem arises in structural dynamic model updating by using the moore penrose generalized inverse and the singular value decomposition svd matrices the solvability condition and the expression for the solution of the problem are presented a numerical algorithm for solving the problem is developed

some neglected problems in irt the paper addresses three neglected questions from irt in section 1 the properties of the measurement of ability or trait parameters and item difficulty parameters in the rasch model are discussed it is shown that the solution to this problem is rather complex and depends both on general assumptions about properties of the item response functions and on assumptions about the available item universe section 2 deals with the measurement of individual change or modifiability based on a rasch test a conditional likelihood approach is presented that yields a an ml estimator of modifiability for given item parameters b allows one to test hypotheses about change by means of a clopper pearson confidence interval for the modifiability parameter or c to estimate modifiability jointly with the item parameters uniqueness results for all three methods are also presented in section 3 the mantel haenszel method for detecting dif is discussed under a novel perspective what is the most general framework within which the mantel haenszel method correctly detects dif of a studied item the answer is that this is a 2pl model where however all discrimination parameters are known and the studied item has the same discrimination in both populations since these requirements would hardly be satisfied in practical applications the case of constant discrimination parameters that is the rasch model is the only realistic framework a simple pearsonx2 test for dif of one studied item is proposed as an alternative to the mantel haenszel test moreover this test is generalized to the case of two items simultaneously studied for dif

the lognormal race a cognitive process model of choice and latency with desirable psychometric properties we present a cognitive process model of response choice and response time performance data that has excellent psychometric properties and may be used in a wide variety of contexts in the model there is an accumulator associated with each response option these accumulators have bounds and the first accumulator to reach its bound determines the response time and response choice the times at which accumulator reaches its bound is assumed to be lognormally distributed hence the model is race or minima process among lognormal variables a key property of the model is that it is relatively straightforward to place a wide variety of models on the logarithm of these finishing times including linear models structural equation models autoregressive models growth curve models etc consequently the model has excellent statistical and psychometric properties and can be used in a wide range of contexts from laboratory experiments to high stakes testing to assess performance we provide a bayesian hierarchical analysis of the model and illustrate its flexibility with an application in testing and one in lexical decision making a reading skill

an inertia formula for hermitian matrices with sparse inverses abstract let a mn be a nonsingular hermitian matrix let g be a chordal graph on vertices 1 n and suppose that g a 1 g then the inertia of a may be expressed in a simple way in terms of the inertias of those principal submatrices of a corresponding to certain readily identifiable sets of vertices of g specifically we prove that the inertia i a satisfies the identity i a c i a s m i a in which c denotes the collection of maximal cliques of g s denotes the collection of minimal vertex separators of g and m is the multiplicity of a separator

local inversion of matrices with sparse inverses abstract we demonstrate that subject to certain regularity conditions any invertible matrix whose inverse is subordinate to a chordal graph g may be inverted via a simple formula involving only the inverses of its principal submatrices corresponding to the maximal cliques and minimal vertex separators of the graph g the resulting formula is reminiscent of known formulae for the determinant and inertia of matrices whose inverses are subordinate to a chordal graph

invertible completions of partial operator matrices the nonsymmetric case abstract we consider nonsymmetric partial operator matrices r whose directed graphs belong to a certain class and for which certain key fully specified principal submatrices are invertible under these circumstances we prove the existence of a unique invertible completion f of r such that f 1 ij is block zero whenever rij is unspecified in this way the existing results of johnson and lundquist are generalized to the nonsymmetric case

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

positive definite completions and determinant maximization abstract a method is described for determining whether a positive definite completion of a given partial hermitian matrix exists and if so for finding the determinant maximizing positive definite completion no assumption is made about the arrangement of the specified entries the method employs iterative application to a modified problem of an explicit formula for the maximum determinant in case there is only one symmetrically placed pair of unspecified entries a robust and fast algorithm based upon this approach is shown to have global convergence to the necessarily unique solution

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

loglinear rasch models for the analysis of stability and change loglinear unidimensional and multidimensional rasch models are considered for the analysis of repeated observations of polytomous indicators with ordered response categories reparameterizations and parameter restrictions are provided which facilitate specification of a variety of hypotheses about latent processes of change models of purely quantitative change in latent traits are proposed as well as models including structural change a conditional likelihood ratio test is presented for the comparison of unidimensional and multiple scales rasch models in the context of longitudinal research this renders possible the statistical test of homogeneity of change against subject specific change in latent traits applications to two empirical data sets illustrate the use of the models

a multidimensional latent trait model for measuring learning and change a latent trait model is presented for the repeated measurement of ability based on a multidimensional conceptualization of the change process a simplex structure is postulated to link item performance under a given measurement condition or occasion to initial ability and to one or more modifiabilities that represent individual differences in change since item discriminations are constrained to be equal within a measurement condition the model belongs to the family of multidimensional rasch models maximum likelihood estimators of the item parameters and abilities are derived and an example provided that shows good recovery of both item and ability parameters properties of the model are explored particularly for several classical issues in measuring change

measuring growth in a longitudinal large scale assessment with a general latent variable model the aim of the research presented here is the use of extensions of longitudinal item response theory irt models in the analysis and comparison of group specific growth in large scale assessments of educational outcomes a general discrete latent variable model was used to specify and compare two types of multidimensional item response theory mirt models for longitudinal data a a model that handles repeated measurements as multiple correlated variables over time and b a model that assumes one common variable over time and additional variables that quantify the change using extensions of these mirt models we approach the issue of modeling and comparing group specific growth in observed and unobserved subpopulations the analyses presented in this paper aim at answering the question whether academic growth is homogeneous across types of schools defined by academic demands and curricular differences in order to facilitate answering this research question a a model with a single two dimensional ability distribution was compared to b a model assuming multiple populations with potentially different two dimensional ability distributions based on type of school and to c a model that assumes that the observations are sampled from a discrete mixture of unobserved populations allowing for differences across schools with respect to mixing proportions for this purpose we specified a hierarchical mixture distribution variant of the two mirt models the latter model c is a growth mixture mirt model that allows for variation of the mixing proportions across clusters in a hierarchically organized sample we applied the proposed models to the pisa i plus data for assessing learning and change across multiple subpopulations the results of this study support the hypothesis of differential growth

determinantal formulae for matrix completions associated with chordal graphs abstract let a be a partial positive definite hermitian matrix and let g a be the undirected graph of the specified off diagonal entries of a when g a is a chordal graph we present an explicit determinantal formula for the unique determinant maximizing positive definite completion maximum entropy completion of a in order to do this we first give a characterization of a chordal graph in terms of the existence of a proper spanning tree t of the intersection graph of its set of maximal cliques the formula for the maximum determinant is then a quotient of products of specified principal minors of a in which each minor in the numerator corresponds to a maximal clique of g a while each minor in the denominator corresponds to the intersection of two maximal cliques associated with an edge of t we next show that the denominator terms can be identified graph theoretically with the minimal vertex separators of g a and that these terms are independent of the particular proper spanning tree t then an alternative formula is given for the maximum determinant as an inclusion exclusion like ratio of principal minors associated with the maximal cliques of g a finally we show that the entries of the determinant maximizing completion may be calculated by solving a sequence of simple one variable maximization problems

the inverse of any two by two nonsingular partitioned matrix and three matrix inverse completion problems a formula for the inverse of any nonsingular matrix partitioned into two by two blocks is derived through a decomposition of the matrix itself and generalized inverses of the submatrices in the matrix the formula is then applied to three matrix inverse completion problems to obtain their complete solutions

submatrix constrained least squares inverse problem for symmetric matrices from the design of vibrating structures abstract given a full column rank matrix x r n m a matrix b r m m and a symmetric matrix a 0 r p p in structural dynamic model updating yuan and dai 2007 considered the matrix equation x t a x b with a leading principal submatrix a 0 constraint for a given matrix a r n n they updated the mass and stiffness matrix in the frobenius norm sense such that the corrected matrices satisfy the generalized eigenvalue equation and orthogonality conditions but due to measurement errors the measured mass and stiffness matrices will not always satisfy these requirements since they still contain some useful information we would like to retrieve their least squares approximations to correct these matrices then we obtain the least squares symmetric solutions of the equation x t a x b with a trailing principal submatrix a 0 constraint by using the matrix differential calculus and canonical correlation decomposition furthermore by applying the generalized singular value decomposition and projection theorem we get the best frobenius norm approximate symmetric solution of this equation according to a given matrix a r n n with a 0 as its trailing principal submatrix finally a numerical algorithm for computing the best approximate solution is established some illustrated numerical examples are also presented

completion problem with partial correlation vines abstract this paper extends the results in d kurowicka r m cooke a parametrization of positive definite matrices in terms of partial correlation vines linear algebra appl 372 2003 225 251 we show that a partial correlation vine represents a factorization of the determinant of the correlation matrix we show that the graph of an incompletely specified correlation matrix is chordal if and only if it can be represented as an m saturated incomplete vine that is an incomplete vine for which all edges corresponding to membership descendents m descendents for short of a specified edge are specified this enables us to find the set of completions and also the completion with maximal determinant for matrices corresponding to chordal graphs

on the relationship between item response theory and factor analysis of discretized variables equivalence of marginal likelihood of the two parameter normal ogive model in item response theory irt and factor analysis of dichotomized variables fa was formally proved the basic result on the dichotomous variables was extended to multicategory cases both ordered and unordered categorical data pair comparison data arising from multiple judgment sampling were discussed as a special case of the unordered categorical data a taxonomy of data for the irt and fa models was also attempted

structural model correlation using large admissible perturbations incognate space a nonlinear perturbation method is developed to solve the problem of correlating a finite element model to a structure for which an incomplete set of natural frequencies and mode shapes and or some static deflections have been measured

rank inequalities for positive semidefinite matrices abstract several inequalities relating the rank of a positive semidefinite matrix with the ranks of various principal submatrices are presented these inequalities are analogous to known determinantal inequalities for positive definite matrices such as fischer s inequality koteljanskii s inequality and extensions of these associated with chordal graphs

comparison of ctt and rasch based approaches for the analysis of longitudinal patient reported outcomes health sciences frequently deal with patient reported outcomes pro data for the evaluation of concepts in particular health related quality of life which cannot be directly measured and are often called latent variables two approaches are commonly used for the analysis of such data classical test theory ctt and item response theory irt longitudinal data are often collected to analyze the evolution of an outcome over time the most adequate strategy to analyze longitudinal latent variables which can be either based on ctt or irt models remains to be identified this strategy must take into account the latent characteristic of what pros are intended to measure as well as the specificity of longitudinal designs a simple and widely used irt model is the rasch model the purpose of our study was to compare ctt and rasch based approaches to analyze longitudinal pro data regarding type i error power and time effect estimation bias four methods were compared the score and mixed models sm method based on the ctt approach the rasch and mixed models rm the plausible values pv and the longitudinal rasch model lrm methods all based on the rasch model all methods have shown comparable results in terms of type i error all close to 5 per cent lrm and sm methods presented comparable power and unbiased time effect estimations whereas rm and pv methods showed low power and biased time effect estimations this suggests that rm and pv methods should be avoided to analyze longitudinal latent variables

resonances in one dimension and fredholm determinants we discuss resonances for schrodinger operators in whole and half line problems one of our goals is to connect the fredholm determinant approach of froese to the fourier transform approach of zworski another is to prove a result on the number of antibound states namely in a half line problem there are an odd number of antibound states between any two bound states

combinatorial matrix analysis an overview abstract we survey recent work in some components of combinatorial matrix analysis including qualitative matrix theory matrix completion problems bounding of matrix parameters the role of the longest simple circuit and attenuation of matrix products some definitional thoughts historical perspectives and unification challenges etc are offered the style is informal following the talk on which this writeup is based

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