the minimum rank of symmetric matrices described by a graph a survey abstract the minimum rank of a simple graph g is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry for i j is nonzero whenever i j is an edge in g and is zero otherwise this paper surveys the current state of knowledge on the problem of determining the minimum rank of a graph and related issues

techniques for determining the minimum rank of a small graph abstract the minimum rank of a simple graph g is defined to be the smallest possible rank over all symmetric real matrices whose ij th entry for i j is nonzero whenever i j is an edge in g and is zero otherwise minimum rank is a difficult parameter to compute however there are now a number of known reduction techniques and bounds that can be programmed on a computer we have developed a program using the open source mathematics software sage to implement several techniques we have also established several additional strategies for computation of minimum rank these techniques have been used to determine the minimum ranks of all graphs of order 7

exponential family mixture models with least squares estimators pour une densite arbitraire d une famille exponentielle a un parametre on montre comment construire une distribution de melange sur le parametre de telle sorte que la distribution resultante soit une famille exponentielle a deux ou plusieurs parametres

inversion of e simple block matrices we recall previous results on inverting matrices the digraph of which is e simple i e such that every edge is contained in at most one simple cycle we present and analyze a finite algorithm for the inversion applications to m matrices are included

transitional regression models with application to environmental time series abstract environmental epidemiologists often encounter time series data in the form of discrete or other nonnormal outcomes for example in modeling the relationship between air pollution and hospital admissions or mortality rates we present a case study examining the association between pollen counts and meteorologic covariates although such time series data are inadequately described by standard methods for gaussian time series they are often autocorrelated and warrant an analysis beyond those provided by ordinary generalized linear models glms transitional regression models trms signifying nonlinear regression models expressed in terms of conditional means and variances given past observations provide a unifying framework for two mainstream approaches to extending the glm for autocorrelated data the first approach models current outcomes with a glm that incorporates past outcomes as covariates whereas the second models individual outcomes with marginal glms and then couples the error term

analytical inversion of symmetric tridiagonal matrices in this paper we present an analytical formula for the inversion of symmetrical tridiagonal matrices the result is of relevance to the solution of a variety of problems in mathematics and physics as an example the formula is used to derive an exact analytical solution for the one dimensional discrete poisson equation with dirichlet boundary conditions

an efficient algorithm for solving general periodic toeplitz systems an efficient algorithm is presented for inverting matrices which are periodically toeplitz i e whose diagonal and subdiagonal entries exhibit periodic repetitions such matrices are not per symmetric and thus cannot be inverted by trench s 1964 method an alternative approach based on appropriate matrix factorization and partitioning is suggested the algorithm provides certain insight on the formation of the inverse matrix is implementable on a set of circularly pipelined processors and as a special case can be used for inverting a set of block toeplitz matrices without requiring any matrix operation

positive semidefinite zero forcing numbers of two classes of graphs abstract the positive semidefinite zero forcing number is a variant of the zero forcing number which is an important parameter in the study of minimum rank maximum nullity problems in this paper we consider two classes of graphs matching chain graphs and claw free graphs we first introduce the propagation decomposition of graphs then we use this decomposition to prove a lower bound for the positive semidefinite zero forcing number of a graph we apply this lower bound to find the positive semidefinite zero forcing number of matching chain graphs we show that the positive semidefinite zero forcing number and the zero forcing number agree for matching chain graphs as a consequence we prove the conjecture about the positive semidefinite zero forcing number of the cartesian product of two paths and partially prove the conjecture about the positive semidefinite zero forcing number of the cartesian product of a cycle and a path for claw free graphs we establish a relationship between the positive semidefinite zero forcing number and the zero forcing number we also prove that it is np complete to find the positive semidefinite zero forcing number of line graphs

on the controllability properties of circulant networks this paper examines the controllability of a group of first order agents adopting a weighted consensus type coordination protocol over a circulant network specifically it is shown that a circulant network with laplacian eigenvalues of maximum algebraic multiplicity q is controllable from q nodes our approach leverages on the cauchy binet formula which in conjunction with the popov belevitch hautus test leads to new insights on structural aspects of network controllability

variance component models with binary response interviewer variability summary interviewer variability in a binary response is an example of a problem requiring variance component estimation in a non normal family the maximum likelihood estimation procedure is derived and used to examine some binary items on a large questionnaire this raises some interesting questions about the use of unbalanced ano va methods with these data two interviewers asking the same question of the same respondent can obtain different responses such interviewer effects create problems in the design and analysis of survey questionnaires collins 1980 gives a review of these problems together with a variety of partial remedies these include pilot studies and careful design to eliminate difficulties with wording of questions and instructions on the questionnaires in spite of these precautions interviewer effects remain questionnaire items are usually categorical and occasionally binary but usually for analysis the questionnaire item is reduced to a binary response it is the variability of the population of interviewers that is of interest and not just the individual interviewers thus this variability is best modelled as a random effect rather than a fixed effect the number of interviews completed by each interviewer varies so that the design is unbalanced thus we have an unbalanced variance component model with a categorical or binary response the intraclass correlation the ratio of the interviewer variance to the sum of all variances in the model is of great interest as a measure of interviewer variability in estimating the variance components the categorical or binary nature of the response is usually ignored and the analysis carried out using analysis of variance this is not maximum likelihood but analysis of variance does have some desirable properties when the error variables are non normal the rule of thumb generally applied is that the analysis of variance is reasonably accurate as long as the proportions in each of the categories of the response are between 0 1 and 0 9 scheffe 1959 p 343 discusses the effects of non normality in a one way balanced model very little theoretical work has been

computation of minimal rank and path cover number for certain graphs for a given undirected graph g the minimum rank of g is defined to be the smallest possible rank over all real symmetric matrices a whose i j th entry is non zero whenever i j and i j is an edge in g the path cover number of g is the minimum number of vertexdisjoint paths occurring as induced subgraphs of g that cover all the vertices of g for trees the relationship between minimum rank and path cover number is completely understood however for non trees only sporadic results are known we derive formulae for the minimum rank and path cover number for graphs obtained from edge sums and formulae for minimum rank of vertex sums of graphs in addition we examine previously identified special types of vertices and attempt to unify the theory behind them

on the graph complement conjecture for minimum rank abstract the minimum rank of a graph has been an interesting and well studied parameter investigated by many researchers over the past decade or so one of the many unresolved questions on this topic is the so called graph complement conjecture which grew out of a workshop in 2006 this conjecture asks for an upper bound on the sum of the minimum rank of a graph and the minimum rank of its complement and may be classified as a nordhaus gaddum type problem involving the graph parameter minimum rank the conjectured bound is the order of the graph plus two other variants of the graph complement conjecture are introduced here for the minimum semidefinite rank and the colin de verdiere type parameter we show that if the graph complement conjecture is true for two graphs then it is true for the join of these graphs related results for the graph complement conjecture and the positive semidefinite version for joins of graphs are also established we also report on the use of recent results on partial k trees to establish the graph complement conjecture for graphs of low minimum rank

on the inverse and determinant of general bordered tridiagonal matrices in the current paper we present a symbolic algorithm that never suffers from breakdown for computing the inverse of general bordered tridiagonal matrices meanwhile an explicit formula of evaluating the determinant of a bordered tridiagonal matrix is also derived the implementation of the proposed algorithm using computer algebra systems cass is straightforward some numerical examples are given in order to demonstrate the performance and validity of the presented results

homogeneity tests against central mixture alternatives abstract a test of homogeneity is derived for a general sampling model the alternative hypothesis is a mixture over a parameter of the sampling model centered at the null hypothesis the test statistic is derived through a score test its functional form is independent of the particular mixing distribution suppose yi i 1 n are independent random variables with respective probability density or mass functions fi yi i under the null hypothesis suppose all i homogeneous and equal to the common value 0 under the alternative hypothesis suppose the i behave as random samples taken from a distribution with mean 0 and finite third moment the score statistic for testing these hypotheses rejects the null for large values of where is the maximum likelihood estimate of 0 under the null hypothesis and when the sample size n is large s is normally distributed under both the null hypothesis and a sequence of alternative hypotheses in which the variance of the mixing distribution tends

random effects models with non parametric priors we discuss the performance of non parametric maximum likelihood npml estimators for the distribution of a univariate random effect in the analysis of longitudinal data for continuous data we analyse generated and real data sets and compare the npml method to those that assume a gaussian random effects distribution and to ordinary least squares for binary outcomes we use generated data to study the moderate and large sample performance of the npml compared with a method based on a gaussian random effect distribution in logistic regression we find that estimated fixed effects are compatible for all approaches but that appropriate standard errors for the npml require adjusting the likelihood based standard errors we conclude that the non parametric approach provides an attractive alternative to gaussian based methods though additional evaluations are necessary before it can be recommended for general use

zero forcing sets and controllability of dynamical systems defined on graphs in this technical note controllability of systems defined on graphs is discussed we consider the problem of controllability of the network for a family of matrices carrying the structure of an underlying directed graph a one to one correspondence between the set of leaders rendering the network controllable and zero forcing sets is established to illustrate the proposed results special cases including path cycle and complete graphs are discussed moreover as shown for graphs with a tree structure the proposed results of the present technical note together with the existing results on the zero forcing sets lead to a minimal leader selection scheme in particular cases

an efficient numerical algorithm for the determinant of a cyclic pentadiagonal toeplitz matrix abstract in the past few years a number of numerical and symbolic algorithms for evaluating the determinants of cyclic pentadiagonal matrices have been developed in this paper we present a fast numerical algorithm for the determinant of an n by n cyclic pentadiagonal matrix with toeplitz structure the algorithm is based upon a certain type of matrix reordering and partitioning and linear transformation some numerical examples are provided and the results are compared with the ones obtained via matlab built in function and two existing algorithms all of the numerical experiments are performed on a computer with the aid of programs written in matlab

analytical inversion of general periodic tridiagonal matrices in this paper we present an analytical forms for the inversion of general periodic tridiagonal matrices and provide some very simple analytical forms which immediately lead to closed formulae for some special cases such as symmetric or perturbed toeplitz for both periodic and non periodic tridiagonal matrices an efficient computational algorithm for finding the inverse of any general periodic tridiagonal matrices from the analytical form is given it is suited for implementation using computer algebra systems such as maple matlab macsyma and mathematica an example is also given to illustrate the algorithm

lower bounds for minimum semidefinite rank from orthogonal removal and chordal supergraphs abstract the minimum semidefinite rank msr of a graph is the minimum rank among positive semidefinite matrices with the given graph the os number is a useful lower bound for msr which arises by considering ordered vertex sets with some connectivity properties in this paper we develop two new interpretations of the os number we first show that os number is also equal to the maximum number of vertices which can be orthogonally removed from a graph under certain nondegeneracy conditions our second interpretation of the os number is as the maximum possible rank of chordal supergraphs who exhibit a notion of connectivity we call isolation preserving these interpretations not only give insight into the os number but also allow us to prove some new results for example we show that msr g g 2 if and only if os g gzsfnc 2

a new lower bound for the positive semidefinite minimum rank of a graph abstract the real positive semidefinite minimum rank of a graph is the minimum rank among all real positive semidefinite matrices that are naturally associated via their zero nonzero pattern to the given graph in this paper we use orthogonal vertex removal and sign patterns to improve the lower bound for the real positive semidefinite minimum rank determined by the os number and the positive semidefinite zero forcing number

minimum rank matrices with prescribed graph abstract we study properties of real symmetric matrices with prescribed graph and lowest possible rank we concentrate on the case where the graph is a tree and give an algorithm for computing the minimum possible rank in this case a more general problem is also considered

negative binomial additive models summary the generalized additive model is extended to handle negative binomial responses the extension is complicated by the fact that the negative binomial distribution has two parameters and is not in the exponential family the methodology is applied to data involving dna adduct counts and smoking variables among ex smokers with lung cancer a more detailed investigation is made of the parametric relationship between the number of adducts and years since quitting while retaining a smooth relationship between adducts and the other covariates

inversion of jacobi s tridiagonal matrix abstract this paper gives a simple algorithm for finding the explicit inverse of a general jacobi tridiagonal matrix a few illustrations are also provided to demonstrate the applicability of our algorithm

a note on overdispersed exponential families abstract the issue of creating overdispersion in a given one parameter one dimensional exponential family by extending it to a two parameter exponential family with the same support is considered an easily verifiable sufficient condition for this is derived it is shown that a large class of families satisfy this condition and that this class includes efron s 1986 and lindsay s 1986 family as special cases this class is also closely related to jorgensen s 1988 exponential dispersion models ump unbiased tests for testing overdispersion are exhibited and it is shown that in this context cox s overdispersion test is a special case a graphical display is developed to select a family of the general class which should be used with a given set of data to overdisperse a given target one parameter family two real data illustrations are given kr

bayesian semiparametric analysis of developmental toxicology data modeling of developmental toxicity studies often requires simple parametric analyses of the dose response relationship between exposure and probability of a birth defect but poses challenges because of nonstandard distributions of birth defects for a fixed level of exposure this article is motivated by two such experiments in which the distribution of the outcome variable is challenging to both the standard logistic model with binomial response and its parametric multistage elaborations we approach our analysis using a bayesian semiparametric model that we tailored specifically to developmental toxicology studies it combines parametric dose response relationships with a flexible nonparametric specification of the distribution of the response obtained via a product of dirichlet process mixtures approach pdpm our formulation achieves three goals 1 the distribution of the response is modeled in a general way 2 the degree to which the distribution of the response adapts nonparametrically to the observations is driven by the data and 3 the marginal posterior distribution of the parameters of interest is available in closed form the logistic regression model as well as many of its extensions such as the beta binomial model and finite mixture models are special cases in the context of the two motivating examples and a simulated example we provide model comparisons illustrate overdispersion diagnostics that can assist model specification show how to derive posterior distributions of the effective dose parameters and predictive distributions of response and discuss the sensitivity of the results to the choice of the prior distribution

explicit inverse of a tridiagonal k toeplitz matrix we obtain explicit formulas for the entries of the inverse of a nonsingular and irreducible tridiagonal k toeplitz matrix a the proof is based on results from the theory of orthogonal polynomials and it is shown that the entries of the inverse of such a matrix are given in terms of chebyshev polynomials of the second kind we also compute the characteristic polynomial of a which enables us to state some conditions for the existence of a 1 our results also extend known results for the case when the residue mod k of the order of a is equal to 0 or k 1 numer math 10 1967 pp 153 161

strong structural controllability of systems on colored graphs this article deals with strong structural controllability of leader follower networks the system matrix defining the network dynamics is a pattern matrix in which a priori given entries are equal to zero while the remaining entries take nonzero values these nonzero entries correspond to edges in the network graph the network is called strongly structurally controllable if for all choices of real values for the nonzero entries in the pattern matrix the system is controllable in the classical sense the novelty of this article is that we consider the situation that prespecified nonzero entries in the system s pattern matrix are constrained to take identical nonzero values these constraints can be caused by symmetry properties or physical constraints on the network restricting the system matrices to those satisfying these constraints yields a new notion of strong structural controllability the aim of this article is to establish graph theoretic conditions for this more general property of strong structural controllability

vertex and edge spread of zero forcing number maximum nullity and minimum rank of a graph abstract the minimum rank of a simple graph g is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry for i j is nonzero whenever i j is an edge in g and is zero otherwise maximum nullity is taken over the same set of matrices the zero forcing number is the minimum size of a zero forcing set of vertices and bounds the maximum nullity from above the spread of a graph parameter at a vertex v or edge e of g is the difference between the value of the parameter on g and on g v or g e rank spread at a vertex was introduced in 4 this paper introduces vertex spread of the zero forcing number and edge spreads for minimum rank maximum nullity and zero forcing number properties of the spreads are established and used to determine values of the minimum rank maximum nullity and zero forcing number for various types of grids with a vertex or edge deleted

constructing a controllable graph under edge constraints abstract this paper is concerned with the problem of constructing a controllable graph subject to some practical edge constraints specifically suppose the total amount of vertices and the upper bounds on the graph diameter or on the vertex degree are given we consider the problem of exploring a class of feasible graphs that satisfy the constraints using the hybrid of a path graph and an antiregular graph we propose a simple and systematic method to generate a class of controllable graphs whose diameters or degrees cover the full possible ranges the method to select the control vector to ensure the controllability of the combined graph is also proposed numerical examples are provided to demonstrate our results

updating the inverse of a matrix the sherman morrison woodbury formulas relate the inverse of a matrix after a small rank perturbation to the inverse of the original matrix the history of these fomulas is presented and various applications to statistics networks structural analysis asymptotic analysis optimization and partial differential equations are discussed the sherman morrison woodbury formulas express the inverse of a matrix after a small rank perturbation in terms of the inverse of the original matrix this paper surveys the history of these formulas and we examine some applications where these formulas are helpful

zero forcing sets and the minimum rank of graphs the minimum rank of a simple graph g is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry for i 6 j is nonzero whenever i j is an edge in g and is zero otherwise this paper introduces a new graph parameter z g that is the minimum size of a zero forcing set of vertices and uses it to bound the minimum rank for numerous families of graphs often enabling computation of the minimum rank

explicit inverse of a tridiagonal p r toeplitz matrix abstract we have named tridiagonal p r toeplitz matrix to those tridiagonal matrices in which each diagonal is a quasi periodic sequence d p j r d j so with period p n but multiplied by a real number r we present here the necessary and sufficient conditions for the invertibility of this kind of matrices and explicitly compute their inverse the techniques we use are related with the solution of boundary value problems associated to second order linear difference equations these boundary value problems can be expressed throughout the discrete schrodinger operator and their solutions can be computed using recent advances in the study of linear difference equations with quasi periodic coefficients the conditions that ensure the uniqueness solution of the boundary value problem lead us to the invertibility conditions for the matrix whereas the solutions of the boundary value problems provides the entries of the inverse matrix

a parametric meta analysis in a meta analysis we assemble a sample of independent nonidentically distributed p values the fisher s combination procedure provides a chi squared test of whether the p values were sampled from the null uniform distribution after rejecting the null uniform hypothesis we are faced with the problem of how to combine the assembled p values we first derive a distribution for the p values the distribution is parameterized by the standardized mean difference smd and the sample size it includes the uniform as a special case the maximum likelihood estimate mle of the smd can then be obtained from the independent nonidentically distributed p values the mle can be interpreted as a weighted average of the study specific estimate of the effect size with a shrinkage the method is broadly applicable to p values obtained in the maximum likelihood framework simulation studies show that our method can effectively estimate the effect size with as few as 6 p values in the meta analyses we also present a bayes estimator for smd and a method to account for publication bias we demonstrate our methods on several meta analyses that assess the potential benefits of citicoline for patients with memory disorders or patients recovering from ischemic stroke

the inverse of a tridiagonal matrix in this paper explicit formulae for the elements of the inverse of a general tridiagonal matrix are presented by first extending results on the explicit solution of a second order linear homogeneous difference equation with variable coefficients to the nonhomogeneous case and then applying these extended results to a boundary value problem a formula for the characteristic polynomial is obtained in the process we also establish a connection between the matrix inverse and orthogonal polynomials in addition the case of a cyclic tridiagonal system is discussed

on minimum rank and zero forcing sets of a graph abstract for a graph g on n vertices and a field f the minimum rank of g over f written as mr f g is the smallest possible rank over all n n symmetric matrices over f whose i j th entry for i j is nonzero whenever ij is an edge in g and is zero otherwise the maximum nullity of g over f is m f g n mr f g the minimum rank problem of a graph g is to determine mr f g or equivalently m f g this problem has received considerable attention over the years in f barioli w barrett s butler s m cioab d cvetkovic s m fallat c godsil w haemers l hogben r mikkelson s narayan o pryporova i sciriha w so d stevanovic h van der holst k v meulen a w wehe aim minimum rank special graphs work group zero forcing sets and the minimum rank of graphs linear algebra appl 428 2008 1628 1648 a new graph parameter z g the zero forcing number was introduced to bound m f g from above the authors posted an attractive question what is the class of graphs g for which z g m f g for some field f this paper focuses on exploring the above question

using a new zero forcing process to guarantee the strong arnold property abstract the maximum nullity m g and the colin de verdiere type parameter g both consider the largest possible nullity over matrices in s g which is the family of real symmetric matrices whose i j entry i j is nonzero if i is adjacent to j and zero otherwise however g restricts to those matrices a in s g with the strong arnold property which means x o is the only symmetric matrix that satisfies a x o i x o and a x o this paper introduces zero forcing parameters z sap g and z vc g and proves that z sap g 0 implies every matrix a s g has the strong arnold property and that the inequality m g z vc g g holds for every graph g finally the values of g are computed for all graphs up to 7 vertices establishing g z g for these graphs

the inverse inertia problem for the complements of partial k trees let f be an infinite field with characteristic not equal to two for a graph g v e with v 1 n let s g f be the set of all symmetric n n matrices a ai j over f with ai j 0 i j if and only if ij e we show that if g is the complement of a partial k tree and m k 2 then for all nonsingular symmetric m m matrices k over f there exists an m n matrix u such that utku s g f as a corollary we obtain that if k 2 m n and g is the complement of a partial k tree then for any two nonnegative integers p and q with p q m there exists a matrix in s g r with p positive and q negative eigenvalues

testing for overdispersion in poisson and binomial regression models abstract in this article a method for obtaining tests for overdispersion with respect to a natural exponential family is derived the tests are designed to be powerful against arbitrary alternative mixture models where only the first two moments of the mixed distribution are specified various tests for extra poisson and extra binomial variation are obtained as special cases the use of a particular test may be motivated by a consideration of the mechanism through which the overdispersion may arise the common occurrence of extra poisson and extra binomial variation has been noted by several authors however the poisson and binomial models remain valid in many instances and because of their simplicity and appeal it is of real interest to ascertain when they apply this paper develops a unifying theory for testing for overdispersion and generalizes tests previously derived including those by fisher 1950 collings and margolin 1985 and prentice 1986 it also shows the pearson statistic to be a sc

positive semidefinite zero forcing abstract the positive semidefinite zero forcing number z g of a graph g was introduced in barioli et al 2010 4 we establish a variety of properties of z g any vertex of g can be in a minimum positive semidefinite zero forcing set this is not true for standard zero forcing the graph parameters tw g tree width z g and z g standard zero forcing number all satisfy the graph complement conjecture see barioli et al 2012 3 graphs having extreme values of the positive semidefinite zero forcing number are characterized the effect of various graph operations on positive semidefinite zero forcing number and connections with other graph parameters are studied

algorithm and hardness results on neighborhood total domination in graphs abstract a set d v of a graph g v e is called a neighborhood total dominating set of g if d is a dominating set and the subgraph of g induced by the open neighborhood of d has no isolated vertex given a graph g min ntds is the problem of finding a neighborhood total dominating set of g of minimum cardinality the decision version of min ntds is known to be np complete for bipartite graphs and chordal graphs in this paper we extend this np completeness result to undirected path graphs chordal bipartite graphs and planar graphs we also present a linear time algorithm for computing a minimum neighborhood total dominating set in proper interval graphs we show that for a given graph g v e min ntds cannot be approximated within a factor of 1 e log v unless np dtime v o log log v and can be approximated within a factor of o log where is the maximum degree of the graph g finally we show that min ntds is apx complete for graphs of degree at most 3

total forcing versus total domination in cubic graphs abstract a set s of vertices in a graph g is a total dominating set of g if every vertex has a neighbor in s the total domination number t g is the minimum cardinality of a total dominating set of g a total forcing set in a graph g is a forcing set zero forcing set in g which induces a subgraph without isolated vertices the total forcing number of g denoted ft g is the minimum cardinality of a total forcing set in g our main contribution is to show that the total forcing number and the total domination number of a cubic graph are related more precisely we prove that if g is a connected cubic graph different from k3 3 then f t g 3 2 t g

generalizing logistic regression by nonparametric mixing abstract logistic regression is a common technique for analyzing the effect of a covariate vector x on the number of successes y in m trials when y has a binomial distribution but at times either the logistic curve does not describe the probability of success p x adequately or m is larger than 1 and y is more variable than the binomial distribution allows overdispersion relative to the binomial distribution is possible if the m trials in a set or litter are positively correlated an important covariate is omitted or x is measured with error a simple way to accommodate departures from the logit link and overdispersion is to introduce a random intercept and thus permit a random propensity toward success when varies between individual binary trials according to a discrete or multimodal distribution p x has smooth steps and y has a binomial m p x distribution when the random is constant for a set of m binary trials and varies between sets of m trials according to a discrete or multimodal

zero forcing linear and quantum controllability for systems evolving on networks we study the dynamics of systems on networks from a linear algebraic perspective the control theoretic concept of controllability describes the set of states that can be reached for these systems our main result says that controllability in the quantum sense expressed by the lie algebra rank condition and controllability in the sense of linear systems expressed by the controllability matrix rank condition are equivalent conditions we also investigate how the graph theoretic concept of a zero forcing set impacts the controllability property if a set of vertices is a zero forcing set the associated dynamical system is controllable these results open up the possibility of further exploiting the analogy between networks linear control systems theory and quantum systems lie algebraic theory this study is motivated by several quantum systems currently under study including continuous quantum walks modeling transport phenomena

generalized linear models with varying dispersion summary generalized linear models are further generalized to include a linear predictor for the dispersion as well as for the mean it is shown how the convenient structure of generalized linear models can be carried over to this more general setting by considering the mean and dispersion structure separately mean and dispersion submodels are formulated for this the dependent variable for the dispersion submodel being the deviance components of the mean submodel the fact that both submodels are essentially generalized linear models themselves is used to derive simple expressions for the likelihood equations and for asymptotic tests estimation algorithms are proposed which have good convergence properties the results apply mainly to the normal inverse gaussian and gamma distributions but can be extended to discrete distributions by using quasi likelihoods the methods developed are applied to a well known data set

double exponential families and their use in generalized linear regression abstract in one parameter exponential families such as the binomial and poisson the variance is a function of the mean double exponential families allow the introduction of a second parameter that controls variance independently of the mean double families are used as constituent distributions in generalized linear regressions in which both means and variances are allowed to depend on observed covariates the theory is applied to two examples a logistic regression and a large two way contingency table in such cases the binomial model of variance is often untrustworthy for example because genuine random sampling was infeasible the subjects may have been obtained in clumps so that the statistician should really be using smaller sample sizes clumped sampling is just one of many possible causes of overdispersion a habitual source of concern to users of binomial and poisson models this article concerns a class of regression families that allow the statistician to model overdispersion while carrying

an extended quasi likelihood function summary wedderburn s original definition of quasi likelihood for generalized linear models is extended to allow the comparison of variance functions as well as those of linear predictors and link functions the relationship between generalized linear models and the use of transformations of the response variable is explored and the ideas are illustrated by three examples generalized linear models by allowing the full distributional assumption about the random component in the model to be replaced by a much weaker assumption in which only the first and second moments were defined in making this extension wedderburn widened the scope of generalized linear models in a way very similar to that of gauss when he replaced the assumption of normality in classical linear models by that of equal variance for generalized linear models with distributions in the exponential family likelihood ratio and score tests are used for testing hypotheses concerning nested subsets of covariates in the linear predictor and for assessing hypothesized link functions these methods are also applicable with wedderburn s form of quasi likelihood however neither of these methods is suitable for the comparison of different variance functions in this paper we introduce an extended quasi likelihood function which allows for the comparison of various forms of all the components of a generalized linear model i e the linear predictor the link function and the variance function we then apply the ideas to the analysis of several sets of data

analytical inversion of general tridiagonal matrices in this paper we give a complete analysis for general tridiagonal matrix inversion for both non block and block cases and provide some very simple analytical formulae which immediately lead to closed forms for some special cases such as symmetric or toeplitz tridiagonal matrices

on a markov chain roulette type game a markov chain on non negative integers which arises in a roulette type game is discussed the transition probabilities are p01 pnj nj pi i w q pi i 1 p 1 q 1 w n 0 1 n w j n and i 1 2 n w using formulae for the determinant of a partitioned matrix a closed form expression for the solution of the markov chain roulette type game is deduced the present analysis is supported by two mathematical models from tumor growth and war with bargaining

diagnostics for overdispersion abstract diagnostic tools are proposed for assessing the dependence of extrabinomial or extra poisson variation on explanatory variables and for comparing several common models for overdispersion these tools are based on tests for regression terms in the dispersion parameter of a generalized linear model using double exponential family and pseudolikelihood formulations score tests do not require the full fitting of models for variation and lead to easy graphical and numerical procedures based on squared residuals deviance or pearson robust modifications of these are motivated by levene s test in linear models the diagnostic tools are intended primarily to ensure prudent modeling of the variance to make correct inferences about parameters in the mean

a technique for computing the zero forcing number of a graph with a cut vertex abstract the zero forcing number of a graph is the minimum size of a zero forcing set this parameter is useful in the minimum rank maximum nullity problem as it gives an upper bound to the maximum nullity results for determining graphs with extreme zero forcing numbers for determining the zero forcing number of graphs with a cut vertex and for determining the zero forcing number of unicyclic graphs are presented

an upper bound for the minimum rank of a graph abstract for a graph g of order n the minimum rank of g is defined to be the smallest possible rank over all real symmetric n n matrices a whose i j th entry for i j is nonzero whenever i j is an edge in g and is zero otherwise we prove an upper bound for minimum rank in terms of minimum degree of a vertex is valid for many graphs including all bipartite graphs and conjecture this bound is true over for all graphs and prove a related bound for all zero nonzero patterns of not necessarily symmetric matrices most of the results are valid for matrices over any infinite field but need not be true for matrices over finite fields

the analytic inversion of any finite symmetric tridiagonal matrix we use the theory of orthogonal polynomials to write down explicit expressions for the polynomials of the first and second kind associated with a given infinite symmetric tridagonal matrix h the green s function is the inverse of the infinite symmetric tridiagonal matrix h zi by calculating the inverse of the finite symmetric tridiagonal matrix we can find the analytical form of the inverse of the finite symmetric tridiagonal matrix

optimally bounded score functions for generalized linear models with applications to logistic regression summary we study optimally bounded score functions for estimating regression parameters in a generalized linear model our work extends results obtained by krasker welsch 1982 for the linear model and provides a simple proof of krasker welsch s first order condition for strong optimality the application of these results to logistic regression is studied in some detail with an example given comparing the bounded influence estimator with maximum likelihood where h q and c are known functions and 0 is a vector of regression parameters models of this type include logistic and probit regression poisson regression linear regression with known variance and certain models for lifetime data our motivation for seeking robust estimators is the same as that in the linear model maximum likelihood estimation is sometimes sensitive to outlying data for logistic regression pregibon 1981 1982 has documented the nonrobustness of the maximum likelihood estimator and expounded the benefits of diagnostics as well as robust or resistant fitting procedures see also johnson 1985 much of the work on robust estimation concerns finding estimators which sacrifice little efficiency at the assumed model while providing protection against outliers and model violations we follow this course finding bounded influence estimators minimizing certain functionals of the asymptotic covariance matrix related work includes that of hampel 1978 krasker 1980 and krasker welsch 1982 when fitting models to data two important issues are identification of outliers and influential cases and accommodation of these observations frequently when influential cases are present the fitted model is not representative of the bulk of the data to rectify this one can simply delete influential cases and refit via standard methods but this

a distance based approach to strong target control of dynamical networks this paper deals with controllability of dynamical networks it is often unfeasible or unnecessary to fully control large scale networks which motivates the control of a prescribed subset of agents of the network this specific form of output control is known under the name target control we consider target control of a family of linear control systems associated with a network and provide both a necessary and a sufficient topological condition under which the network is strongly targeted controllable furthermore a leader selection algorithm is presented to compute leader sets achieving target control

the minimum rank problem over the finite field of order 2 minimum rank 3 our main result is a sharp bound for the number of vertices in a minimal forbidden subgraph for the graphs having minimum rank at most 3 over the finite field of order 2 we also list all 62 such minimal forbidden subgraphs we conclude by exploring how some of these results over the finite field of order 2 extend to arbitrary fields and demonstrate that at least one third of the 62 are minimal forbidden subgraphs over an arbitrary field for the class of graphs having minimum rank at most 3 in that field

residual diagnostics for mixture models abstract a sample is commonly modeled by a mixture distribution if the observations follow a common distribution but the parameter of interest differs between observations for example we observe the lengths but not the ages of a sample offish it may be reasonable to assume that length is normally distributed about an unknown mean that depends on the age of the fish provided there is more than one age class in the sample then the data are distributed as a mixture of normals in this article we assume that the data are a random sample from a mixture of exponential family distributions and that for each observation the parameter of interest is sampled independently from an unknown mixing distribution q the adequacy of a fitted mixture model can be assessed by examining residuals based on the ratio of the observed to expected fit residuals based on the homogeneity model in which q is a one point distribution display a convexity property when the data follow a mixture model this becomes the basis fo

inversion of k tridiagonal matrices with toeplitz structure in this paper we consider an inverse problem with the k tridiagonal toeplitz matrices a theoretical result is obtained that under certain assumptions the explicit inverse of a k tridiagonal toeplitz matrix can be derived immediately two numerical examples are given to demonstrate the validity of our results

explicit inverses of some tridiagonal matrices abstract we give explicit inverses of tridiagonal 2 toeplitz and 3 toeplitz matrices which generalize some well known results concerning the inverse of a tridiagonal toeplitz matrix

complementary bases in symplectic matrices and a proof that their determinant is one new results on the patterns of linearly independent rows and columns among the blocks of a symplectic matrix are presented these results are combined with the block structure of the inverse of a symplectic matrix together with some properties of schur complements to give a new and elementary proof that the determinant of any symplectic matrix is 1 the new proof is valid for any field information on the zero patterns compatible with the symplectic structure is also presented

on the reachability and observability of path and cycle graphs in this technical note we investigate the reachability and observability properties of a network system running a laplacian based average consensus algorithm when the communication graph is a path or a cycle specifically we provide necessary and sufficient conditions based on simple rules from number theory to characterize all and only the nodes from which the network system is reachable respectively observable interesting immediate corollaries of our results are i a path graph is reachable observable from any single node if and only if the number of nodes of the graph is a power of two n 2i i n and ii a cycle is reachable observable from any pair of nodes if and only if n is a prime number for any set of control observation nodes we provide a closed form expression for the unreachable unobservable eigenvalues and for the eigenvectors of the unreachable unobservable subsystem

zero forcing number constrained matchings and strong structural controllability abstract the zero forcing number is a graph invariant introduced to study the minimum rank of the graph in 2008 aazami proved the np hardness of computing the zero forcing number of a simple undirected graph we complete this np hardness result by showing that the non equivalent problem of computing the zero forcing number of a directed graph allowing loops is also np hard the rest of the paper is devoted to the strong controllability of a networked system this kind of controllability takes into account only the structure of the interconnection graph but not the interconnection strengths along the edges we provide a necessary and sufficient condition in terms of zero forcing sets for the strong controllability of a system whose underlying graph is a directed graph allowing loops moreover we explain how our result differs from a recent related result discovered by monshizadeh et al finally we show how to solve the problem of finding efficiently a minimum size input set for the strong controllability of a self damped system with a tree structure

minimum rank problems abstract a graph describes the zero nonzero pattern of a family of matrices with the type of graph undirected or directed simple or allowing loops determining what type of matrices symmetric or not necessarily symmetric diagonal entries free or constrained are described by the graph the minimum rank problem of the graph is to determine the minimum among the ranks of the matrices in this family the determination of maximum nullity is equivalent this problem has been solved for simple trees p m nylen minimum rank matrices with prescribed graph linear algebra appl 248 1996 303 316 c r johnson a leal duarte the maximum multiplicity of an eigenvalue in a matrix whose graph is a tree linear and multilinear algebra 46 1999 139 144 trees allowing loops l m dealba t l hardy i r hentzel l hogben a wangsness minimum rank and maximum eigenvalue multiplicity of symmetric tree sign patterns linear algebra appl 418 2006 389 415 and directed trees allowing loops f barioli s fallat d hershkowitz h t hall l hogben h van der holst b shader on the minimum rank of not necessarily symmetric matrices a preliminary study electron j linear algebra 18 2000 126 145 we survey these results from a unified perspective and solve the minimum rank problem for simple directed trees

an analytical approach explicit inverses of periodic tridiagonal matrices abstract we derive an explicit formula for the inverse of a general periodic tridiagonal matrix our approach is to derive its l u factorization using backward continued fractions bcf which are an essential tool in number theory we then use these formulae to construct an algorithm for inverting a general periodic tridiagonal matrix which we implement in maple 1 finally we present the results of testing the efficiency of our new algorithm against another published implementation and against the library procedures available within maple to invert a general matrix and to compute its determinant

overdispersion diagnostics for generalized linear models abstract generalized linear models glm s are simple convenient models for count data but they assume that the variance is a specified function of the mean although overdispersed glm s allow more flexible mean variance relationships they are often not as simple to interpret nor as easy to fit as standard glm s this article introduces a convexity plot or c plot for short that detects overdispersion and relative variance curves and relative variance tests that help to understand the nature of the overdispersion convexity plots sometimes detect overdispersion better than score tests and relative variance curves and tests sometimes distinguish the source of the overdispersion better than score tests

maximum nullity of outerplanar graphs and the path cover number abstract let g v e be a graph with v 1 2 n define s g as the set of all n n real valued symmetric matrices a a ij with a ij 0 i j if and only if ij e by m g we denote the largest possible nullity of any matrix a s g the path cover number of a graph g denoted p g is the minimum number of vertex disjoint paths occurring as induced subgraphs of g which cover all the vertices of g there has been some success with relating the path cover number of a graph to its maximum nullity johnson and duarte 5 have shown that for a tree t m t p t barioli et al 2 show that for a unicyclic graph g m g p g or m g p g 1 notice that both families of graphs are outerplanar we show that for any outerplanar graph g m g p g further we show that for any partial 2 path g m g p g

forbidden minors for the class of graphs g with ξ g 2 abstract for a given simple graph g s g is defined to be the set of real symmetric matrices a whose i j th entry is nonzero whenever i j and ij is an edge in g in f barioli s fallat l hogben a variant on the graph parameters of colin de verdiere implications to the minimum rank of graphs electron j linear algebra 13 2005 387 404 g is defined to be the maximum corank i e nullity among a s g having the strong arnold property is used to study the minimum rank maximum eigenvalue multiplicity problem for g since is minor monotone the graphs g such that g k can be described by a finite set of forbidden minors we determine the forbidden minors for g 2 and present an application of this characterization to computation of minimum rank among matrices in s g

bounds for minimum semidefinite rank from superpositions and cutsets abstract the real complex minimum semidefinite rank of a graph is the minimum rank among all real symmetric complex hermitian positive semidefinite matrices that are naturally associated via their zero nonzero pattern to the given graph in this paper we give an upper bound on the minimum semidefinite rank of a graph when the graph is modified from the superposition of two graphs by cancelling some number of edges we also provide a lower bound for the minimum semidefinite rank of a graph determined by a given cutset when the complement of the cutset is a star forest these lower and upper bounds coincide and we can compute the minimum semidefinite rank in terms of smaller graphs this result encompasses the previously known case in which the cut set has order two or smaller next we provide results for when the cut set has order three using these results we provide an example where the positive semidefinite zero forcing number is strictly greater than the maximum positive semidefinite nullity

combining incomplete information from independent assessment surveys for estimating masonry deterioration construction materials used in building structures such as masonry wood and reinforced concrete deteriorate over time because of many factors including poor design defective materials or manufacture and poor workmanship this article is concerned with estimates of masonry deterioration and the effects of covariates on the damage to bricks on the walls of five multiple story buildings of a residential complex located in the bronx new york in this case study the damage of primary interest was a spall a physical separation of a portion of the brick face from the body of the brick eventually the face becomes so damaged that portions fall off the result is an unattractive appearance and a hazard to passersby in this study spall damage was assessed by means of three different and independent condition assessment surveys an expensive precise and hence very limited scaffold drop survey and two additional inexpensive but more detailed photographic and visual surveys the photographic survey wa

zero forcing parameters and minimum rank problems the zero forcing number z g which is the minimum number of vertices in a zero forcing set of a graph g is used to study the maximum nullity minimum rank of the family of symmetric matrices described by g it is shown that for a connected graph of order at least two no vertex is in every zero forcing set the positive semidefinite zero forcing number z g is introduced and shown to be equal to g os g where os g is the recently defined ordered set number that is a lower bound for minimum positive semidefinite rank the positive semidefinite zero forcing number is applied to the computation of positive semidefinite minimum rank of certain graphs an example of a graph for which the real positive symmetric semidefinite minimum rank is greater than the complex hermitian positive semidefinite minimum rank is presented

graphs with real algebraic co rank at most two abstract recently there have been found relations between the algebraic co rank and the zero forcing number along with the minimum rank we continue on this direction by giving a characterization of the graphs with real algebraic co rank at most 2 this implies that for any graph with minimum rank at most 3 its minimum rank is bounded from above by its real algebraic co rank this sheds some light on the conjecture that the real minimum rank is bounded from above by the real algebraic co rank

symbolic algorithms for the inverses of general k tridiagonal matrices two symbolic algorithms for inverting k tridiagonal matrices have been recently found by el mikkawy and atlan 2014 2015 these two algorithms are mainly based on the doolittle lu factorization of the k tridiagonal matrix in the current paper we present a new explicit analytic expression for the inverses of general tridiagonal matrices at first by using a block diagonalization technique we then relate k tridiagonal matrix inversion to tridiagonal matrix inversion meanwhile an efficient algorithm is derived for computing the inverses of nonsingular k tridiagonal matrices with the help of any algorithm for computing the inverses of tridiagonal matrices three examples are given in order to illustrate the performance and efficiency of the proposed algorithms

the minimum rank of universal adjacency matrices abstract in this paper we introduce a new parameter for a graph called the minimum universal rank this parameter is similar to the minimum rank of a graph for a graph g the minimum universal rank of g is the minimum rank over all matrices of the form u a i j d where a is the adjacency matrix of g j is the all ones matrix and d is the matrix with the degrees of the vertices in the main diagonal and 0 are scalars bounds for general graphs based on known graph parameters are given as is a formula for the minimum universal rank for regular graphs based on the multiplicity of the eigenvalues of a the exact value of the minimum universal rank of some families of graphs are determined including complete graphs complete bipartite graph paths and cycles bounds on the minimum universal rank of a graph obtained by deleting a single vertex are established it is shown that the minimum universal rank is not monotone on induced subgraphs but bounds based on certain induced subgraphs including bounds on the union of two graphs are given