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

positive semidefinite matrix completion universal rigidity and the strong arnold property this paper addresses the following three topics positive semidenite psd matrix completions universal rigidity of frameworks and the strong arnold property sap we show some strong connections among these topics using semidenite programming as unifying theme our main contribution is a sufcient condition for constructing partial psd matrices which admit a unique completion to a full psd matrix such partial matrices are an essential tool in the study of the gram dimension gd g of a graph g a recently studied graph parameter related to the low psd matrix completion problem additionally we derive an elementary proof of connelly s sucient condition for universal rigidity of tensegrity frameworks and we investigate the links between these two sucient conditions we also give a geometric characterization of psd matrices satisfying the strong arnold property in terms of nondegeneracy of an associated semidenite program which we use to establish some links between the gram dimension gd and the colin de verdi ere type graph parameter

graphs whose positive semi definite matrices have nullity at most two abstract let g v e be a undirected graph containing n vertices and let m g be the set of all hermitian n n matrices m m i j with m i j 0 if i and j are connected by one edge of g with m i j c if i and j are connected by at least two edges with m i j 0 if i j and i and j are not connected by an edge of g and with m i i for i 1 n a real number what is the largest nullity attained by any positive semi definite matrix m m g in this paper we characterize for t 1 and 2 those graphs g for which the maximum nullity is not greater than t

a limit equation associated to the solvability of the vacuum einstein constraint equations by using the conformal method let m g be a compact riemannian manifold on which a trace free and divergence free sigma is an element of w 1 w p and a positive function tau is an element of w 1 w p p n are fixed in thi

reduced thin sandwich equations on manifolds euclidean at infinity and on closed manifolds existence and multiplicity the reduced thin sandwich equations rtse appear within wheeler s thin sandwich approach toward the einstein constraint equations ece of general relativity it is known that these equations cannot be well posed in general but on closed manifolds sufficient conditions for well posedness have been established in particular it has been shown that the rtse are well posed in a neighborhood of umbilical solutions of the constraint equations without conformal killing fields in this paper we will analyze such a set of equations on the manifolds euclidean at infinity in a neighborhood of asymptotically euclidean ae solutions of the ece the main conclusion in this direction is that on ae manifolds admitting a yamabe positive metric the solutions of the rtse parameterize an open subset in the space of solutions of the ece in addition we show that in the case of closed manifolds these equations are well posed around umbilical solutions of the ece admitting killing fields and present some relevant examples finally it will be shown that in the set of umbilical solutions of the vacuum ece on closed manifolds the rtse are generically well posed

observer based control of systems with slope restricted nonlinearities an observer design is presented which makes use of bounds on the slope of system nonlinearities necessary and sufficient conditions are derived for the feasibility of the design a class of state feedback control laws are characterized which when implemented with the observer states ensure global asymptotic stability one such certainty equivalence design is illustrated on an active magnetic bearing example

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

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

brief a receding horizon approach to the nonlinear h control problem the receding horizon rh methodology is extended to the design of a robust controller of h type for nonlinear systems using the nonlinear analogue of the fake h algebraic riccati equation we derive an inverse optimality result for the rh schemes for which increasing the horizon causes a decrease of the optimal cost function this inverse optimality result shows that the input output map of the closed loop system obtained with the rh control law has a bounded l 2 gain robustness properties of the nonlinear h control law in face of dynamic input uncertainty are considered

kochen specker sets and the rank 1 quantum chromatic number the quantum chromatic number of a graph g is sandwiched between its chromatic number and its clique number which are well known np hard quantities we restrict our attention to the rank 1 quantum chromatic number q 1 g which upper bounds the quantum chromatic number but is defined under stronger constraints we study its relation with the chromatic number g and the minimum dimension of orthogonal representations g it is known that g q 1 g g we answer three open questions about these relations we give a necessary and sufficient condition to have g q 1 g we exhibit a class of graphs such that g q 1 g and we give a necessary and sufficient condition to have q 1 g g our main tools are kochen specker sets collections of vectors with a traditionally important role in the study of contextuality of physical theories and more recently in the quantification of quantum zero error capacities finally as a corollary of our results and a result by avis et al on the quantum chromatic number we give a family of kochen specker sets of growing dimension

graph realizations associated with minimizing the maximum eigenvalue of the laplacian in analogy to the absolute algebraic connectivity of fiedler we study the problem of minimizing the maximum eigenvalue of the laplacian of a graph by redistributing the edge weights via semidefinite duality this leads to a graph realization problem in which nodes should be placed as close as possible to the origin while adjacent nodes must keep a distance of at least one we prove three main results for a slightly generalized form of this embedding problem first given a set of vertices partitioning the graph into several or just one part the barycenter of each part is embedded on the same side of the affine hull of the set as the origin second there is an optimal realization of dimension at most the tree width of the graph plus one and this bound is best possible in general finally bipartite graphs possess a one dimensional optimal embedding

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

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

on the difference between the maximum multiplicity and path cover number for tree like graphs abstract for a given undirected graph g the maximum multiplicity of g is defined to be the largest multiplicity of an eigenvalue 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 vertex disjoint paths occurring as induced subgraphs of g that cover all the vertices of g we derive a formula for the path cover number of a vertex sum of graphs and use it to prove that the vertex sum of so called non deficient graphs is non deficient for unicyclic graphs we provide a complete description of the path cover number and the maximum multiplicity and hence the minimum rank and we investigate the difference between path cover number and maximum multiplicity for a class of graphs referred to as block cycle graphs

unitary matrix digraphs and minimum semidefinite rank for an undirected simple graph g the minimum rank among all positive semidefinite matrices with graph g is called the minimum semidefinite rank msr of g in this paper we show that the msr of a given graph may be determined from the msr of a related bipartite graph finding the msr of a given bipartite graph is then shown to be equivalent to determining which digraphs encode the zero nonzero pattern of a unitary matrix we provide an algorithm to construct unitary matrices with a certain pattern and use previous results to give a lower bound for the msr of certain bipartite graphs

brief paper adaptive output control of uncertain nonlinear systems with non symmetric dead zone input this paper deals with the adaptive output feedback control problem of a class of uncertain nonlinear systems with an unknown non symmetric dead zone nonlinearity the nonlinear system considered here is dominated by a triangular system without zero dynamics satisfying polynomial growth in the unmeasurable states an adaptive control scheme is developed without constructing the dead zone inverse the proposed adaptive control scheme requires only the information of bounds of the slopes and the breakpoint of dead zone nonlinearity the novelty of this paper is that a universal type adaptive output feedback controller is numerically constructed by using a sum of squares sos optimization algorithm which ensures the boundedness of all the signals in the adaptive closed loop without knowing the growth rate of the uncertainties an example is presented to show the effectiveness of the proposed approach

nonlinear observers robust to measurement disturbances in an iss sense this paper formulates and studies the concept of quasi disturbance to error stability qdes which characterizes robustness of a nonlinear observer to an output measurement disturbance in essence an observer is qdes if its error dynamics are input to state stable iss with respect to the disturbance as long as the plant s input and state remain bounded we develop lyapunov based sufficient conditions for checking the qdes property for both full order and reduced order observers we use these conditions to show that several well known observer designs yield qdes observers while some others do not our results also enable the design of novel qdes observers as we demonstrate with examples when combined with a state feedback law robust to state estimation errors in the iss sense a qdes observer can be used to achieve output feedback control design with robustness to measurement disturbances as an application of this idea we treat a problem of stabilization by quantized output feedback

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

a unifying integral iss framework for stability of nonlinear cascades we analyze nonlinear cascades in which the driven subsystem is integral input to state stable iss and we characterize the admissible integral iss gains for stability this characterization makes use of the convergence speed of the driving subsystem and allows a larger class of gain functions when the convergence is faster we show that our integral iss gain characterization unifies different approaches in the literature which restrict the nonlinear growth of the driven subsystem and the convergence speed of the driving subsystem the result is used to develop a new observer based backstepping design in which the growth of the nonlinear damping terms is reduced

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

linearly independent vertices and minimum semidefinite rank we study the minimum semidefinite rank of a graph using vector representations of the graph and of certain subgraphs we present a sufficient condition for when the vectors corresponding to a set of vertices of a graph must be linearly independent in any vector representation of that graph and conjecture that the resulting graph invariant is equal to minimum semidefinite rank rotation of vector representations by a unitary matrix allows us to find the minimum semidefinite rank of the join of two graphs we also improve upon previous results concerning the effect on minimum semidefinite rank of the removal of a vertex

embedded in the shadow of the separator eigenvectors to the second smallest eigenvalue of the laplace matrix of a graph also known as fiedler vectors are the basic ingredient in spectral graph partitioning heuristics maximizing this second smallest eigenvalue over all nonnegative edge weightings with bounded total weight yields the absolute algebraic connectivity introduced by fiedler who proved tight connections of this value to the connectivity of the graph our objective is to gain a better understanding of the connections between separators and the eigenspace of this eigenvalue by studying the dual semidefinite optimization problem to the absolute algebraic connectivity by exploiting optimality conditions we show that this problem is equivalent to finding an embedding of the n nodes of the graph in n space so that their barycenter is the origin the distance between adjacent nodes is bounded by one and the nodes are spread as much as possible the sum of the squared norms is maximized for connected graphs we prove that for any separator in the graph at least one of the two separated node sets is embedded in the shadow with the origin being the light source of the convex hull of the separator furthermore we show that there always exists an optimal embedding whose dimension is bounded by the tree width of the graph plus one

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

local existence of solutions of self gravitating relativistic perfect fluids this paper deals with the evolution of the einstein gravitational fields which are coupled to a perfect fluid we consider the einstein euler system in asymptotically flat spacestimes and therefore use the condition that the energy density might vanish or tend to zero at infinity and that the pressure is a fractional power of the energy density in this setting we prove local in time existence uniqueness and well posedness of classical solutions the zero order term of our system contains an expression which might not be a c function and therefore causes an additional technical difficulty in order to achieve our goals we use a certain type of weighted sobolev space of fractional order in brauer and karp j diff eqs 251 1428 1446 2011 we constructed an initial data set for these of systems in the same type of weighted sobolev spaces

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

the mass of an asymptotically flat manifold we show that the mass of an asymptotically flat n manifold is a geometric invariant the proof is based on harmonic coordinates and to develop a suitable existence theory results about elliptic operators with rough coefficients on weighted sobolev spaces are summarised some relations between the mass scalar curvature and harmonic maps are described and the positive mass theorem for n dimensional spin manifolds is proved

well posedness of the einstein euler system in asymptotically flat spacetimes the constraint equations this paper deals with the construction of initial data for the coupled einstein euler system we consider the condition where the energy density might vanish or tend to zero at infinity and where the pressure is a fractional power of the energy density in order to achieve our goals we use a type of weighted sobolev space of fractional order par the common lichnerowicz york scaling method choquet bruhat and york 1980 9 cantor 1979 7 for solving the constraint equations cannot be applied here directly the basic problem is that the matter sources are scaled conformally and the fluid variables have to be recovered from the conformally transformed matter sources this problem has been addressed although in a different context by dain and nagy 2002 11 we show that if the matter variables are restricted to a certain region then the einstein constraint equations have a unique solution in the weighted sobolev spaces of fractional order the regularity depends upon the fractional power of the equation of state

maximum stable set formulations and heuristics based on continuous optimization the stability number g for a given graph g is the size of a maximum stable set in g the lovasz theta number provides an upper bound on g and can be computed in polynomial time as the optimal value of the lovasz semidefinite program in this paper we show that restricting the matrix variable in the lovasz semidefinite program to be rank one and rank two respectively yields a pair of continuous nonlinear optimization problems each having the global optimal value g we propose heuristics for obtaining large stable sets in g based on these new formulations and present computational results indicating the effectiveness of the heuristics

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

examples of ges systems that can be driven to infinity by arbitrarily small additive decaying exponentials examples are given of nonlinear time invariant systems in continuous time discrete time and of hybrid type that have linear sector growth the origin globally exponentially stable and that can be driven to infinity by arbitrarily small additive decaying exponentials resulting observations about additive cascades and lyapunov functions are discussed these observations extend those derived from an example which recently appeared in the literature of a globally asymptotically stable continuous time system destabilized by a piecewise constant integrable disturbance

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

causal geometry of einstein vacuum spacetimes with finite curvature flux one of the central difficulties of settling the l2 bounded curvature conjecture for the einstein vacuum equations is to be able to control the causal structure of spacetimes with such limited regularity in this paper we show how to circumvent this difficulty by showing that the geometry of null hypersurfaces of enstein vacuum spacetimes can be controlled in terms of initial data and the total curvature flux through the hypersurface

boundary harmonic coordinates on manifolds with boundary in low regularity in this paper we prove the existence of h 2 regular coordinates on riemannian 3 manifolds with boundary assuming only l 2 bounds on the ricci curvature l 4 bounds on the second fundamental form of the boundary and a positive lower bound on the volume radius the proof follows by extending the theory of cheeger gromov convergence to include manifolds with boundary in the above low regularity setting the main tools are boundary harmonic coordinates together with elliptic estimates and a geometric trace estimate and a rigidity argument using manifold doubling assuming higher regularity of the ricci curvature we also prove corresponding higher regularity estimates for the coordinates

the boost problem in general relativity we show that any asymptotically flat initial data for the einstein field equations have a development which includes complete spacelike surfaces boosted relative to the initial surface furthermore the asymptotic fall off is preserved along these boosted surfaces and there exists a global system of harmonic coordinates on such a development we also extend former results on global solutions of the constraint equations by virtue of this extension the constraint and evolution parts of the problem fit together exactly several theorems are given which concern the behaviour in the large of general classes of linear and quasilinear differential systems this paper contains in addition a systematic exposition of the functional spaces employed

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

orthogonal representations minimum rank and graph complements orthogonal representations are used to show that complements of certain sparse graphs have positive semidefinite minimum rank at most 4 this bound applies to the complement of a 2 tree and to the complement of a unicyclic graph hence for such graphs the sum of the minimum rank of the graph and the minimum rank of its complement is at most two more than the order of the graph the minimum rank of the complement of a 2 tree is determined exactly

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

exact semidefinite formulations for a class of random and non random nonconvex quadratic programs we study a class of quadratically constrained quadratic programs qcqps called diagonal qcqps which contain no off diagonal terms x j x k for j ne k and we provide a sufficient condition on the problem data guaranteeing that the basic shor semidefinite relaxation is exact our condition complements and refines those already present in the literature and can be checked in polynomial time we then extend our analysis from diagonal qcqps to general qcqps i e ones with no particular structure by reformulating a general qcqp into diagonal form we establish new polynomial time checkable sufficient conditions for the semidefinite relaxations of general qcqps to be exact finally these ideas are extended to show that a class of random general qcqps has exact semidefinite relaxations with high probability as long as the number of constraints grows no faster than a fixed polynomial in the number of variables to the best of our knowledge this is the first result establishing the exactness of the semidefinite relaxation for random general qcqps

certainty equivalence feedback design with polynomial type feedbacks which guarantee iss the purpose of this note is to establish a certainty equivalence feedback design for inverse optimally controlled affine systems in particular it is shown that a class of polynomial type state feedbacks in conjunction with a globally asymptotically convergent observer leads to a globally asymptotically stable closed loop a key step in the proposed certainty equivalence feedback design procedure is the identification of a new class of polynomial type inverse optimal feedbacks which guarantees input to state stability iss with respect to measurement errors as a consequence the proposed certainty equivalence feedback design has the important feature that the state feedback is allowed to contain polynomial nonlinearities of arbitrarily high degree in the unmeasured states this feature is illustrated on an example

rough solutions of the einstein constraints on closed manifolds without near cmc conditions we consider the conformal decomposition of einstein s constraint equations introduced by lichnerowicz and york on a closed manifold we establish existence of non cmc weak solutions using a combination of priori estimates for the individual hamiltonian and momentum constraints barrier constructions and fixed point techniques for the hamiltonian constraint riesz schauder theory for the momentum constraint together with a topological fixed point argument for the coupled system although we present general existence results for non cmc weak solutions when the rescaled background metric is in any of the three yamabe classes an important new feature of the results we present for the positive yamabe class is the absence of the near cmc assumption if the freely specifiable part of the data given by the traceless transverse part of the rescaled extrinsic curvature and the matter fields are sufficiently small and if the energy density of matter is not identically zero in this case the mean extrinsic curvature can be taken to be an arbitrary smooth function without restrictions on the size of its spatial derivatives so that it can be arbitrarily far from constant giving what is apparently the first existence results for non cmc solutions without the near cmc assumption using a coupled topological fixed point argument that avoids near cmc conditions we establish existence of coupled non cmc weak solutions with positive conformal factor w s p where p 1 and s p 1 3 p in the cmc case the regularity can be reduced to p 1 and s p 3 p 1 in the case of s 2 we reproduce the cmc existence results of choquet bruhat 10 and in the case p 2 we reproduce the cmc existence results of maxwell 33 but with a proof that goes through the same analysis framework that we use to obtain the non cmc results the non cmc results on closed manifolds here extend the 1996 non cmc result of isenberg and moncrief in three ways 1 the near cmc assumption is removed in the case of the positive yamabe class 2 regularity is extended down to the maximum allowed by the background metric and the matter and 3 the result holds for all three yamabe classes this last extension was also accomplished recently by allen clausen and isenberg although their result is restricted to the near cmc case and to smoother background metrics and data

the minimum semidefinite rank of a triangle free graph abstract we employ a result of moshe rosenfeld to show that the minimum semidefinite rank of a triangle free graph with no isolated vertex must be at least half the number of its vertices we define a rosenfeld graph to be such a graph that achieves equality in this bound and we explore the structure of these special graphs their structure turns out to be intimately connected with the zero nonzero patterns of the unitary matrices finally we suggest an exploration of the connection between the girth of a graph and its minimum semidefinite rank and provide a conjecture in this direction

a new graph parameter related to bounded rank positive semidefinite matrix completions the gram dimension mathrm gd g of a graph g is the smallest integer k ge 1 such that any partial real symmetric matrix whose entries are specified on the diagonal and at the off diagonal positions corresponding to edges of g can be completed to a positive semidefinite matrix of rank at most k assuming a positive semidefinite completion exists for any fixed k the class of graphs satisfying mathrm gd g le k is minor closed hence it can be characterized by a finite list of forbidden minors we show that the only minimal forbidden minor is k k 1 for k le 3 and that there are two minimal forbidden minors k 5 and k 2 2 2 for k 4 we also show some close connections to euclidean realizations of graphs and to the graph parameter nu g of van der holst combinatorica 23 4 633 651 2003 in particular our characterization of the graphs with mathrm gd g le 4 implies the forbidden minor characterization of the 3 realizable graphs of belk discret comput geom 37 139 162 2007 and belk and connelly discret comput geom 37 125 137 2007 and of the graphs with nu g le 4 of van der holst combinatorica 23 4 633 651 2003

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

the minimum semidefinite rank of the complement of partial k trees abstract for a graph g v e with v 1 n let s g be the set of all real symmetric n n matrices a a i j with a i j 0 i j if and only if ij e we prove the following results if g is the complement of a partial k tree h then there exists a positive semidefinite matrix a s g with rank a k 2 if in addition k 3 or g is k connected then there exist positive semidefinite matrices a s g and b s h such that rank a rank b n 2

graphs associated with vector spaces of even dimension a link with differential geometry abstract we define a new association between graphs and orthonormal bases of even dimensional euclidean vector spaces endowed with an special isomorphism motivated by the recently introduced theory of submanifolds associated with graphs we provide several interesting examples and we analyze the shape of such graphs by proving some general results

rough solutions of the einstein vacuum equations this is the first in a series of papers in which we initiate the study of very rough solutions to the initial value problem for the einstein vacuum equations expressed relative to wave coordinates by very rough we mean solutions which cannot be constructed by the classical techniques of energy estimates and sobolev inequalities following kl ro we develop new analytic methods based on strichartz type inequalities which result in a gain of half a derivative relative to the classical result our methods blend paradifferential techniques with a geometric approach to the derivation of decay estimates the latter allows us to take full advantage of the specific structure of the einstein equations

separation principles for input output and integral input to state stability we present new characterizations of input output to state stability this is a notion of detectability formulated in the iss input to state stability framework equivalent properties are presented in terms of asymptotic estimates of the state trajectories based on the magnitudes of the external input and output signals these results provide a set of separation principles for input output to state stability characterizations of the property as conjunctions of weaker stability notions when applied to the notion of integral iss these characterizations yield analogous results

certainty equivalence output feedback design with circle criterion observers we study how the class of observers introduced in a previous paper here referred to as circle criterion observers can be incorporated in output feedback control due to the absence of a controller observer separation property for nonlinear systems the certainty equivalence implementation of a state feedback design may lead to severe forms of instability we show on the contrary that the state dependent convergence properties of circle criterion observers can prevent such instabilities exploiting these convergence properties we develop a modified circle criterion observer design that guarantees global asymptotic stability for certainty equivalence controllers

non cmc solutions of the einstein constraint equations on compact manifolds with apparent horizon boundaries in this article we continue our effort to do a systematic development of the solution theory for conformal formulations of the einstein constraint equations on compact manifolds with boundary by building in a natural way on our recent work in holst and tsogtgerel class quantum gravity 30 205011 2013 and holst et al phys rev lett 100 16 161101 2008 commun math phys 288 2 547 613 2009 and also on the work of maxwell j hyperbolic differ eqs 2 2 521 546 2005a commun math phys 253 3 561 583 2005b math res lett 16 4 627 645 2009 and dain class quantum gravity 21 2 555 573 2004 under reasonable assumptions on the data we prove existence of both near and far from constant mean curvature cmc solutions for a class of robin boundary conditions commonly used in the literature for modeling black holes with a third existence result for cmc appearing as a special case dain and maxwell addressed initial data engineering for space times that evolve to contain black holes determining solutions to the conformal formulation on an asymptotically euclidean manifold in the cmc setting with interior boundary conditions representing excised interior black hole regions holst and tsogtgerel compiled the interior boundary results covered by dain and maxwell and then developed general interior conditions to model the apparent horizon boundary conditions of dainand maxwell for compact manifolds with boundary and subsequently proved existence of solutions to the lichnerowicz equation on compact manifolds with such boundary conditions this paper picks up where holst and tsogtgerel left off addressing the general non cmc case for compact manifolds with boundary as in our previous articles our focus here is again on low regularity data and on the interaction between different types of boundary conditions while our work here serves primarily to extend the solution theory for the compact with boundary case we also develop several technical tools that have potential for use for other cases

sharp local well posedness results for the nonlinear wave equation this article is concerned with local well posedness of the cauchy problem for second order quasilinear hyperbolic equations with rough initial data the new results obtained here are sharp in low dimension

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

positive radial symmetric solutions to an exterior elliptic robin boundary value problem and application abstract we study the existence uniqueness and behavior of positive radial symmetric solutions of the exterior boundary value problem u h x f u in r n b 0 r 0 u n a u on and u x 0 as x when a 0 the robin boundary condition has a wrong sign and this makes the analysis of the problem interesting and non trivial we determine a sharp condition on the coefficient a for the existence of positive spherically symmetric solutions of the boundary value problem the important model case f u u 1 7 a 1 2 r 0 and 1 2 r 0 is related to the initial data problem in general relativity for this model case as an example of application of our results we obtain the existence and uniqueness of the conformal factor corresponding to the hamiltonian constraint equation in the case of a single black hole in the radial symmetric regime

separation results for the stabilization of nonlinear systems using different high gain observer designs high gain observers have been used in the design of output feedback controllers due to their ability to robustly estimate the unmeasured states while asymptotically attenuating disturbances the available techniques for the design of high gain observers can be classified into three groups pole placement algorithms riccati equation based algorithms and lyapunov equation based algorithms in 1 we presented separation results for globally bounded stabilizing state feedback controllers when the high gain observer is designed using pole placement so as to create a closed loop system with two time scale structure in this paper we show that the separation results of 1 hold for the other observer designs