on the uniqueness of euclidean distance matrix completions the case of points in general position the euclidean distance matrix completion problem edmcp is the problem of determining whether or not a given partial matrix can be completed into a euclidean distance matrix edm in this paper we investigate the necessary and sufficient conditions for the uniqueness of a given edm completion in the case where this edm completion is generated by points in general position we also show that the problem of checking the validity of these conditions can be formulated as a semidefinite programming problem

faces of the cone of euclidean distance matrices characterizations structure and induced geometry in this paper we obtain new characterizations of the faces of the cone of euclidean distance matrices in one case the characterization is based on a special subspace associated with each distance matrix in other case on linear restrictions of coordinate matrices we also relate the faces to the supporting hyperplanes of the cone and we show how the supporting hyperplanes induce certain geometry on the configurations that belong to that face

high accuracy solutions to energy gradient flows from material science models a computational framework is presented for materials science models that come from energy gradient flows the models of interest lead to the evolution of structure involving two or more phases the framework includes higher order derivative models and vector problems solutions are considered in periodic cells and standard fourier spectral discretization in space is used implicit time stepping is used with adaptivity based on local error estimates the implicit system at every time step is solved iteratively with newton s method the resulting linear systems are solved in inner iterations with the conjugate gradient method using a novel preconditioner that is a constant coefficient version of the system taking values for the coefficients at the pure phase states solutions with high spatial and temporal accuracy are obtained the dependence of the condition number of the preconditioned system on the size of the time step and the order parameter in the model that represents the scaled width of transition layers between phases is investigated numerically and with formal asymptotics in a simple setting the asymptotic results require a conjecture on the rank of a modified square distance matrix results from a fast graphical processing unit implementation for a three dimensional model are shown a comparison to time stepping with operator splitting into convex and concave parts that guarantees energy decrease in the numerical scheme is done

a general approach to read s type constructions of operators without non trivial invariant closed subspaces we present a general method for constructing operators without non trivial invariant closed subsets on a large class of non reflexive banach spaces in particular our approach unifies and generalizes several constructions due to read of operators without non trivial invariant subspaces on the spaces l 1 c 0 or circle plus l2 j and without non trivial invariant subsets on l 1 we also investigate how far our methods can be extended to the hilbertian setting and construct an operator on a quasi reflexive dual banach space which has no non trivial w closed invariant subspace

moore penrose inverse of a euclidean distance matrix abstract we obtain expressions for the moore penrose inverse of a euclidean distance matrix edm that are determined only by the positive semidefinite matrix associated with the edm the results complement formulas for the moore penrose inverse of an edm given in balaji and bapat 2007 2 a formula for the inverse of a principal submatrix of an edm is also derived whose expression uses the schur complement of the laplacian of the edm as an application we obtain an expression for the terminal wiener index of a tree

the structure of almost invariant half spaces for some operators abstract given a banach space x and an operator t l x an almost invariant half space aihs is a closed subspace y satisfying t y y f for some finite dimensional subspace f and dim y dim x y in this paper we study the connection between aihs and noncompactness of some subsets of the unit ball chains of t invariant subspaces and different parts of spectrum of t and t we describe a simple template for such subspaces and show that if t is quasinilpotent or weakly compact then it admits an aihs y

a remark on the faces of the cone of euclidean distance matrices a new characterization of the faces of the cone of euclidean distance matrices edms was recently obtained by tarazaga in terms of lgs d a special subspace associated with each edm d in this note we show that lgs d is nothing but the gale subspace associated with edms

on unit spherical euclidean distance matrices which differ in one entry abstract a unit spherical euclidean distance matrix edm d is a matrix whose entries can be realized as the interpoint squared euclidean distances of n points on a unit sphere in this paper given such a d and 1 k l n we present a characterization of the set of all unit spherical edms whose entries agree with those of d except possibly with the entry in the klth and lkth positions as a result we show that this set can be discrete consisting of one or two elements or it can be continuous the results are derived using two alternative approaches the second of which is based on cayley menger matrices

every operator has almost invariant subspaces abstract we show that any bounded operator t on a separable reflexive infinite dimensional banach space x admits a rank one perturbation which has an invariant subspace of infinite dimension and codimension in the non reflexive spaces we show that the same is true for operators which have non eigenvalues in the boundary of their spectrum in the hilbert space our methods produce perturbations that are also small in norm improving on an old result of brown and pearcy

two theorems on euclidean distance matrices and gale transform abstract we present a characterization of those euclidean distance matrices edms d which can be expressed as d e c for some nonnegative scalar and some correlation matrix c where e is the matrix of all ones this shows that the cones cone e e n cone e e n d n where e n is the elliptope set of correlation matrices and d n is the closed convex cone of edms the characterization is given using the gale transform of the points generating d we also show that given points p 1 p 2 p n r r for any scalars 1 2 n such that j 1 n j p j 0 j 1 n j 0 we have j 1 n j p i p j 2 for all i 1 n for some scalar independent of i

block matrices and multispherical structure of distance matrices abstract the block structure of a matrix and its relation to the block structure of the corresponding eigenvectors is investigated a set of points is said to have multispherical structure if they lie on a collection of concentric spheres when the centroid of each of the clusters lies at the common center the associated distance matrix has a block structure with simple relations between the blocks further such block structure may be recognized from the structure of the eigenvectors of the distance matrix a computational procedure is proposed to find the least number of concentric spheres containing the points represented by a distance matrix

the invariant subspace problem for rank one perturbations we show that for any bounded operator t acting on an infinite dimensional banach space there exists an operator f of rank at most one such that t f has an invariant subspace of infinite dimension and codimension we also show that whenever the boundary of the spectrum of t or t does not consist entirely of eigenvalues we can find such rank one perturbations that have arbitrarily small norm when this spectral condition is not satisfied we can still find suitable finite rank perturbations of arbitrarily small norm but not necessarily of rank one

the communicability distance in graphs let g be a simple connected graph with adjacency matrix a the communicabilitygpq between two nodes p and q of the graph is defined as the pq entry of g exp a we prove here that p q gpp gqq 2gpq 1 2 is a euclidean distance and give expressions for it in paths cycles stars and complete graphs with n nodes the sum of all communicabilitydistances in a graph is introduced as a new graph invariant g we compare this index with the wiener and kirchhoff indices of graphs and conjecture about the graphs with maximum and minimum values of this index

completely positive factorizations associated with euclidean distance matrices corresponding to an arithmetic progression abstract euclidean distance matrices corresponding to an arithmetic progression have rich spectral and structural properties we exploit those properties to develop completely positive factorizations of translations of those matrices we show that the minimal translation that makes such a matrix positive semidefinite results in a completely positive matrix we also discuss completely positive factorizations of such matrices over the integers methods developed in the paper can be used to find completely positive factorizations of other matrices with similar properties

graph rigidity via euclidean distance matrices abstract let g v e be an incomplete graph with node set v edge set e and nonnegative weights ij s on the edges let each edge v i v j be viewed as a rigid bar of length ij which can rotate freely around its end nodes a realization of a graph g is an assignment of coordinates in some euclidean space to each node of g in this paper we consider the problem of determining whether or not a given realization of a graph g is rigid we show that each realization of g can be epresented as a point in a compact convex set r m and that a generic realization of g is rigid if and only if its corresponding point is a vertex of i e an extreme point with full dimensional normal cone

minimizing effective resistance of a graph the effective resistance between two nodes of a weighted graph is the electrical resistance seen between the nodes of a resistor network with branch conductances given by the edge weights the effective resistance comes up in many applications and fields in addition to electrical network analysis including for example markov chains and continuous time averaging networks in this paper we study the problem of allocating edge weights on a given graph in order to minimize the total effective resistance i e the sum of the resistances between all pairs of nodes we show that this is a convex optimization problem and can be solved efficiently either numerically or in some cases analytically we show that optimal allocation of the edge weights can reduce the total effective resistance of the graph compared to uniform weights by a factor that grows unboundedly with the size of the graph we show that among all graphs with n nodes the path has the largest value of optimal total effective resistance and the complete graph has the least

predictability time from the seismic signal in an earthquake model the authors compute the maximum lyapunov exponent lambda of an earthquake model which exhibits deterministic chaos and they discuss its relation with the predictability time of the system a method is proposed to estimate lambda by the calculation of the entropy of markov processes which mimic i a poincare map of the model and ii a random map related to the seismic signal the latter map can be obtained using experimental records generated by low dimensional chaotic system where poincare maps are not feasible

new and old bounds for standard quadratic optimization dominance equivalence and incomparability a standard quadratic optimization problem stqp consists in minimizing a quadratic form over a simplex among the problems which can be transformed into a stqp are the general quadratic problem over a polytope and the maximum clique problem in a graph in this paper we present several new polynomial time bounds for stqp ranging from very simple and cheap ones to more complex and tight constructions the main tools employed in the conception and analysis of most bounds are semidefinite programming and decomposition of the objective function into a sum of two quadratic functions each of which is easy to minimize we provide a complete diagram of the dominance incomparability or equivalence relations among the bounds proposed in this and in previous works in particular we show that one of our new bounds dominates all the others furthermore a specialization of such bound dominates schrijver s improvement of lovasz s function bound for the maximum size of a clique in a graph

chaotic and self organized critical behavior of a generalized slider block model the dynamical behavior of two dimensional arrays of slider blocks is considered the blocks are pulled across a frictional surface by a constant velocity driver the blocks are connected to the driver and to each other by springs only one block is allowed to slip at a time and its displacement can be obtained analytically the system is deterministic with no stochastic inputs studies of a pair of slider blocks show that they exibit periodic limit cycle or choatic behavior depending upon parameter values and initial conditions studies of large two dimensional arrays of blocks show self organized criticality positive lyapunov exponents are found that depend upon the stiffness and size of the array

infinite matrices with few non zero entries and without non trivial invariant subspaces in this paper we investigate a family of infinite matrices that act on l1 we derive a condition sufficient to guarantee that a matrix has no non trivial closed invariant subspaces as a result a simplest known operator on l1 without invariant subspaces is obtained all entries of the matrix of the example but one are non negative

a contribution to the study of the metric and euclidean structures of dissimilarities this paper is concerned with the geometric properties of dissimilarity coefficients defined on finite sets and especially with their euclidean nature we present several particular transformations which preserve euclideanarity and we complete through the study of a one parameter family the current knowledge of the metric and euclidean structure of coefficients based on binary data these results are directly deduced from two theorems which prove the positive semi definite status of some quadratic forms which play a large role in some definitions of dissimilarity commonly used

almost invariant and essentially invariant halfspaces abstract in this paper we study sufficient conditions for an operator to have an almost invariant half space as a consequence we show that if x is an infinite dimensional complex banach space then every operator t l x admits an essentially invariant half space we also show that whenever a closed algebra of operators possesses a common aihs then it has a common invariant half space as well

distributed sensor localization in random environments using minimal number of anchor nodes the paper introduces diloc a distributed iterative algorithm to locate m sensors with unknown locations in rm m ges 1 with respect to a minimal number of m 1 anchors with known locations the sensors and anchors nodes in the network exchange data with their neighbors only no centralized data processing or communication occurs nor is there a centralized fusion center to compute the sensors locations diloc uses the barycentric coordinates of a node with respect to its neighbors these coordinates are computed using the cayley menger determinants i e the determinants of matrices of internode distances we show convergence of diloc by associating with it an absorbing markov chain whose absorbing states are the states of the anchors we introduce a stochastic approximation version extending diloc to random environments i e when the communications among nodes is noisy the communication links among neighbors may fail at random times and the internodes distances are subject to errors we show a s convergence of the modified diloc and characterize the error between the true values of the sensors locations and their final estimates given by diloc numerical studies illustrate diloc under a variety of deterministic and random operating conditions

a diffusion approach to network localization the localization of nodes on a network is a challenging research topic it arises in a variety of applications such as communications and sensor network analysis we propose a computational approach to recovering the positions of network nodes given partial and corrupted distance measurements and the positions of a small subset of anchor nodes first we show how to derive geometrically adaptive diffusion bases defined over the entire network given only partial distance measurements second we propose to utilize several diffusion bases simultaneously to derive multiscale diffusion frames last we utilize the diffusion frames to formulate a l1 regression based extension of the anchor points coordinates to the entire network we experimentally show that under a wide range of conditions our method compares favorably with state of the art approaches

on yielding and jointly yielding entries of euclidean distance matrices abstract an n n matrix d is a euclidean distance matrix edm if there exist points p 1 p n in some euclidean space such that d i j p i p j 2 for all i j 1 n let d be an edm and let e i j be the n n symmetric matrix with 1 s in the ij th and ji th entries and 0 s elsewhere we say that l i j u i j is the yielding interval of entry d i j if it holds that d t e i j is an edm if and only if l i j t u i j if the yielding interval of entry d i j has length 0 i e if l i j u i j 0 then d i j is said to be unyielding otherwise if l i j u i j then d i j is said to be yielding let d i j and d i k be two unyielding entries of d we say that d i j and d i k are jointly yielding if d t 1 e i j t 2 e i k is an edm for some nonzero scalars t 1 and t 2 in this paper we characterize the yielding and the jointly yielding entries of an edm d in terms of gale transforms of p 1 p n moreover for each yielding entry we present explicit formulae of its yielding interval finally we specialize our results to the case where points p 1 p n are in general position

the cell matrix closest to a given euclidean distance matrix abstract cell matrix introduced by jaklic and modic 6 is a special euclidean distance matrix edm that has a quite attractive and simple structure it is of interest to approximate an edm by a cell matrix in this paper we consider the problem of finding the cell matrix that is closest to a given edm with respect to the frobenius norm the majorization ordering of the eigenvalues of a cell matrix is also discussed

properties of euclidean and non euclidean distance matrices abstract a distance matrix d of order n is symmetric with elements 1 2 d ij 2 where dii 0 d is euclidean when the 1 2 n n 1 quantities dij can be generated as the distances between a set of n points x n p in a euclidean space of dimension p the dimensionality of d is defined as the least value of p rank x of any generating x in general p 1 and p 2 are also acceptable but may include imaginary coordinates even when d is euclidean basic properties of euclidean distance matrices are established in particular when rank d it is shown that depending on whether etd e is not or is zero the generating points lie in either p 1 dimensions in which case they lie on a hypersphere or in p 2 dimensions in which case they do not the notation e is used for a vector all of whose values are one when d is non euclidean its dimensionality p r s will comprise r real and s imaginary columns of x and r s are invariant for all generating x of minimal rank higher ranking representations can arise only from p 1 r 1 s or p 1 r s 1 or p 2 r 1 s 1 so that not only are r s invariant but they are both minimal for all admissible representations x

concept of complexity in random dynamical systems we introduce a measure of complexity in terms of the average number of bits per time unit necessary to specify the sequence generated by the system in dynamical systems with small random perturbations this indicator coincides with the rate k of divergence of nearby trajectories evolving under two different noise realizations the meaning of k is discussed in the context of the information theory and it is shown that it can be determined from real experimental data in the presence of strong dynamical intermittency the value of k is very different from the standard lyapunov exponent ensuremath lambda mathrm ensuremath sigma computed considering two nearby trajectories evolving under the same realization of the randomness however the former is much more relevant than the latter from a physical point of view as illustrated by some numerical computations for noisy maps and sandpile models textcopyright 1996 the american physical society

ricci flow embedding for rectifying non euclidean dissimilarity data pairwise dissimilarity representations are frequently used as an alternative to feature vectors in pattern recognition one of the problems encountered in the analysis of such data is that the dissimilarities are rarely euclidean while statistical learning algorithms often rely on euclidean dissimilarities such non euclidean dissimilarities are often corrected or a consistent euclidean geometry is imposed on them via embedding this paper commences by reviewing the available algorithms for analysing non euclidean dissimilarity data the novel contribution is to show how the ricci flow can be used to embed and rectify non euclidean dissimilarity data according to our representation the data is distributed over a manifold consisting of patches each patch has a locally uniform curvature and this curvature is iteratively modified by the ricci flow the raw dissimilarities are the geodesic distances on the manifold rectified euclidean dissimilarities are obtained using the ricci flow to flatten the curved manifold by modifying the individual patch curvatures we use two algorithmic components to implement this idea firstly we apply the ricci flow independently to a set of surface patches that cover the manifold second we use curvature regularisation to impose consistency on the curvatures of the arrangement of different surface patches we perform experiments on three real world datasets and use these to determine the importance of the different algorithmic components i e ricci flow and curvature regularisation we conclude that curvature regularisation is an essential step needed to control the stability of the piecewise arrangement of patches under the ricci flow

principal component analysis combined with truncated newton minimization for dimensionality reduction of chemical databases the similarity and diversity sampling problems are two challenging optimization tasks that arise in the analysis of chemical databases as a first step to their solution we propose an efficient projection refinement protocol based on the principal component analysis pca and the truncated newton minimization method implemented by our package tnpack pca tnpack we show that pca can provide the same initial guess as the singular value decomposition svd for the optimization task of solving the distance geometry optimization problem if each column of a database matrix has a mean of zero hence our pca tnpack approach is analogous to the svd tnpack projection refinement protocol that we developed recently for visualizing large chemical databases using pca tnpack and the merck mddr database mdl drug data report we further investigate the projection refinement procedure with regards to the preservation of the original clusters of chemical compounds the accuracy of similarity and diversity sampling of chemical compounds and the potential application in the study of structure activity relationships we also explore by simple experiments accuracy and efficiency aspects of the pca tnpack procedure compared to those of a global optimization algorithm simulated annealing as implemented by the program package simann in terms of producing the projection mapping of a database numerical results show that the 2d pca tnpack mapping can preserve the distance relationships of the original database and is thus valuable as a first step in similarity and diversity applications of course the generation of a global rather than local minimizer and its interpretation in terms of pharameceutical applications remains a challenge since all numerical tests are performed on the merck mddr database results are representative of realistic cases encountered in the field of drug design and may help analyze properties of medicinal compounds

exact reconstruction of euclidean distance geometry problem using low rank matrix completion the euclidean distance geometry problem arises in a wide variety of applications from determining molecular conformations in computational chemistry to localization in sensor networks when the distance information is incomplete the problem can be formulated as a nuclear norm minimization problem in this paper this minimization program is recast as a matrix completion problem of a low rank r gram matrix with respect to a suitable basis the well known restricted isometry property cannot be satisfied in this scenario instead a dual basis approach is introduced to theoretically analyze the reconstruction problem if the gram matrix satisfies certain coherence conditions with parameter nu the main result shows that the underlying configuration of n points can be recovered with very high probability from o nr nu log 2 n uniformly random samples computationally simple and fast algorithms are designed to solve the euclidean distance geometry problem numerical tests on different 3 d data and protein molecules validate the effectiveness and efficiency of the proposed algorithms

approximation by matrices positive semidefinite on a subspace abstract we obtain the best approximation to a matrix by matrices positive semidefinite on a subspace as a by product we present two new characterizations of euclidean distance matrices

a hereditarily indecomposable mathcal l _ infty space that solves the scalar plus compact problem we construct a hereditarily indecomposable banach space with dual space isomorphic to l1 every bounded linear operator on this space is expressible as i k with a scalar and k compact

on almost invariant subspaces and approximate commutation abstract a closed subspace y of a banach space x is almost invariant for a collection s of bounded linear operators on x if for each t s there exists a finite dimensional subspace f t of x such that t y y f t in this paper we study the existence of almost invariant subspaces for algebras of operators we show in particular that if a closed algebra of operators on a hilbert space has a non trivial almost invariant subspace then it has a non trivial invariant subspace we also examine the structure of operators which admit a maximal commuting family of almost invariant subspaces in particular we prove that if t is an operator on a separable hilbert space and if t p p t has finite rank for all projections p in a given maximal abelian self adjoint algebra m then t m f where m m and f is of finite rank

computing the nearest euclidean distance matrix with low embedding dimensions euclidean distance embedding appears in many high profile applications including wireless sensor network localization where not all pairwise distances among sensors are known or accurate the classical multi dimensional scaling cmds generally works well when the partial or contaminated euclidean distance matrix edm is close to the true edm but otherwise performs poorly a natural step preceding cmds would be to calculate the nearest edm to the known matrix a crucial condition on the desired nearest edm is for it to have a low embedding dimension and this makes the problem nonconvex there exists a large body of publications that deal with this problem some try to solve the problem directly and some are the type of convex relaxations of it in this paper we propose a numerical method that aims to solve this problem directly our method is strongly motivated by the majorized penalty method of gao and sun for low rank positive semi definite matrix optimization problems the basic geometric object in our study is the set of edms having a low embedding dimension we establish a zero duality gap result between the problem and its lagrangian dual problem which also motivates the majorization approach adopted numerical results show that the method works well for the euclidean embedding of network coordinate systems and for a class of problems in large scale sensor network localization and molecular conformation

the cone of distance matrices abstract the geometry of the cone of euclidean distance matrices edms is analyzed using a new characterization of an edm the facial structure and the angle that edms of embedding dimension one make with the center ray are found this result follows from a complete analysis of the critical points of the distance function in frobenius norm from the matrix e consisting of zero diagonal and ones elsewhere to the edms of embedding dimension one

a genetic algorithm for circulant euclidean distance matrices this paper presents a fast genetic algorithm to determine three dimensional configurations of points that generate circulant euclidean distance matrices edms a parallel implementation is possible by using the message passing interface mpi standard in addition theoretical results about the polyhedral structure of both the cone of circulant symmetric positive semidefinite matrices and the cone of circulant edms are introduced