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Realizing Euclidean Distance Matrices by Sphere Intersection
Realizing Euclidean distance matrices by sphere intersection Jorge Alencara,∗, Carlile Lavorb, Leo Libertic aInstituto Federal de Educa¸c~ao,Ci^enciae Tecnologia do Sul de Minas Gerais, Inconfidentes, MG, Brazil bUniversity of Campinas (IMECC-UNICAMP), 13081-970, Campinas-SP, Brazil cCNRS LIX, Ecole´ Polytechnique, 91128 Palaiseau, France Abstract This paper presents the theoretical properties of an algorithm to find a re- alization of a (full) n × n Euclidean distance matrix in the smallest possible embedding dimension. Our algorithm performs linearly in n, and quadratically in the minimum embedding dimension, which is an improvement w.r.t. other algorithms. Keywords: Distance geometry, sphere intersection, euclidean distance matrix, embedding dimension. 2010 MSC: 00-01, 99-00 1. Introduction Euclidean distance matrices and sphere intersection have a strong mathe- matical importance [1, 2, 3, 4, 5, 6], in addition to many applications, such as navigation problems, molecular and nanostructure conformation, network 5 localization, robotics, as well as other problems of distance geometry [7, 8, 9]. Before defining precisely the problem of interest, we need the formal defini- tion for Euclidean distance matrices. Let D be a n × n symmetric hollow (i.e., with zero diagonal) matrix with non-negative elements. We say that D is a (squared) Euclidean Distance Matrix (EDM) if there are points x1; x2; : : : ; xn 2 RK (for a positive integer K) such that 2 D(i; j) = Dij = kxi − xjk ; i; j 2 f1; : : : ; ng; where k · k denotes the Euclidean norm. The smallest K for which such a set of points exists is called the embedding dimension of D, denoted by dim(D). -
Arxiv:1701.03378V1 [Math.RA] 12 Jan 2017 Setao Apihe Rusadfe Probability” Free and Groups T Lamplighter by on Supported Technology
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by TUGraz OPEN Library Linearizing the Word Problem in (some) Free Fields Konrad Schrempf∗ January 13, 2017 Abstract We describe a solution of the word problem in free fields (coming from non- commutative polynomials over a commutative field) using elementary linear algebra, provided that the elements are given by minimal linear representa- tions. It relies on the normal form of Cohn and Reutenauer and can be used more generally to (positively) test rational identities. Moreover we provide a construction of minimal linear representations for the inverse of non-zero elements. Keywords: word problem, minimal linear representation, linearization, realiza- tion, admissible linear system, rational series AMS Classification: 16K40, 16S10, 03B25, 15A22 Introduction arXiv:1701.03378v1 [math.RA] 12 Jan 2017 Free (skew) fields arise as universal objects when it comes to embed the ring of non-commutative polynomials, that is, polynomials in (a finite number of) non- commuting variables, into a skew field [Coh85, Chapter 7]. The notion of “free fields” goes back to Amitsur [Ami66]. A brief introduction can be found in [Coh03, Section 9.3], for details we refer to [Coh95, Section 6.4]. In the present paper we restrict the setting to commutative ground fields, as a special case. See also [Rob84]. In [CR94], Cohn and Reutenauer introduced a normal form for elements in free fields in order to extend results from the theory of formal languages. In particular they characterize minimality of linear representations in terms of linear independence of ∗Contact: [email protected], Department of Discrete Mathematics (Noncommutative Structures), Graz University of Technology. -
Left Eigenvector of a Stochastic Matrix
Advances in Pure Mathematics, 2011, 1, 105-117 doi:10.4236/apm.2011.14023 Published Online July 2011 (http://www.SciRP.org/journal/apm) Left Eigenvector of a Stochastic Matrix Sylvain Lavalle´e Departement de mathematiques, Universite du Quebec a Montreal, Montreal, Canada E-mail: [email protected] Received January 7, 2011; revised June 7, 2011; accepted June 15, 2011 Abstract We determine the left eigenvector of a stochastic matrix M associated to the eigenvalue 1 in the commu- tative and the noncommutative cases. In the commutative case, we see that the eigenvector associated to the eigenvalue 0 is (,,NN1 n ), where Ni is the ith principal minor of NMI= n , where In is the 11 identity matrix of dimension n . In the noncommutative case, this eigenvector is (,P1 ,Pn ), where Pi is the sum in aij of the corresponding labels of nonempty paths starting from i and not passing through i in the complete directed graph associated to M . Keywords: Generic Stochastic Noncommutative Matrix, Commutative Matrix, Left Eigenvector Associated To The Eigenvalue 1, Skew Field, Automata 1. Introduction stochastic free field and that the vector 11 (,,PP1 n ) is fixed by our matrix; moreover, the sum 1 It is well known that 1 is one of the eigenvalue of a of the Pi is equal to 1, hence they form a kind of stochastic matrix (i.e. the sum of the elements of each noncommutative limiting probability. row is equal to 1) and its associated right eigenvector is These results have been proved in [1] but the proof the vector (1,1, ,1)T . -
Two-Graphs and Skew Two-Graphs in Finite Geometries
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Two-Graphs and Skew Two-Graphs in Finite Geometries G. Eric Moorhouse Department of Mathematics University of Wyoming Laramie Wyoming Dedicated to Professor J. J. Seidel Submitted by Aart Blokhuis ABSTRACT We describe the use of two-graphs and skew (oriented) two-graphs as isomor- phism invariants for translation planes and m-systems (including avoids and spreads) of polar spaces in odd characteristic. 1. INTRODUCTION Two-graphs were first introduced by G. Higman, as natural objects for the action of certain sporadic simple groups. They have since been studied extensively by Seidel, Taylor, and others, in relation to equiangular lines, strongly regular graphs, and other notions; see [26]-[28]. The analogous oriented two-graphs (which we call skew two-graphs) were introduced by Cameron [7], and there is some literature on the equivalent notion of switching classes of tournaments. Our exposition focuses on the use of two-graphs and skew two-graphs as isomorphism invariants of translation planes, and caps, avoids, and spreads of polar spaces in odd characteristic. The earliest precedent for using two-graphs to study ovoids and caps is apparently owing to Shult [29]. The degree sequences of these two-graphs and skew two-graphs yield the invariants known as fingerprints, introduced by J. H. Conway (see Chames [9, lo]). Two LZNEAR ALGEBRA AND ITS APPLZCATZONS 226-228:529-551 (1995) 0 Elsevier Science Inc., 1995 0024-3795/95/$9.50 655 Avenue of the Americas, New York, NY 10010 SSDI 0024-3795(95)00242-J 530 G. -
Hadamard and Conference Matrices
Hadamard and conference matrices Peter J. Cameron December 2011 with input from Dennis Lin, Will Orrick and Gordon Royle Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. A matrix attaining the bound is a Hadamard matrix. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? A matrix attaining the bound is a Hadamard matrix. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. -
On Sign-Symmetric Signed Graphs∗
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 19 (2020) 83–93 https://doi.org/10.26493/1855-3974.2161.f55 (Also available at http://amc-journal.eu) On sign-symmetric signed graphs∗ Ebrahim Ghorbani Department of Mathematics, K. N. Toosi University of Technology, P.O. Box 16765-3381, Tehran, Iran Willem H. Haemers Department of Econometrics and Operations Research, Tilburg University, Tilburg, The Netherlands Hamid Reza Maimani , Leila Parsaei Majd y Mathematics Section, Department of Basic Sciences, Shahid Rajaee Teacher Training University, P.O. Box 16785-163, Tehran, Iran Received 24 October 2019, accepted 27 March 2020, published online 10 November 2020 Abstract A signed graph is said to be sign-symmetric if it is switching isomorphic to its negation. Bipartite signed graphs are trivially sign-symmetric. We give new constructions of non- bipartite sign-symmetric signed graphs. Sign-symmetric signed graphs have a symmetric spectrum but not the other way around. We present constructions of signed graphs with symmetric spectra which are not sign-symmetric. This, in particular answers a problem posed by Belardo, Cioaba,˘ Koolen, and Wang in 2018. Keywords: Signed graph, spectrum. Math. Subj. Class. (2020): 05C22, 05C50 1 Introduction Let G be a graph with vertex set V and edge set E. All graphs considered in this paper are undirected, finite, and simple (without loops or multiple edges). A signed graph is a graph in which every edge has been declared positive or negative. In fact, a signed graph Γ is a pair (G; σ), where G = (V; E) is a graph, called the underlying ∗The authors would like to thank the anonymous referees for their helpful comments and suggestions. -
Pretty Good State Transfer in Discrete-Time Quantum Walks
Pretty good state transfer in discrete-time quantum walks Ada Chan and Hanmeng Zhan Department of Mathematics and Statistics, York University, Toronto, ON, Canada fssachan, [email protected] Abstract We establish the theory for pretty good state transfer in discrete- time quantum walks. For a class of walks, we show that pretty good state transfer is characterized by the spectrum of certain Hermitian adjacency matrix of the graph; more specifically, the vertices involved in pretty good state transfer must be m-strongly cospectral relative to this matrix, and the arccosines of its eigenvalues must satisfy some number theoretic conditions. Using normalized adjacency matrices, cyclic covers, and the theory on linear relations between geodetic an- gles, we construct several infinite families of walks that exhibits this phenomenon. 1 Introduction arXiv:2105.03762v1 [math.CO] 8 May 2021 A quantum walk with nice transport properties is desirable in quantum com- putation. For example, Grover's search algorithm [12] is a quantum walk on the looped complete graph, which sends the all-ones vector to a vector that almost \concentrate on" a vertex. In this paper, we study a slightly stronger notion of state transfer, where the target state can be approximated with arbitrary precision. This is called pretty good state transfer. Pretty good state transfer has been extensively studied in continuous-time quantum walks, especially on the paths [9, 22, 1, 5, 21]. However, not much 1 is known for the discrete-time analogues. The biggest difference between these two models is that in a continuous-time quantum walk, the evolution is completely determined by the adjacency or Laplacian matrix of the graph, while in a discrete-time quantum walk, the transition matrix depends on more than just the graph - usually, it is a product of two unitary matrices: U = SC; where S permutes the arcs of the graph, and C, called the coin matrix, send each arc to a linear combination of the outgoing arcs of the same vertex. -
Hadamard Matrices Include
Hadamard and conference matrices Peter J. Cameron University of St Andrews & Queen Mary University of London Mathematics Study Group with input from Rosemary Bailey, Katarzyna Filipiak, Joachim Kunert, Dennis Lin, Augustyn Markiewicz, Will Orrick, Gordon Royle and many happy returns . Happy Birthday, MSG!! Happy Birthday, MSG!! and many happy returns . Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. A matrix attaining the bound is a Hadamard matrix. This is a nice example of a continuous problem whose solution brings us into discrete mathematics. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? A matrix attaining the bound is a Hadamard matrix. This is a nice example of a continuous problem whose solution brings us into discrete mathematics. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. -
On D. G. Higman's Note on Regular 3-Graphs
Also available at http://amc.imfm.si ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 99–115 On D. G. Higman’s note on regular 3-graphs Daniel Kalmanovich Department of Mathematics Ben-Gurion University of the Negev 84105 Beer Sheva, Israel Received 17 October 2011, accepted 25 April 2012, published online 4 June 2012 Abstract We introduce the notion of a t-graph and prove that regular 3-graphs are equivalent to cyclic antipodal 3-fold covers of a complete graph. This generalizes the equivalence of regular two-graphs and Taylor graphs. As a consequence, an equivalence between cyclic antipodal distance regular graphs of diameter 3 and certain rank 6 commutative association schemes is proved. New examples of regular 3-graphs are presented. Keywords: Antipodal graph, association scheme, distance regular graph of diameter 3, Godsil- Hensel matrix, group ring, Taylor graph, two-graph. Math. Subj. Class.: 05E30, 05B20, 05E18 1 Introduction This paper is mainly a clarification of [6] — a short draft written by Donald Higman in 1994, entitled “A note on regular 3-graphs”. The considered generalization of two-graphs was introduced by D. G. Higman in [5]. As in the famous correspondence between two-graphs and switching classes of simple graphs, t-graphs are interpreted as equivalence classes of an appropriate switching relation defined on weights, which play the role of simple graphs. In his note Higman uses certain association schemes to characterize regular 3-graphs and to obtain feasibility conditions for their parameters. Specifically, he provides a graph theoretic interpretation of a weight and from the resulted graph he constructs a rank 4 symmetric association scheme and a rank 6 fission of it. -
2020 Ural Workshop on Group Theory and Combinatorics
Institute of Natural Sciences and Mathematics of the Ural Federal University named after the first President of Russia B.N.Yeltsin N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences The Ural Mathematical Center 2020 Ural Workshop on Group Theory and Combinatorics Yekaterinburg – Online, Russia, August 24-30, 2020 Abstracts Yekaterinburg 2020 2020 Ural Workshop on Group Theory and Combinatorics: Abstracts of 2020 Ural Workshop on Group Theory and Combinatorics. Yekaterinburg: N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, 2020. DOI Editor in Chief Natalia Maslova Editors Vladislav Kabanov Anatoly Kondrat’ev Managing Editors Nikolai Minigulov Kristina Ilenko Booklet Cover Desiner Natalia Maslova c Institute of Natural Sciences and Mathematics of Ural Federal University named after the first President of Russia B.N.Yeltsin N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences The Ural Mathematical Center, 2020 2020 Ural Workshop on Group Theory and Combinatorics Conents Contents Conference program 5 Plenary Talks 8 Bailey R. A., Latin cubes . 9 Cameron P. J., From de Bruijn graphs to automorphisms of the shift . 10 Gorshkov I. B., On Thompson’s conjecture for finite simple groups . 11 Ito T., The Weisfeiler–Leman stabilization revisited from the viewpoint of Terwilliger algebras . 12 Ivanov A. A., Densely embedded subgraphs in locally projective graphs . 13 Kabanov V. V., On strongly Deza graphs . 14 Khachay M. Yu., Efficient approximation of vehicle routing problems in metrics of a fixed doubling dimension . -
ON EUCLIDEAN DISTANCE MATRICES of GRAPHS∗ 1. Introduction. a Matrix D ∈ R N×N Is a Euclidean Distance Matrix (EDM), If Ther
Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 26, pp. 574-589, August 2013 ELA ON EUCLIDEAN DISTANCE MATRICES OF GRAPHS∗ GASPERˇ JAKLICˇ † AND JOLANDA MODIC‡ Abstract. In this paper, a relation between graph distance matrices and Euclidean distance matrices (EDM) is considered. It is proven that distance matrices of paths and cycles are EDMs. The proofs are constructive and the generating points of studied EDMs are given in a closed form. A generalization to weighted graphs (networks) is tackled. Key words. Graph, Euclidean distance matrix, Distance, Eigenvalue. AMS subject classifications. 15A18, 05C50, 05C12. n n 1. Introduction. A matrix D R × is a Euclidean distance matrix (EDM), ∈ if there exist points x Rr, i =1, 2,...,n, such that d = x x 2. The minimal i ∈ ij k i − j k possible r is called an embedding dimension (see, e.g., [3]). Euclidean distance matri- ces were introduced by Menger in 1928, later they were studied by Schoenberg [13], and other authors. They have many interesting properties, and are used in various applications in linear algebra, graph theory, and bioinformatics. A natural problem is to study configurations of points xi, where only distances between them are known. n n Definition 1.1. A matrix D = (d ) R × is hollow, if d = 0 for all ij ∈ ii i =1, 2,...,n. There are various characterizations of EDMs. n n Lemma 1.2. [8, Lemma 5.3] Let a symmetric hollow matrix D R × have only ∈ one positive eigenvalue λ and the corresponding eigenvector e := [1, 1,..., 1]T Rn. -
A Submodular Approach
SPARSEST NETWORK SUPPORT ESTIMATION: A SUBMODULAR APPROACH Mario Coutino, Sundeep Prabhakar Chepuri, Geert Leus Delft University of Technology ABSTRACT recovery of the adjacency and normalized Laplacian matrix, in the In this work, we address the problem of identifying the underlying general case, due to the convex formulation of the recovery problem, network structure of data. Different from other approaches, which solutions that only retrieve the support approximately are obtained, are mainly based on convex relaxations of an integer problem, here i.e., thresholding operations are required. we take a distinct route relying on algebraic properties of a matrix Taking this issue into account, and following arguments closely representation of the network. By describing what we call possible related to the ones used for network topology estimation through ambiguities on the network topology, we proceed to employ sub- spectral templates [10], the main aim of this work is twofold: (i) to modular analysis techniques for retrieving the network support, i.e., provide a proper characterization of a particular set of matrices that network edges. To achieve this we only make use of the network are of high importance within graph signal processing: fixed-support modes derived from the data. Numerical examples showcase the ef- matrices with the same eigenbasis, and (ii) to provide a pure com- fectiveness of the proposed algorithm in recovering the support of binatorial algorithm, based on the submodular machinery that has sparse networks. been proven useful in subset selection problems within signal pro- cessing [18–22], as an alternative to traditional convex relaxations. Index Terms— Graph signal processing, graph learning, sparse graphs, network deconvolution, network topology inference.