The Generalized Dedekind Determinant
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Language Reference Manual
MATLAB The Language5 of Technical Computing Computation Visualization Programming Language Reference Manual Version 5 How to Contact The MathWorks: ☎ (508) 647-7000 Phone PHONE (508) 647-7001 Fax FAX ☎ (508) 647-7022 Technical Support Faxback Server FAX SERVER ✉ MAIL The MathWorks, Inc. Mail 24 Prime Park Way Natick, MA 01760-1500 http://www.mathworks.com Web INTERNET ftp.mathworks.com Anonymous FTP server @ [email protected] Technical support E-MAIL [email protected] Product enhancement suggestions [email protected] Bug reports [email protected] Documentation error reports [email protected] Subscribing user registration [email protected] Order status, license renewals, passcodes [email protected] Sales, pricing, and general information Language Reference Manual (November 1996) COPYRIGHT 1994 - 1996 by The MathWorks, Inc. All Rights Reserved. The software described in this document is furnished under a license agreement. The software may be used or copied only under the terms of the license agreement. No part of this manual may be photocopied or repro- duced in any form without prior written consent from The MathWorks, Inc. U.S. GOVERNMENT: If Licensee is acquiring the software on behalf of any unit or agency of the U. S. Government, the following shall apply: (a) for units of the Department of Defense: RESTRICTED RIGHTS LEGEND: Use, duplication, or disclosure by the Government is subject to restric- tions as set forth in subparagraph (c)(1)(ii) of the Rights in Technical Data and Computer Software Clause at DFARS 252.227-7013. (b) for any other unit or agency: NOTICE - Notwithstanding any other lease or license agreement that may pertain to, or accompany the delivery of, the computer software and accompanying documentation, the rights of the Government regarding its use, reproduction and disclosure are as set forth in Clause 52.227-19(c)(2) of the FAR. -
Adjacency and Incidence Matrices
Adjacency and Incidence Matrices 1 / 10 The Incidence Matrix of a Graph Definition Let G = (V ; E) be a graph where V = f1; 2;:::; ng and E = fe1; e2;:::; emg. The incidence matrix of G is an n × m matrix B = (bik ), where each row corresponds to a vertex and each column corresponds to an edge such that if ek is an edge between i and j, then all elements of column k are 0 except bik = bjk = 1. 1 2 e 21 1 13 f 61 0 07 3 B = 6 7 g 40 1 05 4 0 0 1 2 / 10 The First Theorem of Graph Theory Theorem If G is a multigraph with no loops and m edges, the sum of the degrees of all the vertices of G is 2m. Corollary The number of odd vertices in a loopless multigraph is even. 3 / 10 Linear Algebra and Incidence Matrices of Graphs Recall that the rank of a matrix is the dimension of its row space. Proposition Let G be a connected graph with n vertices and let B be the incidence matrix of G. Then the rank of B is n − 1 if G is bipartite and n otherwise. Example 1 2 e 21 1 13 f 61 0 07 3 B = 6 7 g 40 1 05 4 0 0 1 4 / 10 Linear Algebra and Incidence Matrices of Graphs Recall that the rank of a matrix is the dimension of its row space. Proposition Let G be a connected graph with n vertices and let B be the incidence matrix of G. -
Graph Equivalence Classes for Spectral Projector-Based Graph Fourier Transforms Joya A
1 Graph Equivalence Classes for Spectral Projector-Based Graph Fourier Transforms Joya A. Deri, Member, IEEE, and José M. F. Moura, Fellow, IEEE Abstract—We define and discuss the utility of two equiv- Consider a graph G = G(A) with adjacency matrix alence graph classes over which a spectral projector-based A 2 CN×N with k ≤ N distinct eigenvalues and Jordan graph Fourier transform is equivalent: isomorphic equiv- decomposition A = VJV −1. The associated Jordan alence classes and Jordan equivalence classes. Isomorphic equivalence classes show that the transform is equivalent subspaces of A are Jij, i = 1; : : : k, j = 1; : : : ; gi, up to a permutation on the node labels. Jordan equivalence where gi is the geometric multiplicity of eigenvalue 휆i, classes permit identical transforms over graphs of noniden- or the dimension of the kernel of A − 휆iI. The signal tical topologies and allow a basis-invariant characterization space S can be uniquely decomposed by the Jordan of total variation orderings of the spectral components. subspaces (see [13], [14] and Section II). For a graph Methods to exploit these classes to reduce computation time of the transform as well as limitations are discussed. signal s 2 S, the graph Fourier transform (GFT) of [12] is defined as Index Terms—Jordan decomposition, generalized k gi eigenspaces, directed graphs, graph equivalence classes, M M graph isomorphism, signal processing on graphs, networks F : S! Jij i=1 j=1 s ! (s ;:::; s ;:::; s ;:::; s ) ; (1) b11 b1g1 bk1 bkgk I. INTRODUCTION where sij is the (oblique) projection of s onto the Jordan subspace Jij parallel to SnJij. -
Network Properties Revealed Through Matrix Functions 697
SIAM REVIEW c 2010 Society for Industrial and Applied Mathematics Vol. 52, No. 4, pp. 696–714 Network Properties Revealed ∗ through Matrix Functions † Ernesto Estrada Desmond J. Higham‡ Abstract. The emerging field of network science deals with the tasks of modeling, comparing, and summarizing large data sets that describe complex interactions. Because pairwise affinity data can be stored in a two-dimensional array, graph theory and applied linear algebra provide extremely useful tools. Here, we focus on the general concepts of centrality, com- municability,andbetweenness, each of which quantifies important features in a network. Some recent work in the mathematical physics literature has shown that the exponential of a network’s adjacency matrix can be used as the basis for defining and computing specific versions of these measures. We introduce here a general class of measures based on matrix functions, and show that a particular case involving a matrix resolvent arises naturally from graph-theoretic arguments. We also point out connections between these measures and the quantities typically computed when spectral methods are used for data mining tasks such as clustering and ordering. We finish with computational examples showing the new matrix resolvent version applied to real networks. Key words. centrality measures, clustering methods, communicability, Estrada index, Fiedler vector, graph Laplacian, graph spectrum, power series, resolvent AMS subject classifications. 05C50, 05C82, 91D30 DOI. 10.1137/090761070 1. Motivation. 1.1. Introduction. Connections are important. Across the natural, technologi- cal, and social sciences it often makes sense to focus on the pattern of interactions be- tween individual components in a system [1, 8, 61]. -
Diagonal Sums of Doubly Stochastic Matrices Arxiv:2101.04143V1 [Math
Diagonal Sums of Doubly Stochastic Matrices Richard A. Brualdi∗ Geir Dahl† 7 December 2020 Abstract Let Ωn denote the class of n × n doubly stochastic matrices (each such matrix is entrywise nonnegative and every row and column sum is 1). We study the diagonals of matrices in Ωn. The main question is: which A 2 Ωn are such that the diagonals in A that avoid the zeros of A all have the same sum of their entries. We give a characterization of such matrices, and establish several classes of patterns of such matrices. Key words. Doubly stochastic matrix, diagonal sum, patterns. AMS subject classifications. 05C50, 15A15. 1 Introduction Let Mn denote the (vector) space of real n×n matrices and on this space we consider arXiv:2101.04143v1 [math.CO] 11 Jan 2021 P the usual scalar product A · B = i;j aijbij for A; B 2 Mn, A = [aij], B = [bij]. A permutation σ = (k1; k2; : : : ; kn) of f1; 2; : : : ; ng can be identified with an n × n permutation matrix P = Pσ = [pij] by defining pij = 1, if j = ki, and pij = 0, otherwise. If X = [xij] is an n × n matrix, the entries of X in the positions of X in ∗Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA. [email protected] †Department of Mathematics, University of Oslo, Norway. [email protected]. Correspond- ing author. 1 which P has a 1 is the diagonal Dσ of X corresponding to σ and P , and their sum n X dP (X) = xi;ki i=1 is a diagonal sum of X. -
Kung J., Rota G.C., Yan C.H. Combinatorics.. the Rota Way
This page intentionally left blank P1: ... CUUS456-FM cuus456-Kung 978 0 521 88389 4 November 28, 2008 12:21 Combinatorics: The Rota Way Gian-Carlo Rota was one of the most original and colorful mathematicians of the twentieth century. His work on the foundations of combinatorics focused on revealing the algebraic structures that lie behind diverse combinatorial areas and created a new area of algebraic combinatorics. His graduate courses influenced generations of students. Written by two of his former students, this book is based on notes from his courses and on personal discussions with him. Topics include sets and valuations, partially or- dered sets, distributive lattices, partitions and entropy, matching theory, free matrices, doubly stochastic matrices, Mobius¨ functions, chains and antichains, Sperner theory, commuting equivalence relations and linear lattices, modular and geometric lattices, valuation rings, generating functions, umbral calculus, symmetric functions, Baxter algebras, unimodality of sequences, and location of zeros of polynomials. Many exer- cises and research problems are included and unexplored areas of possible research are discussed. This book should be on the shelf of all students and researchers in combinatorics and related areas. joseph p. s. kung is a professor of mathematics at the University of North Texas. He is currently an editor-in-chief of Advances in Applied Mathematics. gian-carlo rota (1932–1999) was a professor of applied mathematics and natural philosophy at the Massachusetts Institute of Technology. He was a member of the Na- tional Academy of Science. He was awarded the 1988 Steele Prize of the American Math- ematical Society for his 1964 paper “On the Foundations of Combinatorial Theory I. -
Dynamical Systems Associated with Adjacency Matrices
DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2018190 DYNAMICAL SYSTEMS SERIES B Volume 23, Number 5, July 2018 pp. 1945{1973 DYNAMICAL SYSTEMS ASSOCIATED WITH ADJACENCY MATRICES Delio Mugnolo Delio Mugnolo, Lehrgebiet Analysis, Fakult¨atMathematik und Informatik FernUniversit¨atin Hagen, D-58084 Hagen, Germany Abstract. We develop the theory of linear evolution equations associated with the adjacency matrix of a graph, focusing in particular on infinite graphs of two kinds: uniformly locally finite graphs as well as locally finite line graphs. We discuss in detail qualitative properties of solutions to these problems by quadratic form methods. We distinguish between backward and forward evo- lution equations: the latter have typical features of diffusive processes, but cannot be well-posed on graphs with unbounded degree. On the contrary, well-posedness of backward equations is a typical feature of line graphs. We suggest how to detect even cycles and/or couples of odd cycles on graphs by studying backward equations for the adjacency matrix on their line graph. 1. Introduction. The aim of this paper is to discuss the properties of the linear dynamical system du (t) = Au(t) (1.1) dt where A is the adjacency matrix of a graph G with vertex set V and edge set E. Of course, in the case of finite graphs A is a bounded linear operator on the finite di- mensional Hilbert space CjVj. Hence the solution is given by the exponential matrix etA, but computing it is in general a hard task, since A carries little structure and can be a sparse or dense matrix, depending on the underlying graph. -
Mathematisches Forschungsinstitut Oberwolfach Analytic Number Theory
Mathematisches Forschungsinstitut Oberwolfach Report No. 50/2019 DOI: 10.4171/OWR/2019/50 Analytic Number Theory Organized by J¨org Br¨udern, G¨ottingen Kaisa Matom¨aki, Turku Robert C. Vaughan, State College Trevor D. Wooley, West Lafayette 3 November – 9 November 2019 Abstract. Analytic number theory is a subject which is central to modern mathematics. There are many important unsolved problems which have stim- ulated a large amount of activity by many talented researchers. At least two of the Millennium Problems can be considered to be in this area. Moreover in recent years there has been very substantial progress on a number of these questions. Mathematics Subject Classification (2010): 11Bxx, 11Dxx, 11Jxx-11Pxx. Introduction by the Organizers The workshop Analytic Number Theory, organised by J¨org Br¨udern (G¨ottingen), Kaisa Matom¨aki (Turku), Robert C. Vaughan (State College) and Trevor D. Woo- ley (West Lafayette) was well attended with over 50 participants from a broad geographic spectrum, all either distinguished and leading workers in the field or very promising younger researchers. Over the last decade or so there has been considerable and surprising movement in our understanding of a whole series of questions in analytic number theory and related areas. This includes the establishment of infinitely many bounded gaps in the primes by Zhang, Maynard and Tao, significant progress on our understand- ing of the behaviour of multiplicative functions such as the M¨obius function by Matom¨aki and Radziwi l l, the work on the Vinogradov -
Adjacency Matrices Text Reference: Section 2.1, P
Adjacency Matrices Text Reference: Section 2.1, p. 114 The purpose of this set of exercises is to show how powers of a matrix may be used to investigate graphs. Special attention is paid to airline route maps as examples of graphs. In order to study graphs, the notion of graph must first be defined. A graph is a set of points (called vertices,ornodes) and a set of lines called edges connecting some pairs of vertices. Two vertices connected by an edge are said to be adjacent. Consider the graph in Figure 1. Notice that two vertices may be connected by more than one edge (A and B are connected by 2 distinct edges), that a vertex need not be connected to any other vertex (D), and that a vertex may be connected to itself (F). Another example of a graph is the route map that most airlines (or railways) produce. A copy of the northern route map for Cape Air from May 2001 is Figure 2. This map is available at the web page: www.flycapeair.com. Here the vertices are the cities to which Cape Air flies, and two vertices are connected if a direct flight flies between them. Figure 2: Northern Route Map for Cape Air -- May 2001 Application Project: Adjacency Matrices Page 2 of 5 Some natural questions arise about graphs. It might be important to know if two vertices are connected by a sequence of two edges, even if they are not connected by a single edge. In Figure 1, A and C are connected by a two-edge sequence (actually, there are four distinct ways to go from A to C in two steps). -
On Structured Realizability and Stabilizability of Linear Systems
On Structured Realizability and Stabilizability of Linear Systems Laurent Lessard Maxim Kristalny Anders Rantzer Abstract We study the notion of structured realizability for linear systems defined over graphs. A stabilizable and detectable realization is structured if the state-space matrices inherit the sparsity pattern of the adjacency matrix of the associated graph. In this paper, we demonstrate that not every structured transfer matrix has a structured realization and we reveal the practical meaning of this fact. We also uncover a close connection between the structured realizability of a plant and whether the plant can be stabilized by a structured controller. In particular, we show that a structured stabilizing controller can only exist when the plant admits a structured realization. Finally, we give a parameterization of all structured stabilizing controllers and show that they always have structured realizations. 1 Introduction Linear time-invariant systems are typically represented using transfer functions or state- space realizations. The relationship between these representations is well understood; every transfer function has a minimal state-space realization and one can easily move between representations depending on the need. In this paper, we address the question of whether this relationship between representa- tions still holds for systems defined over directed graphs. Consider the simple two-node graph of Figure 1. For i = 1, 2, the node i represents a subsystem with inputs ui and outputs yi, and the edge means that subsystem 1 can influence subsystem 2 but not vice versa. The transfer matrix for any such a system has the sparsity pattern S1. For a state-space realization to make sense in this context, every state should be as- sociated with a subsystem, which means that the state should be computable using the information available to that subsystem. -
A Note on the Spectra and Eigenspaces of the Universal Adjacency Matrices of Arbitrary Lifts of Graphs
A note on the spectra and eigenspaces of the universal adjacency matrices of arbitrary lifts of graphs C. Dalf´o,∗ M. A. Fiol,y S. Pavl´ıkov´a,z and J. Sir´aˇnˇ x Abstract The universal adjacency matrix U of a graph Γ, with adjacency matrix A, is a linear combination of A, the diagonal matrix D of vertex degrees, the identity matrix I, and the all-1 matrix J with real coefficients, that is, U = c1A + c2D + c3I + c4J, with ci 2 R and c1 6= 0. Thus, as particular cases, U may be the adjacency matrix, the Laplacian, the signless Laplacian, and the Seidel matrix. In this note, we show that basically the same method introduced before by the authors can be applied for determining the spectra and bases of all the corresponding eigenspaces of arbitrary lifts of graphs (regular or not). Keywords: Graph; covering; lift; universal adjacency matrix; spectrum; eigenspace. 2010 Mathematics Subject Classification: 05C20, 05C50, 15A18. 1 Introduction For a graph Γ on n vertices, with adjacency matrix A and degree sequence d1; : : : ; dn, the universal adjacency matrix is defined as U = c1A+c2D +c3I +c4J, with ci 2 R and c1 6= 0, where D = diag(d1; : : : ; dn), I is the identity matrix, and J is the all-1 matrix. See, for arXiv:1912.04740v1 [math.CO] 8 Dec 2019 instance, Haemers and Omidi [10] or Farrugia and Sciriha [5]. Thus, for given values of the ∗Departament de Matem`atica, Universitat de Lleida, Igualada (Barcelona), Catalonia, [email protected] yDepartament de Matem`atiques,Universitat Polit`ecnicade Catalunya; and Barcelona Graduate -
Matrices Related to Dirichlet Series 2
MATRICES RELATED TO DIRICHLET SERIES DAVID A. CARDON Abstract. We attach a certain n n matrix An to the Dirichlet series × ∞ −s L(s)= k=1 akk . We study the determinant, characteristic polynomial, eigenvalues,P and eigenvectors of these matrices. The determinant of An can be understood as a weighted sum of the first n coefficients of the Dirichlet series L(s)−1. We give an interpretation of the partial sum of a Dirichlet series as a product of eigenvalues. In a special case, the determinant of An is the sum of the M¨obius function. We disprove a conjecture of Barrett and Jarvis regarding the eigenvalues of An. 1. Introduction To the Dirichlet series ∞ a L(s)= k , ks Xk=1 we attach the n n matrix × ∞ Dn = akEn(k), Xk=1 where E (k) is the n n matrix whose ijth entry is 1 if j = ki and 0 otherwise. n × For example, a1 a2 a3 a4 a5 a6 arXiv:0809.0076v1 [math.NT] 30 Aug 2008 a1 a2 a3 a1 a2 D6 = . a1 a1 a1 Since En(k1)En(k2)= En(k1k2) for every k ,k N, formally manipulating linear combinations of E (k) is 1 2 ∈ n very similar to formally manipulating Dirichlet series. However, because En(k) Date: October 28, 2018. 1 MATRICES RELATED TO DIRICHLET SERIES 2 is the zero matrix whenever k > n, the sum defining Dn is guaranteed to converge. Of course, the n n matrix contains less information than the × Dirichlet series. Letting n tend to infinity produces semi-infinite matrices, the formal manipulation of which is exactly equivalent to formally manipulating Dirichlet series.