DOCSLIB.ORG

MATAR, MONA, Ph.D., August, 2019 APPLIED MATHEMATICS

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
MATAR, MONA, Ph.D., August, 2019 APPLIED MATHEMATICS MATAR, MONA, Ph.D., August, 2019 APPLIED MATHEMATICS NODE AND EDGE IMPORTANCE IN NETWORKS VIA THE MATRIX EXPONENTIAL (131 pp.) Director of Dissertation: Lothar Reichel, Omar De la Cruz Cabrera The matrix exponential has been identified as a useful tool for the analysis of undirected networks, with sound theoretical justifications for its ability to model important aspects of a given network. Its use for directed networks, however, is less developed and has been less successful so far. In this dissertation we discuss some methods to identify important nodes in a directed network using the matrix exponential, taking into account that the notion of importance changes whether we consider the influence of a given node along the edge directions (downstream influence) or how it is influenced by directed paths that point to it (upstream influence). In addition, we introduce a family of importance measures based on counting walks that are allowed to reverse their direction a limited number of times, thus capturing relationships arising from influencing the same nodes, or being influenced by the same nodes, without sacrificing information about edge direction. These measures provide information about branch points. This dissertation is also concerned with the identification of important edges in a network, in both their roles as transmitters and receivers of information. We propose a method based on computing the matrix exponential of a matrix associated with a line graph of the given network. Both undirected and directed networks are considered. Edges may be given positive weights. Computed examples illustrate the performance of the proposed method. In addition to the identification of important nodes and edges in unweighted and edge-weighted networks, we study the importance of nodes in node-weighted graphs. To the best of our knowledge, adjacency matrices for node-weighted graphs have not received much attention. This dissertation describes how the line graph associated with a node-weighted graph can be used to construct an edge-weighted graph, that can be analyzed with available methods. Both undirected and directed graphs with positive node weights are considered. We show that when the weight of a node in- creases, the importance of this node in the graph increases as well. Some applications to real-life problems are shown. NODE AND EDGE IMPORTANCE IN NETWORKS VIA THE MATRIX EXPONENTIAL A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Mona Matar August, 2019 Dissertation written by Mona Matar B.S., The Lebanese University, 2004 M.S., The University of Akron, 2014 Ph.D., Kent State University, 2019 Approved by Lothar Reichel , Chairs, Doctoral Dissertation Committee Omar De la Cruz Cabrera , Jing Li , Members, Doctoral Dissertation Committee Jun Li , Austin Melton , Hassan Peyravi , Accepted by Andrew Tonge , Chair, Department of Mathematical Sciences James L. Blank , Dean, College of Arts and Sciences TABLE OF CONTENTS TABLE OF CONTENTS . .v ACKNOWLDGEMENTS . viii 1 Introduction . .1 2 Basic Definitions and Properties . .7 2.1 Graphs . .7 2.2 Adjacency Matrix . .8 2.3 Incidence Matrix of Undirected Graphs . 10 2.3.1 Incidence to Adjacency Matrix for Undirected Graphs . 10 2.3.2 Adjacency to Incidence Matrix for Undirected Graphs . 11 2.4 Incidence Matrix for Directed Graphs . 12 2.4.1 Incidence Matrix for Directed Graphs Using Entries ±1 . 12 2.4.2 Incidence and Exsurgence Matrices for Directed Graphs . 15 2.5 Weights . 15 3 Analysis of Directed Networks Via the Matrix Exponential . 17 3.1 The Matrix Exponential and Other Matrix Functions . 17 3.2 Node Importance and Communicability Using the Matrix Exponential 18 3.2.1 Existing Methods . 18 3.2.2 Aggregate Upstream and Downstream Reachability . 22 3.3 Reverting Walks with a Bounded Number of Reversions . 24 3.4 Examples . 27 v 3.4.1 Small Examples . 27 3.4.2 Real-Life Large Examples . 35 3.4.3 Bus Route Network Targeting Specific Nodes . 38 3.5 Numerical Considerations . 40 4 Edge Importance in a Network Via Line Graphs and the Matrix Exponential 47 4.1 Incidence and Exsurgence Matrices . 47 4.2 Line Graphs . 48 4.2.1 Line Graphs of an Undirected Graph . 48 4.2.2 Line Graphs of a Directed Graph . 48 4.3 Edge Weights . 50 4.3.1 Example . 53 4.4 Computing the Most Important Edges in an Undirected Network by the Matrix Exponential . 53 4.4.1 Review of the Adjacency Matrix Exponential . 53 4.4.2 Exponential of the Line Graph Adjacency Matrix for Undi- rected Graphs . 54 4.4.3 A Comparison of Downdating Methods for Undirected Graphs 56 4.4.4 A Comparison of Downdating Methods for Directed Graphs . 62 4.5 Computing the Most Important Edges in a Directed Unweighted Net- work Using the Matrix Exponential . 68 4.5.1 The Exponential of the Extended Line Graph Adjacency Matrix E+ ................................. 68 4.5.2 The Exponential of the Line Graph Adjacency Matrix E→ .. 70 4.6 Computing the Most Important Edges in a Directed Weighted Network Using the Matrix Exponential . 75 vi 4.6.1 Example . 75 4.6.2 Flight Example II . 76 4.7 Computational Aspects . 77 5 Node Importance in Node-Weighted Networks via Line Graphs and the Matrix Exponential . 79 5.1 The Sensitivity of Node Centrality to Weight Change . 79 5.1.1 Preliminaries . 80 5.1.2 Matrix Perturbation Results . 82 5.1.3 Example on Sensitivity to Weight Change . 88 5.2 Node-Weighted to Edge-Weighted . 89 5.2.1 Edge Weights from Endpoint Node Weights . 91 5.2.2 The Case when h is Factorizable . 93 5.2.3 Node Weights to Line Graph Edge Weights . 95 5.3 Computing Node Importance in Node-Weighted Networks . 99 5.4 Real-Life Examples . 101 5.4.1 Example of Genotype Mutation . 101 5.4.2 Example of Social Networks: Medium and Twitter . 105 5.5 Computational Aspects . 107 5.5.1 The Arnoldi Process . 109 5.5.2 The Nonsymmetric Lanczos Process . 110 5.5.3 Approximations for the Medium-Twitter Example . 111 6 Conclusions . 114 vii ACKNOWLEDGEMENTS The following work would not have been possible without the guidance of my advisors Dr. Lothar Reichel and Dr. Omar De la Cruz Cabrera, and the support of my parents and my husband. To my daughters I say, you inspire me every day to work hard, and be the best version of myself. Thank you! viii CHAPTER 1 Introduction Often a complex system can be modeled as a network: a set of nodes, any two of which might be connected in some fashion by edges. The nature of the nodes and the connections may vary widely from application to application; Like all models, network models leave out many details of reality; however, they are able to capture a substantial part of the complexity of a system in a way that is amenable to math- ematical and computational analysis. Mathematically, we represent a network by a graph, which may be directed or undirected [25, 27, 36, 50]. In spite of their simplicity, which makes mathematical analysis tractable, these concepts can capture much of the complexity of the behavior of the system. Some examples are: • Social networks: Nodes are individuals (human or animal), and edges represent social relationships (e.g., acquaintance, friendship, allegiance). • Online social networks: These are internet-based services in which registered users can establish formalized relationships that modulate sharing of informa- tion (the prototype is Facebook). Nodes are users, and edges represent connec- tions like “friendship” (undirected) or “following” (directed). • Road networks: Each intersection or endpoint is a node, and each road section connecting one node to another one is an edge. • In molecular biology, genes and/or proteins can be regarded as nodes, connected by relationships like regulation (directed) or interaction (undirected) [57]. 1 Network analysis can be carried out on at least three levels: individual vertices and edges, subgraphs, and global properties of the whole network [17, 27, 50]. In this work, we are interested in determining the relative importance of nodes, as well as communicability between pairs of nodes. The importance of a node not only depends on how many edges originate from or end at the node, but also on the importance of the neighboring nodes. For instance, consider a graph in which the nodes represent papers and the edges represent citations. An important paper conveys importance to papers that it cites and, to a lesser extent, papers that cite an important paper also may be important. Network analysis can help determine which nodes (papers) contribute the most in broadcasting or receiving of information through the network. Various measures to quantify the importance of a node in a network have been proposed in the literature; see, e.g., [11, 23, 30, 34, 43, 44]. In the undirected case, quantities that try to capture the intuitive notion of importance have come to be known as notions of centrality [13, 17, 27, 31], based on the idea that important nodes should be reachable from many nodes in fairly few steps. In directed networks, a node can be important in two ways: a node can have high downstream influence (can reach many nodes in fairly few steps along the direction of the edges) or high upstream influence (is reached by many nodes in fairly few steps along the direction of the edges); a node with high centrality should have high influence both upstream and downstream. Each one of these concepts can be of independent interest, depending on the application. Some approaches (see, e.g., [5, 10, 11, 12, 13, 30, 34]) involve the use of matrix functions, like the matrix exponential of the adjacency matrix; see Section 3.1 for further details.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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].
    [Show full text]
  • 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.
    [Show full text]
  • The Generalized Dedekind Determinant
    Contemporary Mathematics Volume 655, 2015 http://dx.doi.org/10.1090/conm/655/13232 The Generalized Dedekind Determinant M. Ram Murty and Kaneenika Sinha Abstract. The aim of this note is to calculate the determinants of certain matrices which arise in three different settings, namely from characters on finite abelian groups, zeta functions on lattices and Fourier coefficients of normalized Hecke eigenforms. Seemingly disparate, these results arise from a common framework suggested by elementary linear algebra. 1. Introduction The purpose of this note is three-fold. We prove three seemingly disparate results about matrices which arise in three different settings, namely from charac- ters on finite abelian groups, zeta functions on lattices and Fourier coefficients of normalized Hecke eigenforms. In this section, we state these theorems. In Section 2, we state a lemma from elementary linear algebra, which lies at the heart of our three theorems. A detailed discussion and proofs of the theorems appear in Sections 3, 4 and 5. In what follows below, for any n × n matrix A and for 1 ≤ i, j ≤ n, Ai,j or (A)i,j will denote the (i, j)-th entry of A. A diagonal matrix with diagonal entries y1,y2, ...yn will be denoted as diag (y1,y2, ...yn). Theorem 1.1. Let G = {x1,x2,...xn} be a finite abelian group and let f : G → C be a complex-valued function on G. Let F be an n × n matrix defined by F −1 i,j = f(xi xj). For a character χ on G, (that is, a homomorphism of G into the multiplicative group of the field C of complex numbers), we define Sχ := f(s)χ(s).
    [Show full text]
  • 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.
    [Show full text]
  • 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).
    [Show full text]
  • 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.
    [Show full text]
  • 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
    [Show full text]
  • 4/2/2015 1.0.1 the Laplacian Matrix and Its Spectrum
    MS&E 337: Spectral Graph Theory and Algorithmic Applications Spring 2015 Lecture 1: 4/2/2015 Instructor: Prof. Amin Saberi Scribe: Vahid Liaghat Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications. 1.0.1 The Laplacian matrix and its spectrum Let G = (V; E) be an undirected graph with n = jV j vertices and m = jEj edges. The adjacency matrix AG is defined as the n × n matrix where the non-diagonal entry aij is 1 iff i ∼ j, i.e., there is an edge between vertex i and vertex j and 0 otherwise. Let D(G) define an arbitrary orientation of the edges of G. The (oriented) incidence matrix BD is an n × m matrix such that qij = −1 if the edge corresponding to column j leaves vertex i, 1 if it enters vertex i, and 0 otherwise. We may denote the adjacency matrix and the incidence matrix simply by A and B when it is clear from the context. One can discover many properties of graphs by observing the incidence matrix of a graph. For example, consider the following proposition. Proposition 1.1. If G has c connected components, then Rank(B) = n − c. Proof. We show that the dimension of the null space of B is c. Let z denote a vector such that zT B = 0. This implies that for every i ∼ j, zi = zj. Therefore z takes the same value on all vertices of the same connected component. Hence, the dimension of the null space is c. The Laplacian matrix L = BBT is another representation of the graph that is quite useful.
    [Show full text]
  • Determinants Associated to Zeta Matrices of Posets
    DETERMINANTS ASSOCIATED TO ZETA MATRICES OF POSETS Cristina M. Ballantine Sharon M. Frechette John B. Little College of the Holy Cross April 1, 2004 t Abstract. We consider the matrix ZP = ZP + ZP , where the entries of ZP are the values of the zeta function of the ¯nite poset P . We give a combinatorial interpreta- tion of the determinant of ZP and establish a recursive formula for this determinant in the case in which P is a boolean algebra. x1. Introduction The theory of partially ordered sets (posets) plays an important role in enumer- ative combinatorics and the MÄobiusinversion formula for posets generalizes several fundamental theorems including the number-theoretic MÄobiusinversion theorem. For a detailed review of posets and MÄobiusinversion we refer the reader to [S1], chapter 3, and [Sa]. Below we provide a short exposition of the basic facts on the subject following [S1]. A partially ordered set (poset) P is a set which, by abuse of notation, we also call P together with a binary relation, called a partial order and denoted ·, satisfying: (1) x · x for all x 2 P (reflexivity). (2) If x · y and y · x, then x = y (antisymmetry). (3) If x · y and y · z, then x · z (transitivity). 1991 Mathematics Subject Classi¯cation. 15A15, 05C20, 05C50, 06A11. Key words and phrases. poset, zeta function, MÄobiusfunction. Typeset by AMS-TEX 1 2 DETERMINANTS OF ZETA MATRICES Two elements x and y are comparable if x · y or y · x. Otherwise they are incomparable. We write x < y to mean x · y and x 6= y.
    [Show full text]
  • GPU Accelerated Approach to Numerical Linear Algebra and Matrix Analysis with CFD Applications
    University of Central Florida STARS HIM 1990-2015 2014 GPU Accelerated Approach to Numerical Linear Algebra and Matrix Analysis with CFD Applications Adam Phillips University of Central Florida Part of the Mathematics Commons Find similar works at: https://stars.library.ucf.edu/honorstheses1990-2015 University of Central Florida Libraries http://library.ucf.edu This Open Access is brought to you for free and open access by STARS. It has been accepted for inclusion in HIM 1990-2015 by an authorized administrator of STARS. For more information, please contact [email protected]. Recommended Citation Phillips, Adam, "GPU Accelerated Approach to Numerical Linear Algebra and Matrix Analysis with CFD Applications" (2014). HIM 1990-2015. 1613. https://stars.library.ucf.edu/honorstheses1990-2015/1613 GPU ACCELERATED APPROACH TO NUMERICAL LINEAR ALGEBRA AND MATRIX ANALYSIS WITH CFD APPLICATIONS by ADAM D. PHILLIPS A thesis submitted in partial fulfilment of the requirements for the Honors in the Major Program in Mathematics in the College of Sciences and in The Burnett Honors College at the University of Central Florida Orlando, Florida Spring Term 2014 Thesis Chair: Dr. Bhimsen Shivamoggi c 2014 Adam D. Phillips All Rights Reserved ii ABSTRACT A GPU accelerated approach to numerical linear algebra and matrix analysis with CFD applica- tions is presented. The works objectives are to (1) develop stable and efficient algorithms utilizing multiple NVIDIA GPUs with CUDA to accelerate common matrix computations, (2) optimize these algorithms through CPU/GPU memory allocation, GPU kernel development, CPU/GPU communication, data transfer and bandwidth control to (3) develop parallel CFD applications for Navier Stokes and Lattice Boltzmann analysis methods.
    [Show full text]