Discrete Differential Forms for Computational Modeling
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Combinatorial Laplacian and Rank Aggregation
Combinatorial Laplacian and Rank Aggregation Combinatorial Laplacian and Rank Aggregation Yuan Yao Stanford University ICIAM, Z¨urich,July 16–20, 2007 Joint work with Lek-Heng Lim Combinatorial Laplacian and Rank Aggregation Outline 1 Two Motivating Examples 2 Reflections on Ranking Ordinal vs. Cardinal Global, Local, vs. Pairwise 3 Discrete Exterior Calculus and Combinatorial Laplacian Discrete Exterior Calculus Combinatorial Laplacian Operator 4 Hodge Theory Cyclicity of Pairwise Rankings Consistency of Pairwise Rankings 5 Conclusions and Future Work Combinatorial Laplacian and Rank Aggregation Two Motivating Examples Example I: Customer-Product Rating Example (Customer-Product Rating) m×n m-by-n customer-product rating matrix X ∈ R X typically contains lots of missing values (say ≥ 90%). The first-order statistics, mean score for each product, might suffer from most customers just rate a very small portion of the products different products might have different raters, whence mean scores involve noise due to arbitrary individual rating scales Combinatorial Laplacian and Rank Aggregation Two Motivating Examples From 1st Order to 2nd Order: Pairwise Rankings The arithmetic mean of score difference between product i and j over all customers who have rated both of them, P k (Xkj − Xki ) gij = , #{k : Xki , Xkj exist} is translation invariant. If all the scores are positive, the geometric mean of score ratio over all customers who have rated both i and j, !1/#{k:Xki ,Xkj exist} Y Xkj gij = , Xki k is scale invariant. Combinatorial Laplacian and Rank Aggregation Two Motivating Examples More invariant Define the pairwise ranking gij as the probability that product j is preferred to i in excess of a purely random choice, 1 g = Pr{k : X > X } − . -
Introduction to Gauge Theory Arxiv:1910.10436V1 [Math.DG] 23
Introduction to Gauge Theory Andriy Haydys 23rd October 2019 This is lecture notes for a course given at the PCMI Summer School “Quantum Field The- ory and Manifold Invariants” (July 1 – July 5, 2019). I describe basics of gauge-theoretic approach to construction of invariants of manifolds. The main example considered here is the Seiberg–Witten gauge theory. However, I tried to present the material in a form, which is suitable for other gauge-theoretic invariants too. Contents 1 Introduction2 2 Bundles and connections4 2.1 Vector bundles . .4 2.1.1 Basic notions . .4 2.1.2 Operations on vector bundles . .5 2.1.3 Sections . .6 2.1.4 Covariant derivatives . .6 2.1.5 The curvature . .8 2.1.6 The gauge group . 10 2.2 Principal bundles . 11 2.2.1 The frame bundle and the structure group . 11 2.2.2 The associated vector bundle . 14 2.2.3 Connections on principal bundles . 16 2.2.4 The curvature of a connection on a principal bundle . 19 arXiv:1910.10436v1 [math.DG] 23 Oct 2019 2.2.5 The gauge group . 21 2.3 The Levi–Civita connection . 22 2.4 Classification of U(1) and SU(2) bundles . 23 2.4.1 Complex line bundles . 24 2.4.2 Quaternionic line bundles . 25 3 The Chern–Weil theory 26 3.1 The Chern–Weil theory . 26 3.1.1 The Chern classes . 28 3.2 The Chern–Simons functional . 30 3.3 The modui space of flat connections . 32 3.3.1 Parallel transport and holonomy . -
Pg**- Compact Spaces
International Journal of Mathematics Research. ISSN 0976-5840 Volume 9, Number 1 (2017), pp. 27-43 © International Research Publication House http://www.irphouse.com pg**- compact spaces Mrs. G. Priscilla Pacifica Assistant Professor, St.Mary’s College, Thoothukudi – 628001, Tamil Nadu, India. Dr. A. Punitha Tharani Associate Professor, St.Mary’s College, Thoothukudi –628001, Tamil Nadu, India. Abstract The concepts of pg**-compact, pg**-countably compact, sequentially pg**- compact, pg**-locally compact and pg**- paracompact are introduced, and several properties are investigated. Also the concept of pg**-compact modulo I and pg**-countably compact modulo I spaces are introduced and the relation between these concepts are discussed. Keywords: pg**-compact, pg**-countably compact, sequentially pg**- compact, pg**-locally compact, pg**- paracompact, pg**-compact modulo I, pg**-countably compact modulo I. 1. Introduction Levine [3] introduced the class of g-closed sets in 1970. Veerakumar [7] introduced g*- closed sets. P M Helen[5] introduced g**-closed sets. A.S.Mashhour, M.E Abd El. Monsef [4] introduced a new class of pre-open sets in 1982. Ideal topological spaces have been first introduced by K.Kuratowski [2] in 1930. In this paper we introduce 28 Mrs. G. Priscilla Pacifica and Dr. A. Punitha Tharani pg**-compact, pg**-countably compact, sequentially pg**-compact, pg**-locally compact, pg**-paracompact, pg**-compact modulo I and pg**-countably compact modulo I spaces and investigate their properties. 2. Preliminaries Throughout this paper(푋, 휏) and (푌, 휎) represent non-empty topological spaces of which no separation axioms are assumed unless otherwise stated. Definition 2.1 A subset 퐴 of a topological space(푋, 휏) is called a pre-open set [4] if 퐴 ⊆ 푖푛푡(푐푙(퐴) and a pre-closed set if 푐푙(푖푛푡(퐴)) ⊆ 퐴. -
Geometric Representations of Graphs
1 Geometric Representations of Graphs Laszl¶ o¶ Lovasz¶ Microsoft Research One Microsoft Way, Redmond, WA 98052 e-mail: [email protected] 2 Contents I Background 5 1 Eigenvalues of graphs 7 1.1 Matrices associated with graphs ............................ 7 1.2 The largest eigenvalue ................................. 8 1.2.1 Adjacency matrix ............................... 8 1.2.2 Laplacian .................................... 9 1.2.3 Transition matrix ................................ 9 1.3 The smallest eigenvalue ................................ 9 1.4 The eigenvalue gap ................................... 11 1.4.1 Expanders .................................... 12 1.4.2 Random walks ................................. 12 1.5 The number of di®erent eigenvalues .......................... 17 1.6 Eigenvectors ....................................... 19 2 Convex polytopes 21 2.1 Polytopes and polyhedra ................................ 21 2.2 The skeleton of a polytope ............................... 21 2.3 Polar, blocker and antiblocker ............................. 22 II Representations of Planar Graphs 25 3 Planar graphs and polytopes 27 3.1 Planar graphs ...................................... 27 3.2 Straight line representation and 3-polytopes ..................... 28 4 Rubber bands, cables, bars and struts 29 4.1 Rubber band representation .............................. 29 4.1.1 How to draw a graph? ............................. 30 4.1.2 How to lift a graph? .............................. 32 4.1.3 Rubber bands and connectivity ....................... -
Digital and Discrete Geometry Li M
Digital and Discrete Geometry Li M. Chen Digital and Discrete Geometry Theory and Algorithms 2123 Li M. Chen University of the District of Columbia Washington District of Columbia USA ISBN 978-3-319-12098-0 ISBN 978-3-319-12099-7 (eBook) DOI 10.1007/978-3-319-12099-7 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: 2014958741 © Springer International Publishing Switzerland 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) To the researchers and their supporters in Digital Geometry and Topology. -
General Topology
General Topology Tom Leinster 2014{15 Contents A Topological spaces2 A1 Review of metric spaces.......................2 A2 The definition of topological space.................8 A3 Metrics versus topologies....................... 13 A4 Continuous maps........................... 17 A5 When are two spaces homeomorphic?................ 22 A6 Topological properties........................ 26 A7 Bases................................. 28 A8 Closure and interior......................... 31 A9 Subspaces (new spaces from old, 1)................. 35 A10 Products (new spaces from old, 2)................. 39 A11 Quotients (new spaces from old, 3)................. 43 A12 Review of ChapterA......................... 48 B Compactness 51 B1 The definition of compactness.................... 51 B2 Closed bounded intervals are compact............... 55 B3 Compactness and subspaces..................... 56 B4 Compactness and products..................... 58 B5 The compact subsets of Rn ..................... 59 B6 Compactness and quotients (and images)............. 61 B7 Compact metric spaces........................ 64 C Connectedness 68 C1 The definition of connectedness................... 68 C2 Connected subsets of the real line.................. 72 C3 Path-connectedness.......................... 76 C4 Connected-components and path-components........... 80 1 Chapter A Topological spaces A1 Review of metric spaces For the lecture of Thursday, 18 September 2014 Almost everything in this section should have been covered in Honours Analysis, with the possible exception of some of the examples. For that reason, this lecture is longer than usual. Definition A1.1 Let X be a set. A metric on X is a function d: X × X ! [0; 1) with the following three properties: • d(x; y) = 0 () x = y, for x; y 2 X; • d(x; y) + d(y; z) ≥ d(x; z) for all x; y; z 2 X (triangle inequality); • d(x; y) = d(y; x) for all x; y 2 X (symmetry). -
Statistical Ranking and Combinatorial Hodge Theory
STATISTICAL RANKING AND COMBINATORIAL HODGE THEORY XIAOYE JIANG, LEK-HENG LIM, YUAN YAO, AND YINYU YE Abstract. We propose a number of techniques for obtaining a global ranking from data that may be incomplete and imbalanced | characteristics that are almost universal to modern datasets coming from e-commerce and internet applications. We are primarily interested in cardinal data based on scores or ratings though our methods also give specific insights on ordinal data. From raw ranking data, we construct pairwise rankings, represented as edge flows on an appropriate graph. Our statistical ranking method exploits the graph Helmholtzian, which is the graph theoretic analogue of the Helmholtz operator or vector Laplacian, in much the same way the graph Laplacian is an ana- logue of the Laplace operator or scalar Laplacian. We shall study the graph Helmholtzian using combinatorial Hodge theory, which provides a way to un- ravel ranking information from edge flows. In particular, we show that every edge flow representing pairwise ranking can be resolved into two orthogonal components, a gradient flow that represents the l2-optimal global ranking and a divergence-free flow (cyclic) that measures the validity of the global ranking obtained | if this is large, then it indicates that the data does not have a good global ranking. This divergence-free flow can be further decomposed or- thogonally into a curl flow (locally cyclic) and a harmonic flow (locally acyclic but globally cyclic); these provides information on whether inconsistency in the ranking data arises locally or globally. When applied to statistical ranking problems, Hodge decomposition sheds light on whether a given dataset may be globally ranked in a meaningful way or if the data is inherently inconsistent and thus could not have any reasonable global ranking; in the latter case it provides information on the nature of the inconsistencies. -
Discrete Calculus with Cubic Cells on Discrete Manifolds 3
DISCRETE CALCULUS WITH CUBIC CELLS ON DISCRETE MANIFOLDS LEONARDO DE CARLO Abstract. This work is thought as an operative guide to ”discrete exterior calculus” (DEC), but at the same time with a rigorous exposition. We present a version of (DEC) on ”cubic” cell, defining it for discrete manifolds. An example of how it works, it is done on the discrete torus, where usual Gauss and Stokes theorems are recovered. Keywords: Discrete Calculus, discrete differential geometry AMS 2010 Subject Classification: 97N70 1. Introduction Discrete exterior calculus (DEC) is motivated by potential applications in com- putational methods for field theories (elasticity, fluids, electromagnetism) and in areas of computer vision/graphics, for these applications see for example [DDT15], [DMTS14] or [BSSZ08]. DEC is developing as an alternative approach for com- putational science to the usual discretizing process from the continuous theory. It considers the discrete mesh as the only thing given and develops an entire calcu- lus using only discrete combinatorial and geometric operations. The derivations may require that the objects on the discrete mesh, but not the mesh itself, are interpolated as if they come from a continuous model. Therefore DEC could have interesting applications in fields where there isn’t any continuous underlying struc- ture as Graph theory [GP10] or problems that are inherently discrete since they are defined on a lattice [AO05] and it can stand in its own right as theory that parallels the continuous one. The language of DEC is founded on the concept of discrete differential form, this characteristic allows to preserve in the discrete con- text some of the usual geometric and topological structures of continuous models, arXiv:1906.07054v2 [cs.DM] 8 Jul 2019 in particular the Stokes’theorem dω = ω. -
Space-Time Finite-Element Exterior Calculus and Variational Discretizations of Gauge Field Theories
21st International Symposium on Mathematical Theory of Networks and Systems July 7-11, 2014. Groningen, The Netherlands Space-Time Finite-Element Exterior Calculus and Variational Discretizations of Gauge Field Theories Joe Salamon1, John Moody2, and Melvin Leok3 Abstract— Many gauge field theories can be described using a the space-time covariant (or multi-Dirac) perspective can be multisymplectic Lagrangian formulation, where the Lagrangian found in [1]. density involves space-time differential forms. While there has An example of a gauge symmetry arises in Maxwell’s been much work on finite-element exterior calculus for spatial and tensor product space-time domains, there has been less equations of electromagnetism, which can be expressed in done from the perspective of space-time simplicial complexes. terms of the scalar potential φ, the vector potential A, the One critical aspect is that the Hodge star is now taken with electric field E, and the magnetic field B. respect to a pseudo-Riemannian metric, and this is most natu- @A @ rally expressed in space-time adapted coordinates, as opposed E = −∇φ − ; r2φ + (r · A) = 0; to the barycentric coordinates that the Whitney forms (and @t @t their higher-degree generalizations) are typically expressed in @φ terms of. B = r × A; A + r r · A + = 0; @t We introduce a novel characterization of Whitney forms and their Hodge dual with respect to a pseudo-Riemannian metric where is the d’Alembert (or wave) operator. The following that is independent of the choice of coordinates, and then apply gauge transformation leaves the equations invariant, it to a variational discretization of the covariant formulation of Maxwell’s equations. -
The Language of Differential Forms
Appendix A The Language of Differential Forms This appendix—with the only exception of Sect.A.4.2—does not contain any new physical notions with respect to the previous chapters, but has the purpose of deriving and rewriting some of the previous results using a different language: the language of the so-called differential (or exterior) forms. Thanks to this language we can rewrite all equations in a more compact form, where all tensor indices referred to the diffeomorphisms of the curved space–time are “hidden” inside the variables, with great formal simplifications and benefits (especially in the context of the variational computations). The matter of this appendix is not intended to provide a complete nor a rigorous introduction to this formalism: it should be regarded only as a first, intuitive and oper- ational approach to the calculus of differential forms (also called exterior calculus, or “Cartan calculus”). The main purpose is to quickly put the reader in the position of understanding, and also independently performing, various computations typical of a geometric model of gravity. The readers interested in a more rigorous discussion of differential forms are referred, for instance, to the book [22] of the bibliography. Let us finally notice that in this appendix we will follow the conventions introduced in Chap. 12, Sect. 12.1: latin letters a, b, c,...will denote Lorentz indices in the flat tangent space, Greek letters μ, ν, α,... tensor indices in the curved manifold. For the matter fields we will always use natural units = c = 1. Also, unless otherwise stated, in the first three Sects. -
The Divergence Theorem Cartan's Formula II. for Any Smooth Vector
The Divergence Theorem Cartan’s Formula II. For any smooth vector field X and any smooth differential form ω, LX = iX d + diX . Lemma. Let x : U → Rn be a positively oriented chart on (M,G), with volume j ∂ form vM , and X = Pj X ∂xj . Then, we have U √ 1 n ∂( gXj) L v = di v = √ X , X M X M g ∂xj j=1 and √ 1 n ∂( gXj ) tr DX = √ X . g ∂xj j=1 Proof. (i) We have √ 1 n LX vM =diX vM = d(iX gdx ∧···∧dx ) n √ =dX(−1)j−1 gXjdx1 ∧···∧dxj ∧···∧dxn d j=1 n √ = X(−1)j−1d( gXj) ∧ dx1 ∧···∧dxj ∧···∧dxn d j=1 √ n ∂( gXj ) = X dx1 ∧···∧dxn ∂xj j=1 √ 1 n ∂( gXj ) =√ X v . g ∂xj M j=1 ∂X` ` k ∂ (ii) We have D∂/∂xj X = P` ∂xj + Pk Γkj X ∂x` , which implies ∂X` tr DX = X + X Γ` Xk. ∂x` k` ` k Since 1 Γ` = X g`r{∂ g + ∂ g − ∂ g } k` 2 k `r ` kr r k` r,` 1 == X g`r∂ g 2 k `r r,` √ 1 ∂ g ∂ g = k = √k , 2 g g √ √ ∂X` X` ∂ g 1 n ∂( gXj ) tr DX = X + √ k } = √ X . ∂x` g ∂x` g ∂xj ` j=1 Typeset by AMS-TEX 1 2 Corollary. Let (M,g) be an oriented Riemannian manifold. Then, for any X ∈ Γ(TM), d(iX dvg)=tr DX =(div X)dvg. Stokes’ Theorem. Let M be a smooth, oriented n-dimensional manifold with boundary. Let ω be a compactly supported smooth (n − 1)-form on M. -
High Order Gradient, Curl and Divergence Conforming Spaces, with an Application to NURBS-Based Isogeometric Analysis
High order gradient, curl and divergence conforming spaces, with an application to compatible NURBS-based IsoGeometric Analysis R.R. Hiemstraa, R.H.M. Huijsmansa, M.I.Gerritsmab aDepartment of Marine Technology, Mekelweg 2, 2628CD Delft bDepartment of Aerospace Technology, Kluyverweg 2, 2629HT Delft Abstract Conservation laws, in for example, electromagnetism, solid and fluid mechanics, allow an exact discrete representation in terms of line, surface and volume integrals. We develop high order interpolants, from any basis that is a partition of unity, that satisfy these integral relations exactly, at cell level. The resulting gradient, curl and divergence conforming spaces have the propertythat the conservationlaws become completely independent of the basis functions. This means that the conservation laws are exactly satisfied even on curved meshes. As an example, we develop high ordergradient, curl and divergence conforming spaces from NURBS - non uniform rational B-splines - and thereby generalize the compatible spaces of B-splines developed in [1]. We give several examples of 2D Stokes flow calculations which result, amongst others, in a point wise divergence free velocity field. Keywords: Compatible numerical methods, Mixed methods, NURBS, IsoGeometric Analyis Be careful of the naive view that a physical law is a mathematical relation between previously defined quantities. The situation is, rather, that a certain mathematical structure represents a given physical structure. Burke [2] 1. Introduction In deriving mathematical models for physical theories, we frequently start with analysis on finite dimensional geometric objects, like a control volume and its bounding surfaces. We assign global, ’measurable’, quantities to these different geometric objects and set up balance statements.