DOCSLIB.ORG

Figure 4. the Fano Plane. 11. Projective Space One Can Approach the Study of Projective Spaces from a Number of Different Angles

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Figure 4. the Fano Plane. 11. Projective Space One Can Approach the Study of Projective Spaces from a Number of Different Angles FINITE PERMUTATION GROUPS AND FINITE CLASSICAL GROUPS 49 Figure 4. The Fano plane. 11. Projective Space One can approach the study of projective spaces from a number of different angles. Since we will be primarily interested in projective spaces of finite dimension over finite fields, our spaces themselves are finite and so fall most naturally it seems to me! into the sphere of combinatorics. Our approach is, therefore, combinatorial. 11.1. Incidence structures. "e define a category IncStruct as follows. Objects# An object in IncStruct is a finite tuple P1,...,P�,I! whereP 1,...,P� are sets, andI isa subset ofP P . 1 × · · · × � Some terminology# An object #% P 1,...,P�,I! is called an incidence structure. &lements from setP i are said to have typei. IfI containsI an element p , . , p ! then, for 1 i,j �, we say thatp is incident 1 � ≤ ≤ i withp j. The incidence structure is called finite ifP 1,...,P� are all finite. To complete the definition of this category we must, of course, define what the arrows are. "e will do this shortly, but first some e(amples. Note that the definition of an incidence structure is extremely general* in most cases incidence structures are only studied subject to extra a(ioms. Example 22. "e have seen the category SimpleGraph earlier in lectures, and we con+ sidered the category Graph in exercises. Both of these categories could be seen as (full! subcategories of IncStruct. In particular we define the category Graph as follows: Objects# An object #% P ,P ,I! is an incidence structure with - types. (Elements G 1 2 ofP 1 are what we thin. of as vertices, elements ofP 2 are edges!. "e require that, for any p P , 2 ∈ 2 p P p , p ! I -. |{ 1 ∈ 1 | 1 2 ∈ }| ≤ Since Graph is a full subcategory of IncStruct, the definition of arrows in Graph is the same as the definition in IncStruct, and we give this definition in a moment. Example 23. An abstract projective plane is an incidence structure P,L,I! such that 1) any two elements ofP are incident with a uni/ue element ofL* -! any two elements ofL are incident with a uni/ue element ofP. 0! there are four elements ofP such that no element ofL is incident with more than two of them. "e call elements ofP points and elements ofL lines. The third axiom listed above 1presence of a /uadrangle2! is there simply to eliminate some degenerate examples. IfP is finite it is easy to see thatL is also finite and, in fact, that P % L . The smallest finite projective plane has P % 3. It is called the Fano plane and is| represented| | | in Figure 4* in this representation elements| | ofP are drawn as points, and elements ofL are drawn as lines (in si( cases) or as a circle; each element ofL is incident with three elements ofP and vice versa!. Example 24. 5etV be a vector space of dimensionn over a fieldk. "e define projective space 67 V !, or 67n 1 k!, to be the incidence structure V1,...,Vn 1,I! where, fori%1, . , n 1, − − − Vi is the set of subspaces ofV of dimensioni and I #% v 1, . , vn 1! V 1 V n 1 v 1 < v2 < <v n 1 . { − ∈ × · · · × − | ··· − } 50 NICK GILL In other words two subspaces are incident if and only if one is contained in the other. This fact allows us to rela( language a little: we say things li.e 1v1 lies onv 22, or 1v2 containsv 12 when we really mean that 1v1 andv 2 are incident2. $ subspaceU V is said to have projective dimension, pdim U!, equal toi 1 and we ∈ i − call elements ofV 1 1points’, elements ofV 2 1lines2, and elements ofV n 1 1hyperplanes’. Ifk is − finite of orderq, then we sometimes write 67 n 1 q! for 67n 1 k!. The incidence structure − − 672 q! is called the Desarguesian projective plane of orderq. (E11.1) Show thatPG 2(2) and the Fano plane are the same incidence structure. (We would do better to write that “PG2(2) and the Fano plane are isomorphic as incidence structures”, but we have not yet defined what we mean by isomorphism.) (E11.2*) Show that, for any prime powerq,PG 2(q) is an abstract projective plane. 11.-. Some counting. "e are interested in calculating the order ofV 1,...,Vn 1 in 67n 1 q!. To do this it is convenient to introduce Gaussian coefficients. 5etq be a prime power,m and−n positive− integers. Then define n n n m 1 n q 1! q q! q q − ! 10! #% − − ··· − . m qm 1! qm q! q m q m 1! � �q − − ··· − − n Lemma 11.1. 1! The number of subspaces of projective dimensionm 1 in 67 n 1 q! is . In − m q n qn 1 − particular, the number of points in 67n 1 q! is % − . − 1 q q 1 � � -! The number of subspaces of projective dimensionm 1 containing− a subspace of projective dimension n l � −� l 1 in 67 q! is − . n 1 m l q − − − Proof. 5etV beann-dimensional� � vector space overk%F q. To prove a! we count the number of linearly independentm-tuples of vectors inV . The first entry in them+tuple can be chosen to be any non-zero vector, there areq n 1 of these; the second must lie outside the span of the first, so there areq n q choices for this, thenq n q−2 for the third and so on. "e conclude that the numerator of the right-hand− side of 10! corresponds− to the number of linearly independentm-tuples of vectors. Now the result is completed by observing using the same reasoning! that the denominator of the right+hand side of 10! corresponds to the number of linearly independentm+tuples of vectors all lying inside any givenm-dimensional subspace ofV. (E11.3)Prove (b). � Gaussian coefficients have properties resembling those of binomial coefficients, to which they tend as q 1. → n n (E11.4)Prove that limq 1 = . → m q m (E11.5)Prove that � � � � n n m+1 n n+1 +q − = . m m 1 m � �q � − �q � �q 11.0. Collineations and the fundamental theorem. "e are ready to complete the definition of the category IncStruct# 1 1 1 2 2 2 F !rro"s# 5et 1 % P1 ,...,P� ,I !, 2 % P1 ,...,P� ,I ! be incidence structures. An arrow 1 2 is a set of� functionsIφ #P 1 P 2 such thatI I → I i i → i p , . , p ! I 1 % φ p !,...,φ p !! I 2. 1 � ∈ ⇒ 1 � ∈ Note that we only define arrows between incidence structures having the same number of types. F 1 2 The notion of isomorphism is now clear: An isomorphism 1 2 is a set of� bijectionsφ i #P i P i such that I → I → p , . , p ! I 1 φ p !,...,φ p !! I 2. 1 � ∈ ⇐⇒ 1 � ∈ FINITE PERMUTATION GROUPS AND FINITE CLASSICAL GROUPS 51 One would expect that an isomorphism 1 F 1 would be .nown as an automorphism, however the termi+ nology for this category is a little different#I → such I a thing is called a collineation. As usual we denote the set of all collineations of an incidence structure by Aut( ). I I Example 25. LetV beann+dimensional vector space over a fieldk. The action of ;L V! on the setV extends naturally to an action on the set of subspaces ofV# ifv 1, . , vk V andg ;5 V !, then we writeU% v , . , v and define ∈ ∈ � 1 n� U g % v g, . , vg . � 1 k� (E11.6*)Prove that this action is well-defined, and that the action preserves the incidence relation for PG(V). This exercise implies that we have a homomorphismφ # ;5 V! Aut 67 V !). The next result asserts that this homomorphism is also surjective, i.e. all collineations→ of 67 V ! are induced by a semilinear transformation. Theorem 11.2. (Fundamental theorem of projective geometry) If dimV 0, then Im φ! % $ut 67 V!!. ≥ Proof. This is omitted. See <Camb, Chapter 1] or <Tay9->. � The first isomorphism theorem of group theory implies, then, that Aut 67 V!! ∼% ;5 V!/ .er φ!. (E11.7) Prove that ker(φ)= αI GL(V) α k . { ∈ | ∈ } 5et us writeK for .er φ! and observe thatK is just the group of scalar transformations in ;L V ). Note thatK is actually a subgroup of 75 V!. "e now define three new groups in terms of this subgroupK. 1) 6;5 V ! #% ;5 V!/K* -! 675 V ! #% 75 V!/K* 0! 6S5 V!#%S5 V!/ K S5 V !). ∩ ForX ;, G, S we write 6XL n k! as a synonym for 6XL V!. Observe∈{ that} the Fundamental theorem of projective geometry could be e(pressed as follows: If dimV 0, then ≥ 6;5 k! % Aut 67 V!. In particular the three groups just defined all act faithfully on 67 V!. (E11.8)Prove thatK is central in GL(V). "an you characteri#e those fieldsk and those vector spacesV for whichK is central in ΓL(V)$ (E11.9*)Prove that 1, ifn is odd; PGLn(R) : PSLn(R) = | | 2, ifn is even. � 11.4. %ualit&. Next we define a variant of the category IncStruct that we call WIncStruct. The objects are the same @ incidence structures @ but we allow more arrows# 1 1 1 2 2 2 F !rro"s# 5et 1 % P1 ,...,P� ,I !, 2 % P1 ,...,P� ,I ! be incidence structures. An arrow 1 2 in WIncStruct isI a permutationπ S ,I and set of� functionsφ #P 1 P 2 such that I → I ∈ k i i → iπ 1 2 p , .
Load more

Recommended publications
  • 1 Lifts of Polytopes
    Lecture 5: Lifts of polytopes and non-negative rank CSE 599S: Entropy optimality, Winter 2016 Instructor: James R. Lee Last updated: January 24, 2016 1 Lifts of polytopes 1.1 Polytopes and inequalities Recall that the convex hull of a subset X n is defined by ⊆ conv X λx + 1 λ x0 : x; x0 X; λ 0; 1 : ( ) f ( − ) 2 2 [ ]g A d-dimensional convex polytope P d is the convex hull of a finite set of points in d: ⊆ P conv x1;:::; xk (f g) d for some x1;:::; xk . 2 Every polytope has a dual representation: It is a closed and bounded set defined by a family of linear inequalities P x d : Ax 6 b f 2 g for some matrix A m d. 2 × Let us define a measure of complexity for P: Define γ P to be the smallest number m such that for some C s d ; y s ; A m d ; b m, we have ( ) 2 × 2 2 × 2 P x d : Cx y and Ax 6 b : f 2 g In other words, this is the minimum number of inequalities needed to describe P. If P is full- dimensional, then this is precisely the number of facets of P (a facet is a maximal proper face of P). Thinking of γ P as a measure of complexity makes sense from the point of view of optimization: Interior point( methods) can efficiently optimize linear functions over P (to arbitrary accuracy) in time that is polynomial in γ P . ( ) 1.2 Lifts of polytopes Many simple polytopes require a large number of inequalities to describe.
    [Show full text]
  • Projective Geometry: a Short Introduction
    Projective Geometry: A Short Introduction Lecture Notes Edmond Boyer Master MOSIG Introduction to Projective Geometry Contents 1 Introduction 2 1.1 Objective . .2 1.2 Historical Background . .3 1.3 Bibliography . .4 2 Projective Spaces 5 2.1 Definitions . .5 2.2 Properties . .8 2.3 The hyperplane at infinity . 12 3 The projective line 13 3.1 Introduction . 13 3.2 Projective transformation of P1 ................... 14 3.3 The cross-ratio . 14 4 The projective plane 17 4.1 Points and lines . 17 4.2 Line at infinity . 18 4.3 Homographies . 19 4.4 Conics . 20 4.5 Affine transformations . 22 4.6 Euclidean transformations . 22 4.7 Particular transformations . 24 4.8 Transformation hierarchy . 25 Grenoble Universities 1 Master MOSIG Introduction to Projective Geometry Chapter 1 Introduction 1.1 Objective The objective of this course is to give basic notions and intuitions on projective geometry. The interest of projective geometry arises in several visual comput- ing domains, in particular computer vision modelling and computer graphics. It provides a mathematical formalism to describe the geometry of cameras and the associated transformations, hence enabling the design of computational ap- proaches that manipulates 2D projections of 3D objects. In that respect, a fundamental aspect is the fact that objects at infinity can be represented and manipulated with projective geometry and this in contrast to the Euclidean geometry. This allows perspective deformations to be represented as projective transformations. Figure 1.1: Example of perspective deformation or 2D projective transforma- tion. Another argument is that Euclidean geometry is sometimes difficult to use in algorithms, with particular cases arising from non-generic situations (e.g.
    [Show full text]
  • Robot Vision: Projective Geometry
    Robot Vision: Projective Geometry Ass.Prof. Friedrich Fraundorfer SS 2018 1 Learning goals . Understand homogeneous coordinates . Understand points, line, plane parameters and interpret them geometrically . Understand point, line, plane interactions geometrically . Analytical calculations with lines, points and planes . Understand the difference between Euclidean and projective space . Understand the properties of parallel lines and planes in projective space . Understand the concept of the line and plane at infinity 2 Outline . 1D projective geometry . 2D projective geometry ▫ Homogeneous coordinates ▫ Points, Lines ▫ Duality . 3D projective geometry ▫ Points, Lines, Planes ▫ Duality ▫ Plane at infinity 3 Literature . Multiple View Geometry in Computer Vision. Richard Hartley and Andrew Zisserman. Cambridge University Press, March 2004. Mundy, J.L. and Zisserman, A., Geometric Invariance in Computer Vision, Appendix: Projective Geometry for Machine Vision, MIT Press, Cambridge, MA, 1992 . Available online: www.cs.cmu.edu/~ph/869/papers/zisser-mundy.pdf 4 Motivation – Image formation [Source: Charles Gunn] 5 Motivation – Parallel lines [Source: Flickr] 6 Motivation – Epipolar constraint X world point epipolar plane x x’ x‘TEx=0 C T C’ R 7 Euclidean geometry vs. projective geometry Definitions: . Geometry is the teaching of points, lines, planes and their relationships and properties (angles) . Geometries are defined based on invariances (what is changing if you transform a configuration of points, lines etc.) . Geometric transformations
    [Show full text]
  • Simplicial Complexes
    46 III Complexes III.1 Simplicial Complexes There are many ways to represent a topological space, one being a collection of simplices that are glued to each other in a structured manner. Such a collection can easily grow large but all its elements are simple. This is not so convenient for hand-calculations but close to ideal for computer implementations. In this book, we use simplicial complexes as the primary representation of topology. Rd k Simplices. Let u0; u1; : : : ; uk be points in . A point x = i=0 λiui is an affine combination of the ui if the λi sum to 1. The affine hull is the set of affine combinations. It is a k-plane if the k + 1 points are affinely Pindependent by which we mean that any two affine combinations, x = λiui and y = µiui, are the same iff λi = µi for all i. The k + 1 points are affinely independent iff P d P the k vectors ui − u0, for 1 ≤ i ≤ k, are linearly independent. In R we can have at most d linearly independent vectors and therefore at most d+1 affinely independent points. An affine combination x = λiui is a convex combination if all λi are non- negative. The convex hull is the set of convex combinations. A k-simplex is the P convex hull of k + 1 affinely independent points, σ = conv fu0; u1; : : : ; ukg. We sometimes say the ui span σ. Its dimension is dim σ = k. We use special names of the first few dimensions, vertex for 0-simplex, edge for 1-simplex, triangle for 2-simplex, and tetrahedron for 3-simplex; see Figure III.1.
    [Show full text]
  • Degrees of Freedom in Quadratic Goodness of Fit
    Submitted to the Annals of Statistics DEGREES OF FREEDOM IN QUADRATIC GOODNESS OF FIT By Bruce G. Lindsay∗, Marianthi Markatouy and Surajit Ray Pennsylvania State University, Columbia University, Boston University We study the effect of degrees of freedom on the level and power of quadratic distance based tests. The concept of an eigendepth index is introduced and discussed in the context of selecting the optimal de- grees of freedom, where optimality refers to high power. We introduce the class of diffusion kernels by the properties we seek these kernels to have and give a method for constructing them by exponentiating the rate matrix of a Markov chain. Product kernels and their spectral decomposition are discussed and shown useful for high dimensional data problems. 1. Introduction. Lindsay et al. (2008) developed a general theory for good- ness of fit testing based on quadratic distances. This class of tests is enormous, encompassing many of the tests found in the literature. It includes tests based on characteristic functions, density estimation, and the chi-squared tests, as well as providing quadratic approximations to many other tests, such as those based on likelihood ratios. The flexibility of the methodology is particularly important for enabling statisticians to readily construct tests for model fit in higher dimensions and in more complex data. ∗Supported by NSF grant DMS-04-05637 ySupported by NSF grant DMS-05-04957 AMS 2000 subject classifications: Primary 62F99, 62F03; secondary 62H15, 62F05 Keywords and phrases: Degrees of freedom, eigendepth, high dimensional goodness of fit, Markov diffusion kernels, quadratic distance, spectral decomposition in high dimensions 1 2 LINDSAY ET AL.
    [Show full text]
  • Projective Planarity of Matroids of 3-Nets and Biased Graphs
    AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 76(2) (2020), Pages 299–338 Projective planarity of matroids of 3-nets and biased graphs Rigoberto Florez´ ∗ Deptartment of Mathematical Sciences, The Citadel Charleston, South Carolina 29409 U.S.A. [email protected] Thomas Zaslavsky† Department of Mathematical Sciences, Binghamton University Binghamton, New York 13902-6000 U.S.A. [email protected] Abstract A biased graph is a graph with a class of selected circles (“cycles”, “cir- cuits”), called “balanced”, such that no theta subgraph contains exactly two balanced circles. A 3-node biased graph is equivalent to an abstract partial 3-net. We work in terms of a special kind of 3-node biased graph called a biased expansion of a triangle. Our results apply to all finite 3-node biased graphs because, as we prove, every such biased graph is a subgraph of a finite biased expansion of a triangle. A biased expansion of a triangle is equivalent to a 3-net, which, in turn, is equivalent to an isostrophe class of quasigroups. A biased graph has two natural matroids, the frame matroid and the lift matroid. A classical question in matroid theory is whether a matroid can be embedded in a projective geometry. There is no known general answer, but for matroids of biased graphs it is possible to give algebraic criteria. Zaslavsky has previously given such criteria for embeddability of biased-graphic matroids in Desarguesian projective spaces; in this paper we establish criteria for the remaining case, that is, embeddability in an arbitrary projective plane that is not necessarily Desarguesian.
    [Show full text]
  • COMBINATORICS, Volume
    http://dx.doi.org/10.1090/pspum/019 PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS Volume XIX COMBINATORICS AMERICAN MATHEMATICAL SOCIETY Providence, Rhode Island 1971 Proceedings of the Symposium in Pure Mathematics of the American Mathematical Society Held at the University of California Los Angeles, California March 21-22, 1968 Prepared by the American Mathematical Society under National Science Foundation Grant GP-8436 Edited by Theodore S. Motzkin AMS 1970 Subject Classifications Primary 05Axx, 05Bxx, 05Cxx, 10-XX, 15-XX, 50-XX Secondary 04A20, 05A05, 05A17, 05A20, 05B05, 05B15, 05B20, 05B25, 05B30, 05C15, 05C99, 06A05, 10A45, 10C05, 14-XX, 20Bxx, 20Fxx, 50A20, 55C05, 55J05, 94A20 International Standard Book Number 0-8218-1419-2 Library of Congress Catalog Number 74-153879 Copyright © 1971 by the American Mathematical Society Printed in the United States of America All rights reserved except those granted to the United States Government May not be produced in any form without permission of the publishers Leo Moser (1921-1970) was active and productive in various aspects of combin• atorics and of its applications to number theory. He was in close contact with those with whom he had common interests: we will remember his sparkling wit, the universality of his anecdotes, and his stimulating presence. This volume, much of whose content he had enjoyed and appreciated, and which contains the re• construction of a contribution by him, is dedicated to his memory. CONTENTS Preface vii Modular Forms on Noncongruence Subgroups BY A. O. L. ATKIN AND H. P. F. SWINNERTON-DYER 1 Selfconjugate Tetrahedra with Respect to the Hermitian Variety xl+xl + *l + ;cg = 0 in PG(3, 22) and a Representation of PG(3, 3) BY R.
    [Show full text]
  • Arxiv:1910.10745V1 [Cond-Mat.Str-El] 23 Oct 2019 2.2 Symmetry-Protected Time Crystals
    A Brief History of Time Crystals Vedika Khemania,b,∗, Roderich Moessnerc, S. L. Sondhid aDepartment of Physics, Harvard University, Cambridge, Massachusetts 02138, USA bDepartment of Physics, Stanford University, Stanford, California 94305, USA cMax-Planck-Institut f¨urPhysik komplexer Systeme, 01187 Dresden, Germany dDepartment of Physics, Princeton University, Princeton, New Jersey 08544, USA Abstract The idea of breaking time-translation symmetry has fascinated humanity at least since ancient proposals of the per- petuum mobile. Unlike the breaking of other symmetries, such as spatial translation in a crystal or spin rotation in a magnet, time translation symmetry breaking (TTSB) has been tantalisingly elusive. We review this history up to recent developments which have shown that discrete TTSB does takes place in periodically driven (Floquet) systems in the presence of many-body localization (MBL). Such Floquet time-crystals represent a new paradigm in quantum statistical mechanics — that of an intrinsically out-of-equilibrium many-body phase of matter with no equilibrium counterpart. We include a compendium of the necessary background on the statistical mechanics of phase structure in many- body systems, before specializing to a detailed discussion of the nature, and diagnostics, of TTSB. In particular, we provide precise definitions that formalize the notion of a time-crystal as a stable, macroscopic, conservative clock — explaining both the need for a many-body system in the infinite volume limit, and for a lack of net energy absorption or dissipation. Our discussion emphasizes that TTSB in a time-crystal is accompanied by the breaking of a spatial symmetry — so that time-crystals exhibit a novel form of spatiotemporal order.
    [Show full text]
  • Octonion Multiplication and Heawood's
    CONFLUENTES MATHEMATICI Bruno SÉVENNEC Octonion multiplication and Heawood’s map Tome 5, no 2 (2013), p. 71-76. <http://cml.cedram.org/item?id=CML_2013__5_2_71_0> © Les auteurs et Confluentes Mathematici, 2013. Tous droits réservés. L’accès aux articles de la revue « Confluentes Mathematici » (http://cml.cedram.org/), implique l’accord avec les condi- tions générales d’utilisation (http://cml.cedram.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation á fin strictement personnelle du copiste est constitutive d’une infrac- tion pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Confluentes Math. 5, 2 (2013) 71-76 OCTONION MULTIPLICATION AND HEAWOOD’S MAP BRUNO SÉVENNEC Abstract. In this note, the octonion multiplication table is recovered from a regular tesse- lation of the equilateral two timensional torus by seven hexagons, also known as Heawood’s map. Almost any article or book dealing with Cayley-Graves algebra O of octonions (to be recalled shortly) has a picture like the following Figure 0.1 representing the so-called ‘Fano plane’, which will be denoted by Π, together with some cyclic ordering on each of its ‘lines’. The Fano plane is a set of seven points, in which seven three-point subsets called ‘lines’ are specified, such that any two points are contained in a unique line, and any two lines intersect in a unique point, giving a so-called (combinatorial) projective plane [8,7].
    [Show full text]
  • Collineations in Perspective
    Collineations in Perspective Now that we have a decent grasp of one-dimensional projectivities, we move on to their two di- mensional analogs. Although they are more complicated, in a sense, they may be easier to grasp because of the many applications to perspective drawing. Speaking of, let's return to the triangle on the window and its shadow in its full form instead of only looking at one line. Perspective Collineation In one dimension, a perspectivity is a bijective mapping from a line to a line through a point. In two dimensions, a perspective collineation is a bijective mapping from a plane to a plane through a point. To illustrate, consider the triangle on the window plane and its shadow on the ground plane as in Figure 1. We can see that every point on the triangle on the window maps to exactly one point on the shadow, but the collineation is from the entire window plane to the entire ground plane. We understand the window plane to extend infinitely in all directions (even going through the ground), the ground also extends infinitely in all directions (we will assume that the earth is flat here), and we map every point on the window to a point on the ground. Looking at Figure 2, we see that the lamp analogy breaks down when we consider all lines through O. Although it makes sense for the base of the triangle on the window mapped to its shadow on 1 the ground (A to A0 and B to B0), what do we make of the mapping C to C0, or D to D0? C is on the window plane, underground, while C0 is on the ground.
    [Show full text]
  • Matroids You Have Known
    26 MATHEMATICS MAGAZINE Matroids You Have Known DAVID L. NEEL Seattle University Seattle, Washington 98122 [email protected] NANCY ANN NEUDAUER Pacific University Forest Grove, Oregon 97116 nancy@pacificu.edu Anyone who has worked with matroids has come away with the conviction that matroids are one of the richest and most useful ideas of our day. —Gian Carlo Rota [10] Why matroids? Have you noticed hidden connections between seemingly unrelated mathematical ideas? Strange that finding roots of polynomials can tell us important things about how to solve certain ordinary differential equations, or that computing a determinant would have anything to do with finding solutions to a linear system of equations. But this is one of the charming features of mathematics—that disparate objects share similar traits. Properties like independence appear in many contexts. Do you find independence everywhere you look? In 1933, three Harvard Junior Fellows unified this recurring theme in mathematics by defining a new mathematical object that they dubbed matroid [4]. Matroids are everywhere, if only we knew how to look. What led those junior-fellows to matroids? The same thing that will lead us: Ma- troids arise from shared behaviors of vector spaces and graphs. We explore this natural motivation for the matroid through two examples and consider how properties of in- dependence surface. We first consider the two matroids arising from these examples, and later introduce three more that are probably less familiar. Delving deeper, we can find matroids in arrangements of hyperplanes, configurations of points, and geometric lattices, if your tastes run in that direction.
    [Show full text]
  • The Projective Geometry of the Spacetime Yielded by Relativistic Positioning Systems and Relativistic Location Systems Jacques Rubin
    The projective geometry of the spacetime yielded by relativistic positioning systems and relativistic location systems Jacques Rubin To cite this version: Jacques Rubin. The projective geometry of the spacetime yielded by relativistic positioning systems and relativistic location systems. 2014. hal-00945515 HAL Id: hal-00945515 https://hal.inria.fr/hal-00945515 Submitted on 12 Feb 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The projective geometry of the spacetime yielded by relativistic positioning systems and relativistic location systems Jacques L. Rubin (email: [email protected]) Université de Nice–Sophia Antipolis, UFR Sciences Institut du Non-Linéaire de Nice, UMR7335 1361 route des Lucioles, F-06560 Valbonne, France (Dated: February 12, 2014) As well accepted now, current positioning systems such as GPS, Galileo, Beidou, etc. are not primary, relativistic systems. Nevertheless, genuine, relativistic and primary positioning systems have been proposed recently by Bahder, Coll et al. and Rovelli to remedy such prior defects. These new designs all have in common an equivariant conformal geometry featuring, as the most basic ingredient, the spacetime geometry. In a first step, we show how this conformal aspect can be the four-dimensional projective part of a larger five-dimensional geometry.
    [Show full text]