Combinatorial Generation of Small Point Configurations and Hyperplane Arrangements

Oriented matroids are combinatorial structures that generalize point configurations, vector configurations, hyperplane arrangements, polyhedra, linear programs, and directed graphs. Oriented matroids have played a key role in combinatorics, computational geometry, and optimization. This paper surveys prior work and presents an update on the search for bounds on the diameter of the cocircuit graph of an oriented matroid. The motivation for our investigations is the complexity of the simplex method and the criss-cross method. We review the diameter problem and show the diameter bounds of general oriented matroids reduce to those of uniform oriented matroids. We give the latest exact bounds for oriented matroids of low rank and low corank, and for all oriented matroids with up to nine elements (this part required a large computer-based proof). For arbitrary oriented matroids, we present an improvement to a quadratic bound of Finschi. Our discussion highlights an old conjecture that states a linear bound for the diameter is possible. On the positive side, we show the conjecture is true for oriented matroids of low rank and low corank, and, verified with computers, for all oriented matroids with up to nine elements. On the negative side, our computer search showed two natural strengthenings of the main conjecture are false.

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Reading the Nicomachean Ethics as an Investigation

History of Philosophy and Logical Analysis ◽

10.30965/26664275-02301011 ◽

2020 ◽

Vol 23 (1) ◽

pp. 167-196

Author(s):

Guy Schuh

Keyword(s):

Nicomachean Ethics ◽

Small Point

Abstract Aristotle tells us that the Nicomachean Ethics is an “inquiry” and an “investigation” (methodos and zētēsis). This paper focuses on an under-appreciated way that the work is investigative: its employment of an exploratory investigative strategy—that is, its frequent positing of, and later revision or even rejection of, merely preliminary positions. Though this may seem like a small point, this aspect of the work’s methodology has important consequences for how we should read it—specifically, we should be open to the possibility that some contradictions in the text are the result of his employment of this investigative strategy. In the paper, I describe this investigative strategy, discuss what motivates Aristotle to employ it in the work, and go through three contradictions that are plausibly identified as examples of its use—specifically, his claims that courageous people do and do not fear death, that friendship is and is not mutually recognized goodwill, and that virtuous people do and do not choose noble actions for their own sake.

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Piles of Cubes, Monotone Path Polytopes, and Hyperplane Arrangements

Discrete & Computational Geometry ◽

10.1007/pl00009404 ◽

1999 ◽

Vol 21 (1) ◽

pp. 117-130 ◽

Cited By ~ 1

Author(s):

C. A. Athanasiadis

Keyword(s):

Hyperplane Arrangements ◽

Monotone Path

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Conditional probabilities via line arrangements and point configurations

Linear and Multilinear Algebra ◽

10.1080/03081087.2021.1912693 ◽

2021 ◽

pp. 1-33

Author(s):

Oliver Clarke ◽

Fatemeh Mohammadi ◽

Harshit J. Motwani

Keyword(s):

Conditional Probabilities ◽

Line Arrangements ◽

Point Configurations

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On polytopes and generalizations of the KLT relations

Journal of High Energy Physics ◽

10.1007/jhep12(2020)057 ◽

2020 ◽

Vol 2020 (12) ◽

Cited By ~ 1

Author(s):

Nikhil Kalyanapuram

Keyword(s):

Scattering Amplitudes ◽

Large Class ◽

Moduli Space ◽

Differential Forms ◽

Natural Generalization ◽

Intersection Theory ◽

Hyperplane Arrangements ◽

Number Of Solutions ◽

Scalar Theories ◽

Double Copy

Abstract We combine the technology of the theory of polytopes and twisted intersection theory to derive a large class of double copy relations that generalize the classical relations due to Kawai, Lewellen and Tye (KLT). To do this, we first study a generalization of the scattering equations of Cachazo, He and Yuan. While the scattering equations were defined on ℳ0, n — the moduli space of marked Riemann spheres — the new scattering equations are defined on polytopes known as accordiohedra, realized as hyperplane arrangements. These polytopes encode as patterns of intersection the scattering amplitudes of generic scalar theories. The twisted period relations of such intersection numbers provide a vast generalization of the KLT relations. Differential forms dual to the bounded chambers of the hyperplane arrangements furnish a natural generalization of the Bern-Carrasco-Johansson (BCJ) basis, the number of which can be determined by counting the number of solutions of the generalized scattering equations. In this work the focus is on a generalization of the BCJ expansion to generic scalar theories, although we use the labels KLT and BCJ interchangeably.

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The Tutte polynomial of symmetric hyperplane arrangements

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216520500042 ◽

2020 ◽

Vol 29 (03) ◽

pp. 2050004

Author(s):

Hery Randriamaro

Keyword(s):

Hyperplane Arrangement ◽

Dual Graph ◽

Tutte Polynomial ◽

Hyperplane Arrangements ◽

Weyl Groups ◽

Reflection Groups ◽

Tutte Polynomials ◽

One Year ◽

The One ◽

Definition Of

The Tutte polynomial is originally a bivariate polynomial which enumerates the colorings of a graph and of its dual graph. Ardila extended in 2007 the definition of the Tutte polynomial on the real hyperplane arrangements. He particularly computed the Tutte polynomials of the hyperplane arrangements associated to the classical Weyl groups. Those associated to the exceptional Weyl groups were computed by De Concini and Procesi one year later. This paper has two objectives: On the one side, we extend the Tutte polynomial computing to the complex hyperplane arrangements. On the other side, we introduce a wider class of hyperplane arrangements which is that of the symmetric hyperplane arrangements. Computing the Tutte polynomial of a symmetric hyperplane arrangement permits us to deduce the Tutte polynomials of some hyperplane arrangements, particularly of those associated to the imprimitive reflection groups.

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Note on Le Moustier Flints from Acton, West Drayton and Iver

Proceedings of the Prehistoric Society of East Anglia ◽

10.1017/s095884180002559x ◽

1927 ◽

Vol 5 (3) ◽

pp. 297-298 ◽

Cited By ~ 2

Author(s):

J. G. Marsden

Keyword(s):

The Other ◽

Raw Material ◽

Small Point

I am exhibiting a number of flakes and implements, which appear to be of Le Moustier date, from Acton on the Taplow Terrace, and West Drayton and Iver on the Boyn Hill Terrace.The Acton specimens are from the immediate vicinity of the working floor discovered by the late Mr. Allen Brown some forty years ago. During recent excavations in Creffield Road, I picked out of the brick-earth, thrown up from a depth of 4 to 6½ feet, seventy humanly struck flints. Forty-six are unpatinated, fresh-looking and unabraded, twenty have a bluey-white patination, amongst which are several with slightly dulled edges, and four are light-ochreous. The freshness of the cortex on several of the unpatinated flints suggests that some of the raw material was derived directly from the chalk. The majority are simple flakes and spalls. The remainder consists of eighteen small Levallois flakes, varying in size from 2 by 1½ to 3½ by 2¼ inches, one small tortoise core, a small flattish core with flaking on one face at right angles to that on the other, a flint pebble which has been used as a hammerstone, one small point and two exceptional pieces. One of these is a fairly typical graver (bee de flûte) with bluey-white patination (Fig. 1); the other may be described as a busked graver, it is made from a thickish external flake and is unpatinated (Fig. 2). A few of the flakes shew slight signs of use.

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