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Hyperbolic geometry
 In Flavors of geometry
, 1997
"... 3. Why Call it Hyperbolic Geometry? 63 4. Understanding the OneDimensional Case 65 ..."
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3. Why Call it Hyperbolic Geometry? 63 4. Understanding the OneDimensional Case 65
LECTURE NOTES ON CHERNSIMONS (SUPER)GRAVITIES
, 2008
"... This is intended to be a broad introduction to ChernSimons gravity and supergravity. The motivation for these theories lies in the desire to have a gauge invariant system –with a fiber bundle formulation – in more than three dimensions, which could provide a firm ground for constructing a quantum t ..."
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This is intended to be a broad introduction to ChernSimons gravity and supergravity. The motivation for these theories lies in the desire to have a gauge invariant system –with a fiber bundle formulation – in more than three dimensions, which could provide a firm ground for constructing a quantum theory of the gravitational field. The starting point is a gravitational action which generalizes the Einstein theory for dimensions D> 4 –Lovelock gravity. It is then shown that in odd dimensions there is a particular choice of the arbitrary parameters of the action that makes the theory gauge invariant under the (anti)de Sitter or the Poincaré groups. The resulting lagrangian is a ChernSimons form for a connection of the corresponding gauge groups and the vielbein and the spin connection are parts of this connection field. These theories also admit a natural supersymmetric extension for all odd D where the local supersymmetry algebra closes offshell and without a need for auxiliary fields. No analogous construction is available in even dimensions. A cursory discussion of the unexpected dynamical features of these theories and a number of open problems are also presented. These notes were prepared for the Fifth CBPF Graduate School, held in Rio de Janeiro in July 2004, published in Portuguese [1]. These notes were in turn based on a lecture series presented at the Villa de Leyva
HyperAutomaton System Applied to Geometry Demonstration Environment. Computer Aided Systems Theory
 EUROCAST’2001  Eigth International Conference
, 2001
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Centralized vs. Marketbased and Decentralized DecisionMaking: A Review of the Evidence in Computer Science and Economics
, 2008
"... Within both economics and computer science, many authors have claimed that decentralized or marketbased approaches to decisionmaking are superior in general to centralized approaches. The contrary claim has also been made. Unfortunately, these claims are often supported only by informal or anecdot ..."
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Within both economics and computer science, many authors have claimed that decentralized or marketbased approaches to decisionmaking are superior in general to centralized approaches. The contrary claim has also been made. Unfortunately, these claims are often supported only by informal or anecdotal evidence. In order to assess these competing claims, we present a review of the literatures in economics and in computer science bearing on these issues. Specifically, we report research findings based on empirical evidence and on simulation studies, and we outline the evidence based on formal deductive proofs or on informal and anecdotal evidence. Our main findings from this literature survey are: (i) for efficiency assessments, that there is wider variance in performance of organizations using MarketBased Control (MBC) than in organizations using Centralized Control (CC); (ii) that MBC and CC have the same efficiency on average; which may explain the observation (iii) that human and computer organizations tend to cycle between CC and Decentralized Control (DC) structures. 1
Contents
, 2006
"... 1 Jim Fahey’s many helpful comments are acknowledged with great gratitude. It was Jim who long ago heard my early (more technical) version of the argument from the core coma cases discussed herein, and provided valuable feedback. I’m indebted as well to Katherine Bringsjord for trenchant objections, ..."
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1 Jim Fahey’s many helpful comments are acknowledged with great gratitude. It was Jim who long ago heard my early (more technical) version of the argument from the core coma cases discussed herein, and provided valuable feedback. I’m indebted as well to Katherine Bringsjord for trenchant objections, and to Heather Hewitt for proofreading, for pointers to pinpointaccurate background reading, and for some exceedingly sharp objections of her own. John Alsdorf read an early draft, and made invaluable suggestions. i
A Euclidean Algorithm for Normal Bases ∗
, 2005
"... Inversion in finite fields GF (2 k) is a critical operation for many applications. A well known representation basis, i.e. normal basis, provides an efficient squaring operation realized as a simple rotation of the operand coefficients. Inversion in normal basis is computed using methods derived fro ..."
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Inversion in finite fields GF (2 k) is a critical operation for many applications. A well known representation basis, i.e. normal basis, provides an efficient squaring operation realized as a simple rotation of the operand coefficients. Inversion in normal basis is computed using methods derived from Fermat’s Little theorem, e.g. the ItohTsujii algorithm or with the aid of basis conversion algorithms using the Extended Euclidean algorithm. In this paper we present alternative normal basis inversion algorithm derived from the polynomial version of the extended Euclidean algorithm. The normal basis Euclidean algorithm has (roughly) the same complexity as the polynomial version of the Euclidean algorithm. The proposed algorithm requires on average a linear number of multiplications. We also present a modification to our algorithm which delays the multiplications to the very end of the computation and thereby gives opportunity for recursive computation using only a logarithmic number of multiplications.
Collaborative Research: Learning Discrete Mathematics and Computer Science via Primary Historical Sources
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Raphael’s School of Athens: A Theorem in a Painting?
"... Raphael’s famous painting The School of Athens includes a geometer, presumably Euclid himself, demonstrating a construction to his fascinated students. But what theorem are they all studying? This article first introduces the painting, and describes Raphael’s lifelong friendship with the eminent ma ..."
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Raphael’s famous painting The School of Athens includes a geometer, presumably Euclid himself, demonstrating a construction to his fascinated students. But what theorem are they all studying? This article first introduces the painting, and describes Raphael’s lifelong friendship with the eminent mathematician Paulus of Middelburg. It then presents several conjectured explanations, notably a theorem about a hexagram (Fichtner), or alternatively that the construction may be architecturally symbolic (Valtieri). The author finally offers his own “null hypothesis”: that the scene does not show any actual mathematics, but simply the fascination, excitement, and joy of mathematicians at their work. Raphael’s famous painting The School of Athens shows among the great Greek thinkers at work a geometer, presumably Euclid himself, demonstrating a construction to his students (Figure 1, front right).1 But exactly what theorem is he proving? In this article I describe the best known candidate. I begin with an overview of Raphael’s career and mathematical expertise, then survey the painting as a whole. Next, I focus down on Euclid’s slate–in the painting tilted so sharply that the exact nature of its figure has been a source of controversy–and test whether that figure can in fact arise from an equilateral hexagram (sixpointed star). 1Raphael’s figures in this article are images obtained through Wikimedia, and are in public domain. For readers who would like to see hard print versions, [18] contains well over a hundred beautiful photographs taken by photographers Felice Bono and Pietro
Why Did Lagrange “Prove ” the Parallel Postulate?
"... doxes by Augustus de Morgan: “Lagrange, in one of the later years of his life, imagined ” that he had solved the problem of proving Euclid’s parallel postulate. “He went so far as to write a paper, which he took with him to the [Institut de France], and began ..."
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doxes by Augustus de Morgan: “Lagrange, in one of the later years of his life, imagined ” that he had solved the problem of proving Euclid’s parallel postulate. “He went so far as to write a paper, which he took with him to the [Institut de France], and began