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At least 145 records · Page 8Linked to original sources
arXiv2008

An Essay on the Application of mathematical Analysis to the theories of Electricity and Magnetism

Green's famous essay (Nottingham, 1828), with which he introduced the potential function, was transcribed from its reprint in Crelle's Journal (1850-54), with several typographical corrections and a reference section added. Green starts with accounts of earlier work, and some introductory remarks motivating his notation and method. Then, he gives a textual summary of later formal calculations, beginning with the general results. Finally, Green applies the results to several problems concerning electricity and magnetism.

physics.hist-ph↗
arXiv2009

History of the formulas and algorithms for pi

Throughout more than two millennia many formulas have been obtained, some of them beautiful, to calculate the number pi. Among them, we can find series, infinite products, expansions as continued fractions and expansions using radicals. Some expressions which are (amazingly) related to pi have been evaluated. In addition, a continual battle has been waged just to break the records computing digits of this number; records have been set using rapidly converging series, ultra fast algorithms and really surprising ones, calculating isolated digits. The development of powerful computers has played a fundamental role in these achievements of calculus.

math.HO↗
arXiv2008

Generalization of the Apollonius Circles

The three Apollonius circles of a triangle, each passing through a triangle vertex, the corresponding vertex of the cevian triangle of the incenter and the corresponding vertex of the circumcevian triangle of the symmedian point, are coaxal. Similarly defined three circles remain coaxal, when the circumcevian triangle is defined with respect to any point on the triangle circumconic through the incenter and symmedian point. Inversion in the incircle of the reference triangle carries these three coaxal circles into coaxal circles, each passing through a vertex of the inverted triangle and centered on the opposite sideline, at the intersection of the orthotransversal with respect to a point on the Euler line of the inverted triangle. A similar circumconic exists in a more general configuration, when the cevian triangle is defined with respect to an arbitrary point, passing through this arbitrary point and isogonal conjugate of its complement.

math.HO↗
arXiv2008

Counterexamples in Cake-Cutting

This article contains counterexamples to theorems and claims in Brams, Jones and Klamler's article "Better Ways to Cut a Cake" in the December 2006 Notices of the American Mathematical Society.

math.HO↗
arXiv2008

Through a Glass Darkly

We consider the question of how mathematicians view themselves and how non-mathematicians view us. What is our role in society? Is it effective? Is it rewarding? How could it be improved? This paper will be part of a forthcoming volume on this circle of questions.

math.HO↗
arXiv2008

A comparative review of recent researches in geometry

Felix Klein's so-called Erlangen Program was published in 1872 as professoral dissertation. It proposed a new solution to the problem how to classify and characterize geometries on the basis of projective geometry and group theory. The given translation was made in 1892 by Dr. M. W. Haskell and transcribed by N. C. Rughoonauth. We replaced bibliographical data in text and footnotes with pointers to a complete bibliography section.

math.HO↗
arXiv2008

A Novel Proof of the Heine-Borel Theorem

Every beginning real analysis student learns the classic Heine-Borel theorem, that the interval [0,1] is compact. In this article, we present a proof of this result that doesn't involve the standard techniques such as constructing a sequence and appealing to the completeness of the reals. We put a metric on the space of infinite binary sequences and prove that compactness of this space follows from a simple combinatorial lemma. The Heine-Borel theorem is an immediate corollary.

math.HO↗
arXiv2014

There are infinitely many prime numbers in all arithmetic progressions with first term and difference coprime

Dirichlet's proof of infinitely many primes in arithmetic progressions was published in 1837, introduced L-series for the first time, and it is said to have started rigorous analytic number theory. Dirichlet uses Euler's earlier work on the zeta function and the distribution of primes. He first proves a simpler case before going to full generality. The paper was translated from German by R. Stephan and given a reference section.

math.HO↗
arXiv2008

Jamming as Information: a Geometric Approach

In this paper I discuss the kinds of information that can be extracted by our enemy if our jamming is too precise. I show geometric solutions for reconstructing linear routes given certain information about them, such as the shortest distance to a point or the times of entering and exiting a circle.

math.HO↗
arXiv2010

Finite Sets and Counting

We start by presenting a theory of finite sets using the approach which is essentially that taken by Whitehead and Russell in Principia Mathematica}, and which does not involve the natural numbers (or any other infinite set). This theory is then applied to prove results about structures which, like the natural numbers, satisfy the principle of mathematical induction, but do not necessarily satisfy the remaining Peano axioms.

math.HO↗
arXiv2008

Fonction constante et dérivée nulle : un résultat si trivial..

We study various proofs of the caracterization of constant functions, more precisely of the theorem: a derivable function, defined on a real interval, is constant if, and only if, its derivative is null. Our aim is to study the relationships of these proofs with the mathematical curriculum of secondary schools and the begining of undergraduate studies in France, from various point of views (epistemological, historical, didactical).

math.HO↗
arXiv2008

Terwilliger Algebras of Wreath Powers of One-Class Association Schemes

In this paper, we study the subconstituent algebras, also called as Terwilliger algebras, of association schemes that are obtained as the wreath product of one-class association schemes $K_n=H(1, n)$ for $n\ge 2$. We find that the $d$-class association scheme $K_{n_{1}}\wr K_{n_{2}} \wr ... \wr K_{n_{d}}$ formed by taking the wreath product of $K_{n_{i}}$ has the triple-regularity property. We determine the dimension of the Terwilliger algebra for the association scheme $K_{n_{1}}\wr K_{n_{2}}\wr ... \wr K_ {n_{d}}$. We give a description of the structure of the Terwilliger algebra for the wreath power $(K_n)^{\wr d}$ for $n \geq 2$ by studying its irreducible modules. In particular, we show that the Terwilliger algebra of $(K_n)^{\wr d}$ is isomorphic to $M_{d+1}(\mathbb{C})\oplus M_1(\mathbb{C})^{\oplus \frac12d(d+1)}$ for $n\ge3$, and $M_{d+1}(\mathbb{C})\oplus M_1(\mathbb{C})^{\oplus \frac12d(d-1)}$ for $n=2$.

math.CO↗
arXiv2008

Desperately seeking mathematical truth

This article discusses epistemological problems in the philosophy of mathematics and issues concerning the reliability of the mathematical literature.

math.HO↗
arXiv2011

The Pythagorean Tree: A New Species

In 1967 the Dutch mathemetician F.J.M. Barning described an infinite, planar, ternary tree*. Seven years later, A. Hall independently discovered the same tree. Both used the method of uni-modular matrices to transform one triple to another. A number of rediscoveries have occurred more recently. In this article we announce the discovery of an entirely different ternary tree, and show how it relates to the one found by Barning and Hall.

math.HO↗