Vladimir Igorevich Arnold
In the following article a Ph.D. student of V.I.~Arnold gives a personal account on his teacher who unexpectedly passed away earlier this year.
math.HO↗
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The recension of the Kepler laws in England
The purpose of this note consists of discrete rational reconstruction which took place during the years 1609-1630 and 1630-1666, ie, the year of the publication of their Astronomia Nova and the year of death of the great German astronomer Johannes Kepler, and the subsequent range from desperate to knowledge of these laws of planetary motion, by the English scientist Sir Isaac Newton.
math.HO↗
When is .999... less than 1?
We examine alternative interpretations of the symbol described as nought, point, nine recurring. Is "an infinite number of 9s" merely a figure of speech? How are such alternative interpretations related to infinite cardinalities? How are they expressed in Lightstone's "semicolon" notation? Is it possible to choose a canonical alternative interpretation? Should unital evaluation of the symbol .999 . . . be inculcated in a pre-limit teaching environment? The problem of the unital evaluation is hereby examined from the pre-R, pre-lim viewpoint of the student.
math.HO↗
The Magic of Permutation Matrices: Categorizing, Counting and Eigenspectra of Magic Squares
Permutation matrices play an important role in understand the structure of magic squares. In this work, we use a class of symmetric permutation matrices than can be used to categorize magic squares. Many magic squares with a high degree of symmetry are studied, including classes that are generalizations of those categorized by Dudeney in 1917. We show that two classes of such magic squares are singular and the eigenspectra of such magic squares are highly structured. Lastly, we prove that natural magic squares of singly-even order of these classes do note exist.
math.HO↗
Splitting primes
We give an elementary introduction, through illustrative examples but without proofs, to one of the basic consequences of the Langlands programme, namely the law governing the primes modulo which a given irreducible integral polynomial splits completely. Some recent results, such as the modularity of elliptic curves over the rationals, or the proof of Serre's conjecture by Khare and Wintenberger, are also illustrated through examples.
math.HO↗
India's Contribution to Arab mathematics
This paper was presented at the Colloque de Saint-Denis de la Reunion, November 3-7, 1997, and published in the book L'Océan Indien au carrefour des mathématiques arabes, chinoises, européennes et indiennes (pp. 211-223), Tournès, Dominique (ed.), Saint-Denis, Réunion, I.U.F.M. de La Réunion, 1998. The article is an introductory survey of the in?uence of India on the mathematics of the Islamic world.
math.HO↗
Viviani Polytopes and Fermat Points
Given a set of oriented hyperplanes $\mathcal{P}=\{p_1, ..., p_k\}$ in $\mathbb{R}^n$, define $v(P)$ for any point $P\in\mathbb{R}^n$ as the sum of the signed distances from $P$ to $p_1$,..., $p_k$. We give a simple geometric characterization of $\mathcal{P}$ so that $v$ is a constant. The characterization leads to a connection with the Fermat point of $k$ points in $\mathbb{R}^n$. Finally, we discuss historically the full content of Viviani's theorem.
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Bertrand's paradox: a physical solution
We present a conclusive answer to Bertrand's paradox, a long standing open issue in the basic physical interpretation of probability. The paradox deals with the existence of mutually inconsistent results when looking for the probability that a chord, drawn at random in a circle, is longer than the side of an inscribed equilateral triangle. We obtain a unique solution by substituting chord drawing with the throwing of a straw of finite length L on a circle of radius R, thus providing a satisfactory operative definition of the associated experiment. The obtained probability turns out to be a function of the ratio L/R, as intuitively expected.
physics.data-an↗
Did Lobachevsky Have A Model Of His "imaginary Geometry"?
The invention of non-Euclidean geometries is often seen through the optics of Hilbertian formal axiomatic method developed later in the 19th century. However such an anachronistic approach fails to provide a sound reading of Lobachevsky's geometrical works. Although the modern notion of model of a given theory has a counterpart in Lobachevsky's writings its role in Lobachevsky's geometrical theory turns to be very unusual. Lobachevsky doesn't consider various models of Hyperbolic geometry, as the modern reader would expect, but uses a non-standard model of Euclidean plane (as a particular surface in the Hyperbolic 3-space). In this paper I consider this Lobachevsky's construction, and show how it can be better analyzed within an alternative non-Hilbertian foundational framework, which relates the history of geometry of the 19th century to some recent developments in the field.
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Chern conjecture and isoparametric hypersurfaces
In this note we will review the most important results and questions related to Chern conjecture and isoparametric hypersurfaces, as well as their interactions and applications to various aspects in mathematics.
math.HO↗
The Delone Peak
One of beautifull mountains in the neighborhood of the Belukha, the tallest summit in the Altai (in Siberia), is called the Delone Peak in the name of Boris Nikolaevich Delone (Delaunay), prominent mathematician. Here we are going to say about the life and science of this extremely suprising man (the paper is a seriously remade and supplemented version of the author's paper "The Delone Peak" published in "Kvant", 1987, No (Russian)).
math.HO↗
The homotopy theory of function spaces: a survey
We survey research on the homotopy theory of the space map(X, Y) consisting of all continuous functions between two topological spaces. We summarize progress on various classification problems for the homotopy types represented by the path-components of map(X, Y). We also discuss work on the homotopy theory of the monoid of self-equivalences aut(X) and of the free loop space LX. We consider these topics in both ordinary homotopy theory as well as after localization. In the latter case, we discuss algebraic models for the localization of function spaces and their applications.
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Things to do with a broken stick
We use the idea of the broken stick problem (which goes back to Poincare) and calculate the corresponding probabilities for the cases in which the three broken part are: the medians in a triangle, the altitudes, radii of excircles, angle bisectors, distances from I or O to the vertices, respectively sides, and some other three elements in a triangle which determine (more or less uniquely) the triangle. For each case we also look at the probability that the triangle that is (more or less uniquely) defined by the elements, being acute and compare to that of being obtuse.
math.HO↗
Various observations on angles proceeding in geometric progression
This is a translation of Euler's 1773 "Variae observationes circa angulos in progressione geometrica progredientes", E561 in the Enestr{ö}m index. I translated this paper as a result of my study of Euler's work on the infinite product $\prod_{k=1}^\infty (1-z^k)$. If one instead considers the finite product $\prod_{k=1}^n (1-z^k)$, one can study its behavior on the unit circle. The absolute value of $\prod_{k=1}^n (1-e^{ikθ})$ is $2^n |\prod_{k=1}^n \sin kθ/2|$. My interest in the product $\prod_{k=1}^n \sin kθ/2$ has inspired me to become acquainted with Euler's papers on trigonometric identities, in particular E447, E561, and E562. E561 says nothing about the product $\prod_{k=1}^n \sin kθ/2$, but it has identities which I had not seen before. The identities have a form similar to Viète's infinite product $\prod_{k=1}^\infty \cos θ/2^k=\frac{\sinθ}θ$.
math.HO↗
Roots of polynomials of degrees 3 and 4
We present the solutions of equations of degrees 3 and 4 using Galois theory and some simple Fourier analysis for finite groups, together with historical comments on these and other solution methods.
math.HO↗
An ode to Phipps' jeep convoys
The jeep problem was first solved by O. Helmer and N.J. Fine. But not much later, C.G. Phipps formulated a more general solution. He formulated a so-called convoy or caravan variant of the jeep problem and reduced the original problem to it. We shall refine the convoy idea of Phipps and subsequently view a more general jeep problem, which we solve for jeep convoys as well as for a single jeep. In the last section we solve Maddex' jeep problem.
math.HO↗
Exploring Mount Neverest
The problem `Exploring Mount Neverest' by Henry Ernest Dudeney is solved. Dudeney formulated the problem in the beginning of the 20th century and gave a non-optimal solution (without estimating the non-optimality).
math.HO↗
Jeep variants
The jeep problem was first solved by O. Helmer and N.J. Fine. But not much later, C.G. Phipps formulated a more general solution. He formulated a so-called convoy or caravan variant of the jeep problem and reduced the original problem to it. The convoy idea of Phipps was refined in [3]. Here we will apply this refined idea to several variants of the jeep problem.
math.HO↗