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At least 109 records · Page 6Linked to original sources
arXiv2008

Walk versus Wait: The Lazy Mathematician Wins

In this recreational mathematics note, we address a simple, yet instructive question: Justin has to travel a distance of d miles along a bus route. Along this route, there are n bus stops i, each spaced at a distance of d_i from the starting point. At each bus stop, Justin is faced with a choice: to walk or to wait. If he walks on, he can still catch a bus at the next bus stop--but if a bus passes him while he walks, he is almost assured a longer wait. We model Justin's decision constraint and completely solve the model in a special case. The answer is intuitive: the optimal strategy is the laziest.

math.HO↗
arXiv2008

A more accurate treatment of the problem of drawing the shortest line on a surface

E727 in the Enestrom index. This is a translation from the Latin original "Accuratior evolutio problematis de linea brevissima in superficie quacunque ducenda" (1779). Given a surface $pdx+qdy+rdz=0$, Euler wants to develop equations that give the geodesics on this surface. I am new to the calculus of variations, so it is not clear to me what steps follow from results that are previously known (like the Euler-Lagrange equation in the calculations) and what steps follow from earlier in this paper. I would appreciate comments from any readers who are familiar with calculus of variations.

math.HO↗
arXiv2008

On Wittgenstein's philosophy of mathematics

Mathematics cannot anymore be assimilated to a linguistic game, where formal proofs are strongly differentiated with conjectural thinking, without building any category of knowledge to understand the passage (Wittgenstein's gist). Nowadays, philosophy has to face with the growing, exponential ramified tree of speculative mathematical thinking. Our main (problematical) theses are: 1. In mathematics, there is no empirical automatism, and no separate, physical-like motricity. 2: The irreversible-synthetical must force to complexify the exegetical game of philosophy; numerical experiments in algebra and in number theory are a kind of letting blow up all possible problems; 4. The nature of mathematical questioning still remains in question.

math.HO↗
arXiv2008

Comments on "Reverse auction: the lowest positive integer game"

In Zeng et al. [Fluct. Noise Lett. 7 (2007) L439--L447] the analysis of the lowest unique positive integer game is simplified by some reasonable assumptions that make the problem tractable for arbitrary numbers of players. However, here we show that the solution obtained for rational players is not a Nash equilibrium and that a rational utility maximizer with full computational capability would arrive at a solution with a superior expected payoff. An exact solution is presented for the three- and four-player cases and an approximate solution for an arbitrary number of players.

math.CO↗
arXiv2008

The origin of infinitely divisible distributions: from de Finetti's problem to Levy-Khintchine formula

The article provides an historical survey of the early contributions on infinitely divisible distributions starting from the pioneering works of de Finetti in 1929 up to the canonical forms developed in the thirties by Kolmogorov, Levy and Khintchine. Particular attention is paid to single out the personal contributions of the above authors that were published in Italian, French or Russian during the period 1929-1938. In Appendix we report the translation from the Russian into English of a fundamental paper by Khintchine published in Moscow in 1937.

math.HO↗
arXiv2009

Sobolev of the Euler School

This is a short overview of the origins of distribution theory as well as the life of Sergei Sobolev (1908--1989) and his contribution to the formation of the modern outlook of mathematics.

math.HO↗
arXiv2009

Transseries for beginners

From the simplest point of view, transseries are a new kind of expansion for real-valued functions. But transseries constitute much more than that--they have a very rich (algebraic, combinatorial, analytic) structure. The set of transseries is a large ordered field, extending the real number field, and endowed with additional operations such as exponential, logarithm, derivative, integral, composition. Over the course of the last 20 years or so, transseries have emerged in several areas of mathematics: asymptotic analysis, model theory, computer algebra, surreal numbers. This paper is an exposition for the non-specialist mathematician. All a mathematician needs to know in order to apply transseries.

math.RA↗
arXiv2008

A Note on Walking Versus Waiting

This mathematical recreation extends the analysis of a recent paper, asking when a traveller at a bus stop and not knowing the time of the next bus is best advised to wait or to start walking toward the destination. A detailed analysis and solution is provided for a very general class of probability distributions of bus arrival time, and the solution characterised in terms of a function analogous to the hazard rate in reliability theory. The note also considers the question of intermediate stops. It is found that the optimal strategy is not always the laziest, even when headways are not excessively long. For the common special case where one knows the (uniform) headway but not the exact timetable, it is shown that one should wait if the headway is less than the walking time (less bus travel time), and walk if the headway is more than twice this much. In between it may be better to wait or to walk, depending on one's confidence in being able to catch up to a passing bus.

math.HO↗
arXiv2008

Perelman, Poincare, and the Ricci Flow

In this expository article, we introduce the topological ideas and context central to the Poincare Conjecture. Our account is intended for a general audience, providing intuitive definitions and spatial intuition whenever possible. We define surfaces and their natural generalizations, manifolds. We then discuss the classification of surfaces as it relates to the Poincare and Thurston Geometrization conjectures. Finally, we survey Perelman's results on Ricci flows with surgery.

math.HO↗
arXiv2008

An upper bound for the number of perfect matchings in graphs

We give an upper bound on the number of perfect matchings in an undirected simple graph $G$ with an even number of vertices, in terms of the degrees of all the vertices in $G$. This bound is sharp if $G$ is a union of complete bipartite graphs. This bound is a generalization of the upper bound on the number of perfect matchings in bipartite graphs on $n+n$ vertices given by the Bregman-Minc inequality for the permanents of $(0,1)$ matrices.

math.CO↗
arXiv2008

Validation of a Model of the Domino Effect?

A recent paper proposing a model of the limiting speed of the domino effect is discussed with reference to its need and the need of models in general for validation against experimental data. It is shown that the proposed model diverges significantly from experimentally derived speed estimates over a significant range of domino spacing using data from the existing literature and this author's own measurements, hence if its use had had economic importance its use outside its range of validity could have led to loses of one sort or another to its users.

math.HO↗
arXiv2008

Various considerations on hypergeometric series

E661 in the Enestrom index. This was originally published as "Variae considerationes circa series hypergeometricas" (1776). In this paper Euler is looking at the asymptotic behavior of infinite products that are similar to the Gamma function. He looks at the relations between some infinite products and integrals. He takes the logarithm of these infinite products, and expands these using the Euler-Maclaurin summation formula. In section 14, Euler seems to be rederiving some of the results he already proved in the paper. However I do not see how these derivations are different. If any readers think they understand please I would appreciate it if you could email me. I am presently examining Euler's work on analytic number theory. The two main topics I want to understand are the analytic continuation of analytic functions and the connection to divergent series, and the asymptotic behavior of the Gamma function.

math.HO↗
arXiv2020

Pressure and Equilibrium States in Ergodic Theory

Our goal is to present the basic results on one-dimensional Gibbs and equilibrium states viewed as special invariant measures on symbolic dynamical systems, and then to describe without technicalities a sample of results they allowed to obtain for certain differentiable dynamical systems. We hope that this contribution will illustrate the symbiotic relationship between ergodic theory and statistical mechanics, and also information theory.

math.DS↗