Recurrent proofs of the irrationality of certain trigonometric values

We use recurrences of integrals to give new and elementary proofs of the irrationality of pi, tan(r) for all nonzero rational r, and cos(r) for all nonzero rational r^2. Immediate consequences to other values of the elementary transcendental functions are also discussed

Irrationality proofs \`a la Hermite

As rewards of reading two great papers of Hermite from 1873, we trace the historical origin of the integral Niven used in his well-known proof of the irrationality of $\pi$, uncover a rarely acknowledged simple proof by Hermite of the irrationality of $\pi^2$, give a new proof of the irrationality of $r\tan r$ for nonzero rational $r^2$, and generalize it to a proof of the irrationality of certain ratios of Bessel functions.Comment: 8 page

Polyhedral billiards, eigenfunction concentration and almost periodic control

We study dynamical properties of the billiard flow on convex polyhedra away from a neighbourhood of the non-smooth part of the boundary, called ``pockets''. We prove there are only finitely many immersed periodic tubes missing the pockets and moreover establish a new quantitative estimate for the lengths of such tubes. This extends well-known results in dimension $2$. We then apply these dynamical results to prove a quantitative Laplace eigenfunction mass concentration near the pockets of convex polyhedral billiards. As a technical tool for proving our concentration results on irrational polyhedra, we establish a control-theoretic estimate on a product space with an almost-periodic boundary condition. This extends previously known control estimates for periodic boundary conditions, and seems to be of independent interest.Comment: 32 pages, a few sections reorganised and a few results adde

Lambert’s proof of the irrationality of Pi: Context and translation

This document gives the first complete English translation of Johann Heinrich Lambert’s memoir on the irrationality of Pi published in 1768, as well as some contextual elements, such as Legendre’s proof and more recent proofs such as Niven’s

一般化されたThue-Morse数列から見た数の加法的表示について

早大学位記番号:新7476早稲田大

The Cycle of Rents: a Model of Rational Bull-and-Bear Cycles in an Efficient Market

A widespread misbelief asserts that an efficient market would arbitrage out any cyclical or otherwise partially-predictable, non-random-walk pattern on the observed market prices time series. Hence, when such patterns are observed, they are often attributed to either irrational behavior or market inefficiency. Yet, strictly speaking, the efficient markets hypothesis only rules such patterns out of the expected (i.e. mean) path, whereas, if the probability diffusion process is asymmetric (as in most economic and financial stochastic models), the observed time series will approximate the median path, which is not subject to such constraint. This paper combines a general imperfect-competition production function specification (i.e. one generating economic rents) with the concept of time-to-build to develop a rational-expectations, efficient-markets model displaying a valuation cycle along its median path. This model may therefore help to explain the bull-and-bear cycles observed in asset markets generating economic rents e.g. real estate, commodities or, for that matter, most if not all of the assets quoted in the stock exchange

Complex numbers from 1600 to 1840

This thesis uses primary and secondary sources to study advances in complex number theory during the 17th and 18th Centuries. Some space is also given to the early 19th Century. Six questions concerning their rules of operation, usage, symbolism, nature, representation and attitudes to them are posed in the Introduction. The main part of the thesis quotes from the works of Descartes, Newton, Wallis, Saunderson, Maclaurin, d'Alembert, Euler, Waring, Frend, Hutton, Arbogast, de Missery, Argand, Cauchy, Hamilton, de Morgan, Sylvester and others, mainly in chronological order, with comment and discussion. More attention has been given tp algebraists, the originators of most advances in complex numbers, than to writers in trigonometry, calculus and analysis, who tended to be users of them. The last chapter summarises the most important points and considers the extent to which the six questions have been resolved. The most important developments during the period are identified as follows: (i) the advance in status of complex numbers from 'useless' to 'useful'. (ii) their interpretation by Wallis, Argand and Gauss in arithmetic, geometric and algebraic ways. (iii) the discovery that they are essential for understanding polynomials and logarithmic, exponential and trigonometric functions. (iv) the extension of trigonometry, calculus and analysis into the complex number field. (v) the discovery that complex numbers are closed under exponentiation, and so under all algebraic operations. (vi) partial reform of nomenclature and symbolism. (vii) the eventual extension of complex number theory to n dimensions