The Product eπe \pi Is Irrational

This note shows that the product $e \pi$ of the natural base $e$ and the circle number $\pi$ is an irrational number.Comment: Twelve Pages. Improved Version. keywords: Irrational number; Natural base, Circle number; Unbounded partial quotients. arXiv admin note: text overlap with arXiv:1212.408

Searching for Apery-Style Miracles [Using, Inter-Alia, the Amazing Almkvist-Zeilberger Algorithm]

Roger Apery's seminal method for proving irrationality is "turned on its head" and taught to computers, enabling a one second redux of the original proof of zeta(3), and many new irrationality proofs of many new constants, alas, none of them is both famous and not-yet-proved-irrational.Comment: 16 pages. Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger, May 2014, and this arxiv.org. Accompanied my Maple package NesApery, available from http://www.math.rutgers.edu/~zeilberg/tokhniot/NesAper

Periodic boxcar deconvolution and diophantine approximation

We consider the nonparametric estimation of a periodic function that is observed in additive Gaussian white noise after convolution with a ``boxcar,'' the indicator function of an interval. This is an idealized model for the problem of recovery of noisy signals and images observed with ``motion blur.'' If the length of the boxcar is rational, then certain frequencies are irretreviably lost in the periodic model. We consider the rate of convergence of estimators when the length of the boxcar is irrational, using classical results on approximation of irrationals by continued fractions. A basic question of interest is whether the minimax rate of convergence is slower than for nonperiodic problems with 1/f-like convolution filters. The answer turns out to depend on the type and smoothness of functions being estimated in a manner not seen with ``homogeneous'' filters.Comment: Published at http://dx.doi.org/10.1214/009053604000000391 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Report on some recent advances in Diophantine approximation

A basic question of Diophantine approximation, which is the first issue we discuss, is to investigate the rational approximations to a single real number. Next, we consider the algebraic or polynomial approximations to a single complex number, as well as the simultaneous approximation of powers of a real number by rational numbers with the same denominator. Finally we study generalisations of these questions to higher dimensions. Several recent advances have been made by B. Adamczewski, Y. Bugeaud, S. Fischler, M. Laurent, T. Rivoal, D. Roy and W.M. Schmidt, among others. We review some of these works.Comment: to be published by Springer Verlag, Special volume in honor of Serge Lang, ed. Dorian Goldfeld, Jay Jorgensen, Dinakar Ramakrishnan, Ken Ribet and John Tat

Conflicts in the learning of real numbers and limits

question: "Is 0.999... (nought point nine recurring) equal to one, or just less than one?". Many answers contained infinitesimal concepts: "The same, because the difference between them is infinitely small." " The same, for at infinity it comes so close to one it can be considered the same." "Just less than one, but it is the nearest you can get to one without actually saying it is one." "Just less than one, but the difference between it and one is infinitely small." The majority of students thought that 0.999... was less than one. It may be that a few students had been taught using infinitesimal concepts, or that the phrase “just less than one ” had connotations for the students different from those intended by the questioner; but it seems more likely that the answers represent the students ’ own rationalisations made in an attempt to resolve conflicts inherent in the students ’ previous experience of limiting processes. Some conscious and subconscious conflicts Most of the mathematics met in secondary school consists of sophisticated idea