Auxiliary functions in transcendental number theory

Abstract

The author wishes to thank Krishna Alladi who suggested that this paper be published in the collection of SASTRA conferences he is editing. The author is also grateful to Claude Levesque and Paul Voutier for a number of clever remarks on a preliminary version of this survey.

Algebraic Numbers as Product of Powers of Transcendental Numbers

“…The complete solution to this problem was found independently by Gel'fond and Schneider (see [5], p. 9) in 1934. Their results can be formulated as the following theorem (the ideas of the Gel'fond-Schneider proof were used partially in, e.g., [6][7][8]). Theorem 1.…”

An asymptotic approach in Mahler's method

“…Mahler's early results focused on degree-1 Mahler functions, his most famous result in this area being the transcendence of the Thue-Morse number T (1/2), which is a special value of the function T (z) satisfying T (z) − (1 − z)T (z 2 ) = 0. According to Waldschmidt [41], after Mahler's initial results his method was forgotten; the resurgence waited nearly forty years, following the publication of Mahler's paper "Remarks on a paper of W. Schwarz" [27] in 1969. Mahler's method was then extended by Kubota, Loxton, Ke.…”

Some aspects of the development of formalism of nonequilibrium thermodynamics for systems based on polyelectrolyte hydrogels

“…Разумеется, для случая, когда система достаточно близка к равновесию, можно использовать формализм, основанный на разложении соответствующих функционалов в ряд Тейлора и использовать представление о малости возмущений [7][8][9][10][11]. Именно по этой схеме и строятся теории, так или иначе связаные с принципом Онзагера [12].…”