The Good Journal

Many mathematicians have studied the algebraic independence over Q of the values of gap series, and the values of lacunary series satisfying functional equations of Mahler type. In this paper, we give a new criterion for the algebraic independence over Q of the values ∑∞n=0t(n)β−n for distinct sequences (t(n))∞n=0 of nonnegative integers, where β is a fixed Pisot or Salem number. Our criterion is applicable to certain power series which are not lacunary. Moreover, our criterion does not use functional equations. Consequently, we deduce the algebraic independence of certain values ∑∞n=0t1(n)β−n,…,∑∞n=0tr(n)β−n satisfying

Hölder Regularity for Degenerate Parabolic Obstacle Problems

Universal extensions arise naturally in the Auslander bijections. For an abelian category having Auslander–Reiten duality, we exploit a bijection triangle, which involves the Auslander bijections, universal extensions and the Auslander–Reiten duality. Some consequences are given, in particular, a conjecture by Ringel is verified.