Linear equations with two variables in Piatetski-Shapiro sequences

Word Equations with Two Variables

The problem whether the set of all equations that are satisfiable in some free semigroup or, equivalently, in an algebra of words with concatenation is recursive (usually called the satisfiability problem for semigroup equations) was first formulated by A.A. Markov in early sixties (see [3]). Special cases of the problem were solved affirmatively by A.A. Markov (see [3]), Yu.I. Khmelevskiı̆ [8],...

Linear equations in quaternionic variables

We study the quaternionic linear system which is composed out of terms of the form ln(x) := ∑n p=1 apxbp with quaternionic constants ap, bp and a variable number n of terms. In the first place we investigate one equation in one variable. If n = 2 the corresponding equation, which is normally called Sylvester’s equation will be treated completely by using only quaternionic algebra. For larger n ...

Rank-One Residues of Eisenstein Series∗ Dedicated to Ilya Piatetski-Shapiro with admiration and affection

The trace-class problem was for some years a rather embarrassing unsolved problem in the theory of automorphic forms. Although the absence of a solution did not seriously obstruct progress, it did undesirably, and in the general view unnecessarily, complicate the statements of various results, especially in the context of the trace formula. Fortunately, it has recently been solved by W. Müller ...

Generalized Rudin – Shapiro sequences

Equations in Two Space Variables

Hölder estimates for spatial second derivatives are proved for solutions of fully nonlinear parabolic equations in two space variables. Related techniques extend the regularity theory for fully nonlinear parabolic equations in higher dimensions.