in this paper, we examine that some simple $k_4$-groups can be determined uniquely by their orders and one or two irreducible complex character degrees.

The Pólya-Schur theory describes the class of hyperbolicity preservers, i.e., the class of linear operators acting on univariate polynomials and preserving real-rootedness. We attempt to develop an analog of Pólya-Schur theory in the setting of linear finite difference operators. We study the class of linear finite difference operators preserving the set of real-rooted polynomials whose mesh (i...

We fix a finitely presented group Q and consider short exact sequences 1 → N → Γ → Q → 1 with Γ finitely generated. The inclusion N ↪→ Γ induces a morphism of profinite completions N̂ → Γ̂. We prove that this is an isomorphism for all N and Γ if and only if Q is super-perfect and has no proper subgroups of finite index. We prove that there is no algorithm that, given a finitely presented, residua...

A multilinear version of Schur’s test is obtained for products of L spaces and is used to derive boundedness for multilinear multiplier operators acting on Sobolev and Besov spaces.

We study the notion of Schur multiplier a pair (N, L) Lie superalgebras and obtain some upper bounds concerning dimensions. Moreover, we characterize pairs finite dimensional (nilpotent) for whichfor t = 0; 1, where dim N (m|n).

The Pólya-Schur theory describes the class of hyperbolicity preservers, i.e., the linear operators on univariate polynomials preserving realrootedness. We attempt to develop an analog of Pólya-Schur theory in the setting of linear finite difference operators. We study the class of linear finite difference operators preserving the set of real-rooted polynomials whose mesh (i.e., the minimal dist...

In the study of Zeilberger’s conjecture on an integer sequence related to the Catalan numbers, Lassalle proposed the following conjecture. Let (t)n denote the rising factorial, and let ΛR denote the algebra of symmetric functions with real coefficients. If φ is the homomorphism from ΛR to R defined by φ(hn) = 1/((t)nn!) for some t > 0, then for any Schur function sλ, the value φ(sλ) is positive...