Archiv der Mathematik最新文献

This article is to shed light on a class of commutative unital complex Banach algebras. We show that these Banach algebras of sequences of generalized bounded variation are all semi-simple, regular, and self-adjoint and that their carrier spaces are homeomorphic to compactifications of the discrete space of positive integers. In particular, we provide necessary and sufficient conditions under which the carrier space may be identified with the one-point compactification or the Stone–Čech compactification of the positive integers. It turns out that the carrier spaces of many of the Banach algebras under consideration are neither of these familiar compactifications.

A base for a permutation group G acting on a set (Omega ) is a sequence ({mathcal {B}}) of points of (Omega ) such that the pointwise stabiliser (G_{{mathcal {B}}}) is trivial. The base size of G is the size of a smallest base for G. We derive a character theoretic formula for the base size of a class of groups admitting a certain kind of irreducible character. Moreover, we prove a formula for enumerating the non-equivalent bases for G of size (lin {mathbb {N}}.) As a consequence of our results, we present a very short, entirely algebraic proof of the formula of Mecenero and Spiga for the base size of the symmetric group (textrm{S}_n) acting on the k-element subsets of ({1,2,3,ldots ,n}.) Our methods also provide a formula for the base size of many product type permutation groups.

Let (R=K[x_1,ldots ,x_n]) denote the polynomial ring in n variables over a field K and I be a polymatroidal ideal of R. In this paper, we provide a comprehensive classification of all unmixed polymatroidal ideals. This work addresses a question raised by Herzog and Hibi (Eur J Comb 27:513–517, 2006).

Assume F is a finite field of order (p^f) and q is an odd prime for which (p^f-1=sq^m), where (m ge 1) and ((s,q)=1). In this article, we obtain the order of the symmetric and the unitary subgroup of the semisimple group algebra (FC_q.) Further, for the extension G of (C_q = langle b rangle ) by an abelian group A of order (p^n) with (C_{A}(b) = {e}), we prove that if (m>1,) or ((s+1) ge q) and (2n ge f(q-1)), then G does not have a normal complement in V(FG).

It is known that the Morrey–Campanato space is a subspace of the dual of some atomic Hardy space. We give relations between these spaces.

Let p be a prime number. A longstanding conjecture asserts that every finite non-abelian p-group has a non-inner automorphism of order p. In this paper, under some conditions on an odd order finite p-group G with cyclic center, we prove that G exhibits a non-inner automorphism of order p. As a consequence, under certain conditions on a finite p-group G ((p>2),) the conjecture is proved for all nilpotency classes except class 2 and maximal class. Moreover, we also settle the conjecture for some non-abelian finite 3-groups of coclass 3,  which is a pending case of the main result of Ruscitti et al. (Monatsh. Math. 183(4):679–697, 2016).

We extend results on value sets of polynomials over finite fields to polynomial images of ‘structured’ finite sets.

Let (0<alpha <n) and (M_{alpha }) be the fractional maximal function. For a locally integrable function b, we denote by (M_{alpha ,b}) and ([b,M_{alpha }]) the maximal commutator and the commutator of (M_{alpha }) with b. In this paper, we consider Bloom-type estimates for (M_{alpha ,b}) and ([b,M_{alpha }]). Some necessary and sufficient conditions to characterize the Bloom-type two-weight norm inequalities are given.

A graph is called integral if its eigenvalues are integers. In this article, we provide necessary and sufficient conditions for a Cayley graph over a finite symmetric algebra R to be integral. This generalizes the work of So who studies the case where R is the ring of integers modulo n. We also explain some number-theoretic constructions of finite symmetric algebras arising from global fields, which we hope could pave the way for future studies on Paley graphs associated with finite Hecke characters.

Let K be an algebraically closed field of characteristic zero, and let A and B be two simple algebras with involution over K. In this note, we study the embedding problem for algebras with involution. More specifically, if the algebra A satisfies the polynomial identities with involution of the algebra B, we investigate whether there exists an embedding of A into B that preserves the involutions.