Archiv der Mathematik最新文献

We establish a general form of Wiener’s lemma for measures on locally compact Abelian groups by using Fourier analysis and the theory of Følner sequences. Our approach provides a unified framework that encompasses both the discrete and continuous cases. We also show a version of Wiener’s lemma for Bochner–Riesz means on both ({mathbb {R}}^d) and ({mathbb {T}}^d).

In this paper, we focus on the following Choquard-type Brézis–Nirenberg problem:

where (Omega ) is a smooth bounded domain in (mathbb {R}^4), (alpha in (0,4)), (4-frac{alpha }{2}) is the upper critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality, and (varepsilon >0) is a small parameter. By applying the reduction argument, we prove the existence of solutions, which blow up and concentrate around the critical points of the Robin function as (varepsilon rightarrow 0).

In this article, some generalizations of Girstmair’s irreducibility criterion have been established for polynomials having integer coefficients. These results for the ring of polynomials over the integers accentuate significant bounds on the number of irreducible factors of the underlying polynomial f apart from irreducibility under some austere factorization and divisibility conditions imposed on the integers f(m) and (f^{(i)}(m)), the i-th formal derivative of f at m strictly superceding the height (H_f) of f.

A compact quantum metric space is a unital (C^*)-algebra equipped with a Lip-norm. We prove that the (infinite) tensor product of compact quantum metric spaces is a compact quantum metric space for any (C^*)-norm on the algebraic tensor product.

The determination of the minimal generator number d(G) of a polycyclic group G is one of the few still open problems in the algorithmic theory of these groups. If (hat{G}) is the profinite completion of G, then it is known that (d(G) = d(hat{G})) or (d(G) = d(hat{G}) +1) holds. The aim here is to introduce a construction for (d(hat{G})) if G is polycyclic and nilpotent-by-finite.

We prove that singular minimal surfaces with constant Gauss curvature are planes, spheres, and cylindrical surfaces. We also classify all singular minimal surfaces with a constant principal curvature and singular minimal surfaces with constant mean curvature.

If a module M has finite projective dimension, then the Ext modules of M against any other module eventually vanish and the projective dimension of M gives a uniform bound for this vanishing. For modules of infinite projective dimension, there can still exist a bound. Such a bound is called the Auslander bound. In this paper, we define a similar bound for complexes and give applications.

Hadwiger’s conjecture in combinatorial geometry states that any n-dimensional convex body can be covered by at most (2^n) smaller bodies homothetic to the original body. We prove Hadwiger’s conjecture for strongly monotypic polytopes by studying a characterization of the set of normals. One of the nice properties of (strongly) monotypic polytopes is that the set of normals decides the combinatorics of the polytope.

The Ma–Qiu index of a group is the minimum number of normal generators of the commutator subgroup. We show that the Ma–Qiu index gives a lower bound of the presentation distance of two groups, the minimum number of relator replacements to change one group to the other. Since many local moves in knot theory induce relator replacements in knot groups, this shows that the Ma–Qiu index of knot groups gives a lower bound of the Gordian distance based on various local moves. In particular, this gives a unified and simple proof of the Nakanishi index bounds of various unknotting numbers, including virtual or welded knot cases.

We prove that the weak Hilbert property ascends along a morphism of varieties over an arbitrary field of characteristic zero, under suitable assumptions.