Archiv der Mathematik最新文献

In this paper, we focus on covering and illumination properties of a specific class of convex polytopes denoted by (mathcal {P}). These polytopes are obtained as the convex hull of the Minkowski sum of a finite subset of (mathbb {Z}^n) and ((1/2)[-1,1]^n). Our investigation includes the verification of Hadwiger’s covering conjecture for (mathcal {P}), as well as the estimation of the covering functional for convex polytopes in (mathcal {P}). Furthermore, we demonstrate that when an integer M is sufficiently large, the elements belonging to (mathcal {P}) that are contained in (M[-1,1]^n) serve as an (varepsilon )-net for the space of convex bodies in (mathbb {R}^n), equipped with the Banach–Mazur metric.

For a given closed nonempty subset E of a Hilbert space H, the singular set (Sigma _E) consists of the points in (Hsetminus E) where the distance function (d_E) is not Fréchet differentiable. It is known that (Sigma _E) is a weak deformation retract of the open set (mathcal {G}_E={xin H: d_{overline{{text {co}}},E}(x)< d_E(x)}). This short paper sheds light on the relationship between the connected components of the three sets (Sigma _Esubset mathcal {G}_Esubseteq H{setminus } E).

Rédei and Megyesi proved that the number of directions determined by a p-element subset of ({mathbb F}_p^2) is either 1 or at least (frac{p+3}{2}). The same result was independently obtained by Dress, Klin, and Muzychuk. We give a new and short proof of this result using a lemma proved by Kiss and the author. The new proof relies on a result on polynomials over finite fields.

In this note, we provide a characterization for the set of extreme points of the Lipschitz unit ball in a specific vectorial setting. While the analysis of the case of real-valued functions is covered extensively in the literature, no information about the vectorial case has been provided up to date. Here, we aim at partially filling this gap by considering functions mapping from a finite metric space to a strictly convex Banach space that satisfy the Lipschitz condition. As a consequence, we present a representer theorem for such functions. In this setting, the number of extreme points needed to express any point inside the ball is independent of the dimension, improving the classical result from Carathéodory.

In this note, we study the regularity of the Berezin transform on the generalized Hartogs triangles. By introducing a rotation invariant weight function, we show the unboundedness of the Berezin transform of weighted Hilbert spaces defined on the generalized Hartogs triangles.

Let p be an odd prime number. We show that the modular isomorphism problem has a positive answer for finite p-groups whose center has index (p^3), which is a strong contrast to the analogous situation for (p = 2).

Denote by ({mathbb {G}}(k,n)) the Grassmannian of linear subspaces of dimension k in ({mathbb {P}}^n). We show that if (varphi :{mathbb {G}}(l,n) rightarrow {mathbb {G}}(k,n)) is a nonconstant morphism and (l not =0,n-1), then (l=k) or (l=n-k-1) and (varphi ) is an isomorphism.

The solvable conjugacy class graph of a finite group G, denoted by (Gamma _{sc}(G)), is a simple undirected graph whose vertices are the non-trivial conjugacy classes of G and two distinct conjugacy classes C, D are adjacent if there exist (x in C) and (y in D) such that (langle x, yrangle ) is solvable. In this paper, we discuss certain properties of the genus and crosscap of (Gamma _{sc}(G)) for the groups (D_{2n}), (Q_{4n}), (S_n), (A_n), and ({{,mathrm{mathop {textrm{PSL}}},}}(2,2^d)). In particular, we determine all positive integers n such that their solvable conjugacy class graphs are planar, toroidal, double-toroidal, or triple-toroidal. We shall also obtain a lower bound for the genus of (Gamma _{sc}(G)) in terms of the order of the center and number of conjugacy classes for certain groups. As a consequence, we shall derive a relation between the genus of (Gamma _{sc}(G)) and the commuting probability of certain finite non-solvable group.

Given an uncountable, compact metric space X, we show that there exists no reproducing kernel Hilbert space that contains the space of all continuous functions on X.