Since Paul J. Cohen's 1963 result, it has been possible to investigate the consequences of the axioms of Zermelo-Fraenkel set theory without the Axiom of Choice. In this paper we examine the relative strengths of a number of finiteness criteria against the background of ZF set theory without AC.

The aim of this paper is to give a metric regularity theorem for multifunctions between metric spaces involving some known results for multifunctions by using the notion of strict (G, δ)-differentiability of multifunctions and a simple convergence procedure.

Let H∞ be the Banach algebra of all bounded analytic functions in the unit disc. We present a complete description of the closed primary (respectively prime) ideals contained in a maximal ideal of the Shilov boundary of H∞. The paper is also concerned with chains of prime ideals in H∞.

For any ring R graded by a finite group, we give a bound on the classical Krull dimension of R in terms of the dimension of the initial component Re. It follows that if Re has finite classical Krull dimension, then the same is true of the whole ring R, too.

In this paper, using a multidimensional Volkenborn integral, we give a p-adic expression of the higher order Bernoulli numbers. This shows immediately the relation to the sums of products of the ordinary Bernoulli numbers of Dilcher in 1996. We also consider the Mahler expansion of several p-adic variables function, and give some examples.

A submodule W of a torsion module M over a discrete valuation domain is called stacked in M if there exists a basis ℬ of M such that multiples of elements of ℬ form a basis of W. We characterise those submodules which are stacked in a pure submodule of M.

Qi and Chapman [‘Two closed forms for the Bernoulli polynomials’, J. Number Theory159 (2016), 89–100] gave a closed form expression for the Bernoulli polynomials as polynomials with coefficients involving Stirling numbers of the second kind. We extend the formula to the degenerate Bernoulli polynomials, replacing the Stirling numbers by degenerate Stirling numbers of the second kind.

In his paper in Mat. Sb. (N.S.) 61 (103) (1963), Zareckiĭ associated with any binary relation α an ordered pair, (Lα Mα), say, of lattices and showed that α is a left [right] divisor of β if and only if We provide an alternative proof of this result by embedding the category of relations in the category of sets. Our approach provides a unified treatment of several hitherto independent results, and gives new results for the category of partial transformations.

The multiple T-value, which is a variant of the multiple zeta value of level two, was introduced by Kaneko and Tsumura [‘Zeta functions connecting multiple zeta values and poly-Bernoulli numbers’, in: Various Aspects of Multiple Zeta Functions, Advanced Studies in Pure Mathematics, 84 (Mathematical Society of Japan, Tokyo, 2020), 181–204]. We show that the generating function of a weighted sum of multiple T-values of fixed weight and depth is given in terms of the multiple T-values of depth one by solving a differential equation of Heun type.

The following result is proved. Let X be a Schunk class and k a positive integer. Then the k–th nilpotent product of any two groups in X¯ is in X¯.

We extend existing methods which treat the semilinear Calderón problem on a bounded domain to a class of complex manifolds with Kähler metric. Given two semilinear Schrödinger operators with the same Dirchlet-to-Neumann data, we show that the integral identities that appear naturally in the determination of the analytic potentials are enough to deduce uniqueness on the boundary up to infinite order. By exploiting the assumed complex structure, this information allows us to apply the method of stationary phase and recover the potentials in the interior as well.

An abstract metrical version of the well-known Ekeland and Brøndsted maximality principle is used to derive a number of stability criteria for a class of (function) contractive semigroups on (complete) metric spaces, extending a number of classical contributions due to Bre'zis and Browder.

We show that if the normal, closure of an element a, of an orderable group, G, is abelian, then G can be embedded in an orderable group, G#, which contains an n-th root of a for every positive integer, n. Furthermore, every order of G extends to an order of G#.

The aim of this paper is three-fold: to fill the gap between different deformation lemmas, to obtain a unifying minimax result where multiple linking situations can occur, and to locate the critical points as solutions of minimisation problems.

This paper deals with the motion of incoherent matter, and hence of test particles, in the presence of fields with an arbitrary energy-momentum tensor. The equations of motion are obtained from Einstein's field equations and are written in the form of geodesic equations of an affine connection. The special cases of the electromagnetic field, the Proca field and a scalar field are discussed.

The existence of solutions to a homogeneous Dirichlet problem for a p-Laplacian differential inclusion is studied via a fixed-point type theorem concerning operator inclusions in Banach spaces. Some meaningful special cases are then worked out.

Under weak hypotheses, the geometric ergodicity of the states of a subcritical branching process with immigration is demonstrated by obtaining an asymptotic representation of the transition probabilities.

It is shown that the functional fails to possess a variational derivative, contrary to what is claimed by Gelfand and Fomin. A modified definition is given with respect to which the functional does possess a variational derivative.