We give the relation between the (rigid) graded Morita duality and the Morita duality on a finitely group-graded ring and the relation between a left Morita ring and some of its matrix rings.

In this paper, we shall prove that any subset of $\overline{\mathbb{Q}}$, which is closed under complex conjugation, is the exceptional set of uncountably many transcendental entire functions with rational coefficients. This solves an old question proposed by Mahler [Lectures on Transcendental Numbers, Lecture Notes in Mathematics, 546 (Springer, Berlin, 1976)].

The main goal of this paper is to give the answer to one of the main problems of the theory of nonautonomous superposition operators acting in the space of functions of bounded variation in the sense of Jordan. Namely, we prove that if the superposition operator maps the space $BV[0,1]$ into itself, then it is automatically locally bounded, provided its generator is a locally bounded function.

In this paper we employ quasi-continuous multifunctions and introduce almost Baire class 1 multifunctions in order to generalize a theorem of Kuratowski and also to answer a question posed by him concerning Baire class 1 multifunctions. We also show that certain multifunctions can be decomposed into mutually singular multifunctions.

For a Banach algebra 𝒜 and a character ϕ on 𝒜, we have recently introduced and studied the notion of ϕ-pseudo-amenability of 𝒜. Here, we give some characterizations of this notion in terms of derivations from 𝒜 into various Banach 𝒜-bimodules.

In this paper we establish two minimax theorems of Sion-type in G-convex spaces. As applications we obtain generalisations of some theorems concerning compatibility of some systems of inequalities.

Oscillation results are obtained for a first order functional differential equation, by transforming it to an equation for which oscillatory information exists in the literature already.

Suppose $a^{2}(a^{2}+1)$ divides $b^{2}(b^{2}+1)$ with $b>a$. We improve a previous result and prove a gap principle, without any additional assumptions, namely $b\gg a(\log a)^{1/8}/(\log \log a)^{12}$. We also obtain $b\gg _{\unicode[STIX]{x1D716}}a^{15/14-\unicode[STIX]{x1D716}}$ under the abc conjecture.

Let R be a finite ring and let ${\mathrm {zp}}(R)$ denote the nullity degree of R, that is, the probability that the multiplication of two randomly chosen elements of R is zero. We establish the nullity degree of a semisimple ring and find upper and lower bounds for the nullity degree in the general case.

Swisher [‘On the supercongruence conjectures of van Hamme’, Res. Math. Sci. 2 (2015), Article no. 18] and He [‘Supercongruences on truncated hypergeometric series’, Results Math. 72 (2017), 303–317] independently proved that Van Hamme’s (G.2) supercongruence holds modulo $p^4$ for any prime $p\equiv 1\pmod {4}$ . Swisher also obtained an extension of Van Hamme’s (G.2) supercongruence for $p\equiv 3 \pmod 4$ and $p>3$ . In this note, we give new one-parameter generalisations of Van Hamme’s (G.2) supercongruence modulo $p^3$ for any odd prime p. Our proof uses the method of ‘creative microscoping’ introduced by Guo and Zudilin [‘A q-microscope for supercongruences’, Adv. Math. 346 (2019), 329–358].

In 1929 Rolf Nevanlinna posed a problem attributed to Bloch which has since been known as the Bloch-Nevanlinna conjecture. It can be stated as follows: Is the derivative of a function of bounded characteristic of bounded characteristic? A variety of different counterexamples have provided negative answers to this question. The purpose of the paper is to survey these counterexamples and then give a truly elementary proof of the following theorem.

In the 1993 Western Number Theory Conference, Richard Guy proposed Problem 93:31, which asks for integers n representable by ${(x+y+z)^3}/{xyz}$ , where $x,\,y,\,z$ are integers, preferably with positive integer solutions. We show that the representation $n={(x+y+z)^3}/{xyz}$ is impossible in positive integers $x,\,y,\,z$ if $n=4^{k}(a^2+b^2)$ , where $k,\,a,\,b\in \mathbb {Z}^{+}$ are such that $k\geq 3$ and $2\nmid a+b$ .

We show that a semigroup satisfying a heterotypical identity of which at least one side has no repeated variable is saturated and find sufficient conditions on a homotypical identity which is not a permutation identity and of which at least one side has no repeated variable, to ensure that any semigroup satisfying the identity is saturated.

This paper resulted from an attempt to answer questions like: does βP have a subspace homeomorphic to βQ and does βQ have a subspace homeomorphic to P, where P denotes the space of all irrational numbers. These questions are answered in the negative by providing the appropriate machinery which can also be applied to other examples. En route we prove that weakly Fréchet realcompact spaces have homeomorphic Stone-Čech compactifications if and only if they are homeomorphic.

Let G be a nondiscrete locally compact abelian group with dual group Γ. For 1 ≤ p ≤ ∞, denote by Ap(G) the space of integrable functions on G whose Fourier transforms belong to Lp(Γ). We investigate multipliers from Ap(G) to Aq(G). If G is compact and 2 < p1, p2 < ∞, we show that multipliers of and multipliers of are different, provided Pl ≠ P2. For compact G, we also exhibit a relationship between lr (Γ) and the multipliers from Ap(G) to Aq(G). If G is a compact nonabelian group we observe that the spaces Ap(G) behave in the same way as in the abelian case as far as the multiplier problems are concerned.

For every positive number w, there exists an odd entire transcendental function

.

with rational integral coefficients ah such that f〈z+w〉 = f〈z〉.

Let Fq be the field with q elements and let G = PGLn(Fq) or PSLn(Fq) act on Fq(x1,…,xn−1), the rational function field of n − 1 variables. Then Fq(x1,…,xn−1)G is purely transcendental over Fq. In fact, a set of n − 1 generators of Fq(x1,…xn−1)G, over Fq is exhibited. The case n = 2 is treated by direct computation.

We prove a Korovkin type approximation theorem for positive linear operators on weighted spaces of continuous real-valued functions on a compact Hausdorff space X. These spaces comprise a variety of subspaces of C (X) with suitable locally convex topologies and were introduced by Nachbin 1967 and Prolla 1977. Some early Korovkin type results on the weighted approximation of real-valued functions in one and several variables with a single weight function are due to Gadzhiev 1976 and 1980.

A concept of synchronicity associated with convex functions in linear spaces and a Chebyshev type inequality are given. Applications for norms, semi-inner products and convex functions of several real variables are also given.

The Schauder estimates for solutions of linear second order parabolic equations with Venttsel initial boundary conditions are proved, and existence and uniqueness of classical solutions under such an initial boundary condition are established. An application to an engineering problem is also given.