Let $A$ be a subset of $\mathbb{N}$ , the set of all nonnegative integers. For an integer $h\geq 2$ , let $hA$ be the set of all sums of $h$ elements of $A$ . The set $A$ is called an asymptotic basis of order $h$ if $hA$ contains all sufficiently large integers. Otherwise, $A$ is called an asymptotic nonbasis of order $h$ . An asymptotic nonbasis $A$ of order $h$ is called a maximal asymptotic nonbasis of order $h$ if $A\cup \{a\}$ is an asymptotic basis of order $h$ for every $a\notin A$ . In this paper, we construct a sequence of asymptotic nonbases of order $h$ for each $h\geq 2$ , each of which is not a subset of a maximal asymptotic nonbasis of order $h$ .

Let $a,b,c$ be a primitive Pythagorean triple and set $a=m^{2}-n^{2},b=2mn,c=m^{2}+n^{2}$ , where $m$ and $n$ are positive integers with $m>n$ , $\text{gcd}(m,n)=1$ and $m\not \equiv n~(\text{mod}~2)$ . In 1956, Jeśmanowicz conjectured that the only positive integer solution to the Diophantine equation $(m^{2}-n^{2})^{x}+(2mn)^{y}=(m^{2}+n^{2})^{z}$ is $(x,y,z)=(2,2,2)$ . We use biquadratic character theory to investigate the case with $(m,n)\equiv (2,3)~(\text{mod}~4)$ . We show that Jeśmanowicz’ conjecture is true in this case if $m+n\not \equiv 1~(\text{mod}~16)$ or $y>1$ . Finally, using these results together with Laurent’s refinement of Baker’s theorem, we show that Jeśmanowicz’ conjecture is true if $(m,n)\equiv (2,3)~(\text{mod}~4)$ and $n<100$ .

Let $R\subset S$ be an extension of integral domains, with $R^{\ast }$ the integral closure of $R$ in $S$ . We study the set of intermediate rings between $R$ and $S$ . We establish several necessary and sufficient conditions for which every ring contained between $R$ and $S$ compares with $R^{\ast }$ under inclusion. This answers a key question that figured in the work of Gilmer and Heinzer [‘Intersections of quotient rings of an integral domain’, J. Math. Kyoto Univ.7 (1967), 133–150].

Let $S\subseteq \mathbb{P}^{d}$ be an anticanonically embedded surface of degree $d\geq 3$ . In this note, we classify stable Ulrich bundles on $S$ of rank two. We also study their moduli spaces.

A group $G$ is said to have the $T$ -property (or to be a $T$ -group) if all its subnormal subgroups are normal, that is, if normality in $G$ is a transitive relation. The aim of this paper is to investigate the behaviour of uncountable groups of cardinality  $\aleph$ whose proper subgroups of cardinality $\aleph$ have a transitive normality relation. It is proved that such a group  $G$ is a $T$ -group (and all its subgroups have the same property) provided that $G$ has an ascending subnormal series with abelian factors. Moreover, it is shown that if $G$ is an uncountable soluble group of cardinality $\aleph$ whose proper normal subgroups of cardinality  $\aleph$ have the $T$ -property, then every subnormal subgroup of $G$ has only finitely many conjugates.

We complete the classification of the finite special linear groups $\text{SL}_{n}(q)$ which are $(2,3)$ -generated, that is, which are generated by an involution and an element of order $3$ . This also gives the classification of the finite simple groups $\text{PSL}_{n}(q)$ which are $(2,3)$ -generated.

A class of abelian topological groups was previously defined to be a variety of topological groups with coproducts if it is closed under forming subgroups, quotients, products and coproducts in the category of all abelian topological groups and continuous homomorphisms. This extended research on varieties of topological groups initiated by the second author. The key to describing varieties of topological groups generated by various classes was proving that all topological groups in the variety are a quotient of a subgroup of a product of groups in the generating class. This paper analyses generating varieties of topological groups with coproducts. It focuses on the interplay between forming products and coproducts. It is proved that the variety of topological groups with coproducts generated by all discrete groups contains topological groups which cannot be expressed as a quotient of a subgroup of a product of a coproduct of discrete groups. It is proved that the variety of topological groups with coproducts generated by any infinite-dimensional Hilbert space contains all infinite-dimensional Hilbert spaces, answering an open question. This contrasts with the result that a variety of topological groups generated by a topological group does not contain any infinite-dimensional Hilbert space of greater cardinality.

In this paper we show that a polygonal quasiconformal mapping always corresponds to a chord-arc curve. Furthermore, we find that the set of curves corresponding to polygonal quasiconformal mappings is path connected in the set of all bounded chord-arc curves.

This paper is concerned with the modified Wigner (respectively, Wigner–Fokker–Planck) Poisson equation. The quantum mechanical model describes the transport of charged particles under the influence of the modified Poisson potential field without (respectively, with) the collision operator. Existence and uniqueness of a global mild solution to the initial boundary value problem in one dimension are established on a weighted $L^{2}$ -space. The main difficulties are to derive a priori estimates on the modified Poisson equation and prove the Lipschitz properties of the appropriate potential term.

The aim of this work is to prove the existence of a positive almost periodic solution to a multifinite time delayed nonlinear differential equation that describes the so-called hematopoiesis model. The approach uses the Hilbert projective metric in a cone. With some additional assumptions, we construct a fixed point theorem to prove the desired existence and uniqueness of the solution.

In this paper we establish new optimal bounds for the derivative of some discrete maximal functions, in both the centred and uncentred versions. In particular, we solve a question originally posed by Bober et al. [‘On a discrete version of Tanaka’s theorem for maximal functions’, Proc. Amer. Math. Soc.140 (2012), 1669–1680].

We study the regularity properties of several classes of discrete maximal operators acting on $\text{BV}(\mathbb{Z})$ functions or $\ell ^{1}(\mathbb{Z})$ functions. We establish sharp bounds and continuity for the derivative of these discrete maximal functions, in both the centred and uncentred versions. As an immediate consequence, we obtain sharp bounds and continuity for the discrete fractional maximal operators from $\ell ^{1}(\mathbb{Z})$ to $\text{BV}(\mathbb{Z})$ .

Because of its practical and theoretical importance in rheology, numerous algorithms have been proposed and utilised to solve the convolution equation $g(x)=(\text{sech}\,\star h)(x)\;(x\in \mathbb{R})$ for $h$ , given $g$ . There are several approaches involving the use of series expansions of derivatives of $g$ , which are then truncated to a small number of terms for practical application. Such truncations can only be expected to be valid if the infinite series converge. In this note, we examine two specific truncations and provide a rigorous analysis to obtain sufficient conditions on $g$ (and equivalently on $h$ ) for the convergence of the series concerned.

In this paper we provide an elementary proof of James’ characterisation of weak compactness for Banach spaces whose dual ball is weak∗ sequentially compact.

Wu [‘An order characterization of commutativity for $C^{\ast }$ -algebras’, Proc. Amer. Math. Soc.129 (2001), 983–987] proved that if the exponential function on the set of all positive elements of a $C^{\ast }$ -algebra is monotone in the usual partial order, then the algebra in question is necessarily commutative. In this note, we present a local version of that result and obtain a characterisation of central elements in $C^{\ast }$ -algebras in terms of the order.

Let $\unicode[STIX]{x1D6E4}$ be a countable discrete group that acts on a unital $C^{\ast }$ -algebra $A$ through an action $\unicode[STIX]{x1D6FC}$ . If $A$ has a faithful $\unicode[STIX]{x1D6FC}$ -invariant tracial state $\unicode[STIX]{x1D70F}$ , then $\unicode[STIX]{x1D70F}^{\prime }=\unicode[STIX]{x1D70F}\circ {\mathcal{E}}$ is a faithful tracial state of $A\rtimes _{\unicode[STIX]{x1D6FC},r}\unicode[STIX]{x1D6E4}$ where ${\mathcal{E}}:A\rtimes _{\unicode[STIX]{x1D6FC},r}\unicode[STIX]{x1D6E4}\rightarrow A$ is the canonical faithful conditional expectation. We show that $(A\rtimes _{\unicode[STIX]{x1D6FC},r}\unicode[STIX]{x1D6E4},\unicode[STIX]{x1D70F}^{\prime })$ has the Haagerup property if and only if both $(A,\unicode[STIX]{x1D70F})$ and $\unicode[STIX]{x1D6E4}$ have the Haagerup property. As a consequence, suppose that $(A\rtimes _{\unicode[STIX]{x1D6FC},r}\unicode[STIX]{x1D6E4},\unicode[STIX]{x1D70F}^{\prime })$ has the Haagerup property where $\unicode[STIX]{x1D6E4}$ has property $T$ and $A$ has strong property $T$ . Then $\unicode[STIX]{x1D6E4}$ is finite and $A$ is residually finite-dimensional.

In this paper we first give a negative answer to a question of Amini-Harandi [‘Best proximity point theorems for cyclic strongly quasi-contraction mappings’, J. Global Optim.56 (2013), 1667–1674] on a best proximity point theorem for cyclic quasi-contraction maps. Then we prove some new results on best proximity point theorems that show that results of Amini-Harandi for cyclic strongly quasi-contractions are true under weaker assumptions.

Let $R=\mathbb{F}_{p}+u\mathbb{F}_{p}+u^{2}\mathbb{F}_{p}+\cdots +u^{d-1}\mathbb{F}_{p}$ , where $u^{d}=u$ and $p$ is a prime with $d-1$ dividing $p-1$ . A relation between the support weight distribution of a linear code $\mathscr{C}$ of type $p^{dk}$ over $R$ and the dual code $\mathscr{C}^{\bot }$ is established.