We consider the uniqueness of an entire function and a linear differential polynomial generated by it. One of our results improves a result of Li and Yang [‘Value sharing of an entire function and its derivatives’, J. Math. Soc. Japan51(4) (1999), 781–799].

For two given graphs $G_{1}$ and $G_{2}$, the Ramsey number $R(G_{1},G_{2})$ is the smallest integer $N$ such that, for any graph $G$ of order $N$, either $G$ contains $G_{1}$ as a subgraph or the complement of $G$ contains $G_{2}$ as a subgraph. A fan $F_{l}$ is $l$ triangles sharing exactly one vertex. In this note, it is shown that $R(F_{n},F_{m})=4n+1$ for $n\geq \max \{m^{2}-m/2,11m/2-4\}$.

We give a sufficient condition under which a semigroup is nonfinitely based. As an application, we show that a certain variety is nonfinitely based, and we indicate the additional analysis (to be presented in a forthcoming paper), which shows that this example is a new limit variety of aperiodic monoids.

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.

In this paper, we introduce the concept of A-proper topological degree in Menger PN-space and study some of its properties. Utilising its properties, we obtain a new fixed point theorem.

The purpose of this paper is to study the existence of periodic solutions and the topological structure of the solution set of first-order differential equations involving the distributional Henstock–Kurzweil integral. The distributional Henstock–Kurzweil integral is a general integral, which includes the Lebesgue and Henstock–Kurzweil integrals. The main results extend some previously known results in the literature.

In 1986, Lampe presented a counterexample to the conjecture that every algebraic lattice with a compact greatest element is isomorphic to the lattice of extensions of an equational theory. In this paper we investigate equational theories of semi-lattices with operators. We construct a class of lattices containing all infinitely distributive algebraic lattices with a compact greatest element and closed under the operation of taking the parallel join, such that every element of the class is isomorphic to the lattice of equational theories, extending the theory of a semilattice with operators.

The first author [J. Brough, ‘On vanishing criteria that control finite group structure’, J. Algebra458 (2016), 207–215] has shown that for certain arithmetical results on conjugacy class sizes it is enough to consider only the vanishing conjugacy class sizes. In this paper we further weaken the conditions to consider only vanishing elements of prime power order.

The Jacobson radical J(K[S]) of the semigroup ring K[S] of a cancellative semigroup S over a field K is studied. We show that, if J(K[S]) ≠ 0, then either S is a reversive semigroup or K[S] has many nilpotents and J(K[P]) ≠ 0 for a reversive subsemigroup P of S. This is used to prove that J(K[S]) = 0 for every unique product. semigroup S.

Let T: B → B be a bounded linear operator on the complex Banach space B and let f(z) be analytic on a domain D containing the spectrum Sp(T) of T. Then f(T) is defined by

where C is a contour surrounding SP(T) and contained in D.

We give the crossing number of the join product $W_{4}+D_{n}$, where $W_{4}$ is the wheel on five vertices and $D_{n}$ consists of $n$ isolated vertices. The proof is based on calculating the minimum number of crossings between two different subgraphs from the set of subgraphs which do not cross the edges of the graph $W_{4}$ and from the set of subgraphs which cross the edges of $W_{4}$ exactly once.

Wu and Shi [‘A note on k-Galois LCD codes over the ring $\mathbb {F}_q + u\mathbb {F}_q$ ’, Bull. Aust. Math. Soc. 104(1) (2021), 154–161] studied $ k $ -Galois LCD codes over the finite chain ring $\mathcal {R}=\mathbb {F}_q+u\mathbb {F}_q$ , where $u^2=0$ and $ q=p^e$ for some prime p and positive integer e. We extend the results to the finite nonchain ring $ \mathcal {R} =\mathbb {F}_q+u\mathbb {F}_q+v\mathbb {F}_q+uv\mathbb {F}_q$ , where $u^2=u,v^2=v $ and $ uv=vu $ . We define a correspondence between the $ l $ -Galois dual of linear codes over $ \mathcal {R} $ and the $ l $ -Galois dual of their component codes over $ \mathbb {F}_q $ . Further, we construct Euclidean LCD and $ l $ -Galois LCD codes from linear codes over $ \mathcal {R} $ . We prove that any linear code over $ \mathcal {R} $ is equivalent to a Euclidean code over $\mathbb {F}_q$ with $ q>3 $ and an $ l $ -Galois LCD code over $ \mathcal {R}$ with $0<l<e$ and $p^{e-l}+1\mid p^e-1$ . Finally, we investigate MDS codes over $ \mathcal {R}$ .

Let a finite group A act coprimely on a finite group G and χ ∈ Irr A(G). Isaacs, Lewis and Navarro proved that if G is nilpotent then the degrees of any two A-primitive characters of A-invariant subgroups of G inducing χ coincide. In this note we aim at extending this result by weakening the hypothesis on G.

The Cauchy problem x′ = f(t, x), x(0) = x0, is considered in a non-reflexive Banach space E, where f is weakly continuous. A local existence theorem is proved using the measure of weak noncompactness.

Let (T, T, μ) be a σ-finite measure space and X a Suslin space. Let A be a class of normal integrands on T × X. We discuss the existence of an essential supremum of A, namely, a normal integrand l with

where A0 is a countable subclass of A, and, for each α ∈ A,

In this way we obtain an extension of the classical essential supremum concept. The applications include a result on measurable selectors of nonmeasurable multifunctions.

Let A be a Banach algebra and let X be a Banach A-bimodule. We consider the Banach algebra ${A\oplus _1 X}$ , where A is a commutative Banach algebra. We investigate the Bochner–Schoenberg–Eberlein (BSE) property and the BSE module property on $A\oplus _1 X$ . We show that the module extension Banach algebra $A\oplus _1 X$ is a BSE Banach algebra if and only if A is a BSE Banach algebra and $X=\{0\}$ . Furthermore, we consider $A\oplus _1 X$ as a Banach $A\oplus _1 X$ -module and characterise the BSE module property on $A\oplus _1 X$ . We show that $A\oplus _1 X$ is a BSE Banach $A\oplus _1 X$ -module if and only if A and X are BSE Banach A-modules.

By using zeros of elliptic integrals we establish an upper bound for the number of limit cycles that emerge from the period annulus of the Hamiltonian XH in the system Xε = XH + ε(P, Q), where H = y2 + x4 and P, Q are polynomials in x, y, as a function of the degrees of P and Q. In particular, if (P, Q) = , with N = 2k + 1 or 2k + 2, this upper bound is k − 1.

Milnor's classic result that the fundamental group of a compact Riemannian manifold of negative sectional curvature has exponential growth is generalised to the case of negative Ricci curvature and non-positive sectional curvature.

A set mapping on a set S is a function f mapping S into the powerset of S such that x ∉ f(x) for each x in S. The set map f has order θ if θ is the least cardinal such that |f(x)| < θ for each x in S. A subset H of S is free for f if x ∉ f(y) for all x, y in H. In this paper we use classical results about set mappings of large order to investigate conditions which ensure a large free set for set mappings of unrestricted order.

Some inequalities involving the binomial coefficients are obtained. They are used to determine the domain of convergence of the Bleimann, Butzer and Hahn approximation process for exponential type functions. An answer to Hermann's conjecture related to the Bleimann, Butzer and Hahn operators for monotone functions is given.