The tropical analogue of the lemma on the logarithmic derivative is generalised for noncontinuous tropical meromorphic functions, that is, piecewise linear functions that may have discontinuities. In addition, two Borel type results are generalised for piecewise continuous functions. With the generalisation of the tropical analogue of the lemma on the logarithmic derivative, several tropical analogues of Clunie and Mohon’ko type results are also automatically generalised for noncontinuous tropical meromorphic functions.

We solve the problem of finding the inverse connection formulae for the generalised Bessel polynomials and their reciprocals, the reverse generalised Bessel polynomials. The connection formulae express monomials in terms of the generalised Bessel polynomials. They enable formulae for the elements of change of basis matrices for both kinds of generalised Bessel polynomials to be derived and proved correct directly.

We show that there is a set $S \subseteq {\mathbb N}$ with lower density arbitrarily close to $1$ such that, for each sufficiently large real number $\alpha $, the inequality $|m\alpha -n| \geq 1$ holds for every pair $(m,n) \in S^2$. On the other hand, if $S \subseteq {\mathbb N}$ has density $1$, then, for each irrational $\alpha>0$ and any positive $\varepsilon $, there exist $m,n \in S$ for which $|m\alpha -n|<\varepsilon $.

We study the problem of determining the holomorphic self maps of the unit disc that induce a bounded composition operator on Dirichlet-type spaces. We find a class of symbols $\varphi $ that induce a bounded composition operator on the Dirichlet-type spaces, by applying results of the multidimensional theory of composition operators for the weighted Bergman spaces of the bi-disc.

Let $\alpha $ be a complex-valued $2$-cocycle of a finite group G with $\alpha $ chosen so that the $\alpha $-characters of G are class functions and analogues of the orthogonality relations for ordinary characters are valid. Then the real or rational elements of G that are also $\alpha $-regular are characterised by the values that the irreducible $\alpha $-characters of G take on those respective elements. These new results generalise two known facts concerning such elements and irreducible ordinary characters of $G;$ however, the initial choice of $\alpha $ from its cohomology class is not unique in general and it is shown the results can vary for a different choice.

We provide explicit bounds for the Riemann zeta-function on the line $\mathrm {Re}\,{s}=1$, assuming that the Riemann hypothesis holds up to height T. In particular, we improve some bounds in finite regions for the logarithmic derivative and the reciprocal of the Riemann zeta-function.

We prove analogues of Schur’s lemma for endomorphisms of extensions in Tannakian categories. More precisely, let $\mathbf {T}$ be a neutral Tannakian category over a field of characteristic zero. Let E be an extension of A by B in $\mathbf {T}$. We consider conditions under which every endomorphism of E that stabilises B induces a scalar map on $A\oplus B$. We give a result in this direction in the general setting of arbitrary $\mathbf {T}$ and E, and then a stronger result when $\mathbf {T}$ is filtered and the associated graded objects to A and B satisfy some conditions. We also discuss the sharpness of the results.

A subgroup A of a group G is said to be hereditarily G-permutable with a subgroup B of G, if $AB^x = B^xA$ for some element $x \in \langle A, B \rangle $. A subgroup A of a group G is said to be hereditarily G-permutable in G if A is hereditarily G-permutable with every subgroup of G. In this paper, we investigate the structure of a finite group G with all its Schmidt subgroups hereditarily G-permutable.

In this paper, we investigate finite solvable tidy groups. We prove that a solvable group with order divisible by at least two primes is tidy if all of its Hall subgroups that are divisible by only two primes are tidy.

We give an example of a pair of real Banach spaces such that they are neither linearly isomorphic nor isomorphic with respect to the structure of Birkhoff–James orthogonality, but have mutually homeomorphic geometric structure spaces.

This paper relies on nested postulates of separate, linear and arc-continuity of functions to define analogous properties for sets that are weaker than the requirement that the set be open or closed. This allows three novel characterisations of open or closed sets under convexity or separate convexity postulates: the first pertains to separately convex sets, the second to convex sets and the third to arbitrary subsets of a finite-dimensional Euclidean space. By relying on these constructions, we also obtain new results on the relationship between separate and joint continuity of separately quasiconcave, or separately quasiconvex functions. We present examples to show that the sufficient conditions we offer cannot be dispensed with.

Arithmetic quasidensities are a large family of real-valued set functions partially defined on the power set of $\mathbb {N}$, including the asymptotic density, the Banach density and the analytic density. Let $B \subseteq \mathbb {N}$ be a nonempty set covering $o(n!)$ residue classes modulo $n!$ as $n\to \infty $ (for example, the primes or the perfect powers). We show that, for each $\alpha \in [0,1]$, there is a set $A\subseteq \mathbb {N}$ such that, for every arithmetic quasidensity $\mu $, both A and the sumset $A+B$ are in the domain of $\mu $ and, in addition, $\mu (A + B) = \alpha $. The proof relies on the properties of a little known density first considered by Buck [‘The measure theoretic approach to density’, Amer. J. Math. 68 (1946), 560–580].

We study curve-shortening flow for twisted curves in $\mathbb {R}^3$ (that is, curves with nowhere vanishing curvature $\kappa $ and torsion $\tau $) and define a notion of torsion-curvature entropy. Using this functional, we show that either the curve develops an inflection point or the eventual singularity is highly irregular (and likely impossible). In particular, it must be a Type-II singularity which admits sequences along which ${\tau }/{\kappa ^2} \to \infty $. This contrasts strongly with Altschuler’s planarity theorem, which shows that ${\tau }/{\kappa } \to 0$ along any essential blow-up sequence.

A subgroup H of a group G is said to be pronormal in G if each of its conjugates $H^g$ in G is already conjugate to it in the subgroup $\langle H,H^g\rangle $. The aim of this paper is to classify those (locally) finite simple groups which have only nilpotent or pronormal subgroups.

As an extension of Sylvester’s matrix, a tridiagonal matrix is investigated by determining both left and right eigenvectors. Orthogonality relations between left and right eigenvectors are derived. Two determinants of the matrices constructed by the left and right eigenvectors are evaluated in closed form.

The Frobenius–Schur indicators of characters in a real $2$-block with dihedral defect groups have been determined by Murray [‘Real subpairs and Frobenius–Schur indicators of characters in 2-blocks’, J. Algebra 322 (2009), 489–513]. We show that two infinite families described in his work do not exist and we construct examples for the remaining families. We further present some partial results on Frobenius–Schur indicators of characters in other tame blocks.

Let $\ell $ and p be (not necessarily distinct) prime numbers and F be a global function field of characteristic $\ell $ with field of constants $\kappa $. Assume that there exists a prime $P_\infty $ of F which has degree $1$ and let $\mathcal {O}_F$ be the subring of F consisting of functions with no poles away from $P_\infty $. Let $f(X)$ be a polynomial in X with coefficients in $\kappa $. We study solutions to Diophantine equations of the form $Y^{n}=f(X)$ which lie in $\mathcal {O}_F$ and, in particular, show that if m and $f(X)$ satisfy additional conditions, then there are no nonconstant solutions. The results apply to the study of solutions to $Y^{n}=f(X)$ in certain rings of integers in $\mathbb {Z}_{p}$-extensions of F known as constant $\mathbb {Z}_p$-extensions. We prove similar results for solutions in the polynomial ring $K[T_1, \ldots , T_r]$, where K is any field of characteristic $\ell $, showing that the only solutions must lie in K. We apply our methods to study solutions of Diophantine equations of the form $Y^n=\sum _{i=1}^d (X+ir)^m$, where $m,n, d\geq 2$ are integers.

Let $\varphi $ be Euler’s function and fix an integer $k\ge 0$. We show that for every initial value $x_1\ge 1$, the sequence of positive integers $(x_n)_{n\ge 1}$ defined by $x_{n+1}=\varphi (x_n)+k$ for all $n\ge 1$ is eventually periodic. Similarly, for all initial values $x_1,x_2\ge 1$, the sequence of positive integers $(x_n)_{n\ge 1}$ defined by $x_{n+2}=\varphi (x_{n+1})+\varphi (x_n)+k$ for all $n\ge 1$ is eventually periodic, provided that k is even.

Formulas evaluating differences of integer partitions according to the parity of the parts are referred to as Legendre theorems. In this paper we give some formulas of Legendre type for overpartitions.