Necklaces are the equivalence classes of words under the action of the cyclic group. Let a transition in a word be any change between two adjacent letters modulo the word’s length. We present a closed-form solution for the enumeration of necklaces in n beads, k colours and t transitions. We show that our result provides a more general solution to the problem of counting alternating (proper) colourings of the vertices of a regular n-gon.

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 find all integer solutions to the equation $x^2+5^a\cdot 13^b\cdot 17^c=y^n$ with $a,\,b,\,c\geq 0$ , $n\geq 3$ , $x,\,y>0$ and $\gcd (x,\,y)=1$ . Our proof uses a deep result about primitive divisors of Lucas sequences in combination with elementary number theory and computer search.

We prove a necessary and sufficient condition for isogenous elliptic curves based on the algebraic dependence of p-adic elliptic functions. As a consequence, we give a short proof of the p-adic analogue of Schneider’s theorem on the linear independence of p-adic elliptic logarithms of algebraic points on two nonisogenous elliptic curves defined over the field of algebraic numbers.

Necessary and sufficient conditions for the equality of the determinant and permanent for all powers of a given matrix are provided. The characterisation is based on a condition on merely one power.

By making use of the Cauchy double alternant and the Laplace expansion formula, we establish two closed formulae for the determinants of factorial fractions that are then utilised to evaluate several determinants of binomial coefficients and Catalan numbers, including those obtained recently by Chammam [‘Generalized harmonic numbers, Jacobi numbers and a Hankel determinant evaluation’, Integral Transforms Spec. Funct. 30(7) (2019), 581–593].

We investigate a class of generalised stochastic complex matrices constructed from the class of all doubly stochastic matrices and a special class of circulant matrices. We determine the exact values of the structured singular values of all matrices in the class in terms of the constant row (column) sum.

For a character $\chi $ of a finite group G, the number $\chi ^c(1)={[G:{\textrm {ker}}\chi ]}/{\chi (1)}$ is called the co-degree of $\chi $ . A finite group G is an ${\textrm {NDAC}} $ -group (no divisibility among co-degrees) when $\chi ^c(1) \nmid \phi ^c(1)$ for all irreducible characters $\chi $ and $\phi $ of G with $1< \chi ^c(1) < \phi ^c(1)$ . We study finite groups admitting an irreducible character whose co-degree is a given prime p and finite nonsolvable ${\textrm {NDAC}} $ -groups. Then we show that the finite simple groups $^2B_2(2^{2f+1})$ , where $f\geq 1$ , $\mbox {PSL}_3(4)$ , ${\textrm {Alt}}_7$ and $J_1$ are determined uniquely by the set of their irreducible character co-degrees.

A subgroup H of a group G is pronormal in G if each of its conjugates $H^g$ in G is conjugate to it in the subgroup $\langle H,H^g\rangle $ ; a group is prohamiltonian if all of its nonabelian subgroups are pronormal. The aim of the paper is to show that a locally soluble group of (regular) cardinality in which all proper uncountable subgroups are prohamiltonian is prohamiltonian. In order to obtain this result, it is proved that the class of prohamiltonian groups is detectable from the behaviour of countable subgroups. Examples are exhibited to show that there are uncountable prohamiltonian groups that do not behave very well. Finally, it is shown that prohamiltonicity can sometimes be detected through the analysis of the finite homomorphic images of a group.

For any (Hausdorff) compact group G, denote by $\mathrm{cp}(G)$ the probability that a randomly chosen pair of elements of G commute. We prove that there exists a finite group H such that $\mathrm{cp}(G)= {\mathrm{cp}(H)}/{|G:F|^2}$ , where F is the FC-centre of G and H is isoclinic to F with $\mathrm{cp}(F)=\mathrm{cp}(H)$ whenever $\mathrm{cp}(G)>0$ . In addition, we prove that a compact group G with $\mathrm{cp}(G)>\tfrac {3}{40}$ is either solvable or isomorphic to $A_5 \times Z(G)$ , where $A_5$ denotes the alternating group of degree five and the centre $Z(G)$ of G contains the identity component of G.

Morgan and Parker proved that if G is a group with ${\textbf{Z}(G)} = 1$ , then the connected components of the commuting graph of G have diameter at most $10$ . Parker proved that if, in addition, G is solvable, then the commuting graph of G is disconnected if and only if G is a Frobenius group or a $2$ -Frobenius group, and if the commuting graph of G is connected, then its diameter is at most $8$ . We prove that the hypothesis $Z (G) = 1$ in these results can be replaced with $G' \cap {\textbf{Z}(G)} = 1$ . We also prove that if G is solvable and $G/{\textbf{Z}(G)}$ is either a Frobenius group or a $2$ -Frobenius group, then the commuting graph of G is disconnected.

Given a finite group G with a normal subgroup N, the simple graph $\Gamma _{\textit {G}}( \textit {N} )$ is a graph whose vertices are of the form $|x^G|$ , where $x\in {N\setminus {Z(G)}}$ and $x^G$ is the G-conjugacy class of N containing the element x. Two vertices $|x^G|$ and $|y^G|$ are adjacent if they are not coprime. We prove that, if $\Gamma _G(N)$ is a connected incomplete regular graph, then $N= P \times {A}$ where P is a p-group, for some prime p, $A\leq {Z(G)}$ and $\textbf {Z}(N)\not = N\cap \textbf {Z}(G)$ .

Let V be an infinite-dimensional vector space over a field F and let $I(V)$ be the inverse semigroup of all injective partial linear transformations on V. Given $\alpha \in I(V)$ , we denote the domain and the range of $\alpha $ by ${\mathop {\textrm {dom}}}\,\alpha $ and ${\mathop {\textrm {im}}}\,\alpha $ , and we call the cardinals $g(\alpha )={\mathop {\textrm {codim}}}\,{\mathop {\textrm {dom}}}\,\alpha $ and $d(\alpha )={\mathop {\textrm {codim}}}\,{\mathop {\textrm {im}}}\,\alpha $ the ‘gap’ and the ‘defect’ of $\alpha $ . We study the semigroup $A(V)$ of all injective partial linear transformations with equal gap and defect and characterise Green’s relations and ideals in $A(V)$ . This is analogous to work by Sanwong and Sullivan [‘Injective transformations with equal gap and defect’, Bull. Aust. Math. Soc. 79 (2009), 327–336] on a similarly defined semigroup for the set case, but we show that these semigroups are never isomorphic.

Let f be analytic in the unit disk $\mathbb {D}=\{z\in \mathbb {C}:|z|<1 \}$ and let ${\mathcal S}$ be the subclass of normalised univalent functions with $f(0)=0$ and $f'(0)=1$ , given by $f(z)=z+\sum _{n=2}^{\infty }a_n z^n$ . Let F be the inverse function of f, given by $F(\omega )=\omega +\sum _{n=2}^{\infty }A_n \omega ^n$ for $|\omega |\le r_0(f)$ . Denote by $ \mathcal {S}_p^{* }(\alpha )$ the subset of $ \mathcal {S}$ consisting of the spirallike functions of order $\alpha $ in $\mathbb {D}$ , that is, functions satisfying

$$\begin{align*}{\mathrm{Re}} \ \bigg\{e^{-i\gamma}\dfrac{zf'(z)}{f(z)}\bigg\}>\alpha\cos \gamma, \end{align*}$$

for $z\in \mathbb {D}$ , $0\le \alpha <1$ and $\gamma \in (-\pi /2,\pi /2)$ . We give sharp upper and lower bounds for both $ |a_3|-|a_2| $ and $ |A_3|-|A_2| $ when $f\in \mathcal {S}_p^{* }(\alpha )$ , thus solving an open problem and presenting some new inequalities for coefficient differences.

Let X be a real Banach space. The rectangular constant $\mu (X)$ and some generalisations of it, $\mu _p(X)$ for $p \geq 1$ , were introduced by Gastinel and Joly around half a century ago. In this paper we make precise some characterisations of inner product spaces by using $\mu _p(X)$ , correcting some statements appearing in the literature, and extend to $\mu _p(X)$ some characterisations of uniformly nonsquare spaces, known only for $\mu (X)$ . We also give a characterisation of two-dimensional spaces with hexagonal norms. Finally, we indicate some new upper estimates concerning $\mu (l_p)$ and $\mu _p(l_p)$ .

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.

Let A and $\tilde A$ be unbounded linear operators on a Hilbert space. We consider the following problem. Let the spectrum of A lie in some horizontal strip. In which strip does the spectrum of $\tilde A$ lie, if A and $\tilde A$ are sufficiently ‘close’? We derive a sharp bound for the strip containing the spectrum of $\tilde A$ , assuming that $\tilde A-A$ is a bounded operator and A has a bounded Hermitian component. We also discuss applications of our results to regular matrix differential operators.

We prove that the restriction of a given orthogonal-complete metric space to the closure of the orbit induced by the origin point with respect to an orthogonal-preserving and orthogonal-continuous map is a complete metric space. Then we show that many existence results on fixed points in orthogonal-complete metric spaces can be proved by using the corresponding existence results in complete metric spaces.

Let G be a locally compact group and let ${\mathcal {SUB}(G)}$ be the hyperspace of closed subgroups of G endowed with the Chabauty topology. The main purpose of this paper is to characterise the connectedness of the Chabauty space ${\mathcal {SUB}(G)}$ . More precisely, we show that if G is a connected pronilpotent group, then ${\mathcal {SUB}(G)}$ is connected if and only if G contains a closed subgroup topologically isomorphic to ${{\mathbb R}}$ .