In this paper, we count a dual set of Stirling permutations by the number of alternating runs and study properties of the generating functions, including recurrence relations, grammatical interpretations and convolution formulas.

A Latin square is reduced if its first row and first column are in natural order. For Latin squares of a particular order $n$, there are four possible different parities. We confirm a conjecture of Stones and Wanless by showing asymptotic equality between the numbers of reduced Latin squares of each possible parity as the order $n\rightarrow \infty$.

We present a sufficient condition for a pair of finite integer sequences to be degree sequences of a bipartite graph, based only on the lengths of the sequences and their largest and smallest elements.

A simple graph $G=(V,E)$ admits an $H$-covering if every edge in $E$ belongs to at least one subgraph of $G$ isomorphic to a given graph $H$. Then the graph $G$ is $(a,d)$-$H$-antimagic if there exists a bijection $f:V\cup E\rightarrow \{1,2,\ldots ,|V|+|E|\}$ such that, for all subgraphs $H^{\prime }$ of $G$ isomorphic to $H$, the $H^{\prime }$-weights, $wt_{f}(H^{\prime })=\sum _{v\in V(H^{\prime })}f(v)+\sum _{e\in E(H^{\prime })}f(e)$, form an arithmetic progression with the initial term $a$ and the common difference $d$. When $f(V)=\{1,2,\ldots ,|V|\}$, then $G$ is said to be super $(a,d)$-$H$-antimagic. In this paper, we study super $(a,d)$-$H$-antimagic labellings of a disjoint union of graphs for $d=|E(H)|-|V(H)|$.

We express the number of points on the Dwork hypersurface $X_{\unicode[STIX]{x1D706}}^{d}:x_{1}^{d}+x_{2}^{d}+\cdots +x_{d}^{d}=d\unicode[STIX]{x1D706}x_{1}x_{2}\cdots x_{d}$ over a finite field of order $q\not \equiv 1\,(\text{mod}\,d)$ in terms of McCarthy’s $p$-adic hypergeometric function for any odd prime $d$.

In this short note, we give a proof, conditional on the generalised Riemann hypothesis, that there exist numbers $x$ which are normal with respect to the continued fraction expansion but not to any base-$b$ expansion. This partially answers a question of Bugeaud.

We improve recent results of Bourgain and Shparlinski to show that, for almost all primes $p$, there is a multiple $mp$ that can be written in binary as

$$\begin{eqnarray}mp=1+2^{m_{1}}+\cdots +2^{m_{k}},\quad 1\leq m_{1}<\cdots <m_{k},\end{eqnarray}$$

$k=6$

$p$

$A=\langle g\rangle \subset \mathbb{F}_{p}^{\ast }$

$g\in \{2,3,5\}$

$|A|\gg p/(\log p)^{3}$

$A\cdot A+A\cdot A$

$\mathbb{F}_{p}$

We revisit Berkovich and Garvan’s two bijections: the first gives symmetry of cranks and the second relates partitions with crank $\leq k$ to those with $k$ in the rank-set of partitions. Using these, we give a combinatorial proof for the relationship between the first positive crank moment and the sum of sizes of Durfee squares. We also study refinements of the first and second positive crank moments.

We prove that a nonreal algebraic number $\unicode[STIX]{x1D703}$ with modulus greater than $1$ is a complex Pisot number if and only if there is a nonzero complex number $\unicode[STIX]{x1D706}$ such that the sequence of fractional parts $(\{\Re (\unicode[STIX]{x1D706}\unicode[STIX]{x1D703}^{n})\})_{n\in \mathbb{N}}$ has a finite number of limit points. Also, we characterise those complex Pisot numbers $\unicode[STIX]{x1D703}$ for which there is a convergent sequence of the form $(\{\Re (\unicode[STIX]{x1D706}\unicode[STIX]{x1D703}^{n})\})_{n\in \mathbb{N}}$ for some $\unicode[STIX]{x1D706}\in \mathbb{C}^{\ast }$. These results are generalisations of the corresponding real ones, due to Pisot, Vijayaraghavan and Dubickas.

Let $G$ be a finite group and $\mathsf{cd}(G)$ denote the set of complex irreducible character degrees of $G$. We prove that if $G$ is a finite group and $H$ is an almost simple group whose socle is a sporadic simple group $H_{0}$ and such that $\mathsf{cd}(G)=\mathsf{cd}(H)$, then $G^{\prime }\cong H_{0}$ and there exists an abelian subgroup $A$ of $G$ such that $G/A$ is isomorphic to $H$. In view of Huppert’s conjecture, we also provide some examples to show that $G$ is not necessarily a direct product of $A$ and $H$, so that we cannot extend the conjecture to almost simple groups.

Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. We determine the structure of $N$ when the diameter of the graph associated to the $G$-conjugacy classes contained in $N$ is as large as possible, that is, equal to three.

The last term of the lower central series of a finite group $G$ is called the nilpotent residual. It is usually denoted by $\unicode[STIX]{x1D6FE}_{\infty }(G)$. The lower Fitting series of $G$ is defined by $D_{0}(G)=G$ and $D_{i+1}(G)=\unicode[STIX]{x1D6FE}_{\infty }(D_{i}(G))$ for $i=0,1,2,\ldots \,$. These subgroups are generated by so-called coprime commutators $\unicode[STIX]{x1D6FE}_{k}^{\ast }$ and $\unicode[STIX]{x1D6FF}_{k}^{\ast }$ in elements of $G$. More precisely, the set of coprime commutators $\unicode[STIX]{x1D6FE}_{k}^{\ast }$ generates $\unicode[STIX]{x1D6FE}_{\infty }(G)$ whenever $k\geq 2$ while the set $\unicode[STIX]{x1D6FF}_{k}^{\ast }$ generates $D_{k}(G)$ for $k\geq 0$. The main result of this article is the following theorem: let $m$ be a positive integer and $G$ a finite group. Let $X\subset G$ be either the set of all $\unicode[STIX]{x1D6FE}_{k}^{\ast }$-commutators for some fixed $k\geq 2$ or the set of all $\unicode[STIX]{x1D6FF}_{k}^{\ast }$-commutators for some fixed $k\geq 1$. Suppose that the size of $a^{X}$ is at most $m$ for any $a\in G$. Then the order of $\langle X\rangle$ is $(k,m)$-bounded.

Let $S$ be a semigroup possibly with no identity and $f:S\rightarrow \mathbb{C}$. We consider the general superstability of the exponential functional equation with a perturbation $\unicode[STIX]{x1D713}$ of mixed variables

$$\begin{eqnarray}\displaystyle |f(x+y)-f(x)f(y)|\leq \unicode[STIX]{x1D713}(x,y)\quad \text{for all }x,y\in S. & & \displaystyle \nonumber\end{eqnarray}$$

$S$

$2$

$\unicode[STIX]{x1D713}$

$$\begin{eqnarray}\displaystyle \biggl|f\biggl(\frac{x+y}{2}\biggr)^{2}-f(x)f(y)\biggr|\leq \unicode[STIX]{x1D713}(x,y)\quad \text{for all }x,y\in S. & & \displaystyle \nonumber\end{eqnarray}$$

Several papers have, as their raison d’être, the exploration of the generalised Lau product associated to a homomorphism $T:B\rightarrow A$ of Banach algebras. In this short note, we demonstrate that the generalised Lau product is isomorphic as a Banach algebra to the usual direct product $A\oplus B$. We also correct some misleading claims made about the relationship between this generalised Lau product and an older construction of Monfared [‘On certain products of Banach algebras with applications to harmonic analysis’, Studia Math. 178(3) (2007), 277–294].

In this short note we present a common characterisation of the logarithmic function and the subspace of all trace zero elements in finite von Neumann factors.

We focus on a question raised by Daws [‘Arens regularity of the algebra of operators on a Banach space’, Bull. Lond. Math. Soc.36 (2004), 493–503] concerning the Arens regularity of $B(X)$, the algebra of operators on a Banach space $X$. Among other things, we show that $B(X)$ is Arens regular if and only if $X$ is ultrareflexive.

We apply the lifting theorem of Searle and the second author to put metrics of almost nonnegative curvature on the fake $\mathbb{R}P^{6}$s of Hirsch and Milnor and on the analogous fake $\mathbb{R}P^{14}$s.

In this paper, we characterise the structure of the eigencone for the Finsler Laplacian corresponding to the first Dirichlet eigenvalue on a compact Finsler manifold with a smooth boundary.

The Gehring–Martin–Tan inequality for two-generator subgroups of $\text{PSL}(2,\mathbb{C})$ is one of the best known discreteness conditions. A Kleinian group $G$ is called a Gehring–Martin–Tan group if the equality holds for the group $G$. We give a method for constructing Gehring–Martin–Tan groups with a generator of order four and present some examples. These groups arise as groups of finite-volume hyperbolic 3-orbifolds.