A positive-definite diagonal quadratic form $a_{1}x_{1}^{2}+\cdots +a_{n}x_{n}^{2}\;(a_{1},\ldots ,a_{n}\in \mathbb{N})$ is said to be prime-universal if it is not universal and for every prime $p$ there are integers $x_{1},\ldots ,x_{n}$ such that $a_{1}x_{1}^{2}+\cdots +a_{n}x_{n}^{2}=p$. We determine all possible prime-universal ternary quadratic forms $ax^{2}+by^{2}+cz^{2}$ and all possible prime-universal quaternary quadratic forms $ax^{2}+by^{2}+cz^{2}+dw^{2}$. The prime-universal ternary forms are completely determined. The prime-universal quaternary forms are determined subject to the validity of two conjectures. We make no use of a result of Bhargava concerning quadratic forms representing primes which is stated but not proved in the literature.

The congruent number elliptic curves are defined by $E_{d}:y^{2}=x^{3}-d^{2}x$, where $d\in \mathbb{N}$. We give a simple proof of a formula for $L(\operatorname{Sym}^{2}(E_{d}),3)$ in terms of the determinant of the elliptic trilogarithm evaluated at some degree zero divisors supported on the torsion points on $E_{d}(\overline{\mathbb{Q}})$.

We prove a new linear relation for multiple zeta values. This is a natural generalisation of the restricted sum formula proved by Eie, Liaw and Ong. We also present an analogous result for finite multiple zeta values.

For positive integers $t_{1},\ldots ,t_{k}$, let $\tilde{p}(n,t_{1},t_{2},\ldots ,t_{k})$ (respectively $p(n,t_{1},t_{2},\ldots ,t_{k})$) be the number of partitions of $n$ such that, if $m$ is the smallest part, then each of $m+t_{1},m+t_{1}+t_{2},\ldots ,m+t_{1}+t_{2}+\cdots +t_{k-1}$ appears as a part and the largest part is at most (respectively equal to) $m+t_{1}+t_{2}+\cdots +t_{k}$. Andrews et al. [‘Partitions with fixed differences between largest and smallest parts’, Proc. Amer. Math. Soc.143 (2015), 4283–4289] found an explicit formula for the generating function of $p(n,t_{1},t_{2},\ldots ,t_{k})$. We establish a $q$-series identity from which the formulae for the generating functions of $\tilde{p}(n,t_{1},t_{2},\ldots ,t_{k})$ and $p(n,t_{1},t_{2},\ldots ,t_{k})$ can be obtained.

We give an algorithmic generalisation of Dickson’s method of classifying permutation polynomials (PPs) of a given degree $d$ over finite fields. Dickson’s idea is to formulate from Hermite’s criterion several polynomial equations satisfied by the coefficients of an arbitrary PP of degree $d$. Previous classifications of PPs of degree at most 6 were essentially deduced from manual analysis of these polynomial equations, but this approach is no longer viable for $d>6$. Our idea is to calculate some radicals of ideals generated by the polynomials, implemented by a computer algebra system. Our algorithms running in SageMath 8.6 on a personal computer work very fast to determine all PPs of degree 8 over an arbitrary finite field of odd order $q>8$. Such PPs exist if and only if $q\in \{11,13,19,23,27,29,31\}$ and are explicitly listed in normalised form.

A polynomial $f$ over a finite field $\mathbb{F}_{q}$ can be classified as a permutation polynomial by the Hermite–Dickson criterion, which consists of conditions on the powers $f^{e}$ for each $e$ from $1$ to $q-2$, as well as the existence of a unique solution to $f(x)=0$ in $\mathbb{F}_{q}$. Carlitz and Lutz gave a variant of the criterion. In this paper, we provide an alternate proof to the theorem of Carlitz and Lutz.

We prove that a reduced and irreducible algebraic surface in $\mathbb{CP}^{3}$ containing infinitely many twistor lines cannot have odd degree. Then, exploiting the theory of quaternionic slice regularity and the normalisation map of a surface, we give constructive existence results for even degrees.

The factorial conjecture was proposed by van den Essen et al. [‘On the image conjecture’, J. Algebra 340(1) (2011), 211–224] to study the image conjecture, which arose from the Jacobian conjecture. We show that the factorial conjecture holds for all homogeneous polynomials in two variables. We also give a variation of the result and use it to show that the image of any linear locally nilpotent derivation of $\mathbb{C}[x,y,z]$ is a Mathieu–Zhao subspace.

This note provides an affirmative answer to Problem 2.6 of Praeger and Schneider [‘Group factorisations, uniform automorphisms, and permutation groups of simple diagonal type’, Israel J. Math. 228(2) (2018), 1001–1023]. We will build groups $G$ (abelian, nonabelian and simple) for which there are two automorphisms $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}$ of $G$ such that the map

$$\begin{eqnarray}T=T_{\unicode[STIX]{x1D6FC}}\times T_{\unicode[STIX]{x1D6FD}}:G\longrightarrow G\times G,\quad g\mapsto (g^{-1}g^{\unicode[STIX]{x1D6FC}},g^{-1}g^{\,\unicode[STIX]{x1D6FD}})\end{eqnarray}$$

Let $T_{n}(\mathbb{F})$ be the semigroup of all upper triangular $n\times n$ matrices over a field $\mathbb{F}$. Let $UT_{n}(\mathbb{F})$ and $UT_{n}^{\pm 1}(\mathbb{F})$ be subsemigroups of $T_{n}(\mathbb{F})$, respectively, having $0$s and/or $1$s on the main diagonal and $0$s and/or $\pm 1$s on the main diagonal. We give some sufficient conditions under which an involution semigroup is nonfinitely based. As an application, we show that $UT_{2}(\mathbb{F}),UT_{2}^{\pm 1}(\mathbb{F})$ and $T_{2}(\mathbb{F})$ as involution semigroups under the skew transposition are nonfinitely based for any field $\mathbb{F}$.

We show that Matui’s HK conjecture holds for groupoids of unstable equivalence relations and their corresponding $C^{\ast }$-algebras on one-dimensional solenoids.

The aim of this paper is to find a broad family of means defined on a subinterval of $I\subset [0,+\infty )$ such that

$$\begin{eqnarray}\mathop{\sum }_{n=1}^{\infty }\mathscr{M}(a_{1},\ldots ,a_{n})<+\infty \quad \text{for all }a\in \ell _{1}(I).\end{eqnarray}$$

$(a_{1},\,a_{2},a_{3}\,,\ldots )\mapsto (a_{1},\,\mathscr{M}(a_{1},a_{2}),\,\mathscr{M}(a_{1},a_{2},a_{3}),\ldots )$

$\ell _{1}(I)$

$f^{-1}(\sum f(a_{i})/n)$

$f:I\rightarrow \mathbb{R}$

$n\in \mathbb{N}$

$a\in I^{n}$

Let ${\mathcal{P}}_{{\mathcal{H}}}^{0}(M)$ denote the class of normalised harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D}$ satisfying $\text{Re}\,(zh^{\prime \prime }(z))>-M+|zg^{\prime \prime }(z)|$, where $h^{\prime }(0)-1=0=g^{\prime }(0)$ and $M>0$. Let ${\mathcal{B}}_{{\mathcal{H}}}^{0}(M)$ denote the class of sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D}$ satisfying $|zh^{\prime \prime }(z)|\leq M-|zg^{\prime \prime }(z)|$, where $M>0$. We discuss the coefficient bound problem, the growth theorem for functions in the class ${\mathcal{P}}_{{\mathcal{H}}}^{0}(M)$ and a two-point distortion property for functions in the class ${\mathcal{B}}_{{\mathcal{H}}}^{0}(M)$.

We prove the global logarithmic stability of the Cauchy problem for $H^{2}$-solutions of an anisotropic elliptic equation in a Lipschitz domain. The result is based on existing techniques used to establish stability estimates for the Cauchy problem combined with related tools used to study an inverse medium problem.

The longest gap $L(t)$ up to time $t$ in a homogeneous Poisson process is the maximal time subinterval between epochs of arrival times up to time $t$; it has applications in the theory of reliability. We study the Laplace transform asymptotics for $L(t)$ as $t\rightarrow \infty$ and derive two natural and different large-deviation principles for $L(t)$ with two distinct rate functions and speeds.

We generalise Sidel’nikov’s theorem from binary codes to $q$-ary codes for $q>2$. Denoting by $A(z)$ the cumulative distribution function attached to the weight distribution of the code and by $\unicode[STIX]{x1D6F7}(z)$ the standard normal distribution function, we show that $|A(z)-\unicode[STIX]{x1D6F7}(z)|$ is bounded above by a term which tends to $0$ when the code length tends to infinity.