Let p be a prime with p\equiv 1\pmod {4}$p\equiv 1\pmod {4}$ . Gauss first proved that 2$2$ is a quartic residue modulo p if and only if p=x^2+64y^2$p=x^2+64y^2$ for some x,y\in \Bbb Z$x,y\in \Bbb Z$ and various expressions for the quartic residue symbol (\frac {2}{p})_4$(\frac {2}{p})_4$ are known. We give a new characterisation via a permutation, the sign of which is determined by (\frac {2}{p})_4$(\frac {2}{p})_4$ . The permutation is induced by the rule x \mapsto y-x$x \mapsto y-x$ on the (p-1)/4$(p-1)/4$ solutions (x,y)$(x,y)$ to x^2+y^2\equiv 0 \pmod {p}$x^2+y^2\equiv 0 \pmod {p}$ satisfying 1\leq x < y \leq (p-1)/2$1\leq x < y \leq (p-1)/2$ .

When a page, represented by the interval [0,1],$[0,1],$ is folded right over left n$n$ times, the right-hand fold contains a sequence of points. We specify these points and the order in which they appear in each fold. We also determine exactly where in the folded structure any point in [0,1]$[0,1]$ appears and, given any point on the bottom line of the structure, which point lies at each level above it.

We generalise two quartic surfaces studied by Swinnerton-Dyer to give two infinite families of diagonal quartic surfaces which violate the Hasse principle. Standard calculations of Brauer–Manin obstructions are exhibited.

We develop some asymptotics for a kernel function introduced by Kohnen and use them to estimate the number of normalised Hecke eigenforms in S_k(\Gamma _0(1))$S_k(\Gamma _0(1))$ whose L-values are simultaneously nonvanishing at a given pair of points each of which lies inside the critical strip.

The aim of this article is to establish the behaviour of partial Euler products for Dirichlet L-functions under the generalised Riemann hypothesis (GRH) via Ramanujan’s work. To understand the behaviour of Euler products on the critical line, we invoke the deep Riemann hypothesis (DRH). This work clarifies the relation between GRH and DRH.

Let p_{\{3, 3\}}(n)$p_{\{3, 3\}}(n)$ denote the number of 3$3$ -regular partitions in three colours. Da Silva and Sellers [‘Arithmetic properties of 3-regular partitions in three colours’, Bull. Aust. Math. Soc. 104(3) (2021), 415–423] conjectured four Ramanujan-like congruences modulo 5$5$ satisfied by p_{\{3, 3\}}(n)$p_{\{3, 3\}}(n)$ . We confirm these conjectural congruences using the theory of modular forms.

Girstmair [‘On an irreducibility criterion of M. Ram Murty’, Amer. Math. Monthly 112(3) (2005), 269–270] gave a generalisation of Ram Murty’s irreducibility criterion. We further generalise these criteria.

We show that the elements of the dual of the Euclidean distance matrix cone can be described via an inequality on a certain weighted sum of its eigenvalues.

Every transitive family of subspaces of a vector space of finite dimension n\ge 2$n\ge 2$ over a field \mathbb {F}$\mathbb {F}$ contains a subfamily which is transitive but has no proper transitive subfamily. Such a subfamily is called minimally transitive. Each has at most n^2-n+1$n^2-n+1$ elements. On {{\mathbb {C}}}^n, n\ge 3${{\mathbb {C}}}^n, n\ge 3$ , a minimally transitive family of subspaces has at least four elements and a minimally transitive family of one-dimensional subspaces has \tau $\tau$ elements where n+1\le \tau \le 2n-2$n+1\le \tau \le 2n-2$ . We show how a minimally transitive family of one-dimensional subspaces arises when it consists of the subspaces spanned by the standard basis vectors together with those spanned by 0$0$ – 1$1$ vectors. On a space of dimension four, the set of nontrivial elements of a medial subspace lattice has five elements if it is minimally transitive. On spaces of dimension 12$12$ or more, the set of nontrivial elements of a medial subspace lattice can have six or more elements and be minimally transitive.

Let R be a finite ring and let {\mathrm {zp}}(R)${\mathrm {zp}}(R)$ denote the nullity degree of R, that is, the probability that the multiplication of two randomly chosen elements of R is zero. We establish the nullity degree of a semisimple ring and find upper and lower bounds for the nullity degree in the general case.

If G is permutation group acting on a finite set \Omega $\Omega$ , then this action induces a natural action of G on the power set \mathscr{P}(\Omega )$\mathscr{P}(\Omega )$ . The number s(G)$s(G)$ of orbits in this action is an important parameter that has been used in bounding numbers of conjugacy classes in finite groups. In this context, \inf ({\log _2 s(G)}/{\log _2 |G|})$\inf ({\log _2 s(G)}/{\log _2 |G|})$ plays a role, but the precise value of this constant was unknown. We determine it where G runs over all permutation groups not containing any {{\textrm {A}}}_l, l> 4${{\textrm {A}}}_l, l> 4$ , as a composition factor.

We show that the automorphism groups of right-angled Artin groups whose defining graphs have at least three vertices are not relatively hyperbolic. We then show that the outer automorphism groups are also not relatively hyperbolic, except for a few exceptional cases. In these cases, the outer automorphism groups are virtually isomorphic to either a finite group, an infinite cyclic group or \mathrm {GL}_2(\mathbb {Z})$\mathrm {GL}_2(\mathbb {Z})$.

A linear étale representation of a complex algebraic group G is given by a complex algebraic G-module V such that G has a Zariski-open orbit in V and \dim G=\dim V$\dim G=\dim V$ . A current line of research investigates which reductive algebraic groups admit such étale representations, with a focus on understanding common features of étale representations. One source of new examples arises from the classification theory of nilpotent orbits in semisimple Lie algebras. We survey what is known about reductive algebraic groups with étale representations and then discuss two classical constructions for nilpotent orbit classifications due to Vinberg and to Bala and Carter. We determine which reductive groups and étale representations arise in these constructions and we work out in detail the relation between these two constructions.

We prove that immersions of planar domains are uniquely specified by their Jacobian determinant, curl function and boundary values. This settles the two-dimensional version of an outstanding conjecture related to a particular grid generation method in computer graphics.

We show that there exists a continuous function from the unit Lebesgue interval to itself such that for any \epsilon \geq 0$\epsilon \geq 0$ and any natural number k, any point in its domain has an \epsilon $\epsilon$ -neighbourhood which, when feasible, contains k mutually disjoint extremally scrambled sets of identical Lebesgue measure, homeomorphic to each other. This result enables a satisfying generalisation of Li–Yorke (topological) chaos and suggests an open (difficult) problem as to whether the result is valid for piecewise linear functions.

We investigate the notion of relatively amenable topological action and show that the action of Thompson’s group T on S^1$S^1$ is relatively amenable with respect to Thompson’s group F. We use this to conclude that F is exact if and only if T is exact. Moreover, we prove that the groupoid of germs of the action of T on S^1$S^1$ is Borel amenable.

We investigate unbounded, linear operators arising from a finite sum of composition operators on Fock space. Real symmetry and complex symmetry of these operators are characterised.