A group is said to have rational growth with respect to a generating set if the growth series is a rational function. It was shown by Parry that certain torus bundle groups of even trace exhibits rational growth. We generalize this result to a class of torus bundle groups with odd trace.

We formulate a conjecture on the special values of zeta functions of regular arithmetic schemes in terms of Weil-étale cohomology…

We prove the existence of a ground state positive solution of Schrödinger–Poisson systems in the plane of the form

\[ -\Delta u + V(x)u + \frac{\gamma}{2\pi} \left(\log|\cdot| \ast u^2 \right)u = b |u|^{p-2}u \quad\text{in}\ \mathbb{R}^2, \]

$p\ge 4$

$\gamma,b>0$

$V$

Sonar systems are frequently used to classify objects at a distance by using the structure of the echoes of acoustic waves as a proxy for the object’s shape and composition. Traditional synthetic aperture processing is highly effective in solving classification problems when the conditions are favourable but relies on accurate knowledge of the sensor’s trajectory relative to the object being measured. This article provides several new theoretical tools that decouple object classification performance from trajectory estimation in synthetic aperture sonar processing. The key insight is that decoupling the trajectory from classification-relevant information involves factoring a function into the composition of two functions. The article presents several new general topological invariants for smooth functions based on their factorisations over function composition. These invariants specialise to the case when a sonar platform trajectory is deformed by a non-small perturbation. The mathematical results exhibited in this article apply well beyond sonar classification problems. This article is written in a way that supports full mathematical generality.

Let $K={\mathbf {Q}}(\theta )$ be an algebraic number field with $\theta$ a root of an irreducible polynomial $x^5+ax+b\in {\mathbf {Z}}[x]$. In this paper, for every rational prime $p$, we provide necessary and sufficient conditions on $a,\,~b$ so that $p$ is a common index divisor of $K$. In particular, we give sufficient conditions on $a,\,~b$ for which $K$ is non-monogenic. We illustrate our results through examples.

In this contribution, we present a modelling and simulation framework for parametrised lithium-ion battery cells. We first derive a continuum model for a rather general intercalation battery cell on the basis of non-equilibrium thermodynamics. In order to efficiently evaluate the resulting parameterised non-linear system of partial differential equations, the reduced basis method is employed. The reduced basis method is a model order reduction technique on the basis of an incremental hierarchical approximate proper orthogonal decomposition approach and empirical operator interpolation. The modelling framework is particularly well suited to investigate and quantify degradation effects of battery cells. Several numerical experiments are given to demonstrate the scope and efficiency of the modelling framework.

We prove new mixing rate estimates for the random walks on homogeneous spaces determined by a probability distribution on a finite group $G$. We introduce the switched random walk determined by a finite set of probability distributions on $G$, prove that its long-term behaviour is determined by the Fourier joint spectral radius of the distributions, and give Hermitian sum-of-squares algorithms for the effective estimation of this quantity.

Narrow escape and narrow capture problems which describe the average times required to stop the motion of a randomly travelling particle within a domain have applications in various areas of science. While for general domains, it is known how the escape time decreases with the increase of the trap sizes, for some specific 2D and 3D domains, higher-order asymptotic formulas have been established, providing the dependence of the escape time on the sizes and locations of the traps. Such results allow the use of global optimisation to seek trap arrangements that minimise average escape times. In a recent paper (Iyaniwura (2021) SIAM Rev. 63(3), 525–555), an explicit size- and trap location-dependent expansion of the average mean first passage time (MFPT) in a 2D elliptic domain was derived. The goal of this work is to systematically seek global minima of MFPT for $1\leq N\leq 50$ traps in elliptic domains using global optimisation techniques and compare the corresponding putative optimal trap arrangements for different values of the domain eccentricity. Further, an asymptotic formula for the average MFPT in elliptic domains with N circular traps of arbitrary sizes is derived, and sample optimal configurations involving non-equal traps are computed.

We consider supercritical site percolation on the $d$-dimensional hypercube $Q^d$. We show that typically all components in the percolated hypercube, besides the giant, are of size $O(d)$. This resolves a conjecture of Bollobás, Kohayakawa, and Łuczak from 1994.

New classes of conditionally integrable systems of nonlinear reaction–diffusion equations are introduced. They are obtained by extending a well-known nonclassical symmetry of a scalar partial differential equation to a vector equation. New exact solutions of nonlinear predator–prey systems with cross-diffusion are constructed. Infinite dimensional classes of exact solutions are made available for such nonlinear systems. Some of these solutions decay towards extinction and some oscillate or spiral around an interior fixed point. It is shown that the conditionally integrable systems are closely related to the standard diffusive Lotka–Volterra system, but they have additional features.

A superelliptic curve over a discrete valuation ring $\mathscr{O}$ of residual characteristic p is a curve given by an equation $\mathscr{C}\;:\; y^n=\,f(x)$, with $\textrm{Disc}(\,f)\neq 0$. The purpose of this article is to describe the Galois representation attached to such a curve under the hypothesis that f(x) has all its roots in the fraction field of $\mathscr{O}$ and that $p \nmid n$. Our results are inspired on the algorithm given in Bouw and WewersGlasg (Math. J. 59(1) (2017), 77–108.) but our description is given in terms of a cluster picture as defined in Dokchitser et al. (Algebraic curves and their applications, Contemporary Mathematics, vol. 724 (American Mathematical Society, Providence, RI, 2019), 73–135.).

We demonstrate that two supersoluble complements of an abelian base in a finite split extension are conjugate if and only if, for each prime $p$, a Sylow $p$-subgroup of one complement is conjugate to a Sylow $p$-subgroup of the other. As a corollary, we find that any two supersoluble complements of an abelian subgroup $N$ in a finite split extension $G$ are conjugate if and only if, for each prime $p$, there exists a Sylow $p$-subgroup $S$ of $G$ such that any two complements of $S\cap N$ in $S$ are conjugate in $G$. In particular, restricting to supersoluble groups allows us to ease D. G. Higman's stipulation that the complements of $S\cap N$ in $S$ be conjugate within $S$. We then consider group actions and obtain a fixed point result for non-coprime actions analogous to Glauberman's lemma.

A $(p,q)$-colouring of a graph $G$ is an edge-colouring of $G$ which assigns at least $q$ colours to each $p$-clique. The problem of determining the minimum number of colours, $f(n,p,q)$, needed to give a $(p,q)$-colouring of the complete graph $K_n$ is a natural generalization of the well-known problem of identifying the diagonal Ramsey numbers $r_k(p)$. The best-known general upper bound on $f(n,p,q)$ was given by Erdős and Gyárfás in 1997 using a probabilistic argument. Since then, improved bounds in the cases where $p=q$ have been obtained only for $p\in \{4,5\}$, each of which was proved by giving a deterministic construction which combined a $(p,p-1)$-colouring using few colours with an algebraic colouring.

In this paper, we provide a framework for proving new upper bounds on $f(n,p,p)$ in the style of these earlier constructions. We characterize all colourings of $p$-cliques with $p-1$ colours which can appear in our modified version of the $(p,p-1)$-colouring of Conlon, Fox, Lee, and Sudakov. This allows us to greatly reduce the amount of case-checking required in identifying $(p,p)$-colourings, which would otherwise make this problem intractable for large values of $p$. In addition, we generalize our algebraic colouring from the $p=5$ setting and use this to give improved upper bounds on $f(n,6,6)$ and $f(n,8,8)$.

We prove a double commutant theorem for separable subalgebras of a wide class of corona C*-algebras, largely resolving a problem posed by Pedersen in 1988. Double commutant theorems originated with von Neumann, whose seminal result evolved into an entire field now called von Neumann algebra theory. Voiculescu later proved a C*-algebraic double commutant theorem for subalgebras of the Calkin algebra. We prove a similar result for subalgebras of a much more general class of so-called corona C*-algebras.