We show that, for any prime p, there exist absolutely simple abelian varieties over $\mathbb {Q}$ with arbitrarily large p-torsion in their Tate-Shafarevich groups. To prove this, we construct explicit $\mu _p$-covers of Jacobians of curves of the form $y^p = x(x-1)(x-a)$ which violate the Hasse principle. In the appendix, Tom Fisher explains how to interpret our proof in terms of a Cassels-Tate pairing.

Oda’s problem, which deals with the fixed field of the universal monodromy representation of moduli spaces of curves and its independence with respect to the topological data, is a central question of anabelian arithmetic geometry. This paper emphasizes the stack nature of this problem by establishing the independence of monodromy fields with respect to finer special loci data of curves with symmetries, which we show provides a new proof of Oda’s prediction.

Interacting particle systems (IPSs) are a very important class of dynamical systems, arising in different domains like biology, physics, sociology and engineering. In many applications, these systems can be very large, making their simulation and control, as well as related numerical tasks, very challenging. Kernel methods, a powerful tool in machine learning, offer promising approaches for analyzing and managing IPS. This paper provides a comprehensive study of applying kernel methods to IPS, including the development of numerical schemes and the exploration of mean-field limits. We present novel applications and numerical experiments demonstrating the effectiveness of kernel methods for surrogate modelling and state-dependent feature learning in IPS. Our findings highlight the potential of these methods for advancing the study and control of large-scale IPS.

In this paper, we consider a delayed discrete single population patch model in advective environments. The individuals are subject to both random and directed movements, and there is a net loss of individuals at the downstream end due to the flow into a lake. Choosing time delay as a bifurcation parameter, we show the existence of Hopf bifurcations for the model. In homogeneous non-advective environments, it is well known that the first Hopf bifurcation value is independent of the dispersal rate. In contrast, for homogeneous advective environments, the first Hopf bifurcation value depends on the dispersal rate. Moreover, we show that the first Hopf bifurcation value in advective environments is larger than that in non-advective environments if the dispersal rate is large or small, which suggests that directed movements of the individuals inhibit the occurrence of Hopf bifurcations.

The sharp bound of the second Hankel determinant of logarithmic coefficients of inverse functions of strongly starlike functions is computed.

We derive an asymptotic expansion for the critical percolation density of the random connection model as the dimension of the encapsulating space tends to infinity. We calculate rigorously the first expansion terms for the Gilbert disk model, the hyper-cubic model, the Gaussian connection kernel, and a coordinate-wise Cauchy kernel.

We investigate the mean-field dynamics of stochastic McKean differential equations with heterogeneous particle interactions described by large network structures. To express a wide range of graphs, from dense to sparse structures, we incorporate the recently developed graph limit theory of graphops into the limiting McKean–Vlasov equations. Global stability of the splay steady state is proven via a generalised entropy method, leading to explicit graph structure-dependent decay rates. We highlight the robustness of the entropy approach by extending the results to the closely related Sakaguchi–Kuramoto model with intrinsic frequency distributions. We also present central examples of random graphs, such as power law graphs and the spherical graphop, and analyse the limitations of the applied methodology.

We introduce a free boundary model to study the effect of vesicle transport onto neurite growth. It consists of systems of drift-diffusion equations describing the evolution of the density of antero- and retrograde vesicles in each neurite coupled to reservoirs located at the soma and the growth cones of the neurites, respectively. The model allows for a change of neurite length as a function of the vesicle concentration in the growth cones. After establishing existence and uniqueness for the time-dependent problem, we briefly comment on possible types of stationary solutions. Finally, we provide numerical studies on biologically relevant scales using a finite volume scheme. We illustrate the capability of the model to reproduce cycles of extension and retraction.

We give a generators-and-relations description of the reduced versions of quiver quantum toroidal algebras, which act on the spaces of BPS states associated to (noncompact) toric Calabi–Yau threefolds X. As an application, we obtain a description of the K-theoretic Hall algebra of (the quiver with potential associated to) X, modulo torsion.

Given a Gromov hyperbolic domain $G\subsetneq \mathbb{R}^n$ with uniformly perfect Gromov boundary, Zhou and Rasila recently proved that for all quasiconformal homeomorphisms $\psi\colon G\to G$ with identity value on the Gromov boundary, the quasihyperbolic displacement $k_G(x,\psi(x))$ for all $x\in G$ is bounded above. In this paper, we generalize this result and establish Teichmüller displacement theorem for quasi-isometries of Gromov hyperbolic spaces in a quantitative way. As applications, we obtain its connections to bilipschitz extensions of certain Gromov hyperbolic spaces.

Let f(x) and g(x) be polynomials in $\mathbb F_{2}[x]$ with ${\rm deg}\text{ } f=n$. It is shown that for $n\gg 1$, there is an $g_{1}(x)\in \mathbb F_{2}[x]$ with ${\rm deg}\text{ } g_{1}\leqslant \max\{{\rm deg}\text{ } g, 6.7\log n\}$ and $g(x)-g_{1}(x)$ having $ \lt 6.7\log n$ terms such that $\gcd(f(x), g_{1}(x))=1$. As an application, it is established using a result of Dubickas and Sha that given $f(x)\in \mathbb F_{2}[x]$ of degree $n\geqslant 1$, there is a separable $g(x)\in 2[x]$ with ${\rm deg}\text{ } g= {\rm deg}\text{ } f$ and satisfying that $f(x)-g(x)$ has $\leqslant 6.7\log n$ terms. As a simple consequence, the latter result holds in $\mathbb Z[x]$ after replacing ‘number of terms’ by the L1-norm of a polynomial and $6.7\log n$ by $6.8\log n$. This improves the bound $(\log n)^{\log 4 +\operatorname{\varepsilon}}$ obtained by Filaseta and Moy.

Let G be a finite solvable group. We prove that if $\chi\in{{\operatorname{Irr}}}(G)$ has odd degree and $\chi(1)$ is the minimal degree of the nonlinear irreducible characters of G, then $G/\operatorname{Ker}\chi$ is nilpotent-by-abelian.

This article studies the dynamical behaviour of classical solutions of a hyperbolic system of balance laws, derived from a chemotaxis model with logarithmic sensitivity, with time-dependent boundary conditions. It is shown that under suitable assumptions on the boundary data, solutions starting in the $H^2$-space exist globally in time and the differences between the solutions and their corresponding boundary data converge to zero as time goes to infinity. There is no smallness restriction on the magnitude of the initial perturbations. Moreover, numerical simulations show that the assumptions on the boundary data are necessary for the above-mentioned results to hold true. In addition, numerical results indicate that the solutions converge asymptotically to time-periodic states if the boundary data are time-periodic.

Let $(A,\mathfrak{m})$ be a Cohen–Macaulay local ring, and then the notion of a $T$-split sequence was introduced in the part-1 of this paper for the $\mathfrak{m}$-adic filtration with the help of the numerical function $e^T_A$. In this article, we explore the relation between Auslander–Reiten (AR)-sequences and $T$-split sequences. For a Gorenstein ring $(A,\mathfrak{m})$, we define a Hom-finite Krull–Remak–Schmidt category $\mathcal{D}_A$ as a quotient of the stable category $\underline{\mathrm{CM}}(A)$. This category preserves isomorphism, that is, $M\cong N$ in $\mathcal{D}_A$ if and only if $M\cong N$ in $\underline{\mathrm{CM}}(A)$.This article has two objectives: first objective is to extend the notion of $T$-split sequences, and second objective is to explore the function $e^T_A$ and $T$-split sequences. When $(A,\mathfrak{m})$ is an analytically unramified Cohen–Macaulay local ring and $I$ is an $\mathfrak{m}$-primary ideal, then we extend the techniques in part-1 of this paper to the integral closure filtration with respect to $I$ and prove a version of Brauer–Thrall-II for a class of such rings.