Given a finite abelian group $G$ and $t\in \mathbb{N}$, there are two natural types of subsets of the Cartesian power $G^t$; namely, Cartesian powers $S^t$ where $S$ is a subset of $G$ and (cosets of) subgroups $H$ of $G^t$. A basic question is whether two such sets intersect. In this paper, we show that this decision problem is NP-complete. Furthermore, for fixed $G$ and $S$, we give a complete classification: we determine conditions for when the problem is NP-complete and show that in all other cases the problem is solvable in polynomial time. These theorems play a key role in the classification of algebraic decision problems in finitely generated rings developed in later work of the author.

In this paper, we provide a characterization for a class of convex curves on the 3-sphere. More precisely, using a theorem that represents a locally convex curve on the 3-sphere as a pair of curves in $\mathbb S^2$, one of which is locally convexand the other is an immersion, we are able of completely characterizing a class of convex curves on the 3-sphere.

For Γ a finite subgroup of $\mathrm{SL}_2(\mathbb{C})$ and $n \geq 1$, we study the fibres of the Procesi bundle over the Γ-fixed points of the Hilbert scheme of n points in the plane. For each irreducible component of this fixed point locus, our approach reduces the study of the fibres of the Procesi bundle, as an $(\mathfrak{S}_n \times \Gamma)$-module, to the study of the fibres of the Procesi bundle over an irreducible component of dimension zero in a smaller Hilbert scheme. When Γ is of type A, our main result shows, as a corollary, that the fibre of the Procesi bundle over the monomial ideal associated with a partition λ is induced, as an $(\mathfrak{S}_n \times \Gamma)$-module, from the fibre of the Procesi bundle over the monomial ideal associated with the core of λ. We give different proofs of this corollary in two edge cases using only representation theory and symmetric functions.

In this paper, we first describe the cohomology theory of Lie supertriple systems by using the cohomology theory of the associated Leibniz superalgebras. Then we focus on Lie supertriple systems with superderivations, called LSTSDer pairs. We introduce the notion of representations of LSTSDer pairs and investigate their corresponding cohomology theory. We also construct a differential graded Lie algebra whose Maurer–Cartan elements are LSTSDer pairs. Moreover, we consider the relationship between a LSTSDer pair and the associated LeibSDer pair. Furthermore, we develop the 1-parameter formal deformation theory of LSTSDer pairs and prove that it is governed by the cohomology groups. At last, we study abelian extensions of LSTSDer pairs and show that equivalent abelian extensions of LSTSDer pairs are classified by the third cohomology groups.

In this paper, we show that for any nonautonomous discrete time dynamical system in a Banach space if its linear part has a dichotomy and the composition of a generalized Green function and the nonlinear term of the system has a weighted integrable Lipschitz constant then the system has the weighted Lipschitz shadowing property for a type of weighted pseudo orbits in the whole phase space. Additionally, if the generalized Green function is the Green function for the dichotomy and the evolution operator restricted to the stable subspace (resp. unstable subspace) tends to 0 in weight as time tends to $+\infty$ (resp. $-\infty$) then the system has the weighted generalized forward (resp. backward) limit shadowing property. By the same approach we prove that a C1 map with a compact hyperbolic invariant set has the weighted Lipschitz shadowing property and the generalized weighted limit shadowing property for weighted pseudo orbits in the hyperbolic set. We also give the parallel results for differential equations.

We prove that the non-properness set of a local homeomorphism $\mathbb R^n \to \mathbb R^n$ cannot be ambient homeomorphic to an affine subspace of dimension n − 2. This particularly provides a partial positive answer to a conjecture of Jelonek, that claims the global invertibility of polynomial local diffeomorphisms having non-properness set with codimension greater than 1. Our reasons to obtain this result lead us to some properties of the non-properness set when it has codimension 2, the heart of Jelonek’s conjecture. We also provide a global injectivity theorem related to this conjecture that turns out to generalize previous results of the literature.

We consider radially symmetric solutions of the degenerate Keller–Segel system

\begin{align*}\begin{cases}\partial_t u=\nabla\cdot (u^{m-1}\nabla u - u\nabla v),\\0=\Delta v -\mu +u,\quad\mu =\frac{1}{|\Omega|}\int_\Omega u,\end{cases}\end{align*}

in balls $\Omega\subset\mathbb R^n$, $n\ge 1$, where m > 1 is arbitrary. Our main result states that the initial evolution of the positivity set of u is essentially determined by the shape of the (nonnegative, radially symmetric, Hölder continuous) initial data u0 near the boundary of its support $\overline{B_{r_1}(0)}\subsetneq\Omega$: It shrinks for sufficiently flat and expands for sufficiently steep u0. More precisely, there exists an explicit constant $A_{\mathrm{crit}} \in (0, \infty)$ (depending only on $m, n, R, r_1$ and $\int_\Omega u_0$) such that if $u_0(x)\le A(r_1-|x|)^\frac{1}{m-1}$ for all $|x|\in(r_0, r_1)$ and some $r_0\in(0,r_1)$ and $A \lt A_{\mathrm{crit}}$ then there are T > 0 and ζ > 0 such that $\sup\{\, |x| \mid x \in \operatorname{supp} u(\cdot, t)\,\}\le r_1 -\zeta t$ for all $t\in(0, T)$, while if $u_0(x)\ge A(r_1-|x|)^\frac{1}{m-1}$ for all $|x|\in(r_0, r_1)$ and some $r_0 \in (0, r_1)$ and $A \gt A_{\mathrm{crit}}$ then we can find T > 0 and ζ > 0 such that $\sup\{\, |x| \mid x \in \operatorname{supp} u(\cdot, t)\,\}\ge r_1 +\zeta t$ for all $t\in(0, T)$.

This paper focuses on quadratic Hom–Leibniz algebras, defined as (left or right) Hom–Leibniz algebras equipped with symmetric, non-degenerate, and invariant bilinear forms. In particular, we demonstrate that every quadratic regular Hom–Leibniz algebra is symmetric, meaning that it is simultaneously a left and a right Hom–Leibniz algebra. We provide characterizations of symmetric (resp. quadratic) Hom–Leibniz algebras. We also investigate the $\mathrm{T}^*$-extensions of Hom–Leibniz algebras, establishing their compatibility with solvability and nilpotency. We study the equivalence of such extensions and provide the necessary and sufficient conditions for a nilpotent quadratic Hom–Leibniz algebra to be isometric to a $\mathrm{T}^*$-extension. Furthermore, through the procedure of double extension, which is a central extension followed by a generalized semi-direct product, we get an inductive description of all quadratic regular Hom–Leibniz algebras, allowing us to reduce their study to that of quadratic regular Hom–Lie algebras. Finally, we construct several non-trivial examples of symmetric (resp. quadratic) Hom–Leibniz algebras.

This paper continues the analysis of Schrödinger type equations with distributional coefficients initiated by the authors in a recent paper in Journal of Differential Equations (425) 2025. Here, we consider coefficients that are tempered distributions with respect to the space variable and are continuous in time. We prove that the corresponding Cauchy problem, which in general cannot even be stated in the standard distributional setting, admits a Schwartz very weak solution which is unique modulo negligible perturbations. Consistency with the classical theory is proved in the case of regular coefficients and Schwartz Cauchy data.

Let $\Gamma$ be a Schottky subgroup of $\mathrm{SL}_2(\mathbb{Z})$ and let $X=\Gamma \backslash {\mathbb{H}}^2$ be the associated hyperbolic surface. We consider the family of Hecke congruence coverings of $X$, which we denote as usual by $ X_0(q) = \Gamma _0(q)\backslash {\mathbb{H}}^2$. Conditional on the Lindelöf Hypothesis for quadratic L-functions, we establish a uniform and explicit spectral gap for the Laplacian on $ X_0(q)$ for “almost” all prime levels $q$. Assuming the generalized Riemann hypothesis for quadratic $L$-functions, we obtain an even larger spectral gap.

In this paper, we study the distribution of the temperature within a body where the heat is transported only by radiation. Specifically, we consider the situation where both emission-absorption and scattering processes take place. We study the initial-boundary value problem given by the coupling of the radiative transfer equation with the energy balance equation on a convex domain $ \Omega \subset {\mathbb{R}}^3$ in the diffusion approximation regime, that is, when the mean free path of the photons tends to zero. Using the method of matched asymptotic expansions, we will derive the limit initial-boundary value problems for all different possible scaling limit regimes, and we will classify them as equilibrium or non-equilibrium diffusion approximation. Moreover, we will observe the formation of boundary and initial layers for which suitable equations are obtained. We will consider both stationary and time-dependent problems as well as different situations in which the light is assumed to propagate either instantaneously or with finite speed.

Locally harmonic manifolds are Riemannian manifolds in which small geodesic spheres are isoparametric hypersurfaces, i.e., hypersurfaces whose nearby parallel hypersurfaces are of constant mean curvature. Flat and rank one symmetric spaces are examples of harmonic manifolds. Damek–Ricci spaces are non-compact harmonic manifolds, most of which are non-symmetric. Taking the limit of an ‘inflating’ sphere through a point p in a Damek–Ricci space as the center of the sphere runs out to infinity along a geodesic half-line $\gamma $ starting from p, we get a horosphere. Similarly to spheres, horospheres are also isoparametric hypersurfaces. In this paper, we define the sphere-like hypersurfaces obtained by ‘overinflating the horospheres’ by pushing the center of the sphere beyond the point at infinity of $\gamma $ along a virtual prolongation of $\gamma $. They give a new family of isoparametric hypersurfaces in Damek–Ricci spaces connecting geodesic spheres to some of the isoparametric hypersurfaces constructed by J. C. Díaz-Ramos and M. Domínguez-Vázquez [17] in Damek–Ricci spaces. We study the geometric properties of these isoparametric hypersurfaces, in particular their homogeneity and the totally geodesic condition for their focal varieties.

In this paper, we establish an asymptotic formula for the twisted second moments of Dirichlet $L$-functions with one twist when averaged over all primitive Dirichlet characters of modulus $R$, where $R$ is a monic polynomial in $\mathbb{F}_q[T]$.

We study the so-called averaging functors from the geometric Langlands program in the setting of Fargues’ program. This makes explicit certain cases of the spectral action which was recently introduced by Fargues-Scholze in the local Langlands program for $\mathrm {GL}_n$. Using these averaging functors, we verify (without using local Langlands) that the Fargues-Scholze parameters associated to supercuspidal modular representations of $\mathrm {GL}_2$ are irreducible. We also attach to any irreducible $\ell $-adic Weil representation of degree n an Hecke eigensheaf on $\mathrm {Bun}_n$ and show, using the local Langlands correspondence and recent results of Hansen and Hansen-Kaletha-Weinstein, that it satisfies most of the requirements of Fargues’ conjecture for $\mathrm {GL}_n$.

This paper develops a geometric and analytical framework for studying the existence and stability of pinned pulse solutions in a class of non-autonomous reaction–diffusion equations. The analysis relies on geometric singular perturbation theory, matched asymptotic method and nonlocal eigenvalue problem method. First, we derive the general criteria on the existence and spectral (in)stability of pinned pulses in slowly varying heterogeneous media. Then, as a specific example, we apply our theory to a heterogeneous Gierer–Meinhardt (GM) equation, where the nonlinearity varies slowly in space. We identify the conditions on parameters under which the pulse solutions are spectrally stable or unstable. It is found that when the heterogeneity vanishes, the results for the heterogeneous GM system reduce directly to the known results on the homogeneous GM system. This demonstrates the validity of our approach and highlights how the spatial heterogeneity gives rise to richer pulse dynamics compared to the homogeneous case.

To investigate multiple effects of the interaction between V. cholerae and phage on cholera transmission, we propose a degenerate reaction-diffusion model with different dispersal rates, which incorporates a short-lived hyperinfectious (HI vibrios) state of V. cholerae and lower-infectious (LI vibrios) state of V. cholerae. Our main purpose is to investigate the existence and stability analysis of multi-class boundary steady states, which is much more complicated and challenging than the case when the boundary steady state is unique. In a spatially heterogeneous case, the basic reproduction number $\mathscr{R}_{0}$ is defined as the spectral radius of the sum of two linear operators associated with HI vibrios infection and LI vibrios infection. If $\mathscr{R}_{0}\leq 1$, the disease-free steady state is globally asymptotically stable. If $\mathscr{R}_{0}\gt 1$, the uniform persistence of phage-free model, as well as the existence of the phage-free steady state, are established. In a spatially homogeneous case, when $\ \;\widetilde{\!\!\!\mathscr{R}}_{0}\gt 1$, the global asymptotic stability of phage-free steady state and the uniform persistence of the phage-present model are discussed under some additional conditions. The mathematical approach here has wide applications in degenerate Partial Differential Equations.

Given $n$ convex bodies in the Euclidean space $\mathbb{R}^d$, we can find their volume polynomial which is a homogeneous polynomial of degree $d$ in $n$ variables. We consider the set of homogeneous polynomials of degree $d$ in $n$ variables that can be represented as the volume polynomial of any such given convex bodies. This set is a subset of the set of Lorentzian polynomials. Using our knowledge of operations that preserve the Lorentzian property, we give a complete classification of the cases for $(n,d)$ when the two sets are equal.

The hard-core model has as its configurations the independent sets of some graph instance $G$. The probability distribution on independent sets is controlled by a ‘fugacity’ $\lambda \gt 0$, with higher $\lambda$ leading to denser configurations. We investigate the mixing time of Glauber (single-site) dynamics for the hard-core model on restricted classes of bounded-degree graphs in which a particular graph $H$ is excluded as an induced subgraph. If $H$ is a subdivided claw then, for all $\lambda$, the mixing time is $O(n\log n)$, where $n$ is the order of $G$. This extends a result of Chen and Gu for claw-free graphs. When $H$ is a path, the set of possible instances is finite. For all other $H$, the mixing time is exponential in $n$ for sufficiently large $\lambda$, depending on $H$ and the maximum degree of $G$.

In this paper, we prove the following result advocating the importance of monomial quadratic relations between holomorphic CM periods. For any simple CM abelian variety A, we can construct a CM abelian variety B such that all non-trivial Hodge relations between the holomorphic periods of the product $A\times B$ are generated by monomial quadratic ones which are also explicit. Moreover, B splits over the Galois closure of the CM field associated with A.

In this paper, we establish variational principles for the metric mean dimension of random dynamical systems with infinite topological entropy. This is based on four types of measure-theoretic ϵ-entropies: Kolmogorov-Sinai ϵ-entropy, Shapira’s ϵ-entropy, Katok’s ϵ-entropy and Brin–Katok local ϵ-entropy. The variational principle, as a fundamental theorem, links topological dynamics and ergodic theory.