Primitive positive constructions of first order structures have been shown to be a very useful tool in universal algebra for the study of constraint satisfaction problems. However, they seemed to be very rarely studied in classical algebra such as group theory. This paper fills in this gaps by looking at structures and obstructions based on permutation groups and giving a full classification in this sub-area. This special case is also very important for the generalization to all first order structures as every permutation group describes an easy-to-check necessary condition for the existence of primitive positive constructions, also between structures that are not at all linked to permutation groups.

Motivated by the vision of making sixth-generation (6G) networks sustainable, we study the sparse antenna/array activation problems in uplink cell-free massive multiple-input multiple-output (CF mMIMO) networks. We first develop an antenna-level optimal bilinear equalizer (OBE) weighting framework, in which each access point-user equipment (AP-UE) pair is assigned a matrix-valued long-term weight to shape the contribution of individual antenna elements, thereby generalizing the conventional large-scale fading decoding (LSFD) strategy from scalar coefficients to antenna-element-aware weighting. Building on this structure, we formulate sparse antenna activation as structured sparsity-inducing mean square error (MSE) minimization problems, and design four activation schemes at two granularities: antenna-level and array-level, each with UE-specific and network-wide (all-UEs) variants. The resulting convex problems are solved efficiently via the proximal method with closed-form group-wise updates, while the network-wide schemes are modeled through hierarchical sparsity and handled by a tree-structured proximal operator. Numerical results under correlated Rician channels and a detailed power consumption model demonstrate that the OBE weighting scheme consistently improves spectral efficiency over the LSFD, with gains increasing with the number of antennas. Meanwhile, the studied sparse activation schemes can achieve substantial energy efficiency improvement and power reduction with controllable spectral efficiency loss.

We study the Abel--Jacobi map over the twistor-$\mathbb{P}^1$ in the context of Scholze's geometrisation of the real local Langlands correspondence. In a similar spirit to a result of Fargues over the Fargues--Fontaine curve, we prove that pullback along the Abel--Jacobi map induces an equivalence on Picard groupoids and use this to recover local class field theory for archimedean local fields.

Motivated by the recent work of Li and Wang on parametric generalizations of Van Hamme's $(C.2)$ supercongruence in the $q$-setting, we establish $q$-analogues of a supercongruence related to Van Hamme's $(B.2)$ supercongruence, recently obtained by the authors. In particular, we derive parametric extensions of these $q$-supercongruences by constructing suitable pairs of hypergeometric functions through the $q$-WZ method.

A longstanding open problem posed by Andrew Pitts is whether every Heyting algebra is the lattice of truth values (i.e., of subterminal objects) of some elementary topos. A positive answer is known for complete Heyting algebras (i.e., locales) via sheaves, and for Boolean algebras via a construction due to Peter Freyd. We extend Freyd's construction to all étale-finite Heyting algebras, in the sense of Evgeny Kuznetsov. These are the Heyting algebras satisfying a generalisation of the law of excluded middle relative to some finite Heyting subalgebra. For every étale-finite Heyting algebra $H$, we use Esakia duality to construct an elementary topos whose lattice of truth values is isomorphic to $H$, thereby extending the class of Heyting algebras for which a positive answer to Pitts' question is known. The toposes we construct are categories of certain compact étale spaces. As a consequence, they are finitely propositional: every object has a finite cover by subterminal objects. We show that a Heyting algebra occurs as the lattice of truth values of some finitely propositional topos if and only if it is étale-finite. This exhibits an obstruction to extending our use of compact étale spaces beyond the étale-finite case.

We present new short exponential sum approximations of length $N$ for $f_1(x)=\frac{1}{a+x}$ with $a>0$ on $[0, \infty)$ and for $f_2(x)= {\mathrm e}^{-x^2/2σ}$ with $σ>0$ on ${\mathbb R}$ with geometric error decay $ρ^{-2N}$ for user-defined $N \ge 2$ and $ρ>1$. The approximations are built over consecutive intervals $[b_j, \, b_{j+1}) \subset [0, \infty)$, $j \in {\mathbb N}_{0}$, with interval lengths that depend on $ρ$ and grow exponentially for $f_1$ and are equidistant for $f_2$. All parameters determining the exponential sum approximations on $[b_j, \, b_{j+1})$ are easily computed from the initial parameters on $[b_0, \, b_{1})$, ensuring numerical stability. Our method is based on Gauss-Laguerre and Gauss-Hermite quadrature, respectively, applied to suitable parametric integral representations of $f_1$ and $f_2$. This technique ensures consistent relative errors across all intervals. Using the obtained exponential sum approximations, we achieve highly accurate approximations of $\log(x)$ on $[1,\infty)$ and of the error function $\mathrm{erf}(x)$ with predictable geometric error decay. Numerical examples for $N=8$ and $N=10$ clearly illustrate the theoretical error estimates.

We present and analyze an embedded Trefftz discontinuous Galerkin method for reaction-diffusion problems on anisotropic meshes. The method is constructed by imposing a relaxed local Trefftz condition via an embedding into a tensor-product DG space, yielding a reduced global system while preserving the approximation properties of the underlying high-order discretization. We prove stability and quasi-optimality on anisotropic, possibly curved, quadrilateral elements, and derive anisotropic a priori error estimates. Numerical experiments for $h$- and $hp$-refinement, including curved-domain examples, validate the theoretical results.

In this paper, we study the non-symmetric system of semilinear damped $σ$-evolution equations with different damping types, where two power exponents of nonlinearities belong to the critical curve, by using moduli of continuity in nonlinear terms. Our goal is to determine the sharp conditions on these moduli of continuity that guarantee the global (in time) existence of Sobolev solutions or, conversely, lead to finite-time blow-up. Furthermore, by employing the analysis introduced in the proof of our blow-up result, we provide a positive answer to an open problem for the symmetric models posed in the literature.

It is well-known that the midpoint rule preserves the dissipation inequality if applied to a certain class of energy-based models. We introduce an appropriate scaling of the state variables such that the symmetric part of the resulting iteration matrix is guaranteed to be positive definite. This allows the application of three-term iteration schemes such as the methods of Widlund and Rapoport. Special emphasis is put on examples where the symmetric part is block diagonal such that the computations decouple. This then leads to efficient dissipation-preserving numerical schemes as illustrated in two numerical examples, namely the biharmonic heat equation and linear poroelasticity.

We uncover a finite-temperature extension of de Bruijn's identity -- the classical relation $\frac{d}{dt}h(X+\sqrt{t}Z)=\frac{1}{2}J(X)$ connecting differential entropy and Fisher information. Our framework is the spectral theory of Blahut--Arimoto (BA) dynamics, recently developed by Wang~\cite{Wang2026} for the analysis of rate-distortion optimization. The central observation is elementary yet profound: for Gaussian sources, the spectral gap $\lam$ of the BA relaxation kernel $\G$ satisfies $\lam = 1/(2βσ^2)$~\cite{Wang2026}, while the Fisher information of the source is $J = 1/σ^2$. Hence \[ {\lam = \frac{J}{2β}} \] for all inverse temperatures $β> 1/(2σ^2)$. This identifies the BA spectral gap as a \emph{finite-temperature regularization of Fisher information}. From this observation we derive an exact finite-temperature de Bruijn identity: \[ \frac{\partial F_β}{\partial σ^2} = \frac{1}{2βσ^2} = \lam, \] where $F_β$ is the BA free energy. This identity holds for all finite $β$ without any limit procedure. The classical de Bruijn identity follows as the exact consequence $β\,\partial F_β/\partialσ^2 = J/2$. The significance is structural: classical de Bruijn is not an isolated fact about Gaussian convolutions, but the $β\to\infty$ shadow of a one-parameter family of exact identities living in the spectral geometry of rate-distortion optimization. We discuss implications for the entropy power inequality, the $χ^2$-dissipation structure of BA dynamics, and the geometric unification of information inequalities.

A finite group $G$ is $\varepsilon$-quasirandom if all its nontrivial irreducible complex representations have degree at least $|G|^\varepsilon$. Building on recent work of Golsefidy-Srinivas, we prove that expansion in a quasirandom group is controlled by expansion in its simple quotients. As a consequence, we remove the product theorem from the hypotheses of the Bourgain-Gamburd expansion machine. Moreover, we combine this result with crown theory to deduce that $1 + \lfloor \varepsilon^{-1} \rfloor$ random elements give an expander Cayley graph with high probability. Finally, generalizing results of Breuillard-Green-Tao and Pyber-Szabó, we prove that the diameter of any connected Cayley graph of a quasirandom group is polylogarithmic.

Switched systems are a class of dynamical systems where trajectories switch between different systems based on a switching rule. This rule can depend on time and/or the position of the trajectory in the state space. The existence of a Lyapunov function implies the existence of a uniformly asymptotically stable equilibrium point at the origin, which we prove in this paper. A method to construct Lyapunov functions for switched systems using meshfree collocation and quadratic programming was described in a related article. We prove that, under suitable assumptions, the algorithm described in this previous work converges as the fill distance between the collocation points tends to zero.

A group is called realizable if it is the group of units in a ring with identity. The classification of realizable groups is a difficult open problem -- originally posed by László Fuchs -- and is an active area of research. Realizable groups seem rare, but their proportion within a fixed class of groups (cyclic, dihedral, finite abelian, etc.) varies. To quantify this proportion, we introduce the realizable density of a class of finite groups as an analog of natural density for subsets of the natural numbers. The realizable finite cyclic groups and the realizable finite abelian $p$-groups for $p$ odd have been classified; we prove that their realizable densities are 1/4 and 0, respectively. The realizable finite abelian 2-groups -- and more generally the realizable finite abelian groups -- have not been fully classified, and these special cases appear quite difficult. Nonetheless, we prove that the realizable density of finite abelian 2-groups is 1 and the realizable density of finite abelian groups is 1/2. Our work combines existing classification theorems for realizable groups with tools from analytic number theory.

For fixed $α>0$, we show that the sequence $\{αn^θ\}$ has Poissonian pair correlation for Lebesgue-almost all $θ\in (0,\frac{3}{5})\cup(3,\infty)$. This improves a result of Technau and Yesha, who proved the same for almost all $θ>7$. The approach of Technau and Yesha was based on a repulsion principle, which roughly allows one to estimate the variance of the pair correlation function using the fourth derivative of the phase. In our approach, we split the $θ$-integration in the variance into many short intervals and show that most of the integrals can be estimated using the first derivative. The problem is then reduced to several counting estimates, which we prove using moments of the Riemann zeta function and exponent pairs.

Let $\mathcal{M}_g$ be the moduli space of smooth curves of genus $g$. The image of a non-constant morphism from a curve $T$ to $\mathcal{M}_g$ is a curve in $\mathcal{M}_g$. By work of González Díez and Harvey, for every integer $g \geq 3$, there exists a complete curve in $\mathcal{M}_g$. Here we generalize the construction to produce new complete curves in $\mathcal{M}_g$. We also find a formula for the genus of each curve $T$ using Galois theory for function fields.

We explicitly calculate Pan's geometric intertwining operator and the Fontaine operator on modular curves at infinite level via $q$-expansions, using Heuer's theory of cusps at infinite level. We prove that these two operators coincide on such expansions up to an explicit constant. As an application, we combine this result with $q$-expansion principles to provide a new proof of Pan's theorem that these operators are equal on the locally analytic vectors of completed cohomology of modular curves.

We prove results on moduli spaces of slope stable bundles of projective spaces on a hyperkähler manifold of Type $K3^{[2]}$. Let $X$ be projective of Type $K3^{[2]}$ and $h$ be a (generic) ample class. We prove that the moduli space $M_{\overline{\bf w}_a}(X,h)$ parametrizing $h$ slope stable bundles with a suitable mock Mukai vector ${\overline{\bf w}}_a$ contains an irreducible component $M_{\overline{\bf w}_a}(X,h)^{*}$ whose normalization $\widetilde{M}_{\overline{\bf w}_a}(X,h)^{*}$ is a (projective) HK manifold of Type $K3^{[a^2+1]}$, and that conversely every projective HK manifold $W$ of Type $K3^{[a^2+1]}$ is isomorphic to $\widetilde{M}_{\overline{\bf w}_a}(X,h)^{*}$ for a suitable $(X,h)$ as above. Moreover the universal bundle of projective spaces on $X\times \widetilde{M}_{\overline{\bf w}_a}(X,h)^{*}$ defines a vector bundle whose $2nd$ Chern class defines a rational Hodge isometry $H^2(X)\to H^2(\widetilde{M}_{\overline{\bf w}_a}(X,h)^{*})$. From this and a result of Markman one gets that the analogue of the Shafarevich conjecture (a special case of the Hodge conjecture) holds for rational Hodge isometries $H^2(W_1) \to H^2(W_2)$ between projective hyperkähler manifolds $W_1,W_2$ of Types $K3^{[a_1^2+1]}$ and $K3^{[a_2^2+1]}$ respectively. We prove results also for $(X,ω)$ a general HK manifold of Type $K3^{[2]}$. In fact one ingredient in our proof is Verbistsky's theory of projectively hyperhomolorphic vector bundles.

We propose a novel method for computing invariant densities of certain multidimensional continued fraction algorithms. Inspired by Rauzy induction, our approach builds on the formalism of simplicial systems developed by Fougeron. We introduce a win-lose induction on a graph that is conjugate to the original algorithm, and construct its natural extension by introducing the notion of a dual graph. This method explicitly reconstructs the complete dynamics of the algorithm, yielding a partition of the invariant domain of the natural extension into pieces that map to one another. We further study the ergodic properties of the algorithms within this framework; in particular, we prove that the Modified Triangle algorithm in any dimension admits a unique ergodic measure equivalent to the Lebesgue measure.

For an integer $k\ge 0$ and a graph $G$, the \emph{token-sliding reconfiguration graph $\mathsf{TS}_k(G)$} has the independent $k$-sets of $G$ as vertices. Two vertices are adjacent if one token can slide along an edge of $G$ and the resulting $k$-set is still independent. We study the following realizability problem: for fixed $k\ge 2$, which graphs are isomorphic to $\mathsf{TS}_k(G)$ for some graph $G$? This inverse viewpoint asks which abstract state spaces can occur exactly under a local token rule. We give positive realizability results for the complement targets $\overline{K_n}$, $\overline{K_{m,n}}$, and $\overline{K_n-e}$, and we determine sharp cutoffs for complements of paths and cycles. We also prove a product formula for token-sliding graphs of disjoint unions and apply it to Cartesian products of complete graphs, paths, and cycles. For every grid $Γ_{m,n}=P_m\square P_n$ with $2\le m\le n$, we realize $Γ_{m,n}$ at token value $m+n-2$ and at every token value $k\ge 4$. At small token values, we prove that $C_4\square C_n$ is not a $\mathsf{TS}_2$-graph for $n\ge 4$, classify ladders $Γ_{2,n}$, and settle the first non-ladder grid: for $k\ge 2$, $Γ_{3,3}$ is realizable if and only if $k\ge 4$.

We study Ulam stability for approximate *-homomorphisms of C*-algebras. We prove stability results for several classes of nuclear C*-algebras with respect to von Neumann algebra targets, including abelian C*-algebras and large classes arising in the Elliott classification program. We also discuss permanence properties, counterexamples, and related stability phenomena. As applications, we obtain rigidity and independence results for corona algebras.

Carlen, Frank and Lieb studied stability estimates for the lowest eigenvalue of a Schrödinger operator by decomposing the problem into a stability estimate for Hölder's inequality and a stability estimate for a Gagliardo--Nirenberg--Sobolev inequality. In this note we point out that, if the Hölder step is replaced by the optimal $L^1$-stability theorem of Leng and Lu in probabilistic form, then one obtains a density-distance version of the Carlen--Frank--Lieb stability theorem. The new formulation measures the $L^1$ distance between the normalized density $V_-^s/\int V_-^s$ induced by the negative part of the potential and the corresponding density induced by an optimal potential, where $s=γ+d/2$. As a geometric application of the same idea, we also derive a density-stability version of the $L_p$ mixed volume inequality. In the case where one of the two convex bodies is centrally symmetric and both bodies are trapped between two concentric Euclidean balls, this gives an averaged stability estimate for the non-evenness of the support function.

The Borel lifting problem asks when a nonsingular near action of a Polish group can be represented by a genuine Borel action. For locally compact Polish groups, a classical theorem of Mackey and Ramsay gives an affirmative answer. At the opposite extreme, Glasner-Tsirelson-Weiss proved that every probability preserving Borel action of a Lévy group is trivial, and asked whether a Lévy group can admit a nontrivial nonsingular Borel action. We prove a fixed-point theorem which gives a negative answer for most of the standard Lévy groups in the literature: if $G$ is a locally Wasserstein group, then every quasi-invariant $σ$-finite Borel measure for a Borel action of $G$ is supported on the fixed points. The proof introduces Wasserstein stability, a local-to-global principle for compact measure metric groups that refines the Gromov-Milman framework for concentration of measure in topological groups. It asserts for a compact measure metric group that the local total variation of the Haar measure tilted by density kernels, controls global Wasserstein distance from the Haar measure. We show that Wasserstein stability implies concentration, and prove a functional inequality using martingales in the spirit of Milman-Schechtman, which is governed by $\ell^{1}$-sums of martingale increments. This gives a geometric criterion by which most of the standard Lévy groups in the literature are Wasserstein or locally Wasserstein, including $L^{0}$-groups with compact targets, measure preserving and nonsingular automorphism groups, full groups of amenable equivalence relations, isometry groups of $L^{p}$-spaces for $p\neq2$, the unitary group of the hyperfinite $\mathrm{II}_{1}$-factor, the infinite dimensional unitary and orthogonal groups, the Cameron-Martin affine group, and the isometry group of the Urysohn space.

We consider the compressible time-dependent Navier-Stokes equations in a bounded perforated domain in dimensions two and three. Provided the perforations are small enough, we show that the limiting equations do not change their form when the perforation size goes to zero while their number increases to infinity. The novelty of this result is the form of the pressure: we consider a hard-sphere pressure law, giving an \emph{a priori} bound for the density while, compared to the barotropic case, having worse regularity for the pressure, therefore causing significant problems in the homogenization procedure. To the best of our knowledge, the homogenization for this kind of pressures has not been addressed in the literature yet.

We find a necessary and sufficient condition for a smooth functional on a subspace to admit a norm-preserving smooth extension to the entire space in polyhedral norms. The characterization is geometric: such an extension exists if and only if the unique absolute norm-attaining point of the smooth functional is an extreme point of both the unit ball of the subspace and that of the ambient space. We show by example that such a result is not true in non-polyhedral norms, even under sufficiently strong hypothesis. Extremity of the norm preserving restrictions of extreme functionals are also discussed.

A classical theorem of Nash-Williams states that if $G$ is a $2$-connected graph on $n$ vertices with minimum degree at least $(n+2)/3$, then for every longest cycle $C$ of $G$, the graph $G-V(C)$ is edgeless. Motivated by a higher-connectivity analogue, Bondy conjectured in 1980 that if $G$ is a $k$-connected graph on $n$ vertices with minimum degree at least $(n+k(k-1))/(k+1)$, then for every longest cycle $C$ of $G$, every path in $G-V(C)$ has at most $k-1$ vertices. This conjecture is known for $k\le 3$ and remains open for all $k\ge 4$. In this paper, we prove Bondy's conjecture for all sufficiently large graphs. The key ingredient is a new Dirac-type theorem that gives a lower bound on the length of a longest cycle in a $k$-connected graph, which also yields a partial solution to a conjecture of Jung from 1990. Along the way, we develop several new tools, including a DFS lemma and an average-degree analogue of the Bondy--Jackson theorem. We conclude with a discussion of related problems and a counterexample to a conjecture of Voss from 1991.

Integrated sensing and communication (ISAC) has emerged as a promising key technology for future wireless networks, enabling the efficient coordination of sensing and communication functions within limited resources. This work investigates a secure ISAC system assisted by an uncrewed aerial vehicle (UAV). By incorporating the extended Kalman filter (EKF), the proposed system is capable of delivering communication services to legitimate users while simultaneously jamming eavesdroppers and performing joint prediction and tracking of the trajectories of both legitimate and illegitimate users. Considering practical constraints such as {sensing beamwidth}, transmit power, and UAV's propulsion energy consumption, the secrecy rate is maximized through the joint design of transmit beamforming and UAV trajectory. To tackle the resulting highly non-convex optimization problem, an efficient iterative algorithm is developed by integrating block coordinate descent, successive convex approximation, and EKF, thereby yielding a high-quality suboptimal solution. Extensive simulation results validate the superior performance of the proposed scheme compared to benchmarks.

We prove that, on a bounded open convex domain $Ω\subset\mathbb{R}^n$, the first Dirichlet eigenfunction of the Laplacian or the Ornstein--Uhlenbeck operator is $α$-logconcave for every $α\in(0,1/2]$. This extends the recent $1/2$-logconcavity theorem of Crasta--Fragalà for the Laplacian to the weighted Gaussian setting and, simultaneously, to a broader range of exponents. More precisely, if $u$ denotes the first eigenfunction normalized by $\|u\|_\infty=1$, then for every $α\in(0,1/2]$, the function $-\bigl(-\log(κu(x))\bigr)^α$ is concave in $Ω$ provided the scaling parameter $κ$ lies below an explicit threshold $κ_α(Ω)\in(0,1)$, which depends on the first Dirichlet eigenvalue and on the diameter of~$Ω$. For the Ornstein--Uhlenbeck operator, $κ_α(Ω)$ also depends on the distance between $Ω$ and the origin. Moreover, we establish a local counterpart: for every $κ\in(0,1)$, the function $\bigl(-\log(κu)\bigr)^α$ is convex on a convex neighborhood $Ω_κ$ of the unique maximum point of~$u$. We also provide counterexamples showing that unscaled $1/2$-logconcavity may fail for the first Dirichlet eigenfunction of a Schrödinger operator with a smooth convex potential, and for the first Dirichlet eigenfunction of a weighted Laplacian associated with an affine log-concave weight.

A $\mathbf{GL}$-variety is a typically infinite dimensional variety equipped with a suitable action of the infinite general linear group $\mathbf{GL}$. In earlier work, we established the unirationality theorem: an irreducible $\mathbf{GL}$-variety admits a dominant map from a particularly simple $\mathbf{GL}$-variety, namely, the product of an irreducible finite-dimensional variety with trivial $\mathbf{GL}$-action and an infinite-dimensional affine space on which $\mathbf{GL}$ acts linearly. The main result of this paper states that this map can in fact be constructed to be surjective rather than merely dominant. An immediate application is that secant varieties to varieties of tensors, which are typically constructed as image closures of certain $\mathbf{GL}$-equivariant maps, are in fact also images of (more complicated) $\mathbf{GL}$-equivariant maps. We derive several consequences of this improved unirationality theorem.

This paper studies the global regularity problem for the two-dimensional incompressible Boussinesq equations with fractional dissipation given by $(-Δ)^{\frac\alpha2}u$ and $(-Δ)^{\frac\beta2} θ$. Attention is focused on the subcritical regime where $α+ β>1$. The case $α>\frac23$ was recently settled in a joint work of the authors [Math. Ann., \textbf{391} (2025), 5965-6012], which established global regularity under this condition. This paper addresses the remaining case $α\leq \frac23$. We obtain the sharpest regularity result by minimizing assumptions on $α$ and $β$. We derive nonlinear lower bounds for the fractional Laplacian operator and implement an iterative procedure.

In this work we introduce a class of non-local differential operators defined through a beta-weighted averaging of the ordinary derivative. We investigate their analytical properties and establish connections with the Caputo and Erdélyi-Kober operators. Differential equations involving the beta-weighted derivative are studied by Mellin transform methods, leading to solutions represented in terms of Barnes G-functions and a new class of G-hypergeometric functions. We also analyze asymptotic properties, Laplace transforms, and the second-order equation involving the sequential beta-weighted derivative. Finally, we present stochastic applications of these results, showing that continuous-time random walks, with waiting times characterized by the beta-weighted derivative, converge to Brownian motions time-changed by a scaled inverse stable subordinator. We compare this anomalous-diffusion model with a time-changed Brownian motion whose one-dimensional distribution solve a heat-type equation with beta-weighted derivative.