https://news.mcmedia.cam/feed/Maths MCMedia News RSS Feed - Maths specific feed https://news.mcmedia.cam/rss-info.html MCMedia RSS Generator v1.0 Fri, 17 Apr 2026 17:30:32 +0000 https://arxiv.org/abs/2604.04682 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.04682v2 Announce Type: replace-cross Abstract: We prove gap theorems for entropy norms on automorphism groups of K3 surfaces, Enriques surfaces, and irreducible holomorphic symplectic manifolds. We also study the achirality of automorphisms of K3 surfaces and Enriques surfaces in terms of genus-one fibrations. Maths/math.GR_(Group_Theory) https://arxiv.org/abs/2511.04296 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2511.04296v3 Announce Type: replace-cross Abstract: Let $G$ be a group acting on a field $L$, and suppose that $L /L^G$ is a finite extension. We show that the category of semilinear representations of $G$ over $L$ can be described in terms of the category of linear representations of $H$, the kernel of the map $G \rightarrow \mathrm{Aut}(L)$. When $G$ is finite and $L$ has characteristic 0 this provides a character theory for semilinear representations of $G$ over $L$, which recovers ordinary character theory when the action of $G$ on $L$ is trivial. Maths/math.GR_(Group_Theory) https://arxiv.org/abs/2602.20613 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2602.20613v2 Announce Type: replace Abstract: In this paper, we introduce a class of blocks which is called hyperfocal abelian Frobenius blocks.This class of blocks is an analogous version of the block with abelian defect group and Frobenius inertial quotient at hyperfocal level and includes the blocks with Klein four hyperfocal subgroups and cyclic hyperfocal subgroups. We show that there is a stable equivalence of Morita type between the hyperfocal subalgebras of the hyperfocal abelian Frobenius blocks and a group algebra of a Frobenius group associated with the hyperfocal subgroup of the block. As applications, we can partially describe some structures of the blocks with Klein four hyperfocal subgroups and cyclic hyperfocal subgroups,such as the structures of their hyperfocal subalgebras in terms of derived categories and the structures of their characters. As a consequence, we show that Broue's abelian defect group conjecture holds for blocks with Klein four hyperfocal subgroups. Maths/math.GR_(Group_Theory) https://arxiv.org/abs/2506.13235 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2506.13235v2 Announce Type: replace Abstract: Given a finitely generated amenable group $H$ satisfying some mild assumptions, we relate isoperimetric profiles of the lampshuffler group $\mathsf{Shuffler}(H)=\mathsf{FSym}(H)\rtimes H$ to those of $H$. Our results are sharp for all exponential growth groups for which isoperimetric profiles are known, including Brieussel-Zheng groups. This refines previous estimates obtained by Erschler and Zheng and by Saloff-Coste and Zheng. The most difficult part is to find an optimal upper bound, and our strategy consists in finding suitable lamplighter subgraphs in lampshufflers. This novelty applies more generally for many examples of halo products, a class of groups introduced recently by Genevois and Tessera as a natural generalisation of wreath products. Lastly, we also give applications of our estimates on isoperimetric profiles to the existence problem of regular maps between such groups. Maths/math.GR_(Group_Theory) https://arxiv.org/abs/2505.05220 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2505.05220v2 Announce Type: replace Abstract: Let F be a non-archimedean local field of characteristic zero whose residue field has at least three elements. Let G be an almost simple linear algebraic group over F, with rank_F(G) >= 2. Let X be a simply connected symmetric space of infinite dimension and finite rank, with non-positive curvature operator. We prove that every continuous action by isometries of G on X has a fixed point. If the group G contains SL_3(F), the result holds without any assumption on the non-archimedean local field F. The result extends to cocompact lattices in G if the cardinality of the residue field of F is large enough, with a bound that depends on rank_F(G). Maths/math.GR_(Group_Theory) https://arxiv.org/abs/2604.15154 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15154v1 Announce Type: cross Abstract: Let $G$ and $H$ be finitely generated groups. In this paper, we prove the quantitative coarse Baum--Connes conjecture for the free product $G* H$ under the assumption that the conjecture holds for both $G$ and $H$. Maths/math.GR_(Group_Theory) https://arxiv.org/abs/2604.14383 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14383v1 Announce Type: cross Abstract: This article describes a natural piecewise Euclidean bi-simplicial cell structure for the space of $n$-element multisets in a fixed Euclidean rectangle. In particular, we highlight some connections with spaces of complex polynomials and permutahedra. Maths/math.GR_(Group_Theory) https://arxiv.org/abs/2604.15303 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15303v1 Announce Type: new Abstract: We prove a strong general-purpose bound for the diameter of a finite group depending only on the diameters of its composition factors and the maximal exponent of a normal abelian section. There are a number of notable applications: (1) if $G$ is a finite soluble group of exponent $e$, $\mathrm{diam}(G) \ll e (\log |G|)^8$, (2) anabelian groups with bounded-rank composition factors have polylogarithmic diameter, (3) transitive soluble subgroups of $S_n$ have diameter $\ll n^5$, and (4) Grigorchuk's gap conjecture holds for any finitely generated group acting faithfully on a bounded-degree rooted tree. Additionally, conditional on Babai's conjecture, (5) any transitive permutation group of degree $n$ has diameter bounded by a polynomial in $n$ (a folkloric conjecture), and (6) Grigorchuk's gap conjecture holds for residually finite groups, and thus the conjecture reduces to the simple case. Maths/math.GR_(Group_Theory) https://arxiv.org/abs/2604.14244 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14244v1 Announce Type: new Abstract: In this paper, we investigate the existence of $C^n$, $n\in \mathbb{N}^+$, solutions for a class of second-order iterative functional equations involving iterates of the unknown function and a nonlinear term. Applying the Fiber Contraction Theorem and Fa\`a di Bruno's Formula, we establish the existence of bounded $C^n$ solutions with bounded derivatives of order from $1$ to $n$. Maths/math.GM_(General_Mathematics) https://arxiv.org/abs/2604.14194 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14194v1 Announce Type: new Abstract: Let $\mathbb{F}$ be a sub-modulus field such that $2 \neq 0$. Let $\mathcal{X}$ be a sub-normed linear space over $\mathbb{F}$. Then we show that \begin{align*} \bigg|\|x\|-\|y\|\bigg|\leq \frac{2}{|2|}\|x+y\|+\frac{2}{|2|}\max\{\|x-y\|, \|y-x\|\}-(\|x\|+\|y\|) \end{align*} and \begin{align*} \bigg|\|x\|-\|y\|\bigg|\leq \|x\|+\|y\|-\frac{2}{|2|}\|x+y\|+\frac{2}{|2|}\max\{\|y-x\|, \|x-y\|\}. \end{align*} Above inequalities are finite field versions of important Tarski-Maligranda inequalities obained by Maligranda [\textit{Banach J. Math. Anal., 2008}]. Maths/math.GM_(General_Mathematics) https://arxiv.org/abs/2604.01100 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.01100v2 Announce Type: replace Abstract: We prove a sharp phase transition in the regularity of the extremal distribution $E^s \oplus E^u$ for $C^\infty$ volume-preserving partially hyperbolic diffeomorphisms on closed $3$-manifolds: if $E^s \oplus E^u$ is Lipschitz, then it is automatically $C^\infty$. This extends the rigidity phenomenon established by Foulon--Hasselblatt for conservative Anosov flows in dimension $3$ to the partially hyperbolic setting. This gain in regularity has several applications to rigidity problems. In particular, we study the relationship between the $\ell$-integrability condition introduced by Eskin--Potrie--Zhang and joint integrability in the conservative setting, yielding rigidity results for $u$-Gibbs measures. We also obtain several $C^\infty$ classification results for partially hyperbolic diffeomorphisms on $3$-manifolds under various assumptions. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2602.08608 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2602.08608v2 Announce Type: replace Abstract: We prove the dynamical Mordell-Lang conjecture for product of endomorphisms of an affine curve and a projective curve over $\overline{\mathbb{Q}}$. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2601.04408 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2601.04408v2 Announce Type: replace Abstract: This research examines the influence of the Coriolis parameter on the behaviour of the geophysical Korteweg-de Vries (KdV) equation. To efficiently approximate its solution, a novel surrogate framework, termed G-KdVNet, is proposed by integrating artificial neural networks with the Adomian decomposition method (ADM). In the proposed approach, ADM is first employed to generate reliable semi-analytical solution data, which are subsequently used to train the neural network model. The developed model demonstrates strong predictive capability in capturing the nonlinear dynamics of the KdV system. Numerical results indicate that the proposed model achieves improved accuracy compared with conventional baseline methods, with absolute errors of the order of e-3 for unseen data. The results suggest that the proposed ANN-ADM surrogate offers an efficient and accurate alternative for solving nonlinear geophysical models, with potential applicability to a broader class of dispersive wave equations. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2512.19430 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2512.19430v3 Announce Type: replace Abstract: We survey recent results and current challenges concerning the growth rate inequality for sphere endomorphisms, and present a number of open problems and conjectures arising in this context. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2511.21324 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2511.21324v4 Announce Type: replace Abstract: Astorg and Boc Thaler studied the dynamics of certain skew-product tangent to the identity on $\mathbb{C}^2$, with two real parameters $\alpha>1$ and $\beta$ derived from its coefficients. They proved that if there exists an increasing sequence of positive integers $(n_k)_{k\geqslant 1}$ such that $(\sigma_k)_{k\geqslant 1}:=(n_{k+1}-\alpha n_k-\beta\ln n_k)_{k\geqslant 1}$ converges, then $f$ admits wandering domains of rank one. They also proved that for $\alpha>1$ with the Pisot property, the condition that $\theta:=\frac{\beta\ln\alpha}{\alpha-1}$ is rational is sufficient for the existence of $(n_k)_{k\geqslant 1}$ such that $(\sigma_k)_{k\geqslant 1}$ converges to a cycle. They asked if this condition is necessary. When $\alpha$ is an algebraic number, we answer the question of Astorg and Boc Thaler in the affirmative. Furthermore, denoting by $P(x)\in\mathbb{Z}[x]$ the minimal polynomial of~$\alpha$, we prove that $\theta\in\frac{1}{P(1)}\mathbb{Z}$ is necessary and sufficient for the existence of $(n_k)_{k\geqslant 1}$ such that $(\sigma_k)_{k\geqslant 1}$ converges. Combined with the work of Astorg and Boc Thaler, our result provides explicit new examples of skew-products on $\mathbb{C}^2$ with wandering domains of rank one. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.15181 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15181v1 Announce Type: cross Abstract: Extrapolative prediction of complex nonlinear dynamics remains a central challenge in engineering. This study proposes a one-shot learning method to identify global frequency-response curves from a single excitation time history by learning governing equations. We introduce MEv-SINDy (Multi-frequency Evolutionary Sparse Identification of Nonlinear Dynamics) to infer the governing equations of non-autonomous and multi-frequency systems. The methodology leverages the Generalized Harmonic Balance (GHB) method to decompose complex forced responses into a set of slow-varying evolution equations. We validated the capabilities of MEv-SINDy on two critical Micro-Electro-Mechanical Systems (MEMS). These applications include a nonlinear beam resonator and a MEMS micromirror. Our results show that the model trained on a single point accurately predicts softening/hardening effects and jump phenomena across a wide range of excitation levels. This approach significantly reduces the data acquisition burden for the characterization and design of nonlinear microsystems. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.14945 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14945v1 Announce Type: cross Abstract: We focus on halo products, a class of groups introduced by Genevois and Tessera, and whose geometry mimics lamplighters. Famous examples are lampshufflers. Motivated by their work on the classifications up to quasi-isometry of these groups, we initiate a more quantitative study of their geometry. Indeed, it follows from the work of Delabie, Koivisto, Le Ma\^itre and Tessera that quantitative orbit equivalence between amenable groups is closely related to their large scale geometry, such a connection being justified by the use, in their main results, of a well-known quasi-isometry invariant: the isoperimetric profile. Inspired by their work on quantitative orbit equivalence between lamplighters, we prove a stability result for orbit equivalence of permutational halo products, going beyond the framework of standard halo products, using a new notion of orbit equivalence of pairs. Combined with our asymptotics of isoperimetric profiles obtained in an earlier article, we prove that most of these constructions are quantitatively optimal. For instance, we show that $\mathsf{Shuffler}(\mathbb{Z}^{k+\ell})$ and $\mathsf{Shuffler}(\mathbb{Z}^{k})$ are $\mathrm{L}^p$ orbit equivalent if and only if $p<\frac{k}{k+\ell}$, thus quantifying how much the geometries of these non-quasi-isometric groups differ. We finally build orbit equivalence couplings using the notion of F{\o}lner tiling sequences. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.14669 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14669v1 Announce Type: cross Abstract: Zeroth-order (ZO) methods are widely used when gradients are unavailable or prohibitively expensive, including black-box learning and memory-efficient fine-tuning of large models, yet their optimization dynamics in deep learning remain underexplored. In this work, we provide an explicit step size condition that exactly captures the (mean-square) linear stability of a family of ZO methods based on the standard two-point estimator. Our characterization reveals a sharp contrast with first-order (FO) methods: whereas FO stability is governed solely by the largest Hessian eigenvalue, mean-square stability of ZO methods depends on the entire Hessian spectrum. Since computing the full Hessian spectrum is infeasible in practical neural network training, we further derive tractable stability bounds that depend only on the largest eigenvalue and the Hessian trace. Empirically, we find that full-batch ZO methods operate at the edge of stability: ZO-GD, ZO-GDM, and ZO-Adam consistently stabilize near the predicted stability boundary across a range of deep learning training problems. Our results highlight an implicit regularization effect specific to ZO methods, where large step sizes primarily regularize the Hessian trace, whereas in FO methods they regularize the top eigenvalue. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.14464 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14464v1 Announce Type: cross Abstract: Given a thin torus $T_K$ around a knot $K\subset \mathbb{R}^3$, we construct Morse models of cord algebra $Cord(T_K)$ with $\mathbb{Z}$ and loop space coefficients. Using the Multiple time scale dynamics we identify $Cord(T_K; \mathbb{Z})$ with $Cord(K; \mathbb{Z})$. In combination with the works of Cieliebak-Ekholm-Latschev-Ng and Petrak this indirectly relates $Cord(T_K)$ to $0$-th degree Legendrian contact homology $LCH_0(\mathcal{L}^\ast_+ T_K)$ of one component of the unit conormal bundle over $T_K$. Our definition of $Cord(T_K)$ is motivated by $J$-holomorphic curves with boundary on the Lagrangian submanifold $L^\ast_+ T_K\cup\mathbb{R}^3$ with an arboreal singularity along the torus $T_K$. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.14346 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14346v1 Announce Type: cross Abstract: In this paper we consider the Toda lattice $(\mathbf{p}(t);\mathbf{q}(t))$ at thermal equilibrium, meaning that its variables $(p_j)$ and $(e^{q_j - q_{j+1}})$ are independent Gaussian and Gamma random variables, respectively. We show under diffusive scaling that the space-time fluctuations for the model's currents converge to an explicit Gaussian limit. As consequences, we deduce, (i) the scaling limit for the trajectory of a single particle $q_0$ is a Brownian motion; (ii) space-time two-point correlation functions for the model decay inversely with time, with explicit scaling distributions predicted by Spohn (Spohn, J. Phys. A 53 (2020), 265004). Our starting point is the notion that the Toda lattice can be thought of as a dense collection of many ``quasi-particles'' that interact through scattering. The core of our work is to establish that the full joint scaling limit of the fluctuations for these quasi-particles is given by a Gaussian process, called a dressed L\'evy-Chentsov field. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.15293 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15293v1 Announce Type: new Abstract: For each $d \in {1,2,3,7,11}$, let $T_d$ be the nearest-integer complex continued fraction map associated with the Euclidean ring $\mathcal{O}*d$, and let $(a_n)$ be its digit sequence. We prove two metric results for this five-system family. First, for every sequence $(u_n)*{n\ge 1}$ with $u_n \ge 1$, the set of points for which $|a_n| \ge u_n$ for infinitely many $n$ has full or zero normalized Lebesgue measure according as $\sum_{n=1}^\infty u_n^{-2}$ diverges or converges. This gives a unified Borel--Bernstein theorem, extending the Hurwitz case $d=1$ to all five Euclidean imaginary quadratic fields. Second, for any infinite set $S \subset \mathcal{O}_d$, if $\tau(S)$ denotes its convergence exponent, then the digit-restricted set $F_d(S)={z:\ a_n(z)\in S\ \text{for all } n,\ |a_n(z)|\to\infty}$ satisfies $\dim_H F_d(S)=\tau(S)/2$. More generally, for any cutoff function $f(n)\to\infty$, the set $F_d(S,f)={z\in F_d(S):\ |a_n(z)|\le f(n)\ \text{for all } n}$ is either empty or has the same Hausdorff dimension $\tau(S)/2$. The proof combines quantitative ergodic properties of the nearest-integer systems with a large-digit conformal iterated function subsystem that is $2$-decaying. We also obtain applications to sparse patterns, shrinking targets, and almost-sure $L'evy$- and Khinchine-type laws. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.15220 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15220v1 Announce Type: new Abstract: We present a mathematical model of a market with $m$ shares traded across $n$ investor groups, each one with similar motivations and trading strategies. The market of each asset consists of a fixed amount of cash and shares (no additions are allowed over time, so the system is closed), and the trading groups are influenced by trend and valuation motivations when buying or selling each asset, but follow a strategy where the purchase of one asset depends on the price of another, while the sale does not. Using these assumptions and basic microeconomic principles, the mathematical model is derived using a dynamic systems approach. We analyze the stability of the model's equilibrium points and determine the parameter conditions for such stability. First, we show that all equilibria are stable in the absence of a clear emphasis on trend-based valuation for each share. Secondly, for systems where the trading group prioritizes the valuation of each stock and the trend of the other for trading purposes, we establish stability conditions and demonstrate with numerical examples that when instability occurs, it manifests as price oscillations in the stocks. Furthermore, we argue for the existence of periodic solutions via a Hopf bifurcation, taking the momentum coefficient as the bifurcation parameter. Finally, we present examples and numerical simulations to support and expand upon the analytical results. One finding in economics and finance is the existence of cyclical behavior in the absence of exogenous factors, as determined by the momentum coefficient. In particular, a stable equilibrium price becomes unstable as trend-based trading increases. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.14964 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14964v1 Announce Type: new Abstract: In this paper, we investigate induced and nonlinear fiber topological pressure for random dynamical systems. We define a non-averaged induced fiber pressure via spanning and separated sets, characterize it as the pseudo-inverse of the classical fiber topological pressure studied previously, and establish the corresponding variational principle. We also define the nonlinear fiber pressure and prove the associated variational principles. Finally, we extend the combined theory to the higher-dimensional setting. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.14948 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14948v1 Announce Type: new Abstract: We investigate expansive solutions of the $N$-body problem in $\mathbb{R}^d$ ($d\ge2$) driven by homogeneous Newtonian potentials of degree $-\alpha$. We establish the existence of half-entire expansive motions with prescribed initial configuration and asymptotic direction for a wide range of homogeneity exponents $\alpha$. Our approach is variational and relies on the minimization of a suitably renormalized Lagrangian action, allowing us to treat in a unified framework the hyperbolic, parabolic, and hyperbolic-parabolic regimes in the sense of Chazy's classification. Beyond existence, we derive refined asymptotic expansions for all classes of expansive solutions, identifying higher-order correction terms and improving previously known growth estimates, including the classical Newtonian case $\alpha=1$. In particular, for hyperbolic-parabolic solutions, we provide a detailed description of the interplay between linear escape of cluster centers and internal parabolic dynamics, extending the cluster scattering picture to general homogeneous potentials. Finally, we interpret these solutions within the geometric framework of the Jacobi-Maupertuis metric and the weak KAM theory. In this perspective, expansive motions correspond to geodesic rays and calibrating curves for the associated Hamilton-Jacobi equation, yielding a dynamical characterization of the boundary at infinity and a refined description of global viscosity solutions. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.14759 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14759v1 Announce Type: new Abstract: While the classical Critical Depth Hypothesis (CDH) effectively explains the onset of blooms as transient instabilities, it does not fully capture the seasonal decoupling of biological rates and the long-term persistence of phytoplankton communities in fluctuating thermal environments. To address these limitations, we introduce a parsimonious framework that leverages the theory of non-autonomous dynamical systems to diagnose the stability of phytoplankton communities throughout the entire annual cycle. By linearizing the dynamics around the extinction equilibrium, we identify the invasion growth rate -formally the Floquet exponent-and derive the critical nutrient requirement ($\gamma$crit) as a bifurcation point for uniform persistence. Using end-of-the-century projections from the GFDL-ESM4 model under a high-emission scenario (SSP5-8.5), we identify a global regime shift characterized by a widespread expansion of metabolic-driven regimes, which increasingly displace regions where stability was historically governed by physical mixing. Relevance to Life Sciences. Quantitative analysis of system stability challenges CDH by demonstrating that metabolic constraints increasingly modulates phytoplankton persistence in a changing ocean. Our results, based on high-emission projections, reveal a profound physical-biological decoupling at the poles: while warming reduces the critical nutrient requirement ($\gamma$crit) facilitating persistence in previously marginal waters, this metabolic expansion is offset at poles. A 1:4 ratio between newly viable niches and ice-free deserts suggests that cryospheric retreat does not guarantee a proportional expansion of life. In addition, we identify the North Atlantic Subpolar Gyre as a ''metabolic refuge'' where mixing dynamics still anchor the ecosystem against global thermalization. By providing a ''radiography'' of the future ocean's complexity, this methodology offers a mechanistic basis to deconstruct how the dynamic balance between environmental energy and metabolic demands may determine the functional integrity of the marine biosphere under extreme anthropogenic forcing. Mathematical Content. The temperature dependence of biological rates is modeled using a thermodynamic equation, coupling population dynamics with seasonal variations in mixed layer depth and temperature. Given the non-autonomous nature of the system under annual forcing, we characterize the stability of the extinction equilibrium through its associated invasion growth rate. This rate is analytically derived as the Floquet exponent $\lambda$P , which provides a rigorous condition for uniform persistence (Theorem 3.2). The numerical analysis of this exponent, projected onto a global scale, quantifies the relative influence of environmental drivers on the stability threshold $\gamma$crit. This allows for the definition of the thermal dominance index (DT ), a metric that identifies the geographic transition from mixing-driven to metabolic-driven ecological control. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2601.20640 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2601.20640v2 Announce Type: replace-cross Abstract: We consider on an arbitrary Riemannian manifold $M$ the \textit{Leibenson equation} $\partial _{t}u=\Delta _{p}u^{q}$, that is also known as a \textit{doubly nonlinear evolution equation}. We prove that if $p>1, q>0$ and $pq\geq 1$ then the Cauchy-problem \begin{equation*} \left\{\begin{array}{ll}\partial _{t}u=\Delta _{p}u^{q} &\text{in}~M\times (0, \infty), \\u(x, 0)=u_{0}(x)& \text{in}~M,\end{array}\right.\end{equation*} has a unique weak solution for any $u_{0}\in L^{1}(M)\cap L^{\infty}(M)$. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2310.20526 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2310.20526v2 Announce Type: replace-cross Abstract: In this paper, we focus on estimating measure upper bounds of nodal sets of solutions to the following boundary value problem \begin{equation*} \left\{ \begin{array}{lll} \Delta u+Vu=0\quad \mbox{in}\ \Omega,\\[2mm] u=0\quad \mbox{on}\ \partial\Omega, \end{array}\right. \end{equation*} where $V\in W^{1,\infty}(\Omega)$ is a potential function, and $\Omega \subset \mathbb{R}^n$ ($n \geq 2$) is a bounded domain whose boundary is of class $C^{1,\alpha}$ for any $0<\alpha<1$. By developing a delicate dividing iteration procedure, we show that upper bound of the $(n-1)$-dimensional Hausdorff measure of the nodal set of $u$ in $\Omega$ is $$C\Big(1+\log\left(\|\nabla V\|_{L^{\infty}(\Omega)}+1\right)\Big)\cdot\left(\|V\|_{L^{\infty}(\Omega)}^{\frac{1}{2}}+\|\nabla V\|_{L^{\infty}(\Omega)}^{\frac{1}{2}}+1\right),$$ provided $V$ is analytic, here $C$ is a positive constant depending only on $n$ and $\Omega$. In particular, if $\|\nabla V\|_{L^{\infty}(\Omega)}$ is small, the upper bound for the measure of the nodal set of $u$ is $C\left(\|V\|^{\frac{1}{2}}_{L^{\infty}(\Omega)}+1\right)$, which is sharp in the sense of a famous conjecture of Yau. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2604.08822 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.08822v2 Announce Type: replace Abstract: We prove that for a residual (and hence dense) subset $\mathcal{G}$ of Riemannian metrics on $S^{n+1}$ in the $C^{3}$ topology, no area-minimizing integral $n$-current that is a boundary admits a singular tangent cone which is linearly stable in the Euclidean sense. The proof proceeds in three stages. First, we develop a perturbation theorem: given any area-minimizer possessing an isolated singularity whose unique tangent cone $C$ is linearly stable, we construct an explicit $C^{3}$-small metric perturbation that destroys the compatibility conditions required for $C$ to persist as a tangent cone. The construction rests on the Hardt--Simon asymptotic expansion near isolated singularities, the spectral theory of the Jacobi operator on the cross-section of $C$, and a surjectivity argument showing that the map from compactly supported metric variations to forcing terms in the linearised minimal-surface equation on $C$ has dense range. Second, we establish that the set of metrics admitting no area-minimizer with a prescribed cone type as tangent cone is open, using compactness of integral currents and upper-semicontinuity of the density function. Third, we assemble these ingredients via a Baire category argument, intersecting countably many open dense sets to obtain the residual set $\mathcal{G}$. An extension to non-isolated singularities is outlined using Federer--Almgren dimension reduction. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2602.10913 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2602.10913v2 Announce Type: replace Abstract: In this paper we classify the low energy $\varepsilon$-harmonic maps from the surfaces of constant curvature with positive genus into the round sphere. We find that all such maps with degree $\pm1$ are all quantitively close to a bubble configuration with bubbles forming at special points on the domain with bubbling radius proportional to $\varepsilon^{1/4}$. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2507.05459 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2507.05459v2 Announce Type: replace Abstract: Llarull's scalar curvature rigidity theorem states that a 1-Lipschitz map $f: M\to S^n$ from a closed connected Riemannian spin manifold $M$ with scalar curvature $\mathrm{scal}\ge n(n-1)$ to the standard sphere $S^n$ is an isometry if the degree of $f$ is nonzero. We investigate if one can replace the condition $\mathrm{deg}(f)\neq0$ by the weaker condition that $f$ is surjective. The answer turns out to be "no" for $n\ge3$ but "yes" for $n=2$. If we replace the scalar curvature by Ricci curvature, the answer is "yes" in all dimensions. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2507.00471 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2507.00471v2 Announce Type: replace Abstract: We prove that a sub-Riemannian manifold equipped with a full-support Radon measure is never $\mathrm{CD}(K,N)$ for any $K\in \mathbb{R}$ and $N\in (1,\infty)$ unless it is Riemannian. This generalizes previous non-CD results for sub-Riemannian manifolds, where a measure with smooth and positive density is considered. Our proof is based on the analysis of the tangent cones and the geodesics within. Secondly, we construct new $\mathrm{RCD}$ structures on $\mathbb{R}^n$, named cone-Grushin spaces, that fail to be sub-Riemannian due to the lack of a scalar product along a curve, yet exhibit characteristic features of sub-Riemannian geometry, such as horizontal directions, large Hausdorff dimension, and inhomogeneous metric dilations. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2412.21050 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2412.21050v2 Announce Type: replace Abstract: We prove parabolic versions of several known gap theorems in classical Yang-Mills theory. On an $\mathrm{SU}(r)$-bundle of charge $\kappa$ over the 4-sphere, we show that the space of all connections with Yang-Mills energy less than $4 \pi^2 \left( |\kappa| + 2 \right)$ deformation-retracts under Yang-Mills flow onto the space of instantons, allowing us to simplify the proof of Taubes's path-connectedness theorem. On a compact quaternion-K\"ahler manifold with positive scalar curvature, we prove that the space of pseudo-holomorphic connections whose $\mathfrak{sp}(1)$ curvature component has small Morrey norm deformation-retracts under Yang-Mills flow onto the space of instantons. On a nontrivial bundle over a compact manifold of general dimension, we prove that the infimum of the scale-invariant Morrey norm of curvature is positive. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2105.10711 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2105.10711v3 Announce Type: replace Abstract: We construct a one-parameter family of embedded doubly periodic minimal surfaces of genus three with four parallel ends. The Weierstrass data for each surface of the family are given and the two dimensional period problem is solved. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2604.15192 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15192v1 Announce Type: cross Abstract: Symplectic and complex toric quasifolds are a generalization of toric manifolds and orbifolds to the nonrational case. In this paper, we reframe these notions from the viewpoint of algebraic geometry. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2604.15087 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15087v1 Announce Type: cross Abstract: This article provides a survey of gauge theory for families, with a particular focus on its applications to diffeomorphism groups of $4$-manifolds that were developed during the period 2021--2025. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2604.15071 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15071v1 Announce Type: cross Abstract: This article surveys gauge theory for families and its applications to the comparison between the diffeomorphism group and the homeomorphism group of $4$-manifolds, up to 2021. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2604.15008 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15008v1 Announce Type: cross Abstract: Semiclassical analysis and noncommutative geometry are two pillars of quantum theory. It's only recently that bridges between them have been emerging. In this monograph, we combine various techniques from functional analysis and spectral theory to obtain semiclassical Weyl laws and extensions of Connes' integration formula for a large class of noncommutative manifolds (i.e., spectral triples). These results generalize and simplify recent results of McDonald-Sukochev-Zanin. In particular, all the regularity assumptions and restrictions on dimension there are removed in our approach. Moreover, the Tauberian condition used by McDonald-Sukochev-Zanin is replaced by a weaker spectral theoretic condition, called Condition (W). That condition holds in fairly greater generality and significantly open the scope of applicability of the main results. We also give Tauberian conditions that imply Condition (W). These Tauberian conditions are easier to check in practice than the Tauberian condition of McDonald-Sukochev-Zanin and are satisfied in numerous examples. The need for these conditions was highlighted by Alain Connes in an online seminar. The main results of this memoire are illustrated by semiclassical Weyl's laws and integration formulas in the following settings: (i) Dirichlet and Neumann problems on Euclidean domains with smooth boundaries; (ii) closed Riemannian manifolds; (iii) open manifolds with conformally cusp metrics of finite volume; (iv) quantum tori; and (v) sub-Riemannian manifolds. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2604.15200 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15200v1 Announce Type: new Abstract: We study the question of whether a sequence of non-instanton Yang-Mills connections can limit to a bubbling configuration composed only of instantons. In the case that the Uhlenbeck limit and the bubbles are of opposite charge, we determine an obstruction coming from deformations of the Uhlenbeck limit. As an application, we prove that instantons are the only solutions of the $\mathrm{SU}(2)$ Yang-Mills equation on $\mathbb{R}^4$ with energy less than $4\pi^2 \left( |\kappa| + 2 \right) + \varepsilon_\kappa,$ where $\kappa$ is the charge. We also prove discreteness of the energy spectrum on the trivial $\mathrm{SU}(2)$-bundle in the range $\left[ 0, 16 \pi^2 \right).$ Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2604.15198 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15198v1 Announce Type: new Abstract: We provide a comparatively simple proof of the dynamical stability of Ricci flow near a linearly stable Ricci-flat ALE metric with integrable deformations. Our proof relies on the equivalence between integrability and an "almost-orthogonality" property of the Ricci-DeTurck tensor, allowing us to analyze the latter directly. We obtain our main results in weighted Holder spaces and then show how to recover the $L^p$-stability theorems of Deruelle-Kroncke and Kroncke-Petersen. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2604.15159 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15159v1 Announce Type: new Abstract: We establish existence and uniqueness results for asymptotically locally Euclidean (ALE) and asymptotically locally flat (ALF) gravitational instantons. In particular, we prove the existence of a unique, Ricci-flat, toric ALE and ALF gravitational instanton, for every admissible rod structure, that is smooth up to possible conical singularites. We also give an elementary proof that any toric ALE or ALF self-dual instanton is a multi-Eguchi-Hanson or multi-Taub-NUT solution. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2604.14804 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14804v1 Announce Type: new Abstract: In this paper, inspired by the work of Guan and Li (2015), we introduce a fourth-order centro-equiaffine invariant curve flow via the affine Minkowski formula. Without any smallness assumptions on the initial curve, we establish the long-time existence of the flow and prove that, as $t \to +\infty$, the evolving curve preserves its enclosed area and converges smoothly to a round circle up to the action of $\mathrm{SL}(2)$. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2604.14704 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14704v1 Announce Type: new Abstract: We prove existence, uniqueness and structure results for complete noncompact 7-dimensional G2-holonomy metrics with ALC (asymptotically locally conical) asymptotics. We regard such spaces as G2-analogues of ALF gravitational instantons in 4-dimensional hyperk\"ahler geometry. Our main results include the existence of a G2-analogue of the Atiyah-Hitchin metric in 4-dimensional hyperk\"ahler geometry, the existence of a good moduli theory for ALC G2-holonomy metrics and rigidity results for ALC G2-metrics in terms of the symmetries of their asymptotic model. The analytic toolkit needed to prove all these results is a robust Fredholm theory for the natural geometric linear elliptic operators on ALC spaces. We provide a self-contained derivation of this Fredholm theory for arbitrary Riemannian manifolds with ALC asymptotics. Since our ALC Fredholm theory does not rely on imposing any holonomy reduction or curvature conditions it may also be of utility beyond the setting of ALC special holonomy metrics. As one such application of our general Fredholm theory we prove some Hodge-theoretic results on general ALC spaces. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2604.14675 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14675v1 Announce Type: new Abstract: Utilizing the Weierstrass representation for embedded doubly periodic minimal surfaces with parallel ends, we construct entire singly periodic graphs of spacelike maximal surfaces with isolated cone-like singularities in the Lorentz-Minkowski 3-space. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2604.14600 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14600v1 Announce Type: new Abstract: This paper studies the large $p$ asymptotics of three geometric quantities on complete noncompact Riemannian manifolds: the $p-$capacity of a compact set, the first Dirichlet $p-$eigenvalue, and the Maz'ya constant, thereby offering a new perspective on the study of such manifolds. We introduce the infinity capacity $\mathcal{C}(\Omega)$, the infinity eigenvalue $\Lambda(M)$, and the Maz'ya limit $\mathcal{M}(M)$, and establish the general inequality, for any $\Omega\subset M$, $$ \mathcal{V}(M) \ge \mathcal{C}(\Omega) \ge \Lambda(M) = \mathcal{M}(M), $$ where $\mathcal{V}(M)$ is the volume entropy. Under geometric conditions such as isoperimetric control of balls, rotational symmetry, or curvature bounds, these quantities coincide and equal $\mathcal{V}(M)$ or the dimension. We also provide examples showing strict inequalities hold. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2604.14537 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14537v1 Announce Type: new Abstract: For any complete and noncompact manifold $M$ with $\mathrm{Ric}\ge 0$, we define a function $\mathrm{RV}(s)$ that describes the growth of relative volume asymptotically $$\mathrm{RV}(s)=\limsup_{r\to\infty} \dfrac{\mathrm{vol} B_{rs}(p)}{\mathrm{vol} B_r(p)},\quad s\ge 1.$$ Then we study the fundamental groups of such manifolds with slow relative volume growth and sublinear diameter growth. We show that if $\mathrm{RV}(s)\ll s^2$ as $s\to\infty$, then $\pi_1(M)$ is almost abelian; if $\mathrm{RV}(s)\ll s^{1+\delta}$ for some $\delta\in (0,1)$ and the Ricci curvature is positive at a point, then $\pi_1(M)$ is finite. These results generalize our previous work on complete manifolds with $\mathrm{Ric}\ge 0$ and linear (minimal) volume growth. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2506.20368 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2506.20368v3 Announce Type: replace-cross Abstract: We establish norm inequalities for fractional powers of degenerate Laplacians, with degeneracy being determined by weights in the Muckenhoupt class $A_2(\mathbb{R}^n)$, accompanied by specific additional reverse H\"older assumptions. This extends the known results for classical Riesz potentials. The approach is based on size estimates for the degenerate heat kernels. The approach also applies to more general weighted degenerate operators. Maths/math.CA_(Classical_Analysis_and_ODEs) https://arxiv.org/abs/2604.14827 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14827v1 Announce Type: cross Abstract: A. Albouy and R. Moeckel in 2000 found some interesting inequalities related to the inverse problem for collinear (Moulton) central configurations: the Pfaffian of a certain matrix is positive since all coefficients of some polynomials are positive, for the Newtonian (interaction potential $1/r$ and $n\leq 6$). They conjectured that for all $n$ such Pfaffians, for the Newtonian case, are positive. In this article we analyze further the problem, and we prove that such inequalities hold true in more general cases (potentials with log-convex derivative, such as those with homogeneity parameters $\alpha>0$, for all even $n\leq 14$). Maths/math.CA_(Classical_Analysis_and_ODEs) https://arxiv.org/abs/2604.14662 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14662v1 Announce Type: new Abstract: We study dimensions of sets projected to an $(n-2)$-dimensional family of hyperplanes in $\mathbb{R}^n$ under curvature conditions. Let $n\ge 3$ and $\Sigma \subset S^{n-1}$ be an $(n-2)$-dimensional $C^2$ manifold such that $\Sigma$ has non-vanishing geodesic curvature ($n=3$)/sectional curvature $>1$ ($n \ge 4)$. Let $Z \subset \mathbb{R}^{n}$ be analytic with $\dim Z \le n-2$ and $0 n-2$, if in addition $\pi_{T_yS^{n-1}}(Z) \le n-2$ for some $y \in S^{n-1}$, we show that $\dim \pi_{T_xS^{n-1}}(Z) = \min\{\dim Z, n-1\}$ for $\mathcal{H}^{n-2}$-a.e. $x \in \Sigma$. Maths/math.CA_(Classical_Analysis_and_ODEs) https://arxiv.org/abs/2604.14658 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14658v1 Announce Type: new Abstract: A new weighted Hardy-type inequality for functions from the Sobolev space $W_{p}^{1}$ is proved. It is assumed that functions vanish on small alternating pieces of the boundary. The proved inequality generalizes the classical known weighted Hardy-type inequalities. Maths/math.CA_(Classical_Analysis_and_ODEs) https://arxiv.org/abs/2604.14429 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14429v1 Announce Type: new Abstract: In this paper we study higher-order difference equations which can be written as follows: $$ \mathbf{J} (y_0,y_1,...)^T = \lambda^N (y_0,y_1,...)^T, $$ where $\mathbf{J}$ is a $(2N+1)$-diagonal bounded banded matrix ($\mathbf{J}=(g_{m,n})_{m,n=0}^\infty$, $| g_{m,n} | 0$; and $g_{k,l}=0$ if $|k-l|>N$), $y_j$s are unknowns, $\lambda$ is a complex parameter, $N\in\mathbb{N}$. It is assumed that all $g_{k,k+N}$ and $g_{l-N,l}$ are nonzero. Two special cases are considered: \noindent \textit{Case A}: The matrix $\mathbf{J}$ is complex symmetric, i.e. $\mathbf{J} = \mathbf{J}^T$. \noindent \textit{Case B}: The matrix $\mathbf{J}$ is such that $g_{k,k+N}=1$, $k=0,1,2,...$. Notice that this condition can be attained by changing $y_j$s by their multiples. In both cases there exists a \textit{positive} matrix measure $M$ on a circle in the complex plane such that polynomial solutions satisfy some orthogonality relations. Namely, in case~A this is related to a $J$-orthogonality in the Hilbert space $L^2(M)$ ($J$ is a complex conjugation). In case~B we have a left $J$-orthogonality in $L^2(M)$. As a tool, a related matrix moment problem is studied. A complex rank-one perturbation of a free Jacobi matrix is discussed. Maths/math.CA_(Classical_Analysis_and_ODEs) https://arxiv.org/abs/2502.06520 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2502.06520v3 Announce Type: replace-cross Abstract: Discrete Morse theory helps us compute the homology groups of simplicial complexes in an efficient manner. A "good" gradient vector field reduces the number of critical simplices, simplifying the homology calculations by reducing them to the computation of homology groups of a simpler chain complex. This homology computation hinges on an efficient enumeration of gradient trajectories. The technique of cancelling pairs of critical simplices reduces the number of critical simplices, though it also perturbs the gradient trajectories. In this article, in a purely combinatorial manner, we derive an explicit formula for computing the modified boundary operators after cancelling a critical pair, in terms of the original boundary operators. The same formula can be obtained through a sequence of elementary row operations on the original boundary operators. Thus, it eliminates the need of enumeration of the new gradient trajectories. We also obtain a similar result for coboundary operators. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2501.17927 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2501.17927v3 Announce Type: replace-cross Abstract: These extended lecture notes survey a novel derivation of anyonic topological order (as seen in fractional quantum Hall systems) on single magnetized M5-branes probing Seifert orbi-singularities ("geometric engineering" of anyons), which we motivate from fundamental open problems in the field of quantum computing. The rigorous construction is non-Lagrangian and non-perturbative, based on previously neglected global completion of the M5-brane's tensor field by flux-quantization consistent with its non-linear self-duality and its twisting by the bulk C-field. This exists only in little-studied non-abelian generalized cohomology theories, notably in a twisted equivariant (and "twistorial") form of unstable Cohomotopy ("Hypothesis H"). As a result, topological quantum observables form Pontrjagin homology algebras of mapping spaces from the orbi-fixed worldvolume into a classifying 2-sphere. Remarkably, results from algebraic topology imply from this the quantum observables and modular functor of abelian Chern-Simons theory, as well as braid group actions on defect anyons of the kind envisioned as hardware for topologically protected quantum gates. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2603.18550 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2603.18550v2 Announce Type: replace Abstract: A generalization of the Borsuk-Ulam theorem to Stiefel manifolds is considered. This theorem is applied to derive bounds on $d$ that guarantee-for a given set of $m$ measures in $\mathbb{R}^d$-the existence of $k$ mutually orthogonal hyperplanes, any $n$ of which partition each of the measures into $2^n$ equal parts. If $n=k$, the result corresponds to the bound obtained in [11], but with the stronger conclusion that the hyperplanes are mutually orthogonal. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2508.14341 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2508.14341v2 Announce Type: replace Abstract: In this paper, we classify the homotopy types of the total spaces of $S^{2k-1}$-bundles (or fibrations) over $S^{2k}$ for $2\leq k\leq 6$. One of the two key new ingredients in the argument is the new necessary and sufficient conditions for a CW complex to be homotopy equivalent to the total space of a sphere bundle (fibration); the other is a formula relating the attaching map of the top cell of the total space and the characteristic map of a sphere bundle for $k=2,4$. When $k=4$, the classification results provide a negative answer to the conjecture in [6]. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2401.14146 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2401.14146v3 Announce Type: replace Abstract: In this note, (rational) Betti numbers of homotopy colimits for toric diagrams and their classifying spaces are described in terms of sheaf cohomology over CW posets. We prove for any $T$-diagram $D$ over any CW poset that Cohen-Macaulayness (over $\mathbb{Q}$) of the $T$-action on $hocolim\ D$ is equivalent to acyclicity for a certain sheaf. The ordinary and bigraded Betti numbers are computed for skeletons of equivariantly formal spaces from this class (in particular, of compact smooth toric manifolds). Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2604.15262 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15262v1 Announce Type: cross Abstract: We develop a framework for detecting regime transitions in dynamical systems using the Mixup Euler Characteristic Profile (Mixup ECP) -- the Euler characteristic of the geometric intersection of ball unions around adjacent delay-embedded trajectory segments, viewed as a function of filtration scale. The Mixup ECP provides a detection statistic with a built-in null and guaranteed stability. We formalize regime detection as a low-side-permutation test, establish its validity and consistency, and introduce a multi-delay extension that automatically selects the most informative dynamical timescale. Complementing the topological signal with Complexity Variance, Higuchi fractal dimension, and a rolling mean baseline, the four-signal combined method achieves $9.50$ days MAE on Indian monsoon onset (Nepal target) -- a $32\%$ improvement over the rolling mean baseline and $9\%$ over CUSUM. Validated on the Lorenz system, logistic map, and three monsoon systems spanning both hemispheres (Indian/Nepal, Indian/Kerala, Western North Pacific), plus ENSO and a synthetic EEG dataset, the framework adds value precisely when the transition is gradual or obscured by noise. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2604.15243 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15243v1 Announce Type: cross Abstract: Given a group $G$ and an integer $n \geq 0$, let $\mathcal{F}_n$ denote the family of all virtually abelian subgroups of $G$ of rank at most $n$. In this article, we show that for each $n \geq 1$, the minimal dimension of a model for the classifying space $E_{\mathcal{F}_n}G$ for the pure braid group of a surface of non-negative Euler characteristic with at least one boundary component or one puncture is equal to the virtual cohomological dimension of $G$ plus $n$. We prove an analogous result for the full braid group of the sphere. As an application, we compute the minimal dimension of a model for the classifying space associated to the family of amenable subgroups of pure surface braid groups. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2604.15142 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15142v1 Announce Type: cross Abstract: We introduce a notion of parity for formal morphisms between invertible objects and use it to prove a corresponding coherence theorem. Parity is conceptually similar to the sign of underlying permutations, but not defined as such. To give complete details, this work includes a thorough treatment of the free permutative category on an invertible generator, its skeletal model, known as the super integers, and an equivalence between them classified by the pair of integers $\pm$1. Our approach is organized and clarified as an application of 2-monadic algebra, particularly the concept of flexibility and the Lack model structure. The final section contains a number of examples applying the main results. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2604.14929 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14929v1 Announce Type: cross Abstract: In this paper, we study homotopical analogues of the Cantor-Bendixson derivative. For each $n\geq 0$, the "$\pi_n$-wild set" $\mathbf{w}_n(X)$ of a topological space $X$ is the subspace of $X$ consisting of the points at which there exists a shrinking sequence of essential based maps $S^n\to X$. Since the operator $\mathbf{w}_n$ permits iteration, every given space $X$ yields a descending transfinite sequence of nested subspaces $\{\mathbf{w}_n^{\kappa}(X)\}_{\kappa}$ that stabilizes at some smallest ordinal $\mathbf{wrk}_n(X)$ called the "$\pi_n$-wild rank" of $X$. We show that the entire transfinite sequence $\{ho(\mathbf{w}_n^{\kappa}(X))\}_{\kappa}$ of homotopy types is a homotopy invariant of $X$ and that $\mathbf{wrk}_n(X)$ can be an arbitrary countable ordinal when $X$ is an $n$-dimensional Peano continuum. It remains open if there exists a continuum $X$ with uncountable $\pi_n$-wild rank. This difficulty motivates the parallel study a basepoint-free version $\mathbf{fwrk}_n(X)$, called the "free $\pi_n$-wild rank" of $X$. We show that for every continuum $X$, $\mathbf{fwrk}_n(X)$ is always countable and can be any countable ordinal. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2604.14536 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14536v1 Announce Type: cross Abstract: Oriented cohomology theories provide a general framework to perform intersection-theory-type calculus. The Chow ring, algebraic $K$-theory, and Levine--Morel's algebraic cobordism are all instances of such theories satisfying $\mathbb A^1$-invariance. Topological Hochschild homology, topological cyclic homology, and Hodge cohomology are important examples of theories without $\mathbb A^1$-invariance. In this paper, we prove an additive blowup formula for oriented cohomology theories in the non-$\mathbb A^1$-invariant category of motivic spectra, developed by Annala, Hoyois, and Iwasa. Then, we specialize to $\mathbb A^1$-invariant theories and give presentations of oriented cohomology rings of the blowup of a smooth scheme along a smooth center. We compute explicit examples of such presentations for the cases of del Pezzo surfaces, the blowup of $\mathbb P^3$ along the twisted cubic, and the blowup of $\mathbb P^5$ along the Veronese surface, which can be identified with the moduli space of complete conics. We demonstrate that one can recover solutions to classical enumerative geometry problems, such as Steiner's $3264$ conics, using arbitrary oriented cohomology theories. Finally, we give a presentation of oriented cohomology rings of $\overline M_{0,n}$, which generalizes Keel's presentation of the Chow ring. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2604.15265 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15265v1 Announce Type: new Abstract: Determining whether two graphs are isomorphic is a fundamental problem with practical applications in areas such as molecular chemistry or social network analysis, yet it remains a challenging task, with exact solutions often being computationally expensive. We address this task using persistent homology built on motif-based filtrations of graphs, a method from topological data analysis that summarizes the shape of data by tracking the persistence of structural features along filtrations. Specifically, we use edge-weighting schemes based on the densities of triangles, chordless squares, and chordless pentagons, which have been shown to be effective for detecting network dimensionality. Our cycle-density filtrations distinguish non-isomorphic graphs perfectly or nearly perfectly across four demanding graph families, many of which exhibit symmetries. We outperform curvature-based, degree-based, and Vietoris--Rips filtrations, and match or exceed the accuracy of egonet-distance methods while incurring a lower computational cost. The expressive power of our filtrations goes beyond isomorphism testing: because they capture rich structural information from graphs, they consistently achieve top performance on property prediction tasks using real-world data, and exhibit high sensitivity to edge rewiring and removal. Together, these findings establish cycle-density filtrations as an effective and computationally tractable framework for graph comparison and characterization, bridging topological data analysis and network science. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2604.14913 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14913v1 Announce Type: new Abstract: If a finite group $G$ acts on a rational homology manifold, then the orbit space is well-known to be a rational homology manifold again. We consider here actions on spaces that may be much more singular. If the $G$-space is a Witt pseudomanifold, which includes all arbitrarily singular complex pure-dimensional algebraic varieties, then we prove that the orbit space is again a Witt pseudomanifold. In the compact oriented situation, this implies that the orbit space possesses characteristic L-classes, as defined by Goresky and MacPherson. We then construct Atiyah-Singer-Zagier type equivariant L-classes for such $G$-pseudomanifolds which serve, as we show by establishing an averaging formula, as a tool to compute the Goresky-MacPherson L-class of the orbit space. The construction of the equivariant class builds on intersection homological transfer properties and on recent joint K-theoretic work with Eric Leichtnam and Paolo Piazza, which established a G-signature theorem on Witt pseudomanifolds. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2604.14840 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14840v1 Announce Type: cross Abstract: We study the minimization problem for eigenvalues of the Dirac operator within a fixed conformal class on a closed spin Riemannian manifold. We establish a criterion for the existence of a minimizer for this variational problem, focusing specifically on the case of closed surfaces. Furthermore, we apply our results to derive isoperimetric inequalities for the Dirac operator on the two-dimensional sphere, providing a complete characterization of its conformal spectrum. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14812 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14812v1 Announce Type: cross Abstract: We present a time-dependent extension of logarithmic perturbation theory for nonrelativistic quantum dynamics governed by the Schr\"odinger equation, in which the logarithm of the wave function is expanded in powers of a coupling constant. The resulting hierarchy of equations defining the perturbative corrections is governed by a gauge-rotated Hamiltonian of the unperturbed system and leads to closed-integral expressions for the time-dependent corrections based on Duhamel's formula. This closed-integral structure of corrections is a hallmark of time-independent logarithmic perturbation theory and is preserved in the present extension. This structure, in particular, provides a computable expression for the instantaneous energy shift. Furthermore, dynamic energy shifts arise naturally within this framework in the form of time-averaged expectation values of pseudopotentials and can be related, for example, to AC Stark shifts and electric polarizabilities. As an illustration, we apply the method to the harmonic oscillator and the hydrogen atom, both driven by a time-dependent laser field. The harmonic oscillator provides a proof of principle for which the exact solution is recovered, while the hydrogen atom illustrates the method applied to atomic systems. Supported by numerical simulations, we demonstrate the applicability to obtain relevant physical observables with high accuracy. The present approach offers a promising alternative for analytical studies of time-dependent multi-photon processes in the perturbative regime. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14752 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14752v1 Announce Type: cross Abstract: We consider a class of stochastic damped semilinear wave equations, in the small-mass limit. It has previously been established that the solution converges to the solution of a stochastic semilinear heat equation. In this work we exhibit strong and weak rates of convergence in this Smoluchowski--Kramers approximation result. The rates depend on the regularity of the driving Wiener process. For instance, for trace-class noise the strong and weak rates of convergence are $1$, whereas for space-time white noise (in dimension $1$) the strong and weak rates of convergence are $1/2$ and $1$ respectively. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14715 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14715v1 Announce Type: cross Abstract: On generalized Heisenberg-type groups $\mathbb{G}(2n,m,\mathbb{U},\mathbb{W})$, we give uniform volume estimates for the ball defined by a large class of Carnot-Carath\'{e}odory distances, and establish weak (1, 1) $O(C^m \, n)$-estimates for associated centered Hardy-Littlewood maximal functions, extending the results in \cite{BLZ25}. As a by-product, we establish uniformly volume doubling property on Heisenberg groups for a class of left-invariant Riemannian metrics. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14393 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14393v1 Announce Type: cross Abstract: We describe a method to prove new integral inequalities for stable minimal hypersurfaces in Euclidean space. As an application, we give a simple proof that complete, two sided, stable minimal hypersurfaces in $\mathbb{R}^4$ are hyperplanes. A core part of the argument hinges on the fact that stable minimal hypersurfaces in non-negatively curved spaces are examples of manifolds with a spectral Ricci curvature lower bound; in particular, we prove a sharp pointwise gradient estimate for the Green kernel on non-parabolic manifolds with spectral Ricci lower bounds, extending previous work by Colding. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14366 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14366v1 Announce Type: cross Abstract: We study the Ricci-Bourguignon flow on warped product manifolds with noncompact base. This setting leads naturally to a parabolic partial differential equation on the space of smooth warping functions, arising from the necessary and sufficient conditions for a warped metric to evolve under the flow. One of our main results establishes a gradient estimate for this equation, providing the analytic input for the geometric applications developed herein and, in particular, recovering classical gradient estimates for the heat equation under the Ricci flow. Furthermore, we show how to construct explicit warped solutions to the Ricci-Bourguignon flow and present examples that are not only of independent interest but also illustrate and support our results Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.15206 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15206v1 Announce Type: new Abstract: The study of Sobolev and Poincar\'e inequalities for differential forms in Carnot groups and in the more general sub-Riemannian setting is still an open problem in its full generality. One may conjecture that, for general Carnot groups, these inequalities are expressed in terms of suitable graded Lebesgue norms. In recent years, many results have been obtained, both in the Euclidean setting and in the Heisenberg groups, as well as for contact manifolds with bounded geometry. There are also some results for general Carnot groups; however, these do not cover the problem in its full generality. In this paper, we consider a particular Carnot group, the so-called Cartan group (a free Carnot group, of step $3$ with $2$ generators), which provides a natural testing ground for these questions, since its step-three structure already exhibits several phenomena that do not occur in the Heisenberg groups. In this setting, we are able to prove global Poincar\'e and Sobolev-Gaffney inequalities for differential forms. With the aim of obtaining sharp estimates, we replace the de Rham complex of differential forms with the Rumin complex. The case $p>1$ is carried out after establishing an $L^p$-Hodge decomposition with homogeneous Sobolev classes. We are able to consider also the endpoint case $p=1$; however, as in Euclidean setting, when $p=1$, the operator we deal with provides only weak-type estimates which do not yield a Hodge decomposition analogous to the case $p>1$. Therefore, in this situation the proof follows a different approach, relying on a recent result proved in \cite{BT}. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.15183 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15183v1 Announce Type: new Abstract: We investigate the asymptotic behavior of the solutions to the Neumann sieve problem for the Poisson equation in a thin, randomly perforated domain. The perforations (sieve-holes) are generated by a stationary marked point process. According to the scaling between the domain thickness and the typical hole size, three distinct limiting regimes emerge. We also identify the optimal stochastic integrability condition on the random hole radii that guarantees stochastic homogenization, even in the presence of clustering holes. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.15053 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15053v1 Announce Type: new Abstract: We improve previous results on dispersive decay for 1D Klein- Gordon equation. We develop a novel approach, which allows us to establish the decay in more strong norms and weaken the assumption on the potential. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14997 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14997v1 Announce Type: new Abstract: We consider the Euler-Poisson system for ions where the electrons are given by a Maxwell-Boltzmann distribution, and we investigate the existence of one-dimensional periodic traveling waves. More precisely, we first establish the existence of a smooth global branch of bifurcation emanating from a constant equilibrium. We then construct a singular traveling wave emerging as the limiting profile at the end of the global curve of bifurcation. Our analysis accommodates a wide class of pressure laws and provides a comprehensive characterization of both smooth and singular traveling waves. A central difficulty in this model arises from the exponential nonlinearity, induced by the nonlocal Poisson-Boltzmann equation, which prevents any explicit representation of the electron field in terms of the ion density. This poses significant obstacles compared to previous studies on related models, where such explicit formulas were crucial for global bifurcation arguments. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14923 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14923v1 Announce Type: new Abstract: We consider the following class of mixed local-nonlocal equations: \begin{align}\label{abs}\tag{$\mathcal{P}$} -\Delta_p u + (-\Delta)_p^s u = V |u|^{p-2}u \text{ in } \Omega, \end{align} where $s \in (0,1), p \in (1, \infty)$, and the weight function $V$ lies in scaling subcritical Lebesgue space $L^q(\Omega)$ where $q>\frac{d}{p}$ when $d>p$ and $q>1$ when $d \le p$. We establish Harnack inequality for weak solution and weak Harnack inequality for weak supersolution to ($\mathcal{P}$). Our approach is based on the De Giorgi-Nash-Moser theory, the expansion of positivity and estimates involving a tail term. Our results also apply to integro-differential operators, with the prototype given by $(-\Delta)_p^s$. This work generalizes some regularity results of Garain-Kinnunen (Trans. Am. Math. Soc., 375(8), 2022) and Garain (Nonlinear Anal., 256, 2025) to the setting of general weight functions. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14916 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14916v1 Announce Type: new Abstract: We consider stationary $p$-Schr\"odinger equations on the whole space with integrable data and potentials that are confining in measure. We introduce asymptotic energy solutions in an asymptotic $L^p$ framework and establish existence and uniqueness in the degenerate range $p\ge2$. The proof relies on a new Rellich$\unicode{x2013}$Kondrachov-type compactness theorem of independent interest, which provides sufficient conditions for families of Sobolev functions to be precompact in asymptotic $L^p$ spaces, without any dimension-dependent restriction on the exponent. For data in the duality regime $L^1(\mathbb{R}^n)\cap L^{p'}(\mathbb{R}^n)$, asymptotic energy solutions coincide with weak energy solutions. We also show that additional compactness assumptions yield localized entropy-type solutions and, under suitable local regularity, distributional solutions. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14911 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14911v1 Announce Type: new Abstract: We analyse the effect of expansion in Newtonian cosmology on the asymptotic behaviour of charged self-interacting plasmas close to Poisson equilibria. To this end, we study the Vlasov-Poisson system on the phase space of a $3$-torus which is expanding with respect to the scale factor $a(t)$. We show that, for $a(t)=t^q$ with $q\in(0,\frac12)$, solutions to this system exhibit nonlinear Landau damping for initial data that is small with respect to a suitably strong Gevrey class, i.e., the charge density contrast of the plasma decays superpolynomially. For larger choices of $q$ within this range, the initial data requirements become stricter while the decay weakens. To our knowledge, this is the first result showing Landau damping in a cosmological setting. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14852 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14852v1 Announce Type: new Abstract: We investigate the focusing and defocusing energy-critical stochastic nonlinear Schr\"odinger equation, subject to random perturbations in the form of either additive or multiplicative (Stratonovich) noise. We establish local well-posedness for random or deterministic initial data $u_0$ in $\dot{H}^1(\mathbb{R}^n)$ or $H^1(\mathbb{R}^n)$, depending on the noise type. In the focusing case we provide quantitative estimates regarding the existence time and probability. Moreover, we derive blow-up criteria for solutions with positive energy in both cases of noise, provided that the noise intensity is sufficiently small, showing that blow-up occurs before a certain given positive time with positive probability, thus, extending deterministic results of Kenig-Merle [24] for the energy-critical NLS equation to the stochastic setting. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14792 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14792v1 Announce Type: new Abstract: We study the behaviour of the solution $u_\varepsilon$ to the Navier-Stokes equations with vanishing viscosity and a non-slip condition in a randomly perforated domain. We consider the space $\mathbb{R}^3$ where we remove $N$ holes that are i.i.d. distributed. The behaviour depends on the particle size $\varepsilon^\alpha=N^{-\alpha/3}$ and the viscosity $\varepsilon^\gamma=N^{-\gamma/3}$ of the fluid. We prove quantitative convergence results to a function $u$, provided that the local Reynolds number is small, in the subcritical ($\alpha+\gamma>3$) and critical ($\alpha+\gamma=3$) regime. In the first case, $u$ solves the Euler equations, whereas in the second case $u$ solves the Euler-Brinkman equations. This extends the results of https://doi.org/10.1088/1361-6544/acfe56 from the periodic to the random setting. We only treat the case $\alpha>2$ so that the particles do not overlap with overwhelming probability. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14775 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14775v1 Announce Type: new Abstract: We study a one-dimensional cross-diffusion system for two interacting populations on the torus, with a linear pressure law and different mobilities. For arbitrary bounded non-negative initial data, we show that any good approximation scheme, yields existence of global weak solutions. More precisely, we introduce a notion of \textit{admissible approximation sequence} and show that any such sequence admits a subsequence converging to a weak solution of the system. The strategy relies on entropy estimates and the div--curl lemma, in the framework of Young measures. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14743 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14743v1 Announce Type: new Abstract: We study the complex Ginzburg-Landau equation posed on possibly unbounded domains, including some singular and saturated nonlinear damping terms. This model interpolates between the nonlinear Schr{\"o}dinger equation and dissipative parabolic dynamics through a complex timederivative prefactor, capturing the interplay between dispersion and dissipation. As a continuation of our previous study on the existence and uniqueness of solutions, we prove here some strong stabilization properties. In particular, we show the finite time extinction of solutions induced by the nonlinear saturation mechanism, which, sometimes, can be understood as a bang-bang control. The analysis relies on refined energy methods. Our results provide a rigorous justification of nonlinear dissipation as an effective stabilization mechanism for this class of complex equations where the maximum principle fails. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14636 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14636v1 Announce Type: new Abstract: We consider an inverse problem for the compressible Euler's equations in polytropic fluid. We show that by taking active measurements near a particle trajectory one can determine the background flow in a set where pressure waves can propagate from and return to the particle trajectory, under the additional assumption that the flow has nonzero vorticity. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14573 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14573v1 Announce Type: new Abstract: This paper is concerned with the spreading speeds of nonlocal dispersal predator-prey systems in shifting habitats under general initial conditions. By employing geometric optics techniques and theory of viscosity solutions, we reformulate the problem into the study of Hamilton-Jacobi equations. Through a detailed analysis of the structure of viscosity solutions, we provide a complete classification of explicit formulas for the spreading speed of the prey population, especially in cases where it invades the habitat more rapidly than predators, yielding two fundamentally distinct ``nonlocal determinacy'' results derived by different mechanisms. We also obtain an upper bound for spreading speed of the predators, incorporating the decay rate of the initial data and the speed of shifting habitats. These findings demonstrate that there are complex connections among spreading speeds, habitat shifting speed and initial conditions, and emphasize the significance of nonlocal dispersal in determining the propagation dynamics of predator-prey systems. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14544 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14544v1 Announce Type: new Abstract: In this paper, we investigate the local boundedness of weak solutions to degenerate parabolic double phase equation of type $$ u_t-\textrm{div}(|Du|^{p-2}Du+a(x,t)|Du|^{q-2}Du)=0\quad \text{in } \Omega_T := \Omega\times (0,T), $$ where $0\leq a(\cdot)\in L^\infty(\Omega_T)$. To this end, we derive the Caccioppoli inequality and a parabolic embedding theorem, which are then utilized in an iteration method. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2508.19889 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2508.19889v4 Announce Type: replace Abstract: For any extension of commutative rings $A\subseteq B$, by using invertible ideals, we first define an Abelian group $\Cl(A,B)$, that we call the ideal class group of this extension. Then we study the main properties of this group. Among them, we prove that the group $\Cl(A,B)$ is indeed the kernel of the natural group morphism $\Pic(A)\rightarrow \Pic(B)$ which is given by $L\mapsto L\otimes_{A}B$. Then we show that both the classical ideal class group and, surprisingly, the Picard group are special cases of this structure. Next, we prove that ... Maths/math.AC_(Commutative_Algebra) https://arxiv.org/abs/2508.08753 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2508.08753v5 Announce Type: replace Abstract: Let $(R,M,k)$ be a complete local ring (not necessarily Noetherian). As the first main result of this article, we prove that in the unequal characteristic case $\Char(R)\neq\Char(k)$, the natural surjective map between the groups of units $R^{\ast}\rightarrow k^{\ast}$ admits a splitting. \\ Next, we reprove by a new method that in the equi-characteristic case $\Char(R)=\Char(k)$, the natural surjective ring map $R\rightarrow k$ admits a splitting. In our proof there is no need for the existence of the coefficient fields for equi-characteristic complete local rings, whose existence is the most difficult part of the known proof. \\ As an application, we show that for any complete local ring $(R,M,k)$ the following short exact sequence of Abelian groups: $$\xymatrix{1\ar[r]&1+M\ar[r]& R^{\ast}\ar[r]&k^{\ast} \ar[r]&1}$$ is always split. In particular, we have an isomorphism of Abelian groups $R^{\ast}\simeq(1+M)\times k^{\ast}$. We also show with an example that the above exact sequence does not split for many incomplete local rings. Maths/math.AC_(Commutative_Algebra) https://arxiv.org/abs/2310.09581 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2310.09581v3 Announce Type: replace Abstract: This article discusses ramification and the structure of relative K\"ahler differentials of extensions of valued fields. We begin by surveying the theory developed in recent work with Franz-Viktor Kuhlmann and Anna Rzepka constructing the relative K\"ahler differentials of extensions of valuation rings in Artin-Schreier and Kummer extensions. We then show how this theory is applied to give a simple proof of Gabber and Ramero's characterization of deeply ramified fields. Section 4 develops the basics of almost mathematics, and should be accessible to a broad audience. Section 5 gives a simple and self contained proof of Gabber and Ramero's characterization of when the extension of a rank 1 valuation of a field to its separable closure is weakly \'etale. In the final section, we consider the equivalent conditions characterizing deeply ramified fields, as they are defined by Coates and Greenberg, and show that they are the same as the conditions of Gabber Ramero for local fields. Maths/math.AC_(Commutative_Algebra) https://arxiv.org/abs/2309.15620 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2309.15620v2 Announce Type: replace Abstract: The main result of this article is a fantastic generalization of a classical result in graded ring theory. In fact, our result states that if $S$ is a multiplicative set of homogeneous elements of an $M$-graded commutative ring $R=\bigoplus\limits_{m\in M}R_{m}$ with $M$ a commutative monoid, then the localization ring $S^{-1}R=\bigoplus\limits_{x\in G}(S^{-1}R)_{x}$ is a $G$-graded ring where $G$ is the Grothendieck group of $M$ and each homogeneous component $(S^{-1}R)_{x}$ is the set of all fractions $f\in S^{-1}R$ such that $f=0$ or it is of the form $f=r/s$ where $r$ is a homogeneous element of $R$ and $x=[\dg(r),\dg(s)]$. As an application, ... Maths/math.AC_(Commutative_Algebra) https://arxiv.org/abs/2212.09528 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2212.09528v5 Announce Type: replace Abstract: We show that any polarization of an Artin monomial ideal defines a triangulated ball. This proves a conjecture of A.Almousa, H.Lohne and the first author. Geometrically, polarizations of ideals containing $(x_1^{a_1}, \ldots, x_n^{a_n})$ define full-dimensional triangulated balls on the sphere which is the join of boundaries of simplices of dimensions $a_1-1, \cdots, a_n-1$. We prove that every full-dimensional Cohen-Macaulay sub-complex of this joined sphere is of this kind, and these balls are constructible. Such a triangulated ball has a dual cell complex which is a sub-complex of the product of simplices of dimensions $a_1-1, \cdots a_n-1$. We prove that this cell complex gives cellular minimal free resolution of this of the Alexander dual ideal of the triangulated ball. When the product of simplices is a hypercube, using these dual cell complexes we classify in a range examples all polarizations of the Artin monomial ideal. We also show that the squeezed balls of G.Kalai \cite{Ka} derive from polarizations of Artin monomial ideals. Maths/math.AC_(Commutative_Algebra) https://arxiv.org/abs/2604.14384 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14384v1 Announce Type: cross Abstract: We construct minimal resolutions of pushforwards of structure sheaves of toric substacks of smooth toric stacks by line bundles as strong deformation retracts of cellular resolutions constructed by Hanlon, Hicks and Lazarev. We also provide a canonical and combinatorial description of the differentials of such minimal resolutions. Two key ingredients are the homological perturbation lemma and the Moore-Penrose inverses. Maths/math.AC_(Commutative_Algebra) https://arxiv.org/abs/2604.14912 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14912v1 Announce Type: new Abstract: We formalize the Wu-Ritt characteristic set method for the triangular decomposition of polynomial systems in the Lean 4 theorem prover. Our development includes the core algebraic notions of the method, such as polynomial initials, orders, pseudo-division, pseudo-remainders with respect to a polynomial or a triangular set, and standard and weak ascending sets. On this basis, we formalize algorithms for computing basic sets, characteristic sets, and zero decompositions, and prove their termination and correctness. In particular, we formalize the well-ordering principle relating a polynomial system to its characteristic set and verify that zero decomposition expresses the zero set of the original system as a union of zero sets of triangular sets away from the zeros of the corresponding initials. This work provides a machine-checked verification of Wu-Ritt's method in Lean 4 and establishes a foundation for certified polynomial system solving and geometric theorem proving. Maths/math.AC_(Commutative_Algebra) https://arxiv.org/abs/2604.14686 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14686v1 Announce Type: cross Abstract: For a given graph \( G \), let \( G^{(j)} \) denote the graph obtained by the deletion of vertex \( v_j \) from \( G \). The difference \( \mathscr{E}(G) - \mathscr{E}(G^{(j)}) \) quantifies the change in the energy of \( G \) upon the removal of \( v_j \), termed as the local energy of \( G \) at vertex $v_j$, as defined by Espinal and Rada in 2024. The local energy of $G$ at vertex $v$ is denoted by \(\mathscr{E}_G(v)\). The local energy of the graph \( G \), therefore, is the summation of these vertex-specific local energies across all vertices in \( V(G) \), expressed by \( e(G) = \sum \mathscr{E}_G(v) \). Two graphs of the same order are defined as locally equienergetic if they have identical local energy. In this paper, we have investigated several pairs of locally equienergetic graphs. Maths/math.SP_(Spectral_Theory) https://arxiv.org/abs/2604.14195 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14195v1 Announce Type: cross Abstract: The \emph{generalized reciprocal distance matrix} of a graph $\mathscr{G}$, denoted by $RD_\alpha(\mathscr{G})$, is defined as $RD_\alpha(\mathscr{G})=\alpha\,RT_r(\mathscr{G})+(1-\alpha)\,RD(\mathscr{G}), \, \alpha\in[0,1],$ where $RT_r(\mathscr{G})$ represents the diagonal matrix of reciprocal vertex transmissions, and $RD(\mathscr{G})$ is the Harary (reciprocal distance) matrix of $\mathscr{G}$. In this paper, we investigate the $RD_\alpha$-spectrum of graphs obtained through the joined union operation. We derive explicit formulas for the characteristic polynomial of $RD_\alpha(\mathscr{G})$ when $\mathscr{G}$ is formed as a joined union of regular graphs. These results provide closed-form expressions for the corresponding spectra of several important graph classes. Moreover, we show that the power graphs of the dihedral group $D_{2n}$ and the generalized quaternion group $Q_{4n}$ admit representations as joined union graphs. Using this structural characterization, we determine the $RD_\alpha$-spectra of power graphs arising from various classes of finite groups, including cyclic groups $\mathbb{Z}_n$, dihedral groups $D_{2n}$, generalized quaternion groups $Q_{4n}$, elementary abelian $p$-groups, and certain non-abelian groups of order $pq$. Maths/math.SP_(Spectral_Theory) https://arxiv.org/abs/2603.26883 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2603.26883v2 Announce Type: replace Abstract: We consider Kleene and Stone algebras defined on the completion DM(RS) of the ordered set of rough sets induced by a reflexive relation. We focus on cases where the completion forms a spatial and completely distributive lattice. We derive the conditions under which DM(RS) is a regular pseudocomplemented Kleene algebra and a completely distributive double Stone algebra. Finally, we describe the reflexive relations for which DM(RS) forms a regular double Stone algebra, which is the same structure as in the case of equivalences. Our results generalise earlier findings on algebras of rough sets induced by equivalences, quasiorders, and tolerance relations. Maths/math.RA_(Rings_and_Algebras) https://arxiv.org/abs/2510.03775 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2510.03775v2 Announce Type: replace Abstract: The classical Noether Normalization Lemma states that if $S$ is a finitely generated algebra over a field $k$, then there exist elements $x_1,\dots,x_n$ which are algebraically independent over $k$ such that $S$ is a finite module over $k[x_1,\dots,x_n]$. This lemma has been studied intensively in different flavors. In 2024, Elad Paran and Thieu N. Vo successfully generalized this lemma for the case when $S$ is a quotient ring of the skew polynomial ring $D[x_1,\dots,x_n;\sigma_1,\dots,\sigma_n]$. In this paper, we investigate this lemma in a more general setting when $S$ is a quotient ring of an iterated skew polynomial ring $D[x_1;\sigma_1,\delta_1]\dots[x_n;\sigma_n,\delta_n]$. We extend several key results of Elad Paran and Thieu N. Vo to this broader context and introduce a new version of Combinatorial Nullstellensatz over division rings. Maths/math.RA_(Rings_and_Algebras) https://arxiv.org/abs/2604.14420 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14420v1 Announce Type: cross Abstract: Platonov in 1991 conjectured that adjoint groups are rational as varieties over arbitrary infinite fields, and as a consequence have weak approximation. The rationality part of the conjecture was disproved by Merkurjev in 1996, but the question about weak approximation remained open. We settle this in the negative. Maths/math.RA_(Rings_and_Algebras) https://arxiv.org/abs/2604.15286 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15286v1 Announce Type: new Abstract: We investigate the problem asking when any square matrix whose entries lie in a finite field of characteristic 2 is decomposable into the sum of a diagonalizable matrix and a nilpotent matrix with index of nilpotency at most 2 and, as a result, we completely resolve this question in the affirmative for any finite field of characteristic 2 having strictly more than three elements. Our main theorem of that type, combined with results from our recent publication in Linear Algebra & Appl. (2026) (see [7]), totally settle this problem for all finite fields different from $\mathbb{F}_2$ and $\mathbb{F}_3$. However, in this paper we also prove that each matrix over $\mathbb{F}_2$ is expressible as the sum of a potent matrix with index of potency not exceeding 4 and a nilpotent matrix with index of nilpotency not exceeding 2, thus substantiating recent examples due to \v{S}ter in Linear Algebra & Appl. (2018) and Shitov in Indag. Math. (2019) (see, respectively, [9] and [8]). Maths/math.RA_(Rings_and_Algebras) https://arxiv.org/abs/2604.15158 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15158v1 Announce Type: new Abstract: Let $F=\mathbb{F}_q$ and let $K=\mathbb{F}_{q^m}$ be a finite extension. An additive group code is an $FG$-submodule of the group algebra $KG$. In this paper, we introduce projector additive group codes and restricted projector additive group codes as additive counterparts of idempotent group codes in the classical theory of group codes. More precisely, they are defined as images and restrictions, respectively, of $FG$-linear projectors on $KG$. This perspective is motivated by the fact that idempotent elements of $KG$ do not yield a sufficiently general and natural algebraic framework for the study of additive left group codes. Projector additive left group codes coincide precisely with the projective left $FG$-submodules of $KG$. Consequently, in the semisimple case every additive left group code arises in this way, whereas in the non-semisimple case the projector construction captures exactly the projective ones. We further relate trace-Euclidean and trace-Hermitian duality to adjoint projectors, establish criteria for the LCD and self-dual cases, study the Murray--von Neumann equivalence of projectors, and interpret quotients by orthogonal codes in terms of module duals. Maths/math.RA_(Rings_and_Algebras) https://arxiv.org/abs/2505.24384 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2505.24384v2 Announce Type: replace-cross Abstract: We develop an estimator-based stochastic fixed-point framework for approximately computing the 2-Wasserstein barycenter of continuous, non-parametric probability measures. Notably, we provide the first rigorous convergence analysis for implementable estimator-based stochastic extensions of the fixed-point iterative scheme proposed by \'Alvarez-Esteban, del Barrio, Cuesta-Albertos, and Matr\'an (2016). In particular, we establish almost sure convergence, and identify sufficient conditions for geometric rates of convergence under controlled errors in optimal transport (OT) map estimation. We subsequently propose a concrete, provably convergent, and computationally tractable stochastic algorithm that accommodates input measures satisfying Caffarelli-type regularity conditions, which form a dense subset of the Wasserstein space. This algorithm leverages a modified entropic OT map estimator to enable efficient and scalable implementation. To facilitate quantitative evaluation, we further propose a novel and efficient procedure for synthetically generating benchmark instances, in which the input measures exhibit non-trivial features and the corresponding barycenters are approximately known. Numerical experiments on both synthetic and real-world datasets demonstrate the strong computational efficiency, estimation accuracy, and sampling flexibility of our approach. Maths/math.PR_(Probability) https://arxiv.org/abs/2603.18748 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2603.18748v2 Announce Type: replace Abstract: We establish annealed and quenched invariance principles for random walks in random conductances lifted to the p-variation rough path topology, allowing for degenerate environments and long-range jumps. Our proof is based on a unified structural strategy where pathwise convergence is viewed as a natural upgrade of the classical theory. This approach decouples the martingale lift from terms involving the integrals with respect to the corrector and the quadratic covariations. In the quenched regime, we show that the existence of a stationary potential for the corrector with $2+\epsilon$ moments is sufficient to ensure the vanishing of the corrector in $p$-variation for any $p>2$. This input, combined with our structural framework, provides a direct and modular pathway to rough path convergence. We further provide a transfer lemma to construct this potential from spatial moment bounds. While presently verified in the literature primarily for nearest-neighbor settings, our formulation isolates the exact analytic input required for pathwise convergence in more general environments. Maths/math.PR_(Probability) https://arxiv.org/abs/2510.04092 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2510.04092v2 Announce Type: replace Abstract: We examine a delayed stochastic interest rate model with super-linearly growing coefficients and develop several new mathematical tools to establish the properties of its true and truncated EM solutions. Moreover, we show that the true solution converges to the truncated EM solutions in probability as the step size tends to zero. Further, we support the convergence result with some illustrative numerical examples and justify the convergence result for the Monte Carlo evaluation of some financial quantities. Maths/math.PR_(Probability) https://arxiv.org/abs/2409.08882 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2409.08882v2 Announce Type: replace Abstract: This paper develops a non-asymptotic approach to mean field approximations for systems of $n$ diffusive particles interacting pairwise. The interaction strengths are not identical, making the particle system non-exchangeable. The marginal law of any subset of particles is compared to a suitably chosen product measure, and we find sharp relative entropy estimates between the two. Building upon prior work of the first author in the exchangeable setting, we use a generalized form of the BBGKY hierarchy to derive a hierarchy of differential inequalities for the relative entropies. Our analysis of this complicated hierarchy exploits an unexpected but crucial connection with first-passage percolation, which lets us bound the marginal entropies in terms of expectations of functionals of this percolation process. Maths/math.PR_(Probability) https://arxiv.org/abs/2406.03058 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2406.03058v3 Announce Type: replace Abstract: We consider strong approximations of $1+1$-dimensional stochastic PDEs driven by additive space-time white noise. It has been long proposed (Davie-Gaines '01, Jentzen-Kloeden '08), as well as observed in simulations, that approximation schemes based on samples from the stochastic convolution, rather than from increments of the underlying Wiener processes, should achieve significantly higher convergence rates with respect to the temporal timestep. The present paper proves this. For a large class of nonlinearities, with possibly superlinear growth, a temporal rate of (almost) $1$ is proven, a major improvement on the rate $1/4$ that is known to be optimal for schemes based on Wiener increments. The spatial rate remains (almost) $1/2$ as it is standard in the literature. Maths/math.PR_(Probability) https://arxiv.org/abs/2202.10999 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2202.10999v3 Announce Type: replace Abstract: For deterministic monotone cellular automata on the $d$-dimensional integer lattice, Toom has given necessary and sufficient conditions for the all-one fixed point to be stable against small random perturbations. The proof of sufficiency is based on an intricate Peierls argument. We present a simplified version of this Peierls argument. Our main motivation is the open problem of determining stability of monotone cellular automata with intrinsic randomness, in which for the unperturbed evolution the local update rules at different space-time points are chosen in an i.i.d. fashion according to some fixed law. We apply Toom's Peierls argument to prove stability of a class of cellular automata with intrinsic randomness and also derive lower bounds on the critical parameter for some deterministic cellular automata. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.14765 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14765v1 Announce Type: cross Abstract: We present a geometric framework for Reinforcement Learning (RL) that views policies as maps into the Wasserstein space of action probabilities. First, we define a Riemannian structure induced by stationary distributions, proving its existence in a general context. We then define the tangent space of policies and characterize the geodesics, specifically addressing the measurability of vector fields mapped from the state space to the tangent space of probability measures over the action space. Next, we formulate a general RL optimization problem and construct a gradient flow using Otto's calculus. We compute the gradient and the Hessian of the energy, providing a formal second-order analysis. Finally, we illustrate the method with numerical examples for low-dimensional problems, computing the gradient directly from our theoretical formalism. For high-dimensional problems, we parameterize the policy using a neural network and optimize it based on an ergodic approximation of the cost. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.14681 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14681v1 Announce Type: cross Abstract: Given a classical gas described by the truncated correlation functions of all orders, we prove convergence of an expansion of the pair interaction part of the (unknown) potential in terms of the truncated correlation functions of all orders, at infinite volume. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.14277 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14277v1 Announce Type: cross Abstract: We study the growth of entanglement and circuit complexity in random passive linear optical networks as a function of the circuit depth. For entanglement dynamics, we start with an initial Gaussian state with all $n$ modes squeezed. For random brickwall circuits, we show that entanglement, as measured by the R\'enyi-2 entropy, grows at most diffusively as a function of the depth. In the other direction, for arbitrary circuit geometries we prove bounds on depths which ensure the average subsystem entanglement reaches within a constant factor of the maximum value in all subsystems, and bounds which ensure closeness of the random linear optical unitary to a Haar random unitary in $L^2$ Wasserstein distance. We also consider robust circuit complexity for random one-dimensional brickwall circuits, as measured by the minimum number of gates required in any circuit that approximately implements the linear optical unitary. Viewing this as a function of the number of modes and the circuit depth, we show the robust circuit complexity for random one-dimensional brickwall circuits scales at most diffusively in the depth with high probability. The corresponding Gaussian unitary $\tilde{\mathcal U}$ for the approximate implementation retains high output fidelity $|\langle\psi|\mathcal U^\dagger \tilde{\mathcal U}|\psi\rangle|^2$ for pure states $|\psi\rangle$ with constrained expected photon-number. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.15226 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15226v1 Announce Type: new Abstract: In this paper, we prove existence and uniqueness of energy solutions for nonlinear Schr\"odinger equations with a multiplicative white noise on $R^d$ with $d\le3$. We rely on an exponential trans-form and conserved quantities for existence of energy solutions. Using paracontrolled calculus, we prove Strichartz inequalities which encode the dispersive properties of the solutions. This allows to obtain local well-posedness for low regularity solutions and uniqueness of energy solutions for various equations. In particular, our results are the first results of propagation without loss of both regularity and localization for such equations in full space as well as the first results on $R^3$ for such singular dispersive SPDEs. We are also obtain local well-posedness in two dimensions for deterministic initial data. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.15152 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15152v1 Announce Type: new Abstract: We revisit the random allocation of $n$ balls into $N$ boxes with probabilities $q_1,\ldots,q_N$, focusing on the proportion $\hat q_r$ of boxes containing exactly $r$ balls. Classical asymptotic results for the expectations, variances, and covariances of these proportions are reformulated in terms of the size distribution of a randomly chosen box. We further derive explicit two-sided bounds for the associated remainder terms, allowing for weaker assumptions than those previously required. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.15146 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15146v1 Announce Type: new Abstract: We consider nested CLE$_4$ in a simply-connected domain and compute the following renormalised probabilities: the probability that two points belong to the same CLE$_4$ gasket and the probability that two points belong to the outermost CLE$_4$ gasket. While the integrability is rooted in the conformal field theory of the Ashkin-Teller (AT) model, we provide a purely probabilistic calculation via Brownian loop soups and the geometry of the 2D continuum Gaussian free field. More generally, we also calculate renormalised probabilities that two points belong to CLE$_4$ gaskets sampled in alternation with certain two-valued sets of the Gaussian free field. These quantities correspond to the two-point function of the conjectured scaling limit of the AT single spins on the critical line. At the decoupling point, our results recover the Ising model correlations and suggest a CLE$_4$-based FK representation of the AT spin model. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.15007 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15007v1 Announce Type: new Abstract: Consider a generalised preferential attachment tree with attachment function $f$, that is a random tree, where at each time-step a node connects to an existing node $v$ with probability proportional to $f(\mathrm{deg}(v))$, where $\mathrm{deg}(v)$ denotes the degree of the node in the existing tree. We provide a counter-example to a conjecture of the author asserting that under the assumption $\sum_{j=1}^{\infty} \frac{1}{f(j)^2} < \infty$ there is a persistent hub in the model, that is, a single node that has the maximal degree for all but finitely many time-steps. The counter-example is a minor modification of a related counter-example due to Galganov and Ilienko. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.14893 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14893v1 Announce Type: new Abstract: This paper investigates a class of generalized mean-reflected McKean-Vlasov type backward stochastic differential equations (BSDEs). Our new framework combines a mean reflection constraint on the solution's expectation with a generalized integral with respect to a continuous non-decreasing process. We establish the existence and uniqueness of the solution. The uniqueness is derived via stability estimates, while the existence is proved by employing a penalization method combined with a smooth approximation of the obstacle. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.14851 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14851v1 Announce Type: new Abstract: We present and study the Pool model in $\mathbb{R}^2$, a rotationally symmetric analogue of Multi-Particle Diffusion-Limited Aggregation (MDLA), in which particles ("droplets") perform continuous-time random walks and are absorbed upon entering a circular pool initially centered at the origin. Each absorbed particle increases the pool's mass, and the pool expands so that its area grows accordingly, yielding a natural mass-preserving dynamics. A central tool which is of independent interest is a version of Kurtz's theorem for this model, depicting the field of particles conditioned on the growth of the pool as an independent non-homogeneous Poisson point process. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.14543 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14543v1 Announce Type: new Abstract: This paper investigates the approximation of invariant measures for McKean-Vlasov stochastic differential equations (SDEs) using the Euler-Maruyama (EM) scheme under a monotonicity condition. Firstly, the convergence of the numerical solution from the EM scheme to its continuous-time counterpart is established. Secondly, we show that the numerical solution admits a unique invariant measure and derive its convergence rate under the Wasserstein metric. In parallel, it is demonstrated that the associated particle system also possesses these properties. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.14427 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14427v1 Announce Type: new Abstract: A criterion for proving a strong form of propagation of chaos on the path space, known as entropy chaos, for a general interacting diffusion system is proposed. Our analysis focuses on the class of conservative diffusions introduced by Carlen, which are characterized by infinitesimal characteristic pairs, that is, a time-marginal probability density and a current velocity field. A key property of this broad class is that the processes remain diffusions under time-reversal. We prove that, given a suitable bound on the relative entropy (with respect to the Wiener measure) and the weak convergence of both drifts and fixed-time marginal densities, strong entropy chaos at the process level is achieved in the infinite particle limit, provided the limit drift satisfies a specific regularity condition. This stochastic framework encompasses various singular interacting particle systems and their related asymptotic scenarios. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.14396 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14396v1 Announce Type: new Abstract: By using a probabilistic technique based on the exponential change of measure we find a precise tail asymptotic behavior of some perpetuities with distributions close to the Dickman distribution. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.14697 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14697v1 Announce Type: cross Abstract: Let $X$ be a Dedekind complete Banach lattice, and let $P\colon X\to X$ be a positive projection for which the largest central operator below $P$ is $\alpha \operatorname{id}_X$, for some $\alpha \ge 0$. Wickstead conjectured that $\alpha $ must either be $0$ or $1/n$, for some $n \in \mathbb{N}$, and proved it for finite-dimensional $X$. In this paper, we show that the conjecture holds in general. As a consequence, we settle the representation problem for Banach lattice algebras: we show that there exist Banach lattice algebras of dimension $2$ that do not admit a faithful representation as regular operators on any Dedekind complete Banach lattice. Maths/math.OA_(Operator_Algebras) https://arxiv.org/abs/2604.14366 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14366v1 Announce Type: cross Abstract: We study the Ricci-Bourguignon flow on warped product manifolds with noncompact base. This setting leads naturally to a parabolic partial differential equation on the space of smooth warping functions, arising from the necessary and sufficient conditions for a warped metric to evolve under the flow. One of our main results establishes a gradient estimate for this equation, providing the analytic input for the geometric applications developed herein and, in particular, recovering classical gradient estimates for the heat equation under the Ricci flow. Furthermore, we show how to construct explicit warped solutions to the Ricci-Bourguignon flow and present examples that are not only of independent interest but also illustrate and support our results Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.15206 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15206v1 Announce Type: new Abstract: The study of Sobolev and Poincar\'e inequalities for differential forms in Carnot groups and in the more general sub-Riemannian setting is still an open problem in its full generality. One may conjecture that, for general Carnot groups, these inequalities are expressed in terms of suitable graded Lebesgue norms. In recent years, many results have been obtained, both in the Euclidean setting and in the Heisenberg groups, as well as for contact manifolds with bounded geometry. There are also some results for general Carnot groups; however, these do not cover the problem in its full generality. In this paper, we consider a particular Carnot group, the so-called Cartan group (a free Carnot group, of step $3$ with $2$ generators), which provides a natural testing ground for these questions, since its step-three structure already exhibits several phenomena that do not occur in the Heisenberg groups. In this setting, we are able to prove global Poincar\'e and Sobolev-Gaffney inequalities for differential forms. With the aim of obtaining sharp estimates, we replace the de Rham complex of differential forms with the Rumin complex. The case $p>1$ is carried out after establishing an $L^p$-Hodge decomposition with homogeneous Sobolev classes. We are able to consider also the endpoint case $p=1$; however, as in Euclidean setting, when $p=1$, the operator we deal with provides only weak-type estimates which do not yield a Hodge decomposition analogous to the case $p>1$. Therefore, in this situation the proof follows a different approach, relying on a recent result proved in \cite{BT}. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.15183 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15183v1 Announce Type: new Abstract: We investigate the asymptotic behavior of the solutions to the Neumann sieve problem for the Poisson equation in a thin, randomly perforated domain. The perforations (sieve-holes) are generated by a stationary marked point process. According to the scaling between the domain thickness and the typical hole size, three distinct limiting regimes emerge. We also identify the optimal stochastic integrability condition on the random hole radii that guarantees stochastic homogenization, even in the presence of clustering holes. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.15053 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15053v1 Announce Type: new Abstract: We improve previous results on dispersive decay for 1D Klein- Gordon equation. We develop a novel approach, which allows us to establish the decay in more strong norms and weaken the assumption on the potential. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14997 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14997v1 Announce Type: new Abstract: We consider the Euler-Poisson system for ions where the electrons are given by a Maxwell-Boltzmann distribution, and we investigate the existence of one-dimensional periodic traveling waves. More precisely, we first establish the existence of a smooth global branch of bifurcation emanating from a constant equilibrium. We then construct a singular traveling wave emerging as the limiting profile at the end of the global curve of bifurcation. Our analysis accommodates a wide class of pressure laws and provides a comprehensive characterization of both smooth and singular traveling waves. A central difficulty in this model arises from the exponential nonlinearity, induced by the nonlocal Poisson-Boltzmann equation, which prevents any explicit representation of the electron field in terms of the ion density. This poses significant obstacles compared to previous studies on related models, where such explicit formulas were crucial for global bifurcation arguments. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14923 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14923v1 Announce Type: new Abstract: We consider the following class of mixed local-nonlocal equations: \begin{align}\label{abs}\tag{$\mathcal{P}$} -\Delta_p u + (-\Delta)_p^s u = V |u|^{p-2}u \text{ in } \Omega, \end{align} where $s \in (0,1), p \in (1, \infty)$, and the weight function $V$ lies in scaling subcritical Lebesgue space $L^q(\Omega)$ where $q>\frac{d}{p}$ when $d>p$ and $q>1$ when $d \le p$. We establish Harnack inequality for weak solution and weak Harnack inequality for weak supersolution to ($\mathcal{P}$). Our approach is based on the De Giorgi-Nash-Moser theory, the expansion of positivity and estimates involving a tail term. Our results also apply to integro-differential operators, with the prototype given by $(-\Delta)_p^s$. This work generalizes some regularity results of Garain-Kinnunen (Trans. Am. Math. Soc., 375(8), 2022) and Garain (Nonlinear Anal., 256, 2025) to the setting of general weight functions. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14916 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14916v1 Announce Type: new Abstract: We consider stationary $p$-Schr\"odinger equations on the whole space with integrable data and potentials that are confining in measure. We introduce asymptotic energy solutions in an asymptotic $L^p$ framework and establish existence and uniqueness in the degenerate range $p\ge2$. The proof relies on a new Rellich$\unicode{x2013}$Kondrachov-type compactness theorem of independent interest, which provides sufficient conditions for families of Sobolev functions to be precompact in asymptotic $L^p$ spaces, without any dimension-dependent restriction on the exponent. For data in the duality regime $L^1(\mathbb{R}^n)\cap L^{p'}(\mathbb{R}^n)$, asymptotic energy solutions coincide with weak energy solutions. We also show that additional compactness assumptions yield localized entropy-type solutions and, under suitable local regularity, distributional solutions. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14911 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14911v1 Announce Type: new Abstract: We analyse the effect of expansion in Newtonian cosmology on the asymptotic behaviour of charged self-interacting plasmas close to Poisson equilibria. To this end, we study the Vlasov-Poisson system on the phase space of a $3$-torus which is expanding with respect to the scale factor $a(t)$. We show that, for $a(t)=t^q$ with $q\in(0,\frac12)$, solutions to this system exhibit nonlinear Landau damping for initial data that is small with respect to a suitably strong Gevrey class, i.e., the charge density contrast of the plasma decays superpolynomially. For larger choices of $q$ within this range, the initial data requirements become stricter while the decay weakens. To our knowledge, this is the first result showing Landau damping in a cosmological setting. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14852 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14852v1 Announce Type: new Abstract: We investigate the focusing and defocusing energy-critical stochastic nonlinear Schr\"odinger equation, subject to random perturbations in the form of either additive or multiplicative (Stratonovich) noise. We establish local well-posedness for random or deterministic initial data $u_0$ in $\dot{H}^1(\mathbb{R}^n)$ or $H^1(\mathbb{R}^n)$, depending on the noise type. In the focusing case we provide quantitative estimates regarding the existence time and probability. Moreover, we derive blow-up criteria for solutions with positive energy in both cases of noise, provided that the noise intensity is sufficiently small, showing that blow-up occurs before a certain given positive time with positive probability, thus, extending deterministic results of Kenig-Merle [24] for the energy-critical NLS equation to the stochastic setting. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14792 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14792v1 Announce Type: new Abstract: We study the behaviour of the solution $u_\varepsilon$ to the Navier-Stokes equations with vanishing viscosity and a non-slip condition in a randomly perforated domain. We consider the space $\mathbb{R}^3$ where we remove $N$ holes that are i.i.d. distributed. The behaviour depends on the particle size $\varepsilon^\alpha=N^{-\alpha/3}$ and the viscosity $\varepsilon^\gamma=N^{-\gamma/3}$ of the fluid. We prove quantitative convergence results to a function $u$, provided that the local Reynolds number is small, in the subcritical ($\alpha+\gamma>3$) and critical ($\alpha+\gamma=3$) regime. In the first case, $u$ solves the Euler equations, whereas in the second case $u$ solves the Euler-Brinkman equations. This extends the results of https://doi.org/10.1088/1361-6544/acfe56 from the periodic to the random setting. We only treat the case $\alpha>2$ so that the particles do not overlap with overwhelming probability. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14775 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14775v1 Announce Type: new Abstract: We study a one-dimensional cross-diffusion system for two interacting populations on the torus, with a linear pressure law and different mobilities. For arbitrary bounded non-negative initial data, we show that any good approximation scheme, yields existence of global weak solutions. More precisely, we introduce a notion of \textit{admissible approximation sequence} and show that any such sequence admits a subsequence converging to a weak solution of the system. The strategy relies on entropy estimates and the div--curl lemma, in the framework of Young measures. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14743 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14743v1 Announce Type: new Abstract: We study the complex Ginzburg-Landau equation posed on possibly unbounded domains, including some singular and saturated nonlinear damping terms. This model interpolates between the nonlinear Schr{\"o}dinger equation and dissipative parabolic dynamics through a complex timederivative prefactor, capturing the interplay between dispersion and dissipation. As a continuation of our previous study on the existence and uniqueness of solutions, we prove here some strong stabilization properties. In particular, we show the finite time extinction of solutions induced by the nonlinear saturation mechanism, which, sometimes, can be understood as a bang-bang control. The analysis relies on refined energy methods. Our results provide a rigorous justification of nonlinear dissipation as an effective stabilization mechanism for this class of complex equations where the maximum principle fails. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14636 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14636v1 Announce Type: new Abstract: We consider an inverse problem for the compressible Euler's equations in polytropic fluid. We show that by taking active measurements near a particle trajectory one can determine the background flow in a set where pressure waves can propagate from and return to the particle trajectory, under the additional assumption that the flow has nonzero vorticity. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14573 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14573v1 Announce Type: new Abstract: This paper is concerned with the spreading speeds of nonlocal dispersal predator-prey systems in shifting habitats under general initial conditions. By employing geometric optics techniques and theory of viscosity solutions, we reformulate the problem into the study of Hamilton-Jacobi equations. Through a detailed analysis of the structure of viscosity solutions, we provide a complete classification of explicit formulas for the spreading speed of the prey population, especially in cases where it invades the habitat more rapidly than predators, yielding two fundamentally distinct ``nonlocal determinacy'' results derived by different mechanisms. We also obtain an upper bound for spreading speed of the predators, incorporating the decay rate of the initial data and the speed of shifting habitats. These findings demonstrate that there are complex connections among spreading speeds, habitat shifting speed and initial conditions, and emphasize the significance of nonlocal dispersal in determining the propagation dynamics of predator-prey systems. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14544 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14544v1 Announce Type: new Abstract: In this paper, we investigate the local boundedness of weak solutions to degenerate parabolic double phase equation of type $$ u_t-\textrm{div}(|Du|^{p-2}Du+a(x,t)|Du|^{q-2}Du)=0\quad \text{in } \Omega_T := \Omega\times (0,T), $$ where $0\leq a(\cdot)\in L^\infty(\Omega_T)$. To this end, we derive the Caccioppoli inequality and a parabolic embedding theorem, which are then utilized in an iteration method. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2508.19889 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2508.19889v4 Announce Type: replace Abstract: For any extension of commutative rings $A\subseteq B$, by using invertible ideals, we first define an Abelian group $\Cl(A,B)$, that we call the ideal class group of this extension. Then we study the main properties of this group. Among them, we prove that the group $\Cl(A,B)$ is indeed the kernel of the natural group morphism $\Pic(A)\rightarrow \Pic(B)$ which is given by $L\mapsto L\otimes_{A}B$. Then we show that both the classical ideal class group and, surprisingly, the Picard group are special cases of this structure. Next, we prove that ... Maths/math.AC_(Commutative_Algebra) https://arxiv.org/abs/2508.08753 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2508.08753v5 Announce Type: replace Abstract: Let $(R,M,k)$ be a complete local ring (not necessarily Noetherian). As the first main result of this article, we prove that in the unequal characteristic case $\Char(R)\neq\Char(k)$, the natural surjective map between the groups of units $R^{\ast}\rightarrow k^{\ast}$ admits a splitting. \\ Next, we reprove by a new method that in the equi-characteristic case $\Char(R)=\Char(k)$, the natural surjective ring map $R\rightarrow k$ admits a splitting. In our proof there is no need for the existence of the coefficient fields for equi-characteristic complete local rings, whose existence is the most difficult part of the known proof. \\ As an application, we show that for any complete local ring $(R,M,k)$ the following short exact sequence of Abelian groups: $$\xymatrix{1\ar[r]&1+M\ar[r]& R^{\ast}\ar[r]&k^{\ast} \ar[r]&1}$$ is always split. In particular, we have an isomorphism of Abelian groups $R^{\ast}\simeq(1+M)\times k^{\ast}$. We also show with an example that the above exact sequence does not split for many incomplete local rings. Maths/math.AC_(Commutative_Algebra) https://arxiv.org/abs/2310.09581 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2310.09581v3 Announce Type: replace Abstract: This article discusses ramification and the structure of relative K\"ahler differentials of extensions of valued fields. We begin by surveying the theory developed in recent work with Franz-Viktor Kuhlmann and Anna Rzepka constructing the relative K\"ahler differentials of extensions of valuation rings in Artin-Schreier and Kummer extensions. We then show how this theory is applied to give a simple proof of Gabber and Ramero's characterization of deeply ramified fields. Section 4 develops the basics of almost mathematics, and should be accessible to a broad audience. Section 5 gives a simple and self contained proof of Gabber and Ramero's characterization of when the extension of a rank 1 valuation of a field to its separable closure is weakly \'etale. In the final section, we consider the equivalent conditions characterizing deeply ramified fields, as they are defined by Coates and Greenberg, and show that they are the same as the conditions of Gabber Ramero for local fields. Maths/math.AC_(Commutative_Algebra) https://arxiv.org/abs/2309.15620 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2309.15620v2 Announce Type: replace Abstract: The main result of this article is a fantastic generalization of a classical result in graded ring theory. In fact, our result states that if $S$ is a multiplicative set of homogeneous elements of an $M$-graded commutative ring $R=\bigoplus\limits_{m\in M}R_{m}$ with $M$ a commutative monoid, then the localization ring $S^{-1}R=\bigoplus\limits_{x\in G}(S^{-1}R)_{x}$ is a $G$-graded ring where $G$ is the Grothendieck group of $M$ and each homogeneous component $(S^{-1}R)_{x}$ is the set of all fractions $f\in S^{-1}R$ such that $f=0$ or it is of the form $f=r/s$ where $r$ is a homogeneous element of $R$ and $x=[\dg(r),\dg(s)]$. As an application, ... Maths/math.AC_(Commutative_Algebra) https://arxiv.org/abs/2212.09528 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2212.09528v5 Announce Type: replace Abstract: We show that any polarization of an Artin monomial ideal defines a triangulated ball. This proves a conjecture of A.Almousa, H.Lohne and the first author. Geometrically, polarizations of ideals containing $(x_1^{a_1}, \ldots, x_n^{a_n})$ define full-dimensional triangulated balls on the sphere which is the join of boundaries of simplices of dimensions $a_1-1, \cdots, a_n-1$. We prove that every full-dimensional Cohen-Macaulay sub-complex of this joined sphere is of this kind, and these balls are constructible. Such a triangulated ball has a dual cell complex which is a sub-complex of the product of simplices of dimensions $a_1-1, \cdots a_n-1$. We prove that this cell complex gives cellular minimal free resolution of this of the Alexander dual ideal of the triangulated ball. When the product of simplices is a hypercube, using these dual cell complexes we classify in a range examples all polarizations of the Artin monomial ideal. We also show that the squeezed balls of G.Kalai \cite{Ka} derive from polarizations of Artin monomial ideals. Maths/math.AC_(Commutative_Algebra) https://arxiv.org/abs/2604.14384 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14384v1 Announce Type: cross Abstract: We construct minimal resolutions of pushforwards of structure sheaves of toric substacks of smooth toric stacks by line bundles as strong deformation retracts of cellular resolutions constructed by Hanlon, Hicks and Lazarev. We also provide a canonical and combinatorial description of the differentials of such minimal resolutions. Two key ingredients are the homological perturbation lemma and the Moore-Penrose inverses. Maths/math.AC_(Commutative_Algebra) https://arxiv.org/abs/2604.14912 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14912v1 Announce Type: new Abstract: We formalize the Wu-Ritt characteristic set method for the triangular decomposition of polynomial systems in the Lean 4 theorem prover. Our development includes the core algebraic notions of the method, such as polynomial initials, orders, pseudo-division, pseudo-remainders with respect to a polynomial or a triangular set, and standard and weak ascending sets. On this basis, we formalize algorithms for computing basic sets, characteristic sets, and zero decompositions, and prove their termination and correctness. In particular, we formalize the well-ordering principle relating a polynomial system to its characteristic set and verify that zero decomposition expresses the zero set of the original system as a union of zero sets of triangular sets away from the zeros of the corresponding initials. This work provides a machine-checked verification of Wu-Ritt's method in Lean 4 and establishes a foundation for certified polynomial system solving and geometric theorem proving. Maths/math.AC_(Commutative_Algebra) https://arxiv.org/abs/2604.14686 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14686v1 Announce Type: cross Abstract: For a given graph \( G \), let \( G^{(j)} \) denote the graph obtained by the deletion of vertex \( v_j \) from \( G \). The difference \( \mathscr{E}(G) - \mathscr{E}(G^{(j)}) \) quantifies the change in the energy of \( G \) upon the removal of \( v_j \), termed as the local energy of \( G \) at vertex $v_j$, as defined by Espinal and Rada in 2024. The local energy of $G$ at vertex $v$ is denoted by \(\mathscr{E}_G(v)\). The local energy of the graph \( G \), therefore, is the summation of these vertex-specific local energies across all vertices in \( V(G) \), expressed by \( e(G) = \sum \mathscr{E}_G(v) \). Two graphs of the same order are defined as locally equienergetic if they have identical local energy. In this paper, we have investigated several pairs of locally equienergetic graphs. Maths/math.SP_(Spectral_Theory) https://arxiv.org/abs/2604.14195 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14195v1 Announce Type: cross Abstract: The \emph{generalized reciprocal distance matrix} of a graph $\mathscr{G}$, denoted by $RD_\alpha(\mathscr{G})$, is defined as $RD_\alpha(\mathscr{G})=\alpha\,RT_r(\mathscr{G})+(1-\alpha)\,RD(\mathscr{G}), \, \alpha\in[0,1],$ where $RT_r(\mathscr{G})$ represents the diagonal matrix of reciprocal vertex transmissions, and $RD(\mathscr{G})$ is the Harary (reciprocal distance) matrix of $\mathscr{G}$. In this paper, we investigate the $RD_\alpha$-spectrum of graphs obtained through the joined union operation. We derive explicit formulas for the characteristic polynomial of $RD_\alpha(\mathscr{G})$ when $\mathscr{G}$ is formed as a joined union of regular graphs. These results provide closed-form expressions for the corresponding spectra of several important graph classes. Moreover, we show that the power graphs of the dihedral group $D_{2n}$ and the generalized quaternion group $Q_{4n}$ admit representations as joined union graphs. Using this structural characterization, we determine the $RD_\alpha$-spectra of power graphs arising from various classes of finite groups, including cyclic groups $\mathbb{Z}_n$, dihedral groups $D_{2n}$, generalized quaternion groups $Q_{4n}$, elementary abelian $p$-groups, and certain non-abelian groups of order $pq$. Maths/math.SP_(Spectral_Theory) https://arxiv.org/abs/2603.26883 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2603.26883v2 Announce Type: replace Abstract: We consider Kleene and Stone algebras defined on the completion DM(RS) of the ordered set of rough sets induced by a reflexive relation. We focus on cases where the completion forms a spatial and completely distributive lattice. We derive the conditions under which DM(RS) is a regular pseudocomplemented Kleene algebra and a completely distributive double Stone algebra. Finally, we describe the reflexive relations for which DM(RS) forms a regular double Stone algebra, which is the same structure as in the case of equivalences. Our results generalise earlier findings on algebras of rough sets induced by equivalences, quasiorders, and tolerance relations. Maths/math.RA_(Rings_and_Algebras) https://arxiv.org/abs/2510.03775 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2510.03775v2 Announce Type: replace Abstract: The classical Noether Normalization Lemma states that if $S$ is a finitely generated algebra over a field $k$, then there exist elements $x_1,\dots,x_n$ which are algebraically independent over $k$ such that $S$ is a finite module over $k[x_1,\dots,x_n]$. This lemma has been studied intensively in different flavors. In 2024, Elad Paran and Thieu N. Vo successfully generalized this lemma for the case when $S$ is a quotient ring of the skew polynomial ring $D[x_1,\dots,x_n;\sigma_1,\dots,\sigma_n]$. In this paper, we investigate this lemma in a more general setting when $S$ is a quotient ring of an iterated skew polynomial ring $D[x_1;\sigma_1,\delta_1]\dots[x_n;\sigma_n,\delta_n]$. We extend several key results of Elad Paran and Thieu N. Vo to this broader context and introduce a new version of Combinatorial Nullstellensatz over division rings. Maths/math.RA_(Rings_and_Algebras) https://arxiv.org/abs/2604.14420 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14420v1 Announce Type: cross Abstract: Platonov in 1991 conjectured that adjoint groups are rational as varieties over arbitrary infinite fields, and as a consequence have weak approximation. The rationality part of the conjecture was disproved by Merkurjev in 1996, but the question about weak approximation remained open. We settle this in the negative. Maths/math.RA_(Rings_and_Algebras) https://arxiv.org/abs/2604.15286 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15286v1 Announce Type: new Abstract: We investigate the problem asking when any square matrix whose entries lie in a finite field of characteristic 2 is decomposable into the sum of a diagonalizable matrix and a nilpotent matrix with index of nilpotency at most 2 and, as a result, we completely resolve this question in the affirmative for any finite field of characteristic 2 having strictly more than three elements. Our main theorem of that type, combined with results from our recent publication in Linear Algebra & Appl. (2026) (see [7]), totally settle this problem for all finite fields different from $\mathbb{F}_2$ and $\mathbb{F}_3$. However, in this paper we also prove that each matrix over $\mathbb{F}_2$ is expressible as the sum of a potent matrix with index of potency not exceeding 4 and a nilpotent matrix with index of nilpotency not exceeding 2, thus substantiating recent examples due to \v{S}ter in Linear Algebra & Appl. (2018) and Shitov in Indag. Math. (2019) (see, respectively, [9] and [8]). Maths/math.RA_(Rings_and_Algebras) https://arxiv.org/abs/2604.15158 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15158v1 Announce Type: new Abstract: Let $F=\mathbb{F}_q$ and let $K=\mathbb{F}_{q^m}$ be a finite extension. An additive group code is an $FG$-submodule of the group algebra $KG$. In this paper, we introduce projector additive group codes and restricted projector additive group codes as additive counterparts of idempotent group codes in the classical theory of group codes. More precisely, they are defined as images and restrictions, respectively, of $FG$-linear projectors on $KG$. This perspective is motivated by the fact that idempotent elements of $KG$ do not yield a sufficiently general and natural algebraic framework for the study of additive left group codes. Projector additive left group codes coincide precisely with the projective left $FG$-submodules of $KG$. Consequently, in the semisimple case every additive left group code arises in this way, whereas in the non-semisimple case the projector construction captures exactly the projective ones. We further relate trace-Euclidean and trace-Hermitian duality to adjoint projectors, establish criteria for the LCD and self-dual cases, study the Murray--von Neumann equivalence of projectors, and interpret quotients by orthogonal codes in terms of module duals. Maths/math.RA_(Rings_and_Algebras) https://arxiv.org/abs/2505.24384 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2505.24384v2 Announce Type: replace-cross Abstract: We develop an estimator-based stochastic fixed-point framework for approximately computing the 2-Wasserstein barycenter of continuous, non-parametric probability measures. Notably, we provide the first rigorous convergence analysis for implementable estimator-based stochastic extensions of the fixed-point iterative scheme proposed by \'Alvarez-Esteban, del Barrio, Cuesta-Albertos, and Matr\'an (2016). In particular, we establish almost sure convergence, and identify sufficient conditions for geometric rates of convergence under controlled errors in optimal transport (OT) map estimation. We subsequently propose a concrete, provably convergent, and computationally tractable stochastic algorithm that accommodates input measures satisfying Caffarelli-type regularity conditions, which form a dense subset of the Wasserstein space. This algorithm leverages a modified entropic OT map estimator to enable efficient and scalable implementation. To facilitate quantitative evaluation, we further propose a novel and efficient procedure for synthetically generating benchmark instances, in which the input measures exhibit non-trivial features and the corresponding barycenters are approximately known. Numerical experiments on both synthetic and real-world datasets demonstrate the strong computational efficiency, estimation accuracy, and sampling flexibility of our approach. Maths/math.PR_(Probability) https://arxiv.org/abs/2603.18748 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2603.18748v2 Announce Type: replace Abstract: We establish annealed and quenched invariance principles for random walks in random conductances lifted to the p-variation rough path topology, allowing for degenerate environments and long-range jumps. Our proof is based on a unified structural strategy where pathwise convergence is viewed as a natural upgrade of the classical theory. This approach decouples the martingale lift from terms involving the integrals with respect to the corrector and the quadratic covariations. In the quenched regime, we show that the existence of a stationary potential for the corrector with $2+\epsilon$ moments is sufficient to ensure the vanishing of the corrector in $p$-variation for any $p>2$. This input, combined with our structural framework, provides a direct and modular pathway to rough path convergence. We further provide a transfer lemma to construct this potential from spatial moment bounds. While presently verified in the literature primarily for nearest-neighbor settings, our formulation isolates the exact analytic input required for pathwise convergence in more general environments. Maths/math.PR_(Probability) https://arxiv.org/abs/2510.04092 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2510.04092v2 Announce Type: replace Abstract: We examine a delayed stochastic interest rate model with super-linearly growing coefficients and develop several new mathematical tools to establish the properties of its true and truncated EM solutions. Moreover, we show that the true solution converges to the truncated EM solutions in probability as the step size tends to zero. Further, we support the convergence result with some illustrative numerical examples and justify the convergence result for the Monte Carlo evaluation of some financial quantities. Maths/math.PR_(Probability) https://arxiv.org/abs/2409.08882 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2409.08882v2 Announce Type: replace Abstract: This paper develops a non-asymptotic approach to mean field approximations for systems of $n$ diffusive particles interacting pairwise. The interaction strengths are not identical, making the particle system non-exchangeable. The marginal law of any subset of particles is compared to a suitably chosen product measure, and we find sharp relative entropy estimates between the two. Building upon prior work of the first author in the exchangeable setting, we use a generalized form of the BBGKY hierarchy to derive a hierarchy of differential inequalities for the relative entropies. Our analysis of this complicated hierarchy exploits an unexpected but crucial connection with first-passage percolation, which lets us bound the marginal entropies in terms of expectations of functionals of this percolation process. Maths/math.PR_(Probability) https://arxiv.org/abs/2406.03058 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2406.03058v3 Announce Type: replace Abstract: We consider strong approximations of $1+1$-dimensional stochastic PDEs driven by additive space-time white noise. It has been long proposed (Davie-Gaines '01, Jentzen-Kloeden '08), as well as observed in simulations, that approximation schemes based on samples from the stochastic convolution, rather than from increments of the underlying Wiener processes, should achieve significantly higher convergence rates with respect to the temporal timestep. The present paper proves this. For a large class of nonlinearities, with possibly superlinear growth, a temporal rate of (almost) $1$ is proven, a major improvement on the rate $1/4$ that is known to be optimal for schemes based on Wiener increments. The spatial rate remains (almost) $1/2$ as it is standard in the literature. Maths/math.PR_(Probability) https://arxiv.org/abs/2202.10999 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2202.10999v3 Announce Type: replace Abstract: For deterministic monotone cellular automata on the $d$-dimensional integer lattice, Toom has given necessary and sufficient conditions for the all-one fixed point to be stable against small random perturbations. The proof of sufficiency is based on an intricate Peierls argument. We present a simplified version of this Peierls argument. Our main motivation is the open problem of determining stability of monotone cellular automata with intrinsic randomness, in which for the unperturbed evolution the local update rules at different space-time points are chosen in an i.i.d. fashion according to some fixed law. We apply Toom's Peierls argument to prove stability of a class of cellular automata with intrinsic randomness and also derive lower bounds on the critical parameter for some deterministic cellular automata. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.14765 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14765v1 Announce Type: cross Abstract: We present a geometric framework for Reinforcement Learning (RL) that views policies as maps into the Wasserstein space of action probabilities. First, we define a Riemannian structure induced by stationary distributions, proving its existence in a general context. We then define the tangent space of policies and characterize the geodesics, specifically addressing the measurability of vector fields mapped from the state space to the tangent space of probability measures over the action space. Next, we formulate a general RL optimization problem and construct a gradient flow using Otto's calculus. We compute the gradient and the Hessian of the energy, providing a formal second-order analysis. Finally, we illustrate the method with numerical examples for low-dimensional problems, computing the gradient directly from our theoretical formalism. For high-dimensional problems, we parameterize the policy using a neural network and optimize it based on an ergodic approximation of the cost. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.14681 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14681v1 Announce Type: cross Abstract: Given a classical gas described by the truncated correlation functions of all orders, we prove convergence of an expansion of the pair interaction part of the (unknown) potential in terms of the truncated correlation functions of all orders, at infinite volume. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.14277 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14277v1 Announce Type: cross Abstract: We study the growth of entanglement and circuit complexity in random passive linear optical networks as a function of the circuit depth. For entanglement dynamics, we start with an initial Gaussian state with all $n$ modes squeezed. For random brickwall circuits, we show that entanglement, as measured by the R\'enyi-2 entropy, grows at most diffusively as a function of the depth. In the other direction, for arbitrary circuit geometries we prove bounds on depths which ensure the average subsystem entanglement reaches within a constant factor of the maximum value in all subsystems, and bounds which ensure closeness of the random linear optical unitary to a Haar random unitary in $L^2$ Wasserstein distance. We also consider robust circuit complexity for random one-dimensional brickwall circuits, as measured by the minimum number of gates required in any circuit that approximately implements the linear optical unitary. Viewing this as a function of the number of modes and the circuit depth, we show the robust circuit complexity for random one-dimensional brickwall circuits scales at most diffusively in the depth with high probability. The corresponding Gaussian unitary $\tilde{\mathcal U}$ for the approximate implementation retains high output fidelity $|\langle\psi|\mathcal U^\dagger \tilde{\mathcal U}|\psi\rangle|^2$ for pure states $|\psi\rangle$ with constrained expected photon-number. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.15226 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15226v1 Announce Type: new Abstract: In this paper, we prove existence and uniqueness of energy solutions for nonlinear Schr\"odinger equations with a multiplicative white noise on $R^d$ with $d\le3$. We rely on an exponential trans-form and conserved quantities for existence of energy solutions. Using paracontrolled calculus, we prove Strichartz inequalities which encode the dispersive properties of the solutions. This allows to obtain local well-posedness for low regularity solutions and uniqueness of energy solutions for various equations. In particular, our results are the first results of propagation without loss of both regularity and localization for such equations in full space as well as the first results on $R^3$ for such singular dispersive SPDEs. We are also obtain local well-posedness in two dimensions for deterministic initial data. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.15152 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15152v1 Announce Type: new Abstract: We revisit the random allocation of $n$ balls into $N$ boxes with probabilities $q_1,\ldots,q_N$, focusing on the proportion $\hat q_r$ of boxes containing exactly $r$ balls. Classical asymptotic results for the expectations, variances, and covariances of these proportions are reformulated in terms of the size distribution of a randomly chosen box. We further derive explicit two-sided bounds for the associated remainder terms, allowing for weaker assumptions than those previously required. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.15146 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15146v1 Announce Type: new Abstract: We consider nested CLE$_4$ in a simply-connected domain and compute the following renormalised probabilities: the probability that two points belong to the same CLE$_4$ gasket and the probability that two points belong to the outermost CLE$_4$ gasket. While the integrability is rooted in the conformal field theory of the Ashkin-Teller (AT) model, we provide a purely probabilistic calculation via Brownian loop soups and the geometry of the 2D continuum Gaussian free field. More generally, we also calculate renormalised probabilities that two points belong to CLE$_4$ gaskets sampled in alternation with certain two-valued sets of the Gaussian free field. These quantities correspond to the two-point function of the conjectured scaling limit of the AT single spins on the critical line. At the decoupling point, our results recover the Ising model correlations and suggest a CLE$_4$-based FK representation of the AT spin model. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.15007 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15007v1 Announce Type: new Abstract: Consider a generalised preferential attachment tree with attachment function $f$, that is a random tree, where at each time-step a node connects to an existing node $v$ with probability proportional to $f(\mathrm{deg}(v))$, where $\mathrm{deg}(v)$ denotes the degree of the node in the existing tree. We provide a counter-example to a conjecture of the author asserting that under the assumption $\sum_{j=1}^{\infty} \frac{1}{f(j)^2} < \infty$ there is a persistent hub in the model, that is, a single node that has the maximal degree for all but finitely many time-steps. The counter-example is a minor modification of a related counter-example due to Galganov and Ilienko. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.14893 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14893v1 Announce Type: new Abstract: This paper investigates a class of generalized mean-reflected McKean-Vlasov type backward stochastic differential equations (BSDEs). Our new framework combines a mean reflection constraint on the solution's expectation with a generalized integral with respect to a continuous non-decreasing process. We establish the existence and uniqueness of the solution. The uniqueness is derived via stability estimates, while the existence is proved by employing a penalization method combined with a smooth approximation of the obstacle. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.14851 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14851v1 Announce Type: new Abstract: We present and study the Pool model in $\mathbb{R}^2$, a rotationally symmetric analogue of Multi-Particle Diffusion-Limited Aggregation (MDLA), in which particles ("droplets") perform continuous-time random walks and are absorbed upon entering a circular pool initially centered at the origin. Each absorbed particle increases the pool's mass, and the pool expands so that its area grows accordingly, yielding a natural mass-preserving dynamics. A central tool which is of independent interest is a version of Kurtz's theorem for this model, depicting the field of particles conditioned on the growth of the pool as an independent non-homogeneous Poisson point process. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.14543 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14543v1 Announce Type: new Abstract: This paper investigates the approximation of invariant measures for McKean-Vlasov stochastic differential equations (SDEs) using the Euler-Maruyama (EM) scheme under a monotonicity condition. Firstly, the convergence of the numerical solution from the EM scheme to its continuous-time counterpart is established. Secondly, we show that the numerical solution admits a unique invariant measure and derive its convergence rate under the Wasserstein metric. In parallel, it is demonstrated that the associated particle system also possesses these properties. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.14427 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14427v1 Announce Type: new Abstract: A criterion for proving a strong form of propagation of chaos on the path space, known as entropy chaos, for a general interacting diffusion system is proposed. Our analysis focuses on the class of conservative diffusions introduced by Carlen, which are characterized by infinitesimal characteristic pairs, that is, a time-marginal probability density and a current velocity field. A key property of this broad class is that the processes remain diffusions under time-reversal. We prove that, given a suitable bound on the relative entropy (with respect to the Wiener measure) and the weak convergence of both drifts and fixed-time marginal densities, strong entropy chaos at the process level is achieved in the infinite particle limit, provided the limit drift satisfies a specific regularity condition. This stochastic framework encompasses various singular interacting particle systems and their related asymptotic scenarios. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.14396 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14396v1 Announce Type: new Abstract: By using a probabilistic technique based on the exponential change of measure we find a precise tail asymptotic behavior of some perpetuities with distributions close to the Dickman distribution. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.14697 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14697v1 Announce Type: cross Abstract: Let $X$ be a Dedekind complete Banach lattice, and let $P\colon X\to X$ be a positive projection for which the largest central operator below $P$ is $\alpha \operatorname{id}_X$, for some $\alpha \ge 0$. Wickstead conjectured that $\alpha $ must either be $0$ or $1/n$, for some $n \in \mathbb{N}$, and proved it for finite-dimensional $X$. In this paper, we show that the conjecture holds in general. As a consequence, we settle the representation problem for Banach lattice algebras: we show that there exist Banach lattice algebras of dimension $2$ that do not admit a faithful representation as regular operators on any Dedekind complete Banach lattice. Maths/math.OA_(Operator_Algebras) https://arxiv.org/abs/2604.04682 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.04682v2 Announce Type: replace-cross Abstract: We prove gap theorems for entropy norms on automorphism groups of K3 surfaces, Enriques surfaces, and irreducible holomorphic symplectic manifolds. We also study the achirality of automorphisms of K3 surfaces and Enriques surfaces in terms of genus-one fibrations. Maths/math.GR_(Group_Theory) https://arxiv.org/abs/2511.04296 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2511.04296v3 Announce Type: replace-cross Abstract: Let $G$ be a group acting on a field $L$, and suppose that $L /L^G$ is a finite extension. We show that the category of semilinear representations of $G$ over $L$ can be described in terms of the category of linear representations of $H$, the kernel of the map $G \rightarrow \mathrm{Aut}(L)$. When $G$ is finite and $L$ has characteristic 0 this provides a character theory for semilinear representations of $G$ over $L$, which recovers ordinary character theory when the action of $G$ on $L$ is trivial. Maths/math.GR_(Group_Theory) https://arxiv.org/abs/2602.20613 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2602.20613v2 Announce Type: replace Abstract: In this paper, we introduce a class of blocks which is called hyperfocal abelian Frobenius blocks.This class of blocks is an analogous version of the block with abelian defect group and Frobenius inertial quotient at hyperfocal level and includes the blocks with Klein four hyperfocal subgroups and cyclic hyperfocal subgroups. We show that there is a stable equivalence of Morita type between the hyperfocal subalgebras of the hyperfocal abelian Frobenius blocks and a group algebra of a Frobenius group associated with the hyperfocal subgroup of the block. As applications, we can partially describe some structures of the blocks with Klein four hyperfocal subgroups and cyclic hyperfocal subgroups,such as the structures of their hyperfocal subalgebras in terms of derived categories and the structures of their characters. As a consequence, we show that Broue's abelian defect group conjecture holds for blocks with Klein four hyperfocal subgroups. Maths/math.GR_(Group_Theory) https://arxiv.org/abs/2506.13235 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2506.13235v2 Announce Type: replace Abstract: Given a finitely generated amenable group $H$ satisfying some mild assumptions, we relate isoperimetric profiles of the lampshuffler group $\mathsf{Shuffler}(H)=\mathsf{FSym}(H)\rtimes H$ to those of $H$. Our results are sharp for all exponential growth groups for which isoperimetric profiles are known, including Brieussel-Zheng groups. This refines previous estimates obtained by Erschler and Zheng and by Saloff-Coste and Zheng. The most difficult part is to find an optimal upper bound, and our strategy consists in finding suitable lamplighter subgraphs in lampshufflers. This novelty applies more generally for many examples of halo products, a class of groups introduced recently by Genevois and Tessera as a natural generalisation of wreath products. Lastly, we also give applications of our estimates on isoperimetric profiles to the existence problem of regular maps between such groups. Maths/math.GR_(Group_Theory) https://arxiv.org/abs/2505.05220 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2505.05220v2 Announce Type: replace Abstract: Let F be a non-archimedean local field of characteristic zero whose residue field has at least three elements. Let G be an almost simple linear algebraic group over F, with rank_F(G) >= 2. Let X be a simply connected symmetric space of infinite dimension and finite rank, with non-positive curvature operator. We prove that every continuous action by isometries of G on X has a fixed point. If the group G contains SL_3(F), the result holds without any assumption on the non-archimedean local field F. The result extends to cocompact lattices in G if the cardinality of the residue field of F is large enough, with a bound that depends on rank_F(G). Maths/math.GR_(Group_Theory) https://arxiv.org/abs/2604.15154 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15154v1 Announce Type: cross Abstract: Let $G$ and $H$ be finitely generated groups. In this paper, we prove the quantitative coarse Baum--Connes conjecture for the free product $G* H$ under the assumption that the conjecture holds for both $G$ and $H$. Maths/math.GR_(Group_Theory) https://arxiv.org/abs/2604.14383 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14383v1 Announce Type: cross Abstract: This article describes a natural piecewise Euclidean bi-simplicial cell structure for the space of $n$-element multisets in a fixed Euclidean rectangle. In particular, we highlight some connections with spaces of complex polynomials and permutahedra. Maths/math.GR_(Group_Theory) https://arxiv.org/abs/2604.15303 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15303v1 Announce Type: new Abstract: We prove a strong general-purpose bound for the diameter of a finite group depending only on the diameters of its composition factors and the maximal exponent of a normal abelian section. There are a number of notable applications: (1) if $G$ is a finite soluble group of exponent $e$, $\mathrm{diam}(G) \ll e (\log |G|)^8$, (2) anabelian groups with bounded-rank composition factors have polylogarithmic diameter, (3) transitive soluble subgroups of $S_n$ have diameter $\ll n^5$, and (4) Grigorchuk's gap conjecture holds for any finitely generated group acting faithfully on a bounded-degree rooted tree. Additionally, conditional on Babai's conjecture, (5) any transitive permutation group of degree $n$ has diameter bounded by a polynomial in $n$ (a folkloric conjecture), and (6) Grigorchuk's gap conjecture holds for residually finite groups, and thus the conjecture reduces to the simple case. Maths/math.GR_(Group_Theory) https://arxiv.org/abs/2604.14244 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14244v1 Announce Type: new Abstract: In this paper, we investigate the existence of $C^n$, $n\in \mathbb{N}^+$, solutions for a class of second-order iterative functional equations involving iterates of the unknown function and a nonlinear term. Applying the Fiber Contraction Theorem and Fa\`a di Bruno's Formula, we establish the existence of bounded $C^n$ solutions with bounded derivatives of order from $1$ to $n$. Maths/math.GM_(General_Mathematics) https://arxiv.org/abs/2604.14194 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14194v1 Announce Type: new Abstract: Let $\mathbb{F}$ be a sub-modulus field such that $2 \neq 0$. Let $\mathcal{X}$ be a sub-normed linear space over $\mathbb{F}$. Then we show that \begin{align*} \bigg|\|x\|-\|y\|\bigg|\leq \frac{2}{|2|}\|x+y\|+\frac{2}{|2|}\max\{\|x-y\|, \|y-x\|\}-(\|x\|+\|y\|) \end{align*} and \begin{align*} \bigg|\|x\|-\|y\|\bigg|\leq \|x\|+\|y\|-\frac{2}{|2|}\|x+y\|+\frac{2}{|2|}\max\{\|y-x\|, \|x-y\|\}. \end{align*} Above inequalities are finite field versions of important Tarski-Maligranda inequalities obained by Maligranda [\textit{Banach J. Math. Anal., 2008}]. Maths/math.GM_(General_Mathematics) https://arxiv.org/abs/2604.01100 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.01100v2 Announce Type: replace Abstract: We prove a sharp phase transition in the regularity of the extremal distribution $E^s \oplus E^u$ for $C^\infty$ volume-preserving partially hyperbolic diffeomorphisms on closed $3$-manifolds: if $E^s \oplus E^u$ is Lipschitz, then it is automatically $C^\infty$. This extends the rigidity phenomenon established by Foulon--Hasselblatt for conservative Anosov flows in dimension $3$ to the partially hyperbolic setting. This gain in regularity has several applications to rigidity problems. In particular, we study the relationship between the $\ell$-integrability condition introduced by Eskin--Potrie--Zhang and joint integrability in the conservative setting, yielding rigidity results for $u$-Gibbs measures. We also obtain several $C^\infty$ classification results for partially hyperbolic diffeomorphisms on $3$-manifolds under various assumptions. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2602.08608 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2602.08608v2 Announce Type: replace Abstract: We prove the dynamical Mordell-Lang conjecture for product of endomorphisms of an affine curve and a projective curve over $\overline{\mathbb{Q}}$. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2601.04408 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2601.04408v2 Announce Type: replace Abstract: This research examines the influence of the Coriolis parameter on the behaviour of the geophysical Korteweg-de Vries (KdV) equation. To efficiently approximate its solution, a novel surrogate framework, termed G-KdVNet, is proposed by integrating artificial neural networks with the Adomian decomposition method (ADM). In the proposed approach, ADM is first employed to generate reliable semi-analytical solution data, which are subsequently used to train the neural network model. The developed model demonstrates strong predictive capability in capturing the nonlinear dynamics of the KdV system. Numerical results indicate that the proposed model achieves improved accuracy compared with conventional baseline methods, with absolute errors of the order of e-3 for unseen data. The results suggest that the proposed ANN-ADM surrogate offers an efficient and accurate alternative for solving nonlinear geophysical models, with potential applicability to a broader class of dispersive wave equations. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2512.19430 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2512.19430v3 Announce Type: replace Abstract: We survey recent results and current challenges concerning the growth rate inequality for sphere endomorphisms, and present a number of open problems and conjectures arising in this context. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2511.21324 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2511.21324v4 Announce Type: replace Abstract: Astorg and Boc Thaler studied the dynamics of certain skew-product tangent to the identity on $\mathbb{C}^2$, with two real parameters $\alpha>1$ and $\beta$ derived from its coefficients. They proved that if there exists an increasing sequence of positive integers $(n_k)_{k\geqslant 1}$ such that $(\sigma_k)_{k\geqslant 1}:=(n_{k+1}-\alpha n_k-\beta\ln n_k)_{k\geqslant 1}$ converges, then $f$ admits wandering domains of rank one. They also proved that for $\alpha>1$ with the Pisot property, the condition that $\theta:=\frac{\beta\ln\alpha}{\alpha-1}$ is rational is sufficient for the existence of $(n_k)_{k\geqslant 1}$ such that $(\sigma_k)_{k\geqslant 1}$ converges to a cycle. They asked if this condition is necessary. When $\alpha$ is an algebraic number, we answer the question of Astorg and Boc Thaler in the affirmative. Furthermore, denoting by $P(x)\in\mathbb{Z}[x]$ the minimal polynomial of~$\alpha$, we prove that $\theta\in\frac{1}{P(1)}\mathbb{Z}$ is necessary and sufficient for the existence of $(n_k)_{k\geqslant 1}$ such that $(\sigma_k)_{k\geqslant 1}$ converges. Combined with the work of Astorg and Boc Thaler, our result provides explicit new examples of skew-products on $\mathbb{C}^2$ with wandering domains of rank one. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.15181 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15181v1 Announce Type: cross Abstract: Extrapolative prediction of complex nonlinear dynamics remains a central challenge in engineering. This study proposes a one-shot learning method to identify global frequency-response curves from a single excitation time history by learning governing equations. We introduce MEv-SINDy (Multi-frequency Evolutionary Sparse Identification of Nonlinear Dynamics) to infer the governing equations of non-autonomous and multi-frequency systems. The methodology leverages the Generalized Harmonic Balance (GHB) method to decompose complex forced responses into a set of slow-varying evolution equations. We validated the capabilities of MEv-SINDy on two critical Micro-Electro-Mechanical Systems (MEMS). These applications include a nonlinear beam resonator and a MEMS micromirror. Our results show that the model trained on a single point accurately predicts softening/hardening effects and jump phenomena across a wide range of excitation levels. This approach significantly reduces the data acquisition burden for the characterization and design of nonlinear microsystems. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.14945 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14945v1 Announce Type: cross Abstract: We focus on halo products, a class of groups introduced by Genevois and Tessera, and whose geometry mimics lamplighters. Famous examples are lampshufflers. Motivated by their work on the classifications up to quasi-isometry of these groups, we initiate a more quantitative study of their geometry. Indeed, it follows from the work of Delabie, Koivisto, Le Ma\^itre and Tessera that quantitative orbit equivalence between amenable groups is closely related to their large scale geometry, such a connection being justified by the use, in their main results, of a well-known quasi-isometry invariant: the isoperimetric profile. Inspired by their work on quantitative orbit equivalence between lamplighters, we prove a stability result for orbit equivalence of permutational halo products, going beyond the framework of standard halo products, using a new notion of orbit equivalence of pairs. Combined with our asymptotics of isoperimetric profiles obtained in an earlier article, we prove that most of these constructions are quantitatively optimal. For instance, we show that $\mathsf{Shuffler}(\mathbb{Z}^{k+\ell})$ and $\mathsf{Shuffler}(\mathbb{Z}^{k})$ are $\mathrm{L}^p$ orbit equivalent if and only if $p<\frac{k}{k+\ell}$, thus quantifying how much the geometries of these non-quasi-isometric groups differ. We finally build orbit equivalence couplings using the notion of F{\o}lner tiling sequences. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.14669 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14669v1 Announce Type: cross Abstract: Zeroth-order (ZO) methods are widely used when gradients are unavailable or prohibitively expensive, including black-box learning and memory-efficient fine-tuning of large models, yet their optimization dynamics in deep learning remain underexplored. In this work, we provide an explicit step size condition that exactly captures the (mean-square) linear stability of a family of ZO methods based on the standard two-point estimator. Our characterization reveals a sharp contrast with first-order (FO) methods: whereas FO stability is governed solely by the largest Hessian eigenvalue, mean-square stability of ZO methods depends on the entire Hessian spectrum. Since computing the full Hessian spectrum is infeasible in practical neural network training, we further derive tractable stability bounds that depend only on the largest eigenvalue and the Hessian trace. Empirically, we find that full-batch ZO methods operate at the edge of stability: ZO-GD, ZO-GDM, and ZO-Adam consistently stabilize near the predicted stability boundary across a range of deep learning training problems. Our results highlight an implicit regularization effect specific to ZO methods, where large step sizes primarily regularize the Hessian trace, whereas in FO methods they regularize the top eigenvalue. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.14464 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14464v1 Announce Type: cross Abstract: Given a thin torus $T_K$ around a knot $K\subset \mathbb{R}^3$, we construct Morse models of cord algebra $Cord(T_K)$ with $\mathbb{Z}$ and loop space coefficients. Using the Multiple time scale dynamics we identify $Cord(T_K; \mathbb{Z})$ with $Cord(K; \mathbb{Z})$. In combination with the works of Cieliebak-Ekholm-Latschev-Ng and Petrak this indirectly relates $Cord(T_K)$ to $0$-th degree Legendrian contact homology $LCH_0(\mathcal{L}^\ast_+ T_K)$ of one component of the unit conormal bundle over $T_K$. Our definition of $Cord(T_K)$ is motivated by $J$-holomorphic curves with boundary on the Lagrangian submanifold $L^\ast_+ T_K\cup\mathbb{R}^3$ with an arboreal singularity along the torus $T_K$. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.14346 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14346v1 Announce Type: cross Abstract: In this paper we consider the Toda lattice $(\mathbf{p}(t);\mathbf{q}(t))$ at thermal equilibrium, meaning that its variables $(p_j)$ and $(e^{q_j - q_{j+1}})$ are independent Gaussian and Gamma random variables, respectively. We show under diffusive scaling that the space-time fluctuations for the model's currents converge to an explicit Gaussian limit. As consequences, we deduce, (i) the scaling limit for the trajectory of a single particle $q_0$ is a Brownian motion; (ii) space-time two-point correlation functions for the model decay inversely with time, with explicit scaling distributions predicted by Spohn (Spohn, J. Phys. A 53 (2020), 265004). Our starting point is the notion that the Toda lattice can be thought of as a dense collection of many ``quasi-particles'' that interact through scattering. The core of our work is to establish that the full joint scaling limit of the fluctuations for these quasi-particles is given by a Gaussian process, called a dressed L\'evy-Chentsov field. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.15293 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15293v1 Announce Type: new Abstract: For each $d \in {1,2,3,7,11}$, let $T_d$ be the nearest-integer complex continued fraction map associated with the Euclidean ring $\mathcal{O}*d$, and let $(a_n)$ be its digit sequence. We prove two metric results for this five-system family. First, for every sequence $(u_n)*{n\ge 1}$ with $u_n \ge 1$, the set of points for which $|a_n| \ge u_n$ for infinitely many $n$ has full or zero normalized Lebesgue measure according as $\sum_{n=1}^\infty u_n^{-2}$ diverges or converges. This gives a unified Borel--Bernstein theorem, extending the Hurwitz case $d=1$ to all five Euclidean imaginary quadratic fields. Second, for any infinite set $S \subset \mathcal{O}_d$, if $\tau(S)$ denotes its convergence exponent, then the digit-restricted set $F_d(S)={z:\ a_n(z)\in S\ \text{for all } n,\ |a_n(z)|\to\infty}$ satisfies $\dim_H F_d(S)=\tau(S)/2$. More generally, for any cutoff function $f(n)\to\infty$, the set $F_d(S,f)={z\in F_d(S):\ |a_n(z)|\le f(n)\ \text{for all } n}$ is either empty or has the same Hausdorff dimension $\tau(S)/2$. The proof combines quantitative ergodic properties of the nearest-integer systems with a large-digit conformal iterated function subsystem that is $2$-decaying. We also obtain applications to sparse patterns, shrinking targets, and almost-sure $L'evy$- and Khinchine-type laws. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.15220 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15220v1 Announce Type: new Abstract: We present a mathematical model of a market with $m$ shares traded across $n$ investor groups, each one with similar motivations and trading strategies. The market of each asset consists of a fixed amount of cash and shares (no additions are allowed over time, so the system is closed), and the trading groups are influenced by trend and valuation motivations when buying or selling each asset, but follow a strategy where the purchase of one asset depends on the price of another, while the sale does not. Using these assumptions and basic microeconomic principles, the mathematical model is derived using a dynamic systems approach. We analyze the stability of the model's equilibrium points and determine the parameter conditions for such stability. First, we show that all equilibria are stable in the absence of a clear emphasis on trend-based valuation for each share. Secondly, for systems where the trading group prioritizes the valuation of each stock and the trend of the other for trading purposes, we establish stability conditions and demonstrate with numerical examples that when instability occurs, it manifests as price oscillations in the stocks. Furthermore, we argue for the existence of periodic solutions via a Hopf bifurcation, taking the momentum coefficient as the bifurcation parameter. Finally, we present examples and numerical simulations to support and expand upon the analytical results. One finding in economics and finance is the existence of cyclical behavior in the absence of exogenous factors, as determined by the momentum coefficient. In particular, a stable equilibrium price becomes unstable as trend-based trading increases. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.14964 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14964v1 Announce Type: new Abstract: In this paper, we investigate induced and nonlinear fiber topological pressure for random dynamical systems. We define a non-averaged induced fiber pressure via spanning and separated sets, characterize it as the pseudo-inverse of the classical fiber topological pressure studied previously, and establish the corresponding variational principle. We also define the nonlinear fiber pressure and prove the associated variational principles. Finally, we extend the combined theory to the higher-dimensional setting. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.14948 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14948v1 Announce Type: new Abstract: We investigate expansive solutions of the $N$-body problem in $\mathbb{R}^d$ ($d\ge2$) driven by homogeneous Newtonian potentials of degree $-\alpha$. We establish the existence of half-entire expansive motions with prescribed initial configuration and asymptotic direction for a wide range of homogeneity exponents $\alpha$. Our approach is variational and relies on the minimization of a suitably renormalized Lagrangian action, allowing us to treat in a unified framework the hyperbolic, parabolic, and hyperbolic-parabolic regimes in the sense of Chazy's classification. Beyond existence, we derive refined asymptotic expansions for all classes of expansive solutions, identifying higher-order correction terms and improving previously known growth estimates, including the classical Newtonian case $\alpha=1$. In particular, for hyperbolic-parabolic solutions, we provide a detailed description of the interplay between linear escape of cluster centers and internal parabolic dynamics, extending the cluster scattering picture to general homogeneous potentials. Finally, we interpret these solutions within the geometric framework of the Jacobi-Maupertuis metric and the weak KAM theory. In this perspective, expansive motions correspond to geodesic rays and calibrating curves for the associated Hamilton-Jacobi equation, yielding a dynamical characterization of the boundary at infinity and a refined description of global viscosity solutions. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.14759 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14759v1 Announce Type: new Abstract: While the classical Critical Depth Hypothesis (CDH) effectively explains the onset of blooms as transient instabilities, it does not fully capture the seasonal decoupling of biological rates and the long-term persistence of phytoplankton communities in fluctuating thermal environments. To address these limitations, we introduce a parsimonious framework that leverages the theory of non-autonomous dynamical systems to diagnose the stability of phytoplankton communities throughout the entire annual cycle. By linearizing the dynamics around the extinction equilibrium, we identify the invasion growth rate -formally the Floquet exponent-and derive the critical nutrient requirement ($\gamma$crit) as a bifurcation point for uniform persistence. Using end-of-the-century projections from the GFDL-ESM4 model under a high-emission scenario (SSP5-8.5), we identify a global regime shift characterized by a widespread expansion of metabolic-driven regimes, which increasingly displace regions where stability was historically governed by physical mixing. Relevance to Life Sciences. Quantitative analysis of system stability challenges CDH by demonstrating that metabolic constraints increasingly modulates phytoplankton persistence in a changing ocean. Our results, based on high-emission projections, reveal a profound physical-biological decoupling at the poles: while warming reduces the critical nutrient requirement ($\gamma$crit) facilitating persistence in previously marginal waters, this metabolic expansion is offset at poles. A 1:4 ratio between newly viable niches and ice-free deserts suggests that cryospheric retreat does not guarantee a proportional expansion of life. In addition, we identify the North Atlantic Subpolar Gyre as a ''metabolic refuge'' where mixing dynamics still anchor the ecosystem against global thermalization. By providing a ''radiography'' of the future ocean's complexity, this methodology offers a mechanistic basis to deconstruct how the dynamic balance between environmental energy and metabolic demands may determine the functional integrity of the marine biosphere under extreme anthropogenic forcing. Mathematical Content. The temperature dependence of biological rates is modeled using a thermodynamic equation, coupling population dynamics with seasonal variations in mixed layer depth and temperature. Given the non-autonomous nature of the system under annual forcing, we characterize the stability of the extinction equilibrium through its associated invasion growth rate. This rate is analytically derived as the Floquet exponent $\lambda$P , which provides a rigorous condition for uniform persistence (Theorem 3.2). The numerical analysis of this exponent, projected onto a global scale, quantifies the relative influence of environmental drivers on the stability threshold $\gamma$crit. This allows for the definition of the thermal dominance index (DT ), a metric that identifies the geographic transition from mixing-driven to metabolic-driven ecological control. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2601.20640 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2601.20640v2 Announce Type: replace-cross Abstract: We consider on an arbitrary Riemannian manifold $M$ the \textit{Leibenson equation} $\partial _{t}u=\Delta _{p}u^{q}$, that is also known as a \textit{doubly nonlinear evolution equation}. We prove that if $p>1, q>0$ and $pq\geq 1$ then the Cauchy-problem \begin{equation*} \left\{\begin{array}{ll}\partial _{t}u=\Delta _{p}u^{q} &\text{in}~M\times (0, \infty), \\u(x, 0)=u_{0}(x)& \text{in}~M,\end{array}\right.\end{equation*} has a unique weak solution for any $u_{0}\in L^{1}(M)\cap L^{\infty}(M)$. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2310.20526 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2310.20526v2 Announce Type: replace-cross Abstract: In this paper, we focus on estimating measure upper bounds of nodal sets of solutions to the following boundary value problem \begin{equation*} \left\{ \begin{array}{lll} \Delta u+Vu=0\quad \mbox{in}\ \Omega,\\[2mm] u=0\quad \mbox{on}\ \partial\Omega, \end{array}\right. \end{equation*} where $V\in W^{1,\infty}(\Omega)$ is a potential function, and $\Omega \subset \mathbb{R}^n$ ($n \geq 2$) is a bounded domain whose boundary is of class $C^{1,\alpha}$ for any $0<\alpha<1$. By developing a delicate dividing iteration procedure, we show that upper bound of the $(n-1)$-dimensional Hausdorff measure of the nodal set of $u$ in $\Omega$ is $$C\Big(1+\log\left(\|\nabla V\|_{L^{\infty}(\Omega)}+1\right)\Big)\cdot\left(\|V\|_{L^{\infty}(\Omega)}^{\frac{1}{2}}+\|\nabla V\|_{L^{\infty}(\Omega)}^{\frac{1}{2}}+1\right),$$ provided $V$ is analytic, here $C$ is a positive constant depending only on $n$ and $\Omega$. In particular, if $\|\nabla V\|_{L^{\infty}(\Omega)}$ is small, the upper bound for the measure of the nodal set of $u$ is $C\left(\|V\|^{\frac{1}{2}}_{L^{\infty}(\Omega)}+1\right)$, which is sharp in the sense of a famous conjecture of Yau. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2604.08822 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.08822v2 Announce Type: replace Abstract: We prove that for a residual (and hence dense) subset $\mathcal{G}$ of Riemannian metrics on $S^{n+1}$ in the $C^{3}$ topology, no area-minimizing integral $n$-current that is a boundary admits a singular tangent cone which is linearly stable in the Euclidean sense. The proof proceeds in three stages. First, we develop a perturbation theorem: given any area-minimizer possessing an isolated singularity whose unique tangent cone $C$ is linearly stable, we construct an explicit $C^{3}$-small metric perturbation that destroys the compatibility conditions required for $C$ to persist as a tangent cone. The construction rests on the Hardt--Simon asymptotic expansion near isolated singularities, the spectral theory of the Jacobi operator on the cross-section of $C$, and a surjectivity argument showing that the map from compactly supported metric variations to forcing terms in the linearised minimal-surface equation on $C$ has dense range. Second, we establish that the set of metrics admitting no area-minimizer with a prescribed cone type as tangent cone is open, using compactness of integral currents and upper-semicontinuity of the density function. Third, we assemble these ingredients via a Baire category argument, intersecting countably many open dense sets to obtain the residual set $\mathcal{G}$. An extension to non-isolated singularities is outlined using Federer--Almgren dimension reduction. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2602.10913 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2602.10913v2 Announce Type: replace Abstract: In this paper we classify the low energy $\varepsilon$-harmonic maps from the surfaces of constant curvature with positive genus into the round sphere. We find that all such maps with degree $\pm1$ are all quantitively close to a bubble configuration with bubbles forming at special points on the domain with bubbling radius proportional to $\varepsilon^{1/4}$. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2507.05459 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2507.05459v2 Announce Type: replace Abstract: Llarull's scalar curvature rigidity theorem states that a 1-Lipschitz map $f: M\to S^n$ from a closed connected Riemannian spin manifold $M$ with scalar curvature $\mathrm{scal}\ge n(n-1)$ to the standard sphere $S^n$ is an isometry if the degree of $f$ is nonzero. We investigate if one can replace the condition $\mathrm{deg}(f)\neq0$ by the weaker condition that $f$ is surjective. The answer turns out to be "no" for $n\ge3$ but "yes" for $n=2$. If we replace the scalar curvature by Ricci curvature, the answer is "yes" in all dimensions. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2507.00471 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2507.00471v2 Announce Type: replace Abstract: We prove that a sub-Riemannian manifold equipped with a full-support Radon measure is never $\mathrm{CD}(K,N)$ for any $K\in \mathbb{R}$ and $N\in (1,\infty)$ unless it is Riemannian. This generalizes previous non-CD results for sub-Riemannian manifolds, where a measure with smooth and positive density is considered. Our proof is based on the analysis of the tangent cones and the geodesics within. Secondly, we construct new $\mathrm{RCD}$ structures on $\mathbb{R}^n$, named cone-Grushin spaces, that fail to be sub-Riemannian due to the lack of a scalar product along a curve, yet exhibit characteristic features of sub-Riemannian geometry, such as horizontal directions, large Hausdorff dimension, and inhomogeneous metric dilations. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2412.21050 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2412.21050v2 Announce Type: replace Abstract: We prove parabolic versions of several known gap theorems in classical Yang-Mills theory. On an $\mathrm{SU}(r)$-bundle of charge $\kappa$ over the 4-sphere, we show that the space of all connections with Yang-Mills energy less than $4 \pi^2 \left( |\kappa| + 2 \right)$ deformation-retracts under Yang-Mills flow onto the space of instantons, allowing us to simplify the proof of Taubes's path-connectedness theorem. On a compact quaternion-K\"ahler manifold with positive scalar curvature, we prove that the space of pseudo-holomorphic connections whose $\mathfrak{sp}(1)$ curvature component has small Morrey norm deformation-retracts under Yang-Mills flow onto the space of instantons. On a nontrivial bundle over a compact manifold of general dimension, we prove that the infimum of the scale-invariant Morrey norm of curvature is positive. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2105.10711 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2105.10711v3 Announce Type: replace Abstract: We construct a one-parameter family of embedded doubly periodic minimal surfaces of genus three with four parallel ends. The Weierstrass data for each surface of the family are given and the two dimensional period problem is solved. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2604.15192 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15192v1 Announce Type: cross Abstract: Symplectic and complex toric quasifolds are a generalization of toric manifolds and orbifolds to the nonrational case. In this paper, we reframe these notions from the viewpoint of algebraic geometry. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2604.15087 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15087v1 Announce Type: cross Abstract: This article provides a survey of gauge theory for families, with a particular focus on its applications to diffeomorphism groups of $4$-manifolds that were developed during the period 2021--2025. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2604.15071 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15071v1 Announce Type: cross Abstract: This article surveys gauge theory for families and its applications to the comparison between the diffeomorphism group and the homeomorphism group of $4$-manifolds, up to 2021. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2604.15008 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15008v1 Announce Type: cross Abstract: Semiclassical analysis and noncommutative geometry are two pillars of quantum theory. It's only recently that bridges between them have been emerging. In this monograph, we combine various techniques from functional analysis and spectral theory to obtain semiclassical Weyl laws and extensions of Connes' integration formula for a large class of noncommutative manifolds (i.e., spectral triples). These results generalize and simplify recent results of McDonald-Sukochev-Zanin. In particular, all the regularity assumptions and restrictions on dimension there are removed in our approach. Moreover, the Tauberian condition used by McDonald-Sukochev-Zanin is replaced by a weaker spectral theoretic condition, called Condition (W). That condition holds in fairly greater generality and significantly open the scope of applicability of the main results. We also give Tauberian conditions that imply Condition (W). These Tauberian conditions are easier to check in practice than the Tauberian condition of McDonald-Sukochev-Zanin and are satisfied in numerous examples. The need for these conditions was highlighted by Alain Connes in an online seminar. The main results of this memoire are illustrated by semiclassical Weyl's laws and integration formulas in the following settings: (i) Dirichlet and Neumann problems on Euclidean domains with smooth boundaries; (ii) closed Riemannian manifolds; (iii) open manifolds with conformally cusp metrics of finite volume; (iv) quantum tori; and (v) sub-Riemannian manifolds. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2604.15200 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15200v1 Announce Type: new Abstract: We study the question of whether a sequence of non-instanton Yang-Mills connections can limit to a bubbling configuration composed only of instantons. In the case that the Uhlenbeck limit and the bubbles are of opposite charge, we determine an obstruction coming from deformations of the Uhlenbeck limit. As an application, we prove that instantons are the only solutions of the $\mathrm{SU}(2)$ Yang-Mills equation on $\mathbb{R}^4$ with energy less than $4\pi^2 \left( |\kappa| + 2 \right) + \varepsilon_\kappa,$ where $\kappa$ is the charge. We also prove discreteness of the energy spectrum on the trivial $\mathrm{SU}(2)$-bundle in the range $\left[ 0, 16 \pi^2 \right).$ Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2604.15198 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15198v1 Announce Type: new Abstract: We provide a comparatively simple proof of the dynamical stability of Ricci flow near a linearly stable Ricci-flat ALE metric with integrable deformations. Our proof relies on the equivalence between integrability and an "almost-orthogonality" property of the Ricci-DeTurck tensor, allowing us to analyze the latter directly. We obtain our main results in weighted Holder spaces and then show how to recover the $L^p$-stability theorems of Deruelle-Kroncke and Kroncke-Petersen. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2604.15159 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15159v1 Announce Type: new Abstract: We establish existence and uniqueness results for asymptotically locally Euclidean (ALE) and asymptotically locally flat (ALF) gravitational instantons. In particular, we prove the existence of a unique, Ricci-flat, toric ALE and ALF gravitational instanton, for every admissible rod structure, that is smooth up to possible conical singularites. We also give an elementary proof that any toric ALE or ALF self-dual instanton is a multi-Eguchi-Hanson or multi-Taub-NUT solution. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2604.14804 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14804v1 Announce Type: new Abstract: In this paper, inspired by the work of Guan and Li (2015), we introduce a fourth-order centro-equiaffine invariant curve flow via the affine Minkowski formula. Without any smallness assumptions on the initial curve, we establish the long-time existence of the flow and prove that, as $t \to +\infty$, the evolving curve preserves its enclosed area and converges smoothly to a round circle up to the action of $\mathrm{SL}(2)$. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2604.14704 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14704v1 Announce Type: new Abstract: We prove existence, uniqueness and structure results for complete noncompact 7-dimensional G2-holonomy metrics with ALC (asymptotically locally conical) asymptotics. We regard such spaces as G2-analogues of ALF gravitational instantons in 4-dimensional hyperk\"ahler geometry. Our main results include the existence of a G2-analogue of the Atiyah-Hitchin metric in 4-dimensional hyperk\"ahler geometry, the existence of a good moduli theory for ALC G2-holonomy metrics and rigidity results for ALC G2-metrics in terms of the symmetries of their asymptotic model. The analytic toolkit needed to prove all these results is a robust Fredholm theory for the natural geometric linear elliptic operators on ALC spaces. We provide a self-contained derivation of this Fredholm theory for arbitrary Riemannian manifolds with ALC asymptotics. Since our ALC Fredholm theory does not rely on imposing any holonomy reduction or curvature conditions it may also be of utility beyond the setting of ALC special holonomy metrics. As one such application of our general Fredholm theory we prove some Hodge-theoretic results on general ALC spaces. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2604.14675 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14675v1 Announce Type: new Abstract: Utilizing the Weierstrass representation for embedded doubly periodic minimal surfaces with parallel ends, we construct entire singly periodic graphs of spacelike maximal surfaces with isolated cone-like singularities in the Lorentz-Minkowski 3-space. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2604.14600 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14600v1 Announce Type: new Abstract: This paper studies the large $p$ asymptotics of three geometric quantities on complete noncompact Riemannian manifolds: the $p-$capacity of a compact set, the first Dirichlet $p-$eigenvalue, and the Maz'ya constant, thereby offering a new perspective on the study of such manifolds. We introduce the infinity capacity $\mathcal{C}(\Omega)$, the infinity eigenvalue $\Lambda(M)$, and the Maz'ya limit $\mathcal{M}(M)$, and establish the general inequality, for any $\Omega\subset M$, $$ \mathcal{V}(M) \ge \mathcal{C}(\Omega) \ge \Lambda(M) = \mathcal{M}(M), $$ where $\mathcal{V}(M)$ is the volume entropy. Under geometric conditions such as isoperimetric control of balls, rotational symmetry, or curvature bounds, these quantities coincide and equal $\mathcal{V}(M)$ or the dimension. We also provide examples showing strict inequalities hold. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2604.14537 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14537v1 Announce Type: new Abstract: For any complete and noncompact manifold $M$ with $\mathrm{Ric}\ge 0$, we define a function $\mathrm{RV}(s)$ that describes the growth of relative volume asymptotically $$\mathrm{RV}(s)=\limsup_{r\to\infty} \dfrac{\mathrm{vol} B_{rs}(p)}{\mathrm{vol} B_r(p)},\quad s\ge 1.$$ Then we study the fundamental groups of such manifolds with slow relative volume growth and sublinear diameter growth. We show that if $\mathrm{RV}(s)\ll s^2$ as $s\to\infty$, then $\pi_1(M)$ is almost abelian; if $\mathrm{RV}(s)\ll s^{1+\delta}$ for some $\delta\in (0,1)$ and the Ricci curvature is positive at a point, then $\pi_1(M)$ is finite. These results generalize our previous work on complete manifolds with $\mathrm{Ric}\ge 0$ and linear (minimal) volume growth. Maths/math.DG_(Differential_Geometry) https://arxiv.org/abs/2506.20368 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2506.20368v3 Announce Type: replace-cross Abstract: We establish norm inequalities for fractional powers of degenerate Laplacians, with degeneracy being determined by weights in the Muckenhoupt class $A_2(\mathbb{R}^n)$, accompanied by specific additional reverse H\"older assumptions. This extends the known results for classical Riesz potentials. The approach is based on size estimates for the degenerate heat kernels. The approach also applies to more general weighted degenerate operators. Maths/math.CA_(Classical_Analysis_and_ODEs) https://arxiv.org/abs/2604.14827 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14827v1 Announce Type: cross Abstract: A. Albouy and R. Moeckel in 2000 found some interesting inequalities related to the inverse problem for collinear (Moulton) central configurations: the Pfaffian of a certain matrix is positive since all coefficients of some polynomials are positive, for the Newtonian (interaction potential $1/r$ and $n\leq 6$). They conjectured that for all $n$ such Pfaffians, for the Newtonian case, are positive. In this article we analyze further the problem, and we prove that such inequalities hold true in more general cases (potentials with log-convex derivative, such as those with homogeneity parameters $\alpha>0$, for all even $n\leq 14$). Maths/math.CA_(Classical_Analysis_and_ODEs) https://arxiv.org/abs/2604.14662 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14662v1 Announce Type: new Abstract: We study dimensions of sets projected to an $(n-2)$-dimensional family of hyperplanes in $\mathbb{R}^n$ under curvature conditions. Let $n\ge 3$ and $\Sigma \subset S^{n-1}$ be an $(n-2)$-dimensional $C^2$ manifold such that $\Sigma$ has non-vanishing geodesic curvature ($n=3$)/sectional curvature $>1$ ($n \ge 4)$. Let $Z \subset \mathbb{R}^{n}$ be analytic with $\dim Z \le n-2$ and $0 n-2$, if in addition $\pi_{T_yS^{n-1}}(Z) \le n-2$ for some $y \in S^{n-1}$, we show that $\dim \pi_{T_xS^{n-1}}(Z) = \min\{\dim Z, n-1\}$ for $\mathcal{H}^{n-2}$-a.e. $x \in \Sigma$. Maths/math.CA_(Classical_Analysis_and_ODEs) https://arxiv.org/abs/2604.14658 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14658v1 Announce Type: new Abstract: A new weighted Hardy-type inequality for functions from the Sobolev space $W_{p}^{1}$ is proved. It is assumed that functions vanish on small alternating pieces of the boundary. The proved inequality generalizes the classical known weighted Hardy-type inequalities. Maths/math.CA_(Classical_Analysis_and_ODEs) https://arxiv.org/abs/2604.14429 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14429v1 Announce Type: new Abstract: In this paper we study higher-order difference equations which can be written as follows: $$ \mathbf{J} (y_0,y_1,...)^T = \lambda^N (y_0,y_1,...)^T, $$ where $\mathbf{J}$ is a $(2N+1)$-diagonal bounded banded matrix ($\mathbf{J}=(g_{m,n})_{m,n=0}^\infty$, $| g_{m,n} | 0$; and $g_{k,l}=0$ if $|k-l|>N$), $y_j$s are unknowns, $\lambda$ is a complex parameter, $N\in\mathbb{N}$. It is assumed that all $g_{k,k+N}$ and $g_{l-N,l}$ are nonzero. Two special cases are considered: \noindent \textit{Case A}: The matrix $\mathbf{J}$ is complex symmetric, i.e. $\mathbf{J} = \mathbf{J}^T$. \noindent \textit{Case B}: The matrix $\mathbf{J}$ is such that $g_{k,k+N}=1$, $k=0,1,2,...$. Notice that this condition can be attained by changing $y_j$s by their multiples. In both cases there exists a \textit{positive} matrix measure $M$ on a circle in the complex plane such that polynomial solutions satisfy some orthogonality relations. Namely, in case~A this is related to a $J$-orthogonality in the Hilbert space $L^2(M)$ ($J$ is a complex conjugation). In case~B we have a left $J$-orthogonality in $L^2(M)$. As a tool, a related matrix moment problem is studied. A complex rank-one perturbation of a free Jacobi matrix is discussed. Maths/math.CA_(Classical_Analysis_and_ODEs) https://arxiv.org/abs/2502.06520 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2502.06520v3 Announce Type: replace-cross Abstract: Discrete Morse theory helps us compute the homology groups of simplicial complexes in an efficient manner. A "good" gradient vector field reduces the number of critical simplices, simplifying the homology calculations by reducing them to the computation of homology groups of a simpler chain complex. This homology computation hinges on an efficient enumeration of gradient trajectories. The technique of cancelling pairs of critical simplices reduces the number of critical simplices, though it also perturbs the gradient trajectories. In this article, in a purely combinatorial manner, we derive an explicit formula for computing the modified boundary operators after cancelling a critical pair, in terms of the original boundary operators. The same formula can be obtained through a sequence of elementary row operations on the original boundary operators. Thus, it eliminates the need of enumeration of the new gradient trajectories. We also obtain a similar result for coboundary operators. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2501.17927 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2501.17927v3 Announce Type: replace-cross Abstract: These extended lecture notes survey a novel derivation of anyonic topological order (as seen in fractional quantum Hall systems) on single magnetized M5-branes probing Seifert orbi-singularities ("geometric engineering" of anyons), which we motivate from fundamental open problems in the field of quantum computing. The rigorous construction is non-Lagrangian and non-perturbative, based on previously neglected global completion of the M5-brane's tensor field by flux-quantization consistent with its non-linear self-duality and its twisting by the bulk C-field. This exists only in little-studied non-abelian generalized cohomology theories, notably in a twisted equivariant (and "twistorial") form of unstable Cohomotopy ("Hypothesis H"). As a result, topological quantum observables form Pontrjagin homology algebras of mapping spaces from the orbi-fixed worldvolume into a classifying 2-sphere. Remarkably, results from algebraic topology imply from this the quantum observables and modular functor of abelian Chern-Simons theory, as well as braid group actions on defect anyons of the kind envisioned as hardware for topologically protected quantum gates. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2603.18550 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2603.18550v2 Announce Type: replace Abstract: A generalization of the Borsuk-Ulam theorem to Stiefel manifolds is considered. This theorem is applied to derive bounds on $d$ that guarantee-for a given set of $m$ measures in $\mathbb{R}^d$-the existence of $k$ mutually orthogonal hyperplanes, any $n$ of which partition each of the measures into $2^n$ equal parts. If $n=k$, the result corresponds to the bound obtained in [11], but with the stronger conclusion that the hyperplanes are mutually orthogonal. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2508.14341 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2508.14341v2 Announce Type: replace Abstract: In this paper, we classify the homotopy types of the total spaces of $S^{2k-1}$-bundles (or fibrations) over $S^{2k}$ for $2\leq k\leq 6$. One of the two key new ingredients in the argument is the new necessary and sufficient conditions for a CW complex to be homotopy equivalent to the total space of a sphere bundle (fibration); the other is a formula relating the attaching map of the top cell of the total space and the characteristic map of a sphere bundle for $k=2,4$. When $k=4$, the classification results provide a negative answer to the conjecture in [6]. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2401.14146 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2401.14146v3 Announce Type: replace Abstract: In this note, (rational) Betti numbers of homotopy colimits for toric diagrams and their classifying spaces are described in terms of sheaf cohomology over CW posets. We prove for any $T$-diagram $D$ over any CW poset that Cohen-Macaulayness (over $\mathbb{Q}$) of the $T$-action on $hocolim\ D$ is equivalent to acyclicity for a certain sheaf. The ordinary and bigraded Betti numbers are computed for skeletons of equivariantly formal spaces from this class (in particular, of compact smooth toric manifolds). Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2604.15262 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15262v1 Announce Type: cross Abstract: We develop a framework for detecting regime transitions in dynamical systems using the Mixup Euler Characteristic Profile (Mixup ECP) -- the Euler characteristic of the geometric intersection of ball unions around adjacent delay-embedded trajectory segments, viewed as a function of filtration scale. The Mixup ECP provides a detection statistic with a built-in null and guaranteed stability. We formalize regime detection as a low-side-permutation test, establish its validity and consistency, and introduce a multi-delay extension that automatically selects the most informative dynamical timescale. Complementing the topological signal with Complexity Variance, Higuchi fractal dimension, and a rolling mean baseline, the four-signal combined method achieves $9.50$ days MAE on Indian monsoon onset (Nepal target) -- a $32\%$ improvement over the rolling mean baseline and $9\%$ over CUSUM. Validated on the Lorenz system, logistic map, and three monsoon systems spanning both hemispheres (Indian/Nepal, Indian/Kerala, Western North Pacific), plus ENSO and a synthetic EEG dataset, the framework adds value precisely when the transition is gradual or obscured by noise. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2604.15243 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15243v1 Announce Type: cross Abstract: Given a group $G$ and an integer $n \geq 0$, let $\mathcal{F}_n$ denote the family of all virtually abelian subgroups of $G$ of rank at most $n$. In this article, we show that for each $n \geq 1$, the minimal dimension of a model for the classifying space $E_{\mathcal{F}_n}G$ for the pure braid group of a surface of non-negative Euler characteristic with at least one boundary component or one puncture is equal to the virtual cohomological dimension of $G$ plus $n$. We prove an analogous result for the full braid group of the sphere. As an application, we compute the minimal dimension of a model for the classifying space associated to the family of amenable subgroups of pure surface braid groups. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2604.15142 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15142v1 Announce Type: cross Abstract: We introduce a notion of parity for formal morphisms between invertible objects and use it to prove a corresponding coherence theorem. Parity is conceptually similar to the sign of underlying permutations, but not defined as such. To give complete details, this work includes a thorough treatment of the free permutative category on an invertible generator, its skeletal model, known as the super integers, and an equivalence between them classified by the pair of integers $\pm$1. Our approach is organized and clarified as an application of 2-monadic algebra, particularly the concept of flexibility and the Lack model structure. The final section contains a number of examples applying the main results. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2604.14929 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14929v1 Announce Type: cross Abstract: In this paper, we study homotopical analogues of the Cantor-Bendixson derivative. For each $n\geq 0$, the "$\pi_n$-wild set" $\mathbf{w}_n(X)$ of a topological space $X$ is the subspace of $X$ consisting of the points at which there exists a shrinking sequence of essential based maps $S^n\to X$. Since the operator $\mathbf{w}_n$ permits iteration, every given space $X$ yields a descending transfinite sequence of nested subspaces $\{\mathbf{w}_n^{\kappa}(X)\}_{\kappa}$ that stabilizes at some smallest ordinal $\mathbf{wrk}_n(X)$ called the "$\pi_n$-wild rank" of $X$. We show that the entire transfinite sequence $\{ho(\mathbf{w}_n^{\kappa}(X))\}_{\kappa}$ of homotopy types is a homotopy invariant of $X$ and that $\mathbf{wrk}_n(X)$ can be an arbitrary countable ordinal when $X$ is an $n$-dimensional Peano continuum. It remains open if there exists a continuum $X$ with uncountable $\pi_n$-wild rank. This difficulty motivates the parallel study a basepoint-free version $\mathbf{fwrk}_n(X)$, called the "free $\pi_n$-wild rank" of $X$. We show that for every continuum $X$, $\mathbf{fwrk}_n(X)$ is always countable and can be any countable ordinal. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2604.14536 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14536v1 Announce Type: cross Abstract: Oriented cohomology theories provide a general framework to perform intersection-theory-type calculus. The Chow ring, algebraic $K$-theory, and Levine--Morel's algebraic cobordism are all instances of such theories satisfying $\mathbb A^1$-invariance. Topological Hochschild homology, topological cyclic homology, and Hodge cohomology are important examples of theories without $\mathbb A^1$-invariance. In this paper, we prove an additive blowup formula for oriented cohomology theories in the non-$\mathbb A^1$-invariant category of motivic spectra, developed by Annala, Hoyois, and Iwasa. Then, we specialize to $\mathbb A^1$-invariant theories and give presentations of oriented cohomology rings of the blowup of a smooth scheme along a smooth center. We compute explicit examples of such presentations for the cases of del Pezzo surfaces, the blowup of $\mathbb P^3$ along the twisted cubic, and the blowup of $\mathbb P^5$ along the Veronese surface, which can be identified with the moduli space of complete conics. We demonstrate that one can recover solutions to classical enumerative geometry problems, such as Steiner's $3264$ conics, using arbitrary oriented cohomology theories. Finally, we give a presentation of oriented cohomology rings of $\overline M_{0,n}$, which generalizes Keel's presentation of the Chow ring. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2604.15265 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.15265v1 Announce Type: new Abstract: Determining whether two graphs are isomorphic is a fundamental problem with practical applications in areas such as molecular chemistry or social network analysis, yet it remains a challenging task, with exact solutions often being computationally expensive. We address this task using persistent homology built on motif-based filtrations of graphs, a method from topological data analysis that summarizes the shape of data by tracking the persistence of structural features along filtrations. Specifically, we use edge-weighting schemes based on the densities of triangles, chordless squares, and chordless pentagons, which have been shown to be effective for detecting network dimensionality. Our cycle-density filtrations distinguish non-isomorphic graphs perfectly or nearly perfectly across four demanding graph families, many of which exhibit symmetries. We outperform curvature-based, degree-based, and Vietoris--Rips filtrations, and match or exceed the accuracy of egonet-distance methods while incurring a lower computational cost. The expressive power of our filtrations goes beyond isomorphism testing: because they capture rich structural information from graphs, they consistently achieve top performance on property prediction tasks using real-world data, and exhibit high sensitivity to edge rewiring and removal. Together, these findings establish cycle-density filtrations as an effective and computationally tractable framework for graph comparison and characterization, bridging topological data analysis and network science. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2604.14913 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14913v1 Announce Type: new Abstract: If a finite group $G$ acts on a rational homology manifold, then the orbit space is well-known to be a rational homology manifold again. We consider here actions on spaces that may be much more singular. If the $G$-space is a Witt pseudomanifold, which includes all arbitrarily singular complex pure-dimensional algebraic varieties, then we prove that the orbit space is again a Witt pseudomanifold. In the compact oriented situation, this implies that the orbit space possesses characteristic L-classes, as defined by Goresky and MacPherson. We then construct Atiyah-Singer-Zagier type equivariant L-classes for such $G$-pseudomanifolds which serve, as we show by establishing an averaging formula, as a tool to compute the Goresky-MacPherson L-class of the orbit space. The construction of the equivariant class builds on intersection homological transfer properties and on recent joint K-theoretic work with Eric Leichtnam and Paolo Piazza, which established a G-signature theorem on Witt pseudomanifolds. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2604.14840 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14840v1 Announce Type: cross Abstract: We study the minimization problem for eigenvalues of the Dirac operator within a fixed conformal class on a closed spin Riemannian manifold. We establish a criterion for the existence of a minimizer for this variational problem, focusing specifically on the case of closed surfaces. Furthermore, we apply our results to derive isoperimetric inequalities for the Dirac operator on the two-dimensional sphere, providing a complete characterization of its conformal spectrum. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14812 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14812v1 Announce Type: cross Abstract: We present a time-dependent extension of logarithmic perturbation theory for nonrelativistic quantum dynamics governed by the Schr\"odinger equation, in which the logarithm of the wave function is expanded in powers of a coupling constant. The resulting hierarchy of equations defining the perturbative corrections is governed by a gauge-rotated Hamiltonian of the unperturbed system and leads to closed-integral expressions for the time-dependent corrections based on Duhamel's formula. This closed-integral structure of corrections is a hallmark of time-independent logarithmic perturbation theory and is preserved in the present extension. This structure, in particular, provides a computable expression for the instantaneous energy shift. Furthermore, dynamic energy shifts arise naturally within this framework in the form of time-averaged expectation values of pseudopotentials and can be related, for example, to AC Stark shifts and electric polarizabilities. As an illustration, we apply the method to the harmonic oscillator and the hydrogen atom, both driven by a time-dependent laser field. The harmonic oscillator provides a proof of principle for which the exact solution is recovered, while the hydrogen atom illustrates the method applied to atomic systems. Supported by numerical simulations, we demonstrate the applicability to obtain relevant physical observables with high accuracy. The present approach offers a promising alternative for analytical studies of time-dependent multi-photon processes in the perturbative regime. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14752 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14752v1 Announce Type: cross Abstract: We consider a class of stochastic damped semilinear wave equations, in the small-mass limit. It has previously been established that the solution converges to the solution of a stochastic semilinear heat equation. In this work we exhibit strong and weak rates of convergence in this Smoluchowski--Kramers approximation result. The rates depend on the regularity of the driving Wiener process. For instance, for trace-class noise the strong and weak rates of convergence are $1$, whereas for space-time white noise (in dimension $1$) the strong and weak rates of convergence are $1/2$ and $1$ respectively. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14715 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14715v1 Announce Type: cross Abstract: On generalized Heisenberg-type groups $\mathbb{G}(2n,m,\mathbb{U},\mathbb{W})$, we give uniform volume estimates for the ball defined by a large class of Carnot-Carath\'{e}odory distances, and establish weak (1, 1) $O(C^m \, n)$-estimates for associated centered Hardy-Littlewood maximal functions, extending the results in \cite{BLZ25}. As a by-product, we establish uniformly volume doubling property on Heisenberg groups for a class of left-invariant Riemannian metrics. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.14393 Fri, 17 Apr 2026 00:00:00 -0400 arXiv:2604.14393v1 Announce Type: cross Abstract: We describe a method to prove new integral inequalities for stable minimal hypersurfaces in Euclidean space. As an application, we give a simple proof that complete, two sided, stable minimal hypersurfaces in $\mathbb{R}^4$ are hyperplanes. A core part of the argument hinges on the fact that stable minimal hypersurfaces in non-negatively curved spaces are examples of manifolds with a spectral Ricci curvature lower bound; in particular, we prove a sharp pointwise gradient estimate for the Green kernel on non-parabolic manifolds with spectral Ricci lower bounds, extending previous work by Colding. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.13889 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13889v1 Announce Type: new Abstract: In this paper, we propose an efficient two-level additive Schwarz method for solving large-scale eigenvalue problems arising from the finite element discretization of symmetric elliptic operators, which may compute efficiently more interior multiple and clustered eigenvalues other than only the first several smallest eigenvalues. The proposed method is parallel in two ways: one is to solve the preconditioned Jacobi-Davidson correction equations by the two-level additive Schwarz preconditioner, the other is to solve different clusters of eigenvalues (see Figure 1 in Introduction) simultaneously. It only requires computing a series of parallel subproblems and solving a small-dimensional eigenvalue problem per iteration for a cluster of eigenvalues. Based on some new estimates and tools, we provide a rigorous theoretical analysis to prove that convergence factor of the proposed method is bounded by $\gamma=c(H)\rho(\frac{\delta}{H},d_{m}^{-},d_{M}^{+})$, where $H$ is the diameter of subdomains, $\delta$ is the overlapping size and $d_{m}^{-},d_{M}^{+}$ are the distances from both ends of the targeted eigenvalues to others (see Figure 2 in Introduction). The positive number $\rho(\frac{\delta}{H},d_{m}^{-},d_{M}^{+})<1$ is independent of the fine mesh size and the internal gaps among the targeted eigenvalues. The $H$-dependent constant $c(H)$ decreases monotonically to 1, as $H\to 0$, which means the more subdomains lead to the better convergence. Numerical results supporting our theory are given. Maths/math.NA_(Numerical_Analysis) https://arxiv.org/abs/2602.15944 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2602.15944v2 Announce Type: replace-cross Abstract: We study constraints imposed by four-dimensional unitarity (formalised as graded unitarity in recent work by the first author) on possible ${\mathcal W}_3$ vertex algebras arising from four-dimensions via the SCFT/VOA correspondence. Under the assumption that the $\mathfrak{R}$-filtration is a weight-based filtration with respect to the usual strong generators of the vertex algebra, we demonstrate that all values of the central charge other than those of the $(3,q+4)$ minimal models are incompatible with four-dimensional unitarity. These algebras are precisely the ones that are realised by performing principal Drinfel'd--Sokolov reduction to boundary-admissible $\mathfrak{sl}_3$ affine current algebras; those affine algebras were singled out by a similar graded unitarity analysis in \cite{ArabiArdehali:2025fad}. Furthermore, these particular vertex algebras are known to be associated with the $(A_2,A_q)$ Argyres--Douglas theories. Maths/math.QA_(Quantum_Algebra) https://arxiv.org/abs/2604.13514 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13514v1 Announce Type: cross Abstract: Applying Gr\"obner basis theory to concrete problems in Lean 4 remains difficult since the current formalization of multivariate polynomials is based on a non-computable representation and is therefore not suitable for efficient symbolic computation. As a result, computing Gr\"obner bases directly inside Lean is impractical for realistic examples. To address this issue, we propose a certificate-based approach that combines external computer algebra systems, such as SageMath or SymPy, with formal verification in Lean 4. Our approach uses a computable representation of multivariate polynomials in Lean to import and verify externally generated Gr\"obner basis computations. The external solver carries out the main algebraic computations, while the returned results are verified inside Lean. Based on this method, we develop automated tactics that transfer polynomial data between Lean and the external system and certify the returned results. These tactics support tasks such as remainder verification, Gr\"obner basis checking, ideal equality, and ideal or radical membership. This work provides a practical way to integrate external symbolic computation into Lean 4 while preserving the reliability of formal proof. Maths/math.AC_(Commutative_Algebra) https://arxiv.org/abs/2512.23839 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2512.23839v2 Announce Type: replace Abstract: In this article, we disprove a conjecture of F. Alarc\'on and D. Anderson and give a complete classification of the prime ideals in the one variable polynomial semiring with coefficients in Boolean semifield. We group the prime ideals of $\mathbb{B}[x]$ into three classes, indexed by integers. Maths/math.AC_(Commutative_Algebra) https://arxiv.org/abs/2504.15981 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2504.15981v2 Announce Type: replace-cross Abstract: Guided by the $Q$-shaped derived category framework introduced by Holm and Jorgensen, we provide a differential module analogue of a classical result that characterises when a finitely generated module over a local commutative noetherian ring has finite injective dimension. As an application, we characterise local Cohen-Macaulay rings using the homological algebra of differential modules. Maths/math.AC_(Commutative_Algebra) https://arxiv.org/abs/2506.23533 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2506.23533v4 Announce Type: replace-cross Abstract: In this paper, we introduce the notion of a characteristic-zero lifting of an object in positive characteristic by means of ``skeletons''. Using this notion, we relate invariants of singularities in positive characteristic to their counterparts in characteristic zero. As an application, we prove that the set of log discrepancies for pairs consisting of a smooth surface and a multi-ideal is discrete. We also show that the set of minimal log discrepancies and the set of log canonical thresholds of such pairs in positive characteristic are contained in the corresponding sets in characteristic zero. Another application is the construction of Campillo's complex model of a plane curve in positive characteristic via the skeleton lifting method. Maths/math.AC_(Commutative_Algebra) https://arxiv.org/abs/2507.21911 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2507.21911v3 Announce Type: replace-cross Abstract: Let $G=\mathrm{GL}_n(\mathbb{F})$, $\mathrm{O}_n(\mathbb{F})$, or $\mathrm{Sp}_{2n}(\mathbb{F})$ be one of the classical groups over an algebraically closed field $\mathbb{F}$ of characteristic $0$, let $\breve{G}$ be the MVW-extension of $G$, and let $\mathfrak{g}$ be the Lie algebra of $G$. In this paper, we classify the closed orbits in the enhanced standard representation $\mathfrak{g}\times E$ of $G$, where $E$ is the natural representation if $G=\mathrm{O}_n(\mathbb{F})$ or $\mathrm{Sp}_{2n}(\mathbb{F})$, and is the direct sum of the natural representation and its dual if $G=\mathrm{GL}_n(\mathbb{F})$. Additionally, for every closed $G$-orbit in $\mathfrak{g}\times E$, we prove that it is $\breve{G}$-stable, and determine explicitly the corresponding stabilizer group as well as the action on the normal space. Maths/math.AC_(Commutative_Algebra) https://arxiv.org/abs/2603.23008 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2603.23008v2 Announce Type: replace-cross Abstract: We compare several classes of biparameter persistence modules: $\gamma$-products of monoparameter modules, hook-decomposable modules, modules admitting a Smith-type structure theorem, and modules of projective dimension at most 1. We determine all logical implications among these classes, providing explicit counterexamples showing that the converses fail when appropriate. In particular, $\gamma$-products (i.e., hook-decomposable modules) form a very small subclass of biparameter modules, precisely the ones for which a structure theorem still holds, thus making explicit the richer structural complexity of the biparameter setting compared to the monoparameter one. Maths/math.AC_(Commutative_Algebra) https://arxiv.org/abs/2604.14024 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14024v1 Announce Type: new Abstract: Koll\'{a}r showed that small deformations of elliptically fibered smooth $K$-torsion varieties with $H^2(X,\mathcal{O}_X)=0$ remain elliptically fibered. We extend this result to any fibered smooth $K$-torsion variety $X$ with $H^2(X,\mathcal{O}_X)=0$, using Hodge theoretic techniques and the $T^1$-lifting criterion of Kawamata--Ran. More generally, our strategy implies that even without the cohomological assumption, small deformations of a semiample line bundle on a smooth $K$-torsion variety remain semiample up to homological equivalence. Maths/math.AG_(Algebraic_Geometry) https://arxiv.org/abs/2604.14040 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14040v1 Announce Type: new Abstract: We prove a lower bound on the Calabi functional for degenerations of polarized varieties, involving the difference of CM degrees between generically isomorphic families. This may be viewed as a discretely valued version of Donaldson's lower bound for models, in the sense of non-Archimedean geometry. In particular, this generalizes a result of Donaldson, who considered a single polarized variety. As a main tool, we develop the theory of GIT height, introduced by Wang, and apply it to the family GIT problem of the Chow variety. Using the GIT height, we also give a numerical proof of separatedness of GIT quotients for general and special linear actions, strengthening prior work of Wang--Xu. Maths/math.AG_(Algebraic_Geometry) https://arxiv.org/abs/2604.14085 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14085v1 Announce Type: new Abstract: Consider a pair of $S$-dual hyperspherical varieties $G\circlearrowright X$ and $G^\vee\circlearrowright X^\vee$ equipped with equivariant quantizations $Q(X)$, $Q(X^\vee)$. Assume that the local conjecture of Ben-Zvi, Sakellaridis and Venkatesh holds for this pair, and also that $X\simeq T^*_\psi(Y)$ is polarized, so that $Q(X)=D_\psi(Y)$. Let $B\subset G$ (resp. $B^\vee\subset G^\vee$) be Borel subgroups. Then we deduce an equivalence between the ${\mathbb Z}/2$-graded $B$-equivariant category $(D_\psi(Y)\operatorname{-mod})^{{\mathbb Z}/2})^B$ and the ${\mathbb Z}/2$-graded unipotent $B^\vee$-monodromic category $(Q(X^\vee)\operatorname{-mod}^{{\mathbb Z}/2})^{B^\vee,\operatorname{mon}}$. Maths/math.AG_(Algebraic_Geometry) https://arxiv.org/abs/2604.14127 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14127v1 Announce Type: new Abstract: On a compact connected Riemann surface $C$ of genus at least $2$, we construct Lagrangian correspondences between moduli spaces of rank-$n$ Higgs bundles (respectively, holomorphic connections) and the Hilbert schemes of points on $T^\ast C$ (respectively, the twisted cotangent bundles of $C$). Central to these constructions are Higgs bundles (respectively, holomorphic connections) which are transversal to line subbundles of the underlying bundles: these naturally induce divisors on $C$ together with auxiliary parameters, namely lifts to divisors on spectral curves for Higgs bundles and residue parameters of apparent singularities for holomorphic connections. We discuss the evidence showing that the Dolbeault geometric Langlands correspondence is generically realized by these Lagrangian correspondences; we expect that the de Rham geometric Langlands correspondence can be realized by their quantization, following Drinfeld's construction of Hecke eigensheaves. We also discuss the relations of our constructions to various topics, including reductions of Kapustin-Witten equations, the conformal limit, separation of variables, and degenerate fields in conformal field theories. Maths/math.AG_(Algebraic_Geometry) https://arxiv.org/abs/2604.13198 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13198v1 Announce Type: cross Abstract: We construct examples of complete Calabi-Yau metrics on smoothings of 3-dimensional Calabi-Yau cones that are not products of lower-dimensional Calabi-Yau cones and that have orbifold singularities away from the vertex. Maths/math.AG_(Algebraic_Geometry) https://arxiv.org/abs/2604.13209 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13209v1 Announce Type: cross Abstract: We give a definition of all-genus reduced Gromov-Witten invariants of symplectic manifolds by using effectively supported multivalued perturbations on derived orbifold/Kuranishi charts, which bypasses the hard analytical result of sharp compactification/ghost bubble censorship of Zinger and Ekholm-Shende. Maths/math.AG_(Algebraic_Geometry) https://arxiv.org/abs/2604.13373 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13373v1 Announce Type: cross Abstract: A noncommutative projective variety is defined, after Artin and Zhang, by a graded coherent algebra A, where the category of coherent sheaves is the quotient qgr(A) of the category of finitely presented graded modules by the subcategory of torsion modules. We consider the categorical and polynomial entropies of the Serre twist, that is, of the degree shift functor on the bounded derived category of qgr(A). These two types of entropy can be viewed as analogues of the dimension of the noncommutative variety. We relate these invariants with the growth of the algebra. For algebras of finite global dimension, the entropies are bounded above by the growth entropy and the Gelfand--Kirillov dimension of the algebra. Moreover, these equalities hold for regular algebras, as well as for coordinate rings of smooth projective varieties. However, the polynomial entropy is zero for monomial algebras of polynomial growth, so in this case the inequality is strict. Maths/math.AG_(Algebraic_Geometry) https://arxiv.org/abs/2604.14037 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14037v1 Announce Type: cross Abstract: Parameter space is not function space for neural network architectures. This fact, investigated as early as the 1990s under terms such as ``reverse engineering," or ``parameter identifiability", has led to the natural question of parameter space symmetries\textemdash the study of distinct parameters in neural architectures which realize the same function. Indeed, the quotient space obtained by identifying parameters giving rise to the same function, called the \textit{neuromanifold}, has been shown in some cases to have rich geometric properties, impacting optimization dynamics. Thus far, techniques towards complete classifications have required the analyticity of the activation function, notably excising the important case of ReLU. Here, in contrast, we exploit the non-differentiability of the ReLU activation to provide a complete classification of the symmetries in the shallow case. Maths/math.AG_(Algebraic_Geometry) https://arxiv.org/abs/2212.07745 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2212.07745v3 Announce Type: replace Abstract: This article surveys various duality statements attached to a pair consisting of a smooth complex quasi-projective variety and a regular function on it. It is dedicated to the memory of Bumsig Kim. Maths/math.AG_(Algebraic_Geometry) https://arxiv.org/abs/2401.12814 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2401.12814v3 Announce Type: replace Abstract: We show that $b$-Hurwitz numbers with a rational weight are obtained by taking an explicit limit of a Whittaker vector for the $\mathcal{W}$-algebra of type $A$. Our result is a vast generalization of several previous results that treated the monotone case, and the cases of quadratic and cubic polynomial weights. It also provides an interpretation of the associated Whittaker vector in terms of generalized branched coverings that might be of independent interest. Our result is new even in the special case $b=0$ that corresponds to classical hypergeometric Hurwitz numbers, and implies that they are governed by the topological recursion of Eynard-Orantin. This gives an independent proof of the recent result of Bychkov-Dunin-Barkowski-Kazarian-Shadrin. Maths/math.AG_(Algebraic_Geometry) https://arxiv.org/abs/2502.12136 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2502.12136v2 Announce Type: replace Abstract: Let $k$ be a perfect field of characteristic $p > 0$. For a strictly semi-stable scheme over $k[[t]]$, we construct the weight spectral sequence in $p$-adic cohomology using the theory of arithmetic $\mathcal{D}$-modules, whose $E_1$ terms are described by rigid cohomologies of irreducible components of the closed fiber and whose $E_\infty$ terms are conjecturally described by the (unipotent) nearby cycle of Lazda-P\'{a}l's rigid cohomology over the bounded Robba ring. We also show its functoriality by pushforward and state the conjecture of its functoriality by pullback and dual. Maths/math.AG_(Algebraic_Geometry) https://arxiv.org/abs/2507.13538 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2507.13538v2 Announce Type: replace Abstract: We study automorphisms of quasi-smooth hypersurfaces in weighted projective spaces, extending classical results for smooth hypersurfaces in projective space to the weighted setting. We establish effective criteria for when a power of a prime number can occur as the order of an automorphism, and we derive explicit bounds on the possible prime orders. A key role is played by a weighted analogue of the classical Klein hypersurface, which we show realizes the maximal prime order of an automorphism under suitable arithmetic conditions. Our results generalize earlier work by Gonz\'alez-Aguilera and Liendo. Maths/math.AG_(Algebraic_Geometry) https://arxiv.org/abs/2510.16417 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2510.16417v2 Announce Type: replace Abstract: We introduce and study the Hesse pencil variety $H_8$, obtained as the Zariski closure in the Grassmannian $G(1,9)$ of the set of pencils generated by a smooth plane cubic and its Hessian. We prove that $H_8$ has dimension $8$ and can be realized as the intersection of $G(1,9)$ with ten hyperplanes corresponding to the Schur module $\mathbb{S}_{(5,1)}\mathbb{C}^3$. Moreover, $H_8$ coincides with the closure of the $SL(3)$-orbit of the pencil $\langle x^3+y^3+z^3,\ xyz\rangle$ and contains eight additional orbits. The variety is singular, and its singular locus is precisely the union of two orbits, $O(\langle x^3,x^2y\rangle)$ and $O(\langle x^2y,x^2z\rangle)$. A key ingredient in our study is a cubic skew-invariant $R\in \bigwedge^3(\mathrm{Sym}^3\mathbb{C}^3)$ defined by $R(l^3,m^3,n^3)=(l\wedge m\wedge n)^3$, whose vanishing characterizes pencils generated by a cubic and its Hessian. This invariant allows us to write explicit equations defining $H_8$. A crucial geometric step in our argument is the fact that through four general points of $\mathbb{P}^2$ there pass exactly six Hesse configurations, which enables us to compute the multidegree of $H_8$ and conclude that it coincides with the variety defined by the invariant $R$. Maths/math.AG_(Algebraic_Geometry) https://arxiv.org/abs/2603.12393 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2603.12393v2 Announce Type: replace Abstract: We prove that, under certain geometric conditions, that only \(m-1\) different non-degenerate \((m+2)\)-secant \(m\)-planes plus one degenerate \((m+2)\)-secant \(m\)-plane to the Kummer variety implies the existence of a curve of ${(m+2)}$-secants to the Kummer variety. This is done by constructing a set of equations in terms of theta functions from the germ of a curve on the described points. The relation between those equations allows to proceed by induction to get the entire desired curve since the first of them is equivalent to the hypothesis that we ask. Maths/math.AG_(Algebraic_Geometry) https://arxiv.org/abs/2604.01842 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.01842v2 Announce Type: replace Abstract: For a one-parameter variation of biextension mixed Hodge structures, Brosnan and Pearlstein showed that the limit of the asymptotic height of the variation is given by a certain limit height of the nilpotent orbit. This limit height depends on the choice of a parameter. In the case of a variation of geometric origin related to Ceresa cycles associated with curves of genus three, after fixing a parameter, we show that this limit height is given by the Deligne splitting of a biextension mixed Hodge structure associated with cycles in the boundary. Maths/math.AG_(Algebraic_Geometry) https://arxiv.org/abs/2308.02155 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2308.02155v3 Announce Type: replace-cross Abstract: We construct many new examples of complete Calabi-Yau metrics of maximal volume growth on certain smoothings of Cartesian products of Calabi-Yau cones with smooth cross-sections. A detailed description of the geometry at infinity of these metrics is given in terms of a compactification by a manifold with corners obtained through the notion of weighted blow-up for manifolds with corners. A key analytical step in the construction of these Calabi-Yau metrics is to derive good mapping properties of the Laplacian on some suitable weighted H\"older spaces. Our methods also produce singular Calabi-Yau metrics with an isolated conical singularity modelled on a Calabi-Yau cone distinct from the tangent cone at infinity, in particular yielding a transition behavior between different Calabi-Yau cones as conjectured by Yang Li. This is used to exhibit many examples where the tangent cone at infinity does not uniquely specify a complete Calabi-Yau metric with exact K\"ahler form. Maths/math.AG_(Algebraic_Geometry) https://arxiv.org/abs/2604.07243 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.07243v2 Announce Type: replace-cross Abstract: We prove that every locally inner (class-preserving) endomorphism of the adjoint Chevalley group $G_{\mathrm{ad}}(\Phi,R)$ and of its elementary subgroup $E_{\mathrm{ad}}(\Phi,R)$ over a commutative ring $R$ is inner for $\Phi\in\{\mathbf A_1,\mathbf A_2,\mathbf B_2\}$ under the assumption $2\in R^\times$, and for $\Phi=\mathbf G_2$ under the stronger assumption $2,3\in R^\times$. Consequently, all these groups are Sha-rigid. Maths/math.AG_(Algebraic_Geometry) https://arxiv.org/abs/2604.13300 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13300v1 Announce Type: new Abstract: For the Keller-Segel system \[ \left\{\, \begin{aligned} u_t &= \Delta u - \nabla \cdot ( u \nabla v ), \\ v_t &= \Delta v - v + u \end{aligned} \right. \tag{$\star$} \] posed in a planar domain $\Omega$ with Neumann boundary conditions, the existence of classical solutions blowing up at some finite time $T$ has long been established. In fact, it has been shown that for every blow-up point $x$ the quantity $\int_{B_R(x)\cap\Omega} u(\cdot,t )\ln(u(\cdot, t))$ is unbounded as $t\nearrow T$ for all $R > 0$ even though the global mass of $u$ is always conserved. The present manuscript provides some quantitative information on the behavior of such localized $L\log L$ expressions by asserting the existence of $\delta_0=\delta_0(\Omega)>0$ such that any solution to the Neumann problem for ($\star$) blowing up at time $T\in (0,\infty)$ satisfies \[ \limsup_{t\nearrow T} \frac{1}{\ln\frac{T}{T-t}}\int_{B_R(x)\cap\Omega} u(\cdot, t)\ln(u(\cdot, t)) \ge \delta_0 \tag{$\star\star$} \] for all $R > 0$ at each blow-up point $x$. This confirms a certain universality property of the blow-up mechanism seen in the particular examples of radial collapsing solutions constructed in the seminal work [16], especially also beyond the realm of symmetry; apart from that, along with a consequence of ($\star\star$) on the corresponding asymptotics of similarly localized $L^p$ norms of $u$ for $p\in (1,\infty]$, this provides some extension of a known result on non-degeneracy of blow-up points that has concentrated on the choice $p=\infty$ here. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.13303 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13303v1 Announce Type: new Abstract: We study regularity properties for solutions to the nakedly degenerate elliptic equation $a_{ij}\partial_{ij}u =0$, where the coefficients satisfy $I \ge a_{ij}(x) \ge \lambda(x) I$ and the only assumption is that $\lambda^{-1} \in L^p$. We prove an improvement of oscillation and a Liouville theorem for $p>d-1$, and a Harnack inequality for $p$ sufficiently large depending on dimension. Along the way, we obtain a new $\log-L^\varepsilon$ Weak Harnack inequality for supersolutions. Then, touching subsolutions by double exponential blow-up barriers, we also derive a logarithmic local maximum principle that is new even in the uniformly elliptic case. Both of these results hold for $p>d-1$. Finally, we construct examples showing that there cannot be Harnack or Weak Harnack inequalities in the regime $p<d-1$, nor can there be power-type $L^\varepsilon$ inequalities in the case of any $p<\infty$. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.13324 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13324v1 Announce Type: new Abstract: In this article, we show that for one-phase free boundary problems in noncollapsed limits of $n$-dimensional manifolds with two-sided Ricci curvature bounds, the Hausdorff dimension of the singular set of the free boundary can be bounded by $n-5$, which is sharp in this context. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.13330 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13330v1 Announce Type: new Abstract: These lecture notes are devoted to solutions of hyperbolic-parabolic systems with persistent oscillations. We consider two examples both from mechanics: (i) The system of viscoelasticity of Kelvin-Voigt type with strain energies involving double well potentials, as employed in phase transitions. (ii) The compressible Navier-Stokes equations for a barotropic gas. For each system we construct solutions with persistent oscillations. In a later part we consider the nonlinear homogenization problem. For the systems of viscoelasticity in one-space dimension in Lagrangian coordinates, and for the compressible Navier-Stokes system for barotropic fluids we show how ideas from the kinetic formulation of conservation laws can be used to derive effective equations. The effective equation consists by a kinetic equation coupled with the macroscopic flow. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.13338 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13338v1 Announce Type: new Abstract: In this investigation, we conduct a systematic computational search for potential singularities in 3D Navier-Stokes flows on a periodic domain $\Omega$ based on the Ladyzhenskaya-Prodi-Serrin conditions. They assert that for a solution $\mathbf{u}(t)$ of the Navier-Stokes system to be regular on an interval $[0,T]$, the integral $\int_{0}^T \|\mathbf{u}(t)\|_{L^q}^p\,dt$, where $2/p+3/q=1,\;q>3$, and the expression $\sup_{t \in [0,T]} \|\mathbf{u}(t)\|_{L^3}$ must be bounded. Flows which might become singular and violate these conditions are sought by solving a family of variational PDE optimization problems where we identify initial conditions $\mathbf{u}_{0}$ with the corresponding flows $\mathbf{u}(t)$ locally maximizing the integral $\int_{0}^T \|\mathbf{u}(t)\|_{L^q}^p\,dt$ for a range of different values of $q$ and $p$ or the norm $\|\mathbf{u}(T)\|_{L^3}$ for different time windows $T$ and increasing sizes $\| \mathbf{u}_0 \|_{L^q}$ of the initial data. We consider two formulations where these expressions are maximized over appropriate Lebesgue spaces $L^q(\Omega)$ or the largest Hilbert-Sobolev spaces $H^s(\Omega)$ embedded in them. The lack of Hilbert-space structure in the first case necessitates development of a novel computational approach to solve the problem. While no evidence of unbounded growth of the quantities of interest, and hence also for singularity formation, was detected, we were able to quantify how "close" the flows realizing such worst-case scenarios come to forming a singularity. A comparison of these results with estimates on the rate of growth of the norms $||\mathbf{u}(t)||_{L^q}$ and of the enstrophy $\mathcal{E}(t)$ indicates that the extreme flows do enter a regime where these quantities are amplified at a rate consistent with singularity formation in finite time, but this growth is not sustained long enough for singularities to form. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.13443 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13443v1 Announce Type: new Abstract: In this paper, we investigate the higher weak differentiability of solutions to a class of mixed local and nonlocal degenerate elliptic equations in the Heisenberg group $\mathbb{H}^n$. Owing to the non-commutative property and two-step nilpotent Lie algebra structure of $\mathbb{H}^n$, we first employ an iterative scheme involving fractional difference quotients to establish the weak differentiability of solutions in the vertical direction. This is subsequently extended to the horizontal and vertical gradients. Then, by coupling a truncation argument with the difference quotient method, we prove the higher weak differentiability of the gradients of solutions. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.13498 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13498v1 Announce Type: new Abstract: This paper aims to extend to Orlicz-Sobolev spaces some results of integral representation for the simultaneous homogenization and dimensional reduction of integral energies defined on fields taking values on a differentiable manifold. Since our functional framework goes beyond the classical Sobolev's spaces, we also prove, via $\Gamma$-convergence, a general integral representation results in the unconstrained Orlicz setting. Due to $\Delta_2$ and $\nabla_2$ conditions verified by the Young function $\Phi$ (which modulated the growth behaviour), we prove that the density of the $\Gamma$-limit is a tangential quasiconvex integrand represented by a cell formula. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.13548 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13548v1 Announce Type: new Abstract: We study the complex Ginzburg-Landau equation posed on possibly unbounded domains, including some singular and saturated nonlinear damping terms. This model interpolates between the nonlinear Schr{\"o}dinger equation and dissipative parabolic dynamics through a complex time-derivative prefactor, capturing the interplay between dispersion and dissipation. Under suitable structural conditions on the complex coefficients, we establish the existence and uniqueness of global solutions. The analysis relies on the delicate proofs that the maximal monotone operator theory can be adapted to this framework, even for unbounded domains. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.13572 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13572v1 Announce Type: new Abstract: In this paper, we study the Boltzmann equation in a close to the hydrodynamic limit regime, set in bounded spatial domains with non-isothermal Maxwell boundary conditions. We establish the existence, uniqueness, and asymptotic stability of a non-equilibrium steady state under suitable smallness assumption on the temperature fluctuations at the boundary. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.13578 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13578v1 Announce Type: new Abstract: This article investigates the existence of closed, star-shaped hypersurfaces for a class of Hessian quotient type curvature equations, in which the operator $\frac{\sigma_k}{\sigma_l}(\Lambda)$ arising in these equations can be viewed as a generalization of the classical Hessian quotient operator. By combining a priori estimates with the continuity method, we establish the existence and uniqueness of $(\mathbf{p}, k)$-convex hypersurfaces for both nonhomogeneous and homogeneous equations of this type. Furthermore, by exploiting the recently discovered ``inverse convexity'' property of the operator $\frac{\sigma_k}{\sigma_l}(\Lambda)$, we prove a constant rank theorem and thereby obtain the existence and uniqueness of strictly convex solutions to these curvature equations. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.13591 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13591v1 Announce Type: new Abstract: In this paper, we investigate a class of critical Ambrosetti-Prodi type problems involving the sub-Laplacian on a Carnot group. Specifically, we consider \[ \left\{ \begin{aligned} -\Delta_{\mathbb{G}} u &= \lambda u + u_{+}^{2_{Q}^{*}-1} + f(\xi) \quad &&\text{in } \Omega,\\[2mm] u &= 0 \quad &&\text{on } \partial\Omega, \end{aligned} \right. \] where $\Delta_{\mathbb{G}}$ is the sub-Laplacian on a Carnot group $\mathbb{G}$, $\Omega \subset \mathbb{G}$ is an open bounded domain with smooth boundary, $\lambda>0$ is a real parameter, $f\in L^{\infty}(\Omega)$, $u_{+}$ denotes the positive part of $u$, and $2_{Q}^{*}$ is the critical Sobolev exponent associated with the homogeneous dimension $Q$. Motivated by the classical Ambrosetti-Prodi problem, we establish existence and multiplicity results for the cases $\lambda \lambda_{1}$, where $\lambda_{k}$ denotes the $k$-th Dirichlet eigenvalue of $-\Delta_{\mathbb{G}}$. We also prove the existence of solutions at resonance when $\lambda=\lambda_{1}$ and show that bifurcation occurs from each eigenvalue $\lambda_{k}, k >1$. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.13595 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13595v1 Announce Type: new Abstract: We investigate standing waves for the energy critical Schr\"odinger system with three waves interaction arising as a model for the Raman amplification in a plasma. Several results are proved: simultaneous existence of stable and unstable standing waves, existence of global solutions, and absence of small data scattering. Our main results show some specific features arising from the three waves interaction differently from the classical energy critical Schr\"odinger equation, and they support some experimental observations on Raman amplification. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.13625 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13625v1 Announce Type: new Abstract: In this paper, we consider the stochastic reaction-diffusion equation $\mathrm{d}u = (\mathcal{A} u + f(u))\mathrm{d}t + \sigma(u)\mathrm{d}W$ on a smooth bounded domain $\mathcal{O}$ with homogeneous Dirichlet boundary conditions. We investigate the long-time behavior of solutions with a strongly dissipative drift nonlinearity and superlinear multiplicative noise in the Bochner space $L^q(\Omega; C_0(\overline{\mathcal{O}}))$, $q \ge 2$. Here $\mathcal{A}$ is a second-order self-adjoint elliptic operator and $W$ is a two-sided trace-class Wiener process. The standard Galerkin method fails to yield energy estimates in $L^q(\Omega; L^q(\mathcal{O}))$ via the It\^o formula for $q > 2$, owing to the interference of projection operators when dealing with nonlinear terms; meanwhile, the classical theory of mild solutions lacks sufficient spatial regularity to apply the It\^o formula directly. To overcome these difficulties, we consider mild solutions and establish a critical regularity estimate for the corresponding stopped process $u_n(t)$ in $W_0^{1,q}(\mathcal{O})$, which rigorously justifies the use of the It\^o formula in the non-Hilbert space $L^q(\Omega; L^q(\mathcal{O}))$. As a result, we derive explicit moment energy estimates and quantitative dissipativity bounds, yielding global existence, uniqueness, and exponential asymptotic decay of solutions in $L^q(\Omega; C_0(\overline{\mathcal{O}}))$. Unlike previous qualitative results in continuous function spaces, our framework provides a fully quantitative theory of global dissipativity. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.13641 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13641v1 Announce Type: new Abstract: In the present paper, we study the diffusion limit of the classical solution to the modified Vlasov-Poisson-Boltzmann (mVPB) System with initial data near a global Maxwellian. Based on the spectral analysis, weprove the convergence and establish the convergence rate of the global strong solution to the mVPB system towards the solution to an incompressible Navier-Stokes-Poisson-Fourier system with the precise estimation on the initial layer. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.13669 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13669v1 Announce Type: new Abstract: In this work we study the large-time behaviour of solutions of the Heat Equation in the hyperbolic space $\mathbb{H}^d$, providing precise speeds of convergence in $L^1$ and $L^\infty$ to their asymptotic profiles by means of an adaptation of entropy estimates. For $L^1$ initial conditions we are able to identify the asymptotic profile in $L^1$, which is not universal but contains a certain memory of the initial distribution of the mass of the solution. We improve thus on previous results, where speed of convergence was absent and asymptotic profiles where not known in the general case, and show a way to adapt entropy estimates employed in the study of diffusion processes to non-compact Riemannian manifolds. The main strategy to prove this is to consider transient profiles as minimizers of the entropy functional. These profiles are time-dependent and encompass the geometric information of the Riemannian manifold. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.13703 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13703v1 Announce Type: new Abstract: The pointwise space-time behavior of the Green's function of the three-dimensional modified Vlasov-Poisson-Boltzmann system is studied in this paper. It is shown that the Green's function has a decomposition of the macroscopic diffusive waves and Huygens waves with the speed $\sqrt{\frac{8}{3}}$ at low-frequency, the singular kinetic wave and the remainder term decaying exponentially in space and time. In addition, we establish the pointwise space-time estimate of the global solution to the nonlinear modified Vlasov-Poisson-Boltzmann system based on the Green's function. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.13732 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13732v1 Announce Type: new Abstract: We study Poincare-Sobolev type inequalities for compactly supported smooth functions which are defined in the Euclidean $n$-space and whose absolute value of gradient are Choquet $\delta /n$-integrable with respect to the $\delta$-dimensional Hausdorff content, $n\geq 2$, $\delta\in (0,n]$. In particular, our results imply a new Sobolev inequality for quasicontinuous functions defined in the Sobolev space $W^{1,1}_0(\mathbb{R}^n)$. As an application we extend a recently introduced superlevel Sobolev inequality into a context of the Hausdorff content. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.13736 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13736v1 Announce Type: new Abstract: In this paper we consider the volume-constrained minimization of a variant of the Ohta-Kawasaki functional with an anisotropic surface energy replacing the standard perimeter. Following and suitably adapting the second variation approach devised in arXiv:1211.0164, we prove local minimality results for the horizontal lamellar configuration, in analogy with the isotropic case, under the assumption that the anisotropy is uniformly elliptic. If instead the Wulff shape of the anisotropy has upper and lower horizontal facets, we prove that the lamella exhibits a rigid behavior and is an isolated local minimizer for all parameter values. We conclude by showing some global minimality results, mostly focusing on the planar case. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.13758 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13758v1 Announce Type: new Abstract: We investigate both qualitative and quantitative issues related to the classification of non-negative energy solutions to the anisotropic critical $p$-Laplace equation in $\mathbb{R}^n$, for $1<p<n$. Specifically, we establish an anisotropic version of Struwe's decomposition, along with the interaction estimate for the family of bubbles in this decomposition. Moreover, we provide a short proof of the classification result as well as a quantitative stability result, proving that every energy solution to a perturbation of the anisotropic critical equation must be closed to a bubble, in the absence of bubbling. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.13770 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13770v1 Announce Type: new Abstract: We determine the explicit value of the optimal constant in the trace inequality for functions of bounded variations in the case the domain has a particular class of singularities. Maths/math.AP_(Analysis_of_PDEs) https://arxiv.org/abs/2604.13629 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13629v1 Announce Type: new Abstract: Given a GKM$_3$ action of a torus $K$ on a manifold $M$ with GKM graph $\Gamma$, we show that for any extension of $\Gamma$ to an abstract GKM graph the corresponding restriction map in equivariant graph cohomology is surjective. While the corresponding statement for extensions of actions is well-known, we observe that this graph-theoretical statement is false in the GKM$_2$ setting. As a corollary, we obtain a description of the equivariant cohomology ring of Hamiltonian and complexity one GKM$_4$ actions in terms of generators and relations. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2604.13750 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13750v1 Announce Type: new Abstract: The purpose of this paper is to investigate the relationship between hairy graph complexes associated to cyclic operads and their counterparts for operads (and, more generally, dioperads). This is based on the author's interpretation of these as Koszul complexes for the associated modules over the respective appropriate twisted downward (walled) Brauer category. The general question of relating such Koszul complexes is addressed by analysing the relationships between the respective twisted Brauer-type categories, proceeding through a direct analysis. The passage from the walled to unwalled context involves functors induced by the disjoint union of finite sets. As an application, for the cyclic operad associated to an operad, this leads to an explicit relation between the respective (hairy) graph homologies. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2503.17149 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2503.17149v3 Announce Type: replace Abstract: In the 1980's, Mahowald and Kane used integral Brown--Gitler spectra to decompose $ku \wedge ku$ as a sum of finitely generated $ku$-module spectra. This splitting, along with an analogous decomposition of $ko \wedge ko,$ led to a great deal of progress in stable homotopy computations and understanding of $v_1$-periodicity in the stable homotopy groups of spheres. In this paper, we construct a $C_2$-equivariant lift of Mahowald and Kane's splitting of $ku \wedge ku$. We also describe the resulting $C_2$-equivariant splitting in terms of $C_2$-equivariant Adams covers and record an analogous splitting for $H\underline{\mathbb{Z}} \wedge H \underline{\mathbb{Z}}$. Along the way, we give complete computations of the $ku_{\mathbb{R}}$ and $H \mathbb{Z}$ operations and cooperations algebras. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2507.21975 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2507.21975v2 Announce Type: replace Abstract: We study Swan modules, which are a special class of projective modules over integral group rings, and their consequences for the homotopy classification of CW-complexes. We show that there exists a non-free stably free Swan module, thus resolving Problem A4 in the 1979 Problem List of C. T. C. Wall. As an application we show that, in all dimensions $n \equiv 3$ mod $4$, there exist finite $n$-complexes which are homotopy equivalent after stabilising with multiple copies of $S^n$, but not after a single stabilisation. This answers a question of M. N. Dyer. We also resolve a question of S. Plotnick concerning Swan modules associated to group automorphisms and, as an application, obtain a short and direct proof that there exists a group with $k$-periodic cohomology which does not have free period $k$. In contrast to the original proof our R. J. Milgram, our proof circumvents the need to compute the Swan finiteness obstruction. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2602.14680 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2602.14680v3 Announce Type: replace Abstract: In this paper, we systematically develop the $m$-contiguity distance between simplicial maps as a discrete approximation framework for homotopical complexity in the category of simplicial complexes. We construct an increasing sequence of invariants that approximate the contiguity distance from below. The fundamental properties of $m$-contiguity distance are established, including its behaviour under barycentric subdivision, under compositions, and a categorical product inequality. As applications of this theory, we define the $m$-simplicial Lusternik-Schnirelmann category and the $m$-discrete topological complexity, proving that each arises naturally as a special case of $m$-contiguity distance. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2603.09903 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2603.09903v2 Announce Type: replace Abstract: We show that basic homotopical notions such as homotopy sets and groups, connected and truncated maps, cellular constructions and skeleta, etc., extend to the setting of $(\infty,\infty)$-categories, as well as to presentable categories enriched in $(\infty,\infty)$-categories under the Gray tensor product. The homotopy posets of an $(\infty,\infty)$-category are indexed by boundaries of categorical disks; in particular, there is a fundamental poset for each pair of objects, which we regard as a oriented point where the source and target objects have opposite orientation. In contrast to the situation in topology, weakly contractible geometric building blocks such as oriented polytopes typically have nontrivial homotopy posets. The homotopy posets assemble to form an oriented analogue of the long exact sequence of a fibration and form the layers of a categorical Postnikov tower, which converges for any $(\infty,n)$-category but not for general $(\infty,\infty)$-categories. We show that the full subcategory consisting of the Postnikov complete $(\infty,\infty)$-categories is obtained by inverting the coinductive equivalences and canonically identifies with the limit of the categories of $(\infty,n)$-categories taken along the truncation functors. We also study truncated morphisms in general oriented categories and connected morphisms in presentable oriented categories. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2511.00473 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2511.00473v3 Announce Type: replace-cross Abstract: For $g\geq 0$, a genus $g$ Kashiwara-Vergne associator, introduced by Alekseev-Kawazumi-Kuno-Naef as a solution to the generalised KV equations in relation to the formality problem of the Goldman-Turaev Lie bialgebra on an oriented surface with a framing, is directly constructed from a genus $g$ analogue of a Drinfeld associator formulated by Gonzalez, which we call a Gonzalez-Drinfeld associator. The proof is based on Massuyeau's work in genus $0$. The framing is determined from the choice of a Gonzalez-Drinfeld associator, and in the case of genus $1$, we show that only particular framings are realised by our construction. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2604.09354 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.09354v2 Announce Type: replace-cross Abstract: We introduce an enriched notion of coalgebras over a V-operad P in a symmetric monoidal V-category C. When C is semicartesian and P is unital, we construct an endofunctor on C associated to P and give conditions under which it is an enriched comonad with co-Eilenberg-Moore V-category equal to the V-category of enriched P-algebras in V. In many cases, this permits computation of V-categories of coalgebras. We give several examples and show that our theorem generalises a well-known theorem of Fox. Maths/math.AT_(Algebraic_Topology) https://arxiv.org/abs/2604.13913 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13913v1 Announce Type: new Abstract: Let $\Theta=\{\theta_n\}, \Lambda=\{\lambda_n\}$ be two sequences of independent and identically distributed uniform random variables on $[0,1]$. The random vector-valued Weierstrass function is given by \[ f_{\Theta,\Lambda}(t)= \left( \sum_{n=0}^{\infty} b^{-\beta n}\cos\bigl(2\pi (b^n t+\theta_n)\bigr),\ \sum_{n=0}^{\infty} b^{-\beta n}\sin\bigl(2\pi (b^n t+\lambda_n)\bigr) \right),\quad t\in[0,1], \] where $b>1, \beta\in (0,1/2)$. We prove that, with probability one, the Hausdorff dimension of the graph of this function is \[ \dim_H G(f_{\Theta,\Lambda})=3-2\beta, \] extending a result of Hunt in 1998. Maths/math.CA_(Classical_Analysis_and_ODEs) https://arxiv.org/abs/2604.13782 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13782v1 Announce Type: cross Abstract: We consider several examples of nonautonomous systems of difference equations coming from semi-classical orthogonal polynomials via recurrence coefficients and ladder operators, with respect to various generalisations of Laguerre and Meixner weights. We identify these as discrete Painlev\'e equations and establish their types in the Sakai classification scheme in terms of the associated rational surfaces. In particular, we find examples which come from different weights and share a common surface type $D_5^{(1)}$ but are inequivalent in two ways. First, their dynamics are generated by non-conjugate elements of $\widehat{W}(A_3^{(1)})$. Second, some of the examples have associated surfaces being non-generic in the sense of having nodal curves. The symmetries of these examples form subgroups of the generic symmetry group, which we compute. In particular, we find $(W(A_1^{(1)})\times W(A_1^{(1)}))\rtimes \mathbb{Z}/2\mathbb{Z}$. These examples give further weight to the argument that any correspondence between different weights and the Sakai classification should make use of the refined version of the discrete Painlev\'e equivalence problem, which takes into account not just surface type, but also the group elements generating the dynamics as well as parameter constraints, e.g. those corresponding to nodal curves. Maths/math.CA_(Classical_Analysis_and_ODEs) https://arxiv.org/abs/2604.13825 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13825v1 Announce Type: cross Abstract: Analytic self-maps of the unit disc whose hyperbolic derivative is uniformly bounded by a constant smaller than one, are called contractive. We describe these maps in terms of their Aleksandrov-Clark measures and in terms of their inner-outer factorization. In addition, we show that contractive inner functions can be described in terms of a certain mixing property of its boundary values. We also present other results on the boundary behavior of contractive inner functions. Maths/math.CA_(Classical_Analysis_and_ODEs) https://arxiv.org/abs/2604.14076 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14076v1 Announce Type: cross Abstract: We present a model for sticky particles in which cluster sizes after a reaction have $\ell$ fewer total particles than the sum of their reactants. The finite particle system is modeled as a Markov process under a mean-field assumption for selecting reactants. The limiting kinetic equations form an infinite system of nonlinear differential equations similar to the Smoluchowski coagulation equations with multiplicative kernel. We show existence and uniqueness for systems whose cluster sizes are either bounded above or below by the emission size $\ell$. When clusters have at most $\ell$ particles, well-posedness can be extended until an exhaustion time in which certain cluster fractions vanish. For clusters with more than $\ell$ particles, we prove short-time well-posedness, along with explicit formulas for cluster sizes and moments. We also conduct numerical experiments which suggest these formulas hold until a gelation time, at which an infinite-sized cluster forms. Maths/math.CA_(Classical_Analysis_and_ODEs) https://arxiv.org/abs/2505.09037 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2505.09037v2 Announce Type: replace Abstract: We prove bilinear $\ell^2$-decoupling and refined bilinear decoupling inequalities for the truncated hyperbolic paraboloid in $\mathbb{R}^3$. As an application, we prove the associated restriction estimate in the range $p>22/7$, matching an earlier result for the elliptic paraboloid. Maths/math.CA_(Classical_Analysis_and_ODEs) https://arxiv.org/abs/2511.08757 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2511.08757v2 Announce Type: replace Abstract: We prove finite-field analogs of Bourgain's projection theorem in higher dimensions. In particular, for a certain range of parameters we improve on an exceptional set estimate by Chen in all dimensions and codimensions. Maths/math.CA_(Classical_Analysis_and_ODEs) https://arxiv.org/abs/2604.13221 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13221v1 Announce Type: new Abstract: Let \( G \) be a graph of order \( n \) with maximum degree $\Delta$, and let $P(G,x)$ denote its chromatic polynomial. We investigate several properties of $P(G,x)$ related to its derivatives and higher-order derivatives. First, we study the monotonicity of $P(G,x)/x^n$. Dong proved that $(x-1)^nP(G,x)\geq x^nP(G,x-1)$ for all real $x\geq n$. In particular, taking $x=n$ establishes the Bartels-Welsh ``shameful conjecture" that $P(G,n)/P(G,n-1)>e$. Fadnavis later showed that the same inequality holds for all real $x\geq 36\Delta^{\frac{3}{2}}$. We improve this bound by proving that it already holds for all real $x\geq 10\Delta^{\frac{3}{2}}$. We then consider a conjecture of Dong, Ge, Gong, Ning, Ouyang, and Tay asserting that \( \frac{d^k}{dx^k} \bigl( \ln[(-1)^n P(G, x)] \bigr) < 0 \) for all \( k \geq 2 \) and \( x \in (-\infty, 0) \). We establish this conjecture for all \( k \geq 2 \) and \( x\leq -2.99\Delta k \). Maths/math.CO_(Combinatorics) https://arxiv.org/abs/2604.13238 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13238v1 Announce Type: new Abstract: We introduce a quadratic form $Q$ on the space of functions on the gap poset $G$ of the numerical semigroup $\langle a,b\rangle$. We prove combinatorially that when evaluated on the indicator function of an upward closed subset $D$, this quadratic form precisely recovers the Gorsky--Mazin $\mathtt{dinv}$ statistic of $D$, viewed as a Young subdiagram of $G$. Furthermore, we prove Theorem~1.2 that when evaluated on a pair of subdiagrams of $G$, the symmetric bilinear form associated with $Q$ is equal to a novel cross-$\mathtt{dinv}$ statistic, which is nonnegative. Combining these, we prove the inequality \[ Q(\mathbf{n})\geq \dfrac{1}{|G|}\,\|\mathbf{n}\|_\infty^2\] if $\mathbf{n}$ is a real-valued decreasing function on $G$, showing an effective positive definiteness of $Q$ on the corresponding cone. Theorem~1.2, the main engine of the paper, was autoformalized in Lean/Mathlib by AxiomProver. Maths/math.CO_(Combinatorics) https://arxiv.org/abs/2604.13402 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13402v1 Announce Type: new Abstract: We study the intersection statistics of affine subspaces in the hypercube $\mathbb{F}_2^n$, motivated by recent work of Alon, Axenovich, and Goldwasser on the intersection statistics of axis-aligned subcubes of an $n$-dimensional cube. Let $d\ge 1$ and $0\le s\le 2^d$ be nonnegative integers. For a subset $A\subseteq \mathbb{F}_2^n$ where $n\ge d$, define $\lambda^*(n,d,s,A)$ to be the fraction of affine $d$-flats in $\mathbb{F}_2^n$ that intersect $A$ at exactly $s$ points. Let $\lambda^*(n,d,s) = \max_{A\subseteq \mathbb{F}_2^n}\lambda^*(n,d,s,A)$ and let $\lambda^*(d,s) = \lim_{n\to \infty}\lambda^*(n,d,s)$. We show that when $s = j\cdot 2^k$ with $j$ odd and $k\ge 1$, we have $\lambda^*(d,s)\to 1-\Theta(2^{-k})$ as $d\to \infty$. This implies that $\lambda^*(d,s)$ is controlled up to constant factors by the $2$-adic valuation of $s$ when $s$ is even. When $s$ is odd, we show that $\lambda^*(d,s)\le \frac{1}{2}$ in contrast to the behavior of axis-aligned subcube statistics. We also present several upper and lower bounds for certain specific values of $s$. Maths/math.CO_(Combinatorics) https://arxiv.org/abs/2604.13434 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13434v1 Announce Type: new Abstract: We determine the vertex-minor Ramsey number $\Rvm(4)=11$, where $\Rvm(k)$ is the smallest~$n$ such that every $n$-vertex graph contains the edgeless graph~$E_k$ as a vertex-minor. We prove this by an exhaustive classification of the graphs on~$10$ and~$11$ vertices under local complementation. At the extremal order $n=10$, exactly six non-isomorphic graphs avoid~$E_4$ as a vertex-minor; up to isomorphism, they represent five LC-equivalence classes, and each labeled LC orbit has cardinality~$8{,}712$. Thus $k=4$ is the first case in which the general upper bound $2^k-1$ is not attained. Using the extremal graphs as building blocks, we derive explicit lower bounds on~$\Rvm(k)$ that surpass the leading term of the asymptotic bound for all $k\leq 9$; in particular, $\Rvm(5)\geq 13$. We also describe structural properties of the six extremal graphs and formulate the next open problem, whether $\Rvm(5)=15$. Maths/math.CO_(Combinatorics) https://arxiv.org/abs/2604.13475 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13475v1 Announce Type: new Abstract: We study intersecting families of words from the Erd\H{o}s-Ko-Rado perspective. When the alphabet size is $2$, a maximum intersecting family is not necessarily a star. However, we prove that every maximum $3$-wise intersecting family is a star. We also present a new proof of the known result for alphabets of size at least $3$: maximum intersecting families of words are exactly the stars. Maths/math.CO_(Combinatorics) https://arxiv.org/abs/2604.13554 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13554v1 Announce Type: new Abstract: We determine the quantum query complexity of oracle identification on the hyperoctahedral group $B_N = \{\pm 1\}^N \rtimes S_N$ with respect to the natural representation: $Q_{LV}(B_N) = 2(N-1)$ for all $N \ge 2$. This is twice the symmetric-group value $Q_{LV}(S_N) = N-1$; the doubling arises from an $\varepsilon$-parity obstruction that restricts the bottleneck representation $\operatorname{sgn}(\sigma)$ to even tensor powers. The proof combines a reduction to $S_N$ Kronecker products via Rademacher moment polynomials with the bipartition distance formula $d_T(((N),\varnothing),(\alpha,\beta)) = 2(N-\alpha_1)-|\beta|$ in the tensor product graph. A closed-form generating function yields the first-appearance multiplicity $(2N-3)!!$. We also show $Q_{\mathrm{decomp}}(\varphi) \le 2\,Q_{\mathrm{signed}}(\varphi)$, with equality on $B_2$, and conjecture a link between the adversary bound and the graph eccentricity. Maths/math.CO_(Combinatorics) https://arxiv.org/abs/2604.13606 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13606v1 Announce Type: new Abstract: Let $k \in \mathbb{N}$ and let $G$ be a simple graph with maximum degree $\Delta$. A $k$-colouring $\varphi$ of $G$ is an assignment of colours from $\{1,2,\ldots,k\}$ to the vertices of $G$. We call $\varphi$ proper if adjacent vertices receive distinct colours, and equitable if the sizes of any two colour classes differ by at most one. The celebrated Hajnal--Szemer\'{e}di theorem states that a proper equitable $k$-colouring exists whenever $k \ge \Delta + 1$. In this paper, we study its tree colouring variant in which each colour class induces a forest. This is closely related to the vertex arboricity which was introduced by Chartrand, Kronk, and Wall. More precisely, we prove that if $n \ge 3\Delta^4$ and $k \ge (\Delta+2)/2$, then every $n$-vertex graph with maximum degree at most $\Delta$ contains an equitable tree $k$-colouring. This confirms a conjecture of Wu, Zhang, and Li when $\Delta$ is even and up to an additive constant of $1$ otherwise for large $n$. We also consider $d$-degenerate colouring in which each colour class induces a $d$-degenerate graph. Maths/math.CO_(Combinatorics) https://arxiv.org/abs/2604.13700 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13700v1 Announce Type: new Abstract: Given a digraph $D$, let $c(D)$ denote the largest integer $k$ such that there exists a collection of $k$ openly disjoint cycles through a vertex, i.e., a collection of directed cycles $C_1,\ldots,C_k$ through a common vertex $v$ such that $C_1-v,\ldots,C_k-v$ are pairwise vertex-disjoint. The famous Caccetta-H\"aggkvist conjecture and its regular variant due to Behzad, Chartrand and Wall from 1970, have motivated the study of degree conditions forcing $c(D)$ to be large. Surprisingly, in 1985 Thomassen constructed digraphs of arbitrarily high minimum out- and in-degree such that $c(D)\le 2$. In 2005, Seymour asked whether in contrast every $r$-regular digraph satisfies $c(D)=r$, which would have implied the aforementioned conjecture of Behzad, Chartrand and Wall. In 2008, Mader answered this negatively for every $r\ge 8$, but conjectured that nevertheless the minimum value $c_r$ of $c(D)$ over all $r$-regular digraphs grows with $r$, i.e. $\lim_{r\rightarrow\infty}c_r=\infty$. In this paper, we prove Mader's conjecture in a strong form by showing that $c_r\ge \left\lceil\frac{3}{22} r\right\rceil$ for every $r\in \mathbb{N}$. We also show that $c_r\le 7\left\lceil \frac{r}{8}\right\rceil$, which improves the previous best upper bound $c_r\le r-\Theta(\sqrt{r})$ due to Mader. Maths/math.CO_(Combinatorics) https://arxiv.org/abs/2604.13726 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13726v1 Announce Type: new Abstract: Let $\gamma$ be a constant such that $0 (2/3+\gamma)n$ for all $v\in V(H)$, where $\rho(N_H(v))$ denotes the spectral radius of $N_H(v)$, then $H$ has a perfect matching. This bound is asymptotically tight. Furthermore, for integer $s$ satisfying $n\geq 3s+3$, we establish that if \[ \rho(N_H(v))>\frac{1}{2}(s-1+\sqrt{(s-1)^2+4s(n-s-1)})\] holds for every $v\in V(H),$ then $H$ admits a fractional matching of size $s+1$. Notably, this second spectral bound is tight. Maths/math.CO_(Combinatorics) https://arxiv.org/abs/2604.13850 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13850v1 Announce Type: new Abstract: Let $F_n$, $W_n$, and $\widehat{K}_n$ be the graphs obtained by joining a vertex to $n$ independent edges, a cycle and a path of order $n-1$, respectively. In this paper, we give new bounds for the Ramsey numbers $R(F_n,F_m)$ and $R(W_n,W_n)$, which improve those due to Chen, Yu, and Zhao [EJC, 2021] and Mao, Wang, Magnant, and Schiermeyer [G&C, 2022], respectively, and establish lower and upper bounds for $R(\widehat{K}_n,\widehat{K}_n)$. Moreover, we present a blow-up technique to establish some new lower bounds for the Ramsey numbers of wheels versus cliques. Maths/math.CO_(Combinatorics) https://arxiv.org/abs/2604.13887 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13887v1 Announce Type: new Abstract: The orbit dimension $\sigma(G)$ (also called the separation number or rigidity index) of a permutation group $G$ with domain $\Omega$ is the minimum cardinality of a subset $S \subseteq \Omega$ such that, for any two distinct elements $\omega,\omega'\in \Omega$, there exists $\alpha\in S$ for which $\omega$ and $\omega'$ lie in distinct orbits of the stabilizer $G_\alpha$. In this paper, we first observe that if $G$ is transitive, then $\sigma(G)\le |\Omega|-r+1$, where $r$ is the rank of $G$, and we obtain strong structural information on the groups for which equality holds. Next, we investigate the orbit dimension in the case where $G$ is the symmetric group of degree $n$, acting on the set of $k$-subsets of $\{1,\ldots,n\}$. In this case, this invariant equals the metric dimension of Johnson graphs. Maths/math.CO_(Combinatorics) https://arxiv.org/abs/2604.13949 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13949v1 Announce Type: new Abstract: In this article, we provide three formulas allowing to compute the minimum amount of initial chips leading to an infinite Chip-firing game. These formulas hold for strongly connected directed loop-free multigraphs and generalize what was already known in the Eulerian case. In addition to the many theoretical aspects, some algorithmic consequences are also investigated. Maths/math.CO_(Combinatorics) https://arxiv.org/abs/2604.14006 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14006v1 Announce Type: new Abstract: Given a graph $G$ and an integer $r\ge 1$, the $r$th power $G^r$ of $G$ is the graph obtained from $G$ by adding edges for all pairs of distinct vertices at distance at most $r$ from each other. We focus on two basic structural properties of the $r$th power of the binomial random graph $G_{n,p}$, namely, the maximum degree $\Delta(G_{n,p}^r)$ and the chromatic number $\chi(G_{n,p}^r)$, and give with high probability (w.h.p.) bounds. In the sparse case that $p=d/n$ for some fixed constant $d>0$, we prove the following. We prove that w.h.p.~$\Delta(G_{n,p}^r) \sim \frac{\log n}{\log_{(r+1)}n}$ (where $\log_{(1)}n=\log n$ and $\log_{(r+1)}n=\log\log_{(r)}n$) and that w.h.p.~$\Delta(G_{n,p}^{\lfloor{r/2}\rfloor})+1 \le \chi(G_{n,p}^r) \le \Delta(G_{n,p}^{r-1})+1$. For $r=2$, we show the upper bound holds with equality. For denser cases, for $d$ satisfying $d=\omega(\log n)$ and $d\le n^{1/r-\Omega(1)}$ as $n\to\infty$, we have $\chi(G_{n,p}^r) = \Theta(d^r/\log d)$ w.h.p. Maths/math.CO_(Combinatorics) https://arxiv.org/abs/2604.14042 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14042v1 Announce Type: new Abstract: This work is concerned with the fundamental scaling laws of quasi-complementary sequence sets (QCSSs) by understanding how large the set size (denoted by $M$) can grow with the flock size ($K$) and the sequence length ($N$). We first establish a geometric framework that transforms a QCSS into a complex unit-norm codebook, through which and by exploiting the density thresholds of the codebooks, certain polynomial upper bounds of the QCSS set size are obtained. Sharp quadratic and cubic scaling laws are then introduced. Specifically, we show that asymptotically optimal QCSSs with tightness factor $\rho=1$ satisfy $M \le (1+o(1))K^2N$, while asymptotically near-optimal QCSSs satisfy $M \le (1+o(1))K^3N^2$ for $\rho < {(1+\sqrt{5})}/{2}$. To validate these upper bounds, we further propose explicit additive-character and mixed-character based constructions for QCSSs that achieve $M = K^2N + K$ and $M = K^3N^2 + 2K^2N + K$, respectively, thereby showing that the quadratic and cubic scaling laws are asymptotically tight. Our proposed constructions admit flexible parameter choices, and their maximum correlation estimates are shown to be tight through explicit extremal examples. Additionally, it is conjectured that the cubic scaling law is universal for all $1<\rho\le 2$, i.e., any asymptotically near-optimal QCSSs should satisfy $M \le (1+o(1))K^3N^2$. This identifies a fundamental cubic barrier for QCSS scalability. Maths/math.CO_(Combinatorics) https://arxiv.org/abs/2604.13431 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13431v1 Announce Type: cross Abstract: We give new explicit constructions of several fundamental objects in linear-algebraic pseudorandomness and combinatorics, including lossless rank extractors, weak subspace designs, and strong $s$-blocking sets over finite fields. Our focus is on the small-field regime, where the field size depends only on a secondary parameter (such as the rank or codimension) and is independent of the ambient dimension. This regime is central to several applications, yet remains poorly understood from the perspective of explicit constructions. In this setting, we obtain the first explicit constructions of lossless rank extractors and weak subspace designs for $r\ll k$, where $r$ denotes the rank (or codimension), over finite fields $\mathbb{F}_q$ with $q \ge \mathrm{poly}(r)$ and $q$ non-prime, with near-optimal parameters. For other finite fields, including prime fields and small fields, we obtain weaker but still improved bounds. As a consequence, we construct explicit strong $s$-blocking sets in $\mathrm{PG}(k-1,q)$ of size $O(s(k-s)q^s)$ for all sufficiently large non-prime fields $q \ge \mathrm{poly}(s)$, matching the best known non-explicit bounds up to constant factors. This significantly improves the previous best bound $2^{O(s^2 \log s)} q^s k$ of Bishnoi and Tomon (Combinatorica, 2026), which requires $q \ge 2^{\Omega(s)}$. Our approach is primarily algebraic, combining techniques from function fields and polynomial identity testing. In addition, we develop a complementary Fourier-analytic framework based on $\varepsilon$-biased sets, which yields improved explicit constructions of strong $s$-blocking sets over small fields. Maths/math.CO_(Combinatorics) https://arxiv.org/abs/2604.13510 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13510v1 Announce Type: cross Abstract: The aim of this paper is to prove a version of Lie's theorem for the supertropical algebra. Maths/math.CO_(Combinatorics) https://arxiv.org/abs/2604.13619 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13619v1 Announce Type: cross Abstract: We study $n$-homomorphisms in the sense of Khudaverdian--Voronov, but generalized to maps from arbitrary rings to arbitrary commutative rings. We show that the sum of an $n$-homomorphism and an $m$-homomorphism is an $\left( n+m\right) $-homomorphism, and that the composition of an $n$-homomorphism and an $m$-homomorphism is an $nm$-homomorphism. The proofs are entirely combinatorial. Maths/math.CO_(Combinatorics) https://arxiv.org/abs/2604.13819 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13819v1 Announce Type: cross Abstract: We extend the notions of finite free convolution and finite free cumulants to the setting of formal power series by introducing their natural analogues, namely $t$-deformed convolution and $t$-deformed cumulants. In this framework, we establish $t$-deformed analogues of the law of large numbers and the central limit theorem, revealing structural parallels with classical, free, and finite free probability theories. We show that the case $t=-1$ recovers classical convolution at the level of moment generating functions, thereby connecting the theory directly to classical probability. We further investigate the infinitesimal generators associated with $\boxplus^t$-continuous semigroups, deriving explicit representation formulas that clarify how these generators describe the infinitesimal evolution of the semigroup. In the case $t = d$, our results yield explicit formulas for finite free infinitesimal generators. In the case $t = -1$, we relate these generators to those of one-dimensional L\'{e}vy processes by identifying the corresponding terms in their representations. This establishes a direct connection between $\boxplus^t$-convolution semigroups and classical L\'{e}vy-Khintchine-type generators. Maths/math.CO_(Combinatorics) https://arxiv.org/abs/2604.14138 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14138v1 Announce Type: cross Abstract: We present a simple trimming algorithm that generates nested uniform binary plane trees by removing leaves one-by-one using a best-of-three-match procedure. While its one-step transition specializes to the Luczak-Winkler & Caraceni-Stauffer coupling, its scaling limit provides a suprising c\`adl\`ag erasure of Brownian trees, reminiscent of SLE theory. Maths/math.CO_(Combinatorics) https://arxiv.org/abs/2210.05915 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2210.05915v3 Announce Type: replace Abstract: We survey work on coloring, list coloring, and painting squares of graphs; in particular, we consider strong edge-coloring. We focus primarily on planar graphs and other sparse classes of graphs. Maths/math.CO_(Combinatorics) https://arxiv.org/abs/2604.14105 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14105v1 Announce Type: new Abstract: We show that the category of (right-)preordered groups is a quasivariety of universal algebras by giving explicit axioms. We then look at lattices of effective equivalence relations, which turn out to be similar to the lattices of equivalence relations in the category of groups. We study internal structures in the category of right-preordered groups, and we especially consider the class $\sS$ of Schreier split epimorphisms. The category of right-preordered groups turns out to be action representable when we restrict our attention to split epimorphisms in $\sS$. Relatively to this class of split epimorphisms, we define the notion of $\sS\mhyphen$precrossed modules, and then of $\sS\mhyphen$crossed modules that correspond exactly to Schreier internal reflexive graphs and Schreier internal categories, respectively. Lastly, we characterize groupoids among Schreier internal categories and give some examples. Maths/math.CT_(Category_Theory) https://arxiv.org/abs/2604.13698 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13698v1 Announce Type: cross Abstract: We introduce a notion of global dimension for a triangulated category relative to a compact silting object. We prove that the finiteness of this dimension is an intrinsic property of the triangulated category itself and, therefore, independent of the choice of the silting object. Focusing on the setup of connective differential graded (dg) algebras, we analyse the behaviour of global dimension under dg algebra homomorphisms and establish explicit bounds. This allows us to deduce a bound for the global dimension of certain dg quiver algebras. We also relate the regularity of the big singularity category of a proper connective dg algebra to the finiteness of its global dimension. Maths/math.CT_(Category_Theory) https://arxiv.org/abs/2603.07773 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2603.07773v2 Announce Type: replace Abstract: In this paper, we justify and make precise an elementary approach that establishes the existence of (co)limits in $\mathbf{Cat}$. This approach, while conceptually evident, has not been made fully explicit or systematically described in the literature. We first demonstrate an equivalence between the existence of the homotopy category functor $h : \mathbf{sSet} \rightarrow \mathbf{Cat}$ and the existence of a specific class of weighted colimits in $\mathbf{Cat}$. We then construct these weighted colimits explicitly by using certain properties of simplicial sets and the nerve functor. Consequentially, the embedding $N : \mathbf{Cat} \hookrightarrow \mathbf{sSet}$ is reflective, and can be used to infer the (co)completeness of $\mathbf{Cat}$. Finally, we use this approach to reformulate the construction of coequalizers and localizations in $\mathbf{Cat}$. Maths/math.CT_(Category_Theory) https://arxiv.org/abs/2506.21879 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2506.21879v2 Announce Type: replace-cross Abstract: For any affine Hopf algebra $H$ which admits a large central Hopf subalgebra, $H$ can be endowed with a Cayley-Hamilton Hopf algebra structure in the sense of De Concini-Procesi-Reshetikhin-Rosso. The category of finite-dimensional modules over any fiber algebra of $H$ is proved to be an indecomposable exact module category over the tensor category of finite-dimensional modules over the identity fiber algebra $H/\mathfrak{m}_{\overline{\varepsilon}}H$ of $H$. For any affine Cayley-Hamilton Hopf algebra $(H,C,\text{tr})$ such that $H/\mathfrak{m}_{\overline{\varepsilon}}H$ has the Chevalley property, it is proved that if the zero locus of a discriminant ideal of $(H,C,\text{tr})$ is non-empty then it contains the orbit of the identity element of the affine algebraic group $\text{maxSpec}C$ under the left (or right) winding automorphism group action. Its proof relies on the fact that $H/\mathfrak{m}_{\overline{\varepsilon}}H$ has the Chevalley property if and only if the $\overline{\varepsilon}$-Chevalley locus of $(H,C)$ coincides with $\text{maxSpec}C$. Then, we provide a description of the zero locus of the lowest discriminant ideal of $(H,C,\text{tr})$. It is proved that the lowest discriminant ideal of $(H,C,\text{tr})$ is of level $\text{FPdim}(\text{Gr}(H/\mathfrak{m}_{\overline{\varepsilon}}H))+1$, where $\text{Gr}(H/\mathfrak{m}_{\overline{\varepsilon}}H)$ is the Grothendieck ring of the finite-dimensional Hopf algebra $H/\mathfrak{m}_{\overline{\varepsilon}}H$ and $\text{FPdim}(\text{Gr}(H/\mathfrak{m}_{\overline{\varepsilon}}H))$ is the Frobenius-Perron dimension of $\text{Gr}(H/\mathfrak{m}_{\overline{\varepsilon}}H)$. Some recent results of Mi-Wu-Yakimov about lowest discriminant ideals are generalized. We also prove that all the discriminant ideals are trivial if $H$ has the Chevalley property. Maths/math.CT_(Category_Theory) https://arxiv.org/abs/2604.13845 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13845v1 Announce Type: cross Abstract: We show for $\kappa \in (4,8)$ that the canonical conformally covariant measure on the conformal loop ensemble (CLE$_\kappa$) gasket, previously constructed indirectly by the first co-author and Schoug, can be realized as the limit of several natural approximation schemes. These include the Euclidean Minkowski content and its box-count variants, the properly renormalized number of dyadic squares that intersect the gasket, and the properly renormalized minimal number of balls of radius $\delta$ necessary to cover the gasket with respect to both its canonical geodesic and resistance metrics. This in particular allows us to identify the CLE$_6$ gasket measure with the conformally covariant measure constructed by Garban-Pete-Schramm as a scaling limit of the number of vertices in a macroscopic critical percolation cluster on the triangular lattice. Along the way, we show that the CLE gasket measure of every fixed compact set has finite moments of all orders; previously this was only known for first moments. Maths/math.CV_(Complex_Variables) https://arxiv.org/abs/2604.14020 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14020v1 Announce Type: cross Abstract: This paper models the theory of abstract harmonic spaces in the syntax of the continuous first-order logic of Banach lattices. It addresses a topological question asking when a one-to-one harmonic map onto smooth manifolds $M^n$ is a diffeomorphism. We give $M^n$ ($n\le 2$) a characterization by $U$-rank and elementary saturation for large cardinals. Polar sets are characterized by several equivalent conditions from the omitting type theorem. Consequently, harmonic measures on the ideal boundary in Martin representation are bijectively mapped to Keisler measures supported on non-principal types. Further problems concerning o-minimality and non-local potentials are finally discussed. Maths/math.CV_(Complex_Variables) https://arxiv.org/abs/2507.13269 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2507.13269v2 Announce Type: replace-cross Abstract: Liouville Brownian motion (LBM) is the canonical diffusion process on a Liouville quantum gravity (LQG) surface. In this work, we establish upper and lower bounds for the heat kernel for LBM when $\gamma=\sqrt{8/3}$ in terms of the $\sqrt{8/3}$-LQG metric which are sharp up to a polylogarithmic factor in the exponential. Maths/math.CV_(Complex_Variables) https://arxiv.org/abs/2604.13111 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13111v1 Announce Type: new Abstract: Consider the following probabilistic contracting on average iterated function system $$\Phi = \left\{f_i (x) = \lambda_i x + d_i,\;i=1,2 ;\;\; p = \left(\frac{1}{2} , \frac{1}{2}\right) \right\},$$ where the contraction ratios $\lambda_1 , \lambda_2$ are such that $0<\lambda_1<1<\lambda_2$ and $\lambda_1\lambda_2<1$. Denote by $\mu_{\lambda_1,\lambda_2}$ its stationary measure. We study the differentiability of $$(\heartsuit)\quad\quad\quad\quad\quad \lambda_1 \mapsto \int_{\mathbb{R}} \phi(x) \,d\mu_{\lambda_1,\lambda_2}(x),$$ where $\phi$ is a suitable test function. We establish three cases where $(\heartsuit)$ is differentiable and show the derivative coincides with the one obtained by taking formal derivative, which can be generalized to the case of multiple maps with different probabilities. We also present sufficient conditions under which there exists a smooth, bounded test function $\phi$ so that $(\heartsuit)$ is not differentiable. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.13124 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13124v1 Announce Type: new Abstract: We investigate iterated function systems (IFS) that randomly alternate between two non-identical one-dimensional maps. Our primary focus is on finite invariant sets exhibiting ``toss-and-catch'' dynamics, in which trajectories alternate between fixed points and periodic orbits of the constituent maps. We derive exact parameter conditions for several toss-and-catch structures in a pair of logistic maps (logistic IFS) and a combination of logistic and tent maps (logistic-tent IFS). Notably, we identify cases in which the invariant set contains bridging points that belong to neither of the invariant sets of the individual maps. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.13189 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13189v1 Announce Type: new Abstract: Let $(X,T)$ be a compact dynamical system. This article proves that if $(X,T)$ has the partial specification property, then it has the average shadowing property. It is also proven that if $(X,T)$ is surjective and has the partial specification property, then the set of ergodic measures of $(X,T)$ is dense in the space of its invariant measures. An example of a compact dynamical system that is not Besicovitch complete is also given. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.13259 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13259v1 Announce Type: new Abstract: When a learning algorithm reshapes the data distribution it trains on, the long-run behavior depends on the joint evolution of the policy, the value estimate, and the data distribution. We study finite-state actor-critic mean dynamics on the enlarged phase space $(\theta,w,\mu)$, where $\theta$ is the actor parameter, $w$ is an auxiliary critic state, and $\mu$ is a state-law variable (the distribution over states induced by the current policy). The state-law coordinate follows the exact controlled-Markov equation $\delta \dot\mu = Q_\theta^*\mu$. Under a softmax actor with box confinement (a smooth proxy for parameter clipping), a uniformly coercive linear critic equation, and a Lipschitz generator family $\theta \mapsto Q_\theta$, we prove that for each $\delta > 0$ the resulting autonomous semiflow possesses a compact global attractor. Under a uniform exponential-mixing assumption, we prove that the invariant-law map $\theta \mapsto \mu_\theta$ is Lipschitz and that the reduced invariant-law system on $(\theta,w)$ is well posed. Under an additional pathwise exponential-stability estimate for the non-autonomous fast state equation, we show that the exact flow tracks the reduced flow on every finite time interval up to the initial layer, and that the exact attractors converge upper semicontinuously to the lifted reduced attractor as $\delta \to 0$. We also give a concrete finite-state reference-state minorization condition implying the pathwise hypothesis. All results are formalized in Lean 4 without custom axioms. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.13376 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13376v1 Announce Type: new Abstract: We develop an abstract operator-theoretic variational principle for asymptotic growth rates arising from subadditive processes driven by Markov operators: for each invariant measure on the base, the growth rate equals the supremum of fiber integrals over invariant lifts to the bundle, and this supremum is attained on an ergodic lift. Applied to (random) linear bundle morphisms, the principle extends the classical projective formulas for sums of Lyapunov exponents, including singular cocycles, and yields new asymptotic representations in terms of conditional annealed growth along individual directions. As an application, we prove that for random linear bundle morphisms driven by Markovian place-dependent noise, the pointwise Lyapunov exponents depend only on the current noise state and initial position, not on the full noise realization. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.13401 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13401v1 Announce Type: new Abstract: We study cohomology of Holder continuous linear cocycles over a hyperbolic dynamical system and regularity of conjugacy between Anosov systems. For cocycles $A$ and $B$ with conjugate periodic data, we establish Holder cohomology under various conditions: the periodic data of $B$ has narrow spectrum and the periodic data conjugacy $C(p)$ is Holder continuous at a periodic point; $B$ is constant and the cocycles are measurably cohomologous; $B$ is constant and diagonalizable over $\mathbb C$ and either its Lyapunov spaces are at most two-dimensional or $C(p)$ is in a bounded set. We also prove that a topological conjugacy between a weakly irreducible hyperbolic automorphism $L$ and an Anosov diffeomorphism $f$ of $\mathbb T^d$ is smooth if their derivative cocycles $L$ and $Df$ are conjugate. Using this and our results on cohomology of cocycles we obtain global periodic data rigidity results for weakly irreducible hyperbolic automorphisms. In the argument we also establish differentiability of stable holonomies in low regularity setting. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.13411 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13411v1 Announce Type: new Abstract: We introduce and analyze a class of neural network models motivated by the Drosophila central complex, designed to capture the emergence and dynamics of orientation-selective activity bumps. Starting from a biologically inspired ring connectivity model, we derive a simplofied reduced model of recurrent neural activity that supports stable localized patterns encoding angular position. We first study the deterministic dynamics and identify parameter regimes ensuring existence and global stability of bump solutions. We then extend the framework to a stochastic setting, incorporating both additive Brownian noise and a Markovian switching mechanism representing time-varying external cues. The resulting system is a switching diffusion with piecewise linear drift, for which we establish well-posedness, characterize the infinitesimal generator, and prove the existence of an invariant measure. Numerical simulations in low and high dimensions illustrate the robustness of the bump attractor under noise and switching stimuli, as well as the convergence toward the predicted stationary states. These results provide a mathematically tractable framework for understanding how population activity in the insect central complex encodes heading direction in the presence of variability. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.13702 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13702v1 Announce Type: new Abstract: We improve the best known upper bounds on the number of Ruelle resonances in disks of large radius for Gevrey uniformly hyperbolic flows. The proof is based on Rugh's approach of dynamical determinants that replaces the study of the flow itself by the analysis of a system of open hyperbolic maps. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.13802 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13802v1 Announce Type: new Abstract: We study the Rokhlin lemma in the context of infinite measure-preserving bijections, and completely classify such bijections up to $\lambda$-approximate conjugacy, where $\lambda$ is the infinite measure which is preserved. This sharpens the classical version of the Rokhlin lemma, which only provides such a classification up to $\mu$-approximate conjugacy where $\mu$ is a probability measure equivalent to $\lambda$. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2411.19270 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2411.19270v1 Announce Type: cross Abstract: Kirchhoff's kinetic analogy relates the equilibrium solutions of an elastic rod or strip to the motion of a spinning top. In this analogy, time is replaced by the arc length parameter in the phase portrait to determine the equilibrium configurations of the rod. Predicted equilibrium solutions from the phase portrait for specific boundary value problems, as well as certain localized solutions, have been experimentally observed. In this study, we employ the kinetic analogy to investigate the equilibrium solutions of planar soft ferromagnetic rods subjected to transverse and longitudinal external magnetic fields. Our analysis reveals a subcritical pitchfork bifurcation in the phase portrait of a ferromagnetic rod subjected to transverse external magnetic field as the axial load is decreased continuously from a large compressive load. Similarly, a supercritical pitchfork bifurcation is observed in the case of longitudinal external magnetic field. We predict equilibrium configurations for a free-standing soft ferromagnetic elastic rod and the same subjected to canonical boundary conditions. Furthermore, we observe novel localized equilibrium solutions arising from homoclinic and heteroclinic orbits, which are absent in the phase portraits of purely elastic rods. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.13250 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13250v1 Announce Type: cross Abstract: We investigate the deformation of symmetry on cotangent bundles from the Euclidean plane to two-dimensional constant-curvature surfaces and the continuation of local dynamics aspects in Hamiltonian systems. For a fixed curvature sign $\sigma\in\{+1,-1\}$, the curved problem is set up either on the sphere $(\sigma=+1)$ or on the hyperbolic plane $(\sigma=-1)$, both with radius $R=1/\varepsilon$, recovering flat space in the limit $\varepsilon\to 0$. The symmetry of these spaces is taken into account by using the In\"on\"u--Wigner contraction of Lie algebras from $\mathfrak{so}(3)$ or $\mathfrak{so}(2,1)$ to $\mathfrak{se}(2)$. We use Riemannian exponential coordinates centred at the North pole together with the pull-back the associated momentum map and the symplectic form. Within this geometric setting we use a local slice construction and prove the persistence from flat to curved spaces of non-degenerate relative equilibria and relative periodic orbits of general cotangent bundle Hamiltonian systems. We apply the resulting framework to the Newtonian $n$-body problem. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.13790 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13790v1 Announce Type: cross Abstract: Ferromagnetic elastic slender structures offer the potential for large actuation displacements under modest external magnetic fields, due to the magneto-mechanical coupling. This paper investigates the phase portraits of the Hamiltonian governing the three-dimensional deformation of inextensible ferromagnetic elastic rods subjected to combined terminal tension and twisting moment in the presence of a longitudinal magnetic field. The total energy functional is formulated by combining the Kirchhoff elastic strain energy with micromagnetic energy contributions appropriate to soft and hard ferromagnetic materials: magnetostatic (demagnetization) energy for the former, and exchange and Zeeman energies for the latter. Exploiting the circular cross-sectional symmetry and the integrable structure of the governing equations, conserved Casimir invariants are identified and the Hamiltonian is reduced to a single-degree-of-freedom system in the Euler polar angle. Analysis of the resulting phase portraits reveals that purely elastic and hard ferromagnetic rods undergo a supercritical Hamiltonian Hopf pitchfork bifurcation, whereas soft ferromagnetic rods exhibit this bifurcation only within a restricted range of the magnetoelastic parameter, $0<\tilde{K}_{dM}<1/8$. Both helical and localized post-buckling configurations are analyzed, and the corresponding load-deformation relationships are systematically characterized across a range of loading scenarios. Localized buckling modes, corresponding to homoclinic orbits in the Hamiltonian phase space, are constructed numerically. In contrast to the purely elastic case, the localized configurations of soft ferromagnetic rods exhibit non-collinear extended straight segments, a geometrically distinctive feature arising directly from the magnetoelastic coupling. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.14108 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14108v1 Announce Type: cross Abstract: Recent work suggests that (stochastic) gradient descent self-organizes near an instability boundary, shaping both optimization and the solutions found. Momentum and mini-batch gradients are widely used in practical deep learning optimization, but it remains unclear whether they operate in a comparable regime of instability. We demonstrate that SGD with momentum exhibits an Edge of Stochastic Stability (EoSS)-like regime with batch-size-dependent behavior that cannot be explained by a single momentum-adjusted stability threshold. Batch Sharpness (the expected directional mini-batch curvature) stabilizes in two distinct regimes: at small batch sizes it converges to a lower plateau $2(1-\beta)/\eta$, reflecting amplification of stochastic fluctuations by momentum and favoring flatter regions than vanilla SGD; at large batch sizes it converges to a higher plateau $2(1+\beta)/\eta$, where momentum recovers its classical stabilizing effect and favors sharper regions consistent with full-batch dynamics. We further show that this aligns with linear stability thresholds and discuss the implications for hyperparameter tuning and coupling. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.14130 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14130v1 Announce Type: cross Abstract: In this paper, we propose a novel framework for the joint identification of system dynamics and noise covariance in linear systems, under general noise distributions beyond Gaussian. Specifically, we would like to simultaneously estimate the dynamical matrix $A$ and the noise covariance matrix $\varSigma$ using state transition data. The formulation builds upon a novel parameterization of the state-transition distribution, which enables more effective use of distributional "shape" information for improved identification accuracy. We introduce two practical estimators, namely the maximum likelihood estimator (MLE) and the score-matching estimator (SME), to solve the joint dynamics-covariance identification problem, and provide rigorous analysis of their statistical properties and sample complexity. Simulation results show that the proposed estimators outperform the ordinary least squares (OLS) baseline. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2509.13986 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2509.13986v3 Announce Type: replace Abstract: We consider a perturbation $f$ of a hyperbolic toral automorphism $L$. We study rigidity related to exceptional properties of the strong and weak stable foliations for $f$. If the strong foliation is mapped to the linear one by the conjugacy $h$ between $f$ and $L$, we obtain smoothness of $h$ along the weak foliation and regularity of the joint foliation of the strong and unstable foliations. We also establish a similar global result. If the weak foliation is sufficiently regular, we obtain smoothness of the conjugacy along the strong foliation and regularity of the joint foliation of the weak and unstable foliations. If both conditions hold then we get smoothness of $h$ along the stable foliation. We also deduce a rigidity result for the symplectic case. The main theorems are obtained in a unified way using our new result on relation between holonomes and normal forms. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2602.00293 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2602.00293v2 Announce Type: replace Abstract: We construct an example of a smooth ($C^\infty$) circle covering map topologically conjugate to the doubling map, such that it has a physical measure supported on a hyperbolic repelling fixed point. By relaxing the smooth condition at a single point, we also construct an example where the basin of the physical measure has full measure. A key technical step is a realization method of independent interest, which gives a canonical way to construct a full branch map given its induced map. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2602.10033 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2602.10033v2 Announce Type: replace Abstract: Let $f$ be a $C^r$ ($r>1$) diffeomorphism on a compact surface $M$ with $h_{\rm top}(f)\geq\frac{\lambda^{+}(f)}{r}$ where $\lambda^{+}(f):=\lim_{n\to+\infty}\frac{1}{n}\max_{x\in M}\log \left\|Df^{n}_{x}\right\|$. We establish an equivalent formula for the topological entropy: $$h_{\rm top}(f)=\lim_{n\to+\infty}\frac{1}{n}\log\int_{M}\left\|Df^{n}_{x}\right\|\,dx.$$ We also characterize the topological entropy via the volume growth of curves and several applications are presented. Our approach builds on the key ideas developed in the works of Buzzi-Crovisier-Sarig (\emph{Invent. Math.}, 2022) and Burguet (\emph{Ann. Henri Poincar\'e}, 2024) concerning the continuity of the Lyapunov exponents. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2603.15716 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2603.15716v3 Announce Type: replace Abstract: In this manuscript, we introduce a family of parametrized non-homogeneous linear complex differential equations on $[1,\infty)$, depending on a complex parameter. We identify a "Rotation number hypothesis" on the non-homogeneous term, which establishes a structural asymmetry: if two solutions with the same initial condition equal to $1$, corresponding respectively to the parameters $s$ and $1-\overline{s}$ lying in the critical strip, are both bounded on $[1,+\infty)$, then $\Re(s) = \tfrac{1}{2}$. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.04665 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.04665v2 Announce Type: replace Abstract: In this paper, we investigate the sharp regularity properties of a special weighted Sobolev space defined on the $ n $-dimensional torus, which is of independent interest. As a key application, we show that for almost all $ n $-dimensional vector fields, the Kolmogorov-Arnold-Moser (KAM) theory holds via this regularity, and in this case, the perturbation must have classical derivatives up to order $ \left[ {n/2} \right] $, yet it can admit unbounded weak derivatives from order $ \left[ {n/2} \right]+1 $ to $ n$. This result may appear surprising within the classical framework of KAM theory. We also provide further discussion of historical KAM theorems and relevant counterexamples. These findings constitute a new step in the long-standing KAM regularity conjecture. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.07043 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.07043v2 Announce Type: replace Abstract: This paper presents a framework for local solutions to time-varying linear differential-algebraic equations (DAEs) with real meromorphic coefficients. The local solutions on compact intervals form a sheaf. This permits a simple definition of controllability in the sense of Jan C. Willems. We prove that this notion is equivalent to the established global notion by giving an algebraic characterization based on the Teichm\"uller-Nakayama form. Finally, we study conditions under which local solutions admit extension, which is necessary for controllability. Maths/math.DS_(Dynamical_Systems) https://arxiv.org/abs/2604.13239 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13239v1 Announce Type: new Abstract: We give a direct proof of the Cotlar-Stein lemma, which does not rely on the power trick. Maths/math.FA_(Functional_Analysis) https://arxiv.org/abs/2604.13310 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13310v1 Announce Type: new Abstract: The question of determining a signal from its higher-order autocorrelation data is of practical interest in fields as varied as X-ray crystallography, image processing, and satellite communications. At the heart of the issue is how much of this autocorrelation data one truly needs. We prove two new upper bounds on the order of data needed to determine a signal on a general (i.e. not necessarily cyclic) finite abelian group depending on some knowledge of the vanishing of the signal's Fourier transform. In investigating lower bounds on the required data, we classify signals on $\mathbb{Z}_6$ not determined by their fifth-order data and provide analogous examples on $\mathbb{Z}_{30}$. Maths/math.FA_(Functional_Analysis) https://arxiv.org/abs/2604.13607 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13607v1 Announce Type: new Abstract: We study strongly continuous and locally equicontinuous families of operators on sequentially complete Hausdorff locally convex spaces. In case of Saks spaces, we relate the general notions to bi-continuity as well as equitightness. In this way, we recover and also generalise known results for special classes of operator families such as bi-continuous ($C$-)semigroups and ($C$-)cosine families by well-known results for the corresponding families in Hausdorff locally convex spaces. Maths/math.FA_(Functional_Analysis) https://arxiv.org/abs/2604.14033 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14033v1 Announce Type: new Abstract: We introduce a family of (nonlinear) pairing measures that ensure the validity of the divergence rule for composite functions $\boldsymbol{B}(x,u(x))$, where $\boldsymbol{B}(\cdot,t)$ is a bounded divergence-measure vector field, and $u$ is a scalar function of bounded variation. The elements of the family depend on the choice of the pointwise representative of $u$ on its jump set. Beyond the standard properties, such as the Coarea and Gauss-Green formulas on sets of finite perimeter, this flexibility allows us to characterize the pairings that ensure the lower semicontinuity of the corresponding functionals along sequences converging in $L^1$ with controlled precise values. We show that these lower semicontinuous pairings arise as the relaxation of integral functionals defined in Sobolev spaces. Maths/math.FA_(Functional_Analysis) https://arxiv.org/abs/2604.14068 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14068v1 Announce Type: new Abstract: In this paper, we study non-reflexive Banach spaces $X$ for which the quotient space $X^{**}/X$ is reflexive. Such spaces were first introduced by James R.~Clark, where they were called coreflexive spaces. We show that a space $X$ is coreflexive if and only if every separable subspace $Y\subseteq X$ is coreflexive, provided that $X$ is w$^*$-sequently dense in its bidual $X^{**}$. We show that coreflexive spaces are stable under $\ell^{p}$-sum for $1<p<\infty$. We show that if $X$ is a coreflexive space such that $X^{**}/X$ is separable, then the space of Bochner $p$-integrable functions, $L^{p}(\mu,X)$ is coreflexive for $1<p<\infty$. We conclude by providing an alternative proof of the fact, in a quasi-reflexive space $X$, w-PC's of the unit ball $X_{1}$ continue to have the same property in all the higher even-order dual unit balls of $X$. Maths/math.FA_(Functional_Analysis) https://arxiv.org/abs/2604.14102 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14102v1 Announce Type: new Abstract: We extend the Daugavet property (and a perfect version) to transfinite cardinals and provide a number of examples. We characterise the transfinite Daugavet $C(K)$ spaces in terms of a cardinal index $\mathfrak r(K)$, which generalises the notion of the reaping number of a Boolean algebra. We study several inheritance results of the transfinite Daugavet properties by almost isometric ideals, absolute sums, and tensor product spaces with a number of applications, including the classification of these properties for $L_1(\mu)$ and $L_\infty(\mu)$ spaces. The perfect version of the Daugavet property for $\omega$ is also characterised in the space of Lipschitz functions $\mathop{Lip}(M)$. Maths/math.FA_(Functional_Analysis) https://arxiv.org/abs/2604.13342 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13342v1 Announce Type: cross Abstract: It is well known that the spectrum of the Dirichlet Laplacian for a two-dimensional waveguide, which is a local deformation of a straight strip, is unstable with respect to waveguide boundary deformations. This means that, when the waveguide is a straight strip, the spectrum of the Dirichlet Laplacian is purely essential. On the other hand, local boundary perturbations of the straight strip produce eigenvalues below the essential spectrum. This paper considers the Dirichlet-Laplace operator with a compactly supported magnetic field. Furthermore, we omit the condition that the boundary perturbation is local. We prove that, in this case, the spectrum of the magnetic Laplacian is stable under small deformations of the waveguide boundary. Maths/math.FA_(Functional_Analysis) https://arxiv.org/abs/2604.14055 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14055v1 Announce Type: cross Abstract: We define 2-indexed $(q,p)$-Schatten quasi-norms for any $q,p > 0$ on operators on a tensor product of Hilbert spaces, naturally extending the norms defined by Pisier's theory of operator-valued Schatten spaces. We establish several desirable properties of these quasi-norms, such as relational consistency and the behavior on block diagonal operators, assuming that $|\frac{1}{q} - \frac{1}{p}| \leq 1$. In fact, we show that this condition is essentially necessary for natural properties to hold. Furthermore, for linear maps between spaces of such quasi-norms, we introduce completely bounded quasi-norms and co-quasi-norms. We prove that the $q \to p$ completely bounded co-quasi-norm is super-multiplicative for tensor products of quantum channels for $q \geq p>0$, extending an influential result of [Devetak, Junge, King, Ruskai, 2006]. Our proofs rely on elementary matrix analysis and operator convexity tools and do not require operator space theory. On the applications side, we demonstrate that these quasi-norms can be used to express relevant quantum information measures such as R\'enyi conditional entropies for $\alpha \geq \frac{1}{2}$ or the Sandwiched R\'enyi Umlaut information for $\alpha < 1$. Our multiplicativity results imply a tensorizing notion of reverse hypercontractivity, additivity of the completely bounded minimum output R\'enyi-$\alpha$-entropy for $\alpha\geq\frac{1}{2}$ extending another important result of [Devetak, Junge, King, Ruskai, 2006], and additivity of the maximum output R\'enyi-$\alpha$ entropy for $\alpha \geq \frac{1}{2}$. Maths/math.FA_(Functional_Analysis) https://arxiv.org/abs/2604.14106 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14106v1 Announce Type: cross Abstract: We prove a new "Toeplitz exactness" theorem for strong convergence. This is a machine to upgrade strong convergence in the general setting of $C^\ast$-correspondences, and has several applications. Maths/math.FA_(Functional_Analysis) https://arxiv.org/abs/2404.18422 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2404.18422v3 Announce Type: replace Abstract: Given $H$ self-adjoint, $V$ symmetric and relatively $H$-bounded, and $f:\mathbb{R}\to\mathbb{C}$ satisfying mild conditions, we show that the Gateaux derivative $$\frac{d^n}{dt^n}f(H+tV)|_{t=0}$$ exists in the operator norm topology, for every natural $n$, give a new explicit formula for this derivative in terms of multiple operator integrals, and establish useful perturbation formulas for multiple operator integrals under relatively bounded perturbations. Moreover, if the $H$-bound of $V$ is less than 1, we obtain sufficient conditions on $f$ which ensure that the Taylor expansion $$f(H+V)=\sum_{n=0}^\infty\frac{1}{n!}\frac{d^n}{dt^n} f(H+tV)\big|_{t=0}$$ exists and converges absolutely in operator norm. Finally, assuming that $V(H-i)^{-p}\in\mathcal{S}^{s/p}$ for $p=1,\ldots,s$ for some $s\in\mathbb{N}$ (for instance, when $H$ is an order 1 differential operator on an $s-1$ dimensional space), we show that the Krein--Koplienko spectral shift functions $\eta_{k,H,V}$, satisfying $${Tr}\left(f(H+V)-\sum_{m=0}^{k-1}\frac{1}{m!}\frac{d^m}{dt^m} f(H+tV)\big|_{t=0}\right)=\int_{\mathbb{R}} f^{(k)}(x)\eta_{k,H,V}(x)dx,$$ exist for every $k=1,2,3,\ldots$, independently of $s$. The latter result (which is significantly stronger than \cite{vNS22}) is completely new also in the case that $V$ is bounded. The proof is based on \cite{PSS}, combined with a generalisation of the multiple operator integral compatible with \cite{HMvN}. We discuss applications of our results to quantum physics and noncommutative geometry. Maths/math.FA_(Functional_Analysis) https://arxiv.org/abs/2412.17448 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2412.17448v2 Announce Type: replace Abstract: In this article, we develop a pseudodifferential calculus on a general filtered manifold M . The symbols are fields of operators $\sigma$(x, $\pi$) parametrised by x $\in$ M and the unitary dual G x M of the osculating Lie group G x M . We define classes of symbols and a local quantization formula associated to a local frame adapted to the filtration. We prove that the collection of operators on M coinciding locally with the quantization of symbols enjoys the essential properties of a pseudodifferential calculus: composition, adjoint, parametrices, continuity on adapted Sobolev spaces. Moreover, we show that the polyhomogeneous subcalculus coincides with the calculus constructed by van Erp and Yuncken via groupoids. Maths/math.FA_(Functional_Analysis) https://arxiv.org/abs/2510.23952 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2510.23952v4 Announce Type: replace Abstract: This paper introduces a new type of asymptotic contraction, called an adaptive relaxed asymptotic contraction, which significantly relaxes the assumptions of Kirk's classical theorem. The condition is formulated using two quantities naturally generated by the map: the T-short distance (an adaptive switching combination of a pointwise minimum and the ordinary iterated distance) and the T-long distance (the maximum of the original distance, the one-step movement distances, and the distance between the images). At each iteration step, the T-short distance is required to be bounded above by a function of the T-long distance, where the bounding functions converge uniformly on bounded sets to a Boyd-Wong comparison function. We prove that any continuous self-mapping of a complete metric space that satisfies this condition and has a bounded orbit must possess a unique fixed point, and all iterates converge to it. Maths/math.FA_(Functional_Analysis) https://arxiv.org/abs/2511.12672 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2511.12672v3 Announce Type: replace Abstract: The classical Banach spaces $L_\infty[0,1]$ and $\ell_\infty$ are isomorphic. We present here some lower and upper bounds for their Banach-Mazur distance. Maths/math.FA_(Functional_Analysis) https://arxiv.org/abs/2603.01466 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2603.01466v5 Announce Type: replace Abstract: Violation of Bell inequalities in bipartite systems represented by mutually-commuting von Neumann algebras has pioneered the study of vacuum entanglement in algebraic quantum field theory. It is unexpected that the maximal violation of Bell inequality can discover algebraic structures. In the paper, we establish the mutually-commuting von Neumann algebra model for entanglement swapping networks and Bell-type inequalities on this model. It generalizes the bipartite case to the ternary case. These algebras are all general von Neumann algebras, which provide a natural perspective to investigate Bell nonlocality in quantum networks in the infinitely-many-degree-of-freedom setting. We determine various bounds for Bell-type inequalities based on the structure of von Neumann algebras, and identify the algebraic structural conditions required for their violation. Finally, we show that the maximal violation of Bell-type inequalities in entanglement swapping networks can be used to determine partially the type classification of the underlying von Neumann algebras. Maths/math.FA_(Functional_Analysis) https://arxiv.org/abs/2512.10410 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2512.10410v4 Announce Type: replace-cross Abstract: We analyze the Namioka-Phelps minimal and maximal tensor products of compact convex sets arising as state spaces of C$^*$-algebras, and, relatedly, study entanglement in (infinite dimensional) C$^*$-algebras. The minimal Namioka-Phelps tensor product of the state spaces of two C$^*$-algebras is shown to correspond to the set of separable (= un-entangled) states on the tensor product of the C$^*$-algebras. We show that these maximal and minimal tensor product of the state spaces agree precisely when one of the two C$^*$-algebras is commutative. This confirms an old conjecture by Barker in the case where the compact convex sets are state spaces of C$^*$-algebras. The Namioka-Phelps tensor product of the trace simplexes of two C$^*$-algebras is shown always to be the trace simplex of the tensor product of the C$^*$-algebras. This can be used, for example, to show that the trace simplex of (any) tensor product of C$^*$-algebras is the Poulsen simplex if and only if the trace simplex of each of the C$^*$-algebras is the Poulsen simplex or trivial (and not all trivial). Maths/math.FA_(Functional_Analysis) https://arxiv.org/abs/2601.07215 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2601.07215v3 Announce Type: replace-cross Abstract: The action potential constitutes the digital component of the signaling dynamics of neurons. But the biophysical nature of the full-time course of the action potential associated with changes in membrane potential is mathematically distinct from its representation as a discrete set of events that encode when action potentials are triggered in a collection of spike trains. In this paper, we develop from first principles a unified functional-analytic framework for neuronal spike trains, grounded in Schwartz distribution theory. We show how this representation provides an exact operational calculus for convolution, distributional differentiation, and distributional support, which enables closed-form analysis of spike train dynamics without discretization, rate approximation, or smoothing. We then analyze the framework in the context of a two-neuron reciprocal circuit with propagation latencies and refractoriness, deriving exact results for synaptic drive, spike timing sensitivity, and causal admissibility of inputs, quantities that are either ill-defined or require approximation in conventional treatments. Maths/math.FA_(Functional_Analysis) https://arxiv.org/abs/2604.12410 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.12410v2 Announce Type: replace-cross Abstract: We analyze the Schwarz inequality and its generalizations, as well as inequalities resulting from the Jensen inequality. They are used in quantum theory to derive the Heisenberg-Robertson (HR) and Schroedinger-Robertson (SR) uncertainty relation for two non-commuting observables and their generalizations to three or more non-commuting observables. Jensen's inequality, in turn, is helpful in deriving various the "sum uncertainty relations" for two or more observables. Using these inequalities, we derive various types of generalized uncertainty relations for more than two non--commuting observables and analyze their properties and critical points. We also study the connections between the generalizations of the HR and SR uncertainty relations for two and more observables and the correlations of these observables in the state of the quantum system under study. In this analysis, we pay special attention to the consequences of the generalized SR uncertainty relation for three non--commuting observables on their correlations in a given state of a quantum system and to the connections of this relation with the appropriate correlation matrix, whose matrix elements are the quantum versions of the Pearson coefficient. We show also that the SR uncertainty relation (including the generalized ones) can be written in an equivalent way using these Pearson coefficients. Maths/math.FA_(Functional_Analysis) https://arxiv.org/abs/2604.13063 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13063v1 Announce Type: new Abstract: The Homotopy Analysis Method (HAM) is a widely used analytical approach for solving nonlinear problems, yet its theoretical foundation lacks rigorous justification, and its intrinsic correlation with perturbation theory remains ambiguous, leading to prevalent confusion in the existing literature. This study demonstrates that the fundamental homotopy deformation equation of HAM can be naturally derived from the weak-nonlinearity perturbation theory. We construct a specific analytical expression and optimize the core parameters (the optimal auxiliary linear operator, convergence-control parameter, and auxiliary function) to mitigate the inherent strong nonlinearity of the nonlinear operator. Extending the small parameter \epsilon of perturbation theory to the interval [0,1] enables a systematic homotopy deformation process, which connects the linear auxiliary system (at \epsilon=0) with the original nonlinear problem (at \epsilon=1) and confirms HAM as a structured, adaptive generalization of classical perturbation theory. Furthermore, this work provides a rigorous proof that the Homotopy Perturbation Method (HPM) is a special case of HAM: HPM can be directly derived by fixing the optimal auxiliary linear operator as the linear component of the nonlinear system and setting the convergence-control parameter and auxiliary function to specific values, thus making HPM a degenerate form of HAM. This study clarifies the perturbation-theoretic origin of HAM, defines the hierarchical subordination of HPM to HAM, unifies the theoretical framework of homotopy-based nonlinear analytical methods, rectifies common misconceptions in the existing literature, and offers valuable guidance for the rational application, comparative analysis, and further development of such methods. Maths/math.GM_(General_Mathematics) https://arxiv.org/abs/2604.13080 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13080v1 Announce Type: new Abstract: In the present article an endeavor is made to solve the variable order fractional diffusion equations using a powerful method viz., Homotopy Analysis method. It is demonstrated how the method can be used while solving approximately two types of variable order fractional diffusion equations having physical importance. Numerical simulation results show that the method is reliable and effective for solving fractional order diffusion equations even when the order of the derivative is varying with respect to space or time or both or it is dependent upon some other parameters. Maths/math.GM_(General_Mathematics) https://arxiv.org/abs/2604.13083 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13083v1 Announce Type: new Abstract: This paper introduces a biharmonic interpolatory subdivision framework on Riemannian manifolds. In the Euclidean setting, the six-point Deslauriers-Dubuc stencil is characterised as the unique minimiser of a discrete curvature-variation energy under symmetric six-point support and degree-five polynomial reproduction conditions, linking a classical interpolatory rule to a first-principles fairness criterion. Exact symbol analysis establishes fourth-order smoothness. The construction extends to the two-sphere and the hyperbolic plane via a second-order reduced governing ODE derived from the biharmonic Euler-Lagrange equation on constant-curvature surfaces. This reduced model yields closed-form insertion rules, and proximity analysis confirms that the manifold scheme satisfies the Wallner-Dyn second-order condition, preserving fourth-order smoothness. A hierarchy of biharmonic stencils achieving higher smoothness orders is also described. Numerical experiments demonstrate that the six-point scheme delivers lower fairness energy and smoother curvature profiles than the classical four-point Dyn-Gregory-Levin scheme, while remaining more local and exhibiting less ringing on non-uniform data than the eight-point variant. Maths/math.GM_(General_Mathematics) https://arxiv.org/abs/2604.13086 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13086v1 Announce Type: new Abstract: We study binomially weighted summation methods given by \[ (x_n)_{n\in \mathbb{N}} \mapsto \left(\sum_{k=0}^n\binom{n}{k}r^k(1-r)^{n-k}x_k\right)_{n\in \mathbb{N}} \] for $r\in (0,1)$, and their behavior under composition with summation methods of the form \[ (x_n)_{n\in \mathbb{N}} \mapsto \left(\sum_{k=0}^n\lambda_k x_{n-k}\right)_{n\in \mathbb{N}}. \] Our main result shows that if the binomially weighted averages of a sequence $(x_n)_{n\in \mathbb{N}}$ converge to a limit then the binomially weighted averages of the sequence $\left(\sum_{k=0}^n\lambda_kx_{n-k}\right)_{n\in \mathbb{N}}$ converge to the same limit whenever $(\lambda_n)_{n\in\mathbb{N}}$ is an absolutely summable sequence with $\sum_{k=0}^{\infty}\lambda_k = 1$. This result disproves a theorem appearing in the literature. Additionally, we discuss applications and extensions of our main result to compositions with weighted Ces\`aro averages. Maths/math.GM_(General_Mathematics) https://arxiv.org/abs/2604.13089 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13089v1 Announce Type: new Abstract: The notion of asymptotic space for an unbounded metric space has been introduced by Micha Gromov in 1980s. It is intended to capture the structure of a metric space at infinity. The most comprehensive definition of asymptotic space is given in the lahguage of Nonstandard Analysis (NSA). It turns out that the asymptotic space depends on the underlying nonstandard extension of the standard universe. This paper contains the exhaustive description of asymptotic spaces of the Lobachevski plane which turns ourt to be an R-tree. However, there turn out to be a plenty of different nonisometric asymptotic spaces, including the spaces of high cardinality. Maths/math.GM_(General_Mathematics) https://arxiv.org/abs/2604.13090 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13090v1 Announce Type: new Abstract: Let $p>1$ be a large prime number and let $x=O((\log p)^2(\log\log p)^5$ be a real number. It is proved that the least consecutive pair of primitive roots $u\ne\pm1, v^2$ and $u+1$ satisfies the upper bound $u\ll x$ in the prime field $\mathbb{F}_p$. Maths/math.GM_(General_Mathematics) https://arxiv.org/abs/2604.13094 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13094v1 Announce Type: new Abstract: Many generalized set models have the same basic form: they assign a value to each object, and the main difference lies in the kind of values that are allowed. This paper studies that common form through scale-valued sets (SV-sets), defined as maps $U\times E\to\Sigma$, where $U$ is a universe, $E$ is a parameter set, and $\Sigma$ is a bounded De Morgan lattice. With a suitable choice of scale, SV-sets include ordinary sets, fuzzy sets, soft sets, bounded multisets, intuitionistic fuzzy sets, $L$-fuzzy sets, and Type-2 fuzzy sets. We study the basic structure of SV-sets. The relation between SV-sets and lattice-valued interval soft sets is also discussed. For complete chains, the SV setting gives a natural topological construction, and for groups, it gives an algebraic structure through SV-subgroups. The applications show how graded suitability and supporting evidence can be kept together in a single model, whereas one-coordinate reductions lose information. Maths/math.GM_(General_Mathematics) https://arxiv.org/abs/2604.13095 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13095v1 Announce Type: new Abstract: The mathematical representation of uncertainty has led to a proliferation of preference structures, such as interval-valued fuzzy sets, intuitionistic fuzzy sets, and various granular models. While these extensions are often studied independently, they share profound geometric and topological foundations. This paper provides a unifying framework by identifying these disparate structures with the simplicial geometry of $n$-dimensional fuzzy sets. We first conduct an extensive revision of both classical and modern preference structures, demonstrating that they are distinct semantic interpretations of the same underlying topological objects within the lattice $L_n$. Building on this unification, we introduce a new, highly interpretable preference structure based on Deck-of-Cards membership functions. This approach generalizes the revised models by providing a flexible mechanism to represent complex membership degrees through monotonic sequences. Furthermore, we establish a formal simplicial structure for the set of multidimensional fuzzy sets $L_\infty$. By employing face and degeneracy maps, we demonstrate how this framework unifies existing models into a single simplicial set, allowing for the consistent transformation of information across different levels of granularity. The examples provided illustrate the utility of this simplicial connection in several contexts, offering a robust topological foundation for future developments in fuzzy set theory. Maths/math.GM_(General_Mathematics) https://arxiv.org/abs/2604.13096 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13096v1 Announce Type: new Abstract: We propose and analyse a class of analytically solvable models of quantum reinforcement learning (QRL), formulated as finite-horizon Markov decision processes in finite-dimensional Hilbert spaces. The models are built around a `unitary-control-then-measure' protocol, in which a learning agent applies unitary transformations to a quantum state and interleaves each control step with a projective measurement onto a prescribed reference basis. Exact closed-form expressions for trajectory probabilities, rewards, and the expected return are derived for four concrete realisations: a closed-chain and an anti-periodic qubit implementation, a qutrit model with ladder coupling, and a four-level two-qubit system. Two structural features of these QRL protocols are rigorously analysed. First, we identify and quantify a two-level reduction in the computational complexity of the expected return, from the nominally exponential $O(e^N)$ scaling in the trajectory length~$N$ to an explicit power-law $O(N^{\mathcal{I}})$: a trajectory-based level, arising from equivalence classes of paths sharing the same unordered state counts and transition frequencies, and a policy-based level, arising from the sparsity of the transition graph enforced by constrained unitary actions. Second, we characterise the degeneracy of optimal policies. The low-dimensional models exhibit unique optima whose asymptotic behaviour with~$N$ is governed by the quantum Zeno effect, while the four-level system displays both plateau-type quasi-degeneracy at large horizons and genuine discrete degeneracy at critical energy parameters -- phenomena with no counterpart in the measurement-free quantum optimal control landscape. Maths/math.GM_(General_Mathematics) https://arxiv.org/abs/2604.13106 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13106v1 Announce Type: new Abstract: Using the log-convexity of the Gamma function and Euler's reflection formula, we give a new proof of a classical weighted sine product inequality. Two different parameter choices yield two competing upper bounds for the same product. We determine precisely, via algebraic criteria, when one bound is sharper than the other. Explicit results are given for the general $n$-angle case, the $2n$-angle case, and for two and three angles. Several sharp corollaries are derived, including $\sin(\pi x)\leq \sin(2\pi x(1-x))$. Maths/math.GM_(General_Mathematics) https://arxiv.org/abs/2604.13113 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13113v1 Announce Type: new Abstract: The sigma index of a graph, defined as the population variance of its degree sequence, is a fundamental measure of structural irregularity. In this paper, we introduce and systematically investigate its natural extension to fuzzy graphs, termed the fuzzy sigma index $$ \sigma^*(\Gamma) = \frac{1}{n} \sum_{v \in V(\Gamma)} \left( d_\Gamma(v) - \frac{2\,\mathrm{ew}}{n}\right)^2, $$ where $d_\Gamma(v)$ denotes the fuzzy degree of a vertex $v$, and $\mathrm{ew}$ represents the fuzzy size of the fuzzy graph $\Gamma=(V,\nu, \mu)$. We establish several fundamental properties of this topological index. In particular, we derive sharp lower and upper bounds. Analyze the behavior of $\sigma^*(\Gamma)$ under standard fuzzy graph operations. This work provides a foundation for further study of variance-based topological indices in fuzzy graph theory. Maths/math.GM_(General_Mathematics) https://arxiv.org/abs/2604.13117 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13117v1 Announce Type: new Abstract: We study the second-order differential operators \(\mathcal D_{\Xi}\) and \(\mathcal D_{\Lambda}\) associated with the rescaled polynomial families \((\widetilde{\Xi}_n)\) and \((\widetilde{\Lambda}_n)\), and more generally the polynomial sequences generated by iterating these operators from an arbitrary linear initial datum \(cx-d\). We establish structural properties of \(\mathcal D_{\Xi}\) and \(\mathcal D_{\Lambda}\), including factorizations into first-order operators, weighted divergence forms, formal self-adjointness, and hypergeometric descriptions of the corresponding formal eigenvalue equations. We also show that both operators preserve hyperbolicity, preserve zeros in \((0,b)\) for \(b\ge 1\), and preserve proper position. For the iterated polynomial sequences, we derive explicit closed formulae in terms of the auxiliary families \((\widetilde{\Xi}_n)\) and \((\widetilde{\Lambda}_n)\), prove strict interlacing of consecutive zeros under explicit conditions on \(d/c\), and obtain asymptotic formulae for the normalized logarithmic derivatives. As a consequence, the associated zero counting measures converge weakly to the same limiting probability measure as in the auxiliary case. Maths/math.GM_(General_Mathematics) https://arxiv.org/abs/2604.13138 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13138v1 Announce Type: new Abstract: Four coframes of invariant 1-forms are explicitly constructed using the Inductive Cartan equivalence method with rank zero corresponding to four distinct branches. These coframes are employed to characterize non-linearizable fourth-order ODEs under point transformation with a five-point symmetry Lie subalgebra. Moreover, we propose a procedure for obtaining the point transformation by using the derived invariant coframes, demonstrated through examples. Maths/math.GM_(General_Mathematics) https://arxiv.org/abs/2603.00017 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2603.00017v2 Announce Type: replace Abstract: We formalize a ten-face triangular wing set on a regular icosahedron under a vertex labeling N, S, U1-U5, L1-L5 with rotation axis NS. The wing faces satisfy: (i) each face is an isosceles 36-36-108 triangle with a 36-degree angle anchored at a pole (N or S); (ii) distinct faces may share vertices but share no edges; and (iii) a natural equatorial cross-section yields a perfectly balanced regular decagon. We derive a closed form for the decagon radius, R = (phi/2)*ell, where ell is the icosahedron edge length and phi is the golden ratio phi = (1 + sqrt(5))/2. Beyond the geometric results, we interpret the ten-face closure as a symmetry-consistent design principle for a pole-anchored wing layout and provide a reproducible construction workflow. Maths/math.GM_(General_Mathematics) https://arxiv.org/abs/2604.13626 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13626v1 Announce Type: new Abstract: In this paper, we introduce the notion of a $\gamma$-density point for Lebesgue-measurable subsets of $\mathbb{R}$, where $\gamma$ is a modulus function, and study its basic measure-theoretic properties. We show that every $\gamma$-density point is a Lebesgue density point, while under Condition~(A) the two notions coincide. Consequently, for such modulus functions, the set of $\gamma$-density points of a measurable set differs from the set itself only by a null set, yielding a modulus version of the Lebesgue Density Theorem. We then define the associated $\gamma$-density topology $\tau_\gamma$ and investigate its structure. In general, $\tau_\gamma$ is contained in the classical Lebesgue density topology, and if $\gamma$ satisfies Condition~(A), then $\tau_\gamma=\tau_d$. We also compare $\tau_\gamma$ with $\psi$-density topologies and establish several topological properties of $\tau_\gamma$, including that countable sets are $\tau_\gamma$-closed and that $(\mathbb{R},\tau_\gamma)$ is nonseparable, nonregular, and nonmetrizable. Finally, we introduce $\gamma$-approximately continuous functions, prove that they form a vector space, and show that the bounded class of such functions is a Banach space under the supremum norm. Maths/math.GN_(General_Topology) https://arxiv.org/abs/2604.13544 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13544v1 Announce Type: cross Abstract: A perforated surface is the complement $\mathring\Sigma:=\Sigma\setminus A$ of a countable dense subset $A$ in a connected paracompact surface $\Sigma$. It is known that the topological type of $\Sigma\setminus A$ is independent of the choice of $A$. Any perforated surface is one-dimensional, connected, locally path connected, and is not semi-locally simply connected at any of its points. In this paper we obtain a classification theorem for perforated surfaces, using the classification theorem for surfaces. We show that any connected covering of a perforated surface $\mathring \Sigma$ arises from a covering of a surface $\Sigma'$ such that $\mathring\Sigma\cong \mathring\Sigma'$. We show that the fundamental group of perforated surfaces are large. We also show that the fundamental groups of $\mathring \Sigma$, the Sierpi\'nski curve and the Menger curve are not Hopfian. Maths/math.GN_(General_Topology) https://arxiv.org/abs/2404.17116 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2404.17116v3 Announce Type: replace-cross Abstract: The notion of ends in an infinite graph $G$ might be modified if we consider them as equivalence classes of infinitely edge-connected rays, rather than equivalence classes of infinitely (vertex-)connected ones. This alternative definition yields the edge-end space $\Omega_E(G)$ of $G$, in which we can endow a natural (edge-)end topology. For every graph $G$, this paper proves that $\Omega_E(G)$ is homeomorphic to $\Omega(H)$ for some possibly another graph $H$, where $\Omega(H)$ denotes its usual end space. However, we also show that the converse statement does not hold: there is a graph $H$ such that $\Omega(H)$ is not homeomorphic to $\Omega_E(G)$ for any other graph $G$. In other words, as a main result, we conclude that the class of topological spaces $\Omega_E = \{\Omega_E(G) : G \text{ graph}\}$ is strictly contained in $\Omega = \{\Omega(H) : H \text{ graph}\}$. Maths/math.GN_(General_Topology) https://arxiv.org/abs/2510.24540 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2510.24540v2 Announce Type: replace-cross Abstract: We consider several questions related to Pontryagin duality in the category of abelian pro-Lie groups. Maths/math.GN_(General_Topology) https://arxiv.org/abs/2604.13898 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13898v1 Announce Type: new Abstract: We prove that if a proper metric space is quasi-isometric to a finitely generated group and to a space with a horoball over a finitely generated group, then that space is quasi-isometric to a rank-one symmetric space or the real line. Maths/math.GR_(Group_Theory) https://arxiv.org/abs/2604.13989 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13989v1 Announce Type: new Abstract: Right-reversing is an algorithm used to compute least common multiples in monoids that admit a right-complemented presentation. The algorithm can either terminate and find a result, fail, or run indefinitely. The correctness of the algorithm can be proved with additional assumptions coming from Garside theory. In the same framework, we prove that a non-terminating run of the algorithm is necessarily cyclic. Stopping when a cycle is detected provides a way of computing a minimal Garside family. Maths/math.GR_(Group_Theory) https://arxiv.org/abs/2604.13809 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13809v1 Announce Type: cross Abstract: We present CubicJordanMatrixAlg, a GAP package for symbolic computation in cubic Jordan matrix algebras and in related Lie-theoretic structures. As an application, we use it to compute certain (commutator) relations in $F_4$-graded groups that were constructed by De Medts and the author from cubic Jordan matrix algebras. Maths/math.GR_(Group_Theory) https://arxiv.org/abs/2604.13953 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13953v1 Announce Type: cross Abstract: In this paper, we exhibit $\textsf{AC}^{3}$ isomorphism tests for coprime extensions $H \ltimes N$ where $H$ is elementary Abelian and $N$ is Abelian; and groups where $\text{Rad}(G) = Z(G)$ is elementary Abelian and $G = \text{Soc}^{*}(G)$. The fact that isomorphism testing for these families is in $\textsf{P}$ was established respectively by Qiao, Sarma, and Tang (STACS 2011), and Grochow and Qiao (CCC 2014, SIAM J. Comput. 2017). The polynomial-time isomorphism tests for both of these families crucially leveraged small (size $O(\log |G|)$) instances of Linear Code Equivalence (Babai, SODA 2011). Here, we combine Luks' group-theoretic method for Graph Isomorphism (FOCS 1980, J. Comput. Syst. Sci. 1982) with the fact that $G$ is given by its multiplication table, to implement the corresponding instances of Linear Code Equivalence in $\textsf{AC}^{3}$. As a byproduct of our work, we show that isomorphism testing of arbitrary central-radical groups is decidable using $\textsf{AC}$ circuits of depth $O(\log^3 n)$ and size $n^{O(\log \log n)}$. This improves upon the previous bound of $n^{O(\log \log n)}$-time due to Grochow and Qiao (ibid.). Maths/math.GR_(Group_Theory) https://arxiv.org/abs/2303.07499 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2303.07499v3 Announce Type: replace Abstract: We construct examples of non-bi-orderable one-relator groups without generalized torsion. This answers a question asked in [2]. Maths/math.GR_(Group_Theory) https://arxiv.org/abs/2408.15614 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2408.15614v2 Announce Type: replace Abstract: We study uniform stability of discrete groups, Lie groups and Lie algebras in the rank metric, and the connections between uniform stability of these objects. We prove that semisimple Lie algebras are far from being flexibly $\mathbb{C}$-stable, and that semisimple Lie groups and lattices in semisimple Lie groups of higher rank are not strictly $\mathbb{C}$-stable. Furthermore, we prove that free groups are not uniformly flexibly $F$-stable over any field $F$. Maths/math.GR_(Group_Theory) https://arxiv.org/abs/2602.19529 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2602.19529v2 Announce Type: replace Abstract: We provide two new characterizations of geometrically infinite actions on Gromov hyperbolic spaces: one in terms of the existence of escaping geodesics, and the other via the presence of uncountably many non-conical limit points. These results extend corresponding theorems of Bonahon, Bishop, and Kapovich--Liu from the settings of Kleinian groups and pinched negatively curved manifolds to discrete groups acting properly on proper Gromov hyperbolic spaces. Maths/math.GR_(Group_Theory) https://arxiv.org/abs/2503.02121 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2503.02121v2 Announce Type: replace-cross Abstract: We axiomatize the theory of the Farey graph and prove that it is $\omega$-stable of Morley rank $\omega$. Maths/math.GR_(Group_Theory) https://arxiv.org/abs/2509.07818 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2509.07818v2 Announce Type: replace-cross Abstract: We give a formula to estimate the number of fixed points of a pseudo-Anosov homeomorphism of a surface. When the homeomorphism satisfies a mild property called strong irreducibility, the log of the number of fixed points is coarsely equal to the Teichmuller translation length. We also discuss several applications, including an inequality relating the hyperbolic volume of a mapping torus to the rank of its Heegaard Floer homology. Maths/math.GR_(Group_Theory) https://arxiv.org/abs/2602.23275 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2602.23275v2 Announce Type: replace-cross Abstract: We introduce a construction that simultaneously yields cusped spaces of relatively hyperbolic groups, and spaces quasi-isometric to Teichmueller metrics. We use this to study Dehn-filling-like quotients of various groups, among which mapping class groups of punctured spheres. In particular, we show that the mapping class group of a five-holed sphere (resp. the braid group on four strands) has infinite hyperbolic quotients (strongly) not isomorphic to hyperbolic quotients of any other given sphere mapping class group (resp. any other braid group). These quotients are obtained by modding out suitable large powers of Dehn twists, and we further argue that the corresponding quotients of the extended mapping class group have trivial outer automorphism groups. We obtain these results by studying torsion elements in the relevant quotients. Maths/math.GR_(Group_Theory) https://arxiv.org/abs/2604.13799 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13799v1 Announce Type: new Abstract: We take a close look at a classical magic trick performed with a string, where a trivial knot is seemingly isotoped into a trefoil, and generalize it to a family of magic tricks for transforming the unknot into other knots. We encode such a trick by depicting the target knot as a special type of knot diagram, which we call a "knotholder diagram". By proving that all knots admit knotholder diagrams, we obtain variants of the trick for producing every knot. Maths/math.GT_(Geometric_Topology) https://arxiv.org/abs/2604.13999 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13999v1 Announce Type: new Abstract: The triple-cup product form $\mu$ is a classical invariant of $3$-manifolds, determining the cohomology ring up to torsion. Given a closed, connected, oriented $3$-manifold $M$, we describe an explicit formula for computing $\mu$ from a Heegaard diagram of $M$. Then, we show that the triple-cup product form $\mu$ can be recovered as a reduction of Turaev's homotopy intersection form $\eta$ of the Heegaard surface. Maths/math.GT_(Geometric_Topology) https://arxiv.org/abs/2505.10629 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2505.10629v3 Announce Type: replace-cross Abstract: The HOMFLY-PT polynomial is a two-parameter knot polynomial that admits a character expansion, expressed as a sum of Schur functions over Young diagrams. The Harer-Zagier (HZ) transform, which converts the HOMFLY--PT polynomial into a rational function, can be applied directly to the characters, yielding hence the HZ character expansion. This illuminates the structure of the HZ functions and articulates conditions for their factorisability, including that non-vanishing contributions should come from hook-shaped Young diagrams. An infinite HZ-factorisable family of hyperbolic knots, that can be thought of as a hyperbolic extension of torus knots, is constructed by full twists, partial full twists and Jucys-Murphy twists, which are braid operations that preserve HZ factorisability. Among them, of interest is a family of pretzel links, which are the Coxeter links for E type Dynkin diagrams. Moreover, when the HZ function is non-factorisable, which occurs for the vast majority of knots and links, we conjecture that it can be decomposed into a sum of factorised terms. In the 3-strand case, this is proven using the symmetries of Young diagrams. Maths/math.GT_(Geometric_Topology) https://arxiv.org/abs/2604.04296 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.04296v2 Announce Type: replace-cross Abstract: We present, in detail and with a modern rigor, the two title proofs. The Weak Jordan Theorem states that the complement of any topological circuit in the plane is disconnected. Maths/math.GT_(Geometric_Topology) https://arxiv.org/abs/2604.13126 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13126v1 Announce Type: new Abstract: Even if it has been less than a decade and a half since Tian introduced his concept of evolution algebras to represent algebraically non-Mendelian rules in Genetics, their study is becoming increasingly widespread mainly due to their applications to many scientific disciplines. In order to facilitate further research on the topic, this paper deals with the past and present research on these kind of algebras, together with the most relevant topics regarding them. Maths/math.HO_(History_and_Overview) https://arxiv.org/abs/2604.13190 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13190v1 Announce Type: new Abstract: This study examines the effectiveness of a structured instructional approach to decomposition and recomposition of large numbers in six primary school classes (three Year 4 and three Year 5, N = 120) using a quasi - experimental design with a control group. The 12 - week intervention is grounded in the Concrete Pictorial Abstract (CPA) progression. The experimental groups achieved average gains of 34.0 points (Year 4) and 29.6 points (Year 5) out of 100, significantly higher than the control groups (16.4 and 11.1 points; p < .001). The Time Group interaction in the mixed ANOVA reached {\eta}^2p = .931. The ANCOVA with the pre - test as covariate estimated an adjusted difference of 18.25 points (F(1,117) = 2,978.10, p < .001, \eta^2p = .962), confirming the robustness of the effect after controlling for baseline differences. Four-week retention exceeded 97% in the experimental group. Internal reliability of the instrument was satisfactory (Cronbach's {\alpha} = .735). Maths/math.HO_(History_and_Overview) https://arxiv.org/abs/2601.18853 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2601.18853v3 Announce Type: replace Abstract: Probabilities is the English translation of the book Probabilit\'es Tome 1 and Tome 2. The mathematic content is authored by Prof. Jean-Yves Ouvrard. The English version has been done by his eldest son Dr. Xavier Ouvrard. In this first version, only the first part is released. Part 1 contains 7 chapters and corresponds to bachelor level. The first part introduces the fundamentals of probability theory across 7 chapters, targeting bachelor level, including event algebras, random variables, independence, conditional probabilities, moments of discrete and continuous random variables, generating functions, and limit theorems. The second part contains 10 chapters and corresponds to master level. Following a brief introduction to measure theory, this part develops more advanced topics: probability measures and their complements, distributions and moments of random variables, modes of convergence, laws of large numbers, conditional expectation, Fourier transforms and characteristic functions, Gaussian random variables, convergence of measures, convergence in distribution, discrete-time stochastic processes, martingales, and Markov chains. The reader's work is greatly facilitated by the inclusion, in every chapter, of numerous exercises, all accompanied by detailed solutions that often provide substantial extensions to the theoretical material. Any feedback is welcome, at probabilities@xerox.mozmail.com The content is released in CC-BY-NC-SA. Maths/math.HO_(History_and_Overview) https://arxiv.org/abs/2604.05056 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.05056v2 Announce Type: replace-cross Abstract: Nested (or reconciled) phylogenetic trees model co-evolutionary systems in which one evolutionary history is embedded within another. We introduce a geometric framework for such systems by defining $\sigma$-space, a moduli space of fully nested ultrametric phylogenetic trees with a fixed leaf map. Generalizing the $\tau$-space of Gavryushkin and Drummond, $\sigma$-space is constructed as a cubical complex parametrised by nested ranked tree topologies and inter-event time coordinates of the combined host and parasite speciation events. We characterise admissible orderings via binary \textit{nesting sequences} and organise them into a natural poset. We show that $\sigma$-space is contractible and satisfies Gromov's cube condition, and is therefore CAT(0). In particular, it admits unique geodesics and well-defined Fr\'echet means. We further describe its geometric structure, including boundary strata corresponding to cospeciation events, and relate it to products of ultrametric tree spaces via natural forgetful maps. Maths/math.MG_(Metric_Geometry) https://arxiv.org/abs/2604.13582 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13582v1 Announce Type: new Abstract: By employing the polar re-formulation, we show that there are no solutions of the Dirac equations in spherical symmetry when the spinor is required to satisfy the same symmetries as the space-time via the Lie derivative. Maths/math-ph_(Mathematical_Physics) https://arxiv.org/abs/2604.13781 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13781v1 Announce Type: new Abstract: We study prethermalization in time-independent quantum many-body systems on a $d$-dimensional lattice with an extensive local Hamiltonian $H=N+\varepsilon P$, in the regime where $\varepsilon \ll 1$. We show that the prethermalization time is exponentially large in $\varepsilon_0/\varepsilon$, where $\varepsilon_0$ is the ratio between an effective spectral gap width and the local norm of $P$. We prove also that for exponentially long times, there exist two quasi-conserved quantities up to exponentially small errors. Maths/math-ph_(Mathematical_Physics) https://arxiv.org/abs/2604.13838 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13838v1 Announce Type: new Abstract: In a previous article we have proved non-existence of certain "solutions" of the cubically nonlinear Schr\"odinger equation in the general case, and presented solutions in the non-generic case. -- In the present article we describe a further family of solutions enlarging the set of non-generic solutions. Maths/math-ph_(Mathematical_Physics) https://arxiv.org/abs/2604.14077 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14077v1 Announce Type: new Abstract: The idea of a $\bigvee$-system was introduced by Veselov in the study of rational solutions of the WDVV equations of associativity. These are algebraic/geometric conditions on the set of covectors that appear in rational solutions to the WDVV equations. Here, this idea is generalized to open WDVV equations, which are an additional set of PDEs originating from open Gromow-Witten Theory. We develop -- for rank-one extensions -- algebraic/geometric conditions on the covectors that supplement the $\bigvee$-system to give rational solutions to the open WDVV equations. Examples, and the relation to superpotentials and to Dubrovin almost-duality, are given. Maths/math-ph_(Mathematical_Physics) https://arxiv.org/abs/2602.06378 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2602.06378v2 Announce Type: cross Abstract: We investigate the relation between the one--dimensional free particle and the harmonic oscillator from a unified viewpoint based on projective geometry, Cayley transformations, and the Schwarzian derivative. Treating time as a projective coordinate on $\mathbb {RP}^1$ clarifies the $SL(2,\mathbb R)\cong Sp(2,\mathbb R)$ conformal sector of the Schr\"odinger--Jacobi symmetry and provides a common framework for two seemingly different correspondences: the Cayley--Niederer (lens) map between the time--dependent Schr\"odinger equations and the conformal bridge transformation relating the stationary problems. We formulate these relations as canonical transformations on the extended phase space and as their metaplectic lifts, identifying the quantum Cayley map with the Bargmann transform. General time reparametrizations induce oscillator--type terms governed universally by the Schwarzian cocycle, connecting the present construction to broader appearances of Schwarzian dynamics. Maths/math-ph_(Mathematical_Physics) https://arxiv.org/abs/2604.11902 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.11902v1 Announce Type: cross Abstract: A simple method is presented for deriving universal formulae for the correlators, frequently denoted $W_{g,n}(\{z_i\}), i=1,..n$, of a wide range of models of physical and mathematical interest. While many alternative methods exist for constructing such correlators, these formulae can be simply written in terms of one defining function and its derivatives. The method has been applied to the Airy and Bessel models, various minimal string and superstring theories, and their associated intersection theory settings, ordinary and various kinds of supersymmetric Weil-Petersson volumes, and more besides. For all these cases, their $W_{g,n}(\{z_i\})$ are just all specializations of the {\it same} universal formulae. A special variant of the method useful for ${N}{=}1$ supersymmetric cases is also presented. It allows for swift derivations of Norbury's three closed-form formulae for the volumes $V_{g,n}$ ($g{=}1,2,3$) of ${ N}{=}1$ supersymmetric Weil-Petersson volumes, and generalizations of them to a wider set of models. Moreover a new closed-form formula for the genus 4 case $V_{4,n}$ is derived. The straightforward method for how to derive such formulae for $g{>}4$ cases is described. Throughout, crucial roles are played by the underlying integrable KdV flows, as well as the Gel'fand-Dikii equation. Maths/math-ph_(Mathematical_Physics) https://arxiv.org/abs/2604.13163 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13163v1 Announce Type: cross Abstract: We study noncommutativity in open strings using the covariant phase space formalism. For tensile open strings in a constant Kalb-Ramond background, we show that the (pre)-symplectic current splits into a bulk kinetic term plus an exact boundary term, recovering the Seiberg-Witten noncommutativity parameter. We then extend the analysis to intrinsically tensionless strings. In the absence of background fields, the reduced phase space is degenerate and carries no intrinsic Poisson structure. In the presence of a constant Kalb-Ramond field, the symplectic current localises entirely on the boundary, so that the physical phase space becomes purely boundary-supported and the endpoint coordinates acquire a noncommutative Poisson algebra. Including a boundary gauge-field coupling similarly leads to a boundary symplectic form governed by the effective Born-Infeld combination on the D-brane. Our results provide a unified description of noncommutativity in both tensile and tensionless open strings. Maths/math-ph_(Mathematical_Physics) https://arxiv.org/abs/2604.13182 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13182v1 Announce Type: cross Abstract: We explore the properties of bilinear products for black-hole quasinormal modes (QNMs) formulated on hyperboloidal foliations. We find that, although QNM solutions are smooth and finite on future-directed hyperboloids, the integrand of the bilinear form with respect to which the modes are orthogonal is still divergent. This is a result of the reflection (equivalently, CPT) transformation required in the definition of the products, which modifies the behaviour of the integrand at the boundaries. We present several regularisation procedures that yield a finite and well-defined bilinear form. In addition, we examine an alternative definition of the bilinear products that incorporates flux contributions, discussing its advantages and limitations. Finally, we define the QNM excitation factors and coefficients within the hyperboloidal framework in terms of the bilinear products, and compute them explicitly for a choice of mode numbers and constant initial data. For concreteness, we work with the QNMs associated to scalar perturbations of the Schwarzschild family of spacetimes. Maths/math-ph_(Mathematical_Physics) https://arxiv.org/abs/2604.13362 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13362v1 Announce Type: cross Abstract: We extend Penrose's quasi-local mass definition to include higher-spin charges associated with the celestial $Lw_{1+\infty}$ symmetries and relate them to traditional definitions of multipoles. The resulting formulae provide explicit expressions that can be computed on finite 2-surfaces, given a choice of null hypersurface. They yield a geometric definition of celestial symmetries and multipoles in generic spacetimes in terms of higher-valence solutions to the twistor equations. This, in turn, gives rise to natural flux-balance laws along the null hypersurface. We also present a first-principles phase-space derivation of these charges, starting from a twistor space action for self-dual gravity that can be identified with the standard gravitational asymptotic phase space at null infinity, performing a large gauge transformation analysis and using the Penrose transform to connect with the corresponding spacetime expressions. Finally, we formulate the spacetime analysis in the Plebanski gauge and relate the celestial symmetries to the integrability of self-dual gravity in the case of a self-dual background. Maths/math-ph_(Mathematical_Physics) https://arxiv.org/abs/2604.13387 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13387v1 Announce Type: cross Abstract: In previous work [AHP24], we proved a finite-time large deviation principle in the Hausdorff metric for multiradial Schramm-Loewner evolution, SLE$(\kappa)$, as $\kappa \to 0$, with good rate function being the multiradial Loewner energy. Here, we extend this result to infinite time in the topology of common-capacity-parameterized curves, and streamline the proof. A main step is to derive detailed escape probability estimates for multiradial SLE$(\kappa)$ curves in the common parameterization, which extend the single-curve estimates achieved in [AP26]. As a by-product, we also get that multiradial SLE$(\kappa)$ curves, with $\kappa \leq 8/3$, are transient at their common terminal point, generalizing [FL15, HL21]. As a corollary to the LDP result, we obtain explicit asymptotics of the Brownian loop measure interaction term for finite-energy radial multichords, which is linear in the capacity-time and coincides with a certain choice of a cocycle for the Virasoro algebra. Maths/math-ph_(Mathematical_Physics) https://arxiv.org/abs/2604.13415 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13415v1 Announce Type: cross Abstract: Membrane wrapping underlies nanoparticle uptake during endocytosis, whereas the reverse process of membrane unwrapping accompanies particle expulsion and membrane fusion events. Existing theoretical descriptions typically focus on adhesion and bending energies within the particle-membrane contact region and often neglect the deformation energy of the membrane outside the contact zone. This approximation is valid only in the limit of vanishing membrane tension, where the non-contact membrane assumes a catenoid-like configuration with negligible bending energy. However, at finite tension the deformation of the non-contact membrane becomes a dominant energetic contribution. Here we show that this tension-dependent non-contact energy governs the progression of particle wrapping. By analyzing the variation of the total membrane energy with wrapping degree, we uncover a competition between particle adhesion, membrane tension and particle size that determines whether wrapping proceeds, stalls, or reverses into spontaneous unwrapping. This framework reveals a stalling boundary separating regimes of particle uptake and expulsion. To capture the non-contact deformation efficiently, we derive a compact analytical approximation that accurately reproduces the full numerical solution of the membrane shape. The resulting energetic map provides a unified physical description of particle wrapping and unwrapping, with implications for endocytosis, membrane fusion, and nanoparticle design. Maths/math-ph_(Mathematical_Physics) https://arxiv.org/abs/2604.13599 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13599v1 Announce Type: cross Abstract: We establish an observability inequality from space-time measurable sets for a class of strongly coupled parabolic systems consisting of two equations, where the observation acts on a single-component. The model is motivated by parabolic equations with complex coefficients and serves as a prototypical example of strongly coupled systems. The main difficulty lies in the fact that, unlike in the scalar and weakly coupled cases, pointwise-in-time interpolation observability estimates fail, as the observed component may exhibit high-frequency oscillatory cancellations induced by the coupling. To overcome this difficulty, we develop a new integral-type interpolation observability inequality based on a Remez-type inequality. With the aid of this integral-type interpolation observability inequality and the strategy developed in [Phung and Wang, JEMS, (2013), 681--703] and [Apraiz, Escauriaza,Wang and Zhang, JEMS, (2014), 2433--2475] for deriving observability from measurable sets, we obtain the desired observability inequality. Maths/math-ph_(Mathematical_Physics) https://arxiv.org/abs/2604.13639 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13639v1 Announce Type: cross Abstract: Heat transport exhibits distinct regimes ranging from ballistic propagation to diffusive relaxation, traditionally described by disparate theoretical frameworks. Here, we introduce a unified first-order operator formulation in which temperature and heat flux are treated as a coupled state vector, yielding a minimal dynamical closure of heat transport. The resulting generator is intrinsically non-Hermitian and gives rise to a spectral structure governed by an exceptional point that separates overdamped diffusion from underdamped wave-like propagation. In this framework, Fourier law emerges as a singular limit of a hyperbolic dissipative system, while the Cattaneo equation arises naturally as the minimal hydrodynamic closure of kinetic theory. We show that the exceptional point induces nonanalytic spectral transitions, nonmodal transient dynamics, and a breakdown of conventional modal decomposition. The theory further generalizes to anisotropic media, where direction-dependent exceptional surfaces enable intrinsic steering of heat flow. Our results establish a unified non-Hermitian dynamical framework for heat transport and reveal exceptional-point physics as a fundamental organizing principle underlying thermal dynamics across scales. Maths/math-ph_(Mathematical_Physics) https://arxiv.org/abs/2604.13852 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13852v1 Announce Type: cross Abstract: We derive the asymptotic first passage time (FPT) distribution for space-dependent variable-order time-fractional diffusion, where the fractional exponent $\alpha(x)$ varies with position. For any sufficiently smooth $\alpha(x)$ on a finite domain with absorbing and reflecting boundaries, we show that the survival probability decays as $\Psi(t)\sim C\,t^{-\alpha_*}/(\ln t)^{\nu}$, where $\alpha_*$ is the minimum value of the fractional exponent and $\nu$ is determined by the location and shape of the minimum. For a constant fractional exponent $\nu=0$ and this provides a theoretical prediction that can identify spatially heterogeneous anomalous transport in experiments. We validate the theory against exact Laplace-space solutions and Monte Carlo simulations for linear and nonlinear profiles of $\alpha(x)$. Maths/math-ph_(Mathematical_Physics) https://arxiv.org/abs/2604.13894 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13894v1 Announce Type: cross Abstract: This paper introduces the concept of the wandering range of a robust symmetry $S$ of a Hamiltonian $H$. This quantity measures how the perturbed time evolution $\mathrm{e}^{\mathrm{i}t(H+\varepsilon V)} S \mathrm{e}^{-\mathrm{i} t(H+\varepsilon V)}$ deviates from its unperturbed counterpart $\mathrm{e}^{\mathrm{i} tH} S\mathrm{e}^{-\mathrm{i} tH} = S$. Although the wandering range does not necessarily scale linearly with the perturbation strength $\varepsilon$, we identify conditions under which this linear behavior is recovered and we obtain explicit nonperturbative bounds. Maths/math-ph_(Mathematical_Physics) https://arxiv.org/abs/2604.14122 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14122v1 Announce Type: cross Abstract: We consider critical site percolation ($p=p_c=1/2$) on the triangular lattice $\mathbf{T}$ in two dimensions. We show that the simple random walk on the clusters of open vertices converges in the scaling limit to a continuous diffusion which lives in the gasket of a conformal loop ensemble with parameter $\kappa = 6$ $\big(\mathrm{CLE}_6\big)$, the so-called $\mathrm{CLE}_6$ Brownian motion. We also show that the intrinsic (i.e., chemical distance) metric converges in the scaling limit to the geodesic $\mathrm{CLE}_6$ metric. As a consequence, we deduce the existence of the chemical distance exponent, the resistance exponent, and the spectral dimension of the critical percolation clusters. Moreover, we show that the exponents satisfy the Einstein relations. Maths/math-ph_(Mathematical_Physics) https://arxiv.org/abs/2507.04988 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2507.04988v3 Announce Type: replace Abstract: We consider the discrete Schr\"odinger operator $H = -\Delta + V$ on $\ell^2(\mathbb{Z}^d)$ with a decaying potential, in arbitrary lattice dimension $d\in\mathbb{N}^*$, where $\Delta$ is the standard discrete Laplacian and $V_n = o(|n|^{-1})$ as $|n| \to \infty$. We prove the absence of singular continuous spectrum for $H$. For the unitary evolution $e^{-i tH}$, we prove that it exhibits ballistic transport in the sense that, for any $r > 0$, the weighted $\ell^2-$norm $$\|e^{-i tH}u\|_r:=\left(\sum_{n\in\mathbb{Z}^d} (1+|n|^2)^{r} |(e^{-i tH}u)_n|^2\right)^\frac12 $$ grows at rate $\simeq t^r$ as $t\to \infty$, provided that the initial state $u$ is in the absolutely continuous subspace and satisfies $\|u\|_r<\infty$. The proof relies on commutator methods and a refined Mourre estimate, which yields quantitative lower bounds on transport for operators with purely absolutely continuous spectrum over appropriate spectral intervals. Compactness arguments and localized spectral projections are used to extend the result to perturbed operators, extending the classical result for the free Laplacian to a broader class of decaying potentials. Maths/math-ph_(Mathematical_Physics) https://arxiv.org/abs/2511.04509 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2511.04509v2 Announce Type: replace Abstract: We have constructed the mean-field trivial solution of the $\varphi^4$ theory $O(N)$ model in four dimensions in two previous papers using the flow equations of the renormalization group. Here we establish a relation between the trivial solutions we constructed and perturbation theory. We show that if an UV-cutoff is maintained, we can define a renormalized coupling constant $g$ and obtain the perturbative solutions of the mean-field flow equations at each order in perturbation theory. We prove the local Borel-summability of the renormalized mean-field perturbation theory in the presence of an UV cutoff and show that it is asymptotic to the non-perturbative solution. Maths/math-ph_(Mathematical_Physics) https://arxiv.org/abs/2603.09535 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2603.09535v2 Announce Type: replace Abstract: We identify a class of left-invariant pseudo-Riemannian metrics on Lie groups for which the Laplace-Beltrami equation reduces to a first-order PDE and admits exact solutions. The defining condition is the existence of a commutative ideal $\mathfrak{h}$ in the Lie algebra $\mathfrak{g}$ whose orthogonal complement satisfies $\mathfrak{h}^\perp\subseteq\mathfrak{h}$. Using the noncommutative integration method based on the orbit method and generalized Fourier transforms, we reduce the Laplace--Beltrami equation to a first-order linear PDE, which can then be integrated explicitly. The symmetry of the reduced equation gives rise, via the inverse transform, to nonlocal symmetry operators for the original equation. These operators are generically integro-differential, contrasting with the polynomial symmetries appearing in previously studied classes. The method is illustrated by two examples: the Heisenberg group $\mathrm{H}_3(\mathbb{R})$ with a Lorentzian metric and a four-dimensional non-unimodular group with a metric of signature $(2,2)$. In the latter, classical separation of variables is not directly applicable, yet the noncommutative approach yields explicit solutions and reveals the predicted nonlocal symmetry. Maths/math-ph_(Mathematical_Physics) https://arxiv.org/abs/2604.12111 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.12111v2 Announce Type: replace Abstract: By an idealized quantum mechanical model, we formally describe the dispersion of nonretarded electromagnetic waves that express charge density oscillations near a fixed plane in three spatial dimensions (3D) at zero temperature. Our goal is to capture the interplay of microscopic scales that include a confinement length in the emergence of the surface plasmon, a collective low-energy charge excitation in the vicinity of the plane. We start with a time-dependent Hartree-type equation in 3D. This model accounts for particle binding to the plane and the repulsive Coulomb interaction associated with the induced charge density relative to the ground state. By linearizing the equation of motion, we formulate a homogeneous integral equation for the scattering amplitude of the particle wave function in the (z-) coordinate vertical to the plane. For a binding potential proportional to a negative delta function and symmetric-in-z wave function, we apply the Laplace transform with respect to positive z and convert the integral equation into a functional equation that involves several values of the transformed solution. The scattering amplitude and dispersion relation are derived exactly in terms of rapidly convergent series via the Mittag-Leffler theorem. In the semiclassical regime, our result furnishes an asymptotic expansion for the energy excitation spectrum. The leading-order term is found in agreement with the prediction of a classical hydrodynamic model based on a projected-Euler-Poisson system. Maths/math-ph_(Mathematical_Physics) https://arxiv.org/abs/2604.13227 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13227v1 Announce Type: new Abstract: This work proposes a hybrid method (ULR) which integrates a rotation-equivariance-aware neural network and a low-rank structure to solve the two dimensional inverse medium scattering problem. The neural network is to model the data corrector which maps the full data to the Born data, and the low-rank structure is to design an inverse Born solver that finds a regularized solution from the perturbed Born data. The proposed rotation-equivariance-aware neural network naturally incorporates the reciprocity relation and the rotation-equivariance in inverse scattering, while the low-rank structure effectively filters high-frequency noise in the output of the neural network and leads to a regularized method supported by theoretical stability in the Born region. For a comparative study, we replace the low-rank inverse Born solver by another rotation-equvariance-aware neural network to propose a two-step neural network (UU). Furthermore, we extend the proposed methods (ULR and UU) to tackle the more challenging case with only limited aperture data. A variety of numerical experiments are conducted to compare the proposed ULR, UU, and a black-box neural network. Maths/math.NA_(Numerical_Analysis) https://arxiv.org/abs/2604.13506 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13506v1 Announce Type: new Abstract: We study an inverse drift problem for a two-dimensional parabolic equation on the unit square with mixed boundary conditions, where the drift coefficient is recovered from terminal observation data $g=u(\cdot,T)$. A monotone operator is constructed whose fixed point coincides with the unknown drift, yielding uniqueness in an admissible class and a constructive iterative reconstruction scheme. Numerical experiments illustrate the monotone convergence and the effectiveness of the proposed method, and show that it remains effective for noisy terminal data under the denoising strategy. Maths/math.NA_(Numerical_Analysis) https://arxiv.org/abs/2604.13754 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13754v1 Announce Type: new Abstract: The paper is concerned with three types of cubic splines over a triangulation that are characterized by three degrees of freedom associated with each vertex of the triangulation. The splines differ in computational complexity, polynomial reproduction properties, and smoothness. With the aim to make them a versatile tool for numerical analysis, a unified representation in terms of locally supported basis functions is established. The construction of these functions is based on geometric concepts and is expressed in the Bernstein--B\'ezier form. They are readily applicable in a range of standard approximation methods, which is demonstrated by a number of numerical experiments. Maths/math.NA_(Numerical_Analysis) https://arxiv.org/abs/2604.13785 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13785v1 Announce Type: new Abstract: The paper deals with the numerical treatment of index-1 stochastic differential-algebraic equations (SDAEs) with nonlinear coefficients that satisfy the local Lipschitz and the Khasminskii conditions. The key challenge here is the presence of a singular and non-autonomous matrix in the equation, which makes the numerical method challenging to analyze. To tackle this challenge, we develop a more general numerical method using a local linearization technique. More precisely, we use the Taylor expansion to decompose locally the drift component of the SDAEs in linear and nonlinear parts. The linear part is approximated implicitly and must resolve the singularity issue of each time step, while the nonlinear part is approximated explicitly. This method is fascinating due to the fact that it is efficient in high dimension. We prove that this novel numerical method converges in the pathwise sense with rate $\frac{1}{2}-\epsilon$, for arbitrary $\epsilon >0$. The implementation of this novel numerical method is also carried out to verify our theoretical result. Maths/math.NA_(Numerical_Analysis) https://arxiv.org/abs/2604.13830 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13830v1 Announce Type: new Abstract: Integro-differential equations arise in a wide range of applications, including transport, kinetic theory, radiative transfer, and multiphysics modeling, where nonlocal integral operators couple the solution across phase space. Such nonlocality often introduces dense coupling blocks in deterministic discretizations, leading to increased computational cost and memory usage, while physics-informed neural networks may suffer from expensive nonconvex training and sensitivity to hyperparameter choices. In this work, we present randomized neural networks (RaNNs) as a mesh-free collocation framework for linear integro-differential equations. Because the RaNN approximation is intrinsically dense through globally supported random features, the nonlocal integral operator does not introduce an additional loss of sparsity, while the approximate solution can still be represented with relatively few trainable degrees of freedom. By randomly fixing the hidden-layer parameters and solving only for the linear output weights, the training procedure reduces to a convex least-squares problem in the output coefficients, enabling stable and efficient optimization. As a representative application, we apply the proposed framework to the steady neutron transport equation, a high-dimensional linear integro-differential model featuring scattering integrals and diverse boundary conditions. Extensive numerical experiments demonstrate that, in the reported test settings, the RaNN approach achieves competitive accuracy while incurring substantially lower training cost than the selected neural and deterministic baselines, highlighting RaNNs as a robust and efficient alternative for the numerical simulation of nonlocal linear operators. Maths/math.NA_(Numerical_Analysis) https://arxiv.org/abs/2604.13921 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13921v1 Announce Type: new Abstract: We present a higher-order extension of the dual cell method for the time-domain Maxwell equations in three spatial dimensions. The approach builds upon a variational reinterpretation of the Finite Integration Technique on dual meshes and generalises a previously developed two-dimensional high-order formulation. The electric and magnetic fields are discretised on mutually dual barycentric grids using curl-conforming polynomial spaces constructed via tensor-product Gauss--Radau interpolation. The resulting semi-discrete formulation yields block-diagonal mass matrices and sparse discrete curl operators, enabling explicit time integration while preserving a discrete energy identity. Special attention is devoted to the construction of compatible approximation spaces on the three-dimensional primal and dual meshes, the reference-to-physical element mappings, and the preservation of tangential continuity. We show that the method achieves arbitrary-order convergence, avoids spurious modes, and maintains optimal sparsity properties. Numerical experiments confirm spectral correctness, high-order accuracy, and computational efficiency on unstructured tetrahedral meshes. Maths/math.NA_(Numerical_Analysis) https://arxiv.org/abs/2604.13982 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13982v1 Announce Type: new Abstract: We consider the simplicial de Rham complex and the \v{C}ech-de Rham complex, two bigraded Hilbert complexes whose Hodge-Laplace problems govern spatially coupled problems in mixed dimension and homogeneous dimension, respectively. The former complex can be realized as a subcomplex of the latter. In this paper, we quantify how close these complexes are to each other by constructing bounded cochain complexes between them, and thus we quantify how close a mixed-dimensional formulation of a problem is to an equidimensionally coupled formulation of the same problem. From this construction, we derive a priori- and a posteriori error estimates between the associated Hodge-Laplace problems on the two complexes. These estimates represent the error which is introduced by treating a spatially coupled problem as mixed-dimensional, rather than an equidimensional problem with thin overlaps. Maths/math.NA_(Numerical_Analysis) https://arxiv.org/abs/2604.14050 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14050v1 Announce Type: new Abstract: The long-standing hypothesis formulated by Goreinov, Tyrtyshnikov and Zamarashkin \cite{GTZ1997} has been recently solved by Sengupta and Pautov \cite{SP2026} in the case of two-column matrices. In this paper, we complement their elegant proof with the equality criteria. Maths/math.NA_(Numerical_Analysis) https://arxiv.org/abs/2604.14107 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14107v1 Announce Type: new Abstract: Simulating infiltration in porous media using Richards' equation remains computationally challenging due to its parabolic structure and nonlinear coefficients. While a wide range of numerical methods for differential equations have been applied over the past several decades, basic higher-order numerical methods often fail to preserve physical bounds on water pressure and saturation, leading to spurious oscillations and poor iterative solver convergence. Instead, low-order, bound-preserving methods have been preferred. The combination of mass lumping and relative permeability upwinding preserves bounds but degrades accuracy to first order in space. Flux-corrected transport is a high-resolution numerical technique designed for combining the bound-preserving property of low-order schemes with the accuracy of high-order methods, by blending the two methods through limited anti-diffusive fluxes. In this work, we extend flux-corrected transport schemes to the nonlinear, degenerate parabolic structure of Richards' equation, verify attainment of second-order convergence on unstructured meshes, and demonstrate applications to stormwater management infrastructure. Maths/math.NA_(Numerical_Analysis) https://arxiv.org/abs/2604.13206 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13206v1 Announce Type: cross Abstract: As Large Language Models (LLMs) are increasingly integrated into agentic workflows, their unpredictability stemming from numerical instability has emerged as a critical reliability issue. While recent studies have demonstrated the significant downstream effects of these instabilities, the root causes and underlying mechanisms remain poorly understood. In this paper, we present a rigorous analysis of how unpredictability is rooted in the finite numerical precision of floating-point representations, tracking how rounding errors propagate, amplify, or dissipate through Transformer computation layers. Specifically, we identify a chaotic "avalanche effect" in the early layers, where minor perturbations trigger binary outcomes: either rapid amplification or complete attenuation. Beyond specific error instances, we demonstrate that LLMs exhibit universal, scale-dependent chaotic behaviors characterized by three distinct regimes: 1) a stable regime, where perturbations fall below an input-dependent threshold and vanish, resulting in constant outputs; 2) a chaotic regime, where rounding errors dominate and drive output divergence; and 3) a signal-dominated regime, where true input variations override numerical noise. We validate these findings extensively across multiple datasets and model architectures. Maths/math.NA_(Numerical_Analysis) https://arxiv.org/abs/2604.13433 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13433v1 Announce Type: cross Abstract: We propose a new sparse matrix format, PackSELL, designed to support diverse data representations and enable efficient sparse matrix-vector multiplication (SpMV) on GPUs. Building on sliced ELLPACK (SELL), PackSELL incorporates delta encoding of column indices and a novel packing scheme that stores each index-delta-value pair in a single word, thereby reducing memory footprint and data movement. This design further enables fine-grained control over the bit allocation between deltas and values, allowing flexible data representations, including non-IEEE formats. Experimental results show that, when configured for half precision (FP16), the PackSELL-based SpMV kernel outperforms the cuSPARSE SELL-based kernel by up to $1.63\times$. Moreover, with configurations using customized formats, PackSELL achieves FP32-level accuracy while exceeding the performance of FP16 cuSPARSE. These benefits extend to sparse linear solvers; for example, a mixed-precision preconditioned conjugate gradient (PCG) solver using PackSELL achieves up to a $2.09\times$ speedup over the standard full-precision PCG. Maths/math.NA_(Numerical_Analysis) https://arxiv.org/abs/2604.13678 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13678v1 Announce Type: cross Abstract: This paper explores the problem of generalized phase retrieval, which involves reconstructing a length-$n$ signal $\bm{x}$ from its $m$ phaseless samples $y_k = \left|\langle \bm{a}_k,\bm{x}\rangle\right|^2$, where $k = 1,2,...,m$, and $\bm{a}_k$ are the measurement vectors. This problem can be reformulated into recovering a positive semidefinite rank-$1$ matrix $\bm{X}=\bm{x}\bm{x}^*$ from linear samples $\bm{y}=\mathcal{A}(\bm{X})\in\mathbb{R}^m$, thereby requiring us to find a rank-$1$ solution of the linear equations. We demonstrate that several existing phase retrieval algorithms, including Wirtinger Flow (WF) and the canonical Riemannian gradient descent (RGD), actually solve the least-squares fitting of this linear equation on the Riemannian manifold of rank-$1$ matrices, but utilize different metrics on this manifold. Nevertheless, these metrics only allow for a stable and far-apart-from-isometric embedding of rank-$1$ matrices to $\mathbb{R}^m$ by $\mathcal{A}$, resulting in a linear convergence with a considerably large convergence factor. To expedite the convergence, we establish a new metric on the rank-$1$ matrix manifold that facilitates the nearly isometric embedding of rank-$1$ matrices into $\mathbb{R}^m$ through $\mathcal{A}$. A RGD algorithm under this new metric, termed Weighted RGD (WRGD), is proposed to tackle the phase retrieval problem. Owing to the near isometry, we prove that our WRGD algorithm, initialized by spectral methods, can linearly converge to the underlying signal $\bm{x}$ with a small convergence factor. Empirical experiments strongly validate the efficiency and resilience of our algorithms compared to the truncated Wirtinger Flow (TWF) algorithm and the canonical RGD algorithm. Maths/math.NA_(Numerical_Analysis) https://arxiv.org/abs/2604.13988 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13988v1 Announce Type: cross Abstract: Objective: Investigate whether hypnogram 'realism' can be used to guide an unsupervised method for handling arbitrary types of signal degradation in mobile sleep monitoring. Approach: Combining a pretrained, state-of-the-art 'u-sleep' model with a 'discriminator' network, we align features from a target domain with a feature space learned during pretraining. To test the approach, we distort the source domain with realistic signal degradations, to see how well the method can adapt to different types of degradation. We compare the performance of the resulting model with best-case models designed in a supervised manner for each type of transfer. Main Results: Depending on the type of distortion, we find that the unsupervised approach can increase Cohen's kappa with as little as 0.03 and up to 0.29, and that for all transfers, the method does not decrease performance. However, the approach never quite reaches the estimated theoretical optimal performance, and when tested on a real-life domain mismatch between two sleep studies, the benefit was insignificant. Significance: 'Discriminator-guided fine tuning' is an interesting approach to handling signal degradation for 'in the wild' sleep monitoring, with some promise. In particular, what it says about sleep data in general is interesting. However, more development will be necessary before using it 'in production'. Maths/math.NA_(Numerical_Analysis) https://arxiv.org/abs/2405.20836 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2405.20836v3 Announce Type: replace Abstract: Solving time-dependent Partial Differential Equations (PDEs) is one of the most critical problems in computational science. While Physics-Informed Neural Networks (PINNs) offer a promising framework for approximating PDE solutions, their accuracy and training speed are limited by two core barriers: gradient-descent-based iterative optimization over complex loss landscapes and non-causal treatment of time as an extra spatial dimension. We present Frozen-PINN, a novel PINN based on the principle of space-time separation that leverages random features instead of training with gradient descent, and incorporates temporal causality by construction. On eight PDE benchmarks, including challenges such as extreme advection speeds, shocks, and high dimensionality, Frozen-PINNs achieve superior training efficiency and accuracy over state-of-the-art PINNs, often by several orders of magnitude. Our work addresses longstanding training and accuracy bottlenecks of PINNs, delivering quickly trainable, highly accurate, and inherently causal PDE solvers, a combination that prior methods could not realize. Our approach challenges the reliance of PINNs on stochastic gradient-descent-based methods and specialized hardware, leading to a paradigm shift in PINN training and providing a challenging benchmark for the community. Maths/math.NA_(Numerical_Analysis) https://arxiv.org/abs/2501.08247 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2501.08247v3 Announce Type: replace Abstract: Diffusion through tubular networks with variable radius arises in a wide range of biological, engineering, and physical applications. The Fick-Jacobs equation is the standard one-dimensional reduction of this problem, briefly derived nearly a century ago in a classical textbook, but was shown to be unstable and inaccurate when the radial gradient is large by Zwanzig in 1992. Three decades of subsequent modifications have failed to resolve this instability because they all inherit a common structural inconsistency introduced by truncation in the original derivation - one that becomes immediately apparent from novel elementary analysis. In this work, we return to the foundations of the Fick-Jacobs derivation and treat it as a locally defined Taylor expansion, recovering a model with geometry-independent error that contrasts directly with the geometry-dependent instability of past corrections. The result is a new geometry-aware expansion of the Fick-Jacobs model, with a numerical discretization that is provably stable and convergent, and the first method known to the authors to converge spatially to the correct geometry-aware solution. Analysis shows that standard corrections from the literature cannot converge to this solution regardless of spatial refinement. We derive efficient numerical schemes for branched networks at equivalent computational cost, and demonstrate that a geometry-aware one-dimensional reduction can faithfully reproduce full three-dimensional results of a neurobiologically relevant problem that the standard reduction cannot achieve. Maths/math.NA_(Numerical_Analysis) https://arxiv.org/abs/2505.03074 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2505.03074v2 Announce Type: replace Abstract: We develop and analyze layer potential methods to represent harmonic functions on finitely-connected tori (i.e., doubly-periodic harmonic functions). The layer potentials are expressed in terms of a doubly-periodic and non-harmonic Green's function that can be explicitly written in terms of the Jacobi theta function or a modified Weierstrass sigma function. Extending results for finitely-connected Euclidean domains, we prove that the single- and double-layer potential operators are compact linear operators and derive the relevant limiting properties at the boundary. We show that when the boundary has more than one connected component, the Fredholm operator of the second kind associated with the double-layer potential operator has a non-trivial null space, which can be explicitly constructed. Finally, we apply our developed theory to obtain solutions to the Dirichlet and Neumann boundary value problems, as well as the Steklov eigenvalue problem. We present numerical results using Nystr\"om discretizations and find approximate solutions to these problems in several numerical examples. Our method avoids a lattice sum of the free-space Green's function, is shown to be spectrally convergent, and exhibits a faster convergence rate than the method of particular solutions for problems on tori with irregularly shaped holes. Maths/math.NA_(Numerical_Analysis) https://arxiv.org/abs/2509.21280 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2509.21280v2 Announce Type: replace Abstract: We propose a reduced-order modeling approach for nonlinear, parameter-dependent ordinary differential equations (ODE). Dimensionality reduction is achieved using nonlinear maps represented by autoencoders. The resulting low-dimensional ODE is then solved using standard integration in time schemes, and the high-dimensional solution is reconstructed from the low-dimensional one. We investigate the architecture of neural networks for constructing effective autoencoders that hold necessary properties to reconstruct the input manifold with exact representation capabilities. We study the convergence of the reduced-order model to the high-fidelity one. Numerical experiments show the robustness and accuracy of our approach in different scenarios, highlighting its effectiveness in highly complex and nonlinear settings without sacrificing accuracy. Moreover, we examine how the reduction influences the stability properties of the reconstructed high-dimensional solution. Maths/math.NA_(Numerical_Analysis) https://arxiv.org/abs/2510.11949 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2510.11949v2 Announce Type: replace Abstract: Exact reconstruction of an image from measurements of its Discrete Fourier Transform (DFT) typically requires all DFT coefficients to be available. However, incorporating the prior assumption that the image contains only integer values enables unique recovery from a limited subset of DFT coefficients. This paper develops both theoretical and algorithmic foundations for this problem. We use algebraic properties of the DFT to define a reduction from two-dimensional recovery to several well-chosen one-dimensional recoveries. Our reduction framework characterizes the minimum number and location of DFT coefficients that must be sampled to guarantee unique reconstruction of an integer-valued image. Algorithmically, we develop reconstruction procedures which use dynamic programming to efficiently recover an integer signal or image from its minimal set of DFT measurements. While the new inversion algorithms still involve NP-hard subproblems, we demonstrate how the divide-and-conquer approach drastically reduces the associated search space. To solve the NP-hard subproblems, we employ a lattice-based framework which leverages the LLL approximation algorithm to make the algorithms fast and practical. Maths/math.NA_(Numerical_Analysis) https://arxiv.org/abs/2601.20119 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2601.20119v2 Announce Type: replace Abstract: Strength-of-connection algorithms play a key role in algebraic multigrid (AMG). Specifically, they determine which matrix nonzeros are classified as weak and so ignored when coarsening matrix graphs and defining interpolation sparsity patterns. The general goal is to encourage coarsening only in directions where error can be smoothed and to avoid coarsening across sharp problem variations. Unfortunately, developing robust and inexpensive strength-of-connection schemes is challenging. The classification of matrix nonzeros involves four aspects: (a) choosing a strength-of-connection matrix, (b) scaling its values, (c) choosing a criterion to classify scaled values as strong or weak, and (d) dropping weak entries which includes adjusting matrix values to account for dropped terms. Typically, smoothed aggregation AMG uses the linear system being solved as a strength-of-connection matrix. It scales values symmetrically using square-roots of the matrix diagonal. It classifies based on whether scaled values are above or below a threshold. Finally, it adjusts matrix values by modifying the diagonal so that the sum of entries within each row of the dropped matrix matches that of the original. While these procedures can work well, we illustrate failure cases that motivate alternatives. The first alternative uses a distance Laplacian strength-of-connection matrix. The second centers on non-symmetric scaling. We then investigate alternative classification criteria based on identifying gaps in the values of the scaled entries. Finally, an alternative lumping procedure is proposed where row sums are preserved by modifying all retained matrix entries (as opposed to just diagonal entries). A series of numerical results illustrates trade-offs demonstrating in some cases notably more robust convergence on matrices coming from linear finite elements on stretched meshes. Maths/math.NA_(Numerical_Analysis) https://arxiv.org/abs/2604.13854 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13854v1 Announce Type: new Abstract: This paper provides a new proof of the $p$-adic Gross--Zagier formula for the $p$-adic $L$-function associated with the base change of a normalised cuspidal eigen-newform $f$ of weight $k \geq 2$ (and families of such) to an imaginary quadratic field $K$. Our results encompass both the classical $p$-ordinary cases and non-ordinary scenarios, including new cases where $k > 2$ and $\mathrm{ord}_p(a_p(f)) > 0$. Unlike the traditional approach of comparing geometric and analytic kernels, we employ a ``wall-crossing'' strategy centred on the BDP formula and the theory of Beilinson--Flach elements. Maths/math.NT_(Number_Theory) https://arxiv.org/abs/2604.13868 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13868v1 Announce Type: new Abstract: Let $\psi\colon \mathbb{N} \to [0,\frac{1}{2})$ be a positive function, and let $\theta\colon \mathbb{N}\to \mathbb{R}$ be a real-valued function. Let $\{A_q\}$ and $\{B_q\}$ be sequences of integers such that $\gcd(A_q, B_q)=1$ and $B_q > 0$ for all $q$. Define $W^{\ast}(\psi,\theta)$ to be the set of $x \in [0,1]$ for which \[ \left| x - \frac{p+\theta(q)}{q} \right|<\frac{\psi(q)}{q} \text{ for infinitely many }(p,q)\in \mathbb{Z}\times \mathbb{N} \text{ with } \gcd(B_qp+A_q,q)=1. \] In this paper, we determine the Fourier dimension of \( W^*(\psi, \theta) \). Our result not only recovers the classical theorems of Kaufman and Bluhm (concerning the homogeneous case \( \psi(q) = q^{-\tau} \), \( \tau \ge 1 \)) but also furnishes a complete inhomogeneous generalization. In addition, it affirmatively resolves the coprime formulation of the Chen and Xiong conjecture. Maths/math.NT_(Number_Theory) https://arxiv.org/abs/2604.14036 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14036v1 Announce Type: new Abstract: We study the distribution modulo one of linear recurrent sequences of real numbers. We prove criteria for the finiteness of the set of limit values of the fractional parts of such a sequence and give lower bounds for the maximal distance between two limit values. Our results generalize theorems of Flatto, Lagarias, Pollington, and Dubickas. Maths/math.NT_(Number_Theory) https://arxiv.org/abs/1909.01764 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:1909.01764v5 Announce Type: replace Abstract: We study the behavior of the Iwasawa invariants of the Iwasawa modules which appear in Kato's main conjecture without $p$-adic $L$-functions under congruences. It generalizes the work of Greenberg-Vatsal, Emerton-Pollack-Weston, B.D. Kim, Greenberg-Iovita-Pollack, and one of us simultaneously. As a consequence, we establish the propagation of Kato's main conjecture for modular forms of higher weight at arbitrary good prime under the assumption on the mod $p$ non-vanishing of Kato's zeta elements. The application to the $\pm$ and $\sharp/\flat$-Iwasawa theory for modular forms is also discussed. Maths/math.NT_(Number_Theory) https://arxiv.org/abs/2310.15628 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2310.15628v2 Announce Type: replace Abstract: We pursue the question how integers can be ordered or partitioned according to their divisibility properties. Based on pseudometrics on $\mathbb{Z}$, we investigate induced preorders, associated equivalence relations, and quotient sets. The focus is on metrics or pseudometrics on $\mathbb{D}_n$, the set of divisors of a given modulus $n\in\mathbb{N}$, that can be extended to pseudometrics on $\mathbb{Z}$. Arithmetic functions can be used to generate such pseudometrics. We discuss several subsets of additive and multiplicative arithmetic functions and various combinations of their function values leading to binary metric functions that represent different divisibility properties of integers. We conclude this paper with numerous examples and review the most important results. As an additional result, we derive a necessary condition for the truth of the odd k-perfect number conjecture. Maths/math.NT_(Number_Theory) https://arxiv.org/abs/2408.01643 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2408.01643v3 Announce Type: replace Abstract: We consider a variant of the strong multiplicity one theorem. Let $\pi_{1}$ and $\pi_{2}$ be two unitary cuspidal automorphic representations for $\mathrm{GL(2)}$ that are not twist-equivalent. We find a lower bound for the lower Dirichlet density of the set of places for which $\left\lvert a_{v}(\pi_{1}) \right\rvert > \left\lvert a_{v}(\pi_{2}) \right\rvert$, where $a_{v}(\pi_{i})$ is the trace of Langlands conjugacy class of $\pi_{i}$ at $v$. One consequence of this result is an improvement on the existing bound on the lower Dirichlet density of the set of places for which $\left\lvert a_{v}(\pi_{1})\right\rvert \neq \left\lvert a_{v}(\pi_{2}) \right\rvert$. Maths/math.NT_(Number_Theory) https://arxiv.org/abs/2504.09411 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2504.09411v2 Announce Type: replace Abstract: The classical Khintchine--Jarn\'ik Theorem provides elegant criteria for determining the Lebesgue measure and Hausdorff measure of sets of points approximated by rational points, which has inspired much modern research in metric Diophantine approximation. This paper concerns the Lebesgue measure, Hausdorff measure and Fourier dimension of sets arising in weighted and multiplicative Diophantine approximation. We provide zero-full laws for determining the Lebesgue measure and Hausdorff measure of the sets under consideration. In particular, the criterion for the weighted setup refines a dimensional result given by Li, Liao, Velani, Wang, and Zorin [arXiv: 2410.18578 (2024)], while the criteria for the multiplicative setup answer a question raised by Hussain and Simmons [J. Number Theory (2018)] and extend beyond it. A crucial observation is that, even in higher dimensions, both setups are more appropriately understood as consequences of the `balls-to-rectangles' mass transference principle. We also determine the exact Fourier dimensions of these sets. The result we obtain indicates that, in line with the existence results, these sets are generally non-Salem sets, except in the one-dimensional case. This phenomenon can be partly explained by another result of this paper, which states that the Fourier dimension of the product of two sets equals the minimum of their respective Fourier dimensions. Maths/math.NT_(Number_Theory) https://arxiv.org/abs/2506.10838 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2506.10838v2 Announce Type: replace Abstract: In this paper, we study three integers arising naturally from B\'{e}zout's identity, the resultant and the reduced resultant of two coprime integer polynomials. We establish several new divisibility relations among them. We also pose two conjectures by making computations. Maths/math.NT_(Number_Theory) https://arxiv.org/abs/2512.10658 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2512.10658v2 Announce Type: replace Abstract: Let $\spt(n)$ be the number of smallest parts in the partitions of $n$. In this paper, we give some generalized Euler-like recursive formulas for the $\spt$ function in terms of Hecke trace of values of special twisted quadratic Dirichlet series. As a corollary, we give a closed form expression of the power series $\sum_{n\geq 0}\spt(\ell n-\delta_{\ell})q^n\pmod{\ell}$, $\delta_{\ell}:=(\ell^2-1)/24$, by Hecke traces for weight $\ell+1 $ cusp forms on $\SL_2(\mathbb{Z})$. We further establish an incongruence result for the $\spt$ function. Maths/math.NT_(Number_Theory) https://arxiv.org/abs/2604.12697 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.12697v2 Announce Type: replace Abstract: For $F \in \mathbb{Z}[s,t]$ a binary quadratic form which is irreducible over $\mathbb{Q}$, and $L$ an abelian number field with class number $1$, we obtain the order of magnitude for the number of values $F(s,t)$ which are a norm from $L$. Our result relies on the fundamental lemma of sieve theory and on geometry of numbers. Maths/math.NT_(Number_Theory) https://arxiv.org/abs/2311.08992 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2311.08992v2 Announce Type: replace-cross Abstract: In this work we investigate the problem of producing iso-dual algebraic geometry (AG) codes over a finite field $\mathbb{F}_q$ with $q$ elements. Given a finite separable extension $\mathcal{M}/\mathcal{F}$ of function fields and an iso-dual AG-code $\mathcal{C}$ defined over $\mathcal{F}$, we provide a general method to lift the code $\mathcal{C}$ to another iso-dual AG-code $\tilde{\mathcal{C}}$ defined over $\mathcal{M}$ under some assumptions on the divisors $D$ and $G$ and on the parity of the involved different exponents. We apply this method to lift iso-dual AG-codes over the rational function field to elementary abelian $p$-extensions, like the maximal function fields defined by the Hermitian, Suzuki, and one covered by the $GGS$ function field. We also obtain long binary and ternary iso-dual AG-codes defined over cyclotomic extensions. Maths/math.NT_(Number_Theory) https://arxiv.org/abs/2405.14395 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2405.14395v3 Announce Type: replace-cross Abstract: For the Tits building B(G) of a finite group of Lie type G(Fq), we study the edge zeta function, which enumerates edge-geodesic cycles in the 1-skeleton. We show that every nonzero edge eigenvalue becomes a power of q after raising to a bounded exponent k depending on the type of G. The proof is uniform across types using a Hecke algebra approach. This extends previous results for type A and for oppositeness graphs to the full edge-geodesic setting and all finite groups of Lie type. Maths/math.NT_(Number_Theory) https://arxiv.org/abs/2604.13272 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13272v1 Announce Type: new Abstract: This paper studies a stochastic algorithm for linearly constrained nonconvex optimization, where the objective function is smooth but only unbiased stochastic gradients with bounded variance are available. We propose a momentum-based augmented Lagrangian method that employs a Polyak-type gradient estimator and requires only one stochastic gradient evaluation per iteration. Under the standard stochastic oracle model and the smoothness condition of the expected objective, we establish a convergence guarantee in terms of the first-order KKT residual of the original constrained problem. In particular, the proposed method computes an $\epsilon$-stationary solution in expectation within $O(\epsilon^{-4})$ stochastic gradient evaluations. Numerical experiments further show that the proposed method achieves competitive iteration complexity and improved wall-clock efficiency compared with representative recursive-momentum baselines. Maths/math.OC_(Optimization_and_Control) https://arxiv.org/abs/2604.13302 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13302v1 Announce Type: new Abstract: We revisit the optimization problem solved in L{\o}kka & Zervos (2008), i.e., the maximization of dividends, in a Brownian risk model, with the possibility (not the obligation) of making capital injections. Following the approach introduced in Alvarez & Shepp (1998), Renaud & Simard (2021), Renaud et al. (2023), we consider instead absolutely continuous (AC) dividend strategies with an affine bound on the payment rates, while singular capital injections are still allowed. In addition, we incorporate a parameter for the cost of ruin or, said differently, a penalty at ruin in the performance function. We show that the solution is a so-called L{\o}kka-Zervos dichotomy: the surplus is never ruined by making bail-out payments, or no capital is injected and bankruptcy can occur; in either case, dividends are paid at full rate when the surplus is above a threshold. Our framework allows us to provide explicit conditions to express the dichotomy, either using the cost of capital injections or the cost of ruin as a criterion, which also exposes the underlying structure of the solution. In particular, for some values of the parameters, we show that it is optimal to liquidate. Moreover, we perform a numerical analysis highlighting the range of values generated under this AC affine-bound structure. Maths/math.OC_(Optimization_and_Control) https://arxiv.org/abs/2604.13357 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13357v1 Announce Type: new Abstract: Non-pharmaceutical interventions are critical for epidemic suppression but impose substantial societal costs, motivating feedback control policies that adapt to time-varying transmission. We formulate an infinite-horizon optimal control problem for a mobility-coupled networked SIQR epidemic model that minimizes isolation burden while enforcing epidemic suppression through a spectral decay condition. From this formulation, we derive a safety-critical Model Predictive Control (MPC) framework in which the spectral certificate is imposed as a hard stage-wise constraint, yielding a tunable exponential decay rate for infections. Exploiting the monotone depletion of susceptible populations, we construct a robust terminal set and safe backup policy. This structure ensures recursive feasibility and finite-horizon closed-loop exponential decay, and it certifies the existence of a globally stabilizing feasible continuation under bounded worst-case transmission rates. Numerical simulations on a 14-county Massachusetts network under a variant-induced surge show that, with administrative rate limits, reactive myopic control fails whereas MPC anticipates the shock and maintains exponential decay with lower isolation burden. Maths/math.OC_(Optimization_and_Control) https://arxiv.org/abs/2604.13378 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13378v1 Announce Type: new Abstract: We revisit the convergence analysis of constant stepsize stochastic approximation (SA) with decision-dependent Markovian noise, with a focus on characterizing the stationary bias against the root of the mean-field equation. We first establish the finite-time $p$-th moment bounds for the SA iterates in a general decision-dependent setting, which serve as a stability foundation for the subsequent analysis. Building on this foundation, and leveraging a local regularity condition termed Poisson--Gateaux differentiability (WD$^\ast$) for the solution to Poisson equation induced by the decision-dependent Markov kernel, we show that the stationary bias is of the order $\mathcal{O}(\alpha)$ for a broad class of decision-dependent settings. Additionally, we establish geometric weak convergence of the joint SA process towards a unique stationary distribution, and a functional central limit theorem. Our relaxed regularity condition enables us to cover cases of non-smooth kernels such as acceptance--rejection mechanisms, projected Langevin dynamics, and clipped state dynamics. Maths/math.OC_(Optimization_and_Control) https://arxiv.org/abs/2604.13388 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13388v1 Announce Type: new Abstract: We propose a novel study of the stochastic proximal gradient method for minimizing the sum of two convex functions, one of which is smooth. Under suitable assumptions and without requiring any boundedness or control of the variance of the random variables, we derive the almost sure convergence and the convergence in the mean of the iterates to a solution of the minimization problem. The results are applied to classification and convex feasibility problems. Maths/math.OC_(Optimization_and_Control) https://arxiv.org/abs/2604.13393 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13393v1 Announce Type: new Abstract: Davis, Drusvyatskiy, and Jiang showed that gradient descent with an adaptive stepsize converges locally at a nearly-linear rate for smooth functions that grow at least quartically away from their minimizers. The argument is intricate, relying on monitoring the performance of the algorithm relative to a certain manifold of slow growth -- called the ravine. In this work, we provide a direct Lyapunov-based argument that bypasses these difficulties when the objective is in addition convex and a has a unique minimizer. As a byproduct of the argument, we obtain a more adaptive variant than the original algorithm with encouraging numerical performance. Maths/math.OC_(Optimization_and_Control) https://arxiv.org/abs/2604.13483 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13483v1 Announce Type: new Abstract: Most non-convex optimization theory is built around gradient dynamics, leaving global convergence largely unexplored. The dominant paradigm focuses on stationarity, certifying only that the gradient norm vanishes, which is often a weak proxy for actual optimization success. In practice, gradient norms can stagnate or even increase during training, and stationary points may be far from global solutions. In this work, we propose a new framework for global non-convex optimization that avoids gradient-based reasoning altogether. We revisit the Ball Proximal Point Method (BPM), a trust-region-style algorithm introduced by Gruntkowska et al. (2025), and propose a novel structural condition - Broximal Alignment - under which BPM provably converges to a global minimizer. Our condition requires no convexity, smoothness, or Lipschitz assumptions, and it permits multiple and disconnected global minima as well as non-optimal local minima. We show that this class generalizes standard non-convex frameworks such as quasiconvexity, star convexity, quasar convexity, and the aiming condition. Our results provide a new conceptual foundation for global non-convex optimization beyond stationarity. Maths/math.OC_(Optimization_and_Control) https://arxiv.org/abs/2604.13566 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13566v1 Announce Type: new Abstract: In nonlinear elasticity, finding the deformation of a material which minimizes a given stored energy density is a challenging calculus of variations problem which may fail to have minimizers: the energy optimal material forms infinitely fine microstructures (wrinkles) rather than deforming smoothly. In the case where the energy function is non-convex but frame indifferent and convex with respect to the Cauchy-Green strain tensor, we use the standard Le Dret-Raoult semidefinite projection formula for the quasiconvex envelope of the energy function to prove that there is no relaxation gap between the original non-convex calculus of variations problem and its linear moment formulation based on occupation measures. This implies convergence of the Lasserre moment-sum-of-squares (SOS) hierarchy and provides a computationally efficient, mesh-free numerical method that, unlike the finite element method, avoids undesirable mesh-dependent artifacts. Under the additional condition that the boundary condition is linear and the function is SOS convex in the strain tensor, we show that the first relaxation of the Lasserre hierarchy is exact. In other words, computing the quasiconvex envelope at a point boils down to solving a small convex semidefinite optimization problem. Maths/math.OC_(Optimization_and_Control) https://arxiv.org/abs/2604.13727 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13727v1 Announce Type: new Abstract: In the context of spacecraft attitude control, parametrizations such as direction vectors or quaternions are often used to avoid singularities in the attitude representation. This, however, complicates the stability analysis of the system since, given the additional unit constraints, the resulting dynamics evolve on non-contractible manifolds. In this paper, we present a framework to verify almost global asymptotic stability of such systems using LaSalle's invariance principle and sum-of-squares programming, simplifying the search for Lyapunov functions. The framework is then applied to two examples: two-axis attitude acquisition utilizing aerodynamics in very low Earth orbits, and three-axis attitude acquisition for a satellite subject to gravity gradient torques in a circular orbit. Maths/math.OC_(Optimization_and_Control) https://arxiv.org/abs/2604.13836 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13836v1 Announce Type: new Abstract: We study potential games on unimodular random graphs of bounded degree, where players interact through the underlying network. Using the unimodular measure, we define a well-posed global potential that captures both finite- and infinite-player games. A key observation is that the mass-transport principle identifies the first variation of this potential with the first-order condition of a representative (root) player. Under suitable convexity assumptions, we prove that minimizers of the potential coincide with quenched Nash equilibria, and conversely. We also establish the thermodynamic limit of the potential along weakly convergent sequences of unimodular measures. Finally, we present examples with semi-explicit equilibrium descriptions. In linear-quadratic games on unimodular graphs, equilibria are expressed in terms of the Green kernel of the simple random walk operator, while in convex settings, equilibria are characterized by solutions to nonlinear Poisson equations. Maths/math.OC_(Optimization_and_Control) https://arxiv.org/abs/2604.13857 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13857v1 Announce Type: new Abstract: In this paper, we consider the design of Model Predictive Control (MPC) algorithms based on Mamba neural networks. Mamba is a neural network architecture capable of sub-quadratic computational scaling in sequence length with state-of-the-art modeling capabilities. We provide a consistent and complete mathematical description of the Mamba neural network is provided. Then, adjustments and optimizations are made to construct a decoder-only Mamba multi-step predictor for MPC and an input-output formulation is given for sequence-to-sequence modeling of dynamical systems. The performance of Mamba-MPC is evaluated on several numerical examples and compared to a Long-Short-Term-Memory based MPC (LSTM-MPC) equivalent. First, a Single-Input-Single-Output (SISO) Van der Pol oscillator is considered, where stability, reference tracking, and noise robustness are evaluated. Then, a MIMO Four Tank setup is introduced where Multiple-Input-Multiple-Output (MIMO) reference tracking is evaluated. Lastly, Mamba-MPC is implemented on a physical Quanser Aero2 setup for closed-loop reference tracking. The results demonstrate that Mamba-MPC is able to stabilize and track a reference for SISO and MIMO systems, both in simulation and on a physical setup. Moreover, Mamba-MPC consistently outperforms LSTM-MPC in predictive control and is significantly computationally faster. Maths/math.OC_(Optimization_and_Control) https://arxiv.org/abs/2604.13870 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13870v1 Announce Type: new Abstract: We consider the well-studied setting of minimizing a convex Lipschitz function using either gradient descent (GD) or its stochastic variant (SGD), and examine the last iterate convergence. By now, it is known that standard stepsize choices lead to a last iterate convergence rate of $\log T/\sqrt{T}$ after $T$ steps. A breakthrough result of Jain et al. [2019] recovered the optimal $1/\sqrt{T}$ rate by constructing a non-standard stepsize sequence. However, this sequence requires choosing $T$ in advance, as opposed to common stepsize schedules which apply for any time horizon. Moreover, Jain et al. conjectured that without prior knowledge of $T$, no stepsize sequence can ensure the optimal error for SGD's last iterate, a claim which so far remained unproven. We prove this conjecture, and in fact show that even in the noiseless case of GD, it is impossible to avoid an excess poly-log factor in $T$ when considering an anytime last iterate guarantee. Our proof further suggests that such (slightly) suboptimal stopping times are unavoidably common. Maths/math.OC_(Optimization_and_Control) https://arxiv.org/abs/2604.13915 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13915v1 Announce Type: new Abstract: A rigid motion in $\mathbb{R}^d$ consists of a proper rotation and a translation, and it can be represented as a matrix in $\mathbb{R}^{(d+1)\times (d+1)}$. The problem of rigid motion synchronization aims to estimate a collection of rigid motions $G^*_1, \dots, G^*_n$ from noisy observations of their comparisons ${G^*_i}^{-1} G^*_j$. Such problems naturally arise in diverse applications across signal processing, robotics, and computer vision, and have thus attracted intense research attention in recent years. Motivated by geometric considerations, this paper develops a novel spectral approach for rigid motion synchronization, called the anchored spectral estimator (ASE). Theoretically, we establish uniform estimation error bounds for the estimators produced by ASE. Empirically, we show that ASE outperforms the widely used two-stage approach, which first estimates the rotations and then the translations. Further numerical experiments on the multiple point-set registration problem are presented to demonstrate the superiority of ASE over state-of-the-art methods. Maths/math.OC_(Optimization_and_Control) https://arxiv.org/abs/2604.13937 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13937v1 Announce Type: new Abstract: We formulate and analyse an optimal control problem for the coagulation-fragmentation equation, where a scalar, time-dependent control modulates the coagulation rate by multiplying the coagulation kernel. The objective functional consists of a quadratic penalisation of the control and a terminal cost depending on the final size distribution. In a weighted $L^1$ framework, we prove weak-to-weak continuity of the control-to-state map under perturbations of the coefficients and obtain existence of optimal controls by the direct method. We then establish $\Gamma$-convergence of the corresponding cost functionals, providing stability of optimal controls and justifying truncation of unbounded kernels in the optimisation setting. For bounded coagulation kernels we show differentiability of the dynamics, derive an adjoint equation, and obtain a Pontryagin-type minimum principle. Lipschitz continuity of the gradient with respect to the control yields, at the continuous level, convergence of a projected-gradient algorithm with Armijo backtracking. A proof-of-concept finite-volume implementation is then used in a numerical study targeting the number of particles within a prescribed size window, demonstrating that a single low-dimensional actuator can effectively reshape an infinite-dimensional particle-size distribution. Maths/math.OC_(Optimization_and_Control) https://arxiv.org/abs/2604.13965 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13965v1 Announce Type: new Abstract: This paper studies the variance dichotomy in continuous simulation optimization (CSO). Existing literature shows a sharp contrast between deterministic CSO and stochastic CSO, with convergence rates in stochastic settings appearing insensitive to the magnitude of the noise variance. However, this asymptotic view does not fully explain the behavior of CSO under finite simulation budgets, especially in low-noise settings. To address this gap, this work develops a minimax lower-bound analysis and shows that the complexity is decided by the maximum of a variance-dependent term and a variance-independent term. When the simulation budget is not very large and the noise variance is low, the variance-independent term dominates, implying that low-noise stochastic CSO has essentially the same complexity as deterministic CSO. As the budget increases, the variance-dependent term becomes dominant, and the convergence behavior of stochastic CSO transitions to a slower regime determined jointly by the noise variance and the simulation budget. Maths/math.OC_(Optimization_and_Control) https://arxiv.org/abs/2604.14017 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14017v1 Announce Type: new Abstract: Under interpolation-type assumptions such as the strong growth condition, stochastic optimization methods can attain convergence rates comparable to full-batch methods, but their performance, particularly for SGD, remains highly sensitive to step-size selection. To address this issue, we propose a unified stochastic trust-region framework that eliminates manual step-size tuning and extends naturally to equality-constrained problems. For unconstrained optimization, we develop a first-order stochastic trust-region algorithm and show that, under the strong growth condition, it achieves an iteration and stochastic first-order oracle complexity of $O(\varepsilon^{-2} \log(1/\varepsilon))$ for finding an $\varepsilon$-stationary point. For equality-constrained problems, we introduce a quadratic-penalty-based stochastic trust-region method with penalty parameter $\mu$, and establish an iteration and oracle complexity of $O(\varepsilon^{-4} \log(1/\varepsilon))$ to reach an $\varepsilon$-stationary point of the penalized problem, corresponding to an $O(\varepsilon)$-approximate KKT point of the original constrained problem. Numerical experiments on deep neural network training and orthogonally constrained subspace fitting demonstrate that the proposed methods achieve performance comparable to well-tuned stochastic baselines, while exhibiting stable optimization behavior and effectively handling hard constraints without manual learning-rate scheduling. Maths/math.OC_(Optimization_and_Control) https://arxiv.org/abs/2604.14075 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14075v1 Announce Type: new Abstract: We introduce Multistage Conditional Compositional Optimization (MCCO) as a new paradigm for decision-making under uncertainty that combines aspects of multistage stochastic programming and conditional stochastic optimization. MCCO minimizes a nest of conditional expectations and nonlinear cost functions. It has numerous applications and arises, for example, in optimal stopping, linear-quadratic regulator problems, distributionally robust contextual bandits, as well as in problems involving dynamic risk measures. The na\"ive nested sampling approach for MCCO suffers from the curse of dimensionality familiar from scenario tree-based multistage stochastic programming, that is, its scenario complexity grows exponentially with the number of nests. We develop new multilevel Monte Carlo techniques for MCCO whose scenario complexity grows only polynomially with the desired accuracy. Maths/math.OC_(Optimization_and_Control) https://arxiv.org/abs/2604.13213 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13213v1 Announce Type: cross Abstract: Rare events such as conformational changes in biomolecules, phase transitions, and chemical reactions are central to the behavior of many physical systems, yet they are extremely difficult to study computationally because unbiased simulations seldom produce them. Transition Path Theory (TPT) provides a rigorous statistical framework for analyzing such events: it characterizes the ensemble of reactive trajectories between two designated metastable states (reactant and product), and its central object--the committor function, which gives the probability that the system will next reach the product rather than the reactant--encodes all essential kinetic and thermodynamic information. We introduce a framework that casts committor estimation as a stochastic optimal control (SOC) problem. In this formulation the committor defines a feedback control--proportional to the gradient of its logarithm--that actively steers trajectories toward the reactive region, thereby enabling efficient sampling of reactive paths. To solve the resulting hitting-time control problem we develop two complementary objectives: a direct backpropagation loss and a principled off-policy Value Matching loss, for which we establish first-order optimality guarantees. We further address metastability, which can trap controlled trajectories in intermediate basins, by introducing an alternative sampling process that preserves the reactive current while lowering effective energy barriers. On benchmark systems, the framework yields markedly more accurate committor estimates, reaction rates, and equilibrium constants than existing methods. Maths/math.OC_(Optimization_and_Control) https://arxiv.org/abs/2604.13525 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13525v1 Announce Type: cross Abstract: The robust low-rank tensor completion problem addresses the challenge of recovering corrupted high-dimensional tensor data with missing entries, outliers, and sparse noise commonly found in real-world applications. Existing methodologies have encountered fundamental limitations due to their reliance on uniform regularization schemes, particularly the tensor nuclear norm and $\ell_1$ norm regularization approaches, which indiscriminately apply equal shrinkage to all singular values and sparse components, thereby compromising the preservation of critical tensor structures. The proposed tensor weighted correlated total variation (TWCTV) regularizer addresses these shortcomings through an $M$-product framework that combines a weighted Schatten-$p$ norm on gradient tensors for low-rankness with smoothness enforcement and weighted sparse components for noise suppression. The proposed weighting scheme adaptively reduces the thresholding level to preserve both dominant singular values and sparse components, thus improving the reconstruction of critical structural elements and nuanced details in the recovered signal. Through a systematic algorithmic approach, we introduce an enhanced alternating direction method of multipliers (ADMM) that offers both computational efficiency and theoretical substantiation, with convergence properties comprehensively analyzed within the $M$-product framework.Comprehensive numerical evaluations across image completion, denoising, and background subtraction tasks validate the superior performance of this approach relative to established benchmark methods. Maths/math.OC_(Optimization_and_Control) https://arxiv.org/abs/2604.13529 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13529v1 Announce Type: cross Abstract: We propose and analyze an experimentally accessible Lindblad master equation for a quantum harmonic oscillator, simplifying a previous proposal to alleviate implementation constraints. It approximately stabilizes periodic grid states introduced in 2001 by Gottesman, Kitaev and Preskill (GKP), with applications for quantum error correction and quantum metrology. We obtain explicit estimates for the energy of the solutions of the Lindblad master equation. We estimate the convergence rate to the codespace when stabilizing a GKP qubit, and numerically study the effect of noise. We then present simulations illustrating how a modification of parameters allows preparing states of metrological interest in steady-state. Maths/math.OC_(Optimization_and_Control) https://arxiv.org/abs/2604.13149 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13149v1 Announce Type: new Abstract: We study the almost surely finite random variable $S$ defined by the distributional fixed-point equation \[ S \stackrel{d}{=} 1 + \max\{US', (1-U)S''\}, \qquad U \sim \mathrm{Unif}(0,1), \] where $S'$ and $S''$ are independent copies of $S$, independent of $U$. This random variable arises as the almost sure limit of the normalized worst-case number of key comparisons used by classical Quickselect with uniformly chosen pivots in the model of Devroye. Our first contribution concerns the right tail of $S$. We prove explicit one-sided bounds for the rate function $-\log \mathbb{P}(S>t)$ and, in particular, identify its first-order asymptotic growth: \[ -\log \mathbb{P}(S>t) = t \log t + O(t \log \log t), \qquad t \to \infty. \] The argument combines a binary-search-tree embedding and a one-level second-moment method with a moment-generating-function comparison inspired by ideas of Alsmeyer and Dyszewski for the nonhomogeneous smoothing transform. As a byproduct, we obtain an explicit pointwise Chernoff majorant for the tail. Our second contribution is a distribution-function scheme for deriving explicit upper bounds on $\mathbb{E}[S]$. Starting from the fixed-point equation at the level of the distribution function, we construct an order-preserving lower iteration and a conservative mesh discretization suited to computer-assisted upper bounds on the mean. We illustrate the latter numerically in floating-point arithmetic, but do not pursue a certified numerical proof here. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.13454 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13454v1 Announce Type: new Abstract: In this paper, we consider the classical spin systems on unbounded lattices given by infinite-dimensional stochastic differential equations (SDEs). We assume that the stochastic forcing acts only on one particle. The other particles are not subject to stochastic forcing directly, but interact with their nearest neighbouring particles. Under the above highly degenerate noise setting, with some mild assumptions on the local interaction of each particle such as weak dissipation, we obtain the existence, uniqueness and the Markovian property of weak martingale solutions. We prove that the one-dimensional noise can propagate to any spin particle in the system in the sense that there exists a unique invariant/periodic measure and geometric ergodicity holds for the Markovian system when restricted to any finite volume. We then prove the finite-dimensional invariant measure and the average of lifted periodic measure are tight, and weak convergent subsequence gives an invariant and periodic measures of the infinite spin systems, respectively, in the time-homogeneous or time-periodic cases. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.13493 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13493v1 Announce Type: new Abstract: We study whether a uniformly random Boolean function $f : \{-1,1\}^p \to \{-1,1\}$ is determined by its Walsh--Fourier coefficients of degree at most $d$. We show that the threshold lies at $p/2$ up to an $O(\sqrt{p \log p})$ window: if \[ d \le \frac{p}{2} - \sqrt{\frac{p}{2}\bigl(\log p + \omega(1)\bigr)}, \] then with probability $1-o(1)$ there exists another Boolean function $g \ne f$ with the same degree-$\le d$ coefficients. Conversely, for every fixed $\eta \in (0,1)$, if \[ d \ge \frac{p}{2} + \sqrt{\frac{p}{2}\log\frac{6p}{\eta^2}}, \] then with probability at least $1-2^{-p}$, the function $f$ is uniquely determined by its degree-$\le d$ coefficients, even among all bounded functions $g : \{-1,1\}^p \to [-1,1]$. This resolves a question of Vershynin. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.13681 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13681v1 Announce Type: new Abstract: The node2vec random walk is a non-Markovian random walk on the vertex set of a graph, widely used for network embedding and exploration. This random walk model is defined in terms of three parameters which control the probability of, respectively, backtracking moves, moves within triangles, and moves to the remaining neighboring nodes. From a mathematical standpoint, the node2vec random walk is a nontrivial generalization of the non-backtracking random walk and thus belongs to the class of second-order Markov chains. Despite its widespread use in applications, little is known about its long-run behavior. The goal of this paper is to begin exploring its fundamental properties on arbitrary graphs. To this aim, we show how lifting the node2vec random walk to the state spaces of directed edges and directed wedges yields two distinct Markovian representations which are key for its asymptotic analysis. Using these representations, we find mild sufficient conditions on the underlying finite or infinite graph to guarantee ergodicity, reversibility, recurrence and characterization of the invariant measure. As we discuss, the behavior of the node2vec random walk is drastically different compared to the non-backtracking random walk. While the latter simplifies on arbitrary graphs when using its natural edge Markovian representation thanks to bistochasticity, the former simplifies on regular graphs when using its natural wedge Markovian representation. Remarkably, this representation reveals that a graph is regular if and only if a certain weighted Eulerianity condition holds. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.13968 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13968v1 Announce Type: new Abstract: We analyze an optimal stopping problem for random walk in random scenery on general graphs, and determine when it has a finite optimum. We use this to extend a theorem of Levine, Murugan, Peres, and Ugurcan [2016]. They proved that on a vertex-transitive graph, the divisible sandpile with i.i.d. initial masses of mean $\mu$ stabilizes almost surely if $\mu 1$, and explodes if $\mu = 1$ with positive finite variance. Their proofs rely on conservation of mean mass under toppling. This conservation extends to unimodular random graphs, but fails on general graphs. We prove explosion for all infinite bounded-degree graphs whenever $\mu \geq 1$, and stabilization for $\mu 3$. Our conditions are nearly sharp: we exhibit unbounded-degree graphs on which sandpiles with $\mu > 1$ stabilize, and for every $p < 3$ we construct bounded-degree graphs on which sandpiles with~$\mu < 1$ and finite $p$-th moment explode. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.13147 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13147v1 Announce Type: cross Abstract: This paper studies continuous-time stochastic control problems whose controlled states are fully non-Markovian and depend on unknown model parameters. Such problems arise naturally in path-dependent stochastic differential equations, rough-volatility hedging, and systems driven by fractional Brownian motion. Building on the discrete skeleton approach developed in earlier work, we propose a Monte Carlo learning methodology for the associated embedded backward dynamic programming equation. Our main contribution is twofold. First, we construct explicit dominating training laws and Radon--Nikodym weights for several representative classes of non-Markovian controlled systems. This yields an off-model training architecture in which a fixed synthetic dataset is generated under a reference law, while the dynamic programming operators associated with a target model are recovered by importance sampling. Second, we use this structure to design an adaptive update mechanism under parametric model uncertainty, so that repeated recalibration can be performed by reweighting the same training sample rather than regenerating new trajectories. For fixed parameters, we establish non-asymptotic error bounds for the approximation of the embedded dynamic programming equation via deep neural networks. For adaptive learning, we derive quantitative estimates that separate Monte Carlo approximation error from model-risk error. Numerical experiments illustrate both the off-model training mechanism and the adaptive importance-sampling update in structured linear-quadratic examples. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.13295 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13295v1 Announce Type: cross Abstract: t-SNE has gained popularity as a dimension reduction technique, especially for visualizing data. It is well-known that all dimension reduction techniques may lose important features of the data. We provide a mathematical framework for understanding this loss for t-SNE by establishing a number of results in different scenarios showing how important features of data are lost by using t-SNE. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.13798 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13798v1 Announce Type: cross Abstract: Using only the characteristic function, we derive short-time at-the-money (ATM) call-price asymptotics for the exponential CGMY model with activity parameter $Y\in(1,2)$. The Lipton--Lewis formula expresses the normalized ATM call price, denoted $c(t,0)$, in terms of the characteristic exponent, which, upon rescaling at the rate $t^{-1/Y}$ from the $Y$-stable domain of attraction, yields $c(t,0) = d_{1} t^{1/Y} + d_{2} t + o(t)$ as $t\downarrow 0$. The first-order coefficient $d_{1}$ is the known stable limit from the domain of attraction of a symmetric $Y$-stable law, and $d_{2}$ is given by an explicit integral involving the characteristic exponent and the limiting stable exponent. We then extract closed-form higher-order coefficients by keeping the full Lipton--Lewis integrand intact and introducing a dynamic cutoff that partitions the domain into inner, core, and tail regions, establishing the expansion with controlled remainder. All coefficients are verified numerically against existing closed-form expressions where available. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.14061 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14061v1 Announce Type: cross Abstract: We show that the maximum expected inner product between a random vector and the standard normal vector over all couplings subject to a mutual information constraint or regularization is equivalent to a truncated integral involving the rate-distortion function, up to universal multiplicative constants. The proof is based on a lifting technique, which constructs a Gaussian process indexed by a random subset of the type class of the probability distribution involved in the information-theoretic inequality, and then applying a form of the majorizing measure theorem. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.14135 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14135v1 Announce Type: cross Abstract: We consider the block withholding attacks on pools, more specifically the state-of-the-art Power Adjusting Withholding (PAW) attack. We propose a generalization called Temporary PAW (T-PAW) where the adversary withholds a fPoW from pool mining at most $T$-time even when no other block is mined. We show that PAW attack corresponds to $T\to\infty$ and is not optimal. In fact, the extra reward of T-PAW compared to PAW improves by an unbounded factor as adversarial hash fraction $\alpha$, pool size $\beta$ and adversarial network influence $\gamma$ decreases. For example, the extra reward of T-PAW is 22 times that of PAW when an adversary targets a pool with $(\alpha,\beta,\gamma)=(0.05,0.05,0)$. We show that honest mining is sub-optimal to T-PAW even when there is no difficulty adjustment and the adversarial revenue increase is non-trivial, e.g., for most $(\alpha,\beta)$ at least $1\%$ within $2$ weeks in Bitcoin even when $\gamma=0$ (for PAW it was at most $0.01\%$). Hence, T-PAW exposes a significant structural weakness in pooled mining-its primary participants, small miners, are not only contributors but can easily turn into potential adversaries with immediate non-trivial benefits. Maths/math.PR_(Probability) https://arxiv.org/abs/2302.14823 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2302.14823v3 Announce Type: replace Abstract: We establish precise upper-tail asymptotics and large deviation principles for the rightmost eigenvalue $\lambda_1$ of Wigner matrices with sub-Gaussian entries. In contrast to the case of heavier tails, where deviations of $\lambda_1$ are due to the appearance of a few large entries, and the sharp sub-Gaussian case that is governed by the collective deviation of entries in a delocalized rank-one pattern, we show that the general sub-Gaussian case is determined by a mixture of localized and delocalized effects. Our key result is a finite-$N$ approximation for the upper tail of $\lambda_1$ by an optimization problem involving \emph{restricted annealed free energies} for a spherical spin glass model. This new type of argument allows us to derive full large deviation principles when the log-Laplace transform of the entries' distribution $\mu$ has bounded second derivative, whereas previous results required much more restrictive assumptions, namely sharp sub-Gaussianity and symmetry, or only covered certain ranges of deviations. We show that the sharp sub-Gaussian condition characterizes measures $\mu$ for which the rate function coincides with that of the Gaussian Orthogonal Ensemble (GOE). When $\mu$ is not sharp sub-Gaussian, at a certain distance from the bulk of the spectrum there is a transition from the GOE rate function to a non-universal rate function depending on $\mu$, and this transition coincides with the onset of a localization phenomenon for the associated eigenvector. Maths/math.PR_(Probability) https://arxiv.org/abs/2410.20167 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2410.20167v2 Announce Type: replace Abstract: We prove that the hydrodynamic limit of the symmetric exclusion process (SEP) is a Fokker-Planck equation in the setting of Poisson random neighborhood graphs approximating a weighted Riemannian manifold with Ricci curvature bounded from below. We also consider the lift of the SEP to a principal bundle, and obtain a Fokker-Planck equation with a weighted horizontal Laplacian as its hydrodynamic limit. Both results significantly extend the geometric settings in which one can prove the hydrodynamic limit from duality combined with convergence of the single particle random walk towards a diffusion process. Maths/math.PR_(Probability) https://arxiv.org/abs/2503.02795 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2503.02795v3 Announce Type: replace Abstract: We prove large deviation principles (LDPs) for full chordal, radial, and multichordal SLE(0+) curves parameterized by capacity. The rate function is given by the appropriate variant of the Loewner energy. There are two key novelties in the present work. First, we strengthen the topology in the known chordal LDPs into the topology of full parameterized curves including all curve endpoints. We also obtain LDPs in the space of unparameterized curves. Second, we address the radial case, which requires in part different methods from the chordal case, due to the different topological setup. We establish our main results via proving an exponential tightness property and combining it with detailed curve escape probability estimates, in the spirit of exponentially good approximations in LDP theory. In the radial case, additional work is required to refine the estimates appearing in the literature. Notably, since we manage to prove a finite-time LDP in a better topology than in earlier literature, escape energy estimates follow as a consequence of the escape probability estimates. Maths/math.PR_(Probability) https://arxiv.org/abs/2505.07648 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2505.07648v2 Announce Type: replace Abstract: Demand for studying queueing systems with multiple servers providing correlated services was created about 60 years ago, motivated by various applications. In recent years, the importance of such studies has been significantly increased, supported by new applications of greater significance to much larger scaled industry, and the whole society. Such studies have been considered very challenging. In this paper, a new Markov modelling approach for queueing systems with servers providing correlated services is proposed. We apply this new proposed approach to a queueing system with arrivals according to a Poisson process and two positive correlated exponential servers, referred to as the $M/M_D/2$ queue. We first prove that the queueing process (the number of customers in the system) is a Markov chain, and then provide an analytic solution for the stationary distribution of the process, based on which it becomes much easier to see the impact of the dependence on system performance compared to the performance with independent services. Maths/math.PR_(Probability) https://arxiv.org/abs/2512.23164 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2512.23164v3 Announce Type: replace Abstract: In 2009, Yano, Yano and Yor proposed the question of studying the infinite divisibility of the $\alpha$-Cauchy variable $\mathcal{C}_\alpha$ for $\alpha > 1$. The particular case $\mathcal{C}_2$ is the well-known standard Cauchy variable, which is infinitely divisible and indeed stable. For $\alpha \neq 2$, the infinite divisibility of $\mathcal{C}_\alpha$ is previously unknown. In this paper, we prove that $\mathcal{C}_\alpha$ is infinitely divisible if and only if $1 < \alpha \leq 2$. Maths/math.PR_(Probability) https://arxiv.org/abs/2408.10610 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2408.10610v5 Announce Type: replace-cross Abstract: We propose an $L^2$ norm for stationary Autoregressive Moving Average (ARMA) models. We look at ARMA models within the Hilbert space of the past with present of a true purely linearly non-deterministic stationary process $X_t$, and compute the $L^2$ norm based on its Wold decomposition. As an application of this $L^2$ norm, we derive bounds on the mean square prediction error for AR(1) models of MA(1) processes, and verify these bounds empirically for sample data. Maths/math.PR_(Probability) https://arxiv.org/abs/2507.12575 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2507.12575v3 Announce Type: replace-cross Abstract: The definition of metastable states is an ubiquitous task in the design and analysis of molecular simulations, and is a crucial input in a variety of acceleration methods for the sampling of long configurational trajectories. Although standard definitions based on local energy minimization procedures can sometimes be used, these definitions are typically suboptimal, or entirely inadequate when entropic effects are significant, or when the lowest energy barriers are quickly overcome by thermal fluctuations. In this work, we propose an approach to the definition of metastable states, based on the shape-optimization of a local separation of timescale metric directly linked to the efficiency of a family of accelerated molecular dynamics algorithms. To realize this approach, we derive analytic expressions for shape-variations of Dirichlet eigenvalues for a class of operators associated with reversible elliptic diffusions, and use them to construct a local ascent algorithm, explicitly treating the case of multiple eigenvalues. We propose two methods to make our method tractable in high-dimensional systems: one based on dynamical coarse-graining, the other on recently obtained low-temperature shape-sensitive spectral asymptotics. We validate our method on a benchmark biomolecular system, showcasing a significant improvement over conventional definitions of metastable states. Maths/math.PR_(Probability) https://arxiv.org/abs/2512.23748 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2512.23748v2 Announce Type: replace-cross Abstract: For complex simulation problems, inferring parameters often precludes the use of classical likelihood-based techniques due to intractable likelihoods. Simulation-based inference (SBI) methods offer a likelihood-free approach to directly learn posterior distributions $p(\bftheta \mid \xobs)$ from simulator outputs. Recently, diffusion models have emerged as promising tools for SBI, addressing limitations of earlier neural methods such as neural likelihood/posterior estimation and normalizing flows. This review examines diffusion-based SBI from first principles to applications, emphasizing robustness in three non-ideal data scenarios common to scientific computing: model misspecification (simulator-reality mismatch), unstructured or infinite-dimensional observations, and missing data. We synthesize mathematical foundations and survey eight methods addressing these challenges, such as conditional diffusion for irregular data, guided diffusion for prior adaptation, sequential and factorized approaches for efficiency, and consistency models for fast sampling. Throughout, we maintain consistent notation and emphasize conditions required for accurate posteriors. We conclude with open problems and applications to geophysical uncertainty quantification, where these challenges are acute. Maths/math.PR_(Probability) https://arxiv.org/abs/2601.04193 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2601.04193v2 Announce Type: replace-cross Abstract: We propose a discrete transport equation on graphs which connects distributions on both vertices and edges. We then derive a discrete analogue of the Benamou-Brenier formulation for Wasserstein-$1$ distance on a graph and as a result classify all $W_1$ geodesics on graphs. Maths/math.PR_(Probability) https://arxiv.org/abs/2602.18310 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2602.18310v2 Announce Type: replace-cross Abstract: In a previous paper (arXiv:2510.19746), we have studied the maximal hard-code model on the square lattice ${\mathbb Z}^2$ from the perspective of recoverable systems. Here we extend this study to the case of the triangular lattice ${\mathbb A}$. The following results are obtained: (1) We derive bounds on the capacity of the associated recoverable system on ${\mathbb A}$; (2) We show non-uniqueness of Gibbs measures in the high-activity regime; (3) We characterize extremal periodic Gibbs measures for sufficiently low values of activity. Maths/math.PR_(Probability) https://arxiv.org/abs/2604.13889 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13889v1 Announce Type: new Abstract: In this paper, we propose an efficient two-level additive Schwarz method for solving large-scale eigenvalue problems arising from the finite element discretization of symmetric elliptic operators, which may compute efficiently more interior multiple and clustered eigenvalues other than only the first several smallest eigenvalues. The proposed method is parallel in two ways: one is to solve the preconditioned Jacobi-Davidson correction equations by the two-level additive Schwarz preconditioner, the other is to solve different clusters of eigenvalues (see Figure 1 in Introduction) simultaneously. It only requires computing a series of parallel subproblems and solving a small-dimensional eigenvalue problem per iteration for a cluster of eigenvalues. Based on some new estimates and tools, we provide a rigorous theoretical analysis to prove that convergence factor of the proposed method is bounded by $\gamma=c(H)\rho(\frac{\delta}{H},d_{m}^{-},d_{M}^{+})$, where $H$ is the diameter of subdomains, $\delta$ is the overlapping size and $d_{m}^{-},d_{M}^{+}$ are the distances from both ends of the targeted eigenvalues to others (see Figure 2 in Introduction). The positive number $\rho(\frac{\delta}{H},d_{m}^{-},d_{M}^{+})<1$ is independent of the fine mesh size and the internal gaps among the targeted eigenvalues. The $H$-dependent constant $c(H)$ decreases monotonically to 1, as $H\to 0$, which means the more subdomains lead to the better convergence. Numerical results supporting our theory are given. Maths/math.NA_(Numerical_Analysis) https://arxiv.org/abs/2602.15944 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2602.15944v2 Announce Type: replace-cross Abstract: We study constraints imposed by four-dimensional unitarity (formalised as graded unitarity in recent work by the first author) on possible ${\mathcal W}_3$ vertex algebras arising from four-dimensions via the SCFT/VOA correspondence. Under the assumption that the $\mathfrak{R}$-filtration is a weight-based filtration with respect to the usual strong generators of the vertex algebra, we demonstrate that all values of the central charge other than those of the $(3,q+4)$ minimal models are incompatible with four-dimensional unitarity. These algebras are precisely the ones that are realised by performing principal Drinfel'd--Sokolov reduction to boundary-admissible $\mathfrak{sl}_3$ affine current algebras; those affine algebras were singled out by a similar graded unitarity analysis in \cite{ArabiArdehali:2025fad}. Furthermore, these particular vertex algebras are known to be associated with the $(A_2,A_q)$ Argyres--Douglas theories. Maths/math.QA_(Quantum_Algebra) https://arxiv.org/abs/2604.13514 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13514v1 Announce Type: cross Abstract: Applying Gr\"obner basis theory to concrete problems in Lean 4 remains difficult since the current formalization of multivariate polynomials is based on a non-computable representation and is therefore not suitable for efficient symbolic computation. As a result, computing Gr\"obner bases directly inside Lean is impractical for realistic examples. To address this issue, we propose a certificate-based approach that combines external computer algebra systems, such as SageMath or SymPy, with formal verification in Lean 4. Our approach uses a computable representation of multivariate polynomials in Lean to import and verify externally generated Gr\"obner basis computations. The external solver carries out the main algebraic computations, while the returned results are verified inside Lean. Based on this method, we develop automated tactics that transfer polynomial data between Lean and the external system and certify the returned results. These tactics support tasks such as remainder verification, Gr\"obner basis checking, ideal equality, and ideal or radical membership. This work provides a practical way to integrate external symbolic computation into Lean 4 while preserving the reliability of formal proof. Maths/math.AC_(Commutative_Algebra) https://arxiv.org/abs/2512.23839 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2512.23839v2 Announce Type: replace Abstract: In this article, we disprove a conjecture of F. Alarc\'on and D. Anderson and give a complete classification of the prime ideals in the one variable polynomial semiring with coefficients in Boolean semifield. We group the prime ideals of $\mathbb{B}[x]$ into three classes, indexed by integers. Maths/math.AC_(Commutative_Algebra) https://arxiv.org/abs/2504.15981 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2504.15981v2 Announce Type: replace-cross Abstract: Guided by the $Q$-shaped derived category framework introduced by Holm and Jorgensen, we provide a differential module analogue of a classical result that characterises when a finitely generated module over a local commutative noetherian ring has finite injective dimension. As an application, we characterise local Cohen-Macaulay rings using the homological algebra of differential modules. Maths/math.AC_(Commutative_Algebra) https://arxiv.org/abs/2506.23533 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2506.23533v4 Announce Type: replace-cross Abstract: In this paper, we introduce the notion of a characteristic-zero lifting of an object in positive characteristic by means of ``skeletons''. Using this notion, we relate invariants of singularities in positive characteristic to their counterparts in characteristic zero. As an application, we prove that the set of log discrepancies for pairs consisting of a smooth surface and a multi-ideal is discrete. We also show that the set of minimal log discrepancies and the set of log canonical thresholds of such pairs in positive characteristic are contained in the corresponding sets in characteristic zero. Another application is the construction of Campillo's complex model of a plane curve in positive characteristic via the skeleton lifting method. Maths/math.AC_(Commutative_Algebra) https://arxiv.org/abs/2507.21911 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2507.21911v3 Announce Type: replace-cross Abstract: Let $G=\mathrm{GL}_n(\mathbb{F})$, $\mathrm{O}_n(\mathbb{F})$, or $\mathrm{Sp}_{2n}(\mathbb{F})$ be one of the classical groups over an algebraically closed field $\mathbb{F}$ of characteristic $0$, let $\breve{G}$ be the MVW-extension of $G$, and let $\mathfrak{g}$ be the Lie algebra of $G$. In this paper, we classify the closed orbits in the enhanced standard representation $\mathfrak{g}\times E$ of $G$, where $E$ is the natural representation if $G=\mathrm{O}_n(\mathbb{F})$ or $\mathrm{Sp}_{2n}(\mathbb{F})$, and is the direct sum of the natural representation and its dual if $G=\mathrm{GL}_n(\mathbb{F})$. Additionally, for every closed $G$-orbit in $\mathfrak{g}\times E$, we prove that it is $\breve{G}$-stable, and determine explicitly the corresponding stabilizer group as well as the action on the normal space. Maths/math.AC_(Commutative_Algebra) https://arxiv.org/abs/2603.23008 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2603.23008v2 Announce Type: replace-cross Abstract: We compare several classes of biparameter persistence modules: $\gamma$-products of monoparameter modules, hook-decomposable modules, modules admitting a Smith-type structure theorem, and modules of projective dimension at most 1. We determine all logical implications among these classes, providing explicit counterexamples showing that the converses fail when appropriate. In particular, $\gamma$-products (i.e., hook-decomposable modules) form a very small subclass of biparameter modules, precisely the ones for which a structure theorem still holds, thus making explicit the richer structural complexity of the biparameter setting compared to the monoparameter one. Maths/math.AC_(Commutative_Algebra) https://arxiv.org/abs/2604.14024 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14024v1 Announce Type: new Abstract: Koll\'{a}r showed that small deformations of elliptically fibered smooth $K$-torsion varieties with $H^2(X,\mathcal{O}_X)=0$ remain elliptically fibered. We extend this result to any fibered smooth $K$-torsion variety $X$ with $H^2(X,\mathcal{O}_X)=0$, using Hodge theoretic techniques and the $T^1$-lifting criterion of Kawamata--Ran. More generally, our strategy implies that even without the cohomological assumption, small deformations of a semiample line bundle on a smooth $K$-torsion variety remain semiample up to homological equivalence. Maths/math.AG_(Algebraic_Geometry) https://arxiv.org/abs/2604.14040 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14040v1 Announce Type: new Abstract: We prove a lower bound on the Calabi functional for degenerations of polarized varieties, involving the difference of CM degrees between generically isomorphic families. This may be viewed as a discretely valued version of Donaldson's lower bound for models, in the sense of non-Archimedean geometry. In particular, this generalizes a result of Donaldson, who considered a single polarized variety. As a main tool, we develop the theory of GIT height, introduced by Wang, and apply it to the family GIT problem of the Chow variety. Using the GIT height, we also give a numerical proof of separatedness of GIT quotients for general and special linear actions, strengthening prior work of Wang--Xu. Maths/math.AG_(Algebraic_Geometry) https://arxiv.org/abs/2604.14085 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14085v1 Announce Type: new Abstract: Consider a pair of $S$-dual hyperspherical varieties $G\circlearrowright X$ and $G^\vee\circlearrowright X^\vee$ equipped with equivariant quantizations $Q(X)$, $Q(X^\vee)$. Assume that the local conjecture of Ben-Zvi, Sakellaridis and Venkatesh holds for this pair, and also that $X\simeq T^*_\psi(Y)$ is polarized, so that $Q(X)=D_\psi(Y)$. Let $B\subset G$ (resp. $B^\vee\subset G^\vee$) be Borel subgroups. Then we deduce an equivalence between the ${\mathbb Z}/2$-graded $B$-equivariant category $(D_\psi(Y)\operatorname{-mod})^{{\mathbb Z}/2})^B$ and the ${\mathbb Z}/2$-graded unipotent $B^\vee$-monodromic category $(Q(X^\vee)\operatorname{-mod}^{{\mathbb Z}/2})^{B^\vee,\operatorname{mon}}$. Maths/math.AG_(Algebraic_Geometry) https://arxiv.org/abs/2604.14127 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14127v1 Announce Type: new Abstract: On a compact connected Riemann surface $C$ of genus at least $2$, we construct Lagrangian correspondences between moduli spaces of rank-$n$ Higgs bundles (respectively, holomorphic connections) and the Hilbert schemes of points on $T^\ast C$ (respectively, the twisted cotangent bundles of $C$). Central to these constructions are Higgs bundles (respectively, holomorphic connections) which are transversal to line subbundles of the underlying bundles: these naturally induce divisors on $C$ together with auxiliary parameters, namely lifts to divisors on spectral curves for Higgs bundles and residue parameters of apparent singularities for holomorphic connections. We discuss the evidence showing that the Dolbeault geometric Langlands correspondence is generically realized by these Lagrangian correspondences; we expect that the de Rham geometric Langlands correspondence can be realized by their quantization, following Drinfeld's construction of Hecke eigensheaves. We also discuss the relations of our constructions to various topics, including reductions of Kapustin-Witten equations, the conformal limit, separation of variables, and degenerate fields in conformal field theories. Maths/math.AG_(Algebraic_Geometry) https://arxiv.org/abs/2604.13198 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13198v1 Announce Type: cross Abstract: We construct examples of complete Calabi-Yau metrics on smoothings of 3-dimensional Calabi-Yau cones that are not products of lower-dimensional Calabi-Yau cones and that have orbifold singularities away from the vertex. Maths/math.AG_(Algebraic_Geometry) https://arxiv.org/abs/2604.13209 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13209v1 Announce Type: cross Abstract: We give a definition of all-genus reduced Gromov-Witten invariants of symplectic manifolds by using effectively supported multivalued perturbations on derived orbifold/Kuranishi charts, which bypasses the hard analytical result of sharp compactification/ghost bubble censorship of Zinger and Ekholm-Shende. Maths/math.AG_(Algebraic_Geometry) https://arxiv.org/abs/2604.13373 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.13373v1 Announce Type: cross Abstract: A noncommutative projective variety is defined, after Artin and Zhang, by a graded coherent algebra A, where the category of coherent sheaves is the quotient qgr(A) of the category of finitely presented graded modules by the subcategory of torsion modules. We consider the categorical and polynomial entropies of the Serre twist, that is, of the degree shift functor on the bounded derived category of qgr(A). These two types of entropy can be viewed as analogues of the dimension of the noncommutative variety. We relate these invariants with the growth of the algebra. For algebras of finite global dimension, the entropies are bounded above by the growth entropy and the Gelfand--Kirillov dimension of the algebra. Moreover, these equalities hold for regular algebras, as well as for coordinate rings of smooth projective varieties. However, the polynomial entropy is zero for monomial algebras of polynomial growth, so in this case the inequality is strict. Maths/math.AG_(Algebraic_Geometry) https://arxiv.org/abs/2604.14037 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2604.14037v1 Announce Type: cross Abstract: Parameter space is not function space for neural network architectures. This fact, investigated as early as the 1990s under terms such as ``reverse engineering," or ``parameter identifiability", has led to the natural question of parameter space symmetries\textemdash the study of distinct parameters in neural architectures which realize the same function. Indeed, the quotient space obtained by identifying parameters giving rise to the same function, called the \textit{neuromanifold}, has been shown in some cases to have rich geometric properties, impacting optimization dynamics. Thus far, techniques towards complete classifications have required the analyticity of the activation function, notably excising the important case of ReLU. Here, in contrast, we exploit the non-differentiability of the ReLU activation to provide a complete classification of the symmetries in the shallow case. Maths/math.AG_(Algebraic_Geometry) https://arxiv.org/abs/2212.07745 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2212.07745v3 Announce Type: replace Abstract: This article surveys various duality statements attached to a pair consisting of a smooth complex quasi-projective variety and a regular function on it. It is dedicated to the memory of Bumsig Kim. Maths/math.AG_(Algebraic_Geometry) https://arxiv.org/abs/2401.12814 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2401.12814v3 Announce Type: replace Abstract: We show that $b$-Hurwitz numbers with a rational weight are obtained by taking an explicit limit of a Whittaker vector for the $\mathcal{W}$-algebra of type $A$. Our result is a vast generalization of several previous results that treated the monotone case, and the cases of quadratic and cubic polynomial weights. It also provides an interpretation of the associated Whittaker vector in terms of generalized branched coverings that might be of independent interest. Our result is new even in the special case $b=0$ that corresponds to classical hypergeometric Hurwitz numbers, and implies that they are governed by the topological recursion of Eynard-Orantin. This gives an independent proof of the recent result of Bychkov-Dunin-Barkowski-Kazarian-Shadrin. Maths/math.AG_(Algebraic_Geometry) https://arxiv.org/abs/2502.12136 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2502.12136v2 Announce Type: replace Abstract: Let $k$ be a perfect field of characteristic $p > 0$. For a strictly semi-stable scheme over $k[[t]]$, we construct the weight spectral sequence in $p$-adic cohomology using the theory of arithmetic $\mathcal{D}$-modules, whose $E_1$ terms are described by rigid cohomologies of irreducible components of the closed fiber and whose $E_\infty$ terms are conjecturally described by the (unipotent) nearby cycle of Lazda-P\'{a}l's rigid cohomology over the bounded Robba ring. We also show its functoriality by pushforward and state the conjecture of its functoriality by pullback and dual. Maths/math.AG_(Algebraic_Geometry) https://arxiv.org/abs/2507.13538 Thu, 16 Apr 2026 00:00:00 -0400 arXiv:2507.13538v2 Announce Type: replace Abstract: We study automorphisms of quasi-smooth hypersurfaces in weighted projective spaces, extending classical results for smooth hypersurfaces in projective space to the weighted setting. We establish effective criteria for when a power of a prime number can occur as the order of an automorphism, and we derive explicit bounds on the possible prime orders. A key role is played by a weighted analogue of the classical Klein hypersurface, which we show realizes the maximal prime order of an automorphism under suitable arithmetic conditions. Our results generalize earlier work by Gonz\'alez-Aguilera and Liendo. Maths/math.AG_(Algebraic_Geometry)