Papers

We compute the pro-étale fundamental group of a connected Nagata J-2 scheme in terms of the étale fundamental groups of the normalizations of its irreducible components and a discrete free group. The result generalizes a formula of E. Lavanda for semi-stable curves and relies on a combination of proper descent techniques for étale morphisms and a combinatorial van Kampen construction for Noohi groups. As a by-product we characterize when a continuous representation of the pro-étale fundamental group factors through a discrete quotient.

In this article, by means of the matrix-weighted grand maximal function we first introduce the variable Hardy space $H^{p(\cdot)}_W$ on $\mathbb{R}^n$ with the $\mathscr{A}_{p(\cdot),\infty}$ matrix weight $W$ and with the variable exponent $p(\cdot)$ having globally log-Hölder continuity, and then via using several different convex body valued maximal functions we establish its various maximal function equivalent characterizations. Combining a refined Whitney decomposition with both the convex body valued maximal function and its corresponding convex-body reducing operator, we obtain the atomic characterization of $H^{p(\cdot)}_W$. As applications, we give its dual space and establish the boundedness of Calderón–Zygmund operators from $H^{p(\cdot)}_W$ to the matrix-weighted variable Lebesgue space $L^{p(\cdot)}_W$ and to itself. This approach to establishing atomic characterization differs from all previous ones.

We give a negative answer to a question of Ciliberto, Knutsen, Lesieutre, Lozovanu, Miranda, Mustopa, and Testa on effective divisors of positive self-intersection on smooth projective surfaces. The main result of this paper is obtained by generative AI, particularly Chatgpt 5.5 pro and the Rethlas system.

This paper studies the stochastic optimal control of jump-diffusion processes and the associated fully nonlinear backward stochastic Hamilton–Jacobi–Bellman (BSHJB) equations. We establish the dynamic programming principle (DPP) via backward semigroups to characterize the value function. To handle non-local integro-differential operators and polynomial growth, we introduce a stochastic viscosity solution framework based on semimartingale test functions and global tangency conditions. Existence is proved using the measurable selection theorem and the generalized Itô–Kunita formula. Finally, under a super-parabolicity condition, we establish a weak comparison principle and prove global uniqueness via localized bounding envelopes and backward induction.

A viscous fluid confined to a thin layer around a curved surface is governed, as the layer thickness vanishes, by an effective viscous operator on the surface. We show that the wall conditions select this operator. The vorticity-free (free) and stress-free (Navier) conditions, which coincide on flat walls but differ on curved ones by the shape operator, yield respectively the Hodge Laplacian and the deformation Laplacian, and these differ universally, on any hypersurface, by twice the Ricci curvature; a one-parameter family of wall conditions joins them, with an effective operator that couples to the extrinsic geometry only in between. We prove this in two forms: formally, by matched asymptotics, on an arbitrary hypersurface, and rigorously, as Mosco convergence of the viscous energy forms, hence with resolvent, semigroup and spectral convergence, on surfaces of revolution. The stress-free limit on general surfaces is due to Miura and is recovered here; the rigorous vorticity-free limit beyond the sphere, via a uniform Gaffney inequality, together with the interpolating family and the spectral packaging, is new, and the analysis makes precise a conflation of the two conditions in the classical spherical treatment. The extension-dependence of the operator found on the ellipsoid is explained as a dependence on the wall condition.

We compute the exact log-asymptotics of the persistence probability, and determine the entropic repulsion profile conditioned on persistence, for general $d$-dimensional stationary Gaussian fields with spectral singularity at the origin of order $\alpha \in [0,d)$. Under mild regularity conditions these are shown to be universal, depending only on $\alpha$ and $d$, and to have explicit formulations in terms of the capacity and equilibrium potential of the $\alpha$-Riesz kernel. This generalises a result of Bolthausen, Deuschel and Zeitouni on the Gaussian free field to a wide class of Gaussian fields with spectral singularity.

Virtual double categories provide an effective framework for formal category theory. Recent work has investigated the question of higher morphisms between virtual double categories, following on from work on higher morphisms between double categories, and building up to Arkor's recent conjecture on exponentiable virtual double categories–those virtual double categories, morphisms out of which can themselves be enriched to a whole virtual double category. In this paper we resolve Arkor's conjecture by providing a number of equivalent characterizations of the exponentiable virtual double categories in terms of existence of decompositions of cells. We also show that virtual double categories of cospans are always exponentiable, as are the virtual double categories arising from pseudo double categories or from exponentiable multicategories, as studied by Pisani. We give conditions under which the virtual double category of virtual double functors admits composites, following work of Paré for the non-virtual case. We base our work on a general approach to exponentiability for categories of models of limit sketches, which we apply to give new treatments of exponentiability for semicategories, categories, multicategories, and their functors.

We answer a question of Kollár and Kovács by constructing a flat projective morphism to a smooth curve whose fibers are Cohen–Macaulay and reduced, whose generic fiber is smooth, and for which the first cohomology of the structure sheaf of the fibers is not constant. The main result of this paper is obtained by generative AI, particularly Chatgpt 5.5 pro and the Rethlas system.

This paper presents a novel multilevel projection-based stabilization method for advection-dominated convection–diffusion problems within the framework of Isogeometric Analysis. The proposed approach extracts and penalizes fine-scale fluctuations using continuous B-spline quasi-interpolants, avoiding both the highly sensitive parameters used in residual-based stabilization methods and the discontinuous auxiliary spaces required by classical Local Projection Stabilization. Stabilization is applied hierarchically across nested levels of the discrete space via explicit mesh-dependent weights. We establish the theoretical foundation of the method by deriving a priori error estimates, supplemented by a discrete inf-sup condition established for the one-dimensional setting with constant advection under a numerically validated stability hypothesis that ensures robust streamline derivative control. Numerical experiments on stringent benchmarks demonstrate the method's ability to significantly reduce spurious oscillations across a variety of regimes, including the limiting cases of pure advection and advection–reaction. Notably, despite being a fully linear formulation, the method achieves significant reduction of undershoots near sharp layers, providing an effective stabilization mechanism without requiring complex nonlinear terms. Furthermore, by utilizing a robust global parameter scaling, the proposed approach significantly alleviates the parameter sensitivity that typically affects residual-based alternatives, reducing the strong dependence on problem-dependent tuning.

We show that there are sets of $n$ points in the plane with $n$ arbitrarily large that contain more than $n^{1.014}$ pairs of points separated by a distance exactly $1$. This improves on very recent work of a team at OpenAI, who proved the same result with an inexplicit exponent greater than $1$, drastically improving on the best previous lower bound and disproving a conjecture of Erdős. The method is number-theoretic, relying on constructing algebraic number fields of large degree and small discriminant with many primes of small norm via a Golod-Shafarevich criterion argument.

We study design-unbiased estimation of the finite-population total $\sum_{i=1}^N y_i$ when each outcome satisfies known bounds $y_i\in[a_i,b_i]$. For any sampling design with inclusion probabilities $\pi_i>0$, we prove a sharp lower bound on the worst-case squared error over the rectangular parameter space. This bound is attained if and only if the unit inclusion indicators are pairwise independent, in which case the minimax estimator is the midpoint-differenced Horvitz-Thompson estimator $\sum_{i=1}^N m_i+\sum_{i\in S}(y_i-m_i)/\pi_i$, with $m_i=(a_i+b_i)/{2}$. We then solve the joint design-and-estimation problem under the constraint $\sum_i \pi_i\le n$. We find that a minimax strategy samples units independently with probabilities $\pi_i^\ast=\min(1,c (b_i-a_i))$ where $c>0$ is chosen so that $\sum_i \pi_i^\ast=n$, and uses the midpoint-differenced estimator. This extends Gabler (1990)'s linear minimax result to the full class of design-unbiased estimators. We also show that the estimator is admissible among unbiased estimators and affine equivariant.

We show that every dense subgroup of a connected Lie group G contains a dense subgroup generated by 2d elements, where d=dim(G). We also give a detailed proof for the quantitive characterization of a contracting projective transformation in terms of the ratio between the two leading terms in its Cartan decomposition.

Papadopoulos showed in 2021: The action of the Lorentz group SO(3,1) on the lightcone of Minkowski space lifts to an SO(3,1)-action on the blow-up of the lightcone. From this, he draws conclusions regarding further hypothetical vacuum states in quantum field theories. The present note shows that the possbility to lift the SO(3,1)-action to the blow-up is a pure mathematical result. It does not depend on the context from physics, in particular it does not make use of the Dirac equation. Instead, it is a simple application of the blow-up theory from algebraic geometry. Lifting is possible because the group action is linear and the exceptional divisor of the blow-up is the projectivized tangent cone.

This study investigates the conditions under which the group inverse of a singular, irreducible, symmetric $M$-matrix retains the $M$-matrix property. By concentrating on a structured subclass derived from rank-one perturbations of a diagonal matrix, and inspired by recoverable complete networks, we obtain explicit analytical results. Utilizing both matrix-theoretic methodologies and potential theory on networks, we establish necessary and sufficient conditions for the $M$-property of the specified network in terms of conductances and associated Doob potentials. This framework facilitates the construction of families of singular, irreducible $M$-matrices whose group inverses maintain the $M$-matrix structure. Our findings offer novel insights into this research domain and enhance the relationship between $M$-matrix theory and network analysis.

Systems moving in low Reynolds number fluid regimes are known to be governed by a “motility map” which linearly relates their shape change rates to they body frame velocity moving through the fluid. A consequence of this is “Purcell's Scallop Theorem” – a locomotion system that undergoes shape changes that follow the same path forward and backward in time (reciprocal body deformations) cannot achieve net displacement, regardless of pacing of those changes.We show that linear-in-velocity motility maps extend to any power law viscosity (a.k.a. Ostwald–de Waele fluid), and therefore to many biological fluids in intermediate shear ranges. We also show that the linear-in-velocity property can be violated in Carreau-Yasuda fluids to produce net motion using an “inchworm” model consisting of two unequal masses with unequal drag coefficients performing reciprocal motions. Interestingly, the direction of motion can be switched by changing speeds. Our results show that the linear motility map of geometric mechaincs can be used to analyze and design locomotion in power-law fluids, and that some nonlinear drag relationships such as Carreau-Yasuda can be exploited to generate net locomotion in seeming violation of the “scallop theorem”.

We address the issue of global convergence in stochastic continuous optimization. For that purpose, we formulate the Canonical Evolutionary Strategy (CES) as a controlled mathematical framework to analyze global convergence in evolutionary algorithms via the semiclassical limit of a Schrödinger-type replicator-mutator equation. We provide a rigorous hierarchy from a discrete individual-based dynamics to a deterministic mean-field limit, demonstrating that global convergence is governed by the principal eigenfunction of the underlying operator. This property, defined as Geometric Selection, naturally prioritizes robust, flat optima over narrow local traps, offering a mathematical justification for the ”survival of the flattest” phenomenon. Moreover, unlike consensus-driven methods that are prone to premature variance collapse when the global minimizer resides outside the initial support, the replicator-mutator dynamics of CES facilitate intrinsic mass transport. High-dimensional benchmarks (d = 30) confirm this advantage, showing that CES achieves lower residual errors in shifted initialization scenarios where standard consensus-driven and gradient-based methods fail to migrate effectively. By shifting the focus from point-wise consensus to spectral concentration, our framework provides a robust theoretical foundation for global convergence in Evolution Strategies (ES) without the need for additional numerical heuristics.

This work presents an end-to-end strategy for solving inverse problems constrained by Partial Differential Equations within a fully differentiable Machine Learning framework. The proposed formulation provides a unified and user-friendly methodology applicable to a wide range of problems, from data assimilation to closure modeling. Our approach combines a baseline differentiable PDE solver, which predicts the state w from the nonlinear system $R(w) = 0$, with a generic additive, parametrized, and differentiable correction $f_\phi(w)$, with trainable parameters $\phi$. We show how to optimize phi within a fully differentiable Python workflow by reformulating the PDE as an implicit layer, enabling its integration into arbitrary objective functions, while leveraging PyTorch's automatic differentiation graph. The method is demonstrated on the Reynolds-Averaged Navier-Stokes equations for compressible flows, where the closure term, or a portion of it, is modeled using trainable parameters or a Neural Network. The first application considers the 2D NASA Wall-Mounted Hump test case, where a production-term parameter is optimized against time-averaged LES data. A second application is carried out on the VKI LS-59 turbine blade, where the Spalart-Allmaras eddy viscosity field is reconstructed through the optimization of a trainable spatial field. A dataset is generated starting from the VKI LS-59 turbine blade geometry using the differentiable BROADCAST solver with the Spalart-Allmaras turbulence model. The results highlight the flexibility of the framework, showing its applicability beyond turbulence modeling to a broader class of physics-informed PDE-constrained problems with data-driven components.

The purpose of this article is to develop and analyze $\mathcal{R}(p,q)-$topological analysis of the classical nuclear space within the general framework of $\mathcal{R}(p,q)-$calculus. We begin by introducing the $\mathcal{R}(p,q)-$Gamma functions, establishing their main properties and their connection with the deformed factorials. We develop a rigorous analytic and functional-analytic framework for holomorphic functions governed by a general $\mathcal{R}(p,q)-$deformation, where $\mathcal{R}(u,v)$ is a meromorphic kernel satisfying $0<q<p\leq 1$, $\mathcal{R}(1,1)=0$, and $\mathcal{R}(p^n,q^n)>0$. A Stirling-type asymptotic expansion is established for the $\mathcal{R}(p,q)-$deformed Gamma function $\Gamma_{\mathcal{R}(p,q)}$, yielding precise exponential quadratic growth estimates driven by the asymptotics of the deformed factorial $\mathcal{R}!(p^n,q^n)\sim \exp(\lambda n^2)$. These asymptotics induce sharp coefficient bounds and Cauchy-type inequalities for $\mathcal{R}(p,q)-$entire functions. Based on these estimates, we introduce $\mathcal{R}(p,q)-$weighted Banach and Fréchet spaces of holomorphic functions, together with deformation dependent pseudo-norms and valuation maps. Within this setting, we define $\mathcal{R}(p,q)-$discs and anisotropic sectors adapted to the deformation geometry and prove $\mathcal{R}(p,q)-$analogues of the Cauchy-Hadamard theorem, the Borel-Carathéodory inequality and Phragmén-Lindelöf type growth principles. These results contribute to the broader program of constructing a consistent functional calculus in $\mathcal{R}(p,q)-$quantum algebras, with potential applications to deformed fractional differential equations, operator theory, spectral problems, and non commutative models arising in mathematical physics.

We prove that the Natsume-Olsen non-commutative spheres $\mathbb{S}^{2n-1}_{\theta}$ dualize for rational deformation parameters to provide examples of quantum branched covers over their respective centers' maximal spectra, embeddable into locally trivial $C^*$ bundles precisely when said spheres are classical; this, despite the bundles in question being of finite type over dimensional strata. The proof of the noted rigidity phenomenon relies on the quantum analogue of the 3-sphere's Heegaard splitting as a pushout of two solid tori.

If $\Gamma$ is a graph for which every edge is in exactly one clique of order $\omega$, then one can form a new graph with vertex set equal to these cliques. This is a generalization of the line graph of $\Gamma$. We discover many general results and classifications related to these clique graphs that will be useful to researchers studying graphs with this property. In particular, we find bounds on the spectrum of $\Gamma$ (with exact results when $\Gamma$ is $k$-regular) and some complete classifications when $\Gamma$ is strongly regular. We apply our results to derive novel information about the existence questions of certain strongly regular graphs. We also examine the critical group of graphs with this property and their associated transformations. Finally, we discuss examples of widely studied families of graphs that have this property, and provide some examples to make the results more concrete.

A signed graph is a pair $\Gs$ in which $G$ is a finite simple graph and $\sigma:\E(G)\to\{+1,-1\}$ is a signature. Following Máčajová–Raspaud- Škoviera and Jin–Kang–Steffen, a proper coloring of $\Gs$ is a map $c:\V(G)\to\Z$ with $c(u)\ne\sigma(uv)\,c(v)$ for every edge $uv$, and $\Gs$ is \emph{signed $k$-choosable} if such a coloring exists from any list assignment $L$ with $|L(v)|\ge k$. In a celebrated two-page note, Thomassen proved that every planar graph is $5$ choosable, and Jin, Kang, and Steffen subsequently extended this to signed planar graphs. Our principal contribution is a short, self-contained, and signature-blind proof of the latter: the inductive bookkeeping inserts one factor of $\sigma(\cdot)$ uniformly into every constraint, so that with $\sigma\equiv +1$ the argument reduces verbatim to Thomassen's original. From the strengthened extension statement (\cref{thm:main}) we deduce the main result (\cref{thm:JKS}: $\chs\Gs\le 5$ for every planar signed graph), the Máčajová–Raspaud–Škoviera signed Five-Color Theorem in the symmetric palette $\Ns{2}=\{-2,-1,0,1,2\}$, the Switching Invariance Lemma, $3$-choosability of outerplanar signed graphs, $1$-defective signed $4$-choosability of planar signed graphs, a sandwich inequality relating $\chs$ to the unsigned and positive” choice numbers, and a polynomial-time list-coloring algorithm. Voigt's planar non-$4$-choosable graph and Mirzakhani's smaller variant show the bound $5$ is best possible. We close with examples illustrating that negative edges genuinely refine unsigned phenomena, a comparison table situating our work in the literature, and several open problems.

Channel foundation models assume access to fully observed channels, an assumption that fails in deployment. We introduce PilotWiMAE, a self-supervised framework whose encoder ingests noisy pilot observations directly and whose attention factorizes along the axis separating temporal from joint space-frequency processing, an inductive bias inspired by the physics of the problem. Pilot input shrinks the observation space by up to two orders of magnitude and also removes the unrealistic assumption of full-CSI availability while incurring lower latency. The factorized design generates robust representations by exploiting the separable channel structure and allows a pretraining mask ratio of $99\%$. We pair patch-normalized reconstruction, which captures small-scale fading structure, with an auxiliary scale loss that recovers the large-scale fading features, and use an AWGN curriculum to match pilot noise at pretraining and deployment. Pretrained solely on $3.5$\,GHz and evaluated at $28$\,GHz across in-distribution and out-of-distribution settings, PilotWiMAE's cross-frequency beam selection and channel characterization beat supervised baselines despite operating on a smaller observation space. To weaken the coupling between decoder capacity and representation quality, we further propose a decoder-centric pretraining stage following the encoder-decoder joint pretraining, which allows PilotWiMAE to demonstrate competitive channel estimation without sacrificing representation quality. To foster further work in this direction, we release the PilotWiMAE pretrained weights and training pipeline, together with CSIGen, our Sionna-based ray-tracing channel-generation tool, and the channel datasets used in this work.

Matrix $B=(b_{ij})$ with $b_{ij}=(2j+1)(2j+2i-1)$ was introduced in \cite{Shi2024} as an additive sieve for odd primes. In this paper we introduce the minimal-anchor function $\pmin(d)$, the least odd prime $p$ such that $p+d$ is prime (sequence A020483 of the OEIS at index $d/2$), whose finiteness for all even $d$ is exactly the weak Polignac (Maillet) conjecture, i.e.\ the statement that every row of $B$ contains a semiprime. We prove that for each fixed $z$ the set $\{d\ \text{even}:\pmin(d)\le z\}$ has density zero, with the asymptotic $(\pi(z)-1)X/\log X$; consequently no fixed finite set of anchor primes can cover a positive proportion of the rows. Density-one coverage of the rows nevertheless holds, by a classical theorem of Lavrik which we restate in the matrix-$B$ framework: almost every row contains the number of semiprimes predicted by the Hardy–Littlewood conjecture. We formulate quantitative conjectures on $\pmin$ and support them with numerical data. Every-row coverage (= weak Polignac) is explicitly left open. We prove unconditional lower bounds for $\pmin$: for every $\psi\to0$, $\pmin(d)>\psi(d)\log d\log\log d$ for almost all even $d$, which is the conjectured typical order; and $\max_{d\le X}\pmin(d)\ge(\frac12+o(1))\log X\log\log X$. The same counting gives the corresponding lower bounds for the least prime in a Goldbach partition.

We develop a general finite-alphabet framework for Euler-type sums based on the notion of a monoidal alphabet. An alphabet of summand letters is called monoidal when it is closed under pointwise multiplication, thereby inducing the usual stuffle, or quasi-shuffle, algebra on the associated nested sums. This viewpoint places classical multiple harmonic numbers, colored harmonic sums, and several generalized Euler sums under a common structural mechanism. We focus on three fundamental families of monoidal alphabets: the ordinary power alphabet generated by $n$, the affine alphabet generated by linear factors $an+b$, and the polynomial-base alphabet generated by polynomial factors $P(n)$. The resulting classes of multiple harmonic numbers, multiple affine harmonic numbers, and multiple polynomial-base harmonic numbers provide systematic containers for a wide range of finite and infinite Euler-type sums. We prove closure and lifting results showing that nested sums whose summands are built from these alphabets, possibly multiplied by harmonic-number factors, reduce to the corresponding finite harmonic-number objects. As consequences, the framework recovers many known Euler-sum identities and produces many new identities in a uniform way. While reduction to simpler functions remains a separate and often difficult problem, the monoidal-alphabet perspective provides a unified algebraic language for organizing, transforming, and extending harmonic-sum identities.

In this paper, we investigate the combinatorial structure and asymptotic distribution of the solution set of the equation $\sigma(n+1) = k\sigma(n)$ for a given integer $k>1$. From a combinatorial perspective, the solutions to this equation are closely related to the concept of $k$-layered numbers, which are a generalization of Zumkeller numbers. In the analytic section, which constitutes the core of this research, we employ the framework of probabilistic number theory and an extension of the classical Kubilius model to study the oscillatory and local behavior of the sum-of-divisors function. Utilizing the truncation technique for arithmetic functions and applying the Chinese Remainder Theorem, the problem is reduced to a synthetic measure space equipped with independent random variables. Subsequently, by applying the optimized version of the Kolmogorov-Rogozin anti-concentration inequality (Petrov's theorem) to the difference of additive variables and finely tuning the error parameters, we prove that the natural density of this set is zero. The main quantitative outcome of this approach is the derivation of the explicit upper bound $A_k(x) \ll_k \frac{x}{\sqrt{\log \log \log x}}$ for the counting function of the solutions. Finally, alongside the zero asymptotic density, relying on the framework of polynomials and Schinzel's H Hypothesis, we establish the conditional infinitude of the solution set for the case $k=2$ and formulate the existential results.

We prove a variety of results about subgroups of Thompson's group $V$. First we prove that every action graph of a finitely generated subgroup of $V$ acting on an orbit in Cantor space is quasi-isometric to a tree. Then we prove that for a broad class of groups of homeomorphisms of the real line, for example Thompson's group $F$, any action on the Cantor space via an embedding into Thompson's group $V$ must be semiconjugate to the standard action on the line. Finally, we use this to establish that many such groups cannot embed into $V$; in particular the Stein group $F_{2,3}$ cannot embed in $V$, answering a question of the third author.

The reconfigurable coupler antenna (RCA), also called the flexible coupler antenna (FCA), is a new technique that aims to improve the performance of wireless communication networks by reconfiguring the positions and rotations of low-cost couplers around fixed-position active antennas to harness mutual coupling. Specifically, different couplers can independently adjust their positions and/or rotations at the transceiver to reshape the induced currents on the couplers for radiation, thereby collaboratively achieving mechanical beamforming for directional signal enhancement or nulling. The position and/or rotation reconfiguration of passive couplers provides a new and cost-effective means of enhancing wireless communication performance, while significantly reducing the antenna and radio-frequency (RF) chain costs of conventional active arrays. The compact and low form-factor structure of the RCA makes it particularly appealing for devices with stringent size, weight, and power (SWAP) constraints. In this article, we provide an overview of RCA to reveal its promising capabilities in wireless networks, including its system modeling, practical implementation, and competitive advantages over existing techniques. We present a variety of RCA-enabled performance enhancements in terms of mechanical beamforming gain, path-loss reduction, fading mitigation, spatial multiplexing gain, interference suppression, and geometric gain. Furthermore, we elaborate on the design challenges of RCA as well as promising solutions, and discuss the key applications of RCA in wireless networks. Finally, numerical results are presented to verify the substantial capacity gains enabled by RCA-aided transmission in wireless networks.

In this paper, we investigate the long-time behavior of the $L^2$-norm of solutions to the Cauchy problem for the strongly damped wave equation on $\mathbb{R}^n$, with particular focus on the low-dimensional cases $n=1$ and $n=2$. Although the energy is dissipative, the $L^2$-norm may grow because of low-frequency effects. We compare the diffusion-wave profile of the strongly damped equation with the corresponding free-wave evolution generated by the same initial velocity. Introducing the difference operator $D(t)$ between these two evolutions, we prove that in one dimension $D(t)$ is controlled by $Ct^{1/4}\|g\|_{L^1}$, showing that the free wave remains an effective asymptotic profile. In contrast, in two dimensions $D(t)$ has a logarithmic lower bound when the mass of the initial velocity is nonzero, implying that the wave approximation fails. Corresponding estimates for the original solution are also obtained.

In this paper, we study the long-time stability behavior of a class of linear stochastic evolution equations in a Hilbert space with multiplicative noise. Explicit sufficient conditions for $p$-th moment and almost sure exponential stability are established, highlighting the interplay between the principal eigenvalue of the governing operator, the drift coefficient, and the noise intensity. The relationship between these two notions of stability is also clarified. Applications to several stochastic partial differential equations are presented. In addition, a fully discrete spectral Galerkin method together with the implicit Euler–Maruyama scheme is shown to preserve these stability properties at the discrete level. Finally, numerical simulations are provided to confirm the theoretical results.

Kernel density estimation risk expansions are commonly expressed through the integrated squared curvature term that enters second-order AMISE and plug-in bandwidth rules. This paper develops a weak-curvature formulation of this classical calculation for densities whose second derivative exists weakly rather than as a continuous classical function. We prove that if a density has square-integrable weak curvature, then the standard second-order AMISE expansion, oracle bandwidth order, and kernel-dependent optimality calculation remain valid with the curvature functional understood in the weak sense. The class $C^{1,1}(\mathbb{R})\setminus C^2(\mathbb{R})$ serves as a concrete and practically relevant subclass: the first derivative is Lipschitz, while curvature may be kinked, discontinuous, or undefined at isolated points. Building on this formulation, we introduce a generalized-curvature plug-in (GCPI) bandwidth selector. The selector estimates the weak-curvature functional by a pilot density-derivative estimator with a leave-one-out U-statistic correction and substitutes this estimate into the AMISE bandwidth formula. We prove first-order oracle equivalence under ratio-consistent weak-curvature estimation and establish consistency of the proposed U-statistic curvature estimator under explicit pilot-bandwidth conditions. We also give a scalar-bandwidth multivariate extension based on weak Hessians and illustrate the theory through nonsmooth density examples, simulations, and a real-data application.