Let $\Omega$ be a bounded open set of class $C^{1,1}$ in $\mathbb{R}^N$ and $s\in(\frac{1}{2}, 1)$. We study a family of fractional Hardy-type inequalities \begin{equation} \frac{c_{N,s}}{2}\displaystyle\iint_{\Omega\times\Omega}\frac{(u(x)-u(y))^2}{|x-y|^{N+2s}}\ dxdy-\displaystyle\lambda\int_{\Omega}u^2\ dx\geq C\displaystyle\int_{\Omega}\frac{u^2}{\delta^{2s}}\ dx,~~~\quad\forall\lambda\in\mathbb{R},~~~~~~~(0.1) \end{equation} with $u\in C_c^\infty(\Omega)$ and $C=C(\Omega,s,N,\lambda)>0$. We show that the best constant in $(0.1)$ is achieved if and only if $\lambda>\lambda^*(s,\Omega)$, for some $\lambda^*(s,\Omega)\in\mathbb{R}$. As a by-product, we derive in particular that the best constant in Hardy inequality $\mu_{N,s}(\Omega)$ is achieved if and only if $\mu_{N,s}(\Omega)<\mathfrak{h}_{N,s}$, with $\mathfrak{h}_{N,s}$ being the best constant for the fractional Hardy inequality in the half space. Moreover, if $\Omega$ is a convex open set, we obtain a lower bound for $\lambda^*(s,\Omega)$ in terms of the volume of $\Omega$. Specifically, we prove that $\lambda^*(s,\Omega)\geq a(N,s)|\Omega|^{-\frac{2s}{N}}$ with an explicit constant $a(N,s)>0$. Finally, for bounded $C^{1,1}$ domains, we prove that, for $s$ sufficiently close to $\frac{1}{2}$, the optimal Hardy constant is independent of both the geometry and the topology of $\Omega$. More precisely, we establish that $\mu_{N,s}(\Omega)=\mathfrak{h}_{N,s}$. This behavior is in sharp contrast with the local case, where the topology/geometry of the domain strongly influences the value of the optimal constant, and reveals a new rigidity phenomenon in the nonlocal setting.

Let C_1, C_2,... be a sequence of separable unital C*-algebras, equipped with faithful tracial states and satisfying a mild condition. Let A be a unital direct limit of one dimensional NCCW complexes, also equipped with a faithful tracial state. Suppose there is a unital trace preserving embedding of A in the Jiang-Su algebra which is an isomorphism on K-theory. (For example, A could be C([0,1]) with Lebesgue measure, or the Jiang-Su algebra itself.) Let D be the infinite reduced free product of the algebras C_n. Then the reduced free product A*D is isomorphic to D. If D is exact and the factors satisfy a blockwise real rank zero condition, then in place of A we can use C(X) for any contractible compact metric space X and any faithful tracial state on C(X). An example consequence is that the reduced free product of infinitely many copies of C([0,1]), with Lebesgue measure, is isomorphic to the reduced free product of infinitely many copies of the Jiang-Su algebra.

It is known that when minimizing smooth Polyak-Łojasiewicz (PL) functions, momentum algorithms cannot significantly improve the convergence bound of gradient descent, contrasting with the acceleration phenomenon occurring in the strongly convex case. To bridge this gap, the literature has proposed strongly quasar-convex functions as an intermediate non-convex class, for which accelerated bounds have been suggested to persist. We show that this is not true in general: the additional structure of strong quasar-convexity does not suffice to guaranty better worst-case bounds for momentum compared to gradient descent. As an alternative, we study PL functions under an aiming condition that measures how well the descent direction points toward a minimizer. This perspective clarifies the geometric ingredient enabling provable acceleration by momentum when minimizing PL functions.

Recent advances in imaging technologies, deep learning and numerical performance have enabled non-invasive detailed analysis of artworks, supporting their documentation and conservation. In particular, automated detection of craquelure in digitized paintings is crucial for assessing degradation and guiding restoration, yet remains challenging due to the possibly complex scenery and the visual similarity between cracks and crack-like artistic features such as brush strokes or hair. We propose a hybrid approach that models crack detection as an inverse problem, decomposing an observed image into a crack-free painting and a crack component. A deep generative model is employed as powerful prior for the underlying artwork, while crack structures are captured using a Mumford--Shah-type variational functional together with a crack prior. Joint optimization yields a pixel-level map of crack localizations in the painting.

The Landau-Lifshitz-Gilbert (LLG) equation, regarded as a gradient flow with manifold constraint, is the fundamental model describing magnetization dynamics in ferromagnetic materials. It is well known that the normalized tangent plane method is able to simultaneously achieve the non-convex manifold constraint and original energy dissipation. However, the associated computational cost of this numerical approach is exceedingly high. By contrast, the projection method is more straightforward to implement, while it often compromises the inherent energy dissipative property of the continuous model, and the error analysis turns out to be even more challenging. In this work, we first construct a linear and fully discrete finite difference numerical scheme, based on the projection method for the LLG equation, which is capable of simultaneously preserving the non-convex manifold constraint \(|\mathbf{m}| = 1\) and an unconditional original energy dissipation. In the error analysis, the classical theoretical technique becomes ineffective, due to the presence of the nonlinear Laplacian term, which in turn poses a significant challenge. To overcome this subtle difficulty, we carefully rewrite the numerical method in an equivalent weak form, in which a point-wise length preserving feature of the numerical solution plays an essential role. As a result of these estimates in the reformulated weak form, an optimal convergence rate could be theoretically established. In our knowledge, this numerical method is the first linear algorithm that preserves the following combined theoretical properties: (i) point-wise length preservation, (ii) unconditional original energy dissipation, (iii) a theoretical justification of convergence analysis and optimal rate error estimate.

We develop a low-order Crouzeix--Raviart--Raviart--Thomas (CR--RT) equilibrated-flux certification workflow for finite element approximations of semilinear diffusion--reaction problems, with particular emphasis on anisotropic mesh settings. Given a computed conforming finite element state $\tilde u_h$, the certification process is reduced to three computable quantities required by a Newton--Kantorovich argument: a dual-norm residual bound, a stability constant for the Fréchet derivative, and a Lipschitz bound for the derivative in a neighborhood of $\tilde u_h$. These components yield an explicit radius $\rho>0$, ensuring that the exact solution exists locally and uniquely within the ball $B(\tilde u_h,\rho)\subset V$. The residual bound is obtained from an $H(\mathrm{div})$-conforming $\mathbb{RT}^0$ certificate flux reconstructed through a Marini-type CR--RT route. The purpose of this route is not to replace general higher-order or local mixed equilibrated reconstructions, but to provide an explicit low-order construction whose algebraic structure is transparent on anisotropic simplicial meshes. Within the certified neighborhood, we further enclose selected quantities of interest $\mathcal J(u)$; the baseline enclosure follows from the verified inclusion, while an adjoint-based correction sharpens the resulting intervals. The numerical experiments report the behavior of the computable certification quantities for monotone semilinear models, including anisotropic mesh tests. Unless interval or outward-rounded scalar post-processing is explicitly used, the reported computations should be understood as floating-point evaluations of the derived rigorous estimators.

In this note we propose a two-layer viscous mechanism for preventing finite time singularity formation in the Moffatt-Kimura model of two counter-rotating vortex rings colliding at a nontrivial angle. In the first layer the scenario is recast within the framework of the study of turbulent dissipation based on a suitably defined `scale of sparseness' of the regions of intense fluid activity. Here it is found that the problem is (at worst) critical, i.e., the upper bound on the scale of sparseness of the vorticity super-level sets is comparable to the lower bound on the radius of spatial analyticity. In the second layer, an additional more subtle mechanism is identified, potentially capable of driving the scale of sparseness into the dissipation range and preventing the formation of a singularity. The mechanism originates in certain analytic cancellation properties of the vortex-stretching term in the sense of compensated compactness in Hardy spaces which then convert information on local mean oscillations of the vorticity direction (boundedness in certain log-composite weighted local bmo spaces) into log-composite faster decay of the vorticity super-level sets.

We establish the existence of smooth solutions for the LYZ equation at the critical phase $\theta =(n-2)\frac{\pi}{2}$, thereby solving the critical case of a problem posed by Collins-Jacob-Yau and Li concerning the solvability for phase $\theta \leq (n-2)\frac{\pi}{2}$. As applications, we solve the 3D Hessian equation $\sigma_2 = 1$ and the 4D Hessian quotient equation $\sigma_3 = \sigma_1$ under weaker assumptions than previously required.

In this note we give formulas for cup product in Tate cohomology in terms of inhomogeneous cochains. Using one of these formulas, for a torus T defined over a non-archimedean local field K and splitting over a cyclic extension of K, we compute explicit cocycles representing all cohomology classes in H^1(K,T).

This paper introduces a boundary integral equation for time-harmonic electromagnetic scattering by composite dielectric objects. The formulation extends the classical Müller equation to composite structures through the global multitrace method. The key ingredient enabling this extension is the use of the Stratton-Chu representation in complementary region, also known as the extinction property, which augments the off-diagonal blocks of the interior representation operator. The resulting block system is composed entirely of second-kind operators. A Petrov-Galerkin (mixed) discretization using Rao-Wilton-Glisson trial functions and Buffa-Christiansen test functions is employed, yielding linear systems that remain well conditioned on dense meshes and at low frequencies without the need for additional stabilization. This reduces computational costs associated with matrix-vector multiplications and iterative solving. Numerical experiments demonstrate the accuracy of the method in computing field traces and derived quantities.

We show that quantum search can be used to compute the hardness to round an elementary function, that is, to determine the minimum working precision required to compute the values of an elementary function correctly rounded to a target precision of $n$ digits for all possible precision-$n$ floating-point inputs in a given interval. For elementary functions $f$ related to the exponential function, quantum search takes time $\tilde O(2^{n/2} \log (1/\delta))$ to return, with probability $1-\delta$, the hardness to round $f$ over all $n$-bit floating-point inputs in a given binade. For periodic elementary functions in large binades, standalone quantum search yields an asymptotic speedup over the best known classical algorithms and heuristics. We then estimate the resources required for a fault-tolerant implementation of the proposed algorithm for the $\sin$ and $\cos$ functions in double precision. We find that, although the algorithm can in principle compete with the fastest known practical method for computing the hardness to round over all binades in the format, it requires qubit coherence times that are unrealistically long for present technology.

Empirical evidence shows that wealthy households have substantially higher saving rates and markedly lower marginal propensity to consume (MPC) than other groups. Existing theory cannot account for this pattern without jointly imposing restrictive assumptions on returns, discounting, and preferences. In this paper, we develop a general theory of optimal savings with preference shocks and identify a novel mechanism through which stochastic risk preferences reshape the asymptotic consumption and saving behavior. Specifically, the mere possibility of becoming less risk averse next period raises the value of carrying wealth forward, since future selves may be more willing to convert wealth into consumption. Unlike the classical precautionary saving motive, which typically arises from resource risks and weakens as wealth increases, this force remains operative even at arbitrarily high wealth levels, generating a persistent incentive to defer consumption and driving the asymptotic MPC to zero (i.e., a 100% asymptotic saving rate). As a result, vanishing MPCs emerge as a generic implication of risk preference dynamics, rather than an artifact of restrictive assumptions, offering a theoretically robust and empirically consistent account of the persistently high saving rates and low MPCs observed among wealthy households.

In this short note, we provide an explicit sufficient condition for non-explosion of Crump--Mode--Jagers branching processes with pure birth reproduction. It shows that the standard sufficient condition for explosion, namely the convergence of the series of reciprocals of the birth rates, is -- at least for rate sequences without excessive oscillations -- remarkably close to being necessary. At the same time, it is not necessary in full generality: we construct a counterexample which also yields a general preferential attachment tree without fitness with an infinite path and no vertices of infinite degree, thereby answering an open question previously raised in the literature.

We study diffusion approximations for a class of interacting stochastic systems with reflection and control. Motivated by interacting stochastic dynamics subject to feedback mechanisms and boundary constraints, we consider diffusion-scaled stochastic processes incorporating stochastic fluctuations, state-dependent interactions, and reflection. Under suitable assumptions, we establish convergence in distribution of the scaled processes to systems of interacting reflected stochastic differential equations of Ornstein-Uhlenbeck type. The limiting dynamics capture key features of constrained multi-agent systems, including mean-reverting behavior, interaction effects, and confinement within bounded domains through Skorokhod reflection. The analysis combines diffusion-scaling arguments, stability estimates, and continuity properties of the Skorokhod map to connect discrete stochastic systems with their reflected diffusion limits. To illustrate the framework, we present numerical examples motivated by crowd dynamics and neural population dynamics. The simulations demonstrate qualitative agreement between the finite stochastic systems and the corresponding reflected diffusion models and illustrate how diffusion approximations can provide tractable descriptions of interacting stochastic systems with constraints.

We develop a generalization of conditional probability for arbitrary ordered vector spaces. A related problem is that of assigning a numerical value to one vector relative to another. We characterize the groups for which these generalized probabilities can be stationary, respectively invariant. Our results deviate from the setting of classical probability and lead to a new criterion for amenability and for fixed points in cones.

We study the topological stability of Voronoi percolation in higher dimensions. We show that slightly increasing p allows a discretization that preserves increasing topological properties with high probability. This strengthens a theorem of Bollobás and Riordan and generalizes it to higher dimensions. As a consequence, we prove a sharp phase transition for the emergence of i-dimensional giant cycles in Voronoi percolation on the 2i-dimensional torus.

Fisher's fiducial probability has recently attracted renewed attention under the notion of epistemic confidence. Epistemic confidence statements can be formulated through extended likelihoods, thereby clarifying several long-standing controversies regarding its fiducial probability properties. It establishes a direct connection between Fisher's epistemic notion of confidence for observed data and Neyman's frequentist aleatory coverage probability for future data, thereby enabling extension of epistemic confidence statements for multidimensional parameters. We demonstrate how higher-order asymptotic theory can be applied to refine the first-order asymptotic epistemic confidence statements of the observed region, as a direct consequence of extended likelihood property.

How can we learn the laws underlying the dynamics of stochastic systems when their trajectories are sampled sparsely in time? Existing methods either require temporally resolved high-frequency observations, or rely on geometric arguments that apply only to conservative systems, limiting the range of dynamics they can recover. Here, we present a new framework that reconciles these two perspectives by reformulating inference as a stochastic control problem. Our method uses geometry-driven path augmentation, guided by the geometry in the system's invariant density to reconstruct likely trajectories and infer the underlying dynamics without assuming specific parametric models. Applied to overdamped Langevin systems, our approach accurately recovers stochastic dynamics even from extremely undersampled data, outperforming existing methods in synthetic benchmarks. This work demonstrates the effectiveness of incorporating geometric inductive biases into stochastic system identification methods.

We classify the inertial Weil-Deligne types arising from elliptic curves over all finite extensions $F/\mathbb Q_p$. Based on this classification, we give a fully explicit description of the types and implement an algorithm that computes all inertial types of elliptic curves defined over a given $F$. As an application, we determine all inertial types arising from elliptic curves over any extension $F/\mathbb Q_p$ of degree at most 3.

We study nonparametric Bayesian binary classification, in the case where the unknown probability response function is possibly spatially inhomogeneous, for example, being generally flat across the domain but presenting localized sharp variations. We consider a hierarchical procedure based on the Besov-Laplace priors from the inverse problems and imaging literature, with a carefully tuned hyper-prior on the regularity parameter. We show that the resulting posterior distribution concentrates towards the ground truth at optimal rate, automatically adapting to the unknown regularity. To implement posterior inference in practice, we devise an efficient Markov chain Monte Carlo (MCMC) algorithm based on recent ad-hoc dimension-robust methods for Besov-Laplace priors. We then test the considered approach in extensive numerical simulations, where we obtain a solid corroboration of the theoretical results.

In this work, we propose a modified Hybrid Parallel Kolmogorov--Arnold Network and Multilayer Perceptron Physics-Informed Neural Network to overcome the high-frequency and multiscale challenges inherent in Physics-Informed Neural Networks. This proposed model features a trainable weighting parameter to optimize the convex combination of outputs from the Kolmogorov--Arnold Network and the Multilayer Perceptron, thus maximizing the networks' capabilities to capture different frequency components. Furthermore, we adopt an overlapping domain decomposition technique to decompose complex problems into subproblems, which alleviates the challenge of global optimization. Benchmark results demonstrate that our method reduces training costs and improves computational efficiency compared with manual hyperparameter tuning in solving high-frequency multiscale problems.

Stochastic Gradient Descent (SGD) often slows in the late stage of training due to anisotropic curvature and gradient noise. We analyze preconditioned SGD in the geometry induced by a symmetric positive definite matrix $\mathbf{M}$, deriving bounds in which both the convergence rate and the stochastic noise floor are governed by $\mathbf{M}$-dependent quantities: the rate through an effective condition number in the $\mathbf{M}$-metric, and the floor through the product of that condition number and the preconditioned noise level. For nonconvex objectives, we establish a preconditioner-dependent basin-stability guarantee: when smoothness and basin size are measured in the $\mathbf{M}$-norm, the probability that the iterates remain in a well-behaved local region admits an explicit lower bound. This perspective is particularly relevant in Scientific Machine Learning (SciML), where achieving small training loss under stochastic updates is closely tied to physical fidelity, numerical stability, and constraint satisfaction. The framework applies to both diagonal/adaptive and curvature-aware preconditioners and yields a simple design principle: choose $\mathbf{M}$ to improve local conditioning while attenuating noise. Experiments on a quadratic diagnostic and three SciML benchmarks validate the predicted rate-floor behavior.

In this paper, we develop a regularization theory for neural network approximations of general ill-posed operator equations with noisy data. Within the framework of iterative regularization, we introduce two expanding neural network methods (ENNs) under different a priori assumptions on the exact solution. Instead of prescribing a fixed architecture, ENNs adaptively select the number of neurons through an a posteriori stopping rule, so that the selected network size serves as a regularization parameter balancing approximation accuracy and stability with respect to data noise. We prove the regularization properties of the proposed ENNs and establish quantitative relationships between the selected network size and the noise level. Within the framework of variational regularization, we propose a neural network-based Tikhonov scheme and derive both convergence and convergence-rate results under mild assumptions. The resulting estimates account for the noise level, the network size, and the underlying smoothness expressed through general variational source conditions, thereby allowing greater flexibility than existing results. Numerical experiments demonstrate the effectiveness and robustness of the proposed algorithms. In particular, they show that, for highly noisy data, relatively small network architectures can already produce stable reconstructions, whereas excessively large architectures may degrade stability due to overfitting.

This paper studies the modal logical aspects of provability predicates and consistency statements for theories of arithmetic. First, we provide an overview of previous works on the correspondence between various derivability conditions for provability predicates and different modal logics. The main technical contribution of the present paper is to establish the arithmetical completeness of the logics $\mathsf{NP}$, $\mathsf{ND}$, $\mathsf{NP4}$, and $\mathsf{ND4}$ by extending Solovay's method and refining Arai's construction of Rosser provability predicates.

In this paper, we introduce a unified framework for nonlinear Krylov subspace methods (\textit{nlKrylov}) to solve systems of nonlinear equations. Building on classical GCR-like/type linear Krylov solvers such as GMRESR, we generalize these approaches to nonlinear problems via nested algorithmic structures. We present rigorous convergence results for problems, relying on relaxed assumptions that avoid the need for exact line searches. The framework is further extended to matrix-valued root finding problems using global nonlinear Krylov approaches. Extensive numerical experiments validate the theoretical insights and demonstrate the robustness and efficiency of our proposed algorithms.

In this paper, we introduce the concept of graded extension dimension for a group graded ring R, denoted by this http URL (R). We prove that when R is strongly graded, its graded extension dimension coincides with the non-graded extension dimension of both R itself and its degree-zero subring Re. Furthermore, we demonstrate that graded equivalence and graded separable equivalence preserve the extension dimension under appropriate conditions.

The approximation of invariant measures for nonlinear ergodic stochastic differential equations (SDEs) is a central problem in scientific computing, with important applications in stochastic sampling, physics, and ecology. We first propose an easily applicable explicit Truncated Euler-Maruyama (TEM) scheme and prove its numerical ergodicity in the $L^p$-Wasserstein distance ($p\geqslant 1$). Furthermore, by combining truncation techniques with the coupling method, we establish a uniform-in-time $1/2$-order convergence rate in moments for the TEM scheme. Additionally, leveraging the exponential ergodicity of both the numerical and exact solutions, we derive a $1/2$-order convergence rate for the invariant measures of the TEM scheme and the exact solution in the $L^1$-Wasserstein distance. Finally, two numerical experiments are conducted to validate our theoretical results.

We construct the first examples of chamber-regular lattices on $\tilde C_2$-buildings. Assuming a conjecture of Kantor, our list of examples becomes a classification for type-preserving, chamber-regular $\tilde C_2$-lattices on locally finite $\tilde C_2$-buildings. The links of special vertices in the buildings we construct, are all isomorphic to the unique generalized quadrangle Q of order (3,5). In particular, our constructions involve chamber-regular actions on Q. These actions on Q are the first and if Kantor's conjecture holds the only chamber-regular actions on a finite generalized quadrangle and therefore interesting in their own right. Moreover Q is not Moufang and therefore none of our examples are Bruhat-Tits buildings and all our lattices are exotic building lattices.

We study the operator algebra of a rank-one boundary defect in a semi-infinite tight-binding chain, $T=U+\varepsilon E$ on $\ell^2(\mathbb{Z}_{\ge 0})$, with $U$ the forward unilateral shift and $E=\langle e_0,\cdot\rangle e_0$. The Lie algebra $\mathcal{A}=\mathrm{span}\{U^aE(U^*)^b,\,U^n\}$ has finitely supported, trace-zero commutators, the noncommutativity confined to the boundary and vanishing on the bulk. To each site we attach a $2$-cochain $\omega_j(X,Y)=\langle e_j,[X,Y]e_j\rangle$; each is a Chevalley--Eilenberg coboundary, yet $H^2(\mathcal{A},\mathbb{C})$ is infinite-dimensional, carried by the abelian bulk and classifying the central extensions. On the polynomial Toeplitz algebra obtained by adjoining $U^*$, the total cochain $\sum_{j}\omega_j(T_f,T_g)$ equals the symbol pairing $\frac{1}{2\pi i}\oint f\,dg$, which for conjugate symbols $g=1/f$ is the Fredholm index; the $\omega_j$ thus form a site-resolved index density, $\omega_j(U^n,(U^*)^n)=-\mathbf{1}_{\{j<n\}}$, localized at the edge. For modulated couplings with $\varepsilon_j\to c$, the index is fixed by the bulk limit and undergoes a topological transition as $|c|$ crosses~$1$, independently of the boundary profile.

Fix a positive prime power $q$, and let $\mathbb{F}_q[t]$ be the ring of polynomials over the finite field $\mathbb{F}_q$ with $\text{char}(\mathbb{F}_q)>2$. Suppose $A \subseteq \{f \in mathbb{F}_q[t]: °f \leq N\}$ contains no pair of elements whose difference is of the form $P-1$ with $P$ irreducible. Adapting Green's approach to Sárközy's theorem for shifted primes in $\mathbb{Z}$ using the van der Corput property, we show that \[ |A| \ll q^{(N+1)(11/12+o(1))}, \] improving upon the bound $O\big(q^{(1-c/\log N)(N+1)}\big)$ due to Lê and Spencer. An important distinction between Green's argument and ours lies in the properties of exponential sums over function fields, which differ in several interesting ways from their number-field counterparts.