Papers

Recovering two-dimensional Ito generators from trajectory data is difficult because drift increments have low signal-to-noise, bivariate weak designs can be ill-conditioned, and unconstrained tensor estimates need not be positive semidefinite. We study WG-SINDy estimator combining covariance-shaped spatial kernels, a ridge-stabilized local-polynomial projection, adaptive-LASSO/STLSQ selection, one in-sample per-component feasible diagonal GLS pass, and a PSD projection–Cholesky read-out with mild isotropic shrinkage. The released estimator uses a data-dependent full-cloud smoother and one in-sample per-component feasible diagonal GLS pass; accordingly, we do not claim exact finite-sample martingale cancellation or a feasible-GLS efficiency theorem for the reported implementation. We evaluate the estimator on 29 synthetic two-dimensional systems: 19 meet their declared per-system recovery contracts, eight are retained as named limits, and two remain scoped reviews. Across the 19 PASS rows, the median central-grid drift metric is 0.204 and the median tensor error is 0.0397. Among the six systems with a finite, non-degenerate off-diagonal target, the median $a_{12}$ cosine is 0.997. Positive-semidefinite validity is imposed by construction. These results are synthetic, in-sample sampled-region diagnostics and do not establish universal or real-data recovery.

We develop a representation-theoretic approach to generalized Reedy categories through a systematic study of standard modules. For a broad class of these categories, we provide a uniform, computable framework that reduces abstract Hom-spaces between standard modules to elementary linear algebra and graph theory via incidence matrices and morphism fibers. As a primary application, we establish a uniform extension of the Dold–Kan correspondence for certain categories arising from rooted trees, and obtain an orthogonal decomposition for finite spiders, including the category of finite sets and partial injections. Crucially, this machinery unifies and provides a single conceptual framework for several classical, seemingly disparate results across algebraic topology and representation theory, including Kuhn's decomposition theorem for vector spaces and the Thévenaz–Webb semisimplicity theorem for Mackey functors.

We study Volterra memory terms with locally integrable completely monotone kernels on a finite time interval $(0,\Tend)$ and ask how much $L^{2}$ coercivity they retain at high temporal frequencies. The main tool is an exact formula for the Rayleigh quotients of the cosine modes $\psi_n(t)=(2/\Tend)^{1/2}\cos(2\pi nt/\Tend)$. It shows that these quotients lie between $(1-\kappa_n/(2\pi n))\,m(2\pi n/\Tend)$ and $m(2\pi n/\Tend)$, where $m$ is the real memory symbol and $\kappa_n\in[1-e^{-2\pi n},1]$ is the best constant valid for all such kernels. Consequently, for non-constant kernels the algebraic decay rate of the quotients does not depend on $\Tend$ and coincides with the decay index $\rho\in[0,2]$ of $m$; any value in $[0,2]$ occurs. The Gaussian kernel $e^{-t^{2}}$, which is of positive type but not completely monotone, shows that this may fail otherwise: its quotients decay like $n^{-4}$, while its symbol decays faster than any power. We also show that the largest Rayleigh quotient on $(0,\Tend)$ is a continuous and strictly increasing function of $\Tend$, so that a kernel of total mass larger than one has exactly one critical horizon. Finally, because the memory operator on $(0,\Tend)$ is compact, it does not contribute to the uniform $L^{2}$ coercivity constant, and for $k_n(t)=ne^{-nt}$ the associated operators converge to the identity strongly but not in norm. The diffusion equation with memory serves as a model throughout.

Starting from the classic result of Wentzell, we derive a conditional forward equation and an associated stochastic Dupire PDE for a local-stochastic-volatility model (LSV). As an application, we obtain a density-weighted Rao–Blackwell estimator for the leverage function in LSV. We also derive an SPDE for a rolling expiry vanilla option, in the spirit of the Musiela parametrization in interest rate modeling.

In this paper, inspired by the idea of Adaptive Gradient Descent (AdGD) by Malitsky and Mishchenko, we propose a method that adaptively provides a stepsize interval, which represents the condition for stepsizes ensuring global convergence. By projecting the Barzilai Borwein stepsize onto this interval, we can ensure global convergence and expect further acceleration of convergence. Furthermore, we propose enlarging the stepsize interval obtained by AdGD and show that global convergence can be guaranteed within this enlarged stepsize interval. This extension of the interval enables us to adopt the full Barzilai-Borwein (BB) stepsize more frequently. We report the results of numerical experiments on the stepsize that combines the proposed interval with the BB stepsize.

Shor relaxations are standard tools for nonconvex quadratic optimization, but they may fail Slater's condition. The singularity degree quantifies this degeneracy by the number of facial-reduction steps needed to reach the minimal face. We determine how the constraints defining a binary feasible set affect this quantity. For every nonempty binary set defined by linear equations and inequalities, the Shor relaxation has singularity degree at most one. It is zero precisely when the linear programming (LP) relaxation contains a point in the open cube that strictly satisfies every inequality, and is one otherwise. As an algorithmic consequence of this characterization, Slater's condition can be restored by solving an LP rather than an auxiliary semidefinite program (SDP). In sharp contrast, for every $n\geq1$, we construct a binary feasible set defined by quadratic equalities whose Shor relaxation has a positive semidefinite matrix variable of order $n+1$ and singularity degree $n$. Thus, singularity degree is uniformly bounded in the linearly constrained case but can grow linearly with the matrix order when quadratic defining constraints are used.

This study extends variable-lattice-density optimization to high-Reynolds-number flows for the design of periodically arranged pin-fin heat sinks. Effective permeability and drag coefficient are identified from unit-cell Reynolds-averaged Navier–Stokes analyses of cylindrical pin-fin arrays and incorporated into a reduced model based on the Darcy–Forchheimer law for macroscopic design exploration. To enable stable optimization under high-Reynolds-number conditions, a dual-mesh framework is introduced, in which the flow field and sensitivities are evaluated on a fine mesh, whereas the design variables are updated on a coarse mesh corresponding to the unit-cell arrangement. For the base condition, the $L_2$ norm of the temperature deviation from the area-averaged temperature is decreased from 4.38 K to 1.00 K in the reduced model, and geometry-resolved analysis of the reconstructed design confirms a reduction from 5.98 K to 1.17 K. Additional calculations with higher inlet velocities and modified outlet locations show that the proposed method captures the dominant flow-redistribution trends under different operating and geometric conditions, although the thermal objective became less accurate when local solid-temperature variations became pronounced. These results indicate that the proposed approach is useful as practical design-exploration for identifying candidate pin-fin heat sink configurations under high-Reynolds-number conditions.

This paper investigates a two-species chemotaxis system with degenerate diffusion in the whole space $\R^d(d \ge 3)$. The diffusion of each species is governed by porous-medium-type operators. Under suitable conditions on the diffusion and aggregation exponents, we establish the global existence of uniformly bounded weak solutions. The proof hinges on a refined energy estimate that exploits the regularizing effect of degenerate diffusion to counteract the chemotactic aggregation. Furthermore, under additional assumptions on the system parameters, we derive exponential convergence rates of the solutions toward the constant steady state. Our results reveal that sufficiently strong degenerate diffusion ensures global boundedness and exponential stabilization in the whole space.

We construct, given $1<p<\infty$, a Banach space $Y$ and a closed, convex and symmetric set $L\subseteq B_Y$ with the following properties: 1) $Y^{**}$ is strongly regular (henceforth, $Y$ is strongly regular). 2) Every non-empty relatively weakly-star open subset of $\overline{L}^{w^*}$ (the $w^*$ closure of $L$ in $Y^{**}$) has radius one. In particular, every non-empty relatively weakly open subset of $L$ has radius $1$. 3) Every non-empty relatively weakly open subset of $L$ has diameter, at least, $2^\frac{1}{p}$. This constitutes an advance to the question whether there exists a strongly regular Banach spaces satisfying that every non-empty relatively weakly open subset of the unit ball has radius $1$. As a partial answer, we get that for every $\varepsilon>0$ there exists a strongly regular Banach spaces where weakly open subsets have radius, at least, $1-\varepsilon$.

Positive but not completely positive maps provide one of the most direct ways to detect entanglement beyond the positive-partial-transpose (PPT) criterion. We introduce and analyze an exactly solvable two-parameter family of sparse bistochastic positive maps on qutrits, in which two coherence channels are independently tuned by parameters $w$ and $z$. The sparse structure makes the full phase diagram analytic: positivity holds exactly on the square $0\le w,z\le2/3$, complete positivity on the smaller square $0\le w,z\le1/3$, and decomposability is lost precisely outside a quarter circle in the corner $w,z\ge1/3$. The indecomposable region is certified by explicit PPT entangled state adapted to the same witness geometry. At the endpoint $W_*=W(2/3,2/3)$ we construct a four-parameter family of PPT edge states of rank type $(5,5)$, derive their analytic detection region, and show that the corresponding rays are exposed faces of the PPT cone. Finally, although $W_*$ is not optimal, we give an explicit optimal refinement whose detection region on this family is strictly larger. The result is an analytically tractable qutrit setting in which positivity, indecomposability, PPT entanglement, optimality, and exposed convex geometry can be studied in a single framework.

We introduce local flag algebras, a variant of Razborov's flag algebra framework in which densities are normalised by the maximum degree $\Delta(G)$ rather than the order $|G|$. The framework supports the same semidefinite-method machinery as the classical version, but is tailored to extremal problems that scale with the maximum degree. As an illustrative first application we bound the number of pentagons in a triangle-free graph $G$ as a function of $|G|$ and $\Delta(G)$.

We consider inhomogeneous random graphs in which vertices are assigned i.i.d.\ random weights, pairs of distinct vertices are connected by an edge independently with a probability that is a bi-variate function of the weights of the vertices, and single vertices are connected to themselves by a self-loop independently with a probability that is a uni-variate function of the weight of the vertex. We apply a renormalisation transformation in which vertices are aggregated into groups of equal size according to a greedy algorithm, namely, distinct groups of aggregated vertices are connected by an aggregated edge if and only if there is at least one edge connecting two constituent vertices across the groups, while a group of aggregated vertices is connected to itself by an aggregated self-loop if and only if there is at least one self-loop at an internal vertex or one edge connecting a pair of distinct internal vertices. We analyse what happens when the renormalisation transformation is iterated. In particular, we show that, starting from appropriately scaled connection functions, the iterated renormalised graphs converge to a two-parameter family of random graphs, acting as an attractor in a universality class. We consider a light-tailed regime, for which the scaling limit is a homogeneous Erdős–Rényi random graph, and a heavy-tailed regime, for which the scaling limit is an inhomogeneous random graph with stable infinite-mean random weights and an exponential disconnection function. Different scalings are needed for the two regimes. Which of the two regimes prevails depends on the choice of the connection functions and the choice of the law of the random weights.

We study flags of subrepresentations of a general representation of a finite quiver. For each representation, its flag scheme is the zero scheme of a section of a quotient bundle on a product of partial flag varieties. We obtain the generic dimension and cycle class and equivariant localization formulas for Euler characteristics and parabolic Schubert coefficients. We further express these coefficients as finite signed sums of semi-invariant dimensions for auxiliary star quivers with inward and outward flag arms. In the acyclic one-step case, this is compatible with the result of Derksen–Schofield–Weyman, and the zero-dimensional case recovers their semi-invariant counting theorem. Examples treat matrix pencils and generalized Kronecker quivers.

Let $G$ be a digraph with vertex set $\{1,2,...,n\}$ and $H_{1},H_{2},...,H_{n}$ be $n$ digraphs. The generalized join digraph $\overrightarrow{G}=G[H_{1},H_{2},...,H_{n}]$ is a digraph obtained from $G$ by replacing each vertex $i$ with $H_{i}$ and for any $u\in V(H_{i})$ and $v\in V(H_{j})$, $(u,v)\in E(\overrightarrow{G})$ if and only if $(i,j)\in E(G)$. In this paper we express the number of oriented spanning trees in $\overrightarrow{G}$ in terms of Laplacian eigenvalues of $H_{1},H_{2},...,H_{n}$ and oriented spanning trees of $G$. Furthermore, we consider the number of oriented spanning trees with a fixed root in $\overrightarrow{G}$. First, we introduce the biclique-directed star transformation formula for counting oriented spanning trees with a fixed root in digraphs. Using it, we give the formula for the total number of oriented spanning trees with roots in a certain $H_{i}$ $(1\leq i \leq n)$ of $\overrightarrow{G}$ in terms of Laplacian eigenvalues of $H_{1},H_{2},...,H_{n}$ and oriented spanning trees of $G$. As applications, when each $H_{i}$ is a given digraph, the enumerative formulas for oriented spanning trees with a fixed root of $\overrightarrow{G}$ are derived from our work.

This paper investigates a regional optimal control problem for a nonlinear spatio-temporal epidemiological model involving fractional diffusion on a bounded domain. In this work, we consider a general form of disease transmission, and two types of control, vaccination and treatment. We establish the existence and uniqueness of global-in-time solutions to our proposed system. We also prove the existence of an optimal control that minimizes the infected individuals as well as the cost of vaccination and treatment. The necessary conditions for optimality are derived. We show that concentrating vaccination in a selected region of the domain can substantially mitigate the spread of the disease. Numerical simulations are given.

Let $f_{r}(d,s_{1},\ldots,s_{r})$ be the least $N$ such that every $N$-point set $P\subseteq\mathbb{R}^{d}$ has an $r$-partition $P=P_{1}\sqcup\cdots\sqcup P_{r}$ with the following property: whenever $C_{i}\supseteq P_{i}$ is a union of at most $s_{i}$ convex sets, one has $\bigcap_{i=1}^{r}C_{i}\ne\emptyset$. A recent breakthrough of Alon and Smorodinsky proved that $f_{r}(d,s,\ldots,s)\le cdr^{2}s^{r}\log r\log(es^{r})$ for an absolute constant $c>0$. In this paper, we determine the asymptotic order in two principal ranges: $f_{2}(2,s,s)=\Theta(s^{2})$, and $f_{r}(d,s,\ldots,s)=\Theta_{d,r}(s^{r}\log{s})$ for every fixed $d,r$ with $d\ge r+2$. The first one determines the order of the extremal function proposed by Kalai from the 1970s. Together, the two results show a sharp dependence on the dimension: for two parts, the logarithmic factor disappears in the plane but is necessary in every fixed dimension $d\ge4$. Beyond these sharp results, when $r\ge d+1$ we improve the upper bound of Alon and Smorodinsky by proving both $f_{r}(d,s,\ldots,s)\le c_{d}rs^{r}\log(ers^{r})$ and $f_{r}(d,s,\ldots,s)\le c_{d}r^{d+2}s^{d+1}\log(ers)$ through a local Helly-type argument. We also prove $f_{r}(d,s,\ldots,s)>s^{\min\{r,d\}}$ for every $d\ge2$. Finally, we study two colored analogues. The direct Bárány–Larman-type extension, in which one seeks $r$ disjoint rainbow sets chosen from $d+1$ color classes, fails as soon as two convex pieces are allowed. Nevertheless, a different extension does hold: given sufficiently many prescribed $r$-point classes, one can split every class completely among the $r$ final parts while retaining the required intersection property.

We give the conjectured solution for the generating function of $k$-convex polyominoes, enumerated by semi-perimeter. The solution was obtained from the analysis of enumeration data that we generated.

We classify the local maximizers of the entrywise fourth-power objective \[ Q\longmapsto \lVert Q\rVert_4^4 =\sum_{i,j=1}^r q_{ij}^4 \] over the real orthogonal group $\mathcal O(r)$. We prove that the signed permutation matrices are the only local maximizers, and hence the only global maximizers, in every dimension. More strongly, every other stationary point has an explicit rank-two tangent direction with strictly positive second variation. The proof is based on a maximum-entry pivot for the orthostochastic matrix $Q^{\circ2}$: the associated full Riemannian Hessian can be evaluated exactly and is positive at a largest nonunit squared entry. The argument is self-contained and handles zeros, repeated magnitudes, reducible support, and Hadamard-type stationary points.

Mixed precision algorithms can significantly enhance the performance of linear algebra solvers by leveraging increasingly powerful low precision hardware while recovering working precision accuracy through, for example, iterative refinement. In this paper, we propose a novel mixed precision algorithm for computing matrix square roots. Our algorithm combines a Schur decomposition approach in low precision with iterative refinement performed through an approximate Newton method. We perform a detailed convergence analysis of the approximate Newton method. For the special case of symmetric positive definite matrices, this analysis implies that one can recover full working precision accuracy under mild conditions. Numerical experiments on x86-64 architectures indicate that our algorithm frequently reduces execution time compared with a fixed working-precision Schur algorithm.

We study strong numerical approximations for reflected stochastic differential equations in possibly unbounded convex domains with super-linearly growing drift and diffusion coefficients. Under a coupled monotonicity condition and polynomial local Lipschitz assumptions, we first establish the well-posedness of the reflected SDE and derive uniform moment bounds for its solution. We then introduce a coupled tamed Euler–Peano scheme, in which the drift and the squared diffusion coefficient are tamed by a common factor and the resulting Euler–Peano path is corrected through the Skorokhod problem. This common taming factor preserves the drift–diffusion coercivity structure and yields uniform moment estimates for the numerical solution. We prove strong convergence of order $1/2$ for both the constrained state process and the boundary regulator, thereby recovering the standard Euler-type strong order in this reflected setting. Numerical experiments for a reflected stochastic Ginzburg–Landau type system illustrate the constraint preservation of the scheme and support the theoretical convergence rate.

We study the control and homogenization properties of a coupled hyperbolic system characterised by a rapidly oscillating coefficient present in the principal term. Due to the high-frequency microstructural variations of the density parameter $\varepsilon$, classical uniform controllability results across all initial data fail under boundary controls alone. To overcome this limitation, we divide the spectrum into low and high-frequency components. Utilizing Ingham-Beurling-type inequalities combined with spectral gap analysis, we demonstrate the uniform null controllability of the projected low-frequency and high-frequency states. Furthermore, we analyse the structural limits as $\varepsilon \to 0$ and prove that the associated sequence of boundary controls converge weakly to a null control for the underlying homogenized system. Finally, we show that by using an additional interior feedback control strategy, full space controllability can be achieved for all initial data in the energy space.

In 1966 Jane Matthews claimed that the conjugacy problem is solvable in the standard restricted wreath product $A \wr B$ of two nontrivial groups $A$ and $B$ if and only if (i) the conjugacy problem is solvable in $A$ and $B$ and (ii) $B$ has a solvable power problem. We show that there should be an additional condition that either $A$ is abelian or $B$ has a solvable order problem. We also show that, if $A$ and $B$ are non-trivial recursively presented groups where $A$ has an infinite number of conjugacy classes and $B$ acts on $B/H$ transitively, then the conjugacy problem in the permutational restricted wreath product $A \wr_{B/H} B$ is solvable if and only if the following hold: (1) the conjugacy problem is solvable in $A$ and in $B$; (2) either $A$ is abelian or the orbit order problem is solvable in $B$; (3) for any $\gamma, \beta \in B$ the membership problem for $H \gamma \langle \beta \rangle$ is solvable; and (4) for any $\beta \in B$ and any finite set of $n$ pairs of elements $(\alpha_i, \gamma_i)$, where $\alpha_i, \gamma_i \in B$ we can determine whether or not $ \big{[} \bigcap_{i=1}^n \alpha^{-1}_iH\gamma_i \langle \beta \rangle\big{]} \cap C_B(\beta) = \emptyset $.

The reconstruction of prehistoric migration routes requires the accurate evaluation of effective distances across complex topographies. Traditional Least-Cost Path Analyses (LCPA) using high-resolution elevation models often lead to dense computational grids, making algorithms like Dijkstra's prohibitively slow for large-scale archaeological modelling. In this paper, we present a novel approach using adaptive wavelet methods to dynamically compress topographic data. By retaining high resolution only in areas of high topographic complexity (e.g., mountain passes) and smoothing homogeneous regions, we significantly accelerate the Dijkstra algorithm without losing essential routing topology. We demonstrate the efficacy of this method using migration scenarios.

Time-periodic problems arise when the desired solution is a periodic steady state rather than a transient trajectory. The periodic parareal algorithm with a periodic coarse problem (PP-PC) is a periodicity-preserving parallel-in-time approach for such problems. Projection-based correction can accelerate convergence of both parareal and PP-PC. In this paper, we propose a Fourier-aware construction of projection spaces and a new correction scheme to further accelerate the convergence of projection-based PP-PC. We develop a convergence analysis of projection-based PP-PC with the discrepancy-based correction scheme for general nonlinear time-periodic problems. For an arbitrary orthogonal projection, we derive a local one-step convergence estimate controlled by the unresolved error and explicit nonlinear contributions. A temporal Fourier decomposition bounds the unresolved error by a tail-leak quantity, which is small when dominant error modes are selected and their coefficients are captured by the projection space. For linear problems, the nonlinear contributions vanish, yielding a globally valid one-step tail-leak convergence estimate under weaker assumptions. Experiments on linear and nonlinear problems show that Fourier-aware PP-PC requires fewer outer iterations than Krylov-enhanced PP-PC. For the linear problems, the errors track the tail-leak bound. For the nonlinear problems, the experiments quantify the unresolved-error and explicit nonlinear contributions in the local one-step estimate and show that the evaluated tail-leak estimate follows the observed decay.

For a vertex algebra $V$ with a finite-order automorphism $g$ satisfying $g^T = 1$ for some $T \in \mathbb{N}$, we construct an associative algebra $\tilde{\mathbf{A}}^{g,\infty}(V)$ and prove that the category of $\frac{1}{T}\mathbb{N}$-graded $g$-twisted $\phi$-coordinated $V$-modules is isomorphic to the category of graded $\tilde{\mathbf{A}}^{g,\infty}(V)$-modules. Furthermore, when $V$ is a vertex operator algebra, we construct associative algebras $\mathbf{A}^{g,\infty}(V)$ and $A^{g,\infty}(V)$, and establish that the categories of admissible $g$-twisted $V$-modules and ordinary $g$-twisted $V$-modules are isomorphic to the categories of graded $\mathbf{A}^{g,\infty}(V)$-modules and graded $A^{g,\infty}(V)$-modules, respectively. By proving that $\tilde{\mathbf{A}}^{g,\infty}(V)$ is isomorphic to $\mathbf{A}^{g,\infty}(V)$, we obtain the equivalence between the category of $\frac{1}{T}\mathbb{N}$-graded $g$-twisted $\phi$-coordinated $V$-modules and the category of admissible $g$-twisted $V$-modules. Let $g_1, g_2, g_3$ be three commuting automorphisms of $V$ of finite order such that $g_1 g_2 = g_3$ and $g_i^T = 1$ for $i = 1, 2, 3$ and some $T \in \mathbb{N}$. Suppose that $W_i$ is a $g_i$-twisted $V$-module for each $i = 1, 2, 3$. We then construct an $A^{g_3,\infty}(V)$-$A^{g_2,\infty}(V)$-bimodule ${A}^{g_3,g_2,\infty}(W_1)$, and prove that the space of intertwining operators of type $\binom{W_3}{W_1 \; W_2}$ is isomorphic to $ \operatorname{Hom}_{A^{g_3,\infty}(V)}\!\left( {A}^{g_3,g_2,\infty}(W_1) \otimes_{A^{g_2,\infty}(V)} W_2, \, W_3 \right). $

We continue the study of initial dialgebras defined in~\cite{DMS2026}. For a binary operad $\Var$ we define the class of initial pre-$\Var$-algebras and the corresponding operad $\pre\Var^{\I}$ in such a way that \[ (\di\Var^{\I})^{!}=(\pre(\Var^{!}))^{\I} \] in the case when $\Var $ is quadratic. We propose an intuitive algorithm for finding the defining relations of the operad $\pre\Var^{\I}$ in the case when $\Var$ is a binary quadratic operad. We also study free initial pre-algebras in the associative and commutative settings. For the nonsymmetric operad $\pre\As^{\I}$, we construct a Grobner–Shirshov basis in the free magma operad, describe a linear basis in terms of admissible decorated planar binary trees, and establish a bijection between these trees and certain combinatorial objects.

We present SinCoTrap (Singularity-Corrected Trapezoidal Rule), a high-order locally corrected trapezoidal method for periodic singular integrals in arbitrary dimension $d$ with kernel $|\boldsymbol{x}|^{-s}$, $0<s<d$. The scheme preserves the uniform tensor grid and modifies only a fixed, small stencil of weights near the singularity. For a correction order $p$, the resulting quadrature attains the error rate $O(h^{2p+2+d-s})$. We derive explicit, mesh-independent limiting correction weights via analytic continuation of a special generalization of the Riemann zeta function, yielding rapidly computable formulas that can be pretabulated for each $(d,s,p)$. This makes SinCoTrap both efficient in application and robust for high-order accuracy across a broad class of periodic singular integrals.

Hierarchical Bayesian inversion with Gaussian random field priors addresses uncertainty in covariance hyperparameters, such as the standard deviation and correlation length. When a Gaussian random field is represented by the Karhunen-Loève (KL) expansion, the basis functions depend on these hyperparameters through an integral eigenvalue problem (IEVP) associated with the covariance kernel. Consequently, the IEVP must be solved repeatedly whenever the hyperparameters are updated, leading to significant computational cost in hierarchical inference. In this paper, we focus on the squared exponential kernel and construct the KL expansion using the analytical solution to a Gaussian-weighted IEVP. This analytical KL expansion offers a computationally efficient alternative to the conventional KL expansion by eliminating the repeated numerical solutions of the IEVP during hyperparameter updates. While the analytical KL expansion is applicable to arbitrary domains and dimensions, it does not have the same mean-square optimality as the conventional KL expansion. To address this limitation, we employ an optimization-based approach that selects the standard deviation of the Gaussian weight function in the IEVP to effectively reduce the truncation error of the KL expansion. Numerical experiments in one- and two-dimensional settings show that this selection strategy provides sufficient accuracy for practical applications. Furthermore, the analytical KL expansion admits closed-form differentiation, enabling efficient posterior sampling via HMC. The proposed framework is applied to Bayesian inversion for a steady Darcy flow model, where the hydraulic conductivity field is successfully estimated using weakly informative hyperpriors.

We study spatial averages of the mild solution to a one-dimensional space–time fractional stochastic heat equation driven by space–time white noise. For fixed \(t>0\), we prove a quantitative central limit theorem for the normalized spatial average over \([-R,R]\): as \(R\to\infty\), its law converges to the standard normal law at rate \(R^{-1/2}\) in total variation distance. The proof relies on the Malliavin–Stein method, combined with precise estimates for the space–time fractional heat kernel and for the Malliavin derivative of the mild solution. We further establish a functional central limit theorem.

In this paper, we determine the probability that a genus $g$ hyperelliptic curve with a Weierstrass point over a number field has good reduction at a given prime of residue characteristic $>2g+1$. We also obtain analogous probability formulas for several other reduction types, including cases with positive toric or unipotent rank. As an application, assuming the Hasse–Weil conjecture and the generalized Riemann hypothesis, we derive an explicit upper bound for the average analytic rank of genus $g$ hyperelliptic curves with a Weierstrass point.