Papers

Physics-informed neural networks (PINNs) have emerged as a promising route to solve partial differential equations, yet they have struggled to reach the precision of classical solvers. The obstacle is increasingly understood to be one of optimisation, owing to the severely ill-conditioned loss landscape. We present $\textbf{DSGNAR}$: Doubly-Sketched Gauss-Newton with Adaptive Ratio, a scalable second-order optimisation framework that confronts this ill-conditioning and, in doing so, obtains unprecedented accuracy and speed. $\textbf{DSGNAR}$ couples a doubly-sketched Gauss-Newton model with a novel strategy that carefully controls both regularisation and step length. Across a suite of problems spanning nonlinear, chaotic, multi-scale, high-dimensional, and Navier-Stokes, the framework greatly improves on the state of the art: able to attain relative $\ell_2$ errors as low as $3\times10^{-16}$ in double precision, improve contemporary results by five orders of magnitude on the canonical Burgers' equation, and as much as eight orders on a high-dimensional Poisson problem, while remaining markedly faster. We further show that, in single precision, solutions at the limit of round-off error can be obtained very quickly: Burgers' equation to $\ell_2^{\text{rel}} = 4.75 \times 10^{-7}$ in under ten seconds. The framework is also robust to the choice of architecture, arithmetic precision, and initial hyperparameters. The code is available at https://www.github.com/wephy/physics-informed-neural-networks

In \cite{Wang1954}, Wang proved (among other things) a sufficiency result for a compact homogeneous manifold $G/H$ to admit a $G$-invariant complex structure. In this note, we give a new Lie theoretic proof of Wang's theorem which relies on nothing more than the familiar properties of the root space decomposition of a compact Lie group. It should be noted that the recent work of Ni and Wallach \cite{NiWallach2025} also revisits the aforementioned theorem of Wang (and others) and offers new Lie theoretic proofs as well. However, the approach of \cite{NiWallach2025} relies on such objects as Borel subalgebras, parabolic subalgebras, and Iwasawa decomposition which may be somewhat less familiar to the working differential geometer.

We prove new $L^p$ boundedness results for Bochner-Riesz means associated with the spectral decomposition of the sub-Laplacian on the Heisenberg group $\mathbb H_n$. Our results hold for a range $1\le p\le p_n$ where $p_n\to 2$ as $n\to\infty$. As shown by the first named author in 1990 a Stein-Tomas type Fourier restriction theorem fails to hold on $\mathbb H_n$ and thus previous results based on the approach by Fefferman and Stein from the Euclidean setting only allowed to cover the cases $p=1$ and $p=\infty$. Our results on Bochner-Riesz means follow from a more general $p$-sensitive spectral multiplier theorem which is the main result of this article. This is obtained as a consequence of $L^p$ estimates for square functions associated with the Heisenberg wave operator.

This paper concerns a three-dimensional inverse acoustic obstacle scattering problem from scattered field or phased/phaseless far-field data. Based on the boundary integral defined on a homothetic surface, we propose a highly efficient iterative approach for obstacle reconstruction that completely avoids dealing with any singularity. Here, the injectivity and dense-range property of the Fréchet derivative have been proved to ensure the solvability of the linearized equivalent data equation. We also prove that the scattered field generated by the homothetic surface can arbitrarily approximate the exact one. Numerical experiments are presented to verify the superiority and robustness of the proposed approach.

We propose and analyze a linearly implicit mass-lumped finite element method for the heat flow of harmonic maps into the unit sphere. The method consists of a linear predictor followed by a nodal projection and therefore preserves the unit-length constraint exactly at all finite element nodes. The predictor is derived from a cross-product reformulation of the equation and is shown to be equivalent to a mass-lumped discretization of the original formulation with a correction term enforcing nodal orthogonality, as well as to a tangent plane scheme. A key ingredient is the consistent use of the discrete inner product in both the mass and stiffness terms. This yields a nodal orthogonality relation implying that the auxiliary solution lies on or outside the unit sphere at every node. Consequently, the projection is well defined and the projected error satisfies a contraction property in the discrete \(L^2\)-norm. On Cartesian rectangular and cuboidal tensor-product meshes, the nodal projection is also nonexpansive in a discrete Dirichlet energy, which gives an unconditional discrete energy dissipation law. For sufficiently smooth solutions, we prove optimal error estimates without any coupling condition between the time step and the mesh size: the method converges with order \(O(\Delta t+h^2)\) in \(\ell^\infty(0,T;L^2)\) and order \(O(\Delta t+h)\) in \(\ell^2(0,T;H^1)\). The proof combines the projected-error contraction, quadrature consistency estimates, edge-based cancellation identities, and a bootstrap argument for controlling nonlinear terms. Numerical experiments confirm the predicted convergence rates and the discrete energy decay.

For an arbitrary reduced root system, we give upper bounds for the Dunkl kernel with regular spectral parameter and its derivatives, which are uniform in the spatial variable. These estimates generalize well-known sharp upper bounds for classical one-variable Bessel functions and for spherical functions of Cartan motion groups. As a consequence, we prove that the representing measure of Dunkl's intertwining operator is absolutely continuous with respect to the Lebesgue measure for multiplicities $k> 1/2$ and generic spectral parameter. This settles a conjecture posed in [RdJ02] at least for $k>1/2$.

We consider the Sherrington-Kirkpatrick spin glass model at the critical inverse temperature $\beta = 1$ with zero external field. We prove that the free energy $F_N = F_{N,\beta=1}$ of this model has variance \[ \mathrm{Var}(F_N) = \frac16 \log N + O(1)\,, \] confirming a physics prediction of Aspelmeier \cite{aspelmeier2008free}, and that the centered and scaled $F_N$ satisfies a Gaussian CLT. We also identify the critical two-replica overlap scale, proving \[ \mathbb{E} \langle R_{1,2}^2\rangle \asymp N^{-2/3}\,, \] as conjectured by Talagrand \cite{talagrand2011mean2}, together with a uniform exponential moment bound for $N^{1/3} |R_{1,2}|$. The key input is a critical reweighted moment method, in the spirit of the “small subgraph conditioning” technique from probabilistic combinatorics, but capable of capturing diverging fluctuations. Through this reweighting, we relate the critical SK model to the BBP critical edge, which determines the overlap and fluctuation scales.

We prove that any generalized extended code is monomially equivalent to the Hermitian dual of a code which is closely related to a second kind of extended code of $\C^{\perp_{\rm H}}$. Every $[n+1,k+1]_{q^2}$ linear code $\D$ with $d(\D^{\perp_{\rm H}})>1$ is monomially equivalent to the generalized extended code $\C({\bf u},a)$ of an $[n,k]_{q^2}$ linear code $\C$ for a fixed $a\in\F_{q^2}^{*}$ and some ${\bf u}\in\F_{q^2}^{n}$. We then characterize the Hermitian hull and Hermitian dual distance of $\C({\bf u},a)$ in terms of the position of ${\bf u}$ relative to $\C+\C^{\perp_{\rm H}}$ and the interaction between ${\bf u}$ and the minimum weight codewords of $\C^{\perp_{\rm H}}$, respectively. We obtain explicit criteria to independently control the expected Hermitian hull dimension and Hermitian dual distance of $\C({\bf u},a)$. In particular, several conditions for simultaneously increasing the Hermitian hull dimension and the Hermitian dual distance of $\C({\bf u},a)$ are derived. Applying these results to the Hermitian construction for EAQECCs gives us $267$ new EA qubit codes of lengths $n \leq 40$ and $14$ new EA qutrit codes of lengths $n \leq 25$ compared to the best-known codes in Grassl's code tables and the imporvements recorded in very recent works in the literature. Among the new parameter sets, we confirm improvements for $236$ qubit and $8$ qutrit codes.

Recent work by Gusakova et al. (Stochastic Process. Appl. 164 (2023) 357-382) has shown a central and a stable limit theorem for the logarithmic volume of random simplices and random convex bodies under an elliptical framework in the high dimensional regime, that is, if p and n tend to infinity in such a way that the ratio tends to \gamma within (0,1). A technical condition (Equation (2.6) of Assumption (B) therein) requires that the population matrix AA* is close in Frobenius norm to a multiple of the identity matrix, which is rather restrictive and rules out various settings for statistical application, such as spiked models and dependent structure models. In this note we offer a general relaxation of this condition, which arrives at a reasonable condition and covers numerous scenarios, as well as consequences for the volume of general random simplices and random convex bodies. In particular, our results covers the Toeplitz/AR(1) covariance structures studied by Jiang and Pham (Ann. Stat. 53 (2025) 907-928), giving a concrete application of our theorem to high-dimensional dependent covariance models.

Let $\mathcal{M}_{g,n}(\mathbf{L})$ be the moduli space of hyperbolic surfaces of genus $g$ with $n \geq 0$ hyperbolic ends of widths $\mathbf{L} \in \mathbb{R}_{\geq 0}^n$. We regard the total mass $|\mu_X^\kappa|$ of the Brownian loop measure with the killing rate $\kappa$ as a random variable on $\mathcal{M}_{g,n}(\mathbf{L})$. Under the condition $|\mathbf{L}|^2 =o(g)$ as $g \to \infty$, we obtain the following two main results: $(1)$ For any $\kappa > 0$, the expected value of $|\mu_X^\kappa|$ on all non-peripheral homotopy classes over $\mathcal{M}_{g,n}(\mathbf{L})$ converges to an explicit function of $\kappa$, which blows up at the rate $ \log \left(\frac{1}{\kappa}\right)$ as $\kappa \to 0^+$. $(2)$ For $\kappa=0$, over $\mathcal{M}_{g,n}(\mathbf{L})$ the expected value of $|\mu_X|$ on homotopy classes of (iterates of) all non-peripheral simple closed geodesics is asymptotically $\frac{1}{2} \log g$.

We survey recent progress on the geometric Bombieri–Lang conjecture over function fields of characteristic zero. We discuss recent work of Xie–Yuan and Guoquan Gao, which together proves the conjecture for varieties admitting finite morphisms to abelian varieties. The guiding idea, developed in joint work with Xinyi Yuan, is that Vojta's dictionary can be made concrete in this setting: from rational points of large height one constructs entire curves on complex fibers.

Mumford curves generalize the Tate uniformization of elliptic curves with split multiplicative reduction and provide p-adic analogues of the uniformization of Riemann surfaces. In this paper, we present several algorithms for hyperelliptic Mumford curves. For a given hyperelliptic Mumford curve $X$ defined over a finite extension of the field of p-adic numbers for some $p\neq 2$, we first describe how to compute a p-adic Schottky group W that uniformizes X; this is based on our extension to Kadziela's approximation theorem. As applications, we explain how to use this uniformization in order to compute p-adic Abelian integrals and $p$-adic Schneider heights on X; the latter uses Werner's formula expressing the p-part of the Schneider height in terms of theta functions. We illustrate our algorithms with numerical examples computed using the computer algebra system SageMath.

We show that, for every fixed graph $H$, every $n$-vertex graph $G$ that excludes $H$ as a minor is $3$-colourable with clustering $O_H(n^{4/9})$. That is, there exists a function $f$ such that for every graph $H$, every $n\ge 1$, every $n$-vertex graph $G$ that excludes $H$ as a minor has a vertex colouring with $3$ colours in which each monochromatic component has size at most $f(H)\cdot n^{4/9}$. This generalizes a recent result of Dujmović, Morin, Norin, and Wood (arXiv:2507.03163) from planar graphs to all proper minor-closed graph classes and is the first improvement on clustered $3$-colouring of proper minor-closed graph classes since the upper bound of $O_H(\sqrt{n})$ due to Linial, Matoušek, Sheffet, and Tardos (Comb. Prob. Comput., 17(4):577–589, 2008).

The Feigin-Semikhatov duality asserts that the Heisenberg cosets of the subregular $W$-algebra of $\mathfrak{sl}_n$ at level $k$ and the one of the principal $W$-superalgebra of $\mathfrak{sl}_{n|1}$ at level $\ell$ coincide when the levels satisfy the Feigin-Frenkel relation $(k+n)(\ell+n-1)=1$. A similar duality holds between the subregular $W$-algebra of $\mathfrak{so}_{2n+1}$ and the principal $W$-superalgebra of $\mathfrak{osp}_{2|2n}$. We study these dualities in the critical/large level limit. We describe the centerless subregular $W$-algebra at the critical level as an orbifold of the large level limit of the principal $W$-superalgebra times a lattice VOA. Our construction yields a functor between certain categories of the two involved vertex algebras. We show that in this set-up one in fact gets block-wise equivalences of categories. Studying the principal block of the large level limit of the principal $W$-superalgebra then gives us the structure of the principal blocks of the subregular $W$-algebras in the category of weight modules (which is much larger than the more common category of lower bounded modules).

This study establishes Liouville-type theorems for indefinite quasilinear elliptic equations in the upper half-space. Additionally, we demonstrate the existence of solutions for this class of problems using the fibering method. Our approach relies on a novel weighted Sobolev embedding developed for the upper half-space.

This paper investigates the relationship between tiles and weak tiles in the context of finite cyclic group $\mathbb{Z}_{pq}$. We prove that weak tiles and translational tiles are equivalent in this group. Our proof employs Fourier analysis, Delsarte parameters, and the Coven-Meyerowitz conditions.

We study the growth spectrum of groups acting on hyperbolic spaces, i.e.\ the set of exponential growth rates achieved by subgroups. For a finitely generated free group or a surface group acting convex-cocompactly on a proper geodesic hyperbolic metric space, we prove that the growth spectrum is the full interval $[0, \omega_G]$. For any hyperbolic group, we prove that the growth spectrum contains a large interval $[0, \omega_{\mathcal{F}}]$ where $\omega_{\mathcal{F}} \geq \omega_G / 2$, with strict inequality when the action is divergent. In the case of the Cayley graph of a free group, we also present an approach via the non-backtracking matrix of the configuration model, connecting the density of growth rates to a spectral concentration result for random graphs.

We note that homeomorphism groups of all pseudo-solenoids, including the pseudo-circle, have non-metrizable universal minimal flows.

We study timestep allocation for score-based diffusion sampling, where a learned reverse-time dynamics is discretized on a finite grid. Uniform and hand-crafted schedules are standard choices, but they rely on fixed prescriptions and can therefore be suboptimal. To address this limitation, we propose Adaptive Reparameterized Time (ART), a continuous-time control formulation that learns a time change by treating the speed of the sampling clock as the control, so that a uniform grid on the learned clock induces adaptive timesteps in the original diffusion time. Based on a leading-order Euler error surrogate, ART provides a principled objective for allocating timesteps along the sampling trajectory. To solve this deterministic control problem, we introduce ART-RL, an auxiliary randomized formulation with Gaussian policies that turns schedule learning into a continuous-time reinforcement learning problem. We prove that the randomized ART-RL formulation is equivalent to ART at the optimizer level, in the sense that its optimal Gaussian policy recovers the optimal ART time-warping rate through its mean. We further establish policy evaluation and policy improvement characterizations and derive trajectory-based moment identities that yield implementable actor–critic updates for learning the schedule. Across experiments ranging from controlled low-dimensional settings to image generation, ART-RL can be plugged into existing diffusion samplers by changing only the timestep grid, consistently improving sample quality over strong baseline schedules at matched budgets while leaving the rest of the sampling pipeline unchanged. The learned schedules also exhibit broad generalization, transferring without retraining across sampling budgets, datasets, solvers, pipelines, and representation spaces.

Let $G$ be an abelian group, and let $\mathcal F (G)$ be the free commutative monoid with basis $G$, and $\mathcal A (G)$ the set consisting of all minimal zero-sum subsequences over $G$. For any subset $\Omega \subset \mathcal F (G)$, we define the universal zero-sum invariant ${\mathsf d}_{\Omega}(G)$ as the minimal positive integer $\ell$ such that every sequence $T$ over $G$ of length $\ell$ contains a subsequence lying in $\Omega$. The classical Davenport constant ${\rm D}(G)$ for $G$ can also be written as ${\mathsf d}_{\mathcal A (G)}(G)$. We give a complete classification of all finite abelian groups for which $\mathcal A(G)$ is a minimal set to represent the Davenport constant. We also investigate the weighted Davenport constant over abelian groups (which may be infinite). Let $F$ and $G$ be abelian groups, and let $\Psi \subseteq \mathrm{Hom}(F,G)$ denote a weight set. We reinterpret the weighted Davenport constant $D_{\Psi}(G)$ in terms of coverings of Cartesian powers $F^n$ by kernels of induced homomorphisms arising from tuples in $\Psi^n$; these homomorphisms are naturally linked to coproducts in the category of abelian groups. This motivates the notion of kernel-cover compactness, a property characterizing when such kernel coverings admit finite subcovers. We establish a correspondence between weighted zero-sum invariants and kernel-cover structures, where the bound $D_{\Psi}(G)\le n$ is equivalent to a canonical kernel-cover property on $F^n$. We further study finite reduction phenomena for infinite weight sets and provide sufficient conditions ensuring uniform kernel-cover compactness. The present work constitutes a follow-up to [G. Wang, Comm. Algebra, 2025].

A crystallographic group is a discrete subgroup of the Euclidean group $\operatorname{E}(n)$ that has a compact fundamental domain. Since such a crystallographic group $\Gamma$ is infinite, computing fundamental domains of $\Gamma$ is algorithmically challenging. We address this difficulty by targeting the computation of Dirichlet cells that can form fundamental domains of $\Gamma$. We show that the half-spaces defining such a Dirichlet cell can be derived from elements of $\Gamma$ acting on $\mathbb{R}^n$ that can be expressed as words of bounded length in a suitable generating set. Based on these results, we design an algorithm for the computation of fundamental domains of crystallographic groups and exploit it to study the construction of topological interlocking assemblies.

Motivated by the concept of almost Yamabe solitons, a special class of generalized $m$-quasi-Einstein manifolds is investigated in this paper. We refer to these Riemannian manifolds as generalized $m$-quasi-Einstein manifolds of Yamabe-type. We study the rigidity properties for the potential (or defining) vector field associated to these manifolds in both the compact and non-compact settings. We show that under certain natural assumptions the potential vector field either vanishes identically or becomes a non-trivial Killing vector field.

At the specialization $t=-q$, $q\geq0$, the shifted $t$-Schur function associated with the modified odd Greaves–Jing–Zhu operator is $Q_\lambda[X+qX]$. Instead of merging the two alphabets $X$ and $qX$, we insert an intermediate strict partition between the two corresponding half-vertex operators. This gives a two-color lift of the shifted Schur measure on pairs $\mu\subseteq\lambda$ with weight \[ Q_\mu(qX)Q_{\lambda/\mu}(X)P_\lambda(Y). \] We compute the normalization and both marginals, identify an explicit Markov transition kernel, prove a semigroup property, and show that the two color volumes $|\mu|$ and $|\lambda|-|\mu|$ are independent. We also realize the model as a two-time shifted Schur process and write its Pfaffian correlation kernel in Vuletić's convention. Rectangular specializations give closed formulas and Gaussian limits for the color volumes.

The Birman-Craggs-Johnson homomorphism is a homomorphism $\sigma \colon \mathcal{I}_g \to \mathbb{B}_3'$ from the Torelli group to a certain $\mathbb{Z}/2\mathbb{Z}$-vector space of Boolean polynomials. In 1983, Johnson computed $H_1(\mathcal{I}_g)$ for $g \geq 3$ and showed, in particular, that the induced homomorphism on $H_1(\mathcal{I}_g)$ is injective when restricted to the subgroup generated by Dehn twists about separating simple closed curves. In this paper, we extend Johnson's result to higher homology groups. Given any collection of pairwise disjoint separating simple closed curves on $\Sigma_g$, the corresponding Dehn twists pairwise commute and determine a homology class in $H_k(\mathcal{I}_g)$ called an abelian cycle. We prove that the pushforward homomorphism restricted to the subgroup of $H_k(\mathcal{I}_g)$ generated by such abelian cycles is injective for $k \leq g-2$.

In this paper, we introduce a weighted derivative histopolation framework on families of intervals. The degrees of freedom consist of one scalar normalization and weighted integral moments of the derivative over a prescribed family of subintervals. We prove that the resulting scheme is unisolvent on $\Pi_N$ when the interval family separates polynomials of degree at most $N-1$ through weighted moments and the normalization is nonzero on constants. Thus, the derivative moments determine the polynomial up to an additive constant, and the scalar normalization fixes this remaining degree of freedom. This gives a sharp criterion for the well-posedness of the interpolation problem and a complete characterization of the admissible scalar normalizations. We then show how admissible families of intervals can be constructed from a fixed grid. When the endpoints of the intervals belong to the grid, admissibility is reduced to the nonsingularity of an interval matrix associated with the family, which depends only on the representation of the intervals in terms of consecutive cells. For Jacobi weights, the associated data matrices have a natural block structure in Jacobi polynomial bases, and the reduced derivative matrix can be expressed in terms of shifted Jacobi moment matrices. We next study Chebyshev configurations in which this structure becomes explicit. For the four classical Chebyshev families, suitable polynomial bases lead to diagonal Gram matrices for the reduced derivative matrices. We show that this diagonal structure depends on the simultaneous choice of the weight, the basis, and the grid. Numerical experiments on equispaced and Chebyshev–Lobatto nodes show the behaviour of the method for different interval families and for different Jacobi parameters.

Graph neural networks (GNNs) have recently been shown to learn algebraic properties of finite groups from their Cayley graphs [1,2]. In this work, we investigate whether such models generalize to infinite finitely generated groups. Motivated by Gromov's theorem [3], a GNN is trained and validated exclusively on finite complete and truncated Cayley graphs, and then evaluated, without retraining, on truncated Cayley graphs of unseen infinite groups. The evaluation includes free abelian groups of various ranks, the discrete Heisenberg group, the infinite dihedral group, free groups, and direct products with both infinite abelian and finite groups. The results show strong generalization across these families, suggesting that finite Cayley graphs encode sufficient local geometric information to transfer to the infinite setting. Overall, this provides evidence that GNNs trained solely on finite groups can capture geometric features related to the growth of infinite finitely generated groups.

The geodesic treewidth of a graph $ G $ is the smallest $k$ for which there is a partition $\mathcal{P}$ into geodesics such that $G/\mathcal{P}$ has treewidth $k$, where $G/\mathcal{P}$ is obtained from $ G $ by contracting each part of $ \mathcal{P} $. Based on this notion, row treewidth was developed and is defined for a graph $ G $ as the smallest $ k $ such that $ G \subseteq H \boxtimes P $ for some graph $ H $ of treewidth $ k $ and a path $ P $. Equivalently, the row treewidth of a graph $ G $ is the smallest $ k $ for which there is a partition $ \mathcal{P} $ into disjoint unions of geodesics that are aligned with respect to some layering such that $ G/\mathcal{P} $ has treewidth $ k $. We separate the two notions by showing that bounded row treewidth does not imply bounded geodesic treewidth and by presenting a polynomial-time algorithm to decide whether a graph of treewidth 2 has geodesic treewidth 1, which is known to be NP-hard for row treewidth [Biedl, Eppstein, Ueckerdt, 2025]. More generally, we provide an algorithm to decide whether a given graph has geodesic treewidth at most $ d $ that is XP in the treewidth, whereas there is no such algorithm for row treewidth, unless P = NP [Biedl, Eppstein, Ueckerdt, 2025]. On the other hand, we show that computing the geodesic treewidth is NP-hard and that every graph with geodesic treewidth 1 has bounded row treewidth. Moreover, we improve the best known lower bound on the geodesic treewidth of planar graphs to 5.

We present a complete classification of integrable Yang-Baxter quantum circuits with open boundary conditions and arbitrary circuit geometries. Starting from the standard transfer-matrix construction with two types of staggered inhomogeneities, we derive a general mapping that determines the arrangement of circuit gates in terms of the inhomogeneities and the system size. We conjecture that time-periodic quantum circuits are integrable whenever the local bulk and boundary gates satisfy the Yang-Baxter equation and the same bulk gate is applied exactly once per period to every nearest-neighbor pair of spins. Our construction also provides an algorithm to detect Yang-Baxter integrability for circuits with arbitrary geometries. Furthermore, we introduce a third type of inhomogeneity, denoted by $\rho$, and demonstrate that the minimum possible circuit depth is four. We show that when these $\rho$-inhomogeneities are placed at the endpoints and in their immediate neighborhood, the resulting boundary gates can be interpreted as single gates acting on multiple sites. Our construction is fully general and applies to regular $R$-matrices, both of difference and non-difference type, together with their associated boundary matrices. As an application, we consider two-qubit gates corresponding to 6- and 8-vertex $R$-matrices of non-difference form satisfying the Yang-Baxter equation, and we construct the associated reflection matrices that generate integrable quantum circuits.

Inspired by Problem 17 from the 2024 American Mathematics Competition (AMC) 10B, this work focuses on enumerating the distinct outcomes of a snail race with specified number of ties of a certain type. We begin by developing a recurrence relation and subsequently derive a closed-form formula for the number of possible outcomes using the exponential generating function method. Two special cases of the problem are considered in detail. Our analysis also explores the connections between the solution to this problem and the ordered Bell numbers, Stirling numbers of the second kind, and partial Bell polynomials.

We study weighted estimates for linear and multilinear integral operators whose kernels satisfy only size conditions. Extending a theorem of E. Stein and its refinement by Soria and Weiss, we prove weighted estimates on Herz and Cesàro type spaces, together with multilinear strong-type and weak-type analogues. As applications, we derive consequences for a range of rough singular integral operators and related variants, including linear, oscillatory, and multilinear settings.