Papers

For connected proper metric spaces $X$, equipped with a locally doubling measure supporting a local $p$-Poincaré inequality, we completely characterize which compact sets $K$ with positive capacity are removable for bounded $p$-harmonic functions, $p>1$. Similar results are proved also for bounded quasiharmonic functions. The characterization is both in geometric and analytic terms. In particular, removability is shown to be equivalent to the validity of a Liouville type theorem in $X\setminus K$. Properties such as local connectedness, sequential annular quasiconvexity, concentration of capacity and $p$-parabolicity are identified as crucial for removability. Along the way, we give a rather elementary proof of the Liouville theorem for quasisuperharmonic functions in $p$-parabolic spaces. Our results apply in particular to manifolds and $\mathbf{R}^n$ equipped with (locally) $p$-admissible weights.

Given an object $A$ in a big tensor-triangulated category, we study the homological and cohomological Bousfield classes of the associated localization: the tensor-triangulated category of $A$-local objects. We show that the homological support classifies the homological Bousfield classes of the $A$-local category precisely when an $A$-relative form of the homological detection property holds. Moreover, we prove that this holds if and only if $A$ is Bousfield equivalent to a coproduct of homological residue fields. The analogous classification of cohomological Bousfield classes by homological cosupport is strictly stronger: it is equivalent to an $A$-relative form of homological stratification. This equivalence between stratification and the classification of cohomological Bousfield classes is new even in the absolute case. A further surprise is that stratification is also equivalent to the classification of homological Bousfield classes together with the statement that every cohomological Bousfield class is homological. Applied to chromatic homotopy theory, these results classify the homological Bousfield classes of any localization of spectra with respect to a coproduct of Morava $K$-theories. This covers many localizations of interest. We also completely characterize when such chromatic localizations are relatively homologically stratified. This yields new examples of cohomological Bousfield classes that are not homological. In particular, it answers a question of Wolcott concerning the category of harmonic spectra. Our examples are produced by exhibiting local spectra with empty homological cosupport.

This paper proposes a fast solver for time-harmonic elastic scattering by multiple particles embedded in layered media, with either Dirichlet or Neumann boundary conditions imposed on the particle surfaces. Such problems arise in many important applications, including composite material optimization, nondestructive testing, and subsurface imaging. They are computationally challenging because of strong multiple scattering interactions among the particles and the layered interface. The proposed method represents the layered medium contribution through Sommerfeld integrals and couples this representation with a well-posed boundary integral formulation for the particle scattering problem. High-order integral equation discretization and scattering matrix are used to handle particles of general shape, while multiple scattering theory provides an efficient description of particle interactions. To reduce the cost for large particle systems, the resulting multiple scattering computation is further accelerated by the fast multipole method. The main formulation is developed in both two and three dimensions. Numerical experiments for rigid and traction-free particles validate the accuracy of the formulation, and demonstrate its flexibility in both direct scattering simulations and inverse scattering applications.

We answer the question whether all Heyting algebras can appear as the lattice of subterminal objects of an elementary topos in the negative. Concretely, we show that the free Heyting algebra on two generators, hence also on N generators for every N greater than 2, cannot be such a Heyting algebra. The mathematical results in this document were obtained with the help of ChatGPT 5.6 Sol, although the document itself was written entirely by us, and we take full responsibility for its contents.

Connected and automated vehicle (CAV) platooning offers a promising approach to improving road safety and traffic capacity. However, platoon control in real-world traffic is challenging due to uncertainty and heterogeneous driving behaviors. Reinforcement learning (RL) has strong potential for addressing such control problems, but its practical deployment raises challenges related to safety and learning efficiency. This paper proposes a generic modeling and simulation framework for investigating CAV platoon joining maneuvers and comparing deep reinforcement learning (DRL)-based control algorithms. The problem is particularly challenging in mixed-traffic environments, where CAVs coexist with human-driven vehicles exhibiting heterogeneous longitudinal and lateral behaviors. The objective is to achieve safe and efficient joining maneuvers by either incorporating penalties for risky behaviors into the learning process or using an external safety controller to constrain the learned policy. An agent-based modeling framework coupled with the Simulation of Urban MObility (SUMO) simulator is used to evaluate Deep Q-Network (DQN), Double Deep Q-Network (DDQN), and Proximal Policy Optimization (PPO). Results show that PPO outperforms DQN and DDQN, achieving a joining success rate of approximately 98 % and a collision rate below 1 %, largely due to risk-related penalties incorporated into the reward function. However, this improved performance requires more decision steps to complete the maneuver, revealing a trade-off between safety, joining effectiveness, and decision efficiency. An external safety controller effectively prevents collisions, although its interventions may reduce joining efficiency. The results highlight the importance of jointly considering safety and efficiency when designing RL-based controllers for CAV platoon joining in mixed traffic.

Let $R$ be a commutative ring with unity such that $\frac{1}{2}\in R$. Let $X$ be a connected finite poset with $|X|>2$ and $I(X,R)$ be the incidence algebra of $X$ over $R$. In this paper, we characterize the forms of linear maps $F_1,F_2,F_3,F_4:I(X,R)\to I(X,R)$ satisfying \[ F_1(f)g+fF_2(g)+F_3(g)f+gF_4(f)=0, \] whenever $fg=gf=0$. We prove that the $F_i$'s are of the so-called standard form if and only if any two edges in the comparability graph of $X$ are contained in one cycle. The ingredients of the proof contain a characterization of $2$-connectedness in comparability graph and the two-sided zero product determined property of incidence algebras.

For the surface growth model under molecular beam epitaxy on $\mathbb{R}$, we show that an inverted parabola is a stable steady solution in the sense that any small perturbation of it decays in $L^{\infty}(\mathbb{R})$ at the rate $O(t^{-1/2}\log t)$. The proof is based on the explicit solution operator generated by the linearized perturbation equation. This result partially justifies the merging of two neighboring islands into a bigger one. Moreover, any derivative of its solution is shown to be decaying in $L^{2}(\mathbb{R})$ for sufficiently small initial data.

The maximum energy dissipation principle is employed to nonlinear chemical thermodynamics in terms of distance variable (generalized displacement) from the global equilibrium, applying the optimal control interpretation to develop a variational formulation. The cost-like functional was chosen to support the suggestion that such a formulation corresponds to the maximum energy dissipation principle. Using this approach, the variational framework was proposed for a nonlinear chemical thermodynamics, including general cooperative kinetics model. The formulation is in good agreement with standard linear non-equilibrium chemical thermodynamics.

Since Ilmanen's pioneering work [J. Differential Geom. 38, 417-461, (1993)] it has been a major open problem to understand the sharp-interface limit of systems of coupled Allen-Cahn equations. We prove - for the first time - a global, unconditional convergence result for such a coupled system and show that the limit is a multiphase mean curvature flow in the sense of Brakke.

In this paper, we establish a small-time annealed path large deviation principle for one-dimensional diffusions in a random environment associated with the generator ${\mathcal L}_W f(x)=e^{-\rho(x,W)}(e^{a(x,W)}f'(x))'$. The coefficients $\{\rho(x,\cdot):x\in\mathbb R\}$ and $\{a(x,\cdot):x\in\mathbb R\}$ are random. We assume that for each fixed realization of the environment, $\rho$ and $a$ are continuous and locally exponentially integrable, and that the support of the associated intrinsic coordinates is compact and non-collapsing. This framework includes the extensively studied Brox diffusion $dX_t=dB_t-\frac12\dot W(X_t)\,dt$, where $B$ is a standard Brownian motion and $W$ is an independent two-sided Brownian motion representing the environment. The Itô–McKean representation of the diffusions and the estimates of the first exit probabilities derived via Moser iteration play a crucial role.

This paper is concerned with the following semilinear elliptic equation involving the fractional Laplacian: $$(-\Delta)^s u+ g|u|^{p-1}u= \lambda \frac{u}{|x|^{2s}}+f(x),$$ in a bounded domain $\Omega$ of $\mathbb{R}^N\,(N>2s)$, subject to the zero Dirichlet condition in $\mathbb{R}^N\setminus \Omega$, where $0<g\in L_{loc}^1(\Omega)$, $p>1$ and $f\in L^{(p+1)/p}_g(\Omega)$. Under certain integrability condition on $g$, the existence of solution is proven for every $\lambda\in \mathbb{R}$. Moreover, the regularity of solution is also obtained.

Many matrices in computational chemistry are neither sparse nor low-rank, making the design of low-scaling algorithms difficult. In this Perspective, we point out that many important matrices in computational chemistry, such as the Coulomb matrix, the electronic repulsion integral tensor, the density matrix, the localized molecular orbital (LMO) coefficient matrix, the Fock matrix, and the nuclear Hessian matrix share the same property: when their basis functions are suitably ordered, their off-diagonal blocks have low numerical ranks (despite that they as a whole have high numerical ranks). This property, termed off-diagonal low rank (ODLR), has been extensively studied in the mathematics community, but has surprisingly found very little use in computational chemistry. This Perspective reviews the existing mathematical literature on how to use the ODLR property of matrices to compactly store, as well as efficiently calculate or use them. Subsequently, we review the use of the ODLR property in computational chemistry, and point out possible future opportunities of devising new low-scaling methods for dense, full-rank matrices, exploiting the ODLR property. In particular, we prove for the first time that Fock matrices and LMO coefficient matrices satisfy the ODLR property, even if the system is gapless (in which case the matrices are dense). This paves the way to linear scaling electronic structure calculations of gapless systems at zero electronic temperature.

We develop a unified approach to constructing the moving frames of the evolute, involute and parallel of a Legendre curve via rotational transformations (rotation, boost and null rotation). The definitions of the evolute, involute and parallel of a Legendre curve in hyperbolic and de Sitter 2-spaces are systematically presented. Furthermore, the connections and dualities among these curves are discussed.

We determine the Hausdorff and packing dimensions of Cartesian products of one-dimensional exact approximation sets. Our main result establishes the exact-approximation counterpart of the recent product theorem of Wang and Wu (2024) for limsup approximation sets, showing that passing to the substantially smaller exact approximation sets (liminf sets) does not reduce the Hausdorff dimension of the Cartesian product. One of the key ingredients is a refinement of the well-distributed-system framework of Bandi–Ghosh–Nandi (2023) by exploiting the fine arithmetic distribution of rational points which then gives the Hausdorff dimension of the product set under a weaker convergence condition.

We investigate the form of the Saito metric on quotients of the reflection representation of an extended affine Weyl group, restricted to an arbitrary stratum of its discriminant. We compute the Saito determinant for all strata and Dynkin types and show that it is divisible by a product of q-analogues of restricted roots for the associated hyperplane arrangement. Our results simultaneously generalise the factorisation in the Weyl denominator formula to higher codimension strata, and provide q-analogue refinements of results of Antoniou-Feigin-Strachan for Coxeter groups.

It is well known that the Einstein equations are tensor equations for a Lorentzian metric. Hence, choosing a suitable gauge condition is crucial for solving them. As a powerful gauge condition, maximal foliations have played a pivotal role in two groundbreaking works in general relativity: the global stability of Minkowski spacetime and the bounded L^2 curvature conjecture. Nevertheless, if the initial hypersurface fails to be maximal, the usual elliptic approach for constructing maximal foliations encounters fundamental difficulties when solving the Einstein equations. The purpose of the paper is to provide a construction of maximal foliations around any initial asymptotically flat Cauchy surface satisfying vacuum Einstein constraint equations, under a suitable smallness condition on the mean curvature.

Let $f,g:\mathbb{N}\to\mathbb{T}$ be completely multiplicative functions with dense images in the complex unit circle $\mathbb{T}$. We prove that, for every non-empty open set $U \subset \mathbb{T}^2$, the set of integers $n$ such that $(f(n),g(n+1)) \in U$ has positive upper logarithmic density, unless the pair $(f,g)$ is of a special form. This result strengthens earlier theorems of Klurman and Mangerel, as well as of Charamaras, Mountakis, and Tsinas, and yields simpler proofs.

Motivated by the Ekedahl-Shepherd-Barron-Taylor conjecture, we study arithmetic criteria for the global and local integrability of foliations in characteristic zero. Our main result is a finite-holonomy theorem for the regular parts of invariant prime divisors of corank one foliations which are \(p\)-closed for almost all primes. As a consequence, we give a complete proof for corank one foliations of the statement proposed by Ekedahl-Shepherd-Barron-Taylor: a foliation on a smooth projective variety which is \(p\)-closed for almost all primes is algebraically integrable whenever it admits a compact leaf. We also introduce a local analytic notion of \(p\)-closedness for foliation germs on normal complex varieties and formulate local conjectures relating it to meromorphic and holomorphic first integrals. We establish holomorphic integrability for analytically \(p\)-closed foliations with canonical singularities on smooth surfaces and on klt surface germs. Combining these local results with the finite-holonomy theorem, we obtain holomorphic first integrals for certain analytically \(p\)-closed non-dicritical surface germs and for analytically \(p\)-closed simple singularities in arbitrary dimension. Finally, we formulate a connection-theoretic counterpart to local holomorphic integrability in terms of flat meromorphic extensions of the Bott partial connection, and verify it in dimension two.

Sea ice dynamics plays a crucial role in the Earth's climate system, making it an important component of weather and climate prediction models. Its numerical simulation remains challenging, however, as it exhibits complex mechanical behaviors due to a nonlinear ice rheology and interactions with external physical forcings. In particular, sea ice presents linear kinematic features (LKFs), i.e., narrow bands of intense deformation associated with processes such as lead opening or pressure-ridge formation. Accurately representing these features is necessary as they affect thermodynamics and ocean-atmosphere exchange. Yet their number, localization, and structure are highly sensitive to spatial resolution and to the chosen discretization of the sea ice velocity. In this work, we investigate the spatial discretization of Hibler's viscous plastic sea-ice model using a fully discontinuous Galerkin (DG) representation of all variables, including the velocity field. The ability of DG elements to represent discontinuities while having high order local polynomial approximation makes them well suited for resolving the complex ice deformation features, and insuring robustness towards mesh-induced numerical artefacts. We assess the fully DG method using an established sea ice dynamics benchmark and compare the obtained sea ice deformation with state-of-the-art discretizations. Additionally, we study the theoretical convergence of the modified Elastic-Viscous-Plastic (mEVP) formulation as a pseudo-time iterative solver for the viscous-plastic (VP) momentum equations. We prove the convergence of the underlying continuous pseudo-time dynamical system towards the viscous plastic formal limit, thereby providing a theoretical foundation for the use of mEVP as an iterative solver for Hibler's VP model.

Let $W$ be an affine Coxeter group of type $\widetilde{A}_n$, that is, the affine symmetric group $\widetilde{S}_N$ with $N=n+1$, let $T$ be its set of reflections, and let $\mathrm{Red}_T(w)$ be the set of reduced reflection factorizations of an element $w\in W$. The braid group acts on $\mathrm{Red}_T(w)$ by the Hurwitz action. For finite Coxeter groups it is known exactly when this action is transitive, namely precisely for the parabolic quasi-Coxeter elements. We address this problem for the affine type $\widetilde{A}_n$ by determining all orbits of $\mathrm{Red}_T(w)$.

We study bubbling for sequences of solutions of the Hamiltonian-perturbed Fueter equation. By Walpuski's compactness theorem, a bounded-energy sequence converges weakly to a Fueter map, with energy loss recorded by a defect measure supported on a codimension 2 rectifiable set. Assuming that the limiting map has no non-removable singularities and the bubbles attach to it on a subset of the bubbling locus of positive measure, we show that the limiting Hamiltonian lies in an exceptional subset of infinite codimension. The proof combines quantitative control of rational sweepout loci in hyperkähler manifolds with a transversality argument. We expect the attachment assumption to be automatic and follow from a suitable no-neck result. This strongly suggests that, in the absence of non-removable singularities, bubbling of Hamiltonian-perturbed Fueter maps is highly nongeneric.

We establish quantitative results on finite-dimensional spectral approximation and ergodic-capacity convergence for continuous Holographic Multiple-Input Multiple-Output (HMIMO) channels with square apertures and physically prescribed circular wavenumber support. The resulting spatial-wavenumber domain mismatch leads to a non-separable square-disk concentration problem for which the classical separable construction based on prolate spheroidal wave functions (PSWFs) cannot be directly applied. We project the continuous operator onto a tensor-product subspace of one-dimensional (1D) PSWFs while preserving the circular wavenumber support, yielding a generally non-diagonal but highly sparse finite-dimensional matrix. We show that the whole-spectrum approximation error, accounting for retained-eigenvalue perturbations and the residual spectral tail, remains controlled by a 1D PSWF eigenvalue-tail envelope despite the loss of separability and induced off-diagonal coupling. Beyond an explicit 1D truncation threshold, this error decays super-exponentially. This analysis further yields an asymptotic upper envelope for the eigenspectrum under the flattened two-dimensional eigenvalue ordering. We further establish a non-asymptotic upper bound on the gap between the actual ergodic capacities of the continuous and tensor-PSWF-truncated channels under their respective transmit-covariance optimizations. Combined with the spectral result, this capacity-gap bound inherits the same super-exponential dependence on the truncation order. Finally, quadrature rules with explicit radial and angular node thresholds are developed for evaluating the projected matrix. Numerical results show that conventional truncation based on spatial degrees of freedom can omit performance-relevant modes, particularly for compact apertures.

We study the deterministic first-order oracle complexity of finding stationary points of the value function in smooth nonconvex-Polyak-Łojasiewicz (NC-PŁ) minimax optimization. We assume that the objective is jointly $\ell$-smooth and satisfies the $\mu$-PŁ condition in the dual variable, and that its value function $\Phi(x):=\max_y f(x;y)$ satisfies $\Phi(0)-\inf_x\Phi(x)\leq\Delta$. When $\kappa:=\ell/\mu\gtrsim 1$ and $0<\epsilon^2\lesssim\ell\Delta$, we prove that every deterministic first-order method requires $\Omega(\ell\Delta\kappa/\epsilon^2)$ oracle queries in the worst case to find $x$ satisfying $\|\nabla\Phi(x)\|\leq\epsilon$. This rate matches the known upper bound in its dependence on $(\ell,\Delta,\kappa,\epsilon)$ [Yang et al., 2022] and shows that the linear dependence on $\kappa$ is unavoidable for deterministic first-order methods.

In this paper, we construct a lifting operator from the Grothendieck group of admissible Harish-Chandra modules of $G =\mathrm{SO}_{2n}(\mathbb C)$ (resp. $\mathrm{Sp}_{2n}(\mathbb C)$) to that of genuine representations of $\mathrm{Spin}_{2n}(\mathbb C)$ (resp. $\mathrm{Spin}_{2n+1}(\mathbb C)$). We determine the lift of the sum of special unipotent representations attached to any ${}^{\vee}\mathcal O \subseteq {}^{\vee}\mathfrak g$ explicitly. In particular, the representations occurring in these lifts, if nonzero, are genuine unipotent representations of complex Spin groups and are unitary. As a consequence, the lifting operator preserves unitarity on a large class of unitary representations.

Let $p>3$ be a prime, $E/\mathbb Q$ be an elliptic curve and $K$ an imaginary quadratic field satisfying the hypotheses of Burungale-Castella-Skinner, and let $L/K$ be the unique $\mathbb{Z}_p^2$-extension. In this note, by combining their integral two-variable main conjecture with the specialisation formula of the first-named author, we obtain a characteristic-ideal identity over every $\mathbb{Z}_p$-line in $L/K$, involving an explicit local factor. With the characteristic ideal of a non-torsion module defined to be zero, this identity also applies when the two-variable Perrin-Riou element specialises to zero. We call such lines exceptional and prove that only finitely many occur. We also prove that the cyclotomic line is non-exceptional with trivial local factor, and that the anticyclotomic line is exceptional. Finally, we bound the number of exceptional lines by the cyclotomic augmentation order and prove that if the $p$-primary Tate-Shafarevich group over $K$ is finite and the cyclotomic $p$-adic height pairing is non-degenerate, then this number is at most ${\rm rank}(E(K))$. In particular, when $E(K)$ has rank one, the anticyclotomic line is the unique exceptional line.

We study size bounds for conjunctive query (CQ) results which in recent years have played a crucial role in database theory. In particular, we compare the so-called entropic bound which is known to be asymptotically tight but not known to be computable, and its computable relaxation called the polymatroid bound, which is generally not tight. We focus here on conjunctive queries under acyclic functional dependencies. These queries are known to be well-behaved in the sense that, in the case without projections, both bounds coincide. We show that this picture changes when projections are allowed: in this case, even for acyclic functional dependencies, there is in general a polynomial gap between the two bounds. We complement this negative result by showing a special case for which the polymatroid bound is tight under acyclic functional dependencies and projections that is characterized by the position of the output variables in a topological order of the query variables.

Let $G$ be a finite simple graph and let $S\subseteq V(G)$. We prove that the minimum number of vertices meeting every cycle that intersects $S$ is at most the maximum number of vertices of $S$ covered by a collection of vertex-disjoint cycles. This answers a question posed by Bowler, Ghorbani, Gut, Jacobs, and Reich [SIAM Journal on Discrete Mathematics 40 (2026), 988–999]. An incidence-based reduction to their bidirected packing–covering theorem preserves the packing value and projects transversals without increasing their cardinality.

Weak bialgebras are natural generalizations of bialgebras that play a fundamental role in quantum groups, tensor categories, and representation theory. In this paper, we systematically classify all (weak) bialgebra structures on the finite-dimensional semisimple algebra \(\Bbbk^{\oplus n}\), with a distinguished basis of primitive orthogonal idempotents. For the bialgebra case, we prove that such structures are in bijection with finite monoids, and that bialgebra isomorphisms correspond exactly to monoid isomorphisms. Extending the result to weak bialgebras, we show that every weak bialgebra structure uniquely determines a finite small category.

Even when all clients optimize the same strongly convex objective, the global model parameters produced by constant-step federated learning may still have a stationary mean that differs from the minimizer. Does increasing the number of clients eliminate this bias, or can some components persist? Answering this question clarifies the limits of client averaging. To this end, we study two federated learning algorithms, \Scaf{} and matched \FedAvg{}, under full participation, a sufficiently smooth objective, and independent bounded additive gradient noise. Analyzing global stationary balance and within-round local fluctuations yields multidimensional expansions with a shared $\gamma/N$ term and explicit client-independent $\gamma^2$ coefficients, where $\gamma$ is the step size and $N$ is the client count. The \Scaf{} coefficient combines the ordinary local-update contribution found in \FedAvg{} with an additional persistent-control contribution. Although zero-sum controls cancel from the direct server increment, they change local second moments, which interact with third-order curvature to shift the stationary mean. For fixed dimension $d$ and local-step count $H$, a remainder of $O_{H,d}(\gamma^2/N+\gamma^3)$, uniform in $N$ for sufficiently small $\gamma$, identifies the second-order coefficients along arbitrary joint small-step, large-client limits after removing the shared leading term. Controlled experiments assess the predicted coefficients. In real-data analyses, finite-tail mean displacement can account for a substantial share of averaged-model parameter error, and a post-hoc comparison shows directional agreement with the derived curvature–noise structure.

In this paper, we characterize compactness in terms of the Berezin transform and obtain an essential norm estimate for operators in the Toeplitz algebra on Bergman spaces of the Siegel upper half-space.