Papers

We identify a physical-space balance-law mechanism underlying bilinear null forms for the free Dirac equation in three space dimensions. For each spatial direction, we decompose a spinor into the two eigenspaces of the directional Dirac symbol. The principal parts of the corresponding modes propagate in opposite directions, while the transverse derivatives and the mass term couple them. After integration over the transverse variables, their charge densities satisfy a pair of one-dimensional balance laws, and a div–curl interaction estimate controls the mixed product of these densities. The algebraic anticommutation condition defining the spinorial null form exchanges exactly the same two eigenspaces. This identifies the algebraic cancellation with the interaction selected by the balance laws. Combined with angular localization, the argument yields frequency-localized $L^2$ spacetime estimates while preserving the natural first-order formulation of the Dirac equation. In the massless case, the estimate has the same lower-frequency scaling as the three-dimensional wave null-form estimate and gains half a derivative over the direct product bound. In the massive case, we obtain a channel-dependent refinement. For the pseudoscalar interaction, anticommutation with the full massive Dirac Hamiltonian gives an additional frequency-to-mass factor $K/m$ for low-frequency interactions within the same energy branch, where $K$ is the frequency scale and $m$ is the mass. This gain is absent in the scalar channel. The bilinear estimates also yield factorized bounds for cubic spinorial null forms arising in nonlinear Dirac models.

We investigate a coupled thermoelastic plate system consisting of a fourth-order displacement equation and a heat evolution equation linked through a spatially varying coupling factor $\alpha(x)$. The model accounts for thermoelastic interactions through the operators $\operatorname{div}(\alpha(x)\nabla \theta)$ and $\operatorname{div}(\alpha(x)\nabla u_t)$. We establish the well-posedness of the direct problem under homogeneous Neumann conditions for $u$ and Dirichlet conditions for $\theta$, deriving optimal energy estimates and demonstrating continuous dependence of solutions on the given data. We further introduce an input-output operator corresponding to the considered inverse problem and show that it is compact and Lipschitz continuous, confirming the ill-posed nature of the associated inverse problem. Using these properties, the inverse problem is formulated as a minimization problem for the Tikhonov functional, and we establish the existence of a minimizer.

For a finite family of representations of a reductive group, we define a Laurent polynomial on the group. The exponential sum associated this Laurent polynomial is called a hypergeometric exponential sum. We introduce an arithmetic hypergeometric $\mathcal D$-module to study the hypergeometric exponential sum. It is an overholonomic arithmetic $\mathcal D$-module with a Frobenius structure so that the trace of the Frobenius at a rational point is the exponential sum. We prove that over the locus where the Laurent polynomial is nondegenerate, the arithmetic hypergeometric $\mathcal {D}$-module defines an $F$-isocrystal overconvergent along the degenerate locus. As an application, we get an estimation of the hypergeometric exponential sum.

We study the Cauchy problem for the three-dimensional barotropic compressible Navier–Stokes equations with a time-independent potential force near a spatially nonconstant stationary state. The potential is controlled in unweighted homogeneous Besov spaces; in particular, no polynomial spatial-weight condition involving $(1+|x|)^j\nabla^j\phi$ is imposed. For initial data relative to the stationary state that are sufficiently small in $\dot H^{\frac12-\delta}\cap\dot H^3$, we establish the existence and uniqueness of a global strong solution in $H^3$, while allowing the initial $L^2$ norm to be arbitrarily large. If the initial data are bounded in $\dot B^s_{2,\infty}$ for $s\in[-\frac32,-1)$, then the solution and its first spatial derivative decay at the optimal rates $(1+t)^{-\frac{k-s}{2}}$ with $k=0$ and $1$, respectively. The analysis relies on refined homogeneous energy estimates and a frequency-localized description for the dissipative and asymptotic structures of the system.

We prove a conjecture of Dubrovin, Liu, Yang, and Zhang that states that the Poisson bracket of the Dubrovin-Zhang hierarchy of a rank-one cohomological field theory is constant if and only if it is given by the triple Hodge class with the parameters satisfying the Calabi-Yau condition.

Multisections were defined by Ben Aribi–Courte–Golla–Moussard as a way to decompose closed manifolds into $1$-handlebodies, which is a generalization of Heegaard splittings and trisections. Previously, their existence was only known in dimensions up to $5$. We show that multisections exist for closed manifolds in arbitrary dimensions. We also show that submanifolds of codimension at least $2$ in multisected manifolds can always be put into an appropriate bridge position, generalizing the existence result on bridge trisections in dimension $4$.

A graph $G$ is $F$-saturated if $G$ contains no copy of $F$ but $G+e$ contains a copy of $F$ for every missing edge $e$ of $G$. The saturation number $\sat(n,F)$ is the minimum number of edges in an $n$-vertex $F$-saturated graph. Motivated by a problem posed by Faudree, Ferrara, Gould, and Jacobson concerning $K_p\cup K_q\cup K_{q+1}$, we determine the saturation number and the unique extremal graph for $K_p\cup K_q\cup K_r$ whenever $2\le p\le q<r<p+q$ and $n$ is sufficiently large. Together with the previously known results for $r\ge p+q$ and for $r=q$, this completes the determination of the saturation number and the extremal graphs for unions of three cliques, for all sufficiently large $n$.

An old conjecture in non-Kähler geometry states that, if a compact Hermitian manifold has constant holomorphic sectional curvature, then the metric must be Kähler (when the constant is non-zero) or Chern flat (when the constant is zero). It is known to be true in complex dimension $2$ by the work of Balas and Gauduchon in 1985 (when the constant is negative or zero) and Apostolov, Davidov and Muskarov in 1996 (in general). In dimension $3$ or higher, the conjecture is only known in some special cases, such as for all twistor spaces by the work of Davidov, Grantcharov, and Muskarov, or for the locally conformally Kähler case (when the constant is negative or zero) by the work of H. Chen, L. Chen and Nie, or for all non-balanced Bismut torsion parallel (BTP) manifolds by the work of S. Chen and Zheng, where they also showed that the conjecture holds for all balanced BTP threefolds, utilizing a classification result by Zhao and Zheng. In this article, we show that the conjecture is valid for all balanced BTP manifolds in complex dimension $4$. The interesting part is that while balanced BTP manifolds are highly restrictive, the classification is still lacking in dimensions $4$ or higher, and this study might shed some light on the structure of balanced BTP manifolds in dimension $4$.

We study when statistically learnable latent structure can also be recovered efficiently, and how membership queries change the answer. An unknown support $A\subseteq\mathbb F_2^n$ has small additive doubling and is observed through a fixed set $B$ satisfying $|A\triangle B|\le\eta|A|$. We seek one linear subspace $V$ such that every compatible support $A$ is covered by few $V$-cosets and satisfies $|V|\le|A|$. For every $\eta<1$, polynomially many uniform samples suffice statistically, with cost polynomial in the doubling constant and proportional to $(1-\eta)^{-1}$; this radius dependence is sharp. Under a specified hardness assumption for learning parities with noise (search-LPN), however, no polynomial-time sample-only learner achieves even constant covering cost, including when the latent support is unique. At fixed structural parameters and the same constant covering budget, adding exact membership queries to $B$ permits polynomial-time recovery. The general query learner constructs a short structural list and uses fresh samples to select one common output through a majority-coverage rule. Persistent structured cores make this candidate construction possible. At doubling one, a complementary distinction appears at $\eta=1/3$: coarse recovery remains polynomial time, while exact recovery requires exponentially many accesses in the worst case when latent cardinality is unknown.

We give a rigorous proof that the hard-core model on a natural family of infinite graphs exhibits a first-order phase transition.

This paper is devoted to an in-depth study of the minimal commutant property for composition operators acting on Hilbert spaces of holomorphic functions in several complex variables, such as the Fock space on $\mathbb{C}^d$, the Hardy space on the Euclidean ball, and the Hardy space on the unit polydisc.

Using Poisson approximation techniques, we derive the detection threshold of the packing test in \cite{Jiang13} when testing spherical uniformity under high-dimensional Fisher–von Mises–Langevin (FvML) and Watson alternatives. Our result rigorously confirms the empirical observation that the packing test is strictly suboptimal for testing uniformity in these two popular models. In the high-dimensional FvML model, its detection threshold is precisely \(\kappa=\Theta\lb p^{3/4}/(\log n)^{1/4}\rb\). In the high-dimensional Watson model, its detection threshold is \(p-2\kappa=\Theta(\sqrt{p\log n})\), or equivalently \(\kappa=p/2-\Theta(\sqrt{p\log n})\). The non-null limiting distributions of the packing test under these two models are derived. We show that the limiting scalings of the largest squared inner product undergo a discontinuous phase transition in the Watson model, whereas no analogous phenomenon occurs in the FvML model.

We study the periodic homogenization of a class of viscous Hamilton–Jacobi equations with fast dependence on the unknown. This problem combines features of first-order Hamilton–Jacobi equations with \(u^\epsilon/\epsilon\)-periodic Hamiltonians and semilinear heat equations with rapidly oscillating positive potentials. In this paper, we prove qualitative homogenization results for $H(y,s,p)=F(s,p)+\eta W(y,s,p)$ when $|\eta|$ is sufficiently small, with the smallness threshold depending on the Lipschitz constant of the initial datum, and establish a large-time averaging result for a general class of evolutionary cell problems.

We determine the bias in higher moments of the Legendre family of elliptic curves in view of the Negative Bias Conjecture of S. J. Miller. We show that every lower order term is either zero or negative on average for both even and odd moments, and explicitly compute the third and fourth moment. The results about Hurwitz class numbers in arithmetic progressions which we prove in the process may be useful for other applications.

A Catalan multijection is a way to partition a set of $\binom{2n}{n}$ objects into equivalence classes of size $n+1$, where each equivalence class has exactly one distinct Catalan object. Isaak and Langley investigated when two multijections produced the same partitions in an unpublished manuscript. We propose some new multijections for which we get different partitions from previously discovered multijections.

We construct a one-parameter family of small self-similar solutions to the three-dimensional one-phase Muskat problem with surface tension. The solutions have the form $\eta_\varepsilon(t,x) = t^{1/3}U_\varepsilon(t^{-1/3}x)$ and emanate from the conical initial data $\eta_\varepsilon(0,x)=\varepsilon|x|$. The profiles are perturbations of the linear capillary regularization of the cone, and we identify the leading quadratic correction. The proof combines a raywise inverse estimate for the linear similarity operator, a favorable high–high-to-low cancellation in the quadratic term, and finite-order tame estimates for the Dirichlet–Neumann operator on asymptotically conical graphs. These estimates yield the solutions by a contraction argument and show that the conical singularity is instantaneously rounded for positive time.

Let $\beta(s)=\sum_{k=0}^{\infty}(-1)^k(2k+1)^{-s}$ and let $G=\beta(2)$ be Catalan's constant. We develop two families of positive series for reciprocal powers $\beta(s)^{-r}$. The first is obtained from the classical Euler transformation and is evaluated at $1/2$; it is valid for real $s>0$. A second transformation, valid for $s\geq2$, gives a faster series evaluated at $1/3$. We derive finite-sum and integral formulas for the base coefficients, together with a positive recurrence and a composition formula for arbitrary reciprocal powers. When $s$ is an integer, all coefficients are rational. We also give explicit remainder estimates and determine the exact root-convergence rates of the two families. As applications, we obtain positive rational series for every reciprocal power of Catalan's constant and for reciprocal powers of $\pi$ arising from odd values of the Dirichlet beta function.

This is an expository paper based on the authors' joint works. The goal is to explain the statements and the ideas behind two conjectures on associated varieties and simple modules of simple affine vertex algebras $L_k(\mathfrak{g})$ for a simple and simply-laced Lie algebra $\mathfrak{g}$ and a integer level $k$ above the critical level.

In 2016, Straub proved that Euler's classic partition identity holds true for partitions with largest hook (perimeter) $n$. This inspired further study of the relationship between classical partitions and fixed perimeter partitions. We extend the study of parity bias inequalities, first introduced by Kim, Kim, and Lovejoy in 2020, to the fixed perimeter setting and show using combinatorial methods that fixed perimeter analogues of many classical parity bias results can be proven and generalized. We also extend these methods to prove similar inequalities for the fixed perimeter analogues of PED and POD partitions. We additionally develop recursive formulas for the number of perimeter $n$ partitions with odd parts distinct and even parts unrestricted and with even parts distinct and odd parts unrestricted.

This paper is devoted to solving one-dimensional backward stochastic differential equations (BSDEs in short) with a general random terminal time $\tau$ taking values in the extended nonnegative real numbers. The generator $g$ of BSDEs satisfies some stochastic growth/continuity conditions in the state variables $(y,z)$, featuring unbounded stochastic coefficients $\mu_\cdot\in\R$ and $\nu_\cdot\in\R_+$ satisfying $\int_0^\tau (|\mu_t|+\nu^2_t) {\rm d}t<+\infty$. For any given real $p>1$, let $\rho_\cdot\geq \mu_\cdot+\frac{\theta}{2(p-1)}\nu_\cdot^2$ (instead of $\rho_\cdot\geq\mu_\cdot+\frac{\theta}{2[1\wedge(p-1)]}\nu_\cdot^2$ used in Zhang, Li, Hu and Fan [2026, arXiv:2603.13873v1]) be a real-valued process for some constant $\theta>1$ such that $\int_0^\tau |\rho_t|{\rm d}t<+\infty$. We work within a weighted $L^p$ space with the weighting factor $e^{\int_0^t \rho_r{\rm d}r}$. Within this framework, we establish several innovative results on the weighted $L^p$ solutions of BSDEs: an existence result, an existence and uniqueness result, an existence and uniqueness result of the minimal (maximal) solution, and two comparison theorems. These findings unify and improve some existing results. Some novel ideas are employed to address the challenges posed by general unbounded stochastic coefficients and general weighted spaces.

Bound states in the continuum (BICs) are localized eigenmodes with their frequencies in the radiation continuum of scattering states. The existence of a BIC implies the loss of uniqueness for scattering problems with given incident waves. Perturbed wave systems close to the ideal ones with a BIC exhibit strong resonance effects that are essential to numerous practical applications. A question of fundamental importance is whether a BIC is robust, i.e., whether it can continue its existence when the structure is slightly perturbed. In an earlier work [Yuan and Lu, Optics Letters, Vol.~42, pp.~4490-4493, 2017], for a class of BICs governed by the two-dimensional (2D) Helmholtz equation, which are not trivially protected by symmetry, we uncovered the conditions that ensure robustness and formally constructed the BIC in perturbed systems using a perturbation method. In this paper, we present a rigorous theory on the robustness of BICs in 2D dielectric structures with a single periodic direction. Specifically, we analyze the solvability and provide estimates for each order in the perturbation series, and prove the convergence of the series.

Whether the Scott space of the complete Boolean algebra $\RO(\bbR)$ of regular open subsets in the real line is sober was explicitly recorded as an open problem by Xu and Zhao and was reiterated in a later survey. We give a negative answer to this problem in this paper.

Nonlocal partial differential equations arise in many applications but are often difficult to model and learn because of the presence of nonlocal operators. We present a flow-map learning (FML) framework for modeling unknown nonlocal PDEs directly from solution data. Rather than learning or approximating the underlying nonlocal operators, the proposed approach learns the finite-time evolution operator in either modal or nodal space. Two complementary formulations are developed for spectral and grid-based solution representations. Numerical experiments on one- and two-dimensional fractional diffusion and wave equations demonstrate accurate and stable long-time prediction using only short observation windows. The proposed approach provides an effective data-driven framework for learning unknown nonlocal dynamics without explicit evaluation of nonlocal operators.

We study the frog model on $\mathbb Z^d$ and on the discrete tori $\mathbb T_L^d$, $d\ge 2$, with a symmetric, translation-invariant, and heavy-tailed transition kernel satisfying \[ Q(x,y)\asymp |x-y|^{-(d+\alpha)}, \qquad \alpha>0. \] Starting from an i.i.d. Poisson$(\lambda)$ number of sleeping particles per site and one active particle at the origin. Active particles perform independent $Q$-random walks and activate the particles they encounter. We first determine the timescale for activating distant vertices. When $\alpha\in(0,d)$, the time required to activate all vertices within distance $L$ of the origin is, with high probability, \[ (\log L)^{\Delta+o(1)}, \qquad \Delta^{-1}:=\log_2\left(\frac{2d}{d+\alpha}\right), \] as $L\to\infty$. This polylogarithmic spreading contrasts sharply with the linear spreading of the classical frog model driven by simple random walks; see Alves, Machado, and Popov (2002) and Ramírez and Sidoravicius (2004). When $\alpha>d$, we recover this classical linear behavior by proving matching linear upper and lower bounds; at $\alpha=d$, we prove a linear upper bound. Finally, we consider the finite-lifespan model on $\mathbb T_L^d$, in which each particle is removed after taking $\ell$ steps. We show that the cover lifespan, defined as the smallest $\ell$ for which the torus is entirely activated, is asymptotic to the cover time of a Poisson$(\lambda L^d)$ cloud of independent stationary random walkers.

In this paper, we study a class of one-dimensional piecewise maps defined by infinitely many smooth expanding branches. This class arises naturally in the context of non-smooth dynamical systems and includes the first-return maps locally defined near sliding Shilnikov connections. By means of the theory of conformal iterated function systems (CIFS), we investigate several dynamical properties of these maps as well as the topological complexity of their invariant sets. In particular, we show that the dynamics restricted to the invariant set is topologically conjugate to the shift on $\mathbb{N}^{\mathbb{N}}$. We also establish the existence of a unique conformal measure that is invariant and ergodic under the map.

We develop structure-preserving dynamical low-rank methods for stochastic Vlasov–Poisson equations with transport noise. We first derive continuous low-rank evolution equations that inherit the mass, momentum, and energy balance laws through fixed velocity modes. We then construct two augmented basis-update Galerkin (BUG) integrators based on Euler–Maruyama discretization of the Itô formulation and Heun-type discretization of the Stratonovich formulation. Both integrators combine standard basis augmentation with conservative rank truncation to preserve the selected velocity moments. We prove pathwise mass conservation and conservation of total momentum in expectation. The energy balances contain distinct stochastic corrections with identical conditional expectations and an explicitly characterized time-discretization defect. Numerical experiments illustrate the conservation properties, rank adaptation, and temporal convergence behavior of the proposed methods.

Let $\Lambda=B[E]$ be the one-point extension algebra of $B$ by the extension module $E$. Using the standard triple description of $\Lambda$-modules, we characterize mixed bricks through an injectivity condition and a scalar-stabilizer condition. Under the assumption that $B$ is $E$-visible finite, we interpret mixed bricks as orbits in Grassmannians and derive a necessary and sufficient criterion for the brick-finiteness of $\Lambda$.

Bandwidth-limited visual communication systems require efficient transmission of high-dimensional data under dynamic wireless conditions. Existing approaches either rely on joint source-channel coding, which tightly couples representation learning with channel models and lacks flexibility across varying environments, or adopt generative reconstruction techniques that may introduce semantically inconsistent outputs. In this paper, we propose RQ-NAC, a channel-agnostic semantic compression framework for visual communication. The proposed method leverages residual quantization to produce scalable discrete semantic representations, enabling fine-grained and predictable control over the rate-distortion tradeoff. To further enhance compression efficiency, we integrate an n-gram-driven arithmetic coding module that exploits contextual dependencies among latent indices for lossless entropy coding. Extensive experiments demonstrate that RQ-NAC achieves over 600$\times$ compression relative to uncompressed visual data while preserving high perceptual quality. The results show that our approach enables efficient, flexible, and reliable semantic transmission under bandwidth-constrained conditions.

Let $(\mathcal{A},G,\alpha)$ be a $C^*$-dynamical system. Unique extension properties for the $C^*$-inclusion $\mathcal{A} \subseteq \mathcal{A} \rtimes_{\alpha,r} G$ have been studied extensively. In this paper, we investigate unique extension properties for some other natural “dynamical” $C^*$-inclusions, namely: $\bullet$ $C_r^*(G) \subseteq \mathcal{A} \rtimes_{\alpha,r} G$; $\bullet$ $\mathcal{A}^G \subseteq \mathcal{A}$, where $\mathcal{A}^G$ is the fixed-point subalgebra; $\bullet$ $Z(\mathcal{A})^c \subseteq \mathcal{A} \rtimes_{\alpha,r} G$, where $Z(\mathcal{A})^c = Z(\mathcal{A})' \cap (\mathcal{A} \rtimes_{\alpha,r} G)$ is the relative commutant of the center. Our hope is that a larger selection of examples will enable progress on some lingering open problems, in particular (1) the relationship between aperiodicity and the unique pseudo-expectation property and (2) the relationship between the faithful unique pseudo-expectation property and norming.

Fix a squarefree integer $d_0>1$, and let $d$ range over the positive squarefree integers coprime to $2d_0$. Although $\mathbb{Q}(\sqrt{-d})$ and $\mathbb{Q}(\sqrt{-d_0d})$ share all variable ramified primes, we prove that their class-group $4$-ranks are asymptotically independent. Over the subfamily $d\le X$, their joint distribution converges in total variation to the product of two copies of the Cohen–Lenstra–Gerth distribution, with error bounded by a negative power of $\log\log X$. We further conjecture that the corrected $2$-primary groups $2\operatorname{Cl}_{\mathbb{Q}(\sqrt{-d})}[2^\infty]$ and $2\operatorname{Cl}_{\mathbb{Q}(\sqrt{-d_0d})}[2^\infty]$ are asymptotically independent, each with the Cohen–Lenstra distribution. Suppose in addition that the class number of $\mathbb{Q}(\sqrt{d_0})$ is odd. For a density-one subset of this family, we prove that extension of ideals to $K(d)=\mathbb{Q}(\sqrt{d_0},\sqrt{-d})$ induces $4\operatorname{Cl}_{K(d)}[2^\infty]\cong 2\operatorname{Cl}_{\mathbb{Q}(\sqrt{-d})}[2^\infty]\oplus 2\operatorname{Cl}_{\mathbb{Q}(\sqrt{-d_0d})}[2^\infty]$. Together with this decomposition, the group-valued conjecture predicts that $4\operatorname{Cl}_{K(d)}[2^\infty]$ is distributed as the direct sum of two independent Cohen–Lenstra $2$-groups, giving a corrected Cohen–Lenstra–Martinet distribution for the biquadratic family. Unconditionally, the $8$-rank of $\operatorname{Cl}_{K(d)}$ has limiting distribution given by the convolution of two copies of the Cohen–Lenstra–Gerth distribution. The proof combines Smith's box method with quantitative truncated Gaussian-binomial moment inversion for diagonally coupled, fixed-width bordered Rédei matrices.