The stochastic theta method is proposed to approximate invariant measures of stochastic differential equations (SDEs), both of whose drift and diffusion coefficients may grow super-linearly. For the numerical solution generated by the stochastic theta method, we show the existence and uniqueness of the numerical invariant measure first. Then, we prove that the numerical invariant measure is convergent to the exact invariant measure of the underlying SDE. We also provide some numerical simulations to illustrate our theoretical results. This work could be regarded as an extension of the results in [Y. Jiang et al, Numer. Algorithms 83(4)(2020), pp. 1531-1553] to the case of super-linearly growing diffusion coefficient. As the backward Euler-Maruyama (EM) method is a special case of the stochastic theta method, the results derived in this work could also be regarded as a generalization of the results for the backward EM method in [W. Liu et al. Appl. Numer. Math. 184(2023), pp. 137-150] to the stochastic theta method.

In this paper, we consider the classical overdetermined capillary problem: \begin{equation*} \begin{cases} \mathrm{div} \left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) - bu =0 &~~\mbox{in}~~ Ω, \partial_ν u=κ&~~\mbox{on}~~\partialΩ, u=c &~~\mbox{on}~~\partialΩ, \end{cases} \end{equation*} where $b$, $c$ and $κ$ are positive constants, and $Ω\subset \mathbb{R}^2$. When $Ω$ is an infinite strip, i.e., a domain bounded by two parallel straight lines, there exists a unique one-dimensional solution (called the trivial solution) to this problem. By means of a bifurcation argument, we establish the existence of a critical period $T_*$ at which a branch of non-trivial solutions bifurcates from the trivial one. These solutions are genuinely two-dimensional and are defined in unbounded periodic domains $Ω$ that are diffeomorphic to an infinite strip, yet whose boundaries are no longer straight lines. This result offers a significant physical interpretation in the context of capillary phenomena.

We establish $L^p$ bounds for operators associated with the quasi-regular representation of the free group on its Gromov boundary. The $p=2$ case recovers Haagerup's inequality, yielding a new interpolation-theoretic proof of the the Rapid Decay property for the free group.

We study the homogenization of the compressible isentropic Navier-Stokes equations in periodically perforated domains where the size of the obstacles is of the same order as the distance between neighboring obstacles. Using the two-scale convergence method, which can be characterized via the unfolding operator, we derive the corresponding macroscopic model determined by Darcy's law. In particular, the macroscopic density satisfies the porous medium equation. The main challenge lies in identifying the pressure term in the limit. We overcome this by establishing the strong two-scale convergence of the densities, which is achieved by controlling the oscillation defect measure of the unfolded densities. A crucial contribution of our work is the development of a methodological framework applicable to more complex compressible fluid models. Furthermore, regarding conservative forces, we extend existing results from the literature to adiabatic constants $γ> \frac95$.

This paper studies a class of distributed optimization problems in networked nonlinear systems (NNSs) subject to input delays and consensus constraints. It introduces input-delay tolerant semiglobal convergence (IDTSC), meaning that for any prescribed compact initial set there exists an admissible delay bound under which the optimal solution is computed within consensus constraints and all node states converge to the solution. Building on a hierarchical design and input-to-state stability analysis, a new semiglobal input-delay tolerant (SIDT) algorithm is developed that practically achieves IDTSC for distributed optimization under the coupling between input delays and nonlinear dynamics. Further, by relaxing strict convexity requirements through the Polyak-Łojasiewicz condition, the SIDT algorithm broadens its applicability to nonconvex optimization. Finally, numerical experiments corroborate the theory on NNSs with input delays.

The questions of Grünbaum and Loewner dated back to the 1960s is completely settled. Our proof rely on spherical harmonics and Morse theory.

Let $n_k$ denote the least integer $n>2k$ such that $(n-k)(n-k+1)\cdots(n-1)$ is not divisible by any prime in the interval $(k,2k)$. Confirming a conjecture of Erdős, we prove that, for all sufficiently large $k$, $$ n_k > e^{\frac{\log^2 k}{20 \log \log k}}. $$

Recent research on Doeblin coefficients has shed light on their usefulness as a multi-way generalization of the Dobrushin contraction coefficient for TV distance, in a separate vein from their classic role in the theory of Markov chain ergodicity. However, strong conditions, such as being bounded away from 0, are typically necessary for Doeblin coefficients to establish the existence of information contraction. Building on recently formulated concepts of nonlinear information contraction, we aim to propose a finer-grained Doeblin-based characterization of multi-way contraction behavior which yields non-vacuous contraction guarantees even for channels whose Doeblin coefficient is 0. To this end, we introduce the notion of a Doeblin curve -- a nonlinear function which quantifies the contraction behavior of a Markov kernel on collections of input distributions at specific levels of divergence and power. Through the course of our analysis, we develop a new variational characterization of Doeblin coefficients, present several properties of Doeblin curves, define several versions of power-constrained Doeblin curves, and derive upper and lower bounds using our aforementioned variational characterization. We then utilize these results in diverse areas, including generalization bounds for noisy iterative optimization, error bounds for reliable computation with noisy circuits, and differential privacy guarantees for online iterative algorithms. In particular, we extend results in these areas to broader domains or group settings, leveraging Doeblin curves to reveal finer-grained contraction phenomena than Doeblin coefficients.

We define a mass invariant adapted to the spectral scalar curvature for asymptotically hyperbolic 3-manifolds with toroidal infinity and show its positivity under a lower bound on the spectral scalar curvature. In addition, we show a rigidity theorem and some band width estimates under similar assumptions.

This paper investigates the one-dimensional Fourier phase retrieval problem for indicator functions of finite unions of intervals. Specifically, we study the recovery of a set $Ω= \bigcup_{j=1}^m I_j \subset\mathbb{R}$ from the magnitude of its Fourier transform $|\widehat{\mathbf{1}_Ω}|$, where each $I_j \subset \mathbb{R}$ is a bounded interval. For $m\le 2$, we prove that $Ω$ is uniquely determined by $ |\widehat{\mathbf{1}_Ω}|$ up to the natural ambiguities of translation and reflection, and we further establish a stability result for this reconstruction. In contrast, for $m\ge 3$, uniqueness fails in general. More precisely, for every $m\ge 3$, we explicitly construct functions $f_m,g_m\in\mathcal{I}_m$ such that $|\widehat{f_m}|=|\widehat{g_m}|,$ while $f_m$ cannot be obtained from $g_m$ by any translation or reflection, where $\mathcal{I}_m$ denotes the class of indicator functions of unions of exactly $m$ intervals. Furthermore, building on the theory of the turnpike problem, in which a finite integer set is uniquely determined by its multiset of pairwise differences under a collision-free condition, we establish an analogous result for finite subsets of $\mathbb{R}$. This, in turn, yields a sufficient condition for recovering indicator functions of finite unions of intervals. These results provide a complete characterization of the Fourier phase retrieval problem for indicator functions of finite unions of intervals and offer new insights into Fourier phase retrieval for indicator functions of more general domains in higher dimensions.

Keevash and Michaeli recently proved that, under the robustness assumption that \(G\) is an \(s\)-connector (i.e. \(\overline G\) is \(K_{s,s}\)-free), \(G\) has essentially the same multicolour Ramsey matching properties as complete graphs, with an additive error \(O(qs)\), where \(q\) is the number of colours. They asked whether the dependence on \(q\) can be removed. We answer this question in a stronger exact form. For \({\bf t}=(t_1,\ldots,t_q)\in\mathbb N_+^q\), let \(R_s({\bf t})\) be the smallest integer \(N\) such that every \(N\)-vertex \(s\)-connector \(G\) satisfies \( G\to (t_1K_2,\ldots,t_qK_2). \) We determine the exact value \[ R_s({\bf t})=\sum_{j\in[q]}(t_j-1)+ \max\left\{2s,\ s+\max_{j\in[q]}t_j\right\}. \] While Keevash and Michaeli's proof uses a compression algorithm based on the Gallai--Edmonds decomposition to reduce the colouring to a structured form, our proof is a direct minimal-counterexample argument together with a new counting method for monochromatic matchings which can be applied to \(s\)-connectors.

We prove several results about Stiefel-Whitney Classes (SWCs) $w_k(π)$ of representations $π$ of $S_n$. First, each SWC is polynomial in the character values of $π$ at involutions. Next, for a fixed $k$, the proportion of irreducible $π$ for which $w_k(π)=0$ approaches $100\%$ as $n \to \infty$. A similar result holds for the top SWCs. We also provide a simple criterion which determines the first nonvanishing SWC for a representation. The first four SWCs are computed explicitly. Finally, we give analogues for alternating groups.

In this paper, we are concerned with the optimal dimension-dependent $\ell^p$ norm of the discrete Riesz Transforms $R_{\text{dis}}^{(k)}$ on $\mathbb{Z}^d$ given by the singular convolution kernel $K_k(m)=c_d m_k/|m|^{d+1}$, where $c_d=Γ(\frac{d+1}{2})/π^{(d+1)/2}$ . We show that for fixed $1<p<\infty$, when $d\to \infty$ $$\|R_{dis}^{\left( k \right)}\|_{\ell ^p\left( \mathbb{Z}^d \right) \rightarrow \ell ^p\left( \mathbb{Z}^d \right)}=2c_d\left( 1+\frac{\left( \sqrt{2}+o\left( 1 \right) \right) d}{2^{\frac{d}{2}}} \right) .$$ The operator norm of $R_{\text{dis}}^{(k)}$ grows super-exponentially as $d\to\infty$ since $c_d\sim(\frac{d-1}{2eπ})^{\frac{d-1}{2}}\sqrt{\frac{d-1}π}$ by Stirling's formula, which gives a negative answer to the conjecture proposed by Bañuelos, Kim and Kwaśnicki in \cite{BKK}. The optimal dimension-dependent $\ell^{1,\infty}$ estimate of $R_{\text{dis}}^{(k)}$ is also established.

The clique adjacency polynomial (CAP), introduced by Soicher (2015), provides a powerful method for bounding the clique numbers of edge-regular graphs. In this paper, we extend the CAP framework to arbitrary finite graphs by expressing the relevant parameters in terms of average vertex degree and average edge-degree over potential cliques. This leads to a generalised CAP bound and an associated clique existence polynomial (CEP), which removes the dependence on an auxiliary integer variable and facilitates computation. We compare the resulting bounds with classical spectral and linear programming bounds, including those of Delsarte, Hoffman, and Haemers. We show that the generalised CAP improves upon these bounds for several families of graphs. In particular, we identify infinite families of edge-regular graphs arising from projective geometry for which the CAP outperforms the Delsarte bound, as well as families of regular and non-regular graphs where the generalised CAP improves upon the Hoffman and Haemers bounds. We also develop techniques for bounding feasible parameter regions, enabling practical application of the method to both structured and unstructured graphs.

We provide a partial answer to Problems 1 and 2 raised in the recent article by Blaschke et al., concerning the analytic continuation of weighted $H$-harmonic Bergman spaces. These are spaces of functions annihilated by the Möbius-invariant Laplacian on the unit ball. More precisely, we identify some of the discrete Wallach sets and show, among others, that structure depends on the parity of the dimension.

We prove that the top wedge power of the Ricci form of a complete non-compact Kähler manifold with positive sectional curvature has finite integral. Using a result of Chen-Zhu, an immediate consequence is the quasiprojectivity of such manifolds under the assumption of bounded sectional curvature. A key new idea to prove Bézout estimates along with a Lipschitz weight with finite Monge-Ampère mass is used in the proof of the main result.

Our goal is to survey the development of positive mass theorem and the Yamabe equation on CR manifolds in recent years. We introduce the notion of the mass in several complex variables or CR geometry. We then consider the Yamabe problem on CR manifolds to find a minimizer for the CR-Sobolev quotient. The positive mass theorem plays a key role in finding a solution to the Yamabe equation with minimum energy for the positive curvature case. We mainly focus on the team works in the following three papers [CMY17], [CMY23] and [CC22], on a positive mass theorem in 3-dimensional CR geometry, the CR-Sobolev quotient of Rossi spheres, and the 5-dimensional situation, respectively.

For a broad class of Delone sets in $\mathbb{R}^n$ that are of significance in both mathematics and physics, we prove a non-uniform Balian-Low type theorem and settle the converse problem on the existence of Gabor frames, for arbitrary dimension $n$. To this end, we introduce a class of Gabor frames, termed full Gabor frames, and prove that the existence of such a frame on the Delone set with Schwartz window functions is equivalent to the condition that the lower Beurling density be strictly greater than one. In fact, the usual Balian-Low direction using window functions from the Feichtinger's algebra can be proven for arbitrary point sets, thereby improving an earlier density theorem by Christensen, Deng, and Heil. The corresponding dual result for Riesz sequences is also obtained. The main technical tools employed in this paper are tiling groupoid constructions and $C^*$-algebraic methods. As a byproduct, we resolve an open question from Ito's thesis concerning the bounded dynamical asymptotic dimension of tiling groupoids. Furthermore, this result allows us to extend the classification theorem of Ito, Whittaker, and Zacharias to the twisted case.

Introduced by Ballantine, Beck, and Merca, the elementary symmetric partition function $\mathrm{pre}_k$, defined on the set of partitions with at least $k$ parts, has been a topic of recent interest. We prove that $\mathrm{pre}_k$ is injective on the set of $m$-ary partitions for positive integers $m \ge k$, generalizing the binary $k = 2$ result of Ballantine, Beck, and Merca, and complementing a result of Hadelyn, Niergarth, Li and Li showing that, for each $k \ge 3$, $\mathrm{pre}_k$ is not injective on partitions of $n$ with length $2k$ for infinitely many $n$. We introduce the skew Schur partition function $\mathrm{prs}_{λ'/μ'}$, prove injectivity results for particular choices of $λ',μ'$, and describe an application to representation theory.

We study dynamic resource allocation in a multicore computing system with a fixed number of processing cores and a stream of {\it malleable} jobs. Each job may adjust its level of parallelism during execution, allowing adaptive redistribution of resources across concurrently active jobs. Jobs belong to one of two observable classes, each characterized by a distinct speed-up function with unknown parameters. The objective is to learn a core-allocation policy that minimizes the long-run mean number of jobs in the system, equivalently the mean response time in steady state. \noindent To address this uncertainty, we develop an iterative learning-and-control framework. The system alternates between estimating the unknown speed-up parameters from observed job completions and solving the associated Markov decision process (MDP) to update the allocation policy. Within each job class, cores are shared equally among active jobs; the fraction of capacity assigned to each class is obtained from the MDP formulation of \cite{berg2017}, evaluated at the current parameter estimates. We construct a maximum likelihood estimator based on state-dependent inter-departure times and prove its strong consistency under a fixed allocation policy. We further propose two learning algorithms that combine this estimation step with dynamic programming-based policy updates, and illustrate their through numerical experiments.

Given a complex reductive group $G$, a representation $V$ of $G$ and a Borel-stable subspace $M \subset V$, we consider the associated Steinberg-type variety $Z$. We prove that, under a certain condition on $(V,M)$, called gluability, the equivariant Borel-Moore homology or $K$-theory of $Z$, equipped with the convolution product, is obtained as the intersection of two copies of the nil-Hecke algebra inside its localization. We also provide a description of these new algebras in terms of poles and residues. Similar results are obtained when $G$ is replaced by its loop group. This generalizes results of Ginzburg, Kapranov and Vasserot describing the affine Hecke algebra and DAHA, as well as a result of Teleman and Gannon--Webster that realizes certain Coulomb branches by gluing two copies of the universal centralizer.

Patterson-Sullivan measures encode the distribution of orbits of discrete group actions near the boundary. In this paper, we prove a global shadow lemma for Patterson-Sullivan measures associated to relatively Morse subgroups of higher-rank semisimple Lie groups. The estimate is uniform for shadows centered at arbitrary points in a Gromov model, including points deep in the cuspidal part. This extends the global shadow lemma of Stratmann-Velani for geometrically finite real hyperbolic groups. As applications, we obtain uniform local estimates for Patterson-Sullivan measures, and we give sufficient conditions under which these measures agree, up to scale, with the Hausdorff measure defined by the associated visual quasi-metric.

We propose a signature-based method to solve the optimal market-making problem under a mean-variance criterion. By exploiting signature linearization techniques, we reduce the market-making problem to a pseudo-linear optimization over the expected signature of an augmented market path, and we develop a signature algorithm named Sig-REINFORCE to learn the optimal bid and ask quotes. We test our method in two scenarios, in which market-order arrivals follow either a Poisson or a self-exciting Hawkes process, and we benchmark it against a Proximal Policy Optimization (PPO) baseline.

We introduce a second-order approximation to the compressible atmospheric Euler equations with gravity that is invariant domain preserving and well-balanced with respect to rest states. The approximation is built upon discrete auxiliary states derived from a hydrostatic reconstruction of the density. These auxiliary states, together with an affine shift of the numerical state, provide local bounds needed for maintaining well-balancing and invariant domain preserving properties of the method. The numerical method is then verified and validated with analytic solutions, well-balancing tests, and typical benchmark problems for atmospheric flows.

We prove an optimal dimension-free sparsification theorem for suprema of centered Gaussian processes. Given a bounded set $T\subseteq\mathbb{R}^n$, we show that the supremum of the canonical Gaussian process on $T$ can be $L^2$-approximated by the supremum of a shifted subprocess indexed by only $\exp(O(1/\varepsilon^2))$ points, with error at most $\varepsilon$ times the Gaussian width of $T$. In particular, the size of the approximating process is independent of both the ambient dimension and the cardinality of the original index set. This improves a recent sparsification theorem of De, Nadimpalli, O'Donnell, and Servedio (2026) by an exponential factor, and we show that the dependence on $\varepsilon$ is tight up to constants in the exponent. As consequences, we obtain an exponentially improved junta theorem for norms over Gaussian space and sharpen results on learning, property testing, and polyhedral approximation of convex sets under the Gaussian measure. The proof is based on an interpolation argument that combines Sudakov's minoration with the Brascamp--Lieb inequality.

The Helmholtzian of a graph $G$ is the Hodge $1$-Laplacian $L_1=L_1^{\mathrm{up}}+L_1^{\mathrm{down}}$ of its clique complex, built from the triangle--edge and edge--vertex boundary operators $\partial_2$ and $\partial_1$. Problem~5.5 of Lu, Shi, Stanić, Wang and Wang asks whether $λ_{\max}(L_1)=μ_1(G)$ for every graph $G$, where $μ_1(G)$ is the largest Laplacian eigenvalue; by the Hodge decomposition this is equivalent to $λ_{\max}(L_1^{\mathrm{up}})\leμ_1(G)$. We recast it as a question about the complement of $G$: localizing $L_1^{\mathrm{up}}$ on the cycle space of $K_n$ turns it into the inequality $λ_{\min}(\bar L|_{Z_1})\ge a(\overline{G})$, where $\bar L$ is the up Laplacian of the missing triangles of $G$ and $a(\overline{G})=n-μ_1(G)$ is the algebraic connectivity of the complement. From this viewpoint, we prove the unconditional bound \[ λ_{\max}\!\big(L_1^{\mathrm{up}}(G)\big)\ \le\ μ_1(G)+\frac13\big(n-μ_1(G)\big), \] which refines the integrality ceiling $λ_{\max}(L_1^{\mathrm{up}})\le n$ of Duval and Reiner and is sharp exactly when that ceiling is attained. We then isolate the single sharp inequality, on the dense part of $\overline{G}$, that stops the method short of Problem~5.5, and we show that the localization, the bound, and this obstruction all persist for the up Laplacian of an arbitrary finite simplicial complex, in every dimension.

The paper extends previous results on the nth root problem to a large class of Hilbert-space operators, namely, the class of all normaloid operators with normaloid parts, which includes the paranormal operators, and also the $k$-paranormal operators. It is shown that if a normaloid operator with normaloid parts has a normal nth power, then it is normal.

We develop a partial differential equation formulation of the Behrens-Fisher problem for two independent normal samples with unknown and unequal variances. An orthogonal decomposition separates mean and residual components (corresponding to the centered within-sample variation left after removal of the mean directions) and recasts the studentized difference of sample means as a scale-invariant geometric constraint. This reduction transforms the distributional problem into the evaluation of spherical wedge probabilities, which are identified with harmonic measure and with the value at the origin of a Laplace-Dirichlet boundary value problem. From this framework, we derive exact finite-sample representations for the cumulative distribution function and the probability density function in terms of beta functions, with dependence only on the sample sizes and the variance ratio. These representations place the Behrens-Fisher law in a standard special-function form that is directly accessible in widely available commercial software -- including Microsoft Excel -- thereby facilitating distributional evaluation and quantile computation. We also obtain a Gegenbauer separation-of-variables expansion for the associated harmonic extension and its threshold derivative, with coefficients in closed Beta-Gamma form, and derive sharp tail expansions with explicit leading constants and higher-order corrections.

For a family of graphs $\mathcal F$, a graph $G$ is said to be $\mathcal F$-free if it contains no member of $\mathcal F$ as a subgraph. A wheel graph $W_k$ is a graph on $k+1$ vertices formed by joining a new vertex to all vertices of a $k$-cycle. Given an integer $k\ge 3$, we consider the problem of determining the maximum number of triangles in a $W_{\geq k}$-free graph, where $W_{\geq k}=\{W_\ell: \ell \geq k\}$. The case $k=3$ was raised by Gallai, who proposed a conjecture for this case (see Erdős [5]. Gallai's conjecture was disproved by Zhou [17] and independently by Füredi, Goemans, and Kleitman [9]. In this paper, we study the case $k=4$. Namely, for every integer $n\ge 3$, we determine the maximum number of triangles in an $n$-vertex $W_{\geq 4}$-free graph and characterize all extremal graphs.

We propose and analyze a structure-preserving numerical method for the $2\tfrac{1}{2}$-dimensional (2.5D) electron magnetohydrodynamics system with fractional dissipation on the periodic torus. The method works directly with the magnetic field components and combines this component formulation with the gradient recovery operator of [T. Chu, H. Guo, and Z. Zhang, SIAM J. Numer. Anal., 63 (2025), pp. 23--53]. We establish discrete energy stability for a semi-implicit structure-preserving formulation and use an explicit-Hall integrating-factor implementation for efficient computation on periodic grids. The fractional dissipation is treated exactly in Fourier space, and the in-plane divergence constraint is enforced by a spectral Hodge projection. Numerical experiments demonstrate second-order spatial convergence and stable Hall-driven dynamics across several benchmark tests.