We introduce Ergodic Deviation-Robust Equilibrium (EDRE), a dynamics-relative equilibrium concept for repeated finite games in which agents learn via entropic mirror descent (EMD). EDRE requires three properties to hold simultaneously for the same profile and learning run: (E1) the limit profile is an $\varepsilon$-Nash equilibrium at a product distribution; (E2) along the entire learning trajectory, every fixed coalition's cumulative aggregate (summed-unilateral) deviation gain is $\tilde{\mathcal{O}}(\sqrt{T})$ with high probability; and (E3) the limit profile is a fixed point of the EMD map, so that it is selected by the dynamics rather than merely certified as an equilibrium. We prove that the $\sqrt{T}$ deviation-regret rate is order-tight, establish existence in exact-potential games (via Nash's theorem, with a constructive proximal route under concavity) together with Lyapunov monotonicity of EMD (and pointwise convergence when the fixed-point set is a singleton), and extend the selection property to monotone polymatrix games through variational inequalities. Although a static EDRE coincides with an $\varepsilon$-Nash equilibrium, its content is dynamic: robust (positive-measure) selection under EMD excludes linearly unstable equilibria, so EDRE acts as a Nash equilibrium equipped with a dynamic certificate rather than a static refinement. On the complexity side, we show that computing EDRE is PPAD-hard in general polymatrix games and belongs to promise-PPAD for potential games. A worked $2\times 2$ coordination-game example illustrates all components of the framework. Additional results, including a bandit-feedback extension, a period-doubling route to Li-Yorke chaos for the two-strategy EMD map at large step size, a linear-program formulation for minimum-cost steering, and supporting simulations, appear in the appendices.

Temporal difference (TD) learning with linear function approximation is a core method for policy evaluation. Its classical continuous-time description is an ordinary differential equation (ODE), which captures the asymptotic mean dynamics but neglects stochastic fluctuations determining the error floor. We introduce a stochastic differential equation (SDE) approximation for linear TD(0) under Markovian noise. The resulting model distinguishes the contraction dynamics governed by the projected Bellman operator from the influence of Markovian sampling. As a consequence, the model explains the constant-stepsize error floor through the interaction between Markovian long-run covariance and the contraction geometry of the projected Bellman operator.

In 1965, Wegner conjectured that every finite family \(\mathcal R\) of axis-parallel rectangles in the plane satisfies \(\tau(\mathcal R) \le 2\nu(\mathcal R)-1\), where \(\tau(\mathcal R)\) denotes the minimum number of points needed to pierce all rectangles in \(\mathcal R\), and \(\nu(\mathcal R)\) denotes the maximum size of a pairwise disjoint subfamily. Over the last six decades, the conjecture has motivated a long line of work: it has been verified for several special classes of rectangle families, and the best known general upper bounds have been progressively improved, but the conjecture itself had remained open. We give an explicit counterexample. More precisely, we construct a triangle-free rectangle-intersection graph on \(n\) vertices whose independence number is at most \(n/4\). Since the graph is triangle-free, no point of the plane can lie in three rectangles; hence every piercing point hits at most two rectangles. Consequently, \(\tau(\mathcal R) \ge n/2 \ge 2\nu(\mathcal R)\), contradicting Wegner's conjectured bound. We also give a slightly more general construction for which \(\tau(\mathcal R) \ge 2.21\nu(\mathcal R)\). This shows that the standard point relaxation, equivalently the clique relaxation, for the Maximum Independent Set of Rectangles problem has integrality gap at least \(2.21\).

Response-adaptive randomization (RAR) in clinical trials aims to improve ethical and statistical efficiency by dynamically allocating patients to treatments based on observed outcomes. While RAR based on a target optimal allocation have been extensively studied for two-arms settings, their extension to multi-treatment experiments ($K \geq 2$) remains theoretically fragmented, with most existing methods focusing on specific algorithms or restricted target allocations. In this paper, we introduce a unified framework for response-adaptive targeting, the $\alpha$-Rebalancing Targeting Strategies ($\alpha$RTS), which generalize the ERADE two-armed strategy of Hu et al. [2009]. We prove that all designs in this family share fundamental asymptotic properties: strong consistency, asymptotic normality of allocation proportions and treatment effect estimators, and asymptotic efficiency. To address sparse target regimes (where some treatments are asymptotically eliminated), we further propose $\alpha$RTS with Forced Exploration, a variant that guarantees infinite sampling for all treatments while preserving the asymptotic guarantees. Extensive simulations illustrate the finite-sample behavior of $\alpha$RTS variants in a 3-armed context, highlighting in particular the critical role of forced exploration in sparse settings.

Given a simple graph on $n$ vertices and a parameter $k$, the triangle-densest-$k$-subgraph problem is known to be computationally hard in the worst case. To circumvent the computational hardness, we study an average-case model where a triangle-dense subgraph on $k$ vertices is planted in an Erdős-Rényi random graph on $n$ vertices. For the recovery of the planted subgraph, we propose a simple spectral algorithm and a semidefinite program, both of which use a graph matrix whose entries are local signed triangle counts. Theoretical guarantees for these algorithms are established through spectral analysis of the graph matrix. Finally, we provide evidence showing a statistical-to-computational gap analogous to that for the planted clique problem. The computational threshold in terms of the subgraph size $k$ is at least $\sqrt{n}$ in the framework of low-degree polynomial algorithms, while the information-theoretic threshold is at most logarithmic in $n$.

Mediation analysis is essential for decomposing the causal effect of a treatment into direct and indirect pathways. However, many practical settings rely on the stringent assumption that recanting witnesses, defined as treatment-induced mediator-outcome confounders, are either absent or fully known a priori. Such a requirement is often untenable, especially when these variables remain unobservable due to measurement difficulties or privacy constraints. In this paper, we leverage proximal causal inference to develop three novel identification strategies to address the challenge of identifying path-specific effects in the presence of unknown recanting witnesses. Building on this, we develop a semiparametric inference framework that derives the efficient influence function and proposes a proximal multiply robust estimator, which remains consistent if at least one set of nuisance models is correctly specified. When all nuisance models are correctly specified and converge at appropriate rates, the estimator is asymptotically normal and achieves the semiparametric efficiency bound. We provide a minimax optimization-based debiased machine learning procedure for point estimation and constructing valid confidence intervals. The performance of the proposed methods is demonstrated by simulation studies and a real data application.

This paper considers the problem of a low-thrust spacecraft pursuit-evasion differential game with an arbitrary terminal relative speed constraint. It addresses the terminal phase of the engagement for two relatively close spacecraft near a circular orbit. The problem is formulated as a linear-quadratic zero-sum differential game, with soft constraints on the terminal relative position and velocity, and running costs on the players' control efforts. An analytical, closed-loop, minimum-fuel-consumption optimal guidance law is derived for each player, forming a saddle-point solution. It is proven that any terminal speed can be achieved by properly choosing the weighting parameters of the cost function. To verify the optimality of the solution, a conjugate point analysis is performed when the cost function velocity weighting matrix is either positive or negative definite. The negative-definite case arises at high terminal speeds and is seldom seen in the literature. The performance of the derived guidance law is evaluated in simulations for different target maneuvers and compared to a state-of-the-art optimal-control-based guidance law. The simulations show that the derived guidance law satisfies the constraints and offers a substantial advantage over the optimal-control-based guidance law when the target is optimally evading.

Prey aggregation is widely regarded as a defense against predation, yet we show that in disease-structured populations subject to predator-induced fear and demographic Allee thresholds, aggregation can paradoxically accelerate ecosystem collapse. We develop and analyze a susceptible-infectious-predator model incorporating dual fear responses -- together with a sublinear aggregation-based predation term and an Allee effect. Critically, we derive an explicit upper bound on the extinction time that decreases as predator pressure increases or aggregation strengthens, quantifying for the first time how behavioral and demographic parameters jointly determine the speed of ecological collapse. This finite-time extinction subsequently triggers a cascade collapse of the infected prey and predator populations, driving the entire ecological community to extinction. Bifurcation analysis reveals transcritical, saddle-node, and Hopf bifurcations as fear intensity, aggregation strength, and Allee threshold vary. Two-parameter continuation further identifies the precise regions of the fear--Allee parameter plane in which stable coexistence, oscillatory coexistence, predator exclusion, and finite-time extinction occur, demonstrating that stronger aggregation monotonically enlarges the finite-time extinction region while weaker aggregation supports a richer landscape of coexistence dynamics. These results demonstrate that behavioral defenses operating at the population level can generate abrupt ecological tipping points when they interact with disease dynamics and demographic vulnerability.

We investigate the learning of interpretable bases in non-negative matrix factorisation (NMF) by regularising the topology of the learned basis functions. Our approach is motivated by the observation that many data modalities can be viewed as non-negative functions on a structured domain, where the quality of a basis is intrinsically linked to its topology. However, naive methods for incorporating the topology of the support are often hindered by discreteness and threshold dependence, rendering them unsuitable for continuous optimisation. We address these challenges by employing persistent homology as a stable, threshold-free topological quantifier and by designing topological scores that integrate into the NMF objective as regularisers. The resulting framework encompasses spatially coherent image components, periodic time-series structures, and clique-like graph signals within a unified modelling language.

We consider the concentration properties of functions of infinitely exchangeable random variables. By conditioning on the de Finetti directing measure, we show that the deviation of any function with bounded-difference constants $c_1, \dots, c_n$ decomposes into a conditional sampling fluctuation and a latent mixture fluctuation. When this latent mixture is $\sigma_{\mathrm{mix}}^2$-subgaussian, we establish a concentration inequality with an effective variance proxy of $\frac{1}{4}\sum_i c_i^2 + \sigma_{\mathrm{mix}}^2$. Crucially, we demonstrate that for zero-sum linear contrasts, such as the difference between a subsample mean and a full population mean, the latent mixture term cancels exactly. This cancellation yields a tight, mixture-free Hoeffding-type bound that provides a direct de Finetti mechanism for the infinite-extendibility limit of recent finite-exchangeable concentration results. We apply this framework to quantify uncertainty in composite AI benchmarks, such as MMLU, where question items naturally exhibit exchangeable dependence across domains. Our results provide both a domain-stratified hierarchical model for bounding the uncertainty of accuracy scores, and a distribution-free, cost-saving statistical guarantee for accurately estimating full benchmark scores from random subsets.

We develop approximation and generalization error estimates for multi-input neural operators, with the output error measured in Sobolev norms. In contrast to standard operator-learning settings with a single input function, our framework allows multiple input functions defined on possibly different domains, with different dimensions and Sobolev regularities. The derived rates explicitly quantify the contribution of each input space to the final error bound. In particular, in the balanced regime, the approximation and generalization rates are governed by the interaction between the input dimensions, regularities, and Sobolev orders, while the dependence on the model complexity retains a \(\log\log/\log\)-type structure. Our analysis provides a general theoretical framework for multi-input operator learning, including Sobolev training, and is applicable to operator learning problems arising from partial differential equations and scientific computing.

Autonomous spacecraft rendezvous and proximity operations (RPO) require controllers that guarantee safety under thrust constraints while minimizing fuel expenditure. Input-constrained control barrier functions (ICCBFs) provide a control method for nonlinear systems with actuation constraints that construct a forward-invariant safe set. Previous work has shown that learning class-$\mathcal{K}$ functions defining the ICCBF recursion via meta reinforcement learning (meta-RL) yields a robust, non-greedy approach to safety-critical control in RPO. This paper extends that framework further by investigating the performance of three recurrent network architectures (Long Short Term Memory (LSTM), Gated Recurrent Unit (GRU), Selective State Space Model (Mamba)) and two training algorithms (Proximal Policy Optimization (PPO) and Soft Actor Critic (SAC)) to identify the best setup for tuning ICCBF class-K functions via meta-RL. In addition to cooperative test cases, performance is evaluated in the presence of adversarial behavior where the target spacecraft behaves in a way that worsens the safety of the chaser spacecraft. Results indicate that state space models such as Mamba when used with PPO achieve superior task completion, safety, and fuel-savings compared to other architectures, across all cooperative and uncooperative scenarios tested.

Motivated by the optimization of bounded binary black-box functions, we study the problem of learning polynomial surrogates over the Boolean hypercube. To ensure that optimizing the surrogate yields good solutions for the underlying objective, we require uniform $L_\infty$-error guarantees rather than the usual $L_2$-type guarantees. We characterize the minimax sample complexity of uniform estimation under subgaussian noise for two classes of bounded polynomials. First, for polynomials of degree at most $d$ on $n$ variables, the sample complexity scales as $n^{d+1}$. Second, for $s$-sparse Fourier-Walsh polynomials with $s \leq n$, it scales as $ns^2$. These rates differ structurally from the noiseless setting, where uniform exact recovery scales as $n^d$ and $ns$, respectively. Our lower bounds hold even for arbitrary adaptive learners, showing that the additional factors are intrinsic to the noisy cases. Standard Fourier-analysis tools for the $L_2$-norm do not naturally extend to the $L_\infty$-setting in a way that yields uniform guarantees. Our proofs overcome this difficulty by relying on suitably chosen auxiliary norms that serve as proxies for controlling the $L_\infty$-error. Together, our results provide a tight characterization of the sample complexity of learning optimization-safe polynomial surrogates.

Real-time trajectory generation for on-orbit robotic servicing is challenging due to the nonlinear coupling between spacecraft bus motion, manipulator dynamics, visibility cone, and trajectory-level safety constraints. This paper studies learning-based warm-starting for sequential convex programming (SCP) in the terminal approach of a space manipulator toward a tumbling target. The proposed framework decomposes the problem into a system center-of-mass translational planning stage and a coupled attitude--manipulator torque-allocation stage, and applies a causal transformer warm-start to the latter, which constitutes the dominant computational bottleneck. Linear and flow matching action decoders are compared under different action-chunking and training dataset sizes, and the resulting warm-starts are evaluated under both cost-optimal and feasibility projection using SCP. Across 300 held-out scenarios, the learned warm-start reduces the second-stage SCP iteration count by up to 28% and the runtime by 23% while preserving the final control-cost distribution. When the learned warm-starts are used for nonconvex feasibility projection, they nearly halve the runtime relative to cost-optimal SCP, while avoiding the catastrophic high-cost tail behavior observed when initialized heuristically. These results indicate that sequence-model warm-starts can improve both the computational efficiency and trajectory robustness of optimization-based terminal guidance for space manipulation.

In this paper we study the class of graphs without cycles of size 4 and independent sets of size 4 as induced subgraphs: $\mathop{Free}\{C_4, 4K_1\}$. This is one of the three minimal minimal open cases for the complexity of the colouring problem when restricted to classes defined by excluding induced subgraphs of order 4. We investigate the clique width of some subclasses of $\mathop{Free}\{C_4, 4K_1\}$. We introduce a new framework: the $(k,l,m)$-decomposition and prove that if all the graphs of a class $\cal G$ are $(k,l,m)$-decomposable, then graphs in $\cal G$ have bounded clique width. We give a few examples of such class, found with the help of a program we designed. We also show, for any graph $G \in \mathop{Free}\{C_4, 4K_1\}$ that is 3 cliques coverable, an infinite family in $\mathop{Free}\{C_4, 4K_1\}$ of supergraphs of $G$ which have unbounded clique width.

In survival analysis, dependent censoring poses significant challenges in accurately estimating model parameters and survival functions. This study introduces a novel framework leveraging Extended Generalized Marshall-Olkin (EGMO) models to address dependent censoring mechanisms. Geometric optimization techniques are employed to develop efficient estimation procedures that capture dependencies between failure and censoring times. We establish their asymptotic properties. Simulation studies and real data applications illustrate the method's robustness and effectiveness.

In this paper, a novel optimal soft-landing guidance law with inequality approach-angle path constraints is analytically derived. The proposed guidance law prevents ground collision and enables approach-angle control by constraining the optimal trajectory to remain within a convex inverted pyramid originating at the landing point. A 3D point-mass linear kinematic model in a constant gravitational field is employed, together with a quadratic control-effort cost and terminal constraints on position and velocity. Analytical open-loop and closed-loop solutions, together with the optimal final time, are derived using Pontryagin's Minimum Principle and the optimality conditions at the transitions between unconstrained and constrained arcs. It is additionally shown that the optimal final time decreases when the path constraints become active. The resulting guidance law is continuous, piecewise linear in time, and nonlinear in the states in closed-loop. When a constraint becomes active, the controller cancels the gravitational component normal to the constraint, causing the trajectory to evolve along the constraint surface. The proposed guidance law is evaluated in simulations under various initial conditions, demonstrating accurate landing performance and consistent satisfaction of the path constraints.

We study sparse semi-continuous tensor data with excess zeros, heavy right tails, and slice-specific dispersion. Such features arise naturally in monetary-valued multi-way data, such as international trade, where most exporter--importer--product--year cells are zero while positive values are continuous and highly variable. To model these data, we propose a Bayesian hierarchical tensor factorization model that places a low-rank CP structure on a latent Poisson rate tensor and couples it with a conditional Gamma model for positive outcomes, with rate parameters that can vary across slices within a mode. The model therefore separates the occurrence and magnitude of positive observations while borrowing strength across all tensor dimensions through a shared low-rank latent structure. To scale posterior inference to large arrays, we develop a hybrid variational--Monte Carlo algorithm that combines efficient coordinate ascent updates with a partially collapsed augmented-data sampler. Applied to approximately 60 million trade flows, the method surfaces multiway dependence across exporters, importers, products, and years that is difficult to recover from gravity-type or pairwise network analyses, which do not jointly model the product and temporal dimensions.

Farrow structures based on linear--phase FIR subfilters provide an efficient realization of variable fractional--delay (VFD) filters with reduced implementation complexity. While the all--linear--phase configuration admits a decoupled least--squares (LS) formulation with an analytical solution, this decoupling fails when branches of mixed types, linear--phase and general FIR, are required, as occurs when a group--delay constraint is imposed. This letter presents a unified LS design for Farrow structures via a nullspace parameterization of the per--branch symmetry constraints, yielding an analytical solution that accommodates arbitrary per--branch types. Numerical results demonstrate that the proposed framework satisfies group--delay constraints that the all linear--phase approach cannot meet, while substantially reducing the number of free parameters relative to the unconstrained general FIR baseline.

Graph neural network architectures based on the graph Laplacian approximate the Laplace-Beltrami operator, thus limiting their application to isotropic operators. As a nonlinear alternative to the Laplace-Beltrami operator, we consider estimates of the Finsler Laplacian on point clouds sampled from a manifold. We prove that these discrete estimates converge to the true operator on the manifold as the number of point samples grows. Moreover, we show that this operator can be expressed as a graph neural network layer, which we use to define a family of Finslerian graph neural networks constrained to express Finsler geometry. We show that Finslerian graph neural networks recover the geometry underlying nonlinear diffusion equations in practice.

Organizations and researchers show increasing interest in using large language models (LLMs) in place of human participants in A/B tests, in the hope of experimenting faster and at lower cost. We study when a treatment effect estimated on LLM outcomes recovers the effect that would have been measured on the human population of interest. Distributional equivalence between LLM and human outcomes would make any standard estimator valid but is unrealistic. We therefore develop a statistical framework that adapts surrogate endpoint theory to LLMs. The framework shows that calibrating LLM outcomes to human outcomes identifies the average treatment effect under surrogacy and comparability conditions that are jointly weaker than distributional equivalence. When these conditions fail, the effect of interest is only partially identified, and we provide diagnostics that can falsify surrogacy on historical experiments together with a bound on the worst-case bias from limited overlap. We further show that the stochasticity inherent to LLMs introduces both bias and variance, but using an average of multiple draws as the surrogate mitigates both. We illustrate the methods and theory in simulations and an application to A/B tests on Upworthy headlines. A central takeaway from our work is that the validity of LLM outcomes as surrogates can only be falsified for past treatments and never verified for new ones, so human experiments remain indispensable for novel interventions. We discuss the role of LLM choice, prompting, and temperature as design variables, and how to size human experiments for validation.

We formulate and analyze a sex- and stage-structured model for Anopheles dynamics under the sterile insect technique (SIT), motivated by the need for tools robust to insecticide resistance and outdoor transmission. The model tracks aquatic stages, adult males, unmated females, and females mated with wild or sterile males; includes egg-laying capacity and larval competition; and uses a refractory period followed by density-dependent mate search. The resulting Holling type-II mating term generates a mate-finding Allee effect. After establishing well-posedness, we prove that this Allee effect makes the mosquito-free equilibrium locally stable for all admissible parameters and globally asymptotically stable when a quick-mate-search reproduction number $R_0^q$ is below one. When $R_0^q>1$, habitat capacity is large, and larval competition is weak, two positive equilibria arise through a saddle-node bifurcation: a stable natural equilibrium and an unstable Allee equilibrium separating persistence from extinction. For a reduced model, a Goh-Volterra Lyapunov functional estimates the persistence basin. We then show how constant and population-responsive sterile-male releases reshape this bistability. Sufficiently large releases annihilate the positive equilibria in a second saddle-node bifurcation, while a sufficiently large constant release drives local elimination from every admissible initial state. Thus SIT need only push the population across the Allee separatrix, after which mate-finding failure can complete extinction. In a free-horizon optimization framework with an Allee-threshold stopping rule, a hybrid release strategy reduces the sterile-male requirement by about $5\%$ relative to the best constant-only strategy and $39\%$ relative to the best population-responsive-only strategy. These results recast the Allee effect as a control lever for vector suppression.

Let $(A,\mathfrak{m},k)$ be a Gorenstein ring of dimension $d\ge 1$, $N$ a perfect module of dimension $t\ge 1$ and $I$ an ideal of definition of $N$. For a non-free maximal Cohen-Macaulay (=MCM) $A$-module $M$ and an integer $i\ge 1$, it is well known that the functions $n \mapsto \ell(Tor_i^A(M,N/I^{n+1}N))$ and $n \mapsto \ell(Ext^i_A(M,N/I^{n+1}N))$ are of polynomial types of degrees $r_i^{I,N}(M)$ and $s_{I,N}^i(M)$, respectively. We prove that $r_i^{I,N}(M)\le t-1$ and $s^i_{I,N}(M)\le t-1$ and when $I$ is the maximal ideal $\mathfrak{m}$, both the inequalities become equalities. We also show that $r_i^{I,N}(M)\le r_1^{I,N}(\Omega^dk)$, $s^i_{I,N}(M)\le s^1_{I,N}(\Omega^dk)$ and $r_i^{I,N}(\Omega^dk)=r_1^{I,N}(\Omega^dk)=s^1_{I,N}(\Omega^dk)=s^i_{I,N}(\Omega^dk)$. \end

We study derived categories associated with log del Pezzo surfaces whose singularities are of type $\frac{1}{3}(1,1)$. For such a surface $X$, we consider the canonical smooth Deligne--Mumford stack $\pi:\mathcal X\to X$ and compare it with the singular coarse surface $X$. Our main result proves that, if $X$ is a complex log del Pezzo surface whose singularities are all of type $\frac{1}{3}(1,1)$, then $D^b(\operatorname{coh}\mathcal X)$ admits a full exceptional collection. The proof combines rationality of log del Pezzo surfaces, Orlov's blow-up formula, and the special McKay correspondence of Ishii--Ueda. We then specialize to a general degree $10$ hypersurface $X_{10}\subset \mathbb P(1,2,3,5)$. The Corti--Heuberger cascade identifies its minimal resolution as $\widetilde{X}_{10}\cong \operatorname{Bl}8\mathbb F_3$, and therefore the canonical stack $\mathcal X_{10}$ has a full exceptional collection of length $13$. We also discuss the singular coarse category through the approach of Karmazyn--Kuznetsov--Shinder.

In this paper, we mainly investigate zero-sum problems over $\mathbb Z/n\mathbb Z$ (with $n>1$) of two new types. Let $s_1(n)$ (resp. $t_1(n)$) be the least positive integer $k$ such that for any integers $a_1,\ldots,a_k$ not divisible by $n$ (resp., relatively prime to $n$), there is an $I\subseteq\{1,\ldots,k\}$ with $|I|=n$ for which the sum $\sum_{i\in I}a_i$ is divisible by $n$ but not divisible by $n^2$. For $n\geqslant 4$, we prove that $2n+1\leqslant s_1(n)\leqslant n^2-2n+2$ and $2n-(-1)^n\leqslant t_1(n)\leqslant (n-1)\varphi(n)+1$. We conjecture that $s_1(n)=2n+1$ and $t_1(n)=2n-(-1)^n$ for any integer $n>2$.

Solutions of hyperbolic conservation laws exhibit both smooth structures across large scales and sharp localized features such as shocks and contact discontinuities, making them difficult to approximate accurately with existing neural operators. The Fourier Neural Operator (FNO) captures long-range interactions well but tends to smear localized structures through excessive numerical dissipation. To address this, we propose a Local-Global Neural Operator (LGNO) that learns a one-step discrete flow map by combining a global FNO branch for representing smooth dynamics at large scales with a local multiresolution branch for enhancing localized discontinuities and nonsmooth features. The model is trained with a one-step loss that combines a physical space prediction term and a spectral penalty on high frequencies to suppress spurious oscillations near steep fronts. On a large collection of benchmarks in one and two dimensions, LGNO consistently outperforms FNO baselines with matched parameter counts, reducing one-step errors by factors of 2-5 and remaining significantly more accurate over long autoregressive rollouts. Most strikingly, although it is trained only on short-time data from a high-order WENO-Z scheme, the long-time rollout of LGNO on a coarse $256^2$ grid exhibits lower numerical dissipation than the same WENO-Z scheme run on a finer $512^2$ grid, while being orders of magnitude cheaper to evaluate. These results suggest that, with an appropriate architecture and training objective, learned operators can effectively learn discrete flow maps. They further suggest that such learned operators have the potential to control long-time numerical dissipation better than the conventional shock-capturing schemes that generate the training data.

We study finite-horizon queue peaks in generalized switches, a standard stochastic-network model in which many queues share constrained service resources. Arrivals may be dependent, time-varying, and adapted to the past; the standing load condition is uniform interior slack, meaning the conditional mean arrival vector stays in a fixed contraction of the capacity region. We show that this slack reshapes the finite-time peak law for drift-minimizing scheduling policies such as MaxWeight. The square-root envelope that is sharp without slack persists only up to a geometry-dependent threshold; beyond that threshold, the running maximum grows only logarithmically with the horizon, both with high probability and in expectation. The mechanism is self-normalization: in the current queue direction, the projected fluctuation scale is normalized by the stabilizing drift scale. This removes capacity geometry from the logarithmic coefficient, while geometry remains in the threshold. Matching lower bounds show that both the logarithmic term and a geometric threshold are unavoidable. When finite-time state-space collapse is available, the threshold can be sharpened using local bottleneck geometry. For generalized input-queued switches, we obtain finite-time peak bounds with tight logarithmic coefficients. Simulations illustrate the two-phase envelope, local geometric refinements, and variance-sensitive improvements predicted by the theory.

We construct non-rigid tensor-triangulated categories with non-distributive lattice of thick tensor-ideals.

We study infinite-horizon optimal stopping for normally reflected two-dimensional diffusions in the positive quadrant with max payoff \(G(x_1,x_2)=x_1\vee\alpha x_2\). The non-smooth payoff produces a singular stopping-gain measure on the kink set \(\Delta=\{x_1=\alpha x_2\}\). We prove $\displaystyle \Gamma^\Delta(dx) = -\frac{n^\top a(x)n}{2\sqrt{1+\alpha^2}}\,\sigma_\Delta(dx)$, with $n=(1,-\alpha)$, so the diagonal component is non-positive and strictly negative under local ellipticity. This implies that every interior kink point lies in the continuation region. We further show that the correct value representation uses the resolvent killed at first entry into the stopping set, $\displaystyle V=G-R_r^{\mathcal C}\Gamma$, and give a closed-form reflected Brownian counter-example showing that the unrestricted reflected resolvent is generally wrong. A reflected Brownian benchmark and numerical experiments illustrate the local-time, resolvent-gap, and diagonal-avoidance mechanisms.

We study two inverse problems for a nonlinear dynamical Schrödinger operator with magnetic and electric potentials. Under suitable analyticity assumptions, we show that the Dirichlet-to-Neumann map uniquely determines time-dependent magnetic and electric potentials. We establish the uniqueness of these potentials from both full data and partial data. In particular, for the partial data problem, the desired uniqueness is established by assuming that the potentials are known near the boundary, and the Neumann data is measured on arbitrarily small open subsets of the boundary. In addition, we establish the well-posedness of the forward problem, where we obtain the optimal Sobolev regularity for solutions.