Papers

Based on the Wigner-Vlasov formalism, this paper investigates the kinematic properties of the Pauli equation. It is shown that the probability current associated with the Pauli equation can be represented as a superposition of two currents with certain expansion coefficients. Each of these currents corresponds to a particular component of the spinor. The expansion coefficients effectively serve as weighting functions that determine the probability contribution of the corresponding spinor component. Therefore, each spin projection corresponds to its own probability flux. A new system of the Hamilton-Jacobi equations and also a system of motion equations in electromagnetic fields are obtained, taking into account the interaction between the spin and the magnetic field. To illustrate how these equations can be applied we have investigated the quantum system kinematics in detail using an exact solution of the Pauli equation in the presence of a uniform magnetic field and an asymmetric quadratic potential.

Movable antennas (MAs) are a promising technology to improve wireless data rates by dynamically adjusting their positions to avoid deep fading. However, finding the optimal MA positions requires full channel state information (CSI) for all possible locations within the movement region, creating massive channel estimation overhead. This paper proposes a deep neural network (DNN)-based learning framework to predict the optimal positions of multiple transmit MAs in a multi-user multiple-input single-output (MISO) system, entirely bypassing explicit channel estimation.First, we analyze a single-user MISO case, revealing a complex, highly nonlinear mapping between the optimal MA positions and the channel power gains from a specific subset of locations in the transmit region to the user. Because this mapping cannot be mathematically characterized for practical channel models, we train a DNN via supervised learning to capture it. The pre-trained DNN can then determine optimized MA positions in real-time relying only on partial power measurements from the transmit region.Extending this to multi-user scenarios is challenging due to complex rate expressions and the lack of globally optimal position solutions to use as training labels. To overcome this, we develop an unsupervised training framework that directly maximizes the multi-user sum-rate. This framework utilizes an attention-based architecture to extract latent features from the partial channel measurements and effectively manage inter-user interference. Simulation results show that our proposed approach achieves near-optimal performance in single-user systems and surpasses conventional CSI-based alternating optimization algorithms in multi-user environments.

Non-negative matrix factorisation (NMF) learns additive representations from data, but non-negativity alone does not ensure interpretable basis functions. We introduce Top-NMF, which guides basis learning through general topological preferences without prescribing the detailed form of the components. Observations and basis vectors are treated as non-negative functions on structured domains. Discrete topological conditions are difficult to optimise directly. Our guiding viewpoint is that persistent homology provides a stable, continuous relaxation of ordinary homological invariants: it tracks connected components and loops across thresholds, allowing structural preferences to be expressed through continuous scores suitable for optimisation. We construct such scores for connected image parts, clique-like graph structure, and periodic time-series components, and incorporate them into a common NMF objective. For graph data, we derive an exact maximum-weight-spanning-tree formula and characterise all local minimisers of the score, showing that their positive-weight edges form disjoint cliques. We establish the regularity of the scores, describe their derivatives, and analyse projected first-order optimisation. Numerical studies on synthetic and real data demonstrate how these priors guide basis learning and examine the balance between structural agreement, atom recovery, and reconstruction accuracy.

Dense Gaussian networks are degree-four algebraic interconnection networks with compact diameter and simple modular routing. This paper studies non-redundant one-to-all broadcast repair in the dense Gaussian network generated by $\alpha=k+(k+1)i$. We propose multi-orientation edge-minimum repair (MOEM), which evaluates a constant-size family of Gaussian broadcast-tree orientations, selects a fault-aware orientation, contracts the fault-pruned tree into healthy components, and reconnects those components using external component-crossing repair edges. The resulting structure is a rooted spanning tree of the healthy subgraph, so each healthy node receives the message exactly once and no faulty node is used. We prove that, for a chosen orientation with $c$ fault-pruned components and a connected healthy component graph, the repair step is non-redundant and uses the minimum possible number $c-1$ of external component-repair edges. We also prove that, for every one- or two-fault placement, the MOEM orientation family contains a repair with depth at most $k+2$. The depth proof combines a certificate framework, an explicit four-case off-axis analysis, and a five-component orthogonal-axis certificate. Exhaustive validation for $k=5,\ldots,10$ and large-scale validation through $k=200$ confirm the implementation and show that random two-fault repairs use approximately two external repair edges.

Perfect resource placement in dense Gaussian networks partitions the network into Lee balls centered at resource nodes. The fault-free placement problem is already classified; this paper studies the complementary post-deployment problem of repairing such placements after resource faults. The paper gives exact local repair theorems for the dense Gaussian placement generated by $t+(t+1)i$; by conjugation and rotation symmetry, the same results hold for the companion generator $(t+1)+ti$. For one failed resource, we prove failure-cell locality, derive the exact replacement number $\rho_G(1)=3$ and $\rho_G(t)=2$ for all $t\ge2$, and prove the sharp minimum-overlap formula $\Omega_G(t)=t+1$ among minimum-size repairs. The overlap lower bound is proved from the corner structure of equal-size Lee balls in the rotated coordinates $u=x+y$ and $v=x-y$, where Gaussian Lee balls become parity-constrained squares. For two failed resources, we prove exact additivity: every pair of failed resource cells requires exactly four local replacements for $t\ge2$, and four always suffice. The two-fault lower bound reduces all relevant resource displacements to two canonical neighboring cases and exhibits four mutually incompatible failed-cell corners in each case. For multi-failure repairs, we prove a general inclusion–exclusion identity for overlap inside the failed region; hence the formula remains exact for arbitrary higher-order dense cores. When a canonical repair instance is certified to have maximum multiplicity three, the identity reduces to the compact correction $\Omega_{\rm extra}=P_2-A-C_3$. A ground-truth audit over 7,494 Gaussian cases recomputes coverage from lattice geometry, verifies all exact formulas, and records reproducible multiplicity witnesses.

The Fomin local rules for Schensted column insertion can be seen as a two-lane box-ball system, in which a carrier moves particles forward or laterally. Running such two-lane dynamics in parallel on a periodic lattice gives rise to a two-dimensional generalization of the box-ball system, which we call the skew column RSK dynamics. Equivalently, this is a deterministic dynamics on pairs of skew semistandard Young tableaux $(P_t,Q_t)_{t \in \mathbb{Z}}$. We prove that this dynamics exhibits solitonic behavior and construct an explicit bijection $(P,Q) \leftrightarrow (H_1,H_2,\kappa,\nu)$ that linearizes the time evolution. The resulting coordinates consist of two horizontally weak tableaux $H_1,H_2$ recording the asymptotic soliton data, integer riggings $\kappa$, and a weakly decreasing sequence of integers $\nu$. A key feature of the construction is an explicit projection from the skew column RSK dynamics to the classical box-ball system; under this projection, the rigging $\kappa$ is precisely the Kerov–Kirillov–Reshetikhin rigging of the associated box-ball configuration. Our proof uses two commuting affine crystal structures on pairs of skew tableaux and a novel connectivity theorem for distinguished subgraphs of tensor products of Kirillov–Reshetikhin crystals. We also derive Greene-type formulas for the soliton lengths in terms of last-passage percolation on the associated cylindrical environment. Finally, by taking generating functions in the linearizing coordinates, we obtain bijective proofs of Cauchy and Kawanaka–Littlewood-type identities for transformed Hall–Littlewood polynomials.

Information-Geometric Optimization (IGO) provides a unified framework for black-box optimization by interpreting the adaptation of a search distribution as a natural gradient update. Despite its conceptual importance, the convergence theory of IGO remains limited: most existing results concern continuous-time idealizations such as the IGO flow, rather than discrete-time updates with non-infinitesimal learning rates. In this paper, we study discrete-time IGO in continuous spaces, formulated as natural gradient updates in the expectation-parameter coordinates of an exponential family. In particular, we analyze IGO over the multivariate Gaussian family on strongly convex quadratic objective functions. Our analysis covers a setting that simultaneously incorporates full covariance adaptation, a fixed positive learning rate, and quantile-based weights. In this setting, we prove that the covariance matrix converges to the zero matrix. We further show that the mean vector converges to the global optimum, provided that the condition number of the appropriately scaled covariance matrix is bounded at sufficiently frequent iterations. These results advance the convergence theory of IGO and help bridge the gap between the mathematical theory of IGO and practical covariance-adaptive search methods such as CMA-ES.

We study an infinite-horizon optimal stopping problem for a normally reflected two-dimensional diffusion in the positive quadrant with nonsmooth max-type reward \(G(x_1,x_2)=x_1\vee \alpha x_2\). The paper develops a conditional measure-theoretic framework for the associated reflected obstacle problem. The main innovation is to show that the stopping gain \(\Gamma=c+rG-\mathcal LG\) is a signed measure, not a function: the kink of \(G\) generates an explicit negative surface measure on \(\Delta=\{x_1=\alpha x_2\}\). We then prove that the correct potential representation uses the resolvent of the reflected diffusion killed on first entry into the stopping set, rather than the unrestricted reflected resolvent. Under explicit monotonicity, regularity, and measure-superharmonicity assumptions, we derive an epigraph representation, a continuation-side boundary-trace condition, and a candidate verification theorem. The framework clarifies hidden regularity and uniqueness assumptions in multidimensional nonsmooth optimal stopping.

In this article, we study the image of the Hitchin morphism for some classical groups over an algebraic surface. The Hitchin morphism is a map from the moduli stack of $G$-Higgs bundles $\mathscr{M}_{X,G}$ to the Hitchin base $\mathscr{A}_{X,G}$, where $X$ is a smooth projective variety. In general, this morphism is not surjective when the dimension of $X$ is greater than one. Chen and Ngô showed that the Hitchin morphism factors through a closed subscheme $\mathscr{B}_{X,G}$ of the Hitchin base, which is called the spectral base. They conjectured that the image of the Hitchin morphism is exactly the spectral base. When $X$ is a smooth projective surface, we prove that this conjecture holds for the special linear algebraic group of odd rank. We also confirm this conjecture for the classical groups ${\rm SL}_n$ and ${\rm Sp}_{2n}$ when $X$ is a product of smooth curves.

In this paper, we study the Schrödinger-type operator associated with geometric stable processes on $\mathbb{R}^{d}$, especially the differentiability of spectral function. Let $\mathcal{H}$ be the generator of the geometric stable process and $\mu$ a smooth measure on $\mathbb{R}^{d}$. Then the spectral function $C(\theta)$ is defined as $C(\theta) = -\inf \sigma(-\mathcal{H} - \theta \mu)$, where $\sigma(\mathcal{A})$ denotes the spectrum of $\mathcal{A}$ and $\theta$ is a real parameter. Since the geometric stable process exhibits severe local singularities in its Lévy measure, its transition semigroup lacks ultracontractivity, which invalidates classical methods for proving the differentiability. To overcome this obstacle, we use the compact embedding of the extended Dirichlet space into $L^2(\mu)$. As a primary application of this differentiability, we establish a large deviation principle for a positive continuous additive functional associated with the smooth measure $\mu$.

We investigate data-driven finite-volume discretizations of the linear advection equation in one dimension. Neural networks for use as numerical advection schemes are constructed adhering to first principles of numerical analysis, allowing us to examine how normalization, training data, and architectural choices influence stability, accuracy, and shape preservation. (i) We show that reconstruction based solely on cell averages leads to a multi-valued learning problem, explaining limited generalization when training data includes widely different curvature regimes. (ii) Numerical stability and good generalization can be achieved by enforcing semilinearity (Lin and Rood 1998) through local stencil-scale normalization, which ensures invariance under affine transformations of the inputs. (iii) A new data-driven flux limiter is introduced that outperforms the classical 'OSTVD3' (Arora and Roe, 1997) scheme in shape preservation by introducing mild antidiffusion in near-linear regimes, while higher-order reconstruction in non-monotonic regions provides limited benefit. (iv) We show that training on polynomial profiles yields stable, high-order accurate discretizations, with the polynomial degree controlling the formal order of accuracy. Together, these results illustrate how the representational, architectural, and training choices govern the stability and generalization of data-driven finite-volume schemes for linear advection.

It is a well-known problem to identify the nontrivial zeros of the Riemann zeta function in terms of an eigenvalue problem. We here find such an eigenvalue problem for second order differential operators on the half-line. In a sense, our analysis pushesthe analysis of the zeta function over to the study of the Jacobi theta function, which may be thought of as the fundamental solution of the heat (or Schrödinger) equation on the unit circle (or the semi-infinite cylinder, if time is added). The eigenvalue problem takes the form $LD u+\alpha Lu=0$, where $L$ and $D$ are first-order differential operators, of which only $L$ involves the theta function. In a formal sense, then, $\alpha$ is an eigenvalue of the twisted operator $-LDL^{-1}$. Based on this formal thinking, we develop the notion of self-adjointness of the pair $(LD,L)$, to adapt the Hilbert-Pólya idea to the spectral problem at hand.

The purpose of this note is to study spectral properties of the horizontal Laplacian of a Riemannian submersion with totally geodesic fibers and an integrable horizontal distribution. We show that the horizontal Laplacian is unitarily equivalent to a twisted Laplacian acting on the space of sections of a certain infinite-rank flat vector bundle over the base manifold of the Riemannian submersion. We give an application of this interpretation to the asymptotic behavior of the scaled first nonzero eigenvalue of the canonical variations introduced by Berard-Bergery and Bourguignon. Our approach enables us to compare the horizontal Laplacian with the usual Laplacian on a Riemannian covering over the base manifold, and, when the holonomy group is infinite and amenable, we prove a coincidence of the essential spectrum, which strengthen, in our special setup, a result due to Kordyukov in the context of geometric analysis on foliated manifolds.

This work introduces and investigates the function \( \mathcal{V}(G) = \frac{\text{Simp}(G)}{|L(G)|} \), where \( \text{Simp}(G) \) denotes the number of simple subgroups and \( |L(G)| \) the total number of subgroups of a finite group \( G \). The function \( \mathcal{V}(G) \), defined on the interval \( [0,1] \), represents the proportion of simple subgroups relative to the total number of subgroups. It serves as a tool for analyzing structural patterns in finite groups, particularly in p-groups and other families.

We develop a new combinatorial large sieve method for sets with bounded algebraic multiplicities. The method exploits algebraic splitting modulo many small primes: local congruence branching produces many modular collisions, while global bounded-multiplicity hypotheses force these collisions to be rare. As a first application, we prove that every Sidon subset $A\subset\{1^2,\ldots,N^2\}$ satisfies \[ |A| \le N\exp\left( -c\frac{\log N}{\log\log N} \right) \] for some absolute constant $c>0$. This gives the first super-polylogarithmic saving for a classical problem of Alon and Erdős. As a second application, we establish new upper bounds for two grid-distance problems. We show that the largest subset of $[N]^2$ with no repeated distance has size at most $N\exp\left(-c\log N/\log\log N\right)$, giving the first progress in over thirty years on a problem of Erdős and Guy. The same method also gives a similar saving for subsets of $[N]^2$ with no isosceles triangles, a problem recently popularized by Ellenberg and by the PatternBoost work of Charton, Ellenberg, Wagner, and Williamson. We then develop an entropic version of the method. This gives bounds for $B_2[g]$-sets in the squares and for analogous bounded-multiplicity problems associated with norm forms over arbitrary number fields. More importantly, this new method also allows us to establish the first nontrivial bounds for $B_3[g]$-sets in the cubes and $B_4[g]$-sets in the fourth powers.

This paper presents a unified framework for exploiting the boundaries of reconfigurable intelligent surfaces (RIS) joint optimization in multi-user wireless systems, where a single RIS accommodates diverse objectives.We first propose an adaptive gradient-scaling mechanism that accelerates the convergence of the underlying optimization algorithm while maintaining stable performance across varying channel and system parameters. The proposed mechanism enables the solver to reach a reasonably good solution rapidly without requiring manual tuning of step sizes or algorithmic hyperparameters when system inputs change. We then propose a low-complexity beamformer recovery method tailored for single-user scenarios, which circumvents the full matrix decomposition required by traditional approaches, thereby significantly reducing computational overhead. Building on these foundations, we develop an element allocation strategy that enables user-specific prioritization through assignment of RIS subsets. This is further extended by a modular add-drop mechanism that supports partial-panel optimization in general multi-user settings. The framework is evaluated across three representative scenarios: (i) signal amplification for all users, (ii) signal suppression for all users, and (iii) selective amplification and suppression. To characterize performance limits, we derive power trade-off boundaries using scalarized joint optimization, which closely align with Monte Carlo simulations. Our unified joint optimization method consistently yield solutions near these boundaries, confirming its near-optimality. Extensive simulations under realistic channel models demonstrate that the proposed approach outperforms conventional semidefinite relaxation techniques, offering a scalable and effective RIS control strategy for cooperative and competitive multi-user environments.

We study the Dirichlet problem for the $p$-Poisson equation in the metric measure space setting equipped with a doubling measure and supporting a $(p,p)$-Poincaré inequality. We prove the existence of the solutions by using a variational approach. We prove the stability and uniqueness of the solutions, when the space is also geodesic.

Tanaka's formula is a classical identity for Brownian motion, and Tsukada (2018) extended it to Lévy processes not necessarily symmetric. From a potential-theoretic point of view, this formula shows that the invariant function for the process killed upon hitting a singleton can be decomposed into the sum of a martingale part and a local time. In this paper, we generalize this singleton setting and derive a Tanaka-type formula for a compact set $B$. To this end, we introduce the equilibrium measure, defined as the rescaled limit of the $q$-capacity measures, and show that the invariant function for the process killed upon hitting $B$ can be represented as the integral, with respect to the equilibrium measure, of the invariant functions associated with processes killed upon hitting singletons, up to an additive constant called the Robin constant. Moreover, when $B$ is an interval, we obtain explicit representations of the equilibrium measure, the Robin constant, and the martingale part for recurrent stable processes as well as for recurrent spectrally negative Lévy processes. Finally, we discuss how an analogous Tanaka-type formula can also be established for transient Lévy processes.

The rapid growth of AI workloads is substantially increasing data center electricity demand and carbon emissions, motivating the development of carbon-aware scheduling methods. However, effective scheduling is challenging because renewable generation and AI workloads are subject to forecast errors, while training and inference workloads exhibit heterogeneity in computational characteristics. This paper proposes a contextual robust optimization framework for AI data center operation. The proposed model explicitly captures the heterogeneous computational characteristics of AI training and inference workloads. To deal with renewable generation and workload forecast errors, we develop loss-based uncertainty learning models that directly map contextual features to covariate-dependent uncertainty sets. The resulting contextual joint chance-constrained scheduling problem is reformulated into a tractable robust optimization problem, and a calibration algorithm is developed to provide finite-sample probabilistic feasibility guarantees for multiple joint chance constraints. Numerical experiments based on real-world AI workload traces and renewable generation data show that the proposed method reduces operating costs by an average of 5.57% compared to benchmark methods while maintaining reliable feasibility and strong computational scalability.

Neural operators are widely used as surrogate solution maps for partial differential equations (PDEs), but full-size models can be costly to store, deploy, and evaluate in many-query scientific workflows. This work introduces Operator Boosting, a stagewise residual-learning framework for constructing compact neural-operator surrogates directly, rather than training a large model and compressing it afterward. Starting from the empirical mean predictor in normalized output coordinates, the method trains a sequence of tiny same-family neural operators on residual fields and incorporates each correction through validation-selected shrinkage. We instantiate the framework with Fourier neural operators (FNOs), DeepONets, and convolutional neural operators (CNOs), and compare boosted tiny stacks against full-size monolithic baselines across one-, two-, and three-dimensional PDE benchmarks from PDEBench, APEBench, and The Well. Across 30 dataset-architecture pairs, 21 show positive mean accuracy gains and 17 have positive confidence intervals, while all boosted stacks reduce trainable parameter count by approximately 72-95%. Best-model comparisons show empirical Pareto improvements on 7 of 10 completed PDE benchmarks, including two-dimensional Navier-Stokes, shallow-water dynamics, Darcy flow, one-dimensional transport and reaction systems, and three-dimensional compressible Navier-Stokes. These results show that Operator Boosting often improves the empirical accuracy-parameter Pareto frontier of neural PDE surrogates, while also exposing PDE- and architecture-dependent regimes where residual boosting fails to offset compression.

Superhedging supermartingales, introduced by the authors in previous work, are non-probabilistic processes defined via subadditive outer integrals that carry a purely financial interpretation in terms of superhedging cost. Building on the Leinert-König theory of non-lattice integration, the present paper establishes several results that are classical in probability theory but whose non-probabilistic proofs require fundamentally new arguments: (i) a tower inequality for the conditional outer integral \overline{\sigma}_j applied at stopping times, reducing to equality when the integrand is conditionally integrable; (ii) three versions of Doob's optional stopping theorem, organised by the class of supermartingale and the range of the stopping times; and (iii) Dubins' upcrossing inequality in both finite- and infinite-time horizons. A key structural result, property (K)-a.e., identifies conditions under which the two superhedging operators \overline{\sigma}_j and \overline{I}_j coincide on non-negative functions, extending the scope of all preceding results to the positive operator \overline{I}_j. None of the proofs invoke classical measure-theoretic tools; in particular, (classical) integrability and measurability are not assumed. The analogues of classical stochastic results acquire a purely financial interpretation and, in this way, gain depth and generality by providing a context that is independent of any a priori probabilistic structure.

The study of principal symmetric ideals (PSIs) in ambient polynomial rings was complicated by the combinatorial instability of minimal generators for ideal powers. We resolve this instability in the two variable case by translating the problem into the arithmetic geometry of symmetric affine plane curves. By working topdown within the Dedekind domain of a symmetric coordinate ring, we establish a precise geometric dictionary for PSIs. We prove that the prime factorization of a PSI is strictly determined by the $S_2$-orbits of its symmetric intersection locus, and that ramification corresponds exactly to tangential intersections, which are detected globally by a novel Symmetric Discriminant ideal. Crucially, we demonstrate that the ideal class of any PSI is a $2$-torsion element in the Ideal Class Group. This establishes that the powers of a PSI exhibit strict periodicity, alternating between being principal and requiring exactly two generators. Finally, we localize this arithmetic obstruction to the axis of symmetry, culminating in a Parity Criterion that determines principality based on intersection multiplicities along the diagonal.

In recent work of Dombos, the set of integer partitions of $n$ wherein the parts are either divisible by 4 or congruent to $\pm 1 \pmod{6}$ arose in a natural way. In this work, we will denote the function which counts the number of such partitions of $n$ by $dp(n)$. Using elementary generating function manipulations and classical $q$–series results, we prove several congruences satisfied by $dp(n)$. As an example, we prove that, for all $\alpha \geq 1$ and all $n \geq 0$, \begin{equation*} dp \left( 3^{2\alpha + 1}n + \frac{7 \cdot 9^\alpha + 1}{4} \right) \equiv 0 \pmod{3}. \end{equation*}

Let $L =-\Delta +V$ be a Schrödinger operator on $\mathbb{R}^n$, $n \geq 3$, with the potential $V$ being nonnegative and belonging to the reverse Hölder class $RH_q$ for some $q >n/2$. For $0< \alpha <2$, the Lipschitz space $\Lambda_L^\alpha(\mathbb{R}^n)$ adapted to $L$ is defined as the space of all measurable functions $f$ on $\mathbb{R}^n$ such that \[ \|f\|_{\Lambda_L^\alpha}:= \|\rho(\cdot)^{-\alpha}f(\cdot)\|_{L^\infty}+ \sup_{z \in \mathbb{R}^n \backslash \{0\}} \frac{\|f(\cdot + z) + f(\cdot -z) -2 f(\cdot)\|_{L^\infty}}{|z|^\alpha} <\infty, \] where $\rho$ is the critical radius function related to $L$. In this paper, we provide characterizations of $\Lambda^\alpha_L(\mathbb{R}^n)$ in terms of Littlewood-Paley-type decompositions and Carleson measures, for $0< \alpha < 2 -(n /q)$.

We study extension of Ulrich bundles from a smooth nondegenerate subvariety $X$ of $\mathbb{P}^n$. If $X$ is a complete intersection of dimension $\geq 2$, we show that the extension is not possible except in the trivial case. For an arbitrary $X$, we characterize when the extension is possible, assuming some condition on the extended vector bundle. As an application, we generalize previous results of López and Zamora. We also give several classes of examples of Ulrich bundles on curves that extend to the ambient projective space.

This work analyzes revises the interior penalty (IP) discontinuous Galerkin (DG) method imposed in [Chen, G., Qiu, W., & Xu, L. (2021). Analysis of an interior penalty DG method for the quad-curl problem. IMA Journal of Numerical Analysis, 41(4), 2990-3023.] for the quad-curl problem in a nonconvex polyhedral domain, while introducing a piecewise constant coefficient matrix. We derive two main results: Under minimal regularity assumptions, we prove that the numerical solutions converge strongly to the true solution in the $ H(\text{curl}) \times H^1(\Omega)$ norm. Under slightly higher regularity, we establish the optimal estimate of the convergence rate depending on the regularity of the solution. These two results, serving as a complement to the existing literature, completely answer how the concerned IP DG method performs on quad-curl problems with low regularity.

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.

In this paper, we establish a general comparison theorem for the order statistics of the edge eigenvectors for generalized Wigner matrices. Consequently, we derive the Gumbel law for the maximal edge eigenvector component and prove the universality of the Gaussian fluctuations of the order statistics in an intermediate regime close to the maximum. In addition, our comparison result also implies a quantitative first order estimate for moderately small order statistics.

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.