Papers

We study the pathwise behavior of the three-dimensional attractive one-point interaction diffusion whose law was constructed by Cranston, Koralov, Molchanov and Vainberg, corresponding to the singular Schrödinger Hamiltonian \[ \frac12\Delta+\frac{\beta}{2}\delta_0, \qquad \beta>0. \] We identify a local stochastic differential equation satisfied by the process away from the origin and use it to construct a natural submartingale whose increasing component in the Doob-Meyer decomposition is supported on the set of times at which the process visits the origin. In particular, we show that the process visits the origin with positive probability and that the law conditioned on avoiding the origin is three-dimensional Wiener measure.

In our previous work \cite{Omar-Shah2}, we initiated the study of $\CSX$- and $\XT$-groups associated with a fixed variety $\X$. A group belongs to the former class if all of its maximal $\X$-subgroups are malnormal, and to the latter if any two $\X$-subgroups with nontrivial intersection generate an $\X$-subgroup. In general, $\CSX$ does not imply $\XT$, however as shown in \cite{Omar-Shah2}, some varieties do satisfy this implication. In this article, we provide additional examples of varieties for which $\CSX$ implies $\XT$.

Hypercube graphs are fundamental model spaces of positive curvature in discrete comparison geometry. Let $G$ be a finite, connected, simple, unweighted graph with Bakry–Émery curvature bounded below by $K$. We call $G$ Lichnerowicz-sharp if its first non-zero non-normalized Laplacian eigenvalue $\lambda_1=K$. We prove that all regular Lichnerowicz-sharp graphs are hypercube bundles with constant Bakry-Émery curvature $2$. A hypercube bundle is a graph bundle whose fiber graphs are hypercubes. As applications, we show that any $d$-regular Lichnerowicz-sharp graph with the multiplicity $m_K(G)$ of $\lambda_1=K$ at least $d/2$ can split off a hypercube of certain dimension. Moreover, we characterize all $d$-regular Lichnerowicz-sharp graphs with $m_{K}(G)\geq d-3$. Interestingly, our result leads to the following spectral rigidity theorem of hypercubes. For a graph $G$ with maximum degree $\Delta$, if the multiplicity $m_K(G)\geq \Delta-1$, then $G$ is a $\Delta$-dimensional hypercube. This improves, in the unweighted setting, the multiplicity condition $m_K(G)\geq \Delta$ appearing in the hypercube rigidity theorem of Liu, Münch, and Peyerimhoff. This improvement is optimal.

Let $M^4$ be a closed, orientable $4$–manifold carrying a transversely oriented $C^2$ codimension–one foliation whose leaves are diffeomorphic to $\mathbb{R}^3$. We prove that $M^4$ is homeomorphic to the $4$–torus $\mathbb{T}^4$. We also show that, whenever the original smooth structure on $M$ admits a smooth defining $1$–form, the conclusion sharpens to a diffeomorphism $M\cong\mathbb{T}^4$.

We investigate the canonical and $c^\infty$-diffeologies on Hausdorff locally convex spaces and their applications to Schwartz distributions. We prove that a Hausdorff locally convex space, endowed with its canonical diffeology, is convenient if and only if the canonical map to its internal tangent space at each point is a linear isomorphism. This yields a geometric characterization of Mackey completeness. We also compare several natural diffeologies on locally convex spaces and identify conditions under which they are preserved under completion, dualization, and the formation of inductive limits. As an application, we realize the space of microlocally multipliable distributions as a diffeological colimit and show that Hörmander-admissible multiplication is smooth. This establishes a framework for nonlinear distribution theory beyond the classical manifold setting.

We give a reformulation of tangential morphisms (which is a generalization of Deligne's tangential base point) via log geometry.

We investigate the joint distribution of the trivariate statistics (des, maj, inv) on classical permutations, Andre permutations of the first and second kinds, and Simsun permutations. By decomposing permutations according to the position of the smallest element, we obtain explicit recurrence relations for the generating functions of these statistics. In the classical permutation setting, our recurrence relation yields the generating function for the trivariate statistics (des, maj, inv) due to Gessel, which is typically proved using MacMahon's technique.

It was recently shown by Huang that the category of $C_1$-cofinite modules for any vertex operator algebra $V$ admits a natural braided monoidal structure. Here, we show that this structure extends uniquely to a vertex algebraically natural braided monoidal structure on the completion of the category of $C_1$-cofinite $V$-modules under filtered colimits, within the ambient category of all generalized $V$-modules. A critical point here is that we do not assume the category of $C_1$-cofinite $V$-modules is abelian or that $C_1$-cofinite modules are compact in the cocompletion, since these properties are not known to hold in general. Our results have many applications in the representation theory of vertex operator algebra extensions, since many vertex operator algebras can be realized as objects in the filtered colimit completion of the category of $C_1$-cofinite modules for a vertex operator subalgebra. Generalizing from the specific vertex algebraic setting, we also establish existence and uniqueness for extensions of monoidal structures along a dense inclusion $\mathscr{C}_0 \to \mathscr{C}$ from an abstract, essentially small monoidal category into a well-structured cocomplete target.

This study investigates a statistical property of Lagrange multipliers in constrained Maximum Likelihood Estimation (MLE) and Least Squares (LS) problems from the perspective of numerical optimization. Building on large-sample theory, we show that the associated Lagrange multipliers converge to zero as the sample size increases, provided the distribution is correctly specified in MLE or the residuals are normally distributed in LS. Although this asymptotic behavior has long been recognized in statistics, it has received little explicit attention in numerical optimization and has rarely been exploited in algorithmic design. Importantly, the insight extends beyond classical low-dimensional settings: even in modern high-dimensional applications, such as deep learning, where the number of parameters may exceed the sample size, the same reasoning applies provided the generalization performance is good. This observation has two main implications. First, many constrained optimization algorithms, including the Augmented Lagrangian Method, Sequential Quadratic Programming, and Interior Point methods, require initial values for the multipliers, and choosing zero is statistically justified. Numerical experiments for constrained regressions and dynamic discrete choice model estimations support this implication by showing that initializing multipliers at zero usually lead to stable and efficient performance. Second, penalty-based approaches that convert constrained problems into unconstrained ones can perform well when the true multipliers are small. This helps explain why penalty-based methods often perform well in practice.

This paper makes several improvements to existing level set based approaches to computing shape optimizers for the Dirichlet eigenvalues subject to a volume constraint. The most notable changes in formulation include an overhaul of the classical level set construction and root-finding procedures as well the use of a regularized approximation to the standard objective function. Our resulting computational minimizers are either comparable to or improvements on the best known minimizers from the literature. We conclude with a survey of subproblems within the field that may benefit from numerical experiments; these include the existence of cusps on the boundary, the end-behavior of eigenfunction weights in the p-parameterized problem, and the nature of Weyl asymptotics as they relate to the Pólya conjecture.

In this paper, we prove the local boundedness of solutions to systems of partial differential equations in divergence form. More specifically, we consider systems that include the first variations of functionals depending on the spatial variable and exhibiting nonstandard growth with respect to the gradient, such as $$\int_{\Omega} \left( 1+ h(|Du|)\right) ^{\alpha(x)} \, d x,$$ where the convex function $h=h(t)$ does not satisfy the so-called $\Delta_2$ property and does not exhibit the conventional polynomial growth behavior.

Master equations in mean field game theory characterize feedback value functions that depend on time, state (space), and the population distribution. Their numerical approximation is challenging because the unknown is defined on a space of probability measures and the equation involves intrinsic measure derivatives and nonlocal population terms. This paper proposes a measure-consistent operator learning method (MCOL) for infinite-dimensional master equations. The population distribution is represented by an empirical measure and encoded through a symmetric pooling structure, so that the network input is built directly from the particles representing the measure. The same particles are used in the empirical quadrature of the nonlocal residual terms, avoiding additional quadrature grids or auxiliary integration points. A key feature is that the intrinsic derivative appearing in the residual is induced by the same measure-dependent representation that defines the approximation of the value function. Consequently, the value function, its measure derivative, and the empirical residual are tied to a common measure representation, leading to a structurally coupled value-derivative approximation. We also introduce an error decomposition separating neural approximation error from empirical discretization error. Numerical experiments on several master equations show that MCOL accurately approximates the value function, intrinsic measure derivatives, and feedback quantities, and remains robust under changes in the input measures.

We study the relation type of ideals of finite reduced sets of points in the projective plane; for a given ideal, this is the maximal $T$-degree of a minimal generator of the defining ideal of the Rees algebra. Our main focus is on point configurations whose defining ideals are not necessarily linearly presented, with an emphasis on almost collinear configurations. We prove that $\rt(X)\in\{1,3\}$ whenever $X\subseteq \PP^2_k$ is a finite set of at most ten points; and we characterize the configurations of relation type $3$ in this range. We then show that a configuration of eleven points in generic position has relation type $5$, thereby yielding the first occurrence of relation type larger than $3$. Finally, we exhibit a configuration of $17$ points with relation type $4$ and we formulate some questions regarding the spectrum of admissible relation types of point configurations.

This paper presents a novel frequency-robust perfectly matched layer (PML) boundary integral equation (BIE) method for solving two-dimensional electromagnetic scattering problems involving periodic arrays of obstacles. In periodic scattering problems, standard BIE formulations based on the quasi-periodic Green's function require the evaluation of lattice sums or challenging Sommerfeld-type integrals, which diverge at Rayleigh–Wood (RW) anomalies. An alternative is to use BIE formulations based on the Helmholtz free-space Green's function, but these are defined on unbounded unit-cell boundaries and therefore require suitable truncation strategies, such as the Windowed Green Function (WGF) method. Although such approaches avoid the use of expensive quasi-periodic Green's functions, they also suffer from breakdowns at RW anomalies unless an appropriate mode correction is incorporated. Similarly, the direct application of PML-BIE techniques to periodic structures experiences comparable difficulties near RW anomalies due to the destruction of exponential convergence near RW anomalies for fixed PML parameters. To overcome this challenge, we propose a modified PML-BIE method that combines the PML technique with a finite-mode correction, ensuring both high accuracy and robustness at and around RW-anomalies. Convergence of the PML-truncated boundary integral operators is proved and several numerical examples are presented to validate the efficiency and performance of the proposed method.

In this paper, we study the B-model categorical enumerative invariants (CEI) associated with derived categories of coherent sheaves on smooth projective Calabi-Yau $3$-folds. We first prove the analogs of the dilaton, string, and divisor equations of CEI in a general context. Then we use these equations and the Givental quantization formula to prove that the B-model CEI for any miniversal family of smooth projective Calabi-Yau $3$-folds satisfies the holomorphic anomaly equations introduced by Bershadsky-Cecotti-Ooguri-Vafa. This provides strong evidence that CEI may be taken as a rigorous mathematical definition of the B-model topological string partition function.

For integers $a,b\ge 2$, put \[ g_n:=\gcd(a^n-1,b^n-1)\qquad(n\ge 1). \] We prove that $(g_n)$ satisfies a constant-coefficient linear recurrence if and only if $a$ and $b$ are multiplicatively dependent; the same conclusion holds if only a tail of $(g_n)$ is assumed to satisfy such a recurrence. More generally, if $a$ and $b$ are multiplicatively independent, then for every fixed shift $N\ge 0$, every integer sequence $(W_n)_{n\ge 1}$ that satisfies a constant-coefficient linear recurrence and the pointwise divisibilities \[ W_n\mid a^{n+N}-1 \qquad\text{and}\qquad W_n\mid b^{n+N}-1 \qquad(n\ge 1) \] is eventually periodic. If $(W_n)$ satisfies a recurrence with nonzero trailing coefficient, then it is periodic from the first term. In particular, every common integer linear divisibility-sequence factor of $a^n-1$ and $b^n-1$, in the convention adopted here, is periodic. The proof combines the Bugeaud–Corvaja–Zannier bound with the structure of subexponential linear recurrences and a local valuation argument.

We study the Minkowski shape of pure number fields \[ K_a=\mathbb Q(\theta),\qquad \theta^n=a. \] For admissible parameters satisfying an explicit local hypothesis at the primes dividing $n$, we prove a discrete–archimedean factorization \[ \mathrm{sh}(K_a)=\bigl[C(a)^{\mathsf T}\mathrm{diag}\bigl(s_1(a),\dots,s_{n-1}(a)\bigr)C(a)\bigr], \] where the $s_m(a)$ arise from normalized monomials and $C(a)\in\mathrm{GL}_{n-1}(\mathbb Q)$ comes from a normalized integral basis. This yields a uniform odd/even rigidity dichotomy: for every odd $n\geq 3$, the Minkowski shape is a complete invariant among admissible pure degree-$n$ fields, whereas for $n=2r$ it determines the core field $\mathbb Q(|a|^{1/r})$; on the squarefree admissible subfamily it is complete up to sign, although infinitely many non-isomorphic pairs $K_a$ and $K_{-a}$ have the same shape. We also derive explicit formulas for $|\mathrm{disc}(K_a)|$, including exponent-vector and divisor-lattice factorizations. Finally, we show that the pure-field shape locus is supported on rational diagonal leaves in shape space: unconditionally it lies in a countable union of closed leaves, while under the same local hypothesis only finitely many leaves occur in each fixed degree. On a fixed normalized stratum, the shape depends only on ratio variables, whereas discriminant growth is governed by independent product variables.

Fréchet regression provides a versatile framework for modeling responses in metric spaces with Euclidean predictors, yet current methodologies rely almost exclusively on frequentist approaches. We propose a Bayesian framework for Fréchet regression that offers a principled way of incorporating prior information into nonlinear global Fréchet regression. By targeting a novel Fréchet Bayes rule, we reduce the object-valued regression problem to a collection of tractable scalar regression tasks. Our approach allows for a controlled interpolation between the prior and the data-driven frequentist estimate, facilitating effective shrinkage toward informed values. While initially derived under Gaussian assumptions, we demonstrate that our framework is robust to model misspecification by establishing its validity under moment conditions via weak conditional expectations. The numerical properties of the proposed methodology are demonstrated in simulation studies and an application to microbiome compositional data, where we show that leveraging an auxiliary cohort to inform the prior significantly enhances predictive performance in a targeted, small-scale study

For $m \in \mathbb{N}$, let $q_m := \frac{\log(2m+1)}{2\log(m+1)}$. The max-convolution inequality \begin{align*} \sum_{k=0}^{2m}\left(\max_{i+j=k} x_i y_j \right)^{q_m} &\ge \left(\sum_{i=0}^{m} x_i\right)^{q_m} \left(\sum_{j=0}^{m} y_j\right)^{q_m} \end{align*}for arbitrary sequences $x_0 \ge x_1 \ge ... \ge x_m \ge 0, y_0 \ge y_1 \ge ... \ge y_m \ge 0$ implies an affirmative answer to a question of Bourgain, Dilworth, Ford, Konyagin, and Kutzarova \cite{BDFKK} on the sizes of sumsets in product sets. This inequality was proven for $m = 2$ by Becker, Ivanisvili, Krachun, and Madrid \cite{BIKM} by reducing the general case to the geometric block case via a max-tie analysis. We prove the geometric block case $x = (1, t, ..., t^{r}, 0, ..., 0)$ and $y = (1, t, ..., t^s, 0, ..., 0)$, $t \in [0, 1]$, for all $m \in \mathbb{N}$ via a comparison of Stolarsky means. Some perturbations are also verified. Finally, we prove the above inequality when one sequence has only two non-zero terms.

It is known that lower powerdomains preserve and reflect both continuity and quasicontinuity, while the preservation of quasicontinuity by upper and convex powerdomains had long been open. Directed spaces provide a topological extension framework for dcpos. Powerdomains of dcpos can be characterized as the $D$-completions of the corresponding directed powerspaces. Using this observation, the authors proved in 2024 that upper powerdomains do not preserve quasicontinuity. In this paper, we prove that directed convex powerspaces and convex powerdomains do not preserve quasicontinuity, but they do reflect both continuity and quasicontinuity. We also prove that upper powerspaces and upper powerdomains reflect quasicontinuity. Together with the known results for lower and upper powerdomains, these arguments give the complete preservation and reflection profile of the lower, upper and convex powerdomains for continuity and quasicontinuity.

We address the inverse problem of reconstructing the state at a final time $T_0$ of a parabolic equation with dynamic boundary conditions of Wentzell type, from a distributed interior observation on a subdomain $\omega \subset \Omega$. Our approach combines (i) a Carleman-type observability inequality for the bulk-surface system, with (ii) a Tikhonov-regularized optimal-control reformulation of the data assimilation problem, and (iii) a fully implementable spatio-temporal discretization based on a Lie splitting that decouples the bulk and surface dynamics. We prove an exact reconstruction theorem for the continuous problem, derive a stability inequality, and analyze a penalized discrete scheme. We then propose an adaptive post-processing step that optimally combines the regularized reconstruction with the observation itself; under noisy data, the resulting estimator automatically selects between full post-processing and the raw reconstruction without prior knowledge of the noise level. Extensive numerical experiments on a two-dimensional Wentzell heat equation validate the method. A semilinear extension to the Allen-Cahn nonlinearity is treated via a Picard outer loop with Schauder fixed-point convergence; two complementary experiments reveal an intrinsic observability-nonlinearity trade-off in the bulk-surface setting.

We analyze the performance of joint source-channel codes in an unknown-channel framework, where the true channel is unknown but the source distribution is known. We derive achievability bounds for a family of mismatched-design joint source-channel codes constructed for a design channel $Q_{Y|X}$ and operated over a possibly different true channel $P_{Y|X}$. Our one-shot achievability bound allows for standard Borel alphabets for the source, reproduction, channel input and channel output. The subsequent block coding result based on the normal approximation applies to stationary memoryless sources and memoryless, possibly nonstationary channels under regularity and moment conditions. The achievability bound is given in terms of the rate-distortion and rate-dispersion functions, as well as two channel-dependent quantities that we call the mismatched-design rate and mismatched-design rate-dispersion. We use a family of Gibbs posteriors parameterized by a single scalar as decoder-side kernels, and the envelope of the corresponding achievable rates recovers the generalized mutual information. In the stationary matched setting covered by our assumptions, our result recovers the achievability part of Kostina and Verdú's 2013 Gaussian approximation result and improves its third-order term. We also formalize a notion of a second-order universal family of source-channel codes under which there is no first- or second-order asymptotic penalty. We then construct two channel-blind families of source-channel codes: one that is second-order universal over a regular class of nonstationary block erasure channels and another that is second-order universal over stationary Gaussian channels. Our code construction uses Poisson functional representations of suitable conditional probability measures to produce the encoder and decoder outputs.

3D Gaussian Splatting (3DGS) has become an efficient explicit representation for radiance field reconstruction and real-time novel view synthesis. However, its standard photometric loss treats flat and structure-rich regions similarly, which may limit the recovery of sharp contours and fine details. Edge-Guided Gaussian Splatting (EGGS) improves structure awareness through edge-guided weighting, but mainly relies on first-order gradient responses and linear weighting. In this paper, we propose LEGS, a Laplacian-Enhanced Gaussian Splatting method with a nonlinearly weighted loss. LEGS replaces first-order gradient guidance with second-order Laplacian structural guidance and maps the normalized Laplacian response into pixel-wise weights through nonlinear response-to-weight functions. The proposed loss improves structure-aware Gaussian optimization while keeping the original 3DGS rendering pipeline unchanged. Experiments on the full Tanks&Temples and Mip-NeRF360 datasets show that LEGS improves peak signal-to-noise ratio (PSNR) by up to 1.68 dB over 3DGS and up to 0.52 dB over EGGS. Incorporating the proposed second-order nonlinear weighting strategy into FastGS and FasterGS further improves PSNR by up to 1.69 dB, demonstrating its effectiveness as a general loss-level extension for Gaussian Splatting pipelines with potential applications in AR/VR, immersive visualization, and real-time 3D content generation.

We prove a variance-aware pointwise majorizing-measure theorem for centered Gaussian processes. Classical generic chaining characterizes the scalar quantity $\mathbb E\sup_{x\in T}X_x$; the theorem here gives a simultaneous high-probability envelope for the entire field. For an ambient prior $\mu$, the envelope at $x$ is governed by a pointwise Fernique-Talagrand functional \[\Phi_\mu(x):=\int_0^{4\sigma(x)}\sqrt{\log\frac{1}{\mu(B_d(x,\varepsilon))}}\,d\varepsilon,\] together with the corresponding Gaussian tail term. The theorem provides a reusable field-level refinement of classical generic chaining and a Gaussian-process counterpart of pointwise empirical-process bounds for deep neural networks. We also record a Bayesian algorithmic lower envelope from the interactive Fano/data-processing principle. For a known prior $\pi$, an observation channel, and a concrete estimator $\widehat t(Y)$, the lower bound is expressed through the exact ghost small-ball mass $\mathbb E_{Y\sim Q}\pi(B_d(\widehat t(Y),\Delta))$, rather than a worst-case covering number. In Gaussian location experiments, comparison decoders convert Bayes location error into lower bounds on decision-aligned Gaussian ranges. We then construct an elementary example separating the usual Fano relaxation, the Bayesian algorithmic lower envelope, the pointwise Gaussian envelope, and the full-class minimax risk. Together, these results show that algorithmic lower bounds provide local-geometric validations of pointwise complexity for fixed estimators in overparameterized ambient classes, precisely in regimes where classical minimax theory becomes either too coarse or oracle-dependent. This separation can also be recast in minimax language as penalty-range information relaxation, highlighting an important question of algorithmic robustness for classical high-dimensional models and regularized algorithms.

Inorganic structural chemistry leads naturally to a theory for crystal structure prediction formalized by a generalized disjunctive programming (GDP), which is formulated using continuous and Boolean variables to involve the algebraic equations, disjunctions, and logic propositions. Since the feasibilities of continuous variables change drastically depending on Boolean variables and vice versa, iterative optimization of continuous and Boolean variables efficiently transforms a randomly generated initial structure into an optimal solution. Boolean variables are introduced to elucidate the “combinatorial backbone” of the original continuous optimization problem, which corresponds to the graphs describing crystal structures with clear chemical meanings. Since many subgraphs are discovered in numerous graphs generated through structural optimizations, on-the-fly learning of the feasibilities of them accelerates the discovery of the optimal solutions. This theory enables designing a wide variety of crystal structures with small computations based on only the atomic radii and feasible coordination numbers of each atom.

We study the finite-lattice statistical-mechanical model whose nearest-neighbor bond potential is the reciprocal cost $J(e^\varepsilon)=\cosh\varepsilon-1$, selected by the d'Alembert functional equation under the stated regularity and calibration assumptions. The structural inputs are stated explicitly; once they are fixed, the analysis is rigorous mathematics about the bond action $V(\Delta\phi)=\cosh(\Delta\phi)-1$ on finite boxes in $\mathbb Z^3\times\mathbb Z/8\mathbb Z$. Our main result pairs a negative and a positive statement about reflection positivity. For the continuous noncompact model the natural temporal kernel $K(u)=\exp[-(\cosh u-1)]$ fails the Bochner positive-definiteness test: an interval-certified quadrature gives $\widetilde K(3)<0$. Thus the standard Bochner route to Osterwalder-Schrader reflection positivity is obstructed. For a finite-alphabet variant, with field values restricted to a finite symmetric set $\Phi=v_0\{-N,\ldots,N\}$, reflection positivity holds whenever the finite crossing-bond Toeplitz matrix $(K_{\Phi(v_0,N)})_{a,b}:=K(b-a), a,b\in\Phi$, is positive semidefinite. For $v_0\in\{1.2,1.5,2.5\}$, this is discharged by a rigorous diagonal-dominance certificate uniform in $N$, and the associated one-step transfer operator is then positive and self-adjoint in an explicit reflection-positivity inner product. These finite-volume results do not provide a continuum Wightman theory, Osterwalder-Schrader reconstruction, LSZ scattering, or a continuum mass gap.

Path planning in environments containing obstacles has numerous practical applications. The problem is challenging because it is inherently nonlinear and nonconvex. Consequently, a variety of techniques have been developed to address this problem, among which machine learning and optimal control (or optimization) have emerged as two prominent approaches. In general, machine learning methods do not require a high-fidelity model, and a trained agent can often generate a feasible path in real time. However, the resulting path is not necessarily optimal with respect to performance objectives such as minimizing path length or travel time. In contrast, optimal control and optimization methods typically rely on high-fidelity models and often require computational effort that may not satisfy real-time constraints. Nevertheless, these methods are more likely to produce optimal or near-optimal solutions. To overcome the limitations of each approach while exploiting their respective strengths, this paper proposes a framework that combines reinforcement learning with an arc-search interior-point method for path planning. Numerical simulations demonstrate that the proposed approach effectively integrates the real-time decision-making capability of reinforcement learning with the optimization performance of the arc-search interior-point method, resulting in improved path-planning performance.

Green asked the following question concerning structures in sumsets: Suppose that $\mathbb{F}_2^n$ is partitioned into sets $A_1, \dots, A_K$. Does $A_i+A_i$ contain a coset of codimension $O_K(1)$ for some $i$? An answer is not known even in the case of $K = 3$. We resolve this question in the affirmative in two special cases: (1) when $A_1, A_2$ are Hamming balls of radius $r < n/2 - 7$ relative to different bases, and (2) when $A_1$ is a Hamming ball of sufficiently small constant density.

We characterize finite-type $\mathbb{G}$-principal $\mathbb{U}$-equivariant bundles on normal $\mathbb{U}$-spaces for compact Lie groups $\mathbb{U}$ and $\mathbb{G}$, in several ways, including (a) their extensibility across the $\mathbb{U}$-equivariant compactification $\beta_{\mathbb{U}}X$ and (b) their becoming finite-type upon extending the structure group along at least one $\mathbb{U}$-equivariant compact-Lie-group embedding $\mathbb{G}\le \mathbb{K}$. This generalizes non-equivariant results of Phillips and the author's characterizing finite-type matrix-algebra bundles, upon specializing $\mathbb{G}$ to projective unitary groups. When the $\mathbb{U}$-action on $X$ has virtually abelian isotropy, matrix-algebra equivariant bundles are also finite-type precisely when, locally over a finite open $\mathbb{U}$-cover, they are tensor factors of trivial matrix bundles. In a $K$-theoretic offshoot we prove that for $\mathbb{U}$-actions with finite isotropy groups on compact Hausdorff spaces $X$ equivariant vector bundles $\mathcal{E}\to X$ are factors of trivial bundles $K$-theoretically: there is a class $a\in K_{\mathbb{U}}(X)$ with $[\mathcal{E}]a$ the class of a bundle induced by a $\mathbb{U}$-representation (which furthermore can be chosen so as to restrict to isotropy groups to multiples of the regular representations). This generalizes a result of Donovan and Karoubi.

Polynomial chaos expansions provide surrogate models for stochastic systems, with coefficients typically derived using Galerkin projection, stochastic collocation, or least squares approximation. These traditional approaches often fail to accurately capture statistical moments without resorting to high-order approximations. We propose a constrained optimization framework that modifies standard techniques to determine polynomial chaos coefficients that precisely recover the first two statistical moments. The effectiveness of our approach is demonstrated on several candidate algebraic functions of random variables, showing significant improvements in statistical accuracy even with low-order approximations.