Papers

We extend the trace–determinant framework of Nayak, Sharma, and Mishra~\cite{nayak2026} for bounding the H-eigenvalues of symmetric positive definite tensors. First, we replace the Arithmetic–Geometric Mean (AM–GM) relaxation underlying previous bounds by the exact solution of the associated constrained optimization problem, yielding sharp upper and lower bounds that are attained on the admissible spectral variety. Second, we incorporate higher-order power sums as additional spectral invariants and prove a structural theorem showing that any extremizer over a $K$-invariant feasibility region has at most $K$ distinct spectral values. This reduces the problem to a finite collection of low-dimensional polynomial systems and yields a hierarchy of increasingly tight bounds. For the four-invariant case $(T,S,p_3,D)$, we develop a complete theory including solution-count estimates, a multistart Newton algorithm, and sharpness conditions. We also derive closed-form bounds in small dimensions, establish perturbation estimates, and obtain refined Lyapunov region-of-attraction bounds. Numerical experiments for dimensions up to $d=100$ show that the sharp three-invariant bound reduces the median relative overestimation gap from $53\%$ to $6\%$ while maintaining low computational cost. The framework is validated on tensors with real H-spectrum.

A group-theoretic interpretation of the periodic system of elements is given within the framework of the weight diagram of the Lie algebra $\mathfrak{so}(4,4)$ of the fourth rank, where the four quantum numbers $n$, $l$, $m$, $s$ correspond to the eigenvalues (weights) of the Cartan generators of the maximal Abelian subalgebra (the maximal torus of the group SO(4,4)). It is shown that the root system of the algebra $\mathfrak{so}(4,4)$ forms a regular four-dimensional self-dual polyhedron (24-cell). The action of the fourth Cartan generator associated with spin leads to a splitting of the Cartan-Weyl basis of the algebra $\mathfrak{so}(4,4)$ into two structurally identical bases, each of which is isomorphic to the Yao basis of the subalgebra $\mathfrak{so}(4,2)$ (the Lie algebra of the conformal group). At this point, a four-dimensional 24-cell is projected onto two three-dimensional cuboctahedra, each of which defines the root system of the subalgebra $\mathfrak{so}(4,2)$. This splitting physically corresponds to spin doubling (two-valuedness). The structure of the energy levels of a periodic system is studied, the states of which (chemical elements) are represented as nodes of the weight diagram of the group algebra $\mathfrak{so}(4,4)$. The structure of the double SO(4,2)-towers of Mendeleev, Seaborg, and 10-periodic extension is examined in detail. The period doubling associated with the sequence of period lengths 2, 8, 8, 18, 18, 32, 32, $\ldots$ of the periodic system of elements is explained by the action of the fourth Cartan generator. It is shown that antimatter (Mendeleev anti-table consisting of antihydrogen, antihelium, antilitium, $\ldots$) is naturally included in the general group-theoretic scheme of description of the periodic table.

We define the dual lattice function of a rational polytope $P$ via the discrete Laplace transform of the exponential of its support function. This definition is a discrete analogue of the dual volume function of a polytope that the authors studied in previous work. We show that the dual lattice function is valuative, and by multiplying with the torus form, it becomes the canonical form of the exponential polytope $\mathrm{exp}(P)$ as a positive geometry. This result suggests the study of the class of toric polytopes, which are certain semialgebraic subsets of projective toric varieties. Our work is a first step towards discretization of positive geometries in the simplest case of polytopes.

Inference for spectral edges of large covariance matrices is a fundamental problem in high-dimensional statistics. A major difficulty is that the largest non-spiked sample eigenvalues, which serve as natural estimators of the edge, fluctuate on the Tracy–Widom scale. Consequently, valid inference requires accurate centering by the deterministic spectral edge together with a precise scaling constant, both of which are often difficult to estimate in practice under general unknown population covariance structures. In this paper, we propose a bias-corrected multiplier bootstrap procedure for inference on the deterministic edge of the bulk spectrum. The key idea is to introduce a carefully calibrated multiplier perturbation that regularizes the edge fluctuation to a slightly larger scale at which Gaussian approximation becomes tractable. The resulting confidence interval is constructed directly from bootstrap eigenvalues, together with a data-driven recentering step that corrects the bootstrap-induced shift of the deterministic edge. On the theoretical side, we show that, after bias correction and rescaling, the largest few non-spiked bootstrap eigenvalues are asymptotically Gaussian conditionally on the data. Building on this result, we establish the asymptotic validity of the proposed confidence interval, whose length is only slightly larger than the Tracy–Widom scale, and prove vanishing coverage under alternatives in which additional spikes separate from the bulk at a local scale larger than $n^{-1/6}$. As a consequence, the same confidence interval yields a threshold-free estimator for the number of spikes, without requiring the spikes to be distinct or very large. Equivalently, the procedure yields a data-driven and theoretically justified cutoff for the scree plot.

We address optimal placement of multi-type sensors for Bayesian inverse problems governed by partial differential equations (PDEs). The proposed framework allows for sensors with different accuracies and observation types. We formulate the optimal experimental design (OED) problem as a knapsack-constrained binary optimization problem for maximizing expected information gain (EIG). To approximately solve the resulting optimization problems, we propose a stochastic cost-benefit greedy algorithm, which admits theoretical guarantees for monotone submodular set functions. Specifically, these guarantees apply in the case of linear Gaussian inverse problems with uncorrelated measurement errors, where the EIG admits a convenient closed-form expression. For nonlinear inverse problems, we develop a non-intrusive approach that uses the Bayesian approximation error framework to define an observation model with an error-corrected global linear model. We show that the corresponding approximate EIG is a lower bound for the exact EIG and thus provides a principled surrogate objective for the OED problem. The effectiveness of the proposed methods is demonstrated in two model inverse problems governed by PDEs.

This work is concerned with an inverse medium problem for elastic waves, in which unknown inhomogeneities are reconstructed from time-resolved boundary measurements. We propose a novel time-domain direct sampling method for locating scatterers from a single incident source, without imposing specific assumptions on the temporal profile of the excitation. In particular, the imaging functional introduces a time-shifted correlation strategy that replaces the traditional $P$-$S$ wave decomposition with a travel-time alignment mechanism, thereby enabling direct imaging from the coupled elastic wave field. To analyze the proposed time-domain imaging functional, we employ Parseval's identity for the Fourier–Laplace transform and reformulate the functional in the frequency domain. By exploiting properties of modified Bessel functions, we characterize the asymptotic behavior of the imaging functional and show that it attains its maximum at the target location, which enables reliable identification of the scatterer. Rigorous theoretical justifications are provided to substantiate the effectiveness of the proposed method. Numerical experiments are also presented to demonstrate its performance and applicability.

We compare two Polish presentations of separable real Banach lattices: closed sublattices of a universal lattice and closed ideals of $\operatorname{FBL}[\ell_1]$. We construct translations preserving lattice isometry in both directions, continuous and injective from sublattices to ideals and Borel in the reverse direction. Natural generation and Fremlin tensor operations are $\boldsymbol\Sigma^0_2$-measurable with $G_\delta$ graphs. Isometric lattice isomorphism in either universal coding is Borel bireducible with a complete orbit equivalence relation of Polish-group actions. Uniform monotonicity and order uniform smoothness are $\boldsymbol\Pi^0_3$-complete in both codings, whereas strict monotonicity, order smoothness, and order continuity are coanalytic. Abstract $L_p$ and AM-spaces form closed classes of sublattice codes. We also give explicit Borel bounds for atom counts, compact order intervals, and the lattice-isometry classes of $L_p[0,1]$, $\ell_p$, and $c_0$.

In 1962, Erdős characterized the maximum size of nonhamiltonian graphs of order $n$ with minimum degree at least $k$. Later, Ning and Peng [Combin. Probab. Comput. 29 (2020) 128-136] extended Erdős's results to the clique condition and provided the maximum clique number for nonhamiltonian graphs of order $n$ with minimum degree at least $k$. Recently, Zhang [European J. Combin. 112 (2023) 103728] determined the maximum number of $s$-cliques in nonhamiltonian graphs with prescribed order and minimum degree. A natural extension is to characterize the maximum number of $s$-cliques under other graph properties. Notably, disjoint path cover problems are closely related to Hamiltonicity. In this paper, we generalize results on Hamiltonicity and establish sufficient conditions for a graph to possess one-to-one, one-to-many and many-to-many $t$-disjoint path covers in terms of the number of cliques and the $\alpha$-spectral radius, respectively. Furthermore, we characterize the extremal graphs that attain these bounds respectively.

We consider a family of semilinear parabolic equations with homogeneous Neumann boundary conditions on a family of varying non-smooth domains $\{\Omega_\mu\}_{\mu \in \Lambda} \subset \mathbb{R}^n $. Assuming only that the domains have uniformly bounded volumes, bounded away from zero, and satisfy a uniform Jones condition, together with the usual structural requirements on the linear and nonlinear terms, we establish the well-posedness of the problem in an appropriate scale of fractional Banach spaces and prove the existence of global attractors. The attractors themselves are shown to be uniformly bounded in $L^{\infty}(\Omega_\mu)$ by an elementary truncation argument, which uses no property of the domains beyond the weak formulation of the Neumann condition. A Moser–Alikakos bootstrap iteration, in conjunction with the uniform Gronwall lemma and the uniform properties of the Jones extension operator, then yields the stronger statement that bounded sets of $L^{q}(\Omega_\mu)$ are absorbed, in finite time and uniformly in $\mu$, into a fixed ball of $L^{\infty}(\Omega_\mu)$; this iteration uses only first-order Sobolev embeddings, which is what makes it available on domains where the domain of the elliptic operator is not contained in $H^{2}$. Finally, assuming the volume convergence of the domains, $|\Omega_\mu \triangle \Omega_0| \to 0$, we construct a framework of connecting maps, show that the uniform Jones condition upgrades this convergence in measure to Mosco convergence of the associated bilinear forms, and prove that the family of attractors is upper semicontinuous at $\mu = 0$ in $L^2(\Omega_\mu)$ and hence, by interpolation, in $H^s(\Omega_\mu)$ for every $s<1$.

We study finite-time breakdown of classical solutions to the Euler-alignment system through the degeneration of the associated Lagrangian flow. This approach allows us to characterize singularity formation in terms of the loss of local invertibility of the flow and the resulting concentration of density along characteristics. For the case of constant communication kernels, we derive an explicit formula for the flow and obtain an exact pointwise breakdown criterion in arbitrary dimension. In two dimensions, this criterion admits a closed-form reformulation in terms of the symmetric part of the initial velocity gradient and the initial vorticity. For general non-constant kernels, we derive sufficient conditions for finite-time degeneracy by combining a leading compressive mechanism with perturbative control of the nonlocal remainder. These conditions provide quantitative supercritical breakdown criteria in arbitrary dimension, complementing the existing subcritical global-regularity theory for multidimensional Euler-alignment systems.

There has recently been considerable progress relating o-minimality to complex analytic geometry. Yet almost nothing is known about coherent cohomology or the classification of vector bundles, even for curves. In $\mathbb{R}_{\mathrm{an}}$ and similar structures, we show that cohomology of noncompact curves is concentrated entirely at punctures. As an application, we compute the cohomology of the structure sheaf on the affine line and describe a connection to Diophantine approximation. Finally, we use similar techniques to characterize which definable Riemann surfaces have definable compactifications. The proofs are based on a careful analysis of boundary behavior for definable holomorphic functions.

We study the normalized Frobenius trace associated with the Legendre family of elliptic curves over $\mathbb F_p$ from the point of view of Fourier complexity. If \[ f(t)=\frac{a_p(E_t)}{\sqrt p}, \qquad E_t:\ y^2=x(x-1)(x-t), \] with $f(0)=f(1)=0$, then \[ \frac{\|\widehat f\|_1}{\|\widehat f\|_2}\asymp \sqrt p. \] More precisely, the Fourier transform of $f$ has squared $\ell^2$ norm of order $p$ while its individual coefficients remain uniformly bounded. It follows that no Fourier model supported on fewer than a sufficiently small constant multiple of $p$ frequencies can approximate $f$ in $\ell^2$ with error smaller than a fixed proportion of $\|f\|_2$. We also show that the Fourier magnitude profile of $f$ supports a family of at least $\exp(cp)$ real-valued functions with identical Fourier magnitudes and identical Fourier ratio, any two of which are separated by at least $c\sqrt p$ in $\ell^2$. Consequently, every deterministic reconstruction procedure that recovers all members of this family from bounded-precision point evaluations must use at least $c_Bp$ samples, where $c_B>0$ depends only on the number of bits used to encode each observation. The arithmetic input is unconditional and relies only on the Weil bound for mixed character sums, the evaluation of the quadratic Gauss sum, and elementary character identities.

We construct a set of points $\left\{x_1, \dots, x_n\right\} \subset \mathbb{S}^2$ such that $$ \sum_{i \neq j} \frac{1}{\|x_i - x_j\|^2} \leq \frac{n^2 \log{n}}{4} + cn^2 + O(n^{11/6} \log{n}),$$ where the constant $c \sim -0.085768\dots$ is given in closed form and matches the constant that was conjectured by Brauchart-Hardin-Saff to be optimal. The point set is motivated by the crystallization conjecture and consists of pieces of the hexagonal lattice projected onto the sphere in a tightly interlocked way.

Angular radio maps describe the received-power distribution over the angle of arrival and underpin beam selection and receiver localization in sixth-generation (6G) networks. Predicting the angular power spectrum (APS) from geometry is difficult, because the mapping is ill-posed in non-line-of-sight (NLOS) conditions and must generalize to unseen environments. Distortion-minimizing regressors return the conditional mean, which over-smooths the spectrum and erases the multipath structure that downstream tasks need. We cast the task as a perception-distortion problem and propose RadioDiff-v2, a dual-branch one-dimensional diffusion transformer trained with flow matching. It couples periodic angular encoding, adaptive layer-normalization conditioning, a Fourier angular mixer, and joint velocity and clean-signal heads. A per-metric estimator portfolio reads every deployment quantity from this single model, so that samples carry the distribution, the clean-signal head supplies a regression-grade point estimate, Bayes-optimal rules select beams, and the conditional likelihood localizes the receiver. We prove that a concentrated conditional yields a straight probability-flow trajectory that one step integrates exactly, identifying deterministic transport as the correct inductive bias. On a zero-shot test of 99 environments and one million links, RadioDiff-v2 leads every baseline on every metric, with a 0.39 dB Wasserstein-1 distance, per-bin error below the regression baseline, a 2.43 dB eight-beam NLOS sweep loss, and a 20.6-pixel localization error with four base stations. Code is available at https://github.com/UNIC-Lab/RadioDiff-v2.

Primal-dual first-order methods are widely used for large-scale semidefinite programming (SDP), but their ability to compute highly accurate solutions is not well explained by global convergence theory alone. We study the local convergence of the primal-dual hybrid gradient (PDHG) method applied to a standard primal–dual SDP pair. We show that PDHG converges eventually (R-)linearly whenever the limiting KKT point satisfies either strict complementarity or primal–dual nondegeneracy. The proof views PDHG as a preconditioned proximal point method for the KKT inclusion and combines its descent inequality with a local error bound. Under strict complementarity, the error bound follows from the local spectral geometry of the positive semidefinite cone; under primal-dual nondegeneracy, it follows from strong regularity of the KKT mapping. We also give a simple SDP instance where both regularity conditions fail and PDHG can converge only sublinearly. This contrasts with linear programming, where PDHG admits a local linear convergence regime even for degenerate instances. Numerical experiments support the theory and identify difficult SDP instances where PDHG struggles to reach high accuracy.

In this paper, we study the following elliptic system \begin{equation}\label{main_1} \begin{cases} -\Delta u = |v|^{p-1} v + \epsilon (\alpha u + \beta_1 v), & \text{in } \Omega, \\ -\Delta v = |u|^{q-1} u + \epsilon (\beta_2 u + \alpha v), & \text{in } \Omega, \\ u = v = 0, & \text{on } \partial \Omega, \end{cases} \tag{*} \end{equation} where \(\Omega\) is the unit ball in $\mathbb{R}^N$, \(\epsilon\) is a small parameter, \(\alpha\), \(\beta_1\) and \(\beta_2\) are real numbers, \((p, q)\) is a pair of positive numbers lying on the critical hyperbola \begin{equation} \frac{1}{p+1} + \frac{1}{q+1} = \frac{N-2}{N}.\nonumber \end{equation} Under suitable assumptions and suitable restrictions on $(p,q)$ and $N$, we construct infinitely many sign-changing solutions to \eqref{main_1} which look like a positive radial solution to \eqref{main_1} crowned by $k$ negative bubbles arranged on a regular polygon of a suitable radius, whose energy can be arbitrarily large.

This paper develops principled updating rules for possibilistic inference, where uncertainty about a fixed parameter is represented by a possibility function, the maxitive analogue of a probability distribution, and comparisons are made pointwise via a partial order. From two complementary foundations, an information-conservation viewpoint and an axiomatic viewpoint, we derive the same canonical update: the posterior is the prior-likelihood product followed by supremum normalisation. The two derivations agree for an arbitrary loss, differing only in where the learning-rate parameter enters. This parameter controls epistemic strength and is not identifiable from the normalising evidence alone, clarifying the role of analogous learning-rate parameters in generalised Bayesian updating.

We establish the exact exponent for smoothing the max-relative entropy on arbitrary von Neumann algebras. The proof first develops the required large-deviation and smoothing estimates in the semifinite setting and then passes to general von Neumann algebras through Haagerup-Junge-Xu reduction. We next study catalytic quantum information decoupling when the reference system is described by an arbitrary von Neumann algebra. A key ingredient is an operator layer-cake formula valid on arbitrary von Neumann algebras. This formula leads to a convex-split estimate for normal states with a general von Neumann algebraic reference system. Combining this estimate with the smoothing exponent result, we obtain lower and upper bounds on the catalytic-decoupling reliability function. These bounds coincide below the corresponding critical rate, yielding the exact decoupling reliability function in that regime.

We combine neural network means with exact Matérn kernel regressions of their residuals and of their learned features, and evaluate the pairing on two public emulation problems with published baselines: the structural-mechanics benchmark of de Hoop et al. and the OCO-2 radiative-transfer emulator of Lamminpää et al. On structural mechanics the combination reaches 4.55% test error, matching the best published architecture, and 5.38% against a published 6.49% in the low-data regime. On OCO-2 it improves on the published Gaussian-process emulator on that problem's own test points, outright on two of the three spectral bands; the same kernel that trails the network tenfold on the raw state overtakes it on the network's features, and we measure why (the target's squared native-space norm drops about fortyfold at fixed effective dimension) and prove the mechanism. Where the two families tie instead, the residuals of every architecture we train correlate above 0.86 and their shared component is flat in diversity and sample size, which reads the published plateau as a property of the data. Supporting results include a second-moment identity that predicts stacking outcomes from measured correlations, an optimal-recovery certificate, and a distribution-free coverage band, the only uncertainty signal that survives our tests.

A key question in operator learning is how to design surrogate operators with provable approximation guarantees in reasonable computational time. Whereas smooth operators can be approximated efficiently, i.e., with at least algebraic convergence in the amount of training data, learning finitely regular operators is known to be less efficient. The reason is an intrinsic curse of sample complexity, which allows only subalgebraic sample complexity rates. This fact makes it all the more important to develop algorithms which provably achieve these rates. In this work, we present a fully data-driven algorithm, termed Hermite-PCA approximation, for learning Gaussian Sobolev operators with near-optimal sample complexity. It employs principal component analysis and weighted least-squares methods and is therefore computationally efficient. Moreover, it is spectral, in the sense that it achieves faster (and near-optimal) convergence the higher the Sobolev regularity. We provide a full error analysis of this algorithm, taking into account all sources of error, along with numerical experiments that verify our theoretical results and empirically confirm the efficacy of Hermite-PCA approximation for learning Sobolev operators.

Real-image diffusion inversion is governed by a tight quality-cost trade-off, with costs incurred in computation, storage, or per-image optimization. We study this trade-off through the forward Gaussian noise anchor that defines a diffusion trajectory and isolate two mechanisms behind effective stored-noise inversion. First, diffusion noise exhibits an element-wise compression asymmetry: int8 full-dimensional anchors preserve reconstruction, whereas low-dimensional subspace summaries are much less reliable, often collapsing even at comparable or smaller payloads; the element-wise over subspace ordering persists across five stored-noise inversion methods. Second, inversion is trajectory-bound and score-prior coupled: the matched forward anchor and a trained score network are both necessary, arguing against a purely algebraic-identity explanation. Together, these findings specify what to store and how to use it. They lead to Noise-Anchored Reverse Correction (NARC), a training-free inversion primitive that stores a single int8 latent anchor and reuses it with a fixed, noise-level-dependent anchor-weight schedule: strong anchoring when the reverse trajectory is noise-dominated, then relaxed anchoring as image detail emerges. On PIE-Bench++ with Stable Diffusion 1.5, NARC outperforms five modern non-exact baselines and improves PSNR by +3.24 dB over PnP DirectInv while using about 400x less inversion storage than PnP DirectInv. The compression asymmetry, anchor specificity, and editing plug-in also transfer to SDXL 1024^2.

Decentralized federated learning (DFL) dispenses with the central server of classical FL by utilizing peer-to-peer model exchanges among edge devices. This server-free architecture enables ad-hoc, flexible distributed learning in large device-to-device (D2D) networks. However, wireless DFL converges slowly because peer-to-peer model aggregation incurs high delays and errors. Each DFL training round involves many-to-many gradient sharing over wireless channels, resulting in uncoordinated channel access, large communication errors from stragglers, and slow model consensus, especially in large-scale D2D networks with pronounced clustering structures. We address these aggregation bottlenecks by provisioning a few reliable backhaul links at straggling nodes to enhance network connectivity. Building on this idea, our budget-aware, cluster-centric DFL framework first partitions the network into densely connected clusters, and then allocates the limited backhaul budget to selected cluster heads. The resulting two-tier protocol executes fast, parallel model aggregation within clusters and infrequent inter-cluster exchanges among the heads, yielding an O(1/t) convergence rate in t iterations. Numerical experiments on image-classification tasks confirm that our approach accelerates convergence compared to state-of-the-art DFL baselines with only a few strategically placed backhaul links.

Redundancy in non-standard numeration systems is often associated with robustness, but its practical value in finite digital arithmetic depends on how representation, storage, and repair are defined. We study intrinsic redundancy in finite beta-expansion systems using a bounded-window model that separates semantic non-uniqueness from canonical codebook admissibility. The model distinguishes arithmetic canonicalization from corruption repair and evaluates structural detectability, value-preserving repair of the observed state, and survival of the original value. For the golden-ratio system and related multinacci bases, we prove that a genuine single-digit corruption in a canonically injective finite codebook cannot be semantically recovered by exact repair without external information. Semantic survival under exact structural repair is possible only for localized multi-digit perturbations corresponding to algebraic rewrite identities, such as the equivalence of 100 and 011. Experiments comparing standard binary, signed-digit non-adjacent form, and multinacci systems quantify trade-offs among codebook sparsity, fault visibility, canonicalization cost, residual error, and boundary loss. The results show that intrinsic beta-redundancy is a constrained-language resource for structural digital integrity rather than a substitute for classical error-control redundancy.

The hat guessing number $HG(G)$ of a graph is defined by the following game: each player is placed on a vertex and assigned a hat with one of $k$ colors. Each vertex can see only the hat color of the other vertices it is connected to in $G$. All vertices guess, simultaneously, the color of their own hat. The hat guessing number $HG(G)$ is the largest $k$ such that the players can guarantee that at least one of them guesses correctly. In this paper, we show a general bound on the hat guessing number of a graph $G$ as a function of its order $n$ and its maximum degree $\Delta$. This is the first nontrivial upper bound on $HG(G)$ as a function of $\Delta$ and $n$ when $\Delta \geq \frac{n}{e}$. From this result we also obtain that the hat guessing number of the random graph $G_{n,1/2}$ is at most asymptotically $cn$ for $c\sim 0.809$, and that graphs with maximum degrees of $ (1-\varepsilon )n$ for fixed $\varepsilon>0$ cannot have $HG(G)=(1-o(1))n$.

A subset $S$ of a group $G$ is said to be sum-free (resp. $\Delta$-free) if there are no solutions to $a+b=c$ (resp. $a+b+c=0$) with $a,b,c\in S$. For a convex region $R\subset\mathbb{R}^d$, let $\sigma(R)$ denote the maximal proportion of the volume of $R$ that a sum-free subset of $R$ can occupy. We prove that $\sigma([-1,1]^d)=1/2$. Our proof employs a careful application of the Brunn-Minkowski inequality. Moreover, for the $d$-dimensional Euclidean ball $\mathbb{B}^d(0,1)$, we show that $\sigma(\mathbb{B}^d(0,1))\leq 1/2+o_d(1)$. We present two arguments for this. The first combines some routine harmonic analysis on the sphere with known bounds on values of the ultraspherical polynomials. The second more elementary argument proceeds by establishing that the maximal $\Delta$-free subset of the unit sphere $\mathbb{S}^{d-1}$ occupies $1/2+O(d^{-1})$ of the sphere's surface measure. This answers a question raised by Bukh.

In this paper, we present solutions to the asymptotic Plateau problem in the hyperbolic space $\mathbb{H}^3$. In this context, we exhibit solutions for curves that are invariant under the action of a one-parameter subgroup of isometries of $\mathbb{H}^3$. To achieve this, we prove the existence of foliations of $\mathbb{H}^3$ by minimal surfaces that are properly embedded, complete, and invariant under these subgroups, which are then used to solve the problem.

Diffusion-based policies have recently emerged as powerful policy parameterizations for reinforcement learning, representing state-conditioned action distributions as terminal laws of diffusion processes with parameterized drifts. This terminal-law representation has shown substantial expressive flexibility in practice, enabling diffusion policies to model complex, multimodal, and highly non-Gaussian action distributions; however, it remains unclear what mathematically drives this expressivity and how to fully exploit it when the policy is learned from finite data. In this paper, we identify the drift Lipschitz budget $K$ as a central quantity governing the expressivity and statistical behavior of diffusion policies. We quantify expressivity through approximation: diffusion policies with $K$-Lipschitz drifts can concentrate near optimal deterministic policies and achieve value approximation error of order $1/K$; moreover, we prove a matching lower bound under nondegenerate diffusion noise. This increased expressivity comes with a statistical cost. When the drift is parameterized by neural networks, increasing $K$ improves approximation but increases statistical complexity. Balancing these two terms yields a finite-sample performance gap of order $\tilde{O}(n^{-2/(m+6)})$ for generic neural-network drifts, and a sharper rate $\tilde{O}(n^{-2/(m+4)})$ for one-sided dissipative drift classes, where $n$ is the sample size and $m$ is the dimension of the state space. Numerical experiments provide empirical evidence for the sample-dependent trade-off in $K$, supporting both theoretical regimes. Our framework also suggests a practical implementation principle: choose the diffusion budget $K$ according to the available sample size, and then select a neural-network architecture with the corresponding fixed Lipschitz coefficient.

We extend some results of weak KAM theory to Lagrangians that are defined only on the horizontal distribution of a subriemannian manifold and depend on the unknown function.

A few decades ago some possible remedies to an inaccurate enforcement of Neumann or Robin conditions prescribed on the boundary of a smooth domain, owing to its approximation by the union of straight-edged triangles or tetrahedra in a fitted mesh, were addressed. These studies, due to authors such as Barrett and Elliott (1988), advocated the use of isoparametric finite elements with a single curvilinear edge or face fitting the true boundary at two or three vertexes and also at additional points on those curves or curved surfaces, so as to define non polynomial piecewise solutions with the optimal order of approximation. In this work we adopt a different approach, whose main feature is the use of a fitted mesh consisting only of straight-edged elements, upon which the solution is a polynomial. Lost optimal orders of convergence, by virtue of the domain's approximation by a polytope, are recovered by means of the addition of terms to the bilinear form. These account for natural boundary conditions of the same type as the true ones, though to be prescribed on the approximating boundary instead. The new technique is applied to the case of triangular Lagrange finite elements, for which we give a rigorous reliability study in the solution of reaction-diffusion equations. Numerical experimentation is supplied in support of the theoretical results.

In this work, I analyze the zero modes of a one-loop semiclassical Yang-Mills theory in \(3+1\)d. I find that zero modes are generated by gauge redundancy of the background field. Proper gauge fixing, achieved by introducing a bosonic ghost term, together with zeta-function regularization, yields a finite one-loop effective action in closed form that reproduces the well-known one-loop Yang-Mills beta function.