Papers

Let $G_1$ denote the incidence graph of the complete graph $K_{q+1}$. We study limited augmented Zarankiewicz numbers in this family through structured 0–1 optimization models for admissible 2-edge augmentation. For the original framework, we combine exact ILP formulations for the smallest instances with constructive search followed by exact admissibility verification for larger instances. This yields \[ {z_L(6,4)=14,\qquad z_L(10,5)=26,\qquad z_L(15,6)\ge 43,\qquad z_L(21,7)\ge 64,\qquad z_L(28,8)\ge 88.} \] The first two values are exact, whereas the latter three are rigorous lower bounds obtained from explicitly verified admissible families. We then formulate the global weak problem determined by \((S),(W2),(W2'),(W3)\). On the larger incidence-family cases, the corresponding global exact optimization problem is currently computationally out of reach, and our computations provide the certified lower bounds \[ {z_{WL}(6,4)=14,\quad z_{WL}(10,5)\ge 29,\quad z_{WL}(15,6)\ge 46,\quad z_{WL}(21,7)\ge 65,\quad z_{WL}(28,8)\ge 90.} \] On the fourteen literal-grid benchmarks from $(4,3)$ through $(7,7)$, by contrast, the global weak problem is solved exactly. Across this exact benchmark block, the weak framework is never worse than the strong framework and is strictly better in several cases. These computations improve the corresponding classical Zarankiewicz numbers and therefore strengthen available lower bounds for $\operatorname{BSR}(m,n)$ within this family.

For a conditional expectation operator $T$ on a Dedekind complete Riesz space, we give representations of the $T$-strong duals of $L^1(T)$ and $L^\infty(T)$. The representation for the $T$-strong dual of $L^1(T)$ follows from the known result for $L^2(T)$. To describe the $T$-strong dual of $L^\infty(T)$, we introduce charges on components of weak order units and develop a corresponding integration theory.

In federated language modeling, $K$ nodes each hold $n$ samples but cannot pool data or exchange full-precision gradients or weights. We study the minimax rate at which a conditional distribution over $V$ tokens can be estimated when each node may upload at most $B$ bits per query in a public probe set. In federated probe-logit distillation (FPLD), each node transmits a scalar-quantized logit vector on the probe set, and an aggregator distills a global parametric student. Prior work (Dubey and Huo, 2026) establishes a high-probability KL rate $O(d/(Kn) + \rho\sqrt{V \log V / m} + K^{-1} \cdot 2^{-2B/V})$ plus optimization slack, with the bandwidth term in its trace-sharpened form. Whether this bandwidth-term rate is tight, and how the upper bound generalizes to heterogeneous per-node bandwidths, are left open. We close both gaps. First, the dithered FPLD construction has a matching single-round lower bound $\Omega(K^{-1} \cdot 2^{-2B/V})$ under non-degeneracy, pinning the bandwidth-axis rate at $\Theta(K^{-1} \cdot 2^{-2B/V})$. $T$-round sequential refinement with nested/scaled residual quantizers achieves $O(K^{-1} \cdot 2^{-2TB/V})$; vanilla FPLD's $T$-independent bandwidth term is suboptimal for every $T > 1$. Second, we establish a heterogeneous-bandwidth upper bound for per-node budgets $B_i$, paired with a closed-form optimal allocation $B_i^* = B_{\mathrm{tot}}/K + (V/2) \log_2(w_i / \bar{w}_g)$, a log-tilted water-filling rule that is the per-node analogue of reverse water-filling for distortion-rate optimization. A plug-in adaptive variant estimates the weights from a short warm-up phase and attains $1 + O(\sqrt{\log(K/\delta)/(m T_0)})$ relative suboptimality. Synthetic n-gram simulations confirm that empirical KL is bracketed by the upper and lower bounds and that the optimal allocation strictly dominates uniform and inverse-weighted baselines under heterogeneous clipping.

In data centers, tasks are dispatched to various servers to evenly distribute the workload. When a data center considers implementing a new scheduling algorithm, it typically conducts an A/B test prior to deployment to assess the real-world impact of this new method. However, a straightforward A/B test might be interfered with so-called “Markovian” interference. We utilized the Differences-in-Q estimator, as developed by Farias et al. (2022), and introduced mixed Differences-in-Q estimators grounded in Little's Law. We show that our A/B testing methods significantly reduce bias and variance when testing various scheduling policies. Extensive simulations were conducted under scenarios like non-stationary arrival rates, heterogeneous service rates, and communication delays. These simulations highlight the robustness and efficacy of our A/B testing approach.

In this paper, we study a structured class of nonconvex constrained stochastic problems with difference-of-convex (DC) regularization, where the feasible set is possibly nonconvex and the concave part of the DC regularizer is allowed to be nonsmooth. The fundamental challenge lies in maintaining feasibility for nonconvex constraints while achieving favorable oracle complexity. Although single-loop algorithms efficiently solve unconstrained DC optimization problems, their potential for constrained optimization with DC structure remains largely unexplored. To address this gap, we develop MoSSP, a Momentum-based Single-loop Stochastic Penalty method for such problems with provable complexity guarantees. The key idea is to apply a single stochastic proximal-gradient step to the Moreau envelope of the penalty plus the convex DC part, with the concave part's proximal mapping computed in parallel. We derive two algorithm variants: a Polyak-momentum version with $O(\varepsilon^{-4})$ oracle complexity for finding stochastic $\varepsilon$-KKT points, and an improved $O(\varepsilon^{-3})$ version incorporating recursive momentum. Experimental results demonstrate the effectiveness of the proposed algorithms.

We derive the local and central limit theorems for the Stirling numbers of the second kind by elementary means, obtaining as corollaries effective asymptotic estimates for the Bell numbers and for the moments of the distribution. We also develop asymptotic expansions along several directions, all based on a novel finite-differencing approach; this provides the first self-contained elementary justification of such expansions.

This paper concerns the barotropic compressible Navier-Stokes equations in a two-dimensional half-space subject to Navier-slip boundary conditions with vacuum or non-vacuum far-field density. The global existence and large-time behavior of weak and strong solutions are established under the assumption that the bulk viscosity coefficient is sufficiently large. It should be remarked that this result is obtained without any restrictions on the size of the initial data. For strong solutions, we derive some a priori decay estimates for the spatial gradient of the velocity field that are uniform with respect to the bulk viscosity coefficient, which play a crucial role in establishing the time-uniform upper bound for the density. Furthermore, we prove that, as the bulk viscosity coefficient tends to infinity, the solutions of the compressible Navier-Stokes equations converge to those of the inhomogeneous incompressible Navier-Stokes equations. In particular, the incompressible limit for weak solutions holds without requiring the initial velocity to be divergence-free.

Pinching-antenna systems (PASS) have emerged as a promising flexible-antenna architecture capable of dynamically reconfiguring wireless channels by activating dielectric particles along waveguides. The sum rate maximization problem in multi-waveguide PASS is investigated in this study. Both in-waveguide propagation loss and coupling effects are explicitly modeled. To tackle the optimization problem, a hierarchical user scheduling (HUS) algorithm is proposed. The HUS algorithm minimizes the sum of squared distances between users and their associated waveguides to mitigate path loss. Additionally, spatially separated users are assigned within each time slot to reduce inter-user interference. Furthermore, a joint optimization framework integrating power allocation and pinching-antenna (PA) positioning is developed to further improve system sum rate. Specifically, PAs' positions are optimized via one-dimensional search, while the power allocation problem is solved by using the Lagrangian duality and fractional programming. Numerical results show that the HUS algorithm clearly outperforms random pairing, and the proposed power allocation algorithm shows a marked performance improvement over the maximum ratio transmission algorithm. Moreover, the results explicitly demonstrate the considerable impact of in-waveguide propagation loss and coupling effects on the performance of PASS.

In this paper, we study plane quintic curves whose automorphism groups have order greater than 10, as well as those with cyclic automorphism groups of order 8 and 10. The latter two cases are represented as one-parameter families, where their superspeciality can be explicitly described in terms of a truncation of certain Gaussian hypergeometric series. Applying this characterization, we determine the exact number of isomorphism classes of superspecial plane quintic curves with automorphism groups $\cong \mathbb{Z}/10\mathbb{Z}$. We also provide an efficient algorithm to enumerate such curves with automorphism groups $\cong \mathbb{Z}/8\mathbb{Z}$, and provide the computational results for the range $13 < p < 10000$.

In this work, we establish the existence of mass-conserving weak solutions to a nonlinear collision-induced breakage equation in which binary collisions may trigger particle breakup. The result is proved for a class of product-type collision kernels whose small-size behavior is controlled by a power-law function of the form $\omega_0(x)\le A_1\,x^\ell$, while no growth restriction is imposed on the large-size factor $\omega_\infty$. The qualitative behavior of the solutions depends crucially on the exponent $\ell$ near the origin. Sublinear growth corresponding to $\ell<\tfrac12$ yields existence only on finite time intervals, whereas superlinear growth corresponding to $\ell>\tfrac12$ ensures global-in-time existence.

The Wolff dynamics is a non-local Markov chain widely used for simulating the Ising model due to its effectiveness in reducing critical slowing down compared to the Glauber dynamics. Despite extensive algorithmic and numerical studies, a rigorous probabilistic understanding remains limited. In this paper, we take a first step toward addressing this gap. For the one-dimensional (1D) Ising model, we first derive the transition probabilities of the Wolff dynamics and show that, at the critical point, it converges to the two fully aligned configurations and subsequently oscillates between them. This behavior is absent in the Glauber dynamics. Second, we establish a log-Sobolev inequality with an explicit constant for the Wolff dynamics in the entire subcritical regime and derive quantitative bounds on its ergodic averages. As a by-product, at infinite temperature, the obtained constant coincides with the classical log-Sobolev constant of the random walk on the hypercube. Finally, we apply these results to analyze the spectrum of the sample covariance matrix generated by the Wolff dynamics, which was used by Chen et al. to study condensation of eigen microstate. We prove that the spectral behavior agrees with their simulations in the 1D Ising model, thereby providing theoretical support for their findings.

We present a high-order implicit-explicit discontinuous Galerkin (IMEX-DG) solver for the compressible Euler equations to account for rotational effects within a fully compressible atmospheric framework. Time integration follows a second-order additive Runge-Kutta scheme, treating stiff acoustic modes implicitly and advective terms explicitly. The solver is built on the deal.II finite element library, combining matrix-free operator evaluation, adaptive non-conforming meshes capabilities, and distributed-memory parallelism. Two alternative treatments of the rotational and gravitational source terms within the solution strategy, based on nonlinear fixed-point iterations, are introduced and compared in terms of accuracy, robustness, and computational efficiency. A discrete analysis of the rotational operator is also carried out in order to derive a formulation suitable for efficient matrix-free implementation and to avoid inconsistent naive discretisations. The proposed formulation is validated through convergence studies on rotating inertia-gravity wave benchmarks and further assessed in fully three-dimensional simulations of stratified flow over orography on both uniform and adaptive meshes. The numerical results show that the rotating IMEX-DG framework has the expected accuracy and stability properties while correctly capturing the asymmetry and wave structures induced by rotation in large-scale atmospheric flows.

We study a determinantal Coulomb gas in the complex plane associated with the external potential $$ Q(z)=\frac{1}{1-\tau^2}\big(|z|^2-\tau \text{Re } z^2\big)-2c\log|z-a|, $$ where $\tau\in[0,1)$, $c\ge0$, and $a\ge0$. In the regimes where the associated droplet is simply or doubly connected, we derive the free energy expansion up to and including the constant term, with all coefficients computed explicitly, thereby extending recent results in the isotropic case $\tau=0$. In particular, we identify the constant term with the Liouville action associated with the droplet. Our result admits a natural interpretation in terms of asymptotic expansions of moments of characteristic polynomials for the elliptic Ginibre ensemble. The proof is based on a deformation framework involving both the singularity location $a$ and the anisotropy parameter $\tau$, relating variations of the free energy to refined asymptotics of planar orthogonal polynomials. The asymptotic analysis relies on the foliation flow method of Hedenmalm and Wennman, providing an alternative to the Riemann–Hilbert approach used in the isotropic setting. The present work suggests a general framework connecting free energy expansions, refined asymptotics of planar orthogonal polynomials, and conformally invariant geometric functionals, with several intermediate results already formulated for general algebraic Hele-Shaw potentials.

The author and Kawakami revealed that the Picard little theorem, the Carathéodory–Montel theorem and the Fujimoto theorem – phenomena concerning omitted values in value distribution theory, normal family theory and the theory of Gauss maps of minimal surfaces, respectively – are not isolated results but can be discussed within a unified framework. We call this theoretical framework the Bloch–Ros principle. Furthermore, the value-distribution-theoretic properties of Gauss maps hold not only for minimal surfaces but also for other classes of surfaces that admit singularities, such as maxfaces in the Lorentz–Minkowski $3$-space and improper affine fronts in the affine $3$-space, thereby further extending this theoretical framework. In this paper, we provide a unified approach to phenomena concerning totally ramified values among value distribution theory, normal family theory and the theory of Gauss maps of these classes of surfaces.

We give a randomized algorithm that samples a nearly uniform Eulerian tour of a directed Eulerian multigraph with $m$ arcs in $\widetilde O(m^{3/2})$ time. The guarantee is worst-case, applies to arbitrary directed Eulerian multigraphs, and breaks the $mn$-type arborescence-sampling barrier on sparse graphs. The core case is a $2$-in/$2$-out graph. We introduce a new local Markov chain, the flip–repair walk: one step locally splits a tour into two circuits and then chooses uniformly among the local flips that repair the state to one tour. We prove that this walk mixes in nearly linear many steps and implement the walk using a dynamic chord data structure. A pointwise degree-reduction wrapper extends the sampler from this degree-two core to arbitrary degrees while preserving the $\widetilde O(m^{3/2})$ total running time. The high-level algorithmic plan, the switching-network reduction, and the dynamic data-structure argument were devised by the author. The author conjectured the mixing theorem underlying the analysis, and GPT 5.5 Pro Extended produced its linear-algebra proof. Codex assisted with manuscript assembly and typesetting.

We derive the first-moment Shallow Water Exner Moment model with sediment entrainment and deposition (SWEMED1) and show that the full source term has a fully-settled water-at-rest equilibrium manifold. We prove that the model is only weakly hyperbolic at this equilibrium, which prevents the use of Yong's structural stability framework. However, a linear spectral analysis and numerical results do not indicate instability. Based on numerical results, we introduce a fast-slow scaling of the source term, and for the fast limit, we derive a new suspended water-at-rest equilibrium manifold, which has a different structure but is still only weakly hyperbolic. Our results show that the remaining obstruction is linked to the transport closure of the SWEMED1, and we give a constructive direction for the derivation of new closures leading to models with more desirable analytical properties.

We determine the sharp threshold for Hamilton cycles in randomly perturbed sparse graphs. For any $\alpha=\alpha(n)=o(1)$, let $G_{\alpha}$ be an $n$-vertex graph with minimum degree $\delta(G_{\alpha})\ge\alpha n$. We prove that if $$p\ge(1+\varepsilon)\frac{\log(1/\alpha)}{n},$$ then the union $G_{\alpha}\cup G(n,p)$ is Hamiltonian asymptotically almost surely. This significantly strengthens a recent result of Hahn-Klimroth, Maesaka, Mogge, Mohr, and Parczyk by improving the leading constant from 6 to the optimal value of 1. Crucially, we show that this bound on $p$ is best possible when $\alpha n\rightarrow\infty$, thereby establishing the exact probability threshold for Hamiltonicity in this sparse regime. Our proof relies on a robust random expansion lemma, Pósa's booster lemma, and a sprinkling argument.

Deep learning optimization relies heavily on the assumption of smooth loss landscapes, a condition systematically violated by modern architectures due to non-smooth components such as ReLU activations and quantization operators. In such non-smooth regimes, adaptive optimizers such as Adam suffer from gradient chattering, violent oscillations caused by conflicting signals within the Clarke subdifferential, leading to poor convergence and suboptimal generalization. To address this, we introduce Singularity-aware Adam (S-Adam), a novel optimizer that stabilizes training by dynamically modulating step sizes based on local geometric instability. Our key contribution is the Local Geometric Instability (LGI) metric, a computationally efficient estimator of the Clarke subdifferential diameter derived from the variance of randomized directional derivatives. S-Adam incorporates an adaptive damping mechanism exp(-$\lambda$$\rho$) that decelerates updates in high-instability regions while preserving fast convergence in smooth basins. We provide a rigorous convergence analysis using differential inclusions, proving that S-Adam converges almost surely to ($\delta$,$\epsilon$)-Clarke stationary points at the optimal O(1/$\sqrt(T)$) rate. Empirical evaluations on Quantization-Aware Training (QAT) and high-noise small-batch learning demonstrate that S-Adam consistently outperforms AdamW and Prox-SGD, achieving accuracy gains of up to +4.54% on CIFAR-100 and +4.27% on TinyImageNet while effectively mitigating gradient oscillations.

Let $(R,\frak m)$ be a generalized Cohen-Macaulay local ring of prime characteristic $p$. In this paper we give a sharp bound for the Frobenius test exponent of parameter ideals. Namely, we prove that $$\mathrm{Fte}(R) \le \lceil \log_p(2n_0)\rceil + \mathrm{HSL}(R),$$ where $n_0$ is the integer such that $\frak m^{n_0} \, H^i_{\frak m}(R) = 0$ for all $i < \mathrm{dim}(R)$, and $\lceil x\rceil$ is the smallest integer that is greater than or equal to $x$.

We study combinatorial structures arising from finite-time transition probabilities of the Totally Asymmetric Simple Exclusion Process with open boundary conditions. While much of the existing combinatorial theory regarding the TASEP concerns the steady-state distribution, we focus instead on the transient dynamics. We first show that the enumeration of transition sequences between two configurations of the open TASEP is equivalent to the enumeration of standard Young tableaux of a family of non-classical shapes which have been of recent interest in the combinatorial literature. This extends to the open-boundary setting the correspondence between the TASEP with periodic boundaries and cylindric tableaux. We then introduce a family of tableau-like objects associated with Young diagrams in which repetitions of cells are allowed, subject to the partial order induced by the diagram. For each diagram, we collect the numbers of these objects into an exponential generating function. We prove that the entries of the homogeneous open TASEP transition matrix can be expressed as signed sums of such generating functions over suitable families of diagrams. This gives a combinatorial and order-theoretic interpretation of finite-time transition probabilities for the open TASEP, analogous to the combinatorial mappings known for steady-state probabilities.

Estimating excursion set confidence regions seeks to identify regions where a function may exceed some threshold with a given confidence level. This paper focuses on estimating such confidence regions in cases where the function has random inputs and a functional output that is returned all at once. We develop a surrogate-based approach for estimating the confidence region, combining principal component analysis and Gaussian process regression. An active learning strategy is also introduced, based on a max-min criterion that selects new samples which are likely to reduce the uncertainty in the confidence region. This strategy leverages efficient sampling of the Gaussian process through a Karhunen-Loève expansion. The proposed approach is applied to estimate the confidence regions of three case studies: a synthetic function, the surface pressure coefficient distribution of a hypersonic vehicle, and the glide-back trajectory of a reusable launcher first stage. The method demonstrates efficiency in accurately estimating the confidence region while reducing sources of modeling uncertainties. It is benchmarked against reference methods from the literature. Relevant metrics for assessing the confidence region estimation performance are discussed.

We study the birational geometry of hypersurfaces in projective varieties of the form $\mathbf{P}^1\times Z$, where $Z$ satisfies mild assumptions. Building on recent results of Herrera–Laface–Ugaglia, we study their Cox rings (when finitely generated) in terms of the Cox ring of $Z$. In particular, we obtain a complete picture when $Z$ is a Fano variety of dimension at least 3, and with class group of rank 1.

We develop an interacting extension of the Double Covariance Model (DCM), a stochastic subquantum framework in which macroscopic quantum dynamics emerge through coarse-graining of correlated microscopic fluctuations. Starting from local stochastic differential equations on subsystem Hilbert spaces, we derive a closed evolution equation for a coarse-grained double covariance operator using multi-scale Itô calculus and sliding-window averaging. The construction explicitly incorporates two separated temporal scales: a fast microscopic fluctuation scale governing subquantum stochastic processes and a slower macroscopic observation scale associated with coarse-grained dynamics. Within the hydrodynamic limit, where the ratio between microscopic correlation time and averaging-window scale vanishes, rapidly fluctuating corrections disappear and the effective dynamics converges to a deterministic macroscopic transport equation. We show that the emergent macroscopic dynamics has the exact Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) form: coherent Hamiltonian evolution arises from deterministic subquantum flow, while dissipative channels emerge from quadratic noise correlations. The framework further demonstrates how non-separable interaction Hamiltonians can arise from strictly local, state-dependent stochastic feedback fields. In the fluctuation-free limit, the model reduces naturally to the standard von Neumann equation, providing a unified stochastic foundation for both open and closed quantum dynamics.

It is well known that band inversion across a straight interface in a periodic medium gives rise to interface modes that are localized near the interface and propagate along it inside the bulk spectral gap. This phenomenon constitutes the key mechanism underlying the valley-Hall effect. In this paper, we address the long-standing problem of the robustness of such interface modes. We prove that, when the interface is bent through an angle of $\frac{2\pi}{3}$, the interface modes persist for every frequency in the bulk spectral gap where the group velocity is non-vanishing, except for a finite exceptional set. We also show that corner-localized modes, if they occur, can appear only at these exceptional frequencies and have finite multiplicity. To the best of our knowledge, this is the first rigorous mathematical theory of the bending immunity of valley-Hall interface modes.

We study the existence of area-minimizing homotopies between homotopic curves in the plane. While the classical Plateau problem establishes the existence of least-area surfaces spanning a single Jordan curve, the corresponding existence theory for homotopies between curves is more subtle and is not directly covered by the same framework. Existing results in the plane are mainly based on combinatorial and algebraic methods, such as decomposing curves into self-overlapping subcurves. These methods are highly effective in the planar setting, but they are often tied to special classes of curves and rely strongly on the local structure of the self-intersections, frequently assuming transverse crossings. In contrast, our approach is geometric and variational, and does not depend on the local structure of the self-intersections. In this paper, we develop a variational existence theory for minimum-area homotopies of immersed planar curves. Our approach adapts classical minimal surface methods by lifting an immersed planar curve with self-intersections into higher co-dimension, where it becomes embedded. For such a lifted curve, we apply Douglas's solution of the Plateau problem to obtain an area-minimizing disk. For closed curves of class $C^1$, we prove uniform convergence of the Douglas minimizers and show that the limiting map minimizes area among all $C^1$ spanning maps of the original planar curve. We then extend the construction to closed Lipschitz curves using approximation and Sobolev compactness arguments. Since the limiting minimizing disk lies in the original plane, it directly produces a null homotopy whose swept area is minimal among all admissible homotopies of the original curve. In this way, the construction connects Plateau theory with minimal homotopy area minimization.

We modify the approach to the arithmetical form of the large sieve by relying on the Parseval identity rather than on an approximate Bessel inequality and as a consequence, improve on the weighted large sieve inequality beyond what was heuristically accessible by the earlier approach. We also discuss the optimality of this approach.

This paper investigates fractional Riesz-Bessel equations with random initial conditions that exhibit either classical or cyclic long-range dependence. It studies zoom-in asymptotics for the corresponding solutions and establishes multiscaling limit theorems. It is known that for similar problems, non-degenerate multiscaling limits may not exist in general. The paper develops a kernel-smoothing approach for these equations and obtains non-degenerate limit fields under suitable normalisation and rescaling. It proves that the kernel-smoothed solutions converge weakly to Gaussian random fields, which are non-stationary in both time and space. Their spectral representations and covariance functions are derived. The paper also analyses the regularity and dependence structure of the limit fields. In particular, under appropriate general assumptions on the smoothing kernel, the limits exhibit long-range dependence in time and short-range dependence in space. Numerical examples for the case of Matérn-type kernels are provided to illustrate the theoretical results.

Given a group $G$, the model $\mathcal{G}(G,p)$ denotes the probability space of all Cayley graphs of $G$ where each element of $G$ is included in the generating set independently at random with probability $p$. In this article, we investigate the threshold probabilities for the diameter of random graphs in this model. Specifically, let $d_N = (1-\gamma)\sqrt{\frac{\log{N}}{2\log{\log{N}}}}$, where $\gamma \in (0,1)$ is any fixed real number. We show that for any $\varepsilon > 0$, any family of groups $G_k$ of order $N_k$ for which $N_k \to \infty$, and any integer $2 \leqslant d\leqslant d_{N_k}$, a graph $\Gamma_k \in \mathcal{G}(G_k,p)$ with high probability has diameter at most $d$ if $p \geqslant \sqrt[d]{(1+\varepsilon) d! \frac{\log{N_k}}{N_k^{d-1}}}$, and diameter greater than $d$ if $p \leqslant \sqrt[d]{\frac{1-\varepsilon}{2^d} \frac{\log{N_k}}{N_k^{d-1}}}$. Up to a constant factor, these thresholds are similar to those for the usual Erdős-Rényi random graphs. However, the precise thresholds in our model depend on the underlying family of groups. We provide specific examples of group families demonstrating that both of our bounds are best possible.

Lloyd algorithm is the standard iterative method for computing quantizers and codebooks in source coding and vector quantization. In this article, we study the dynamical and stability properties of the Lloyd map on the unit circle $\mathbb S^1$ using von Mises distributions. We construct the Lloyd iteration as a discrete dynamical system on the configuration space of ordered point sets modulo rotational symmetry. Also, we study the rotational equivarience of the Lloyd map. Further, we derive an explicit representation of the Jacobian matrix and prove that it possesses a circulant structure for the equally spaced configuration. Also, we study the bifurcation characteristics based on Lloyd map analysis. In the end, we provide the numerical algorithms for stability diagrams, Lyapunov spectrum estimation, and residue analysis, purely for empirical visualization. Our results provide a dynamical systems framework for Lloyd quantization on $\mathbb S^1$ for studying stability properties.

Neural scaling laws appraise data through dataset size, while the Vendi Score uses quantum entropy to measure dataset value. We show both that common neural-scaling-law objectives and the Vendi Score are submodular. We further show that the Vendi Score is a special case of a broader class of submodular objectives that we call matrix spectral functions. This also includes determinantal (DPP) objectives, as well as many others. We also introduce weakly matrix monotone functions and show how they lead to weakly submodular matrix spectral functions, yielding a broad family of practical objectives for data appraisal. We develop secular-equation-based updates that avoid repeated eigendecompositions during greedy optimization, reducing marginal-gain evaluation for $m$-dimensional embeddings by an $O(m)$ factor relative to oracle queries. This yields an average empirical speedup of about 35,000x, making direct optimization of the Vendi Score feasible on ImageNet-1K-scale datasets. Thus enabled, we compare how well several objectives predict the value of training subsets for held-out test performance under fixed-size, class-balanced, and fixed training-budget regimes, including the Vendi Score, DPPs, facility location, and three new matrix spectral variants. Across multiple datasets, facility location performs the best. Direct optimization also reveals that, while the Vendi Score is predictive over moderate score ranges, pushing the objective to higher values can make it a poor downstream performance proxy. We also find that uniformly at random fixed-size subsets, both unconstrained and class-balanced, are remarkably concentrated in both appraisal scores and held-out performance. Finally, we show that size, class balance, and training budget do not alone determine data value: even when controlling for these factors, performance ranges smoothly from good to bad.