Central Limit Theorems for integrals of third degree Hermite polynomials of Berry's random wave model on increasingly large domains are proved in dimensions 2 and 3. These were the missing cases for a complete description of limit theorems for integral functionals of monochromatic random waves based on the Wiener chaos decomposition.

We prove modified scattering for small data solutions to the Vlasov-Poisson system in a functional framework where the initial data, scattering states, and asymptotic convergence are measured in the same topology. In addition, we show that the corresponding wave operators define homeomorphisms between the spaces of initial and scattering data, while enjoying a local Lipschitz continuity property in weaker norms. As a consequence, in the repulsive case, large spherically symmetric solutions are asymptotically stable. The proof relies in particular on the introduction of a suitable system of dynamic coordinates adapted to the asymptotic nonlinear flow.

Given multiple data matrices, many problems in statistics and data science rely on estimating a common subspace that captures certain structure shared by all the data matrices. In this paper we investigate the statistical and computational limits for the common subspace model in which one observes a collection of symmetric low-rank matrices perturbed by noise, where each low-rank matrix shares the same common subspace. Our main results identify several regimes of the signal-to-noise ratio (SNR) such that estimation and inference are statistically or computationally optimal, and we refer to these regimes as weak SNR, moderate SNR, strong estimation SNR, and strong inference SNR. First, we propose an estimator based on projected gradient descent initialized via spectral sum of squares and show that it achieves the optimal $\sinΘ$ error rate under strong estimation SNR. These results are complemented by both statistical and computational lower bounds identifying the weak and moderate estimation SNR regimes. Next, we turn to statistical inference for the $\sinΘ$ distance itself, and we show that our estimator has an asymptotically Gaussian distribution in the strong inference SNR regime. Based on this limiting result we propose confidence intervals and show that they are adaptively minimax optimal in the strong inference SNR regime, where adaptivity is measured in terms of the SNR. Finally, we show that adaptive confidence intervals are information-theoretically impossible below the strong inference SNR regime. Consequently, our results unveil a novel phenomenon: despite the SNR being ``above'' the computational limit for estimation, adaptive statistical inference may still be information-theoretically impossible.

Let $S$ be the set of unit norm linear classifiers $θ\in \mathbb{R}^d$ which correctly classify every point of a labeled dataset $(X_i,y_i)_{i=1}^n$, $X_i \in \mathbb{R}^d$, $y_i \in \{-1,+1\}$, with a possibly negative margin $κ$ fixed in advance. Under two natural data-generating distributions of the $(X,y)$ pairs -- a Gaussian mixture model and a logistic model with Gaussian features -- and in the proportional regime $n/d \to α$ with small enough $α$, we establish a large deviation principle on the event that a point $θ$ chosen uniformly at random from $S$ achieves a given generalization error, with high probability over the choice of the data. The associated large deviation rate function is deterministic and describes the proportion, at the exponential scale in $d$, of interpolating classifiers having a given desired performance. As a consequence, we establish the following concentration phenomenon: all but an exponentially small fraction of interpolating classifiers have approximately the same generalization performance given by the unique maximizer of this rate function. We numerically compare this maximizer to the performance of empirical risk minimization by gradient descent and to the performance of a natural linear program, both finding a point in $S$, and deduce that in the overparametrized regime of small $α$, these efficient procedures outperform the vast majority of interpolators, pointing to their nontrivial benign overfitting in this setting.

We study the defocusing $k$-dispersion generalized Benjamin-Ono equation. For every even integer $k\geq 4$, we prove that solutions with initial data in the energy space $H^{\fracα{2}}$ are global in time and scatter. The proof combines the concentration-compactness-rigidity method of Kenig and Merle with techniques based on the Caffarelli-Silvestre extension and Tao's monotonicity formula adapted to the fractional dispersion setting.

High-rate quantum low-density parity-check (qLDPC) codes are a leading candidate for fault-tolerant quantum computing. They feature higher encoding rates than planar alternatives such as the surface code, but their implementation often entails significant hardware hurdles like the need for long-range couplers. We leverage the flexibility of a trapped-ion quantum computer to demonstrate nine quantum error-correcting codes with starkly different qubit connectivity requirements on a single device without any hardware reconfiguration. These experiments span three families of quantum error-correcting codes: qLDPC codes, topological codes, and concatenated codes. With a qLDPC code encoding 4 logical qubits into 18 physical qubits, we achieve a logical error rate up to $9\times$ better than a previous demonstration of a similar code on superconducting solid-state qubits. Moreover, our implementation exhibits breakeven performance, with some instances achieving qubit lifetimes comparable to or slightly exceeding that of our trapped-ion qubits. We use a novel implementation of the optical-metastable-ground (OMG) architecture for addressable mid-circuit measurement and reset, which enables us to perform these experiments without any ion transport or dedicated coolant ions, requirements that typically consume a large fraction of the runtime or ion count of trapped-ion quantum computers.

We study maximum matching problems in temporal graphs whose underlying graph is a tree. We consider two temporal models. In a $Δ$-matching, selected time edges sharing an endpoint must have time ticks differing by at least $Δ$. In a $γ$-matching, the selected objects are blocks of $γ$ consecutive appearances of the same underlying edge. We also consider the related ordered static problem of $d$-distance matchings. We show that maximum $Δ$-matching remains NP-hard on temporal trees for every $Δ\geq 2$, even in the sparse case where each edge appears at most twice. Using a reduction between the temporal models, we obtain the analogous result for maximum $γ$-matching on temporal trees, even when each edge admits at most two $γ$-edges. We also show, via a reduction from $d$-distance matching, that maximum $γ$-matching is APX-hard even when the underlying graph is bipartite. Complementing these hardness results, we identify several tractable cases. We prove that maximum $Δ$-matching is polynomial-time solvable on temporal trees in which every edge appears exactly once, and that maximum $γ$-matching is polynomial-time solvable when each edge admits at most one $γ$-edge. We also give dynamic-programming algorithms under bounded local-use and local-sparsity assumptions, and derive polynomial-time solvability of maximum $d$-distance matching when the input bipartite graph is a tree. Finally, we prove that both maximum $Δ$-matching and maximum $γ$-matching admit polynomial-time approximation schemes on temporal trees.

We study an inverse source problem for fully nonlinear elliptic equations of the form \[ F(D^2u)=f \quad \text{in } Ω. \] The question is whether the source term can be recovered from the Dirichlet-to-Neumann map. In two dimensions, the first linearization does not immediately give uniqueness: it leaves a natural conformal ambiguity in the linearized coefficients. For homogeneous nonlinearities $F$ with injective differential $DF$, we show that this ambiguity has a precise meaning at the level of the equation itself, namely that the source is determined up to an explicit scalar factor. The main point of the paper is to show how this remaining factor can be removed. We use the second linearization to extract information which is invisible at first order, and combine it with an algebraic nondegeneracy condition on the nonlinearity. Under this condition, the residual ambiguity is forced to be trivial, and the Dirichlet-to-Neumann map uniquely determines the source. The result applies, in particular, to homogeneous admissible Hessian equations of Monge--Ampère type and related examples.

We study an inverse problem for translation-invariant symmetric stable operators of the form \begin{equation*} L_a u(x)=\mathrm{P.V.}\int_{\mathbb R^n}(u(x)-u(y))\frac{a((x-y)/|x-y|)}{|x-y|^{n+2s}}\,dy, \quad 0<s<1, \end{equation*} where the unknown is the even angular density $a$ on $\mathbb Sn$. For a bounded open set $Ω\subset\mathbb R^n$, with $Ω_e=\mathbb R^n\setminus\overlineΩ$, we consider restricted exterior Dirichlet-to-Neumann maps $Λ_a^{W_1,W_2}$, where exterior data are supported in $W_1\SubsetΩ_e$ and the nonlocal Neumann data are observed on $W_2\SubsetΩ_e$. We prove three recovery results for the leading angular density. In the overlapping regime $W_1\cap W_2\ne\emptyset$, the exterior diagonal singularity determines every smooth elliptic angular density. In the separated regime $\overline W_1\cap\overline W_2=\emptyset$, where this singularity is absent, we prove uniqueness in the finite harmonic angular class by an exact factorization of the stable symbol. We also prove separated-data uniqueness for real-analytic angular densities when the source and observation sets lie in the unbounded exterior component, using analytic continuation of the off-diagonal Dirichlet-to-Neumann kernel and a far-field asymptotic argument.

The goal of this article is to show the categorical links between on the one hand the category of reductive Lie algebra pairs $\mathcal{RLP}$ and on the other hand the category of Lie-Yamaguti algebras $\mathcal{LY}$. The fact that the well-known construction of an enveloping algebra associating to a Lie-Yamaguti algebra a reductive Lie algebra pair is not functorial leads us to the main construction of the article, namely a left adjoint to the natural restriction functor $G:\mathcal{RLP}\to\mathcal{LY}$. As a final result we observe that the construction of the enveloping algebra becomes functorial when one restricts the morphisms of the categories $\mathcal{RLP}$ and $\mathcal{LY}$ to the surjective ones. Then it becomes a right adjoint to the restriction functor.

We present a novel approach to eigenvector universality for generalized Wigner matrices. Our main consequences are asymptotic normality of joint eigenvector projections everywhere in the spectrum as well as a quantitative lower bound on the largest entry of an eigenvector. In the case of smooth entries, we are able to obtain joint normality of an explicit growing number of eigenvector projections, and we are also able to obtain an explicit rate of convergence in Kolmogorov distance. This is based on a new analysis of the Dyson vector flow which does not rely on the eigenvector moment flow.

Lens spaces are useful test examples in spectral geometry because their spin Dirac eigenspaces admit explicit congruence descriptions. We use these descriptions to study equivariant $η$ invariants for three-dimensional lens spaces with the round metric and the standard coordinate-torus action, retaining the spin-Fourier character of each eigenspace rather than only the ordinary scalar $η$ value. For the square family $L(\ell^2,\ell-1)$ and $L(\ell^2,2\ell-1)$, with $\ell\geq 5$ odd, we obtain a residual-circle equivariant $η$ separation: the ordinary $η$ values agree, and the first derivative of the residual $η$ germ vanishes by symmetry, but the second derivative is nonzero. For $L(25,4)$ versus $L(25,9)$, the normalized second derivative is $-6080$. Thus, the residual-circle equivariant $η$ germ detects a distinction invisible to the ordinary $η$ invariant. The calculation uses spin-Fourier residues directly; perturbative Hessian signs serve only as motivation and are not part of the invariant.

Let $\mathcal{A}$ be a (not necessarily rational) conformal net with a faithful action of a finite group $G$. Let $\text{Rep}^G(\mathcal{A})$ be the $G$-crossed balanced $\mathrm{W}^*$-tensor category of $G$-twisted representations of $\mathcal{A}$ as introduced in arXiv:2606.03623. We show that there is an equivalence of balanced $\mathrm{W}^*$-tensor categories $(\text{Rep}^G(\mathcal{A}))^G\cong \text{Rep}(\mathcal{A}^G)$ between the $G$-equivariantization of $\text{Rep}^G(\mathcal{A})$ and the category of representations of the fixed-points conformal net $\mathcal{A}^G$. This generalizes to the non-rational case the equivalence of braided tensor categories $(\text{Rep}^G(\mathcal{A}))^G\cong \text{Rep}(\mathcal{A}^G)$ for $\mathcal{A}$ rational appearing (in the language of localized endomorphisms) in arXiv:math/0403322, and it also includes the balances.

Classical chance-constrained programs are solved by safe approximations based on the empirical CVaR, which uses a uniform measure over scenarios and systematically underweights tail events under heavy-tailed distributions. We introduce \emph{q-CCP}, a non-extensive safe approximation grounded in the Riemannian geometry of the Tsallis statistical manifold: the rank-based q-CVaR escort weights are the $g^{(q)}$-geodesic projection onto the tail simplex face, and the q-CCP feasible set is a Tsallis-divergence ball (Proposition~12). This geometric foundation yields three results. First, q-CCP is a provable strict tightening of CVaR-CCP for all $q > 1$ (Theorem~7). Second, the empirical violation ratio satisfies $ρ(q) = [1-(1-\varepsilon)^{q+1}]/\varepsilon$, independent of the tail index $ν$ (Proposition~10). Third, the feasible-region volume cost is monotone increasing in $q$ and $ν$ (Proposition~11), providing a data-adaptive safety knob. The formulation inherits convexity and coherence from the q-CVaR functional and admits an iterative LP reformulation converging in 2--3 iterations. Experiments on 15 Ibovespa equities confirm the theory (violation ratio $0.241$, $q^* = 1.50$); an M5 inventory newsvendor experiment generalises the method to supply chain ($q^* = 1.88$, cost premium $1.155\times$, zero OOS stockout violations).

The extension of the analytic fractal curve trees of (2601.17490} to analytic surface patch trees reveals a new geometric structure: branch points are replaced by interface curves that transmit the full analytical state of parent patches to their children. These interfaces prove to be central in determining the topology of the surface patch trees, including for the conditions for self-similarity of the interfaces, the patches and thus the trees. We establish the analytic conditions for the integrability and well-posedness of the surface patch trees and introduce further restrictions for conformality. We demonstrate that patch trees have a natural foliation that slices the trees into one dimensional curve trees, each of which has their own Hausdorff dimension, jointly creating a smooth dimension field. We extend the two dimensional surface model to arbitrary dimensions $n$ where $n-1$ interface manifolds transport the $n$ field state of the parent patches to their child branches. We note that the balance or discrepancy between patch field dimension and the dimensions in which the branches may evolve, determine the analytical regime from essentially geometrical to essentially operational.

We construct a periodic weighted graph whose discrete Laplacian has a spectrum with precisely $n$ gaps. Moreover, we show that by an appropriate choice of the weights, the endpoints of these gaps, as well as the upper edge of the spectrum, attain the prescribed values. The underlying graph has a brush-like geometry: it consists of an infinite chain of vertices, each of which is connected to $n$ additional pendant vertices by extra edges. Semi-explicit formulae for the weight coefficients are provided: some of the coefficients are determined explicitly, while others are given as roots of an explicitly determined polynomial.

The sub-Gaussian parameter (also called the variance proxy) of a mean-zero random variable $X$ is defined as $ξ^2_* = \sup_{λ\in \mathbb{R}} L(λ)$ where $L(λ) = \frac{2}{λ^2} \log \mathbb{E} e^{λX}$ is a weighted cumulant generating function. Despite the ubiquity of sub-Gaussian random variables, the estimation of $ξ^2_*$ has received little attention and is not yet well understood. In this work, we study a natural estimator of $ξ^2_*$ based on constrained maximization of the empirical analogue of $L$. We prove that the estimator is consistent bound the rates of convergence under assumptions on $L$: if $L$ has an maximizer, then our bound is $O_p(n^{-1/2 + \varepsilon})$ for any $\varepsilon > 0$; if the argmax of $L$ is also bounded, then the bound improves to $O_p(n^{-1/2})$. We show that our assumptions on $L$ are necessary by proving that the minimax risk over all sub-Gaussian distributions is $Ω(1)$; imposing increasingly strong assumptions on the tail growth of $L$ yields a continuum of classes whose minimax lower bound interpolates between $Ω(1/\log n)$ and $Ω(1)$. Root-n rate is possible if we restrict to a subclass of distributions where $L$ attains its supremum in a bounded region, in which case our estimator is minimax optimal. If the underlying distribution is not sub-Gaussian, we show that our estimator goes to infinity with a divergence rate controlled by the tail of the distribution. Finally, we apply our estimator in a Gene Ontology (GO) enrichment study to construct p-values for a large-scale permutation test, showing that it can serve as a reliable alternative to the peaks-over-threshold approach, particularly in regimes where the peaks-over-threshold method is of uncertain validity.

In the noncommutative polydisc, we first prove a positive sum of squares formula for a non-negative hereditary rational nc-function. The number of summands is finite. This result is used to derive a finite-dimensional realization formula for contractive nc-rational functions, where the colligation matrix is contractive. It is unitary if and only if the function is inner. Finally, we apply these results to generalize Carathéodory's classical theorem - approximating holomorphic self-maps of the unit disc by finite Blaschke products - to the setting of holomorphic functions on the noncommutative polydisc. This is in sharp contrast with the commutative situation where Carathéodory's approximation is known for Schur classes only in the unit disc and the unit bidisc.

Order theory is increasingly relevant in applications where data is naturally structured as a partially ordered set (poset), often requiring meaningful notions of distance over posets. In this paper, we introduce a new family of extended metrics on path-connected and fence-connected posets that do not require additional structure. Unlike many existing distances, these metrics are not induced by valuations, but instead arise as a type of shortest-path distance determined by both path length and the number of alternations. For discrete posets, we show that these metrics converge to a type of shortest-fence metric. Our main result establishes that these metrics characterize most discrete path-connected posets up to isomorphism, and up to duality for modular posets. Finally, we prove that this family defines interleaving distances when posets are viewed as thin categories.

Modern semiparametric estimation often relies on flexible black-box machine learning methods to estimate nuisance functions, raising a fundamental question: how do nuisance estimation errors propagate into inference for low-dimensional target parameters? The dominant paradigm, exemplified by double machine learning (DML), yields error bounds in which nuisance estimation errors enter multiplicatively. While widely adopted, it remains unclear whether this multiplicative-rate dependence is optimal for black-box models. In this paper, we start by revisiting the partial linear model $Y = μ_0(X)+T\cdotβ_0+\varepsilon$ under a structure-agnostic setting, where the nuisance function $μ_0$ is estimated using a generic machine learning model, with approximation error $δ^a_μ$ and stochastic error $δ_μ^s$. We show that the standard DML rate is not optimal in the regime where the auxiliary function $\mathbb{E}[T|X=x]$ cannot be consistently estimated. We propose a new estimator for $β_0$ that achieves a sharper rate of $n^{-1/2}+δ^a_μ+(δ_μ^s)^2$ and establish a matching lower bound demonstrating its optimality. Our results reveal a new principle: the first-order stochastic error of nuisance estimation can be eliminated without imposing any additional assumptions. This also leads to a revised tuning strategy favoring under-smoothing, where $δ^a_μ\asymp(δ_μ^s)^2$, rather than the classical bias-variance trade-off $δ^a_μ\asymp δ_μ^s$. Under mild additional conditions, the estimator is asymptotically normal with minimal asymptotic variance. The proposed method extends to a broad class of semi-parametric linear functional estimation problems, including average treatment effect estimation. Our results imply that popular orthogonal score methods in semiparametric estimation with black-box nuisance learners can be substantially improved.

In this paper, we study the structure of double cyclic codes of length $(γ,δ)$ over $\mathbb F_q+u\mathbb F_q, u^2=0$. We also study the dual of double cyclic code of length $(γ,δ)$ and give a minimal spanning set of double cyclic codes. Moreover, we study the necessary and sufficient conditions for a double cyclic code to be reversible and reversible-complement double cyclic code and with the help of these codes, we constructed DNA codes over $\mathbb F_4+u\mathbb F_4, u^2=0$. We also constructed some optimal codes to support our results.

We construct an explicit equivariant quantum Satake map for Grassmannians, which enables us to express their torus-equivariant quantum cohomology in terms of that of projective space. We then consider the exterior algebra of the latter, which admits a canonical identification with a Clifford algebra. We describe the resulting action in several complementary ways: first, from a geometric perspective via push-pull maps, and second, in terms of the shuffle product, which also arises in the simplest cohomological Hall algebra associated with the $A_1$-quiver. Exploiting the Clifford algebra structure, we derive new recurrence relations among equivariant Gromov-Witten invariants, yielding a new method for their computation in terms of Wick's Theorem. As an application, we provide combinatorial proofs of Graham positivity for both equivariant quantum Pieri rules, and in one case extend these results to quantum triple Schubert calculus.

This paper develops a unified probabilistic framework based on distributional derivatives and Dirac delta representations for the analysis of conditional and transportation-related quantities. General identities are established for arbitrary random variables, encompassing absolutely continuous, discrete, and mixed distributions. The proposed approach yields unified formulas for conditional expectations, conditional distributions, hazard functions, and improper distributions, revealing a common localization mechanism underlying these classical concepts. The framework is further combined with copula methods to investigate transportation and dispersion functionals through dependence structures. Exploiting the extremal properties of the Fréchet--Hoeffding bounds together with expectation inequalities induced by $Δ$-antitonic functions, sharp bounds are derived for absolute difference moments under fixed marginals. These results lead to concise derivations of quantile representations for the Wasserstein distance and a corresponding upper transportation functional, as well as survival-function representations and bounds for generalized absolute difference moments. As a particular case, new representations are obtained for the bivariate Gini mean difference and the associated bivariate Gini index. Applications are given to Wasserstein-type functionals arising in the normal approximation of standardized counting distributions, including Poisson, Binomial, and Negative Binomial models, for which explicit quantile representations are derived. Overall, the results establish explicit links among conditional structures, dependence modeling, dispersion measures, normal approximation, and optimal transport, providing a unified perspective on several fundamental constructions in probability and mathematical statistics.

We study deconfined quantum critical points (DQCP) associated with Lieb-Schultz-Mattis (LSM) anomalies in one-dimensional spin chains. Our starting point is a structural characterization of the LSM anomaly in the Lyndon-Hochschild-Serre spectral sequence: $ω_{\mathrm{LSM}}\in E_\infty^{1,2}= H^1(\mathbb Z_{\mathrm{trans}},H^2(G_{\mathrm{int}},\mathrm{U}(1)))\subseteq H^3(G_{\mathrm{int}}\rtimes_ρ\mathbb Z_{\mathrm{trans}},\mathrm{U}(1))$. Physically, this class decorates a translation defect with a projective representation of the internal symmetry $G_\mathrm{int}$. We show that gauging the internal symmetry in the presence of an $E_\infty^{1,2}$-type anomaly necessarily produces a non-invertible dual symmetry. This gives a general mechanism for type-II DQCP: in contrast to type-I examples with $E_\infty^{2,1}$-type anomalies which are dual to ordinary group-like symmetry breaking, type-II transitions are dual to spontaneous breaking of a non-invertible symmetry. We illustrate the mechanism using a spin-$1/2$ chain with an anomalous $D_8$ LSM symmetry. We construct a dimer-to-ferromagnet DQCP candidate, provide numerical evidence for a critical theory with central charge $c\approx 1$, and show, using both category theory and explicit lattice constructions, that gauging the internal symmetry yields the non-invertible $\mathrm{Rep}(H_8)$ dual symmetry.

Asymptotic e-values are emerging as a powerful alternative to asymptotic p-values, particularly in post-hoc inference and multiple testing, where significance levels may be data-dependent. Existing asymptotic e-values, however, suffer from the ``missing factor,'' a scaling inefficiency resulting in overly conservative inference. Drawing on the framework of near-optimal concentration inequalities developed by Bentkus in the 2000s, we introduce Bentkus-type asymptotic e-values and prove that they successfully eliminate the missing factor. We also demonstrate both theoretically and empirically that Bentkus-type e-values consistently deliver sharper inference than existing alternatives, leading to tighter post-hoc confidence intervals and higher rejection rates in multiple testing procedures.

We determine asymptotic results for the average size of Selmer groups arising from certain isogenies related to Jacobians of hyperelliptic curves of genus $g\geq 2$. We do so by combining Bhargava's geometry-of-numbers methods with new parametrisations coming from Vinberg theory, arising from representations related to the Dynkin diagrams of type $B$ and $C$. We additionally prove some lower bounds on the average size of these isogeny Selmer groups by using a formula of Greenberg--Wiles.

Time-dependent high-dimensional partial differential equations (PDEs) with spatially localised and dynamically evolving solutions pose a fundamental challenge for physics-informed neural networks (PINNs), as uniform collocation sampling becomes increasingly ineffective in high-dimensional spatiotemporal domains. In this work, a deep adaptive sampling framework for PINNs is extended to the time-dependent setting by treating space and time as a unified domain without any explicit time marching. A normalising flow neural network model effectively learns the distribution induced by the PDE residual and generates new collocation points concentrated in regions where the solution is most difficult to learn. Unlike conventional adaptive strategies that require explicit time stepping or moving meshes, high-residual regions are automatically identified and tracked across both space and time, driven purely by the PDE residual distribution. The effectiveness of the proposed strategy is assessed on a range of benchmark problems, from sharp and moving features in two spatial dimensions to localised structures in up to eight spatial dimensions.

We introduce a new algorithm for computing persistent homology of Vietoris--Rips filtrations, which in many cases offers a considerable speedup over the existing implementation of the persistence pairing algorithm. The key innovation, called active enumeration, is made possible by a new theoretical framework of Reduction to Zero Differentials (hence RedZeD) in which to view persistent homology.

This paper develops a stochastic-geometry framework for high-altitude platform station (HAPs) networks in which platforms execute cyclic patrol trajectories anchored to designated service regions. We introduce two small-circle ring Cox process models on the spherical Earth. In the small-circle ring Poisson Cox process (SCR-PCP), platforms form one-dimensional Poisson point processes on localized patrol rings, whereas in the small-circle ring binomial Cox process (SCR-BCP), each ring contains a fixed number of uniformly distributed platforms. We establish the isotropy of both models and derive spatial statistics, including the distributions of the nearest-anchor, nearest-ring, and nearest-HAPs distances, together with the joint serving distance and serving ring angle distribution required for SCR-BCP analysis. Building on these results, we derive coverage probability expressions under nearest-HAPs association by decomposing aggregate interference into same-ring and other-ring components and characterizing their conditional Laplace transforms. To account for the flight dynamics of patrol-based HAPs, we integrate a steady circular flight propulsion model with the communication analysis and introduce a coverage energy efficiency (CEE) metric. This yields an analytical condition for the energy-optimal patrol radius that balances coverage performance against the propulsion cost of circular flight. Numerical results reveal fundamental differences between intensity-driven (SCR-PCP) and finite-fleet (SCR-BCP) deployments and demonstrate that patrol geometry, platform density, and cruising velocity should be jointly optimized to achieve energy-efficient HAPs operation.

A \emph{$k$-cactus} generalizes forests and cacti by allowing each edge to lie on at most $k$ cycles. The maximum number of edges is classical for forests and cacti, but for $k$-cacti was known only for $k\le 4$. In this note we treat general $k$. The key idea is that bounding the cycles through each edge forces a $k$-cactus to exclude a large complete minor; in particular, the class of $k$-cacti is minor-closed. From this we prove that every $n$-vertex $k$-cactus has $O\!\left(\frac{\log k}{\sqrt{\log\log k}}\,n\right)$ edges for all sufficiently large $k$, and a construction shows this is optimal up to a factor of $\sqrt{\log\log k}$.