Papers

We study third-order linear differential equations through their associated Pfaffian and projective foliations. In the regular-singular setting, we introduce a Fuchs-adapted Pfaffian model in which the indicial polynomial appears directly in the special-fiber singular scheme, and Frobenius resonance acquires a projective geometric interpretation.

Extreme events in chaotic systems are difficult to learn from short trajectories because they are controlled by transient finite-time instability rather than by frequently observed bulk dynamics. We propose a mechanism-aware conditioning plug-in framework that turns a nudged coarse ensemble into a non-intrusive sensor of local instability geometry. In the small-noise regime, the ensemble covariance aggregates the same finite-time deformation kernels that govern local instability, providing a Jacobian-free proxy for the local amplification structure around a synchronized coarse trajectory. A small FiLM module injects statistics of this ensemble geometry into an otherwise unchanged backbone while leaving the coarse simulator unchanged. We demonstrate this interface in two distinct pipelines: a Transformer-style residual-attention corrector for a controlled low-dimensional chaotic system and a probabilistic recurrent STORN corrector for topographic two-layer quasi-geostrophic (QG) flow. In the low-dimensional benchmark, ensemble covariance directions co-activate with OTD modes and FiLM conditioning improves 99th-percentile exceedance-frequency errors over an identical no-context Transformer baseline. In QG, a fixed ensemble-conditioned FiLM-STORN model trained on only \(50\) time units substantially improves long-horizon rare-event statistics in the data-limited regime, including density-tail errors, exceedance frequencies, and spatial exceedance-area distributions relative to an unconditioned STORN trained on the same data; on averaged high-threshold exceedance diagnostics, it also outperforms the baseline STORN trained with $20$ times more high-resolution data. These results show that local instability geometry is not merely interpretable post hoc, but an actionable conditioning signal for data-efficient rare-event emulation.

Combining $p$-values is a fundamental procedure in global hypothesis testing. In modern high-dimensional settings, however, component $p$-values often exhibit complex dependence and rely on asymptotic approximations rather than exact finite-sample uniform distributions. This paper establishes a unified asymptotic theory for weighted transformation statistics that decouples marginal finite-sample approximation error from the joint dependence structure. We also provide sufficient conditions based on conditional probability bounds to verify the joint-tail conditions. Utilizing this framework, we derive explicit dimension-growth and correlation rates for test statistics operating under asymptotic Gaussian and chi-square calibrations. For non-exact finite-sample statistics, we analyze standardized weighted sums, demonstrating how Cramér moderate deviations control relative tail error. Analytical examples demonstrate why both marginal and joint conditions are mathematically indispensable for valid global inference under dependence, and numerical experiments confirm that our asymptotic framework maintains accurate finite-sample size control at extreme significance levels.

We initiate the study of prophet trading, an online trading model in which a trader interacts with sellers and buyers who arrive in a uniformly random order and whose prices are drawn independently from a common known distribution. The trader aims to maximize the expected gain from trade (GFT), with performance measured against an omniscient prophet that knows the realized arrival order and all prices in advance. Unlike the classical prophet inequality problem, the trader must make both purchasing and selling decisions while managing the inventory, which makes both the prophet benchmark and the analysis of online algorithms substantially more intricate. For arbitrary numbers of buyers and sellers, we propose a simple threshold-based algorithm and prove a distribution-free competitive ratio of~2, which is the best possible. Our main technical contribution is an exact characterization of the inventory process induced by threshold trading. By expressing the algorithm's expected GFT in terms of the cumulative holding probability at buyer arrivals, we derive an exact formula for this inventory term, which serves as the foundation of our analysis and yields sharper guarantees in several important special cases. For balanced trading (with $n$ sellers and $n$ buyers), we improve the competitive ratio to $(2n^2-n)/((n+4^{-n}-1)(n+1))$, which is strictly less than 2 for every $n>1$. For the single-seller case, we sharpen the analysis of the fixed-threshold algorithm to obtain an asymptotic competitive ratio of $2e^2/(e^2+1)\approx1.76$. Furthermore, by exploiting the additional structure of this setting, we design an adaptive-threshold algorithm whose competitive ratio approaches $e/(e-1)\approx 1.58$. Due to a symmetry property of our general algorithmic idea, the same $1.76$-competitive guarantee also holds for the single-buyer case.

We prove that the energy functional associated with surfaces of prescribed mean curvature in $\mathbb{R}^3$ exhibits a convexity property when restricted to maps from the unit $2$-disk having small Dirichlet energy and a fixed boundary value, provided that the $3$-form $H \cdot \text{vol}_{\mathbb{R}^3}$ has a primitive satisfying certain bounds. Under milder assumptions on $H:\mathbb{R}^3 \to \mathbb{R}$, we show that an analogous convexity estimate holds along the heat flow of the functional (the $H$-surface flow) when the initial Dirichlet energy is sufficiently small. This is done first for classical solutions, and then extended by approximation to weak solutions using a quantitative uniqueness result adapted from previous work on the harmonic map heat flow. As a consequence of the convexity estimate, we show that the $H$-surface flow with small-energy initial map of class $C^0 \cap W^{1, 2}$ on the unit $2$-disk converges uniformly to a unique limit at infinite time, which solves the corresponding stationary problem (the $H$-surface system) with the same boundary value.

We consider stochastic nonlinear optimization problems with deterministic equality constraints. While unconstrained stochastic optimization is well understood, the interplay between optimality and feasibility in the constrained setting poses significant challenges. Moreover, existing theoretical guarantees for constrained stochastic methods predominantly rely on bounded-variance assumptions, leaving the heavy-tailed noise regime largely unexplored. To address this gap, we propose a novel trust-region method within the stochastic sequential quadratic programming framework, termed TR-SSQP. Our method employs a normal-tangential decomposition in the step computation to balance optimality and feasibility. In addition, we incorporate a normalization mechanism in the design of the trust-region radius, together with Polyak momentum for gradient estimation, ensuring stable updates without gradient clipping. When the trust-region radius and the momentum parameter decay at appropriate rates, we establish global almost-sure convergence of the method. To the best of our knowledge, this is the first asymptotic convergence result for constrained stochastic optimization under heavy-tailed noise. We demonstrate the promising performance of the proposed method through extensive numerical experiments, including comparisons among its variants and with existing constrained stochastic optimization methods.

We study null geodesics in Reissner-Nordström-de Sitter spacetime and in its $\omega=-2/3$ Kiselev deformation through the elliptic curves defined by the radial equation. The relation of these curves to the non-critical $E_7$ Seiberg-Witten family yields differential equations for finite-distance light deflection, and explains the boundary source term through a suitable gauge of the Seiberg-Witten differential. This geometry separates the extremal point $Q^2=M^2$ from the Argyres-Douglas point $Q^2=9M^2/8$. We also determine the Kiselev critical and horizon loci, and show that, at fixed nonzero angular momentum, the Argyres-Douglas and ultracold loci meet only at zero Killing energy.

We extend the Schatten-duality principle from orthonormal systems to (almost-orthogonal) Bessel families and apply it to prove sharp and improved Bessel-family Strichartz estimates for the Kadomtsev–Petviashvili, Zakharov–Kuznetsov, and radial Schrödinger equations. In particular, for the Schrödinger equation, we establish an improved estimate in a larger radial range of space-time exponents, not from dispersive estimates as in the KP/ZK models, but from direct robust Schatten estimates, using some vector-valued multi-product trace formula and multilinear weighted fractional integral formula. To the best of our knowledge, this is the first radial improvement for systems, and the almost-orthogonal estimates are also new to the literature. We also derive necessary conditions.

The integration of large-scale inverter-based resources (IBRs) into the grid presents two key challenges in terms of optimal power generation and transmission system stability. This work proposes concurrent transmission parameter learning and optimal power generation and current control for the grid-following IBRs. The proposed approach estimates transmission parameters online using the current measurements, enabling the IBRs to adapt their control actions to evolving grid conditions. The learned parameter estimates are incorporated into an optimization-based power distribution strategy to determine the optimal power setpoints. In parallel, a current control scheme is designed to regulate the grid-following (GFL) IBRs current output based on the estimates and received active and reactive power references from the high-level optimizer, thereby improving the dynamic response to the grid events. The effectiveness of the proposed framework is validated through a simulation in a MATLAB-Simulink environment. From the results, the impact of the concurrent estimator and current controller during the grid events can be seen.

This paper considers convex envelopes of disjunctive submodular functions—functions that are lattice family submodular over faces of a hypercube—and constructs the first strongly polynomial algorithm for their separation when there are two facial disjunctions. Submodular functions, whose convex envelopes are characterized by the Lovász extension, have occupied a fundamental role in constructing relaxations for combinatorial and nonlinear optimization problems. However, disjunctive submodular function envelopes have not been explored besides the use of ellipsoid algorithm, which remains practically intractable. Our algorithm is derived in three steps by expressing the disjunctive function as a minimum of two extended submodular functions, introducing a variable lifting technique, and constructing the sublinear envelope in the lifted space. The paper also makes several other contributions. First, we provide a disjunctive formulation for the case where each submodular function admits a linear programming formulation. Second, we derive the closed-form sublinear envelope characterization for intersecting submodular functions, yielding new structural insights into a multi-product inventory sales maximization problem. Third, we fully characterize the convex envelope of a bilinear function defined over a cycle graph in the original variable space. Finally, we show computationally that the cycle inequalities close approximately 60% of the gap for complete and Hadamard graphs, over 30% of the gap for complete bipartite graphs, and over 80% of the gap for sparse graphs such as cactus and Halin graphs. The resulting relaxations are also more efficient to solve than previous extended space formulations.

We study an elliptic optimal control problem governed by Darcy's equation in heterogeneous porous media, with pointwise box constraints on the control. The objective functional is formulated via a Radon measure, which allows the desired pressure state to be tracked on observation sets of varying dimension, including points, curves, and subdomains, within a single formulation. The state and adjoint equations are discretized by a symmetric interior penalty discontinuous Galerkin method, yielding a locally mass-conservative approximation that is robust across strong permeability discontinuities, while the control is approximated by piecewise constants. The state, adjoint, and control are retained as primary unknowns in a single coupled optimality system and solved monolithically by a primal-dual active set strategy. We establish stability and well-posedness of the discretization and derive a priori $L^2$ error estimates for both variables. The principal difficulty is the reduced regularity of the adjoint state induced by the measure-valued tracking data. The analysis controls the resulting adjoint-control coupling through an intermediate adjoint driven by the continuous optimal state. Numerical experiments confirm the predicted convergence rates for point, curve, and subdomain observations, exhibit mesh-independent primal-dual active set iteration counts, and demonstrate local mass conservation.

Using ideas from optimal transport theory, we introduce a notion of measured coarse equivalence for metric measure spaces and define a corresponding variant of the uniformly finite homology of Block and Weinberger, called weighted $\ell^\infty$ homology, for large-scale doubling metric measure spaces. We prove that this homology is invariant under measured coarse equivalence and that the vanishing of its zeroth homology is equivalent to weighted non-amenability. The proofs combine techniques from optimal transport theory and the disintegration of measures.

Let $X: M^{n}\to \mathbb{R}^{n+1}$ be a complete self-shrinker with constant squared norm of the second fundamental form $S$. In this paper, we prove that if $S\leq \frac{10}{7}$, then $S=1$ or $S=0$, and the self-shrinker is isometric to either a round sphere $\mathbb{S}^n(\sqrt{n})$ with the center at the origin, or a cylinder $\mathbb{S}^k(\sqrt{k})\times \mathbb{R}^{n-k},~~1\leq k\leq n-1$, or a plane through the origin.

We study convergence rates when periodic homogenization and vanishing viscosity occur at the same scale in Hamilton–Jacobi equations whose momentum Hessians are positive definite at every point. For bounded Lipschitz initial data and a class of smooth Hamiltonians, we establish an optimal rate $O(\varepsilon|\log\varepsilon|)$ on fixed time intervals. The second-order term plays a key role: the elliptic cell problem provides higher derivative bounds for the effective Hamiltonian $\bar{H}$ and its Legendre dual $\bar{L}$. The proof is purely PDE, based on corrector expansions and viscosity comparison, while its guiding idea comes from the control interpretation of the Hopf–Lax formula. For a fixed target $(x,t)$, a minimizing origin selects a characteristic velocity and its dual momentum, these guide the construction of smooth comparison profiles, whose gradients supply the momentum argument of the first-order corrector. Thus we never need to differentiate the effective solution, even at points where it is nonsmooth.

Ensuring operational safety in threat-prone environments remains a critical challenge for multi-UAV networks serving as aerial base stations. This paper proposes an efficient framework to maximize global energy efficiency (EE) while promoting safe operation through threat-aware clustering and reward-based safety enforcement. The proposed framework is executed in three steps. First, a threat-aware K-means (TAKM) algorithm determines the minimum required UAVs and computes safe initial placements. Second, an optimal matching stage assigns physical UAVs to these centroids to minimize energy expenditure. Third, a threat-aware multi-agent twin delayed deep deterministic policy gradient (MATD3) algorithm dynamically optimizes trajectories, power, and user associations. Simulation results show that the proposed framework achieves zero observed safety violations in the considered scenarios while achieving superior EE and faster convergence than other learning methods and non-clustering baselines. Compared to heuristic optimization, the proposed framework outperforms the greedy particle swarm optimization (GPSO) and achieves performance comparable to that of the optimized PSO (OPSO), while incurring significantly lower online deployment computational complexity. Furthermore, the proposed framework demonstrates effective generalization to unseen user distributions, large UAV fleets, and different threat geometries, while maintaining zero safety violations.

In 2018, Soloveychik, Xiang and Tarokh considered a pseudo-random symmetric circulant matrix constructed from binary Golomb sequences of length $n=2^m-1$, and claimed a proof that its empirical spectral distribution converges almost surely to the semicircle law as $n$ grows to infinity by the method of moments. In this comment note we show that their argument contains several technical flaws and inappropriate applications of technical lemmas. Instead we demonstrate that the eigenvalues of the matrix are simply a normalized twisted Kloosterman sum varying over the multiplicative character, to which we apply Katz's result directly to establish the asymptotically semicircle spectral distribution deterministically.

We investigate fully nonlinear water waves and fluid-particle dynamics generated by the horizontal motion of a seabed obstacle with prescribed time-dependent velocity. The governing equations are the full Euler equations, formulated in a time-dependent conformal domain that simultaneously maps the moving free surface and seabed onto fixed boundaries. The main contribution of this work is a Lagrangian formulation for computing particle trajectories in the resulting genuinely unsteady conformal domain. We derive a closed-form trajectory system in the canonical domain in which the time dependence of the conformal map is entirely represented by the real and imaginary parts of an analytic function that can be evaluated spectrally from the surface. We apply the formulation to waves generated by horizontal submarine landslide motion and characterize particle displacements throughout the fluid as functions of the initial particle position and Froude number. Our findings identify distinct regions in which particle motion is predominantly associated with the moving seabed, the generated wave, or the combined action of both. The results reveal a transition in the dominant mechanism driving particle motion: at low Froude numbers, particle displacements are primarily associated with the moving seabed, whereas at high Froude numbers the generated wave becomes increasingly dominant, particularly near the free surface and away from the obstacle path.The numerical predictions are benchmarked against laboratory data available in the literature, showing good agreement and providing a quantitative assessment of the model accuracy. Moreover, laboratory topographies beyond those considered here can be readily incorporated into the numerical framework through a Hermite interpolation procedure.

We prove that the product of three connected noncompact smooth manifolds without boundary admits a complete metric of uniformly positive scalar curvature. The proof uses the additivity of Morse indices on products and an orientation-free version of Das's open Morse–surgery construction. We also give an example of two open manifolds whose product admits no complete metric of nonnegative scalar curvature.

In this article, we study the reduction of the one-dimensional biharmonic nonlinear Schrödinger equation to the cubic nonlinear Schrödinger equation in the vanishing higher-order dispersion limit. In the intermediate regularity regime $0<s<2$, where the energy conservation law does not control the $H^s$–norm, we prove the long-time $L^2$–convergence of $H^s$–solutions with an exponential-in-time approximation bound. The argument relies mainly on persistence of regularity. This result provides a simple example of its use in limit problems without relying on higher-order conservation laws.

We proved a log-star comparison between the expectation threshold $q(\mathcal F)$ and the fractional expectation threshold $q_f(\mathcal F)$ for any nontrivial increasing family $\mathcal F$ on a finite ground set $V$ of size $|V|=N$. Specifically, we show that $$q_f(\mathcal F)\le64\log_2^*(N+2)\,q(\mathcal F),$$ where $\log_2^* x$ is the iterated logarithm of base-2. Using similiar methods, one can extend this result to the following stronger form: there exists a universal constant $C>0$ such that \[ q_f(\mathcal F)\le Cq(\mathcal F)\log_2^*(l(\mathcal F)),\quad q_f(\mathcal F)\le Cq(\mathcal F)\log_2^*(\operatorname{VC}(\min\mathcal F)), \] where $l(\mathcal F):=\max\{2,\max_{S\in\min\mathcal F}|S|\}$ and $\operatorname{VC}(\min\mathcal F)$ denotes the VC-dimension of $\min\mathcal F$. On October 7, 2026, OpenAI released a result proving the equivalence of fractional and integral thresholds, which supersedes the findings in this note. The authors are keeping this note for historical reference.

We prove that the support of every double Grothendieck polynomial is an $M^\natural$-convex set. Our main new tool is a rigidity result for the $K$-classes of multiprojective varieties with rational singularities.

We prove that, as the lattice spacing $h$ tends to zero, general solutions to the infinite Fermi–Pasta–Ulam–Tsingou (FPUT) system can be approximated in $L^2$ by two counter-propagating Korteweg–de Vries (KdV) waves on time intervals of order $\log(1/h)$. This resolves an open question raised by the first author and collaborators~\cite{HKY2021}. Our proof combines the FPUT conservation law with an $h$-uniform local well-posedness theory in $L^2$ and persistence of Sobolev regularity. We also introduce a frequency-localized auxiliary equation to overcome the difficulty of comparing the Fourier restriction norms associated with the FPUT and KdV flows.

When a uniform steady supersonic oncoming flow impinges on a straight wedge, if the wedge angle is smaller than the detachment angle, two types of steady oblique shocks satisfying the entropy condition will form in the flow field: weak shocks with supersonic or subsonic downstream flow, and strong shocks with subsonic downstream flow. There have been many results on the stability of oblique shocks with subsonic or supersonic downstream flow. However, much less is known about the stability of oblique shocks with sonic downstream flow, since the flow behind the shock wave is very sensitive to disturbances. This paper investigates the stability of oblique sonic shocks under the assumption of a straight wedge with non-uniform incoming flow. We reduce the problem to a degenerate hyperbolic free boundary problem. A local Lipschitz-continuous solution with sonic-supersonic downstream flow to the free boundary problem is constructed, using the classical Ascoli-Arzelà theorem and a diagonal argument. The main difficulty of this free boundary problem lies in the hyperbolic degeneracy on the sonic line. By adopting a weighted characteristic decomposition method, we establish a delicate estimate of the downstream flow near the sonic line.

For any $t\in\mathbb{R}$, consider the Affine twist map $\rm{Aff_t}:\mathbb{T}^2\to\mathbb{T}^2$ given by $$\rm{Aff_t}(x,y)=(x+y \text{ mod 1}, y+t \text{ mod 1}).$$ The map $\rm{Aff_t}$ clearly possesses an invariant foliation by horizontal curves. If $t$ is rational, each leaf is periodic, and if $t$ is irrational, the orbit of each leaf is dense in the torus. In both cases, the vertical rotation set of the proper lift of $\rm{Aff_t}$ to the vertical cylinder is reduced to $\{t\}.$ Now, for $k\in\mathbb{R},$ define $$f_{k,t}(x,y):=(x+y+k\sin (2\pi x)\text{ mod 1},y+k\sin (2\pi x)+t\text{ mod 1}).$$ We show that: 1) when $t=p/q$ for integers $p$ and $q>0$, KAM theory implies the existence of a constant $k_{p/q}>0$ such that for $|k|<k_{p/q}$, the vertical rotation set of an adequate lift of $f_{k,p/q}$ is just $\{p/q\}.$ 2) when $t$ is irrational, for any $k\neq 0$ the vertical rotation set of the adequate lift of $f_{k,t}$ is a non-degenerate interval which contains $t$ in its interior. In other words, it is not easy to build area-preserving twist maps whose vertical rotation sets are reduced to a single irrational number. From Theorem A of \cite{eujul}, such a map needs to have an invariant foliation by Lipschitz graphs over the horizontal coordinate. In particular, all its iterates must satisfy a twist condition. This is precisely what does not hold for $f_{k,t}$, for all non-zero values of $k$.

We prove a rigidity theorem for stationary and uniformly rotating solutions of the two-dimensional incompressible Euler equation with vorticity $ \omega_0\in C^1(\mathbb{R}^2)\cap L^1(\mathbb{R}^2) \cap L^\infty(\mathbb{R}^2). $ If the angular velocity $\Omega$ satisfies \[ \Omega\leq\frac12\inf_{\mathbb{R}^2}\omega_0 \qquad\text{or}\qquad \Omega\geq\frac12\sup_{\mathbb{R}^2}\omega_0, \] then $\omega_0$ is radially symmetric, in particular, every stationary vorticity of one sign in the class $C^1(\mathbb{R}^2)\cap L^1(\mathbb{R}^2) \cap L^\infty(\mathbb{R}^2)$ must be radially symmetric about some point. Furthermore, for $\Omega\neq0$, the center is necessarily the origin; The proof is based on the analysis for level-set geometry of a normalized stream function. A componentwise Bernoulli formula, together with the isoperimetric inequality and Pohozaev identities, yields a nonnegative defect measure encoding both geometric defects and topological branching. An analysis at infinity forces this measure to vanish, reducing the problem to a semilinear elliptic equation with a bounded nonnegative Borel nonlinearity. Radial symmetry then follows from a theorem of P.-L.\ Lions.

The Kahane–Salem–Zygmund inequality provides unimodular $m$-linear forms with small supremum norm. We consider its dimension-free symmetric constant $C_m^{\mathrm{sym}}$ over the real scalar field, defined by the requirement that, for every $n$, some real symmetric unimodular $m$-linear form $A:(\ell_\infty^n)^m\to\mathbb R$ satisfy \[ \|A\|\le C_m^{\mathrm{sym}}n^{(m+1)/2}. \] For complex scalars, Boas obtained under permutation symmetry an upper bound of order at most $\sqrt{m\log m}\,\sqrt{m!}$; over the real scalar field, an elementary argument gives the sharper order $\sqrt m\,\sqrt{m!}$. We prove \[ C_m^{\mathrm{sym}}\ge \left(\sqrt{\frac2e}+o(1)\right)\frac{\sqrt{m!}}m, \] using the square-free Walsh spectrum of the diagonal polynomial. In the opposite direction, we establish \[ C_m^{\mathrm{sym}}\le C_0\sqrt{m!}, \] where $C_0$ is absolute, removing the factor $\sqrt m$ from the real upper bound. This estimate follows from a geometric argument in which rigidity for Gram permanents reduces the relevant configurations to sets controlled by Gaussian width. For unrestricted unimodular forms, we also obtain a quantitative rectangular estimate from truncated Hadamard matrices; in equal dimension, the normalized minimum is at most $1+o(1)$ whenever $m=o(n^{5/6})$.

This paper studies constraint certification in synchronous distributed model predictive control (DMPC) when neighboring predictions change between sampling instants. Before the parallel local solves, each agent communicates a shifted prediction and an announced update budget. A hard trajectory trust region makes that budget enforceable, while an edge-wise feasibility cap computed from the shifted packets keeps the fallback feasible without using any current optimizer output. Distance and relative-attitude constraints are tightened with explicit Lipschitz constants and two budget layers: one accounts for the simultaneous neighbor update and the other retains a checkable shift reserve. We prove hard pairwise constraint satisfaction and recursive feasibility under stated nominal-execution and terminal assumptions, give the additional residual caused by execution error, and derive a local practical value-decrease bound. A spacecraft formation example uses hard terminal and pairwise constraints, a geodesic relative- attitude constraint on $\SO$, and reproducible terminal-set checks. Comparisons with fixed, trajectory-only, and windowed online margins show that the proposed budget reduces conservatism while preserving a positive shifted-feasibility margin.

This paper addresses long-range sensing in bistatic low-Earth-orbit integrated sensing and communication (LEO-ISAC). Conventional OFDM sensing fails in LEO-ISAC because millisecond-scale propagation delays destroy the transmitter–receiver symbol alignment and, with multiple targets, make the receive window longer than the transmitted block. We propose a window-grid (WG) processing framework that exploits the known beam-intersection swath to resolve both problems purely at the receiver, leaving the 3GPP symbol structure unmodified: the swath delay span is folded into cyclic-prefix-sized slices, each handled by one WG. With 103.2 MHz bandwidth at bistatic ranges beyond 640 km, simulations demonstrate meter-level ranging of two targets, and 1000 Monte-Carlo trials over random target–satellite geometries show that, with an ephemeris-based satellite-selection rule, every target is detected with a 90th-percentile range error of 2.4 m.

In this paper, we consider a random walk in a non-homogeneous random environment on two types of random trees and non-negative integers. The random trees here have progeny distributions that depend on the parent's generation. We provide a sharp threshold for the type of the random walk in a random environment on the random trees and the non-negative integers, up to the critical case. Then, we find the speed of the random walk on the non-negative integers and on random spherically symmetric trees. We use the method based on the relationship between the mean exit time from a vertex and speed. Meanwhile, we discuss several special cases that could be derived from those results. We also include many limiting formulas for sequences and series in the appendix, which are helpful for this study and others.

We introduce the categories $\CM(X)$ and $\SCM(X)$ of reflexive sheaves on a minimal partial resolution $f\colon X \to \Spec R$ of a rational surface singularity $\Spec R$. The main result of this paper establishes that $\SCM(X)$ possesses a natural Frobenius structure, serving as a geometric counterpart to the algebraic Frobenius structure on special Cohen–Macaulay $R$-modules, introduced by Iyama–Wemyss and Iyama–Kalck–Wemyss–Yang. Utilizing this geometric framework, we establish an exact equivalence between $\SCM(X)$ and the category of special Cohen–Macaulay $R$-modules equipped with a specific exact structure, which induces a triangle equivalence between their stable categories. Consequently, this provides a direct, geometric proof of the Iyama–Kalck–Wemyss–Yang equivalence and yields a Buchweitz-type equivalence $\underline{\SCM}(X) \simeq D_{\sg}(X)$.