Papers

We characterize the oracle complexity of finding first-order stationary points in smooth nonconvex–strongly-concave (NC–SC) minimax optimization when each query provides the full Hessian in addition to the function value and gradient. For $\Phi(x)=\max_y f(x,y)$, assume that $f$ has an $L$-Lipschitz gradient, a $\rho$-Lipschitz Hessian, and is $\mu$-strongly concave in $y$. Let $\kappa=L/\mu$, let $\Delta$ denote the initial value gap, and define $r=\rho\kappa^{3/2}\varepsilon/L^2$. We establish matching deterministic upper bounds and randomized lower bounds, up to logarithmic factors and additive initialization costs. When the leading term dominates initialization, the resulting minimax complexity is \[ \widetilde{\Theta}\!\left( \frac{\Delta L}{\varepsilon^2}\Psi_\kappa(r) \right), \qquad \Psi_\kappa(r)= \begin{cases} r^{1/2}, & 0<r\le 1,\\ r^{2/7}, & 1<r<\kappa^{7/4},\\ \sqrt{\kappa}, & r\ge\kappa^{7/4}. \end{cases} \] Despite the stronger oracle, the optimal complexity exhibits a nontrivial dependence on accuracy, conditioning, and Hessian smoothness. In particular, these regimes exhibit accuracy dependencies $\varepsilon^{-3/2}$, $\varepsilon^{-12/7}$, and $\varepsilon^{-2}$, respectively. The upper bounds combine second-order regularity estimates for a saddle Moreau envelope with certified accelerated reduced-dual solves and amortized warm starts. The lower bounds combine an all-derivative lift with a serial composition of private-handle constructions, using hard instances chosen independently of the algorithm's realized random seed. The resulting characterization quantifies improvements over the classical first-order benchmark under gradient smoothness alone and identifies the regime in which that benchmark is recovered.

Digital Twin (DT) combined with reinforcement learning (RL) can support adaptive resource allocation and security-aware decision-making by comparing expected and observed network behavior. This paper presents a simulation-based DT reference and a belief-guided Q-learning (BGQL) for joint jammer identification and beam-power control in a site-specific 28-GHz network. For each action, the same channel simulator is evaluated under an attack-free and an attacked configuration. Receiver-level residuals are combined with spatial jamming footprints and accumulated to form a normalized suspicion vector. BGQL incorporates this vector into an aggregated state, a belief-concentration reward, and exploratory beam-selection rule. The BGQL scheme is compared with SARSA, Greedy, and Random under matched attacker sets. Across 30 paired Monte Carlo runs in four configurations, with the number of active jammers supplied to every method, BGQL achieves mean aggregate full-band link-rate scores of approximately 437.4–564.2~Mbit/s and rate-to-device-power scores of 60.37–76.79~Mbit/J. Its detection accuracy is 100% in one-jammer configurations and approximately 78%–81% in two-jammer configurations. BGQL requires approximately 80–130~s per run, compared with 5–14~s for SARSA, reflecting evaluation of 625 beam pairs during exploratory steps. These findings provide a matched-model simulation proof of concept for DT-referenced belief-guided learning for security-aware beam-power control in 6G networks.

The rapid growth of Artificial Intelligence (AI) data centers is introducing dynamic loads into power transmission systems as GPU clusters switch between computational phases with distinct power demands. These dynamics are naturally modeled by switched affine differential-algebraic equations (DAEs), where each mode induces a different equilibrium and consistency space. We study their stability using control and hybrid systems tools. Differential projectors derived from the quasi-Weierstrass form yield an equivalent hybrid representation, from which we characterize the $\omega$-limit set and establish Lyapunov conditions for practical stability with an explicit $\mathcal{KL}$ bound accounting for switching-induced resets. For stochastic inference workloads, we use Foster–Lyapunov techniques to establish probabilistic uniform global recurrence under mode-dependent dwell times and transition probabilities. The framework is validated on the IEEE 39-bus transmission network with co-located AI data centers using high-fidelity electromagnetic transient simulations of deterministic training and stochastic inference workloads.

Let $M$ be a connected Hamiltonian stationary Lagrangian surface of constant Gaussian curvature $K$ in a two-dimensional complex space form $\Mtilde^2(4\epsilon)$ of constant holomorphic sectional curvature $4\epsilon$, and let $H$ denote its mean curvature vector field. We prove that either $K=\epsilon$ or $K=0\ne\epsilon$ and $|H|$ is constant. In particular, if $|H|$ is non-constant, then $K=\epsilon$.

We propose a subgradient-based block coordinate descent (BCD-SubG) framework for nonconvex optimization over products of probability simplices and a closed feasible set, motivated by semi-nonnegative independent component analysis (Semi-NNICA). Elementary column transformations are used to preserve simplex feasibility, while first-order information guides block and pair selection. We establish the global convergence of BCD-SubG using a constrained Lojasiewicz inequality; additional eventual-interiority conditions yield stationary limits and convergence rates. We then specialize BCD-SubG to the direct-fit and indirect-fit Semi-NNICA models and derive explicit updates that ensure global convergence without further assumptions. For both models, eventual interiority also yields stationary limits and convergence-rate estimates. Experiments on synthetic mixtures and simulated magnetic resonance spectroscopy data show generally lower recovery errors and shorter wall-clock times than the tested Semi-NNICA baselines. Joint diagonalization experiments further demonstrate lower final objective values.

We study Galois coinvariants of the geometric étale fundamental group and étale cohomology of varieties over a finitely generated field $k$. We establish finiteness properties of these coinvariants for (possibly infinite) Galois extensions $L/k$ such that $\operatorname{Gal}(L/k)^{\mathrm{ab}}$ has finite exponent. As an application, we obtain a finiteness result for étale fundamental groups of arithmetic schemes over number fields.

We study finite involutive multivalued groups associated with Schur rings. Their reduced valency is shown to equal the least common multiple of the sizes of the basic sets of the corresponding ring. We prove that the double-coset group associated with a complemented subgroup of a finite group is isomorphic to a Schur multivalued group over the complement. In particular, every coset multivalued group is a Schur multivalued group. Using affine Schur rings, we construct families of pairwise nonisomorphic commutative multivalued groups over elementary abelian groups. We obtain asymptotic bounds for the sizes of these families and prove that the proportion of double-coset groups in them tends to zero.

Bergeron, Garsia, Haiman, and Tesler conjectured that $(-1)^{n-\ell(\mu)}\nabla\widetilde H_\mu[X;0,t]$ and $\nabla\omega \widetilde H_\mu[X;0,t^{-1}]$ are Schur positive, and that $\nabla\omega \widetilde H_\mu[X;0,t^{-1}]-\nabla\omega \widetilde H_\nu[X;0,t^{-1}]$ is Schur positive when $\mu$ dominates $\nu$. The first statement follows from the compositional shuffle conjecture of Haglund, Morse, and Zabrocki, proved by Carlsson and Mellit. We prove the other two statements. All three statements also hold for $\nabla^r$ with $r\geq1$. The main new step is an identity: when $\mu$ covers $\nu$, the difference $\omega \widetilde H_\mu[X;0,t^{-1}]-\omega \widetilde H_\nu[X;0,t^{-1}]$ is minus a power of $t$ times one LLT polynomial. Using the Catalanimals and the Cauchy identity of Blasiak, Haiman, Morse, Pun, and Seelinger, we show that $\nabla^r$ of this LLT polynomial is, up to sign, a positive sum of LLT polynomials. We also extend the main results to the Schiffmann algebra setting.

We improve the Plesken–Souvignier algorithm for computing automorphism groups and isometries of positive definite integer quadratic lattices by exploiting characteristic sublattices and refining vector enumeration. For lattices with roots, we also use the Weyl group to accelerate the computation of canonical forms. Experiments with concrete implementations demonstrate substantial speedups and show that canonical forms can be computed at a cost comparable to that of computing automorphism groups.

The classical Hurwitz bound gives a linear upper bound for automorphism groups of algebraic function fields in characteristic zero, whereas algebraic function fields in positive characteristic can have much larger automorphism groups. In this article, we firstly investigate solvable subgroups and $p$-subgroups of $\operatorname{Aut}(F/\mathbb{F}_q)$ where $p=\operatorname{char}(\mathbb{F}_q)$ and $F/\mathbb{F}_q$ is a function field of genus $g_F\ge2$ with full constant field $\mathbb{F}_q$. We prove that there are explicit constants $C_q^{(\mathrm{sol})}$ and $C_q^{(p)}$, depending only on $q$, such that, for every solvable subgroup $G\le\operatorname{Aut}(F/\mathbb{F}_q)$ and every $p$-subgroup $P\le\operatorname{Aut}(F/\mathbb{F}_q)$, \[ |G|\le C_q^{(\mathrm{sol})}(g_F-1), \qquad |P|\le C_q^{(p)}(g_F-1). \] In particular, one may take $C_q^{(\mathrm{sol})}=12q^2(q-1)$ for $q\ge4$ and $C_q^{(p)}=5q$ for every $q$. In the second part of this article , we give asymptotic lower bounds for solvable automorphism groups. For every $q$ there is an explicit constant $L_q>0$ and a sequence of function fields $F_i/\mathbb{F}_q$ with $g_{F_i}\to\infty$, together with solvable subgroups $G_i\le\operatorname{Aut}(F_i/\mathbb{F}_q)$, such that \[ |G_i|\ge L_q(g_{F_i}-1). \] For $q\ge5$ one may take $L_q=2q(q-1)/(q-2)$. If $q=r^2$ with $r\ge3$, an unramified tower construction over the Hermitian curve gives the stronger choice $L_q=2r^3(r-1)/(r-2)$. Moreover, as an application of our tower construction above, we improve the asymptotic lower bound of Ma and Xing \cite{MaXing2019} for subgroups of $\operatorname{Aut}(F/\mathbb{F}_q)$ whose order is relatively prime to $p$. As another application, we extend our method for bounding solvable automorphism groups to $p$-solvable subgroups, obtaining explicit linear bounds in terms of the genus.

We study two sharp competition thresholds in the Lotka-Volterra competition-diffusion system: the zero-speed threshold for bistable traveling fronts and the linear-determinacy threshold for monostable traveling fronts. Building on our previous results on propagation direction and the classification of traveling waves, we derive explicit Gaussian integral representations of both critical parameters, without involving unknown traveling-wave profiles or speeds. These formulas provide exact characterizations of the optimal competition parameters separating the corresponding propagation regimes. The derivation of these representations was assisted by ChatGPT (OpenAI) Pro, while the underlying theoretical foundations were established in our earlier works.

We study endpoint Schatten class properties of commutators $[I_\theta^\epsilon,M_x]$ on quantum tori of dimension $d\geq2$, where $I_\theta^\epsilon$ is the Riesz potential of order $\epsilon$ and $M_x$ denotes the left multiplication operator by an operator $x$ on quantum tori $\mathbb T_\theta^d$. For $-\frac{d}{2}<\epsilon<1$, $\epsilon\neq0$, and $x\in L_1(\mathbb T_\theta^d)$, we prove that $[I_\theta^\epsilon,M_x]\in \mathcal L_{\frac{d}{1-\epsilon},\infty}$ if and only if $x\in\dot W_{\frac{d}{1-\epsilon}}^1(\mathbb T_\theta^d)$. We also determine the singular value asymptotics of this commutator for $x\in\dot W_{\frac{d}{1-\epsilon}}^1(\mathbb T_\theta^d)$. The necessary Sobolev condition holds throughout $1-d<\epsilon<1$, $\epsilon\neq0$. For $\epsilon\leq-\frac{d}{2}$, we characterise Hilbert–Schmidt membership, with a logarithmically weighted Fourier condition at $\epsilon=-\frac{d}{2}$. In order to get the singular value asymptotics of commutators, we prove singular value asymptotics for pseudodifferential operators of arbitrary negative order on quantum tori using $C^*$-algebraic principal symbols. This asymptotic result confirms Conjecture~7.1 of McDonald and Ponge (Adv. Math. 412 (2023), 108815).

Generalizing the setting of the second author's work on indecomposable modules without nonzero hollow factors, we characterize quivers $Q$ for which the dual algebra $R$ of the path coalgebra of $Q$, equivalently the complete path algebra, has no nonzero uniform right ideals: this happens exactly when every vertex of $Q$ can reach a vertex emitting at least two distinct arrows. Equivalently, the injective hull $E(R)$ of the right regular module is superdecomposable. These conditions are also equivalent to the path coalgebra having no nonzero hollow factor modules as a left module over its dual algebra.

Let $f$ be a homeomorphism of a closed orientable surface of genus at least one that is isotopic to a pseudo-Anosov homeomorphism, possibly relative to a finite invariant set. We construct an intrinsic monotone factor by collapsing the maximal continua whose iterates have uniformly bounded diameter in the universal cover. The quotient is homeomorphic to the original surface and the factor is isotopic to $f$ and has several strongly chaotic dynamical features. It is infinitely continuum-wise expansive, tight, and topologically mixing, and it has a dense family of rotational horseshoes, dense periodic points, and a single topological homoclinic class. The factor can be taken as an area-preserving homeomorphism. The quotient preserves stable translation length on the fine curve graph and the applicable homological rotation sets. When $f$ is $C^\infty$, a companion paper gives a unique measure of maximal entropy for the factor; it has full support and is Bernoulli.

We study one-dimensional marginals of an isotropic, log-concave probability measure in $\mathbb{R}^n$. We show that the Kolmogorov distance between a typical marginal and the standard Gaussian distribution is at most $C/n$, for a universal constant $C > 0$. This bound is optimal in general, up to the value of the universal constant. It improves the Gaussian approximation bound of Bobkov, Chistyakov and Götze by a logarithmic factor in the log-concave setting.

We present a statically and dynamically arbitrage-free European option market model with smooth interpolation between option prices, based on the SANOS framework, and prove that its dynamics admit a unique strong solution. In spirit, it is a stochastic model for local volatility when the latter is used as a representation of a European option price surface, not as a pricing model. We illustrate the strength of our approach on S&P Index Option data from January 2022 to August 2025. We show that the model's initial state fits within bid/ask for more than 95% of relevant market option prices. We then train the model dynamics under $\mathbb{P}$ and $\mathbb{Q}$ and show that the model is indeed free of arbitrage while fitting the dynamics of the market well. We also propose a range of BHW-tests to test for absence of arbitrage.

The canonical component of the character variety of a hyperbolic 2-bridge link complement M is an algebraic surface, from the work of Thurston. If we fix a boundary component T of M and perform a Dehn-filling on T, we get a 1-cusped manifold whose character variety is commonly a generalized curve that is naturally embedded in the character variety of M. If we perform distinct Dehn-fillings on M, we get various generalized curves on the canonical surface. In this paper, we analyze how these curves intersect with each other in the surface and study the topological implications for these intersection points. We also classify the intersection points that correspond to reducible characters, using the twisted Alexander polynomials associated to the Dehn-fillings. Throughout the paper, we use the Whitehead link as a motivating example to illustrate our core methodology, while some generalizations to other 2-bridge links will also be discussed.

We prove the existence of minimal models for smooth co-rank one foliations of general type.

Membranes are sufficiently smooth vector-valued functions defined on the unit square, widely used in applications such as interpolation of image data. The signature of a membrane is a collection of nonlinear features that capture its essential geometry. We index these features in the graded Hopf algebra of two-parameter shuffle products and study the algebraic varieties for each graded component. At each truncation level, we obtain the dimension of this variety via a novel Witt-type formula. We propose a Chen-Chow theorem for membranes, provably at the second truncation level, and conjecturally for higher levels with extensive computational evidence in the computer algebra system OSCAR.

We consider newly designed matrix algebras associated with Cauchy orthogonal transforms. In this work, we are interested in checking the approximation of Toeplitz matrices using algebras of matrices, where each matrix is diagonalized by a fixed Cauchy orthogonal transform, in the spirit of fast algorithm based on preconditioning. We use the Korovkin theory for checking the Toeplitz approximability. Unfortunately and somehow surprisingly, the Korovkin test is negative, in contrast with the successful test for all the Fourier-like and trigonometric algebras, widely used in the relevant literature. More in detail, for the Cauchy algebras, the Frobenius optimal approximation fails to lead to the desired clustering, as also confirmed by ad hoc numerical experiments. It remains open to understand which kind of structured matrices can be approximated by these new algebras.

Three-dimensional conjugate heat-transfer simulations provide detailed predictions for microchannel heat sinks but can be computationally demanding when many geometries and operating conditions are evaluated. This study develops a versatile two-layer model (VTLM) that reduces the three-dimensional problem to two coupled two-dimensional fields while retaining the in-plane geometry of the flow passages and fins. Effects associated with the unresolved channel-height direction are incorporated through hydraulic and thermal closures that account for wall friction and the fluid- and fin-side contributions to interlayer heat transfer. The closures are evaluated from the representative channel geometry, flow-development condition, and fin configuration. The VTLM was assessed against three-dimensional reference models across broad ranges of channel aspect ratios and operating conditions, including uniform and nonuniform fin arrangements. Compared with the existing two-layer models, the VTLM showed improved overall agreement with the three-dimensional reference results. The results showed that prediction accuracy was governed mainly by the flow-development condition and channel aspect ratio, from which a conservative screening guideline was proposed for the examined straight-channel configurations. The assessment of finned expansion–contraction channels further supported the applicability of the VTLM to the investigated passages with abrupt width changes. Overall, the VTLM provides a reduced-dimensional framework for thermohydraulic analysis across the range of microchannel heat-sink configurations examined in this study without explicitly resolving the channel-height direction.

Inspired by Steinitz's celebrated result (1913) and recent online vector-balancing results of Kulkarni et al. (2024) and Aden-Ali (2026), we introduce the online Steinitz problem: $T$ vectors in $\mathbb R^d$, each of $\ell_2$-norm at most $1$ and with total sum zero, arrive one at a time and must be assigned immediately and irrevocably to unused positions of an array. The goal is to keep every prefix sum of the resulting order small. We first obtain deterministic prefix-discrepancy bounds of $O(\log T)$ for $d=1$ and $O(\sqrt T)$ for $d\ge2$. Our main contribution is a general reduction that converts deterministic and randomized online vector-balancing algorithms, which assign signs to arriving vectors, into online ordering algorithms. Combined with recent vector-balancing results, this yields a randomized algorithm with prefix discrepancy $O(\log T+\sqrt{d\log T})$ and running time $O(dT\log T)$. The guarantee holds with high probability against an oblivious adversary. The reduction also gives bounds for sparse vectors and extends to inputs in arbitrary compact convex bodies. Finally, we prove matching lower bounds in both the deterministic and randomized cases.

The binding number $b(G)$ of a graph $G$ is the minimum value of $|N_{G}(X)|/|X|$ taken over all nonempty sets $X\subseteq V(G)$ such that $N_{G}(X)\neq V(G)$. A graph $G$ is 1-binding if $b(G)\geq 1$. Fan and Lin [Electron. J. Combin. 31 (2024) P1.30] characterized the unique spectral extremal graphs among 1-binding graphs with $\delta(G)\geq k$ that contain no $k$-factor for $k=1$, $2$. Subsequently, Tang and Zhang solved the problem for $k\geq 3$. Note that $\delta(G)\geq a$ is a necessary condition for the existence of an $[a,b]$-factor in a graph $G$. For $1\leq a<b$, Fan et al. [Discrete Math. 349 (2026) 115103] determined the unique spectral extremal graph $H_n^{a,b}$ among connected graphs with $\delta(G)\geq a$ that contain no $[a,b]$-factor. Note that $b(H_n^{a,b})<1$. This motivates the corresponding spectral extremal problem for 1-binding graphs. Note that every 1-binding graph with $\delta(G)\geq 1$ contains a $[1,b]$-factor for $b>1$. For $2\leq a<b$, we in this paper determine the unique extremal graph $\mathcal{B}_n^{a,b}$ with the maximum spectral radius among 1-binding graphs of order $n$ with $\delta(G)\geq a$ that contain no $[a,b]$-factor. The toughness of a non-complete graph $G$ is $t(G)=\min\{\frac{|X|}{c(G-X)}\mid X\subseteq V(G)\, \text{and}\, c(G-X)\geq 2\}$. A graph $G$ is 1-tough if $c(G-X)\leq |X|$ for every $X\subseteq V(G)$ with $c(G-X)\geq 2$. Every 1-tough graph of order $n\geq2$ is 1-binding. Note that $t(\mathcal{B}_n^{a,b})<1$. This leads us to investigate the corresponding spectral extremal problem for 1-tough graphs. Note that every 1-tough graph with $\delta(G)\geq 1$ contains a $[1,b]$-factor for $b>1$. For $2\leq a<b$, we in this paper characterize the unique extremal graph with the maximum spectral radius among 1-tough graphs of order $n$ with $\delta(G)\geq a$ containing no $[a,b]$-factor.

This paper considers a distributed estimation system where edge nodes transmit averaged features to a center node for parameter estimation. A key challenge arises when the center node possesses auxiliary side information: how to design local extractors that are universally optimal for any downstream task, while avoiding redundancy with the side information. To tackle this problem, we orthogonally decompose the joint function space and demonstrate in the weak-dependence regime that the modal decomposition of the canonical dependence kernel's conditional-additive component yields optimal, task-independent features. We construct a distributed nested H-score to learn this decomposition and show that, under a limited feature budget, the mean-squared error (MSE) gap to the best task-specific features is determined solely by the task's alignment with the discarded, least informative modes, and vanishes to leading order for every square-integrable task once the budget is sufficient.

For an algebraic stack of finite type, we construct a finite stratification of an atlas such that every universally closed open substack pulls back to a union of strata. As a result, there are only finitely many open substacks admitting a proper adequate moduli space.

In the tensor principal components analysis (tensor PCA) problem, we are asked to recover an unknown vector in $\{\pm1\}^N$ from a noisy observation of its $k$-th tensor power, for a fixed $k\ge3$. Tensor PCA exhibits a well-studied information-computation gap: global algorithms such as spectral methods and sum-of-squares work above an algorithmic threshold, below which we know matching lower bounds (against these and some other restricted models of computation). In contrast, local algorithms such as Markov chains were only known to succeed from a random initialization at a polynomially larger signal-to-noise ratio, and were believed to be inferior to global algorithms for tensor PCA. We show that Glauber dynamics, when supplemented with annealing and ergodic averaging, achieves polynomial-time strong recovery above the algorithmic threshold from a uniformly random initialization, demonstrating that local Markov chains can succeed all the way down to the algorithmic threshold from an uninformative cold start.

Under the stated regularity assumptions, the full orthogonality functionals of the five finite para-families considered here are classical for the respective affine, quadratic, $q$-quadratic, $q$-linear and alternating calculi. For each of the para-Krawtchouk, para-Racah, $q$-para-Racah and $q$-para-Krawtchouk families, two finite solutions of one global structural equation have $N+1$ nodes altogether and common moments through degree $N-1$. Their relative mass changes only the central recurrence coefficients. We prove regularity for every normalised combination with both component masses non-zero and recover the full published functionals and recurrences. The alternating construction identifies para-Bannai-Ito within the same structural framework. For the four non-alternating families, equal positive combinations give persymmetric Jacobi matrices; the known spectral gap criterion then decides perfect state transfer. Explicit parameter correspondences locate the classical components and resonant limits of the full systems. A positive seven-point example shows how further named families can be manufactured without a new governing principle. Counterexamples to mathematical assertions in three of four recent preprints expose the consequences of insufficient hypotheses for finite representations and physical models. In an era of AI-assisted symbolic production, we call for a standard that rewards structural explanation and verification rather than the proliferation of examples.

We give a new upper bound on the piercing number $\tau(\mathcal{F})$ of a finite family $\mathcal{F}$ of axis-parallel rectangles in the plane when every $p$ members contain an intersecting pair. Namely, we prove $\tau(\mathcal{F}) = O(p(\log\log p)(\log\log\log p))$, improving the factor $(\log\log p)^2$ in the best previously known bound to $(\log\log p)(\log\log\log p)$. For each $d>2$, when $\mathcal{F}$ is a finite family of axis-parallel boxes in $\mathbb{R}^d$, this yields $\tau(\mathcal{F}) = O(p(\log p)^{d-2}(\log\log p)(\log\log\log p))$. We also study an analogous problem replacing $\mathbb{R}^d=\mathbb{R}\times\cdots\times\mathbb{R}$ with the Cartesian product $T_1\square\cdots\square T_d$ of trees $T_1,\ldots,T_d$, and boxes with objects of the form $S_1\square\cdots\square S_d$ where $S_i$ is a subtree of $T_i$. For a finite family of such objects under the same intersection assumption as above, we establish an upper bound $\tau(\mathcal{F})=O(p(\log p)^{d-1})$ for each $d$. We also construct examples showing that this upper bound cannot be improved in order of magnitude.

This paper is concerned with space inhomogeneous Fermi-Dirac-Fokker-Planck equation posed on a classical kinetic phase space. The nonlinear factor $f(1-f)$ appears both in the transport term and in the collison part of the Fokker-Planck operator, accounting for the exclusion principle of fermions. Solutions are subject to the physical constraint $0\leq f\leq 1$. Previous existing results have mainly been obtained under suitable smallness or close-to-equilibrium assumptions. A main difficulty stems from the nonlinear and degenerate parabolic structure of the equation. We prove the existence and uniqueness of global solutions without any smallness or close-to-equilibrium assumptions. To construct solutions, we use a vanishing viscosity approximation by adding a Laplacian in the spatial variable. We derive uniform estimates for the regularized problem and obtain compactness in the space of functions of bounded variation $(BV),$ allowing us to pass to the limit as the spatial viscosity vanishes. Since the limiting equation is degenerate with a nonlinear transport, we formulate an appropriate notion of entropy solutions incorporating the kinetic structure of the equation. We prove uniqueness within this class and show that every vanishing-viscosity limit satisfies the entropy formulation. Our result provides a global well-posedness theory beyond the perturbative regime.

The introduction of Reconfigurable Intelligent Surfaces (RIS) technology marks a transformative era in modern communication systems, enhancing coverage and facilitating higher data transmission rates. While recent research has delved into the integration of RIS into satellite networks, a critical and practical gap persists concerning its implementation in the satellite-Very Small Aperture Terminal (VSAT) downlink, specifically amplifying contributions from the VSAT side. This paper addresses this gap by proposing a simple and engineering methodology to optimize RIS position and orientation, incorporating considerations for the antennas' radiation patterns and the system's topology to streamline optimization variables. The primary objective of this paper is to alleviate the double fading constraint in the reflect link without the need for active elements, achieved by leveraging the Fresnel zone of the RIS. The improvement power ratio depends on the frequency and the VSAT 's diameter. The results show that for a VSAT with diameter 1m the gain is up to 9dB at 2 GHz (S band) and up to 14dB at 4 GHz (C band). Additionally, the RIS channel equation is refined, encompassing the tropospheric attenuation and accounting for electromagnetic wave polarization.