A bicirculant is a regular graph that admits an automorphism having two vertex-orbits of the same size. A bicirculant can be described as follows. Given an integer $m \ge 1$ and sets $R, S, T \subseteq \mathbb Z_m$ such that $R=-R$, $T=-T$, $0 \not\in R \cup T$ and $0 \in S$, the graph $B(m;R,S,T)$ has vertex set $V=\{u_0,\dots,u_{m-1},v_0,\dots,v_m-1\}$ and edge set $E=\{u_iu_{i+j}| \ i \in\mathbb Z_m, j \in R\} \cup \{v_iv_{i+j}| \ i \in\mathbb Z_m, j \in T\} \cup\{u_iv_{i+j}| \ i \in\mathbb Z_m, j \in S\}.$ Bicirculant graphs with $R=T=\emptyset$ are known as cyclic Haar graphs. In 2025 we conjectured that the only non-hamiltonian graphs among regular connected bicirculants of degree more than one are the generalized Petersen graphs $G(m,2)$ with $m \equiv 5 \pmod 6$. Recently we have verified the conjecture for bicirculants with $|S|\le 2$ and for bicirculants with $|R|=|T|$ odd. In this paper we show that the conjecture holds for all bicirculants with $|S| \le 3$ and for all bicirculants with $|S| \ge 4$ and $m/\gcd(m, S)$ even. As a byproduct of our results, we prove that every connected bicirculant graph on $2m$ vertices with $|S| \ge 4$ is hamiltonian for even $m< 9\, 240$, and for odd $m< 3\,465$. Finally, we show that the existence of a hamilton cycle in every connected cyclic Haar graph of valence at least $4$ implies that every connected bicirculant graph of valence at least $4$ is hamiltonian.

We extend the entropy-stable oscillation-eliminating discontinuous Galerkin spectral element method (ES-OEDG) on curvilinear meshes to adaptive mesh refinement (AMR) grids with nonconforming interfaces. The formulation targets two-dimensional curvilinear quadrilateral meshes under a 2:1 refinement constraint, allowing a single level of hanging nodes. Elementwise volume discretization and geometric mapping are retained, while oscillation elimination and interface coupling are adapted for nonconforming interfaces. A central contribution is the design and analysis of numerical fluxes for such interfaces. We construct an entropy-stable flux that ensures global conservation and a semi-discrete entropy inequality. However, for polynomial degree N >= 2, negative entries in nonconforming interpolation operators lead to loss of formal high-order consistency. To address this, we propose a mortar-based flux that preserves high-order accuracy by interpolating at the solution level and evaluating standard two-point fluxes on fine-side mortars, at the cost of losing provable entropy stability. We also extend the Zhang--Shu positivity-preserving framework to curvilinear AMR meshes. Under forward Euler time stepping and a suitable CFL condition, the scheme using either flux preserves positivity of cell-average density and pressure. Combined with the Zhang--Shu limiter, this yields a fully discrete scheme maintaining admissibility at all nodal points. We further incorporate shock-indicator-based AMR and a conservative, positivity-preserving data transfer procedure between successive meshes, resulting in a robust and efficient algorithm. Numerical experiments on Cartesian and curvilinear AMR grids confirm high-order accuracy and robustness.

This paper investigates the long-time dynamics of solutions for an abstract nonlinear stochastic hydrodynamic-type equation driven by multiplicative Lévy noise. The framework encompasses several key hydrodynamical models, including the stochastic 2D Navier-Stokes equations, magnetohydrodynamic equations, the magnetic Bérnard problem, as well as various stochastic shell models of turbulence. Under the assumption that the nonlinear noise coefficients satisfy local Lipschitz and linear growth conditions, we first establish global well-posedness using a truncation technique. Then, by introducing a mean random dynamical system, we prove the existence and uniqueness of weak pullback mean random attractors for the system. Furthermore, when the external force is time-independent, we study the existence of invariant measures for the corresponding autonomous system, as well as the double limiting behavior of invariant measures with respect to the intensities of Gaussian and Lévy noise. Finally, under additional assumptions on the bilinear nonlinear term (e.g., as in the Navier-Stokes equations), we examine the existence and uniqueness of pullback measure attractors, along with the asymptotically autonomous stability of such attractors as the time parameter tends to negative infinity. It is worth noting that the results of this paper are new even for the single stochastic 2D Navier-Stokes equations.

Graphs whose maximum clique size exceeds half of the total number of vertices satisfy a classical property: the family of their maximum sized cliques can be pierced by a single vertex. This result dates back to a 1965 theorem by Hajnal. Motivated by this theorem, Jung, Keszegh, Pálvölgyi, and Yuditsky recently conjectured that an analogous result should hold for hypergraphs of larger uniformity, with an appropriate constant replacing the threshold $1/2$. In this paper we refute this conjecture in a strong form. We show that for any constant $c<1$ and integers $k\ge 3$ and $t\ge 1$, there exist $k$-uniform hypergraphs $G$ whose maximum clique size exceeds $c|V(G)|$, yet the family of maximum size cliques of $G$ cannot be pierced by $t$ vertices. This demonstrates that no universal constant threshold guarantees bounded piercing number for maximum cliques in uniform hypergraphs. We discuss further questions concerning the relationship between clique size and piercing maximum cliques in hypergraphs, and introduce a geometric variant of the problem using Helly's Theorem.

Cell-free multiple-input multiple-output (CF-MIMO) architecture significantly enhances wireless network performance, offering a promising solution for delay-sensitive applications. This paper investigates the resource allocation problem in CF-MIMO systems, aiming to maximize energy efficiency (EE) while satisfying delay violation rate constraint. We design a Proximal Policy Optimization (PPO) with a primal-dual method to solve it. To address the low sample efficiency and safety risks caused by cold-start of the designed safe deep reinforcement learning (DRL) method, we propose a novel offline pretraining framework based on virtual constrained Markov decision process (CMDP) modeling. The virtual CMDP consists of reward and cost prediction module, initial-state distribution module and state transition module. Notably, we propose an evidence-aware conditional Gaussian Mixture Model (EA-CGMM) inference approach to mitigate data sparsity and distribution drift issues in state transition modeling. Simulation results demonstrate the effectiveness of CMDP modeling and validate the safety and efficiency of the proposed pretraining framework. Specifically, compared with non-pretrained baseline, the agent pretrained through our proposed framework achieves twice the initial EE and maintains a low delay constraint violation rate of $1\%$, while ultimately converging to an EE that is $4.7\%$ higher with a $50\%$ reduction in exploration steps. Additionally, our proposed pretraining framework implementation exhibits comparable performance to the SOTA diffusion model-based implementation, while achieving a $14$-fold reduction in computational complexity.

We derive the classical Gibbs measure on $\mathbb{T}^2$ associated with the fractional Bessel interaction potential $\widehat{v}_β(k)=\langle k\rangle^{-β}$ from a renormalized grand-canonical quantum Bose gas with the same interaction. Our result covers the whole range $\frac32<β\leq2$, where $\widehat{v}_β(k)$ is not summable and the quantum model cannot be written in the usual density-square form, as the associated self-energy diverges. We therefore need to renormalize the zero mode by a centered number-fluctuation term and then develop a detailed analysis for the high-frequency remainders. All this allows us to implement a low-frequency localization and obtain the convergence of the quantum relative free energy to the classical fractional-Bessel free energy, as well as the convergence of the reduced density matrices to the limiting Gibbs measure.

Let $-Δ_{\mathbb{H}}+V$ be the Schrödinger operator on $\mathbb{H}$ where $V \in L^p(\mathbb{H}) \cap L^\infty(\mathbb{H})$ for some $p > 0$. If $(X_n)$ is a uniformly discrete sequence of compact hyperbolic surfaces with a uniform spectral gap that Benjamini-Schramm converges to $\mathbb{H}$, we prove quantum mixing for the eigenfunctions of $-Δ_{X_n}+V_n$ in any sufficiently large spectral window $I$, where $V_n$ is the potential on $X_n$ induced by $V$. These apply to large degree lifts of a potential on a base surface such as congruence covers of arithmetic surfaces, with high probability to random hyperbolic surfaces in the Weil-Petersson model of large genus, and to Hartree one-particle operators arising in thermodynamic limit of many-body Bose gas on hyperbolic surfaces. The proof uses the Duhamel formula for the hyperbolic wave equation together with exponential mixing of the geodesic flow on $T^1 X_n$.

This paper studies a multiple-input multiple-output (MIMO) radar system for sensing the unknown and random angular location (angle) of a point target, based on the target-reflected echo signals and known prior distribution information about the target's angle specified by a probability density function (PDF). We consider a challenging yet practical scenario where the knowledge of such PDF is imperfect, due to the inaccuracy in PDF acquisition or unpredicted change of target appearance pattern; while the real (actual) PDF is modeled as an unknown perturbed version of the imperfect known PDF bounded by a given uncertainty radius. Such PDF imperfection motivates us to study the robust transmit beamforming design to optimize the worst-case sensing performance among all possible real PDFs. Since the sensing mean-squared error (MSE) is difficult to be characterized explicitly, we adopt the worst-case posterior Cramér-Rao bound (PCRB) as the performance metric. We formulate the beamforming optimization problem to minimize the maximum PCRB among all possible real PDFs, which is highly non-trivial since the PCRB has a complex intractable expression over the real PDF, and there are infinite constraints corresponding to the continuous set of real PDFs bounded by the uncertainty radius. To address these challenges, we derive a tractable quadratic approximation of the PCRB via second-order Taylor expansion, and leverage the S-procedure to equivalently transform the infinite constraints into a linear matrix inequality, based on which the problem is reformulated into a convex optimization problem solvable with polynomial time complexity. The obtained solution approaches the globally optimal robust beamforming solution as the uncertainty radius decreases. Numerical results validate the effectiveness of our proposed robust beamforming design.

This paper is dedicated to the analysis of infinite horizon optimal control problems subject to semilinear parabolic equations with constraints on the controls and discounted cost functionals. The discount factors on the cost and the state components are allowed to differ from each other. First-order as well as second-order optimality conditions are derived and the importance of allowing different discount factors for the second-order analysis for the class of nonlinearities under consideration is demonstrated. Finally convergence and rate of convergence for the approximation of the infinite horizon problem by a family of finite horizon problems is proven.

For a graph $G$ and partition $\mathcal{U}$ of its vertex set, an independent transversal of $(G, \mathcal{U})$ is an independent set of $G$ that contains one vertex from each block of $\mathcal{U}$. Buys, Kang, and Ozeki studied when a reconfiguration graph on independent transversals of $(G,\mathcal{U})$ is connected, meaning any independent transversal can be transformed into any other one through a sequence of one-vertex modifications while always maintaining an independent transversal. Analogous to a theorem of Haxell, they proved that this is the case if $G$ has maximum degree $Δ$ and each block of $\mathcal{U}$ has size at least $2Δ$, except if the union of some $k \ge 1$ blocks of $\mathcal{U}$ induces $k$ disjoint copies of the complete bipartite graph $K_{Δ, Δ}$ in $G$. Solving one of their problems, we exactly characterize the partition structure in the latter exceptional instances of their theorem, showing that there is a rich variety of them but they are generated by a simple constructive procedure.

We analyze a dual mixed nonconforming discretization of a generalized Darcy-Forchheimer model. Compared to the analogous scheme proposed by Girault and Wheeler, we consider general, i.e., nonquadratic, Forchheimer nonlinearities; we admit mixed, inhomogeneous boundary conditions; we allow for more general, i.e., with lower Lebesgue regularity, permeability tensors; we construct general-order schemes; we prove convergence to the exact solution under low regularity assumptions, based on novel Sobolev-trace inequalities for broken spaces; we derive error estimates of general-order assuming extra regularity of the exact solution and data; we present numerical results assessing the performance of the proposed schemes for different types of nonlinearity and nonlinear solvers.

In this paper, we investigate the non-symmetric Strong Spectral Property (nSSP) from a combinatorial perspective. To zero-nonzero patterns of matrices we associate directed graphs and study when they require or allow the nSSP, providing a framework that avoids verifying the nSSP for individual matrices. A new combinatorial method is introduced and used to recognise several patterns that require the nSSP. It is shown that loop assignments in double paths play a critical role in establishing this property, and we show that an open question regarding irreducible tridiagonal patterns has a negative answer. We also investigate whether the minimum number of arcs in a directed graph on $n$ vertices that requires the nSSP, is equal to $2n-1$, and confirm this minimum for several specific digraph families.

Given a hypergraph $F$, what is the largest chromatic number that an $F$-free hypergraph can have? In the case of graphs, this question is easy to answer: the chromatic number is unbounded if $F$ contains a cycle, and the largest chromatic number of $F$-free graphs is $k-1$ if $F$ is a forest on $k$ vertices. The situation is more complicated for hypergraphs. The strong coloring of a hypergraph is a coloring of the vertices such that every hyperedge is rainbow. The weak coloring of a hypergraph is a coloring of the vertices such that no hyperedge is monochromatic. The strong/weak chromatic number of a hypergraph is the minimum number of colors in a strong/weak coloring of the hypergraph. Our question has been completely answered for the weak chromatic number, similarly to the graph case. We characterize the hypergraphs $F$ such that $F$-free hypergraphs have bounded strong chromatic number. The only remaining case is when $F$ is the 3-uniform expansion $S_k^+$ of a star with $k$ edges. Concerning the strong chromatic number of $S_k^+$-free hypergraphs, we give bounds that are asymptitically sharp as $k\rightarrow\infty$. We also consider the same problem when the Berge copies of a graph $F$ are forbidden. We characterize when the strong/weak chromatic numbers are bounded in this case, and obtain sharp results or bounds for specific trees. In particular, when $F$ is a path, we give a tight bound when $r=3$ and an asymptotically sharp bound when $r=4$.

The matrix completion problem provides a unifying lens through which many fundamental problems in coding theory can be viewed. In this paper, we investigate Locally Recoverable Codes (LRCs) with Maximal Recoverability (MR) and Maximum Distance Profile (MDP) convolutional codes in the framework of matrix completion. In particular, we present techniques that are general enough to provide constructions for both types of codes. A common feature of our code constructions is the sparsity of their generator matrices and the property that a large number of the entries of the generator matrices are elements of a small subfield of a larger extension field.

In this paper, we propose a class of super-schemes for efficiently solving nonlinear unconstrained optimization problems. The proposed approach introduces two novel choices of step-size parameters, leading to efficient descent directions without requiring second-order information. We develop one-step, two-step, and three-step iterative schemes (denoted by SS1, SS2, and SS3) and establish that these methods achieve higher-order convergence of orders two, four, and six, respectively. Despite their high convergence rates, the computational complexity of the proposed methods remains comparable to existing gradient-based methods, with a cost of $\mathcal{O}(n^2)$ per iteration. The proposed methods are simple to implement and do not require complicated line-search procedures. Their effectiveness is demonstrated through extensive numerical experiments on a wide range of problems, including large-scale and ill-conditioned cases. The results show that the proposed methods significantly outperform classical methods, such as the Barzilai-Borwein method and other gradient-based approaches, in terms of iteration count and computational efficiency. Finally, the numerical results are consistent with the theoretical analysis, confirming the stability of the proposed schemes for test optimization problems.

The internal low-energy symmetry of the massless Lorentz-invariant Dirac Hamiltonian in $2+1$ dimensions is known to be $O(2N)$, where $N$ is the number of two-component Dirac fermions. Here we point out that there exists an analogous internal symmetry of the single-particle quadratic-band-touching Hamiltonian in two spatial dimensions, and it is the unitary symplectic group, $USp(2N)$. All fermionic bilinears belong to one of the three small irreducible representations of this group. The interacting theory that respects the $USp(2N)$ symmetry and the spatial rotations is constructed and found to allow two independent interaction terms. When these interactions are infrared-relevant the symplectic symmetry either remains preserved or becomes spontaneously broken to $USp(N) \times USp(N)$. The symmetry in the lattices such as honeycomb to infinite order in the dispersion's expansion in powers of local momentum is given by the overlap of the symplectic and the orthogonal groups. We show that this overlap is $O(2N) \bigcap USp(2N) = U(N)$.

We construct the first smooth embedded compact special Legendrian surfaces in \(\mathbb S^5\) of genus greater than one. More precisely, for every sufficiently large integer \(k\), we construct an embedded special Legendrian surface whose conformal structure is the Fermat curve of degree \(k\) and genus \(\tfrac12(k-1)(k-2)\). Our approach combines an elementary implicit function theorem with the description of special Legendrian surfaces via loop algebra-valued meromorphic connections and a characterization of the unitarizability locus in the ${SL}_{3}(\mathbb C)$-character variety of the thrice-punctured sphere.

We study bubbling for sequences of Yang-Mills connections on closed four-manifolds and we derive a compatibility of Pohozaev type between the weak limit connection and the bubble formed at a concentration point, involving the Weyl tensor of the background metric. This yields obstruCtions to bubbling extending earlier results of Yin beyond the locally conformally flat case. As an application, we rule out certain bubbling configurations on CP2.

For $0<δ,τ<1$ and $1\le s\le \frac{n}{n-δ}$, we prove that for a given $s$-John domain $Ω\subset \mathbb{R}^n$, the following Boxing inequality holds for every Lebesgue measurable set $U\subsetΩ$ with $|U|/|Ω|\leγ<1$: \[ \mathcal{H}^{s(n-δ)}_{\infty}(U\setminus\mathcal{N}_U)\le C(1-δ)\int_Ω\int_{|x-y|<τ\operatorname{dist}(y,\partialΩ)}\frac{|χ_U(x)-χ_U(y)|}{|x-y|^{n+δ}}\,dx\,dy, \] where $\mathcal{H}^{s(n-δ)}_{\infty}(U)$ denotes the $s(n-δ)$-dimensional Hausdorff content of $U$, $\mathcal{N}_U$ is a set of Lebesgue measure zero and the constant $C$ depends only on $n,τ,s,γ$, the John constant and the diameter of $Ω$. Moreover, we establish the functional formulation of the above Boxing inequality and discuss the equivalence between these two formulations. Based on the Boxing inequality, we prove the fractional Poincaré--Wirtinger trace inequality on $s$-John domains, of which the fractional Sobolev--Poincaré inequality and fractional Hardy-type inequality are special cases. Notably, we prove all of the aforementioned inequalities with the Bourgain--Brezis--Mironescu (BBM) factor $1-δ$. Furthermore, with the aid of the Bourgain--Brezis--Mironescu formula, we recover the Poincaré--Wirtinger trace inequality. Finally, by showing that, under the separation property, any domain supporting the Boxing inequality is necessarily a John domain, we conclude that the John domain condition is essentially sharp for the above inequalities. All the above inequalities with the BBM factor are new even for Lipschitz domains.

We study the efficient generation of random graphs with a prescribed expected degree sequence, focusing on rank-1 inhomogeneous models in which vertices are assigned weights and edges are drawn independently with probabilities proportional to the product of endpoint weights. We adopt a temporal viewpoint, adding edges to the graph one at a time up to a fixed time horizon, and allowing for self-loops or duplicate edges in the first stage. Then, the simple projection of the resulting multigraph recovers exactly the simple Norros--Reittu random graph, whose expected degrees match the prescribed targets under mild conditions. Building on this representation, we develop an exact generator based on \textit{edge-arrivals} for expected-degree random graphs with running time $O(n+m)$, where $m$ is the number of generated edges, and hence proportional to the output size. This removes the typical vertex sorting used by widely-used fast generator algorithms based on \textit{edge-skipping} for rank-1 expected-degree models, which leads to a total running time of $O(n \log n + m)$. In addition, our algorithm is simpler than those in the literature, easy to implement, and very flexible, thus opening up to extensions to directed and temporal random graphs, generalization to higher-order structures, and improvements through parallelization.

A $k$-regular graph of girth $g$ is called vertex-girth-regular if every vertex is contained in the same number of cycles of length $g$. For integers $n, k, g$ and $λ$, we denote such a graph on $n$ vertices in which every vertex lies on exactly $λ$ cycles of length $g$ by a $\text{vgr}(n,k,g,λ)$-graph. It is well-known that any vertex-girth-regular graph satisfies $λ\le \frac{k(k-1)^{\left\lfloor \frac{g}{2} \right\rfloor}}{2}$. Graphs for which $λ$ is close to this bound are of particular interest in connection with the cage problem, since requiring many girth cycles through every vertex is a natural way to isolate highly structured candidates for small regular graphs of prescribed girth. In this paper, we prove that for every $k\ge 3$ and every integer $0< \varepsilon \leq \frac{k-1}{2}$, there does not exist a $\text{vgr}(n,k,5,\frac{k(k-1)^2}{2}-\varepsilon)$-graph. Previous non-existence results had already settled all odd girths at least $7$ and very recently also girth $3$, leaving girth $5$ as the only girth for which no non-trivial non-existence result was known. Thus, our result resolves the final remaining case and completes the picture for odd girths.

A thin-sheet (TS) volume integral equation (VIE) formulation incorporating generalized sheet transition conditions (GSTCs) is presented for the simulation of three-dimensional (3D) bianisotropic metasurfaces. The metasurface is represented as an equivalent TS, with its constitutive tensors derived from the GSTC susceptibility tensors. Invoking the TS approximation, the governing VIEs are reduced to surface integral equations (SIEs), in which tangential and normal flux density components are treated as distinct sets of unknowns and discretized using Rao-Wilton-Glisson and pulse basis functions, respectively. In contrast to conventional GSTC approaches based on conventional SIEs, which represent only tangential fields, the proposed framework rigorously enforces the bianisotropic GSTCs, including normal field interactions, while retaining the flux-based VIE character of the formulation. Numerical examples demonstrate the accuracy and robustness of the proposed TS-VIE-GSTC solver for polarization rotation, perfect reflection, multi-directional attenuation, and oblique phase-shift transformation.

We determine putative optimal packings of regular spherical polygons via optimization on smooth manifolds. For several cases, we establish maximality by extending the Lovász theta number to Cayley graphs on the special orthogonal group ${\rm SO}(3)$. To this end, we introduce an algebraic criterion characterizing when congruent regular spherical polygons have disjoint interiors, leading to a unified formulation of the packing constraints. Using harmonic analysis on ${\rm SO}(3)$, we reduce the theta number to a trigonometric sum-of-squares problem, which can be solved via semidefinite programming.

The Halpern algorithm is a powerful fixed point approximation method for finding the closest point in the fixed point set of a nonexpansive mapping to the initial point. However, in practice, it is not necessarily true that this algorithm can be applied to large-scale fixed point problems, since the computation of the nonexpansive mapping is expensive. In this paper, we present mini-batch stochastic Halpern algorithm to resolve the issue caused by the computational difficulty of the mapping. We preform a convergence analysis demonstrating that the algorithm with diminishing step sizes and increasing batch sizes converges in mean square to the closest point in the fixed point set to the initial point. We also perform a convergence rate analysis demonstrating that convergence speed of the algorithm depends on the settings of the diminishing step sizes.

Motivated by Sarnak's conjecture on Möbius orthogonality, we investigate the general problem of orthogonality for a bounded sequence to topological models of characteristic classes of measure-preserving automorphisms. Our main observation is that whenever a strong form of such orthogonality holds in a system $(X,T)$ then the orthogonality holds for all topological systems in which each ergodic measure yields an automorphism that is measure-theoretically isomorphic to one arising from an ergodic measure in $(X,T)$. This leads us to study two purely dynamical problems: the existence of universal topological models for characteristic classes of measure-preserving automorphisms and the existence of a common ergodic extension for a measurable family of ergodic automorphisms. We show that the class of automorphisms with relative discrete spectrum over the identity factor--as well as several related classes including the weakly mixing case--admit universal models. We also highlight potential applications to the orthogonality phenomena. Moreover, we show that if the set of all measure-theoretic eigenvalues of a zero entropy system $(X,T)$ is countable, then $(X,T)$ satisfies Sarnak's conjecture along a subsequence of full logarithmic density.

We study existence and uniqueness for one-dimensional generalized stochastic differential equations with singular coefficients, including distributional drift and degenerate, possibly discontinuous, diffusion coefficients. Such singularities naturally encode changes in the dynamics at thresholds, including reflecting, skew, or sticky interface behavior. We develop two directions. We provide sufficient conditions for pathwise uniqueness, under weak existence and uniqueness in law, without assuming uniform ellipticity or continuity of the diffusion coefficient. We also investigate a pasting approach for generalized stochastic differential equations that transfers strong existence and pathwise uniqueness, as well as weak existence and uniqueness in law, from local component equations to a global solution. To the best of our knowledge, this provides the first explicit pasting theorem yielding pathwise uniqueness in the setting of generalized stochastic differential equations. As an application, we establish the first existence and uniqueness results for a class of skew sticky threshold Cox-Ingersoll-Ross-type diffusions, including the threshold Chan-Karolyi-Longstaff-Sanders process.

This paper considers constrained stochastic nonsmooth minimax optimization problem of the form $\min_{\mathbf{x}\in\mathcal{X}}\max_{\mathbf{y}\in\mathcal{Y}}f\left(\mathbf{x},\mathbf{y}\right)=\mathbb{E}[F(\mathbf{x},\mathbf{y};\mathbfξ)]$, where the objective $f(\mathbf{x},\mathbf{y})$ is concave in $\mathbf{y}$ but possibly nonconvex in $\mathbf{x}$, the stochastic component $F(\mathbf{x},\mathbf{y};\mathbfξ)$ indexed by random variable $\mathbfξ$ is mean-squared Lipschitz continuous, and the feasible sets $\mathcal X$ and $\mathcal Y$ are convex and compact. We introduce the notion of $(η_x,η_y,δ,ε)$-Goldstein saddle stationary point (GSSP) to characterize the convergence for solving constrained nonsmooth minimax problems. We then develop projected gradient-free descent ascent methods for finding $(η_x,η_y,δ,ε)$-GSSPs of the objective function $f(\mathbf{x},\mathbf{y})$ with non-asymptotic convergence rates. We further propose nested-loop projected gradient-free descent ascent methods to establish the non-asymptotic convergence for finding $(η,δ,ε)$-generalized Goldstein stationary points (GGSP) [Liu et al., 2024] of the primal function $Φ(\mathbf{x})\triangleq\max_{\mathbf{y}\in\mathcal{Y}}{f}\left(\mathbf{x},\mathbf{y}\right)$. It is worth noting that our algorithm designs and theoretical analyses do not require additional assumptions such as the weak convexity used in prior works on nonsmooth minimax optimization [Lin et al., 2025, Boţ and Böhm, 2023].

Zeroth-order (ZO) optimization is indispensable for complex non-convex tasks where explicit gradients are computationally prohibitive or strictly inaccessible. For deploying ZO methods over distributed heterogeneous networks, the gradient tracking technique is often employed to eliminate structural data biases. However, the inherent variance of derivative-free estimators is also amplified. To overcome this problem, we propose Zeroth-Order Momentum Gradient Tracking (ZO-MGT), which integrates momentum-based variance reduction with dynamic gradient tracking. Specifically, ZO-MGT that requires exactly two function queries per iteration can avoid costly batch sampling and prevent variance explosion, while eliminating structural biases. Moreover, by utilizing Rademacher perturbations, it preserves optimal query efficiency and enables bitwise hardware acceleration. We theoretically analyze the convergence of ZO-MGT and establish an $\mathcal{O}(1/T)$ convergence rate. Furthermore, we prove that a large momentum factor can aggressively suppress the heterogeneity-induced bias floor at a remarkable quadratic rate of $\mathcal{O}((1-β)^2)$. Numerical experiments under extreme data heterogeneity verify that ZO-MGT can effectively overcome traditional tracking failures with accelerated convergence guarantees, while achieving significantly tighter consensus.

Let $X$ be a smooth complex projective curve. We prove that there exists a surjective commutative forgetful diagram from the chain of $\mathbb{C}^{*}$-flips of the moduli spaces of $\mathcal{O}_{X}$-twisted rank 2 constrained framed Hitchin pairs on $X$ to the chain of $\mathbb{C}^{*}$-flips of the moduli spaces of rank 2 framed modules on $X$.

We study some geometric properties of the excursion set of a slope field alpha associated to a smooth, planar, centered, Gaussian field f. That, is we consider the set of all points such that the value of alpha is at most l where l is a real parameter called the level. We restrict our attention to the levels l that are supercritical. We show that for almost such l, in the sense of the Lebesgue measure, then with high probability the chemical distance between two points connected in the excursion set at level l is comparable to the usual Euclidean distance between those two points. This result is in the spirit of the Antal Pisztora theorem for Bernoulli percolation. However, many new difficulties arise such as the fact that alpha is a continuous field (not differentiable everywhere) with long range correlations and whose law is still not well understood.