Papers

The Monge-Ampère equation is a fundamental fully nonlinear elliptic partial differential equation that finds extensive applications across multiple disciplines. This study proposes a novel physics-informed neural network integrated with attention feature expansion (PINN-AFE) for its numerical solution. A multi-head attention enhanced feature pool is constructed to enable adaptive nonlinear feature representation, and input convex neural networks are adopted to impose strict convexity of solutions with rigorous theoretical guarantees. Meanwhile, a dynamically weighted loss function combined with hybrid optimization is formulated to accelerate training convergence. Comprehensive numerical experiments validate the accuracy and computational efficiency of the developed framework. The PINN-AFE paradigm is further extended to image processing tasks, delivering high-quality and physically consistent results in both image enhancement and medical image registration scenarios.

We prove an analog of Aldous' spectral gap conjecture in the generalized symmetric groups $G\wr S_n$ where $G$ is an arbitrary finite group. Moreover, we show that Caputo's extension of the conjecture to hypergraphs transfers to these groups whenever it holds in the ordinary symmetric group.

We consider the following problem: Let $H$ and $F$ be two graphs on $k$ vertices and assume $F \neq H$. We say that $H$ and $F$ are incomparable if neither $F$ nor $H$ contains the other. Let $H$ be a graph on $k$ vertices and let $G$ be a graph on at least $k$ vertices. Then $G$ is said to be $H$-exact if any induced subgraph of $G$ on $k$ vertices is either isomorphic to $H$ or incomparable with $H$. Exact($H$) is the family of all graphs $G$ which are $H$-exact. We pose the following problem: For a graph $H$ on $k$ vertices, determine or estimate $f(H) = \max \{n: \exists G \in \text{Exact}(H), |V (G)| = n\}$. Among the many results obtained in this paper the following are representatives concerning trees and matchings: 1. For a tree on $k \geq 3$ vertices, $ (k - 1)(\left \lceil \frac{k}{2} \right \rceil -1 ) \leq f(T) \leq ( k-1)^2$. 2. For $k \geq 4$, $f(K_{1,k-1}) = (k-1)(k-2)$. 3. For $k \geq 5$, $f(P_k) = \frac{(k-1)^2}{2}$ if $k$ is odd and $f(P_k) = \frac{(k-1)(k-2)}{2}+1$ if $k$ is even. 4. $f(nK_2) = 3n$ for $n = 2, 3$ and $f(nK_2) = 4n - 4$ for $n \geq 4$.

We have generalised the notion of categorical theory in model theory to the context of coherent theories. We prove a duality result between the full sub-2-category of pretopoi which are categorical, and the 2-category of profinite monoids. We also study the geometry of profinite monoids via the classifying topos construction, and show it identifies them as a full sub-2-category of the 2-category of topoi.

This paper investigates the joint optimization of power allocation and antenna activation in sparse extremely large aperture array systems operating under power amplifier non-linearities. We first derive an analytical expression for the achievable spectral efficiency (SE) of point-to-point MIMO channels affected by non-linear distortions using the Bussgang decomposition. To address the combinatorial and non-convex nature of the energy-efficiency (EE) maximization problem, we employ an unsupervised deep neural network (DNN) that learns the non-linear mapping between the channel state information and the optimal EE operating point. The DNN jointly predicts distortion-aware power allocation, total transmit power scaling, and modular sub-array activation based on singular-value and geometric channel features. Numerical results demonstrate that the proposed DNN-based arrays achieve significant EE gains over the conventional sparse arrays.

We provide a complete description of the automorphism group $\Aut (W)$ of a Coxeter group $W$ admitting a star-shaped finite Coxeter diagram. We prove that each automorphism decomposes as a product of inner and diagram automorphisms, along with three additional types: transvections and two families of partial conjugations. Furthermore, we investigate the natural short exact sequence $1 \to \Inn (W) \to \Aut (W) \to \Out (W) \to 1$. Using Moussong's criteria for hyperbolicity, we show that these groups possess the $R_\infty$-property. Finally, we establish rigidity properties for these groups using known techniques and provide a solution to the isomorphism problem within the class of star-shaped Coxeter systems.

In this paper, we investigate the structure of the fine Mordell-Weil groups over the intermediate subextensions of a given $\mathbb{Z}_p$-extension $F_\infty$ of $F$.

Chatterjee's rank correlation is a directed measure of association designed to detect whether one variable can be predicted as a function of another. While the original coefficient is naturally defined for real-valued data, circular data poses additional difficulty. Applying the usual construction requires cutting each circle at an arbitrary point and treating it as a line. Different choices of cut points can lead to different finite-sample values, even though the underlying circular relationship is unchanged. This paper proposes a circular version of Chatterjee's coefficient that removes this arbitrary choice. The population construction averages over response cuts in circular rank space, and the finite-sample construction averages over sample cut gaps and reduces to a simple statistic based only on cyclic ranks. The resulting coefficient is intrinsic to the circular ordering of the data, remains directed, and retains the key interpretation of Chatterjee's original coefficient. Under non-atomic circular marginals, it is zero exactly under independence and one exactly when the circular response is a measurable function of the circular predictor. We prove consistency and derive its distribution-free null behavior under independence. Simulations show that the proposed coefficient is especially useful for detecting multi-winding circular relationships, such as cases where the response goes around the circle twice or four times as the predictor goes around once, where standard circular correlations can be nearly blind.

We study the distribution of closed geodesics in short intervals on random hyperbolic surfaces of large genus, and compare it with the classical problem of primes in short intervals. Viewing the surface $M$ as a random point in moduli space equipped with the Weil–Petersson measure, we investigate the random variable $\Psi_M(x;H)$ counting closed geodesics with norms in the interval $[X, X+H]$, weighted by primitive length, where $H=o(X)$. This is analogous to the Chebyshev function in prime number theory. Our main result establishes that in the large genus limit, \[ \lim_{g\to \infty}\mathrm{Var}(\Psi_M(X;H)) \sim 2\,H \log X, \] when $X\to \infty$, $H=o(X)$. Goldston and Montgomery related the variance for primes in short intervals to the form factor associated with zeros of the Riemann zeta function, and conjectured that it is asymptotic to \[ H\log(X/H). \] We show that for automorphic L-functions of degree $d>1$, the early-time GUE form factor already follows from the Riemann Hypothesis, thereby recovering the variance $H\log X$ in the very short interval regime predicted by Bui, Keating and Smith. In the geometric setting, the appearance of $\log X$ reflects the much higher spectral density of Laplace eigenvalues relative to zeros of finite-degree $L$-functions, while the additional factor of $2$ is explained by the expected GOE statistics for the Laplace spectrum of generic hyperbolic surfaces.

Stacked intelligent metasurfaces (SIMs) and fluid antenna systems (FAS) are emerging technologies for wave-domain and spatial signal manipulation, respectively.This letter proposes a novel joint SIM-FAS communication model in which transmission and reception are performed by a SIM and an FAS, respectively. Using the block-diagonal matrix approximation (BDMA), a closed-form expression for the outage probability is derived, and the SIM phase shifts are optimized to minimize outage. Numerical results validate the analytical accuracy and demonstrate substantial performance gains over conventional benchmark schemes.

We give negative answers to both parts of a question of Mauri and Moraga on log Calabi-Yau pairs whose boundary decomposes into big divisors. The main result of this paper is obtained by generative AI, particularly Chatgpt 5.5 pro and the Rethlas system.

In the context of geometric measure theory, Llorente-Winter introduced the (average) fractal Euler number as a notion of the Euler characteristic for fractals embedded in Euclidean space. However, the class of fractals to which it is applicable remains very limited. In the present paper, we modify this notion by applying perspectives of persistent homology and partly the theory of magnitude, which have recently come from applied topology and category theory. We then demonstrate concrete calculation of our average ph-fractal Euler number for some classically well-known fractals, especially the Cantor dust and Menger sponge which are excluded from Llorente-Winter's approach.

In this paper, we provide complete characterizations for the spectrum, essential spectrum, and point spectrum of the generators of weighted composition $C_0$-semigroups induced by hyperbolic semiflows on Bergman spaces. We give an explicit example showing that the spectral properties can be influenced by the behavior of the semigroup near non-fixed self-contact points.

We show that the cohomology of the structure sheaf of smooth and proper schemes over a complete non-archimedean field $K$ of characteristic zero, can be refined to an $\mathbf{A}^1$-invariant cohomology theory of smooth (not necessarily proper) schemes over $K$ with values in $\mathcal{O}_K$-lattices, and the same holds for $K$ of positive characteristic in dimensions at most $3$. As one application, we obtain that the automorphism group of the function field of a proper smooth variety $X$ of dimension at most 3 over a field of positive characteristic acts quasi-unipotently on the cohomology of the structure sheaf of $X$. The construction of the lattices relies on a variant of the tame cohomology of Hübner–Schmidt with coefficients in a twisted version of the tame structure sheaf and uses results from rigid analytic geometry on the cohomology of twisted integral rigid structure sheaves due to Bartenwerfer and van der Put.

This paper addresses the challenge of obtaining strong optimality guarantees in constrained nonsmooth nonconvex optimization under mild regularity conditions, namely local Lipschitz continuity and existence of directional derivatives. While standard methods typically ensure weak stationarity notions, achieving directional (d-)stationarity remains nontrivial. We show that a random direction exploration step is sufficient to attain d-stationarity. The proposed approach augments any feasible base method with one additional randomized exploration step per iteration, which samples a direction and step size and accepts the candidate based on a function value comparison. The resulting scheme guarantees that all accumulation points are d-stationary almost surely, independently of the behavior of the underlying method. Moreover, under conditions on the base method, it preserves convergence rates of the base method, as established for the difference-of-convex algorithm (DCA), prox-linear-type schemes, and interpolated normalized gradient descent (INGD). The theoretical results are complemented by numerical experiments illustrating the effect and guarantees of the exploration step.

Fluid antenna systems (FASs) are emerging as a reconfigurable-aperture technology that expands physical-layer design beyond fixed, rigid antenna geometries. While the fading diversity of FASs – which exploits spatial channel fluctuations for signal enhancement and interference avoidance – has been widely studied, the geometry diversity created by reconfigurable port placement remains far less understood, particularly for planar architectures under finite-aperture constraints. This paper develops a systematic analytical framework for finite-aperture planar fluid antenna arrays (FAAs). First, we derive a closed-form characterization of the minimum inter-port distance under uniform random placement over a rectangular aperture and show that it follows a Rayleigh law. Its mean scales as $\mathcal{O}(M^{-1})$, in sharp contrast to the $\mathcal{O}(M^{-2})$ behavior in the linear case in which $M$ represents the number of candidate ports, revealing a fundamentally more favorable packing geometry in two dimensions. Secondly, we establish a universal Cramér-Rao bound (CRB) for joint elevation-azimuth estimation, governed by a $2\times 2$ geometric inertia matrix whose determinant and eigenstructure fully capture the role of port placement in estimation precision. We further prove that both the trace and determinant of this matrix are invariant to the azimuth look direction. Third, we uncover an intrinsic precision–ambiguity trade-off: maximizing the geometric determinant to minimize the CRB drives ports toward the aperture boundary, but simultaneously increases sidelobe-induced spatial ambiguity.

In a companion paper arXiv:2510.27676, we introduced a non-perturbative classical renormalisation group (RG) flow equation as a novel method for treating strongly interacting problems in general relativity, with a prominent application to the two-body problem. While we demonstrated that it reproduces perturbation theory, via the Post-Minkowskian (PM) expansion, and its computational efficiency in reproducing the 1PN Post-Newtonian action, its derivation was heuristic. In this work, we place this flow equation on a firm formal foundation. In particular, we demonstrate that a Legendre transform maps the classical analogue of the Polchinski equation precisely to our classical RG equation. This establishes a duality between equivalent, exact RG equations for the gravitational effective action. The result, combined with the successful applications in arXiv:2510.27676, solidifies the classical RG framework as a powerful and rigorous new approach to the general relativistic two-body problem and gravitational wave physics.

We propose an AC-independent proof of the existence of a non-measurable set as a consequence of the Hahn-Banach theorem of functional analysis which is known to be strictly weaker than AC.

The asymptotic behavior of the integrated density of states (IDS), \(N(E)\), is investigated for random Schrödinger operators with a single-site potential \(V\) satisfying \(\mathrm{essinf}\, V = -\infty\). Under the assumption that the underlying point process is a Gibbs point process with repulsive pairwise interactions, the leading term of \(\log N(E)\) as \(E \to -\infty\) is determined using a periodic approximation method. It is shown that repulsive pairwise interactions lead to a significantly faster decay of \(N(E)\) compared to the Poisson case. Furthermore, configurations with multiple clusters can provide the dominant contribution to the IDS in the Gibbs setting, contrasting with the single-cluster dominance typically observed in Poisson models. Finally, refined estimates of the leading constants are provided for specific classes of potentials, including those with multiple singularities.

In this article, we compute both the algebraic and the analytic Brauer groups of a homogeneous space under the action of a connected, simply connected, semisimple complex algebraic group, where the stabilizer subgroup is closed and connected.

This article investigates the stability of pantograph delay differential equations, in which the delayed argument is proportional to the present time. We derive analytic criteria that partition the parameter plane into unstable, asymptotically stable, and delay-dependent stability regions. The theoretical results are supported by numerical simulations that illustrate the sharpness of the stability boundaries. We also formulate a proportional-delay analogue of the Mackey–Glass chaotic delay differential equation and examine the resulting dynamical behaviour.

We prove first-order convergence of semi-discrete monotone finite difference schemes for Hamilton–Jacobi equations on the Wasserstein space over a finite graph. A central challenge is the boundary degeneracy of the Wasserstein simplex, which prevents the direct use of the standard $L^1$ adjoint method and limits doubling-of-variables arguments to the suboptimal rate $\mathcal O(h^{\frac 12})$ \cite{CDM25}. We address this issue by introducing a weighted $L^1$ framework with a boundary-vanishing weight and by analyzing the corresponding weighted adjoint equation for the linearized operator of the scheme, featuring a new geometric drift term. Our proof relies on uniform bounds for the weighted adjoint variable and the mesh-parameter derivative of the numerical solution. These estimates are derived from discrete gradient and semi-concavity bounds, obtained through a bootstrap argument for two classes of monotone Hamiltonians.

We study solutions to the Navier-Stokes equations in the class $L^\infty_t C^\alpha_x$. Landau and Lifshitz [LL87] predicted that the Eulerian and Lagrangian temporal structure functions for turbulence exhibit $1/3$ and $1/2$ scaling laws, respectively. These laws were justified for the Euler equations in [Ise23,Ise25], assuming the spatial structure functions satisfies a $1/3$ scaling law. We demonstrate them in a viscous setting by proving that the $C^\alpha_{t,x}$-norm of the solution and the $C^{1/(1-\alpha)}$-norm of any fluid trajectory can be estimated by the $L^\infty_tC^\alpha_x$-norm independently of the viscosity parameter $\nu>0$, for times bounded away from zero by a positive power of $\nu$.

For $l > 1$, the $l$-edge-connectivity $\kappa'_l(G)$ of a connected graph $G$ is defined as the minimum number of edges whose removal leaves a graph with at least $l$ components. A graph is minimally $(k,l)$-edge-connected if $\kappa'_l(G)\geq k$ but for any edge $e\in E(G)$ satisfies that $\kappa'_l(G-e)< k$. Motivated by two foundational extremal problems: Brualdi and Solheid's problem [SIAM J. Algebra Discrete Methods (1986)] for graphs of fixed order: determine sharp upper bounds for the spectral radius over graph families and characterize extremal graphs; and its fixed size analogue proposed by Brualdi and Hoffman [Linear Algebra Appl. (1985)], we resolve both problems for minimally $(k,k)$-edge-connected graphs. Building on the structural framework of Hennayake, Lai, Li, and Mao [J. Graph Theory (2003)], we combine edge-switching method and double eigenvectors skill to characterize the graphs maximizing the spectral radius among all minimally $(k,k)$-edge-connected graphs of prescribed order or size. Our results generalize the $k=2$ cases established by Lou, Min, and Huang [Electron. J. Comb. (2023)] and Chen and Guo [Discrete Math. (2019)].

We introduce a notion of K-stability for adjoint foliated structures via test configurations and the foliated Donaldson-Futaki invariant. We prove reduction to special test configurations for adjoint Fano foliated structures by showing that the mixed Donaldson-Futaki invariant is non-increasing along the birational procedure. We also introduce a notion of Ding stability for adjoint Fano foliated structures which we show is equivalent to our notion of K-stability. We then introduce mixed alpha, beta and delta-invariants and use the reduction theorem to establish valuative criteria for the K-stability of adjoint Fano foliated structures. To conclude, as an application, we show that K-semistable adjoint Fano foliated structures with bounded volume form a bounded family.

In this paper, using extension theory and cohomological approach we introduce the notion of the obstruction class for an inner post-Lie algebra being induced by a Rota-Baxter operator, and show that an inner post-Lie algebra is induced by a Rota-Baxter operator if and only if the obstruction class is trivial. Similarly, we introduce the notion of the obstruction class for an inner post-group being induced by a Rota-Baxter operator, and prove a parallel result. Finally, we give some applications of inner post-Lie algebras and inner post-groups.

We study a two degrees of freedom Hamiltonian system describing the motion of a particle in a potential field of the form of $S^1$ symmetric double well, namely $V = - (x_1^2 + x_2^2) + (x_1^2 + x_2^2)^2$, known also as a champagne bottle potential. This system is completely integrable. The champagne bottle is the simplest member of a class of integrable systems that have no global action variables due to a non-trivial monodromy, Bates (1991). Beyond that, the geometric and dynamical properties of the system near the equilibrium are of primary interest. We calculate the Birkhoff normal form and the nontrivial action near the focus-focus singularity and obtain the semi-global symplectic invariant near focus-focus point, which is introduced by Vũ Ngọc (2003). Examples of such calculations are still few. We compare our result with the semi-global symplectic invariant of the spherical pendulum, calculated by Dullin (2013).

We investigate the category of “matricial order operator spaces,” which generalize operator systems, being equipped with both matricial norms and matricial order. For these objects, we develop duality theory. Taking a cue from the theory of ordered normed spaces, we introduce two important properties describing the interplay between order and norm – “normality” and “generation,” and show that they are dual to each other. As examples, we consider operator systems (in particular, C*-algebras), and Schatten spaces. We also describe the minimal and maximal matricial order structures (which, again, turn out to be in duality), and show how Banach lattices can be equipped with such structures.

In this work, we fully explore three refined convergence structures of the lowest-order rectangular Raviart-Thomas element in solving the Laplace eigenvalue problem. Firstly, the scheme possesses a property of supercloseness between the discrete eigenfunctions and the interpolated ones, so that post-processing can be easily constructed to improve the accuracy at most by one order. The essentially skillful method is the integral expansion for interpolation terms. Secondly, based on the supercloseness property, we derive the error expansions for not only simple eigenvalues but also multiple eigenvalues, and provide a rigorous proof for them, based on which Richardson extrapolation can be performed. As a byproduct, we prove that all eigenvalues converge from above. Moreover, by utilizing the supercloseness property and Rayleigh quotient analysis, we give a rigorous proof for the convergence behavior for multiple eigenvalues on uniform meshes for the problem on the square domain. Thirdly, the equivalence between the lowest-order rectangular Raviart-Thomas element and the enriched rotated bilinear element is also indicated. At the last of this work, several numerical experiments are designed to demonstrate our theory.

In this paper, we introduce and investigate a novel subclass $\Sigma(\theta, \lambda, \gamma)$ of meromorphic functions defined in the punctured unit disk ${D}^*$. This class is constructed utilizing a specialized generalized operator $W_{\alpha, \beta}$ associated with Wright function. We derive the exact integral representation and establish necessary and sufficient convolution (Hadamard product) conditions. Furthermore, sufficient conditions involving strict inequalities are provided for functions to be members of this class $\Sigma(\theta, \lambda, \gamma)$. Additionaly, by employing the properties of Carathéodory functions and the principle of mathematical induction, we establish coefficient estimates for functions belonging to this new class. Finally, as an applications, we the established coefficient bounds, we determine the precise radii of meromorphic starlikeness and meromorphic convexity of order $\rho$. The results presented in this study generalize several existing outcomes in geometric Function Theory.