Papers

This paper presents a method to approximate regions of attraction of unknown nonlinear dynamical systems from data. Assuming point-wise evaluations of the vector field and known Lipschitz bounds, a polyhedral uncertainty set of admissible dynamics is constructed. This uncertainty description enables the synthesis of a continuous piece-wise affine Lyapunov candidate via a linear program, enforcing a robust decrease condition for all admissible vector fields. The approach allows certification of a region of attraction consistent with the available data. Numerical examples illustrate the effectiveness of the proposed method in extracting certified regions of attraction from sparse data.

Infinite-dimensional orthonormal basis expansions play a central role in representing and computing with function spaces due to their favorable linear algebraic properties. However, common bases such as Fourier or wavelets are fixed and do not adapt to the structure of a given problem or dataset. In this paper, we aim to represent these bases with neural networks and optimize them. Our key idea is that any target infinite-dimensional orthonormal basis can be viewed either as a point on the Lie manifold of the orthogonal group, or equivalently, as the endpoint of a continuous path on that manifold that connects a reference basis, e.g. Fourier, to that target. Paths on the Lie manifold satisfy ordinary differential equations (ODEs) governed by skew-adjoint integral operators. Using neural networks to define finite-rank generators of such ODEs allows us to parameterize and optimize orthonormal bases in function space. While relying on finite-rank generators to model infinite operators might seem restrictive, we prove a universality result: even with a rank-2 generator, the integrated solutions of the ODE are dense in the orthogonal group under the appropriate operator topology. In other words, for any target orthonormal basis, there exists a path originating from a reference basis and driven by finite-rank generators that gets arbitrarily close to that target basis. We demonstrate the flexibility of our framework by transforming the Fourier basis into the principal components of a functional dataset, eigenfunctions of linear operators, or dynamic modes of energy-preserving physical simulations.

We present a novel parallel implementation for large-scale three-dimensional electromagnetic inversion based on a Gauss-Newton framework combined with a rational near-best approximation of the matrix exponential for transient simulations. The method employs parallel direct solvers for the shifted linear systems arising from the partial fraction representation of the rational approximation and demonstrates efficient parallel execution on a shared-memory architecture using MPI. A key property of the approach is that the time dependence is entirely contained in the residuals of the employed rational functions, such that the computation of forward responses and sensitivities becomes effectively independent of the number of desired observation times. Model regularization is done with smoothness constraints, formulated with Raviart-Thomas elements. The linearized inverse problems are solved using LSQR, using an implicit parallel Jacobian operator. Numerical experiments demonstrate the successful recovery of a synthetic 3-D conductivity structure with approximately 700,000 degrees of freedom. The study further discusses computational bottlenecks related to memory consumption and shared-memory scalability arising from the simultaneous storage of multiple sparse matrix factorizations. Possible improvements based on preconditioned iterative solvers and distributed high-performance computing architectures are outlined. The implementation in the Julia programming language is released as open-source software to support reproducible research and further development by the geophysical inversion community.

We investigate normalized groundstates for mixed $(p,2)$-Laplacian equations \begin{align*} \begin{cases} -\Delta_p u-\Delta u+\lambda u=f(u) & \text{in } \mathbb{R}^2, \displaystyle \int_{\mathbb{R}^2}|u|^2\,\mathrm{d}x=m, u\in H^1(\mathbb{R}^2)\cap D^{1,p}(\mathbb{R}^2), \end{cases} \end{align*} where $\Delta_p$ denotes the $p$-Laplacian with $1<p<2$, $\lambda\in\mathbb{R}$ represents a Lagrange multiplier and the nonlinerity $f$ exhibits exponential critical growth. Compared to the single-Laplacian case, the lack of regularity here precludes the Pohozaev identity, and the exponential critical growth severely compromises the restoration of compactness. To address these issues, we introduce a refined Moser iteration technique adapted to exponential critical growth, which establishes the Pohozaev identity for weak solutions under the mere assumption of $C_{\mathrm{loc}}^{1,\alpha}$-regularity. By combining constrained minimization on the Pohozaev manifold within a closed $L^2$-ball with a minimax characterization, we prove the existence of normalized groundstates for any prescribed mass $m>0$. Notably, our approach works independently of the sign of the Lagrange multiplier $\lambda$, thereby surmounting the fundamental barrier in recovering compactness for mixed Laplacian problems.

Gradient flows of unit vector fields arise in a wide range of physical models such as harmonic map heat flows, nematic liquid crystals, and magnetization dynamics. Designing numerical schemes that simultaneously preserve the unit length constraint and dissipate energy is essential for reliable simulations of such systems. Although projection methods can effectively enforce the unit length constraint, ensuring energy dissipation under projection, especially in high-order schemes, remains challenging. Unlike traditional implicit-explicit Runge-Kutta (IMEX-RK) methods, in this work we propose a general methodology for constructing product-type IMEX-RK schemes that offers greater adaptability to various models with the goal of designing structure-preserving numerical schemes. For gradient flows of unit vector fields with Dirichlet energy, we design a linear and second-order numerical scheme that simultaneously preserves energy dissipation and the unit length constraint by using product-type IMEX-RK methods and projection techniques. Numerical experiments verify the accuracy, stability, and structure-preserving properties of the scheme. According to our best knowledge, this is the first second-order linear scheme that can preserve both the unit length and the original Dirichlet energy for harmonic map heat flows.

Stochastic infection processes are continuous-time Markov chains on graphs that assign each vertex one of multiple states, such as susceptible, infected, or recovered. Depending on the model, vertices change their state based on random transition rates and the states of their neighbors, resulting in a variety of complex dynamics. The body of rigorous literature is rich for processes that consider a single infection. In contrast, the setting with at least two infections, where the same state exists for different types, allows for far more transition combinations, leaving several interesting models entirely unexplored. We address this shortcoming in the literature by defining the IRIR process, in which two SIR processes run on the same graph and each vertex is immune only to its most recent infection. We study the survival time of the IRIR process, that is, the time until no infected vertex remains, with mathematical rigor. Our main result is a tight threshold, known as epidemic threshold, where the survival time rapidly changes from at most quasi-linear in the graph size $n$ to at least super-polynomial in $n$. This result is applicable to perfectly mixed graphs, which are graphs where the density of edges between each non-empty subset of vertices is a given value $p \in (0, 1]$. Our super-polynomial lower bound extends to jumbled graphs, which allow for some more flexibility in the density. In particular, this includes with high probability Erdos-Renyi graphs with an average degree of $k\in \omega(\ln^2(n))$. Our proof for the lower bound is based on a potential that transforms the configurations of the IRIR process to a supermartingale with drift in a large region, implying the lower bound. We detail how to systematically derive such a potential, based on a Lyapunov function of the transition equilibrium of the process.

We survey two new compactification methods for the KSBA moduli space of general type surfaces so that both of them admit a perfect obstruction theory. Virtual fundamental classes exist on these two moduli spaces, and tautological invariants can be defined on KSBA moduli spaces. This is the starting point to do enumerative geometry on KSBA moduli spaces, and we include some discussions in this direction.

In this paper, a novel dual-field structure-preserving mixed finite element discretization for the incompressible Hall magnetohydrodynamics (MHD) equations is introduced. It preserves pointwise mass conservation, magnetic Gauss law, and conservation of current density. It also satisfies a discrete energy law that exactly captures the correct energy dissipation rate in the dissipative regime and recovers energy conservation in the ideal limit. Numerical experiments are presented to verify the temporal and spatial accuracy and to illustrate the structure-preserving properties of the scheme.

This paper develops a time-inconsistent and path-dependent singular control framework incorporating a running minimum process. We derive a verification theorem that characterizes equilibria under substantially weaker regularity conditions than those imposed in the existing literature, and we obtain a stronger notion of equilibrium by enlarging the class of feasible perturbations. We first establish the mathematical foundations of the framework by proving the existence and uniqueness of strong solutions to a class of Skorokhod reflection problems involving the running minimum and by characterizing admissible singular control laws. We further demonstrate the existence of an equilibrium through a dividend problem, where the running minimum leads to a highly coupled and nonlinear differential-algebraic system. For this problem, we prove the monotonicity and local concavity of the dividend boundary, thereby providing a mathematical explanation for dividend smoothing and scarring effects. Numerical simulations confirm the robustness of the equilibrium across a wide range of parameter values.

The simulation of quantum transport in nanodevices requires the solution of the Dyson and Keldysh equations, a task dominated by the inversion of massive, block-tridiagonal matrices. While the Recursive Green's Function (RGF) method has long been the standard $O(N)$ solver for quasi-1D systems, its formulation has typically been restricted to sequential execution and nearest-neighbor interactions. In this work, we carefully reformulate RGF through the lens of Domain Decomposition and Schur Complement theory. This allows us to extend the recursive formalism to block $n$-diagonal systems (handling higher-order stencils) and to derive a parallel algorithm, Domain-Decomposition based RGF (DDRGF), which stitches macroscopic domains via reduced interface systems. We explore data dependencies in DDRGF in detail, by means of block-sparse structures and tracing back to the desired output as a block tridiagonal approximation, giving a clear, reproducible and extensible formulation. We validate these algorithms using LibNEGF.jl, a Julia-based implementation, demonstrating that the structural insights of domain decomposition provide a robust pathway for high-performance quantum transport simulations on modern multi-core clusters. The theory presented here lays down the base for tackling the Keldysh problem, to be similarly handled in future stages of our work. Although the target here is the acceleration of kernels in the non-equilibrium Green's function method, the algorithms and the implementations presented can be immediately used in any application involving block $n$-diagonal systems.

The Fornberg-Whitham (FW) equation was introduced by Fornberg and Whitham [Fornberg and Whitham, Phil. Trans. R. Soc. Lond. A (1978)] as a nonlocal model for unidirectional shallow water waves capable of capturing wave steepening and breaking. Despite its similarities with integrable shallow-water equations, the FW equation is not completely integrable. Nevertheless, the FW equation is part of the family of peakon-type models as it supports peaked traveling wave solutions. In this paper, we consider smooth solitary wave solutions to the FW equation. We use a variational approach to show that some are orbitally stable.

The set of tritangent planes to smooth tropical space sextic curves has 15 connected components, recording continuous displacements of planes preserving the tritangency condition. These 15 tritangent classes are polyhedral complexes in $\mathbf{R}^3$, and each of them contains the tropicalization of precisely eight tritangent planes to any smooth space sextic curve with the given tropicalization. Prior joint work of the authors with Len confirms that each tropical tritangent plane has 0, 1, 2, 4 or 8 lifts to classical tritangent planes defined over the algebraic closure of the field over which the original algebraic curve is defined. Our main theorem states that when the input classical curve is generic, then only six out of the ten possible partitions of 8 into powers of 2 arise from lifting multiplicities of tritangent classes. Furthermore, we show that these partitions are completely determined by the dimension of a suitable connected subcomplex of the class and the existence of a member with a tropical tangency of a predetermined combinatorial type.

We derive a recursive formula for the moments of the number of flips using a possibly biased coin to produce a prescribed finite binary string $S$ when $S$ is either a run of heads or a run of heads followed by a tails. Our recursive formula involve certain sums, which we simplify by using a one-parameter extension of the well-studied Eulerian number, which belongs to the two-parameter family of numbers introduced by Graham, Knuth, and Patashnik. We also use the Goulden–Jackson cluster method and Faà di Bruno's formula to establish a closed formula for the moments in a more general situation where a die having an arbitrary number of faces with possibly different probabilities is rolled repeatedly until a prescribed finite word occurs.

This paper studies a uniform projection criterion for space-filling designs under the stratified $L_2$-discrepancy. The criterion, denoted by $\Phi_{SD}$, is the average squared stratified $L_2$-discrepancy over all two-dimensional projections. For U-type $(n,m,s^p)$ designs, we derive an explicit formula for $\Phi_{SD}$ in terms of row-pairwise weighted hierarchical distances, and we establish sharp lower and upper bounds with equality conditions. We further show that many known optimal constructions attain the lower bound of $\Phi_{SD}$, and that designs attaining the lower bound of the full stratified $L_2$-discrepancy also attain the lower bound of $\Phi_{SD}$. The criterion can be evaluated in $O(n^2m)$ time, with a modest reduction in arithmetic operations compared with direct projection-wise evaluation. Numerical studies illustrate the theoretical results and show that $\Phi_{SD}$ is effective for assessing low-dimensional projection uniformity.

We explore log Manin's conjecture for integral points and its connections to $\mathbb A^1$-connectedness. We prove log Manin's conjecture for Campana rational curves and for $\mathbb A^1$-curves on split toric varieties. Our arguments combine the Cox ring description of the moduli space of rational curves with Batyrev's heuristic-type counting arguments. As our proofs are geometric in nature, they give a geometric explanation of the mysterious leading constant for Campana points proposed by Chow–Loughran–Takloo-Bighash–Tanimoto.

We show that a generic analytic strongly convex billiard is "maximally chaotic" in the sense that, for every rational number $\frac{p}{q} \in \mathbb{Q} \cap (0,1)$, all intersections between the stable and unstable manifolds of maximizing periodic orbits with rotation number $\frac{p}{q}$ are transverse.

Monitoring the integrity of elastic structures using ultrasonic waves requires the efficient identification of material parameters from measured surface displacements. The displacement field is governed by Cauchy's equation of motion, i.e., an elastic wave equation. Consequently, defect localization leads to a high-dimensional spatial parameter identification problem for a hyperbolic system with given initial and boundary conditions. Stable parameter reconstructions typically rely on regularization techniques such as the iteratively regularized Gauss-Newton method (IRGNM). However, its practical application is computationally demanding due to the high-dimensional nature of the problem. To address this bottleneck, we propose a reduced-order modeling approach that simultaneously reduces the state and parameter spaces using adaptively constructed reduced-basis spaces. This yields online-efficient surrogate models for both the forward and adjoint evaluations required in derivative-based optimization. To ensure reliability, the IRGNM iteration is embedded into an adaptive, trust-region framework that provides accuracy of the reduced-order approximations. The approach extends our recent contributions, which focus on elliptic and parabolic problems, to the hyperbolic setting. We demonstrate the reliability and effectiveness of the method for defect detection through numerical experiments.

Motivated by multimodal estimation, we study a multi-view spiked Wigner model in which several noisy matrix observations contain correlated latent spikes. We derive a spectral estimator for the latent spikes by linearizing approximate message passing (AMP). Our main result is an explicit sharp transition formula for its spectrum: for $L \geq 2$ views, letting $\lambda$ be the $L$-dimensional vector of spike strengths and $B$ the $L\times L$ limiting Gram matrix of the spikes, the critical parameter is $\mathsf{SNR}(\lambda,B)=\lambda_{\max}[\mathrm{Diag}(\sqrt{\lambda}) (B \odot B) \mathrm{Diag}(\sqrt{\lambda})]$. When $\mathsf{SNR}(\lambda,B)<1$, the linearized AMP matrix has no outlier beyond the right edge of its bulk spectrum. When $\mathsf{SNR}(\lambda,B)>1$, an informative outlier is pinned at the distinguished point $1$, and the associated eigenvector has explicit, nontrivial overlaps with the latent signals. Thus $\mathsf{SNR}(\lambda,B)=1$ gives the exact spectral weak-recovery threshold for the linearized AMP method. To establish our results, we analyze the correlated Gaussian noise matrix through a matrix Dyson equation and combine this deterministic description with finite-rank perturbation arguments adapted to the multi-view spike structure. We also show that, for a broad class of spike priors, the spectral threshold $\mathsf{SNR}(\lambda,B)=1$ coincides with the information-theoretic threshold for weak recovery, ruling out a statistical-computational gap for this class of priors.

Reaction-diffusion epidemic models with vital dynamics are an important framework for describing the spatial and temporal spread of infectious diseases. In this work, we present a constraint-aware, physics-informed neural network (PINN) approach to an SEIR reaction-diffusion system with homogeneous Neumann boundary conditions. Due to the scarcity of spatial epidemiological datasets, we generate synthetic benchmark data using structure-preserving implicit-explicit nonstandard finite difference (NSFD) schemes that ensure positivity, boundedness, and numerical stability. The PINN framework integrates PDE residuals, observational data, boundary conditions, and epidemiological constraints within a unified optimization procedure. Specifically, the loss function incorporates the non-negativity of compartment populations and the admissibility of epidemiological parameters. We apply the method to forward simulation and inverse parameter estimation in one- and two-dimensional settings. Numerical experiments demonstrate the framework's ability to accurately reconstruct spatiotemporal epidemic dynamics and reliably identify parameters, even when data is sparse or noisy. These results underscore the potential of constraint-aware PINNs as a robust, data-driven methodology for spatial epidemic modeling.

Assume that $X$ is a connected, open, oriented smooth surface, $B$ is a compact Euclidean neighbourhood retract, and $\mathscr{J}=\{J_b\}_{b\in B}$ is a continuous family of complex structures on $X$ of local Hölder class $\mathscr{C}^\alpha$ for some $0<\alpha<1$. We construct a continuous family of $J_b$-conformal minimal immersions $u_b:X\to \mathbb{R}^3$, $b\in B$, properly projecting to $\mathbb{R}^2$ and having an arbitrary given family of flux homomorphisms ${\rm Flux}_{u_b}:H_1(X,\mathbb{Z})\to\mathbb{R}^3$. In particular, there are continuous families of proper $J_b$-holomorphic null immersions $X\to \mathbb{C}^3$ and of proper $J_b$-holomorphic immersions $X\to\mathbb{C}^2$, $b\in B$.

Many barred galaxies, including the Large Magellanic Cloud (LMC), display strong lopsidedness and off-centre bars. The dynamical connection between bar-disc misalignments, internal mass asymmetries, and arm morphology is not yet fully characterised. We investigate how internal mass imbalances within the bar and global offsets between the bar and the centre of mass of the system modify the equilibrium-point structure and the invariant manifolds that organise arms. We construct a barred galaxy model which includes an off-centred and asymmetric in shape bar. Using numerical continuation, we track the position, stability, and bifurcations of the Lagrangian equilibrium points as functions of the displacement of the asymmetric mass component along the bar and of the offset between the bar and the system's centre of mass. For representative configurations we compute the invariant manifolds of planar Lyapunov orbits around unstable points and analyse the resulting arm structures. Internal bar lopsidedness and modest bar-disc offsets that keep the centre of mass inside the bar preserve the classical configuration with five equilibrium points, but strongly distort the associated invariant manifolds, producing two arms with different densities and shapes. The bar-disc offset reaches a threshold at the point at which the galactic centre of mass exits the bar ellipsoid, in which a pitchfork bifurcation removes the collinear unstable points and the system transitions to a three-equilibrium-point configuration in which a single unstable point and its associated manifold supports one arm. This framework is compatible with the observed correlation between off-centre bars and photometric lopsidedness, and it provides a dynamical explanation for the strongly asymmetric arm morphology of galaxies such as the LMC.

In 1994, Orlik and Terao introduced a commutative Artinian analog S/I(A) of the Orlik-Solomon algebra of a hyperplane arrangement A to answer a question of Aomoto. A central topic of investigation in the study of Artinian algebras is the Weak Lefschetz Property (WLP). We analyze WLP for the Artinian Orlik-Terao algebra of graphc arrangements. Even for chordal graphs (which give rise to Koszul algebras) WLP sometimes fails; conversely an analysis of the state polytope shows WLP can hold even when WLP fails for all possible initial ideals. More generally, for any algebra with a tensor product decomposition, we construct canonical elements in the kernel of the multiplication map, refining previous results in the literature.

Given a decorated planar graph $(G,\omega)$, where $G$ is a planar graph and $\omega\in H^1(|\mathcal{Q}G|,\mathbb{Z})$ with $\mathcal{Q}G$ the directed medial graph of $G$, we call some angular functions $\omega$-compatible and study two distinct but related directed graphs: $\mathcal{L}(G,\omega)$, which is the directed graph of such functions, and $BMS(G,\omega)$, the directed graph of BMS states which are some pairs of $\omega$-compatible functions plus additional data. We give sufficient conditions for $\mathcal{L}(G,\omega)$ to be a graded distributive lattice, recovering Kauffman's Clock Theorem when $G$ is a knot diagram. We also define a potential on $\mathcal{Q} G$ and associate a representation of the corresponding quiver with potential to every BMS state. Under suitable assumptions, this construction yields an isomorphism between $\mathcal{L}(G,\omega)$ and the lattice of subrepresentations of a maximal representation, generalizing a result of Bazier-Matte–Schiffler.

This note records a common threshold/surplus decomposition for single-threshold stopping rules in the classical prophet inequality. The same decomposition is used to certify several deterministic thresholds, including the median, half-mean, and balanced-surplus thresholds, and to give an averaged certificate for randomized thresholds distributed as the maximum.

Neural operators are increasingly used as accelerators inside classical numerical methods, but it is rarely clear which architectural ingredients matter for which application. We answer this question for one important use case: the coarse-space correction inside a two-level preconditioner for discretised linear partial differential equations. We systematically vary four DeepONet-like architectures along two design axes: input discretisation (sampling versus integration against a basis) and source-term linearity. In doing this, we show that the favourable corner of this 2x2 design is occupied by a single architecture, the Neural Green's Operator (NGO), and that moving away from it produces predictable failure modes: highly non-normal preconditioned systems, breakdown of preconditioned conjugate gradients on self-adjoint problems, and stagnation on non-self-adjoint ones. Used as a coarse-space correction, the NGO matches the iteration count of an exact coarse solve on 2D diffusion and advection-diffusion problems. The principle generalises: integrating inputs against the basis used for the output is what allows a neural operator to serve as a Galerkin-type coarse-space correction.

We show that $E$-theory for locally compact Hausdorff spaces constitutes a six-functor formalism which is equivalent to the six-functor formalism of $\mathrm{E}$-valued sheaves. We furthermore show that the $E$-theory category for locales that can be written as unions of finite open sublocales is equivalent to the category of $\mathrm{E}$-valued cosheaves.

A linear code is said to be $\Delta$-divisible if the Hamming weights of all its codewords are divisible by $\Delta$. The $p$-adic valuation of a code is defined as the greatest integer $t$ such that the code is $p^t$-divisible. In this paper, we establish a divisibility criterion for trace codes. Specifically, this criterion provides a systematic method to determine the $p$-adic valuation of the associated trace code, thereby extending Ward's classical divisibility criterion from standard generating sets (or matrices) to generalized generator matrices over an extension field. Furthermore, we present two applications of our framework. The first application provides a concise proof of the celebrated divisibility results on semisimple Abelian codes established by Delsarte and McEliece. The second application establishes several explicit lower bounds on the $p$-adic valuation of the number of solutions over $\mathbb{F}_{q^m}$ (where $q = p^e$) to the Artin–Schreier type equation $ f(x_1,\ldots,x_k)=y^q-y $. In particular, under the coprime condition $\left(d,\frac{q^m-1}{q-1}\right)=1$, we determine the exact minimum $p$-adic valuation of the number of solutions when $f$ is restricted to homogeneous polynomials of degree $d$.

Deep learning (DL) has been widely used in future 6G physical layer communications, but task-specific DL models are difficult to generalize across different physical layer tasks. Recently emerging wireless foundation models demonstrate strong generalization capability. However, existing methods mainly adapt pretrained language/vision models or rely on CSI reconstruction objectives for pretraining, with limited use of channel knowledge, and thus have limited performance. To address this limitation, we propose SPA-MAE, a physics-guided wireless foundation model by exploiting the adapted MAE backbone and channel knowledge. A physical prior module is developed to provide two complementary guidance signals in the pretraining stage. Specifically, the parameter-aware guidance branch extracts features from explicit multipath parameters and encourages the encoder output to align them, while the structure-aware guidance branch encourages the encoder to capture the sparse transformed-domain CSI structure obtained after a 2D FFT. After end-to-end learning, the MAE encoder will be retained for downstream tasks. Experiments on four wireless tasks show that SPA-MAE outperforms state-of-the-art CSI foundation models with smaller number of parameters, especially under low-SNR and limited-data conditions.

Let $G$ be a group and $N$ its normal subgroup. On the mixed commutator subgroup $[G,N]$, the mixed stable commutator length $\mathrm{scl}_{G,N}$ and the restriction of the ordinary stable commutator length $\mathrm{scl}_{G}$ are defined. We characterize when they are bi-Lipschitz equivalent by the vanishing of a certain $\mathbb{R}$-linear space $\mathrm{W}(G,N)$ related to invariant quasimorphisms. For the proof, we obtain a refined version of the generalized mixed Bavard duality theorem, and perform functional analysis on the completion of a certain space of $1$-chains.

In this paper, we introduce and study the notion of $n$-ary S-hyperideals in a Krasner $(m,n)$-hyperring