Papers

We consider the dynamics of an idealised cross-diffusion model of biological invasion in which the diffusion of an invading population is inhibited by a resident population, which is in turn degraded by the former. We formulate and numerically solve the problem that describes the self-similar smoothing of an initially piecewise-constant invading population. In the limit the height ahead of the initial discontinuity vanishes, the speed of propagation also vanishes, but only logarithmically slowly. This result is confirmed by a matched asymptotic analysis. The singular nature of this limit indicates that the system does not permit the existence of compactly supported solutions that exhibit a moving front or interface, which has important implications for the simulation of more complex models that also feature this degenerate diffusive mechanism.

In this paper, we establish spreading speeds for a nonlocal Fisher-KPP equation with measurable time-dependent asymmetric kernels, without imposing an inequality between growth and diffusion or requiring the kernel to be positive near the origin. The kernel has a time independent upper bound which is thin-tailed. Both the kernel and the linear growth rate have uniform means. Under these assumptions, we determine the exact rightward and leftward spreading speeds and prove convergence to the positive equilibrium inside the spreading interval. The proof is based on comparison arguments and a family of compactly supported subsolutions. Explicit examples show that time-dependent dispersal kernels with the same time average can exhibit different propagation behaviors.

This paper develops a novel staggered discontinuous Galerkin (SDG) method for linear elasticity based on the Hellinger-Reissner variational principle. We construct symmetric stress spaces with normal continuity across element boundaries on arbitrary polytopal meshes, while approximating the displacement field using piecewise polynomial functions defined on the same meshes. The method is locking-free and satisfies a local balance of linear momentum and angular momentum. We present a comprehensive theoretical analysis, including proofs of stability and error estimates. The formulation admits a hybridizable structure, which significantly simplifies the numerical implementation. Numerical experiments validate the theoretical results and demonstrate the effectiveness of the proposed approach.

We study the problem of reconstructing the scattering and absorption coefficients in the radiative transport equation using internal data in quantitative photo-acoustic tomography (QPAT). In practical settings, however, this internal data is only partially available near the boundary, owing to medium's strong absorption and limitations of the measurement equipment. Our main contribution is the development of a method to recover these coefficients within a subregion where the internal data can be obtained with sufficient reliability.

In this work, we tackle the following question: Can neural networks trained with gradient-based methods achieve the optimal computational-statistical tradeoff in learning Gaussian single-index models? Prior research has shown that any polynomial-time algorithm under the statistical query (SQ) framework requires $\Omega(d^{s^\star/2}\lor d)$ samples, where $s^\star$ is the generative exponent representing the intrinsic difficulty of learning the underlying model. However, it remains unknown whether neural networks can achieve this sample complexity. Inspired by prior techniques such as label transformation and landscape smoothing for learning single-index models, we propose a unified gradient-based algorithm for training a two-layer neural network in polynomial time. Our method is adaptable to a variety of loss and activation functions, covering a broad class of existing approaches. We show that our algorithm learns a feature representation that strongly aligns with the unknown signal $\theta^\star$, with sample complexity $\widetilde{O} (d^{s^\star/2} \lor d)$, matching the SQ lower bound up to a polylogarithmic factor for all generative exponents $s^\star\geq 1$. Furthermore, we extend our approach to the setting where $\theta^\star$ is $k$-sparse for $k = o(\sqrt{d})$ by introducing a novel weight perturbation technique that leverages the sparsity structure. We derive a corresponding SQ lower bound of order $\widetilde{\Omega}(k^{s^\star})$, matched by our method up to a polylogarithmic factor. Our framework, especially the weight perturbation technique, is of independent interest, and suggests potential gradient-based solutions to other problems such as sparse tensor PCA.

We prove an exact inverse theorem for damped waves on $X=\mathbb{T}\times Y$ with arbitrary bounded measurable transverse damping $a=a(x)\geq 0$. Let $\Theta_a(r)$ be the least average of $a$ on radius-$r$ intervals. Then $\liminf_{r\downarrow 0}\Theta_a(r)>0$ characterizes exponential stability, while for every $L(S)=\ell(\log(eS))$ with $\ell:[1,\infty)\to[1,\infty)$ nondecreasing, unbounded, and eventually doubling, $\Theta_a(r)\gtrsim L(1/r)^{-1}$ is equivalent to stationary- and generator-resolvent bounds $O(|s|^{-1}L(|s|))$ and $O(L(|s|))$. One resonant scalar block per dyadic octave already recovers this multiscale mass bound. Every such critical profile occurs for indicator damping on an open dense set of arbitrarily small measure, whereas the stationary resolvent-to-mass implication fails at every sublinear power gauge. For the cusp $a(x)=(\log(e/|x|))^{-A}$, $A>0$, near its isolated damping well, we prove the sharp obstruction is genuine spectrum. Blow-up yields $-\partial_y^2+i\log|y|$, whose spectrum is a simple interlaced vertical ladder. Every fixed finite ladder portion transfers to damped-wave eigenvalues with complete blockwise algebraic count and two-term asymptotics. Combined with the resolvent theorem, these eigenvalues give the sharp regularized decay scale $\exp[-t^{1/(A+1)}]$.

We study the hyperbolic $O(N)$ linear sigma model, i.e. a system of $N$ interacting stochastic damped nonlinear wave equations (SdNLW) with coupled cubic nonlinearities, posed on the two-dimensional torus and indexed by a parameter $\varepsilon > 0$. We show that as $\varepsilon$ goes to zero (Smoluchowski-Kramers approximation) and $N$ goes to infinity (mean-field limit), each component of the solution to the SdNLW system converges to the solution to the stochastic nonlinear heat equation (SNLH) with a mean-field nonlinearity. We prove such convergence via two regimes: first with $\varepsilon$ going to zero to obtain the parabolic $O(N)$ linear sigma model, i.e. a system of $N$ coupled SNLH, and then with $N$ going to infinity; or first with $N$ going to infinity for each component to obtain the mean-field SdNLW and then with $\eps$ going to zero. As a result, we obtain a commutative diagram regarding the convergence from the hyperbolic $O(N)$ linear sigma model to the mean-field SNLH.

We develop new tools for determining the existence of models of specific Scott ranks under countability conditions. Using these, we improve a result of Sacks by showing that any counterexample to Vaught's conjecture must have at least two models of every parameterized Scott rank – a result that contrasts with the unparameterized case, where minimal counterexamples have only one model at many ranks. We further prove that theories with fewer than continuum many models have trivial Scott spectra and provide a general, systematic classification of low Scott rank models when only countably many $\Sigma_\alpha$-types are realized. Additionally, we classify the Scott complexity spectra for many Ehrenfeucht theories, and prove the $\omega$-Vaught's conjecture in this setting, answering an infinitary strengthening of a question of Pillay and Tanović. We demonstrate that the Scott complexity of prime models for $\omega$-stable first-order theories is commensurate with the complexity of the theory itself. Along the way, we apply our methods to concrete theories like p-groups, trees, and Boolean algebras, answering questions of Harris–Montalbán and Alvir–Csima–MacLean regarding specific structures.

We examine the newly defined class of {\it $n$-$UQ$ rings} described by the condition that $u^n - 1 \in QN(R)$ for every unit $u \in U(R)$, where $QN(R)$ denotes the set of quasi-nilpotent elements (see \cite{Tien}). This class naturally extends the recently defined class of rings in \cite{daoa} and \cite{dam}, as well as expectedly generalizes previously explored concepts such as $UJ$, $UU$ and $UQ$ rings. We conduct here a comprehensive structural analysis of these $n$-$UQ$ rings and study their stability under various ring-theoretic constructions including matrix rings, group rings, trivial extensions and power series rings. As a result, several new characterizations are established, thus revealing relevant connections between $n$-$UQ$ rings and fundamental classes of rings such as reduced, clean, exchange, semi-regular and potent rings, respectively. Moreover, we prove that the classes of $n$-$UJ$ and $n$-$UU$ rings are properly contained in the class of $n$-$UQ$ rings. These achievements not only unify and expand existing theories in this branch, but also provide a robust framework for possible further investigations into the interplay between the unit behavior and quasi-nilpotency in noncommutative ring theory.

In this paper, we study the existence of invariant sublinear expectations of Markovian semigroups on sublinear expectation spaces. To achieve this, we establish a complete metric space of sublinear expectations, on which we extend Harris' method to the nonlinear setting on the convergence of sublinear semigroups. We then explore two cases of $G\text{-}$diffusions by studying the Lyapunov function and the local Doeblin condition. One is the $G\text{-}$Brownian motion on the unit circle which is the case studied in Feng and Zhao(2021), but with the new method. Another is the multidimensional $G\text{-}$SDEs on the whole space $\mathbb{R}^d$. We establish, for the first time in the literature, the existence of the invariant sublinear expectation for $G\text{-}$SDEs under the non-degenerate and weakly dissipative assumption. For this, we prove that for a class of $G\text{-}$SDEs, the $G\text{-}$expectation can be represented as the supremum of the semigroup of a family of SDEs, of which the regularity is obtained by considering { oscillations and coupling of the nonlinear Markovian semigroup,} and the Denis-Hu-Peng representation for the distribution of $G\text{-}$Brownian motions.

We propose a Koopman-enhanced physics-informed neural network (K–PINN) framework for parameter inference and forecasting in nonlinear epidemic models. This method combines Koopman operator theory and physics-informed learning. It maps epidemic states into a latent observable space where the dynamics evolve approximately linearly while satisfying the governing epidemic equations through automatic differentiation. This integration improves interpretability, parameter identifiability, and long-term predictive stability. We apply the proposed framework to a normalized SEIRSD epidemic model and evaluate it using synthetic monkeypox (Mpox) data and real-world datasets from Germany, Morocco, and Sweden for the SARS-CoV-2 virus. Synthetic trajectories are generated using a structure-preserving, nonstandard finite difference scheme to ensure reliable training data. Numerical results demonstrate that K–PINN achieves more accurate parameter estimation, trajectory reconstruction, and long-term forecasting than classical PINNs and Koopman-EDMD approaches. These results suggest that K–PINN is an effective machine learning framework for epidemic modeling that can be extended to more complex systems.

We consider highness in the context of computable structure theory and, particularly, the Scott rank of a structure. We define highness for Scott rank and highness for computably defined Scott rank $\alpha$ and characterize them in terms of the ability to compute $\Delta^0_\beta$ sets for appropriate $\beta$. We close with a discussion of the index sets of structures with a given Scott rank or computably defined Scott rank and a few words about highness for noncomputable Scott ranks.

This paper studies local controllability of nonlinear control-affine systems subject to state-dependent box constraints that strictly exclude the zero input. Such constraints arise naturally in cable-driven robots and other systems with strictly positive actuation, but fall outside classical small-time local controllability theory and existing frameworks for positive or cone-constrained controls. We introduce the admissible balancing set, an input-space object that classifies reference states without requiring the control distribution to have full rank. When an admissible balancing input lies in the interior of the input set, a locally uniform input shift recovers a symmetric-control system, allowing classical accessibility and small-time local controllability criteria to be applied. When no admissible balancing input exists, the feasible-velocity set is strictly separated from the origin. We show that the resulting separating covector defines a local barrier functional that increases at a uniform positive rate along every admissible trajectory, thereby providing a quantitative obstruction to small-time local controllability. This obstruction does not exclude finite-time reachability through trajectories leaving the barrier neighborhood, which motivates the notion of admissible excursions. The framework is illustrated on an underactuated planar cable-driven parallel robot, for which the barrier is certified numerically over a prescribed state neighborhood.

In this article, we prove a variant of domain monotonicity property for the Steklov eigenvalues of the Laplacian on strictly star-shaped bounded domains contained in $\mathbb{R}^n$. As an application of this result, we obtain an upper and a lower bound for Steklov eigenvalues on convex domains.

The moduli spaces $\mathcal{S}_g^-$ parametrise odd spin curves of genus $g$. These are pairs $[C, \eta]$ where $C$ is a smooth genus $g$ curve of and $\eta$ is a line bundle on $C$ such that $\eta^{\otimes 2} = \omega_C$ and $h^0(C, \eta)$ is odd. The main result of this work is the tautology of the Chow ring of $\mathcal{S}_5^-$. Our method of proof revolves around an analysis of the geometry of canonical genus 5 curves and totally tangent hyperplanes. In the course of establishing our main result, we also prove the rationality of the closely related differential stratum in $\mathcal{M}_{5, 4}$ dominating $\mathcal{S}_5^-$.

We prove the existence of forces and smooth initial data such that the associated Leray-Hopf solution of the 3D Navier-Stokes equations is unique, global-in-time, satisfies the energy equality, and has infinitely many (countable) blow-up instants. Different classes of forces and blow-up strengths are considered. All of them are sharp compared to the boundedness statements in literature. Our method also enables us to construct examples of blow-up for the forced 3D Euler equations.

We prove that the Opdam-Cherednik kernel, also known as the nonsymmetric Opdam hypergeometric function, can be written as the Laplace transform of a positive measure supported on the convex hull of the Weyl group orbit of its argument. As a consequence, the trigonometric Dunkl intertwining operator is positivity preserving. The main ingredient in the proof is a new formula for the Opdam-Cherednik kernel as a degeneration of nonsymmetric Macdonald polynomials. As a further application, we prove majorization inequalities for Macdonald polynomials and Heckman-Opdam hypergeometric functions associated with arbitrary root systems.

Dynamic metasurface antennas (DMAs) promise to relieve massive multiple-input multiple-output architectures from their high energy consumption and hardware costs. This paper proposes a two-timescale design for downlink multiuser transmission via DMAs, a design that balances pilot overhead, complexity, and spectral efficiency. At the onset of each frame, the DMA coefficients are configured based only on statistical channel-state information (CSI), a process for which the paper introduces an optimization framework that is shown to outperform the widely used stochastic successive convex approximation method. Then, within each frame, the digital precoder is updated at each slot, based on the optimized DMA coefficients and the effective lower-dimensional instantaneous CSI. The weighted minimum mean-squared error method is applied for this short-term optimization and, for the special case of single-user transmission, a closed-form solution for the digital precoder is provided. Performance evaluations demonstrate that the proposed two-timescale design can be an attractive ingredient for future wireless networks.

This letter revisits the classical PBH test through the lens of finite-horizon reachability. By casting state transfer as a minimum energy, primal optimization problem, we show that unreachable state-space maneuvers admit dual infeasibility certificates. These certificates are computable without forming the controllability matrix meaning that uncontrollability can be efficiently certified. We prove that any such certificate is a linear combination of uncontrollable generalized eigenvectors, thereby providing a spectral interpretation without a global eigendecomposition. We also devise algorithms based on Krylov sub-space methods that extract some of the uncontrollable PBH modes from a certificate and demonstrate favorable scaling on large dynamic networks with thousands of nodes.

In this paper, we focus on the de Moivre random experiment, which allows us to introduce the $s$-Bernoulli distribution and the bi$^{s}$nomial distribution. We present some probabilistic properties such as the expectation, the variance, the skewness and kurtosis coefficients, the moments, the cumulants and the generating functions; the moments are expressed through extended Eulerian numbers and partial Bell polynomials, and the cumulants through logarithmic polynomials. Then we establish that, for an integer $s\geq 2$, the bi$^{s}$nomial distribution converges to a Poisson limiting distribution when the expectations converge, and that, for fixed parameters with $p\neq q$, it satisfies a central limit theorem as $n\rightarrow\infty$.

The underdamped Langevin equation is a fundamental model in statistical mechanics for sampling Gibbs measures and simulating molecular dynamics, for which numerical methods with uniform-in-time weak convergence are essential for accurately reproducing long-time statistical observables and invariant measures of the underlying dynamics. Currently, such uniform-in-time weak convergence is established for implicit schemes, but remains unknown for explicit ones under polynomially growing potentials. To improve efficiency in long-time simulations, we propose the first explicit numerical method for the underdamped Langevin equation with polynomially growing potentials that is proven to achieve uniform-in-time weak convergence. The explicit numerical method is constructed by introducing a dissipativity on the scalar auxiliary variable (SAV), which we call the DSAV method. The proposed DSAV method enables the approximation of the invariant measure for the underdamped Langevin equation with a precision of $\varepsilon$ at a significantly reduced computational cost of $\mathcal{O}(\varepsilon^{-1} \log(\varepsilon^{-1}))$. In addition, we establish the existence and positivity of the density function of the numerical solution without using the Malliavin calculus. Numerical experiments are performed to verify the theoretical findings and demonstrate the long-time stability of the proposed numerical method.

We develop a generalized long-wave limit method for constructing higher-order (degenerate) lump solutions of $(2+1)$-dimensional integrable systems. Starting from the multi-soliton tau function, we introduce wavenumber rescalings, spectral perturbations, and higher-order phase modulations in a single small parameter, and choose their coefficients so that the lower-order Taylor coefficients vanish. Applying this framework to the Kadomtsev–Petviashvili I equation, we explicitly derive second-order two- and three-lump solutions as well as third-order three-, four-, five-, and six-lump solutions. The long-time asymptotics of the peak positions and amplitudes describe anomalous scattering of these degenerate lump solutions. Extensions to hybrid higher-order lump solutions associated with multiple distinct spectral parameters are also discussed.

We define and investigate some preliminary properties of an integral of vector valued functions and its canonically associated indefinite capacity. More precisely, we adapt the Dunford integral to the Choquet integration setting for Banach-valued functions.

We consider the two-dimensional incompressible Navier-Stokes equations with supercritical fractional dissipation in the vorticity formulation. In self-similar variables, we analyze the linearized operator in weighted spaces, prove a spectral gap, and construct a finite-dimensional local slow invariant manifold for small solutions. As a consequence, solutions are attracted to this manifold and admit an explicit long-time asymptotic expansion determined by the leading eigenmodes; additional moment conditions yield faster decay. We also show Lipschitz dependence of the manifold on the dissipation exponent, and recover the classical Navier-Stokes dynamics in the limit as the exponent approaches one.

This article presents the numerical analysis for the biharmonic wave equation with clamped boundary conditions employing two variants of the {hybrid high-order} method for the space discretization and two implicit time-stepping schemes for the time discretization. The Newmark scheme directly discretizes the second-order time derivative, while the Crank-Nicolson scheme discretizes a reformulated system where we introduce velocity as an independent variable to create coupled first-order equations. Optimal orders of convergence in space and time are achieved for both schemes. The numerical experiments validate the theoretical convergence rates and show the effectiveness of the proposed methods. To the best of our knowledge, this is the first work in literature that addresses hybrid-high order method and implicit time schemes for the biharmonic wave equation.

We prove that every natural number is a finite sum of distinct unit fractions whose denominators are semiprimes (products of two distinct primes). This is the $\omega=2$ integer case of a problem of Erdős and Graham, stated only as a conjecture by Butler, Erdős and Graham (Integers 15 (2015), A51), who proved the $\omega=3$ analogue. Counterintuitively the problem hardens as $\omega$ decreases – the induction's feed thins – so $\omega=2$ is the hard case; our proof adapts the Butler-Erdős-Graham induction to this thin-feed regime, where the entire content of the induction step reduces to an explicit onset inequality $Y_0(N)\le\min\{\beta(N),\beta'(N)\}$, proved for all $N\ge10$ by Olson's addition theorem and elementary Chebyshev bounds above a finite, machine-checked base range. The same engine extends to the rationals: for every squarefree $b$, every $a/b$ above an explicit threshold $\min\{B_{N_b}/6,\,1/5\}$ is $\omega=2$ representable, unconditionally. As an application we give the first complete proof of the rational $\omega=3$ statement – every $a/b$ with squarefree $b$ is a sum of distinct sphenic unit fractions – that Butler, Erdős and Graham conjectured but left unpublished; a descent settles every $\omega\ge3$. What remains open is the $\omega=2$ regime below this threshold, which we reduce to a single explicit conjecture – that the gap-free floor of a semiprime subset-sum set tends to zero. This work is a human-AI collaboration: AI tools (notably Anthropic's Claude, used through Claude Code) contributed substantially to the Lean formalisation, the experiments, and the writing; correspondingly, every result is machine-checked in Lean 4 / Mathlib (no sorry; two cited classical axioms, plus the native_decide compiler-trust base for the finite computations), so its correctness is independent of the tools used.

We develop a uniform skew Murnaghan–Nakayama theory for graded Hopf dual pairs equipped with a nondegenerate Hopf pairing. Using the completed Cauchy element, its grouplike factorization, and the resulting partial contraction operators, we establish a general skew Cauchy identity together with an abstract skew Murnaghan–Nakayama rule. Specializing this framework recovers and extends the classical skew Murnaghan–Nakayama rule for symmetric functions, and yields new skew Murnaghan–Nakayama formulas in several settings, including the dual pairs $(\mathrm{NSym}, \mathrm{QSym}) $ and $(\Lambda^{(k)}, \Lambda_{(k)}) $ arising in $k$-Schur theory, as well as the type $C$ affine Grassmannian context. As applications, we obtain generating functions for irreducible characters of Ariki–Koike algebras, including their type $A$ and type $B$ specializations, as well as Hecke–Clifford algebras and $\mathfrak q$-rook monoid algebras. We also give ribbon-tableau expansions for skew $(q,t)$-Kostka polynomials and for the entries of the inverse transition matrix, thereby answering a question of Carbonara (1998). Finally, by specializing the auxiliary alphabet $Y$ to sums of powers of primitive roots of unity, we derive a skew plethystic Murnaghan–Nakayama formula together with a Schur expansion for skew modular Schur functions; as a further consequence, we confirm Walker's conjecture (1994) by showing that if the transition from the modular Schur functions to the Schur basis is trivial in the row indexed by $\lambda$, then $\lambda$ must be a $k$-core.

The main result of this paper is the construction of new sequences of rational approximations to the Lerch function. This construction is based on a generalization of the "remainder Pade approximants" method introduced by the author in 1996. More recently, this method has been applied, in the form of remainder Pade-type approximants, to the approximation of Stieltjes constants.

In this paper we establish a means of using the combinatorics associated to a general monomial ideal $I$ in a polynomial ring $R$ to find a regular sequence of linear forms on $R/I$. The sequence of linear forms provides an effective lower bound on ${\rm{depth}}(R/I)$. When $I$ is the edge ideal of a graph, we provide conditions under which this bound is an equality, allowing the realization of the depth via a regular sequence of linear polynomials. In addition, we explicitly describe the minimal primes of $(I, f_1, \ldots, f_q)$, when $f_1, \ldots, f_q$ are homogeneous polynomials of degree one with pairwise disjoint support and $I$ is any monomial ideal. Finally, we propose a conjecture on the form of all associated primes of the ideal $(I, f_1, \ldots, f_q)$, when $I$ is the edge ideal of a graph and $f_i$ are disjoint stars on $I$.

We study the elliptic family $E_t:Y^2=X^3+4X^2+16t^2$ arising from rational $2$-cycles of $x^3+bx+a$. For every $t\in\mathbb{Q}^\times$, we prove that $P_t=(0,-4t)$ has infinite order, so every nonsingular rational fiber has positive rank. We determine its rational torsion subgroup: it is cyclic of order $2$ precisely when $t=u(u^2+4)/4$ for some $u\in\mathbb{Q}^\times$, and is trivial otherwise. No such fiber admits a rational $3$- or $5$-isogeny. An explicit birational dictionary then shows that, for every fixed $a\in\mathbb{Q}^\times$, infinitely many $b\in\mathbb{Q}$ yield a rational $2$-cycle of $x^3+bx+a$. The uniform non-torsion assertion follows from Nagell–Lutz integrality. The torsion exclusions combine elementary $2$-descent with explicit genus-$3$ curves, an unconditional rank-zero Prym argument, and two-cover descent with elliptic Chabauty. Exact Magma and SageMath certificates accompany the computer-assisted steps.