We consider series expansions in bases of classical orthogonal polynomials. When such a series solves a linear differential equation with polynomial coefficients, its coefficients satisfy a linear recurrence equation. We interpret this equation as the numerator of a fraction of linear recurrence operators. This interpretation lets us give a simple and unified view of previous algorithms computing these recurrences, with a noncommutative Euclidean algorithm as the algorithmic engine. Finally, we demonstrate the effectiveness of our approach on various examples.

We introduce Hyper Input Convex Neural Networks (HyCNNs), a novel neural network architecture designed for learning convex functions. HyCNNs combine the principles of Maxout networks with input convex neural networks (ICNNs) to create a neural network that is always convex in the input, theoretically capable of leveraging depth, and performs reliable when trained at scale compared to ICNNs. Concretely, we prove that HyCNNs require exponentially fewer parameters than ICNNs to approximate quadratic functions up to a given precision. Throughout a series of synthetic experiments, we demonstrate that HyCNNs outperform existing ICNNs and MLPs in terms of predictive performance for convex regression and interpolation tasks. We further apply HyCNNs to learn high-dimensional optimal transport maps for synthetic examples and for single-cell RNA sequencing data, where they oftentimes outperform ICNN-based neural optimal transport methods and other baselines across a wide range of settings.

It is a fundamental question in epidemiology to estimate, model and predict the growth rate of a pandemic. Analogously, analysing the diffusion of innovation, (fake) news, memes, and rumours is of key importance in the social sciences. The resulting epidemic growth curves can be classified according to their growth rates. These have been found to range from exponential to both faster super-exponential curves and slower subexponential or polynomial curves. Previous research has lacked a unified explanatory framework capable of accommodating super-exponential, (stretched) exponential, and polynomial growth patterns within the same contact network. In this paper we propose a simple agent-based network model that can capture all these phases. We provide such a framework by modelling how transmission rates depend on spatial distance and on individuals' numbers of contacts. By comparing the growth rate of spreading processes with or without degree-dependent and/or distance-dependent contact rates through data-driven and synthetic simulations on real and modelled networks with underlying geometry, we find evidence that even a 'sublinear presence' of these causes may cause a significant slow down of the growth rate on the same underlying network. We find that the growth rate is governed by a combination of three factors: geometry, the prevalence of weak ties, and superspreaders. We confirm our results with rigorous proofs in a theoretical model, using a spatial multiscale-argument in long-range heterogeneous first passage percolation. Our results give a plausible explanation of why the consecutive waves of a single pandemic can differ in their growth even if their spreading mechanisms are similar.

We establish that $C^\infty$ three-dimensional flows with positive topological entropy admit only finitely many ergodic measures of maximal entropy, even when singularities (zero-velocity points) are present. Furthermore, every ergodic measure of maximal entropy is rapid mixing for such flows within a $C^\infty$ open and dense subset. To prove this, we develop a novel symbolic coding system for flows with singularities, which serves as a fundamental tool in this work. We also define the strong positive recurrence (SPR) property for singular flows and verify that SPR flows can be coded by suspension flows of SPR symbolic systems. This framework extends to other singular flows, including star flows, and to equilibrium states.

We introduce SILAS, a data-driven framework for discovering polynomial ordinary differential equations (ODEs) with provably bounded trajectories. Boundedness is certified by compact absorbing sets defined via polynomial Lyapunov functions. We jointly identify the ODE vector field and the Lyapunov function using a well-posed nonconvex optimization problem built using polynomial optimization tools. We solve this problem using an alternating block-coordinate optimization scheme with convex subproblems, whose feasibility is ensured by a novel model-agnostic initialization that identifies a candidate Lyapunov function from data. Our methods extend prior approaches for quadratic ODEs with absorbing ellipsoids to a significantly broader class of ODEs and absorbing sets. A suite of over 100 examples demonstrates that SILAS can recover accurate and provably bounded ODE models for a broad range of nonlinear dynamical systems.

The Alternating Direction Method of Multipliers (ADMM) is a widely used method for structured convex optimization, and its practical performance depends strongly on the choice of penalty and relaxation parameters. Motivated by settings such as Model Predictive Control (MPC), where one repeatedly solves related optimization problems with fixed structure and changing parameter values, we propose learning online updates of the relaxation parameter to improve performance on problem classes of interest. This choice is computationally attractive in OSQP-like architectures, since adapting relaxation does not trigger the matrix refactorizations associated with penalty updates. We establish convergence guarantees for ADMM with time-varying penalty and relaxation parameters under mild assumptions, and show on benchmark quadratic programs that the resulting learned policies improve both iteration count and wall-clock time over baseline OSQP.

Cell invasion and spatial pattern formation are two distinct manifestations of cellular self-organisation in development, regeneration, and disease. Here, we develop and analyse a unified theoretical framework that links these two seemingly different behaviours within a single mechanistic model for adhesion-mediated self-organisation in growing cell populations. Using a multiscale analysis, we show that the balance between cell-cell adhesion, self-diffusion, and proliferation controls the emergence of distinct collective dynamics. We find that for weak adhesion, tissues invade through stable monotone fronts. As adhesion increases, invasion slows, fronts become unstable, leading to aggregates and spatial patterns emerging behind the advancing edge. In two spatial dimensions, these instabilities generate fingering morphologies reminiscent of dysregulated invasion in cancer. Crucially, we show that density-dependent regulation of adhesion suppresses these instabilities and restores cohesive tissue expansion. Together, our results identify adhesion strength and its regulation as key determinants of whether tissues invade cohesively or fragment into patterns, and provide a unified framework for understanding collective migration, morphogenesis, and dysregulated growth.

In Orabona and Pál [2016], we introduced the shifted KT potentials, to remove the $\ln \ln T$ factor in the parameter-free learning with expert bound. In this short technical note, I show that this is equivalent to changing the prior in the Krichevsky--Trofimov algorithm. Then, I show how to use the same idea to remove the $\ln \ln T$ factor in the data-independent bound for the Squint algorithm.

In the context of studying $C^*$-algebras generated by Toeplitz operators acting on the poly-Bergman space $\mathcal{A}^2_{n}(Π)$ of the upper half-plane $Π$, we introduce a system of all-but-one orthogonal projections in generic position. We show that the $C^*$-algebra generated by these orthoprojections is closely related to the $C^*$-algebra generated by all Toeplitz operators with vertical symbols satisfying boundary conditions. This result suggests a new approach in the study of Toeplitz operators acting on other reproducing kernel Hilbert spaces. Furthermore, the range of one of the orthoprojections herein has a reproducing kernel expressed in terms of the digamma and the Nielsen's beta functions. The harmonic function also emerges in this development.

Optimization of quadratic functions and the quotient of those are relevant in subspace and iterative optimization methods. In this paper, the calculation of the generalized operator norm and extremal generalized Rayleigh quotient is considered. In contrast to recent works an unconstrained sampling approach on the entire sphere for the random search direction in each iteration is proposed. Furthermore, the link to zeroth-order methods for Riemannian first- and second-order optimization methods is provided in the sense that the Riemannian gradient and Hessian are estimated by the specific surrogates. Even though the tangent space is not used in this construction the optimal step size problem can be computed in a closed form. The subproblems of this and recent works are illuminated in the context of sub-generalized Rayleigh quotient problems on specific Gram matrices. Together the achieved theory allows to construct an accelerated algorithm which shows state-of-the-art behavior and outperforms recent works.

This paper is a continuation of the work in \cite{kimzhang2024} concerning Hele-Shaw flow with both drift and source terms. We prove that, in a local neighborhood, if the free boundary is Lipschitz continuous with a sufficiently small Lipschitz constant, then the free boundary is $C^{1}$. As a corollary, we also consider the 2D vertical Hele-Shaw (or one-phase Muskat) problem with an advection term. We show that, provided the initial data and the advection term are small and the propagation speed is large, the free boundary becomes uniformly $C^1$ after a finite time.

This paper considers the homogeneous Neumann initial-boundary value problem for Alopecia Areata chemotaxis model with weakly singular sensitivity. For any appropriately regular initial conditions,it is shown that the problem admits a global boundedness of classical solutions in two spatial dimensions. Moreover, through the explicit construction of Lyapunov functions, we establish that the globally bounded solution converges exponentially to a constant steady state. The paper concludes with numerical experiments that serve to visually illustrate and corroborate some of the theoretically derived findings.

If $H$ is a numerical semigroup (that is, a cofinite subset of the non-negative integers closed under addition), then the non-empty subsets of $H$ form a semigroup $\mathcal P(H)$ under the sumset operation induced by addition in $H$. Moreover, if $0 \in H$, then $\mathcal P(H)$ is a monoid with identity element $\{0\}$, and the family $\mathcal P_0(H)$ of all subsets of $H$ containing $0$ is a submonoid of $\mathcal P(H)$. We show that the automorphism group of $\mathcal P(H)$ is trivial, and the same holds for $\mathcal P_0(H)$ when $0 \in H$. The proofs blend ideas from combinatorics and semigroup theory.

We prove pathwise convergence of the layerwise evolution of tokens in a finite-depth, finite-width transformer model with MultiLayer Perceptron (MLP) blocks to a continuous-time stochastic interacting particle system. We also identify the stochastic partial differential equation describing the evolution of the tokens' distribution in this limit and prove propagation of chaos when the number of such tokens is large. The bounds we establish are quantitative and the limits we consider commute. We further prove that the limiting stochastic model displays synchronization by noise and establish exponential dissipation of the interaction energy on average, provided that the common noise is sufficiently coercive relative to the deterministic self-attention drift. We finally characterize the activation functions satisfying the former condition.

Continuous data assimilation (CDA) nudges observational data into governing equations to recover the underlying flow and improve predictions. Existing rigorous CDA analyses focus primarily on incompressible flows, yet no physical flow is perfectly incompressible. Approximating a slightly compressible flow with an incompressible model introduces non-negligible model errors. Data assimilation for compressible flows remains challenging due to strong nonlinearities and the presence of shocks. We design an algorithm that addresses the limitations of velocity-only nudging for slightly compressible flow. This work incorporates both velocity and pressure data from the slightly compressible flow and nudges both quantities into the incompressible Navier--Stokes equations. Our analysis shows that the model error decays exponentially in the initial error, with an asymptotic residual of order $\mathcal{O}(H)$, where H denotes the observation resolution. The analysis also identifies a scaling for the pressure nudging parameter $μ_1 = O(1/H^2)$ that ensures effective assimilation. We validate the theoretical results through a suite of numerical experiments: a convergence study confirming optimal rates, a modified Taylor--Green vortex benchmark demonstrating synchronization of energy, enstrophy, and pressure, and an acoustic wave propagation test that isolates the role of pressure nudging and achieves a $97.9\%$ reduction in pressure error relative to velocity-only assimilation. Together, these results provide a foundation for discrete error estimates and realistic compressible applications.

We prove a generalization of the infinite quantum Ramsey theorem of Kennedy et al. (arXiv:1711.09526), showing that it follows from an archetypical "selective" pattern satisfied by certain families of projections in an infinite-dimensional Hilbert space.

We develop efficient algorithms for a fundamental network design problem arising in potential-based flow models, which are central to many energy transport networks (e.g., hydrogen and electricity). In contrast to classical network flow problems, the nonlinearities inherent in potential-based networks introduce significant new challenges. We address these challenges through intricate reductions to classical combinatorial optimization problems, such as (constrained) shortest path problems, enabling the application of well-established algorithmic techniques to compute exact and approximate solutions efficiently. Finally, we complement these algorithmic results with matching complexity results concerning the hardness and non-approximability of the considered problem variants.

We study the entropy solution for a class of systems of nonlocal conservation laws in which the convective flux is convoluted with a kernel in both spatial and temporal variables. This formulation models the flux dependence on the solution within its spatial neighbourhood (nonlocal in space) as well as on prior states in time (nonlocal in time), thereby incorporating memory effects. In addition, employing a convergent finite volume approximation, the existence of the entropy solution is discussed. The uniqueness of such entropy solutions is also established. In addition, we analyze the asymptotic behaviour of the solutions as the support of the temporal convolution kernel shrinks, demonstrating the "memory-to-memoryless" effect and convergence to the entropy solution of the corresponding nonlocal conservation law without memory (i.e., nonlocal only in space). Convergence rate estimates are derived. In addition, the proposed numerical approximations are shown to be asymptotically compatible with this passage to the memoryless limit by deriving the corresponding asymptotic convergence rate estimates. The analysis is carried out in a very general setting, without imposing any geometric restrictions such as the convexity of the spatial and temporal convolution kernels, unlike the existing literature on the asymptotic analysis of nonlocal-in-space only conservation laws. To the best of our knowledge, this provides the first convergence and asymptotic analysis for finite volume schemes applied to nonlocal conservation laws with memory. Numerical experiments are included to illustrate the theory.

Differential-difference matrix Lax representations (Lax pairs), gauge transformations, and discrete Miura-type transformations (MTs) belong to the main tools in the theory of (nonlinear) integrable differential-difference equations. For a given equation, two matrix Lax representations (MLRs) are said to be gauge equivalent if one of them can be obtained from the other by applying a matrix gauge transformation. Generalizing and extending several previous works on MLRs and MTs, we present new results on the following problems: - When and how can one simplify a given MLR by means of gauge transformations? - How can one use MLRs and gauge transformations for constructing MTs? - A MLR is called fake if it is gauge equivalent to a trivial MLR. How to determine whether a given MLR is not fake? We consider the general (1+1)-dimensional evolutionary differential-difference case when a MLR can depend on any shifts of dependent variables and can be non-autonomous. As applications and illustrations of the presented general theory, we construct several new two-component integrable equations (with new MLRs) connected by new MTs to known integrable equations from the papers [S. Konstantinou-Rizos, A.V. Mikhailov, P. Xenitidis, J. Math. Phys. 2015], [E. Mansfield, G. Mari Beffa, Jing Ping Wang, Found. Comput. Math. 2013]), including non-autonomous examples.

We consider the free boundary problem for relativistic plasma--vacuum interfaces in two and three spatial dimensions. The plasma flow is governed by the equations of ideal relativistic magnetohydrodynamics, while the vacuum magnetic and electric fields satisfy Maxwell's equations. The plasma and vacuum magnetic fields are tangential to the interface, which moves with the plasma flow. This yields a nonlinear, multidimensional hyperbolic problem with a free boundary that is characteristic of variable multiplicity. We identify a quantitative stability condition and establish the linear stability of three-dimensional relativistic plasma--vacuum interfaces in the sense that the variable-coefficient linearized problem satisfies energy estimates in anisotropic Sobolev spaces. In estimating tangential derivatives, we exploit an intrinsic cancellation effect to convert the boundary term into an instant integral. We then separate the estimate involving spatial derivatives from that involving time derivatives, so that the instant integral can be mainly absorbed by the instant energy under the stability condition. Moreover, we prove the local-in-time existence and uniqueness of solutions to the nonlinear problem in two-dimensional space, provided that the plasma and vacuum magnetic fields do not vanish simultaneously at any point of the initial interface. The proof combines the solvability and tame estimate of the linearized problem with a suitable modified Nash--Moser iteration. In particular, to establish its solvability, the two-dimensional linearized problem is reduced to a transport equation for the interface function and a hyperbolic boundary problem with maximally nonnegative boundary conditions.

We consider a system of two coupled first-order linear hyperbolic partial differential equations modeling heat transport in a counter-flow heat exchanger: one equation describes the transport of a hot fluid, and the other the transport of a cold fluid in the opposite direction. For this system, we design a boundary observer that uses only the temperature of the cold fluid measured at one boundary. Our approach is spectral: by assigning the spectrum of the operator governing the observation error dynamics to a prescribed region within the open left-half complex plane, we can freely tune the convergence rate of the observation error to zero in the $L^2$ norm. The main technical contribution is the proof that spectral stability, that is, the location of the spectrum in the open left-half plane, is equivalent to $L^2$ exponential stability of the origin for the observation error dynamics. This equivalence is established by showing that the operator governing the observation error dynamics satisfies the so-called spectral mapping property.

The purpose of this paper is to determine all Rota-Baxter operators on dual quaternion algebra $\mathcal{H}_d$ over the reals.

The classification of maximal algebras of square block Toeplitz matrices is a considerably more difficult problem and has received relatively little attention in the existing literature. In this work, we approach the problem under the assumption that the entries belong to the Schur algebra. Within these settings, we obtain a complete classification of all maximal algebras of such block Toeplitz matrices.

The paper studies constitutive modelling of Korteweg fluids. Thermodynamic consistency, i.e. compatibility with entropy balance law, is achieved using Liu's method of multipliers. Appropriate constitutive assumptions facilitated inclusion of the capillary effects in the specific entropy. Korteweg stresses are derived from the equilibrium conditions -- vanishing of the entropy production and its minimization in equilibrium. Material parameter in Korteweg stresses is allowed to depend on temperature, which turns out to be consistent with kinetic-theory results and leads to cross-coupling of mechanical and thermal effects. The generalized Gibbs' relation, which inherits the capillary effects, is derived as consequence, which is a peculiar feature of the Liu's method.

Conventional multiple-input multiple-output (MIMO) systems mainly rely on fixed antenna arrays, which limits their capability to adapt the effective channel matrix to the propagation environment. Rotatable antennas (RAs), which enable mechanical or electronic adjustment of antenna boresight directions, introduce a new orientation-domain degree of freedom for channel reconfiguration. In this paper, we investigate an RA-aided MIMO communication system for channel capacity enhancement. We first establish an orientation-dependent MIMO channel model. Then, we formulate a capacity maximization problem by jointly optimizing the transmit covariance matrix and the transmit/receive RA orientations under practical spherical-cap constraints. To solve this non-convex problem, we develop an alternating optimization algorithm, where the transmit covariance matrix is updated via eigenmode transmission and water-filling, while each RA orientation is optimized through a Riemannian Frank-Wolfe method. We further investigate the low-SNR regime and derive simplified designs for multiple-input single-output (MISO) and single-input multiple-output (SIMO) special cases. Numerical results show that the proposed RA-aided MIMO design significantly improves the channel capacity compared with the fixed-orientation benchmark, demonstrating the benefits of orientation-domain channel reconfiguration.

We establish the rate at which the renormalized stress--energy tensor of a massless minimally coupled scalar field in the in-vacuum state of a collapsing null-shell spacetime approaches the corresponding Unruh-state value. At finite exterior radius, we establish the upper bound \[ |Δ\langle T_{μν}\rangle|\leq C(r)\,t_s^{-3} \] from the Cauchy-surface decomposition of the Hadamard difference and the branch-cut structure of the retarded Green function. At future null infinity, we show that the leading coefficient in the late-time expansion \[ Δ\langle T_{uu}\rangle\sim C_{uu}\,u_s^{-3} \] is nonzero, by computing the branch-cut residue explicitly at small frequency and using the Planck suppression of the thermal spectrum at large frequency to show that the dominant contribution to $C_{uu}$ has a definite sign. The result gives \[ Δ\langle T_{uu}\rangle\big|_{\Iscr^+}(u_s) \sim C_{uu}\,u_s^{-3}, \qquad u_s\to\infty, \] with $C_{uu}\neq 0$. The exponent is determined by the $ω^2\lnω$ branch-point singularity in the Wronskian of the $\ell=0$ radial wave equation, the same structure responsible for Price's law. The sign $C_{uu}<0$ is supported by a physical argument and by the numerical mode data of Gholizadeh Siahmazgi, Anderson, and Fabbri. The result confirms their conjecture that the approach is a power law. We conjecture that the same mechanism gives an analogous $t_s^{-7}$ bound for gravitational perturbations ($\ell_{\min}=2$), though the extension to the spin-2 case involves gauge issues not addressed here.

The notion of a two-dimensional word arises naturally in the study of combinatorics on words, while the iterative construction of pedal triangles results in a rich dynamical system in the study of geometry. At first, these two classes of objects seem to be unrelated. However, it is known that for all $n \geq 1$, the number of primitive two-dimensional words of dimension $2 \times n$ over a binary alphabet agrees with the number of triangles whose first similar pedal triangle is their $n$th pedal triangle. We construct a finite four-symbol coding of the sorted pedal map and use the resulting branch itineraries to give a bijection between these two classes.

Hill functions, dominant in gene regulatory network modeling, carry fundamental limitations: at non-integer cooperativity exponents, routine when fitting dose-response data, derivatives diverge at the origin, complex arithmetic corrupts ODE trajectories, and zero output at zero activation traps models in off-states. This paper employs logistic-based models that are globally $C^\infty$, real-valued, and strictly positive at zero concentration, resolving all three pathologies while preserving sigmoidal dynamics. Using the delay-coupled two-gene mutual-activation and self-repression network of Vinoth et al.\ as a concrete model, we analyze two reformulations: linear additive activation with logistic self-repression, and a fully sigmoidal form with both terms logistic. A closed-form matching relation $λ= n/θ$ follows from equating slopes at half-maximal points. Closed-form parameters of the weighted logistic formulation are derived by matching basal rates and local slopes to the Hill-linear hybrid model. The unique biologically feasible equilibrium is computed in each case; it is lower in the weighted logistic case, the reduction arising from saturation of the bounded activation term. In the delay-free case ($τ=0$), local asymptotic stability holds in both formulations since the Jacobian trace is strictly negative for all positive parameters; stability persists for $τ\in(0,τ_c)$ and is lost via Hopf bifurcation at the critical delay $τ_c$. Numerical solution of the full transcendental system locates $τ_c$, with higher-order bifurcations characterised numerically in each case. Replacing linear additive with weighted logistic activation substantially reduces both the global Lipschitz constant of the right-hand side and that of its Jacobian, permitting larger integration steps.

We study the inverse eigenvector centrality problem on connected undirected graphs, namely, whether a given positive vector can be realized by assigning suitable edge weights. We provide a complete characterization in terms of stable sets and their neighborhoods, showing that the undirected case requires nontrivial global constraints absent in the directed setting.

The implementation of the \ac{isac} feature in \ac{6g} networks is most likely to be based on the framework of \ac{ofdm}. Input distribution design, or constellation design, is a crucial technique in \ac{ofdm}-\ac{isac} systems enabling a favorable balance between communication rate and sensing performance. In this treatise, we propose a computationally efficient input distribution design approach for \ac{ofdm}-\ac{isac} under frequency-selective channels, following the theoretical framework of capacity distortion. We highlight that under practical sensing constraints, the optimal strategy is to treat the kurtosis of constellations as a resource, and allocate it appropriately over subcarriers.