Papers

We consider time continuous multispecies birth and death processes in a regime of large populations. The jump rates depend on a large scaling parameter K modeling the charge capacity. When K tends to infinity, the process is close (in finite time) to a dynamical system containing a non zero global attracting equilibrium and zero as unstable equilibrium. For each fixed K, extinction in finite time occurs almost surely and a quasi-stationary distribution occurs naturally in the study of the statistics over times scales which are large but smaller than the extinction time scale. Before this catastrophic event the process makes many unsuccessful large deviations attempts with time scales corresponding to how far it deviates from the quasi-equilibrium. The paper concerns the statistical description of these typical trajectories starting from the quasi-stationary distribution until extinction. An unusual mixing property yields large time scale behavior for the process starting from a fixed state. We give a precise statistical description of the successive exit times of the process rescaled by K from a neighborhood of the equilibrium of the dynamical system in a clumping time scale and prove their asymptotic Poisson distribution. We also give a precise description of the asymptotic distribution of the successive records until extinction.

The filtering method applied at the level of the Euler-Maxwell-Two-Fluid system produces a Fast Limit Model (FLM) which captures up to the electron depth essential features of plasma dynamics. In the case of prepared data, the discussion reduces to the eXtended MagnetoHydroDynamic (XMHD) framework of physicists, which involves the density __, the velocity u and the magnetic field B as state variables. By contrast, for unprepared data, an electric field E is created by resonances, and it participates to the time evolution. It turns out that FLM is a well-posed system on (__, u, E, B), extending XMHD, and implying a mechanism of interactions between (__, u, B) and E which can convert a part of the energy carried by (__, u, B) into electric energy.

In this article, we investigate the relationship between the fundamental group of a space and its continuous transformations. To be more precise, we show that if a continuous action of a Lie group on a space admits a simply connected cross-section, then we can build the universal covering of the space using an extension of the Lie group by a discrete group.

Let \(F\) and \(G\) be \(r\)-uniform hypergraphs, and let \(f_{F,G}(n)\) be the largest integer \(m\) such that every \(n\)-vertex \(G\)-free \(r\)-graph contains an induced \(F\)-free subgraph on \(m\) vertices. We prove that, for \(r\ge3\) and \(2\le k\le r-1\), if \(F\) is nonempty, \(G\) is \(k\)-tightly connected, and there is no homomorphism from \(G\) to \(F\) (that is, \(G\not\to F\)), then \[ f_{F,G}(n)\le C(\log n)^{\beta_F^{(k)}}, \qquad \beta_F^{(k)}= \max_{\emptyset\ne P\subseteq\partial_kF} \frac{e(P)}{v(P)-1}. \] The case \(r=3\) of our result resolves a conjecture of He and Nie. As a consequence, we obtain the Ramsey lower bound \(r(G,K_n^r)\ge2^{\Omega\bigl(n^{(r-1)/\binom rk}\bigr)}\) for every \(k\)-tightly connected non-\(r\)-partite \(r\)-graph \(G\). This extends a result of Conlon, Fox, Gunby, He, Mubayi, Suk, Verstraëte and Yu from the \(3\)-uniform setting.

We study the size of the largest component in Pittel's preferential attachment process without vertex growth. Starting from the empty graph on a fixed vertex set $[n]$, edges are added one by one with probabilities proportional to $(d_u+\alpha)(d_v+\alpha)$, where $d_u$ and $d_v$ are the current degrees of $u$ and $v$, and $\alpha>0$. Let $L_1$ denote the size of the largest component, and set $m_c:=\frac{\alpha n}{2(\alpha+1)}.$ We prove that if $m=m_c(1-\varepsilon), \varepsilon=\varepsilon(n)\to0, \varepsilon^3 n\to\infty,$ then \[ L_1=(1+o_p(1))\frac{2(\alpha+2)}{\alpha+1}\varepsilon^{-2}\log(\varepsilon^3 n) \] for every fixed $\alpha>0$. Moreover, the same asymptotic holds whenever $\alpha=\alpha(n)\to a\in(0,\infty]$. In particular, the constant $2(\alpha+2)/(\alpha+1)$ converges to the Erdős–Rényi value $2$ as $\alpha\to\infty$. If $m=\left\lfloor \frac n2(1-\varepsilon)\right\rfloor$ and $\alpha\varepsilon\to\infty$, then \[ L_1=(2+o_p(1))\varepsilon^{-2}\log(\varepsilon^3 n). \] The subcritical asymptotics for \(L_1\) resolve the problem left open by Janson and Warnke. The upper bound argument relies on the fact that, after conditioning on the degree sequence, the graph can be treated through the corresponding configuration model, the lower bound follows from tree component asymptotics and a second moment argument.

A graph is balanced if its clique-matrix contains no square submatrix of odd order with exactly two $1$'s in each row and in each column. Although it is known that a graph is balanced if and only if it contains no induced extended odd sun, a characterization of balanced graphs by minimal forbidden induced subgraphs is still unknown. In this work, we prove that, within the class of distance-hereditary graphs, balanced graphs are exactly the hereditary clique-Helly graphs. Equivalently, they are characterized by a single forbidden induced subgraph, namely $\overline{3K_2}$. From this result, we derive an explicit linear-time algorithm that decides balancedness within the class of distance-hereditary graphs and returns an induced $\overline{3K_2}$ when the input graph is not balanced.

We consider the vortex model on the plane, focusing on the case of vortices with the same sign and, for simplicity, assuming all vortices possess equal circulation. In particular we are interested at the time average of the vorticity density, i.e. the empirical measure associated to the vortices. We conjecture that, for a.e. initial data, the time average of the empirical density is radial. We prove the result for N=3 vortices by exploiting the integrability of the system. For N > 3 vortices we motivate the conjecture by transforming the problem into the independence of ergodic stroboscopic averages from initial data along a single trajectory, when using a suitable rotation angle as the independent variable instead of the time variable.

This article is devoted to the study of the Davis–Wielandt shell and the Davis–Wielandt radii of elements in a $C^*$-algebra. Utilizing a state-space approach, several geometric properties of the algebraic Davis–Wielandt shell are established. Upper and lower bounds for the algebraic Davis–Wielandt radii are obtained including the Davis–Wielandt radius of the sum of $k$ elements. We also explore the relationship between norm-parallelism and the Davis–Wielandt radii of elements.

In this paper, we present a linearly implicit, second-order block-centered finite difference (BCFD) prediction-then-projection scheme for the multi-species Keller-Segel chemotaxis system on non-uniform spatio-temporal grids. The proposed scheme integrates a standard Crank-Nicolson time-marching algorithm with an $L^2$ projection step to enforce positivity and mass conservation. The use of variable time stepsize and time-staggered discretization fully decouples the solutions of the multi-species cell density variables and the chemoattractant concentration variable while facilitating linearization, thereby greatly enhancing computational efficiency. Notably, the variable time-stepping algorithm and non-uniform grid BCFD discretization jointly enable adaptive resolution and local refinement near blow-up, thereby improving efficiency and accuracy without compromising the desired physical property-preserving in the simulation. Furthermore, using the mathematical induction method and the energy analysis approach, the unique solvability of the proposed scheme is rigorously proved, and we show that cell densities achieve second-order convergence in both time and space in the discrete $L^2$ norm, while the chemoattractant concentration achieves second-order convergence in the discrete $H^1$ norm. Representative numerical experiments are presented to validate the theoretical findings and demonstrate the reliability of the proposed scheme in simulating the blow-up phenomenon.

Jin and Rubinstein asked whether the fixed-level equivariant Tian's alpha invariant equals the fixed-level equivariant global log canonical threshold, and proved this equality for toric varieties. In this paper we provide a positive answer to Jin and Rubinstein's question in full generality. The main result of this paper was obtained by Chatgpt 5.5 pro, and the Danus system based on the Rethlas system.

We introduce a chain map from quandle homology to relative group homology, and construct several quandle cocycles through the chain map. We also relate this chain map to triangulations of Seifert (hyper)surfaces of 1- and 2-dimensional links.

We prove that an action by $C^1$-diffeomorphisms of a lattice in a simple $p$-adic group on a compact manifold is finite, provided that the dimension of the manifold is less than the rank of the group. We extend this statement to lattices in totally disconnected $S$-arithmetic groups, where the critical dimension is the maximal rank of its simple factors. This uses the machinery developed by Brown, Fisher, and Hurtado.

A long-standing conjecture of Lapidus asserts that under certain conditions a self-similar fractal set is not Minkowski measurable if and only if it is of lattice-type. For self-similar sets in $\mathbb{R}$, the Lapidus conjecture has been confirmed. However, in higher dimensions, it remains unclear whether all lattice-type self-similar sets are not Minkowski measurable. This work presents families of lattice-type subsets in $\mathbb{R}^2$ that are not Minkowski measurable, hence providing further support for the conjecture.

This work presents a goal-oriented a posteriori error estimator based on the Dual Weighted Residual (DWR) method together with space-time mesh adaptivity for the Navier–Stokes equations. The resulting nonlinear algebraic systems on the space-time slabs are solved by Newton's method with GMRES, preconditioned by a slab-wise geometric multigrid method. This combination yields reliable control of target quantities on computationally feasible space-time meshes together with a robust and efficient solution of the algebraic systems. The implementation is based on a MPI-parallel programming model in the deal.II library. Further ingredients are a discontinuous Galerkin discretization in time and inf-sup stable finite element pairs with discontinuous pressure on tensor-product meshes. The performance of the approach is investigated in benchmark computations with regard to accuracy, efficiency, and stability.

This paper studies an inverse boundary value problem for a coupled Stokes-Darcy system modeling fluid-porous medium interaction, with an unknown solid object embedded in the free-flow region. We simultaneously recover the viscosity coefficient $\mu$, the interface $\Gamma$, and the internal object $D$ from localized boundary Cauchy data. A novel method based on the construction of an interior transmission problem is introduced, which can amplify the singularity of solutions. We establish a global uniqueness theorem, showing that all three unknowns are uniquely determined by the boundary measurements.

Deep neural networks show great promise for high-dimensional PDEs, yet enforcing essential boundary conditions remains challenging, especially as penalty parameters require problem-specific retuning with increasing dimensionality. In this work, we extend the Natural Deep Ritz Method (NatDRM) [H. Yu and S. Zhang, J. Comput. Phys., 537 (2025)] to a unified framework for all dimensions $d \geq 2$ based on the de Rham complex and its penalty-free boundary decomposition: curl-type operators act on scalar potentials in 2D, vector potentials in 3D, and antisymmetric second-order tensor potentials in $d \geq 4$, respectively. This method converts Dirichlet constraints into three coupled natural (Neumann-type) subproblems with corresponding Ritz-type losses, eliminating the need for a boundary penalty parameter $\beta$. We derive dimension-unified discrete losses, lightweight boundary-based gauge-fixing regularizations to resolve curl-kernel non-uniqueness, and a joint training procedure; extensions to variable-coefficient elliptic and semilinear Poisson problems are formulated at the first subproblem level. Numerical experiments on smooth benchmarks up to 6D show that NatDRM, without any penalty tuning, matches or exceeds the accuracy of optimally tuned DRM and PINN in most cases. It converges stably in 6D where penalized DRM fails for most penalty values, and exhibits synchronous decay of interior and boundary errors, resolving the inherent imbalance of penalty-based methods.

In this article, we study the Pólya group of a new family of quintic fields, namely Lecacheux quintic fields. We show that the associated Pólya groups can be arbitrarily large elementary abelian \(5\)-groups. Using density arguments, we prove that for every positive integer $k$, the set of odd integers $s$ such that the $5-$rank of the Pólya group of the corresponding Lecacheux quintic field is at least $k$ has a positive density. Combining this with a result of Golod and Shafarevich, we see that for a positive proportion of $s$, the corresponding Lecacheux quintic fields admit an infinte $5-$class field tower. We also establish an upper bound for the Pólya numbers of these fields in terms of the orders of their corresponding Pólya groups. In addition, we prove that several fields in this family are non-monogenic despite having index one.

We extend the analysis of nonrepetitive sequences of Entringer et al. [Journal of Combinatorial Theory, 1974] to relaxations of equality testing under nonstandard equivalence relations, in particular parameterized equivalence and order-preserving equivalence. For this setting, we introduce $\ell^+$-squares, defined as squares whose total length is at least $2\ell$. We obtain an infinite $3^+$-parameterized-square-free ternary word and an infinite $3^+$-order-preserving-square-free binary word through an approach based on combinatorics on words. In addition, we report the longest $\ell^+$-square-free words across several equivalence relations.

Autoencoders (AEs) have emerged as powerful tools for non-linear dimensionality reduction, often surpassing traditional linear methods such as Proper Orthogonal Decomposition (POD) in scenarios characterized by slowly decaying Kolmogorov $n$-widths. In the realm of Reduced-Order Modelling (ROM), these models are increasingly utilized to learn low-dimensional representations of solution manifolds associated with parametric Partial Differential Equations (PDEs). However, the high expressivity of AEs presents a challenge: although trained networks typically minimize reconstruction error, they often struggle to capture the essential properties necessary for building accurate and robust ROMs. Recent works by arXiv:2307.15288v2 and arXiv:2506.11641v1 have tackled this challenge in fully connected AEs by proposing representation-consistent architectures, which preserve some of the properties belonging to POD. This study builds upon that concept by extending representation consistency for convolutional layers. We introduce a novel class of symmetric Convolutional AutoEncoders (CAEs) designed to embody the primary properties of manifold parametrization mappings. When integrated into a ROM framework, this architecture demonstrates significantly improved predictive capabilities. Specifically, we compared the performance of the ROMs based on classical and symmetric CAEs on three one dimensional academic test cases, namely the Linear Advection, the Viscous Burger and the Kuramoto Sivashinsky equation. Numerical results demonstrate that our proposed symmetric approach consistently yields more accurate latent trajectories, lower reconstruction errors, and enhanced model robustness.

The Sinkhorn algorithm is the de facto standard method for numerically solving entropy-regularized optimal transport problems over finite sets. In this work, we investigate a phenomenon arising when Sinkhorn is applied with a small regularization parameter $\tau$: the evolution of the dual variables (the logarithm of the scaling factors) is approximately piecewise-linear, while the primal variables (the approximate transport plans) exhibit a saddle-to-saddle type behavior. We prove that as $\tau \to 0$, the Sinkhorn iterates indeed converge to a continuous-time curve consistent with these observations, when time is rescaled as $t = \tau k$, and we characterize the limiting "cold Sinkhorn" dynamics explicitly. In particular, we show that it acts as a dual optimization dynamics for the unregularized problem with properties analogous to the simplex algorithm. Notably, this dynamics converges in finite time to an unregularized solution, implying a novel guarantee for the Sinkhorn algorithm itself: it achieves $\tilde{O}(\tau)$ dual suboptimality in $k = O(\tau^{-1})$ iterations, instead of $k = O(\tau^{-2})$ as existing analyses would suggest.

This paper studies the asymptotic behaviour of a structured population model on the space of nonnegative Radon measures. Such formulations naturally arise when solutions develop concentration phenomena or when the population is represented by discrete cohorts. Asynchronous exponential convergence of measure solutions towards a one-dimensional global attractor is established. While such results are classical in the $L^1$ setting, their extension to measure spaces requires different compactness and spectral arguments. We identify conditions under which the classical asymptotic behaviour persists in the space of Radon measures endowed with the flat metric, thereby extending the theory of asynchronous exponential growth beyond the classical $L^1$ framework.

The resonant nature of scattering in a waveguide with two barriers is proven in the case of a sufficiently arbitrary deformation of the region between the barriers. The problem is considered as an interior boundary value problem with boundary conditions defined by a Poincare-Steklov operator. A spectral problem is considered whose eigenvalue determines the resonant scattering frequency.

We study the Schrödinger problem on metric graphs and its different formulations. Starting from a static version, we introduce an equivalent reformulation as entropic optimal transport and show $\Gamma$-convergence towards static optimal transport. We then rigorously derive a Benamou-Brenier type dynamic version of the Schrödinger problem, thereby extending known results from ${\rm RCD}^*(K,N)$-spaces. With this equivalence at hand, we conclude that the minimum values of the dynamic Schrödinger problem converge towards the squared Wasserstein distance, and minimizers converge to Wasserstein geodesics. We also extend the dynamic formulation to a more general class of initial and final data and show existence of solutions in this setting using the direct method. Lastly, we illustrate our analytical findings by a numerical investigation.

It was recently established by Perepechko and Zaidenberg that the automorphism group of a normal affine surface is finite-dimensional if and only if the surface admits no non-trivial action of the additive group of the base field. We extend this result to non-normal affine surfaces.

We provide a recurrence for computing the terms of the OEIS sequence A393920, introduced in \cite{KS}. We also describe a surprising connection between A393920 and the Fibonacci sequence A000045, obtain non-trivial lower and upper exponential bounds for its growth, and investigate connections with partial orders, Catalan numbers, and convex topologies on finite chains. For the representation-theoretic lattice underlying A393920, we describe its atoms, coatoms, join-irreducible and meet-irreducible elements.

In a previous work, we defined (type A) c-Birkhoff polytopes and showed that they were unimodularly equivalent to order polytopes of heap posets. In this note we answer the question: What about type B?

Multi-task regression aims at jointly solving multiple regression problems, called tasks. Compared to solving each task separately, better performances can be achieved as long as the tasks are sufficiently related. Full-conformal prediction is a framework that formulates a data-dependent prediction-region containing the unknown output-vector at any prescribed confidence level. However, explicit computation of this prediction-region is intractable in general since it requires training infinitely many predictors. The present work focuses on multi-task regression in a Reproducing Kernel Hilbert Space (RKHS) of vector-valued functions. This computational issue is addressed by designing an approximating predictionregion containing the full-conformal one. This construction is carried out in two scenarios: piq when the inter-task covariance-matrix is known, and piiq when this matrix is estimated. In terms of volume, the tightness of this approximation is assessed theoretically by means of an upper-bound in the first scenario. It is also empirically proved to improve upon the split-conformal prediction on synthetic data in both scenarios.

This paper develops a data-enabled primal-dual framework for learning optimal control policies for unknown linear discrete-time systems from online data. The proposed approach views the data-dependent control synthesis problem as a time-varying semidefinite program (SDP) whose coefficients are recursively updated from online closed-loop measurements. Instead of repeatedly solving a full SDP as new data arrive, the policy is updated online through lightweight primal-dual iterations, each consisting of a linear equation solve and a projection onto the positive semidefinite cone. The framework applies to both direct and indirect data-driven formulations and covers a broad class of control objectives, including LQR, $H_\infty$ control, and safety-critical control. To characterize the coupling between online optimization and closed-loop data generation, we introduce two data-dependent quantities: the Sim-to-Real Gap, which measures the mismatch between noisy and noiseless data-induced SDPs, and the Difference-of-Signal, which measures the temporal variation of the SDP coefficients. Under persistency of excitation, suitable SDP regularity conditions, and sufficiently slow data variation, we establish a local linear tracking result up to residual terms governed by the latter two quantities. A global ergodic convergence bound is also derived for arbitrary initialization. Numerical examples on LQR, $H_\infty$ control, and safe exploration demonstrate that the proposed method can efficiently improve control performance from online data while accommodating SDP constraints beyond the well-explored LQR policy-gradient formulations.

The notion of depth two and higher mock modular forms have found important applications in mathematical physics and enumerative geometry since their inception through indefinite theta functions with general signature. These theta functions generalize Zwegers' work on Lorentzian signature lattices and the framework of mock modular forms that emanated from it. Mock modular forms can also be studied through Eisenstein and Poincaré series. The interaction of this second point of view with the indefinite theta function approach yields a wealth of tools to unearth the rich structure behind mock modular forms. For mock modular forms of higher depth, on the other hand, indefinite theta functions and their variants largely remained the only available approach. In this paper, we show that one can indeed get mock modular forms of depth two by "coupling" a pair of Eisenstein series that yield depth one mock modular forms, thereby providing a new and independent approach to higher depth mock modular forms. We exemplify this new perspective on a depth two object that appeared in the context of Vafa-Witten invariants.

We consider the inverse boundary value problem in an elasticity system. It is proved that the density function $\rho$ and its derivatives at the boundary can be uniquely determined from the local Cauchy data. Furthermore, if the density function is analytic, we can uniquely determine the internal buried objects, as well as the unknown boundary and the boundary conditions imposed on it. Our methods mainly based on a precise characterization for the principal part of the difference between a special first-order singular solution and the fundamental solution in the $H^m$ norm, and the blow-up property for the boundary Sobolev norms of the volume potential corresponding to the fundamental solution.