Papers

In this work, we develop a weak-form version of Extended Dynamic Mode Decomposition that we call WEDMD. We establish a number of analytic results about the method and show explicitly how the weak form is able to mitigate the impacts of noise in linear stochastic differential equations. In nonlinear systems, we likewise show how the method is able to generate excellent approximations to Koopman modes without making recourse to sparsity promoting approaches. This excellent approximation property allows for meaningful forecasting even in the presence of noisy data.

We study additive functionals of stationary Markov chains whose observables depend on a fixed finite block of consecutive states. Such block observables arise naturally in sliding-window statistics, pattern counts, and local dependence analysis. In the independent setting, additive functionals of overlapping finite blocks are known to have a covariance operator with integer spectrum \(0,1,\ldots,k\), and the eigenvalue-one component represents the information first detectable at block length \(k\). We study the corresponding problem when the underlying sequence is a stationary Markov chain. For a fixed block observable \(f(X_t,\ldots,X_{t+k-1})\), we introduce a Hilbert-space decomposition that separates information already contained in shorter consecutive blocks from the genuinely new block-\(k\) component, called the incremental component. We show that this component persists as an eigenvalue-one component under Markovian dependence: on this space the Green–Kubo covariance converges trivially and the covariance operator acts as the identity. More generally, the integers \(1,\ldots,k-1\) are shown to arise hierarchically as eigenvalues of the Green–Kubo operator. In the reversible case, the remaining covariance structure is described through a boundary-corrected one-coordinate marginal spectrum determined by the base transition operator. Reversible two-state chains and Gaussian AR(1) models illustrate the theory through concrete spectral formulas.

The authors' primary goal in this paper is to extend some important results related to the liminf-convergence and $\mathcal{QS}$-convergence in domain theory to the setting of $T_0$-spaces. To that end, we study the quasi-liminf convergence in $T_0$-spaces and introduce a new kind of $T_0$-spaces — weakly locally hypercompact spaces (shortly WLH-spaces). It is proved that every locally hypercompact $T_0$-space is a WLH-space, and a $T_0$-space $(X, \tau)$ is a WLH-space iff the quasi-liminf convergence in $(X, \tau)$ is topological. Hence the quasi-liminf convergence in a locally hypercompact space is topological, and for a quasicontinuous poset $P$, the quasi-liminf convergence is topological and agrees with convergence in the Lawson topology $\lambda(P)$. We also show that a $T_0$-space $(X,\tau)$ is locally hypercompact iff the $\mathcal{QS}$-convergence in $(X,\tau)$ coincides with the convergence in the topology $\tau$. Therefore, a poset $P$ is quasicontinuous iff $\mathcal{QS}$-convergence in the Scott space of $P$ is topological iff $\mathcal{QS}$-convergence coincides with convergence in the Scott topology $\sigma (P)$. Using the quasi-liminf convergence, we give several characterizations of $C$-spaces and continuous posets.

We compute the Hochschild cohomology of the partially wrapped Fukaya category of the symmetric square of an annulus with stops. Using an explicit generating set in this category, we give a description of its dg endomorphism algebra $\widetilde{\mathcal{A}}_{n_1,n_2}$ via a quiver with relations. We show that when one boundary component has a single stop, the dg algebra is formal; however, when both boundaries contain at least two stops, it is not formal, and its minimal $A_\infty$-model carries a nontrivial operation $m_3$. This allows us to compute its Hochschild cohomology via reduction systems and spectral sequences, and to construct a family of dg deformations associated to the resulting Hochschild cocycles.

In this paper, we study semilinear elliptic systems with sublinear coupling terms, showing that, under competition-type interactions, solutions typically have dead cores, i.e., they vanish on certain open subsets of the domain. We apply our results to a large class of solutions treated in the literature, for instance to ground states and least energy sign-changing solutions, under Dirichlet boundary conditions or in the whole space. The proofs are based on a general result stating that subsolutions to a certain sublinear equation have dead cores, and on uniform Hölder bounds.

We use families of generalized Seifert matrices to define algebraic concordance groups for links with non trivial Alexander polynomial. We then use these groups to recover the invariance by concordance of the signature, the Fox-Milnor condition for concordant links and the invariance by concordance of the Witt class of the Blanchfield pairing for links. Finally, we use these invariants to investigate the isomorphism types of these groups.

We prove that the size $\# \{ k(m-k) : 1\leq k\leq m/2\} \cap \{ l(n-l) : 1\leq l\leq n/2\}$ is $(mn)^{o(1)}$, but can be arbitrarily large. This resolves Erdős problem $\# 443$.

Let $\mathbb{M}_\kappa^n$ be the space form of sectional curvature $\kappa\in\{-1,0,1\}$, so that $\mathbb{M}_{-1}^n=\mathbb{H}^n, \mathbb{M}_{0}^n=\mathbb{R}^n, \mathbb{M}_{1}^n=\mathbb{S}^n$. Let $\Omega\subset\mathbb{M}_\kappa^n$ be a nonempty bounded open set with Lipschitz boundary, and assume that $0<|\Omega|<|\mathbb{S}^n|$ when $\kappa=1$. Write $0=\mu_0(\Omega)\leq\mu_1(\Omega)\leq\cdots$ for the Neumann spectrum, and let $B_R^\kappa\subset\mathbb{M}_\kappa^n$ be a geodesic ball of volume $\vert\Omega\vert/2$. We prove the sharp shifted reciprocal inequality \[ \sum_{j=2}^{n+1}\frac1{\mu_j(\Omega)} \geq \frac{n}{\mu_1(B_R^\kappa)} = \frac{n}{\mu_2(B_R^\kappa\sqcup B_R^\kappa)}. \] Equality holds if and only if $\Omega$ is the disjoint union of two equal geodesic balls. This gives an affirmative answer to the conjecture of \cite[Remark~11]{BucurMartinetNahon2025}.

Given a subset $A \subseteq \mathbb F_2^n$, we can consider the distribution of the intersection size of $A$ with a uniformly random $d$-flat $F$. Motivated by the edge statistics problem and the hypercube statistics problem, the affine subspace statistics problem concerns the maximum of $\mathbb{P}[|F\cap A|=s]$ among $A \subseteq \mathbb F_2^n$ for any fixed $s\in\{1,\dots,2^d\}$ over a uniformly random $d$-flat $F$. We use $\lambda^*(d,s)$ to denote the limit of the maximum when $n$ goes to infinity. In this note, we prove tight bounds for $\lambda^*(d,s)$ in two different regimes. For $s=j2^k$ where $j$ is a positive odd integer, the best known lower bound construction achieving $\lambda^*(d,s)\ge 1-2^{-k}$ is due to taking $A$ as the union of $j$ parallel $(n-d+k)$-flats in $\mathbb F_2^n$. Our main result is a matching upper bound with an additive error term of $O(2^{-3k/2})$. We also study the case $s=1$, where we determine $\lambda^*(d,1)$ exactly. We show that the random construction where each point is included with probability $2^{-d}$ is optimal.

In this paper, we prove the existence of two completely independent spanning trees in the $n$-dimensional dual-cube $F_n$, a variant of the hypercube, for every $n \geq 5$. To this end, we use the hypercube structure of the clusters of $F_n$ to extend the construction of CIST from the $(n-1)$-dimensional hypercube to the dual-cube. In addition, we provide a recursive algorithm that builds the two trees while improving their diameters. Finally, we propose a conjecture regarding the existence of $k$ completely independent spanning trees in the dual-cube.

A nonnegative integer $n$ is $d$-translated unit sensitive when appending $d$ zeros to the right of the number and changing the unit digit to any possible unit digit (so long as the resulting number is not $n$) results in a composite number. We find an arithmetic progression of nonnegative integers that are $d$-translated unit sensitive for any nonnegative integer $d$. We construct the arithmetic progression such that there exists $k$ consecutive primes in the progression for any positive integer $k$. The first known prime that is $d$-translated unit sensitive for any nonnegative integer $d$ is found as a consequence. We also refine the arithmetic progression such that the integers in the progression are $d$-translated unit sensitive for any nonnegative integer $d$ and are Brier numbers. The refined arithmetic progression contains $k$ consecutive primes for any choice of positive integer $k$.

We investigate large solutions of linear and semi-linear equations involving the fractional Laplacian on domains that are becoming unbounded in some, but not all, directions. Solutions that blow up on the boundary of a domain are commonly called large solutions. For local operators, such behaviour arises only in the presence of lower-order nonlinear terms. However, for nonlocal operators, the existence of such solutions is more subtle. In contrast to the local case, the nonlocal nature of the operator gives rise to different boundary blow-up phenomena even in the case of linear equations. In this article, we study the existence and qualitative behaviour of boundary blow-up solutions to fractional linear and semi-linear equations on finite cylindrical domains and employ these properties to construct large solutions on infinite cylinders.

We illustrate in several examples that even neural networks of infinite width (specifically, Barron functions) may encounter substantial obstacles when used as a model class for problems in the calculus of variations. An instance of practical relevance concerns the bending, stretching and folding of a thin elastic shell with anchored or clamped boundary conditions where elastic energy could be reduced by folding along a circular line, but the neural networks can only describe straight folds along entire lines. Conversely, we show that there is no gap between the energy that Barron functions and Lipschitz functions can achieve for a large class of integral first-order functionals.

We study the intersection cohomology of minimally compactified Shimura varieties of PEL type AC using Igusa stacks and the work of Fargues-Scholze. More precisely, we construct a sheaf on the moduli stack of $G$-bundles on the Fargues-Fontaine curve, which recovers this intersection cohomology after applying a Hecke operator in the sense of geometric Langlands. We show that this sheaf has several desirable properties; for example, it is Verdier self-dual and perverse. This leads to several applications to intersection cohomology, including a version of the Mantovan product formula, as well as torsion-vanishing and Eichler-Shimura relations. Along the way, we investigate the interaction between Baily-Borel and Newton stratifications on minimally compactified Igusa stacks, and we study perverse t-structures on stratified v-stacks.

In this paper, we survey important results on the dynamics of permutable transcendental entire functions from 1958 to 2025. We have discussed the forms of transcendental entire functions that could be permutable. We have also discussed the dynamical properties of permutable entire functions. For instance, how the Fatou and Julia sets are related. We have also considered the relation of the subsets which are classified based on the dynamics of the orbit of a point in the complex plane. They are the filled Julia set $K(f)$ (set of points whose orbits remain bounded), the escaping set $I(f)$ (set of points whose orbits escape to infinity) and the bungee set $BU(f)$ (set of points which are neither bounded nor escaping to infinity). The escaping set is further classified based on the speed of escape, namely, the fast escaping set $A(f)$ or other sets such as $I_0(f)$ and $T(f)$, whose definitions are mentioned in this article. We have also discussed the dynamical relations of these sets. The bungee set sometimes contains a wandering domain, and in this article we have discussed the dynamical relations and conditions that dictates whether wandering domains exists or if it is contained within the bungee set.

We consider the problem of state reconstruction for a nonlinear dynamical system from observations of a linear function of the state. We present a design method for a tunable observer and provide a general theorem which under certain conditions guarantees exponential convergence of the observer regardless of initial error. Additional results are provided that apply this theorem to chemical reaction network models. Moreover, these results are illustrated via examples of mass action form of chemical reaction networks where a subset of the species concentrations are observed. Numerical results are provided to show the efficacy of our proposed observer. Numerical results are also shown for the case of noisy observations and our observer is compared favorably with the particle filter when the observation noise is small.

In this paper we prove that a process we call tropical relaxation gives a convergent method for numerically approximating the Monge–Ampère Dirichlet problem on bounded convex planar domains. We approximate the prescribed probability measure by point clouds and construct the least concave tropical roof with zero boundary values and a corner at each point, using repeated updates of affine planes with integral gradients. For universally generic clouds of $N$ points in a fixed compact subset of the interior, convergence of the empirical measures implies uniform convergence of the roofs, divided by $\sqrt{N}$, to the Aleksandrov solution. The limiting measure may be singular. Euler's and Pick's formulas relate curvature to point counts, with an $O(N^{-1/2})$ discrepancy against compactly supported $C^1$ tests. A removal estimate gives almost-sure uniform convergence for independent samples from any absolutely continuous probability law on the whole domain. For generic clouds in a fixed interior compact set of a rational-slope polygon, suitably rescaled sandpile odometers converge to the same solution along meshes chosen sufficiently fine for each cloud. Their deficit measures converge to its negative Laplacian.

Let $K\subset \RN$ be a convex body containing the origin in its interior, and let $\hK{\cdot}$ be its Minkowski functional. In this paper, we develop an identity-based framework for sharp anisotropic $L^2$-Caffarelli-Kohn-Nirenberg inequalities associated with the anisotropic radial derivative $$ \mathcal R_K(u)(x)=\frac{x\cdot\nabla u(x)}{\hK{x}}, \quad x\in\RN\setminus\{o\}. $$ A key point of the present work is that $K$ is not assumed to be origin-symmetric. Consequently, the Minkowski functional $\hK{\cdot}$ need not be even, and the usual norm-based anisotropic arguments do not apply directly. The main tools are anisotropic $L^2$-Hardy and $L^2$-Caffarelli-Kohn-Nirenberg identities with explicit nonnegative remainders. These identities yield sharp anisotropic $L^2$-Caffarelli-Kohn-Nirenberg inequalities whose best constants depend on the parameter region of $(a,b)\in\mathbb R^2$. We also study the attainability of the sharp constants in a natural completion space and obtain the corresponding extremal functions. As further consequences, we derive sharp anisotropic Heisenberg-type uncertainty principles and max-type anisotropic gradient inequalities. When $K$ is the Euclidean unit ball, our results recover the classical Euclidean $L^2$ theory; when $K$ is origin-symmetric, they are consistent with the usual norm-based anisotropic framework. In particular, the present results extend the sharp $L^2$-Caffarelli-Kohn-Nirenberg theory to general convex bodies containing the origin in their interiors, for which the Minkowski functional may be non-even.

Many standalone frameworks for numerical solvers have been developed to tackle the simulation of specific single- or multi-physics problems. For solving coupled problems, common approaches are to extend existing solvers, to develop an entirely new solver or to couple two existing solvers by using the interface provided by a coupling library or framework. However, to the best of our knowledge, there is not yet a framework that allows the easy and direct development of coupled solvers. In this work, we prototype a portable reproducible cross-language framework for the development of coupled parallel adaptive solvers using Trixi$.$jl and deal$.$II for the numerical simulation of coupled multi-physics problems. Currently, this is tightly entangled with an example coupled solver. We show its usability by developing a partitioned strongly-coupled multi-physics solver for the dynamics of Newtonian self-gravitational gases. For the coupled solver, we validate the expected order of convergence, physical sensibility of the results and mesh adaptivity. Finally, we investigate its parallel scaling. A publicly accessible reproducibility repository for the numerical results and code is available.

The Julia programming language aims to provide a modern approach to develop high-performance computing (HPC) applications. It tries to achieve this by combining a high-level, dynamic interface with just-in-time compilation to native machine code, thereby facilitating high developer productivity and native code performance at the same time. While this approach has already been shown to work well for serial applications, it is not clear if it readily translates to traditional, massively parallel HPC work loads. In this paper, we fill this gap by analyzing the parallel performance of the numerical computational fluid dynamics simulation code Trixi$.$jl, written in Julia, and compare it to the Fortran code FLUXO. We show some of the challenges of using Julia at scale and discuss possible solutions, specifically with respect to code loading and compilation at startup. Finally, we demonstrate the parallel scaling of our Julia code on up to 61440 CPU cores.

The degree (or winding number) of a sufficiently regular map $f:\mathbb{T} \to \mathbb {S^1}$ is given in terms of its Fourier coefficients by $$ \operatorname{deg} f = \sum_{n\in\mathbb{Z}} n\,|\widehat{f}(n)|^2. $$ At lower regularity the series may diverge, but, as shown by Kahane, when $f$ is $\alpha$-Hölder continuous with $\alpha>1/3$, then the degree can be recovered by a universal linear summation process. We show that no summation process satisfying Brezis's natural axioms can recover the degree universally for $\alpha=1/3$, thereby resolving an open problem by Brezis.

Recently, the authors introduced and studied a singularity invariant of a complex algebraic variety $Z$, written $\HRH(Z)$ (for “Hodge rational homology” manifold level). In this paper, we focus on local complete intersection subvarieties. We relate $\HRH(Z)$ to various well-known invariants, like Bernstein–Sato polynomials and the Dimca-Maisonobe-Saito spectrum. In the hypersurface case it turns out that $\HRH(Z)$ can be completely characterized by these invariants, though higher codimension case is more subtle.

In this paper, we establish the optimal \(L^2(\mathbb{R}^2)\to L^{10/3}(\mathbb{R}^2)\) endpoint estimate for the spectral projection operator associated with the Hermite operator on \(\mathbb{R}^2\). This completes a long-standing line of inquiry into sharp eigenfunction bounds for the Hermite operator, developed through the works of Thangavelu, Karadzhov, Koch–Tataru, and others. In higher dimensions \(d\ge 3\), the corresponding endpoint estimates \[ L^2(\mathbb{R}^d)\to L^{\frac{2(d+3)}{d+1}}(\mathbb{R}^d) \] were recently established by the present authors. Together with these earlier results, the present work fully resolves the problem of optimal \(L^2\to L^q\) eigenfunction bounds for Hermite spectral projections in all dimensions. Although our approach builds on our previous method, we overcome its limitations through a multiscale decomposition in space and time relative to the degeneracy set, combined with an asymmetric refinement on the input side and almost orthogonality.

In classical statistical physics, the Mori-Zwanzig projection operator technique is used to derive generalized Langevin equations for a random variable (observable). Standard derivations implicitly assume the existence of solutions to the so-called orthogonal dynamics equation as well as the validity of the variation of constants formula (Dyson identity). It was pointed out by Givon, Hald and Kupferman that the existence is a subtle issue for infinite-rank projections such as Zwanzig's projection [D. Givon, O. H. Hald, R. Kupferman, Israel Journal of Mathematics, 145 (221-241), 2005]. The authors proved the existence of weak solutions for Zwanzig's projection in the context of stationary Hamiltonian systems. To this date, this is the only existence proof that allows for an infinite-rank projection, whereas the uniqueness and regularity remain open problems. In this article, we generalize the existence proof by Givon et al. to nonstationary non-Hamiltonian systems whose time evolution is given by a quasicontraction semigroup. We establish growth bounds as well as the uniqueness for sufficiently regular solutions (if existent). Finally, we apply our results to Zwanzig's projection using the damped harmonic oscillator as an example.

Topological classification of physical vector fields conventionally relies on field normalization and homotopy-based invariants. However, when field amplitudes vanish, normalization becomes ill-defined, preventing a direct topological characterization. Here, we introduce a general framework for the topological classification of non-normalizable $n$-dimensional vector fields with compactifiable base spaces by transforming them into $(n+1)$-dimensional normalized vector fields. This construction extends homotopy-based classification to fields containing amplitude zeros. We explicitly demonstrate the approach for one-, two-, and three-dimensional non-normalized vector fields and derive the corresponding topological invariants. The resulting topological charges are robust under continuous deformations and can change only when the embedding structure becomes singular. Our framework provides a unified route to the topological characterization of non-normalizable fields and opens the door to the study of topological phenomena in a broad range of systems, including magnetic textures, ferroelectrics, electromagnetic fields, and wave systems.

It is a classical result of D. F. Holt that an uncountable, locally finite discrete group possesses exactly one end. While traditional approaches rely on cohomological methods and the combinatorial manipulation of almost-invariant subsets, we establish this result through coarse geometry via the topological structure of the Higson corona. By interpreting the boundaries of coarsely clopen sets in terms of subgroup invariance, we evaluate their associated characteristic functions on the large-scale geometry of the group. Utilizing a Skolem-hull extraction and a periodic coarse rerouting technique, we force a geometric rigidity that prohibits multiple ends.

We investigate the long-wave linear stability and instability of the two-dimensional viscous, resistive Magnetohydrodynamic (MHD) equations, in vorticity-current formulation, on the periodic domain ${\mathbb T}_\alpha\times {\mathbb T} = \Big( {\mathbb R}/(\frac{2 \pi}{\alpha} {\mathbb Z}) \times {\mathbb R}/(2 \pi {\mathbb Z}) \Big)$, around a periodic shear flow $(U(y),0)$ coupled with a constant background magnetic field ${\bf b}=({\rm b}_1,{\rm b}_2)$. It is a non-trivial extension of a recent paper for the Navier-Stokes equations by Colombo, Dolce, Montalto & Ventura to the MHD setting in the spirit of the classical works of Kolmogorov, Meshalkin, Sinai and Yudovich. We establish explicit conditions on the shear flow profile $U(y)$ involving the viscosity $\nu$, the resistivity $\eta$ and the components of the background magnetic field ${\bf b}$ to obtain linear long-wave stability and instability in the regime $\alpha\ll 1$. The proof combines a non-perturbative normal form transformation decoupling the zero Fourier mode from the non-zero modes with sharp asymptotic expansions of the eigenvalues bifurcating from the zero unperturbed eigenvalue with respect to the parameter $\alpha$. As a dynamical consequence, we obtain a splitting of the phase space into unstable and stable subspaces, on which solutions grow or decay exponentially in Sobolev norm.

The class of FS-domains is known to be closed under Scott function spaces when both the source and target are FS-domains. This paper investigates what remains true under one-sided FS assumptions, with particular emphasis on the role of Plotkin's tie. We establish two complementary continuity theorems. First, whenever \(X\) is an FS-domain, the Scott function space \([X\to T]\) is a continuous dcpo. The proof introduces finite-layer truncation maps on Plotkin's tie, which generate directed families of way-below approximants below every Scott-continuous map. Secondly, whenever \(L\) is an FS-domain, the Scott function space \([T\to L]\) is again a continuous dcpo. Here the argument is based on finitely separating approximate identities, together with a finite-control analysis of the two-branch order structure of Plotkin's tie. These two approximation mechanisms are conceptually different but both produce the directed families of way-below approximants required for continuity. To determine the limits of these positive results, we consider the Lawson closed-disk domain. Although \(\Disk^{\top}\) is an FS-domain, the Scott function space \([\Disk^{\top}\to T]\) is shown to be continuous but not itself an FS-domain. This establishes that preservation of continuity is strictly weaker than preservation of the FS property. The paper concludes by identifying the boundaries of the present methods and proposing a unified approximation principle that may provide a general characterization of continuity for Scott function spaces.

In this paper we develop the $(\rho,\delta)$ calculus of bisingular pseudodifferential operators on the product compact Lie groups, which extends the one that was earlier introduced by the second and third authors in the $(1,0)$ case. The $(\rho,\delta)$ case is the calculus suitable to construct parametrices of hypoelliptic operators on $G_1\times G_2$.

We establish an ultimate Ishii-Pastur theorem for whole-line ergodic block Jacobi operators. We prove that, for almost every realization, the restriction of a maximal spectral measure to the region where the smallest nonnegative Lyapunov exponent is strictly positive is carried by a Borel set of zero capacity. In the scalar case, this settles the whole-line problem formulated by Damanik and Fillman \cite[Problem~4.7.18]{Damanik-Fillman-book}.