Papers

In this paper, we investigate a stochastic averaging principle for a large class of mechanical systems in the presence of random vibrations. We show that a known deterministic averaging principle for affine connection control systems with large-amplitude high-frequency inputs also holds for non-periodic stochastic inputs. The randomly vibrating mechanical system is described by a stochastic differential equation whose solutions do not depend on its interpretation in the sense of Ito, Stratonovich, or Sussmann. We also show that solutions of this stochastic differential equation can be directly computed from a single ordinary differential equation. We illustrate our theoretical results by the example of stochastic source seeking with a nonholonomic vehicle.

Given a finite collection of Lagrangian submanifolds $\mathscr L$ in a compact symplectic manifold $X$, we construct a cyclic, filtered, strictly unital curved $A_{\infty}$ category $\mathcal L$ and develop Floer theory of closed-open maps and open-closed maps. Using them, we prove that, whenever the map from the quantum cohomology of $X$ to the Hochschild cohomology of the Fukaya category $\mathcal L$ with objects $\mathscr L$ is injective, the following consequences follow: (1) any other Lagrangian submanifold equipped with a weak bounding cochain lies in the category split-generated by $\mathscr L$, and (2) the Hochschild homology and cohomology of the Fukaya category are isomorphic to quantum cohomology. In the exact case a similar result was obtained in [Ab]. We also provide some applications.

Diffuse-interface models are a widely used framework for interfacial dynamics in complex fluids, in which interfaces are represented through smooth transition layers and capillary effects are encoded by a free-energy functional. For incompressible mixtures with more than two phases, however, robust computation is substantially more difficult because the numerical method should preserve the balance structure of the continuum model, maintain the saturation constraint, dissipate energy, and treat all phases symmetrically even when density ratios are arbitrary. Existing structure-preserving methods are largely developed for binary flows or for formulations that distinguish a reference phase, so a genuinely symmetric N-phase discretization remains lacking. The practical problem is therefore to construct a fully-discrete method for N-phase incompressible Navier–Stokes–Cahn–Hilliard mixture models that retains the key thermodynamic and conservation properties of the continuum equations for arbitrary density ratios. Here we propose a symmetric fully-discrete method for the N-phase incompressible Navier–Stokes–Cahn–Hilliard mixture model with arbitrary density ratios. The method yields a fully-discrete problem in which every solution satisfies exact phase volume conservation, phase mass conservation, total volume conservation, total mass conservation, and a discrete energy-dissipation law. In addition, if the volume-saturation constraint holds for the initial data, then it is preserved at every time step. We numerically verify these structure-preserving properties and demonstrate the robustness of the method in representative multiphase flow problems. The resulting scheme provides a computational framework for incompressible N-phase mixture flows with complex interfacial dynamics and arbitrary density contrasts.

We resolve a question of Bailey, Hollars, Li and Luo. For all sufficiently large $n$, let $r=\lfloor(n-1)/2\rfloor$. We prove that the edges of any Hamiltonian Berge cycle in a simple $n$-vertex $r$-uniform hypergraph, together with any one additional edge, contain Berge cycles of every length from $2$ to $n$. In odd order we prove a stronger prescribed-unused-edge theorem using rigidity of large subsets of odd cyclic groups and an alternating matching exchange. In even order we introduce a two-gap edge-reassignment method. Split locks cover all lengths outside a seven-term middle band. The absence of the central length forces an exact reflected translation-wave structure, which is eliminated by an additive covering theorem derived from sum-free stability. The remaining near-central lengths follow from a two-defect recurrence and bounded-run forcing.

We study general properties of the perfectoidization of finite algebras over a perfectoid ring, which helps to understand some precise and explicit descriptions. For example, we prove that if $A=R[t]/(m(t))$ where $m(t)$ is monic, $R$ is perfectoid and the discriminant $d$ of $m(t)$ is a non-zero divisor of $R$ satisfying a bounded torsion condition, then $dA_{\mathrm{pfd}}\subset A$. We also prove a density criterion reducing the construction of the perfectoidization to adjoining suitable $p$-power roots modulo $p$. In the second part of the paper, we compute perfectoidizations in several families of examples, including Kummer-type extensions and split finite algebras.

Origami-inspired metamaterials exploit the interplay between geometry and elasticity to achieve programmable mechanical responses. Yet the origin and tunability of snap-through instabilities in non-rigidly foldable patterns remain poorly understood. Here we show that the Mars tessellation, a degree-4 vertex origami pattern composed of alternating square and rhombic faces, is not rigidly foldable because the folding-speed ratios required for vertex compatibility cannot be propagated consistently across neighboring units. This geometric incompatibility forces the facets to bend during folding, giving rise to a reproducible snap-through discontinuity in the force-displacement curve with a mean force drop of about 92.6 +/- 5.5 %, marking a transition between metastable states. Laser scoring of additional diagonal creases, guided by strain-field simulations, enables continuous tuning of the snap magnitude. These results reveal a general mechanism by which geometric frustration can be harnessed to program multistability in thin-sheet metamaterials.

In this paper, we prove the asymptotic stability of the critical constant steady state for a simplified parabolic–elliptic Keller–Segel system in $\mathbb{R}^N$ ($N \ge 3$), which admits a one-parameter family of constant steady states. Although the stability threshold for constant steady states is known, the critical case has remained open. We also show that the convergence rate in the critical case differs from the rates obtained for previously studied subcritical constant steady states.

In this article, we relate any Farey triangle in the extended upper half-plane to a variant of Colmez–Fontaine's fundamental lemma in $p$-adic Hodge theory. In particular, their original fundamental lemma corresponds to the fundamental Farey triangle $(\frac{1}{0},\frac{1}{1},\frac{0}{1})$.

We compute the stable homology of complex braid groups of types $B(e,e,n)$ and $B(2e,e,n)$ for fixed $e\ge2$ and increasing $n$. This accounts for the stable homology of all infinite families of complex braid groups. We achieve this by explicitly computing a quillenization of their stable classifying spaces. In particular, we provide a proof of an identification of the stable homology of Artin groups of type $D$ claimed by Fuchs in the '70s.

We study metric spaces that in some sense thin out at infinity. We define and investigate a measure of sparsity that is a quasi-isometry invariant, and introduce an analogue of topological ends for sparse spaces that is also invariant under quasi-isometries. We study some 51F30examples arising in various contexts.

Brouwer's Laplacian conjecture states that the sum of the largest $k$ eigenvalues of a graph's Laplacian is less than or equal to the number of edges plus $\binom{k+1}{2}$. We give a proof of this conjecture. Our proof relies on the Grone–Merris–Bai theorem for split graphs. We also show the converse, thereby establishing an equivalence between Brouwer's conjecture and the Grone–Merris–Bai theorem.

We get bounds on the Hausdorff dimension of the set where the Minkowski question mark function has infinite derivative.

We show that for any $A\subseteq \mathbb{F}_q^n$ lacking non-trivial solutions to a translation-invariant linear equation of genus one, meaning that no nonempty proper subset of the coefficients sums to $0$, there is a set $B\subseteq \mathbb{F}_q^m$ in some higher dimension which also lacks non-trivial solutions, such that \[|B|^{1/m}>|A|^{1/n}.\] In particular, this implies that no fixed cap set in $\mathbb{F}_3^n$ gives an asymptotically optimal lower bound by direct products alone.

In this paper, we develop a continuity method for the Dirichlet problem of Hessian equations on Riemannian manifolds. Such equations, introduced by Caffarelli, Nirenberg and Spruck, are defined in terms of the eigenvalues of the Hessian and a given pair $(f,\Gamma)$, where $f$ is a symmetric function defined in a symmetric cone $\Gamma\subset\mathbb{R}^n$, and $\Gamma$ specifies the set of admissible eigenvalues for the solution. Our method combines techniques from Morse theory with a characterization of the pair $(f,\Gamma)$. More precisely, in the type 2 case, we first construct admissible functions using Morse theory, and then solve the Dirichlet problem without any additional assumptions on the boundary or the subsolution. Building on this characterization of the pair, we can approximate the type 1 equation by a family of type 2 equations.

In this paper a general framework for convex integration is developed, in order to construct weak solutions to the Cauchy problem, by building on ideas from [C. De Lellis and L. Székelyhidi, Arch. Ration. Mech. Anal., 195 (2010)] and [S. Markfelder, Nonlinearity, 37 (2024)]. This framework may be applied to a large family of partial differential equations in order to construct weak solutions in $L^\infty ((0,T) \times \Omega)$ (for a bounded domain $\Omega)$ which are weakly continuous in time with respect to the weak topology of $L^r (\Omega)$ for some $r \in (1,\infty)$. This allows us to construct solutions which obey an energy inequality. In the second part of the paper we apply the framework to several inviscid models appearing in the field of geophysical fluid mechanics in order to show existence of weak solutions for all initial data, and to prove that there exist initial data for which there are infinitely many solutions which satisfy an energy inequality. We first consider the incompressible and the barotropic compressible Euler equations to recover the corresponding results from the literature. In addition, the framework allows us to prove a new result for the incompressible Euler equations, namely the global existence for the Cauchy problem in $L^\infty$. Moreover, we use the framework in the context of the hydrostatic Euler equations (also known as the incompressible inviscid primitive equations), which leads to the first convex integration approach which is able to construct admissible solutions with the natural energy for this system. A crucial ingredient in the proof of this result is the computation of a large subset of the convex hull. Finally, we apply the framework to the compressible inviscid primitive equations and to the inviscid quasi-geostrophic equations to obtain the first results on existence of wild data for these two geophysical models.

We construct a simple, separable, unital, nuclear C$^*$-algebra without uniform property $\Gamma$. The construction is based on a new topological obstruction arising from the Thom-Porteous theory of degeneracy loci. Constructions of pathological nuclear C$^*$-algebras over the past 30 years have used Chern class calculations introduced by Villadsen to obstruct the existence of large trivial subbundles. Here, by contrast, we use determinantal Schur classes to force every bundle map between certain equal-rank vector bundles to vanish somewhere on the base space. A quadratic Schubert calculus computation shows that this obstruction can persist across an inductive system and ultimately obstructs the comparison of projections by traces in the uniform tracial completion. The relevant Thom-Porteous classes live in degree proportional to the square of the forced rank loss, which in turn forces dimension growth of the same order in the constituent homogeneous C$^*$-algebras of our example. This identifies a new geometric threshold in the structure theory of nuclear C$^*$-algebras, linking the presence or absence of uniform property $\Gamma$ to quadratic dimension growth.

We develop new inference procedures for a linear IV model that are robust to identification strength and heteroskedasticity of unknown form, and nonparametric with respect to the first-stage equation. Our first test is tailored for inference on parameters of endogenous explanatory variables. Our new statistic modifies that of Antoine and Lavergne (2003) to directly account for heteroskedasticity of unknown form. As a result, it is asymptotically pivotal, so that inference is greatly facilitated in practice. We also develop (i) an identification-robust subvector inference procedure that does not rely on the knowledge of identification strength for the remaining parameters, and (ii) a pure specification test. In both cases, the tests are conservative but powerful. We show that our procedures are computationally friendly and competitive with existing ones in simulations and an application.

We present almost sure polynomial bounds for Hölder norms of solutions of the 1d periodic fractional Benjamin-Bona-Mahony (BBM) equation. Namely, we apply quantitative quasi-invariance of certain Gaussian measures with energy cutoff using the strategy from Tzvetkov (2015) and the globalization argument from Bourgain (1994) in order to extend, almost surely, the $L^2$-based deterministic control to the $L^{\infty}$-based setting.

Identifying the governing equations of complex dynamical systems remains a fundamental challenge across science and engineering. While early approaches relied on empirical data and heuristics, modern data-driven methods offer greater flexibility and fewer assumptions. However, data acquisition in real-world settings is often expensive. This work addresses this challenge by introducing an active learning strategy for dynamics discovery in the ultra-low data limit. Rather than sampling randomly, our method iteratively prioritizes regions that are most informative for model identification. This approach builds on Sparse Identification of Nonlinear Dynamics (SINDy), and utilizes an ensemble extension, E-SINDy, to estimate epistemic uncertainty and guide the sampling for both ordinary and partial differential equations (ODEs/PDEs). For ODEs, an exhaustive analysis is conducted on the Lorenz system across varying data budgets and noise levels. For PDEs, two systems with contrasting dynamical characteristics are examined: the Burgers' equation, where a sharp shock front creates a distinction between informative and uninformative regions, and the Kuramoto-Sivashinsky equation, which presents a more spatially complex sampling landscape. Across all scenarios, the proposed method accurately identifies the governing dynamics with significantly fewer data samples than random sampling.

We prove the group version of the Matrix Spencer conjecture. For every finite group $G$, there exist signs $\varepsilon\in\{\pm1\}^G$ such that $$\left\| \sum_{g\in G} \varepsilon_g\rho(g) \right\|\leq C\, \sqrt{|G|},$$ where $\rho$ is the left regular representation of $G$ and $C$ is a universal constant. This conjecture was posed in [BKMZ24], which settled it for simple groups; we establish it for all finite groups, combining the Peter–Weyl decomposition with the intrinsic-freeness inequalities of [BBvH23] in an iterated partial-coloring argument.

A large body of work studies the problem of learning an approximation to an implicit matrix $A\in \mathbb{R}^{m\times n}$ that is only accessible implicitly via matrix-vector product queries (matvec queries) of the form ${x} \rightarrow {A}{x}$ or ${x} \rightarrow {A}^T{x}$. Of particular interest are methods that learn a near-optimal approximation with a fixed sparsity pattern. For example, we might want to learn a near-optimal diagonal, banded, or arrow-head approximation to an implicit matrix $A$. Naturally, the number of matvec queries required to solve this problem depends on the sparsity pattern, which can be encoded as a binary matrix ${S}\in \{0,1\}^{m\times n}$. The query complexity of previous algorithms scales with quantities like the total number of ones in ${S}$, its maximum column/row sparsity, or the chromatic number of a its "conflict graph". These quantities are incomparable: for a given ${S}$, parameterizing by one might yield lower query complexity than another. In this work, we unify and tighten these prior results by providing a nearly sharp characterization of the matvec query complexity of sparse matrix approximation. Generalizing a definition from graph algorithms, let the degeneracy, ${degen}({S})$, denote the smallest number $k$ so that, if we iteratively delete all rows and columns of ${S}$ with $\leq k$ ones, we are left with an empty matrix. We show that a near-optimal approximation to $A$ with sparsity pattern $S$ can be learned with $\tilde{O}({degen}({S}))$ matrix-vector product queries, and $\Omega({degen}({S}))$ queries are necessary, for any sparsity pattern ${S}$. Moreover, unlike prior work based on graph coloring, all of our methods run in polynomial time.

Let $M \subset \{0,\pm1\}^n$ be a set such that $(m,m)=4$ for every $m\in M$, and $(m_1,m_2)\in\{-4,-3,-2,-1,0,3\}$ for any two distinct vectors $m_1,m_2\in M$. We determine the maximum possible cardinality of such a set $M$ for all sufficiently large $n$.

Moving horizon estimation (MHE) provides high precision state estimation for nonlinear systems, but it is often limited by the substantial computational demands of solving a nonlinear optimization problem at every sampling step. To address this issue, we develop an efficient MHE scheme based on linear parameter-varying (LPV) formulation, where the scheduling parameters are given by the estimated states of the system and used to construct inexact Jacobians. Due to the LPV representation, the Jacobian can be pre-specified offline in a structured form and then updated in the quadratic programming (QP) subproblem, which reduces computational cost commonly used in standard nonlinear programming (NLP) systems. We illustrate the performance by numerical simulations.

This paper develops a decoupled low-order conforming finite element method for a fourth-order elliptic singular perturbation problem in three dimensions. By means of a generalized Helmholtz decomposition, the problem is reduced to two second-order elliptic problems and a system of generalized singularly perturbed Stokes-type equations subject to a curl-free constraint. The former are discretized by standard linear finite elements. For the latter, we employ the MINI element and show that, after adding an $L^2$ term involving a Lagrange multiplier, the resulting discretization becomes robust with respect to the perturbation parameter. We further establish an error estimate of order $h^{1/2}$ uniform with respect to the perturbation parameter. Numerical experiments are included to support the theory.

The problem of pricing mobility services has attracted significant attention. In most studies, uncertain demand is modeled as an exogenous random variable with known distribution. This assumption overlooks the likely effect of prices on user adoption decisions. To address this dependency, we formulate the pricing problem as a stochastic program with decision-dependent demand uncertainty. Specifically, we make the non-standard assumption that the probability distribution of demand depends on pricing decisions. We show that the problem can be written as a mixed-integer linear program whose size is exponential in the input parameters. To find exact numerical solutions we specialize the L-shaped method for stochastic programs with decision-dependent uncertainty. In particular, we devise efficient separation routines by proving closed-form primal and dual solutions to the involved subproblems. In addition, we develop problem-specific valid inequalities and cut-sharing mechanisms which significantly improve convergence. We show that the method outperforms by far a commercial solver used to solve the monolithic formulation. Furthermore, in a case study based on a real-world carsharing system, we show that incorporating decision-dependent uncertainty improves expected profits by 8.39% compared to a benchmark that considers deterministic price-elastic demand, and by 8.53% compared to a benchmark that considers exogenous random demand, on average. In addition, we evaluate the performance of preventive pricing and relocation decisions under two vehicle allocation policies. The results suggest that a controlled allocation of vehicles to customers can improve service rates while only marginally affecting profits.

Port-Hamiltonian systems provide a structured framework for modeling physical systems by explicitly capturing their energy storage, dissipation, and exchange. However, deriving such models often requires detailed physical insight and precise knowledge of system parameters, which may not be available in practice. In this paper, we propose a kernel-based framework for the identification of port-Hamiltonian systems from input-state-output data. In contrast to conventional parametric approaches, the maps defining the port-Hamiltonian system are represented in suitably chosen reproducing kernel Hilbert spaces. This leads to an infinite-dimensional optimization problem over the corresponding function spaces. Our main result establishes a representer theorem that reduces this problem to a tractable finite-dimensional one. Since the reduced problem is non-convex, we further provide an algorithm for its solution and prove its convergence.

Accurate modeling of tsunamis (such as those generated by landslides) requires capturing both wave dispersion in the deep ocean and wave breaking near the shore. The shallow water equations are often preferred for working with tsunamis, but neglect dispersion and may be inaccurate in scenarios where dispersive effects are significant. In this work, we develop an approach that seeks to incorporate the best aspects of both hyperbolic and dispersive models by combining either of two hyperbolic reformulations of the Serre-Green-Naghdi equations away from the shore with the non-dispersive shallow water equations near the shore. The model is discretized and implemented within the GeoClaw software, and incorporates adaptive mesh refinement as well as shared-memory parallelism. We validate it through comparison with benchmarks and real tsunami data. The results and performance compare favorably with the existing dispersive water wave solvers, including a speedup of about 2x relative to GeoClaw's existing dispersive solver for a large-scale tsunami simulation.

We study a class of continuous stochastic flows driven by a space-time white noise and characterize their dual flows by explicit stochastic differential equations. A key ingredient of the proof is the convergence of solutions under coefficient approximations. As an application, we derive the dual flows in two illustrative examples, the squared Bessel flow and the Jacobi flow. We also introduce a new model of polynomially self-repelling (PSR) flow and show that it enjoys a self-duality property.

The transition to next-generation mobile communication networks, particularly 6G, demands advanced technologies to meet the requirements for ultra-reliable, low-latency communication, massive connectivity, and intelligent applications. Reconfigurable antennas (RAs) play a crucial role in achieving these objectives by enabling dynamic adjustments to the radio frequency (RF) characteristics of antennas, such as gain, radiation pattern, impedance, and polarization. Unlike traditional fixed-position antennas, RAs can alter both their radiation patterns and positions, offering flexibility in response to varying communication environments. This paper presents a comprehensive survey and tutorial on RAs, with a focus on fluid antennas (FAs), movable antennas (MAs), pinching antennas (PAs), and reconfigurable holographic antennas (RHAs), examining their potential in next-generation mobile networks. We explore the channel modelling and estimation, performance analysis, resource allocation strategies, and their synergy with other emerging wireless technologies for each type of RA. Finally, we provide a comparative analysis of different RAs and discuss the open challenges and future research directions, offering insights and guidance for future investigations in the exciting research area.

We study a mean-field limit for a $N$-particle system in which each particle follows a diffusion and interacts with other particles through a weight on each directed edge. Each weight evolves according to its own nonlinear SDE driven by a Brownian motion, with coefficients involving the states of the two endpoint particles of the edge. The initial vertex and edge variables are assumed to have a dissociated Aldous–Hoover form. We construct the limiting nonlinear SDE by averaging the interaction over an independent neighbor and an edge input, prove its well-posedness, and show that the dissociated vertex-edge structure is propagated by the dynamics. This propagation property is an analogue of propagation of chaos in the case where the weight of each edge may remain correlated with the states of the two endpoint particles. Under either a bounded-observable assumption or a sub-Gaussian edge-input condition, the finite system converges to this limit through quantitative coupling estimates for a typical particle and a typical edge. We also prove the convergence of the empirical measure of particle's state pairs and their interaction weights.