Papers

We study upper bounds on the length of $\mathbb F_q$-linear QMDS codes in the folded Hamming distance relative to their other parameters, especially the field size $q$. Via a correspondence between such codes and families of subspaces, we relate the length problem to that of upper bounding $1$-subspace packings with respect to the other parameters, especially the field size. Our main result is a reduction from these families to partial spreads, which allows us to import sharp bounds from finite geometry, including results of Drake-Freeman, Năstase-Sissokho, and Honold-Kiermaier-Kurz. As a consequence, we recover the Griesmer-type upper bound on the length of QMDS codes by Ball et al. and obtain tighter upper bounds in several parameter regimes.

Given an isomorphism between the fundamental groups of two compact hyperbolic manifolds of dimension at least three, we consider the associated equivariant harmonic lift. We establish a sharp pointwise gradient estimate for this lift. This yields a harmonic-map proof of Mostow rigidity.

We highlight a simple congruence transform that shifts coupling parameters for Dirac operators with shell interactions. As one of the consequences, this leads to new observations concerning the self-adjointness and the infinite mass interpretation of generalized MIT bag models.

We present the GPU acceleration and large-scale performance assessment of HORSES3D, an open-source high-order discontinuous Galerkin solver for computational fluid dynamics. The solver is ported to NVIDIA GPU architectures using OpenACC directives, preserving the original Fortran code structure while enabling GPU-resident execution of the main computational kernels. The implementation exploits the element-local structure of discontinuous Galerkin spectral element methods by mapping element-level loops to GPU gangs and nodal operations to vector-level parallelism. The GPU version is verified using the method of manufactured solutions and validated on canonical turbulent-flow benchmarks. Its performance is assessed on the MareNostrum 5 accelerated partition using NVIDIA H100 GPUs. Taylor-Green vortex benchmarks show that solver efficiency improves with polynomial order and that near-ideal strong and weak scaling is obtained when the workload exceeds approximately 16,000 to 20,000 elements per GPU. The solver is further evaluated on the High-Lift Common Research Model wing-body configuration, which involves a complex geometry, realistic boundary conditions, and unstructured meshes with up to 20.8 million hexahedral elements. Simulations with polynomial orders up to $P=7$ reach approximately $10.7 \times 10^9$ degrees of freedom and scale efficiently to 2,048 GPUs. The results demonstrate that HORSES3D preserves its performance characteristics for industrially relevant configurations and can exploit modern GPU-based supercomputers for billion-degree-of-freedom high-order CFD simulations.

We describe three problems in Hermitian geometry related to the scalar curvature of the Chern connection, aimed at identifying and constructing canonical metrics. They are chosen as toy examples where different techniques naturally interact: analytic methods for partial differential equations, the notion of momentum map, and variational methods.

We extend the Brenier-Strassen theorem about projections in convex order to non-flat spaces with curvature bounded from above. Precisely, for probability measures $\mu$, $\nu$ of finite second moment on a complete separable CAT(0) space, we prove that $\mu$ admits a unique W 2 -projection \bar{\mu} to the set of probability measures dominated by $\nu$ in convex order. Moreover, the unique optimal coupling from $\mu$ to \bar{\mu} is induced by a 1-Lipschitz map, without any absolute-continuity assumption on $\mu$. Our proof identifies the projection problem with a weak optimal transport problem whose cost is the squared distance to the set of convex means. We also establish a localized version on CAT(kappa) spaces with kappa \geq 0, where the optimal map is 1/2-Hölder continuous. Finally, we give a Strassen-type characterization of one-step barycentric martingales on proper CAT(0) spaces.

We consider the multi-dimensional Hamer model for radiating gases in its coupled hyperbolic–elliptic formulation. By means of energy estimates, we establish the global well-posedness for small initial data in hybrid Besov spaces with distinct regularity exponents at low and high frequencies. This framework enables us to relax the regularity assumptions required in \cite{Duan_Klem_Zhu_2010,Duan_Ruan_Zhu_2012}. In addition, we establish optimal time-decay estimates for solutions with initial data in the critical Besov space $\dot{B}_{2,\infty}^{-d/2}(\mathbb{R}^d)$, thus extending previous results obtained under the stronger assumption $L^1(\mathbb{R}^d)$. We discuss the optimality of these decay rates and derive improved decay rates under a zero-mass cancellation condition, corresponding to initial data in the larger negative Besov space $\dot B^{-d/2-1}_{2,\infty}(\mathbb R^d)$.

We propose an intrinsic stabilized Morley finite element method for the stream-function formulation of the surface Stokes problem on closed surfaces. The method is posed directly on a polyhedral approximation of the surface. A parameter-free jump stabilization is introduced to recover coercivity, and a discrete Korn's inequality for the trace-free Hessian is established. The main analytical contribution is a normal-separated geometric estimate for piecewise linearly approximated closed surfaces. It shows that a broad class of normal-dependent geometric consistency errors is in fact second order, even though a direct treatment suggests only first-order control. The missing order is recovered through an integral cancellation in the first normal variation. This estimate refines the standard $P_h\nu$-type estimate and yields second-order consistency for the trace-free Hessian, the Stokes-type tensor Green identity, and the conormal fluxes arising in the discretization. Together with the discrete Korn's inequality, these estimates yield, under the natural $H^3$ regularity, optimal-order convergence: first order in the broken $H^2$ norm and second order in the broken $H^1$ norm. As a direct consequence, the recovered tangential velocity admits a second-order $L^2$ estimate. Numerical experiments are provided to support the theoretical results.

Consider a nonnegative harmonic function $u$ on a domain $D\subset\mathbb{R}^{d-1}$. We establish boundedness of Jones' variational integral on an ultradense, $d-1$ dimensional subset of the boundary of $C^{1,\mathrm{Dini}}$ domains. This result was previously only known for the unit ball.

We obtain a new construction of a sequence of invariant differential operators on a Riemannian manifold $(M,g)$ that governs the linearized deformation theory of $g$. Starting from an explicit linear connection on a natural bundle $\mathcal AM\to M$, we construct a twisted de Rham sequence and then apply an analog of the construction of BGG sequences. If $g$ has constant sectional curvature, both sequences are complexes which compute the cohomology of the sheaf of local Killing fields, which are equivalent to parallel sections of $\mathcal AM$. In a second step, we relate the construction to the description of $(M,g)$ as a (torsion-free) Cartan geometry $(\mathcal OM,\omega)$, where $\mathcal OM$ is the orthonormal frame bundle of $M$. This provides a manifest relation of the twisted de Rham sequence to the deformation theory of the Cartan connection $\omega$ (which is easier do deal with than the deformation theory of $g$). The BGG-like construction can then be nicely viewed as interpreting the linearized deformation theory of torsion free Cartan geometries in terms of the underlying Riemannian metric.

It is well-known that the investigation of vortex filaments (i.e., moving curves) in the Euclidean 3-space $\mathbb R^3$ is an attractive topic both in physics and mathematics. The theory consists mainly of the three vortex models, up to the third-order approximation. Such a theory has been successfully extended to Hermitian symmetric Lie algebras in mathematics with physical and geometrical backgrounds. This article is devoted to developing it to Hermitian reductive Lie algebras in a purely geometric way. The three vortex models obtained in this article fulfill that when the Hermitian reductive Lie algebra ${\mathfrak g}$ equi-collapses to a Hermitian symmetric Lie algebra ${\mathfrak h}$, they revert respectively to those in ${\mathfrak h}$.

Let a finitely generated group $G$ act weakly branch on a locally finite rooted tree $T$ with boundary $\partial T$. The rooted tree structure is encoded by the overlap function, which is our name for the Gromov product on the boundary: \[ c(\xi,\eta)=|\xi\wedge\eta|. \] We axiomatise this function and show that when it is `admissible', one can recover the rooted tree. By the boundary rigidity theorem of Lavreniuk and Nekrashevych, $\operatorname{Aut}(G)$ acts canonically on $\partial T$. We therefore form the automorphism symmetrisation of the overlap function: \[ \widehat c(\xi,\eta) =\inf_{\alpha\in\operatorname{Aut}(G)}c(\alpha \cdot\xi,\alpha \cdot\eta). \] We prove that $ \widehat c $ is again an admissible $G$-invariant overlap function and that its associated tree $ \widehat T $ is locally finite. The action of $G$ on $ \widehat T $ is faithful and weakly branch, and is branch if and only if the original action on $T$ is branch. Moreover, \[ N_{\operatorname{Aut}(\widehat T)}(G) \cong \operatorname{Aut}(G). \] In particular, $ \operatorname{Aut}(G) $ is weakly branch. We also describe the finite weakly branch extensions of $ G $: they are precisely the pullbacks of finite subgroups of $ \operatorname{Out}(G) $. If $ G $ is branch, all these extensions are branch. In both cases, they act on the same tree $ \widehat T $.

Fiorini, Joret, and Wood (2013) showed that tree minors satisfy the so-called Erdős-Pósa property with a linear bound: For every tree $T$ there exists a constant $c \geq 1$ such that, for every graph $G$ and integer $k\geq 0$, either $G$ contains $k$ vertex-disjoint subgraphs each containing a $T$-minor, or $G$ has a set $X$ of at most $c k$ vertices such that $G-X$ has no $T$-minor. In this paper, we prove that the same result remains true if, given a subset $S$ of vertices of $G$, one only considers $T$-minors of $G$ that are rooted in $S$. Here, a $T$-minor is rooted in $S$ if there is a minor-model of $T$ where each branch set contains a vertex from $S$. This result can be seen as a generalization of the classical $S$-Path Theorem of Gallai, which corresponds to the case $T=K_2$. The upper bound on the size of $X$ is best possible up to the value of the constant $c$, and improves on an earlier $O(k^2)$ bound due to Hodor, La, Micek, and Rambaud (2026).

A theorem of Burić, Elezović and Vuk\v sić states that translating the argument in an asymptotic expansion, $f(x)\sim\sum(-1)^na_nx^{-n-1}$, to $f(x+t)$, replaces the constant coefficients $a_n$ by the Appell polynomials $R_n(t)$ generated by $(a_n)$. Their motivating examples came from the gamma and polygamma functions, where Bernoulli polynomials occur. We extend this construction beyond that setting and identify the Borel–Laplace representation $L_A(x)=\int_0^\infty e^{-xs}A(-s)\,\dd s$, whenever the integral exists. For the Gaussian kernel $A(-s)=e^{-s^2/2}$, this representation gives the Mills–Hermite expansion $M(x+t)\sim\sum(-1)^n\He_n(t)x^{-n-1}$, extending the classical scalar expansion. We establish an explicit remainder estimate and derive finite-difference formulas that remain in the Hermite polynomial algebra. As an application, we obtain a large-threshold expansion of the non-central Gaussian tail with explicit polynomial dependence on the mean. If $A(-s)$ is itself the Laplace transform of a positive measure, then the coefficients form a Stieltjes moment sequence. The associated Padé convergents give a systematic hierarchy of two-sided bounds in which successive lower and upper approximants incorporate the moments one at a time.

This paper studies a class of mixed regular-singular control problems under mean-variance criteria, where the drift and diffusion are allowed to depend on both the regular control and the level of singular control. We seek time-consistent equilibrium strategies in an intrapersonal game setting and propose a novel equilibrium notion where both regular and singular controls are unified via control laws, or equivalently, mappings on the augmented state space. Under which, we derive a verification theorem and necessary conditions providing a full mathematical characterization of the equilibrium. We apply the theory to a reinsurance problem. The equilibrium solution turns out to be nontrivially coupled, where the regular control depends on the level of singular control, and the free boundary of the singular control switches dynamically in accordance with the variation of the regular control expression. The free boundary is characterized in a piecewise semi-explicit manner and is proved to be $C^{1}$ across the switching point. In the degenerate case $\alpha_2=0$, the coupled solution reduces to the combination of two independent single-control equilibria and coincides with the limit as the parameter tends to zero.

A pyramid is a generalisation of metric measure spaces (mm-spaces) introduced by M.~Gromov (Birkhäuser 1999) to establish a geometric framework for measure-concentration problems. An extended metric measure space (emm-space) is another generalisation of mm-spaces introduced by Ambrosio–Gigli–Savaré (Invent.~Math.~2014) to extend Sobolev calculus and optimal transport theory to a broader extent. We prove that every pyramid has a representation by an emm-space through $1$-Lipschitz order. Furthermore, if pyramids are concentrated, this correspondence is unique up to isomorphism. This shows, for the first time, that all pyramids can be realised by concrete geometric spaces. Based on this representation, we introduce a new notion, a concentrated emm-space. We then define the observable distance for concentrated emm-spaces and establish three equivalent characterisations of concentration in terms of pyramids, Lipschitz observables and the observable distance. Furthermore, by developing an fibration approach, we show the $\Gamma$-$\limsup$ inequality of Cheeger energies as well as the stability of the log-Sobolev inequality and the Poincaré inequality under the weak convergence of pyramids associated with emm-spaces. Our results provide a new geometric approach for studying both the convergence of emm-spaces and concentration-of-measure phenomena across a broad class of infinite-dimensional models that have so far remained largely beyond the reach of existing geometric methods. Many of the significant examples arise in probability theory, including the Wiener space, the configuration space, Gaussian fields such as massive Gaussian free fields, spatial white noise and massive bi-Laplacian fields.

Networks with nearly identical degree distributions can place their hubs in sharply different neighborhoods. We develop a model diagnostic based on the mean degree of the neighbors of a degree-$k$ vertex. Under rank-one inhomogeneous random graphs, this statistic has degree-invariant centering and $k^{-1/2}$ fluctuations. Under non-rank-one kernels, posterior uncertainty about the root type can instead determine both centering and scale. Under linear preferential attachment, the statistic grows as $(m+\delta)\log k$. We turn these model-specific limits into goodness-of-fit tests for specified sparse-graph nulls and a weighted log-degree slope test for residual hub-neighborhood trends. Simulations evaluate null calibration, degree-distribution misspecification, and power against degree-matched preferential-attachment alternatives. Applications to high-school contact and arXiv coauthorship networks show that the method separates level misspecification from disassortative and positive residual trends. Reddit interaction networks provide a further appendix example.

Primal-dual active set strategies (PDAS) are popular iterative solvers for mixed complementarity problems such as constrained optimization problems with pointwise inequality constraints. Examples include the reduced-space active set algorithm vinewtonrsls found in PETSc. When applied to discretized infinite-dimensional problems, PDAS exhibit local superlinear convergence thanks to their equivalence to a semismooth Newton method (SSN). However, for many problem classes the number of iterations, to reach convergence, grows without bound under mesh refinement. In this paper we numerically study PDAS iteration counts on uniformly refined meshes for obstacle problems, Signorini problems, and related models. As the mesh size tends to zero, PDAS applied to Signorini-type problems lose their local superlinear convergence, resulting in linear growth of the iteration count (adding some iterations with each refinement). For obstacle problems, PDAS stagnates, leading to exponential iteration growth (asymptotically doubling with each refinement). We explain these phenomena by (i) proving that, for obstacle problems, nodal degrees of freedom only peel away from the obstacle layer-by-layer during the deactivation phase, (ii) deriving a general global convergence rate for PDAS that depends on the magnitude of dual feasibility violation, and (iii) demonstrating why, in the infinite-dimensional setting, this leads to a well-defined solver, but without local superlinear convergence, for some problems yet divergence for others.

We study the dynamics of periodic waves in a damped-driven nonlinear Schrödinger (NLS) equation perturbed by a small spatially dependent transport term. This equation arises in nonlinear optics as a model for light propagation in a passive optical cavity driven by a bichromatic laser source. Our main result shows that, for initial data close to a stable stationary wave of the translation-invariant unperturbed problem, the solution of the perturbed problem remains close to translated copies of this wave, with the translation parameter evolving according to an effective ordinary differential equation. If this effective equation admits a stable equilibrium, we prove that the solution of the perturbed NLS converges to a stationary state associated with that equilibrium. In particular, the asymptotically selected state attracts solutions with initial data that are not necessarily close to it, showing that its basin of attraction extends far beyond a small neighborhood of the state. The analysis is complicated by the loss of derivatives caused by the heterogeneous transport and combines the renormalization group method with a refined bootstrap argument that exploits parity-induced cancellations of quadratic nonlinear terms. Numerical simulations illustrate the analytical results.

We investigate the long-time behaviour of nonlinear kinetic Fokker–Planck equations with porous medium diffusion in a non-perturbative setting. Under a priori conditional bounds on macroscopic quantities, we establish exponential convergence to equilibrium in $L^1$. These bounds are automatically satisfied if the initial data is trapped between two global equilibrium profiles. Our approach combines the entropy-entropy dissipation structure and some techniques from $L^2$-hypocoercivity.

The Comtet numbers $b(n,k)$ and $B(n,k)$ of the first and second kind arise from the powers of $(1+x)\log(1+x)$ and of its compositional inverse, respectively. By interpreting both families as special values of weighted Stirling polynomials, we answer Lehmer's question of whether $B(n,k)$ satisfies a recurrence with a fixed number of terms by proving the four-term recurrence $kB(n+1,k+1)=B(n,k-1)-(n-k)B(n,k)-B(n+1,k)$. The corresponding relation for the unsigned array was conjectured by M. Kurkov, but not proved, in OEIS A354794. We further derive explicit formulas, convolution identities, and differential-difference relations for the Touchard-type polynomials attached to the two families.

Let $D$ be a digraph of order $n$ and size $m$ with the property that no out-neighborhood or in-neighborhood of any vertex is acyclic. We show that, if $D$ is strongly connected, then $m\geq 7n/3$, and, if $D$ is strongly $2$-connected, then $m\geq 8n/3$. Both results are best-possible.

In this paper, we study Holder regularity of the Hartman-Grobman theorem on vector bundles, considering regularity both along the fiber direction and along the base direction. While the main argument follows standard approaches, additional effort is devoted to constructing a continuous bundle norm that is equivalent to the original norm and with respect to which the linear part becomes hyperbolic at the first iterate. For the local version of the theorem, we further need to construct a suitable cut-off function on the vector bundle, which is achieved by employing partition of unity.

Graph $k$-coloring is one of the classic NP-complete problems. Previous work has studied its average time complexity, defined to be the average runtime of computing a $k$-coloring over the set of all $k$-colorable graphs on $n$ vertices. A highly influential result of Dyer-Frieze from 1989 gave an algorithm with $O(n^2)$ average runtime for constant $k$. This quadratic runtime appeared natural (and possibly even optimal) since almost all $k$-colorable graphs have $\Theta(n^2)$ edges, so one needs at least this time in order to read the (entire) input. However, this was later improved by Kučera in 1995 to average runtime $O(n^2/k)$ for every $k \leq n^{c}$ where $c \in (0, 1)$. Nevertheless, in the most interesting case of $k = O(1)$, the best-known bound remained quadratic in $n$. The true average complexity of the $k$-coloring problem has remained elusive for the last three decades. We break the longstanding quadratic barrier. Our main result in this paper shows that the exact average-case complexity of this fundamental problem is $\Theta(nk)$ for every $k \leq n^{c'}$ and some $c' \in (0, 1)$. For $k = O(1)$, this reveals the average sublinear nature of $k$-colorability: the average-case complexity is linear in $n$, and thus sublinear in the size of the input. We further show that our $\Theta(nk)$ average runtime is optimal, since a simple bound proves that every algorithm that correctly $k$-colors all $k$-colorable graphs requires $\Omega(n k)$ average runtime. Our proofs draw on ideas from sublinear and local algorithms and also yield a local computation algorithm (LCA) for $k$-coloring with average-case probe complexity $\text{poly}(k)$. A key new ingredient in our algorithm is a method for certifying the unique colorability of random subgraphs, using tools from the theory of graph regularity.

Motivated by thermal diffusion models, a controlled quasi-linear parabolic equation can be formally replaced by its linear approximation, when the initial datum, target and control actions are sufficiently small. This paper makes this approximation precise at the level of optimal control problems. First, we provide verification criteria which distinguish whether an optimal control is an interior point or a boundary point of the admissible control set. We then prove that the differences between the optimal controls and the corresponding optimal trajectories for the quasi-linear and linear parabolic equations are higher-order infinitesimals with respect to the small initial datum and target appearing in the cost functional. Consequently, for small data, the optimal control problem governed by the quasi-linear equation can be approximated by the corresponding linear problem. In the interior-point case, the resulting approximation order is sharp.

We study when a finitely generated group of bounded type contains the AF alternating group of the tail groupoid of its defining Bratteli diagram. This containment is an input to the full group completion theorem of Part I. We give sufficient conditions for this containment, in which selectors separate inherited diagonal actions, and an exact criterion for finite regular coverings. In the covering case, the intersection with the AF alternating group is the centralizer of the deck transformations commuting with the adjoined AF permutations. The formula classifies all subgroups between the canonical lift of the base AF alternating group and the AF alternating group of the covering. In particular, the lift is maximal exactly when the deck group has prime order, and cyclic coverings yield finitely generated groups whose intersections with the AF alternating group of the covering realize these subgroups. We identify fragmentations that produce the required selectors. For periodic fragmentations of the directed generator in the modified Fabrykowski–Gupta group, surjectivity of the activity subspace onto each pair of consecutive coordinates gives containment with absorption delay at most two. A five-periodic example also satisfies this condition, although its activity subspace has dual distance two. The original four-periodic fragmentation group has index four in its topological full group.

We establish the well-posedness of stochastic scalar conservation laws on moving hypersurfaces driven by Brownian motion. To handle the interaction between stochastic forcing and evolving geometry, we derive an Itô formula on moving surfaces and introduce the notion of generalized entropy solutions incorporating the relevant stochastic interaction terms. A martingale entropy solution is constructed via the vanishing-viscosity method, based on a uniform $L^\infty$-bound in space and time, an $L^1$-estimate for the spatial gradient, an $L^1$-continuity estimate in time, and a suitable tightness argument. Pathwise uniqueness is established by adapting Kruzhkov's doubling-of-variables method to moving hypersurfaces, yielding an $L^1$-contraction property. Finally, together with the Yamada-Watanabe theorem, these results yield the well-posedness of the problem.

Let $E(\mathcal{M},\tau)$ and $F(\mathcal{M},\tau)$ be two Calkin operator spaces affiliated with a semifinite von Neumann algebra $\mathcal{M}$ equipped with a semifinite faithful normal trace $\tau $. We show that if $\mathcal{M}$ is atomless, $\tau$ is finite, and $E(v,\tau)\not\subseteq F(\mathcal{M},\tau)$, then every order-measure continuous and disjointness-preserving mapping $T:E(\mathcal{M},\tau)\xrightarrow{\rm into} F(\mathcal{M},\tau)$ is identical to the zero mapping, which establishes a noncommutative version of Abramovich's theorem. We also show that every positive isometry $T$ from a normed $\mathcal{M}$-bimodule $E(\mathcal{M},\tau)$ of $\tau$-measurable operators into another $F(\mathcal{M},\tau)$ preserves disjointness provided that the norm of $F(\mathcal{M},\tau)$ is strictly monotone. As an application, we obtain the general form of $T$, which extends and unifies several results due to Abramovich, de Jager, Conradie, Veksler and Sukochev et al. \cite{SV,HSZ20,Abra1991,vek,dC20}.

We study optimization under performative prediction, where deploying a model affects the future data distribution. For this setting, several gradient-based approaches have been proposed. However, they typically assume specific data distributions or loss functions, which limit their practical applicability. To overcome these limitations, we propose a gradient-based optimization method with convergence guarantees under substantially weaker assumptions. Our method explicitly estimates the induced distribution shift through finite differences. It enables higher-dimensional optimization across broader classes of loss functions and data distributions. We also propose a practical variant that reduces the number of samples required. Numerical experiments demonstrate that our proposed algorithms converge faster and more consistently than existing ones.

In this study, we compare forward solutions computed with finite element methods in Zeffiro Interface and DUNEuro, using the different source models they provided. We compared two of the source models from DUNEuro, called Whitney basis and Local subtraction, and the divergence-conforming model of Zeffiro Interface. For source estimation, we applied sparsity-promoting standardized hierarchical adaptive L1 regression (SHAL1R), standardized Kalman filtering (SKF), classical sLORETA, and dipole scanning. Analyses include Earth Mover's Distance, depth bias scatter plots, and qualitative assessments of amplitude distribution and focality. Preliminary experiments with source interpolation for each method revealed that Local subtraction closely matches expectations for the local behavior of the lead field at various depths. The main results reveal that the success of an inverse method depends strongly on the compatibility between its assumptions about the focality of the source and the chosen source model, with point-source models performing best when paired with methods designed for such sources, i.e., sparsity-promoting methods and methods that scan for a single source. Moreover, source models that admit patch sources with a wide distribution are more sensitive to additional noise.