Papers

We revisit Serrin's interior spatial regularity criterion for distributional solutions to the Navier-Stokes equations in $\mathbb R^3$ and considerably relax the hypotheses in two main directions. More precisely, we show that if $u\in{L_t^{s'}L_x^s}$ locally is a distributional solution to the Navier-Stokes equations with $\frac2{s'}+\frac3s=1$ for $s'\in[4,\infty)$, then $u\in L^q_t(C_x^\infty)$ locally for all $q\in(2,s')$. If $s'\in(2,4)$, the same conclusion holds provided that in addition $u\in L_t^4(L_x^p)$ locally, for some $p>1$. In particular, we remove any integrability hypothesis on the vorticity, and we reduce the requirement of integrability in time all the way to $L^4$ from $L^\infty$. To achieve this, we employ a new bootstrap argument, distinct from Serrin's, and we argue that a reduction of the exponent in time integrability does not follow from Serrin's original argument.

We introduce and study a one-parameter family of algebras that naturally generalize the Schur algebras. We show the Schur algebra is canonically a quotient when the parameter is a nonnegative integer, characterize when they are semisimple, show they are based quasi-hereditary, and that their category of representations is a highest weight category that can be identified as a subcategory of parabolic category $\mathcal{O}$ for the general linear Lie algebra.

We consider a natural CQMS structure on a twisted transformation groupoid $C^{\ast}$-algebra coming from stratified $_{\text {c}}$Lip-norm introduced by Austad. We obtain upper bounds of metric dimension of reduced $C^{\ast}$-algebra of a transformation groupoid $\Gamma\rtimes X$ and its cocycle twist for a suitably chosen CQMS structure, provided $(X,d)$ is a compact metric space of finite Kolmogorov dimension and $\Gamma$ is a discrete group of polynomial growth. When $\Gamma$ has exponential growth, we prove that the dimension is generically $+\infty$ proving that the dichotomy between polynomial growth and exponential growth of groups survive even after considering cocycle twists of transformation groupoids.

Accurate massive MIMO channel state information (CSI) acquisition with low pilot overhead is critical in dynamic propagation environments. Exploiting temporal correlation is key to reducing pilot overhead, yet most existing methods often rely on impractical assumptions. The approximate message passing with side information (AMP-SI) algorithm, built upon a birth-death-drift (BDD) model, represents a significant step in this direction. However, its practical deployment is hindered by three major limitations: reliance on i.i.d. Gaussian sensing matrices, need for perfect BDD parameter knowledge, and a statistically approximate treatment of temporal information. To address these limitations, we introduce BDD-VAMP-EM, a fully automated algorithm that relies on the BDD model, vector AMP (VAMP), and expectation-maximization (EM) in a unified framework. Simulations show that BDD-VAMP-EM consistently outperforms existing benchmarks, particularly under model parameter mismatch, confirming its practical viability.

For Baker's Borel-rectangle measure $\mu$ and $1\le p<\infty$, we classify the continuous vectors of each one-parameter translation group on $L^p(\mu)$. For each $v\in\mathbb{R}^\mathbb{N}$, the continuous vectors form a projection band, which coincides with the whole space when $v$ has finite support, is trivial when $v\notin\ell^1$, and is nonzero and proper otherwise. We also establish in ZFC the isometric classification considered in [Isometric classification of the $L^{p}$-spaces of infinite dimensional Lebesgue measure, Banach J. Math. Anal. 20 (2026), no. 1, Paper No. 7], showing that both $L^p(\mu)$ and the kernel of restriction to integer cubes are lattice isometric to $\ell^p(\mathfrak{c},L^p[0,1])$. When $v\in\ell^1\setminus c_{00}$, the continuous band and its disjoint complement have the same lattice-isometric structure, with the latter contained in the cube-restriction kernel.

We give a new algorithm to attack RSA with small private exponent, when some partial information of $p+q$ is given.The algorithm is a very simple modification of original Wiener's attack with continued fractions, and allows to factor $n$ whenever $d<n^{(1+\delta)/4}$ if we know a $\delta$-fraction of the most significant bits of $p+q$. The algorithm is unconditional, which is not the case in previous improvements that use Coppersmith method. As an example, ouir algorithm can be applied to break any criptosystem with modulus of $512$ bits and $d<n^{0.3}$, giving an improvement in the original attack of Wiener.

We introduce $n$-almost symmetric Datalog and study $n$-almost symmetric linear arc monadic Datalog. We characterize the finite relational structures whose constraint satisfaction problem is solved by this Datalog fragment as those that can be primitive positively constructed from the transitive tournament on $n+2$ vertices. We also give characterizations in terms of a certain homomorphism duality (which we call $n$-fixed unfolded caterpillar duality) and in universal-algebraic terms (the existence of $k$-absorptive operations and of operations forming an elevator chain of length $n+1$). This article generalizes the results from Bodirsky and Starke about symmetric linear arc monadic Datalog.

The celebrated Cesàro operator is a well-known operator with interesting connections to a variety of objects in operator theory. Generalizations have been made for Cesàro-type operators acting on weighted Hardy spaces but constructing analogs of the Cesàro operator for function spaces of several complex variables such as the Drury-Arveson space has yet to be achieved. In this article, we posit a definition we belief is the correct generalization to several variables and establish a few of its basic properties.

Hypertoric (or toric hyperkähler) varieties are a class of symplectic singularities and their resolutions, obtained as Hamiltonian reductions of a symplectic vector space acted on by a torus. In physics, they appear as Higgs (and Coulomb) branches of 3d $\mathcal{N}=4$ supersymmetric quantum field theories with abelian gauge group. In this work, we construct an $\hbar$-adic (in the sense of microlocalisation) sheaf of vertex operator superalgebras over a given smooth hypertoric variety. Its global sections give the $A$-twisted boundary of the corresponding 3d gauge theory. We use this to prove that the associated affine variety of this hypertoric vertex operator superalgebra recovers the singular hypertoric variety. This proves the 3d Higgs branch conjecture for a large class of boundary vertex operator superalgebras. In particular, these vertex operator superalgebras are quasi-lisse. This is in contrast to the (purely even) hypertoric vertex operator superalgebras (and their $\hbar$-adic localisations) constructed previously by Kuwabara as global sections of sheaves on families of universal Poisson deformations of the hypertoric varieties. These are generally not quasi-lisse. We show that the vertex operator superalgebras defined in this paper are (fermionic) simple-current extensions of those defined by Kuwabara, and investigate the consequences for symplectic duality and characters. We observe that the latter are upgraded from partial (or false) theta functions to quasimodular forms.

A convexity space is an ordered pair $(V,\mathcal{C})$, where $V$ is an arbitrary set and $\mathcal{C}$ is a family of subsets of $V$, called convex, which contains $\{\emptyset,V\}$ and is closed under intersections and nested unions of its elements. For any $S\subseteq V$, the convex hull of $S$ is the inclusion-wise minimum convex set $C\in \mathcal{C}$ such that $S\subseteq C$. For a convex set $C\in \mathcal{C}$, an element $p\in C$ is an extreme of $C$ if $p$ does not belong to the convex hull of $C\setminus\{p\}$. A convexity $\mathcal{C}$ defined over $V$ is a convex geometry if any convex set $C\in \mathcal{C}$ is the convex hull of its extreme elements. Given an oriented graph $D = (V,A)$, the family $\mathcal{C}$ of subsets of $V$ is the $\overrightarrow{P_3}$-convexity defined over $D$ if $\mathcal{C}$ is formed by all (convex) sets $C\subseteq V$ such that no vertex $v\in V\setminus C$ is the central vertex of a directed path $P=(u,v,w)$ with $\{u,w\} \subseteq C$, while in the $\overrightarrow{P_3^*}$-convexity defined over $D$, we have that no vertex $v\in V\setminus C$ is the central vertex of a directed path $P=(u,v,w)$ such that $\{u,w\} \subseteq C$ and $(u,w)\notin A$. In this work, we present necessary and sufficient conditions over an oriented graph $D$ so that the $\overrightarrow{P_3}$-convexity over $D$ is geometric, or the $\overrightarrow{P_3^*}$-convexity over $D$ is geometric. While the first case implies a polynomial-time algorithm to decide whether the $\overrightarrow{P_3}$-convexity over $D$ is a geometric, we show that it is coNP-complete to decide whether the $\overrightarrow{P_3^*}$-convexity over $D$ is a convex geometry. We also present a family termed acyclic indifference oriented graphs and demonstrate that deciding whether the $\overrightarrow{P_3^*}$-convexity in this class is geometric can be solved in polynomial-time.

In this work, we study the initial boundary value (IBV) problems for the sine-Gordon (sG) equation in the light-cone coordinates $u_{xt}=\sin u$ in the quarter planes $x> 0$, $t>0$ and $x< 0$, $t>0$ assuming a suitable decay as $x\to +\infty$ or as $x\to -\infty$. Employing the Riemann–Hilbert (RH) problem framework, we demonstrate that these two IBV problems differ significantly with respect to the boundary data required for well-posedness. Specifically, the solution of the “right problem” ($x\ge 0$) is uniquely determined by the initial data $u(x,0)$, $x\ge 0$ alone whereas for the “left problem” ($x\le 0$), the boundary data $u(0,t)$ has to be prescribed in addition to the initial data in order to obtain a well-posed problem.

In this paper we consider a M/G/1 queue for which we want to minimize the expected response time. We show how to compute indices from $n$ samples of the job size distribution such that the corresponding index policy is asymptotically optimal as $n$ grows. This construction is based on a discretization of the bounded support of the job size distribution and a shift of the samples to their nearest discrete point to the right. We show that the Gittins index of the empirical distribution of these shifted samples is close to the Gittins index of the original distribution. This translates to the asymptotic optimality of the corresponding index policy for minimizing the expected response time. Numerical comparison with other approaches further confirm the efficiency of our approach.

We consider the problem of efficiently computing the matrix step function of a large dense symmetric matrix. To this end, we introduce a recursive polynomial expansion method in which a composite polynomial of high degree is built recursively from component polynomials of degree eight. The component polynomial used in each iteration is designed to achieve strong amplification of the spectral gap across the step while favorably positioning the updated gap for subsequent iterations. A key ingredient is a novel evaluation scheme for arbitrary matrix polynomials of degree exactly eight requiring only three matrix-matrix multiplications and three matrices in memory. This scheme makes available a substantially larger class of component polynomials than previously possible within a three-multiplication budget, thereby expanding the class of composite polynomials that can be generated. Together with our polynomial selection strategy, this leads to a significant and consistent reduction in the number of matrix-matrix multiplications required to compute the matrix step function compared to existing recursive expansion methods.

Let $\Omega\subseteq\mathbb{R}^{n}$ ($2\leq n\in\mathbb{N}$) be a domain and let $\zeta\in\{0,\infty\}$ be an isolated point of the boundary of $\Omega$ in the one-point compactification of $\mathbb{R}^{n}$ with the ideal point $\infty$. Under some further conditions, we study Fuchsian-type singularity at $\zeta$ for the Finsler $p$-Laplace equation with a potential $$-\mathrm{div}\mathcal{A}(x,\nabla u)+V|u|^{p-2}u=0\quad (1<p<\infty)\qquad \mbox{in } \Omega,$$ where $\mathcal{A}(x,\xi)\triangleq\nabla_{\xi}(H(x,\xi)^{p}/p)$ for almost all $x\in\Omega$ and all $\xi\in\mathbb{R}^{n}$, $H$ is a family of norms on $\mathbb{R}^{n}$ ($n\geq 2$) parameterized by points $x\in\Omega$, and $V$ belongs to a local Morrey space. In particular, we investigate asymptotic behaviors of positive solutions of the equation near $\zeta$ and asymptotic behaviors of their quotients.

When solving the Einstein vacuum equations, a very helpful feature is that they reduce simply to the vanishing of the Ricci tensor. In Finsler gravity, a promising extension of general relativity that can describe the gravitational field of kinetic gases from a phase space perspective in terms of Finsler geometry, such a reduction is not as straightforward. In this article, we identify precise conditions under which the scalar Finsler gravity vacuum equation (in either its purely metric or its Palatini formulation) reduces to the vanishing of the Finslerian Ricci curvature. Through analytic arguments, we find that this happens if there exists some power $F^n$ of the Finsler function $F$ that is sufficiently regular and whose associated Finsler metric is non-degenerate on the light cones. Moreover, the Landsberg term in the scalar equation must vanish. This result significantly generalizes the findings of [Villasenor2024], where a reduction theorem was established under the quite strong assumption that $F^2$ is regular, which is not satisfied by many examples currently under consideration. We demonstrate the impact of our findings by applying them to solve the Finsler gravity equations for homogeneous and isotropic Finsler spacetime functions of Landsberg type.

We prove that for all $c\in\mathbb N$ and nonnull graphs $H_1,\ldots,H_t$, there exists $n\in\mathbb N$ such that if $G$ is a $c$-edge-colored complete graph with no monochromatic induced copy of the complete join of $H_1,\ldots,H_t$, then $V(G)$ is the union of $n$ sets $V_1,\ldots,V_n$ such that within each set $V_j$ with $|V_j|\neq 1$, the edges of some color form a graph that excludes at least one of $H_1,\ldots,H_t$ as an induced subgraph. In fact, the same holds even if the colors overlap, and with a different list of graphs $H_1,\ldots,H_t$ assigned to each color. When $H_1,\ldots,H_t$ each have a single vertex, this is Ramsey's theorem, and when $c=2$, this is the "excluding pairs of graphs" theorem of Chudnovsky, Scott, and Seymour.

We propose a sufficient condition for the non-periodicity of a set of Wang tiles. It applies to sets of Wang tiles whose tiles have vertical or horizontal stripes. The proof is based on a geometric argument involving a quadrilateral circumscribed to a parabola from which we conclude the irrationality of the densities of the vertical and horizontal stripes. We apply the sufficient condition to propose new proofs of non-periodicity of known sets of Wang tiles, including an encoding of Penrose tilings into 24 Wang tiles and the family of metallic mean Wang tiles. Conversely, for every pair $(\alpha,\beta)\in[0,1]^2$ of irrational numbers in the same quadratic number field, we construct a finite aperiodic set of Wang tiles with stripes that admits a valid tiling whose density of vertical stripes is $\alpha$ and density of horizontal stripes is $\beta$.

Fix an integer $m$ bigger than 2. We prove that if there exists a finite group with $mp+1$ Suylow $p$-subgroups, where $p$ is large enough, then $mp+1$ is prime. This improves on a theorem of M. Hall and is a partial answer to Brauer's Problem 26. Our proof uses techniques from analytic number theory, and it also raises new questions in that area.

In this paper, we classify the hypersurfaces of $\mathbb{H}^2\times\mathbb{H}^2$ with constant sectional curvature. In contrast to the case of $\mathbb{S}^2\times\mathbb{S}^2$, the resulting examples for $\mathbb{H}^2\times\mathbb{H}^2$ exhibit greater diversity, and we construct a special example with non-constant product angle function. For $\mathbb{S}^2\times\mathbb{S}^2$, however, the product angle function of any constant sectional curvature hypersurface is identically zero. As a byproduct, we also classify the hypersurfaces of $\mathbb{H}^2\times\mathbb{H}^2$ with constant product angle function and constant mean curvature (or constant scalar curvature).

Fix an abelian group $A$, a graph $G$, and nowhere-zero $A$-flows $f'$ and $f''$ on $G$. Now $f'$ and $f''$ are \emph{$A$-flow-adjacent} if there exists a cycle $C$ in $G$ such that $f'(e)-f''(e)=0$ for all edges $e\notin E(C)$. And $f'$ and $f''$ are \emph{$A$-flow-equivalent} if there exists a sequence $f_0,\ldots,f_s$ of $A$-flows such that $f_0=f'$, $f_s=f''$, and $f_i$ and $f_{i-1}$ are $A$-flow-adjacent for all $i\in[s]$. Given a group $A$, we seek conditions on a graph $G$ such that all $A$-flows on $G$ are pairwise $A$-flow-equivalent; in this case, we say that $G$ is \emph{$A$-flow-connected}. Analogously, we define $k$-flow-connectedness for nowhere-zero (integer) $k$-flows. The notions of $A$-flow-connectedness and $k$-flow-connectedness were first investigated by Esperet et al., who showed, among other results, that every $2$-edge-connected graph is $A$-flow-connected whenever $A=\mathbb{Z}_2^8$ or $|A| \ge 1.15\times 10^{694}$. In this paper, we first characterize the graphs that are $\mathbb{Z}_3$-flow-connected and that are $3$-flow-connected. We show that every 2-edge-connected graph is $A$-flow-connected if and only if this is true for every 2-edge-connected cubic graphs. We show that all cubic bipartite graphs are $\mathbb{Z}_4$-flow-connected, and construct other cubic graphs that are and are not $\mathbb{Z}_4$-flow-connected. We conjecture that every Eulerian graph is $k$-flow-connected and $A$-flow-connected whenever $k$ or $|A|$ is even; and provide evidence for this conjecture. Finally, we consider $4$-edge-connected graphs $G$. Here, we show that $G$ is $A$-flow-connected whenever $|A|\ge 5.3\times 10^6$.

We study compact Lagrangian self-similar immersions in the unit ball with Legendrian boundary. The Liouville form naturally defines a relative de Rham class, and we prove that the vanishing of this relative Liouville class forces the immersion to be a diffeomorphism onto an equatorial Lagrangian disk. In particular, this gives rigidity for exact immersions with connected boundary. We also establish a boundary flux identity and a localized boundary unique continuation theorem, yielding rigidity when a nonempty relatively open boundary portion satisfies the free-boundary condition, when the cosine of the contact angle has a fixed sign, or when the Legendrian capillary boundary is connected. In the two-dimensional minimal case, these results recover the smooth disk theorem of Li–Wang–Weng and the free-boundary theorem of Luo–Sun, while extending the underlying rigidity mechanisms to higher dimensions and the self-similar setting. Finally, in every dimension $n\geq 2$, we construct compact embedded exact Lagrangian examples in the minimal, self-shrinking, and self-expanding cases with two Legendrian capillary boundary components, nonzero relative Liouville class, and supplementary contact angles.

We characterize the kernel of the linearization $R$ of the minimal surface problem about the Euclidean metric in a bounded smooth domain $\Omega\subset\mathbb{R}^n$, $n\ge2$, with the background minimal surfaces being the Euclidean planes. We show that, in the whole-space Euclidean decomposition, the kernel consists of potential fields and TT fields. For bounded domains, a similar phenomenon appears with additional boundary coupling conditions; in particular, the TT part may be coupled to a harmonic conformal component.

We give an account of Kleene's Type 2 recursion theory modelled on Turing machines. We apply this account to observe that the feedback computation of \cite{AFL2020} is an example of Kleene Recursion in $^2\mathsf{E}$. An application of Gandy Selection in the feedback setting solves questions there raised on uniformly finding indices for recursive unions etc. of feedback semi-computable sets; further it allows for effective choice and other principles.

We study the isomonodromic deformation problem associated with rank-two meromorphic connections on the Riemann sphere having one regular singularity and one irregular singularity of even order at infinity, corresponding to the even Painlevé IV hierarchy. We show that the symmetry $\Psi(-\lambda)= \sigma_1 \Psi(\lambda) \sigma_1$ defines an invariant submanifold whose induced isomonodromic dynamics coincides with the Flaschka-Newell Painlevé II hierarchy. Under this identification, the corresponding Lax matrices, Darboux coordinates and Hamiltonian structures can be matched explicitly. In particular, the Hamiltonians of the first members of the Flaschka-Newell hierarchy are recovered from the even Painlevé IV hierarchy. This provides a geometric interpretation of the Flaschka-Newell hierarchy as a symmetry reduction of an isomonodromic deformation problem, complementing its classical description as a similarity reduction of the modified Korteweg-de Vries hierarchy.

Phase-field models play a central role in the continuum description of phase separation, in which the bulk free-energy density and the interfacial thickness parameter determine pattern formation and microstructural evolution. In practice, these constitutive quantities are rarely known a priori and must be inferred from limited dynamical observations. In this work, an extended pseudo-spectral physics-informed neural network (ESPINN) framework is developed for the inverse identification of phase-field models from transient snapshot data. It enables the simultaneous recovery of both the bulk chemical potential and unknown gradient coefficients. Numerical experiments on the one-dimensional Cahn-Hilliard equation demonstrate accurate and statistically stable reconstruction in the noiseless regime, with substantial constitutive information recoverable from even a single snapshot pair. In the presence of noise, reconstruction accuracy degrades gracefully, and increasing the number of snapshots improves robustness by reducing variance across runs. These results establish ESPINN as a data-efficient and physically consistent approach for learning free-energy structure in continuum models of phase separation.

Observer design for Lur'e-type systems typically reduces to solving a linear matrix inequality (LMI). In certain cases, the observer gain may grow unbounded with the coupling strength. We propose reconstructing key nonlinear terms via a bank of second-order sliding-mode observers and injecting them into a Luenberger observer as additional measurements. This feeds a second correction channel that attenuates the nonlinear coupling in the error dynamics. The convergence of the combined observer is guaranteed under a new LMI which contains the classical one as a special case. We exhibit parameter regimes where classical designs require impractically large gains, while the proposed design maintains moderate gains. A stability analysis of the proposed observer characterizes the trade-off between the two designs and identifies the noise regimes in which the proposed design is preferable. We illustrate the approach on a Wilson-Cowan network showing the trade-off between the nonlinear coupling strength, observer gain, and noise.

The highest rank of a string C-group representation of the alternating group $A_n$ is known for each $n$, but no self-dual representations attaining this highest rank are known when $n > 12$. Motivated by computational results for alternating groups of small degree, we examine a vertex-gluing construction for permutation representation graphs. We establish conditions under which gluing two string C-groups produces another string C-group, and use this construction to obtain infinite families of self-dual representations of alternating groups. In particular, for every $n = 4m+3 \geq 15$, we construct $\left \lfloor \frac{n+9}{8} \right \rfloor$ distinct self-dual string C-groups of rank $2m$ isomorphic to $A_{n}$. These representations have rank one below the maximum possible rank of string C-group representations for $A_n$, and to the authors' knowledge are the highest-rank self-dual representations currently known for alternating groups.

This work presents a new release of the lymph 2.0 library, an open-source MATLAB framework for high-order discontinuous Galerkin discretizations on general polytopal meshes. The lymph 2.0 version is extended to support discretizations with element-wise polynomial approximation degrees, which allows the design of $p$-adaptive strategies based on a posteriori error indicators. In addition, the library introduces a unified assembly framework that abstracts the construction of discrete operators from the underlying physical model, improving code modularity, parallelism, maintainability, and extensibility. Moreover, the proposed approach enables shared-memory parallelism through dedicated parallel tools. Several numerical examples demonstrate the effectiveness of the proposed developments in reducing the computational cost while preserving approximation accuracy.

Finite-time control is central to quantum technologies, yet rigorous limits on reachable targets and optimal control times remain largely unknown. We develop a framework for finite-time reachability and optimal-time certificates in constrained quantum control based on moment relaxations with implicitly time-dependent differential constraints. For fixed control horizons and control constraints, the method yields rigorous upper bounds on achievable terminal fidelities, lower bounds on the optimal control times required to reach them, and certificate gaps for benchmarking explicit control pulses. We demonstrate the versatility of our framework in three use cases: entangled-state preparation in two and three qubits, one-qubit gate synthesis across different control geometries, and excitation transfer in an $N$-qubit $XX$ chain. Our work establishes differential moment hierarchies as a practical tool for certifying reachability limits and optimal control times in quantum control, providing hardware-aware quantum speed limits while highlighting structure exploitation as a key ingredient for scalable certification.

In this paper, we propose a generalized framework for the design and analysis of suboptimal and reduced-order nonlinear Model Predictive Control (MPC) architectures. The proposed framework manages real-time operation of MPC schemes by (i) computing the control action suboptimally, i.e., by running a generic optimal control algorithm for a finite number of iterations, and (ii) relying on a reduced-order model that neglects part of the plant dynamics (accounting for, e.g., unmodeled dynamics or a low-level compensator). To rigorously handle the interplay between optimization error and model mismatch, we treat the sampling time as a tunable design parameter. We analyze the resulting closed-loop system, comprising the full-order physical plant interconnected with the iterative optimization algorithm (treated as a dynamical system), by leveraging tools from timescale separation. We prove that operating at a sufficiently fast sampling rate ensures that the closed-loop system maintains recursive feasibility and achieves an exponentially stable equilibrium point. The effectiveness of the proposed framework is validated on an underactuated two-link robotic arm through virtual experiments in the high-fidelity MuJoCo physics engine.