Papers

We present optimal block diagonal and triangular preconditioners for a mass-conserving mixed stress formulation of Stokes flow. The algebraic formulation leads to a double saddle point system with unknowns corresponding to discrete stress, velocity, vorticity, and pressure. MINRES equipped with a block diagonal preconditioner for an augmented Lagrangian formulation of this system is analyzed and shown to be optimal, in the sense that the convergence rate is independent of key parameters such as mesh size and kinematic viscosity. GMRES equipped with a block triangular preconditioner is also analyzed using a field-of-values approach. Finally, we present numerical results for both two- and three-dimensional model problems to validate the parameter robustness of the proposed preconditioners.

We prove that no Dirichlet $L$-function (and more generally, no nontrivial finite linear combination of Dirichlet $L$-functions) can vanish at every zero of a fixed $L(s,\chi_0)$. At the heart of the proof is a short-window asymptotic for the twisted discrete moment $\sum_{\rho} x^{\rho} L(\rho,\chi_1)$, where $\chi_0\ne\chi_1$ are primitive Dirichlet characters, $\rho=\beta+i\gamma$ runs over zeros of $L(s,\chi_0)$ with $T-\Delta<\gamma\le T$, and $x\in\mathbb{Z}$. The asymptotic is unconditional, assuming no hypothesis of GRH type, and it holds for every window width $\Delta\in[T e^{-C\sqrt{\log T}},\,T/\log T]$, thus reaching windows shorter than $T(\log T)^{-A}$ for any fixed $A$. Notably, the main term $\frac{\chi_1(x)}{2\pi}\,\Delta\log T$ depends on $x$ only through the single value $\chi_1(x)$. Since distinct primitive characters are distinguished by their values, varying $x$ isolates the contribution of each $L$-function within a linear combination, and we deduce that for a positive density of $x\in\mathbb{N}$, every nontrivial combination is nonzero at some zero of $L(s,\chi_0)$ in any sufficiently high short window. The proof combines contour integration of $-\frac{L'}{L}(1-s,\overline{\chi}_0)\,L(s,\chi_1)$ with short-interval estimates for the Dirichlet convolution $(\chi_0\Lambda)*\chi_1$, which derive from the classical de la Vallée Poussin zero-free region.

A pattern is a fundamental object used in the proof of Duron, Mählmann and Toruńczyk to show that hereditary 2-wqo graph classes have bounded clique-width. We answer a question in that paper by exhibiting a pattern-free hereditary graph class that is not 2-wqo.

We study the behavior of currents on flag-manifolds under actions of Anosov subgroups of complex semisimple Lie groups G. Given a current T of bidimension (k, k) on the full flag-manifold F = G/B, we average it under the group action via a construction analogous to the construction of Patterson-Sullivan measures. We show that under certain genericity conditions (in the case of currents of integration over subvarieties), the limiting current is a Gibbs current, i.e. is given by integration (with respect to a Gibbs measure) over the flag-limit set of suitable multiples of currents of integration along Schubert varieties based at the limit points of the group. We prove that the same equidistribution result (but without any genericity assumptions) for currents defined by smooth forms on F. We also prove a form of Axiom A property for the suspension flow of the group action on F.

We compute the K-theory of weighted blowups of smooth stacks satisfying the resolution property along smooth centers. As an application, we determine the K-theory of the stack of stable genus 1 curves with 2 marked points. Furthermore, we express the Lambda polynomial of the tangent complex of the blowup morphism in terms of the blowup data. Along the way, we generalize the Anderson-Payne construction of equivariant operational K-theory from torus actions to actions of smooth affine algebraic groups.

For an oriented graph $G$ the oriented chromatic number of $G$, written $\chi_o(G)$, is the least integer $t$ such that $G$ has a homomorphism to a tournament on $t$ vertices. The oriented chromatic number of a simple graph $H$ is the maximum oriented chromatic number over all orientations of $H$. Borodin, Kostochka, Nešetřil, Raspaud, and Sopena proved in 1999 that for all $\epsilon>0$, graphs with maximum average degree less than $4-\epsilon$ have bounded oriented chromatic number. This is in some sense optimal, because $1$-subdivisions of cliques demonstrate that there exists graphs with maximum average degree strictly less than $4$ and oriented chromatic number $\Omega(\sqrt{n})$. We prove that for every positive integer $d$, every $d$-degenerate graph $G$ with sufficiently large order satisfies $\chi_o(G) \leq 6(1+\frac{9d^2}{4})^{\frac{1}{2}}d 8^{d}\sqrt{n}$. This implies for a fixed $r\geq 4$, the optimal bound for the oriented chromatic number of graphs with maximum average degree less than $r$ is $\Theta(\sqrt{n})$. This complements a bound of Wood, who showed that for all $n$ vertex graphs $\chi_o \leq 2\Delta\sqrt{n-1}$ where $\Delta$ is the maximum degree.

For $\rho \in (0,1]$, let $R(\rho) = \mathbb{Z}[[z]] \cap O(B(0,\rho))$ denote the ring of power series with integer Taylor coefficients converging on the open disk $B(0,\rho)$. We prove that these rings are pairwise non-isomorphic as abstract rings. Three ingredients drive the proof: the ideal $(z)$ is the unique principal ideal with quotient $\mathbb{Z}$, so any isomorphism sends $z$ to a generator $g$ of $(z)$; every isomorphism is substitution by $g$, because it respects the $(z)$-adic filtration; and a Hadamard gap series with a natural boundary at $|w| = \rho_1$ forces the image $g(B(0,\rho_2))$ into $B(0,\rho_1)$, after which the Schwarz lemma and integrality of coefficients force $g = \pm z$ and $\rho_1 = \rho_2$.

We consider a model for the spread of an infection within an interacting particle system on $\mathbb{Z}^d$, generalizing a framework introduced by Kesten and Sidoravicius to a zero-range process in equilibrium with density $\rho > 0$. In our model, at any time, individuals are either healthy or infected. The infection spreads instantaneously whenever infected and healthy particles occupy the same site, while infected particles heal independently at rate $\delta > 0$. We investigate the extinction-survival phase diagram of this process starting from a configuration where only the particles at the origin are infected. For every fixed positive healing rate, we prove that the infection becomes extinct almost surely if the density $\rho$ is sufficiently small and survives with positive probability if $\rho$ is sufficiently large, establishing the existence of a non-trivial critical density. At sufficiently high densities, survival occurs even under instantaneous healing. We also show that, for every positive density, the infection survives when the healing rate is sufficiently small.

We study adaptive gradient descent for continuously differentiable, possibly nonconvex objectives under one-sided Hölder regularity. Unlike classical Hölder- or Lipschitz-gradient assumptions, which control the full gradient variation, our condition bounds only the directional term appearing in the descent inequality. This can allow less conservative step sizes when large gradient changes are orthogonal to, or favorable along, the update direction. We propose an adaptive scalar-step method based on an estimate of positive one-sided Hölder curvature, combined with a simple sufficient-decrease safeguard. For nonconvex objectives on a convex region containing the accepted update segments, we prove an explicit best-iterate stationarity bound with a rate determined by the Hölder exponent. Unlike predetermined diminishing step-size schemes, the method adapts to the local descent geometry. We evaluate the approach on two full-batch benchmarks designed to separate directional curvature from full gradient variation. On a binary classification problem, the method achieves the lowest final cross-entropy, objective value, and gradient norm, together with the largest classification margin among the compared scalar gradient methods. On a nonconvex Hölder regression problem, it attains the lowest final objective gap and gradient norm. These results indicate that one-sided Hölder curvature is an effective adaptive step-size signal when full-gradient variation is inflated by directions that do not hinder descent.

The solution of large-scale symmetric positive definite (SPD) linear systems arising from discretizations of second-order elliptic equations is challenging, especially in applications with highly heterogeneous coefficients, such as flow in porous media, which can lead to severely ill-conditioned systems. In this context, multiscale methods have recently been used to accelerate Krylov subspace methods, owing to their favorable parallel scalability. In this work, we present AlgMortar, a fully algebraic realization of the Multiscale Mortar Mixed Finite Element Method (MMMFEM). The method uses only information extracted from the fine-grid system matrix, which facilitates its implementation in existing solvers. AlgMortar uses graph partitioning to define a domain decomposition directly from the matrix graph. On each subdomain, it builds local linear systems that mimic Dirichlet problems, and couples the resulting local solutions through an algebraic interface condition that recovers the weak flux-continuity mechanism of MMMFEM. We prove that the method is well posed when the matrix is SPD. Numerical experiments on fine-grid linear systems arising from finite-volume discretizations of Darcy flow problems show that, when used as a preconditioner for the conjugate gradient method, the proposed approach exhibits good scalability and is competitive with state-of-the-art algebraic multigrid (AMG) methods for challenging heterogeneous, high-contrast test cases, including highly irregular corner-point grids. Additional tests with SPD matrices arising from problems other than finite-volume discretizations of Darcy flow showed competitive performance with AMG.

We show that the tadpole Nahm sum is essentially a specialization of a series studied by Bartlett and Warnaar. This overlooked connection was first discovered by the internal model mathvision-harness-0.3 at MathVision AI. We then use Macdonald-type identities for the Bartlett-Warnaar series to express the principal tadpole Nahm sum and its twisted version in terms of a Wronskian of generalized theta series. The principal case confirms a conjecture of Milas and Wang.

We introduce a parameterized computational framework for the evolution of strategic behavior in continuous games. We consider the collective dynamics of players through a time-dependent probability density over the strategy space, representing the likelihood of each strategy being chosen at any given time. Instead of directly solving the high-dimensional Fokker-Planck equation that governs this evolution, we represent the probability density as the pushforward of a reference distribution with parameterized pushforward maps and consider the evolution of the parameterized equations. The motivation for this work comes from a limitation of the parameterized Wasserstein gradient flow (PWGF) framework when it is applied to game dynamics. PWGF provides a parameterized approach for evolution equations of probability density that can be formulated as Wasserstein gradient flows. However, not all game dynamics admit such a gradient flow formulation. We generalize the parameterized pushforward map framework to the non-gradient flows and apply it to the game dynamics with provable error bound in Wasserstein metric. Numerical experiments with several non-gradient systems in economics are provided to demonstrate the effectiveness of this new framework.

We consider \[ \min_x f(x)+g(x), \] where $f$ is smooth but its value and gradient are expensive, while $g$ is nonsmooth and has a cheap proximal map. Douglas–Rachford splitting (DRS) then requires a sequence of smooth proximal solves. These solves have different centers but share the same nonlinear curvature. We introduce curvature-recycling DRS (CR-DRS). The method transports an old exact residual to the new proximal center, reuses a quasi-Newton model, and moves the old proximal state before the first new gradient is evaluated. We give paired direct and inverse BFGS and safeguarded symmetric-rank-one realizations. A certified variant accepts a transported quasi-Newton step only when it passes descent and residual tests; otherwise it uses a fixed number of safe gradient steps. Under strong convexity and smoothness, a small-gain condition gives global linear convergence with a constant number of new gradients per outer iteration. A transported Dennis–Moré condition gives local superlinear reduction of the new proximal error. We show that this condition is not automatic, derive full-model and active-subspace sufficient conditions, and relate the active subspace to the local DRS dynamics induced by the nonsmooth proximal map. Experiments on $\ell_1$- and total-variation-regularized problems show large reductions in expensive-gradient calls relative to restarted and curvature-only quasi-Newton proximal solves. For directly proximal $\ell_1$ models, we also compare with FISTA.

Given a strongly convex function $u$, equip $R^d$ with a Riemannian metric given by the Hessian $\nabla^2 u$. This is a so-called Hessian manifold. Given a probability density $\mu$ one may run a Langevin diffusion intrinsic to the manifold with stationary distribution $\mu$. Such (Hessian) manifold-valued Langevin diffusions are called Mirror Langevin diffusions (MLD) which have recently become popular. One of the questions we explore is whether, given $\mu$, one can choose $u$ to get an exponential convergence to equilibrium for the MLD, especially if $\mu$ is not strongly log-concave. Our results are based on Lyapunov function methods and give sufficient conditions for a Poincaré or a log-Sobolev inequality to hold for the MLD. These, in turn, imply exponential convergence. We also introduce a Markov chain approximation to the MLD given by a two step Gibbs sampler with stationary distribution $\mu$. This Markov chain is a variant of the Sinkhorn Markov chain introduced in arXiv:2307.16421 that is conjectured to converge to a time-inhomogeneous generalization of the MLD. Under suitable assumptions, we prove that the Markov chain has a guaranteed convergence rate in $\chi^2$ that is consistent with the diffusion time scale. Our proofs are based on ideas from entropic optimal transport and strong data processing inequalities.

Let $\Omega \Subset \mathbb{R}^{n}$ be a strictly convex bounded domain. Suppose $\mathbf{A} : \mathbb{R}^{Nn} \longrightarrow \mathbb{R}^{Nn}$, $\mathbf{B}: \mathbb{R}^{Nn} \longrightarrow \mathbb{R}^{N}$, $\mathbf{C}: \mathbb{R}^{N} \longrightarrow \mathbb{R}^{N}$ are linear maps, where $\mathbf{A}$ is symmetric and non-negative definite. Given $f \in L^2(\Omega, \mathbb{R}^N)$, we consider the problem of existence of solutions $u: \Omega \longrightarrow \mathbb{R}^N$ to the PDE system \[ \left\{ \begin{array}{rl} \displaystyle\sum_{\beta = 1}^{N}\sum_{i, j = 1}^{n} \mathbf{A}_{\alpha i \beta j}\mathrm{D}_{ij}^{2}u_{\beta} + \sum_{\beta = 1}^{N} \sum_{i=1}^{n} \mathbf{B}_{\alpha \beta i}\mathrm{D}_{i}u_{\beta} + \sum_{\beta = 1}^{N} \mathbf{C}_{\alpha \beta}u_{\beta} = f_\alpha, &\text{ in $\Omega$}, \\ u = 0,\ \,& \text{ on $\partial \Omega$}. \end{array} \right. \] This is a linear degenerate elliptic system, and it has not been considered before without the assumption of strict rank-one convexity. In general, it may not possess not even distributional solutions. By introducing some natural structural assumptions, we prove the existence of an appropriately defined unique generalised solution $u\in L^2(\Omega, \mathbb{R}^N)$, satisfying additional partial regularity properties. This paper extends earlier work of the first appearing author [N. Katzourakis, On linear degenerate elliptic PDE systems with constant coefficients, Adv.\ in Calc.Var.\ 9:3, 283-291 (2016)] to include lower-order terms.

We investigate finite-horizon, zero-sum controller-stopper games in which the stopper observes the state process and implements a feedback stopping rule. Building on a nonlinear Snell envelope representation for a related game established in a companion paper, we prove that our game admits a value by extending the associated first-contact principle. Our approach is purely probabilistic and yields an optimal feedback stopping rule for path-dependent systems while allowing for degeneracy in the underlying stochastic differential equation (SDE). As an application, we consider nonzero-sum controller-stopper games and show that the optimal feedback stopping rule for a zero-sum game appears as a component of a $\varepsilon$-Nash equilibrium for every $\varepsilon>0$.

This paper considers the general problem of the design of boundary controllers for distributed parameter systems. The control objectives are the asymptotic stability of the closed-loop system, as well as the disturbance attenuation of exogenous disturbances affecting the measurements and boundary conditions. More specifically, this work studies input-to-state stability (ISS) and stabilization for systems governed by coupled linear ordinary differential equations (ODEs) and homogeneous linear hyperbolic partial differential equations (PDEs), where actuation, sensing, and disturbance inputs are located at the system boundaries. First, we extend a previously established Lyapunov-based ISS condition for a similar class of systems to the H$_1$ setting under mild assumptions on the disturbance inputs. This extension not only ensures that the H$_1$-norm of the system states driven by non-vanishing disturbances and compatible initial conditions are bounded, but also provides an upper bound on the system H$_1$-norm. Subsequently, these theoretical results are applied to derive stability analysis and control design conditions expressed in terms of linear matrix inequalities. Numerical examples are provided to illustrate the effectiveness and potential of the proposed approach.

We establish dispersive decay estimates for two-dimensional Dirac equations with bounded and unbounded domain walls, together with estimates for the analogous one-dimensional problem. These are paradigmatic models of bulk-edge correspondence in topological insulators, yet their dispersive dynamics have not been previously studied. We show that topologically equivalent models may exhibit qualitatively different dynamics. Our new approach allows us to analyze models which are beyond the usual framework of localized perturbations of constant-coefficient Dirac operators.

Clustering of large deviations events in a stationary stochastic process depends critically on the interplay between the tail behavior of the marginal distribution and the strength of temporal dependence. In the class of doubly infinite moving average processes, when the memory is short, previous work has established a sharp contrast in the clustering patterns between light- and heavy-tailed settings, governed by conspiracy and catastrophe principles respectively. It has also been described how long memory interacts with the conspiracy principle to affect clustering in the light-tailed case. This paper addresses the interaction of long memory with the catastrophe principle in the heavy-tailed case. It turns out that long memory generally allows for a wider range of catastrophes to play a role and leads to longer and qualitatively different clustering patterns.

Akbari, Elphick, Kumar, Pragada and Tang [Discrete Math. 349 (2026) 114953] conjectured that for every connected graph G, the line graph of G has at most one more positive than negative adjacency eigenvalue; equivalently, the signature of a connected line graph is at most 1. We refute the conjecture with two independently found counterexamples: a 14-vertex cactus consisting of two pentagons attached by bridges to adjacent vertices of a square, whose line graph has inertia (9,0,7) by an exact characteristic-polynomial certificate, and a 48-vertex triangle-free graph found by simulated annealing and verified in exact rational arithmetic. Indeed, chaining copies of the 14-vertex graph yields connected graphs on 14k vertices whose line graphs have signature k+1 for every k >= 1. The signature of connected line graphs is therefore unbounded, and no constant-bound repair of the conjecture is possible.

Finding Nash equilibria in the pure coordination game is trivial when every player interacts with every other player evenly. However, when players have a relational structure, the question becomes harder to answer. Here, we show that Nash equilibria to the structured coordination game represent a local notion of the Min-cut partition by way of the potential function. Using the standard relationship between partitions and subgraphs in the planar dual, we define a quasimetric space for which Nash equilibria are local minimizers of a global objective. When the game is restricted to only two strategies, these results allow us to construct a dynamic subgraph process which recapitulates the Myopic Best Response dynamics of the coordination game.

This paper studies proximal-type relations and maximal equicontinuous factors for non-autonomous dynamical systems on compact metric spaces. Two classes of systems are considered: systems generated by uniformly convergent sequences of maps and periodic systems. For uniformly convergent generators, collective convergence and uniform rigidity imply that the regional proximal relation for the non-autonomous system is equal to the regional proximal relation for the limit autonomous system and if the limit system is equicontinuous, then the proximality relation for the non-autonomous system is equal to the proximal relation for the autonomous limit system. In the same setting, the kernel of the maximal equicontinuous factor is identified as the smallest closed equivalence relation containing the autonomous equicontinuous structure relation of the corresponding autonomous system and preserved by all generators. For periodic systems, regional proximality and Banach proximality are determined by the period map, yielding a criterion in terms of mean equicontinuity and the relation of sensitivity in the mean. Examples show that the hypotheses used in these results are necessary.

A C-complex is a union of Seifert surfaces for the components of a link which intersect each other in clasps. The clasp number of a link is the minimal number of clasps amongst all C-complexes it bounds. It gives a measure of how far a link is form being a boundary link. This paper provides a new lower bound for the number of clasps of all C-complexes bounded by a given 3-component link improving results of Amundsen-Anderson-D.-Guyer. Furthermore, we construct links that achieve these bounds. In order to do so, we express the triple linking numbers as the area bounded by three curves in the plane, called word curves, and then perform the geometry and discrete optimization needed to minimize the length of these curves.

We prove the generalized total rank conjecture over regular rings in characteristic $2$: if $R$ is a regular Noetherian domain of characteristic $2$ and $P$ is a differential $R$-module admitting a finite projective flag and having nonzero homology $H(P)$, then $\rank_R(P)\ge2^{\codim_RH(P)}$. In particular, we prove Carlsson's conjecture for elementary abelian $2$-groups in every rank. We also obtain sharp homology bounds for arbitrary continuous actions of such groups and for perfect complexes over finite group algebras; the sphere rank conjecture follows. The proof identifies the homology of a chain model for the $C_2$-Tate construction on $P\otimes_RP$ with the Frobenius pullback of $H(P)$, and compares lengths by deforming the Tate differential.

Strongly correlated quantum matter is fundamentally defined by the tension between non-commuting quantum operators. Hamiltonians exhibiting macroscopic degeneracies are of general interest in this field because they imply an infinite susceptibility to any non-commuting perturbation. In moiré heterostructures, engineering such extensive degeneracies in the kinetic Hamiltonian creates a fertile garden for exotic strongly correlated phases of matter to emerge from the resulting flat bands. Here, we introduce a prescription to construct an infinite family of exact flat band Hamiltonians supported on the faces of arbitrary graphs. We demonstrate this algorithm on the faces of four Bravais lattices. Using a discrete graph generalization of the Atiyah-Singer index theorem, we prove that the extensive degeneracies of such face-graph Hamiltonians are protected by the local topology of the face-graph connectivity. The resulting macroscopic null spaces yield compact localised states that remain localised over time due to frustration in hopping pathways. We discuss the broad implications of such non-dispersing quantum modes in diverse settings, from arrested dynamics in quantum networks and quantum machine learning algorithms to Majorana-free topological quantum computation.

Let $F(n)$ be the largest size of a subset of $\{0,1,\ldots,n-1\}^2$ containing no nondegenerate isosceles right triangle. We give a modified Salem–Spencer-type construction over the Gaussian integers showing that $F(n)=\Omega(n^{1.3})$. The best known upper bound is $F(n)\ll n^2/(\log n)^{1+c}$ for some absolute constant $c>0$, so there is still a large gap between the bounds.

We prove two-weight estimates for fractional sparse forms on spaces of homogeneous type. The relevant transformed weights are assumed to belong to weak-$A_\infty$. The weak reverse Hölder inequality is normalized on enlarged balls, while sparse testing is normalized on dyadic cubes. To measure this loss, we introduce $\mathfrak d_Q^w:=w(Q)/w(2\Lambda B_Q)$. A packing estimate for the defect levels, combined with two-weight testing, gives the required mixed weak-$A_\infty$ bounds.

In this work, we investigate the shape optimization of a Swiss alphorn to achieve resonance frequencies as close as possible to prescribed target notes. We construct an accurate geometric model of the alphorn based on both experimental measurements and imaging techniques. The latter are used to reconstruct geometric information where direct measures are not available. The resulting 3D model is meshed using a fully structured approach. Then, we introduce a geometry-parametrized finite element (FE) formulation of the Helmholtz equation enabling the application of the Reduced Basis Method (RBM) to efficiently generate a large dataset of simulations, in which each set of geometric parameters corresponds to a specific set of resonance frequencies (notes). This dataset enables the training of machine learning (ML) models for forward prediction of resonance frequencies from geometric parameters, as well as inverse design, where geometries are estimated to achieve specified target notes.

This paper investigates the discrepancy of a family of random sampling methods obtained by perturbing the grid $\frac{1}{M}\mathbb{Z}^{d}\cap\left[ -1/2,1/2\right)^{d}$, where $M$ is a large positive integer. Parameterized by an arbitrary probability measure $\mu$, this family encompasses several classical methods for evaluating the quality of an $N$-point set in $\mathbb{T}^{d}$, where $N=M^{d}$. We show that all probability measures, except for Dirac measures, behave like the Lebesgue measure in the Monte Carlo discrepancy. This represents a limiting case where the measure $\mu$ depends on $M$. In this latter context, we prove that, up to a constant, the lowest possible discrepancy is achieved when the support of $\mu$ has diameter $\leq c/M$, and that this upper bound is sharp.

Quasiprobability distributions for systems endowed with SU(1,1) dynamical symmetry have received surprisingly little attention, despite the central role of this symmetry in two-photon physics, squeezed states, and nonlinear interferometry. Here, we fill this gap by constructing a full covariant family of $s$-ordered quasiprobability distributions defined on the two-sheeted hyperboloid, or equivalently, on the Poincaré unit disk via stereographic projection. A key result is that the Wigner function is strictly positive for all Perelomov SU(1,1) coherent states, in sharp contrast to the SU(2) case. This positivity endows Wigner negativity with an unambiguous operational meaning: any negative volume is a direct signature of genuinely quantum behavior. We further examine the stellar rank of SU(1,1) states, defined through the zeros of the Husimi $Q$-function, and show how it compares with Wigner negativity as a geometry-adapted witness of nonclassicality in this setting. We further introduce a hierarchy of generalized multipoles through a harmonic expansion of the density operator on the hyperboloid, providing a complementary framework for probing quantumness. This offers a comprehensive toolkit for characterizing and quantifying quantum resources in SU(1,1) systems.