The Odd Hadwiger number of a graph $G$ is the largest integer $r$ such that $G$ has a clique of size $r$ as an odd minor. In this paper, we investigate how large is the Odd Hadwiger number of the product of two graphs, when considering any of the four standard graph products: Cartesian, direct, lexicographic, strong. We provide an optimal lower bound in the cases of the strong and lexicographic products.

This paper proposes an interior-point framework for constrained optimization problems whose decision variables evolve on matrix Lie groups. The proposed method, termed the Matrix Lie Group Interior-Point Method (MLG-IPM), operates directly on the group structure using a minimal Lie algebra parametrization, avoiding redundant matrix representations and eliminating explicit dependence on Riemannian metrics. A primal-dual formulation is developed in which the Newton system is constructed through sensitivity and curvature matrices. Also, multiplicative updates are performed via the exponential map, ensuring intrinsic feasibility with respect to the group structure while maintaining strict positivity of slack and dual variables through a barrier strategy. A local analysis establishes quadratic convergence under standard regularity assumptions and characterizes the behavior under inexact Newton steps. Statistical comparisons against Riemannian Interior-Point Methods, specifically for optimization problems defined over the Special Orthogonal Group SO(n) and Special Linear Group SL(n), demonstrate that the proposed approach achieves higher success rates, fewer iterations, and superior numerical accuracy. Furthermore, its robustness under perturbations suggests that this method serves as a consistent and reliable alternative for structured manifold optimization.

We construct the first weakly special surfaces that are not Campana-special, including the complement of the plane curve $x^2y^3 = 1$ in $\mathbb{A}^2$. We prove that the set of $\mathcal{O}_{K,S}$-integral points on this surface is non-dense for every number field $K$ and finite set $S$ of finite places of $K$ if and only if Campana's Orbifold Mordell conjecture holds for $(\mathbb{G}_m, \tfrac{1}{2}[1])$. This basic example carries a natural $\mathbb{G}_m$-action, and the quotient stack is an Artin stack parametrizing points on a C-pair. This leads to the introduction of ``Campana stacks'', which encode morphisms of C-pairs in a manner analogous to the role of root stacks for integral points satisfying prescribed divisibility conditions.

We prove a general upper bound on the $k$-th adjacency eigenvalue of a graph. For $k\ge 2$, we show that \[ λ_k(G)\le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1 \] for every graph $G$ on $n$ vertices. We build on a recent approach that addresses the case $k=3$ and generalize the upper bound for all $k \geq 3$ by using the positivity of Gegenbauer polynomials. The upper bound is tight for $k \in \{2,3,4,8,24\}$. We also highlight the close relation of $λ_k(G)$ to questions about equiangular lines.

We study dispersive decay for non-autonomous Hamiltonian systems. While the general theory for dispersion in such non-autonomous systems is largely open, it was shown \cite{kraisler2025time} that there exists a time-periodically forced one-dimensional Dirac equation with unusually slow dispersive decay rate of $t^{-1/5}$. It is to be expected that such behavior is not generic and requires a very particular forcing term; we provide a more general ansatz and systematic procedure to construct such an equation with a dispersive decay rate no faster than $t^{-1/10}$. Our limitations are purely algebraic and it stands to reason that arbitrarily slow decay, $t^{-\varepsilon}$ for every $\varepsilon > 0$, should be achievable.

Fully implicit tensor-product space-time discretizations of time-dependent $(p,δ)$-Navier-Stokes models yield, on each time step, large nonlinear monolithic saddle-point systems. In the shear-thinning regime $1<p<2$, especially as $p\downarrow 1$ and $δ\downarrow 0$, the decisive difficulty is the constitutive tangent: its ill-conditioning impairs Newton globalization and the preconditioning of the arising linear systems. We therefore develop a scalable monolithic modified Newton framework for tensor-product space-time finite elements in which the exact constitutive tangent in the Jacobian action is replaced by a better-conditioned surrogate. Picard and exact Newton serve as reference linearizations within the same algebraic framework. Scalability is achieved through matrix-free operator evaluation, a monolithic multigrid V-cycle preconditioner, order-preserving reduced Gauss-Radau time quadrature, and an inexact space-time Vanka smoother with single-time-point coefficient freezing in local patch matrices. We prove coercivity of the linearized viscous-Nitsche term in the uniformly elliptic regime $ν_\infty>0$ and consistency of the reduced time quadrature. Numerical tests demonstrate robustness with respect to model parameters, nonlinear and linear iteration counts, and scalable parallel performance.

In this paper, we consider the Ising model on random $d$-regular graphs (with $d\ge3$) and Erdös-Rényi graphs $G(n,d/n)$ (with $d>1$) at the critical temperature. We prove that the \textit{magnetization}, i.e.\ the sum of the spins of a configuration, is typically of order $n^{3/4}$ and when multiplied by $n^{-3/4}$ converges in distribution to a non-trivial random variable, whose density we describe. In the regular graph case, the Small Subgraph Conditioning Method applies, and the limiting density is of the form $\frac1{Z}\exp(-C_d z^4)$. Surprisingly, in the Erdös-Rényi case, while the ratio of the second moment and first moment squared is bounded, the short cycle count is not enough to explain the fluctuations of the partition function restricted to a particular magnetization. We identify the additional source of randomness as path counts of slowly diverging length. This quantity is motivated by the heuristic that correlations between distant vertices are proportional to their local branching rate. Augmenting the Small Subgraph Conditioning Method with these path counts allows us to prove convergence of the magnetization to a non-deterministic limiting distribution. To our knowledge, the need to condition on graph observables beyond the cycle counts is a new phenomenon for spin systems. As further corollaries, we derive a polynomial lower bound on the mixing time of the stochastic Ising model on sparse random graphs at the critical temperature complementing recent upper bounds. Moreover, we establish the fluctuations of the free energy in the Erdös-Rényi case, answering a recent question of Coja-Oghlan et. al.

We study Gaussian wave packet solutions for Maxwell's equations in an isotropic, inhomogeneous medium and derive a system of ordinary differential equations that captures the leading-order correction to geodesic motion. The dynamical quantities in this system are the energy centroid, the linear and angular momentum, and the quadrupole moment. Furthermore, the system is closed to first order in the inverse frequency. As an immediate consequence, the energy centroids of Gaussian wave packets with opposite circular polarisations generally propagate in different directions, thereby providing a mathematical proof of the spin Hall effect of light in an inhomogeneous medium.

In this paper we develop a theory for Patterson--Sullivan measures for non-Borel Anosov groups on the Furstenberg boundary. Previously, such a theory has been successfully developed for measures supported on the partial flag manifold associated to the Anosov condition, which coincides with the Furstenberg boundary only under the strongest Anosov condition, Borel Anosov. We establish existence, uniqueness, and ergodicity results for the measures on the Furstenberg boundary under arbitrary Anosov conditions; we show ergodicity of Bowen--Margulis--Sullivan measures on the homogeneous space; and we establish strict convexity results for the critical exponent associated to functionals on the entire Cartan subspace. Using this strict convexity, we establish an entropy rigidity result for Anosov groups with Lipschitz limit set. A key tool we develop is a new sufficient condition for the existence of a measurable boundary map associated to a Zariski dense representation. This result not only applies to Anosov groups, but also to transverse groups, mapping class groups, and discrete subgroups of the isometry groups of Gromov hyperbolic spaces.

In higher rank, there is a well-studied theory of Patterson--Sullivan measures supported on partial flag manifolds. However, establishing the existence and uniqueness of such measures is a difficult question. In this paper, we develop a theory for Patterson--Sullivan measures supported on certain vector-valued horofunction boundaries of the associated symmetric space, where existence is straightforward. We also introduce a notion of shadows for this compactification and establish a shadow lemma. For transverse groups, we prove uniqueness and ergodicity results.

We give a model-independent construction of directed univalent cocartesian fibrations of $(\infty,1)$-categories, and prove a straightening equivalence against such fibrations. The key step is showing that cocartesian fibrations descend along localisations, which we accomplish by analysing mapping spaces of localisations. Along the way we introduce a directed version of the join construction, giving a sequential colimit description of the full image of any functor.

Coarse-grained models of chaotic systems neglect unresolved degrees of freedom, inducing structured model error that limits predictability and distorts long-term statistics. Typical data-driven closures are trained to minimize error over a single time step, implicitly assuming Markovian dynamics and often failing to capture long-term behavior. Recent approaches instead optimize losses over finite trajectories. However, when such trajectory-based training is carried out with deterministic pointwise losses, it introduces a fundamental mathematical degeneracy. We prove that optimizing pointwise deterministic losses such as mean squared error over chaotic trajectories suppresses predictive variance, with corresponding loss of physical variability in long integrations. In contrast, strictly proper scoring rules avoid this degeneracy. By targeting forecast distributions rather than realized trajectories, they remove the penalty against predictive spread and align the long-lead optimum with the invariant measure. Using quasi-geostrophic turbulence as a canonical chaotic system, we validate this theory: one-step-trained closures fail to capture stable coarse-grained dynamics, while deterministic closures optimized over trajectories exhibit the variance-loss tendency predicted by our analysis. Stochastic closures calibrated over trajectories using the energy score, however, overcome both structural limitations, yielding skillful ensemble forecasts and realistic long-term statistics. Our results establish that both stochastic modeling and trajectory-based calibration are essential for faithfully representing the dynamics of coarse-grained systems.

Quantum trajectories are Markov chains modeling quantum systems subjected to repeated indirect measurements. Their stationary regime depends on what observables are measured on the probes used to indirectly measure the system. In this article we explore the properties of quantum trajectories when the choice of probe observable is randomized. The randomization induces some regularization of the quantum trajectories. We show that non-singular randomization ensures that quantum trajectories purify and therefore accept a unique invariant probability measure. We furthermore study the regularity of that invariant measure. In that endeavour, we introduce a new notion of ergodicity for quantum channels, which we call multiplicative primitivity. It is a priory stronger than primitivity but weaker than positivity improving. Finally, we compute some invariant measures for canonical quantum channels and explore the limits of our assumptions with several examples.

We investigate the pushed-to-pulled transition for a minimal model for invasive fronts influence by ``aerotaxis,'' that is, when organisms follow oxygen gradients. We consider two singular reaction-advection-diffusion models for this. The version of primary interest arises as a hydrodynamic limit of a system of branching, rank-based interacting Brownian particles and features a nonlinear, nonlocal advection. The second version is introduced here as a local counterpart. We establish well-posedness for both models, with the local case requiring a novel use of the ``shape defect function.'' We further characterize the front location up to $O(1)$ precision in all cases, including the delicate boundary ``pushmi-pullyu'' case.

We revise the analysis of the acoustic wave equation, addressing the question whether the classical well-posedness implies the existence of an isomorphism between prescribed solution and data spaces. This question is of interest for the design and the analysis of discretization methods. Expanding on existing results, we point out that established choices of solution and data space in terms of classical Bochner spaces must be expected to be incompatible with the existence of such an isomorphism, because of resonant waves. We formulate this observation in the language of the so-called inf-sup theory, with the help of an eigenfunction expansion, which reduces the original partial differential equation to a system of ordinary differential equations. We further verify that an isomorphism can be established, for each equation in the system, upon equipping the data space with a suitable resonance-aware norm. In the appendix, we extend our results to other time-dependent linear PDEs.

Preconditioning for overdetermined least-squares problems has received comparatively little attention, and designing methods that are both effective and memory-efficient remains challenging. We propose a class of ILU-based preconditioners built around a row-splitting strategy that identifies a well-conditioned square submatrix via an incomplete LU factorization and combines its incomplete factors with algebraic corrections from the remaining rows. This construction avoids forming the normal equations and is well suited to problems for which the normal matrix is ill-conditioned or relatively dense. Numerical experiments on test problems arising from practical applications illustrate the effectiveness of the proposed approach when used with a Krylov subspace solver and demonstrate it can outperform preconditioners based on incomplete Cholesky factorization of the normal equations, including for sparse-dense problems, where the splitting naturally isolates dense rows.

In this paper, a class of linear evolution equations with time delay is studied in which the presence of continuous spectrum on the imaginary axis obstructs the analysis of long-time dynamics. To address it, a generalized spectral framework on a Gelfand triple is utilized. When the spectral measure of the unperturbed term (a skew-adjoint operator) admits some analyticity condition, the resolvent is extended to a generalized resolvent. Called generalized spectrum, the collection of singularities on the Riemann surface of the generalized resolvent may differ from the spectrum in the usual sense because of the change of topology via the Gelfand triple. It is shown that under some compactness assumption, the generalized spectrum consists only of isolated generalized eigenvalues (resonance poles). This structure allows contour deformation in the inverse Laplace representation and yields exponential decay in a weak topology. As an application, we analyze the continuum limit of the Kuramoto-Daido model with time delay and prove linear stability of the incoherent state in the weak coupling regime.

This paper studies the numerical approximation of divergence-free vector fields by linearized shallow neural networks, also referred to as random feature models or finite neuron spaces. Combining the stable potential lifting for divergence-free fields with the scalar Sobolev integral representation theory via ReLU$^k$ networks, we derive a core integral representation of divergence-free Sobolev vector fields through antisymmetric potentials parameterized by linearized ReLU$^k$ neural networks. This representation, together with a quasi-uniform distribution argument for the inner parameters, yields optimal approximation rates for such linearized ReLU$^k$ neural networks under an exact divergence-free constraint. Numerical experiments in two and three spatial dimensions, including $L^2$ projection and steady Stokes problems, confirm the theoretical rates and illustrate the effectiveness of exactly divergence-free conditions in computation.

Adapting Bauer and Furuta's constructions of the refinement of the Seiberg-Witten invariants, we establish the analogous stable cohomotopy refinement of the $Pin^{-}(2)$ monopole invariants proposed by Nakamura \cite{nakamura2015pin}, and give the corresponding connected sum formula.

Stochastic resetting has attracted significant attention in recent years due to its wide-ranging applications across physics, biology, and search processes. In most existing studies, however, resetting events are governed by an external timer and remain decoupled from the system's intrinsic dynamics. In a recent Letter by Biswas et al, we introduced threshold resetting (TR) as an alternative, event-driven optimization strategy for target search problems. Under TR, the entire process is reset whenever any searcher reaches a prescribed threshold, thereby coupling the resetting mechanism directly to the internal dynamics. In this work, we study TR-enabled search by $N$ non-interacting diffusive searchers in a one-dimensional box $[0,L]$, with the target at the origin and the threshold at $L$. By optimally tuning the scaled threshold distance $u = x_0/L$, the mean first-passage time can be significantly reduced for $N \geq 2$. We identify a critical population size $N_c(u)$ below which TR outperforms reset-free dynamics. Furthermore, for fixed $u$, the mean first-passage time depends non-monotonically on $N$, attaining a minimum at $N_{\mathrm{opt}}(u)$. We also quantify the achievable speed-up and analyze the operational cost of TR, revealing a nontrivial optimization landscape. These findings highlight threshold resetting as an efficient and realistic optimization mechanism for complex stochastic search processes.

Given any irrational number $α$, we show that for any $0<θ<6/17$, there are infinitely many $y$-smooth (friable) numbers $n$ such that $$\|nα\| < n^{-θ},$$ where $(\log n)^C\leq y\leq n$ for some large constant $C>0$. This improves the previous work of Baker, who obtained the exponent $1/3-2/(3C)+o(1)$ in the case of $y\geq (\log n)^C$, and that of Yau, who obtained the exponent $1/3$ when $y=n^{o(1)}$. Our proof is based on the dispersion method together with arithmetic inputs coming from the average bounds for Kloosterman sums over smooth numbers.

An exponent of distribution 1/16 is established for square-free palindromes. The main input is an upper bound for the number of palindromes, in arithmetic progressions to large moduli, divisible by large squares. Our argument combines a simplifying reformulation with exponential-sum estimates, recent work on 6-almost-prime palindromes, and the large sieve with square moduli of Baier-Zhao.

An important distinction in our understanding of capacities of classical versus quantum channels is marked by the following question: is there an algorithm which can compute (or even efficiently compute) the capacity? While there is overwhelming evidence suggesting that quantum channel capacities may be uncomputable, a formal proof of any such statement is elusive. We initiate the study of the hardness of computing quantum channel capacities. We show that, for a general quantum channel, it is QMA-hard to compute its quantum capacity, and that the entanglement-assisted zero-error capacity under some restrictions is uncomputable; indicative of the fact that quantum channel capacities may generally be undecidable.

In this paper, we introduce the SPINNs (stochastic physics-informed neural networks) in a systematic manner. This provides a mathematical framework for approximating the solution of stochastic differential equations (SDEs) driven by Levy noise using artificial neural networks.

We prove that if two very general cubic fourfolds are L-equivalent then they are isomorphic, and we observe that there exist special cubic fourfolds which are L-equivalent but not isomorphic. When the cubic fourfolds are very general in certain Hassett divisors, we prove that if they are L-equivalent then they are also Fourier-Mukai partners. We also provide further examples in support of the fact that L-equivalent hyper-Kähler manifolds should be D-equivalent, as conjectured by Meinsma.

We present a broad family of high-order finite element algorithms for simulating the flow of electroneutral electrolytes. The governing partial differential equations that we solve are the electroneutral Navier--Stokes--Onsager--Stefan--Maxwell (NSOSM) equations, which model momentum transport, multicomponent diffusion and electrical effects within the electrolyte. Our algorithms can be applied in the steady and transient settings, in two and three spatial dimensions, and under a variety of boundary conditions. Moreover, we allow for the material parameters (e.g. viscosity, diffusivities, thermodynamic factors and density) to be dependent on the local state of the mixture and thermodynamically non-ideal. The flexibility of our approach requires us to address subtleties that arise in the governing equations due to the interplay between boundary conditions and the equation of state. We demonstrate the algorithms in various physical configurations, including (i) electrolyte flow around a microfluidic rotating disk electrode and (ii) the flow in a Hull cell of a cosolvent electrolyte mixture used in lithium-ion batteries.

We study the dynamics of a mean-field Ising model whose coupling depends on the magnetization via a linear feedback function. A key feature of this linear feedback Ising model (FIM) is the possibility of temperature-induced bistability, where a temperature increase can favor bistability between two phases. We show that the linear FIM provides a minimal model for a transcritical bifurcation as the temperature varies. Moreover, there can be two or three critical temperatures when the external magnetic field is non-negative. In the bistable region, we identify a Maxwell temperature where the two phases are equally probable, and we find that increasing the temperature favors the lower phase. We show that the probability distribution becomes non-Gaussian on certain time intervals when the magnetization converges algebraically at either zero temperature or critical temperatures. Near critical points in the parameter space, we derive a Fokker-Planck equation, construct the families of equilibrium distributions, and formulate scaling laws for transition rates between two stable equilibria. The linear FIM offers considerable flexibility in controlling steady-state bifurcations and their associated equilibrium distributions, which can be desirable for modeling feedback systems across various disciplines.

We consider the Anderson Hamiltonian (AH) and the parabolic Anderson model (PAM) with white noise and Dirichlet boundary condition on a bounded planar domain $D\subset\mathbb R^2$. We compute the small time asymptotics of the AH's exponential trace up to order $O(\log t)$, and of the PAM's mass up to order $O(t\log t)$. Our proof is probabilistic, and relies on the asymptotics of intersection local times of Brownian motions and bridges in $\mathbb R^2$. Applications of our main result include the following: (i) If the boundary $\partial D$ is sufficiently regular, then $D$'s area and $\partial D$'s length can both be recovered almost surely from a single observation of the AH's eigenvalues. This extends Mouzard's Weyl law in the special case of bounded domains (Ann. Inst. H. Poincaré Probab. Statist. 58(3): 1385-1425). (ii) If $D$ is simply connected and $\partial D$ is fractal, then $\partial D$'s Minkowski dimension (if it exists) can be recovered almost surely from the PAM's small time asymptotics. (iii) The variance of the white noise can be recovered almost surely from a single observation of the AH's eigenvalues.

The Local Structure Theorem (LST) for Graph Minors roughly states that for every $H$-minor-free graph $G$ that contains a sufficiently large wall $W$, there is a small vertex subset $A,$ whose removal yields a graph that admits an "almost embedding" $δ$ on a surface $Σ$ on which $H$ does not embed. By almost embedding, we mean that there exists a hypergraph $\mathcal{H}$ whose vertex set is a subset of the vertex set of $G - A$ and an embedding of $\mathcal{H}$ on $Σ$ such that the drawing of each hyperedge of $\mathcal{H}$ corresponds to a cell of $δ,$ the boundary of each cell intersects only the vertices of the corresponding hyperedge, and all remaining vertices and edges of $G - A$ are drawn in the interior of cells. The cells corresponding to hyperedges of arity at least $4$, called vortices, are few in number and have small "depth", while "most" of the wall $W$ is disjoint from the vortices and is "grounded" in the embedding $δ$. Suppose that the subgraphs drawn inside each of the non-vortex cells are equipped with some finite index, i.e., each such cell is assigned a color from a finite set. We prove a version of the LST in which the set $C$ of colors assigned to the non-vortex cells exhibits "large" bidimensionality: $G - A$ contains a minor model of a large grid $Γ$ such that, for every color $α\in C$, the model of each vertex of $Γ$ contains the subgraph drawn within an $α$-colored cell. Moreover, $Γ$ can be chosen in a way that is "well-connected" to the original wall $W$.

This work builds on the notion of record of rooted trees. We provide an alternative definition of parking functions, derive from it a record-preserving bijection between rooted trees and parking functions, and establish a join equidistribution result between a 5-tuple of statistics on rooted trees and a corresponding 5-tuple of statistics on parking functions. Some enumerative questions are also considered.