arXiv preprints from January 1, 2026 through September 8, 2026 — 22:18:29 EST
We study in which settings we have a composition AKE principle, i.e. when the theory of the coarsening $(K,w)$ and the theory of the induced valuation $(Kw,\overline{v})$ determine the theory of the composition $(K,v)$. We show that this is the case when $(K,w)$ is tame of equal characteristic, and provide counterexamples in mixed...
The phase field crystal (PFC) method is an efficient technique for simulating the evolution of crystalline microstructures at atomistic length scales and diffusive time scales. Due to the high-order derivatives (sixth-order) and the strongly nonlinear term (locally Lipschitz), developing high-order stable schemes and establishing...
We present randomized algorithms for estimating the log-determinant of regularized symmetric positive semi-definite matrices. The algorithms access the matrix only through matrix vector products, and are based on the introduction of a preconditioner and stochastic trace estimator. We claim that preconditioning as much as we can and...
The algebraic connectivity $a(G)$, defined as the second smallest eigenvalue of the Laplacian matrix $L(G)$, admits a well-known variational characterization involving the minimization of a quadratic form subject to an $\ell_{2}$-norm constraint. In a recent work, Andrade and Dahl (2024) proposed an analogous formulation based on the...
Edge expansion is a parameter indicating how well-connected a graph is. It is useful for designing robust networks, analysing random walks or information flow through a network and is an important notion in theoretical computer science. Modularity is a measure of how well a graph can be partitioned into communities and is widely used...
We study the large-scale behaviour of a class of driven diffusive systems modelled by a Stochastic Partial Differential Equation, the Stochastic Burgers Equation (SBE) with general nonlinearity, at the critical dimension and in infinite volume. Our main result shows that, under a logarithmically superdiffusive space-time scaling, it...
Recent progress in LLM-driven algorithm discovery, exemplified by DeepMind's AlphaEvolve, has produced new best-known solutions for a range of hard geometric and combinatorial problems. This raises a natural question: to what extent can modern off-the-shelf global optimization solvers match such results when the problems are...
We develop a new method for the computation of $(3,3)$-isogenies between principally polarized abelian surfaces. The idea is to work with models in $\mathbb{P}^8$ induced by a symmetric level-$3$ theta structure. In this setting, the action of three-torsion points is linear, and the isogeny formulas can be described in a simple way as...
Let $S$ be a smooth irreducible curve over $\overline{\mathbb{Q}}$, and let $\mathcal{A} \to S$ be an abelian scheme with a curve $C \subset \mathcal{A}$, both defined over $\overline{\mathbb{Q}}$. In 2020, Barroero and Capuano proved that if $C$ is not contained in a proper subgroup scheme, then the intersection of $C$ with the union...
We study the thermodynamic formalism associated with the Schneider map on the p-adic integers $p\mathbb{Z}_p$ . By introducing a geometric potential that captures the expansion of cylinder sets generated by the map, we define a Lyapunov exponent adapted to this non-Archimedean setting. We investigate the corresponding Lyapunov...
We prove an invariance principle for a general class of continuous time critical branching processes with finite variance (non-local) branching mechanism. We show that the genealogical trees, viewed as random compact metric measure spaces, converge under rescaling to the Brownian continuum random tree in the Gromov-Hausdorff-weak...
We provide a comparison test for meromorphic extensions, i.e., if two series are ``close enough" then the existence of a meromorphic extension of one to the entire complex plane ensures a similar extension for the other. We use this result to generate new examples of Dirichlet series admitting meromorphic extensions. Moreover, we...
We study diffusion approximations for a class of interacting stochastic systems with reflection and control. Motivated by interacting stochastic dynamics subject to feedback mechanisms and boundary constraints, we consider diffusion-scaled stochastic processes incorporating stochastic fluctuations, state-dependent interactions, and...
Twin-width is a graph parameter introduced in the context of first-order model checking, and has since become a central parameter in algorithmic graph theory. While many algorithmic problems become easier on arbitrary classes of bounded twin-width, graph isomorphism on graphs of twin-width 4 and above is as hard as the general...
We compute the low degree $\ell$-adic intersection cohomology of symplectic local systems on the Satake compactification of the moduli space $A_g$ of principally polarized abelian varieties. We prove that only a small finite list of irreducible Galois representations can appear in the low degree cohomology of any nonsingular toroidal...
We provide two constructions for $t$ edge-disjoint maximal outerplanar graphs on every number of $n \geq 4t$ vertices. The bound on the minimum number of vertices is tight. These constructions yield the existence of optimal outerthickness-$t$ graphs for every $t \in \mathbb{N}$. While one of the constructions works for all values of...
We study existence of solutions in the variational sense for a class of stochastic phase-field models describing moving boundary problems. The models consist of stochastic reaction-diffusion equations with singular diffusion forced by a phase-field. We investigate both the case of an independently evolving phase-field and of coupled...
For a $C^1_{t,x}$ solution $u$ to the incompressible 3D Euler equations, the helicity $H(u(t))=\int_{\mathbb{T}^3} u \cdot \textrm{curl}\, u$ is constant in time. For general low-regularity weak solutions, it is not always clear how to define the helicity, or whether it must be constant in time in the case that there is a clear...
Sequential Bayesian optimal experimental design (SBOED) for PDE-governed inverse problems is computationally challenging, especially for infinite-dimensional random field parameters. High-fidelity approaches require repeated forward and adjoint PDE solves inside nested Bayesian inversion and design loops. We formulate SBOED as a...
We prove a version of the Poincaré-Bendixson theorem for certain classes of curves on the 2-sphere which are not required to be the trajectories of an underlying flow or semiflow on the sphere itself. Using this result we extend the Poincaré-Bendixson theorem to the context of continuous semiflows on compact subsets of the 2-sphere...
We consider an optimal control problem governed by a rate-inde\-pendent system with non-convex energy. The state equation is approximated by means of viscous regularization w.r.t.\ to hierarchy of two different Hilbert spaces. The regularized problem corresponds to an optimal control problem subject to a non-smooth ODE in Hilbert...
Given any countable group $G$, we construct uncountably many quasi-isometry classes of proper geodesic metric spaces with quasi-isometry group isomorphic to $G$. Moreover, if the group $G$ is a hyperbolic group, the spaces we construct are hyperbolic metric spaces. We make use of a rigidity phenomenon for quasi-isometries exhibited...
Given a set $A=\{a_1,\ldots,a_n\}$ of real numbers and real coefficients $b_1,\ldots,b_n$, consider the distribution of the sum obtained by pairing the $a_i$'s with the $b_i$'s according to a uniformly random permutation. A recent theorem of Pawlowski shows that as soon as the coefficients are not all equal, this distribution is...
We describe the (bigraded) Hochschild cohomology of graded gentle algebras along with the Gerstenhaber bracket and cup product. In particular, this yields a description of the Hochschild cohomology of partially wrapped Fukaya categories of surfaces in the sense of Haiden-Katzarkov-Kontsevich which have at least one stop. Our results...
As network data has become ubiquitous in the sciences, there has been growing interest in network models whose structure is driven by latent node-level variables in a (typically low-dimensional) latent geometric space. These "latent positions" are often estimated via embeddings, whereby the nodes of a network are mapped to points in...