arXiv preprints from January 1, 2026 through September 5, 2026 — 17:56:10 EST
Structure-preserving discontinuous Galerkin (DG) methods typically improve the robustness of high order simulations of real fluids. In addition to conservation, key structures include the preservation of pressure equilibrium and satisfaction of at least one entropy inequality. In this work, we investigate conservative discretizations...
By work of Enomoto, Jackson, Katerinis, and Saito from 1985, every $2$-tough graph has a $2$-factor, and this toughness bound is best possible: for every $\varepsilon>0$, there exist $(2-\varepsilon)$-tough graphs with no $2$-factor. It is natural to ask whether the latter statement remains true for triangle-free graphs. Bauer, van...
Hybrid dealiasing is an FFT-based method for computing linear convolutions of complex-valued data that reduces the cost of dealiasing by performing zero padding implicitly. We develop two new algorithms that extend hybrid dealiasing to real-valued convolutions. The first algorithm exploits conjugate symmetries in the transformed...
The Shimura curve of discriminant $D$ for $D=14, 15$ is uniformized by a subgroup of an arithmetic quadrilateral Fuchsian group $(2, 2, 2, q)$, where $q=4, 6$. We relate the generator of the ring of quaternionic modular forms on this Shimura curve to explicit Heun functions for the quadrilateral group. We also discuss how the...
For a real $f\in \mathbb{N}^\mathbb{N}$, it is in general not the case that every set which is both $Σ^1_1(f)$ and $Π^1_1(f)$ is the $f$-section of a $Δ^1_1$ set. However, there are reals $f$ for which this, in fact, does happen; we say that such a real strongly relativizes $Δ^1_1$. In this paper, we will prove that the reals which...
Let $M$ be a $\mathrm{II}_1$ factor, $N$ a tracial von Neumann algebra, and $Φ: M \rightarrow N$ a subtracial completely positive map. For an irreducible $\mathrm{II}_1$ subfactor $P \subseteq M$, we characterize when $Φ$ exhibits a flattening property under conjugation by unitaries in $P$. To be specific, we show that the failure of...
Let $G$ be an isotropic reductive group and $X$ be a projective $G$-homogeneous variety. Using results from Bialynicki-Birula, Hesselink and Iversen, Brosnan showed that if $G$ is of inner type, the motive of $X$ can be expressed as a direct sum of Tate twists of motives of projective homogeneous varieties for the anisotropic kernel...
We develop an arbitrary-degree geometry-conforming finite element (GC-FE) framework for two-dimensional elliptic boundary value and interface problems on curved domains. Using the Frenet--Serret transformation, curved-boundary and interface-fitted segments are represented exactly, while polynomials in Frenet coordinates generate...
We show that an inductive AH-system with diagonal connecting maps describes an action of the canonical AF-groupoid on the spectrum of the canonical C$^*$-diagonal, and that the canonical groupoid model is given by the transformation groupoid associated to this action. This divides the groupoid structure into two distinct aspects: the...
We study minimax-optimal designs and estimators for estimating the sample average treatment effect in finite population randomized experiments, where both design and estimator are unrestricted. For binary potential outcomes, we show this minimax risk is equivalent to the minimax risk $ρ_n^*$ of an estimation problem with $2$ unknown...
We introduce the notion of a "compressive domain" for the action of a rational function on the Berkovich projective line over a complete nontrivially-valued algebraically closed nonarchimedean field. We prove that such a domain always contains a classical fixed point, and we leverage this fact to give a sharp upper bound for the...
We show that a mass-Hom bound for a numerical pre-stability condition on a projective scheme over a field implies the support property with respect to the full numerical Grothendieck group. Combined with the recent construction of stability conditions on projective schemes, this yields stability conditions with full support property...
A mixed identity of a group is a nontrivial word with constants that vanishes under every substitution of its variables. We derive lower bounds for the length of mixed identities in finite simple groups of Lie type, and characterise exactly those families of such groups of bounded rank which satisfy mixed identities of bounded length....
We study a general class of gradient interface models with Hamiltonian $H=β\sum V(\nablaφ)$, $β>0$, assuming essentially that the potential $V$ is even, $V'(s)\ge αs$ on $[0,\infty)$ for some $α>0$, and $-M\leq V''\le C$. We establish a Helffer-Sjöstrand representation for these models, and use it to prove that their scaling limits...
In this paper, a quantitative condition for optimality for distributed optimal control problems with box-constraints that are subject to a semilinear elliptic equation is considered. An important property of the investigated optimal control problems is the absence of a Tikhonov regularization. It is well known that at a given control,...
We present the mathematical foundations of linear independent component analysis (ICA) models based on standard literature in a self-contained note. It is aimed at readers with a background in measure-theoretic probability theory. We first develop the theory of the characteristic functions of probability measures on $\mathbb{R}^d$,...
Biochemical reaction networks have become a central theoretical framework for implementing molecular computation. A key challenge is finite time computational accuracy, as computation outputs are encoded in limiting steady states (LSSs) of species concentrations while practical implementations operate for only finite time. This work...
Active-quiescent switching occurs in biological populations in which growth is confined to a proliferative active state, while cells may reversibly enter a nonproliferative quiescent state. Experiments often observe only part of this process, through active-state markers, aggregate population measurements, or aggregate data...
Let $f_{\ell}(n)$ be the largest size of a family $\mathcal{A}\subseteq2^{[n]}$ such that no member has size divisible by $\ell$, while the intersection of every two distinct members has size divisible by $\ell$, and let $ω(\ell)$ denote the number of distinct prime divisors of $\ell$. For any prime power $\ell$, the classical answer...
Problem 3.6 in the $\mathrm{K3}$ problem list of Baykur, Kirby and Ruberman asks whether every cofinal tower \[ M_0\longleftarrow M_1\longleftarrow M_2\longleftarrow\cdots \] of finite covers of a finite-volume hyperbolic $3$-manifold satisfies \[ \lim_{n\to\infty} \frac{\log|\operatorname{Tor} H_1(M_n;\mathbb Z)|}...
In this article, we study two consequences of abstract six-functor formalisms. Firstly, we show that an abstract six-functor formalism can be extended to specific Ind- and Pro- categories of geometric setups. As an application, we can define the motivic stable homotopy theory for ind-pro-algebraic stacks such as the Hecke stack....
We formulate several open questions regarding the tropicalization of linear ODEs, aiming primarily to develop methods for calculating the radius of convergence of their classical solutions. To this aim it is of foremost importance to characterize the classes of equations that admit a finite differential tropical basis, as introduced...
We construct quintic surfaces in the three-dimensional projective space $\mathbb P^3$ with $18$ ordinary cusps. Our starting point is the Barth--Rams description of quintics containing a $3$-divisible set of $12$ cusps. A specialization in which the two contact cubics are singular along two skew lines produces a family with $16$...
We resolve the ellipsoid fitting conjecture of Saunderson, Chandrasekaran, Parrilo, and Willsky up to a vanishing factor. Concretely, for $m$ independent Gaussian points in dimension $d$, we show that with high probability, for $m \leq (1-o_d(1)) \cdot d^2/4$, there exists a centered ellipsoid passing through all $m$ points; for...
A stationary Poisson hyperplane process in $\mathbb{R}^d$ is characterized by an intensity parameter and an even probability measure on the unit sphere called the directional distribution. In this work, we investigate the stability of the zero cell, i.e., the random convex polytope of the induced hyperplane tessellation containing the...