arXiv preprints from January 1, 2026 through September 6, 2026 — 21:42:17 EST
We connect modal set theory with Boolean-valued models by developing an \emph{internal} Kripke semantics for modal formulas whose atomic propositions are set-theoretic sentences. Given a complete Boolean algebra $B$, we view its elements as ``local perspectives on truth'' inside the Boolean-valued universe $V^{(B)}$ and interpret the...
We have previously given a factorization of a symplectic column under the action of the parity involution which enabled to explicitly have written the inverse of the reduction map in the quantum Littlewood-Richardson bijection. Watanabe has written the reduction map as a composition of several maps, among them, combinatorial...
Algebraic concordance of knots can be understood from the perspective of Seifert matrices, Blanchfield forms, and homology surgery. We initiate a systematic study of algebraic concordance for links from each of these viewpoints. The present article is concerned with algebraic concordance from the perspective of homology surgery and...
We prove a series of new results in Hamiltonian dynamics on a general closed symplectic manifold $(M, ω)$, including: 1. If $M$ admits a Hamiltonian diffeomorphism which is either a pseudo-rotation or has finite order, then $M$ is geometrically uniruled. This resolves a variant of Problem 24 in McDuff--Salamon's list, which predicts...
We prove Pólya's conjectured lower bound for the Neumann counting function of every Euclidean ball. If $d\ge2$, $R>0$, and $E\ge0$, then \begin{equation*} N_{B_R^d}^{<}(E) \ge \frac{ω_d}{(2π)^d}|B_R^d|E^{d/2} = \frac{(R\sqrt E)^d}{2^dΓ(\frac d2+1)^2}. \end{equation*} The radial boundary condition in dimensions $d\ge3$ is a Dini...
An antichain is a family of rectangles in which no member contains another. Given a family $\mathcal{E}$ of rectangles, let $h_{\mathcal{E}}$ be the sum of the indicator functions of its members. We show that there exist constants $c, C > 0$ such that for every sparse antichain $\mathcal{E}$ of dyadic rectangles in $\mathbb{R}^2$ one...
We establish necessary and sufficient conditions for the uniform boundedness of the multi-parameter singular exponential sum $$ \sum_{|t_1|\le N_1,\dots,|t_k|\le N_k} \frac{e^{2πi P(t_1,\dots,t_k)}}{t_1\cdots t_k}, $$ where $P:\mathbb{Z}^k\to\mathbb{R}$ is a polynomial of the form $ P(t)=\sum_{\mathfrak{m}\in Λ} c_{\mathfrak{m}}\,...
In this work, we develop a weak-form version of Extended Dynamic Mode Decomposition that we call WEDMD. We establish a number of analytic results about the method and show explicitly how the weak form is able to mitigate the impacts of noise in linear stochastic differential equations. In nonlinear systems, we likewise show how the...
As in Bouchard et al. (2010) and Bouchard and Nutz (2014), we study a utility maximization problem with expectation constraint. We first consider a uniformly elliptic case in which the endogenous state boundary associated with the constraint in expectation is proved to be smooth. This allows one to derive a proper Dirichlet condition...
By Godunov's theorem, linear second-order finite-difference schemes for the Fokker-Planck equation cannot preserve positivity. The Diagonal Frog (DF) framework previously bypassed this barrier using eventual positivity, but required a strict minimum time step. This paper resolves the small-step limitation using a nonlinear extension...
We study the long time dynamics of a selection-mutation integro-differential Lotka-Volterra system of $N$ populations. In our model, fitness depends on a continuous phenotypic trait, but the effect of one population on another is independent of this trait. We establish that, under some usual assumptions on the interactions between...
Let $V$ and $U$ be independent standard normal random variables. For any Borel map $φ\colon\mathbb{R}\to\mathbb{R}$, set $Y_φ=φ(V)+U$, and define $P_φ(y)=\mathbb{E}[V\mid Y_φ=y]$ and $F_φ(x)=\mathbb{E}[P_φ(x+U)]$. We prove that, if for every $v\in\mathbb{R}$, the quantity $φ(v)$ maximises $x(v-F_φ(x))$ over $x\in\mathbb{R}$, then $φ$...
We illustrate in several examples that even neural networks of infinite width (specifically, Barron functions) may encounter substantial obstacles when used as a model class for problems in the calculus of variations. An instance of practical relevance concerns the bending, stretching and folding of a thin elastic shell with anchored...
We propose a randomized iterative method for the ordinary least-squares estimation problem in large-scale linear statistical models, namely the Sequential Preconditioned Conjugate Gradient Method (SPCG). SPCG constructs a sequence of sketched least-squares subproblems with increasing sketch sizes, applies PCG as the inner solver, and...
We show that any complete shrinking gradient Kähler-Ricci soliton on a resolution of a Kähler cone is necessarily asymptotically conical. From a result of Esparza, it then follows that up to pullback by biholomorphism, there exists at most one complete shrinking gradient Kähler-Ricci soliton on such a resolution. This confirms a...
Graph expansion has long been recognised as an important and desirable property with applications in a wide range of areas in computer science and mathematics. A particular form of expansion known as `sublinear expansion' has recently been used particularly effectively in extremal graph theory, leading to the resolution of many...
We develop a residue-theoretic framework for studying inverse eigenvectors of a square matrix, defined by the nonlinear equation $Mα=α^{-1}$. Our main result is an inverse analogue of the spectral theorem: under natural transversality and properness assumptions, the identity operator admits an explicit decomposition into rank-one...
Let $d\ge 2$ be an integer and let $c_0(t),\dots, c_{d-2}(t)\in\bar{\mathbb{Q}}[t]$. We consider the family of normalized polynomials $f_λ(z):=z^d+\sum_{i=0}^{d-2} c_i(λ)\cdot z^i$ parameterized by $λ\in\bar{\mathbb{Q}}$; the generic element of our family of polynomials is $f_t(z):=z^d+\sum_{i=0}^{d-2}c_i(t)\cdot z^i\in...
For the algebra $\mathbb{R}[x^1,\ldots,x^d]$ of polynomials in $d\geqslant 1$ variables, regard the complete generalised Wronskian $W_d^k$ of differential order $k\geqslant 1$ over $\mathbb{R}^d$ as the $N=\tbinom{d+k}{d}$-ary Lie bracket. Take an $N$-tuple of polynomials, calculate their Wronskian, and keep re-using the newly-created...
A nonempty subfamily of a $k$-uniform hypergraph is an \emph{even cover} if every vertex lies in an even number of its hyperedges; for $k=2$ these are edge-disjoint unions of cycles, so the minimum size of an even cover is the natural hypergraph analogue of girth. We prove Feige's 2008 conjecture on the hypergraph Moore bound: there...
We previously constructed closed embeddings of Kac-Moody affine Grassmannian slices using fundamental monopole operators. These spaces are defined via the Braverman-Finkelberg-Nakajima construction of Coulomb branches for quiver gauge theories, and the embeddings do not quantize in general. However, there is variant of the BFN...
A derangetropy operator reweighs a probability density by a fixed profile of its own cumulative distribution function, acting through ranks alone. We prove that these operators are precisely the transformations of absolutely continuous laws equivariant under monotone changes of variable, and that they compose through interval maps,...
An antidirected path is an oriented path in which every vertex sees either just incoming or just outgoing edges. We prove that every oriented graph with minimum semidegree at least $k$ contains an antidirected path of length $2 k -1$. This confirms a conjecture of Stein.
We prove that area-minimizing submanifolds in mod $2$ homology are not generically smooth, except in the case of geodesics, minimal surfaces and minimal hypersurfaces. This settles a conjecture of White that asks the generic smoothness of area-minimizing submanifolds in mod $2$ homology. We furthermore establish a lower bound on the...
The cycle double cover conjecture of Szekeres and Seymour, the proof of which was recently announced by OpenAI, states that every bridgeless graph has a collection of cycles covering every edge exactly twice. We study the counting version of this statement for cubic graphs, where we count circuit double covers --- collections of...