arXiv preprints from January 1, 2026 through September 21, 2026 — 22:05:48 EST
Weil's criterion equates the Riemann Hypothesis with the positivity of an explicit quadratic form $Q(f)$. For test functions supported in a window $[-L,L]$ we study the profile $λ^*(L)=\inf Q(f)/\|f\|_2^2$ from both sides. A one-stroke reduction converts window positivity into positive semidefiniteness of a finite matrix; executing it...
We show that the operadic structure on the cohomology of the poset of noncrossing 2-partitions is isomorphic to the Brace operad.
Jungin Lee in 2018 proved a necessary and sufficient condition that an integral quadratic form $\sum_{i=1}^{m} a_iX_i^2$ is universal over $M_2(\mathbb{Z})$. For a positive integer $n \geq 2$, Lee defined $f(n)$ to be the smallest positive integer $m$ such that for every pairwise coprime $a_1, a_2, \ldots a_m \in \mathbb{Z}$,...
We show Lagrange and Hermite interpolation are possible using Stieltjes polynomials, linear combinations of iterated Stieltjes integrals of a constant function. We introduce divided differences via Newton interpolation and provide an explicit error formula of Peano kernel type via a new Taylor formula. Finally, we use the properties...
For $N$ lattices in $\R^d$ with volume $1$ and pairwise trivial intersections, every nonzero common tiling function has support diameter $Ω(N^{1/d})$, while for lattice families whose fundamental domains have uniformly bounded diameters, the standard convolution construction gives an $O(N)$ upper bound, leaving a gap that has remained...
The oscillation of M(x), the partial sum of the Möbius function has been in the focus of researchers in the theory of primes since the famous conjecture of Mertens in 1905 (formulated in a weaker form by Stieltjes in 1885 in a letter to Hermite). The average order of the modulus of M(x) in an interval of type [0,Y] is clearly in...
Let $\mathcal{A}$ be a finite central real hyperplane arrangement and let $\mathcal{G}(\mathcal{A})$ be its tope graph. Koizumi recently proved that the crossing-graded magnitude homology of $\mathcal{G}(\mathcal{A})$ is torsion-free and is freely indexed by face flags. We refine his result by proving that a fixed crossing vector and...
We study the Ising model on $\mathbb{Z}^d$ with $d\geq 4$ and derive near-critical bounds on the truncated two-point function $\langleσ_0;σ_x\rangle_{β,h} := \langleσ_0σ_x\rangle_{β,h} - \langleσ_0\rangle_{β,h}\langleσ_x\rangle_{β,h}$ at parameters $β\leqβ_c$ and $h\geq 0$. As a corollary, we obtain that the associated mass (or...
For $N\ge2$ and $β>0$, consider the canonical 2D Coulomb gas ensemble \begin{equation} \mathrm{d}\mathbb{P}_{N,β}(Z) =\mathsf{Z}_{N,β}^{-1}e^{-β|Z|^2/2} \prod_{i<j}|z_i-z_j|^{β/N}\,\mathrm{d}Z \qquad\text{on }\mathbb{C}^N. \end{equation} The factor $N^{-1}$ in the pair exponent places the ensemble at the diffusive, or...
Let $G$ be a graph of order $n$ with eigenvalues $λ_1(G) \geq \dots \geq λ_n(G)$, and let $s_+(G)=\sum_{λ_i(G)>0}λ_i(G)^2.$ Recently Liu, Tang, and Zhang proved the positive square-energy strengthening of Turán's theorem \[\sqrt{s_+(G)}\leq \left(1-\frac1r\right)n.\] where $r=ω(G)$ is the clique number of $G$. We characterize the...
Direct policy optimization is widely used in reinforcement learning and control, but generally leads to nonconvex optimization problems. For state-feedback $H_\infty$ control, the policy objective is also nonsmooth, despite possessing a benign landscape whose hidden convexity can be revealed by the recently developed extended convex...
We establish a strong averaging principle for fast-slow stochastic differential equations with a time-dependent scale-separation parameter $(\varepsilon_t)_{t \geq 0}$ satisfying $\varepsilon_t \to 0$ as $t \to \infty$. In contrast to approaches based on noise-induced smoothing or elliptic regularity, our approach relies on...
Triangle-based measures, commonly referred to as badness functions, are widely employed to quantify the extent to which a distance matrix deviates from an ideal geometric configuration. Different formulations of these functions may capture distinct facets of local non-uniformity, and their behavior is often influenced by the...
We consider a Turán-type problem arising in phylogenetics: determining the maximum size of a 2-weakly compatible split system. This compatibility condition arises in the reconstruction of phylogenetic networks from quartet weights. It was previously shown that a 2-weakly compatible split system has size at most \[ ...
We develop a stochastic framework for a broad class of heterogeneous SIR epidemic models. In the finite-population construction, each initially susceptible individual is assigned a fixed nonnegative susceptibility, and infection occurs when the accumulated population-level infection pressure exceeds an individual random threshold....
We establish the existence of a stochastic flow on $L^{\infty} (\mathbb{T})$ for the stochastic Allen-Cahn equation with multiplicative noise \[ (\partial_t - \partial_x^2) u = u - u^3 + σ(u) ξ\quad \text{on} \quad \mathbb{R}_+ \times \mathbb{T}, \] where $ξ$ is space-time white noise and $σ: \mathbb{R} \rightarrow \mathbb{R}$ is...
We prove Dennin's conjecture (Conjecture 8.9 of arXiv:2506.21052) that his variant of dual RSK correspondence is symmetric when restricted to biGrassmannian permutations. For a binary matrix $A$, let $A^\dagger$ denote its transpose-complement, and let $\operatorname{ins}(A)$ and $\operatorname{rec}(A)$ denote its insertion and...
For every integer $d\geq 1$, we construct a countably infinite set $X_d\subset\mathbb{R}^d$ in affine general position, with all pairwise distances rational. When $d$ is odd, $X_d$ may also be chosen so that no $d+2$ points lie on a common sphere. The construction is uniform in $d$: positive Chebyshev square decompositions produce...
We investigate which 3-manifolds bound 4-manifolds that are homeomorphic but not stably diffeomorphic, where stabilising means taking connected sum with copies of $S^2\times S^2$. We show that every closed, orientable 3-manifold admits such fillings, as do certain families of nonorientable 3-manifolds. In contrast we show that for a...
We establish a general structure theorem for local quantum Cuntz--Krieger families associated to finite quantum graphs whose relations are analogous to those of the partial isometries generating an ordinary graph $C^*$-algebra, with the quantum edge correspondence governing composition. We apply this structure theorem to the quantum...
We prove the wall-crossing formula conjectured by Gross--Joyce--Tanaka for equivariant enumerative invariants of CY4 categories equipped with framing functors. We also establish a version for stable pairs with fixed-determinant obstruction theories, as used in earlier applications by the first-named author. The main technical...
The probabilistic formulation of principal component analysis promises statistically grounded solutions to the problem of selecting the number of principal components. However, developing tractable model selection methods is complicated by the fact that the probabilistic principal component analysis (PPCA) model exhibits non-standard...
We establish a Demailly-Kollár type continuity theorem for plurisubharmonic functions with respect to adapted measures on normal complex analytic klt pairs. As applications, we prove the equality of the analytic and divisorial versions of the alpha and delta invariants on compact normal Kähler klt pairs, thereby completing a program...
Let $λ(n)=(-1)^{Ω(n)}$ be the Liouville function. Pilatte proved a fixed power saving for the logarithmically weighted two-point correlation at shift one. More recently, Tao and Teräväinen obtained power-logarithmic two-point estimates uniform over polylogarithmically growing shifts and coefficients outside a common exceptional set of...
An element of a finite group is non-vanishing if no irreducible complex character vanishes on it. We prove that every non-vanishing element of a finite solvable group belongs to the Fitting subgroup. The proof combines the reduction of Isaacs, Navarro, and Wolf with an argument using finite symplectic spaces and Clifford theory.