arXiv preprints from January 1, 2026 through September 22, 2026 — 03:37:49 EST
Let a finite population of n labelled examples carry a class-weighted loss, with pi*n in a rare positive class weighted by N0/N1. We study estimation of total risk from a subsample K << n under designs allocating K0 and K1 draws to the two strata. We derive the exact finite-population variance of the weighted risk estimator under...
We prove a necessary and sufficient condition for an abstract polyhedron $Π$ to be realized as a right-angled, ideal polyhedron in hyperbolic space.
For $2 \times 2$ matrices $A_1,\dots,A_k$, write $[A_1,\dots,A_k]$ for the left-normed iterated commutator $[\dots[[A_1,A_2],A_3],\dots,A_k]$, and $I_k$ for the ideal, in the $3k$-variable reduced-coordinate polynomial ring $R_k$, cutting out its vanishing locus. We prove, for every $k \geq 2$ over any field of characteristic $\neq...
We determine, up to a factor of $2^{o(k)}$, the number of $k$-sets $A \subset \{1, \ldots, n\}$ such that $|A + A| \leq m$, where $k = Θ(\log n)$ and $m \leq k^{1 + α}$, for small $α> 0$, answering a question of Green and Morris.
We prove a weak-type $(1,1)$ estimate for the centered Hardy--Littlewood maximal function with respect to Euclidean balls with dimensional dependence $O(\sqrt{n} \log n)$. This improves the order of growth in the classical $O(n)$ estimate of Stein and Strömberg. The proof goes through a pointwise bound of the Hardy--Littlewood maximal...
Arithmetic terms are fixed finite compositions of additions, multiplications, subtractions, divisions with remainder and integer exponentiations. An arithmetic term in natural numbers $A$, $B$, $n$, obtained by refining the general method of the second author (arXiv:2608.22049) for elliptic curves in Weierstraß normal form, counts the...
McDuff and Segal proved homological stability for unordered configuration spaces of connected manifolds with non-empty boundary. Later, Church proved representation stability for ordered configuration spaces on compact manifolds. These stability patterns extend to surface braid groups and pure surface braid groups, respectively,...
Klaja and Ransford exhibited a complex unital Banach algebra in which the exponential spectra of \(ab\) and \(ba\) differ away from zero, and asked whether this can occur in an algebra of bounded operators. We answer their question affirmatively. For a Bourgain--Delbaen space \(X\) obtained from Motakis' construction with Calkin...
We study finite-sample valid tests of the one-sided mean hypothesis $H_0:μ\leq 1$ against $H_1:μ>1$ for nonnegative random variables. To do so, we develop a leave-one-out dual certificate framework, where certain pointwise inequalities imply p-value validity under the conditional mean null $\mathbb{E}[X_i\mid\mathbf{X}_{-i}]\leq 1$,...
A subtraction game is played on a heap of tokens. The players take turns removing s tokens for some s in a fixed set S of positive integers, and the player who cannot move loses. The sequence of Sprague-Grundy values of such a game is eventually periodic. We ask when it is purely periodic, meaning periodic from the very start, for the...
In this note we study a two-regime representation of a loss random variable under quadratic error. For a finite law we compute exactly the change of the optimal risk when one atom crosses the cut. This turns the problem into a convexity question in cumulative-mass coordinates. On an equally spaced support, log-concavity gives this...
Data-Driven Computational Mechanics (DDCM) replaces traditional phenomenological constitutive models by directly reformulating boundary-value problems in terms of local material state data obtained from experiments or fine-scale simulations. Standard DDCM formulations, only enforcing equilibrium and compatibility, do not inherently...
We construct a framework for higher order unbounded rough drivers defined by rough paths over vector fields. This allows us to consider a purely Eulerian perspective for rough transport equations where the driving rough path vector field is allowed to have temporal $\mathfrak{p}$-variation for any $\mathfrak{p} \in (1,\infty)$. We use...
The Koebe-Andreev-Thurston theorem assigns a 3-dimensional hyperbolic reflection group to each combinatorial polyhedron. A natural question is: which of them are arithmetic? In 2016, Kontorovich-Nakamura conjectured that all arithmetic reflection groups obtained in this way are commensurable to those obtained from the tetrahedron,...
We prove that if a real-valued function $f\in L^1$ on the complex unit circle has a gap of width at least $π$ in its essential range, then $\exp(\widetilde f)$ is not integrable, where $\widetilde f$ is the conjugate function. More generally, the exponential function can be replaced by any nonnegative convex function $Y$ satisfying $$...
We consider collisions of simple random walks on comb graphs $\mathrm{Comb}(\mathbb{Z},H)$, which are obtained by attaching vertical segments of the form $[0,H_x] \cap \mathbb{Z}$ to any point $x$ of the integer axis. For $\mathrm{Comb}(\mathbb{Z},H)$ with profile $H_x(x) = |x| \log^γ(|x| \vee 1)$, we show that two independent simple...
We study multiplicative inequalities among the coefficients of Lorentzian polynomials through the notion of \emph{bounded ratios}. Our main result completely characterizes the cone of bounded ratios for Lorentzian polynomials of degree $n$ in $k$ variables. This dual characterization is expressed in terms of equivalence classes of...
For a finite set $S$ of positive integers, put $K(S):=-\min_{x\in\mathbb T}\sum_{s\in S}\cos(2πsx)$. Bedert recently proved the uniform lower bound $K(S)\geq |S|^{1/5-o(1)}$. We remove the subpolynomial loss and prove that $K(S)\geq c|S|^{1/5}$ for an absolute constant $c>0$. The proof combines two estimates from Bedert's argument...
We present a Hessian-based approach to numerically continue periodic orbits in dynamical systems. A loop (periodic orbit candidate) is parametrized as a Fourier series; a loss function is defined based on the deviation of the loop from the physical differential equations. Unlike previous work relying on hand-derived Jacobians, our...
Consider $n$ independent random vectors sampled from uniform distribution on $(p-1)$-dimensional unit sphere. This paper investigates the limiting joint distribution for the minimum and the maximum values of their pairwise angles. It proves that the minimum and the maximum angles are asymptotically independent when both $n$ and $p$...
Glyphosate is a widely used herbicide whose persistence and toxicity in aquatic and terrestrial environments demand efficient removal strategies. Here we employ GFN2 xTB calculations with implicit ALPB aqueous solvation and automated docking to investigate the adsorption of all five pH dependent ionized forms of glyphosate (G1-G5) on...
Non-reciprocal interactions are ubiquitous in active, biological, and disordered systems, generically driving them out of equilibrium. Here, we introduce a hierarchy of non-reciprocally interacting Ornstein-Uhlenbeck (NROU) models governed by a tunable non-reciprocity parameter $g$. At a special point $g=g^*$, the drift matrix becomes...
Advanced optical quantum technologies demands high quality quantum light generation at very high rates. Here, we report on a deterministic single-photon source that simultaneously combines high excitation rates with high system efficiency to reach over 500 MHz of in-fibre single-photon flux. The source delivers optical power of over...
The advancement of spintronic nanoscale devices hinges on the ability to flexibly engineer magnetic spin structures in thin-film stacks with precision and control. Meeting this demand remains a challenge for stable non-collinear spin configurations and, more specifically, vertical spin spirals in thin films. Innovative methods are...
Three-photon states generated via cascaded spontaneous parametric downconversion provide a direct route to multipartite entanglement. However, current implementations require careful spectral matching between successive nonlinear stages, which constrains the choice of downconversion sources. In this work, we show that...