arXiv preprints from January 1, 2026 through September 7, 2026 — 03:14:57 EST
In this paper, I utilize a variant of the Selberg--Delange method to find quantitative estimates of the sums \[M_{k,ω}(x,y)=\sum_{\substack{p_{1}(n)> y\\ n\leq x} } μ(n) {ω(n)-1\choose k-1},\] where $y$ can grow with $x$ but we must have $y\leq Y_0\exp(\mathscr{p}\frac{\log x}{(\log\log (x+1))^{1+ε}})$ with $Y_0,\mathscr{p},ε>0$....
Differential Privacy (DP) provides a rigorous framework for releasing statistics while protecting individual information present in a dataset. Although substantial progress has been made on differentially private linear regression, existing methods almost exclusively address the item-level DP setting, where each user contributes a...
We determine the complete structure of the symmetry algebras associated with the N-body Calogero-Moser system and its maximally superintegrable discretization. We prove that the discretization naturally leads to a nontrivial deformation of the continuous symmetry algebra, with the discretization parameter playing the rôle of a...
We consider the problem of locating the source (starting vertex) of a simple random walk, given a snapshot of the set of edges (or vertices) visited in the first $n$ steps. Considering lattices $\mathbb{Z}^d$, in dimensions $d \geq 5$, we show that the source can be identified (a) with probability bounded away from $0$ using one...
We introduce a universal Bochner formula for scalar curvature that contains, as special cases, the stability inequality for minimal slicings, a Schrödinger-Lichnerowicz-type formula, and a higher-dimensional version of Stern's level-set identity.
We prove Gromov's conjecture on the total mean curvature of fill-ins in various cases. Our methods are based on surgery to reduce the statement to fill-ins of spheres, which can be treated by instances of the positive mass theorem. For spin fill-ins, where we permit the mean curvature to take negative values, we build on a classical...
The manuscript is concerned with the Rudin-Keisler order of ultrafilters on measurable cardinals. The main theorem proved read as follows: Given regular cardinals $λ\leq κ$, the following theories are equiconsistent modulo ZFC: (1) $κ$ is a measurable cardinal with $o(κ)=λ^+$ (resp. $o(κ)=κ$). (2) The Rudin-Keisler order restricted to...
In this paper, we study three classes of subvarieties inside the variety of left-symmetric algebras. We show that these subvarieties are naturally related to some well-known varieties, such as alternative, assosymmetric and Zinbiel algebras. For certain subvarieties of the varieties of alternative and assosymmetric algebras, we...
This work builds upon the recent monograph [5] on self-similar Markov trees. A self-similar Markov tree is a random real tree equipped with a function from the tree to $[0,\infty)$ that we call the decoration. Here, we construct local time measures $L(x,dt)$ at every level $x>0$ of the decoration for a large class of self-similar...
We establish a Schur--Horn type inequality for symmetric hyperbolic polynomials. As an immediate consequence, we resolve a conjecture of Nam Q. Le on Hadamard-type inequalities for hyperbolic polynomials. Our argument is based on the Schur--Horn theorem, the Birkhoff theorem, and Gårding's concavity theorem for hyperbolicity cones....
We establish a comparison principle for viscosity solutions of a class of nonlinear partial differential equations posed on the space of nonnegative finite measures, thereby extending recent results for PDEs defined on the Wasserstein space of probability measures. As an application, we study a controlled branching McKean-Vlasov...
The well-known Zalcman conjecture, which implies the Bieberbach conjecture, states that the coefficients of univalent functions $f(z) = z + \sum\limits_2^{\infty} a_n z^n$ on the unit disk satisfy $|a_n^2 - a_{2n-1}| \le (n-1)^2$ for all $n > 2$, with equality only for the Koebe function and its rotations. The conjecture was proved by...
Let $R$ be a finite local ring. We prove a quantitative universality statement for the cokernel of random matrices with i.i.d. entries valued in $R$. Rather than use the moment method, we use the Lindeberg replacement technique. This approach also yields a universality result for several invariants that are finer than the cokernel,...
We deform block triangular Jacobi matrices appearing in geometry, look at multi-scale Barycentric limits of geometries and droplet boundary manifolds in Potts networks.
We provide a one-parameter family of Lorentz-Riemann signature-change models of metric manifolds. This family generalizes the Kossowski's signature type-changihg stablished in [9]. Simple local expressions are sought around the hypersurface of change.
We disprove the natural domain extension of the Berger--Coburn heat-flow conjecture for Toeplitz operators on the Bargmann space and identify the failure mechanism as a gap between pointwise and uniform control of a Gaussian averaging of the squared modulus of the symbol, a gap that is invisible to the linear form $T_g$. We establish...
This paper describes the vector bundle on the elliptic modular curve that is associated to a vertex operator algebra $V$ (VOA) or more generally a quasi-vertex operator algebra (QVOA), with a view towards future applications aimed at studying the characters of VOAs. We explain how the modes of sections of $V$ give rise naturally to...
Let $\mathfrak{o}$ be the ring of integers of a non-archimedean local field with residue field of odd characteristic, $\mathfrak{p}$ be its maximal ideal and let $\mathfrak{o}_\ell = \mathfrak{o}/\mathfrak{p}^\ell$ for $\ell\ge 2$. In this article, we study real-valued characters and real representations of the finite groups...
Spectral interference, the frequency counterpart of the beating phenomenon in the time domain, can severely distort time-frequency representations (TFRs) in physical applications. We study this phenomenon for the short-time Fourier transform (STFT) with a Gaussian window and for nonlinear refinements based on the reassignment method,...
We introduce the notion of a Nachbin proximity on a bounded archimedean $\ell$-algebra (bal-algebra), and show that Gelfand duality lifts to yield a dual equivalence between the category of uniformly complete bal-algebras equipped with a closed Nachbin proximity and that of Nachbin spaces (compact ordered spaces). The key ingredients...
Many real-world dynamics exhibit chaos, a phenomenon in which neighboring trajectories in the state space of a dynamical system diverge exponentially over time. A common measure used for quantifying the degree of this divergence is the maximal Lyapunov exponent, which relies on pairwise Euclidean distances between the trajectories at...
In this paper, we give a uniform upper bound on the rational points of bounded height provided by conics in a cubic surface. For this target, we give a generalized version of the global determinant method of Salberger by Arakelov geometry.
Recently, in the context of walks of hexagonal circle packings, interest has emerged in the family of skew Dyck paths with two variants of down-steps. These paths have steps $U, D_g, D_b, L=D_r$. Using generating functions, the kernel method and (in)finite linear systems, contributions to the (average) height and other enumerations...
The recently established spectral Favard theorem for bounded banded matrices admitting a positive bidiagonal factorization is applied to a broader class of Markov chains with bounded banded transition matrices, extending beyond the classical birth-and-death setting, to those that allow a positive stochastic bidiagonal factorization....
The cross-ratio degree problem asks for the number of configurations of $n$ points in $\mathbb{P}^1$ that satisfy $n-3$ specified cross-ratio conditions. It is known that the maximal cross-ratio degree for 8 points is at least 4. In this paper, we will see that the maximal cross-ratio degree for 8 points in $\mathbb{P}^1$ is equal to 4.