Related papers: Modular hyperbolas and Beatty sequences

The aim of this manuscript is to derive bounds on the moduli of eigenvalues of special type of rational matrices of the form $T(\lambda) = \displaystyle -B_0 +I\lambda +\frac{B_1}{\lambda-\alpha_1}+ \dots+ \frac{B_m}{\lambda-\alpha_m}$,…

We establish a quasi-monotonicity formula {for an intrinsic frequency function related to solutions to} thin obstacle problems with zero obstacle driven by quadratic energies with Sobolev $W^{1,p}$ coefficients, with $p$ bigger than the…

It is proved that certain discrete analogues of maximally modulated singular integrals of Stein-Wainger type are bounded on $\ell^p(\mathbb{Z}^n)$ for all $p\in (1,\infty)$. This extends earlier work of the authors concerning the case…

We establish the existence and uniqueness of the solution to the Dirichlet problem for the variable exponent $p$-Laplacian on a bounded, smooth domain $\Omega \subset {\mathbb R}^n$, where the boundary datum belongs to $W^{1,p}(\Omega)$.…

We establish finite-step probabilistic upper bounds on the contraction ratios $\rho_k = \Delta_{k+1}/\Delta_k$ for iterated Pearson correlation dynamics. Let $(P_k)_{k\ge 0}$ be the sequence generated by the Pearson update. Define $\Delta_k…

We investigate the behavior of the solutions of a class of certain strictly hyperbolic equations defined on $(0,T]\times \mathbb{R}^n$ in relation to a class of metrics on the phase space. In particular, we study the global regularity and…

We give bounds for the number and the size of the primes $p$ such that a reduction modulo $p$ of a system of multivariate polynomials over the integers with a finite number $T$ of complex zeros, does not have exactly $T$ zeros over the…

A well-known result of Shalom says that lattices in SO$(n,1)$ are $\mathrm{L}^p$ measure equivalent for all $p<n-1$. His proof actually yields the following stronger statement: the natural coupling resulting from a suitable choice of…

The eigenvalues of the Laplace-Beltrami operator and the integrals of products of eigenfunctions must satisfy certain consistency conditions on compact Riemannian manifolds. These consistency conditions are derived by using spectral…

Two results concerning the number of threshold functions $P(2, n)$ and the probability ${\mathbb P}_n$ that a random $n\times n$ Bernoulli matrix is singular are established. We introduce a supermodular function $\eta^{\bigstar}_n : 2^{{\bf…

Consider a nonuniformly hyperbolic map $ T $ modelled by a Young tower with tails of the form $ O(n^{-\beta}) $, $ \beta>2 $. We prove optimal moment bounds for Birkhoff sums $ \sum_{i=0}^{n-1}v\circ T^i $ and iterated sums $ \sum_{0\le…

Fix a poset $P$ and a natural number $n$. For various commutative local rings $\Lambda$, each of Loewy length $n$, consider the category $\textrm{sub}_\Lambda P$ of $\Lambda$-linear submodule representations of $P$. We give a criterion for…

We consider the constrained-degree percolation (CDP) model on the hypercubic lattice. This is a continuous-time percolation model defined by a sequence $(U_e)_{e\in\mathcal{E}^d}$ of i.i.d. uniform random variables and a positive integer…

We prove new upper bounds on homotopy and homology groups of o-minimal sets in terms of their approximations by compact o-minimal sets. In particular, we improve the known upper bounds on Betti numbers of semialgebraic sets defined by…

This paper gives a sharp upper bound for the Betti numbers of a finitely generated multigraded $R$-module, where $R=\Bbbk [x_{1},...,x_{m}]$ is the polynomial ring over a field $\Bbbk$ in $m$ variables. The bound is given in terms of the…

Let $M$ be a connected complete noncompact $n$-dimensional Riemannian manifold with a base point $p \in M$ whose radial sectional curvature at $p$ is bounded from below by that of a noncompact surface of revolution which admits a finite…

A pair of probability distributions over $\{0,1\}^n$ is said to be $(k,\delta)$-wise indistinguishable if all of the size $k$ marginals are within statistical distance at most $\delta$. Previous works introduced this concept and study when…

We prove a surprising higher regularity for solutions to the nonlinear elliptic autonomous Beltrami equation in a planar domain $\Omega$, \[ f_\zbar = {\cal A}(f_z) \hskip15pt a.e.\;\; z\in \Omega, \] when ${\cal A}$ is linear at $\infty$.…

Buchbinder and Feldman recently gave a deterministic $(1-1/e-\varepsilon)$-approximation for maximizing a non-negative monotone submodular function subject to a matroid constraint, with query complexity $\widetilde{O}_\varepsilon(nr)$.…

Let $\mathcal{C}(S_{g,p})$ denote the curve complex of the closed orientable surface of genus $g$ with $p$ punctures. Masur-Minksy and subsequently Bowditch showed that $\mathcal{C}(S_{g,p})$ is $\delta$-hyperbolic for some…