相关论文: Variance estimates in Linnik's problem

Let $\Omega \subset \mathbb R^N$, $N \geq 2$, be a smooth bounded domain. We consider a boundary value problem of the form $$-\Delta u = c_{\lambda}(x) u + \mu(x) |\nabla u|^2 + h(x), \quad u \in H^1_0(\Omega)\cap L^{\infty}(\Omega)$$ where…

We prove a bound for the geodesic diameter of a subset of the unit ball in $\mathbb{R}^n$ described by a fixed number of quadratic equations and inequalities, which is polynomial in $n$, whereas the known bound for general degree is…

We consider the problem of prescribing the $\sigma_k$-curvature on the standard sphere $\mathbb{S}^n$ with $n \geq 3$. We prove existence and compactness theorems when $k \geq n/2$. This extends an earlier result of Chang, Han and Yang for…

The variance of the number of lattice points inside the dilated bounded set rD with random position in R^d has asymptotics r^(d-1) if the rotational quadratic average of the modulus of the Fourier transform of the set is O(r^(-d-1)). The…

We prove matching asymptotic lower and upper bounds on the variances of the intrinsic volumes and the number of $k$-faces of $d$-dimensional random beta-polytopes. Using Stein's methods, we establish central limit theorems for the intrinsic…

The Gauss Circle Problem concerns finding asymptotics for the number of lattice point lying inside a circle in terms of the radius of the circle. The heuristic that the number of points is very nearly the area of the circle is surprisingly…

In quantum theory on curved backgrounds, Heisenberg's uncertainty principle is usually discussed in terms of ensemble variances and flat-space commutators. Here we take a different, preparation-based viewpoint tailored to sharp position…

We find a choice of variables for the 3+1 formulation of general relativity which casts the evolution equations into (flux-conservative) symmetric-hyperbolic first order form for arbitrary lapse and shift, for the first time. We redefine…

A class of elliptic-hyperbolic equations is placed in the context of a geometric variational theory, in which the change of type is viewed as a change in the character of an underlying metric. A fundamental example of a metric which changes…

We prove asymptotic lower bounds on the variance of the number of vertices and missed area of random disc-polygons in convex discs whose boundary is $C_+^2$ smooth. The established lower bounds are of the same order as the upper bounds…

We compute the statistics of $SL_{d}(\mathbb{Z})$ matrices lying on level sets of an integral polynomial defined on $SL_{d}(\mathbb{R})$, a result that is a variant of the well known theorem proved by Linnik about the equidistribution of…

Rudnick and Wigman (Ann. Henri Poincar\'{e}, 2008; arXiv:math-ph/0702081) conjectured that the variance of the volume of the nodal set of arithmetic random waves on the $d$-dimensional torus is $O(E/\mathcal{N})$, as $E\to\infty$, where $E$…

For fixed $d\geq 3$, we construct subsets of the $d$-dimensional lattice cube $[n]^d$ of size $n^{\frac{3}{d + 1} - o(1)}$ with no $d+2$ points on a sphere or a hyperplane. This improves the previously best known bound of…

We bound the variance and other moments of a random vector based on the range of its realizations, thus generalizing inequalities of Popoviciu (1935) and Bhatia and Davis (2000) concerning measures on the line to several dimensions. This is…

We present a method to obtain upper bounds on covering numbers. As applications of this method, we reprove and generalize results of Rogers on economically covering Euclidean $n$-space with translates of a convex body, or more generally,…

In nonparametric statistical problems, we wish to find an estimator of an unknown function f. We can split its error into bias and variance terms; Smirnov, Bickel and Rosenblatt have shown that, for a histogram or kernel estimate, the…

Gardner, Gronchi and Zong posed the problem to find a discrete analogue of M. Meyer's inequality bounding the volume of a convex body from below by the geometric mean of the volumes of its slices with the coordinate hyperplanes. Motivated…

Since the publication of the important work of Rauch and Taylor (Potential and scattering theory on wildly perturbed domains, JFA, 1975) a lot has been done to analyse wild perturbations of the Laplace operator. Here we present results…

In this paper we show that the number of distinct distances determined by a set of $n$ points on a constant-degree two-dimensional algebraic variety $V$ (i.e., a surface) in $\mathbb R^3$ is at least $\Omega\left(n^{7/9}/{\rm polylog}…

We consider the occupancy problem where balls are thrown independently at infinitely many boxes with fixed positive frequencies. It is well known that the random number of boxes occupied by the first n balls is asymptotically normal if its…