Related papers: Falconer-type estimates for dot products
In this paper, exact rate of approximation of functions by linear means of Fourier series and Fourier integrals and corresponding $K$-functionals are expressed via special moduli of smoothness. . Introduction is given in $\S 1$. In $\S2$…
We prove the following. Let $\mu_{1},\ldots,\mu_{n}$ be Borel probability measures on $[-1,1]$ such that $\mu_{j}$ has finite $s_j$-energy for certain indices $s_{j} \in (0,1]$ with $s_{1} + \ldots + s_{n} > 1$. Then, the multiplicative…
We study a variant of Erd\H os' unit distance problem, concerning dot products between successive pairs of points chosen from a large finite point set. Specifically, given a large finite set of $n$ points $E$, and a sequence of nonzero dot…
For a sample of Exponentially distributed durations we aim at point estimation and a confidence interval for its parameter. A duration is only observed if it has ended within a certain time interval, determined by a Uniform distribution.…
In this article we present very intuitive, easy to follow, yet mathematically rigorous, approach to the so called data fitting process. Rather than minimizing the distance between measured and simulated data points, we prefer to find such…
It is solved a problem of construction of separately continuous functions on the product of compacts with a given discontinuity points set. We obtaine the following results. 1. For arbitrary \v{C}ech complete spaces $X$, $Y$ and a separable…
We report on the calculation of the threshold soft function for heavy quark pair production in e+ e- annihilation at two-loop order. Our main result is a generalization of the familiar Drell-Yan threshold soft function to the case of…
When data contains measurement errors, it is necessary to make assumptions relating the observed, erroneous data to the unobserved true phenomena of interest. These assumptions should be justifiable on substantive grounds, but are often…
In this paper we prove asymptotically sharp weighted "first-and-a-half" $2D$ Korn and Korn-like inequalities with a singular weight occurring from Cartesian to cylindrical change of variables. We prove some Hardy and the so-called "harmonic…
This paper naturally extends and generalizes our previous work "Thue-Morse constant is not badly approximable", arXiv:1407.3182 [math.NT]. Here we consider the Laurent series $f_d(x) = \prod_{n=0}^\infty (1 - x^{-d^n})$, $d\in\mathbb{N}$,…
We study the Fourier spectrum of functions $f\colon \{0,1\}^{mk} \to \{-1,0,1\}$ which can be written as a product of $k$ Boolean functions $f_i$ on disjoint $m$-bit inputs. We prove that for every positive integer $d$, \[ \sum_{S \subseteq…
We first prove a weighted inequality of Moser-Trudinger type depending on a parameter, in the two-dimensional Euclidean space. The inequality holds for radial functions if the parameter is larger than -1. Without symmetry assumption, it…
Let $B$ be a $d$-dimensional Gaussian process on $\mathbb{R}$, where the component are independents copies of a scalar Gaussian process $B_0$ on $\mathbb{R}_+$ with a given general variance function…
We establish various forms of the following certainty principle: a set $S \subset \mathbb{R}^{n}$ contains a given finite linear pattern, provided that $S$ is a support of the Fourier transform of a sufficiently singular probability measure…
Our study concerns the Euclidean distance function in case of complex plane curves. We decompose the ED discriminant into components which are responsible for three types of behavior of the Morse points. Besides the traditional focal…
Suppose ${\bf b}=\{b_n\}_{n=1}^{\infty}$ is a sequence of integers bigger than 1 and ${\bf D}=\{{\mathcal D}_{n}\}_{n=1}^{\infty}$ is a sequence of consecutive digit sets. Let $\mu_{{\bf b},{\bf D}}$ be the Cantor-Moran measure defined by…
We study pseudo-Riemannian Einstein manifolds which are conformally equivalent with a metric product of two pseudo-Riemannian manifolds. Particularly interesting is the case where one of these manifolds is 1-dimensional and the case where…
Let $0 \leq s \leq 1$, and let $\mathbb{P} := \{(t,t^{2}) \in \mathbb{R}^{2} : t \in [-1,1]\}$. If $K \subset \mathbb{P}$ is a closed set with $\dim_{\mathrm{H}} K = s$, it is not hard to see that $\dim_{\mathrm{H}} (K + K) \geq 2s$. The…
It is shown by constructing Rohlins canonical measures that for a strictly stationary, d-dimensional vector-valued process X there exists another strictly stationary d-dimensional process U with uniform one-dimensional marginals and with…
We improve the Peres-Schlag result on pinned distances in sets of a given Hausdorff dimension. In particular, for Euclidean distances, with $$\Delta^y(E) = \{|x-y|:x\in E\},$$ we prove that for any $E, F\subset{\Bbb R}^d$, there exists a…