Related papers: Distance between arithmetic progressions and perfe…
Given an arbitrary long but finite sequence of observations from a finite set, we construct a simple process that approximates the sequence, in the sense that with high probability the empirical frequency, as well as the empirical one-step…
We prove sharp homogeneous improvements to $L^1$ weighted Hardy inequalities involving distance from the boundary. In the case of a smooth domain, we obtain lower and upper estimates for the best constant of the remainder term. These…
In solving diffusion problems, it is common to consider the finite difference equation to be an approximation to the differential equation. Nevertheless, history shows that the finite difference equation is primitive and that the…
We give an equivalent form of the Twin prime conjecture relating to a symmetric property that is observed for terms present in a certain sequence of arithmetic progressions defined for a pair of co-prime integers.
We introduce and study arithmetic polygons. We show that these arithmetic polygons are connected to triples of square pyramidal numbers. For every odd $N\geq3$, we prove that there is at least one arithmetic polygon with $N$ sides. We also…
The Fourier transform is approximated over a finite domain using a Riemann sum. This Riemann sum is then expressed in terms of the discrete Fourier transform, which allows the sum to be computed with a fast Fourier transform algorithm more…
We investigate finite and infinite nested square root formulas convergent to unity.
An open problem about finite geometric progressions in syndetic sets leads to a family of diophantine equations related to the commutativity of translation and multiplication by squares.
In this paper, we present a comprehensive system for the treatment of the topic of limits--conceptually, computationally, and formally. The system addresses fundamental linguistic flaws in the standard presentation of limits, which attempts…
We investigate whether there exists an arithmetic progression or geometric progression consisting only palindromic numbers. In this paper we show that the answer to this question is NO. Given the first and final term we will also give an…
We give a metric characterization of the scalar curvature of a smooth Riemannian manifold, analyzing the maximal distance between $(n+1)$ points in infinitesimally small neighborhoods of a point. Since this characterization is purely in…
The Procrustes distance is used to quantify the similarity or dissimilarity of (3-dimensional) shapes, and extensively used in biological morphometrics. Typically each (normalized) shape is represented by N landmark points, chosen to be…
In this article, we study the complexity involved in the computation of the reach in arbitrary dimension and in particular the computation of the critical spherical curvature points of an arbitrary algebraic variety. We present properties…
A permutation is said to be a square if it can be obtained by shuffling two order-isomorphic patterns. The definition is intended to be the natural counterpart to the ordinary shuffle of words and languages. In this paper, we tackle the…
The set ${1, 25, 49}$ is a 3-term collection of integers which forms an arithmetic progression of perfect squares. We view the set ${(1,1), (5,25), (7,49)}$ as a 3-term collection of rational points on the parabola $y=x^2$ whose…
We give a mathematical structure on an arithmetic surface, that has algebraic meanings over finite places and can estimate the canonical norm for a relative differential form on the arithmetic surface. This will give a lower bound for the…
Spatial approximations have been traditionally used in spatial databases to accelerate the processing of complex geometric operations. However, approximations are typically only used in a first filtering step to determine a set of candidate…
It is known that if a subset of $\mathbb{R}$ has positive Lebesgue measure, then it contains arbitrarily long finite arithmetic progressions. We prove that this result does not extend to infinite arithmetic progressions in the following…
The purpose of this article is twofold. On the one hand, we prove asymptotic formulas for the quantitative distribution of rational points on any smooth non-split projective quadratic surface. We obtain the optimal error term for the real…
The Wasserstein distance between two probability measures on a metric space is a measure of closeness with applications in statistics, probability, and machine learning. In this work, we consider the fundamental question of how quickly the…