Arithmetic purity of the Hardy-Littlewood property and geometric sieve for affine quadrics

The second geometric-arithmetic index for trees and unicyclic graphs

Let $G$ be a finite and simple graph with edge set $E(G)$. The second geometric-arithmetic index is defined as $GA_2(G)=sum_{uvin E(G)}frac{2sqrt{n_un_v}}{n_u+n_v}$, where $n_u$ denotes the number of vertices in $G$ lying closer to $u$ than to $v$. In this paper we find a sharp upper bound for $GA_2(T)$, where $T$ is tree, in terms of the order and maximum degree o...

investigating the feasibility of a proposed model for geometric design of deployable arch structures

deployable scissor type structures are composed of the so-called scissor-like elements (sles), which are connected to each other at an intermediate point through a pivotal connection and allow them to be folded into a compact bundle for storage or transport. several sles are connected to each other in order to form units with regular polygonal plan views. the sides and radii of the polygons are...

On the Hardy - Littlewood majorant

Generalising the Hardy-Littlewood method for primes

The Hardy-Littlewood method is a well-known technique in analytic number theory. Among its spectacular applications are Vinogradov’s 1937 result that every sufficiently large odd number is a sum of three primes, and a related result of Chowla and Van der Corput giving an asymptotic for the number of 3-term progressions of primes, all less than N . This article surveys recent developments of the...

Applications of the Hardy - Littlewood Circle Method

1 Waring’s Problem Problem 1. (Waring’s) For every natural number k ≥ 2 there exists a positive integer s such that every natural number is the sum of at most s k powers of natural number (for example, every natural number is the sum of at most 4 squares, or 9 cubes, or 19 fourth powers, etc.). The affirmative answer, known as the Hilbert-Waring theorem, was provided by Hilbert in 1909. Let A =...