Related papers: Bounded gaps between primes
Examples of small contingency tables on binary random variables with large integer programming gaps on the lower bounds of cell entries were constructed by Sullivant. We argue here that the margins for which these constructed large gaps…
I explain so-called quantum nonlocality experiments and discuss how to optimize them. Statistical tools from missing data maximum likelihood are crucial. New results are given on CGLMP, CH and ladder inequalities. Open problems are also…
This is an updated version of the lectures notes for a course on condensed mathematics taught in the summer term 2019 at the University of Bonn. The material presented is joint work with Dustin Clausen. This is intended as a stable citable…
We show that a positive proportion of all gaps between consecutive primes are small gaps. We provide several quantitative results, some unconditional and some conditional, in this flavour.
In this article, a relation between a gap $d_{k}$ and divisors of composite numbers between $p_{k}$ and $p_{k+1}$ is established.
Recently delivered lectures on Self-Referential Mathematics, [2], at the Department of Mathematics and Applied Mathematics, University of Pretoria, are briefly presented. Comments follow on the subject, as well as on Inconsistent…
These are lecture notes mainly aimed at graduate students on selected aspects of generalized geometry: in particular generalized complex and Kaehler structures and generalized holomorphic bundles. They are based on lectures given in March…
In the present work the existence of some patterns of primes is shown which generalize the celebrated result of Green and Tao according to which there are arbitrarily long arithmetic progressions in the sequence of primes
These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new…
This note is an extended version of my presentation at the "Focused Research Workshop on Exterior Di{\S}erential Systems and Lie Theory" held at the Fields Institute in Toronto, December 9-13, 2013. Some ideas and constructions of this note…
We close a gap appearing at the same time in the author's thesis "Iterated rings of bounded elements and generalizations of Schm\"udgen's theorem" [1] and in the author's article "Iterated rings of bounded elements and generalizations of…
In this paper, we show some results about the gap between a prime number and its consecutive prime number for large enough prime numbers. We show that the gap between a prime number $p_n$ and its consecutive prime number is not larger than…
We investigate the approximation to the number of primes in arithmetic progressions given by Vaughan. Instead of averaging the expected error term over all residue classes to modules in a given range, here we only consider subsets of…
This document collects the lecture notes from my course "Communication Complexity (for Algorithm Designers),'' taught at Stanford in the winter quarter of 2015. The two primary goals of the course are: 1. Learn several canonical problems…
In a recent work Friedlander studied the problem of how large consecutive prime gaps should be in order that the sum of the reciprocals should be divergent. Supposing a very deep Hypothesis, a generalization of the Hardy--Littlewood prime…
This note discusses the existence of prime numbers in short intervals. An unconditional elementary argument seems to prove the existence of primes in the short intervals [x, x + y], where y >= x^(1/2)(log x)^e, e > 0, and a sufficiently…
These lecture notes were prepared for a special topics course in the Department of Statistics at the University of Washington, Seattle. They comprise the first eight chapters of a book currently in progress.
These are the notes from my courses on the arithmetic of quadratic forms.
We use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that a positive proportion of consecutive primes are within a…
This is a self-contained set of lecture notes covering various aspects of the theory of open quantum system, at a level appropriate for a one-semester graduate course. The main emphasis is on completely positive maps and master equations,…