scholarly journals Lattices and the Geometry of Numbers

2021 ◽  
pp. 77-92
Author(s):  
Sourangshu Ghosh
Keyword(s):  
Functional Analysis ◽  
Number Theory ◽  
Diophantine Equations ◽  
Algebraic Numbers ◽  
Geometry Of Numbers ◽  
Geometrical Properties ◽  
The Subject ◽  
Broad Overview

In this paper we discuss about properties of lattices and its application in theoretical and algorithmic number theory.  This result of Minkowski regarding the lattices initiated the subject of Geometry of Numbers, which uses geometry to study the properties of algebraic numbers. It has application on various other fields of mathematics especially the study of Diophantine equations, analysis of functional analysis etc. This paper gives an elementary introduction to the field of geometry of numbers. In this paper we shall first give a broad overview of the concept of lattice and then discuss about the geometrical properties it has and its applications.

scholarly journals Lattices and the Geometry of Numbers

2021 ◽  
Author(s):  
Sourangshu Ghosh
Keyword(s):  
Functional Analysis ◽  
Number Theory ◽  
Diophantine Equations ◽  
Algebraic Numbers ◽  
Geometry Of Numbers ◽  
Geometrical Properties ◽  
The Subject ◽  
Broad Overview

In this paper, we discuss the properties of lattices and their application in theoretical and algorithmic number theory. This result of Minkowski regarding the lattices initiated the subject of Geometry of Numbers, which uses geometry to study the properties of algebraic numbers. It has an application in various other fields of mathematics especially the study of Diophantine equations, analysis of functional analysis, etc. This paper gives an elementary introduction to the field of the geometry of numbers. In this paper, we shall first give a broad overview of the concept of lattice and then discuss the geometrical properties it has and its applications.

An application of the Thue–Siegel–Roth theorem to elliptic functions

1971 ◽  
Vol 69 (1) ◽  
pp. 157-161 ◽  
Author(s):  
J. Coates
Keyword(s):  
Number Theory ◽  
Diophantine Equations ◽  
Elliptic Functions ◽  
Rational Numbers ◽  
Algebraic Numbers ◽  
Number Of Solutions ◽  
Transcendental Numbers ◽  
Linearly Independent ◽  
Image Position ◽  
Effective Proof

Let α1, …, αn be n ≥ 2 algebraic numbers such that log α1,…, log αn and 2πi are linearly independent over the field of rational numbers Q. It is well known (see (6), Ch. 1) that the Thue–Siegel–Roth theorem implies that, for each positive number δ, there are only finitely many integers b1,…, bn satisfyingwhere H denotes the maximum of the absolute values of b1, …, bn. However, such an argument cannot provide an explicit upper bound for the solutions of (1), because of the non-effective nature of the theorem of Thue–Siegel–Roth. An effective proof that (1) has only a finite number of solutions was given by Gelfond (6) in the case n = 2, and by Baker(1) for arbitrary n. The work of both these authors is based on arguments from the theory of transcendental numbers. Baker's effective proof of (1) has important applications to other problems in number theory; in particular, it provides an algorithm for solving a wide class of diophantine equations in two variables (2).

Euler's contribution to number theory

2007 ◽  
Vol 91 (522) ◽  
pp. 453-461 ◽  
Author(s):  
Peter Shiu
Keyword(s):  
Eighteenth Century ◽  
Number Theory ◽  
Scientific Output ◽  
Scholarly Work ◽  
Long Period ◽  
Peter The Great ◽  
Complete Work ◽  
Leonhard Euler ◽  
The Subject ◽  
Historic Background

Individuals who excel in mathematics have always enjoyed a well deserved high reputation. Nevertheless, a few hundred years back, as an honourable occupation with means to social advancement, such an individual would need a patron in order to sustain the creative activities over a long period. Leonhard Euler (1707-1783) had the fortune of being supported successively by Peter the Great (1672-1725), Frederich the Great (1712-1786) and the Great Empress Catherine (1729-1791), enabling him to become the leading mathematician who dominated much of the eighteenth century. In this note celebrating his tercentenary, I shall mention his work in number theory which extended over some fifty years. Although it makes up only a small part of his immense scientific output (it occupies only four volumes out of more than seventy of his complete work) it is mostly through his research in number theory that he will be remembered as a mathematician, and it is clear that arithmetic gave him the most satisfaction and also much frustration. Gazette readers will be familiar with many of his results which are very well explained in H. Davenport's famous text [1], and those who want to know more about the historic background, together with the rest of the subject matter itself, should consult A. Weil's definitive scholarly work [2], on which much of what I write is based. Some of the topics being mentioned here are also set out in Euler's own Introductio in analysin infinitorum (1748), which has now been translated into English [3].

scholarly journals Riesz triple probabilisitic of almost lacunary ces$\acute{A}$ro $C_{111}$ statistical convergence of $\chi^{3}$ defined by a Musielak Orlicz function

2018 ◽  
Vol 36 (4) ◽  
pp. 23-32 ◽  
Author(s):  
Dr Vandana ◽  
_ Deepmala ◽  
N. Subramanian ◽  
Vishnu Narayan Mishra
Keyword(s):  
Functional Analysis ◽  
General Framework ◽  
Statistical Convergence ◽  
Metric Spaces ◽  
Orlicz Function ◽  
The Subject

In this paper we study the concept of almost lacunary statistical Ces$\acute{a}$ro of $\chi^{3}$ over probabilistic $p-$ metric spaces defined by Musielak Orlicz function. Since the study of convergence in PP-spaces is fundamental to probabilistic functional analysis, we feel that the concept of almost lacunary statistical Ces$\acute{a}$ro of $\chi^{2}$ over probabilistic $p-$ metric spaces defined by Musielak in a PP-space would provide a more general framework for the subject.

scholarly journals Algebraic Properties of Chromatic Roots

10.37236/6578 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Peter J. Cameron ◽  
Kerri Morgan
Keyword(s):  
Algebraic Integer ◽  
Tutte Polynomial ◽  
Chromatic Polynomial ◽  
Galois Groups ◽  
Algebraic Numbers ◽  
Isaac Newton ◽  
Isaac Newton Institute ◽  
Chromatic Root ◽  
The Subject ◽  
Chromatic Roots

A chromatic root is a root of the chromatic polynomial of a graph.  Any chromatic root is an algebraic integer. Much is known about the location of chromatic roots in the real and complex numbers, but rather less about their properties as algebraic numbers. This question was the subject of a seminar at the Isaac Newton Institute in late 2008.  The purpose of this paper is to report on the seminar and subsequent developments.We conjecture that, for every algebraic integer $\alpha$, there is a natural number n such that $\alpha+n$ is a chromatic root. This is proved for quadratic integers; an extension to cubic integers has been found by Adam Bohn. The idea is to consider certain special classes of graphs for which the chromatic polynomial is a product of linear factors and one "interesting" factor of larger degree. We also report computational results on the Galois groups of irreducible factors of the chromatic polynomial for some special graphs. Finally, extensions to the Tutte polynomial are mentioned briefly.

scholarly journals Spectral statistics of permutation matrices

2014 ◽  
Vol 372 (2007) ◽  
pp. 20120508 ◽  
Author(s):  
Idan Oren ◽  
Uzy Smilansky
Keyword(s):  
Number Theory ◽  
Point Of View ◽  
Spectral Form ◽  
Permutation Matrices ◽  
Divisor Functions ◽  
Point Correlation ◽  
Point Correlation Function ◽  
The Subject ◽  
Spectral Statistics ◽  
The Mean

We compute the mean two-point spectral form factor and the spectral number variance for permutation matrices of large order. The two-point correlation function is expressed in terms of generalized divisor functions, which are frequently discussed in number theory. Using classical results from number theory and casting them in a convenient form, we derive expressions which include the leading and next to leading terms in the asymptotic expansion, thus providing a new point of view on the subject, and improving some known results.

scholarly journals The Brocard and Tucker Circles of a Cyclic Quadrilateral

1917 ◽  
Vol 36 ◽  
pp. 61-83
Author(s):  
Frederick G. W. Brown
Keyword(s):  
Geometrical Properties ◽  
The Subject ◽  
Logical Treatment

The application of the geometrical properties of the Brocard and Tucker circles of a triangle to a quadrilateral appears never to have been adequately worked out, as far as the author can discover. Hence, the object of this paper.Some of the problems involved have been published, under the author's name, as independent questions for solution, and where, in the author's opinion, solutions other than his own have seemed more satisfactory for the logical treatment of the subject, these solutions have been employed, with due acknowledgments to their authors.

scholarly journals Melioration Terminology in Ukrainian Scientific Picture of the World

2019 ◽  
Vol 10 (2) ◽  
pp. 337-348 ◽  
Author(s):  
Tetyana Petrova
Keyword(s):  
Functional Analysis ◽  
Quantitative Analysis ◽  
Theoretical Basis ◽  
Comparative Method ◽  
Natural Sciences ◽  
Scientific Cognition ◽  
World Outlook ◽  
The World ◽  
Structured System ◽  
The Subject

The subject of the study is the features, functions and structure of the Ukrainian scientific picture of the world and its component – the picture of the world of melioration science. The theoretical basis of the research is the thesis of the foreign and Ukrainian scholars on the scientific picture of the world as a consolidated and structured system of universal scientific knowledge. The methods of the research are the following: a method of structural and functional analysis of the concept, a method of definition analysis, a comparative method and a method of quantitative analysis. It has been found that the scientific picture of the world is a system of concepts that represents the scientific cognition of the world. The functions of the scientific picture of the world are usually identified as heuristic, synthetic, methodological, systematising and world outlook creating. The unified scientific picture of the world consists of the following types: humanitarian, natural, social and technical ones. The format of the scientific picture of the world expressed by the terminology of melioration belongs to the natural sciences picture of the world. The contributors to the melioration picture of the world are the subsystems of its sub-sectors, namely: water, land, chemical, technical, phytomelioration and others. The modern melioration picture of the world is largely formed by means of international terms (about 70%), and to a lesser extent by the genuine (national) vocabulary (30%).

scholarly journals Christopher Hooley. 7 August 1928—13 December 2018

2020 ◽  
Vol 69 ◽  
pp. 225-246
Author(s):  
D. R. Heath-Brown
Keyword(s):  
Number Theory ◽  
World Wide ◽  
Exponential Sums ◽  
Analytic Number Theory ◽  
Biographical Sketch ◽  
Artin's Conjecture ◽  
Analytic Number ◽  
The Subject ◽  
Primitive Roots ◽  
Modern Era

Christopher Hooley was one of the leading analytic number theorists of his day, world-wide. His early work on Artin’s conjecture for primitive roots remains the definitive investigation in the area. His greatest contribution, however, was the introduction of exponential sums into every corner of analytic number theory, bringing the power of Deligne’s ‘Riemann hypothesis’ for varieties over finite fields to bear throughout the subject. For many he was a figure who bridged the classical period of Hardy and Littlewood with the modern era. This biographical sketch describes how he succeeded in applying the latest tools to famous old problems.

Sign in / Sign up

Export Citation Format

Share Document