A further study for the upper bound of the cardinality of Farey vertices and application in discrete geometry
The aim of the paper is to bring new combinatorial analytical properties of the Farey diagrams of order $(m,n)$, which are associated to the $(m,n)$-cubes. The latter are the pieces of discrete planes occurring in discrete geometry, theoretical computer sciences, and combinatorial number theory. We...
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doaj-192b7ee5169c483596674096ba6ecca62020-11-24T22:16:39ZengYildiz Technical UniversityJournal of Algebra Combinatorics Discrete Structures and Applications2148-838X2148-838X2015-09-012316919010.13069/jacodesmath.79416A further study for the upper bound of the cardinality of Farey vertices and application in discrete geometry Daniel Khoshnoudirad The aim of the paper is to bring new combinatorial analytical properties of the Farey diagrams of order $(m,n)$, which are associated to the $(m,n)$-cubes. The latter are the pieces of discrete planes occurring in discrete geometry, theoretical computer sciences, and combinatorial number theory. We give a new upper bound for the number of Farey vertices $FV(m,n)$ obtained as intersections points of Farey lines ([14]): $$\exists C>0, \forall (m,n)\in\mathbb{N}^{*2},\quad \Big|FV(m,n)\Big| \leq C m^2 n^2 (m+n) \ln^2 (mn)$$ Using it, in particular, we show that the number of $(m,n)$-cubes $\mathcal{U}_{m,n}$ verifies: $$\exists C>0, \forall (m,n)\in\mathbb{N}^{*2},\quad \Big|\mathcal{U}_{m,n}\Big| \leq C m^3 n^3 (m+n) \ln^2 (mn)$$ which is an important improvement of the result previously obtained in [6], which was a polynomial of degree 8. This work uses combinatorics, graph theory, and elementary and analytical number theory.http://eds.yildiz.edu.tr/AjaxTool/GetArticleByPublishedArticleId?PublishedArticleId=2169Combinatorial number theoryFarey diagramsTheoretical computer sciencesDiscrete planesDiophantine equationsArithmetical geometryCombinatorial geometryDiscrete geometryGraph theory in computer sciences |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Daniel Khoshnoudirad |
spellingShingle |
Daniel Khoshnoudirad A further study for the upper bound of the cardinality of Farey vertices and application in discrete geometry Journal of Algebra Combinatorics Discrete Structures and Applications Combinatorial number theory Farey diagrams Theoretical computer sciences Discrete planes Diophantine equations Arithmetical geometry Combinatorial geometry Discrete geometry Graph theory in computer sciences |
author_facet |
Daniel Khoshnoudirad |
author_sort |
Daniel Khoshnoudirad |
title |
A further study for the upper bound of the cardinality of Farey vertices and application in discrete geometry |
title_short |
A further study for the upper bound of the cardinality of Farey vertices and application in discrete geometry |
title_full |
A further study for the upper bound of the cardinality of Farey vertices and application in discrete geometry |
title_fullStr |
A further study for the upper bound of the cardinality of Farey vertices and application in discrete geometry |
title_full_unstemmed |
A further study for the upper bound of the cardinality of Farey vertices and application in discrete geometry |
title_sort |
further study for the upper bound of the cardinality of farey vertices and application in discrete geometry |
publisher |
Yildiz Technical University |
series |
Journal of Algebra Combinatorics Discrete Structures and Applications |
issn |
2148-838X 2148-838X |
publishDate |
2015-09-01 |
description |
The aim of the paper is to bring new combinatorial analytical properties of the Farey diagrams of order $(m,n)$, which are associated to the $(m,n)$-cubes. The latter are the pieces of discrete planes occurring in discrete geometry, theoretical computer sciences, and combinatorial number theory. We give a new upper bound for the number of Farey vertices $FV(m,n)$ obtained as intersections points of Farey lines ([14]): $$\exists C>0, \forall (m,n)\in\mathbb{N}^{*2},\quad \Big|FV(m,n)\Big| \leq C m^2 n^2 (m+n) \ln^2 (mn)$$ Using it, in particular, we show that the number of $(m,n)$-cubes $\mathcal{U}_{m,n}$ verifies: $$\exists C>0, \forall (m,n)\in\mathbb{N}^{*2},\quad \Big|\mathcal{U}_{m,n}\Big| \leq C m^3 n^3 (m+n) \ln^2 (mn)$$ which is an important improvement of the result previously obtained in [6], which was a polynomial of degree 8. This work uses combinatorics, graph theory, and elementary and analytical number theory. |
topic |
Combinatorial number theory Farey diagrams Theoretical computer sciences Discrete planes Diophantine equations Arithmetical geometry Combinatorial geometry Discrete geometry Graph theory in computer sciences |
url |
http://eds.yildiz.edu.tr/AjaxTool/GetArticleByPublishedArticleId?PublishedArticleId=2169 |
work_keys_str_mv |
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