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A further study for the upper bound of the cardinality of Farey vertices and application in discrete geometry
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نویسنده
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Khoshnoudirad Daniel
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منبع
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journal of algebra combinatorics discrete structures and applications - 2015 - دوره : 2 - شماره : 3 - صفحه:169 -190
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چکیده
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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]): ᴲc>0, v(m,n) ∈n^2*, ifv (m,n)i≤cm^2n^2(m+n)in^2(mn)using it, in particular, we show that the number of (m; n)-cubes um;n verifies: ᴲc>0, v(m,n) ∈n^2*, iu_m,ni≤cm^2n^2(m+n)in^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.
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کلیدواژه
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Combinatorial number theory ,Farey diagrams ,Theoretical computer sciences ,Discrete planes ,Diophantine equations ,Arithmetical geometry ,Combinatorial geometry ,Discrete geometry ,Graph theory in computer sciences
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آدرس
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پست الکترونیکی
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daniel.khoshnoudirad@hotmail.com
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Authors
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