scholarly journals Vertices of paths of minimal lengths on suborbital graphs

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 913-923 ◽  
Author(s):  
Ali Değer
Keyword(s):  
Modular Group ◽  
Rational Numbers ◽  
Suborbital Graphs

The Modular group ? acts on the set of extended rational numbers ?Q transitively. Here, our main purpose is to examine some properties of hyperbolic paths of minimal lengths in the suborbital graphs for ?. We characterize all vertices of these hyperbolic paths in the suborbital graphs which are trees.

scholarly journals Directed suborbital graphs on the Poincare disk

2018 ◽  
pp. 555-560
Keyword(s):  
Modular Group ◽  
Congruence Subgroup ◽  
Directed Graphs ◽  
Suborbital Graphs ◽  
Poincaré Disk ◽  
Special Congruence

In this paper we investigate suborbital graphs of a special congruence subgroup of modular group. And this directed graphs is drawn in Poincare disk.

On Suborbital Graphs for the Extended Modular Group $${\hat{\Gamma}}$$

2012 ◽  
Vol 29 (6) ◽  
pp. 1813-1825 ◽  
Author(s):  
Serkan Kader ◽  
Bahadır Özgür Güler
Keyword(s):  
Modular Group ◽  
Suborbital Graphs

ON SUBORBITAL GRAPHS FOR THE MODULAR GROUP

2001 ◽  
Vol 33 (6) ◽  
pp. 647-652 ◽  
Author(s):  
M. AKBAS
Keyword(s):  
Modular Group ◽  
Suborbital Graphs ◽  
Suborbital Graph

This paper proves a conjecture of G. A. Jones, D. Singerman and K. Wicks, that a suborbital graph for the modular group is a forest if and only if it contains no triangles.

scholarly journals Joint Universality of the Zeta-Functions of Cusp Forms

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2161
Author(s):  
Renata Macaitienė
Keyword(s):  
Cusp Form ◽  
Modular Group ◽  
Zeta Functions ◽  
Rational Numbers ◽  
Cusp Forms ◽  
Algebraic Numbers ◽  
Joint Universality ◽  
Universality Theorem ◽  
Linearly Independent ◽  
Joint Universality Theorem

Let F be the normalized Hecke-eigen cusp form for the full modular group and ζ(s,F) be the corresponding zeta-function. In the paper, the joint universality theorem on the approximation of a collection of analytic functions by shifts (ζ(s+ih1τ,F),⋯,ζ(s+ihrτ,F)) is proved. Here, h1,⋯,hr are algebraic numbers linearly independent over the field of rational numbers.

scholarly journals Suborbital graphs of a power subgroup of the modular group

2018 ◽  
Vol 1 (6) ◽  
pp. 139-144 ◽  
Author(s):  
Zeynep Şanli ◽  
Tuncay Köroğlu ◽  
Bahadir Özgür Güler
Keyword(s):  
Modular Group ◽  
Suborbital Graphs

scholarly journals Forests of complex numbers

2016 ◽  
Vol 13 (01) ◽  
pp. 15-25 ◽  
Author(s):  
Melvyn B. Nathanson
Keyword(s):  
Binary Tree ◽  
Fundamental Domain ◽  
Modular Group ◽  
Rational Numbers ◽  
Complex Numbers ◽  
Positive Integers ◽  
Positive Complex

The Calkin–Wilf tree is an infinite binary tree whose vertices are the positive rational numbers. Each number occurs in the tree exactly once and in the form [Formula: see text], where [Formula: see text] and [Formula: see text] are relatively prime positive integers. In this paper, certain subsemigroups of the modular group are used to construct similar trees in the set [Formula: see text] of positive complex numbers. Associated to each subsemigroup is a forest of trees that partitions [Formula: see text]. The fundamental domain and the set of cusps of the subsemigroup are defined and computed.

scholarly journals A generalization of the suborbital graphs generating Fibonacci numbers for the subgroup Г3

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 631-638
Author(s):  
Seda Öztürk
Keyword(s):  
Modular Group ◽  
Discrete Group ◽  
Fibonacci Numbers ◽  
Suborbital Graphs

The Modular group ? is the most well-known discrete group with many applications. This work investigates some subgraphs of the subgroup ?3, defined by {(ab cd)??:ab+cd ?0 (mod 3)}. In [1], the subgraph F1,1 of the subgroup ?3 ? ? is studied, and Fibonacci numbers are obtained by means of the subgraph of F1,1. In this paper, we give a generalization of the subgraphs generating Fibonacci numbers for the subgroup ?3 and some subgraphs having special conditions.

scholarly journals Chromatic Numbers of Suborbital Graphs for the Modular Group and the Extended Modular Group

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
Wanchai Tapanyo ◽  
Pradthana Jaipong
Keyword(s):  
Modular Group ◽  
Chromatic Numbers ◽  
Suborbital Graphs ◽  
Forest Conditions ◽  
Research Studies

This research studies the chromatic numbers of the suborbital graphs for the modular group and the extended modular group. We verify that the chromatic numbers of the graphs are2or3. The forest conditions of the graphs for the extended modular group are also described in this paper.

scholarly journals SL$ _n(\mathbb{Z}) $-normalizer of a principal congruence subgroup

2022 ◽  
Vol 7 (4) ◽  
pp. 5305-5313
Author(s):  
Guangren Sun ◽  
◽  
Zhengjun Zhao
Keyword(s):  
Modular Group ◽  
Congruence Subgroup ◽  
Rational Numbers ◽  
Principal Congruence ◽  
Common Denominator ◽  
Rational Matrix ◽  
Prime Decomposition ◽  
Principal Congruence Subgroup ◽  
Conjugate Subgroup

<abstract><p>Let SL$ _n(\mathbb{Q}) $ be the set of matrices of order $ n $ over the rational numbers with determinant equal to 1. We study in this paper a subset $ \Lambda $ of SL$ _n(\mathbb{Q}) $, where a matrix $ B $ belongs to $ \Lambda $ if and only if the conjugate subgroup $ B\Gamma_q(n)B^{-1} $ of principal congruence subgroup $ \Gamma_q(n) $ of lever $ q $ is contained in modular group SL$ _n(\mathbb{Z}) $. The notion of least common denominator (LCD for convenience) of a rational matrix plays a key role in determining whether <italic>B</italic> belongs to $ \Lambda $. We show that LCD can be described by the prime decomposition of $ q $. Generally $ \Lambda $ is not a group, and not even a subsemigroup of SL$ _n(\mathbb{Q}) $. Nevertheless, for the case $ n = 2 $, we present two families of subgroups that are maximal in $ \Lambda $ in this paper.</p></abstract>

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