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Reduced Quadratic Irrational Numbers and Types of G-Circuits with Length Four by Modular Group


Affiliations
1 Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore − 54590, Pakistan
 

Objectives: To classify the types of G-circuits with length four in G-orbits αG where α is a reduced quadratic irrational number and G is the modular group. Methods/Statistical Analysis: G-orbits of real quadratic fields are evaluated using coset diagrams of modular group. Findings: There are five distinct types of the G-circuits in all. The number of disjoint G-orbits containing G-circuits of two types out of these five is four and for the remaining three types of G-circuits corresponding number of disjoint G-orbits is two. Application/Improvements: With the help of classification of G-circuits of length four we can find the structure of G-orbits of real quadratic fields.
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  • Reduced Quadratic Irrational Numbers and Types of G-Circuits with Length Four by Modular Group

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Authors

M. Aslam Malik
Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore − 54590, Pakistan
Sajjad Ali
Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore − 54590, Pakistan

Abstract


Objectives: To classify the types of G-circuits with length four in G-orbits αG where α is a reduced quadratic irrational number and G is the modular group. Methods/Statistical Analysis: G-orbits of real quadratic fields are evaluated using coset diagrams of modular group. Findings: There are five distinct types of the G-circuits in all. The number of disjoint G-orbits containing G-circuits of two types out of these five is four and for the remaining three types of G-circuits corresponding number of disjoint G-orbits is two. Application/Improvements: With the help of classification of G-circuits of length four we can find the structure of G-orbits of real quadratic fields.

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DOI: https://doi.org/10.17485/ijst%2F2018%2Fv11i30%2F127391