Abstract
Objectives: To classify the types of G-circuits with length four in G-orbits G α 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. Keywords: Coset Diagrams, Modular Group, Reduced Quadratic Irrational Numbers, Types of G-circuits of Length Four
Highlights
Introduction and PreliminariesWe give a brief outline of the definitions and known results. , where a1, b1, c1, d1 are integers and a1d1 − b1c1 = 1 , is called modular group and it has presentation
Introduction and Preliminaries we give a brief outline of the definitions and known results.The set of linear fractional transformations of the form z →a1z + b1 c1z + d1, where a1, b1, c1, d1 are integers and a1d1 − b1c1 = 1, is called modular group and it has presentationG= x, y : x=2 y=3 1
G-orbits α G, (−α )G, (α )G and (−α )G are distinct whereas we prove that all the four G-orbits α G, (−α )G, (α )G and (−α )G involving G-circuits of the types ( p, q, r, s) and ( p, p, r, s) are mutually distinct
Summary
We give a brief outline of the definitions and known results. , where a1, b1, c1, d1 are integers and a1d1 − b1c1 = 1 , is called modular group and it has presentation. Whereas if α satisfies the conditions α > 1 and −1 < α < 0 , it is called reduced quadratic irrational number. Reduced Quadratic Irrational Numbers and Types of G-circuits with Length Four by Modular Group we mean the circuit containing one vertex, fixed by are equivalent as they satisfy the condition to start from the number of triangles having one vertex outside the circuit and end at the number of triangles having one vertex inside the circuit. We have discussed properties of reduced quadratic irrational numbers and classified G-circuits of length two in[7]. It can be deduced from the result in[6] that for a given sequence of positive integers, there exist a circuit which has period of length 2l , where l divides k. Theorem 1.1 (Aslam and Sajjad): An ambiguous number a + n where c > 0 is reduced number if and c only if, | b + c |< 2a
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