Properties of circuits in coset diagrams by modular group

Muhammad Aamir Javed

doi:10.17485/ijst/v13i14.7

Abstract

Background/Objectives: Graham Higman gave the idea of coset diagrams for the action of modular group PSL(2;Z) on real quadratic irrationals. These special types of graphical figures are composed of closed paths known as Circuits. These circuits can be classified into certain types of even length with respect to the number of inside\\outside triangles. This study is to discuss different properties of reduced numbers in coset diagrams of the type (p;q). Methods: In this study, we have investigated different properties of type (p; q) using reduced quadratic irrationals and continued fractions. We have categorized reduced numbers in accordance with their position in the real line. Distance between two ambiguous numbers and reduced numbers is introduced in this article which will help the reader to understand the structural significance of reduced numbers in a circuit. We have explored different conditions under which certain reduced numbers have the same circuit. Moreover, continued fractions have been used to assist the foundation laid by modular group action and different general results have been derived in this context. Findings: It was possible to define new notions of equivalent, cyclically equivalent and similar circuits using partitions of n and discuss various properties of reduced numbers included in coset diagrams of circuits with length up to four.

Highlights

Our universe is full of unexplored beauties of nature which hide different symmetries and patterns in it
Modular group known as PSL(2, Z) is an eminent group which is generated by two linear fractional transformations x : r → −1/r and y : r → 1 − 1/r and satisfy the relations x2 = y3 = 1
Qcl∗o(s√ednp) aistheqkunaolwton as circuit and the number of that each orbit closed paths in the coset diagram under the action of−. This action of modular group on different subsets has been discussed by Malik in(5)

Summary

Introduction

Our universe is full of unexplored beauties of nature which hide different symmetries and patterns in it. Qcl∗o(s√ednp) aistheqkunaolwton as circuit and the number of that each orbit closed paths in the coset diagram under the action of−. This action of modular group on different subsets has been discussed by Malik in(5). We use above mentioned technique to make coset diagrams and corresponding circuits of length two and four that is in line with repeated part of continued fraction expression of a reduced number. The behaviors of different ambiguous numbers and reduced numbers, in this particular type (p, q) will be discussed as real numbers and in the context of continued fractions. We divide all the circuits of type (p, q) into three genres

Ambiguous distance

Equivalence classes and classification of g-circuits of length four

Conclusion