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Coset diagrams of the modular group and continued fractions
Comptes Rendus Mathematique ( IF 0.8 ) Pub Date : 2019-08-01 , DOI: 10.1016/j.crma.2019.07.002
Ayesha Rafiq , Qaiser Mushtaq

Abstract The coset diagram for each orbit under the action of the modular group on Q ( n ) ⁎ = Q ( n ) ∪ { ∞ } contains a circuit C i . For any α ∈ Q ( n ) , the path leading to the circuit C i and the circuit itself are obtained through continued fractions in this paper. We show that the structure of the continued fractions of a reduced quadratic irrational element is weaved with the structure or type of the circuit. The three types of circuits of the action of V 4 on Q ( n ) ⁎ are also interconnected with the structure of continued fractions. The action of the modular group on Q ( 5 ) ⁎ is chosen specifically because a circuit of it is related to the ratio of the Fibonacci numbers being the solution to the continued fractions of the golden ratio.

中文翻译:

模群和连分数的陪集图

摘要 Q ( n ) ⁎ = Q ( n ) ∪ { ∞ } 上模群作用下每个轨道的陪集图包含一个回路C i 。对于任何α ∈ Q ( n ) ,本文通过连分数获得通向电路C i 的路径和电路本身。我们证明了简化的二次无理元的连分数结构与电路的结构或类型交织在一起。V 4 对Q ( n ) ⁎ 作用的三种电路也与连分数结构相互关联。模群对 Q ( 5 ) ⁎ 的作用是特别选择的,因为它的回路与作为黄金比例连分数解的斐波那契数的比值有关。
更新日期:2019-08-01
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