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Qaiser Mushtaq

Publications and source records attributed to Qaiser Mushtaq.

5 recordsLinked to original sources
arXiv2018

All januarials constructed from Hecke groups

Professor Graham Higman defined januarial as a special instance of map constructed from embedding of a coset diagram for an action of $Δ(2,\ell ,k)$, on finite sets yielding exactly two orbits of the product of the two generators, having equal sizes. In this paper we determine a condition for the existence of a januarial from $Δ(2,\ell ,k),$ the quotients of Hecke groups $H_{Λ_{\ell }},$ when acting on the projective lines over finite fields $PL(F_{q})$. We develope a method to find all the januarials from Hecke groups $H_{Λ_{\ell }}$, when the triangle group $Δ(2,\ell ,k)$ acts on $PL(F_{q})$. We evelove a formula for calculating genus of coset diagram depending on the fixed points. By using it, we determine genus of the januarials.

math.GR↗
arXiv2018

Januarials of simple and general type

Januarials were defined by Graham Higman in his last series of lectures. In this paper we answer some questions posed by Higman in these lectures.

math.GR↗
arXiv2011

Coset Diagram for the Action of Picard Group on Q(i,\surd3)

The Picard group Γ is PSL(2,Z[i]). We have defined coset diagram for the Picard group. It has been observed that some elements of Q(i,/surd3) of the form ((a+b/surd3)/c) and their conjugates ((a-b/surd3)/c) over \mathbb{Q}(i) have different signs in the coset diagram for the action of Γ on the biquadratic field Q(i,/surd3), these are called ambiguous numbers. We have noticed that ambiguous numbers in the coset diagram for the action of Γ on \mathbb{Q}(i,/surd3) form a unique pattern. It has been shown that there are finite number of ambiguous numbers in an orbit Γα, where α is ambiguous, and they form a closed path and it is the only closed path in the orbit Γα. We have devised a procedure to obtain ambiguous numbers of the form ((a+k/surd3)/c), where k is a positive integer.

math.GR↗