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- Mizan R. Khan, Igor E. Shparlinski, Christian L. Yankov
- Experimental Mathematics
- 2008

In this paper we give upper and lower bounds as well as a heuristic estimate on the number of vertices of the convex closure of the set Gn = {(a, b) : a, b ∈ Z, ab ≡ 1 (mod n), 1 ≤ a, b ≤ n − 1} . The heuristic is based on an asymptotic formula of Rényi and Sulanke. After describing two algorithms to determine the convex closure, we 1 compare the numeric… (More)

We investigate the distribution of n−M(n) where M(n) = max { |a− b| : 1 ≤ a, b ≤ n− 1 and ab ≡ 1 (mod n)} . Exponential sums provide a natural tool for obtaining upper bounds on this quantity. Here we use results about the distribution of integers with a divisor in a given interval to obtain lower bounds on n−M(n). We also present some heuristic arguments… (More)

- József Beck, Mizan R. Khan
- Periodica Mathematica Hungarica
- 2002

- Mizan R. Khan, Igor E. Shparlinski
- Periodica Mathematica Hungarica
- 2003

- David Ciplet, Juhanna T Roberts, Mizan R. Khan
- Global environmental politics
- 2013

- Peter B. Goodell, Peter C. Ellsworth, +171 authors G. Frank Zalom
- 2008

- Mizan R. Khan, Michael Avidon
- The American Mathematical Monthly
- 2003

Given an integer n 2, let H n be the set H n = {(a, b) : ab ≡ 1 (mod n), 1 a, b n − 1} and let M (n) be the maximal difference of b − a for (a, b) ∈ H n. We prove that for almost all n, n − M (n) = O n 1/2+o(1). We also improve some previously known upper and lower bounds on the number of vertices of the convex closure of H n .

- Prachi Juyal, Mizan R. Khan, Vaibhav Dixit
- 2010

Let R be a ring with unity. Let α be an endomorphism of R and RR be an α-compatible module. Then the formal power series ring R[[x,α]] is right p.q. Baer iff R is right p.q. Baer and every countable subset of right semicentral idempotents has a generalized countable join. Mathematics Subject Classification: 16D80, 16S36, 16W60

- Ali Gholi Ramin, Sardar Jafari Shoorijeh, +16 authors Angelo Ramina
- 2008