User:Mmhassany/sandbox

Mehdi Hassani
Born	(1979-07-23) 23 July 1979 Zanjan, Iran
Nationality	Iran
Known for	Analytic number theory
Scientific career
Fields	Mathematics
Institutions	University of Zanjan
Doctoral advisors	Mehrdad Mirshams Shahshahani, Jean-Marc Deshouillers

Mehdi Hassani (born on July 23, 1979) in Zanjan) is an Iranian mathematician, specializing in analytic number theory. He is an associate professor at the University of Zanjan.

Education

Hassani received his PhD in 2010 at the Institute for Advanced Studies in Basic Sciences and University of Bordeaux.

Contributions

In 2003 he showed that the number of derangements of an n-Element Set (sequence A000166 in the OEIS) is given by

$d_{n}={\begin{cases}\lfloor {\frac {n!}{e}}+r_{1}\rfloor ,&n{\text{ is odd}},\quad \ r_{1}\in [0,{\frac {1}{2}}],\\\lfloor {\frac {n!}{e}}+r_{2}\rfloor ,&n{\text{ is even}},\quad r_{2}\in [{\frac {1}{3}},1],\end{cases}}$

where $\left\lfloor x\right\rfloor$ is the floor function. So, for any integer n ≥ 1, and for any real number r ∈ [1/3, 1/2],

$d_{n}=\left\lfloor {\frac {n!}{e}}+r\right\rfloor ,\quad \ n\geq 1,\quad r\in \left[{\frac {1}{3}},{\frac {1}{2}}\right].$

Therefore, as e = 2.71828…, 1/e ∈ [1/3, 1/2], then

$d_{n}=\left\lfloor {\frac {n!+1}{e}}\right\rfloor ,\quad \ n\geq 1.$

Also, he proved that

$d_{n}=\left\lfloor (e+e^{-1})n!\right\rfloor -\lfloor en!\rfloor ,\quad n\geq 2.$

In 2020 he found an asymptotic expansion for the number of derangements in terms of Bell numbers. Given any positive integer $m$ he proved that

$d_{n}={\frac {n!}{e}}+\sum _{k=1}^{m}\left(-1\right)^{n+k-1}{\frac {B_{k}}{n^{k}}}+O\left({\frac {1}{n^{m+1}}}\right),$

where $B_{k}$ denotes the $k$-th Bell number and the constant of $O$-term does not exceed $B_{m+1}$.

Category:20th-century Iranian mathematicians Category:21st-century Iranian mathematicians Category:Living people Category:1979 births