Author Archives: Behnam Esmayli

About Behnam Esmayli

I started PhD in Mathematics at Pitt in Fall 2015. I have come to grow a passion for metric spaces -- a set and a distance function that satisfies the triangle inequality -- simple and beautiful! These spaces when equipped with other structures, such as a measure, becomes extremely fun to play with!

Ingredients of a Class Activity

Non-lecture math education seems to be getting its advocates in the math communities at all levels. Every alternative to the traditional math education involves some sort of class activity, where students are given tasks to complete in groups or pairs. … Continue reading

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A Eulogy Of Lipschitz Maps

A Lipschitz map (/function) is one that does not extend distances by more than a pre-assigned factor: $f: X \longrightarrow Y$ is Lipschitz if there exists an $L \in \mathbb{R}$ such that $$ \forall x, \ \  \forall y \ … Continue reading

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Intersection of a Chain of Subsets

Assume $\{F_x\}_{x \in \Gamma}$ is a collection of subsets (of a not-so important set!) such that every two are comparable, i.e for any $x$ and $y$, either $F_x \subset F_y \ \ $ or $\ \ F_x \supset F_y \ … Continue reading

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Real Numbers Base…Factorials! And A By-product

PROPOSITION 1:  For a real number  x  there exists a sequence $ x_1, x_2, x_3,…$ of integers such that $ \hspace{4cm} x=x_1 +\frac{x_2}{2!}+\frac{x_3}{3!} + \cdots + \frac{x_n}{n!} + \cdots,  \hspace{2cm} (*) $ where $x_1$ can be any integer, but for … Continue reading

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SMART

As math graduate students, we often get teaching assignments from our departments, some of us tutor independently for financial reasons, and some volunteer to teach at local schools or libraries. Teaching is an inseparable part of our job/life. But that is where … Continue reading

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