This site is devoted to mathematics and its applications. Created and run by **Peter Saveliev**.

# Peter Saveliev

### From Mathematics Is A Science

Dr. Saveliev is a professor of mathematics at Marshall University, Huntington WV, USA. He has been involved in research in algebraic topology and several other fields. The non-academic projects have been: digital image analysis, automated fingerprint identification, and image matching for missile navigation/guidance. He holds a mathematics Ph.D. from the University of Illinois at Urbana-Champaign.

*Topology Illustrated*, book, 2016*Calculus Illustrated*, book, in progress.

In part, the book is about *Discrete Calculus*, which is based on a simple idea:
$$\lim_{\Delta x\to 0}\left( \begin{array}{cc}\text{ discrete }\\ \text{ calculus }\end{array} \right)= \text{ calculus }.$$
This idea is based, in part, on an even simpler idea:
$$\lim_{\Delta x\to 0}\left( \begin{array}{cc}\text{ algebra over }\\ \text{ the integers }\end{array} \right)= \begin{array}{cc}\text{ algebra over }\\ \text{ the reals }\end{array}.$$

- Once upon a time, I took a better look at the poster of
*Drawing Hands*by Escher hanging in my office and realized that what is shown isn't symmetric! To fix the problem I made my own picture called*Painting Hands*:

Such a symmetry is supposed to be an involution of the $3$-space, $A^2=I$; therefore, its diagonalized matrix has only $\pm 1$ on the diagonal. These are the three cases:

- One $-1$: mirror symmetry, then pen draws pen. No!
- Two $-1$s: $180$ degrees rotation, the we have two right (or two left) hands. No!
- Three $-1$s: central symmetry. Yes!

- The political spectrum is a circle, an essay based on the very last section of the topology book

Note: I am frequently asked, what should "Saveliev" sound like? I used to care about that but got over that years ago. The one I endorse is the most popular: "Sav-leeeeeev". Or, simply call me *Peter*.