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

# Peter Saveliev

Hello! My name is Peter Saveliev. I am a professor of mathematics at Marshall University, Huntington WV, USA.

My current projects are these two books:

*Topology Illustrated*, published in 2016*Calculus Illustrated*, volume 1 published in 2019

In part, the latter book is about *Discrete Calculus*, which is based on a simple idea:
$$\lim_{\Delta x\to 0}\left( \begin{array}{cc}\text{ discrete }\\ \text{ calculus }\end{array} \right)= \text{ calculus }.$$
I have been involved in research in algebraic topology and several other fields but nowadays I think this is a pointless activity. My non-academic projects have been: digital image analysis, automated fingerprint identification, and image matching for missile navigation/guidance.

- 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:

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

- -Why is discrete calculus better than infinitesimal calculus? -Why? -Because it can be integer-valued! -And? -And the integer-valued calculus can detect if the space is non-orientable! Read Integer-valued calculus, an essay making a case for discrete calculus by appealing to topology and physics.

- -The political “spectrum” might be a circle!- So? -Then there can be no fair decision-making system! Read 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*.