History

In 1984 under leadership of professor Eduard Danielian I was parachuted to the desert of Optimal Control and Extremal Problems in a quest for the Holy Grail. It took enormous number of coffee cups, tea and cigarettes consumed actively or passively until we smell it. Unlike other travelers in the desert it did not take 40 years but only a year and a half until we came to a crazy idea for a solution of extremal problems.

 Here,  are fixed numbers. Imagine that we know that a subset  possesses a nice property: any point from  is the solution of the General Problem for  . Then, to state that the solution of General Problem always belongs to the set A we should be sure that the set A is rich enough to possess an element  satisfying (1) for arbitrary .  The incredible is that this simple idea works. We only should understand what does it mean to be rich enough. Yes, it should be n-parametric, open and compact in some sense. In a lot of cases the class   is determined in a natural way. For example, for a class   of all cumulative distribution functions (CDF) on a finite interval   and the functionals  being the moments of CDF, the set  is a set of all staircase CDF with  degree of freedom (that is, the plot is made of  segments connecting  and ) . To go further we need to be sure that in  the image of an open set for a continuous injective map to  is open too. But, is it? 

I did not know. Then, I started to ask white bone mathematicians specializing in Calculus, Functional Analysis, General Topology, et cetera. To our surprise, they did not answer to this question. I left with no choice but to prove it myself. Naturally, I started with the case  . After superhuman efforts I managed to transform it to Jordan Theorem that was proved by, I suppose, Jordan. I clearly understood than the case   is above my capacity. It was unfamiliar mathematics I have no idea about. But I got a clue. A Jordan Theorem. Where it seats? Not in Calculus, General Topology or any “white bone” mathematics. It seats in incomprehensible branch of math called Algebraic Topology. Being young and full of energy I tried to learn it in a couple of weeks. Failure. It is like learning Chinese Mandarine. Everything is different. So, professor Eduard Danielian had a better idea to speak to professor Mirzakhanian, a specialist in Algebraic Topology. Being blind, professor Mirzakhanian said that yes, such theorem exists. Then he called his wife, asked her to bring a book by Albrecht Dold “Lectures on Algebraic Topology”, open it and read it loudly. After turning few pages forward and back she finally read the theorem. Yes, it exists. It is called Invariance of Domain. The consultation lasted around 30 minutes. I left it being the happiest man, I knew the root to the Holy Grail and I became a mathematician.

Then it was a routine of writing articles and even a book. Publishing some of them in prestigious journals. It is funny to mention that as the most controversial in our study the reviewers were finding the theorem on Invariance of Domain. The reaction to it was from “this theorem is evident for any student” to “it is false and, consequently, Dold’s book is false too“. Anyway, it was an academic discussion and at the end the reviews were always positive. Almost … maybe with one important exception. After successfully publishing an article in a very prestigious journal where we presented a narrow aspect of what we done, we sent them second article with more general view and results on the topic. After three rounds of a discussion the reviewer finally said in a private conversation: “My wife is waiting for naturalization. I do not want problems.” So, I was introduced to the notion of feudal hierarchy that controls scarce drops of funds into mathematics.

Thus, we wrote multiple articles published in prestigious magazines, wrote a book. Success? I would say the opposite. Failure. The problem was in a fact that no one read them. The index of citation of our articles is either zero or tends to zero. Even after thirty years being passed from the publications. Loving the life and having obligations to my family I could not afford to be a theoretical mathematician. So, I worked for different interesting projects in applied math but mainly for a respectful international organization acting in war zones and making diplomacy in world capitals. Now, I am back. As a mathematician I became older, more lazy and much, much less encyclopedic. However, I hope I became wiser.

At the time, we were research mathematicians who was above computational methods. We knew that our research opens doors for applications. But it was not our business to do that. Our business was mathematically strictly formulate and prove theorems we publish even if it was extremely difficult to read them.

Time passed and I felt to other extremity. I developed/discovered an original and quite unusual computational method that matches the problems we solved. In fact, it is in some way an extension of “bisection method”. So, the tool calculating that I called Monotone Calculator. I do not care why Monotone Calculator is working as long as it works. If someone wants to understand the logic of Monotone Calculator or to teach the Calculator to solve new problems one should start with the articles we published 30 years ago. The important is that we are now in a possession of a final product. We are in favor to introduce more capitalism in mathematics.

Monotone Calculator has a commercial value. We now in a process to evaluate it. It will take some time. That is why here we will not present the codes but will demonstrate the capacity of Monotone Calculator, the added value it brings to Statistics and Probability and some new results.