This post is not an obituary. This is an ode to a living Teacher.
Mukhtarbai Otelbaev
Early years
Mukhtarbai Otelbaev (I will call him MO for short) surprised us at the first meeting. It was a time when we had more strength than experience and knowledge, and everything was still ahead. In the fourth year of our bachelor program at the Kazakh State University, MO came to give an elective. More precisely, he ran into the classroom, followed by several graduate students who immediately sat down at their desks. No greeting nor anything about the course content. He began to tell theorems from functional analysis, five of them, one after another, without proof, saying that they were obvious. He often stood cross-legged by the blackboard and touched his nose and cheeks, smearing them with chalk. I don't know if MO understood from our eyes that we had never seen such teachers before. Other teachers gave the theory in small pieces, and there were some who, during the lecture, peeped into their notes, and no one had such a flight of thought and imagination.
I was then a Lenin Scholar and the best in the cohort, but I concluded for myself that I knew nothing. Functional analysis is what I need to study, and MO is a person to hold on to. When choosing a specialization, I settled on the Department of Differential Equations, because functional analysis was read there and Tulebai Amanov worked there. When I came to the Institute of Mathematics and Mechanics for practice, Amanov took me to his laboratory and asked: who wants to lead this young man? MO replied: I do, and this marked the beginning of my long ascent to real mathematics.
Arriving from Moscow, he was already an unusually strong mathematician. Those who judged by degrees thought: he had just defended his Candidate’s dissertation, he still had to work a lot to mature, and they were wrong. MO had already set his sights on a Doctor’s dissertation. During pre-defense advertising, he continued to prove new theorems publishing and adding them to his dissertation. Therefore, Boris Levitan said: we have to let him defend as soon as possible. Pyotr Lizorkin highly appreciated MO's approach to weighted Sobolev spaces and specially published two papers with him to popularize his method. Lizorkin supported doctoral defenses of several Asians, including me. With this in mind, Sergei Nikolsky later said: Lizorkin is a cosmopolitan.
In 1978 I tried to read all of MO’s articles. It was the first and last attempt of its kind. But I saw how original and profound they were. Sometimes he would let me type his articles. His manuscripts were scribbled, and I got the impression that he never rewrote them. I was especially impressed by his papers on weighted Sobolev spaces. (He wrote one jointly with Vladimir Maz’ya. In it, MO proposed one of the variants of his function q* and learned from Maz’ya the theory of capacities and techniques for working with the Strichartz seminorm. He applied and developed this knowledge further in another work, where he obtained two-sided estimates for the Kolmogorov widths.) I studied these articles in detail and, when it was not clear, I asked MO for explanations. What he explained in a couple of sentences sometimes required a long proof, and he never had any notes.
Peak of creativity
MO, together with Tynysbek Kalmenov, organized a seminar that for many years was the center of attraction for mathematicians in Kazakhstan. The talks were on a variety of topics. The discussion was lively and frank. Later I had the opportunity to compare the atmosphere of this seminar with the seminars in the USA. In the US, people speak very cautiously. Some are afraid to show their ignorance, others do not want to offend the speaker. The value of the seminar is greatly reduced by this. MO, after complex talks, explained the essence in a few phrases. It was very helpful to watch him think. He immediately saw the idea and could intelligibly explain it. In general, in his explanations, he always went straight to the idea and never talked about technical details. To understand him, you need a certain level. I think he's a great teacher for doctoral students, bad for bachelors, and terrible for high school students.
MO never followed the beaten track in his research. Even where he applied his previous inventions, he thought over everything anew. A good example is his q* function. He proposed its first version in his Candidate’s dissertation. Then there were more variants in articles on spectral theory and Sobolev spaces mentioned above. Later, Ryskul Oinarov breathed new life into it by inventing a generalization serving three weights.
The speed of thinking of MO is simply amazing. I remember once thinking about the connection between one fact from the spectral theory and another fact from the theory of functions for three days. Having understood what was the matter, I decided to test MO. When I came to the laboratory, he was playing blitz chess with someone. I asked him: what do you think about this? He thought for three seconds and said, oh, that's obvious, and continued playing. Then I said: if it is obvious, then prove it. He thought for another three seconds and told me the idea of the proof. Three days in my case and six seconds in his, and even playing chess – such a difference can only be explained by a natural gift. Once, in a conversation with me, he said that among the mathematicians with whom he met, only three thought as quickly (Evgenii Nikishin, Shavkat Alimov, and I don’t remember the third).
One of his most important admonitions to me was this: constantly solve new problems. He himself never wrote more than two articles on the same topic (except for the announcements in the Doklady of the Academy of Sciences of the USSR) and never tried to inflate the volume of an article by writing out all sorts of corollaries and applications. Once he told me: if I cannot find a solution to a problem in three days, then I drop it. Some mathematicians master just one theory and then spend their whole lives looking for only such problems where it can be applied. MO did the opposite: he took on completely new problems and developed methods from scratch that aimed specifically at them. When you see this, you get the impression that you are dealing with a higher being. The combination of the generality of the problem statement and the specificity of the approaches was one of his main lessons for me.
The exception to the three-day rule was the Navier-Stokes problem. He took it up in the early 1990s. There was a beautiful operator part and there was a piece where he hoped to get the result he needed by picking numbers. To check his calculations, he sat me down next to him. He counted in his mind, and often called the numbers wrong and still got the correct result. I counted on the calculator and lagged behind him. He gave me the original version of the article to check. To my shame, I lost the manuscript. He has been working on the Navier-Stokes problem for more than 20 years, which is surprising given his impatience. The result was published, caused a strong resonance in the world, because a million-dollar prize was promised for it, and the method of proof was refuted.
A prime example of MO's innovative approach was his paper on the Vekua theory. Ilia Vekua and his students posed the problem of solving an equation, which defines generalized analytic functions, in the spaces Lp. MO posed and solved a more general problem: to find the widest space in which the properties of this equation are valid. The article was published in a modest collection of conference proceedings and is not even listed on the website www.mathnet.ru, which has web pages of all Soviet mathematicians. Despite this, the article was noticed and appreciated in the US and an English translation was published.
MO’s research was the result of a balance between two of his traits: mathematical vanity and impatience. MO and Oinarov have published two excellent papers on the theory of integral operators. Oinarov told me how they wrote the second article. Oinarov obtained some result, showed it to MO and asked: can you do this? MO, who lost interest after the first article, caught fire again, and the second article was born.
Our work on the monograph on weighted Sobolev spaces played a special role in my development as a mathematician. I wrote the whole book, reading the articles of MO and consulting with him as needed. I didn't want to drag out the writing process for several years and set myself the task of writing at least 5 pages a day. This taught me to work 10-12 hours a day, regardless of my physical condition, and developed my imagination, which is indispensable in mathematics. There was more benefit from one year of work on the book than from the previous 5.
Outside math
MO once told me that he physically can't say things that don't make sense. This quality of his is directly related to his directness and honesty. Knowing that he is the best mathematician in Kazakhstan, he could not diplomatically bypass sharp corners in communication with our academicians. When Orymbek Zhautykov told him that it was necessary to support our Izvestia of the Academy of Sciences of the KazSSR with his articles, MO replied: who needs this journal? Naturally, Zhautykov, who put a lot of effort into the creation of our Academy of Sciences and the journal, was offended. During the seminar of Umirzak Sultangazin, a small dispute arose. MO said something on the topic of the report, Sultangazin objected to him, and MO said: in mathematics, I can always argue with you. Sultangazin knew that MO was right, but he held a grudge. MO also managed to offend several other superior comrades. Because of this, he was not allowed to become a corresponding member of the Academy of Sciences for a long time.
MO generously shared ideas, and that’s why he had many students. Sometimes he made the main contribution and simply gave the article to his student, so that the student could check the proofs and publish it as a joint one. There was a time when MO gushed with articles. In one particular year he published 14 papers in the central journals (the equivalent of the current rated journals), more than the rest of the Institute of Mathematics and Mechanics. According to the existing rule, he was supposed to receive a bonus, but Yengvan Kim, whom MO also managed to offend, began to create obstacles. This was considered at a meeting of the trade union committee, I was present there, and in the end the issue was resolved positively.
A couple of times MO held high administrative posts. I believe that an administrator should be a good politician, weigh the pros and cons and be persistent in order to achieve the desired result. MO is prone to one-sided opinions (like most mathematicians), does not know how to maneuver and is not patient and persistent enough. Therefore, I think that he was a bad administrator, and if he succeeded, it was only because he had good assistants.
In the 1990s, he toyed with the idea of purifying the air in Almaty with a giant pipe that he thought should be laid in the Small Almaty Gorge to pull the dirty air up into the mountains. Several factors played a role here: his self-esteem and the continued unrecognition of his talent by officials, his one-sidedness in evaluating his idea and neglect of other factors, such as the exorbitant cost and underdevelopment of the plan, as well as the decreased ability of MO to concentrate on mathematics.
This year 2022, MO turns 80 years old. Mathematicians create beauty that only they can see. I am grateful to fate for allowing me to watch how a real master creates.
and 
are the sets of real and natural numbers respectively.
is a relation 
satisfying the following conditions for arbitrary 
:
(reflexivity);
and 
then 
(anti-symmetry);
then 
(transitivity).
on it we call a partially ordered set.
(do not confuse it with the usual symbol of inequality). Moreover, we denote 
if 
and 
.
if 
.
if 
, where 
are subsets of 
(the set of all subsets of 
means "
divides 
", where 
. Then 
is a partial order on 
define 
(
is smaller than 
). Consequently we obtain a partial order on 
if 
for all ![t \in [a,b]](https://s0.wp.com/latex.php?latex=t+%5Cin+%5Ba%2Cb%5D&bg=ffffff&fg=000000&s=0 "t \in [a,b]")
.
we can introduce a partial order putting 
if 
and 
.
is a partial order on the English alphabet.
and 
. However they do not imply 
at all. Reflexivity and transitivity of this relation are obvious.
. We call an element
the largest (greatest) in 
, if 
for every 
(
is greater than any other element of 
the smallest (least) in 
for every 
is smaller than any other element of 
gives 
(there is no element of 
gives 
(there is no element of 
and 
. Then observe that 
is the smallest element in 
in 
are comparable under 
.
on 
with subsets 
and 
.
and 
in 
and 
defined on the interval ![[0, 2]](https://s0.wp.com/latex.php?latex=%5B0%2C+2%5D&bg=ffffff&fg=000000&s=0 "[0, 2]")
.
and 
in 
be sets.
. Likewise, 
for every set 
. Consequently, we accept the notation 
.
. Moreover, for any nonempty set 
.
of sets 
is defined as a set of all ordered pairs 
, where 
and 
.
. Then we have 
. Further, note that 
, but 
.
we call a relation between 
.
, and denote this fact by 
means 
.
instead of 
are the sets of reals and integers respectively.
. This is an equivalence relation on 
define 
is even. Consequently we obtain an equivalence relation on 
is not an equivalence relation since it is not transitive.
induces an equivalence relation on 
.
is the equivalence class determined by 
under the equivalence relation 
of all equivalence classes we call the quotient set of 
.
;
;
.
for every 
.
. Then 
and 
imply 
, showing 
.
. Then 
analogously. Therefore, 
.
, and let 
, which implies 
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