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Jun 22

Ode to a living Teacher

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.

Tags: Mukhtarbai Otelbaev, Navier-Stokes equation, Otelbaev

Dec 20

Order Relations

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Order Relations

Nurlan Abiev

The present post we devote to order relations. For preliminaries see our previous post Equivalence Relations. Recall that $\mathbb{R}$ and $\mathbb{N}$ are the sets of real and natural numbers respectively.

1 Partially ordered sets

Definition 1. A partial order on a set $X$ is a relation $R \subseteq X\times X$ satisfying the following conditions for arbitrary $x,y,z \in X$ :
i) $xRx$ (reflexivity);
ii) If $xRy$ and $yRx$ then $y=x$ (anti-symmetry);
iii) If $xRy$ and $yRz$ then $xRz$ (transitivity).

Respectively, given a set $X$ with a chosen partial order $R$ on it we call a partially ordered set.

We often denote a partial order $R$ by the symbol $\preceq$ (do not confuse it with the usual symbol of inequality). Moreover, we denote $x\prec y$ if $x\preceq y$ and $x\ne y$ .

Remark 1. The corresponding definitions of the concepts partial order and equivalence relation are distinguished each from other only with respect to the symmetricity property (compare Definition 1 here with Definition 4 in Equivalence Relations).

Example 1. Examples of partially ordered sets:

i) We can define a partial order on any set $X$ assuming $a\preceq b$ if $a=b$ .

ii) Let $X$ be any given set. Assuming $A\preceq B$ if $A\subseteq B$ , where $A,B$ are subsets of $X$ , we obtain a partial order on the power set $2^X$ (the set of all subsets of $X$ ).

iii) Suppose that $n\preceq m$ means " $n$ divides $m$ ", where $n,m\in \mathbb{N}$ . Then $\preceq$ is a partial order on $\mathbb{N}$ .

iv) For $x,y\in \mathbb{R}$ define $x\preceq y$ if $x\le y$ ( $x$ is smaller than $y$ ). Consequently we obtain a partial order on $\mathbb{R}$ .

v) The following is a partial order on the set of all real valued functions defined on $\mathbb{R}$ :
$f\preceq g$ if $f(t)\le g(t)$ for all $t \in [a,b]$ .

vi) On $\mathbb{R}^2$ we can introduce a partial order putting $(x_1,x_2)\preceq (x_2,y_2)$ if $x_1\le x_2$ and $y_1\le y_2$ .

vii) The alphabetical order of letters $a\preceq \cdots \preceq z$ is a partial order on the English alphabet.

viii) The lexicographical order on the set of English words is a partial order.

Remark 2. Not every relation may be a partial order.

Example 2. On the set of positive integers assume $n\preceq m$ if any prime divisor of $n$ also divides $m$ . Such a relation is not a partial order, since it is not anti-symmetric.
Indeed we have true expressions $4\preceq 8$ and $8\preceq 4$ . However they do not imply $4=8$ at all. Reflexivity and transitivity of this relation are obvious.

Definition 2. Assume that $X$ is a partially ordered set and $A\subseteq X$ . We call an element

i) $b\in A$ the largest (greatest) in $A$ , if $x \preceq b$ for every $x\in A$ ( $b$ is greater than any other element of $A$ ).

ii) $a\in A$ the smallest (least) in $A$ , if $a \preceq x$ for every $x\in A$ ( $a$ is smaller than any other element of $A$ ).

iii) $b\in A$ a maximal in $A$ if $x\in A$ and $b \preceq x$ gives $x=b$ (there is no element of $A$ greater than $b$ ).

iv) $a\in A$ a minimal in $A$ if $x\in A$ and $x \preceq a$ gives $x=a$ (there is no element of $A$ smaller than $a$ ).

Picture 1

Remark 3. The words "largest" and "maximal" seem to us to be synonyms. Accordingly, we identify the words "smallest" and "minimal". Such an identification, of course, is legal on the set of real numbers, for instance. But in general, there is subtle difference between these concepts. To clarify them let's consider the following interesting example (see also https://bit.ly/36v3ns8).

Example 3. Let $X$ be a set of boxes. In $X$ introduce the partial order accepting $x\preceq y$ if a box $y$ contains a box $x$ or if they are the same box. Then under the largest box we should mean the box which contains all other boxes, but to be maximal means the property of a box when it can not be contained in any other box.

Picture 2

Case 1. Assume now that $X$ consists of 3 elements (boxes) as shown in Picture 1. Obviously, the red box is the largest element and the green box is the smallest element.

Case 2. As another case consider $X$ consisting of 4 elements (boxes) as in Picture 2. Clearly, there is no largest element. Red and purple boxes are maximal elements. There is no smallest element. Green and purple boxes are minimal elements.

Case 3. Finally, let $X$ be consisting of 4 elements (boxes) (Picture 3). There is no largest element. Red and purple boxes are maximal elements. The green box is the smallest element.

Picture 3

Example 4. On $\mathbb{R}^2$ introduce a partial order $(x_1,x_2)\preceq (x_2,y_2)$ if $x_1\le x_2$ and $y_1\le y_2$ . Consider the subset $A$ of $\mathbb{R}^2$ defined by the inequalities $0\le x\le 1$ and $0\le y \le 1-x$ . Then observe that $(0,0)$ is the smallest element in $A$ , but $A$ has no largest element. There is an infinite family of maximal elements $(x,1-x)$ in $A$ but no distinct pair of them is comparable.

Remark 4. As follows from the definitions and examples above it is quite possible to have more than one maximal (respectively minimal) element of a set, even though the largest (respectively the smallest) element doesn't exist. By the property of anti-symmetry the largest and the smallest elements of a set are both unique (if they exist, of course). Moreover, the largest (respectively, the smallest) element is maximal (respectively, minimal). The maximal element is not always the largest. Likewise, the minimal element need not to be the smallest.

2 Linearly ordered sets. Well ordered sets

Finally, recall some useful definitions.

Definition 3. A partially ordered set $X$ is said to be linearly ordered if any two elements $x,y\in X$ are comparable under $\preceq$ , in other words, one of the following conditions holds: either $x\preceq y$ or $y\preceq x$ .

For example, the set of reals $\mathbb{R}$ is linearly ordered under the partial order $\le$ on $\mathbb{R}$ .

Example 5. Are the sets considered in Example 1 linearly ordered?

i) No. Any two distinct elements are not comparable.

ii) No. Consider $X=\{a,b,c\}$ with subsets $\{a,b\}$ and $\{b,c\}$ .

iii) No. It suffices to take elements $4$ and $5$ in $\mathbb{N}$ .

iv) Yes. Any two elements of $\mathbb{R}$ are comparable under the partial order $\le$ .

v) No. Take functions $x^2$ and $2-x^2$ defined on the interval $[0, 2]$ .

vi) No. It suffices to take elements $(1,0)$ and $(0,1)$ in $\mathbb{R}^2$ .

vii) Yes. The alphabetical order is linear.

viii) Yes. The lexicographical order is linear as well.

Example 6. As you can observe from Example 3 the set in case 1 is linearly ordered. But the sets in cases 2 and 3 are not.

Definition 4. A partially ordered set $X$ is said to be well ordered if every nonempty subset of $X$ has the least element.

Example 7. $\mathbb{N}$ is well ordered under the usual order $\le$ introduced in $\mathbb{R}$ (this fact we will prove in the future).

Tags: linearly ordered set, partially ordered set, product set, well ordered set

Oct 20

People need real knowledge

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People need real knowledge

Traffic analysis

The number of visits to my website has exceeded 206,000. This number depends on what counts as a visit. An external counter, visible to everyone, writes cookies to the reader's computer and counts many visits from one reader as one. The number of individual readers has reached 23,000. The external counter does not give any more statistics. I will give all the numbers from the internal counter, which is visible only to the site owner.

I have a high percentage of complex content. After reading one post, the reader finds that the answer he is looking for depends on the preliminary material. He starts digging it and then has to go deeper and deeper. Hence the number 206,000, that is, one reader visits the site on average 9 times on different days. Sometimes a visitor from one post goes to another by link on the same day. Hence another figure: 310,000 readings.

I originally wrote simple things about basic statistics. Then I began to write accompanying materials for each advanced course that I taught at Kazakh-British Technical University (KBTU). The shift in the number and level of readership shows that people need deep knowledge, not bait for one-day moths.

For example, my simple post on basic statistics was read 2,300 times. In comparison, the more complex post on the Cobb-Douglas function has been read 7,100 times. This function is widely used in economics to model consumer preferences (utility function) and producer capabilities (production function). In all textbooks it is taught using two-dimensional graphs, as P. Samuelson proposed 85 years ago. In fact, two-dimensional graphs are obtained by projection of a three-dimensional graph, which I show, making everything clear and obvious.

The answer to one of the University of London (UoL) exam problems attracted 14,300 readers. It is so complicated that I split the answer into two parts, and there are links to additional material. On the UoL exam, students have to solve this problem in 20-30 minutes, which even I would not be able to do.

Why my site is unique

My site is unique in several ways. Firstly, I tell the truth about the AP Statistics books. This is a basic statistics course for those who need to interpret tables, graphs and simple statistics. If you have a head on your shoulders, and not a Google search engine, all you need to do is read a small book and look at the solutions. I praise one such book in my reviews. You don't need to attend a two-semester course and read an 800-page book. Moreover, one doesn't need 140 high-quality color photographs that have nothing to do with science and double the price of a book.

Many AP Statistics consumers (that's right, consumers, not students) believe that learning should be fun. Such people are attracted by a book with anecdotes that have no relation to statistics or the life of scientists. In the West, everyone depends on each other, and therefore all the reviews are written in a superlative degree and streamlined. Thank God, I do not depend on the Western labor market, and therefore I tell the truth. Part of my criticism, including the statistics textbook selected for the program "100 Textbooks" of the Ministry of Education and Science of Kazakhstan (MES), is on Facebook.

Secondly, I have the world's only online, free, complete matrix algebra tutorial with all the proofs. Free courses on Udemy, Coursera and edX are not far from AP Statistics in terms of level. Courses at MIT and Khan Academy are also simpler than mine, but have the advantage of being given in video format.

The third distinctive feature is that I help UoL students. It is a huge organization spanning 17 universities and colleges in the UK and with many branches in other parts of the world. The Economics program was developed by the London School of Economics (LSE), one of the world's leading universities.

The problem with LSE courses is that they are very difficult. After the exams, LSE puts out short recommendations on the Internet for solving problems like: here you need to use such and such a theory and such and such an idea. Complete solutions are not given for two reasons: they do not want to help future examinees and sometimes their problems or solutions contain errors (who does not make errors?). But they also delete short recommendations after a year. My site is the only place in the world where there are complete solutions to the most difficult problems of the last few years. It is not for nothing that the solution to one problem noted above attracted 14,000 visits.

Fourthly, my site is unique in terms of the variety of material: statistics, econometrics, algebra, optimization, and finance.

The average number of visits is about 100 per day. When it's time for students to take exams, it jumps to 1-2 thousand. The total amount of materials created in 5 years is equivalent to 5 textbooks. It takes from 2 hours to one day to create one post, depending on the level. After I published this analysis of the site traffic on Facebook, my colleague Nurlan Abiev decided to write posts for the site. I pay for the domain myself, $186 per year. It would be nice to make the site accessible to students and schoolchildren of Kazakhstan, but I don't have time to translate from English.

Once I was looking at the requirements of the MES for approval of electronic textbooks. They want several copies of printouts of all (!) materials and a solid payment for the examination of the site. As a result, all my efforts to create and maintain the site so far have been a personal initiative that does not have any support from the MES and its Committee on Science.

Tags: AP Statistics, EC2020 Elements of econometrics, matrix algebra, optimization, quantitative finance, Raising the Bar, University of London International Programmes, uol affiliate centre, UoL International Programmes

Oct 20

Equivalence Relations

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Equivalence Relations

We welcome our new author Nurlan Abiev. He intends to cover topics in Mathematics.

Under a set we mean an undefined term thought intuitively as a collection of objects of common property.

Definition 1. Let $A$ and $B$ be sets.
i) $B$ is said to be a subset of $A$ if each element of $B$ belongs to $A$ . Usually, this fact we denote by $B \subseteq A$ . Likewise, $A\subseteq A$ for every set $A$ .

ii) $A$ and $B$ are equal if they consist of the same elements. We say that $B$ is a proper subset of $A$ , if $B \subseteq A$ and $B \ne A$ . Consequently, we accept the notation $B\subset A$ .

iii) Obviously, a set consisting of no elements we call the empty set $\emptyset$ . Moreover, for any nonempty set $A$ we accept the agreement $\emptyset\subset A$ .

Definition 2. The product set $X\times Y$ of sets $X$ and $Y$ is defined as a set of all ordered pairs $(x,y)$ , where $x\in X$ and $y\in Y$ .

Example 1. Let $X=\{a,b,c\}$ and $Y=\{0,1\}$ . Then we have $X \times Y=\{(a,0), (a,1), (b,0), (b,1), (c,0), (c,1)\}$ . Further, note that $(b,1) \in X \times Y$ , but $(1,b) \notin X \times Y$ .

Definition 3. Any subset $R$ of $X \times Y$ we call a relation between $X$ and $Y$ , in other words $R \subseteq X\times Y$ .

Furthermore, we say that $x$ is in the relation $R$ to $y$ if $(x,y)\in R$ , and denote this fact by $xRy$ . Respectively, the negation $\overline{xRy}$ means $(x,y)\notin R$ .

Definition 4. An equivalence relation on a set $X$ is a relation $R \subseteq X\times X$ satisfying the following conditions for arbitrary $x,y,z \in X$ :
i) $xRx$ (reflexivity);
ii) If $xRy$ then $yRx$ (symmetry);
iii) If $xRy$ and $yRz$ then $xRz$ (transitivity).

In cases when $R$ is an equivalence relation we will use the notation $x\sim y$ instead of $xRy$ .

Example 2. Assume that $X$ is the set of all people in the world. Consider some relations on $X$ .

i) Descendant relation. A relation $x\sim y$ if $x$ is a descendant of $y$ is transitive, but not reflexive nor symmetric.

ii) Blood relation. A relation $x\sim y$ if $x$ has an ancestor who is also an ancestor of $y$ is reflexive and symmetric, but not transitive (I am not in a blood relation to my wife, although our children are).

iii) Sibling relation. Assuming $x\sim y$ if $x$ and $y$ have the same parents we define an equivalence relation.

Recall that $\mathbb{R}$ and $\mathbb{Z}$ are the sets of reals and integers respectively.

Example 3. Examples of equivalence (non equivalence) relations:

i) For $x,y\in \mathbb{R}$ define a relation $x\sim y$ if $x-y\in \mathbb{Z}$ . This is an equivalence relation on $\mathbb{R}$ .

ii) Let $x\sim y$ if $x\le y$ . This relation is not symmetric.

iii) For $x,y\in \mathbb{Z}$ define $x\sim y$ if $x-y$ is even. Consequently we obtain an equivalence relation on $\mathbb{Z}$ .

iv) A relation $x\sim y$ if $x-y$ is odd can not be an equivalence relation on $\mathbb{Z}$ because it is not reflexive and not transitive.

v) Let $x,y\in \mathbb{R}$ . Clearly, $x\sim y$ if $|x-y|\le 1$ is not an equivalence relation since it is not transitive.

vi) Any function $f\colon X\rightarrow Y$ induces an equivalence relation on $X$ setting $x\sim y$ if $f(x)=f(y)$ .

Definition 5. Assume that $R$ is an equivalence relation on a given set $X$ . We say that $R_a:=\left\{x\in X \; | ~\; xRa \right\}$ is the equivalence class determined by $a\in X$ under the equivalence relation $R$ .

The set $\{R_a~|~ a\in X\}$ of all equivalence classes we call the quotient set of $X$ over $R$ , in symbols $X/ R$ .

Theorem 1. Every equivalence relation on a given set provides a partition of this set to a disjoint union of equivalence classes, in other words, if $R$ is an equivalence relation on $X$ then
i) $X=\bigcup_{x\in X}R_x$ ;
ii) $aRb \Rightarrow R_a=R_b$ ;
iii) $\overline{aRb} ~\Rightarrow~ R_a\cap R_b=\emptyset$ .

Thus any two equivalence classes coincide if they admit a nonempty intersection.

Proof.
i) By reflexivity $x\in R_x$ for every $x\in X$ . So $X= \bigcup_{x\in X}R_x$ .

ii) Let $x\in R_a$ . Then $xRa$ and $aRb$ imply $xRb \:\Leftrightarrow\: x\in R_b$ , showing $R_a\subseteq R_b$ .
By symmetry we also have $bRa$ . Then $R_b \subseteq R_a$ analogously. Therefore, $R_a=R_b$ .

iii) Suppose that $R_a \cap R_b \ne \emptyset$ , and let $x$ be their common element. According to symmetry and transitivity it follows then $xRa$ and $xRb$ , which implies $aRb$ .
Theorem 1 is proved.

The converse of Theorem 1 is also true.

Theorem 2. Every partition of a set $X$ to a disjoint union of its subsets defines on $X$ an equivalence relation $x\sim y$ if $x$ and $y$ belong to the same subset of $X$ .

Tags: equivalence relation, power set, product set, set

Apr 20

Microsoft Teams is a jail

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Microsoft Teams is a jail

With the quarantine due to COVID-19, my universities had to switch to online teaching. One uses Google Classroom, the other Microsoft Teams. I have something to compare and I am enraged with the Teams. I wanted to post this review on the Internet but could not find a good place. All sites are blowing their trumpets to Microsoft:

https://www.pcmag.com/reviews/microsoft-teams

https://www.trustradius.com/products/microsoft-teams/reviews

https://www.techradar.com/reviews/microsoft-teams

Firstly, judging by their terminology, Teams has been developed for enterprises. I have been using it for one month and I still don't know what a channel means. Is it a communication channel? If yes, why don't you call it email or chat? Is it a combination of activities, like a class? Then why not name it a class?

The first time you look for it, they suggest you to download Teams. If you do, it will constantly load on Windows startup without even asking you. Hell, I don't use it every day and I don't want it to take up my computer's resources. Then I learned that there is a browser version of Teams. Even then there are features I don't like. With Google, you log into your email account. Then every Google app that you want to use knows that it is you and the apps you need are constantly open. With Teams, you have to sign into Teams first and then use everything else. I want to use the Teams app (for seeing team membership), OneDrive (for reading what I have uploaded) and Assignments app at the same time. But the damn program periodically logs me out.

The learning curve is steep. On the toolbar there is a chat button. To start a message, you need to type @ first. If my colleague didn't tell me that, I would to have to read the online intro. Sorry, don't have time for that. There is a profusion of apps you can use with Teams. Each time you have to try before you understand what it is good for. Descriptions are short, just one sentence. Again, don't have time for that.

All of this is just minor growl. Here is the main problem. Teams takes control of my computer! Next time I start it, I see a prompt to log in! This is my home computer, there are no secrets and I hate typing a password and even a pin. So I log in, then look on the Internet for ways to get rid of the login prompt. Most of those advices are outdated and the ones that work require me to dance around a lot. The menu in Windows 10 settings has changed, many options have been removed or renamed.

Moreover, Teams thinks that my computer belongs to my school! I am working typing something and all of a sudden my computer shuts down and restarts. The message I see is that my computer was restarted following the policies set by my school!

Of course, none of this happens with Google Classroom. This is where I am now. I want to remove the Microsoft account from my computer and I want to use it only in the browser, like with Google. One site says I have to find it in the settings and press the Remove button. There is no such a button! There is the Manage button which takes you to the Internet. There you can say Remove but then all your tabs in the browser related to Teams log you out. I deeply regret the day I started Teams. It behaves like one of those invasive species that wreak havoc around the world.

Update. I posted the above in April 2020. Hoping that by August 2020 it has become better I installed Teams again. Same story: Teams turned my home computer into a corporate one. The interface is as crazy as it was. I want to get a list of my team members in the Excel format on my computer. Seemingly Microsoft has powerful means to give me such an opportunity: Office 365 and OneNote. But no, you have to install PowerShell and work with scripts. I am an advanced user but I am horrified by the description of the procedure.

Update on August 11, 2022. I wanted to schedule a meeting in personal Teams. The meeting link does not work. The advice on the Internet suggests to delete Teams cache located somewhere in the folder AppData. There is no teams folder! Well, maybe they moved the cache location, so I just reinstalled the program to avoid searching for it. Again, the meeting link does not work! Goodbye Teams, hello Google Meet!

Tags: Microsoft Teams is a jail, Review Microsoft Teams

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