# Sergei Yakovenko's blog: on Math and Teaching

## Monday, January 29, 2018

### Exam

Filed under: lecture,Rothschild course "Analysis for high school teachers" — Sergei Yakovenko @ 5:05
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## Exam

The exam is posted online on Jan 30, 2017, and must be submitted on March 16, 2018 (the first day of the new semester).

Its goals are, besides testing your newly acquired skills in the Analysis, to teach you a few extra things and check your ability for logical reasoning, not your proficiency in performing long computations.
If you find yourself mired in heavy computations, double check whether you understand the formulation of the problem correctly. Remember, small details sometimes matter!

Please provide argumentation, better in the form of logical formulas, not forgetting explicit or implicit quantifiers. They really may change the meaning of what you write!

Problems are subdivided into items. The order of these items is by far non-random, you have to solve them from the first through the last, (solution of one item may be a building block for the next one). A complete solution of a problem is a proof of some important theorem in Analysis, so you will be discovering these results on your own. The Remarks will help you to place the freshly proved theorem on the general picture.

To get the maximal grade 100, it is not necessary to solve all problems. Problems are of varying length, variable complexity, various level of abstraction. No apriori points are assigned for solution of each problem, no summation at the end. You can get extra points for short and elegant argument or have some points removed for writing an obviously stupid things (honest errors will simply bring you zero points). You have all the time, try to solve as many problems as you can, we will appreciate and assess the results as objectively and honestly as possible.

You are assumed to work individually, which is, of course, impossible to verify, but please in any case avoid submitting isomorphic solutions: this is a bad taste for take-home exams.

You are absolutely free to write in English (easier for me) or Hebrew, submit handwritten pages or compuscripts, in a hard copy or by email (even scans will work). I we will encounter difficulties reading your submission, we’ll let you know.

If you believe you found an error or crucial omission in the formulation of a problem, please write me. If this will be indeed the case (errare humanum est), the problem will be either edited (in case of minor omissions) or cancelled (on my account). In case of any doubt don’t hesitate to leave your questions as talkbacks to this post, so that other people will be able to follow. Asking questions is never penalized!

Good luck to everybody!

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### Lectures 11-13 (Jan 16, 23, 30)

Filed under: lecture,Rothschild course "Analysis for high school teachers" — Sergei Yakovenko @ 4:53

## Infinite series

These three lectures were devoted to the general theory of numeric series and the issues of their convergence. Then we switched to studying power series (both formal and convergent). Finally, we used the convergent power series to move into the complex domain and discover fascinating phenomena there.

There two separate sets of lecture notes, on infinite series and operations on them and on functions of complex variable.

## Integration and antiderivation

Again, no new notes were prepared this time.

## Linear algebra and differentiable maps

The revised notes on the linear algebra are available here.

No new notes concerning differentiability are added this year. The only thing you need to understand about it is the idea that linear (affine!) approximations, if available, carry an important bit of information about maps that admit such approximations. Use the notes from the past years.

## Continuity and limits

In these four lectures (sorry for the delay with posting the notes) we have introduced and discussed the notions of continuous functions. Contrary to the usual practice, we switch immediately to the case of functions of several variables, where pictures are much more illustrative.

We discuss the first topological notions: open/closed sets, accumulation/interior points, limits of functions as a way to extend functions continuously beyond the “natural” way of their definition by formulas.

Then we pass to more involved notions as compactness, connectivity (in two flavors) and finally end up by our first really nontrivial topological result, the fixed point theorem for 2-dimensional disk.

The (preliminary) lecture notes are available here: please note that there are over 30 problems approximately of the same sort that will appear on the exam. A more extended version will appear later, besides, you are always invited to recycle the lecture notes from the past years, available from this blog.

## Real numbers as solutions to infinite systems of equalities

In the past we already extended our number system by adding “missing” elements which are assumed to satisfy certain equations, based only on knowing what these equations are. It turns out that we may extend the set of rational numbers $\mathbb Q$ to a much larger set of real numbers $\mathbb R$ by adding solutions to (infinite numbers) of inequalities. As before, the properties of these new numbers could be derived only from the properties of inequalities between the rational numbers.

On one leg, the idea can be explained as follows. Since for any two rational numbers $r,s\in\mathbb Q$ one and only one relation out of three is possible, $r$ < $s$, $r=s$ or $r$ > $s$, we can uniquely define any, say, positive rational unknown number $x$ by looking at the two sets, $L=\{l\in\mathbb Q: 0\le l\le x\}$ and $R=\{r\in\mathbb Q: x\le r\}$. (You don’t have to be too smart at this moment: $x$ is the only element in the intersection $L\cap R$ 😉

However, sometimes the analogous construction leads to problems. For instance, if $L=\{l\in\mathbb Q: l\ge 0, l^2\le 2\}$ and $R=\{r\in\mathbb Q: r^2\ge 2\}$, then $L\cap R=\varnothing$, since the square root of two is not a rational number, but $L\cup R=\mathbb Q_+$, i.e., for any positive rational number we can say whether is smaller or larger the missing number $\sqrt 2$. This allows to derive all properties of $\sqrt 2$, including its approximation with any number of digits.

Proceeding this way, we introduce (positive) real numbers by indication, what is their relative position to all rational numbers. This allows to describe the real numbers completely.

The details can be found here.

## A didactic digression

Some of you complained about insufficient number of problems that are discussed during the tutorials. Everybody knows that problems and questions for self-control are the most important elements of study mathematics, especially in comparison with other disciplines. The rationale behind is the assumption that a student who understands the subject, should be able to answer these questions immediately or after some reflection. Composing such problems is an easy thing: you any mathematical argument you can stop for a second and ask yourself: “why I can do as explained?” or “under what conditions are my actions justified?”. In the lecture notes (see the link above) tens of such problems are explicitly formulated. Similar problems will await you on the exam.

However, remember one simple thing. If you already know how to solve a problem, this is not a problem but rather a job. Unless you solve these problems yourselves, there is no sense in memorizing their solutions: knowing solution of one such problem won’t help you with solving another problem unless you really understand what’s going on. There are no “typical problems”: each one of them is of its own sort, though, of course, some problems can be solved by similar methods.

A practical advice: you should not expect that all problems that appear on the exam will be discussed at length at the tutorials. There are no ready recipes to memorize. Only to understand honestly. Believe me, this is easier than memorize by heart endless formulas and algorithms.

## Numbers

The basic set theory allows us to construct a set $\mathbb N=\{|,||,|||,||||,|||||,\dots\}$ with a function “next”, denoted by $\mathrm{Succ}:\mathbb N\to\mathbb N\smallsetminus\{|\}$, which is bijective. This set describes the process of counting objects and is the most basic structure. Starting from a distinguished element denoted by 1, we construct an infinite number of elements $2=\mathrm{Succ}(1),\ 3=\mathrm{Succ}(2),\ 4=\mathrm{Succ}(3)$ etc. There are two axioms guaranteeing that the set $\mathbb N$ indeed coincides with what we call the set of natural numbers:

1. $\forall x\in\mathbb N\ \mathrm{Succ}(x)\ne 1$
2. Any element $x\in\mathbb N$ is obtained by the iteration of $\mathrm{Succ}$: $x=(\mathrm{Succ}\circ\cdots\circ\mathrm{Succ})(1)$.

Using this function and its partial inverse  one can define on $\mathbb N$ the order and the operations of addition (as repeated addition of 1 which is just evaluation of $\mathrm{Succ}$) and multiplication (repeated addition).

However, not all equations of the form $x+a=b$ or $x\cdot a=b$ are solvable. One can enlarge $\mathbb N$ by adding solutions of all such equations, obtaining the set of integer numbers $\mathbb Z$ which is a commutative group with respect to the operation of addition, and finally the set of rational numbers $\mathbb Q$ in which division is available by any nonzero number.

Division by zero is impossible: if we add “solution of the equation $0\cdot x=1$” as a new imaginary element, then we will not be able to do some arithmetic operations on it. Still, if we are ready to pay this price, then the rational numbers can be extended by a new element so that, say, the function $f(x)=1/x$ would be everywhere defined and continuous.

Details are available in the lecture notes here.

# שלום כיטה א!

The main feature that distinguishes the Calculus (or Mathematical Analysis) from other branches of mathematics is the repeated use of infinite constructions and processes. Without infinity even the simplest things, like the decimal representation of the simple fraction $\frac13=0.333333\dots$ becomes problematic.

Yet to deal with infinity and infinite constructions, we need to make precise our language, based on the notions of sets and functions (maps, applications, – all these words are synonymous).

Look at the first section of the lecture notes here.

You are most welcome to start discussions in the comments to this (or any other) post. Don’t be afraid of asking questions that may look stupid: this never harms! Write in any language (besides Hebrew/English, I hope that Ghadeer will take care of questions in Arabic, and I promise to deal with French/Spanish/Catalan/Italian/Russian/Ukrainian questions) 😉 Subscribe for updates on this site with your usual emails, to be independent from any dependence 😉

Looking forward for a mutually beneficial interaction in the new semester!

## Monday, February 1, 2016

### Finally, exam!

Filed under: lecture,Rothschild course "Analysis for high school teachers" — Sergei Yakovenko @ 3:41
Tags:

# Exam

The exam is posted online on Feb 1, 2016, and must be submitted on the last day of the exams’ period, February 26. Its goals are, besides testing your acquired skills in the Analysis, to teach you a few extra things and see your ability for logical reasoning, not your proficiency in performing long computations. If you find yourself involved in heavy computations, better double check whether you understand the formulation of the problem correctly. Remember, small details sometimes matter!

Please provide argumentation, better in the form of logical formulas, not forgetting explicit or implicit quantifiers $\forall$ and $\exists$. They really may change the meaning of what you write!

Problems are often subdivided into items. The order of these items is not accidental, try to solve them from the first till the last, and not in a random order (solution of one item may be a building block for the next one).

To get the maximal grade, it is not necessary to solve all problems, but it is imperative not to write stupid things. Please don’t try to shoot in the air.

The English version is the authoritative source, but if somebody translates it into Hebrew (for the sake of the rest of you) and send me the translation, I will post it for your convenience, but responsibility will be largely with the translator.

If you believe you found an error or crucial omission in the formulation of a problem, please write me. If this will be indeed the case (errare humanum est), the problem will be either edited (in case of minor omissions) or cancelled (on my account).

That’s all, folks!© Good luck to everybody!

Yes, and feel free to leave your questions/talkbacks here, whether addressed to Michal/Boaz/me or to yourself, if you feel you want to ask a relevant question.

# Corrections

## Correction 1

The formulation of Problem 1 was indeed incorrect. The set $A'$ was intended to be the set of accumulation points for a set $A\subseteq [0,1]$. The formal definition is as follows.

Definition. A point $p\in [0,1]$ belongs to to the set of limit points $A'$ if and only if $\forall\varepsilon$>0 the intersection $(p-\varepsilon,p+\varepsilon)\cap A$ is infinite. The point $p$ itself may be or may be not in $A$.

Isolated points of $A$ are never in $A'$, but $A'$ may contain points $p\notin A$.

Apologies for the hasty formulation.

## Correction 2: Problem 3(b) cancelled!

The statement requested to prove in Problem 3(b) is wrong, and I am impressed how fast did you discover that. Actually, the problem was taken from the textbook by Zorich, vol. 1, where it appears on p. 169, sec. 4.2.3, as Problem 4.

The assertion about existence of the common fixed point of two commuting continuous functions $f,g\colon [0,1]\to[0,1]$ becomes true if we require these functions to be continuously differentiable on $[0,1]$ (in particular, for polynomials), but the proof of this fact is too difficult to be suggested as a problem for the exam.

Thus Problem 3(b) is cancelled.

# Questions and answers

Questions concerned integrability of discontinuous functions, notions of improper integrals (how and when they can be defined), topological properties (equivalent definitions of compactness, connectedness etc.)

Here are some textbooks that I recommend for preparing when working on the exam. Keep them on your virtual bookshelf: they cover much more that I explained in the course, but who knows what questions related to analysis you might have.

1. V. Zorich, vol. 1: Chapters 1-6, pp.1-371.
2. V. Zorich, vol. 2: Parts of Chapter 9 (continuous maps) and the first part of Chapter 18 on Fourier series.
3. W. Rudin: Chapters 1-6, pp.1-165.

The problems for exam will be posted on February 1st (at least the English version).

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