# Sergei Yakovenko's blog: on Math and Teaching

## Monday, January 29, 2018

### 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!

## Tuesday, March 28, 2017

Filed under: conference,schedule — Sergei Yakovenko @ 5:22
Tags: ,

# Galois Meets Newton: Algebraic and Geometric aspects of Singularity Theory

## Celebrating the 70th birthday of Prof. Askold Khovanskii

A terrific conference will be held in the Weizmann Institute in July 3-7, 2017. For details go to the conference site and register there if you plan to attend the event.

## Sunday, February 19, 2017

### Volunteers to moonlight, any?

Filed under: research announcement — Sergei Yakovenko @ 5:36
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## Israeli Ministry of Education Needs You!

As part of the matriculation exam in math (5th level), Israeli high school students are (sometimes?) required to write a “final project“, עבודת גמר.

They are looking for people qualified to read, check these compilations and grade them based on certain criteria. In particular, this presumes a 45 min meeting with the students. Each case will bring you 600 shekels. The required qualification allows to hire PhD students (and faculty members, of course ;-). Needless to say, everything is in Hebrew.

If you are interested in such a job, please write me or (better) directly to Dr. Ilya Tyomkin from the Ben Gurion University, who is overseeing this program.

### Lecture notes

Filed under: Calculus on manifolds course — Sergei Yakovenko @ 4:57
Tags: , ,

## Lecture notes for the course

The set of notes, including extra bibliography and the exam problems, is available here.

These are very raw, extremely informally written and mostly very sketchy notes, consume with moderation at your own risk. Perhaps, one day they will be turned into something more reliable and close to the standards.

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