Monday, March 25, 2024

Analysis for High School Teachers 2023/2024: Exam

Filed under: lecture,Rothschild course "Analysis for high school teachers" — Sergei Yakovenko @ 6:35
Tags: exam

The exam problems accompanied by some auxiliary matter (necessary definitions, remarks, hints) is available by this link.

It is a take-home exam and you have one month to solve the problems and submit solutions. More details in the file. If you have any questions, please leave them as comments below or mail to Peleg/Sergei.

Good luck and חג פורים שמח

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Saturday, February 10, 2024

Intermezzo (summary of several lectures)

Filed under: Rothschild course "Analysis for high school teachers" — Sergei Yakovenko @ 5:08
Tags: contiuuity, derivative, differentiability, differential

For various reasons I failed to publish the summaries of the lessons for the few past weeks. In these lessons we mainly addressed the idea of continuity of a function of one or several real variables (both globally on the entire domain of its definition $A\subseteq\mathbb R^n$ and locally near a point $a\in A$ . The intuitive notion of continuity is very simple: the images $f(a),f(b)\in\mathbb R^m$ of two close points $a,b\in A\subseteq\mathbb R^n$ in the domain are close to each other in the target space of a function $f:A\to\mathbb R^m$ . Yet this formulation lacks quantifiers and a careful look at how they can be placed reveals several close but not identical definitions (e.g., continuity at every point of the domain is weaker than the uniform continuity on the domain).

Having spelled out that, we can address the notion of a limit which traditionally precedes the notion of a continuity and is believed to be very technical. Yet in practice existence of a limit (say, of a function $f:A\to\mathbb Rm$ at a point $a\notin A$ ) is nothing more than the possibility of extending the “natural” domain of this function (given, say, by an explicit function) by assigning the value $f(a)$ so that the extended function retains continuity on the larger set $A\cup\{a\}$ . Usually such procedure is used to “avoid” division by zero in the expressions like $f(x,y)=\displaystyle \frac{p(x)-p(y)}{x-y}$ on the diagonal $x=y$ or $g(x)=\displaystyle \frac{\sin x}{x}$ at the origin, that will appear when introducing derivatives later.

Continuity of functions is a rather strong property, especially when their domains of continuity possess certain properties. Yet to unleash the full power, it is convenient to raise the abstraction level one step more and talk about general topological spaces, not just the Euclidean spaces $\mathbb R^n$ or their proper subsets. This is where the topology with its specific language comes to help and one can introduce and study various properties of continuous functions in the general context. Thus, for instance, one can describe “bounded” (compact) sets in absence of any distance function, and prove that this class is closed by actions of continuous functions (maps). In a similar way one can formalize the notion of a connected set (meaning that this set consists of a “single piece”) and show that continuous functions cannot destroy continuity by “tearing apart” such sets. The proofs in this “abstract context” are very simple, often one-liners and may seem to be a shallow playing with abstract words, but it was a cautious crafting of numerous definitions that made the proofs easy, and in practical applications one needs to verify that all functions and sets satisfy exactly the appropriate definition. Henry Poincarè once quipped, that any mathematical truth is born as a paradox and ends up as a triviality.

After playing around with continuity we shifted to the study of differentiability. Note that the notion of continuity of a function $f(x)$ at a point $a$ can be interpreted as a fact that this function can be reasonably approximated by the constant function $c(x)\equiv c= f(a)\in\mathbb R^m$ : the closer a point $x$ approaches $a$ , the smaller is the error of the approximation $|f(x)-c(x)|=|f(x)-f(a)|$ . Of course, this completely transparent phrase of a human language requires quantifiers to measure “proximity” and “smallness” and how one implies the other.

Constant functions (i.e., real numbers or vectors in the case of functions of several variables) form a very simple class of functions, yet they can study some properties of the continuous functions. One is naturally tempted to look for larger classes of functions which will be on one hand easy to operate with, on the other hand would give a finer and more detailed information about functions that admit approximation by these functions.

This class is called affine (sometimes linear, see below) functions. The origin of this notion is in the Algebra, more precisely, in the so called Linear Algebra. A linear (or vector) space over a field $\mathbb R$ or $\mathbb C$ is a set $V$ equipped with two operations, the vector sum/difference $\pm$ , a binary commutative invertible operation (making $V$ into a commutative group) and the multiplication by the scalars (numbers) from the field. Of course, the distributive and associative laws are assumed. The standard examples are (arithmetic) vector spaces $\mathbb R^n,\ n=\dim V\geqslant 1$ .

A function $L:V\to W$ between two (in general, different) vector spaces is called linear, if it “respects” both linear operations. In particular, it must map the zero vector of $V$ to the zero vector of $W$ . If $\dim V=\dim W=1$ , then all linear functions have a simplest form $L(x)=\lambda x$ for some number $\lambda\in\mathbb R$ (possibly, zero), in the general multidimensional case a linear map is determined by $mn$ real numbers which are naturally arranged in the form of a matrix, an $m\times n$ -table.

However, the class of linear maps is not sufficiently large. For instance, the shift map $T=T_c:\mathbb R^n\to\mathbb R^n, \ T_c(x)=x+c$ , is not linear ( $T(0)\ne 0$ unless $c=0$ , in which case the shift becomes a dull identical map). Thus we need to make one last step and consider the smallest class of maps that would contain all linear maps and all shifts (parallel translations) and closed by compositions. This class is called the class of affine maps: reducing similar terms, one can show that by definition, an affine map $A:\mathbb R^n\to\mathbb R^m$ is a composition, $A=TL$ , where $L$ is a linear map and $T$ a translation in the target space. Explicitly, we have $Ax=Lx+c$ (“linear non-homogeneous function”). Note that there is a well-established tradition not to enclose the argument of linear maps in the parentheses and write $Ax$ rather than $A(x)$ : this is because one can treat $Ax$ as a result of some binary operation, “matrix multiplication” of a matrix $A$ and a column vector $x$ .

How can one construct explicitly an affine approximation for a function $f:\mathbb R^n\to\mathbb R^m$ ? Let us start from the simplest case $m=n=1$ (functions of one real variable). If $f$ were already an affine function, then we would have $\forall x\in\mathbb R\quad f(x)=\lambda x +c$ for some two constants $\lambda,c\in\mathbb R$ , which could be found using the values of $f(x),f(a)$ at any two different points $x\ne a\in\mathbb R$ by the formulas

$\lambda = \displaystyle \frac{f(x)-f(a)}{x-a}, \qquad c= f(a)-\lambda a=f(x)-\lambda x.$

Note that in the affine case the values of $\lambda,c$ do not depend on the choice of the two points $x,a$ .

Proposition. Assume that for a function $f$ as above, and for some point $a\in\mathbb R$ there exists the limit $\lambda=\displaystyle\lim_{x\to a}\frac{f(x)-f(a)}{x-a}\in\mathbb R$ . Then the function $f$ admits an affine approximation by the function $A(x)=\lambda(x-a)+c, \quad c=f(a)$ , in the following sense: the relative error of approximation $E(x)=\displaystyle\frac{|f(x)-A(x)|}{|x-a|}$ tends to zero as $x\to a$ , $\displaystyle\lim_{x\to a}E(x)=0$ .

Definition. The function $f$ is said to be differentiable at the point $a$ . The real number $\lambda$ is called the derivative of $f$ at $a$ , and the linear map $L(v)=\lambda v$ the differential of $f$ at $a$ .

Note that we on purpose denoted the argument of the differential by a new letter $v$ . If $f$ is differentiable at every point $a$ of its domain, then the derivative $\lambda$ and the differential will depend explicitly on $a$ . Thus from a given function $f(x)$ we have “derived” the derivative function (usually denoted by $a\mapsto f'(a)$ ), while the differential will become a function of two independent arguments $a,v$ (usually denoted by $df(a)v$ without extra braces, or simply $df$ , omitting both arguments.

There are no general reasons for functions to be differentiable, the more so be differentiable at all points of their domain of definition: it is a strong condition. In some sense, “most” functions are nowhere differentiable. For instance, the stock exchange rates give an example of such functions, and in fact measurements of almost every microscopic physical quantities also behave like that.

Yet in an absolutely surprising way, the functions most important for description of our World, turn out to be differentiable everywhere or almost everywhere. Polynomials, algebraic, trigonometric, exponential functions, … all are differentiable, possibly except for some singular points. Moreover, the Laws of Nature, as discovered by Newton, are written in the language of Differential Equations, identities connecting unknown functions and their derivatives. And even the equations of the General Relativity have the form of differential equations, though these equations are partial (i. e., involve partial derivatives of the unknown functions). Yet this subject is way aside from our main line of exposition.

Supplementary material

Here are links to the transparencies from the several past lectures.

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Monday, January 15, 2024

Lectures 5-6, Jan 9-16,2024

Filed under: Rothschild course "Analysis for high school teachers" — Sergei Yakovenko @ 8:14
Tags: closed sets, compactness, connected set, continuity, open sets, topology

The simplifying language: Topology

Constructions involving numerous quantifiers are different to grasp. A good theory splits such constructions introducing appropriate notions and building a suitable language. Let $A\subseteq\mathbb R^n$ be a subset in the Euclidean space and $\mathrm{dist}(x,y)$ a metric on it which we (only for simplicity!) will assume translation invariant and denote $|y-x|=|x-y|$ . Everywhere below $B_r(a)$ will denote the open ball of radius $r >0$ centered at a point $a\in\mathbb R^n$ : $B_r(a)=\{x\in\mathbb R^n:|x-a|<r\}$ .

Definitions. A point $a\in A$ is called interior point, if $\exists r>0$ such that the open ball $B_r(a)=\{x\in\mathbb R^n: |x-a|<r\}\subseteq\mathbb R^n$ lies in $A$ , i.e., $B_r(a)\subseteq A$ . This is the same as saying that $\exists r>0\ \forall x\in\mathbb R^n\ |x-a|\leqslant r\implies x\in A$ .

A point $b\notin A$ is called exterior point, if it is interior for the complement $\mathbb R^n\smallsetminus A$ . The points that are neither interior nor exterior are called the boundary points of $A$ .

The sets of all interior and boundary points of $A$ are denoted $\mathrm{int}\, A$ and $\partial A$ respectively.

A point $a\in \mathbb R^n$ is called an accumulation point (for $A$ ), if $\forall \varepsilon >0\ \exists x\in A$ such that $|x-a|<\varepsilon$ . The set of all accumulation points for $A$ is called its closure and denoted by $\mathrm{clos}\,A$ . Sometimes the notation $\overline{A}$ is used for the closure.

Exercise. Prove that $\mathrm{int}\, A\cup\partial A=\mathrm{clos}\,A$ . Prove that the exterior of $A$ is $\mathbb R^n\smallsetminus \mathrm{clos}\, A$ .

Exercise. Prove that the notions of interior, exterior and boundary do not depend on the choice of the distance function in $\mathbb R^n$ .

Definitions. A subset $A\subseteq \mathbb R^n$ is called open, if it coincides with its own interior, $A=\mathrm{int}\,A$ . The subset is closed, if it coincides with its own closure $A=\mathrm{clos}\,A$ .

Exercise. Assume that $n=1$ and $A=[0,1)\subseteq\mathbb R=\{0\leqslant x<1\}$ . Describe interior, closure and boundary of this segment. Is it open? closed? neither?

Exercise. Show that the complement of an open subset is closed and vice versa, the complement of a closed subset is open. Show that Are there other subsets that are both open and closed?

Theorem 1.

$A=\mathbb R^n$ and $A=\varnothing$ are both open and closed simultaneously.
Union of any family (infinite or even uncountable) of open sets is open.
Finite intersection of open sets is open.
Intersection of any Union of any family (infinite or even uncountable) of closed sets is closed.
Finite union of closed sets is closed.

Continuity as a topological notion

Consider first the case of maps (functions) defined on the entire Euclidean space, $f:\mathbb R^n\to\mathbb R^m$ .

Temporary Defininion. A map as above will be called an O-map¹, if the preimage of any open set $V\subseteq\mathbb R^m$ is an open subset $U\subseteq\mathbb R^n$ .

Lemma 1. A map $f$ continuous at all points of $\mathbb R^n$ , is an O-map.

Proof. Consider any open set $V\subset\mathbb R^m$ and its preimage $U=f^{-1}(V)\subseteq\mathbb R^n$ . Let $a\in U$ be any point in this preimage: by definition, this means that $b=f(a)\in V$ . Since $V$ is open in $\mathbb R^m$ , there exists a ball $B_\delta(b)$ of positive radius $\delta>0$ which lies in $V$ . By continuity of $f$ at $a$ , there exists $\varepsilon >0$ such that $|x-a|<\varepsilon\implies |f(x)-b|<\delta$ , that is, all points of the ball $B_\varepsilon(a)$ are mapped inside $B_\delta(b)$ , hence inside $V$ . Therefore the preimage $U=f^{-1}(V)$ together with the point $a$ contains a small ball around $a$ , that is, $a$ is an interior point for $U$ . Since $a$ was chosen arbitrarily, this means that all points of $U$ are interior points, hence $U$ is open. Since $V$ was chosen arbitrary, we have proved that $f$ is an O-map. Q.E.D.

Lemma 2. An O-map is continuous at every point $a\in \mathbb R^n$ .

Proof. Let $a\in\mathbb R^n$ be an arbitrary point, and denote $b=f(a)\in \mathbb R^m$ . Consider an arbitrary open ball $V=B_\delta(b)$ of positive radius $\delta>0$ . To prove the continuity of $f$ , we need to find an open ball around $B_\varepsilon(a)$ such that its $f$ -image is inside $V$ . But since $V$ is open², its preimage $U=f^{-1}(V)$ is also open in $\mathbb R^n$ by the definition of an O-map applied to $f$ . The openness of $U$ means that each its point, in particular, the point $a$ , is interior for $U$ and hence the ball $B_\varepsilon(a)$ with the required property exists. Q.E.D.

These two lemmas together prove that at least for maps whose domain is the entire Euclidean space, the property “Preimages of open sets are open” (as stated in the Temporary Definition) is fully equivalent to the property of being continuous on the entire domain.

How this result should be modified for maps $f:A\to\mathbb R^m$ whose domains are only proper subsets of the Euclidean subspace, $A\subseteq \mathbb R^n$ ? The answer is simpler than you might imagine. You don’t need to modify the definition of O-maps, you need to twist the definition of open sets, making it relative to the arbitrary domain $A$ of definition of the map $f$ .

Definition. Let $A\subseteq\mathbb R^n$ be an arbitrary (not necessarily open) subset of the Euclidean space. A subset $U\subseteq A$ is called open relative to $A$ , if there exists an open (in the original sense) subset $U'\subseteq \mathbb R^n$ such that $U=U'\cap A$ .

One can immediately and easily check (by just passing from open sets to relatively open obtained by intersection with any subset $A$ , that:

Theorem 1 above remains valid in the relative sense, with the only required correction that the “absolute” $A$ should replace $\mathbb R^n$ as a set which is both relatively open and relatively closed.
The proofs of both Lemmas 1 and 2 remain literally true if we replace the (ordinary, absolute) openness by the openness relative to $A$ : indeed, for $x\notin A$ the value $f(x)$ is simply undefined hence cannot violate any inequality or inclusion.

As a result, we obtain the following reformulation of continuity for maps defined on proper subsets of the Euclidean space.

Theorem 2. A map $f:A\to \mathbb R^m$ defined on a subset $A\subseteq \mathbb R^n$ is continuous on its domain of definition, if and only if the preimage $f^{-1}(V)$ of any open subset $V\subseteq\mathbb R^m$ is an open subset relative to $A$ .

Note that this equivalent definition of continuity of a map at all points of its domain formally requires only one quantifier, assuming that the notion of an open set is sufficiently familiar to the reader: indeed, it asserts that $f:A\to\mathbb R^m$ is continuous if and only if

$\forall V\text{ open in }\mathbb R^m\quad f^{-1}(V)\text{ is relatively open in }A.$

But there is much more to gain from the topological approach.

Topological spaces

The topological language that we introduces in a very particular settings (for subsets of the Euclidean sets) actually works in a much broader context. Indeed, Theorem 1 above is a pretty good motivation for the following definition.

Definition. A topological space $X$ is an abstract set (eventually very large, much larger than subsets of $\mathbb R^n$ such that some of its subsets are distinguished by bearing a noble name of open sets $U_\alpha$ . There are only three axioms these open sets must obey:

The total space $X$ itself and the empty set $\varnothing$ are open.
Union of any number of open sets is again open.
Intersection of any finite number of open sets is open.

Note that the axioms do not specify any way concrete way how open sets should be defined in any concrete example. Only their algebraic properties in the Boolean algebra are important. This is dangerous (examples may challenge our intuition) but provides great versatility. In particular, Theorem 2 above allows to define the continuity for any map $f:X\to Y$ between any two topological spaces, with an immediate trivial corollary that composition of any two continuous maps (when defined) will again be continuous. This becomes a trivial observation (why?), although the proof in the “classical” case is also very easy.

What we (on our rather down-to-earth) level can gain from so abstract constructions? Quite a lot, even if we consider only topological spaces embedded in $\mathbb R^n$ with the supply of open spaces through the definition of relative openness.

Connected spaces

Using only topological terms, we can formulate one of the most basic properties of sets, the fact that they do not fall apart as unions of smaller sets. It is instrumental in the study: if something is built from smaller components that do not interact with each other, then one can study these components separately and then “mechanically” bring the results together.

Definition. A topological space $X$ is called disconnected (or disconnect), if it can be represented as a disjoint union of two open sets, $X=U\cup V$ with $U\cap V=\varnothing$ . If such representation is impossible, we call the space connected. Examples are numerous: the Euclidean spaces of all (finite) dimensions are connected, yet the set $A=(-1,0)\cup (0,1)\subseteq\mathbb R^1$ is disconnected, as the two relatively open subsets provide the partition.

Remark. The property of connectedness is very closely related to the completeness of the real numbers. One can consider the rational numbers $\mathbb Q$ as a topological space and define open and closed sets relative to them. Then the sets $\{q\in\mathbb Q: q^2<2\}$ and $\{q\in\mathbb Q: q^2>2\}$ are obviously (relatively) open and disjoint from each other, but their union is the whole of $\mathbb Q$ .

Theorem 3. A continuous map $f:X\to Y$ preserves connectedness: if $X$ is connected, then so is $Y$ .

Proof. Assume that $U,V$ are two disjoint open subsets such that $Y=U\cup V$ . Then their preimages $f^{-1}(U)$ and $f^{-1}(V)$ are open by continuity of $f$ , obviously disjoint and their union gives $X$ in contradiction with the assumption on $X$ . Q.E.D.

Exercise. Describe subsets of $\mathbb R^1$ which are connected topological spaces with respect to the relative topology inherited from $\mathbb R^1$ . Derive from Theorem 4 the familiar Theorem on intermediate value: if a function continuous on a segment $I\subseteq\mathbb R$ (finite or infinite, doesn’t matter) takes two different values $y_1<y_2$ , then it takes also all intermediate values $\{x: y_1\leqslant x \leqslant y_2\}$ .

Warning. One should be very careful and never confuse between preimages and images. The preimage of the connected interval $(1,4)\subseteq\mathbb R$ by the continuous map $f:\mathbb R\to\mathbb R$ , $f(x)=x^2$ , is the disconnected union $(-2,-1)\cup(1,2)$ .

Another example of a useful notion that is of purely topological nature, is that of an isolated point.

Definition. A point $a\in X$ is an isolated point of a topological space $X$ (e.g., a subset $A\subseteq\mathbb R^n$ with the topology defined by the relatively open sets), if the one-point subset $\{a\}\subseteq X$ is both open and closed³.

Proposition. Any map $f:X\to Y$ is automatically continuous at all isolated points of $X$ . Q.E.D.

Compact sets

Another purely topological property of topological spaces (in particular, subsets of $\mathbb R^n$ with the inherited relative topology) is a mighty generalization of some finiteness property. Recall that finite collections (say, of positive numbers) allow to choose a minimal element, which will still be positive: infinite collections of positive numbers, like the set $\{1/n: n\in\mathbb N\}\subseteq\mathbb R^1$ do not allow such choice: the only nonnegative element that is smaller than all number in the above set, is zero which is non-positive.

Definition. A collection (finite or infinite) of sets $\{U_\alpha\subseteq X\}_{\alpha\in\mathscr A}$ is an open covering of the topological space $X$ , if:

All sets $U_\alpha, \alpha\in\mathscr A$ are open, and
$X=\bigcup _{\alpha\in\mathscr A} U_\alpha$ .

When dealing with subsets of Euclidean spaces $A\subseteq\mathbb R^n$ we can assume that a covering $\mathscr U$ is a collection of open subsets $U_\alpha\subseteq\mathbb R^n$ in $\mathbb R^n$ , which contain $A$ in their union, $A\subseteq \bigcup_{\alpha\in \mathscr A} U_\alpha$ .

A subcovering is a subcollection $\{U_\alpha:\alpha\in\mathscr B,\ \mathscr B\subseteq\mathscr A\}$ , that is, a collection of open sets which still cover $X$ obtained by rarefying $\mathscr A$ , that is, discarding (throwing away) some open sets from the initial covering.

Example. Let $f:A\to\mathbb R^1$ is a function continuous at all points of its domain. Then for every point $a\in A$ there exists an open set $U_a\subseteq A$ such that $f(U_a)\subseteq B_1(f(a))$ . The collection $\{U_a:a\in A\}$ is an open covering of $A$ . Another example of the covering is the representation of the real line $\mathbb R^1$ as the union of open sets,

$\mathbb R^1=\bigcup\limits_{n\in\mathbb Z}U_n,\qquad (n-\tfrac 13,n+1+\tfrac13)$ .

The second covering is minimal in the sense that removing of any of the sets $U_n$ is not covering of $\mathbb R^1$ anymore: the middle third of the corresponding segment $[n,n+1]$ will become uncovered. Yet some of the coverings are definitely non-minimal, and one can safely remove some of the open sets which were used to cover.

Definition. A topological space $X$ (i.e., a set $A\subseteq \mathbb R^n$ with the inherited topology) is called compact, if any open covering can be decimated to produce a finite open subcovering.

Make no mistake: compactness does not mean that there simply exists finite open covering: any subset can be covered by just one open set (e.g., the space $X$ itself). Compactness means that a finite covering can be achieved by discarding all but finitely many open sets from any open covering. This definition is rather technical, it is somewhat difficult to digest (people rarely have any working intuition with coverings and their finite subcoverings), yet the idea is quite transparent: compact sets possess some hidden “finiteness”. Yet in a very surprising way sometimes compactness can be achieved by adding some points to a non-compact spaces. For instance, the non-compact real interval $\{0<x<1\}$ (it is non-compact because the infinite open covering $(0,1)=\bigcup_{n\in\mathbb N}(\tfrac 1n, 1-\tfrac1n)$ cannot be reduced to a finite subcovering) can be compactified by adding two endpoints $x=0$ and $x=1$ . The explanation “on one leg” of this phenomenon is simple: adding the extra points imposes additional requirement on the collection of open sets to be a covering, i.e., to cover the extra point as well.

Exercise. An unbounded set $A\subseteq\mathbb R^n$ cannot be finite. Indeed, consider the union of all open balls of radius 1 around all points of $A$ , $\bigcup_{a\in A}B_1(a)$ . This is a covering, since each point belongs to “its” ball. Yet no finite subcovering can be selected from this covering: the union of finitely many balls of radius 1 must be bounded. A similar easy argument shows that a set which is not closed, also cannot be compact.

Proposition. The real closed segment $[0,1]\subseteq\mathbb R^1$ is compact.

Proof. Consider an arbitrary open covering $\mathscr U=\{U_\alpha\}$ and let $M\subseteq [0,1]$ be the set of all points $a\in[0,1]$ such that the subsegment $[0,a]$ admits a finite subcovering selected from $\mathscr U$ . Since $0$ is covered, the set $M$ contains some positive number. Denote by $b$ the supremum of points in $M$ , $b=\sup\limits_{a\in M}a\leqslant 1$ . We claim that $b=1$ . Indeed, if $b<1$ , then $b\in U_\alpha$ for some open set $U_\alpha\in\mathscr U$ . But since $U_\alpha$ is open, for some sufficiently small $\varepsilon >0$ the point $b+\varepsilon$ would still be in $U_\alpha$ and hence the same finite subcovering will still “serve” the point $b+\varepsilon$ . This contradicts our choice of $b<1$ as the exact supremum. This leaves the only possibility that $b=1$ , that is, the entire segment $[0,1]$ admits a finite subcovering selectable from $\mathscr U$ . Q.E.D.

Remark. Compactness of the closed segment $[0,1]$ uses the fact that any bounded set set of real numbers has the exact supremum. Indeed, the “rational closed segment $I=\{q\in\mathbb Q: 0\leqslant q\leqslant q\}$ is not compact. To see this, let us enumerate all points of $I$ by natural numbers, $I=\{q_1,q_2,\dots,q_n,\dots\}$ (this is possible, since $\mathbb Q$ is countable!) and consider the open covering $U$ which covers each point $q_n\in I$ by the interval (open ball) of radius $\bigl(\tfrac13\bigr)^n >0$ centered at this point. This infinite covering does not admit a finite subcovering of $I$ , since the sum of lengths of any finite number of intervals from $\mathscr U$ is less than $\tfrac23$ which is less than $1$ , so at least one point of $I$ will remain “unserved”.

Compactness and continuity

Let $X,Y$ be topological spaces and $f:X\to Y$ a continuous map.

Theorem. If $X$ is compact, then its image is $f(X)$ is compact in $Y$ .

Proof. Let $\mathscr V=\{V_\alpha\}$ be an arbitrary open covering of $Y$ . Consider the sets $U_\alpha=f^{-1}(V_\alpha)$ . By continuity of $f$ , these sets are open and together give a covering $\mathscr U$ of $X$ . Since $X$ is assumed compact, there exists a finite subcovering of $\mathscr U$ . The corresponding finitely many sets $V_\alpha$ give a finite covering of $Y$ . Q.E.D.

Combining this Theorem with the Exercise above, we see that any continuous function is bounded on any compact topological space. Note that the preimage of a compact set may well be noncompact (consider any constant function on $\mathbb R$ ).

Problem. Prove that any closed subset of a compact topological space is also compact.

The following result, which we will not prove, describes all compact subsets of finite-dimensional Euclidean space.

Theorem. A subset $A\subseteq\mathbb R^n$ is compact, if and only if it is bounded (i.e., $\sup_{a\in A}|x|<+\infty$ ) and closed ( $\mathrm{clos}\,A=A$ ). Q.E.D.

Warning

The simplicity obtained by carefully crafting the definitions may well be misleading. Open and closed subsets of $\mathbb R^n$ provide a rich basis for our finitely-dimensional intuition. Yet the general notion of a topological space $X$ without assuming that its topology is inherited from an embedding of $X$ in some space $\mathbb R^n$ (recall that by “topology” we mean a rule that allows to declare some subsets of $X$ open) allows for some surprising results.

Example. Consider the real line $\mathbb R$ with the origin $x=0$ deleted, but with two distinct imaginary points $0^\pm$ added instead. We can introduce a “perverse topology” on this set $\mathbb R^{\bigstar}$ by declaring that the open sets are the open sets in the former $\mathbb R$ by replacing $0$ by only one of the two artificially created points $0^\pm$ . This rule describes all open subsets of $\mathbb R^\bigstar$ which (check it at home) are consistent with claiming that $X=\mathbb R^\bigstar$ is a topological space.

What is “wrong” with this space? The answer is very easy: the two distinct points, $0^+$ and $0^-$ , cannot be separated by disjoint open sets: any two sets $U^\pm$ open in the topology of $\mathbb R^\bigstar$ and containing the points $0^+$ and $0^-$ respectively, necessarily intersect: $U^+\cap U^-\ne\varnothing$ . This means that the topology of $X=\mathbb R^\bigstar$ cannot be generated by any distance function on $X$ . This is an example of the so called non-Hausdorff topology: it happens quite often when dealing with the topological spaces of algebraic origin.

Example. The same space $\mathbb R^1$ can be equipped with a non-standard topology, the so called Zariski topology: namely, declare closed only finite sets and (and the line itself). Hence open will be complements to finite point sets (and the empty set). This topology is also non-Hausdorff: any two non-empty open sets intersect.

Footnotes

Of course, the letter O should remind you about the Openness definition. ↩︎
Here we use the obvious fact (can you prove it ;-)?) that an open ball in $\mathbb R^n$ is an open set! ↩︎
Such a space cannot be connected if it has at least one other point $b\ne a$ . ↩︎

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Sunday, January 7, 2024

Lecture 4, Jan 2, 2024. Happy New Year!

Filed under: Rothschild course "Analysis for high school teachers" — Sergei Yakovenko @ 9:29
Tags: continuity, distance, limit, quantifiers

Continuity

Usually discussion of limits in different forms precedes discussion of continuity. I find this somewhat illogical and difficult to comprehend. Continuity is intuitively very simple property, whereas the theory of limits with its cumbersome sequences of qualifiers is much less transparent.

Our main object of study will be functions (maps) defined on a subset of the Euclidean space of finite dimension $A \subseteq\mathbb R^n$ and taking values in another space $\mathbb R^m$ . The simplest case $n=m=1$ is “too narrow” to talk in proper geometric terms.

The intuitive definition of continuity of $f$ is the following: “For any two points $x,a\in A$ sufficiently close to each other, their images $y=f(x)$ and $b=f(a)$ in $\mathbb R^m$ will be also close to each other”. Rephrasing, “Small change of the argument implies small change of the value of the function”. This definition, however, requires thorough inspection, since the words occurring have little sense by themselves and some quantifiers “for any” and “exists” are clearly missing.

Proximity and distance

To talk about close points one needs to make precise the notion of proximity. For this sake we need to select a distance function on pairs of points $(x,y)\in \mathbb R^n\times\mathbb R^n$ . This function should be far from arbitrary:

$\textrm{dist}(x,y)\geqslant 0$ for any pair of points; the distance is zero if and only if the two points coincide, $x=y$ ;
$\textrm{dist}(x,y)=\textrm{dist}(y,x)$ , that is, the distance is a symmetric function;
The triangle inequality holds: for any three points $x,y,z\in\mathbb R$ we have $\textrm{dist}(x,z)\leqslant\textrm{dist}(x,y)+\textrm{dist}(y,z)$

Obviously, for $n=m=1$ the function $\textrm{dist}(x,y)=|y-x|=|x-y|$ satisfies all these axioms. (Don’t think that this is the only function with such properties! The function $|x^3-y^3|$ also satisfies all of them.) Yet in $\mathbb R^n$ with $n\geqslant 2$ there is a whole family of distance functions, $\textrm{dist}(x,y)=\sqrt[p]{\sum_{i=1}^n(x_i-y_i)^p}$ for any positive $p\ge 1$ . The case $\textrm{dist}(x,y)=\max_{i=1,\dots,n}|x_i-y_i|$ appears as the limit case when $p$ grows to infinity, the case $p=2$ corresponds to the usual Euclidean metric.

Our constructions will not depend on the choice of any of these metrics, though some are more convenient for computations. In any case the sets of the form $B_a(r)=\{x:\textrm{dist}(x,a) < r\}$ we will call open balls centered at $a\in\mathbb R^n$ of radius $r >0$ .

Notation. Moreover, in order to save on keystrokes, we will use the notation $|x-y|$ for any of the above distance functions on $\mathbb R^n$ . It is justified by the fact (that can be easily verified) that $\textrm{dist}(x,y)$ for all $p$ depends only on the difference, $\textrm{dist}(x,y)=\textrm{dist}(x-y,0)$ (invariance by translations). Not all functions satisfying the above three properties possess this invariance.

Quantifiers

The pre-definition of continuity drafted above may now be written in the form of the implication,

$|x-y| \text{ small }\implies |f(x)-f(y)|\text{ small}$ .

But there are no “small” and “large” numbers, even if we restrict to positive numbers. Of course, $0$ is the smallest nonnegative number (the distance must be nonnegative!), but the claim $|x-y| = 0 \implies |f(x)-f(y)|=0$ is trivially valid for any function and the axiom of distance! Besides, it is not clear, for which pairs $(x,y)\in A\times A\subseteq\mathbb R^n$ this claim must hold.

The smallness can be quantified. Let $\varepsilon >0$ be any positive number. Then the set of all numbers $\{z: 0\le z\le \varepsilon\}$ can be called $\varepsilon$ -small, if $\varepsilon$ is chosen as a yardstick (a distance measured in miles may be small, but the same distance measured in yards or inches can be quite large, depending on what one is going to do, sail, swim or kiss).

We have two “small” quantities in the implication above. There is no reason to assume that they could be somehow related to each other: first of all, the corresponding distances are measured in different Euclidean spaces! So we replace the two instances of “smallness” by two quantified terms, $\varepsilon$ -small and $\delta$ -small, with two positive numbers $\varepsilon,\delta >0$ , getting the implication

$|x-y| < \varepsilon \implies |f(x)-f(y)|<\delta$ .

Now we have a logical claim $\mathscr C(x,y,\varepsilon,\delta)$ that involves four “free variables”, $x,y,\in A,\ \varepsilon, \delta >0$ . What shall we do with them? There are two options, either to tie down each variable by one of the two quantifiers $\forall, exists$ or to designate them as a “parameter” that has to be specified in advance.

Let us first address the variables $x,y\in A$ . They clearly play a symmetric role, so it would be natural to assign the same quantifier in front of each of them, and this quantifier is obviously “for all”, $\forall x,y\in A$ (warning: see below). As for the two remaining “scale variables” $\varepsilon, \delta >0$ , we can play around with the two quantifiers $\lozenge_1 \varepsilon$ and $\lozenge_2 \delta$ , where $\lozenge_{1,2}$ are independently either $\forall$ or $\exists$ , and placed in a different order: altogether this yields 8 various possible combinations (some of them identical).

It is an excellent exercise to understand the meaning of all the resulting properties. It turns out that the only one that fits our intuitive understanding of continuity is the the following:

$\forall \varepsilon>0\ \exists\delta>0\text{ such that }\forall (x,y)\in A\times A \quad |x-y| < \varepsilon \implies |f(x)-f(y)|<\delta$ .

The verbose definition corresponding to this phrase, using the terms $\epsilon$ -closeness (resp., $\delta$ -closeness) introduced earlier, is as follows.

Definition. A function $f:A\to\mathbb R^m$ defined on the set $A\subseteq\mathbb R^n$ is uniformly continuous on $A$ , if for any requested proximity measure $\varepsilon>0$ in the target space one can find the proximity measure $\delta >0$ in the domain such that for any two $\delta$ -close points $x,y\in A$ their images $f(x),f(y)$ are $\varepsilon$ -close in the target space.

Variations: continuity at a point $a\in A$

The above definition is aesthetically nice but sometimes need to be relaxed. It implicitly involves some dependence: to establish the continuity, given an arbitrary $latex \varepsilon >0$, one needs to present a suitable $latex \delta>0$ which will serve all pairs $x,y\in A$ simultaneously. This might be a challenge.

Example. Consider the function $f:\mathbb R\to \mathbb R,\ f(x)=x^2$ . Then if $|x-y|<\delta$ , then for any fixed $\delta$ the distance $|f(x)-f(y)|=|x-y|\cdot|x+y|=\delta|x+y|$ will eventually exceed any given $\varepsilon>0$ , if the sum $|x+y|$ is large enough, so the quadratic function, the nicest of all nonlinear function, turns out to be not uniformly continuous on its natural domain. This happens because the natural domain $\mathbb R$ is non-compact (unbounded): the dependence of $\delta$ on $\varepsilon$ must depend also on where exactly are located two “close” points $x,y$ .

To deal with this problem, we need to localize the statement, tying down one of the points, say, $y=a\in\mathbb R^n$ and considering it as a parameter. The corresponding definition (stripped of the word “uniformly“, looks as follows.

Definition. A function $f:A\to\mathbb R^m$ defined on the set $A\subseteq\mathbb R^n$ is continuous at a point $a\in A$ , if for any requested proximity measure $\varepsilon>0$ in the target space one can find the proximity measure $\delta >0$ , depending on $a\in A$ such that for any point $x\in A$ such that $|x-a|<\delta$ its image $f(x)$ is $\varepsilon$ -close in the target space to $b=f(a)$ , $|f(x)-f(a)|<\varepsilon$ .

Note that instead of the “homologous” notation” $x,y$ we switched to a notation $x,a$ , stressing the difference between the point $a\in A$ which is now an external parameter of the statement (for some $a$ the claim may be true, for some false).

Digression on quantifiers

Quantifiers are a subtle issue. If you think accurately, their meaning depends on a lot of hidden data. For instance, if you see a herd of black horses in a field, you might be prompted to conclude that all horses are black. This is patently not true, since there are other horses elsewhere. But even if you add a qualification that all horses in this herd are black, it will be not exactly true: I am not sure whether horses which are black on one side and white on the other exist in the nature, but until this non-existence axiom is added to my logic, I can only conclude that all horses in this herd have a black side that was facing me at the moment of observation. Don’t think that is a stupid game: mathematics requests that all arguments pertinent to a judgement are explicitly put on the table. Yet absent other considerations, for any statement $\mathscr C(x,y)$ that depends on two logical “free variables” $x,y$ , the two claims,

$\forall x\ \exists y\quad \mathscr C(x,y) \qquad \text{or}\quad \exists y\ \forall x\quad \mathscr C (x,y)$

are non-equivalent: оne is stronger and implies the other. Sleep over this wisdom, it will help you a lot when parsing the math code with numerous quantifiers.

The mnemonics is pretty simple: “For every person there is a hat suiting him” and “There is a hat that suits every person”. The first is (allegedly) true, the second is patently wrong. Mind that!

The experience shows that three or more alternating quantifiers may cause problems even for professional mathematicians, not to say about beginning students. For instance, the full expansion of the sentence “the sequence $\{a_n\}$ converges” takes the form involving the horrible four (!) quantifiers:

$\exists A\in \mathbb R \ \forall \varepsilon >0\ \exists N\in\mathbb N\ \forall n>N\quad |a_n-A|<\varepsilon$

No surprise that it takes so strong efforts to digest this construction when you see it for the first time. A proven way to grapple with this problem is to construct appropriate definitions involving the minimal possible numbers of quantifiers. For instance, the above sentence can be reformulated as follows, $\exists A\ \lim_{n\to\infty}a_n=A$ , where the claim $\lim_{n\to\infty}a_n=A$ , involving the number $A$ as external parameter, expands using only three quantifiers, the usual definition of the limit of sequence.

Enclosures

The marked slides with lecture notes: https://drive.google.com/file/d/1YozuLkx8hCP1d9PysXCS5ncFnL1xJQz4/view?usp=sharing

The zoom record: https://weizmann.zoom.us/rec/share/6cP-3gTgT0yWo16KYzKqslE7oyP7uysgQEY5qBgZDqFp_HjG1uJRT7Nc_LWOB_Eh.qavB11mPx7W8RSMx?startTime=1704183044000
Passcode: Prr6Kz

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Saturday, January 6, 2024

Lecture 3, Dec 26, 2023

Filed under: Rothschild course "Analysis for high school teachers" — Sergei Yakovenko @ 9:52
Tags: Dedekind cuts, inequalities, real numbers

Real numbers $\mathbb R$ : why do we need them and how we “construct” them?

In the previous classes we discussed how the number system $\mathbb Q$ can be extended by adding irrational roots of polynomial equations (one or several at once). This, however, does not solve our problems with the circumference $2\pi$ of the unit circle or the $e=y(1)$ , where $y(x)$ is the solution of the simplest differential equation $y'=y$ with the initial condition $y(0)=1$ .
The idea is largely similar: we need to extend $\mathbb Q$ by solutions of systems of two-sided inequalities of the form $l\leqslant x \leqslant r$ with rational left and right bounds $l,r\in\mathbb Q$ . Of course, we have to assume that the system of inequalities is self-consistent and defines not more than one “number”. Consistency means that all left bounds are not exceeding any right bound and vice versa. Uniqueness requires (at least!) that no two different rational numbers satisfy all inequalities forming the system. Obviously, any rational number $q\in\mathbb Q$ satisfies the trivial inequality $l=q\le q\le q=r$ , so the “new” numbers will automatically include all “old” rational numbers.

To that end, we consider the maximal system of inequalities, such that any rational number appears either as a left bound or as a right one (eventually both, but this is possible only in the “trivial case” as above). Thus we have a partition $\mathbb Q=L\cup R, \ L,R\ne\varnothing$ , such that $L\leqslant R$ and $L\cap R$ consists of at most one point. Any such partition (called the Dedekind cut) is intuitively associated with a “number” $x$ (which may not be in $\mathbb Q$ ) which separates the two sets, $L\leqslant x\leqslant R$ .

The “defining inequalities” allow to extend the arithmetic operations on the “new” numbers called real numbers and denoted by $\mathbb R$ : this is easy to do with addition, requires minimal efforts for defining $-x=(-1)\cdot x$ (because the inequalities will be inverted if multiplied by the negative number), then we easily define subtraction as addition of $-x$ . Some technical efforts are required to define product and ratio of numbers in $\mathbb R$ , since again positive and negative numbers should be treated separately. But this is mere technicality.

The most important property of these new numbers is their completeness: unlike the rational numbers that have infinitely many holes between them, the real numbers form a continuous line (complete ordered set), see the lecture notes.

Enclosure

The crisscrossed set of lecture notes (accumulated backlog from previous classes)

The zoom record: https://weizmann.zoom.us/rec/share/KuJ-WIRXE4a2XLYRQgeWNB2xodfye302p9g8CdAhi-6fArs2UKB8sQpqT8SjddTM.XpUELQXydKeybVO-
Passcode: %D8keP

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Friday, December 22, 2023

Lecture 2, Dec 19, 2023

Filed under: Rothschild course "Analysis for high school teachers" — Sergei Yakovenko @ 5:19
Tags: field completion, field extension, number systems, numbers

Extensions of the number systems: adding solutions of non-solvable equations

Starting from the basic object, the natural numbers $\mathbb N$ (with or without zero), we can try and extend this number system to add some “missing elements”. These missing elements are required to solve the equations that have no natural solutions. Two special cases of examples are most important: those of the form $a + x = b$ with $a,b\in\mathbb N$ and $a \leqslant b$ and those of the form $ax=b$ with $b$ not divisible by $a$ . It turns out that in both cases one can add “new” numbers (negative in the first case, fractions in the second) which admit ~~natural~~ easy understandable interpretations in the commercial problems of exchange and trade and which still obey the same laws of arithmetic. These laws follow from the rules of transformation of the equations defining these “unnatural” numbers. Such extension, which in fact leads to the field of rational numbers $\mathbb Q$ , was (implicitly, of course) constructed by many civilizations independently. Al-Khwarismi, the great thinker of the 8th century (probably the darkest times of the European Middle age) was a great contributor to the area, yet absent the concept of negative numbers, he coined rather sophisticated rules on how to solve the linear equations of the form $ax+b =c$ depending on interrelations between the integer coefficients. His way of presenting these rules was much later named after him by the word “algorithm” which now stands for any deterministic rule of performing manipulations.

Another step is related to the discovery of Pythagoras that the equation $x\cdot x=1+1$ does not have rational solutions, while geometrically the expected solution is yielded by the diagonal of the unit square. The Greeks turned to the geometric constructions by ruler and compass as the “source” for legally accepted numbers. Today the class of such numbers is called the class of \emph{quadratic irrationals}. These quadratic irrationals, however, are not sufficient to solve the three famous problems, duplication of the cube, trisection of the angle and squaring the circle.

We discuss how the extension $\mathbb Q(\sqrt 2)$ can be constructed as the set of formal expressions $a+b\,\square$ with $a,b\in\mathbb Q$ closed by all arithmetic operations. This set can be ordered by an order $>$ compatible with the arithmetic operations, assuming that $\square >0$ .

Inspired by this construction, we immediately jump to construction of yet another extension by adding to $\mathbb Q$ the root of the equation $x^2+1=0$ (also absent over the rationals): to that end, we consider all expressions of the form $a+b\,\blacksquare$ and show that it is closed by all arithmetic operations assuming that $\blacksquare\cdot\blacksquare = -1\in\mathbb Q$ . However, unlike in the previous case, the set $\mathbb Q(\blacksquare)$ is a field that cannot be ordered.

One should not think that this formal construction allows to “solve” all equations that have no solutions in a given number system. For instance, the equation $0\cdot x = 1$ must remain unsolvable unless we agree to say good bye to the laws of arithmetic. One needs a special motivation and very special constellation of properties to make this process available.

Enclosed:
the whiteboard slides;
the video record (speakers’ view on Zoom): https://weizmann.zoom.us/rec/share/mW_FTruJsxa1hqi1XuYNjrrvAB0Cvf7TbH3HGOStgmvlNFdLDLOeYx53oUW9ZfuS.Zm7jYp5GV1muyeAc?startTime=1702970133000
Passcode: b1wkwj

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Thursday, December 14, 2023

Lecture 1, Dec 12, 2023

Filed under: Rothschild course "Analysis for high school teachers" — Sergei Yakovenko @ 10:41
Tags: axioms, functions, set theory, sets

‘שלום כיתה א

The first meeting, as expected, has had some technical problems, which hopefully will be solved soon. In particular, the Zoom record that I received, was a black screen with only voices heard in the background and no shared content was seen. I hope the FGS computer gurus will provide me with appropriate links to Panopto.

Meanwhile we discussed the concept of a set as a universal “undefined term” in the modern Mathematics, which is shared across all the areas (Algebra, Geometry, Analysis). We did not discuss the accurate axiomatic Set theory, since all the time we will remain in the so called Naïve Set theory where the “natural operations” like inclusion ∊, union, intersection, complement are unambiguously defined, as well as quantifiers ∀ (for all) and ∃ (exists) can be carefully used. Other elements of the notation include braces { and } e.a. All this will be used in exactly the same way as it is used in the high school.

It turns out that this language allows to define formally what is a function, or a map from one set to another. The “school” definition defines a function as “a rule” which associates with each element a ∈ A of the source set A a unique element b = f(a) ∈ B in the target set. The word “rule” is not in the vocabulary of the language of the Set theory, yet we managed to avoid using it in giving a formal accurate definition.

Another takeaway from the lecture was description of the set of $\mathbb N$ of Natural numbers using the set of Peano axioms. These axioms aim at formalizing the intuitive process of counting ordered objects through a function called the successor. It turns out that using only the successor function, one can rigorously define the arithmetic operations of addition (repeated passing to the successor) and multiplication (as repeated addition).

Enclosed

Whiteboards from the lecture

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Monday, October 9, 2023

War, מלחמה, guerre, война

Filed under: opinion — Sergei Yakovenko @ 3:30

Dear friends, colleagues, друзья, amies et amis, חברים

I was absolutely overwhelmed by the flood of emails, what’s-app and telegram messages full of anxiety for our well-being in all languages. I am physically unable to answer them all, so I use this blog platform to answer all of you. Many thanks for your warmth and concern!

To clarify the current situation. What happened on October 7 in the early morning was our analog of Pearl Harbor, 9/11 or a deja vu of the event that happened precisely 50 years ago, the beginning of Yom Kippur war when Israel was surprised by the attack of Egypt and Syria. That was the bloodiest war among all waged by Israel, but half a century ago the price in blood was paid by the troops guarding the Suez channel and the Golan heights.

This time the main blow was suffered by the civil population. While the country was under barrage of rockets of unprecedented intensity for long four hours, this was relatively harmless because of the incredible efficiency of the “Iron Dome” air defense system: we had to spent this time in bomb shelters, it was not something unusual (events like this happen to us every few years and result mostly in accidental wounds to half a dozen of people).

This time the most horrible events took place near the Gaza border fence: under the cover of the rocket salvos about 1,000 armed terrorists on jeeps and motorbikes burst through the border fence, killing a few border guards, and spread over a couple of dozen of civil communities within few kilometers from the border. What followed then was a massacre of incredible cruelty. Squads of few dozens of monsters walked on streets killing everybody on their way. Then they started breaking into closed houses, murdering entire families at point blank.

Yet the worst was the bloodbath. There was a music festival under the open skies, attended by several thousands of (mostly young) people. They were murdered by the automatic gunfire in huge numbers.

The current body count is close to 1,000 killed (mostly civilians, but a few policemen and soldiers who happened to take the fight), 2,500 wounded (of them a couple of hundreds in the critical state). Some 700 people are yet unaccounted for. Even worse is the fact that about 150 civilians (including elderly people, women and children of age 3 and above) were taken hostages to the Gaza strip. Their prospects are very gloom, unfortunately.

There are many questions, – why it took our army the infinite long 6 hours to bring the special forces to the scene and use the aviation (helicopters e.a.) to stop the carnage, where was our intelligence which had no cue about this operation, obviously very thoroughly planned… We will learn, of course, but later.

For the moment the initial shock is behind, the country resumed its stamina, the army is planning a massive operation to destroy the military power of HAMAS (something that never was done during the 18 years of uncontested HAMAS rule in Gaza, despite numerous rocket attacks). There will be more victims among our military (and for me “military” means our children, our students, our younger friends, – in 24 hours some 300,000 people were called up from the reserves). But for the rest (“usually civilians like us and most our friends) the danger is essentially extremely low (assuming standard safety precautions).

I will add here more if things will take an unpredictable turn.

Once again, – we very much appreciate and value your moral support. Thank you! Спасибо большое, תודה רבה!

P.S. In Israel there are very strict rules concerning publication of the names of fallen soldiers, victims of terror e.a.: it is believed inconceivable if relatives learn about the death or serious wounding from the media and not from an army messangers (accompanied by psychologists, social workers, to help absorbing such a terrible blow). On the contrary, once this sad custom is observed, all papers, TV, radio publish the names of the fallen and their photos.

This link leads to the gallery of the slain military/policemen, those whose death is the easiest to confirm by the nature of their duty. Look at these smiling faces, sometimes uniformed, sometimes in plain clothes, to see whom our country had just lost. And this is only the tiny fraction of the much longer martyrologue, – most victims are civilian and require identification (burned and dismembered or mutilated corpses).

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Wednesday, March 9, 2022

Math from three to seven

Filed under: books,links,Rothschild course "Geometry" — Sergei Yakovenko @ 9:09
Tags: downloads, Hebrew math books, preschool math, zvonkine, ילדים ומתמטיקה

A professional mathematician, Alexandre (Sasha) Zvonkin, once wrote a book summarizing his experience with teaching math to pre-schoolers, which enjoyed an enormous success among parents interested in developing math skills in their kids in a joyful matter. This book was translated into English and freely available from Sasha’s site.

Thanks to efforts of Rachel Paran, now the Hebrew translation is available, titled ילדים ומתמטיקה. Even if you teach mathematics in the high school, you will certainly enjoy the stories of Sasha, many of them eye-openers. And if you have children or grand-children of the appropriate age, this book is simply a “must” for your bookshelf.

The download is free, don’t hesitate to circulate this information.

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Tuesday, March 8, 2022

Call to all members of the Israeli mathematical community

Filed under: opinion — Sergei Yakovenko @ 4:39

Dear colleagues, friends,

With utmost horror we watch the events unfolding now in Ukraine. The atrocities committed by the Russian aggressor army against the civil population were thought unimaginable in the 21st century.

Yet the Russian side, while bearing full responsibility for the war crimes, is not monolith. There are many people who do not want to be part in this war. They are fleeing Russia, which with the lightning speed morphs into GULAG 2.0.

Currently flights to Israel by El Al are essentially the only escape route from Russia. Many of the prospective refugees are Jewish or fall under the provisions of the Law of Return, but even gentiles deserve some help in this tragic situation.

Below follows the letter that I wrote to the Weizmann Institute leadership team headed by professor Alon Chen, the President of WIS. I think that a similar call could be addressed to all Israeli universities. Some colleagues already expressed their wish to help the refugees from Russia on a private basis (zillion thanks!)

Don’t hesitate to share the link to this post to anybody who might help in saving the innocent people from the other side: although their homes are not ruined yet, their lifestyle is definitely ruined.

May I recall that the Jews who tried to escape from the Nazis in Germany, knocked the doors of foreign embassies and consulates with German passports. Don’t confuse the criminal rule with the small fraction of the population that is fiercely opposite to this rule and frightened by its actions.

Dear Alon,

You certainly follow the awful aggression unleashed by Putin’s Russia against Ukraine. The Israeli academia together with all Israelis is united in an attempt to help Ukrainians, especially Ukrainian Jews, especially those who decided to repatriate to Israel.

Yet there is another side to this sad story. The authoritarian rule in Russia in two weeks became an openly oppressive regime: independent media are closed, spreading truth is criminalized by up to 15 years in jail, participants in the street protests are arrested. Mere public use of the word “war” is now a criminal offence, it should be referred to as a “special operation”. What is worse, there is a forced induction to the army of people aged 18-50 and it is used as a punishment for protesters.

Add to it the Iron Curtain 2.0: there are very few escape routes from Russia now. The European airspace is closed for Russian-bound and outgoing flights, border crossings are still available but have very limited throughput.

Connection to Israel is essentially the only lifeline available for the escape from Russia for those who do not want to be the part of the criminal policy and spend the rest of their lives behind the iron curtain. I hope that the wise policy of proclaimed neutrality will allow Israel to maintain this lifeline as long as it is logistically possible.

Among those who flee Russia there are many Jews and people eligible for the Law of Return, many of them excellent scholars. I don’t expect the repetition of the huge immigration wave as in the early 1990-ies, but the Israeli academia should prepare for a sharp increase of scientists of the highest world level that would suddenly look for jobs in Israel.

I believe that the least the WIS should do in this situation is a sharp increase in the number of visiting positions, ranging from a couple of months and up to a year. In parallel we should be ready that a number of the highest quality scientists would become interested in permanent positions. I know that similar arrangements were already made regarding the Ukrainian students and postdocs (Jews and gentiles alike). In my view, it’s time to extend such measures to established scholars, and include also the refugees from Russia who flee the war from the other side of the battlefront.

Being not an administrator, I have no idea on how this can be implemented logistically. I just remember that in 1990-1993 by efforts of a few far-looking scientists (may them be blessed), Israel came prepared to the wave of emigration from fSU much better than USA and many other European countries, which allowed an amazingly large number of ex-fSU scholars to land in Israel and contributed to the subsequent Israeli breakthroughs in STEM. I very much hope that today’s tragic developments in the Eastern Europe could lead to similar benefits for our country. In contrast with that past experience, today we know pretty well who is who in Russian/Ukrainian science and can avoid gambling/errors of judgement while sticking to the highest WIS hiring standards.

Please don’t hesitate if you need any clarifications or particular information: these days I try to maintain the most close contacts with my Russian colleagues and friends.

With the best personal regards,

–Sergei.

Post Scriptum

The response from the Presidents of WIS and Hebrew University was quick and energetically supportive. G-d help them and us to act quickly and efficiently.

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Monday, March 25, 2024

Saturday, February 10, 2024

Supplementary material

Monday, January 15, 2024

The simplifying language: Topology

Continuity as a topological notion

Topological spaces

Connected spaces

Compact sets

Compactness and continuity

Warning

Footnotes

Sunday, January 7, 2024

Continuity

Proximity and distance

Quantifiers

Variations: continuity at a point

Digression on quantifiers

Enclosures

Saturday, January 6, 2024

Real numbers : why do we need them and how we “construct” them?

Enclosure

Friday, December 22, 2023

Extensions of the number systems: adding solutions of non-solvable equations

Thursday, December 14, 2023

‘שלום כיתה א

Enclosed

Monday, October 9, 2023

Wednesday, March 9, 2022

Tuesday, March 8, 2022

Post Scriptum

אלף נכנסים למקרא, מאה למשנה, עשרה לתלמוד, ואחד יוצא להוראה.

What is here? מה יש פה Что тут есть?

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Real numbers $\mathbb R$ : why do we need them and how we “construct” them?