Math Problems Vault

Problem III.4.1

Let

P(z)=a_{0}+a_{1} z+\ldots +a_{m} z^{m} \quad (z \in \mathbb {C}) .

Prove that the image $P(F)$ of any closed set $F \subset \mathbb {C}$ is again a closed set. Is this true for an arbitrary polynomial in two variables?

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Problem III.4.2

Let

\Gamma _{0}=\left\{ z \in \mathbb {C}\left| \right| z \mid =1, z \neq -1\right\} ; \quad \varphi (z)=\frac{\operatorname {Im} z}{1+\operatorname {Re} z} .

Prove that $\varphi$ maps $\Gamma_{0}$ bijectively onto $\mathbb {R}$ . Find $\varphi^{-1}$ .

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Problem III.4.3

Does there exist a partition of the segment $\left[0,1\right]$ into two disjoint sets $A$ and $B$ , each of which can be mapped onto the other bijectively and bicontinuously (homeomorphically)?

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Problem III.4.4

On the interval $\left[0,1\right]$ define functions $\varphi$ and $\psi$ as follows. Let $t \in [0,1], t=\sum 3^{-k} \alpha_{k}$ , where $\alpha_{k} \in \left\{ 0,1,2\right\}$ . Then $\varphi (t)=\sum 3^{-k} \beta_{k}$ , where $\beta_{1}=\alpha_{1}, \beta_{2}=\alpha_{3}$ , if $\alpha_{2}$ is an even number, $\beta_{2}=2-\alpha_{3}$ , if $\alpha_{2}$ is an odd number, $\beta_{n}=\alpha_{2 n-1}$ , if $\alpha_{2}+\alpha_{4}+\ldots +\alpha_{2 n-2}$ is an even number, and $\beta_{n}=2-\alpha_{2 n-1}$ , if $\alpha_{2}+\alpha_{4}+\ldots +\alpha_{2 n-2}$ is an odd number ( $n \geqslant 2$ ). Similarly, $\psi (t)=\sum 3^{-k} \gamma_{k}$ , where $\gamma_{1}=\alpha_{2}$ , if $\alpha_{1}$ is an even number, $\gamma_{1}=2-\alpha_{2}$ , if $\alpha_{1}$ is an odd number, $\gamma_{n}=\alpha_{2 n}$ , if $\alpha_{1}+\alpha_{3}+\ldots +\alpha_{2 n-1}$ is an even number, and $\gamma_{n}=2-\alpha_{2 n}$ , if $\alpha_{1}+\alpha_{3}+\ldots +\alpha_{2 n-1}$ is an odd number.

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(a)

Prove that the functions $\varphi$ and $\psi$ are well defined and continuous.

(b)

Let $f:\left[0,1\right] \rightarrow \mathbb {R}^{2}$ be the map with coordinate functions $\varphi$ and $\psi$ ( $f(t)=(\varphi (t), \psi (t))$ for $t \in [0,1])$ . Prove that $f$ maps the interval $\left[0,1\right]$ onto the square $\left[0,1\right]^{2}$ . The map $f$ is described schematically in Fig. 7. Each of the intervals $\Delta_{k}(k=1,2, \ldots , 9)$ is mapped onto the square with the same number (Fig. $7, a$ ). Fig. 7,b shows the traversal of the unit square upon dividing the interval into $9^{2}$ parts.

Fig. 7 Fig. 7

(c)

Prove that the system of equations

\varphi (t)=\xi , \quad \psi (t)=\eta

has at most four solutions for any $\xi , \eta \in [0,1]$ . Indicate pairs $\xi , \eta$ for which this system has four solutions.

(d)

Prove that the sets of constancy of the functions $\varphi$ and $\psi$ (i.e., the sets $\varphi^{-1}(\left\{ c\right\} )$ and $\psi^{-1}(\left\{ c\right\} ), c \in \mathbb {R}$ ) have no isolated points.

(e)

Prove that $\varphi , \psi \in \operatorname {Lip}_{1 / 2}(\left[0,1\right])$ , but $\varphi , \psi \notin \operatorname {Lip}_{\alpha }(\left[0,1\right])$ , if $\alpha >\frac{1}{2}$ .

Problem III.4.5

A continuous map of an interval onto a square is called a Peano curve. Let $u$ and $v$ be the coordinate functions of an arbitrary Peano curve defined on the interval $\left[0,1\right]$ . Prove that if $u, v \in \operatorname {Lip}_{\alpha }(\left[0,1\right])$ , then $\alpha \leqslant \frac{1}{2}$ . (Thus, the smoothness of the coordinate functions of the Peano curve constructed in the previous problem is the best possible).

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Problem III.4.6

Prove that any two generalized Cantor sets are homeomorphic, i.e., a one-to-one and mutually continuous correspondence can be established between them.

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Problem III.4.7

Let $X, Y \subset (0,1)$ be countable sets that are everywhere dense in $\left[0,1\right]$ . Construct a diffeomorphism of the interval $\left[0,1\right]$ (a mapping onto itself, continuously differentiable together with its inverse mapping) carrying $X$ onto $Y$ .

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Problem III.4.8

(a)(b)(c)

Let $A \subset \mathbb {R}^{n}, x \in \mathbb {R}^{n}$ . A point $y_{0}$ of $A$ is called a best approximation to $x$ in $A$ , if $\left\| x-y_{0}\right\| \leqslant \left\| x-y\right\|$ for all $y \in A$ .

Prove that if $A$ is a convex compact set, then

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(a)

for every point $x \in \mathbb {R}^{n}$ there exists a unique best approximation in $A$ ;

(b)

the metric projection $T$ , assigning to a point its best approximation in $A$ , is a nonexpansive mapping: $\left\| T(x)-T\left(x^{\prime }\right)\right\| \leqslant \leqslant \left\| x-x^{\prime }\right\|$ for any $x, x^{\prime } \in \mathbb {R}^{n}$ ;

(c)

if $\operatorname {Int}(A) \neq \varnothing$ and $A$ is contained in a compact set $B$ , then $T(\partial B)=\partial A$ , where $\partial A$ and $\partial B$ are the boundaries of the sets $A$ and $B$ .