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**Summary:**Functional Analysis. Project Management. Set Theory. Group Theory. Lie Theory. Countability. Banach Algebra. Stochastic. Function Theory. Calculus.

**1.**Prove that ##F\, : \,L^2([0,1])\longrightarrow (C([0,1]),\|.\|_\infty )## defined as $$F(x)(t):=\int_0^1 (t^2+s^2)(x(s))^2\,ds$$ is compact.

**2.**A project manager has ##n## workers to finish the project. Let ##x_i## be the workload of the ##i-##th person, and

$$

x\in S:= \left\{ x\in \mathbb{R}^n \, \mid \, \sum_{i=1}^nx_i=1\, , \, x_i\geq 0\right\}

$$

a possible partition of work. Let ##X_i## be the set of partitions, which person ## i ## agrees upon. We may assume that he automatically agrees if ##x_i=0, ## that ##X_i## is closed, and that there is always at least one person which agrees to a given partition, i.e. ##\bigcup_{i=1}^n X_i=S.## Prove that there is one partition that all workers agree upon.

**3.**Assume the axiom schema of separation for any predicate ##P(x)##

$$

\forall A\, : \,\exists M\, : \,\forall x\, : \,\left(x\in M \Longleftrightarrow x\in A \wedge P(x)\right)

$$

Show that ##|A|<|\mathcal{P}(A)|## where ##\mathcal{P}(A)## is the power set of ##A.##

**4.**(solved by @fishturtle1 ) Let ##\sigma_1,\ldots,\sigma_n## be homomorphisms from a group ##G## into the multiplicative group ##\mathbb{F}^*## of a field ##\mathbb{F}.## Show that they are ##\mathbb{F}-##linearly independent if and only if they are pairwise distinct.

**5.**(solved by @Office_Shredder ) Prove that general Heisenberg (Lie-)algebras ##\mathfrak{H}## are nilpotent.

**6.**(solved by @Office_Shredder and @Not anonymous ) Prove that the polynomial ##\mathbb{N}_0^2\stackrel{P}{\longrightarrow }\mathbb{N}_0## defined as

$$

P(x,y)=\dfrac{1}{2}\left((x+y)^2+3x+y\right)

$$

is a bijection.

**7.**Prove that the spectrum of every element of a complex Banach algebra ##B## with ##1## is nonempty. Conclude that if ##B## is a division ring, then ##B\cong \mathbb{C}.##

**8.**Let ##X_1,\ldots,X_n## be independent random variables, such that almost certain ##a_i\leq X_i-E(X_i)\leq b_i##, and let ##0<c\in \mathbb{R}##. Prove that

$$

\operatorname{Pr}\left(\sum_{i=1}^n (X_i-E (X_i))\geq c\right)\leq \exp\left(\dfrac{-2c^2}{\sum_{i=1}^n(b_i-a_i)^2}\right).

$$

**9.**Let ##G\subseteq \mathbb{C}## be a non-empty, open, connected subset, and ##f,g## holomorphic functions on ##G.## Show that the following statements are equivalent:

- ##f(z)=g(z)## for all ##z\in G.##
- ##\{z\in G\,|\,f(z)=g(z)\}## has a limit point.
- There is a ##z\in G## such that ##f^{(n)}(z)=g^{(n)}(z)## for all ##n\in \mathbb{N}_0.##

**10.**(solved by @julian ) Prove that ##\pi^2## is irrational.

**11.**(solved by @ANB3010 ) Find all functions ##f,g## such that

\begin{align*}

f,g\, &: \,\mathbb{R} \backslash \{-1,0,1\} \longrightarrow \mathbb{R}\\

xf(x)&=1+\dfrac{1}{x}g\left(\dfrac{1}{x}\right)\, \text{ and } \,

\dfrac{1}{x^2}f\left(\dfrac{1}{x}\right)=x^2g(x)

\end{align*}

**Extra: (open)**Determine a number ##r\in \mathbb{R}## such that ##|f(x)-f(x_0)|<0.001## whenever ##|x-x_0|<r## and ##x_0=2,## and explain why there is no such number if we choose ##x_0=1## even if we artificially define some function value for ##f(1).##

**12.**(open) Solve the following equation system in ##\mathbb{R}^3##

\begin{align*}

x^2+y^2+z^2=1 \quad\wedge \quad x+2y+3z=\sqrt{14}

\end{align*}

**Extra: (open)**Give an alternative solution in case you have the additional information that the solution is unique.

**13.**(solved by @Not anonymous ) If ##(x_n)_{n\in \mathbb{N}}\subseteq \mathbb{R}_{>0}## is a monotone decreasing sequence of positive real numbers such that for every ##n\in \mathbb{N}##

$$

\dfrac{x_1}{1}+\dfrac{x_4}{2}+\dfrac{x_9}{3}+\ldots+\dfrac{x_{n^2}}{n}\leq 1

$$

prove that for every ##n\in\mathbb{N}##

$$

\dfrac{x_1}{1}+\dfrac{x_2}{2}+\dfrac{x_3}{3}+\ldots+\dfrac{x_{n}}{n}\leq 3

$$

**Extra: (open)**Prove that both sequences converge to ##0.##

**14.**(solved by @Not anonymous ) Solve the following equation system for real numbers:

\begin{align*}

(1)\qquad x+xy+xy^2&=-21\\

(2)\qquad y+xy+x^2y&=14\\

(3)\qquad \,\,\phantom{y+xy} x+y&=-1

\end{align*}

**Extra:**(solved by @Not anonymous )

**15.**(solved by @Not anonymous ) Find all real numbers ##m\in \mathbb{R}##, such that for all real numbers ##x\in \mathbb{R}## holds

$$

f(x,m):=x^2+(m+2)x+8m+1 >0 \qquad (*)

$$

and determine the value of ##m## for which the minimum of ##f(x,m)## is maximal. What is the maximum?

**Extra: (open)**The set of all intersection points of two perpendicular tangents is called orthoptic of the parabola. Prove that it is the directrix, the straight parallel to the tangent at the extremum on the opposite side of the focus.

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