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**Summary:**Analysis. Projective Geometry. ##C^*##-algebras. Group Theory. Markov Processes. Manifolds. Topology. Galois Theory. Linear Algebra. Commutative Algebra.

**1.a.**(solved by @nuuskur ) Let ##C\subseteq \mathbb{R}^n## be compact and ##f\, : \,C\longrightarrow \mathbb{R}^n## continuous and injective. Show that the inverse ##g=f^{-1}\, : \,f(C)\longrightarrow \mathbb{R}^n## is continuous.

**1.b.**(solved by @nuuskur ) Let ##S:=\{x+tv\,|\,t\in (0,1)\}## with ##x,v\in \mathbb{R}^n,## and ##f\in C^0(\mathbb{R}^n)## differentiable for all ##y\in S.## Show that there is a ##z\in S## such that

$$

f(x+v)-f(x)=\nabla f(z)\cdot v\,.

$$

**1.c.**(solved by @MathematicalPhysicist ) Let ##\gamma \, : \,[0,\pi]\longrightarrow \mathbb{R}^3## be given as

$$

\gamma(t):=\begin{pmatrix}

\cos(t)\sin(t)\\ \sin^2(t)\\ \cos(t)

\end{pmatrix}\, , \,t\in [0,\pi].

$$

Show that the length ##L(\gamma )>\pi.##

**2.**(solved by @mathwonk ) Let ##g,h## be two skew lines in a three-dimensional projective space ##\mathcal{P}=\mathcal{P}(V)##, and ##P## a point that is neither on ##g## nor on ##h##. Prove that there is exactly one straight through ##P## that intersects ##g## and ##h.##

**3.**(solved by @QuantumSpace ) Let ##(\mathcal{A},e)## be a unital ##C^*##-algebra. A self-adjoint element ##a\in \mathcal{A}## is called positive, if its spectral values are:

$$

\sigma(a) :=\{\lambda \in \mathbb{C}\,|\,a-\lambda e \text{ is not invertible }\}\subseteq \mathbb{R}^+:=[0,\infty).

$$

The set of all positive elements is written ##\mathcal{A}_+\,.## A linear functional ##f\, : \,\mathcal{A}\longrightarrow \mathbb{C}## is called positive, if ##f(a)\in \mathbb{R}^+## for all positive ##a\in \mathcal{A}_+\,.##

Prove that a positive functional is continuous.

**4.**Prove that the following groups ##F_1,F_2## are free groups:

**4.a.**(solved by @nuuskur ) Consider the functions ##\alpha ,\beta ## on ##\mathbb{C}\cup \{\infty \}## defined by the rules

$$

\alpha(x)=x+2 \text{ and }\beta(x)=\dfrac{x}{2x+1}.

$$

The symbol ##\infty ## is subject to such formal rules as ##1/0=\infty ## and ##\infty /\infty =1.## Then ##\alpha ,\beta ## are bijections with inverses

$$

\alpha^{-1}(x)=x-2\text{ and }\beta^{-1}(x)=\dfrac{x}{1-2x}.

$$

Thus ##\alpha ## and ##\beta ## generate a group of permutations ##F_1## of ##\mathbb{C}\cup \{\infty \}.##

**4.b.**(solved by @martinbn and @mathwonk ) Define the group ##F_2:=\langle A,B \rangle ## with

$$

A:=\begin{bmatrix}1&2\\0&1 \end{bmatrix} \text{ and }

B:=\begin{bmatrix}1&0\\2&1 \end{bmatrix}

$$

**5.**We model the move of a chess piece on a chessboard as a timely homogeneous Markov chain with the ##64## squares as state space and the position of the piece at a certain (discrete) point in time as a state. The transition matrix is given by the assumption, that the next possible state is equally probable. Determine whether these Markov chains ##M(\text{piece})## are irreducible and aperiodic for

**(a)**king,

**(b)**bishop,

**(c)**pawn, and

**(d)**knight.

**6.**Prove that a ##n##-dimensional manifold ##X## is orientable if and only if

(a) there is an atlas for which all chart changes respect orientation, i.e. have a positive functional determinant,

(b) there is a continuous ##n##-form which nowhere vanishes on ##M.##

**7.**(solved by @nuuskur ) A topological vector space ##E## over ##\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}## is normable if and only if it is Hausdorff and possesses a bounded convex neighborhood of ##\vec{0}.##

**8.a.**(solved by @kmitza ) Determine the minimal polynomial of ##\pi + e\cdot i## over the reals.

**8.b.**(solved by @jbstemp ) Show that ##\mathbb{F}:=\mathbb{F}_7[T]/(T^3-2)## is a field, calculate the number of its elements, and determine ##(T^2+2T+4)\cdot (2T^2+5),## and ##(T+1)^{-1}.##

**8.c.**(solved by @mathwonk ) Consider ##P(X):=X^{7129}+105X^{103}+15X+45\in \mathbb{F}[X]## and determine whether it is irreducible in case

$$

\mathbb{F} \in \{\mathbb{Q},\mathbb{R},\mathbb{F}_2,\mathbb{Q}[T]/(T^{7129}+105T^{103}+15T+45)\}

$$

**8.d.**(solved by @mathwonk ) Determine the matrix of the Frobenius endomorphism in ##\mathbb{F}_{25}## for a suitable basis.

**9.**(solved by @mathwonk ) Let ##V## and ##W## be finite-dimensional vector spaces over the field ##\mathbb{F}## and ##f\, : \,V\otimes_\mathbb{F}W\longrightarrow \mathbb{F}## a linear mapping such that

\begin{align*}

\forall \,v\in V-\{0\}\quad \exists \,w\in W\, &: \,f(v\otimes w)\neq 0\\

\forall \,w\in W-\{0\}\quad \exists \,v\in V\, &: \,f(v\otimes w)\neq 0

\end{align*}

Show that ##V\cong_\mathbb{F} W.##

**10.**(solved by @mathwonk ) Let ##R:=\mathbb{C}[X,Y]/(Y^2-X^2)##. Describe ##V_\mathbb{R}(Y^2-X^2)\subseteq \mathbb{R}^2,## determine whether ##\operatorname{Spec}(R)## is finite, calculate the Krull-dimension of ##R,## and determine whether ##R## is Artinian.

**High Schoolers only**

11.Let ##a\not\in\{-1,0,1\}## be a real number. Solve

11.

$$

\dfrac{(x^4+1)(x^4+6x^2+1)}{x^2(x^2-1)^2}=\dfrac{(a^4+1)(a^4+6a^2+1)}{a^2(a^2-1)^2}\,.

$$

**12.**Define a sequence ##a_1,a_2,\ldots,a_n,\ldots ## of real numbers by

$$

a_1=1\, , \,a_{n+1}=2a_n+\sqrt{3a_n^2+1}\quad(n\in \mathbb{N})\,.

$$

Determine all sequence elements that are integers.

**13.**For ##n\in \mathbb{N}## define

$$

f(n):=\sum_{k=1}^{n^2}\dfrac{n-\left[\sqrt{k-1}\right]}{\sqrt{k}+\sqrt{k-1}}\,.

$$

Determine a closed form for ##f(n)## without summation. The bracket means: ##[x]=m\in \mathbb{Z}## if ##m\leq x <m+1.##

**14.**Solve over the real numbers

\begin{align*}

&(1)\quad\quad x^4+x^2-2x&\geq 0\\

&(2)\quad\quad 2x^3+x-1&<0\\

&(3)\quad\quad x^3-x&>0

\end{align*}

**15.**Let ##f(x):=x^4-(x+1)^4-(x+2)^4+(x+3)^4.## Determine whether there is a smallest function value if ##f(x)## is defined ##(a)## for integers, and ##(b)## for real numbers. Which is it?#

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