- #1

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**Summary:**Lie algebras, Hölder continuity, gases, permutation groups, coding theory, fractals, harmonic numbers, stochastic, number theory.

**1.**Let ##\mathcal{D}_N:=\left\{x^n \dfrac{d}{dx},|\,\mathbb{Z}\ni n\geq N\right\}## be a set of linear operators on smooth real functions. For which values of ##N\in\mathbb{Z}\cup\{\pm \infty \}## do they generate a real Lie algebra and are there isomorphic ones among them? Note that any linear combination of basis vectors has only finitely many nonzero coefficients.

**2.**Let ##c\in (0,1)##. Show that the function ##f:[0,c]\longrightarrow \mathbb{R}##

$$

f(x)=\begin{cases} -\dfrac{1}{\log x}&\text{ if }0< x \leq c \\ 0&\text{ if }x=0\end{cases}

$$

is uniformly continuous, but not Hölder continuous.

**3.**Consider the equation ##pV-C(A-B\sqrt{p}+T)=0## where ##A,B,C## are constant parameters, ##p=p(T,V)## vapor pressure, ##V=V(T,p)## molar volume, and ##T=T(p,V)## absolute temperature. Prove by three

**different**methods that

$$

\left(\dfrac{\partial V}{\partial T}\right)_p\cdot \left(\dfrac{\partial T}{\partial p}\right)_V\cdot \left(\dfrac{\partial p}{\partial V}\right)_T=-1

$$

**4.**Calculate

$$

\left(\dfrac{\partial V}{\partial T}\right)_p\text{ and }\left(\dfrac{\partial V}{\partial p}\right)_T

$$

for ##V=V(T,p)## from the equation of state $$

\left(p+\dfrac{a}{V^2}\right)(V-b)=R\cdot T\,; \,a,b,R>0

$$

**5.**Let ##\sigma \in \operatorname{Aut}(S_n)## be an automorphism of the symmetric group ##S_n\;(n\geq 4)## such that ##\sigma ## sends transpositions to transpositions, then prove that ##\sigma ## is an inner automorphism. Determine the inner automorphism groups of the symmetric and the alternating groups for ##n\geq 4.##

**6.**Consider a code ##C\subseteq \mathbb{F}_q^n## with minimal Hamming distance ##d>n\cdot \dfrac{q-1}{q}##.

Prove that the number of possible codewords is restricted by

$$

c:=\#C\leq \dfrac{d}{d-n \cdot \dfrac{q-1}{q}}

$$

**7.**Prove that the Cantor dust on the real line contains uncountable infinitely many points and that it is a fractal by calculating its Hausdorff-Besicovitch dimension.

**8.**Define the harmonic number ##H(p)=1+\dfrac{1}{2}+\dfrac{1}{3}+\ldots +\dfrac{1}{p-1}=\dfrac{a}{b}.##

Show that ##p^2\,|\,a## for primes ##p>3##.

**9.**An ideal coin is thrown three times in a row and then an ideal dice is thrown twice in a row. Each time you toss a coin you get one point if the coin shows "tails" and two points if the coin shows "heads". If you add the total of the two dice rolls to this number of points, you get the total number of points. Furthermore, let A be the event "the total number of points achieved is odd", B be the event "the total of the two dice rolls is divisible by 5", and C the event "the number of points achieved in the three coin tosses is at least 5". Investigate whether A, B, C are pairwise stochastically independent. Also investigate whether A, B, C are stochastically independent.

**10.**Show

$$

C_n :=\binom{2n}{n}-\binom{2n}{n+1}=\prod_{k=1}^n \dfrac{4k-2}{k+1}

$$

and determine all primes in ##\{C_n\}.##

**High Schoolers only (until 26th)**

11.Check whether there is a natural number ##n\in\mathbb{N}## such that ##\sqrt{n}+\sqrt{n+4}\in\mathbb{Q}.## Note that zero is no natural number.

11.

**12.**Assume that ##n\in \mathbb{N}## is odd, and ##\{a_1,a_2,\ldots a_n\}=\{1,2,\ldots,n\}.##

Prove that

$$

(a_1-1)\cdot(a_2-2)\cdot \ldots \cdot (a_{n-1}-(n-1))\cdot (a_n-n)

$$

is always even.

**13.**Show that for every natural number ##n\in \mathbb{N}## there is a ##c=c(n)\in\mathbb{R}## such that for all real numbers ##a>0##

$$

a+a^2+a^3+\ldots +a^{2n-1}+a^{2n} \leq c(n)\cdot \left(1+a^{2n+1}\right).

$$

Show that there is a smallest solution among all possible values ##c(n)## and determine it.

**14.**Given an integer ##k,## determine all pairs ##(x,y)\in \mathbb{Z}^2## such that

$$

x^2+k\cdot y^2=4 \text{ and }k\cdot x^2 - y^2 =2

$$

**15.**Prove for every natural number ##n\in \mathbb{N}##

$$

\dfrac{1\cdot 3\cdot 5 \cdot \ldots \cdot (2n-1)}{2\cdot 4 \cdot 6\cdot \ldots \cdot 2n}< \dfrac{1}{\sqrt{2n+1}}

$$