- #1

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**Questions**

1.)(disclosed by @Demystifier ) Using the notion of double integrals prove that $$B(m,n) = \frac{\Gamma (m) \Gamma (n)}{\Gamma (m + n)}\; \;(m \gt 0\,,\, n\gt 0)$$ where ##B## and ##\Gamma## are the

1.)

*Beta*and

*Gamma*functions respectively.

**2.)**(solved by @Math_QED ) Show that the Fourier series of an even function cannot contain sine.

**3.)**(solved by @Bosko ) Show that $$\int_{0}^{1} dx \int_{x}^{\sqrt{x}} f(x,y)\,dy = \int_{0}^{1} dy \int_{y^2}^{y} f(x,y)\,dx\,.$$

**4.)**The graphic of eye colors shows us the probability of baby's eye color in dependency of the parents'. This yields the following multiplication table:

\begin{align*}

x \cdot x &= (3/4) x + (3/16) y + (1/16) z\\

x \cdot y &=(1/2) x+(3/8) y+(1/8) z\\

x \cdot z &=(1/2) x+(1/2) z\\

y \cdot y &=(3/4) y+(1/4) z\\

y \cdot z &=(1/2) y+(1/2) z\\

z \cdot z &= z

\end{align*}

which we extend to a real, commutative, distributive, three dimensional algebra ##A##.

**a)**(solved by @fbs7 ) Is ##A## an associative algebra?

**still open:****b)**Prove, that ##A## is a baric algebra, i.e. show that there is a non trivial algebra homomorphismus ##\omega\, : \,A\longrightarrow \mathbb{R}##, the weight function.

**c)**Determine a basis for ##\operatorname{ker}\omega## and rewrite the multiplication table according to this new basis.

**d)**Prove that there is an ideal ##N## of codimension one in ##A##, such that ##A^2 \nsubseteq N##.

**e)**A algebra is called genetic, if there is a basis ##\{\,u_i\,\}## such that the structure constants ##\lambda_{ijk}## defined by

$$

u_i \cdot u_j = \sum_{k=1}^n \lambda_{ijk}u_k

$$

fulfill the following conditions:

- ##\lambda_{111}=1##
- ##\lambda_{1jk}=0## for all ##j>k##
- ##\lambda_{ijk}=0## for all ##i,j > 1## and ##k \leq \operatorname{max}\{i,j\}##

**f)**Show that ##A## is not a genetic algebra.

**g)**Determine all idempotent elements of ##A##. Is there a basis of ##A## with idempotent elements?

**5.)**(solved by @Bosko ) Prove that starting with ##\frac{1}{1}## the following binary tree

$$

\begin{array}{ccccc}

&&\frac{a}{b}&&\\

&\swarrow &&\searrow \\

\frac{a}{a+b}&&&&\frac{a+b}{b}

\end{array}

$$

defines a counting of all positive rational numbers without repetition and all quotients canceled out.

**6.)**(solved by @Cthugha ) Who said what here? (What is the task, who a bonus.)

**7.)**(solved by @lpetrich ) Calculate ##I:=\int_0^\infty \dfrac{\sqrt{x^{e-2}}}{x^e+1}\,dx##

**8.)**An algebra ##A## is a vector space with a binary distributive multiplication. An example are group algebras, i.e. the distributive extension of the formal basis vectors ##g\in G## such as $$A :=\mathbb{R}[S_3]=\mathbb{R}\cdot(1)+\mathbb{R}\cdot(12)+\mathbb{R}\cdot(13)+\mathbb{R}\cdot(23)+\mathbb{R}\cdot(123)+\mathbb{R}\cdot(132)$$

**a.)**Find the center ##Z(A)=\{\,z \in A\,|\,zv=vz\text{ for all }v\in A\,\}## of ##A##, and

**b.)**determine the structure of ##A##, i.e. its decomposition into direct factors and the corresponding isomorphisms.

**9.)**Let ##B\subseteq \mathbb{R}^n## be measurable and ##P=(a_1,\ldots,a_n,b)\in \mathbb{R}^{n+1}## a point with ##b>0## and ##C_B=\{\,P+t(Q-P)\,|\,Q\in B\times \{0\}_{n+1}\, , \,t\in [0,1]\,\}## the cone above the basis ##B## with the peak ##P\,.##

Prove the measure formula

$$

\lambda^{n+1}(C_B)=\dfrac{b}{n+1}\cdot \lambda^n(B)

$$

**10.)**Show that $$

x \cdot y = \frac{2xy-x-y}{xy-1}$$

defines a one dimensional, real, local Lie group ##G## around ##0 \in \mathbb{R}## and compute the vector field of left multiplication by an element ##g \in \mathbb{R}##.

**1.)**(solved by @fbs7 ) Find the limit of the sequence ##a_n = 2\cdot \sqrt[3]{4}\cdot \sqrt[9]{16}\cdot \sqrt[27]{256} \,\cdots \, \sqrt[3^{n-1}]{2^{2^{n-1}}}##.

**2.)**Let ##(b_n)## be a sequence with ##b_n = \dfrac{nn^n}{(n!)^2}##.

**a.)**Show that ##(b_n)## converges and find its limit.

**b.)**Using the above find the limit of the sequence ##(a_n)## with ##a_n = \dfrac{1^n + 2^n + 3^n + \dots + n^n}{(n!)^2}##.

**3.)**(solved by @fbs7 ) Two numbers ##a,b## are called amicable, if the sum of all proper divisors of one is the other number (##1## is included). The smallest example is

$$

(a,b) = (220,284)=(1+2+4+71+142,1+2+4+5+10+11+20+22+44+55+110)

$$

Let ##n\in \mathbb{N}## and ##(x,y,z)=\left(3\cdot 2^n-1\, , \,3\cdot 2^{n-1}-1\, , \,9\cdot 2^{2n-1}-1\right)\,.##

Prove that if ##x,y,z## are all odd primes, then ##(a,b)=\left( 2^n\cdot x\cdot y\,,\, 2^n\cdot z \right)## are amicable numbers.

**Hint:**First find a formula for the sum of all divisors ##\sigma(n)## given the prime decomposition of ##n##.

**4.)**(solved by @fbs7 ) A number is called perfect, if it equals the number of all its divisors except itself, e.g. ##6=1+2+3## and ##28=1+2+4+7+14## are perfect.

Prove: If ##2^k-1## is a prime number, then ##2^{k-1}(2^k-1)## is a perfect number and every even perfect number has this form.

**5.)**

**a.)**(solved by @fbs7 ) What is the smallest five-digit number ##n## such that ##n## and ##2n## together contain all ##10## digits from ##0## to ##9##?

**b.)**(solved by @fbs7 ) On how many zeros does the number ##1000\,!## end?

**c.)**(solved by @fbs7 ) For which six-digit number ##ABCDEF## do we have:

##ABCDEF \cdot 1 = ABCDEF##

##ABCDEF \cdot 3 = BCDEFA##

##ABCDEF \cdot 2 = CDEFAB##

##ABCDEF \cdot 6 = DEFABC##

##ABCDEF \cdot 4 = EFABCD##

##ABCDEF \cdot 5 = FABCDE##

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