- #1

- 17,607

- 18,183

**Questions**

1.(solved by @benorin ) Let ##1<p<4## and ##f\in L^p((1,\infty))## with the Lebesgue measure ##\lambda##. We define ##g\, : \,(1,\infty)\longrightarrow \mathbb{R}## by

1.

$$

g(x)=\dfrac{1}{x}\int_x^{10x}\dfrac{f(t)}{t^{1/4}}\,d\lambda(t).

$$

Show that there exists a constant ##C=C(p)## which depends on ##p## but not on ##f## such that ##\|g\|_2 \leq C\cdot \|f\|_p## so ##g\in L^2((1,\infty)).## (FR)

**2.**We define ##\mathbb{R}^\infty =\mathbb{R}^{(\mathbb{N})}=\{\,(x_1,x_2,\ldots)\,|\,x_i\stackrel{a.a.}{=}0\,\}## and equip ##\mathbb{R}^\infty## with the Euclidean metric ## d((x_1,x_2,\ldots),(y_1,y_2,\ldots)) = \sqrt{\sum_{i=1}^\infty \left|x_i-y_i\right|^2}.## which defines a topology ## \mathcal{S}:=\{\,U\subseteq \mathbb{R}^\infty\,|\, \forall\,p\in U\, \exists\, \varepsilon>0\,:\,B_\varepsilon(p)\subseteq U\,\}## with the open ball ##B_\varepsilon(p)=\{\,q\in\mathbb{R}^\infty\,|\,d(p,q)<\varepsilon\,\}.## (FR)

**a.)**Show that the function

\begin{align*}

\alpha\, : \,(\mathbb{R}^\infty,\mathcal{S})&\longrightarrow (\mathbb{R},\mathcal{E})

(x_1,x_2,\ldots) &\longmapsto \sum_{i=1}^\infty 2^i\cdot x_i \end{align*}

is not continuous, where ##\mathcal{E}## is the usual Euclidean topology on ##\mathbb{R}.##

**b.)**Let ##B## be the diagonal matrix where the diagonal entries are ##2^i## for ##i=1,2,\ldots,## i.e.

$$ B=\begin{bmatrix}2&0&0&\ldots\\0&4&0&\ldots\\0&0&8&\ldots\\ \vdots&\vdots&\vdots&\ddots \end{bmatrix}$$

Show that ##\beta\, : \,(\mathbb{R}^\infty,\mathcal{S}) \longrightarrow (\mathbb{R}^\infty,\mathcal{S})## defined by ##\beta(x)=Bx## is not continuous.

**c.)**Define a topology ##\mathcal{T}## on ##\mathbb{R}^\infty## such that the inclusion maps

\begin{align*}

\iota_n \, : \, (\mathbb{R}^n,\mathcal{E}) & \longrightarrow (\mathbb{R}^\infty,\mathcal{T}) \\

(x_1,\ldots,x_n)&\longmapsto (x_1,\ldots,x_n,0,\ldots) \end{align*}

are continuous for any ##n\in \mathbb{N}_0.##

**3.**(solved by @cbarker1, @benorin , alternative solution possible) Calculate (FR) $$\int_{-\infty}^{+\infty}\dfrac{\cos(\alpha x)}{1+x^2}\,dx \quad (\alpha \geq 0).$$

**4.**Calculate $$\int_0^1 \sin(\pi x)\,x^x\,(1-x)^{1-x}\,dx.$$

Hint: You may use calculators to determine residues. (FR)

**5.**(FR) The ##p##-Prüfer group is defined as

$$

G:=\mathbb{C}_{p^\infty} =\{\,\exp(2n\pi i/p^m )\,|\,n\in \mathbb{Z},m\in \mathbb{N}\,\}\cong\mathbb{Z}\left[\frac{1}{p}\right]/\mathbb{Z}

$$

Show that ##G## is isomorphic to the factor group ##F/R## of the free Abelian group over an countably infinite basis ##\{\,a_1,a_2,\ldots ,a_n,\ldots \,\}## with the subgroup of relations ##R## generated by ##\{\,pa_1,a_1-pa_2,\ldots , a_n-pa_{n+1},\ldots\,\}##, so

$$

G = \langle x_1,x_2,\ldots\,|\,x_1^p=1,x_2^p=x_1,x_3^p=x_2,\ldots \rangle

$$

**6. a.)**(solved by @benorin ) (FR) Let ##u_1,\ldots,u_n## be solutions of the one dimensional heat equation ##\dfrac{du}{dt}-\dfrac{d^2u}{dx^2}=0\;(x\in \mathbb{R},t>0).## Show that $$u(x_1,\ldots,x_n,t):=\displaystyle{\prod_{k=1}^n}u_k(x_k,t)$$ is a solution of the ##n## dimensional heat equation ##\dfrac{\partial u}{\partial t}-\Delta u=0.##

**b.)**(solved by @benorin ) (FR) Calculate a solution for

$$

\begin{cases}

\dfrac{\partial u}{\partial t}(x,t)-\Delta u(x,t)=0 &\text{ for } x\in \mathbb{R}^3,\,t>0 \\[6pt]

u(x,0)=x_1^2x_2^2x_3&\text{ for } x=(x_1,x_2,x_3)\in \mathbb{R}^3

\end{cases}

$$

**Hint:**use part

**a.)**

7.(solved by @mathwonk ) Give an example of a quotient ##R-##module ##M/N## which is Artinian although neither the ring ##R## nor the modules ##M,N## are. (FR)

7.

**8.**Prove and give an example of a solvable group which is not supersolvable. (FR)

**9.**Let ##\gamma## be a non-zero limit ordinal. Consider the order topology on ##[0, \gamma]##. Show that this topological space is compact. (MQ)

**10.**Can you completely cover a disk of diameter ##10## with nine ##1\times 10## rectangles? (IR)

**High Schoolers only**

11.(solved by @etotheipi ) For which natural numbers is ##1!+\ldots + n!## a square number? ##n!=1\cdot 2\cdot \ldots \cdot n\,.##

11.

**12.**(solved by @physion ) Determine ##\{\,(x,y)\in \mathbb{N}_0\times \mathbb{N}_0\,|\,x^3+8x^2-6x+8-y^3=0\,\}\,.##

**13.**Given two different, coprime, positive natural numbers ##a,b \in \mathbb{N}##. Then there are two natural numbers ##x,y \in \mathbb{N}## such that ##ax-by=1\,.##

**14.**How many moves do the towers of Hanoi require to solve by an optimal strategy?

**15.**(solved by @etotheipi ) Among six people are always three who know each other or three who don't. Why?

Last edited: