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fresh_42

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Summer is coming and brings a new basic math challenge! Enjoy! For more advanced problems you can check our other basic level math challenge thread!

a) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. Solutions will be posted around 15th of the following month.

b) It is fine to use nontrivial results without proof as long as you cite them and as long as it is "common knowledge to all mathematicians". Whether the latter is satisfied will be decided on a case-by-case basis.

c) If you have seen the problem before and remember the solution, you cannot participate in the solution to that problem.

d) You are allowed to use google, wolframalpha or any other resource. However, you are not allowed to search the question directly. So if the question was to solve an integral, you are allowed to obtain numerical answers from software, you are allowed to search for useful integration techniques, but you cannot type in the integral in wolframalpha to see its solution.

e) Mentors, advisors and homework helpers are kindly requested not to post solutions, not even in spoiler tags, for the challenge problems, until 16th of each month. This gives the opportunity to other people including but not limited to students to feel more comfortable in dealing with / solving the challenge problems. In case of an inadvertent posting of a solution the post will be deleted.

We have quite a couple of old problems, which are still open. As we think that most of them aren't too difficult, we want to give them another try. One reason is, we want to find out why they have been untouched. So in case you need a hint or two in order to tackle them, don't hesitate to ask! We've also some new questions. Our special thanks go to @Infrared who contributed some new questions.

\begin{align*}

\phi\, &: \, \mathbb{R}^2\longrightarrow \mathbb{R}^3\\

\phi(x,y)&=\dfrac{1}{x^2+y^2+1} \begin{bmatrix}

2x\\2y\\x^2+y^2-1

\end{bmatrix}

\end{align*}

Since ##\mathfrak{A(\mathcal{H})}## is a Lie algebra and $$[X,\alpha]=[\operatorname{ad}(X),\alpha]=\alpha(X)\circ \alpha - \alpha \circ \operatorname{ad(X)}$$ a Lie multiplication, we can define

\begin{align*}

\mathcal{H}_0 &:= \mathcal{H}\\

\mathcal{H}_{n+1} &:= \mathcal{H}_n \ltimes \mathfrak{A(\mathcal{H}_n)}

\end{align*}

and get a series of subalgebras $$\mathcal{H}_0 \leq \mathcal{H}_1 \leq \mathcal{H}_2 \leq \ldots$$

Show that

[(X,\alpha),(Y,\beta)]=([X,Y],[\alpha,\beta]+[\operatorname{ad}(X),\beta]-[\operatorname{ad}(Y),\alpha])$$ However, it is not really needed here. Calculate ##\mathfrak{A}(\mathcal{H})=\mathfrak{A}(\mathcal{H}_0)## and find a copy of ##\mathfrak{sl}(2)## in it, i.e. a ##2 \times 2## block with zero trace. Then note that all ##\mathcal{H}_n## have a central element (= commutes with all others), and consider its implication for ##\mathfrak{A}(\mathcal{H}_n)\,.## Proceed by induction.

This example requires some introduction for all members who aren't familiar with the matter, so let me first give some background information.

The action on a classical particle is the integral of an orbit ##\gamma\, : \,t \rightarrow \gamma(t)## $$ S(\gamma)=S(x(t))= \int \mathcal{L}(t,x,\dot{x})\,dt $$ over the Lagrange function ##\mathcal{L}##, which describes the system considered. Now we consider smooth coordinate transformations

\begin{align*}

x &\longmapsto x^* := x +\varepsilon \psi(t,x,\dot{x})+O(\varepsilon^2)\\

t &\longmapsto t^* := t +\varepsilon \varphi(t,x,\dot{x})+O(\varepsilon^2)

\end{align*}

and we compare $$ S=S(x(t))=\int \mathcal{L}(t,x,\dot{x})\,dt\text{ and }S^*=S(x^*(t^*))=\int \mathcal{L}(t^*,x^*,\dot{x}^*)\,dt^* $$

Since the functional ##S## determines the law of motion of the particle, $$S=S^*$$ means, that the action on this particle is unchanged, i.e. invariant under these transformations, and especially

\begin{equation*}

\dfrac{\partial S}{\partial \varepsilon}=0 \quad\text{ resp. }\quad \left. \dfrac{d}{d\varepsilon}\right|_{\varepsilon =0}\left(\mathcal{L}\left(t^*,x^*,\dot{x}^*\right)\cdot \dfrac{dt^*}{dt} \right) = 0 ~~(*)

\end{equation*}

Emmy Noether showed exactly hundred years ago, that under these circumstances (invariance), there is a conserved quantity ##Q##. ##Q## is called the Noether charge. $$S=S^* \Longrightarrow \left. \dfrac{d}{d\varepsilon}\right|_{\varepsilon =0}\left(\mathcal{L}\left(t^*,x^*,\dot{x}^*\right)\cdot \dfrac{dt^*}{dt} \right) = 0 \Longrightarrow \dfrac{d}{dt}Q(t,x,\dot{x})=0$$

with $$Q=Q(t,x,\dot{x}):= \sum_{i=1}^N \dfrac{\partial \mathcal{L}}{\partial \dot{x}_i}\,\psi_i + \left(\mathcal{L}-\sum_{i=1}^N \dfrac{\partial \mathcal{L}}{\partial \dot{x}_i}\,\dot{x}_i\right)\varphi = \text{ constant}$$

The general way to proceed is:

A. Determine the functions ##\psi,\varphi##, i.e. the transformations, which are considered.

B. Check the symmetry by equation (*).

C. If the symmetry condition holds, then compute the conservation quantity ##Q## with ##\mathcal{L},\psi,\varphi\,.##
Example: Given a particle of mass ##m## in the potential ##U(\vec{r})=\dfrac{U_0}{\vec{r\,}^{2}}## with a constant ##U_0##. At time ##t=0## the particle is at ##\vec{r}_0## with velocity ##\dot{\vec{r}}_0\,.##

a) Give a reason why the energy of the particle is conserved, and what is its energy?

b) Consider the following transformations with infinitesimal ##\varepsilon##

$$\vec{r} \longmapsto \vec{r}\,^*=(1+\varepsilon)\,\vec{r}\,\, , \,\,t\longmapsto t^*=(1+\varepsilon)^2\,t$$

and verify the condition (*) to E. Noether's theorem.

c) Compute the corresponding Noether charge ##Q## and evaluate ##Q## for ##t=0##.

Use ##\psi_i=0 , \varphi=1## for the time invariant energy and consider $$\left. \dfrac{d}{d\varepsilon}\right|_{\varepsilon =0}\left(\mathcal{L}\left(t^*,x^*,\dot{x}^*\right)\cdot \dfrac{dt^*}{dt} \right)$$ For the last part we have ##\partial_x \psi=x\; , \;\partial_y \psi=y\; , \;\partial_z \psi=z## and ##\varphi =2t\,.##

$$

\sigma_1=\begin{bmatrix}0&1\\1&0\end{bmatrix}\, , \,\sigma_2=\begin{bmatrix}0&-i\\i&0\end{bmatrix}\, , \,\sigma_3=\begin{bmatrix}1&0\\0&-1\end{bmatrix}

$$

Now we define an operation on ##V:=\mathbb{C}_2[x,y]##, the vector space of all complex polynomials of degree less than three in the variables ##x,y## by

\begin{align*}

\varphi(\alpha_1\sigma_1 +\alpha_2\sigma_2+\alpha_3\sigma_3)&.(a_0+a_1x+a_2x^2+a_3y+a_4y^2+a_5xy)= \\

&= x(-i \alpha_1 a_3 +\alpha_2 a_3 - \alpha_3 a_1 )+\\

&+ x^2(2i\alpha_1 a_5 +2 \alpha_2 a_5 + 2\alpha_3 a_2 )+\\

&+ y(-i\alpha_1 a_1 -\alpha_2 a_1 +\alpha_3 a_3 )+\\

&+ y^2(2i\alpha_1 a_5 -2\alpha_2 a_5 -2\alpha_3 a_4 )+\\

&+ xy(-i\alpha_1 a_2 -i\alpha_1 a_4 +\alpha_2 a_2 -\alpha_2 a_4 )

\end{align*}

Show that

$$

\varphi(\alpha_1,\alpha_2,\alpha_3):=\varphi(\alpha_1\cdot (i\sigma_1),\alpha_2\cdot (i\sigma_2),\alpha_3\cdot (i\sigma_3))

$$

and decompose ##V## into three invariant subspaces. To determine the weights, consider the ##\mathbb{C}-##basis $$H=\sigma_3,X=\dfrac{1}{2}\sigma_1+\dfrac{1}{2}i\sigma_2,Y=\dfrac{1}{2}\sigma_1-\dfrac{1}{2}i\sigma_2$$

Y := \{\,(x_1,x_2)\in\mathbb{R}^2\,|\,x_1,x_2\geq 0\, , \,x_1+x_2\leq 1\,\}

$$

Show that for any ##\alpha_1\, , \,\alpha_2 > 0##

$$

\int_Y f(x_1+x_2)x_1^{\alpha_1}x_2^{\alpha_2}\,d\lambda(x_1,x_2) = \left[\int_0^1 f(u)u^{\alpha_1+\alpha_2+1}\,d\lambda(u) \right]\cdot \left[\int_0^1 v^{\alpha_1}(1-v)^{\alpha_2}\,d\lambda(v) \right]

$$

\begin{align*}

&|x|=0 \Longleftrightarrow x=0 \\

&|xy| = |x|\;|y|\\

&|x+y| \leq |x|+|y|

\end{align*}

It is called Archimedean, if for any two elements ##a,b\,\,(a\neq 0)## there is a natural number ##n## such that ##|na|>|b|\,.## We consider the rational numbers. The usual absolute value

$$

|x| = \begin{cases} x &\text{ if }x\geq 0 \\ -x &\text{ if }x<0\end{cases}

$$

is Archimedean, whereas the trivial value

$$

|x|_0 = \begin{cases} 0 &\text{ if }x = 0 \\ 1 &\text{ if }x\neq 0\end{cases}

$$

is not.

Determine all non-trivial and non-Archimedean value functions on ##\mathbb{Q}\,.##

__RULES:__a) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. Solutions will be posted around 15th of the following month.

b) It is fine to use nontrivial results without proof as long as you cite them and as long as it is "common knowledge to all mathematicians". Whether the latter is satisfied will be decided on a case-by-case basis.

c) If you have seen the problem before and remember the solution, you cannot participate in the solution to that problem.

d) You are allowed to use google, wolframalpha or any other resource. However, you are not allowed to search the question directly. So if the question was to solve an integral, you are allowed to obtain numerical answers from software, you are allowed to search for useful integration techniques, but you cannot type in the integral in wolframalpha to see its solution.

e) Mentors, advisors and homework helpers are kindly requested not to post solutions, not even in spoiler tags, for the challenge problems, until 16th of each month. This gives the opportunity to other people including but not limited to students to feel more comfortable in dealing with / solving the challenge problems. In case of an inadvertent posting of a solution the post will be deleted.

We have quite a couple of old problems, which are still open. As we think that most of them aren't too difficult, we want to give them another try. One reason is, we want to find out why they have been untouched. So in case you need a hint or two in order to tackle them, don't hesitate to ask! We've also some new questions. Our special thanks go to @Infrared who contributed some new questions.

__QUESTIONS:__

**1.**(solved by @Citan Uzuki )- Prove for any ##\mathbf X \in \mathbb R^{\text{ n x n }}## there exists some ##\mathbf Z## such that ##\mathbf {XZX} = \mathbf X##
- Prove that a satisfying ##\mathbf Z## may be chosen to obey

##\text{trace}\big(\mathbf {ZX}\big) = \text{rank}\big(\mathbf X\big)##

##\text{trace}\big(\mathbf {ZX}^3\big) = \text{trace}\big(\mathbf {X}^2\big)##

**2.**We consider the vector field ##X\, : \,\mathbb{R}\longrightarrow \mathbb{R}^2## given by ##X(p) := \left(p,\begin{pmatrix} 1\\0 \end{pmatrix}\right)\,.##- Compute the derivative ##d\phi\, : \,T\mathbb{R}^2\longrightarrow T\mathbb{R}^3## of the stereographic projection to the north pole, i.e. plane to sphere with ##\phi(0,0)=(0,0,-1)##, and describe the tangent bundle ##T\mathbb{S}^2## of ##\mathbb{S}^2##. Show that position vectors and tangent vectors are orthogonal.
- Compute the vector field ##d\phi(X)## on ##\mathbb{S}^2##. How is it related to the curves ##\gamma(t)=\phi(t,y_0)\,?##
- Is ##d\phi(X)## a continuous vector field on ##\mathbb{S}^2## without zeros?

**Hint:**The stereographic projection to the north pole is given by\begin{align*}

\phi\, &: \, \mathbb{R}^2\longrightarrow \mathbb{R}^3\\

\phi(x,y)&=\dfrac{1}{x^2+y^2+1} \begin{bmatrix}

2x\\2y\\x^2+y^2-1

\end{bmatrix}

\end{align*}

**3.**A covering space ##\tilde{X} ## of ##X## is a topological space together with a continuous surjective map ##p\, : \,\tilde{X} \longrightarrow X\,,## such that for every ##x \in X## there is an open neighborhood ##U\subseteq X## of ##x,## such that ##p^{-1}(U)\subseteq \tilde{X}## is a union of pairwise disjoint open sets ##V_\iota## each of which is homeomorphically mapped onto ##U## by ##p##. A Deck transformation with respect to ##p## is a homeomorphism ##h\, : \,\tilde{X} \longrightarrow \tilde{X}## with ##p \circ h=p\,.## Let ##\mathcal{D}(p)## be the set of all Deck transformations with respect to ##p##.- Show that ##\mathcal{D}(p) ## is a group.
- If ##\tilde{X}## is a connected Hausdorff space and ##h \in \mathcal{D}(p)## with ##h(\tilde{x})=\tilde{x}## for some point ##\tilde{x}\in \tilde{X}\,.## then ##h=\operatorname{id}_{\tilde{X}}\,.##

**Hint:**Show that ##\mathcal{D}(p)## is closed under inversion and multiplication. Then consider ##A:=\{\,\tilde{x}\in \tilde{X}\, : \,h(\tilde{x})=\tilde{x}\,\}\,.##**4.**Given the Heisenberg algebra $$\mathcal{H}=\left\{\,\begin{bmatrix} 0&x&z\\0&0&y\\0&0&0 \end{bmatrix}\,\right\}=\langle X,Y,Z\,:\,[X,Y]=Z \rangle $$ and $$\mathfrak{A(\mathcal{H})}=\{\,\alpha\, : \,\mathcal{H}\longrightarrow \mathcal{H}\, : \,[\alpha(X),Y]=[\alpha(Y),X]\,\forall\,X,Y\in \mathcal{H}\,\} $$Since ##\mathfrak{A(\mathcal{H})}## is a Lie algebra and $$[X,\alpha]=[\operatorname{ad}(X),\alpha]=\alpha(X)\circ \alpha - \alpha \circ \operatorname{ad(X)}$$ a Lie multiplication, we can define

\begin{align*}

\mathcal{H}_0 &:= \mathcal{H}\\

\mathcal{H}_{n+1} &:= \mathcal{H}_n \ltimes \mathfrak{A(\mathcal{H}_n)}

\end{align*}

and get a series of subalgebras $$\mathcal{H}_0 \leq \mathcal{H}_1 \leq \mathcal{H}_2 \leq \ldots$$

Show that

- ##\mathfrak{sl}(2)<\mathcal{H}_n## is a proper subalgebra for all ##n\ge 1##
- ##\dim \mathcal{H}_{n} \ge 3 \cdot (2^{n+1}-1)## for all ##n\ge 0##, i.e. the series is infinite and doesn't get stationaryl

**Hint:**The multiplication in ##\mathcal{H}_n## is given by $$[(X,\alpha),(Y,\beta)]=([X,Y],[\alpha,\beta]+[\operatorname{ad}(X),\beta]-[\operatorname{ad}(Y),\alpha])$$ However, it is not really needed here. Calculate ##\mathfrak{A}(\mathcal{H})=\mathfrak{A}(\mathcal{H}_0)## and find a copy of ##\mathfrak{sl}(2)## in it, i.e. a ##2 \times 2## block with zero trace. Then note that all ##\mathcal{H}_n## have a central element (= commutes with all others), and consider its implication for ##\mathfrak{A}(\mathcal{H}_n)\,.## Proceed by induction.

**5.***On the occasion of the centenary of Emmy Noether's theorem.*This example requires some introduction for all members who aren't familiar with the matter, so let me first give some background information.

\begin{align*}

x &\longmapsto x^* := x +\varepsilon \psi(t,x,\dot{x})+O(\varepsilon^2)\\

t &\longmapsto t^* := t +\varepsilon \varphi(t,x,\dot{x})+O(\varepsilon^2)

\end{align*}

and we compare $$ S=S(x(t))=\int \mathcal{L}(t,x,\dot{x})\,dt\text{ and }S^*=S(x^*(t^*))=\int \mathcal{L}(t^*,x^*,\dot{x}^*)\,dt^* $$

Since the functional ##S## determines the law of motion of the particle, $$S=S^*$$ means, that the action on this particle is unchanged, i.e. invariant under these transformations, and especially

\begin{equation*}

\dfrac{\partial S}{\partial \varepsilon}=0 \quad\text{ resp. }\quad \left. \dfrac{d}{d\varepsilon}\right|_{\varepsilon =0}\left(\mathcal{L}\left(t^*,x^*,\dot{x}^*\right)\cdot \dfrac{dt^*}{dt} \right) = 0 ~~(*)

\end{equation*}

Emmy Noether showed exactly hundred years ago, that under these circumstances (invariance), there is a conserved quantity ##Q##. ##Q## is called the Noether charge. $$S=S^* \Longrightarrow \left. \dfrac{d}{d\varepsilon}\right|_{\varepsilon =0}\left(\mathcal{L}\left(t^*,x^*,\dot{x}^*\right)\cdot \dfrac{dt^*}{dt} \right) = 0 \Longrightarrow \dfrac{d}{dt}Q(t,x,\dot{x})=0$$

with $$Q=Q(t,x,\dot{x}):= \sum_{i=1}^N \dfrac{\partial \mathcal{L}}{\partial \dot{x}_i}\,\psi_i + \left(\mathcal{L}-\sum_{i=1}^N \dfrac{\partial \mathcal{L}}{\partial \dot{x}_i}\,\dot{x}_i\right)\varphi = \text{ constant}$$

The general way to proceed is:

A. Determine the functions ##\psi,\varphi##, i.e. the transformations, which are considered.

B. Check the symmetry by equation (*).

C. If the symmetry condition holds, then compute the conservation quantity ##Q## with ##\mathcal{L},\psi,\varphi\,.##

**Hint:**The Lagrange function with ##\vec{r}=(x,y,z,t)=(x_1,x_2,x_3,t)## of this problem is $$ \mathcal{L}=T-U=\dfrac{m}{2}\,\dot{\vec{r}}\,^2-\dfrac{U_0}{\vec{r\,}^{2}} $$a) Give a reason why the energy of the particle is conserved, and what is its energy?

b) Consider the following transformations with infinitesimal ##\varepsilon##

$$\vec{r} \longmapsto \vec{r}\,^*=(1+\varepsilon)\,\vec{r}\,\, , \,\,t\longmapsto t^*=(1+\varepsilon)^2\,t$$

and verify the condition (*) to E. Noether's theorem.

c) Compute the corresponding Noether charge ##Q## and evaluate ##Q## for ##t=0##.

**6.**Consider the Lie algebra of skew-Hermitian ##2\times 2## matrices ##\mathfrak{g}:=\mathfrak{su}(2,\mathbb{C})## and the Pauli matrices (note that Pauli matrices are not a basis!)$$

\sigma_1=\begin{bmatrix}0&1\\1&0\end{bmatrix}\, , \,\sigma_2=\begin{bmatrix}0&-i\\i&0\end{bmatrix}\, , \,\sigma_3=\begin{bmatrix}1&0\\0&-1\end{bmatrix}

$$

Now we define an operation on ##V:=\mathbb{C}_2[x,y]##, the vector space of all complex polynomials of degree less than three in the variables ##x,y## by

\begin{align*}

\varphi(\alpha_1\sigma_1 +\alpha_2\sigma_2+\alpha_3\sigma_3)&.(a_0+a_1x+a_2x^2+a_3y+a_4y^2+a_5xy)= \\

&= x(-i \alpha_1 a_3 +\alpha_2 a_3 - \alpha_3 a_1 )+\\

&+ x^2(2i\alpha_1 a_5 +2 \alpha_2 a_5 + 2\alpha_3 a_2 )+\\

&+ y(-i\alpha_1 a_1 -\alpha_2 a_1 +\alpha_3 a_3 )+\\

&+ y^2(2i\alpha_1 a_5 -2\alpha_2 a_5 -2\alpha_3 a_4 )+\\

&+ xy(-i\alpha_1 a_2 -i\alpha_1 a_4 +\alpha_2 a_2 -\alpha_2 a_4 )

\end{align*}

Show that

- an adjusted ##\varphi## defines a representation of ##\mathfrak{su}(2,\mathbb{C})## on ##\mathbb{C}_2[x,y]##
- Determine its irreducible components.
- Compute a vector of maximal weight for each of the components.

**Hint:**This is an easy example of a ##\mathfrak{su}(2,\mathbb{C})## representation which shall demonstrate how the ladder up and down operators actually work. Choose ##(1,x,y,x^2,xy,y^2)## as ordered basis for the representation space ##V=\mathbb{C}_2[x,y]## and verify ##[\varphi(\alpha_1,\alpha_2,\alpha_3),\varphi(\alpha'_1,\alpha'_2,\alpha'_3)]=\varphi([(\alpha_1,\alpha_2,\alpha_3),(\alpha'_1,\alpha'_2,\alpha'_3)])## with the adjusted transformation$$

\varphi(\alpha_1,\alpha_2,\alpha_3):=\varphi(\alpha_1\cdot (i\sigma_1),\alpha_2\cdot (i\sigma_2),\alpha_3\cdot (i\sigma_3))

$$

and decompose ##V## into three invariant subspaces. To determine the weights, consider the ##\mathbb{C}-##basis $$H=\sigma_3,X=\dfrac{1}{2}\sigma_1+\dfrac{1}{2}i\sigma_2,Y=\dfrac{1}{2}\sigma_1-\dfrac{1}{2}i\sigma_2$$

**7.**Let ##f\, : \,(0,1)\longrightarrow \mathbb{R}## be Lebesgue integrable and $$Y := \{\,(x_1,x_2)\in\mathbb{R}^2\,|\,x_1,x_2\geq 0\, , \,x_1+x_2\leq 1\,\}

$$

Show that for any ##\alpha_1\, , \,\alpha_2 > 0##

$$

\int_Y f(x_1+x_2)x_1^{\alpha_1}x_2^{\alpha_2}\,d\lambda(x_1,x_2) = \left[\int_0^1 f(u)u^{\alpha_1+\alpha_2+1}\,d\lambda(u) \right]\cdot \left[\int_0^1 v^{\alpha_1}(1-v)^{\alpha_2}\,d\lambda(v) \right]

$$

**Hint:**Consider ##\phi\, : \,(0,1)^2\longrightarrow \mathbb{R}^2## with ##\phi(u,v)=(vu,(1-v)u)\,.## and apply Fubini's theorem.**8.**A function ##|\,.\,|\, : \,\mathbb{F}\longrightarrow \mathbb{R}_{\geq 0}## on a field ##\mathbb{F}## is called a value function if\begin{align*}

&|x|=0 \Longleftrightarrow x=0 \\

&|xy| = |x|\;|y|\\

&|x+y| \leq |x|+|y|

\end{align*}

It is called Archimedean, if for any two elements ##a,b\,\,(a\neq 0)## there is a natural number ##n## such that ##|na|>|b|\,.## We consider the rational numbers. The usual absolute value

$$

|x| = \begin{cases} x &\text{ if }x\geq 0 \\ -x &\text{ if }x<0\end{cases}

$$

is Archimedean, whereas the trivial value

$$

|x|_0 = \begin{cases} 0 &\text{ if }x = 0 \\ 1 &\text{ if }x\neq 0\end{cases}

$$

is not.

Determine all non-trivial and non-Archimedean value functions on ##\mathbb{Q}\,.##

**Hint:**This is indeed a bit tricky. Since ##|\,.\,|## is non-Archimedean, there are elements ##a,b## with ##|n|<\frac{|b|}{|a|}## for all ##n\in \mathbb{N}\,.## If ##|n| > 1## for a natural number, then ##|n^k|=|n|^k## goes to infinity and cannot be bounded. Thus ##|n|\leq 1## for all ##n\in \mathbb{N}\,.## Note that ##|.|## is non-trivial. Pick a smallest natural number ##n_0=ab## and investigate it.**9.**(solved by @lpetrich ) Let ##G## be a group of order ##p^n## for some natural number ##n>1##. Show that ##\text{Aut}(G)## contains an element of order ##p##.**10.**(solved by @nuuskur ) Let ##(X,d)## be a metric space. For open sets ##U\subset X##, let ##*U## denote the interior of the complement of ##U##. Give an example where ##**U\neq U##. Show that ##***U=*U## always holds.
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