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fresh_42

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Summer is coming and brings a new intermediate math challenge! Enjoy! If you find the problems difficult to solve don't be disappointed! Just check our other basic level math challenge thread!

1) 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.

2) 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.

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

4) 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.

5) 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 by @fresh_42

**RULES:**1) 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.

2) 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.

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

4) 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.

5) 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 by @fresh_42

**QUESTIONS:**- Let's consider complex functions in one variable and especially the involutions

$$\mathcal{I}=\{\, z\stackrel{p}{\mapsto} z\; , \; z\stackrel{q}{\mapsto} -z\; , \;z\stackrel{r}{\mapsto} z^{-1}\; , \;z\stackrel{s}{\mapsto}-z^{-1}\,\}$$

We also consider the two functions $$\mathcal{J}=\{\,z\stackrel{u}{\longmapsto}\frac{1}{2}(-1+i \sqrt{3})z\; , \;z\stackrel{v}{\longmapsto}-\frac{1}{2}(1+i \sqrt{3})z\,\}$$

and the set ##\mathcal{F}## of functions which we get, if we combine any of them: ##\mathcal{F}=\langle\mathcal{I},\mathcal{J} \rangle## by consecutive applications. We now define for ##\mathcal{K}\in \{\mathcal{I},\mathcal{J}\}## a relation on ##\mathcal{F}## by $$ f(z) \sim_\mathcal{K} g(z)\, :\Longleftrightarrow \, (\forall \,h_1\in \mathcal{K})\,(\exists\,h_2\in \mathcal{K})\,: f(h_1(z))=g(h_2(z)) $$

a) Show that ##\sim_\mathcal{K}## defines an equivalence relation.

b) Show that ##\mathcal{F}/\sim_\mathcal{I}## admits a group structure on its equivalence classes by consecutive application.

c) Show that ##\mathcal{F}/\sim_\mathcal{J}## does not admit a group structure on its equivalence classes by consecutive applications.

**(by**@fresh_42**)**

- There are ##r## sports 'enthusiasts' in a certain city. They are forming various teams to bet on upcoming events. A pair of people dominated last year, so there are new rules in place this year. The peculiar rules are:

A. each team must have an odd number of members

B. each and every 2 teams must have an even number of members in common.

For avoidance of doubt, nothing in the rules say a given player can only be on one team.

With these rules in place, is it possible to form more than ##r## teams?**(by**@StoneTemplePython**)**

- 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)\,.##

a) 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.

b) Compute the vector field ##d\phi(X)## on ##\mathbb{S}^2##. How is it related to the curves ##\gamma(t)=\phi(t,y_0)\,?##

c) Is ##d\phi(X)## a continuous vector field on ##\mathbb{S}^2## without zeros?

**(by**@fresh_42**)**

- (solved by @julian and @lpetrich ) A body is attracted to a constant point ##O## by a force which is inversely proportional to its distance from point ##O##. If the body is set free without initial velocity, calculate the time it needs to reach ##O##.
**(by**@QuantumQuest**)**

- a) Prove for any ##\mathbf X \in \mathbb R^{\text{ n x n }}## there exists some ##\mathbf Z## such that ##\mathbf {XZX} = \mathbf X## further,

b) 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)##

**(by**@StoneTemplePython**)**

- 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##.

a) Show that ##\mathcal{D}(p) ## is a group.

b) 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}}\,.##**(by**@fresh_42**)**

- (solved by @lpetrich ) Let ##a##,##b## and ##c## be three different integers and ##P## a polynomial which has integer coefficients. Show that ##P(a) = b##, ##P(b) = c## and ##P(c) =a## can't hold true.
**(by**@QuantumQuest**)**

- 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

a) ##\mathfrak{sl}(2)<\mathcal{H}_n## is a proper subalgebra for all ##n\ge 1##

b) ##\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

As a counterexample, if we started with ##\mathcal{H}=\mathfrak{su}(2)\text{ or }\mathfrak{su}(3)## we would get ##\mathcal{H}_n=\mathcal{H}_0## and we were stationary right from the start, which can easily be seen by solving the corresponding system of linear equations.**(by**@fresh_42**)**

- (solved by @julian ) Show that ##\int_{0}^{\infty} e^{-a\lambda^{2}}\cos \beta\lambda d\lambda = \frac{1}{2}\sqrt{\frac{\pi}{a}} e^{\frac{-\beta^{2}}{4a}}##
**(by**@QuantumQuest**)**

**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.

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\,.##

*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##.

**(by**@fresh_42**)**

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