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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.

$$

\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

See https://www.physicsforums.com/insights/pantheon-derivatives-part-v/

$$

F(x)=\begin{bmatrix}(x_2^4+2x_2^2x_3^2)x_1\\(x_3^4+2x_1^2x_3^2)x_2 \\(x_1^4+2x_1^2x_2^2)x_3 \end{bmatrix}

$$

and calculate the integral ##\int_{\partial B}F\cdot N \,d\mathbb{S}^2##

$$

\operatorname{div}(hF)=h \operatorname{div}F + \nabla h\cdot F

$$

$$

\int_{B^n} f \Delta g \,dB^n = -\int_{B^n} \nabla f \cdot \nabla g \,\,dB^n + \int_{\partial B^n} f \nabla g \cdot N \,d\mathbb{S}^{N-1}

$$

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*}

O_n(\mathbb{R})&=\{\,A\in \mathbb{M}(n,\mathbb{R})\,|\,

\langle Av,Aw\rangle = \langle v,w \rangle \text{ for all }v,w\in \mathbb{R}^n\,\}\\

&=\{\,A\in \mathbb{M}(n,\mathbb{R})\,|\,A^\tau A = A A^\tau =1\,\}

\end{align*}

be the orthogonal group of ##n\times n## matrices which operate per matrix multiplication on ##\mathbb{R}^n\,(n\in {N})\,.##

Show that

\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}\,.##

be the space of all bounded functions on ##X\,.## We define a metric on ##\mathcal{B}(X)## by ##d(f,g)=||f-g||_\infty\,.##

__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.

__QUESTIONS:__**1.**(solved by @Citan Uzuki ) Let ##R## be a ring with identity element ##1## and ##r \in R## an element without left inverse but at last one right inverse ##r\cdot a_0=1##. Prove that there are infinitely many right inverses to ##r##.**2.**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:*##\mathfrak{su}(2,\mathbb{C}) \cong \mathfrak{sl}(2,\mathbb{C})##*leads to a more obvious basis for the weight spaces.*

**3.**(solved by @Infrared ) Let ##(X\;,\;||\,.\,||\,)## be a normed vector space. Prove that ##X## is complete if and only if for each sequence with ##\sum_{n=1}^{\infty}||x_n|| < \infty## the series ##\sum_{n=1}^{\infty}x_n## converges as well in ##X##.**4.**(solved by @Infrared ) Gauß' Divergence Theorem: ##\iiint_V (\nabla F)\,dV = \iint_{\partial V}(F\cdot N)d(\partial V)##.See https://www.physicsforums.com/insights/pantheon-derivatives-part-v/

**a.)**Let ##B=B_1(0)## the unit sphere in ##\mathbb{R}^3## and consider the vector field$$

F(x)=\begin{bmatrix}(x_2^4+2x_2^2x_3^2)x_1\\(x_3^4+2x_1^2x_3^2)x_2 \\(x_1^4+2x_1^2x_2^2)x_3 \end{bmatrix}

$$

and calculate the integral ##\int_{\partial B}F\cdot N \,d\mathbb{S}^2##

**b.)**Let ##U \subseteq \mathbb{R}^n## be open and ##h\in C^1(U)\; , \;F\in C^1(U,\mathbb{R}^n)\,.## Show that on ##U## we have$$

\operatorname{div}(hF)=h \operatorname{div}F + \nabla h\cdot F

$$

**c.)**Let ##B^n \subseteq \mathbb{R}^n## be the closed unit ball and ##f,g \in C^2(B^n)\,.## Show that with the unit normal vector field ##N##$$

\int_{B^n} f \Delta g \,dB^n = -\int_{B^n} \nabla f \cdot \nabla g \,\,dB^n + \int_{\partial B^n} f \nabla g \cdot N \,d\mathbb{S}^{N-1}

$$

**5.**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]

$$

**6.**(solved by @Infrared ) Finite Groups.- Let ##U\subsetneq G## be a proper subgroup of a finite group.

Show that ##\bigcup_{g\in G}gUg^{-1}\subsetneq G## is a proper subset. - Let ##G\neq \{\,1\,\}## be a finite group which operates transitive on ##X## which has at least two elements ##|X|>1\,.## Transitive means all elements of ##X## can be reached by the group operation from a single ##x\in X\,.## Show that there is a group element ##g\in G## such that ##g.x \neq x## for all ##x\in X\,.##

**7.**Let\begin{align*}

O_n(\mathbb{R})&=\{\,A\in \mathbb{M}(n,\mathbb{R})\,|\,

\langle Av,Aw\rangle = \langle v,w \rangle \text{ for all }v,w\in \mathbb{R}^n\,\}\\

&=\{\,A\in \mathbb{M}(n,\mathbb{R})\,|\,A^\tau A = A A^\tau =1\,\}

\end{align*}

be the orthogonal group of ##n\times n## matrices which operate per matrix multiplication on ##\mathbb{R}^n\,(n\in {N})\,.##

- (solved by @lpetrich ) Determine the orbit of ##x\in \mathbb{R}^n## under ##O_n(\mathbb{R})\,.##
- (solved by @lpetrich ) Determine the stabilizer ##\operatorname{Stab}_x(O_n(\mathbb{R}))=\{\,A\in O_n(\mathbb{R})\,|\,A.x=x\,\}## of ##x = (0,0,\ldots,1)\in \mathbb{R}^n## in ##O_n(\mathbb{R})\,.##
- (solved by @Infrared ) Determine a bijection ##\mathbb{S}^{n-1} \stackrel{1:1}{\longleftrightarrow} O_n(\mathbb{R})/O_{n-1}(\mathbb{R})## between the unit sphere in ##\mathbb{R}^n## and the factor of two consecutive orthogonal groups.

**8.**We define an equivalence relation on the topological two-dimensional unit sphere ##\mathbb{S}^2\subseteq \mathbb{R}^3## by ##x \sim y \Longleftrightarrow x \in \{\,\pm y\,\}## and the projection ##q\, : \,\mathbb{S}^2 \longrightarrow \mathbb{S}^2/\sim \,.## Furthermore we consider the homeomorphism ##\tau \, : \,\mathbb{S}^2 \longrightarrow \mathbb{S}^2## defined by ##\tau (x)=-x\,.## Note that for ##A \subseteq \mathbb{S}^2## we have ##q^{-1}(q(A))= A \cup \tau(A)\,.##Show that

- ##q## is open and closed.
- ##\mathbb{S}^2/\sim ## is compact, i.e. Hausdorff and covering compact.
- Let ##U_x=\{\,y\in \mathbb{S}^2\,:\,||y-x||<1\,\}## be an open neighborhood of ##x \in \mathbb{S}^2\,.## Show that ##U_x \cap U_{-x} = \emptyset \; , \;U_{-x}=\tau(U_x)\; , \;q(U_x)=q(U_{-x})## and ##q|_{U_{x}}## is injective. Conclude that ##q## is a covering.

**9.**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}\,.##

**10.**(solved by @nuuskur ) For a set ##X## let $$\mathcal{B}(X) = \{\,f\, : \,X\longrightarrow \mathbb{R}\,:\,\sup_{x\in X}\{\,|f(x)|\,\}=:||f||_\infty <\infty\,\}$$be the space of all bounded functions on ##X\,.## We define a metric on ##\mathcal{B}(X)## by ##d(f,g)=||f-g||_\infty\,.##

- Show that ##(\mathcal{B}(X)\, , \,d)## is complete.
- If ##(X,d)## is a metric space and ##a\in X\,.## Prove that the function

$$ \phi_a\, : \,X \longrightarrow \mathcal{B}(X)\, , \,\phi_a(x)=d(x,.)-d(a,.) $$

is an isometry of ##X## in ##\mathcal{B}(X)\,.## - Show that the closure of ##\operatorname{im}(\phi_a)## is a completion of ##X \sim \phi_a(X)\,.##

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