- #1

- 17,247

- 17,244

Merry Christmas to all who celebrate it today!

$$

h(p)=h(x,y,z)=\begin{bmatrix}1&a&c\\0&1&b\\0&0&1\end{bmatrix} \cdot \begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}x+a\\y+b\\z+ay+c\end{bmatrix}

$$

Show that the Heisenberg manifold ##\mathbb{R}^3/H## is orientable.

$$

T(y)(t) := 1+\int_1^t \dfrac{y(s)}{2s}\,ds

$$

has at least one fixed point and determine them.

Show that the Lie algebra ##\mathfrak{gl}(V)## of all endomorphisms of a finite dimensional complex vector space is reductive.

**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.a)

**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.**Questions:**

1.Given the surface $$f(t,\varphi) = ((1+t^2)\cos \varphi \; , \;(1+t^2)\sin \varphi\; , \;t)\quad (t\in \mathbb{R}\; , \;0\leq \varphi \leq 2\pi)$$1.

- Compute the first fundamental form of this surface.
- Compute the second fundamental form and the Gauss curvature of this surface.
- Compute the geodesic curvature ##\kappa_g## and the normal curvature ##\kappa_n## of the circular latitude at ##t=1##.

**2.**(solved by @Young physicist ) Three pirates are stranded on an island and find that there are only a few monkeys besides drinking water and coconuts. After collecting coconuts for a whole day, they want share them the next morning. At night, one of the pirates awakes and hides his third of the coconuts. But since an odd number of nuts is left, he gives one to a monkey. The second pirate awakens shortly afterwards and hides his third of the remaining coconuts. Again an odd number of coconuts remains, so he gives one to a monkey. The third does the same thing a short time later and gives a leftover nut to a monkey. The next morning they divided the few remaining coconuts among each other. Now the question: How many coconuts did the three pirates at least collect the day before and how are they distributed on each?**3.**(solved by @PeroK and @Charles Link )and ) A cyclist drives along a railway track. Every ##30## minutes, he is overtaken by a train and every ##20## minutes he is met by a train. At which frequency do the trains travel on this connection?**4.**The Heisenberg group ##H=\left\{\begin{bmatrix}1&a&c\\0&1&b\\0&0&1\end{bmatrix}\,:\, a,b,c\in \mathbb{Z}^3 \right\}## operates discontinuously on ##\mathbb{R}^3## by$$

h(p)=h(x,y,z)=\begin{bmatrix}1&a&c\\0&1&b\\0&0&1\end{bmatrix} \cdot \begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}x+a\\y+b\\z+ay+c\end{bmatrix}

$$

Show that the Heisenberg manifold ##\mathbb{R}^3/H## is orientable.

**5.**(solved by @Hiero ) Solve ##x'(t)=\dfrac{2t+2x(t)}{3t+x(t)}\; , \;x(2)=0\,.##**6.**(solved by @Hiero ) Show that ##T\, : \,C([1,2])\longrightarrow C([1,2])## defined by$$

T(y)(t) := 1+\int_1^t \dfrac{y(s)}{2s}\,ds

$$

has at least one fixed point and determine them.

**7.**(solved by @Ibix ) Compute ##\exp(tA)## where ##A=\begin{bmatrix}1&0&1\\0&1&0\\-1&0&1\end{bmatrix}## and determine the behavior of ##\det(\exp(tA))## for ##t \to \pm \infty\,.##**8.**(solved by @Periwinkle ) Let ##G## be a group generated by ##\sigma,\varepsilon,\delta## with ##\sigma^7=\varepsilon^{11}=\delta^{13}=1 ##.- Show that there is no transitive operation of ##G## on a set with ##8## elements.
- Is there are group ##G## with the above properties, that operates transitively on a set with ##12## elements?

**9.**(solved by @Math_QED ) Let ##R,S## be rings and ##\varphi\, : \,R \longrightarrow S## a ring epimorphism. Further let ##J \subseteq S## be an ideal.- Define an ideal ##I \subseteq R## such that ##R/I \cong S/J\,.##
- Is the preimage of the center of ##S## equal to the center of ##R\,?##

**10.**A lie algebra ##\mathfrak{g}## is called reductive, if ##\mathfrak{g}=\mathfrak{Z(g)} \oplus [\mathfrak{g},\mathfrak{g}]## is the direct sum of its center and its derived algebra. (This is an important class of Lie algebras, as they are exactly those whose representations split into a direct sum of irreducible representations. Semisimple and in particular the simple, classical matrix Lie algebras are reductive.)Show that the Lie algebra ##\mathfrak{gl}(V)## of all endomorphisms of a finite dimensional complex vector space is reductive.

Last edited: