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fresh_42

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**Questions**

**1. a.**Let ##(\mathfrak{su}(2,\mathbb{C}),\varphi,V)## be a finite dimensional representation of the Lie algebra ##\mathfrak{g}=\mathfrak{su}(2,\mathbb{C})##.

Calculate ##H\,^0(\mathfrak{g},\varphi)## and ##H\,^1(\mathfrak{g},\varphi)## for the Chevalley-Eilenberg complex in the cases

**(i)**##(\varphi,V)= (\operatorname{ad},\mathfrak{g})##

**(ii)**##(\varphi,V)= (0,\mathfrak{g})##

**(iii)**##(\varphi,V)= (\pi,\mathbb{C}^2)## is the natural representation on ##\mathbb{C}^2##.

**b.**Consider the Heisenberg algebra ##\mathfrak{g}=\mathfrak{h}=\left\{ \left.\begin{pmatrix}0&a&c\\0&0&b\\0&0&0\end{pmatrix}\right|a,b,c\in \mathbb{R}\right\}## and calculate ##H\,^0(\mathfrak{h},\operatorname{ad})## and ##H\,^1(\mathfrak{h},\operatorname{ad})\,.##

**2.**Show that the dihedral group ##D_{12}## of order twelve is the finite reflection group of the root system of type ##G_2##.

**3.**(solved by @Periwinkle ) Consider the set

$$

\mathcal{P}_n := \{\,\{2\},\{4\},\ldots,\{2n\}\,\} \subseteq \mathcal{P}(\mathbb{N})

$$

and determine the ##\sigma-##algebra ##\mathcal{A}_\sigma(\mathcal{P}_n) \subseteq \mathcal{P}(\mathbb{N})##, and show that ##\bigcup_{n\in \mathbb{N}}\mathcal{A}_\sigma(\mathcal{P}_n)## isn't a ##\sigma-##algebra.

**4.**(solved by @Periwinkle ) Linear Operators. (Only solutions to both count!)

**a.**Show that eigenvectors to different eigenvalues of a self-adjoint linear operator are orthogonal and the eigenvalues real.

**b.**Given a real valued, bounded, continuous function ##g \in C([0,1])## with

m = \inf_{t\in [0,1]} g(t)\; , \;M = \sup_{t\in [0,1]} g(t)

$$

and an operator ##T_g(f)(t):= g(t)f(t)## on the Hilbert space ##\mathcal{H}=L^2([0,1])\,.##

Calculate the spectrum of ##T_g\,.##

**5.**Let ##\mathbb{F}## be a field. Then for a polynomial ##f \in \mathbb{F}[X_1,\ldots,X_n]## we define ##D(f)=\{\,q\in \mathbb{A}^n(\mathbb{F})\,|\,f(q)\neq 0\,\}##.

Show that these sets build a basis of the Zariski topology on ##\mathbb{A}^n(\mathbb{F})##, and decide whether finitely many of them are sufficient to cover a given open set.

**6.**Let ##R := \mathbb{Q}[x,y]/\langle x^2+y^2-1 \rangle## and ##\varphi \in \operatorname{Der}(R)## a ##\mathbb{Q}-##linear derivation such that ##\varphi(x)=y\; , \;\varphi(y)=-x\,.##

A derivation ##\varphi\, : \,R \longrightarrow R ## of an algebra ##R## is a linear function with ##\varphi(p\cdot q)=\varphi(p)\cdot q + p\cdot \varphi(q)\,.##

**a.**Determine the kernel of ##\varphi\,.##

**b.**Solve ##\varphi^2 + \operatorname{id} = 0\,.##

**c.**Since ##x^2+y^2=1## we can apply Thales' theorem and identify ##(x,\alpha),(y,\alpha)## with the sides of a right triangle with hypotenuse (diameter) ##1## according to an angle ##\alpha\,.##

Show that

$$(x,\alpha +\beta ) = (x,\alpha)(y,\beta) + (x,\beta)(y,\alpha)

$$

**7.**(solved by @Periwinkle ) Prove that for all ##a,b,c \in \mathbb{R}## holds $$a>0\wedge b>0\wedge c>0 \Longleftrightarrow a+b+c>0\wedge ab+ac+bc>0\wedge abc>0\,.$$

**8.**(solved by @Periwinkle ) Let ##a,b \in L^2\left( \left[ -\frac{\pi}{2},+\frac{\pi}{2} \right] \right)## given as

$$

a(x)=11\sin(x) + 8\cos(x) \; , \;b(x)=4\sin(x) + 13\cos(x)

$$

Calculate the angle ##\varphi = \sphericalangle (a,b)## between the two vectors.

**9.**(solved by @Couchyam) Let ##\varepsilon_k :=\begin{cases}1&,\text{ if the decimal representation of }k\text{ has no digit }9\\0&, \text{ otherwise }\end{cases}##

Show that ##\sum_{k=1}^\infty \dfrac{\varepsilon_k}{k}## converges.

**10.**(solved by @Periwinkle ) Let ##x_0\in [a,b]\subseteq \mathbb{R}## and ##f\, : \,[a,b]\longrightarrow \mathbb{R}## continuous and differentiable on ##[a,b]-\{x_0\}##.

Furthermore exists the limit ##c:=\lim_{x \to x_0}f\,'(x)\,.## Then ##f(x)## is differentiable in ##x_0## with ##f\,'(x_0)=c\,.##

__Proof:__Let ##x\in [a,b]-\{x_0\}##. According to the mean value theorem for differentiable functions there is a

\xi(x) \in (\min\{x,x_0\},\max\{x,x_0\})

$$

with ##f\,'(\xi(x))=\dfrac{f(x)-f(x_0)}{x-x_0}\,.## Because ##\lim_{x \to x_0}\min\{x,x_0\}=\lim_{x \to x_0}\max\{x,x_0\}=x_0## we must have ##\lim_{x \to x_0}\xi(x)=x_0## and by assumption ##\lim_{x \to x_0}f\,'(\xi(x))=c##, hence ##\lim_{x \to x_0}\dfrac{f(x)-f(x_0)}{x-x_0}=c##.

What has to be regarded in this proof, and is there a way to avoid this hidden assumption?

**11.**A house ##H## and a rosary ##R## are near a circular lake ##L##.

The Gardener walks with two watering cans from the house to the lake, fills the cans and goes to the rosary. We assume ##\overline{HR}\cap L=\emptyset##.

At which point ##S## of the shore does he have to get water, so that his path length is minimal, and why?

**12.**How long is the distance on a direct flight from London to Los Angeles and where is its most northern point?

How long will it last by an assumed average speed of ##494## knots over ground? We neglect the influence of weather, esp. wind.

We take the values ##51°\,28'\,39''\,N,\,0°\,27'\,41''\,W## for LHR in London,

##\,33°\, 56'\,33''\,N,\,118°\,24'\,29''\,W## for LAX in Los Angeles, and a radius of ##3,958## miles for earth.

**13.**Trial before an American district court.

The witness claims he saw a blue cab drive off after a night accident. The judge decides to test the reliability of the witness.

Result: The witness recognizes the color correctly in the dark in ##80\%## of all cases.

A survey also found that ##85\%## of taxis in the city are green and ##15\%## are blue.

With which probability has the taxi actually been blue?

**14.**A monk climbs a mountain.

He starts at ##8\,##a.m. on ##1000\,##m above sea level and reaches the peak at ##8\,##p.m. at ##3000\,##m.

After a bivouac on top of the mountain, he returns to the valley the next morning and again starts at ##8\,##a.m. and returns at ##8\,##a.m.

**a.**If he wants to avoid being at the same time of day at the same place as the day before when he climbed upwards, which strategy must he use downwards, and why?

**b.**Assume he climbed at a rate of height ##u(t)## proportional to the square root of time, determine his path dependent on hourly noted time ##t##.

**c.**Assume he follows the same path downwards and the height of his path is given by ##d_1(t)= \dfrac{125}{9}(t-20)^2+1000## in the first three hours and ##d_2(t)=-125\,t+3500## for the rest of his way. When will he be at the same point as the day before and at which height.

**15.**I'm annoyed by my two new alarm clocks. They both are powered by the grid.

One leaps two minutes an hour and the other one runs a minute an hour too fast. Yesterday I took the effort and set them to the correct time. This morning, I assume there was a power loss, one clock showed exactly ##6\,a.m## while the other one showed ##7\,a.m.##

When did I set the clocks and how long did they run?

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