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Stochastic
Topology
Algebra
Analysis
Questions
1. (solved by @tnich ) Show that ##\sin\frac{\pi}{m} \sin\frac{2\pi}{m}\sin\frac{3\pi}{m}\cdots \sin\frac{(m  1)\pi}{m} = \frac{m}{2^{m  1}}## for ##m## = ##2, 3, \dots## (@QuantumQuest)
2. (solved by @PeroK ) Show that when a quantity grows or decays exponentially, the rate of increase over a fixed time interval is constant (i.e. it depends only on the time interval and not on the time at which the interval begins). (@QuantumQuest)
3. (solved by @Antarres ) We define the weighted Höldermean as
$$
M_w^p :=\left(\sum_{k=1}^n w_kx_k^p\right)^{\frac{1}{p}} , \,M_w^0:=\lim_{p \to 0}M_w^p = \prod_{k=1}^n x_k^{w_k}
$$
for positive, real numbers ##x_1,\ldots ,x_n > 0## and a weight ##w=(w_1,\ldots,w_n)## with ##w_1+\ldots +w_n=1\, , \,w_k>0## and a ##p \in \mathbb{R}\{\,0\,\}##.
Prove ##M_w^r \leq M_w^s## whenever ##r<s\,.##
Hint: Use Jensen's theorem for convex functions (see October 2019 / 8a). (@fresh_42)
4. Let ##q## be a rational number such that ##\sin(\pi q)## is rational. Show that ##2\sin(\pi q)## is an integer. (@Infrared)
5. (solved by @Fred Wright and @tnich ) Evaluate ##\displaystyle{\int_0^{2\pi}} e^{\cos(\theta)}\cos(\sin(\theta))d\theta.## (@Infrared)
6. Let ##\{X_n\}_{n=1}^\infty## be a sequence of independent random variables on a probability space ##(\Omega, \mathcal{F}, \mathbb{P})##.
(1) Show that ##\mathbb{P}\left(\sum_{n=1}^\infty X_n \mathrm{\ converges \ in \ \mathbb{R}} \right) \in \{0,1\}##
(2) If in addition ##\mathbb{E}[X_n^2] < \infty, \mathbb{E}[X_n] = 0## for all ##n \geq 1## and ##\sum_{n=1}^\infty \operatorname{Var}(X_n) < \infty##, show that the above probability is ##1##.
(@Math_QED)
7. Consider the real ##n##dimensional projective space ##\mathbb{R}P^{n}## which is ##\left(\mathbb{R}^{n+1}\setminus \{(0,\dots, 0)\}\right)/\sim##
where ##\sim## is the equivalence relation given by
$$(x_1, \dots, x_{n+1}) \sim (y_1, \dots, y_{n+1}) \iff \exists \lambda \neq 0: (x_1, \dots, x_{n+1}) = \lambda(y_1, \dots, y_{n+1})$$
Give ##\mathbb{R}P^{n}## the quotient topology. Show that this topological space is Hausdorff and second countable.
(@Math_QED)
8. Let ##D## be a division ring. Consider the matrix ring ##R = M_n(D)## of ##n \times n##matrices with coefficients in ##D##. Prove that there exists a simple left ##R##module and that up to isomorphism, it is unique.
Hint: Composition series and the JordanHölder theorem. (@Math_QED)
9. (solved by @PeroK ) Show that ##\exp(y) \leq 1+y+y^2/2## for ##y < 0##. Since this is elementary, you cannot use any other inequalities without proof.
(@Math_QED)
10. (solved by @PeroK ) If ##f## has the real Fourier representation $$f(x)=\dfrac{a_0}{2}+\sum_{k=1}^\infty (a_k\cos kx + b_k\sin kx)$$ prove
$$
\dfrac{1}{\pi} \int_{\pi}^{\pi}f(x)^2\,dx = \dfrac{a_0^2}{2}+\sum_{k=1}^{\infty}\left(a_k^2+b_k^2\right)
$$
(@fresh_42)
High Schoolers only
11. (solved by @Not anonymous ) A kid throws small balls upwards. It throws each ball when the previous thrown one is at the maximum height of its course. What is the height that balls reach if the kid throws two of them per second?
12. (solved by @etotheipi ) Which rain drops fall faster, small or big ones and why?
13. (solved by @Not anonymous ) A radio station at point ##A## transmits a signal which is received by the receivers ##B## and ##C##. A listener located at ##B## is listening to the signal via his receiver and after one second hears the signal via receiver ##C## which has a strong loudspeaker. What is the distance between ##B## and ##C##?
14. (solved by @Not anonymous ) Mr. Smith on a full up flight with 50 passengers on a CRJ 100 had lost his boarding pass. The flight attendant tells him to sit anywhere. All other passengers sit on their booked seats, unless it is already occupied, in which case they randomly choose another seat just like Mr. Smith did. What are the chances that the last passenger gets the seat printed on his boarding pass?
15. (solved by @Not anonymous ) On the first flight day of a little island hopper there was no wind during the return flight. How does the total flight duration from outward and return flight change if, instead, a strong headwind blows on the way to the neighboring island  and on the way back, an equally strong tailwind?
1. (solved by @tnich ) Show that ##\sin\frac{\pi}{m} \sin\frac{2\pi}{m}\sin\frac{3\pi}{m}\cdots \sin\frac{(m  1)\pi}{m} = \frac{m}{2^{m  1}}## for ##m## = ##2, 3, \dots## (@QuantumQuest)
2. (solved by @PeroK ) Show that when a quantity grows or decays exponentially, the rate of increase over a fixed time interval is constant (i.e. it depends only on the time interval and not on the time at which the interval begins). (@QuantumQuest)
3. (solved by @Antarres ) We define the weighted Höldermean as
$$
M_w^p :=\left(\sum_{k=1}^n w_kx_k^p\right)^{\frac{1}{p}} , \,M_w^0:=\lim_{p \to 0}M_w^p = \prod_{k=1}^n x_k^{w_k}
$$
for positive, real numbers ##x_1,\ldots ,x_n > 0## and a weight ##w=(w_1,\ldots,w_n)## with ##w_1+\ldots +w_n=1\, , \,w_k>0## and a ##p \in \mathbb{R}\{\,0\,\}##.
Prove ##M_w^r \leq M_w^s## whenever ##r<s\,.##
Hint: Use Jensen's theorem for convex functions (see October 2019 / 8a). (@fresh_42)
4. Let ##q## be a rational number such that ##\sin(\pi q)## is rational. Show that ##2\sin(\pi q)## is an integer. (@Infrared)
5. (solved by @Fred Wright and @tnich ) Evaluate ##\displaystyle{\int_0^{2\pi}} e^{\cos(\theta)}\cos(\sin(\theta))d\theta.## (@Infrared)
6. Let ##\{X_n\}_{n=1}^\infty## be a sequence of independent random variables on a probability space ##(\Omega, \mathcal{F}, \mathbb{P})##.
(1) Show that ##\mathbb{P}\left(\sum_{n=1}^\infty X_n \mathrm{\ converges \ in \ \mathbb{R}} \right) \in \{0,1\}##
(2) If in addition ##\mathbb{E}[X_n^2] < \infty, \mathbb{E}[X_n] = 0## for all ##n \geq 1## and ##\sum_{n=1}^\infty \operatorname{Var}(X_n) < \infty##, show that the above probability is ##1##.
(@Math_QED)
7. Consider the real ##n##dimensional projective space ##\mathbb{R}P^{n}## which is ##\left(\mathbb{R}^{n+1}\setminus \{(0,\dots, 0)\}\right)/\sim##
where ##\sim## is the equivalence relation given by
$$(x_1, \dots, x_{n+1}) \sim (y_1, \dots, y_{n+1}) \iff \exists \lambda \neq 0: (x_1, \dots, x_{n+1}) = \lambda(y_1, \dots, y_{n+1})$$
Give ##\mathbb{R}P^{n}## the quotient topology. Show that this topological space is Hausdorff and second countable.
(@Math_QED)
8. Let ##D## be a division ring. Consider the matrix ring ##R = M_n(D)## of ##n \times n##matrices with coefficients in ##D##. Prove that there exists a simple left ##R##module and that up to isomorphism, it is unique.
Hint: Composition series and the JordanHölder theorem. (@Math_QED)
9. (solved by @PeroK ) Show that ##\exp(y) \leq 1+y+y^2/2## for ##y < 0##. Since this is elementary, you cannot use any other inequalities without proof.
(@Math_QED)
10. (solved by @PeroK ) If ##f## has the real Fourier representation $$f(x)=\dfrac{a_0}{2}+\sum_{k=1}^\infty (a_k\cos kx + b_k\sin kx)$$ prove
$$
\dfrac{1}{\pi} \int_{\pi}^{\pi}f(x)^2\,dx = \dfrac{a_0^2}{2}+\sum_{k=1}^{\infty}\left(a_k^2+b_k^2\right)
$$
(@fresh_42)
High Schoolers only
11. (solved by @Not anonymous ) A kid throws small balls upwards. It throws each ball when the previous thrown one is at the maximum height of its course. What is the height that balls reach if the kid throws two of them per second?
12. (solved by @etotheipi ) Which rain drops fall faster, small or big ones and why?
13. (solved by @Not anonymous ) A radio station at point ##A## transmits a signal which is received by the receivers ##B## and ##C##. A listener located at ##B## is listening to the signal via his receiver and after one second hears the signal via receiver ##C## which has a strong loudspeaker. What is the distance between ##B## and ##C##?
14. (solved by @Not anonymous ) Mr. Smith on a full up flight with 50 passengers on a CRJ 100 had lost his boarding pass. The flight attendant tells him to sit anywhere. All other passengers sit on their booked seats, unless it is already occupied, in which case they randomly choose another seat just like Mr. Smith did. What are the chances that the last passenger gets the seat printed on his boarding pass?
15. (solved by @Not anonymous ) On the first flight day of a little island hopper there was no wind during the return flight. How does the total flight duration from outward and return flight change if, instead, a strong headwind blows on the way to the neighboring island  and on the way back, an equally strong tailwind?
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