 #1
fresh_42
Mentor
 13,245
 10,195
Summary:

1.  2. posed and moderated by @QuantumQuest
3.  8. posed and moderated by @Math_QED
9.  10. posed and moderated by @fresh_42
keywords: calculus, abstract algebra, measure theory, mechanics, dynamical systems
Main Question or Discussion Point
Questions
1. (solved by @Piis3 )The maximum value of ##f## with ##f(x) = x^a e^{2a  x}## is minimal for which values of positive numbers ##a## ?
2. (solved by @KnotTheorist ) Find the equation of a curve such that ##y''## is always ##2## and the slope of the tangent line is ##10## at the point ##(2,6)##.
3. (solved by @nuuskur ) Let ##F_q## be the finite field with ##q## elements and ##n \geq 1## with ##(n,q)=1##. What is the smallest field extension of ##F_q## containing an ##n##th root of unity?
4. (solved by @nuuskur ) Show that there is no infinite countable ##\sigma##algebra.
5. (solved by @nuuskur ) Classify all groups of order ##6,8,## and ##12##.
6. (solved by @nuuskur ) Let ##G## be a group of order ##2m## with ##m>1## an odd number. Show that ##G## is not simple, i.e. has at least one non trivial normal subgroup.
7. (a) (solved by @nuuskur ) Show that a group ##G## is abelian if and only the map ##g \mapsto g^{1}## is a group homomorphism.
7. (b) (solved by @nuuskur ) Suppose that ##G## is a finite group and that we have ##\sigma \in \operatorname{Aut}(G)## with the neutral element ##e## as unique fixed point. If ##\sigma^2=\operatorname{id}_G##, prove that ##G## is abelian.
Hint: Prove that every element of ##G## can be written in the form ##x^{1} \sigma(x)## and apply ##\sigma## to such an expression.
(Credits will only be given for solving both subquestions.)
8. (solved by @Piis3 ) Show that there is no triple ##(a,b,c) \in \mathbb{Z}^3\setminus \{(0,0,0)\}## such that ##a^2 + b^2 = 3 c^2##.
Hint: Find all squares in ##\mathbb{Z}/4\mathbb{Z}##.
9. (solved by @nuuskur ) Three identical airplanes start at the same time at the vertices of an equilateral triangle with side length ##L##. Let's say the origin of our coordinate system is the center of the triangle. The planes fly at a constant speed ##v## above ground in the direction of the clockwise next airplane. How long will it take for the planes to reach the same point, and which are the flight paths?
10. (solved by @nuuskur ) The Schwarzian derivative of a holomorphic function ##f## is given by
$$
S_f(z)=\{\,f,z\,\} := \dfrac{d}{dz}\left( \dfrac{f^{''}(z)}{f^{'} (z)} \right)\dfrac{1}{2} \left( \dfrac{f^{''}(z)}{f^{'}(z)} \right)^2= \dfrac{f^{'''}(z)}{f^{'} (z)} \dfrac{3}{2} \left( \dfrac{f^{''}(z)}{f^{'}(z)} \right)^2
$$
Prove a chain rule for the Schwarzian derivative and show that
$$
\{\,f,z\,\} <0 \,\wedge \,\{\,h,z\,\}<0 \Longrightarrow \{\,f\circ h,z\,\}<0
$$
Schwarzian derivatives are used in dynamical systems to investigate attractors, in flows of surfaces, or in the theory of SchwarzChristoffel mappings.
11. (solved by @archaic ) Show geometrically that ##\sin^{1}(\frac{x}{\sqrt{x^2 + 1}}) = \tan^{1} x## for ##x \geq 0##.
12. (solved by @archaic and @Piis3 ) Give an example of a continuous function ##f(x)## on ##[0,1]## for which the conclusion of Rolle's theorem fails.
Rolle's theorem: If ##f## is differentiable on ##(0,1)## and ##f(0)=f(1)## then there is a point ##c\in (0,1)## such that ##f'(c)=0##.
13. (solved by @etotheipi ) David drives to work every working day by car. Outside towns he drives at an average speed of ##180\,\text{km/h}##. On the
##10\,\text{km}## in town, he drives at an average speed of ##40\,\text{km/h}##. As a result, he is often too fast and gets a ticket. Meanwhile he has realized that things can not go on like this and he decides to reduce his average speed by ##20\,\text{km/h}## in town as well as outside. How long is his way to work, if this reduces his average speed by ##40\,\text{km/h}## on total?
14. Show that ##2x^6+3y^6=z^3## has no other rational solutions than ##x=y=z=0\,.##
15. Let ##x,yz \in \mathbb{R}\{\,0\,\}## such that
$$
x+\dfrac{y}{z}=2\;\; , \;\;y+\dfrac{z}{x}=2\;\; , \;\;z+\dfrac{x}{y}=2
$$
Show that ##s:=x+y+z## can only have the values ##3## or ##7##.
You do not need to solve the equation system.
1. (solved by @Piis3 )The maximum value of ##f## with ##f(x) = x^a e^{2a  x}## is minimal for which values of positive numbers ##a## ?
2. (solved by @KnotTheorist ) Find the equation of a curve such that ##y''## is always ##2## and the slope of the tangent line is ##10## at the point ##(2,6)##.
3. (solved by @nuuskur ) Let ##F_q## be the finite field with ##q## elements and ##n \geq 1## with ##(n,q)=1##. What is the smallest field extension of ##F_q## containing an ##n##th root of unity?
4. (solved by @nuuskur ) Show that there is no infinite countable ##\sigma##algebra.
5. (solved by @nuuskur ) Classify all groups of order ##6,8,## and ##12##.
6. (solved by @nuuskur ) Let ##G## be a group of order ##2m## with ##m>1## an odd number. Show that ##G## is not simple, i.e. has at least one non trivial normal subgroup.
7. (a) (solved by @nuuskur ) Show that a group ##G## is abelian if and only the map ##g \mapsto g^{1}## is a group homomorphism.
7. (b) (solved by @nuuskur ) Suppose that ##G## is a finite group and that we have ##\sigma \in \operatorname{Aut}(G)## with the neutral element ##e## as unique fixed point. If ##\sigma^2=\operatorname{id}_G##, prove that ##G## is abelian.
Hint: Prove that every element of ##G## can be written in the form ##x^{1} \sigma(x)## and apply ##\sigma## to such an expression.
(Credits will only be given for solving both subquestions.)
8. (solved by @Piis3 ) Show that there is no triple ##(a,b,c) \in \mathbb{Z}^3\setminus \{(0,0,0)\}## such that ##a^2 + b^2 = 3 c^2##.
Hint: Find all squares in ##\mathbb{Z}/4\mathbb{Z}##.
9. (solved by @nuuskur ) Three identical airplanes start at the same time at the vertices of an equilateral triangle with side length ##L##. Let's say the origin of our coordinate system is the center of the triangle. The planes fly at a constant speed ##v## above ground in the direction of the clockwise next airplane. How long will it take for the planes to reach the same point, and which are the flight paths?
10. (solved by @nuuskur ) The Schwarzian derivative of a holomorphic function ##f## is given by
$$
S_f(z)=\{\,f,z\,\} := \dfrac{d}{dz}\left( \dfrac{f^{''}(z)}{f^{'} (z)} \right)\dfrac{1}{2} \left( \dfrac{f^{''}(z)}{f^{'}(z)} \right)^2= \dfrac{f^{'''}(z)}{f^{'} (z)} \dfrac{3}{2} \left( \dfrac{f^{''}(z)}{f^{'}(z)} \right)^2
$$
Prove a chain rule for the Schwarzian derivative and show that
$$
\{\,f,z\,\} <0 \,\wedge \,\{\,h,z\,\}<0 \Longrightarrow \{\,f\circ h,z\,\}<0
$$
Schwarzian derivatives are used in dynamical systems to investigate attractors, in flows of surfaces, or in the theory of SchwarzChristoffel mappings.
11. (solved by @archaic ) Show geometrically that ##\sin^{1}(\frac{x}{\sqrt{x^2 + 1}}) = \tan^{1} x## for ##x \geq 0##.
12. (solved by @archaic and @Piis3 ) Give an example of a continuous function ##f(x)## on ##[0,1]## for which the conclusion of Rolle's theorem fails.
Rolle's theorem: If ##f## is differentiable on ##(0,1)## and ##f(0)=f(1)## then there is a point ##c\in (0,1)## such that ##f'(c)=0##.
13. (solved by @etotheipi ) David drives to work every working day by car. Outside towns he drives at an average speed of ##180\,\text{km/h}##. On the
##10\,\text{km}## in town, he drives at an average speed of ##40\,\text{km/h}##. As a result, he is often too fast and gets a ticket. Meanwhile he has realized that things can not go on like this and he decides to reduce his average speed by ##20\,\text{km/h}## in town as well as outside. How long is his way to work, if this reduces his average speed by ##40\,\text{km/h}## on total?
14. Show that ##2x^6+3y^6=z^3## has no other rational solutions than ##x=y=z=0\,.##
15. Let ##x,yz \in \mathbb{R}\{\,0\,\}## such that
$$
x+\dfrac{y}{z}=2\;\; , \;\;y+\dfrac{z}{x}=2\;\; , \;\;z+\dfrac{x}{y}=2
$$
Show that ##s:=x+y+z## can only have the values ##3## or ##7##.
You do not need to solve the equation system.
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