 10,304
 7,020
 Summary

1. Lie Algebra. (3 points)
2. Commutative Algebra. (1 point)
3. Differential Equation. (3 points)
4. Schwartzian Derivative. (2 points)
5. Free Fall. (3 points)
6. Predator  Prey. (1 point)
7. Weights. (2 points)
8. Rotation Invariance. (3 points)
9. Differential Forms. (2 points)
10. Limits. (2 points)
11.  15. High School Questions.
We have a prize this month donated by one of our most valued members, and that's what the points are for. The first who achieves 6 points, will win a Gold Membership.
Questions
1. Let ##\mathfrak{g}## be a Lie algebra. Define
$$
\mathfrak{A(g)} = \{\,\alpha\, : \,\mathfrak{g}\longrightarrow \mathfrak{g}\,\,\forall \,X,Y \in \mathfrak{g}\, : \,0=[\alpha(X),Y]+[X,\alpha(Y)]\,\}
$$
Show that ##\mathfrak{A(g)}## is a Lie algebra and ##X.\alpha (Y)=[X,\alpha(Y)]\alpha([X,Y])## defines a representation of ##\mathfrak{g}## on ##\mathfrak{A}(g)##.
2. Let ##R## be a commutative ring with ##1## and ##I## an ideal. Show that ##R/I## is an integral domain if and only if ##I## is a prime ideal, and that ##R/I## is a field if and only if ##I## is a maximal ideal.
3. (solved by @Bullington ) Solve ##x^2y''+xy'y=x^3## for positive ##x##.
4. Show that the Schwarzian Derivative
$$
(Sf)(z) := \left( \dfrac{f''(z)}{f'(z)} \right)' \dfrac{1}{2} \left( \dfrac{f''(z)}{f'(z)} \right)^2
$$
vanishes if and only if ##f(z)=\dfrac{az+b}{cz+d}\,\,## is a Möbius transformation.
5. (solved by @Bullington ) Free Fall. Let ##x(t)## be the height at time ##t##, measured positively on the downward direction. If we consider only gravity, then
##\ddot{x}(t)=\dfrac{d^2x}{dt^2}=a## is a constant, denoted ##g##, the acceleration due to gravity. Note that ##F = ma =
mg##. Air resistance encountered depends on the shape of the object and other things, but under most circumstances, the most significant effect is a force opposing the motion which is proportional to a power of the velocity ##v(t)=\dot{x}(t)##. So
$$
\ddot{x}(t) \cdot m = m\cdot g  k\dot{x}(t)^n
$$
which is a second order differential equation, but there is no ##x## term. So it is first order in ##\dot{x}##.
Therefore,
$$
\dfrac{dv}{dt} = g \dfrac{k}{m}v^n
$$
This is not easy to solve, so we will make the simplifying approximation that ##n = 1## (if ##v## is small, there is not much difference between ##v## and ##v^n##). Therefore, we have to solve
$$
\dfrac{dv}{dt} +\dfrac{k}{m} v = g
$$
6. Consider a land populated by foxes and rabbits, where the foxes prey upon the rabbits. Let ##x(t)## and ##y(t)## be the number of rabbits and foxes, respectively, at time ##t##. In the absence of predators, at any time, the number of rabbits would grow at a rate proportional to the number of rabbits at that time. However, the presence of predators also causes the number of rabbits to decline in proportion to the number of encounters between a fox and a rabbit, which is proportional to the product ##x(t)y(t)##. Therefore, ##dx/dt = axbxy## for some positive constants ##a## and ##b##. For the foxes, the presence of other foxes represents competition for food, so the number declines proportionally to the number of foxes but grows proportionally to the number of encounters. Therefore ##dy/dt = cy + dxy## for some positive constants ##c## and ##d##. The system
$$
\dot{x}(t)=\dfrac{dx}{dt} = ax(t)bx(t)y(t) \; , \;\dot{y}(t)=\dfrac{dy}{dt} = cy(t)+dx(t)y(t)
$$
is our mathematical model. Eliminate the time parameter and find the relation between the population of foxes and the number of rabbits for parameters ##a=10\, , \,b=2\, , \,c=7\, , \,d=1\,.##
7. (solved by @Periwinkle ) Five vessels contain ##100## balls each. Some vessels contain only balls of ##10\, g## mass, while the other vessels contain only balls of ##11\, g## mass. How can we determine with a single weighing which results in a mass, which vessels contain balls of ##10\, g## and which contain balls of ##11\, g##? (It is allowed to remove balls from the vessels.)
8. Let ##f \in L^1(\mathbb{R}^3)## be rotation symmetric, i.e. ##f(Rx)=f(x)## for all ##R \in \operatorname{SO}(3)##. Show that the Fourier transform ##\mathcal{F}f## is rotation symmetric, too, and calculate ##\mathcal{F}f## of ##f\, : \,\mathbb{R}^3\longrightarrow \mathbb{R}## defined by
$$
f(x)=\dfrac{1}{x} \chi_{B_1(0)}(x)
$$
with the Euclidean norm ##\,.\,##, the unit ball ##B_1(0)## around the origin, and the characteristic function ##\chi##.
9. (solved by @cbarker1 ) Solve ##(3x^2y^2+x^2)\,dx+(2x^3y+y^2)\,dy=0\,.##
10. (solved by @cbarker1 ) Calculate ##\lim_{x \to 0}\dfrac{\cos^2 x1}{\sinh^2 x}## and ##\lim_{x \to 0}\dfrac{e^x+e^{x}2x^2}{(\cos x 1)^2}\,.##
11. There are two bands in front of you. The two bands are of different lengths and made of different materials. But both take exactly an hour to burn from one end to the other. The burning speed is not constant, so the tape can burn fast at the beginning, then slower and faster, or randomly. You only have a box of matches and you should measure exactly ##45## minutes with the help of the tapes. You must not cut the tapes, use a watch, etc.!
12. At the end of a one round chess tournament in which all players played once against each other we have the following result:
1. Alan
2. Bernie
3. Chuck
4. David
5. Ernest
The ranking is unambiguous, i.e. all have different scores, and as usual, a victory gets ##1## point, a draw ##1/2##. Bernie is the only one who didn't lose, Ernest the only one who didn't win.
Who played whom with which result?
13. A unit ##e## is an element for which there is a multiplicative inverse, i.e. there is an ##e'## such that ##e\cdot e'=e'\cdot e =1\,.## Units are divisors of ##1\,.##
An irreducible element ##n\neq 0## is an element, which cannot be written as ##n=a\cdot b## unless either ##a## or ##b## is a unit.
A prime ##p## is an element, which is not a unit and if ##p\,\,a\cdot b## then either ##p\,\,a## or ##p\,\,b\,.##
Show that primes are irreducible, and irreducible elements are either units or primes.
Bonus: Which essential property of the integers do we need?
14. The border collie Boy is at the end of a 1 km flock of sheep, which moves forward at a constant speed. As a control he now walks  with a greater constant speed than the herd  from the end to the top of the herd and back to his place at the end of the flock. When he arrives back, the flock of sheep has walked exactly one kilometer further. Which distance did Boy run?
15. What is the smallest limit ##L> \dfrac{\pi}{6}## such that
$$
\int_{\pi/6}^{L}\, \dfrac{dx}{\sin^2 x}= \int_{\pi/6}^{L}\, \dfrac{dx}{1\cos x} + \int_{\pi/6}^{L}\, 6\,\dfrac{\cot x}{\sin x}\,dx
$$
Questions
1. Let ##\mathfrak{g}## be a Lie algebra. Define
$$
\mathfrak{A(g)} = \{\,\alpha\, : \,\mathfrak{g}\longrightarrow \mathfrak{g}\,\,\forall \,X,Y \in \mathfrak{g}\, : \,0=[\alpha(X),Y]+[X,\alpha(Y)]\,\}
$$
Show that ##\mathfrak{A(g)}## is a Lie algebra and ##X.\alpha (Y)=[X,\alpha(Y)]\alpha([X,Y])## defines a representation of ##\mathfrak{g}## on ##\mathfrak{A}(g)##.
2. Let ##R## be a commutative ring with ##1## and ##I## an ideal. Show that ##R/I## is an integral domain if and only if ##I## is a prime ideal, and that ##R/I## is a field if and only if ##I## is a maximal ideal.
3. (solved by @Bullington ) Solve ##x^2y''+xy'y=x^3## for positive ##x##.
4. Show that the Schwarzian Derivative
$$
(Sf)(z) := \left( \dfrac{f''(z)}{f'(z)} \right)' \dfrac{1}{2} \left( \dfrac{f''(z)}{f'(z)} \right)^2
$$
vanishes if and only if ##f(z)=\dfrac{az+b}{cz+d}\,\,## is a Möbius transformation.
5. (solved by @Bullington ) Free Fall. Let ##x(t)## be the height at time ##t##, measured positively on the downward direction. If we consider only gravity, then
##\ddot{x}(t)=\dfrac{d^2x}{dt^2}=a## is a constant, denoted ##g##, the acceleration due to gravity. Note that ##F = ma =
mg##. Air resistance encountered depends on the shape of the object and other things, but under most circumstances, the most significant effect is a force opposing the motion which is proportional to a power of the velocity ##v(t)=\dot{x}(t)##. So
$$
\ddot{x}(t) \cdot m = m\cdot g  k\dot{x}(t)^n
$$
which is a second order differential equation, but there is no ##x## term. So it is first order in ##\dot{x}##.
Therefore,
$$
\dfrac{dv}{dt} = g \dfrac{k}{m}v^n
$$
This is not easy to solve, so we will make the simplifying approximation that ##n = 1## (if ##v## is small, there is not much difference between ##v## and ##v^n##). Therefore, we have to solve
$$
\dfrac{dv}{dt} +\dfrac{k}{m} v = g
$$
6. Consider a land populated by foxes and rabbits, where the foxes prey upon the rabbits. Let ##x(t)## and ##y(t)## be the number of rabbits and foxes, respectively, at time ##t##. In the absence of predators, at any time, the number of rabbits would grow at a rate proportional to the number of rabbits at that time. However, the presence of predators also causes the number of rabbits to decline in proportion to the number of encounters between a fox and a rabbit, which is proportional to the product ##x(t)y(t)##. Therefore, ##dx/dt = axbxy## for some positive constants ##a## and ##b##. For the foxes, the presence of other foxes represents competition for food, so the number declines proportionally to the number of foxes but grows proportionally to the number of encounters. Therefore ##dy/dt = cy + dxy## for some positive constants ##c## and ##d##. The system
$$
\dot{x}(t)=\dfrac{dx}{dt} = ax(t)bx(t)y(t) \; , \;\dot{y}(t)=\dfrac{dy}{dt} = cy(t)+dx(t)y(t)
$$
is our mathematical model. Eliminate the time parameter and find the relation between the population of foxes and the number of rabbits for parameters ##a=10\, , \,b=2\, , \,c=7\, , \,d=1\,.##
7. (solved by @Periwinkle ) Five vessels contain ##100## balls each. Some vessels contain only balls of ##10\, g## mass, while the other vessels contain only balls of ##11\, g## mass. How can we determine with a single weighing which results in a mass, which vessels contain balls of ##10\, g## and which contain balls of ##11\, g##? (It is allowed to remove balls from the vessels.)
8. Let ##f \in L^1(\mathbb{R}^3)## be rotation symmetric, i.e. ##f(Rx)=f(x)## for all ##R \in \operatorname{SO}(3)##. Show that the Fourier transform ##\mathcal{F}f## is rotation symmetric, too, and calculate ##\mathcal{F}f## of ##f\, : \,\mathbb{R}^3\longrightarrow \mathbb{R}## defined by
$$
f(x)=\dfrac{1}{x} \chi_{B_1(0)}(x)
$$
with the Euclidean norm ##\,.\,##, the unit ball ##B_1(0)## around the origin, and the characteristic function ##\chi##.
9. (solved by @cbarker1 ) Solve ##(3x^2y^2+x^2)\,dx+(2x^3y+y^2)\,dy=0\,.##
10. (solved by @cbarker1 ) Calculate ##\lim_{x \to 0}\dfrac{\cos^2 x1}{\sinh^2 x}## and ##\lim_{x \to 0}\dfrac{e^x+e^{x}2x^2}{(\cos x 1)^2}\,.##
11. There are two bands in front of you. The two bands are of different lengths and made of different materials. But both take exactly an hour to burn from one end to the other. The burning speed is not constant, so the tape can burn fast at the beginning, then slower and faster, or randomly. You only have a box of matches and you should measure exactly ##45## minutes with the help of the tapes. You must not cut the tapes, use a watch, etc.!
12. At the end of a one round chess tournament in which all players played once against each other we have the following result:
1. Alan
2. Bernie
3. Chuck
4. David
5. Ernest
The ranking is unambiguous, i.e. all have different scores, and as usual, a victory gets ##1## point, a draw ##1/2##. Bernie is the only one who didn't lose, Ernest the only one who didn't win.
Who played whom with which result?
13. A unit ##e## is an element for which there is a multiplicative inverse, i.e. there is an ##e'## such that ##e\cdot e'=e'\cdot e =1\,.## Units are divisors of ##1\,.##
An irreducible element ##n\neq 0## is an element, which cannot be written as ##n=a\cdot b## unless either ##a## or ##b## is a unit.
A prime ##p## is an element, which is not a unit and if ##p\,\,a\cdot b## then either ##p\,\,a## or ##p\,\,b\,.##
Show that primes are irreducible, and irreducible elements are either units or primes.
Bonus: Which essential property of the integers do we need?
14. The border collie Boy is at the end of a 1 km flock of sheep, which moves forward at a constant speed. As a control he now walks  with a greater constant speed than the herd  from the end to the top of the herd and back to his place at the end of the flock. When he arrives back, the flock of sheep has walked exactly one kilometer further. Which distance did Boy run?
15. What is the smallest limit ##L> \dfrac{\pi}{6}## such that
$$
\int_{\pi/6}^{L}\, \dfrac{dx}{\sin^2 x}= \int_{\pi/6}^{L}\, \dfrac{dx}{1\cos x} + \int_{\pi/6}^{L}\, 6\,\dfrac{\cot x}{\sin x}\,dx
$$
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