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 8,432
Summary:

Topology
Calculus
Group Theory
Main Question or Discussion Point
Questions
1. Let ##(X,d)## be a metric space. The open ball with center ##z\in X## of radius ##r > 0## is defined as
$$
B_r(z) :=\{\,x\in X\,\,d(x,z)<r\,\}
$$
a.) Give an example for
$$
\overline{B_r(z)} \neq K_r(z) :=\{\,x\in X\,\,d(x,z)\leq r\,\}
$$
Does at least one of the inclusions ##\subseteq## or ##\supseteq## always hold?
b.) What are the answers in the previous case, if we additionally assume that ##(X,d)## has an inner metric?
An inner metric ##d_0## associated to ##d## is defined as the infimum of all lengths of rectified curves between two points:
Let ##\sigma \, : \,[0,1]\longrightarrow X## with ##\sigma(0)=x\, , \,\sigma(1)=y## a rectified curve with length
$$
L(\sigma)=\sup \left\{ \left. \sum_{k=1}^n d(\sigma(t_{k1}),\sigma(t_k) )\,\right\,0=t_0<t_1<\cdots < t_n=1\, , \,n\in \mathbb{N} \right\}
$$
Then ##d_0(x,y)=\inf L(\sigma)\,.##
2. Let ##f(z)=\dfrac{7z51}{z^212z+27}## be a complex function.
a.) Determine the Laurent series of ##f(z)## and their radius of convergences around ##z=3## in the cases where ##0## is in the area of convergence, and ##10## is in the area of convergence.
b.) Determine ##\lim_{z \to 3}f(z)\, , \,\operatorname{Res}(f,3)## and the kind of singularity in ##z=3\,.##
3. Write the following groups as amalgamated products of cyclic groups:
a.) ##G=\langle x,y\,\, x^3y^{3},y^6 \rangle##
b.) ##H=\langle x,y\,\, x^{30}, y^{70},x^3y^{5} \rangle##
4. Prove that there are uncountably many groups, which are generated by two elements, and not finitely presented.
Hint: There are uncountably many nonisomorphic groups with two generators [Bernhard Neumann, 1937].
5. (solved by @archaic ) Let ##f : [1,\infty) \longrightarrow [0,\infty)## be a continuously differentiable function. Write ##S## for the solid of revolution of the graph ##y = f(x)## about the ##x##axis. If the surface area of ##S## is finite, then so is the volume.
6. Calculate ##\sum_{k,j=1}^\infty \dfrac{1}{kj(k+j)^2}##
7. Calculate ##S:= \displaystyle{\sum_{n=0}^\infty}\,\displaystyle{\sum_{k=0}^n}\,\dfrac{3^k(2n2k)!(2k)!}{2^k8^n[(nk)!]^2[k!]^2(2n(1+2k)+(14k^2))}##
8. (solved by @Mastermind01 ) Solve ##y'xy=\sqrt{x^2y^2}##
9. Let ##(a_n)_{n\in \mathbb{N}}\, , \,(b_n)_{n\in \mathbb{N}} \subseteq \mathbb{R}_{\geq 0}## be two sequences of nonnegative numbers, where not all sequence elements vanish, and be ##p,q\in \mathbb{R}## with ##1<p,q<\infty\, , \,\frac{1}{p}+\frac{1}{q}=1\,.## Prove
$$
\sum_{n=1}^\infty \sum_{m=1}^\infty \dfrac{a_nb_m}{n+m} < \dfrac{\pi}{\sin (\pi/p)} \cdot \left(\sum_{n=1}^\infty a_n^p\right)^{\frac{1}{p}} \cdot \left(\sum_{m=1}^\infty b_m^q\right)^{\frac{1}{q}}
$$
10. Let ##f\, : \,\mathbb{R}_{\geq 0} \longrightarrow \mathbb{R}_{\geq 0}## be an integrable function and ##p>1\,.## Prove
$$
\int_0^\infty\left(\dfrac{1}{x}\int_0^x f(t)\,dt\right)^p\,dx \leq \left(\dfrac{p}{p1}\right)^p \int_0^\infty (f(x))^p\,dx
$$
Hint: Substitute ##t=xu^{p/r}## and at the end ##r=p1\,.##
High Schoolers only
11. (solved by @Not anonymous ) Choose any odd prime, square it and subtract one. Show that the result is always divisible by twentyfour except for three.
What can be said, if we take the prime up to the power four, and subtract one?
12. In a square of side length ##4##, there is a circle of radius ##1## in each corner. In the center of the square is another circle that touches the other four. Analogously, in the threedimensional case, in the center of a cube of edge length ##4##, there would be a sphere which would touch eight spheres of radius ##1## placed in the corners of the cube. In which dimension does the central hypersphere become so large that it touches all sides of the hypercube?
13. There is only one rule at Christmas at the world's richest family: The gifts have to be expensive, heavy and glamorous. So they all present statues of pure gold. It may be large figure, a tiger sculpture or an opulent candlestick. The eldest son who doesn't live at home anymore receives gifts of nine tons total, but none of which is heavier than a ton. He wants to bring home all of them, but only could rent trucks which can load three tons maximal. How many trucks are needed to at least be able to transport all gifts of gold at the same time?
14. Prove ##\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\geq \dfrac{3}{2}## for ##a,b,c >0##
15. Let ##\mathbf{x}=(x_1,\ldots ,x_n)\, , \,\mathbf{y}=(y_1,\ldots ,y_n)## be tuples of positive numbers. Prove
$$
\prod_{k=1}^{n} (x_k+y_k)^{1/n} \geq \prod_{k=1}^{n} x_k^{1/n} + \prod_{k=1}^{n} y_k^{1/n}
$$
1. Let ##(X,d)## be a metric space. The open ball with center ##z\in X## of radius ##r > 0## is defined as
$$
B_r(z) :=\{\,x\in X\,\,d(x,z)<r\,\}
$$
a.) Give an example for
$$
\overline{B_r(z)} \neq K_r(z) :=\{\,x\in X\,\,d(x,z)\leq r\,\}
$$
Does at least one of the inclusions ##\subseteq## or ##\supseteq## always hold?
b.) What are the answers in the previous case, if we additionally assume that ##(X,d)## has an inner metric?
An inner metric ##d_0## associated to ##d## is defined as the infimum of all lengths of rectified curves between two points:
Let ##\sigma \, : \,[0,1]\longrightarrow X## with ##\sigma(0)=x\, , \,\sigma(1)=y## a rectified curve with length
$$
L(\sigma)=\sup \left\{ \left. \sum_{k=1}^n d(\sigma(t_{k1}),\sigma(t_k) )\,\right\,0=t_0<t_1<\cdots < t_n=1\, , \,n\in \mathbb{N} \right\}
$$
Then ##d_0(x,y)=\inf L(\sigma)\,.##
2. Let ##f(z)=\dfrac{7z51}{z^212z+27}## be a complex function.
a.) Determine the Laurent series of ##f(z)## and their radius of convergences around ##z=3## in the cases where ##0## is in the area of convergence, and ##10## is in the area of convergence.
b.) Determine ##\lim_{z \to 3}f(z)\, , \,\operatorname{Res}(f,3)## and the kind of singularity in ##z=3\,.##
3. Write the following groups as amalgamated products of cyclic groups:
a.) ##G=\langle x,y\,\, x^3y^{3},y^6 \rangle##
b.) ##H=\langle x,y\,\, x^{30}, y^{70},x^3y^{5} \rangle##
4. Prove that there are uncountably many groups, which are generated by two elements, and not finitely presented.
Hint: There are uncountably many nonisomorphic groups with two generators [Bernhard Neumann, 1937].
5. (solved by @archaic ) Let ##f : [1,\infty) \longrightarrow [0,\infty)## be a continuously differentiable function. Write ##S## for the solid of revolution of the graph ##y = f(x)## about the ##x##axis. If the surface area of ##S## is finite, then so is the volume.
6. Calculate ##\sum_{k,j=1}^\infty \dfrac{1}{kj(k+j)^2}##
7. Calculate ##S:= \displaystyle{\sum_{n=0}^\infty}\,\displaystyle{\sum_{k=0}^n}\,\dfrac{3^k(2n2k)!(2k)!}{2^k8^n[(nk)!]^2[k!]^2(2n(1+2k)+(14k^2))}##
8. (solved by @Mastermind01 ) Solve ##y'xy=\sqrt{x^2y^2}##
9. Let ##(a_n)_{n\in \mathbb{N}}\, , \,(b_n)_{n\in \mathbb{N}} \subseteq \mathbb{R}_{\geq 0}## be two sequences of nonnegative numbers, where not all sequence elements vanish, and be ##p,q\in \mathbb{R}## with ##1<p,q<\infty\, , \,\frac{1}{p}+\frac{1}{q}=1\,.## Prove
$$
\sum_{n=1}^\infty \sum_{m=1}^\infty \dfrac{a_nb_m}{n+m} < \dfrac{\pi}{\sin (\pi/p)} \cdot \left(\sum_{n=1}^\infty a_n^p\right)^{\frac{1}{p}} \cdot \left(\sum_{m=1}^\infty b_m^q\right)^{\frac{1}{q}}
$$
10. Let ##f\, : \,\mathbb{R}_{\geq 0} \longrightarrow \mathbb{R}_{\geq 0}## be an integrable function and ##p>1\,.## Prove
$$
\int_0^\infty\left(\dfrac{1}{x}\int_0^x f(t)\,dt\right)^p\,dx \leq \left(\dfrac{p}{p1}\right)^p \int_0^\infty (f(x))^p\,dx
$$
Hint: Substitute ##t=xu^{p/r}## and at the end ##r=p1\,.##
High Schoolers only
11. (solved by @Not anonymous ) Choose any odd prime, square it and subtract one. Show that the result is always divisible by twentyfour except for three.
What can be said, if we take the prime up to the power four, and subtract one?
12. In a square of side length ##4##, there is a circle of radius ##1## in each corner. In the center of the square is another circle that touches the other four. Analogously, in the threedimensional case, in the center of a cube of edge length ##4##, there would be a sphere which would touch eight spheres of radius ##1## placed in the corners of the cube. In which dimension does the central hypersphere become so large that it touches all sides of the hypercube?
13. There is only one rule at Christmas at the world's richest family: The gifts have to be expensive, heavy and glamorous. So they all present statues of pure gold. It may be large figure, a tiger sculpture or an opulent candlestick. The eldest son who doesn't live at home anymore receives gifts of nine tons total, but none of which is heavier than a ton. He wants to bring home all of them, but only could rent trucks which can load three tons maximal. How many trucks are needed to at least be able to transport all gifts of gold at the same time?
14. Prove ##\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\geq \dfrac{3}{2}## for ##a,b,c >0##
15. Let ##\mathbf{x}=(x_1,\ldots ,x_n)\, , \,\mathbf{y}=(y_1,\ldots ,y_n)## be tuples of positive numbers. Prove
$$
\prod_{k=1}^{n} (x_k+y_k)^{1/n} \geq \prod_{k=1}^{n} x_k^{1/n} + \prod_{k=1}^{n} y_k^{1/n}
$$
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