 #1
fresh_42
Mentor
 13,240
 10,183
Summary:

Functional Analysis, Topology, Differential Geometry, Analysis, Physics
Authors: Math_QED (MQ), Infrared (IR), Wrobel (WR), fresh_42 (FR).
Main Question or Discussion Point
1. (solved by @nuuskur ) Let ##V## be an infinite dimensional topological vector space. Show that the weak topology on ##V## is not induced by a norm. (MQ)
2. The matrix groups ##U(n)## and ##SL_n(\mathbb{C})## are submanifolds of ##\mathbb{C}^{n^2}=\mathbb{R}^{2n^2}##. Do they intersect transversely? (IR)
3. (solved by @zinq ) Let ##f:\mathbb{R}\to \mathbb{R},\quad f>0## be a continuous and ##1##periodic function. Show that
$$\int_0^1\frac{f(x+a)}{f(x)}dx\ge 1$$ for any ##a\in\mathbb{R}##. (WR)
4. (solved by @nuuskur ) Prove that ##C[0,1]## is not dual to a Banach space. (WR)
5. (solved by @julian , @mathwonk ) Let ##(M,g)## be a Riemannian manifold. Let ##f:M\to\mathbb{R}## be a smooth function such that ##\nabla f=1## everywhere on ##M##. Show that all integral curves of ##\nabla f## are geodesics. (IR)
6. (solved by @Incand ) Let ##f: (a,b) \to \mathbb{R}## be a continuous function that is midpoint convex, i.e. ##f(\frac{1}{2}x + \frac{1}{2}y) \leq \frac{1}{2}f(x) + \frac{1}{2}f(y)## for all ##x,y \in (a,b)## . Show that ##f## is convex, i.e. ##f(tx + (1t)y) \leq tf(x) + (1t)f(y)## for all ##x,y \in (a,b)## and ##0 \leq t \leq 1##. (MQ)
7. (solved by @nuuskur ) Consider ##D:=\{f: \mathbb{R} \to \mathbb{R}\mid \mathrm{\ f \ is \ not \ continuous}\}##. What is the cardinality of ##D##? (MQ)
8. (solved by @zinq ) Let ##X## be a compact manifold such that ##\pi_1(X)## (the fundamental group of ##X##) is finite and nontrivial. Show that ##\pi_k(X)## is also nontrivial for some ##k\geq 2.## (IR)
9. (solved by @mathwonk ) Let ##(X,d)## be a compact metric space and ##f:X\to X## be a mapping onto. Assume that ##d(f(x),f(y))\le d(x,y),\quad \forall x,y\in X.## Show that ##d(f(x),f(y))= d(x,y),\quad \forall x,y\in X.## (WR)
10. (solved by @etotheipi ) Calculate the electrostatic potential ##U(a)## of a surface ##S=\{\,(x,y,z)\in \mathbb{R}^3\,\,x^2+y^2=z^2,\,0\leq z\leq 1\,\}## charged with a field of homogeneous density ##\rho## at the point ##a=(0,0,1)##. (FR)
High Schoolers only
11. (solved by @Isaac0427 , @ItsukaKitto ) Prove that the product of a finite number of sums of two integers squares is again a sum of two integers squared.
$$
(a_1^2+b_1^2)\cdot (a_2^2+b_2^2)\cdot \ldots \cdot (a_n^2+b_n^2)=a^2+b^2
$$
12. (solved by @etotheipi ) Given a positive integer in decimal representation without zeros. We build a new integer by concatenation of the number of even digits, the number of odd digits, and the number of all digits (the sum of the former two). Then we proceed with that number.
Determine whether this algorithm always comes to a halt. What is or should be the criterion to stop?
13. (solved by @Lament ) List all real functions ##f\, : \,\mathbb{R}\longrightarrow \mathbb{R}## with the following properties:
\begin{align*}
f(xy)&=f(x)f(y)f(x)f(y)+2\\
f(x+y)&=f(x)+f(y)+2xy1\\
f(1)&=2
\end{align*}
14. (solved by @ItsukaKitto ) Find all real solutions ##(x,y)## such that
$$
\sin^4x = y^4+x^2y^24y^2+4\, , \,\cos^4x=x^4+x^2y^24x^2+1
$$
15. (solved by @etotheipi , @Adesh) Prove
$$
\dfrac{(2n)!}{(n!)^2}>\dfrac{4^n}{n+1}
$$
for all natural numbers ##n>1.##
2. The matrix groups ##U(n)## and ##SL_n(\mathbb{C})## are submanifolds of ##\mathbb{C}^{n^2}=\mathbb{R}^{2n^2}##. Do they intersect transversely? (IR)
3. (solved by @zinq ) Let ##f:\mathbb{R}\to \mathbb{R},\quad f>0## be a continuous and ##1##periodic function. Show that
$$\int_0^1\frac{f(x+a)}{f(x)}dx\ge 1$$ for any ##a\in\mathbb{R}##. (WR)
4. (solved by @nuuskur ) Prove that ##C[0,1]## is not dual to a Banach space. (WR)
5. (solved by @julian , @mathwonk ) Let ##(M,g)## be a Riemannian manifold. Let ##f:M\to\mathbb{R}## be a smooth function such that ##\nabla f=1## everywhere on ##M##. Show that all integral curves of ##\nabla f## are geodesics. (IR)
6. (solved by @Incand ) Let ##f: (a,b) \to \mathbb{R}## be a continuous function that is midpoint convex, i.e. ##f(\frac{1}{2}x + \frac{1}{2}y) \leq \frac{1}{2}f(x) + \frac{1}{2}f(y)## for all ##x,y \in (a,b)## . Show that ##f## is convex, i.e. ##f(tx + (1t)y) \leq tf(x) + (1t)f(y)## for all ##x,y \in (a,b)## and ##0 \leq t \leq 1##. (MQ)
7. (solved by @nuuskur ) Consider ##D:=\{f: \mathbb{R} \to \mathbb{R}\mid \mathrm{\ f \ is \ not \ continuous}\}##. What is the cardinality of ##D##? (MQ)
8. (solved by @zinq ) Let ##X## be a compact manifold such that ##\pi_1(X)## (the fundamental group of ##X##) is finite and nontrivial. Show that ##\pi_k(X)## is also nontrivial for some ##k\geq 2.## (IR)
9. (solved by @mathwonk ) Let ##(X,d)## be a compact metric space and ##f:X\to X## be a mapping onto. Assume that ##d(f(x),f(y))\le d(x,y),\quad \forall x,y\in X.## Show that ##d(f(x),f(y))= d(x,y),\quad \forall x,y\in X.## (WR)
10. (solved by @etotheipi ) Calculate the electrostatic potential ##U(a)## of a surface ##S=\{\,(x,y,z)\in \mathbb{R}^3\,\,x^2+y^2=z^2,\,0\leq z\leq 1\,\}## charged with a field of homogeneous density ##\rho## at the point ##a=(0,0,1)##. (FR)
High Schoolers only
11. (solved by @Isaac0427 , @ItsukaKitto ) Prove that the product of a finite number of sums of two integers squares is again a sum of two integers squared.
$$
(a_1^2+b_1^2)\cdot (a_2^2+b_2^2)\cdot \ldots \cdot (a_n^2+b_n^2)=a^2+b^2
$$
12. (solved by @etotheipi ) Given a positive integer in decimal representation without zeros. We build a new integer by concatenation of the number of even digits, the number of odd digits, and the number of all digits (the sum of the former two). Then we proceed with that number.
Determine whether this algorithm always comes to a halt. What is or should be the criterion to stop?
13. (solved by @Lament ) List all real functions ##f\, : \,\mathbb{R}\longrightarrow \mathbb{R}## with the following properties:
\begin{align*}
f(xy)&=f(x)f(y)f(x)f(y)+2\\
f(x+y)&=f(x)+f(y)+2xy1\\
f(1)&=2
\end{align*}
14. (solved by @ItsukaKitto ) Find all real solutions ##(x,y)## such that
$$
\sin^4x = y^4+x^2y^24y^2+4\, , \,\cos^4x=x^4+x^2y^24x^2+1
$$
15. (solved by @etotheipi , @Adesh) Prove
$$
\dfrac{(2n)!}{(n!)^2}>\dfrac{4^n}{n+1}
$$
for all natural numbers ##n>1.##
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