1. There will be no POW this week due to 추석 (thanksgiving) break. POW will resume next week.
2. The submission due for POW2019-12 is extended to Sep. 18 (Wed.).
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1. There will be no POW this week due to 추석 (thanksgiving) break. POW will resume next week.
2. The submission due for POW2019-12 is extended to Sep. 18 (Wed.).
Let \(I, J\) be connected open intervals such that \(I \cap J\) is a nonempty proper sub-interval of both \(I\) and\(J\). For instance, \(I = (0, 2)\) and \(J = (1, 3)\) form an example.
Let \(f\) (\(g\), resp.) be an orientation-preserving homeomorphism of the real line \(\mathbb{R}\) such that the set of points of \(\mathbb{R}\) which are not fixed by \(f\) (\(g\), resp.) is precisely \(I\) (\(J\), resp.).
Show that for large enough integer \(n\), the group generated by \(f^n, g^n\) is isomorphic to the group with the following presentation
\[ <a, b | [ab^{-1}, a^{-1}ba] = [ab^{-1}, a^{-2}ba^2] = id>. \]
Find the smallest prime number \( p \geq 5 \) such that there exist no integer coefficient polynomials \( f \) and \( g \) satisfying
\[
p | ( 2^{f(n)} + 3^{g(n)})
\]
for all positive integers \( n \).
For the 10th problem for POW this year, I added a condition that we only consider the group topologies which make the given group a Hausdorff space. Since the problem has been modified, I decided to extend the deadline for this problem. Please hand in your solution by 12pm on Friday (May 31st).
Let \(G\) be a group. A topology on \(G\) is said to be a group topology if the map \(\mu: G \times G \to G\) defined by \(\mu(g, h) = g^{-1}h\) is continuous with respect to this topology where \(G \times G\) is equipped with the product topology. A group equipped with a group topology is called a topological group. When we have two topologies \(T_1, T_2\) on a set S, we write \(T_1 \leq T_2\) if \(T_2\) is finer than \(T_1\), which gives a partial order on the set of topologies on a given set. Prove or disprove the following statement: for a give group \(G\), there exists a unique minimal group topology on \(G\) (minimal with respect to the partial order we described above) so that \(G\) is a Hausdorff space?
Suppose that \( X \) is a discrete random variable on the set \( \{ a_1, a_2, \dots \} \) with \( P(X=a_i) = p_i \). Define the discrete entropy
\[
H(X) = -\sum_{n=1}^{\infty} p_i \log p_i.
\]
Find constants \( C_1, C_2 \geq 0 \) such that
\[
e^{2H(X)} \leq C_1 Var(X) + C_2
\]
holds for any \( X \).
Let \(G\) be a group acting by isometries on a proper geodesic metric space \(X\). Here \(X\) being proper means that every closed bounded subset of \(X\) is compact. Suppose this action is proper and cocompact,. Here, the action is said to be proper if for all compact subsets \(B \subset X\), the set \[\{g \in G | g(B) \cap B \neq \emptyset \}\] is finite. The quotient space \(X/G\) is obtained from \(X\) by identifying any two points \(x, y\) if and only if there exists \(g \in G\) such that \(gx = y\), and equipped with the quotient topology. Then the action of \(G\) on \(X\) is said to be cocompact if \(X/G\) is compact. Under these assumptions, show that \(G\) is finitely generated.
Suppose that \( f: \mathbb{R} \to \mathbb{R} \) is differentiable and \( \max_{ x \in \mathbb{R}} |f(x)| = M < \infty \). Prove that \[ \int_{-\infty}^{\infty} (|f'|^2 + |f|^2) \geq 2M^2. \]
Compute the following integral \[ \int_{0}^{\pi/2} \log{ (2 \cos{x} )} dx \].
Let \( p_n \) be the \(n\)-th prime number, \( p_1 = 2, p_2 = 3, p_3 = 5, \dots \). Prove that the following series converges:
\[
\sum_{n=1}^{\infty} \frac{1}{p_n} \prod_{k=1}^n \frac{p_k -1}{p_k}.
\]