2019-12 Groups generated by two homeomorphisms of the real line

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

\[ <a, b | [ab^{-1}, a^{-1}ba] = [ab^{-1}, a^{-2}ba^2] = id>. \]

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Solution: 2019-11 Smallest prime

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 \( n \).

The best solution was submitted by 김태균 (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2019-11.

Other solutions were submitted by 고성훈 (2018학번, +3), 조재형 (수리과학과 2016학번, +3), 채지석 (수리과학과 2016학번, +3), 최백규 (생명과학과 2016학번, +3).

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Solution: 2019-10 Is there canonical topology for topological groups?

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?

The best solution was submitted by 이정환 (수리과학과 2015학번). Congratulations!

Here is his solution of problem 2019-10.

An incomplete solutions were submitted by 채지석 (수리과학과 2016학번, +2).

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2019-10 Is there canonical topology for topological groups?

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?

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Solution: 2019-09 Discrete entropy

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 \( X \).

The best solution was submitted by 길현준 (2018학번). Congratulations!

Here is his solution of problem 2019-09.

Alternative solutions were submitted by 최백규 (생명과학과 2016학번, +3). Incomplete solutions were submitted by, 이정환 (수리과학과 2015학번, +2), 채지석 (수리과학과 2016학번, +2).

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2019-09 Discrete entropy

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 \( X \).

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Solution: 2019-08 Group action

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.

The best solution was submitted by 이정환 (수리과학과 2015학번). Congratulations!

Here is his solution of problem 2019-08.

Alternative solutions were submitted by 조재형 (수리과학과 2016학번, +3), 채지석 (수리과학과 2016학번, +3), 김태균 (수리과학과 2016학번, +2).

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Solution: 2019-07 An inequality

Suppose that \( f: \mathbb{R} \to \mathbb{R} \) is differentiable and \( \max_{ x \in \mathbb{R}} |f(x)| = M < \infty \). Prove that \[ \int_{-\infty}^{\infty} (|f'|^2 + |f|^2) \geq 2M^2. \]

The best solution was submitted by 채지석 (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2019-07.

Other solutions were submitted by 고성훈 (2018학번, +3), 길현준 (2018학번, +3), 김기택 (수리과학과 2015학번, +3), 김민서 (2019학번, +3), 김태균 (수리과학과 2016학번, +3), 박재원 (2019학번, +3), 오윤석 (2019학번, +3), 윤영환 (한양대학교, +3), 이본우 (수리과학과 2017학번, +3), 이원용 (2019학번, +3), 이정환 (수리과학과 2015학번, +3), 정의현 (수리과학과 대학원생, +3), 최백규 (생명과학과 2016학번, +3).

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