# 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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# 2019-11 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 $$n$$.

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# Extension of the due date for 10th problem

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).

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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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# Solution: 2019-06 Simple but not too simple integration

Compute the following integral  $\int_{0}^{\pi/2} \log{ (2 \cos{x} )} dx$

The best solution was submitted by 김건우 (수리과학과 2017학번). Congratulations!

Here is his solution of problem 2019-06.

Other solutions were submitted by 강한필 (전산학부 2016학번, +3), 고성훈 (2018학번, +3), 길현준 (2018학번, +3), 김기수 (수리과학과 2018학번, +3), 김민서 (2019학번, +3), 김태균 (수리과학과 2016학번, +3), 김희주 (2015학번, +3), 서준영 (수리과학과 대학원생, +3), 이본우 (수리과학과 2017학번, +3), 이원용 (2019학번, +3), 이정환 (수리과학과 2015학번, +3), 이종서 (2019학번, +3), 조재형 (수리과학과 2016학번, +3), 채지석 (수리과학과 2016학번, +3), 최백규 (생명과학과 2016학번, +3), 홍진표 (서울대학교 재료공학부 2013학번, +3).

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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.