Let \( A_N \) be an \( N \times N \) matrix whose entries are i.i.d. Bernoulli random variables with probability \( 1/2 \), i.e.,
\[\mathbb{P}( (A_N)_{ij} =0) = \mathbb{P}( (A_N)_{ij} =1) = \frac{1}{2}.\]
Let \( p_N \) be the probability that \( \det A_N \) is odd. Find \( \lim_{N \to \infty} p_N \).
POW will resume on Apr. 30.
Let \(\mathcal{A}_n\) be the collection of all sequences \( \mathbf{a}= (a_1,\dots, a_n) \) with \(a_i \in [i]\) for all \(i\in [n]=\{1,2,\dots, n\}\). A nondecreasing \(k\)-subsequence of \(\mathbf{a}\) is a subsequence \( (a_{i_1}, a_{i_2},\dots, a_{i_k}) \) such that \(i_1< i_2< \dots < i_k\) and \(a_{i_1}\leq a_{i_2}\leq \dots \leq a_{i_k}\). For given \(k\), determine the smallest \(n\) such that any sequence \(\mathbf{a}\in \mathcal{A}_n\) has a nondecreasing \(k\)-subsequence.
The best solution was submitted by 고성훈 (수리과학과 2018학번, +4). Congratulations!
Here is his solution of problem 2021-06.
Another solution was submitted by 강한필 (전산학부 2016학번, +3).
Prove or disprove that if all elements of an infinite group G has order less than n for some positive integer n, then G is finitely generated.
The best solution was submitted by 김기수 (수리과학과 2018학번, +4). Congratulations!
Here is his solution of problem 2021-05.
Other solutions were submitted by 강한필 (전산학부 2016학번, +3), 고성훈 (수리과학과 2018학번, +3), Late solutions are not graded.
Let \(\mathcal{A}_n\) be the collection of all sequences \( \mathbf{a}= (a_1,\dots, a_n) \) with \(a_i \in [i]\) for all \(i\in [n]=\{1,2,\dots, n\}\). A nondecreasing \(k\)-subsequence of \(\mathbf{a}\) is a subsequence \( (a_{i_1}, a_{i_2},\dots, a_{i_k}) \) such that \(i_1< i_2< \dots < i_k\) and \(a_{i_1}\leq a_{i_2}\leq \dots \leq a_{i_k}\). For given \(k\), determine the smallest \(n\) such that any sequence \(\mathbf{a}\in \mathcal{A}_n\) has a nondecreasing \(k\)-subsequence.
For an \( n \times n \) matrix \( M \) with real eigenvalues, let \( \lambda(M) \) be the largest eigenvalue of \( M\). Prove that for any positive integer \( r \) and positive semidefinite matrices \( A, B \),
\[[\lambda(A^m B^m)]^{1/m} \leq [\lambda(A^{m+1} B^{m+1})]^{1/(m+1)}.\]
The best solution was submitted by 고성훈 (수리과학과 2018학번, +4). Congratulations!
Here is his solution of problem 2021-04.
Another solutions was submitted by 김건우 (수리과학과 2017학번, +3),
Prove or disprove that if all elements of an infinite group G has order less than n for some positive integer n, then G is finitely generated.
POW 2021-04 is still open and anyone who first submits a correct solution will get the full credit.