Recent content by Euge

Given a complex matrix ##A\in M_n(\mathbb{C})##, let ##X_A## be the subspace of ##M_n(\mathbb{C})## consisting of all the complex matrices ##M## commuting with ##A## (i.e., ##MA = AM##). Suppose ##A## has ##n## distinct eigenvalues. Find the dimension of ##X_A##.

Find, with proof, the sum of the alternating series $$\sum_{n = 0}^\infty \frac{(-1)^n}{(2n+1)^3}$$

Note that ##M## is to depend only on ##K##, but your ##M## depends on ##u##.

For a suggestion, try to prove first that if ##u\in \mathcal{O}(\mathbb{D}) \cap L^2(\mathbb{D})##, then for every compact ##K \subset \mathbb{D}##, there is a constant ##M = M(K) > 0## such that ##\sup_{z\in K} |u(z)| \le M\|u\|_{L^2(\mathbb{D})}##.

Let ##\mathbb{D}## be the open unit disk in ##\mathbb{C}##. Show that the space ##\mathcal{O}(\mathbb{D}) \cap L^2(\mathbb{D})## of square-integrable, holomorphic functions on ##\mathbb{D}## is a closed subspace of ##L^2(\mathbb{D})##.

If ##p## is a prime, and ##G## is a finite non-cyclic ##p##-group, show that there is a normal subgroup ##N## of ##G## such that ##G/N## is isomorphic to ##\mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z}##.

Let ##k## be a field, and let ##X## be a subscheme of ##\mathbb{P}_k^2## defined by a single homogeneous equation ##f(x_0, x_1, x_2) = 0## of degree ##d##. Show that $$\dim_k H^1(X, \mathcal{O}_X) = \frac{(d-1)(d-2)}{2}$$

Let ##D_r\subset \mathbb{R}^2## be the disk of radius ##r## centered at the origin. If ##f : \mathbb{R}^2 \to [0,\infty)## is uniformly continuous such that ##\sup_{0 < r< \infty} \iint_{D_r} f(x,y)\, dx\, dy < \infty##, show that ##f(x,y) \to 0## as ##x^2 + y^2 \to \infty##.

Hi @anuttarasammyak, in this problem ##X## is not assumed to have a density.

Suppose ##X## is a nonnegative random variable and ##p\in (0,\infty)##. Show that ##\mathbb{E}[X^p] < \infty## if and only if ##\sum_{n = 1}^\infty n^{p-1}P(X \ge n) < \infty##.

Let ##M## be a real ##n \times n## matrix. If ##M + M^T## is positive definite, show that $$\det\left(\frac{M + M^T}{2}\right) \le \det M$$

Let ##0 < \alpha < 1##. Find a necessary and sufficient condition for the function ##f : [0,1] \to \mathbb{R}##, ##f(x) = \sqrt{x}##, to belong to the class ##C^{0,\alpha}([0,1])##.

Prove that the series $$\sum_{k = 1}^\infty \frac{(-1)^{k-1}}{|x| + k}$$ converges for all ##x\in \mathbb{R}## to a Lipschitz function on ##\mathbb{R}##.