Let $ \gamma$ be a limit ordinal with cofinality $ \kappa > \omega$ . Is there a stationary subset of $ \gamma$ (a subset of $ \gamma$ that meets every closed unbounded subset of $ \gamma$ ) of size $ \kappa$ ?Read more
Let $ \gamma$ be a limit ordinal with cofinality $ \kappa > \omega$ . Is there a stationary subset of $ \gamma$ (a subset of $ \gamma$ that meets every closed unbounded subset of $ \gamma$ ) of size $ \kappa$ ?Read more
I want to prove or disprove that the Fourier transform $ \mathcal F \colon (\mathcal S(\mathbb R^d), \lVert \cdot \rVert_1) \to L^1(\mathbb R^d)$ is unbounded, where $ \lVert\cdot \rVert_1$ denotes the $ L^1(\mathbb R^d)$ -norm. Having thought about this for a moment, I believe it is indeed unbounded. So I tried to find a sequenceRead more
I was observing this interesting paper and conjecture this result, $ $ \sum_{n=0}^{\infty}\frac{{2n \choose n}}{2^{4n+1}}^2\cdot \frac{n(6n-1)}{(2n-1)^2(2n+1)}=\frac{C}{\pi} \tag1$ $ Where C is Catalan’s constant $ =0.9156965…$ I am unable to present a prove of $ (1)$ . How do we go about to prove its?Read more
Let $ (Z,W)$ be a compact Hausdorff space and $ \tilde X\subseteq Z$ an open subset of $ Z$ . Furthermore let $ h:(\tilde X, W_{|\tilde X}) \to (X, \mathcal T)$ be a homeomorphism. $ Y:=X \cup \{\infty\}$ How can I show that $ f: Z \to Y, f(x)=h(x)$ for $ x \in \tilde X$Read more
Picture of the problem Hi, I think I understand the problem, but I’m not sure on how to properly express the solution. If I am right, it’s asking me for all vectors that are not a linear combination of two given vectors, and therefore not in span S. In other words its asking me toRead more
Let $ X$ , $ Y$ be two independent random variables with $ N(0,1)$ distribution. How can I find the following conditional expected value $ $ \mathbb{E}(X | X^2 + Y^2)?$ $Read more
Does anyone think that tetration by a non-integer will ever be defined … really properly? Great mathematicians struggled with finding an implementation of the non-integer factorial for a long time to no avail … and then eventually Leonhardt Euler devised a means of doing it, & by a sleight-of-mind that was just so slick &Read more
Let $ f\in\mathbb{Q}[x]$ a monic polynomial such that $ f$ has degree $ n$ . Let $ E_f$ be the splitting field of $ f$ over $ \mathbb{Q}$ . I would like to show that there exists a monic polynomial in $ \mathbb{Z}[x]$ such that it has the same splitting field. I don’t even knowRead more
Basically my question is why, when we go from classical solutions to weak solutions, we have to use energy inequalities instead of equalities. More preciscely, consider the Navier-Stokes equations \begin{equation} \partial _t \rho + \text{div}(\rho u) = 0 \ \ \ \text{in } I \times \Omega \end{equation} \begin{equation} \partial _t (\rho u) + \text{div}(\rho uRead more
Problem $ f(x)$ is defined over $ [a,b]$ and differentiable over $ (a,b)$ , where $ b-a\geq 4.$ Prove that there exists $ \xi \in (a,b)$ such that $ f'(\xi)<1+f^2(\xi)$ . My Proof Since $ b-a \geq 4$ ，we can obtain $ $ \exists x_1,x_2 \in (a,b):x_2-x_1>\pi.$ $ Denote$ $ F(x):=\arctan f(x).$ $ Obviously, $Read more