MathOverflow

$\def\R{\mathscr{R}}\def\O{\mathcal{O}}$Let $X$ be a topological space and let $\R$ be a sheaf of unital non-commutative rings over $X$.
When $\R$ is commutative, there is much literature on existence of K-flat resolutions and unbounded derived tensor product of $\R$-modules [S; L, 2.2.5; GW, 21.94; SP, 06Y7]....

Consider a number field $K$. How to classify or caracterize those $K$ intersecting the real line only in the rationals: $K\cap \mathbb{R}=\mathbb{Q}$ ?
An example are quadratic imaginary fields.
In general, we can wright $K=\mathbb{Q}[\alpha]\cong \mathbb{Q}[X]/\langle P\rangle$, where $P$ is the minimal polynomial of $\alpha$. A necessary condition is that $P$ has only non-real roots (in particular its degree is even). But this condition is...

The following version of a de la Vallée Poussin - criterion would be very helpful to me if it would be true. Can you say something about the truth value or give a reference?
Given a positive random variable $X$ with $\mathbb{E}[X] < \infty$, is there a increasing, convex function $\phi$ with $\lim_{x \to \infty} \frac{\phi(x)}{x} = \infty$ such that $\mathbb{E}[\phi(X)] < \infty$ and it holds for constants $c_1,...

I study the following non-linear dynamical system$$dX_t=\lambda_1 \left((-\beta_1 X_t +\beta_2 \sqrt{Y_t})^2 -X_t \right)$$$$dY_t=\lambda_2 \left((-\beta_1 X_t +\beta_2 \sqrt{Y_t})^2 -Y_t \right)$$$$\text{with }Y_0>0 \text{ and } -\beta_1 X_0 +\beta...