All Questions

Let $C_0(\mathbb{R})$ denote the analytic functions $f : \mathbb{R} \rightarrow \mathbb{R}$. I wonder whether there a functions $f \in C_0(\mathbb{R})$ with $f \neq 0$, such that there is a constant $...

Let $u = (u_1, u_2, u_3): \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a divergence-free analytic vector field for $n =3$ or $n =4$, i.e., $u_i : \mathbb{R}^n \rightarrow \mathbb{R}$ are analytic ...

Let $M$ be the set of smooth divergence-free vector fields $u$ on $\mathbb{R}^3$ with $$|\partial_x^{\alpha} u(x)| \leq C_{\alpha K}(1+|x|)^{-K}$$ on $\mathbb{R}^3$ for any $\alpha,K$. Further, we ...

The following differential equation has two independent solutions, one of the two is decreasing exponentially at infinity (k-Bessel function). $$(x^2y')'-x^2y=\lambda \;y$$ Now for a higher-degree ...

What is the most general form of the Sturm oscillation theorem? So far I have only seen cases that treat either unbounded intervals or weighted $L^2$ spaces. I would be especially interested in ...

Assume we are given for a transition between two time points $t_0 = 0$ and $t_1$ a matrix relationship, eventually describing the solution of a system of linear with non-constant coefficients, $$Y(t_1)...