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Are there two algebraically independent irrational numbers $\alpha,\beta$ s.t. $\alpha^\beta$ is a rational number?

A colleague asked me the following question: "What can one do with the following norm on $\ell^1$: $|x|=\int_1^2 |x|_pdp$ where $| \;\; |_p$ is the standard norm on $\ell_p$?" This interesting norm i …

Inspired by The set of all limits of sub-series of an absolute convergent series is the following true?: Let $a_n$ be a strictly decreasing sequence and $\sum_1^\infty a_n=\ell<\infty$ is a converge …

I have already asked this question on MSE; now I repeat it on MO. https://math.stackexchange.com/questions/4132346/on-which-subspace-w-subset-c-infty0-1-is-df-xfx-a-bounded-operator First we introd …

Let $f:\mathbb{R}\to \mathbb{R}$ be a smooth function which is flat at $0\in \mathbb{R}$. That is $f^{(k)}(0)=0,\; k=0,1,2,\ldots $. Assume that $|f''(x)|\leq M|f(x)|\quad \forall x\in \mathbb{R}$ …

Let ${\mathbb{R}^2}^+=\{(x,y)\in \mathbb{R}^2\mid x>0, y>0\}$. Let $P:{\mathbb{R}^2}^+\to \mathbb{R}$ be the function with $P(a,b)=$ $\text{The perimeter of ellipse}\;\; \frac{x^2}{a^2}+\frac{y^2}{b^ …

Edit: According to the comment by LSpice we come back to the initial version of this question Let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a smooth function such that for every $x\in \mathbb{R}^n$ the der …

Updated: A polynomial $P(x,y)\in \mathbb{R}[x,y]$ is called an elliptic polynomial if its last homogeneous part does not vanish on $\mathbb{R}^2\setminus\{0\}$.The two answers to this post provide a …

For every positive real number $x$ we define $$E(x)= \int_0^{\infty} x^t/t!\,\mathrm dt$$ where $t!=\Gamma(t+1)$. This is motivated by classical exponential function. Is this function well defined (t …

Let $E\to M$ be a smooth vector bundle over a smooth manifold $M$. Let $D$ be a differential operator defined on the space $\Gamma(E)$ of smooth sections of $E$. We fix a section $s\in \Gamma(E)$. Ass …

This is not an answer, but is a comment. (I can not give comment since I am under 50 reputation). Linear vector fields are always complete vector field so they do not satis …

For $n=2$ we define $M$ as follows: $M$ is the union of the following sets: 1)The intersection with $x\_$ axis for lines not parallel to this axis. 2)The intersection with $y\_$axis for lines perp …

Edit: After the answer of Prof. Eremenko to the previous version, I realized that a weaker assumption works for the main motivation of this post. so I revise the question. The unit d …

Assume that $f:\mathbb{R}^{2}\to \mathbb{R}$ is a continuous function such that each level set $f^{-1}(c)$ is a convex set. To what extent such functions are studied? In particular: Is there …

Assume that $H$ is a Lebesgue measurable additive subgroup of $\mathbb{R}$. Is $H$ necessarily a Borel subset of $\mathbb{R}$?