This theorem is well-known (maybe it can be called Morera’s theorem): A continuous function satisfying the mean value property on balls is harmonic. I was recently surprised to hear in a talk that the conclusion still holds if you only check the mean value property on three (I think) radii. I also can’t remember if […]

Consider a vector space $X$ over the field $\mathbb{F}$ of real or complex numbers and a set $S\subset X$. In this Wikipedia article about absorbing sets, $S$ is called absorbing if for all $x\in X$ there exists a real number $r$ such that for all $\alpha\in\mathbb{F}$ with $\vert \alpha \vert \geq r$ we have $$ […]

Is anyone aware of any books/papers that discuss the details of the indefinite (special) orthogonal groups $SO(n,m)$, their universal covers, representation theory, etc. (possibly some connections with physics, if any)? My searches thus far have hardly come up with much… I suppose that I should add that I would be particularly interested in the case […]

I’m studying the change of coordinates in Fulton’s Algebraic curves: Fulton’s book is sometimes a little “dry”, I’m confused, intuitive speaking what exactly is $F^T$? anyone could give me a concrete example of $T, T’$ and $T”$? Anyone knows more detailed materials about this stuff? I really need help. Thanks a lot.

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be Probability space and let $\{(X_t) : 0 \leq t \leq T \}$ be a continuous semimartingale on it. Let $\xi$ be $\mathcal{F}_T^X$ measurable and bounded. Does it mean that there exists a process $H_s$ adapted to $\mathcal{F}_t^X$ and constant $x \in \mathbb{R} $ such that $$ \xi = x + […]

I know that the following question is quite broad for this forum. But I am interested in references or any other ideas. Can anyone provide some examples of applications of monotone or pseudomonotone operators in mathematics or applied science in current research. I am aware of the use in proving the existence of solutions to […]

I’m tutoring high school students in Math for a local College and Career prep program and would like to have a reference book on hand that I can consult. I’m a Comp Sci graduate so I have a pretty strong background in Math but it’s been a while since I used high school level Algebra […]

Laplacian operator is defined well on Riemannian manifold, denoted by $\Delta$. Therefore people can study PDE $\Delta f=0$ on manifold. So is there any analogy to heat equation or wave equation on manifold? And what book is recommended for beginner to read about this field. Thank you.

Since it is not easy to determine the integral points of a Mordell curve $$y^2=x^3+n$$ with integer $n\ne 0$, I came to the following questions : $1)$ What is the smallest (in absolute value) integer $n$ , such that it is unknown whether the Mordell-curve $y^2=y^3+n$ has an integral point ? $2)$ What is the […]

If $G$ is a finite group and $F^*(G)$ is the generalized Fitting subgroup we say that $G$ has characteristic $p$ ($p$ is a prime that divides $|G|$) if $$F^*(G)=O_p(G)$$ Moreover $G$ is a characteristic-$p$-type group if all its $p$-local subgroups have characteristic $p$. I need some reference for this argument, in particular I need a […]

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