Articles of reference request

Determining a function is harmonic from mean value property for just three(?) radii.

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 […]

Different definitions of absorbing sets from the Wikepedia

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 $$ […]

Reference Request Regarding Representation Theory of SO(n,m)

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 […]

Change of coordinates (algebraic variety)

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.

Ito Integral representation for bounded claims

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 + […]

mononotone and pseudomonotone operators in current research

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 […]

Reference books for highschool Algebra and Geometry?

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 […]

Heat Equation on Manifold

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.

For which small $n$ is unknown whether the Mordell-curve has an integral point?

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 […]

characteristic-$p$-type groups, and the Borel-Tits theorem for $PSL(V)$

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 […]