# All Questions

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### Can (how) one distinguish germs of continuous functions by a countable set of params?

Continuous functions can be distinguished by their values at say rational points of [0 1]. Germs of analytic functions can be distinguished by derivatives at a point. So in both cases we see ...
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### identity of equation [on hold]

We have the equation ($\partial_{\mu}\partial_{\nu}$-$\eta_{\mu\nu}\Box$)$\phi=0$, where $\phi$ is a scalar field, $\Box=\partial_{\mu}\partial^{\mu}$ is a standart Dalamber operator, $\eta_{\mu\nu}$ ...
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### College - Ecuation to solve in 3 variables (p,q,r) in [1,2] [on hold]

This is the image containing the ecuation
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### orthonormal basis or Parseval frame for Sobolev spaces

Consider the uni-variate Sobolev space of order $m$: $$\mathcal{W}_2^m=\{ f:[0,1] \rightarrow \mathbb{R}|f,f^{(1)},\dots ,f^{(m-1)}\, \text{are absolutely continuous and}\, f^{(m)}\in L^2\}.$$ It is ...
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### Calculation in From Seiberg Witten to pseudo-holomorphic curve

I am reading the Taubes's paper: From From Seiberg Witten to pseudo-holomorphic curve. I don't know how to get the result (2.17) \begin{eqnarray*} ...
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### scheme of commuting matrices

Let $k$ be any field. Let $r$ and $n$ be two positive integers. Consider the functor $F$ from the category of $k$-schemes to the category of sets which sends a $k$-scheme $T$ to the set of matrices ...
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### Properties of solutions of Parabolic type equations

Assume $u\in C([0,1],L^{2})$ satisfies the following Schrodinger equation $$\partial_t u=i(\Delta u+Vu), \text{in} ~\mathbb{R}^n\times[0,1],\\ u(0)=u_{0}.$$ with $V=V_1(x)+V_2(x,t)$, where $V_{1}$ ...
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### In a finite field with characteristic 2, how can I check if a given polynomial is divisible by (x^2+1)? [on hold]

Given the 0 or 1 coefficients of a very high degree polynomial $P(x)$ over GF(2), where x is an element of $GF(1024)$, is there a simple algorithmic way to find out if this polynomial is divisible ...
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### is there an analogy between fractals and automorphic forms? [on hold]

Disclaimer: this question is rather vague and thus might not be suitable for this site. Still, as I already asked a similar question on MSE and got no feedback, I finally decided to take the plunge ...
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### Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?

Richard Stanley keeps a famous running compilation of different guises of the celebrated Catalan numbers. The number of vertices of the associahedron is one instantiation among the multitude, and the ...
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### Where can I find the classification of groups of order 8p? [on hold]

I need to classify the groups of order $8p$ up to isomorphism. We know that one of these groups is $G=\langle a,b| a^p=b^8=1, b^{-1}ab=a^{-1}\rangle$. Can I find other groups of this classification ...
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### Canonical functions in set theory and their applications

Given regular cardinal $\kappa>\omega,$ we can define the canonical functions $f_\alpha: \kappa\to \kappa,$ for $\alpha<\kappa^+.$ Some of their properties are presented in Chapter 22 of the ...
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### Is the kissing number in $n$ dimensions always divisible by $n$? And what is the base of exponential growth of the kissing number?

And why are the kissing numbers for 1, 2, 3, 4 and 8 dimensions all highly composite numbers?
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### How to extend index theorem to infinite dimensional manifolds?

I was thinking about the following "obvious" construction the other day. Let $$\mathbb{S}^{\infty}_{2}=\bigcup_{i=1}^{\infty}\mathbb{S}^{2i}$$ Then because we know that $\chi(\mathbb{S}^{2i})=2$ for ...
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### Separating points in the plane II

Let A be a set of $2m$ points on the plane so that no open set of diameter $2$ has more than m of them. Define $A+A+...+A$ ($k$ times) to be the multiset of $k$-sums from $A$. That is, we consider all ...
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### Parity of $\lfloor 1/(x y) \rfloor$ not equally distributed

A curious puzzle for which I would appreciate an explanation. For $x$ and $y$ both uniformly and independently distributed in $[0,1]$, the value of $\lfloor 1/(x y) \rfloor$ has a bias toward odd ...
### Do the algebras for a $\infty$-monad form a stable $\infty$-category?
I'm wondering if a monad $T$ on a stable $\infty$-category $\cal C$ has a stable $\infty$-category of algebras, provided $T$ preserves finite limits/colimits. Is this true? Edit: Is something ...
### Is the notation ${}^t g$ for the transpose of a linear transformation intended to be suggestive?
The notation ${}^t g$ for the transpose of a linear transformation is, in my view, quite unusual: otherwise (at least in many areas of math), one almost never sees subscripts or superscripts appearing ...