数学笔记

A Proof of Trigonometric Formulas in the Plane of Constant Curvature

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$\newcommand{\pr}{\left.\frac{\partial}{\partial r}\right|_{\gamma(t)}}
\newcommand{\pt}{\left.\frac{\partial}{\partial\theta}\right|_{\gamma(t)}}
\newcommand{\dt}{\frac{\mathrm d}{\mathrm{d} t}}
\newcommand{\prr}{\frac{\partial}{\partial r}}
\newcommand{\ptt}{\frac{\partial}{\partial \theta}}
$

摘要 . In this paper, we solve the geodesics equation in the geodesic polar coordinates of a two dimensional Riemannian manifolds of constant sectional curvature. The relation between edges and angles of geodesic triangle has obtained and as a result the trigonometric formulae has been derived, that is the law of sines, the law of cosines.

1. Induced Connection Along a Mapping Suppose $M$ and $N$ be two smooth manifolds, and $\phi\mathpunct{:}N\to M$ is a smooth mapping. A vector field $X$ along $\phi$ is an assignment which corresponding each $x\in N$ to a vector $X(x)\in T_{\phi(x)}M$. In particular, for any vector field $V$ on $N$, $\phi_\ast V$ may not be a vector field on $M$, but it is a vector field along $\phi$. Clearly, the collection of vector fields along $\phi$ is a vector space, with the natural defined addtion and scalar multiplication.
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The Peter-Weyl theorem

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From Tao’s Blog: The Peter-Weyl theorem, and non-abelian Fourier analysis on compact groups.

Let $G$ be a compact group. (Throughout this post, all topological groups are assumed to be Hausdorff.) Then $G$ has a number of unitary representations, i.e. continuous homomorphisms $\rho: G \rightarrow U(H)$ to the group $U(H)$ of unitary operators on a Hilbert space $H$, equipped with the strong operator topology. In particular, one has the left-regular representation $\tau: G \rightarrow U(L^2(G))$, where we equip $G$ with its normalised Haar measure $\mu$ (and the Borel $\sigma$-algebra) to form the Hilbert space $L^2(G)$, and $\tau$ is the translation operation
$$
\tau(g) f(x) := f(g^{-1} x).
$$ Continue Reading →

Weak Convergence in Sobolev Spaces

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Suppose $\Omega\subset\R^n$ and donote $W^{1,p}:=W^{1,p}(\Omega)$ be the sobolev space for some $1< p< +\infty$. Recall that $f_i\in W^{1,p}$ convergent weakly to $f\in W^{1,p}$, if for any $\phi$ in the dual space of $W^{1,p}$, we have $\inner{f_i,\phi}\to\inner{f,\phi}$, denote as $f_i\weakto f$. This is distinguished by strongly convergence, as we use the dual normal instead of $W^{1,p}$ normal.

命题 1. If $f_i\weakto f$ in $W^{1,p}$, then $f_i\to f$ in $L^p$.

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Overview of moduli spaces,review of G-bundles and connections

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All the contents are from this wikisite, which is aimed to have a E-version lecture notes of the seminar given by Prof. Mrowka. All rights are reserved by the original wikisite, any reprint should be indicate this.
Main Contributor:Christian

These lecture notes are based on notes from the 18.999 geometry seminar class taught by Tomasz Mrowka and from the IAS/Park city Mathematic series book Gauge theory and the topology of four-manifolds 1. Moduli spaces A moduli space can be viewed as a geometric object which classifies the solutions of some problem. For example, there is the moduli space of $n$-tuples of points in $I=[0,1]$,that is, points in $I^{n}$ modulo symmetric transformations:$I^{n}/ Sym_{n}$. Given a complex vector space $V$ of dimension $n$, we can look at the space of endomorphisms of $ V$ modulo isomorphisms:
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图示一个三角函数(Triangle function)和阶梯函数(Step function)的卷积

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和以前一样， 你可以随意移动定位点， 对于远离原点的定位点， 你可以使用绘图区域的不同选项来观察整个图形与局部细节。