16×9 Beamer setting

To use 16×9 aspect ratio for Beamer, we can simply change \documentclass{beamer} to \documentclass[aspectratio=169]{beamer} However, Bakoma won’t display correctly afterward. Go to Option -> Document Properties -> Paper. Just change paper width and paper height to 454bp and 255bp should make things work.

Skew-symmetric matrix representation of cross product

We can define Note that we have and Triple product expansion We will show the above with the triple product expansion: Proof: (1)   Similarly for the and components. Proof of Note that for any     thus Proof of For any (2)   Thus,

Rodrigues’ rotation formula

The rotation matrix for rotating an object along normal direction with angle is given by where such that We can easily validate that the equation is correct, note that as desired. And for any vector perpendicular to as desired as well. Compute and from  Note that  Thus, . Moreover, since , we can compute as…

俾阿囡睇嘅粵語配音動畫

仙巴歷險記 双语 粤语 日语 (似乎link唔work) 【小肥肥一族】(TVB)粤语全52集 【720P】飘零燕【粤语】 猫猫日记  【336P/TVB粤语无字幕】妈妈是小学四年生/超时空保 湯姆歷險記【粤语】 – 哔哩哔哩 【宫崎骏/粤语版】高立的未来世界 我們這一家【粤语版】 第一季 钢琴之森·粤语版 长腿叔叔 全集 冰果

Update to PHP 7.4

WordPress recommends upgrading to PHP 7.4. The installation is quite easy (basically just apt install). However, pages and posts are gone after I switch from 7.2 to 7.4 for my lab website. Yet, there is no problem upgrading my blog site. After several hours, I gave up. No idea why it has such weird behavior….

Formal definition of Lie algebra

Lie algebra is a vector space $latex g$ with a map $latex [\cdot, \cdot]:g\times g \rightarrow g$ such that $latex [\cdot,\cdot]$ is bilinear $latex [x,x]=0 $ Jacobi inequality: $latex [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0,\forall x,y,z$. Note that 2) $latex \Rightarrow [x,y]=-[y,x]$ since $latex 0=[x+y,x+y]=[x,x]+[y,y]+[x,y]+[y,x]$. Note that the converse is true most of time as well since that implies $latex…

Lie algebra of sl(2)

The “S” in $latex SL(2,\mathbb{C})$ stands for special, meaning that $latex A \in SL(2,\mathbb{C})$, then $latex det(A)=1$. $latex sl(2,\mathbb{C})=\left\{\begin{pmatrix}a&b\\c&d\end{pmatrix}: \det\exp \left(t\begin{pmatrix}a&b\\c&d\end{pmatrix} \right)= 1,\forall t \right\}$ $latex \det\exp \left(t\begin{pmatrix}a&b\\c&d\end{pmatrix} \right)=\det \left(t\begin{pmatrix}1+ta&tb\\tc&1+td\end{pmatrix} + O(t^2) \right)$ $latex = 1+t(a+d)+O(t^2)=1, \forall t$. The condition obviously requires $latex a+d=0$. It turns out that the converse is true as well and…