“I am trying to include long URLs in a bibliography, but those often overflow into the margin of the document.” The following works like a charm. \usepackage{url} % breakurl \def\UrlBreaks{\do\/\do-} \usepackage{breakurl} \usepackage[breaklinks]{hyperref}

## 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…

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## ARIMA

It is ARMA after applying lap-1 and lap-M difference to data.

## Bundle adjustment

A very good chinese post on bundle adjustment and source code is available as well.

## 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…