Complex reflection groups

 

 

 

 

 

 

 

 

표준적인 도서 및 추천도서

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위키링크

• http://en.wikipedia.org/wiki/

참고할만한 자료

On Coxeter Diagrams of complex reflection groups

Authors:Tathagata Basak

http://arxiv.org/abs/0809.2427

 

Title: The complex Lorentzian Leech lattice and the bimonsterAuthors: Tathagata Basak

http://front.math.ucdavis.edu/0508.5228

 

Title: The complex Lorentzian Leech lattice and the bimonster (II)

Authors: Tathagata Basak

http://front.math.ucdavis.edu/0811.0062

 

Regular polyhedral groups and reflection groups of rank four

Mitsuo Kato [[1]]^,a and Jiro Sekiguchi [[2]]^,b http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WDY-4B0WHXW-1&_user=4420&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000059607&_version=1&_urlVersion=0&_userid=4420&md5=188db4d982dbbcd13fb099e37f43bc91  

 

26 Implies the Bimonster*1

John H. Conwaya^,1 and Christopher S. Simonsb^,2

http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WH2-45B5NX1-K&_user=4420&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000059607&_version=1&_urlVersion=0&_userid=4420&md5=e01e884479d443f48357724a9b672e9f

 

 

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LaTeX 명령예

 

• \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\)

1. x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

• \(e^{i \pi} +1 = 0\)

1. e^{i\pi}+1=0

• \(2\pi-3\times\frac{3\pi}{5}=\frac{\pi}{5}\)

1. 2\pi-3\times\frac{3\pi}{5}=\frac{\pi}{5}

• \(\frac{\sqrt{3}}{5}\)

1. \frac{\sqrt{3}}{5}

• \(720\div12=60\)

1. 720\div12=60

• \(\large f^\prime(x)\ = \lim_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x}\)

1. \large f^\prime(x)\ =         \lim_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x}

• \(\Large A\ =\ \large\left( \begin{array}{c.cccc}&1&2&\cdots&n\\ \hdash1&a_{11}&a_{12}&\cdots&a_{1n}\\ 2&a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ n&a_{n1}&a_{n2}&\cdots&a_{nn}\end{array}\right)\)

1. \Large A\ =\ \large\left(         \begin{array}{c.cccc}&1&2&\cdots&n\\         \hdash1&a_{11}&a_{12}&\cdots&a_{1n}\\         2&a_{21}&a_{22}&\cdots&a_{2n}\\         \vdots&\vdots&\vdots&\ddots&\vdots\\         n&a_{n1}&a_{n2}&\cdots&a_{nn}\end{array}\right)

• \(\LARGE\tilde y=\left\{ {\ddot x\text{ if $\vec x$ odd}\atop\hat{\,\bar x+1}\text{ if even}}\right.\)

1. \LARGE\tilde y=\left\{  {\ddot x\text{ if $\vec x$ odd}\atop\hat{\,\bar x+1}\text{ if even}}\right.

• \(\Large\left.\begin{eqnarray} x+y+z&=&3\\2y&=&x+z\\2x+y&=&z\end{eqnarray}\right\}\)

1. \Large\left.\begin{eqnarray}    x+y+z&=&3\\2y&=&x+z\\2x+y&=&z\end{eqnarray}\right\}

• \(\int e^{-\frac{x^2}{2}} dx\)

1. \int%20e^{-\frac{x^2}{2}}%20dx

\(e^x=\lim_{n\to\infty} \left(1+\frac~xn\right)^n\)

1. e^x=\lim_{n\to\infty} \left(1+\frac~xn\right)^n

• \(\Large\begin{array}{rccclBCB} &f&\longr[75]^{\alpha:{\normalsize f\rightar~g}}&g\\ \large\gamma&\longd[50]&&\longd[50]&\large\gamma\\ &u&\longr[75]_\beta&v\end{array}\)

1. \Large\begin{array}{rccclBCB}    &f&\longr[75]^{\alpha:{\normalsize f\rightar~g}}&g\\    \large\gamma&\longd[50]&&\longd[50]&\large\gamma\\    &u&\longr[75]_\beta&v\end{array}

• \(\Large\overbrace{a,...,a}^{\text{k a^,s}}, \underbrace{b,...,b}_{\text{l b^,s}}\hspace{10} \large\underbrace{\overbrace{a...a}^{\text{k a^,s}}, \overbrace{b...b}^{\text{l b^,s}}}_{\text{k+l elements}}\)

1. \Large\overbrace{a,...,a}^{\text{k a^,s}},    \underbrace{b,...,b}_{\text{l b^,s}}\hspace{10}    \large\underbrace{\overbrace{a...a}^{\text{k a^,s}},    \overbrace{b...b}^{\text{l b^,s}}}_{\text{k+l elements}}

• \(\normalsize \left(\large\begin{array}{GC+23} \varepsilon_x\\\varepsilon_y\\\varepsilon_z\\\gamma_{xy}\\ \gamma_{xz}\\\gamma_{yz}\end{array}\right)\ {\Large=} \ \left[\begin{array}{CC} \begin{array}\frac1{E_{\fs{+1}x}} &-\frac{\nu_{xy}}{E_{\fs{+1}x}} &-\frac{\nu_{\fs{+1}xz}}{E_{\fs{+1}x}}\\ -\frac{\nu_{yx}}{E_y}&\frac1{E_{y}}&-\frac{\nu_{yz}}{E_y}\\ -\frac{\nu_{\fs{+1}zx}}{E_{\fs{+1}z}}& -\frac{\nu_{zy}}{E_{\fs{+1}z}} &\frac1{E_{\fs{+1}z}}\end{array} & {\LARGE 0} \\ {\LARGE 0} & \begin{array}\frac1{G_{xy}}&&\\ &\frac1{G_{\fs{+1}xz}}&\\&&\frac1{G_{yz}}\end{array} \end{array}\right] \ \left(\large\begin{array} \sigma_x\\\sigma_y\\\sigma_z\\\tau_{xy}\\\tau_{xz}\\\tau_{yz} \end{array}\right)\)

1. \normalsize         \left(\large\begin{array}{GC+23}         \varepsilon_x\\\varepsilon_y\\\varepsilon_z\\\gamma_{xy}\\         \gamma_{xz}\\\gamma_{yz}\end{array}\right)\ {\Large=}         \ \left[\begin{array}{CC}         \begin{array}\frac1{E_{\fs{+1}x}}         &-\frac{\nu_{xy}}{E_{\fs{+1}x}}         &-\frac{\nu_{\fs{+1}xz}}{E_{\fs{+1}x}}\\         -\frac{\nu_{yx}}{E_y}&\frac1{E_{y}}&-\frac{\nu_{yz}}{E_y}\\         -\frac{\nu_{\fs{+1}zx}}{E_{\fs{+1}z}}&         -\frac{\nu_{zy}}{E_{\fs{+1}z}}         &\frac1{E_{\fs{+1}z}}\end{array} & {\LARGE 0} \\         {\LARGE 0} & \begin{array}\frac1{G_{xy}}&&\\         &\frac1{G_{\fs{+1}xz}}&\\&&\frac1{G_{yz}}\end{array}         \end{array}\right]         \ \left(\large\begin{array}         \sigma_x\\\sigma_y\\\sigma_z\\\tau_{xy}\\\tau_{xz}\\\tau_{yz}         \end{array}\right)

 

1. \sum_{k=1}^{\infty}\frac{1}{k^2}=\frac{\pi^2}{6}

1. \int_{a}^{b}f(x)dx=F(b)-F(a)

1. \exists c \in (a,b) \quad \mathbf{s.t.} \quad f'(c)=\frac{f(b)-f(a)}{b-a}

•  

1. E=-N\frac{d\Phi}{dt}

1. \mathbf{X}=\left(\begin{array}{ccc}x_{11} & x_{12} & \ldots } & x_{22} & \ldots & \vdots & \ddots\end{array} \right)