2018-12-29 00:46:47 +08:00
\documentclass [a4paper] { scrartcl}
%% Deutsche Anpassungen %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage [ngerman] { babel}
\usepackage [ansinew] { inputenc}
\usepackage { ulem}
\usepackage { mathtools}
\usepackage { esint}
%\usepackage{txfonts}
\usepackage [landscape] { geometry}
\usepackage { amsmath}
\usepackage { amssymb}
\newcommand { \cbrt } [1]{ \sqrt [3] { #1} }
\newcommand { \bbC } { \mathbb { C} }
\newcommand { \bbH } { \mathbb { H} }
\newcommand { \bbN } { \mathbb { N} }
\newcommand { \bbP } { \mathbb { P} }
\newcommand { \bbQ } { \mathbb { Q} }
\newcommand { \bbZ } { \mathbb { Z} }
\newcommand { \bbR } { \mathbb { R} }
\newcommand { \Angstrom } { \r { A} }
\newcommand { \lefrighttharpoons } { \rightleftharpoons }
\newcommand { \subsetnot } { \not \subset }
\newcommand { \argmax } { \operatorname * { arg\, max} }
\newcommand { \va } { \vec { a} }
\newcommand { \vr } { \vec { r} }
\newcommand { \vR } { \vec { R} }
\newcommand { \argmin } { \mbox { arg\: min} }
\newcommand { \uul } [1]{ \uuline { #1} }
\newcommand { \ool } [1]{ \overline { \overline { #1} } }
\newcommand { \arrow } [1]{ \overrightarrow { #1} }
\begin { document}
\begin { itemize}
\item \textbf { std dev:} \[ \sigma _ x = \sqrt { \langle ( x - \langle x \rangle ) ^ 2 \rangle } = \sqrt { \frac { 1 } { N - 1 } \cdot \left ( \sum _ { i = 1 } ^ N { x _ i } ^ 2 - \frac { 1 } { N } \cdot \left ( \sum _ { i = 1 } ^ Nx _ i \right ) ^ 2 \right ) } \]
\item \textbf { std dev 2:} \[ \sigma _ x = \sqrt { \langle ( x - \langle x \rangle ) ^ 2 \rangle } = \sqrt { \frac { 1 } { N - 1 } \cdot \left ( \sum _ { i = 1 } ^ Nx _ i ^ 2 - \frac { 1 } { N } \cdot \left ( \sum _ { i = 1 } ^ Nx _ i \right ) ^ 2 \right ) } \]
\item \textbf { rotation matrix:} \[ \mathrm { \mathbf { M } } ( \alpha ) = \left ( \begin { matrix } \cos ( \alpha ) + n _ x ^ 2 \cdot ( 1 - \cos ( \alpha ) ) & n _ x \cdot n _ y \cdot ( 1 - \cos ( \alpha ) ) - n _ z \cdot \sin ( \alpha ) & n _ x \cdot n _ z \cdot ( 1 - \cos ( \alpha ) ) + n _ y \cdot \sin ( \alpha ) \\ n _ x \cdot n _ y \cdot ( 1 - \cos ( \alpha ) ) + n _ z \cdot \sin ( \alpha ) & \cos ( \alpha ) + n _ y ^ 2 \cdot ( 1 - \cos ( \alpha ) ) & n _ y \cdot n _ z \cdot ( 1 - \cos ( \alpha ) ) - n _ x \cdot \sin ( \alpha ) \\ n _ z \cdot n _ x \cdot ( 1 - \cos ( \alpha ) ) - n _ y \cdot \sin ( \alpha ) & n _ z \cdot n _ y \cdot ( 1 - \cos ( \alpha ) ) + n _ x \cdot \sin ( \alpha ) & \cos ( \alpha ) + n _ z ^ 2 \cdot ( 1 - \cos ( \alpha ) ) \end { matrix } \right ) \]
\item \textbf { like in label at bottom (no MM):} \[ \left ( \left [ \sqrt { 2 \pi \cdot \int _ { - \infty } ^ \infty f ( x ) \; \mathrm { d } x } \right ] \right ) \]
\item \textbf { like in label at bottom (MM):} \[ \left ( \left [ \sqrt { 2 \pi \cdot \int _ { - \infty } ^ \infty f ( x ) \; \mathrm { d } x } \right ] \right ) \]
\item \textbf { decoration:} \[ \vec { x } \vec { X } \vec { \psi } - - \dot { x } \dot { X } \dot { \psi } - - \ddot { x } \ddot { X } \ddot { \psi } - - \overline { x } \overline { X } \overline { \psi } - - \underline { x } \underline { X } \underline { \psi } - - \hat { x } \hat { X } \hat { \psi } - - \tilde { x } \tilde { X } \tilde { \psi } - - \uul { x } \uul { X } \uul { \psi } - - \ool { x } \ool { X } \ool { \psi } - - \bar { x } \bar { X } \bar { \psi } - - \arrow { x } \arrow { X } \arrow { \psi } \]
\item \textbf { mathtest:} \\ This is normal text: $ this is math: \langle r ^ 2 ( \tau ) \rangle = \left \langle ( \vec { r } ( t ) - \vec { r } ( t + \tau ) ) ^ 2 \right \rangle \ \ \ g ( \tau ) = \frac { 1 } { N } \cdot \left ( 1 + \frac { 2 } { 3 } \frac { \langle r ^ 2 ( \tau ) \rangle } { w _ { xy } ^ 2 } \right ) ^ { - 1 } \lfloor \rfloor \lceil \rceil \langle \rangle \left \{ \va \left | \| \va \| _ 2 \geq 2 \right . \right \} \vr \vR $ \\ $ \frac { \sqrt { \sqrt { \sqrt { \sum _ { i = 0 } ^ \infty \hat { i } ^ 2 } + y ^ \alpha } + 1 } } { \dot { v } \equiv \ddot { r } } \argmin _ { \vec { k } } \sum _ { \sqrt { i } = 0 } ^ { N } \int _ { x _ 0 } ^ { x _ 1 } \left ( \left ( \left ( x \right ) \right ) \right ) \underbrace { \left [ \left \{ \frac { \partial f } { \partial x } \right \} \cdot \frac { 1 } { 2 } \right ] } { \text { underbraced text $ \hbar $ } } \cdots \frac { \sqrt { \sum _ { i = 0 } ^ 2 \hat { i } ^ 2 } + y ^ \alpha } { \dot { v } \equiv \ddot { r } } , \hat { t } \hat { T } \overbrace { \left | \sqrt { x \cdot Y } \right | } { \propto \bbN \circ \bbZ } $ \\ $ \left < \arrow { x ( \tau ) } \cdot \vec { R } ( t + \bar { \tau } ) \right > \alpha \beta \gamma \delta \epsilon \Gamma \Delta \Theta \Omega \left \lfloor \left \lceil \cbrt { \hbar \omega } \right \rceil \right \rfloor $
\item \textbf { chi2 test:} \[ \vec { p } ^ \ast = \argmax \limits _ { \vec { p } } \chi ^ 2 = \argmax \limits _ { \vec { p } } \sum \limits _ { i = 1 } ^ N \left | \frac { \hat { f } _ i - f ( x _ i; \vec { p } ) } { \sigma _ i } \right | ^ 2 \]
\item \textbf { upper/lower parantheses test:} \[ \text { bblabla } \frac { 1 } { 2 } \cdot \left ( \frac { 1 } { \mathrm { e } ^ x + \mathrm { e } ^ { - x } } \right ) \cdot \left ( \frac { 1 } { \frac { 1 + 2 } { 5 + x } } \right ) \cdot \left ( \frac { 1 } { \exp \left [ - \frac { y ^ 2 } { \sqrt { x } } \right ] \cdot \exp \left [ - \frac { 1 } { \frac { 1 } { 2 } } \right ] } \right ) \]
\item \textbf { ACF test:} \[ g _ { rg } ^ { ab } ( \tau ) = \frac { 1 } { N } \cdot \left ( 1 + \frac { 2 } { 3 } \frac { \langle r ^ 2 ( \tau ) \rangle } { w _ { xy } ^ 2 } \right ) ^ { - 1 } \cdot \left ( 1 + \frac { 2 } { 3 } \frac { \langle r ^ 2 ( \tau ) \rangle } { w _ { xy } ^ 2 } \right ) ^ { - \frac { 1 } { 2 } } \]
\item \textbf { MSD test:} \[ \mathrm { MSD } ( \tau ) \equiv \langle r ^ 2 ( \tau ) \rangle = \left \langle ( \vec { r } ( t ) - \vec { r } ( t + \tau ) ) ^ 2 \right \rangle = 2 n \cdot \frac { K _ \alpha } { \Gamma ( 1 + \alpha ) } \cdot \tau ^ \alpha \]
\item \textbf { math: blackboard:} \[ \mathbb { ABCDEFGHIJKLMNOPQRSTUVWXYZ 120 } \]
\item \textbf { math: bf:} \[ \mathbf { ABCDEFGHIJKLMNOPQRSTUVWXYZ 120 } \]
\item \textbf { math: rm:} \[ \mathrm { ABCDEFGHIJKLMNOPQRSTUVWXYZ 120 } \]
\item \textbf { math: cal:} \[ \mathcal { ABCDEFGHIJKLMNOPQRSTUVWXYZ 120 } \]
\item \textbf { subscript test:} \[ r _ { 123 } \ \ r _ { \frac { 1 } { 2 } } \]
\item \textbf { subscript0 test:} \[ r _ { 123 } \]
\item \textbf { subscript1 test:} \[ r _ { 123 } \ \]
\item \textbf { subscript2 test:} \[ r _ { 123 } \ \ \]
\item \textbf { subscript3 test:} \[ r _ { 123 } r _ { \frac { 1 } { 2 } } \]
\item \textbf { superscript test:} \[ r ^ { 123 } \ \ r ^ { \frac { 1 } { 2 } } \]
\item \textbf { superscript0 test:} \[ r ^ { 123 } \]
\item \textbf { superscript1 test:} \[ r ^ { 123 } \ \]
\item \textbf { superscript2 test:} \[ r ^ { 123 } \ \ \]
\item \textbf { superscript3 test:} \[ r ^ { 123 } r ^ { \frac { 1 } { 2 } } \]
\item \textbf { asuperscript test:} \[ a ^ { 123 } \ \ a ^ { \frac { 1 } { 2 } } \]
\item \textbf { asuperscript0 test:} \[ a ^ { 123 } \]
\item \textbf { gsuperscript1 test:} \[ g ^ { 123 } \ \]
\item \textbf { gsuperscript2 test:} \[ g ^ { 123 } \ \ \]
\item \textbf { gsuperscript3 test:} \[ g ^ { 123 } g ^ { \frac { 1 } { 2 } } \]
\item \textbf { frac test:} \[ \frac { a } { b } + \frac { g } { a } - \frac { a ^ 2 } { b ^ 2 } \cdot \frac { a ^ 2 } { b ^ { \frac { 1 } { 2 } } } \]
\item \textbf { tfrac test:} \[ \tfrac { a } { b } + \tfrac { g } { a } - \tfrac { a ^ 2 } { b ^ 2 } \cdot \tfrac { a ^ 2 } { b ^ { \tfrac { 1 } { 2 } } } \]
\item \textbf { dfrac test:} \[ \dfrac { a } { b } + \dfrac { g } { a } - \dfrac { a ^ 2 } { b ^ 2 } \cdot \dfrac { a ^ 2 } { b ^ { \dfrac { 1 } { 2 } } } \]
\item \textbf { stackrel test:} \[ \stackrel { a } { b } + \stackrel { g } { a } - \stackrel { a ^ 2 } { b ^ 2 } \cdot \stackrel { a ^ 2 } { b ^ { \stackrel { 1 } { 2 } } } \]
\item \textbf { brace5 test: ( )} \[ \left ( \left ( \left ( r ^ { 123 } \right ) \right ) \right ) - - \left ( \left ( \left ( r ^ { 123 } \right ) \right ) \right ) \]
\item \textbf { brace6 test: [ ]} \[ \left [ \left [ \left [ r ^ { 123 } \right ] \right ] \right ] - - \left [ \left [ \left [ r ^ { 123 } \right ] \right ] \right ] \]
\item \textbf { brace7 test: { } } \[ \left \{ \left \{ \left \{ r ^ { 123 } \right \} \right \} \right \} - - \left \{ \left \{ \left \{ r ^ { 123 } \right \} \right \} \right \} \]
\item \textbf { brace8 test: || ||} \[ \left \| \left \| \left \| r ^ { 123 } \right \| \right \| \right \| - - \left \| \left \| \left \| r ^ { 123 } \right \| \right \| \right \| \]
\item \textbf { brace9 test: | |} \[ \left | \left | \left | r ^ { 123 } \right | \right | \right | - - \left | \left | \left | r ^ { 123 } \right | \right | \right | \]
\item \textbf { brace10 test} \[ \left \{ \left [ \left ( r ^ { 123 } \right ) \right ] \right \} - - \left \{ \left [ \left ( r ^ { 123 } \right ) \right ] \right \} \]
\item \textbf { brace11 test: floor} \[ \left \lfloor \left \lfloor \left \lfloor r ^ { 123 } \right \rfloor \right \rfloor \right \rfloor - - \left \lfloor \left \lfloor \left \lfloor r ^ { 123 } \right \rfloor \right \rfloor \right \rfloor \]
\item \textbf { brace12 test: ceil} \[ \left \lceil \left \lceil \left \lceil r ^ { 123 } \right \rceil \right \rceil \right \rceil - - \left \lceil \left \lceil \left \lceil r ^ { 123 } \right \rceil \right \rceil \right \rceil \]
\item \textbf { sub-, superscript test} \[ r ^ { 1234 } _ { 321 } r _ { 321 } ^ { 1234 } - - r ^ { 1234 } _ { 321 } r _ { 321 } ^ { 1234 } - - \kappa ^ 2 - - \kappa _ 2 - - \kappa _ 2 ^ 2 \]
\item \textbf { super-, subscript test} \[ r ^ { 123 } _ { 4321 } r _ { 4321 } ^ { 123 } - - r ^ { 123 } _ { 4321 } r _ { 4321 } ^ { 123 } - - \kappa ^ 2 - - \kappa _ 2 - - \kappa _ 2 ^ 2 \]
\item \textbf { math 1:} \[ f ( x ) = \int _ { - \infty } ^ xe ^ { - t ^ 2 } \; \mathrm { d } t \]
\item \textbf { math 2:} \[ \sum _ { i = 1 } ^ \infty \frac { - e ^ { i \pi } } { 2 ^ n } \]
\item \textbf { math 3:} \[ \mbox { det } \begin { pmatrix } 1 & x _ 1 & \ldots & x _ 1 ^ { n - 1 } \\ 1 & x _ 2 & \ldots & x _ 2 ^ { n - 1 } \\ \vdots & \vdots & \ddots & \vdots \\ 1 & x _ n & \ldots & x _ n ^ { n - 1 } \end { pmatrix } = \prod _ { 1 \leq i < j \leq n } ( x _ j - x _ i ) \]
\item \textbf { math 4:} \[ \sqrt { 1 + \sqrt { 1 + \sqrt { 1 + \sqrt { 1 + \sqrt { 1 + \sqrt { 1 + x } } } } } } \]
\item \textbf { math 5:} \[ \left ( \stackrel { p } { 2 } \right ) = x ^ 2 y ^ { p - 2 } - \frac { 1 } { 1 - x } \frac { 1 } { 1 - x ^ 2 } \]
\item \textbf { math 6:} \[ a _ 0 + \frac { 1 } { a _ 1 + \frac { 1 } { a _ 2 + \frac { 1 } { a _ 3 + \frac { 1 } { a _ 4 } } } } \]
\item \textbf { math 7:} \[ \left ( \frac { \partial ^ 2 } { \partial x ^ 2 } + \frac { \partial ^ 2 } { \partial y ^ 2 } \right ) \left | \varphi ( x + \mathrm { i } y ) \right | ^ 2 = 0 \]
\item \textbf { math 8:} \[ 2 ^ { 2 ^ { 2 ^ { x } } } \]
\item \textbf { math 9:} \[ \iint _ Df ( x,y ) \; \mathrm { d } x \; \mathrm { d } y \]
\item \textbf { math 10 (overbrace):} \[ \overbrace { x + x + ... + x } { k \ \mathrm { times } } \]
\item \textbf { math 11 (underbrace):} \[ \underbrace { x + x + ... + x } { k \ \mathrm { times } } \]
\item \textbf { math 12 (under/overbrace):} \[ \underbrace { \overbrace { x + x + ... + x } { k \ \mathrm { times } } \overbrace { x + x + ... + x } { k \ \mathrm { times } } } { 2 k \ \mathrm { times } } \]
\item \textbf { math 13:} \[ y _ 1 '' \ \ \ y _ 2 ''' \]
\item \textbf { math 14:} \[ f ( x ) = \begin { cases } 1 / 3 & \mathrm { if } \ 0 \leq x \leq 1 \\ 2 / 3 & \mathrm { if } \ 3 \leq x \leq 4 \\ 0 & \mathrm { elsewhere } \end { cases } \]
\item \textbf { math 15:} \[ \Re { z } = \frac { n \pi \dfrac { \theta + \psi } { 2 } } { \left ( \dfrac { \theta + \psi } { 2 } \right ) ^ 2 + \left ( \dfrac { 1 } { 2 } \log \left \lvert \dfrac { B } { A } \right \rvert \right ) ^ 2 } . \]
\item \textbf { math 16:} \[ \sum _ { m = 1 } ^ \infty \sum _ { n = 1 } ^ \infty \frac { m ^ 2 \, n } { 3 ^ m \left ( m \, 3 ^ n + n \, 3 ^ m \right ) } \]
\item \textbf { math 17:} \[ \phi _ n ( \kappa ) = \frac { 1 } { 4 \pi ^ 2 \kappa ^ 2 } \int _ 0 ^ \infty \frac { \sin ( \kappa R ) } { \kappa R } \frac { \partial } { \partial R } \left [ R ^ 2 \frac { \partial D _ n ( R ) } { \partial R } \right ] \, dR \]
\item \textbf { math 18:} \[ { } _ pF _ q ( a _ 1 , \dots ,a _ p;c _ 1 , \dots ,c _ q;z ) = \sum _ { n = 0 } ^ \infty \frac { ( a _ 1 ) _ n \cdots ( a _ p ) _ n } { ( c _ 1 ) _ n \cdots ( c _ q ) _ n } \frac { z ^ n } { n ! } \]
\item \textbf { math 19 (overset):} \[ X \overset { = } { def } Y \ \ \ \ \ X \overset { = } { ! } Y \ \ \ \ \ X \overset { \rightarrow } { f } Y \ \ \ \ \ \frac { f ( x + \Delta x ) - f ( x ) } { \Delta x } \overset { \longrightarrow } { \Delta x \to 0 } f' ( x ) \]
\item \textbf { math 20 (underset):} \[ X \underset { = } { \text { def ( 5 ) } } Y \ \ \ \ \ X \underset { \rightarrow } { f } Y \ \ \ \ \ \frac { f ( x + \Delta x ) - f ( x ) } { \Delta x } \underset { \longrightarrow } { \Delta x \to 0 } f' ( x ) \]
\item \textbf { axiom of power test:} \[ \forall A \, \exists P \, \forall B \, [ B \in P \iff \forall C \, ( C \in B \Rightarrow C \in A ) ] \]
\item \textbf { De Morgan's law:} $ \neg ( P \land Q ) \iff ( \neg P ) \lor ( \neg Q ) $ or $ \overline { \bigcap _ { i \in I } A _ { i } } \equiv \bigcup _ { i \in I } \overline { A _ { i } } $ or $ \overline { A \cup B } \equiv \overline { A } \cap \overline { B } $
\item \textbf { quadratic formula:} \[ x = \frac { - b \pm \sqrt { b ^ 2 - 4 ac } } { 2 a } \]
\item \textbf { combination:} \[ \binom { n } { k } = \frac { n ( n - 1 ) ... ( n - k + 1 ) } { k ( k - 1 ) \dots 1 } = \frac { n ! } { k ! ( n - k ) ! } \]
\item \textbf { Sophomore's dream 1:} \[ \int _ 0 ^ 1 x ^ { - x } \, dx = \sum _ { n = 1 } ^ \infty n ^ { - n } ( \scriptstyle { = 1 . 29128599706266354040728259059560054149861936827 \dots ) } \]
\item \textbf { Sophomore's dream 2:} \[ \int _ 0 ^ 1 x ^ x \, dx = \sum _ { n = 1 } ^ \infty ( - 1 ) ^ { n + 1 } n ^ { - n } = - \sum _ { n = 1 } ^ \infty ( - n ) ^ { - n } ( \scriptstyle { = 0 . 78343051071213440705926438652697546940768199014 \dots } ) \]
\item \textbf { divergence 1:} \[ \operatorname { div } \vec { F } = \nabla \cdot \vec { F } = \frac { \partial U } { \partial x } + \frac { \partial V } { \partial y } + \frac { \partial W } { \partial z } \]
\item \textbf { divergence 2:} \[ \overrightarrow { \operatorname { div } } \, ( \mathbf { \underline { \underline { \epsilon } } } ) =
\begin { bmatrix}
\frac { \partial \epsilon _ { xx} } { \partial x} +\frac { \partial \epsilon _ { yx} } { \partial y} +\frac { \partial \epsilon _ { zx} } { \partial z} \\
\frac { \partial \epsilon _ { xy} } { \partial x} +\frac { \partial \epsilon _ { yy} } { \partial y} +\frac { \partial \epsilon _ { zy} } { \partial z} \\
\frac { \partial \epsilon _ { xz} } { \partial x} +\frac { \partial \epsilon _ { yz} } { \partial y} +\frac { \partial \epsilon _ { zz} } { \partial z}
\end { bmatrix} \]
\item \textbf { lim, sum ...:} \[ \lim _ { x \to \infty } f ( x ) = \binom { k } { r } + \frac { a } { b } \sum _ { n = 1 } ^ \infty a _ n + \displaystyle { \left \{ \frac { 1 } { 13 } \sum _ { n = 1 } ^ \infty b _ n \right \} } . \]
%\item\textbf{array test:} \[ f(x) := \left\{\begin{array}[ll] x^2\sin\frac{1}{x} & \text{if} x \ne 0, \\ 0 & \text{if } x = 0 . \end{array}\right. \]
\item \textbf { Schwinger-Dyson:} \[ \left \langle \psi \left | \mathcal { T } \{ F \phi ^ j \} \right | \psi \right \rangle = \left \langle \psi \left | \mathcal { T } \{ iF _ { ,i } D ^ { ij } - FS _ { int,i } D ^ { ij } \} \right | \psi \right \rangle . \]
\item \textbf { Schr<EFBFBD> dinger's equation:} \[ \left [ - \frac { \hbar ^ 2 } { - 2 m } \frac { \partial ^ 2 } { \partial x ^ 2 } + V \right ] \Psi ( x ) = i \hbar \frac { \partial } { \partial t } \Psi ( x ) \]
\item \textbf { Cauchy-Schwarz inequality:} \[ \left ( \sum _ { k = 1 } ^ n a _ k b _ k \right ) ^ 2 \leq \left ( \sum _ { k = 1 } ^ n a _ k ^ 2 \right ) \left ( \sum _ { k = 1 } ^ n b _ k ^ 2 \right ) \]
\item \textbf { Maxwell's equations:} \[ \begin { aligned } \nabla \times \vec { \mathbf { B } } - \, \frac { 1 } { c } \, \frac { \partial \vec { \mathbf { E } } } { \partial t } & = \frac { 4 \pi } { c } \vec { \mathbf { j } } \\ \nabla \cdot \vec { \mathbf { E } } & = 4 \pi \rho \\ \nabla \times \vec { \mathbf { E } } \, + \, \frac { 1 } { c } \, \frac { \partial \vec { \mathbf { B } } } { \partial t } & = \vec { \mathbf { 0 } } \\ \nabla \cdot \vec { \mathbf { B } } & = 0 \end { aligned } \]
\end { itemize}
2015-07-11 18:56:02 +08:00
\end { document}
