From 02c1d40bc9af5d72e80b595ac8633b82b299f7e3 Mon Sep 17 00:00:00 2001 From: "matth@ultra" Date: Sat, 16 Nov 2024 16:25:57 +0100 Subject: [PATCH] work --- src/atom.tex | 2 +- src/calculus.tex | 55 ------------------------ src/condensed_matter.tex | 59 ++++++++++++++++++++++++++ src/electrodynamics.tex | 1 + src/main.tex | 15 +++---- src/{ => maths}/analysis.tex | 58 ++++++++++++++++++++++++- src/{ => maths}/geometry.tex | 0 src/{ => maths}/linalg.tex | 0 src/{ => maths}/probability_theory.tex | 55 +++++++++++++++++++++--- src/quantum_mechanics.tex | 6 +++ src/readme.md | 8 ++++ src/statistical_mechanics.tex | 43 ++++++++++++++----- src/svgs/convertToPdf.sh | 0 src/util/environments.tex | 43 ++++++++++++++----- src/util/macros.tex | 2 + 15 files changed, 253 insertions(+), 94 deletions(-) delete mode 100644 src/calculus.tex mode change 100644 => 100755 src/main.tex rename src/{ => maths}/analysis.tex (73%) rename src/{ => maths}/geometry.tex (100%) rename src/{ => maths}/linalg.tex (100%) rename src/{ => maths}/probability_theory.tex (79%) mode change 100755 => 100644 src/svgs/convertToPdf.sh diff --git a/src/atom.tex b/src/atom.tex index 37b3fa8..6b4c059 100644 --- a/src/atom.tex +++ b/src/atom.tex @@ -28,7 +28,7 @@ \end{formula} \begin{formula}{wave_function} - \desc{Wave function}{}{} + \desc{Wave function}{}{$R_{nl}(r)$ \fqEqRef{qm:h:radial}, $Y_{lm}$ \fqEqRef{qm:spherical_harmonics}} \desc[german]{Wellenfunktion}{}{} \eq{\psi_{nlm}(r, \theta, \phi) = R_{nl}(r)Y_{lm}(\theta,\phi)} \end{formula} diff --git a/src/calculus.tex b/src/calculus.tex deleted file mode 100644 index 3593a9c..0000000 --- a/src/calculus.tex +++ /dev/null @@ -1,55 +0,0 @@ -\Part[ - \eng{Analysis} - \ger{Analysis} - ]{ana} - - \Subsection[ - \eng{Convolution} - \ger{Faltung / Konvolution} - ]{conv} - \begin{ttext} - \eng{Convolution is \textbf{commutative}, \textbf{associative} and \textbf{distributive}.} - \ger{Die Faltung ist \textbf{kommutativ}, \textbf{assoziativ} und \textbf{distributiv}} - \end{ttext} - \begin{formula}{def} - \desc{Definition}{}{} - \desc[german]{Definition}{}{} - \eq{(f*g)(t) = f(t) * g(t) = int_{-\infty}^\infty f(\tau) g(t-\tau) \d \tau} - \end{formula} - \begin{formula}{notation} - \desc{Notation}{}{} - \desc[german]{Notation}{}{} - \eq{ - f(t) * g(t-t_0) &= (f*g)(t-t_0) \\ - f(t-t_0) * g(t-t_0) &= (f*g)(t-2t_0) - } - \end{formula} - \begin{formula}{commutativity} - \desc{Commutativity}{}{} - \desc[german]{Kommutativität}{}{} - \eq{f * g = g * f} - \end{formula} - - \begin{formula}{associativity} - \desc{Associativity}{}{} - \desc[german]{Assoziativität]}{}{} - \eq{(f*g)*h = f*(g*h)} - \end{formula} - - \begin{formula}{distributivity} - \desc{Distributivity}{}{} - \desc[german]{Distributivität}{}{} - \eq{f * (g + h) = f*g + f*h} - \end{formula} - - \begin{formula}{complex_conjugate} - \desc{Complex conjugate}{}{} - \desc[german]{Komplexe konjugation}{}{} - \eq{(f*g)^* = f^* * g^*} - \end{formula} - - \Subsection[ - \eng{Fourier analysis} - \ger{Fourieranalyse} - ]{fourier} - diff --git a/src/condensed_matter.tex b/src/condensed_matter.tex index eaec023..cdaa504 100644 --- a/src/condensed_matter.tex +++ b/src/condensed_matter.tex @@ -2,6 +2,7 @@ \eng{Condensed matter physics} \ger{Festkörperphysik} ]{cm} + \TODO{Bonds, hybridized orbitals, tight binding} \Section[ \eng{Bravais lattice} \ger{Bravais-Gitter} @@ -84,6 +85,17 @@ \end{tabularx} \end{adjustbox} \end{table} + +family of plane that are equivalent due to crystal symmetry + \begin{formula}{miller} + \desc{Miller index}{}{} + \desc[german]{Millersche Indizes}{}{} + \eq{ + (hkl) & \text{\GT{plane}}\\ + [hkl] & \text{\GT{direction}}\\ + \{hkl\} & \text{\GT{millerFamily}} + } + \end{formula} \Section[ @@ -395,3 +407,50 @@ \centering \includegraphics[width=\textwidth]{img/cm_mbe_english.pdf} \end{minipage} + +\Section[ + \eng{Superconductivity} + \ger{Supraleitung} +]{sc} + \begin{ttext} + \eng{ + Materials for which the electric resistance jumps to 0 under a critical temperature. + \\\textbf{Type I}: Has a single critical magnetic field at which the superconuctor becomes a normal conductor. + \\\textbf{Type II}: Has two critical + } + \ger{Materialien, bei denen der elektrische Widerstand beim unterschreiten einer kritischen Temperatur auf 0 springt.} + \end{ttext} + + \begin{formula}{meissner_effect} + \desc{Meißner-Ochsenfeld effect}{Perfect diamagnetism}{} + \desc[german]{Meißner-Ochsenfeld Effekt}{Perfekter Diamagnetismus}{} + \ttxt{ + \eng{Blabla } + \ger{Blubb blubb } + } + \end{formula} + \Subsection[ + \eng{London equation} + \ger{London-Gleichungen} + ]{london} + \begin{formula}{first} + % \vec{j} = \frac{nq\hbar}{m}\Grad S - \frac{nq^2}{m}\vec{A} + \desc{First London Equation}{}{$\vec{j}$ current density, $n$, $m$, $q$ density, mass and charge of superconduticng particles} + \desc[german]{Erste London-Gleichung}{}{$\vec{j}$ Stromdichte, $n$, $m$, $q$ Dichte, Masse und Ladung der supraleitenden Teilchen} + \eq{ + \partical_t \vec{j} = \frac{nq^2}{m}\vec{E} + } + \end{formula} + \begin{formula}{second} + \desc{Second London Equation}{}{$\vec{j}$ current density, $n$, $m$, $q$ density, mass and charge of superconduticng particles} + \desc[german]{Zweite London-Gleichung}{}{$\vec{j}$ Stromdichte, $n$, $m$, $q$ Dichte, Masse und Ladung der supraleitenden Teilchen} + \eq{ + \Rot \vec{j} = -\frac{nq^2}{m} \vec{B} + } + \end{formula} + \begin{formula}{penetration_depth} + \desc{London penetration depth}{}{} + \desc[german]{London Eindringtiefe}{}{} + \eq{\lambda_\textrm{L} = \sqrt{\frac{m}{\mu_0 nq^2}}} + \end{formula} + diff --git a/src/electrodynamics.tex b/src/electrodynamics.tex index 7a82139..a97d133 100644 --- a/src/electrodynamics.tex +++ b/src/electrodynamics.tex @@ -32,6 +32,7 @@ \Rot \vec{H} &= \vec{j} + \odv{\vec{D}}{t} } \end{formula} + \TODO{Polarization, Magnetisation} \Section[ \eng{Fields} diff --git a/src/main.tex b/src/main.tex old mode 100644 new mode 100755 index b5cd6a6..2fd220c --- a/src/main.tex +++ b/src/main.tex @@ -145,13 +145,10 @@ \input{util/translations.tex} -\input{linalg.tex} - -\input{geometry.tex} - -\input{analysis.tex} - -\input{probability_theory.tex} +\input{maths/linalg.tex} +\input{maths/geometry.tex} +\input{maths/analysis.tex} +\input{maths/probability_theory.tex} \input{mechanics.tex} @@ -164,9 +161,9 @@ \input{condensed_matter.tex} -% \input{topo.tex} +\input{topo.tex} -% \input{quantum_computing.tex} +\input{quantum_computing.tex} % \input{many-body-simulations.tex} diff --git a/src/analysis.tex b/src/maths/analysis.tex similarity index 73% rename from src/analysis.tex rename to src/maths/analysis.tex index 2ca4dd8..3cc720f 100644 --- a/src/analysis.tex +++ b/src/maths/analysis.tex @@ -3,6 +3,7 @@ \ger{Analysis} ]{cal} + \Subsection[ \eng{Convolution} \ger{Faltung / Konvolution} @@ -63,7 +64,7 @@ \end{formula} \Eng[real]{real} \Ger[real]{reellwertig} - \begin{formula}{coefficient} + \begin{formula}{coefficient-complex} \desc{Fourier coefficients}{Complex representation}{} \desc[german]{Fourierkoeffizienten}{Komplexe Darstellung}{} \eq{ @@ -112,14 +113,69 @@ \end{enumerate} + \Subsection[ + \eng{Misc} + \ger{Verschiedenes} + ]{misc} + + \begin{formula}{stirling-approx} + \desc{Stirling approximation}{}{} + \desc[german]{Stirlingformel}{}{} + \eq{\ln (N!) \approx N \ln(N) - N + \Order(\ln(N))} + \end{formula} + + \begin{formula}{error-function} + \desc{Error function}{\erf: \C \to \C}{} + \desc[german]{Fehlerfunktion}{Error function: \erf: \C \to \C}{} + \eq{ + \erf(x) &= \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \d t \\ + \erfc(x) &= 1 - \erf(x)\\ + &= \frac{2}{\sqrt{\pi}} \int_x^\infty e^{-t^2} \d t + } + \end{formula} + + +\Section[ + \eng{Logarithm} + \ger{Logarithmus} +]{log} + \begin{formula}{identities} + \desc{Logarithm identities}{}{} + \desc[german]{Logarithmus Identitäten}{Logarithmus Rechenregeln}{} + \eq{ + \log(xy) &= \log(x) + \log(y) \\ + \log \left(\frac{x}{y}\right) &= \log(x) - \log(y) \\ + \log \left(x^d\right) &= d\log(x) \\ + \log \left(\sqrt[y]{x}\right) &= \frac{\log(x)}{y} \\ + x^{\log(y)} &= y^{\log(x)} + } + \end{formula} + \Section[ \eng{List of common integrals} \ger{Liste nützlicher Integrale} ]{integrals} + \begin{formula}{spherical-coordinates} + \desc{Spherical coordinates}{}{} + \desc[german]{Kugelkoordinaten}{}{} + \eq{ + x &= r \sin\phi,\cos\theta \\ + y &= r \cos\phi,\cos\theta \\ + z &= r \sin\theta + } + \end{formula} + \begin{formula}{spheical-coordinates-int} + \desc{Integration in spherical coordinates}{}{} + \desc[german]{Integration in Kugelkoordinaten}{}{} + \eq{\iiint\d x \d y \d z= \int_0^{\infty} \!\! \int_0^{2\pi} \!\! \int_0^\pi \d r \d\phi\d\theta \, r^2\sin\theta} + \end{formula} + \begin{formula}{riemann_zeta} \desc{Riemann Zeta Function}{}{} \desc[german]{Riemannsche Zeta-Funktion}{}{} \eq{\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \frac{1}{(1-2^{(1-s)})\Gamma(s)} \int_0^\infty \d\eta \frac{\eta^{(s-1)}}{\e^\eta + 1}} \end{formula} + + diff --git a/src/geometry.tex b/src/maths/geometry.tex similarity index 100% rename from src/geometry.tex rename to src/maths/geometry.tex diff --git a/src/linalg.tex b/src/maths/linalg.tex similarity index 100% rename from src/linalg.tex rename to src/maths/linalg.tex diff --git a/src/probability_theory.tex b/src/maths/probability_theory.tex similarity index 79% rename from src/probability_theory.tex rename to src/maths/probability_theory.tex index 90dca10..5951da5 100644 --- a/src/probability_theory.tex +++ b/src/maths/probability_theory.tex @@ -4,21 +4,27 @@ ]{pt} \begin{formula}{mean} - \desc{Mean}{}{} - \desc[german]{Mittelwert}{}{} + \desc{Mean}{Expectation value}{} + \desc[german]{Mittelwert}{Erwartungswert}{} \eq{\braket{x} = \int w(x)\, x\, \d x} \end{formula} \begin{formula}{variance} - \desc{Variance}{}{} - \desc[german]{Varianz}{}{} - \eq{\sigma^2 = (\Delta \hat{x})^2 = \braket{\hat{x}^2} - \braket{\hat{x}}^2 = \braket{(x - \braket{x})^2}} + \desc{Variance}{Square of the \fqEqRef{pt:std-deviation}}{} + \desc[german]{Varianz}{Quadrat der\fqEqRef{pt:std-deviation}}{} + \eq{\sigma^2 = (\Delta \hat{x})^2 = \Braket{\hat{x}^2} - \braket{\hat{x}}^2 = \braket{(x - \braket{x})^2}} + \end{formula} + + \begin{formula}{covariance} + \desc{Covariance}{}{} + \desc[german]{Kovarianz}{}{} + \eq{\cov(x,y) = \sigma(x,y) = \sigma_{XY} = \Braket{(x-\braket{x})\,(y-\braket{y})}} \end{formula} - \begin{formula}{std_deviation} + \begin{formula}{std-deviation} \desc{Standard deviation}{}{} \desc[german]{Standardabweichung}{}{} - \eq{\sigma = \sqrt{(\Delta x)^2}} + \eq{\sigma = \sqrt{\sigma^2} = \sqrt{(\Delta x)^2}} \end{formula} \begin{formula}{median} @@ -192,3 +198,38 @@ } \end{ttext} +\Section[ + \eng{Propagation of uncertainty / error} + \ger{Fehlerfortpflanzung} +]{error} + \begin{formula}{generalised} + \desc{Generalized error propagation}{}{$V$ \fqEqRef{pt:covariance} matrix, $J$ \fqEqRef{ana:jacobi-matrix}} + \desc[german]{Generalisiertes Fehlerfortpflanzungsgesetz}{$V$ \fqEqRef{pt:covariance} Matrix, $J$ \fqEqRef{ana:jacobi-matrix}}{} + \eq{V_y = J(x) \cdot V_x \cdot J^{\T} (x)} + \end{formula} + + \begin{formula}{uncorrelated} + \desc{Propagation of uncorrelated errors}{Linear approximation}{} + \desc[german]{Fortpflanzung unabhängiger fehlerbehaftete Größen}{Lineare Näherung}{} + \eq{u_y = \sqrt{ \sum_{i} \left(\pdv{y}{x_i}\cdot u_i\right)^2}} + \end{formula} + + \begin{formula}{weight} + \desc{Weight}{Variance is a possible choice for a weight}{$\sigma$ \fqEqRef{pt:variance}} + \desc[german]{Gewicht}{Varianz ist eine mögliche Wahl für ein Gewicht}{} + \eq{w_i = \frac{1}{\sigma_i^2}} + \end{formula} + + \begin{formula}{weighted-mean} + \desc{Weighted mean}{}{$w_i$ \fqEqRef{pt:error:weight}} + \desc[german]{Gewichteter Mittelwert}{}{} + \eq{\overline{x} = \frac{\sum_{i} (x_i w_i)}{\sum_i w_i}} + \end{formula} + + \begin{formula}{weighted-mean-error} + \desc{Variance of weighted mean}{}{$w_i$ \fqEqRef{pt:error:weight}} + \desc[german]{Varianz des gewichteten Mittelwertes}{}{} + \eq{\sigma^2_{\overline{x}} = \frac{1}{\sum_i w_i}} + \end{formula} + + diff --git a/src/quantum_mechanics.tex b/src/quantum_mechanics.tex index d59605a..ef487e4 100644 --- a/src/quantum_mechanics.tex +++ b/src/quantum_mechanics.tex @@ -379,6 +379,12 @@ } \end{formula} + \begin{formula}{c_a_matrices} + \desc{Matrix forms}{}{} + \desc[german]{Matrix-Form}{}{} + \eq{\TODO{TODO}} + \end{formula} + \Subsubsection[ \eng{Harmonischer Oszillator} \ger{Harmonic Oscillator} diff --git a/src/readme.md b/src/readme.md index 18c523a..3abd4f6 100644 --- a/src/readme.md +++ b/src/readme.md @@ -20,10 +20,18 @@ The `:...:` will be defined as `fqname` (fully qu - figure: `fig` - parts, (sub)sections: `sec` +### Reference functions +Functions that create a hyperlink (and use the translation of the target element as link name): +- `\fqSecRef{}` +- `\fqEqRef{}` + + ## Multilanguage All text should be defined as a translation (`translations` package, see `util/translation.tex`) and then used using the `gt` or `GT` macros. The english translation of any key must be defined, because it will also be used as fallback. +Lower case macros are relative to the current `fqname`, while upper case macros are absolute. + Never make a macro that would have to be changed if a new language was added, eg dont do ```tex % 1: key, 2: english version, 3: german version diff --git a/src/statistical_mechanics.tex b/src/statistical_mechanics.tex index b630f60..13d8f19 100644 --- a/src/statistical_mechanics.tex +++ b/src/statistical_mechanics.tex @@ -332,7 +332,7 @@ \desc[german]{Entropiedichte}{}{$s = \frac{S}{N}$} \eq{ \lim_{T\to 0} s(T) &= 0 \\ - \shortintertext{\GT{and_therefore_also}} \\ + \shortintertext{\GT{and_therefore_also}} \lim_{T\to 0} c_V &= 0 } \end{formula} @@ -376,21 +376,21 @@ \desc[german]{Innere Energie}{}{} \eq{\d U(S,V,N) = T\d S -p\d V + \mu\d N} \end{formula} + \begin{formula}{free_energy} + \desc{Free energy / Helmholtz energy }{}{} + \desc[german]{Freie Energie / Helmholtz Energie}{}{} + \eq{\d F(T,V,N) = -S\d T -p\d V + \mu\d N} + \end{formula} \begin{formula}{enthalpy} \desc{Enthalpy}{}{} \desc[german]{Enthalpie}{}{} \eq{\d H(S,p,N) = T\d S +V\d p + \mu\d N} \end{formula} \begin{formula}{gibbs_energy} - \desc{Gibbs energy}{}{} - \desc[german]{Gibbsche Energie}{}{} + \desc{Free enthalpy / Gibbs energy}{}{} + \desc[german]{Freie Entahlpie / Gibbs-Energie}{}{} \eq{\d G(T,p,N) = -S\d T + V\d p + \mu\d N} \end{formula} - \begin{formula}{free_energy} - \desc{Free energy / Helmholtz energy }{}{} - \desc[german]{Freie Energie / Helmholtz Energie}{}{} - \eq{\d F(T,V,N) = -S\d T -p\d V + \mu\d N} - \end{formula} \begin{formula}{grand_canon_pot} \desc{Grand canonical potential}{}{} \desc[german]{Großkanonisches Potential}{}{} @@ -398,6 +398,27 @@ \end{formula} \TODO{Maxwell Relationen, TD Quadrat} + \begin{formula}{td-square} + \desc{Thermodynamic squre}{}{} + \desc[german]{Themodynamisches Quadrat}{Guggenheim Quadrat}{} + \content{ + \begin{tikzpicture} + \draw[thick] (0,0) grid (3,3); + \node at (0.5, 2.5) {$-S$}; + \node at (1.5, 2.5) {\color{blue}$U$}; + \node at (2.5, 2.5) {$V$}; + \node at (0.5, 1.5) {\color{blue}$H$}; + \node at (2.5, 1.5) {\color{blue}$F$}; + \node at (0.5, 0.5) {$-p$}; + \node at (1.5, 0.5) {\color{blue}$G$}; + \node at (2.5, 0.5) {$T$}; + \end{tikzpicture} + \begin{ttext} + \eng{The corners opposite from the potential are the coefficients and each coefficients differential is opposite to it.} + \ger{Die Ecken gegenüber des Potentials sind die Koeffizienten, das Differential eines Koeffizienten ist in der Ecke gegenüber.} + \end{ttext} + } + \end{formula} \Section[ \eng{Ideal gas} @@ -537,8 +558,8 @@ \end{formula} \begin{formula}{lennard_jones} - \desc{Lennard-Jones potential}{Potential between two molecules. Attractive for $r > \sigma$, repulsive for $r < \sigma$}{} - \desc[german]{Lennard-Jones-Potential}{Potential zwischen zwei Molekülen. Attraktiv für $r > \sigma$, repulsiv für $r < \sigma$}{} + \desc{Lennard-Jones potential}{Potential between two molecules. Attractive for $r > \sigma$, repulsive for $r < \sigma$.\\ In condensed matter: Attraction due to Landau Dispersion \TODO{verify} and repulsion due to Pauli exclusion principle.}{} + \desc[german]{Lennard-Jones-Potential}{Potential zwischen zwei Molekülen. Attraktiv für $r > \sigma$, repulsiv für $r < \sigma$.\\ In Festkörpern: Anziehung durch Landau Dispesion und Abstoßung durch Pauli-Prinzip.}{} \figeq{img/potential_lennard_jones.pdf}{V(r) = 4\epsilon \left[\left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6}\right]} \end{formula} @@ -658,7 +679,7 @@ \eng{Bosons} \ger{Bosonen} ]{bos} - \begin{formula}{partition_sum} + \begin{formula}{partition-sum} \desc{Partition sum}{}{$p \in\N_0$} \desc[german]{Zustandssumme}{}{$p \in\N_0$} \eq{Z_\text{g} = \prod_{p} \frac{1}{1-\e^{-\beta(\epsilon_p - \mu)}}} diff --git a/src/svgs/convertToPdf.sh b/src/svgs/convertToPdf.sh old mode 100755 new mode 100644 diff --git a/src/util/environments.tex b/src/util/environments.tex index 4236ff5..26b5e92 100644 --- a/src/util/environments.tex +++ b/src/util/environments.tex @@ -15,6 +15,13 @@ \def\descwidth{0.3\textwidth} \def\eqwidth{0.6\textwidth} + +% +% FORMULA ENVIRONMENT +% The following commands are meant to be used with the formula environment +% + +% Name in black and below description in gray % [1]: minipage width % 2: fqname of name % 3: fqname of a translation that holds the explanation @@ -133,32 +140,54 @@ } +% 1: key \newenvironment{formula}[1]{ - % key + % [1]: language + % 2: name + % 3: description + % 4: definitions/links \newcommand{\desc}[4][english]{ % language, name, description, definitions \dt[#1]{##1}{##2} \ifblank{##3}{}{\dt[#1_desc]{##1}{##3}} \ifblank{##4}{}{\dt[#1_defs]{##1}{##4}} } + % 1: equation for align environment \newcommand{\eq}[1]{ \insertEquation{#1}{##1} } + % 1: equation for alignat environment \newcommand{\eqAlignedAt}[2]{ \insertAlignedAt{#1}{##1}{##2} } + % 1: equation for flalign environment \newcommand{\eqFLAlign}[1]{ \insertFLAlign{#1}{##1} } + % 1: file path + % 2: equation \newcommand{\figeq}[2]{ \insertEquationWithFigure{#1}{##1}{##2} } + % 1: any content \newcommand{\content}[1]{ \NameLeftContentRight{#1}{##1} } + % 1: content for the ttext environment + \newcommand{\ttxt}[1]{ + \NameLeftContentRight{#1}{ + \begin{ttext}[#1:desc] + ##1 + \end{ttext} + } + } + }{\ignorespacesafterend} +% +% QUANTITY +% \newenvironment{quantity}[5]{ % key, symbol, si unit, si base units, comment (key to translation) \newcommand{\desc}[3][english]{ @@ -166,15 +195,6 @@ \DT[qty:#1]{}{##1}{##2} \ifblank{##3}{}{\DT[qty:#1_desc]{##1}{##3}} } - \newcommand{\eq}[1]{ - \insertEquation{#1}{##1} - } - \newcommand{\eqAlignedAt}[2]{ - \insertAlignedAt{#1}{##1}{##2} - } - \newcommand{\eqFLAlign}[1]{ - \insertFLAlign{#1}{##1} - } \edef\qtyname{#1} \edef\qtysign{#2} @@ -190,6 +210,9 @@ +% +% DISTRIBUTION +% \def\distrightwidth{0.45\textwidth} \def\distleftwidth{0.45\textwidth} diff --git a/src/util/macros.tex b/src/util/macros.tex index 3ea773d..46ace73 100644 --- a/src/util/macros.tex +++ b/src/util/macros.tex @@ -37,6 +37,8 @@ \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\const}{const} \DeclareMathOperator{\erf}{erf} +\DeclareMathOperator{\erfc}{erfc} +\DeclareMathOperator{\cov}{cov} % diff, for integrals and stuff % \DeclareMathOperator{\dd}{d} \renewcommand*\d{\mathop{}\!\mathrm{d}}