\chapter{Glossary of notations}
\section{General}
\begin{itemize}
	\ii $\forall$: for all
	\ii $\exists$: there exists
	\ii $\sign(\sigma)$: sign of permutation $\sigma$
	\ii $X \implies Y$: $X$ implies $Y$
\end{itemize}
\section{Functions and sets}
\begin{itemize}
	\ii $f\im(S)$ is the image of $f : X \to Y$ for $S \subseteq X$.
	\ii $f\inv(y)$ is the inverse for $f : X \to Y$ when $y \in Y$.
	\ii $f\pre(T)$ is the pre-image for $f : X \to Y$ when $T \subseteq Y$.
	\ii $f \restrict{S}$ is the restriction of $f : X \to Y$ to $S \subseteq X$.
	\ii $f^n$ is the function $f$ applied $n$ times
\end{itemize}

Below are some common sets.
These may also be thought of as groups,
rings, fields etc.\ in the obvious way.
\begin{itemize}
	\ii $\CC$: set of complex numbers
	\ii $\RR$: set of real numbers
	\ii $\NN$: set of positive integers
	\ii $\QQ$: set of rational numbers
	\ii $\ZZ$: set of integers
	\ii $\varnothing$: empty set
\end{itemize}

Some common notation with sets:
\begin{itemize}
	\ii $A \subset B$: $A$ is any subset of $B$
	\ii $A \subseteq B$: $A$ is any subset of $B$
	\ii $A \subsetneq B$: $A$ is a \emph{proper} subset of $B$
	\ii $S \times T$: Cartesian product of sets $S$ and $T$
	\ii $S \setminus T$: difference of sets $S$ and $T$
	\ii $S \cup T$: set union of $S$ and $T$
	\ii $S \cap T$: set intersection of $S$ and $T$
	\ii $S \sqcup T$: disjoint union of $S$ and $T$
	\ii $\left\lvert S \right\rvert$: cardinality of $S$
	\ii $S / {\sim}$: if $\sim$ is an equivalence relation on $S$,
	this is the set of equivalence classes
	\ii $x + S$: denotes the set $\{x+s \mid s \in S\}$.
	\ii $xS$: denotes the set $\{xs \mid s \in S\}$.
\end{itemize}

\section{Abstract and linear algebra}
Some common groups/rings/fields:
\begin{itemize}
	\ii $\Zc n$: cyclic group of order $n$
	\ii $\Zm n$: set of units of $\Zc n$.
	\ii $S_n$: symmetric group on $\{1, \dots, n\}$
	\ii $D_{2n}$: dihedral group of order $2n$.
	\ii $0$, $1$: trivial group (depending on context)
	\ii $\FF_p$: integers modulo $p$
\end{itemize}
Notation with groups:
\begin{itemize}
	\ii $1_G$: identity element of the group $G$
	\ii $N \normalin G$: subgroup $N$ is normal in $G$.
	\ii $G/N$: quotient group of $G$ by the normal subgroup $N$
	\ii $Z(G)$: center of group $G$
	\ii $N_G(H)$: normalizer of the subgroup $H$ of $G$
	\ii $G \times H$: product group of $G$ and $H$
	\ii $G \oplus H$: also product group,
	but often used when $G$ and $H$ are abelian
	(and hence we can think of them as $\ZZ$-modules)
	\ii $\Stab_G(x)$: the stabilizer of $x \in X$, if $X$ is acted on by $G$
	\ii $\FixPt g$, the set of fixed points by $g \in G$ (under a group action)
\end{itemize}
Notation with rings:
\begin{itemize}
	\ii $R/I$: quotient of ring $R$ by ideal $I$
	\ii $(a_1, \dots, a_n)$: ideal generated by the $a_i$
	\ii $R^\times$: the group of units of $R$
	\ii $R[x_1, \dots, x_n]$: polynomial ring in $x_i$,
	or ring obtained by adjoining the $x_i$ to $R$
	\ii $F(x_1, \dots, x_n)$: field obtained by adjoining $x_i$ to $F$
	\ii $R^d$: $d$th graded part of a graded (pseudo)ring $R$
\end{itemize}
Linear algebra:
\begin{itemize}
	\ii $\id$: the identity matrix
	\ii $V \oplus W$: direct sum
	\ii $V^{\oplus n}$: direct sum of $V$, $n$ times
	\ii $V \otimes W$: tensor product
	\ii $V^{\otimes n}$: tensor product of $V$, $n$ times
	\ii $V^\vee$: dual space
	\ii $T^\vee$: dual map (for $T$ a vector space)
	\ii $T^\dagger$: conjugate transpose (for $T$ a vector space)
	\ii $\left< -,-\right>$: a bilinear form
	\ii $\Mat(V)$: endomorphisms of $V$, i.e.\ $\Hom_k(V,V)$
	\ii $\ee_1$, \dots, $\ee_n$: the ``standard basis'' of $k^{\oplus n}$
\end{itemize}
\section{Quantum computation}
\begin{itemize}
	\ii $\ket{\psi}$: a vector in some vector space $H$
	\ii $\bra{\psi}$: a vector in some vector space $H^\vee$, dual to $\ket{\psi}$.
	\ii $\braket{\phi|\psi}$: evaluation of an element $\bra{\phi} \in H^\vee$ at $\ket{\phi} \in H$.
	\ii $\zup$, $\zdown$: spin $z$-up, spin $z$-down
	\ii $\xup$, $\xdown$: spin $x$-up, spin $x$-down
	\ii $\yup$, $\ydown$: spin $y$-up, spin $y$-down
\end{itemize}

\section{Topology and real/complex analysis}
Common topological spaces:
\begin{itemize}
	\ii $S^1$: the unit circle
	\ii $S^n$: surface of an $n$-sphere (in $\RR^{n+1}$)
	\ii $D^{n+1}$: closed $n+1$ dimensional ball (in $\RR^{n+1}$)
	\ii $\RP^n$: real projective $n$-space
	\ii $\CP^n$: complex projective $n$-space
\end{itemize}
Some topological notation:
\begin{itemize}
	\ii $\partial Y$: boundary of a set $Y$ (in some topological space)
	\ii $X/S$: quotient topology of $X$ by $S \subseteq X$
	\ii $X \times Y$: product topology of spaces $X$ and $Y$
	\ii $X \amalg Y$: disjoint union of spaces $X$ and $Y$
	\ii $X \vee Y$: wedge product of (pointed) spaces $X$ and $Y$
\end{itemize}
Real analysis (calculus 101):
\begin{itemize}
	\ii $\liminf$: limit infimum
	\ii $\limsup$: limit supremum
	\ii $\inf$: infimum
	\ii $\sup$: supremum
	\ii $\ZZ_p$: $p$-adic integers
	\ii $\QQ_p$: $p$-adic numbers
	\ii $f'$: derivative of $f$
	\ii $\int_a^b f(x) \; dx$: Riemann integral of $f$ on $[a,b]$
\end{itemize}
Complex analysis:
\begin{itemize}
	\ii $\int_\alpha f \; dz$: contour integral of $f$ along path $\alpha$
	\ii $\Res(f;p)$: the residue of a meromorphic function $f$ at point $p$
	\ii $\Wind(\gamma, p)$: winding number of $\gamma$ around $p$.
\end{itemize}
\section{Measure theory and probability}
\begin{itemize}
	\ii $\SA\cme$: the $\sigma$-algebra of Caratheory-measurable sets
	\ii $\SB(X)$: the Borel space for $X$
	\ii $\mu\cme$: the induced measure on $\SA\cme$.
	\ii $\lambda$: Lebesgue measure
	\ii $\mathbf{1}_A$: the indicator function for $A$
	\ii $\int_\Omega f \; d\mu$: the Lebesgue integral of $f$
	\ii $\lim_{n \to \infty} f_n$: pointwise limit of $f_n$
	\ii $\wh G$: Pontryagin dual for $G$
\end{itemize}

\section{Algebraic topology}
\begin{itemize}
	\ii $\alpha \simeq \beta$: for paths, this indicates path homotopy
	\ii $\ast$: path concatenation
	\ii $\pi_1(X) = \pi_1(X, x_0)$: the fundamental group of (pointed) space $X$
	\ii $\pi_n(X) = \pi_n(X, x_0)$: the $n$th homotopy group of (pointed) space $X$
	\ii $f_\sharp$: the induced map $\pi_1(X) \to \pi_1(Y)$ of $f : X \to Y$
	\ii $\Delta^n$: the standard $n$-simplex
	\ii $\partial\sigma$: the boundary of a singular $n$-simplex $\sigma$
	\ii $H_n(A_\bullet)$: the $n$th homology group of the chain complex $A_\bullet$
	\ii $H_n(X)$: the $n$th homology group of a space $X$
	\ii $\wt H_n(X)$: the $n$th reduced homology group of $X$
	\ii $H_n(X, A)$: the $n$th relative homology group of $X$ and $A \subseteq X$
	\ii $f_\ast$: the induced map on $H_n(A_\bullet) \to H_n(B_\bullet)$
	of $f : A_\bullet \to B_\bullet$,
	or $H_n(X) \to H_n(Y)$ for $f : X \to Y$
	\ii $\chi(X)$: Euler characteristic of a space $X$
	\ii $H^n(A^\bullet)$: the $n$th cohomology group of a cochain complex $A^\bullet$
	\ii $H^n(A_\bullet; G)$: the $n$th cohomology group of the cochain complex
	obtained by applying $\Hom(-,G)$ to $A_\bullet$
	\ii $H^n(X; G)$: the $n$th cohomology group/ring of $X$ with $G$-coefficients
	\ii $\wt H^n(X; G)$: the $n$th reduced cohomology group/ring of $X$ with $G$-coefficients
	\ii $H^n(X,A ; G)$: the $n$th relative cohomology group/ring of $X$ and $A \subset X$ with $G$-coefficients
	\ii $f^\sharp$: the induced map on $H^n(A^\bullet) \to H^n(B^\bullet)$
	of $f : A^\bullet \to B^\bullet$,
	or $H^n(X) \to H^n(Y)$ for $f : X \to Y$
	\ii $\Ext(-,-)$: the Ext functor
	\ii $\phi \smile \psi$: cup product of cochains $\phi$ and $\psi$
\end{itemize}

\section{Category theory}
Some common categories (in alphabetical order):
\begin{itemize}
	\ii $\catname{Grp}$: category of groups
	\ii $\catname{CRing}$: category of commutative rings
	\ii $\catname{Top}$: category of topological spaces
	\ii $\catname{Top}_\ast$: category of pointed topological spaces
	\ii $\catname{Vect}_k$: category of $k$-vector spaces
	\ii $\catname{FDVect}_k$: category of finite-dimensional vector spaces
	\ii $\catname{Set}$: category of sets
	\ii $\catname{hTop}$: category of topological spaces,
	whose morphisms are homotopy classes of maps
	\ii $\catname{hTop}_\ast$: pointed version of $\catname{hTop}$
	\ii $\catname{hPairTop}$: category of pairs $(X,A)$ with morphisms
	being pair-homotopy equivalence classes
	\ii $\Opens(X)$: the category of open sets of $X$, as a poset
\end{itemize}
Operations with categories:
\begin{itemize}
	\ii $\obj \AA$: objects of the category $\AA$
	\ii $\AA\op$: opposite category
	\ii $\AA \times \BB$: product category
	\ii $[\AA, \BB]$: category of functors from $\AA$ to $\BB$
	\ii $\ker f : \Ker f \to B$: for $f : A \to B$, categorical kernel
	\ii $\coker f : A \to \Coker f$: for $f : A \to B$, categorical cokernel
	\ii $\img f : A \to \Img f$: for $f : A \to B$, categorical image
\end{itemize}

\section{Differential geometry}
\begin{itemize}
	\ii $Df$: total derivative of $f$
	\ii $(Df)_p$: total derivate of $f$ at point $p$
	\ii $\fpartial{f}{e_i}$: $i^{\text{th}}$ partial derivative
	\ii $\alpha_p$: evaluating a $k$-form $\alpha$ at $p$
	\ii $\int_c \alpha$: integration of the differential form $\alpha$ over a cell $c$
	\ii $d\alpha$: exterior derivative of a $k$-form $\alpha$
	\ii $\phi^\ast \alpha$: pullback of $k$-form $\alpha$ by $\phi$
\end{itemize}

\section{Algebraic number theory}
\begin{itemize}
	\ii $\ol \QQ$: ring of algebraic numbers
	\ii $\ol \ZZ$: ring of algebraic integers
	\ii $\ol F$: algebraic closure of a field $F$
	\ii $\NK(\alpha)$: the norm of $\alpha$ in extension $K/\QQ$
	\ii $\TrK(\alpha)$: the trace of $\alpha$ in extension $K/\QQ$
	\ii $\OO_K$: ring of integers in $K$
	\ii $\ka+\kb$: sum of two ideals $\ka$ and $\kb$
	\ii $\ka\kb$: ideal generated by products of elements in ideals $\ka$ and $\kb$
	\ii $\ka \mid \kb$: ideal $\ka$ divides ideal $\kb$
	\ii $\ka\inv$: the inverse of $\ka$ in the ideal group
	\ii $\Norm(I)$: ideal norm
	\ii $\Cl_K$: class group of $K$
	\ii $\Delta_K$: discriminant of number field $K$
	\ii $\mu(\OO_K)$: set of roots of unity contained in $\OO_K$
	\ii $[K:F]$: degree of a field extension
	\ii $\Aut(K/F)$: set of field automorphisms of $K$ fixing $F$
	\ii $\Gal(K/F)$: Galois group of $K/F$
	\ii $D_\kp$: decomposition group of prime ideal $\kp$
	\ii $I_\kp$: inertia group of prime ideal $\kp$
	\ii $\Frob_\kp$: Frobenius element of $\kp$ (element of $\Gal(K/\QQ)$)
	\ii $P_K(\km)$: ray of principal ideals of a modulus $\km$
	\ii $I_K(\km)$: fractional ideals of a modulus $\km$
	\ii $C_K(\km)$: ray class group of a modulus $\km$
	\ii $\left( \frac{L/K}{\bullet} \right)$: the Artin symbol
	\ii $\Ram(L/K)$: primes of $K$ ramifying in $L$
	\ii $\kf(L/K)$: the conductor of $L/K$
\end{itemize}

\section{Representation theory}
\begin{itemize}
	\ii $k[G]$: group algebra
	\ii $V \oplus W$: direct sum of representations $V = (V, \rho_V)$
	and $W = (W, \rho_W)$ of an algebra $A$
	\ii $V^\vee$: dual representation of a representation $V = (V, \rho_V)$
	\ii $\Reg(A)$: regular representation of an algebra $A$
	\ii $\Homrep(V,W)$: algebra of morphisms $V \to W$ of representations
	\ii $\chi_V$: the character $A \to k$ attached to an $A$-representation $V$
	\ii $\Classes(G)$: set of conjugacy classes of $G$
	\ii $\FunCl(G)$: the complex vector space of functions $\Classes(G) \to \CC$
	\ii $V \otimes W$: tensor product of representations $V = (V, \rho_V)$ and $W = (W, \rho_W)$
	of a \emph{group} $G$ (rather than an algebra)
	\ii $\Ctriv$: the trivial representation
	\ii $\Csign$: the sign representation
\end{itemize}

\section{Algebraic geometry}
\begin{itemize}
	\ii $\VV(-)$: vanishing locus of a set or ideal
	\ii $\Aff^n$: $n$-dimensional (complex) affine space
	\ii $\sqrt I$: radical of an ideal $I$
	\ii $\CC[V]$: coordinate ring of an affine variety $V$
	\ii $\OO_V(U)$: ring of rational functions on $U$
	\ii $D(f)$: distinguished open set
	\ii $\CP^n$: complex projective $n$-space (ambient space for projective varieties)
	\ii $(x_0 : \dots : x_n)$: coordinates of projective space
	\ii $U_i$: standard affine charts
	\ii $\Vp(-)$: projective vanishing locus.
	\ii $h_I$, $h_V$: Hilbert function of an ideal $I$ or projective variety $V$
	\ii $\pi^\sharp$ or $\pi^\sharp_U$: the pullback $\OO_Y \to \OO_X(\pi\pre(U))$ obtained from $\pi \colon X \to Y$
	\ii $\SF_p$: the stalk of a (pre-)sheaf $\SF$ at a point $p$
	\ii $[s]_p:$ the germ of $s \in \SF(U)$ at the point $p$
	\ii $\OO_{X,p}$: shorthand for $(\OO_X)_p$.
	\ii $\SF\sh$: sheafification of pre-sheaf $\SF$
	\ii $\alpha_p : \SF_p \to \SG_p$: morphism of stalks obtained from $\alpha : \SF \to \SG$
	\ii $\km_{X,p}$: the maximal ideal of $\OO_{X,p}$
	\ii $\Spec A$: the spectrum of a ring $A$
	\ii $S\inv A$: localization of ring $A$ at a set $S$
	\ii $A[1/f]$: localization of ring $A$ away from element $f$
	\ii $A_\kp$: localization of ring $A$ at prime ideal $\kp$
	\ii $f(\kp)$: the value of $f$ at $\kp$, i.e.\ $f \pmod \kp$
	\ii $\kappa(\kp)$: the residue field of $\Spec A$ at the element $\kp$.
	% \ii $\Proj R$: the projective scheme of a graded ring $S$
	\ii $\pi^\sharp_{\kp}$: the induced map of stalks in $\pi^\sharp$.
\end{itemize}

\section{Set theory}
\begin{itemize}
	\ii $\ZFC$: standard theory of ZFC
	\ii $\ZFC^+$: standard theory of ZFC, plus the sentence
	``there exists a strongly inaccessible cardinal''
	\ii $2^S$ or $\PP(S)$: power set of $S$
	\ii $A \land B$: $A$ and $B$
	\ii $A \lor B$: $A$ or $B$
	\ii $\neg A$: not $A$
	\ii $V$: class of all sets (von Neumann universe)
	\ii $\omega$: the first infinite ordinal, also the set of nonnegative integers
	\ii $V_\alpha$: level of the von Neumann universe
	\ii $\On$: class of ordinals
	\ii $\bigcup A$: the union of elements inside $A$
	\ii $A \approx B$: sets $A$ and $B$ are equinumerous
	\ii $\aleph_\alpha$: the aleph numbers
	\ii $\cof \lambda$: the cofinality of $\lambda$
	\ii $\MM \vDash \phi[b_1, \dots, b_n]$: model $\MM$ satisfies sentence $\phi$
	with parameters $b_1$, \dots, $b_n$
	\ii $\Delta_n$, $\Sigma_n$, $\Pi_n$: levels of the Levy hierarchy
	\ii $\MM_1 \subseteq \MM_2$: $\MM_1$ is a substructure of $\MM_2$
	\ii $\MM_1 \prec \MM_2$: $\MM_1$ is an elementary substructure of $\MM_2$
	\ii $p \parallel q$: elements $p$ and $q$ of a poset $\Po$ are compatible
	\ii $p \perp q$: elements $p$ and $q$ of a poset $\Po$ are incompatible
	\ii $\Name_\alpha$: the hierarchy of $\Po$-names
	\ii $\tau^G$: interpretation of a name $\tau$ by filter $G$
	\ii $M[G]$: the model obtained from a forcing poset $G \subseteq \Po$
	\ii $p \Vdash \varphi(\sigma_1, \dots, \sigma_n)$: $p \in \Po$ forces the sentence $\varphi$
	\ii $\check x$: the name giving an $x \in M$ when interpreted
	\ii $\dot G$: the name giving $G$ when interpreted
\end{itemize}

% Consider adding:
% lim (convergence)
% group presentation


\RequirePackage{amsmath}
\RequirePackage{fix-cm}
\documentclass{svmono}

\def\ColoredLinks{}
\input{Macros.tex}
\usepackage{xcolor}
\usepackage{lmodern}
\usepackage{wrapfig}
\textwidth160mm
\textheight220mm
\oddsidemargin0mm
\evensidemargin0mm
\topmargin0mm

\def\cmd#1{\textbf{"\texttt{#1}"}}
\def\cm#1{\textbf{\texttt{#1}}}

\begin{document}

\hypersetup{pageanchor=false}
\thispagestyle{empty}
\vskip5cm {\Huge\textbf{Distribution XML reference for \vskip.1cm Warteschlangensimulator}}
\vskip.5cm \hrule
\vskip.5cm {\large \textsc{Alexander Herzog} (\href{mailto:alexander.herzog@tu-clausthal.de}{alexander.herzog@tu-clausthal.de})}
\vskip.25cm {\color{gray}
This reference refers to version \input{../Version.tex} of Warteschlangensimulator.\\
Download address: \href{https://github.com/A-Herzog/Warteschlangensimulator/}{https://github.com/A-Herzog/Warteschlangensimulator/}.
}
\vskip.5cm \hrule
\vskip.5cm

When storing distribution settings to xml files the following xml tags will be used:

English version:\\
\cm{<ModelElementDistribution>distribution name (parameters)</ModelElementDistribution>}

German version:\\
\cm{<ModellElementVerteilung>distribution name (parameters)</ModellElementVerteilung>}

The English or German version will be used when storing xml data in Warteschlangensimulator. When reading xml files Warteschlangensimulator will always understand both versions.

In the following sections the possible values for "distribution name" and the corresponding "parameters" will be listed. Distribution parameters "mean" and "sd" correspond directly to the stochastic characteristics mean and standard deviation.

\section*{Empirical data}

\cm{Empirical data (point1;point2;point3;...)}~\\
\cm{Empirische Daten (point1;point2;point3;...)}

The data points are interpreted as pdf values at equidistant points on the predefined range of the distribution.

\section*{One point distribution}

\cm{One point distribution (point)}~\\
\cm{Ein-Punkt-Verteilung (point)}

Conversion between distribution parameters and stochastic characteristics
\begin{eqnarray*}
mean&=&point\\
sd&=&0
\end{eqnarray*}

\section*{Uniform distribution}

\cm{Uniform distribution (lower;upper)}~\\
\cm{Gleichverteilung (lower;upper)}

Conversion between distribution parameters and stochastic characteristics
\begin{eqnarray*}
mean&=&(lower+upper)/2\\
sd&=&(upper-lower)/\sqrt{12}\\
lower&=&mean-sd\cdot\sqrt{12}/2\\
upper&=&mean+sd\cdot\sqrt{12}/2\\
\end{eqnarray*}

\section*{Exponential distribution}
\cm{Exponentialverteilung (mean)}~\\
\cm{Exponential distribution (mean)}

Conversion between distribution parameters and stochastic characteristics
\begin{eqnarray*}
sd&=&mean
\end{eqnarray*}

\section*{Normal distribution}
\cm{Normal distribution (mean;sd)}~\\
\cm{Normalverteilung (mean;sd)}

\section*{Lognormal distribution}
\cm{Lognormal distribution (mean;sd)}~\\
\cm{Lognormalverteilung (mean;sd)}

\section*{Erlang distribution}
\cm{Erlang distribution (shape;scale)}~\\
\cm{Erlang-Verteilung (shape;scale)}

Conversion between distribution parameters and stochastic characteristics
\begin{eqnarray*}
mean&=&shape\cdot scale\\
sd&=&\sqrt{shape}\cdot scale\\
scale&=&sd^2/mean\\
shape&=&mean^2/sd^2
\end{eqnarray*}

\section*{Gamma distribution}
\cm{Gamma distribution (shape;scale)}~\\
\cm{Gamma-Verteilung (shape;scale)}

Conversion between distribution parameters and stochastic characteristics
\begin{eqnarray*}
mean&=&shape\cdot scale\\
sd&=&\sqrt{shape}\cdot scale\\
scale&=&sd^2/mean\\
shape&=&mean^2/sd^2
\end{eqnarray*}

\section*{Beta distribution}
\cm{Beta distribution (alpha;beta;lower;upper)}~\\
\cm{Beta-Verteilung (alpha;beta;lower;upper)}

Conversion between distribution parameters and stochastic characteristics
\begin{eqnarray*}
mean&=&alpha/(alpha+beta)\cdot(upper-lower)+lower\\
sd&=&(upper-lower)^2\cdot alpha\cdot beta/(alpha+beta)^2/(1+alpha+beta)
\end{eqnarray*}

\section*{Cauchy distribution}
\cm{Cauchy-Verteilung (median;scale)}~\\
\cm{Cauchy distribution (median;scale)}

\section*{Weibull distribution}
\cm{Weibull distribution (scaleInvers;shape)}~\\
\cm{Weibull-Verteilung (scaleInvers;shape)}

Conversion between distribution parameters and stochastic characteristics
\begin{eqnarray*}
mean&=&scale\cdot\Gamma(1+(1/shape))\\
sd&=&scale\cdot\sqrt{\Gamma(1+2/shape)-(\Gamma(1+1/shape))^2}
\end{eqnarray*}

\section*{Chi distribution}
\cm{Chi distribution (degreesOfFreedom)}~\\
\cm{Chi-Verteilung (degreesOfFreedom)}

Conversion between distribution parameters and stochastic characteristics
\begin{eqnarray*}
mean&=&\sqrt{2}\cdot\Gamma((degreesOfFreedom+1)/2)/\Gamma(degreesOfFreedom/2)\\
sd&=&\Big[(2\cdot\Gamma(degreesOfFreedom/2)*\Gamma(1+degreesOfFreedom/2)-\\
~&~&(\Gamma((degreesOfFreedom+1)/2))^2)/\Gamma(degreesOfFreedom/2)\Big]^{\frac{1}{2}}
\end{eqnarray*}

\section*{Chi$^2$ distribution}
\cm{Chi\^{}2 distribution (degreesOfFreedom)}~\\
\cm{Chi\^{}2-Verteilung (degreesOfFreedom)}

Conversion between distribution parameters and stochastic characteristics
\begin{eqnarray*}
mean&=&degreesOfFreedom\\
sd&=&\sqrt{2\cdot degreesOfFreedom}
\end{eqnarray*}

\section*{F distribution}
\cm{F distribution (NumeratorDegreesOfFreedom;DenominatorDegreesOfFreedom)}~\\
\cm{F-Verteilung (NumeratorDegreesOfFreedom;DenominatorDegreesOfFreedom)}

Conversion between distribution parameters and stochastic characteristics\\
(let $m:=NumeratorDegreesOfFreedom$ and $n:=DenominatorDegreesOfFreedom$)
\begin{eqnarray*}
mean&=&
\frac{n}{n-2}\\
sd&=&
\sqrt{2\cdot n^2\cdot\frac{m+n-2}{m\cdot(n-2)\cdot(n-2)\cdot(n-4)}}\\
m&=&
round\left(2\cdot (2\cdot\widetilde m)^2\cdot\frac{2\cdot\widetilde m-2}{sd^2\cdot (2\cdot\widetilde m-2)^2\cdot (2\cdot\widetilde m-4)-2\cdot (2\cdot\widetilde m)^2}\right) ~\text{with}~ \widetilde m:=mean/(mean-1)\\
n&=&
round(2\cdot mean/(mean-1))
\end{eqnarray*}

\section*{Johnson SU distribution}
\cm{Johnson SU distribution (gamma;xi;delta;lambda)}~\\
\cm{Johnson-SU-Verteilung (gamma;xi;delta;lambda)}

Conversion between distribution parameters and stochastic characteristics
\begin{eqnarray*}
mean&=&xi-lambda\cdot\exp(1/(2\cdot delta^2))\cdot\sinh(gamma/delta)\\
sd&=&\sqrt{lambda^2/2\cdot(\exp(1/delta^2)-1)\cdot(\exp(1/delta^2)*\cosh(2*gamma/delta)+1)}
\end{eqnarray*}

\section*{Triangular distribution}
\cm{Triangular distribution (lowerBound;mostLikelyX;upperBound)}~\\
\cm{Dreiecksverteilung (lowerBound;mostLikelyX;upperBound)}

Conversion between distribution parameters and stochastic characteristics
\begin{eqnarray*}
mean&=&(lowerBound+mostLikelyX+upperBound)/3\\
sd&=&\Big[(lowerBound^2+upperBound^2+mostLikelyX^2-lowerBound\cdot upperBound-\\
~&~&lowerBound\cdot mostLikelyX-upperBound\cdot mostLikelyX)/18\Big]^{\frac{1}{2}}\\
lowerBound&=&mean-sd\cdot\sqrt{6}\\
mostLikelyX&=&mean\\
upperBound&=&mean+sd\cdot\sqrt{6}
\end{eqnarray*}

\section*{Pert distribution}
\cm{Pert distribution (lowerBound;mostLikelyX;upperBound)}~\\
\cm{Pert-Verteilung (lowerBound;mostLikelyX;upperBound)}

Conversion between distribution parameters and stochastic characteristics
\begin{eqnarray*}
mean&=&(lowerBound+4\cdot mostLikelyX+upperBound)/6\\
sd&=&\Big[((lowerBound+4\cdot mostLikelyX+upperBound)/6-lowerBound)\cdot\\
~&~&(upperBound-(lowerBound+4\cdot mostLikelyX+upperBound)/6)/7\Big]^{\frac{1}{2}}\\
lowerBound&=&mean-sd\cdot\sqrt{7}\\
mostLikelyX&=&mean\\
upperBound&=&mean+sd\cdot\sqrt{7}
\end{eqnarray*}

\section*{Laplace distribution}
\cm{Laplace distribution (mean;b)}~\\
\cm{Laplace-Verteilung (mean;b)}

Conversion between distribution parameters and stochastic characteristics
\begin{eqnarray*}
sd&=&b\cdot\sqrt{2}
\end{eqnarray*}

\section*{Pareto distribution}
\cm{Pareto distribution (xmin;alpha)}~\\
\cm{Pareto-Verteilung (xmin;alpha)}

Conversion between distribution parameters and stochastic characteristics
\begin{eqnarray*}
mean&=&alpha\cdot xmin/(alpha-1)\\
sd&=&xmin^2\cdot alpha/(alpha-1)^2/(alpha-2)\\
alpha&=&mean/(mean-xmin)
\end{eqnarray*}

\section*{Logistic distribution}
\cm{Logistic distribution (mean;s)}~\\
\cm{Logistische Verteilung (mean;s)}

Conversion between distribution parameters and stochastic characteristics
\begin{eqnarray*}
sd&=&s\cdot\pi/\sqrt{3}
\end{eqnarray*}

\section*{Inverse gaussian distribution}
\cm{Inverse gaussian distribution (lambda;mu)}~\\
\cm{Inverse Gauß-Verteilung (lambda;mu)}

Conversion between distribution parameters and stochastic characteristics
\begin{eqnarray*}
mean&=&mu\\
sd&=&mu\cdot\sqrt{mu/lambda}\\
lambda&=&mean^3/sd^2
\end{eqnarray*}

\section*{Rayleigh distribution}
\cm{Rayleigh distribution (mean)}~\\
\cm{Rayleigh-Verteilung (mean)}

Conversion between distribution parameters and stochastic characteristics
\begin{eqnarray*}
sd&=&\sqrt{(4-\pi)/2}\cdot\sqrt{2/\pi}\cdot mean
\end{eqnarray*}

\section*{Log-logistic distribution}
\cm{Log-logistic distribution (alpha;beta)}~\\
\cm{Log-Logistische Verteilung (alpha;beta)}

Conversion between distribution parameters and stochastic characteristics
\begin{eqnarray*}
mean&=&alpha\cdot\pi/beta/\sin(\pi/beta)\\
sd&=&alpha\cdot\sqrt{2\cdot\pi/beta/\sin(2\cdot\pi/beta)-\pi^2/beta^2}/sin(\pi/beta)
\end{eqnarray*}

\section*{Power distribution}
\cm{Power distribution (a;b;c)}~\\
\cm{Potenzverteilung (a;b;c)}

Conversion between distribution parameters and stochastic characteristics
\begin{eqnarray*}
mean&=&a+(b-a)\cdot c/(c+1)\\
sd&=&(b-a)/(c+1)\cdot\sqrt{c/(c+2)}
\end{eqnarray*}

\section*{Gumbel distribution}
\cm{Gumbel distribution (mean;sd)}~\\
\cm{Gumbel-Verteilung (mean;sd)}

\section*{Fatigue life distribution}
\cm{Fatigue life distribution (mu;beta;gamma)}~\\
\cm{Fatigue-Life-Verteilung (mu;beta;gamma)}

Conversion between distribution parameters and stochastic characteristics
\begin{eqnarray*}
mean&=&mu+beta\cdot (1+gamma\cdot gamma/2)\\
sd&=&beta\cdot gamma\cdot\sqrt{1+5\cdot gamma^2/4}
\end{eqnarray*}

\section*{Frechet distribution}
\cm{Frechet distribution (delta;beta;alpha)}~\\
\cm{Frechet-Verteilung (delta;beta;alpha)}

Conversion between distribution parameters and stochastic characteristics
\begin{eqnarray*}
mean&=&delta+beta\cdot gamma(1-1/alpha)\\
sd&=&beta\cdot\sqrt{gamma(1-2/alpha)-gamma(1-1/alpha)^2}
\end{eqnarray*}

\section*{Hyperbolic secant distribution}
\cm{Hyperbolic secant distribution (mean;sd)}~\\
\cm{Hyperbolische Sekanten-Verteilung (mean;sd)}

\section*{Left sawtooth distribution}
\cm{Left sawtooth distribution (a;b)}~\\
\cm{Linke Sägezahnverteilung (a;b)}
Conversion between distribution parameters and stochastic characteristics
\begin{eqnarray*}
mean&=&(2\cdot a+b)/2\\
sd&=&(b-a)^2/18
\end{eqnarray*}

\section*{Right sawtooth distribution}
\cm{Right sawtooth distribution (a;b)}~\\
\cm{Rechte Sägezahnverteilung (a;b)}
Conversion between distribution parameters and stochastic characteristics
\begin{eqnarray*}
mean&=&(a+2\cdot b)/2\\
sd&=&(b-a)^2/18
\end{eqnarray*}

\end{document}


\chapter{Buoyancy}

When you put a boat into water, it will sink into the water until
the mass of the water it displaces is equal to the mass of the
boat. We think of this in terms of forces. Gravity pulls the mass of
the boat down. The \newterm{buoyant force} pushes the boat up. A boat
dropped into the water will bob up and down a bit before reaching an
equilibrium where the two forces are equal.
% ADD: Explain Action Reaction Pairs in previous chapter
% ADD: Archimedes principle
% KA: https://www.khanacademy.org/science/in-in-class9th-physics-india/in-in-gravity/in-in-pressure-in-liquids-archimedes-principle/v/archimedes-principle-buoyancy-fluids-physics-khan-academy

The buoyant force pushes things up -- against the force of
gravity. The force is equal to the weight of the fluid being
replaced. So, for example, a cubic meter of freshwater has a mass of
about 1000kg.  If you submerge anything with a volume of one meter in
freshwater on earth, the buoyant force will be about 9800 newtons.

For some things, like a block of styrofoam, this buoyant force will be
sufficient to carry it to the surface. Once it reaches the surface, it
will continue to rise (displacing less water) until the mass of the
water it displaces is equal to its mass. And then we say ``It floats!''

\includegraphics[width=0.8\textwidth]{Buoyancy_Displacement_Diagram.png}

For some things, like a block of lead, the buoyant force is not
 sufficient to lift it to the surface, and thus we say ``It sinks!''

This is why a helium balloon floats through the air. The air
that it displaces weighs more than the balloon and the helium itself. (It is easy to forget that air has a mass, but it does.)

\begin{Exercise}[title={Buoyancy}, label=buoyancy]
  You have an aluminum box that has a heavy base, so it will always
  float upright. The box and its contents weigh 10 kg. Its base is 0.3 m x 0.4 m. It is 1m tall.

  When you drop it into freshwater ($1000 kg/m^3$), how far will it sink
  before it reaches equilibrium.

\end{Exercise}
\begin{Answer}[ref=buoyancy]
  Equilibrium will be achieved when the box has displaced 10 kg of water. That is, when it has displaced $0.01$ cubic meters.

  The area of the base of the box is 0.12 square meters.  So if the
  box sinks $x$ meters into the water it will displace $0.12 x$ cubic
  meters.

  Thus at equilibrium $x = \frac{0.01}{0.12} \approx 0.083$ m.  So,
  the box will sink 8.3 cm into the water before reaching equilibrium.
\end{Answer}

\section{The Mechanism of Buoyancy}

As you dive down in the ocean, you will experience greater and
greater pressure from the water. And if you take a balloon with you, you
will gradually see it get smaller as the water pressure compresses the
air in the balloon.

Let's say you are 3 meters below the surface of the water. What is the
pressure in Pascals (newtons per square meter)? You can think of the
water as a column of water crushing down upon you. The pressure over
a square meter is the weight of 3 cubic meters of water pressing down.

$$p = (3)(1000)(9.8) = 29,400 \text{ Pa }$$

This is called \newterm{hydrostatic pressure}. The general rule for
hydrostatic pressure in Pascals $p$ is

$p = d g h$

Where  $d$ is the density of the fluid
in kg per cubic meter, $g$ is the acceleration due to gravity in
$m/s^2$, and $h$ is the height of the column of fluid above you.

So, where does buoyant force come from? Basically, the pressure pushing up on the
deepest part of the object is higher than the pressure pushing down on
the shallowest part of the object. That is where bouyancy comes from.

\includegraphics[width=0.8\textwidth]{Buoyancy_Diagram.png}

% KA: https://www.khanacademy.org/science/physics/fluids/buoyant-force-and-archimedes-principle/a/buoyant-force-and-archimedes-principle-article

\begin{Exercise}[title={Hydrostatic Pressure}, label=mars_pressure]

  You dive into a tank of olive oil on Mars. How much more
  hydrostatic pressure does your body experience at 5 meters deep than
  it did at the surface?

  The density of olive oil is about 900 kg per square meter. The
  acceleration due to gravity on Mars is 3.721 $m/s^2$.

\end{Exercise}
\begin{Answer}[ref=mars_pressure]
$$p = d g h = (900)(3.721)(5) = 16,744.5 \text{ Pa}$$
\end{Answer}

Notice that although the pressure is increasing as you go deeper, the
buoyant force will \emph{not increase} because the buoyant force is always equal
to the weight of the fluid that is displaced, regardless if that is 1
meter or 100 meters underwater.

Also, saltwater is denser then freshwater. That is why people float
better in the sea than they do in a river.

And, lipids, like fats and oils, are less dense than water. That is why
people with a lot of body fat tend to float better than people with
less body fat. And why oil floats in a glass of water.

\includegraphics[width=0.8\textwidth]{Oil_Water.png}

% Image: https://image.shutterstock.com/image-photo/mixture-olive-oil-water-glass-260nw-1576851145.jpg


\section{Applications}
Please check exam schedule! Also, a sample exam is posted. This is the payoff day. All this stuff about Poincar\'e duality has got to be good for something. Recall:
\begin{theorem}[Fully relative duality]
Let $M$ be a $R$-oriented $n$-manifold. Let $L\subseteq K\subseteq M$ be compact ($M$ need not be compact). Then $[M]_K\in H_n(M,M-K)$, and capping gives an isomorphism:
$$\cHH^p(K,L;R)\xrightarrow{\cap[M]_k,\cong}H_{n-p}(M-L,M-K;R)$$
\end{theorem}
Today we'll think about the case $L=\emptyset$, so this is saying:
$$\cHH^p(K;R)\xrightarrow{\cap[M]_k,\cong}H_{n-p}(M,M-K;R)$$
\begin{corollary}
$\cHH^q(K;R)=0$ for $q>n$.
\end{corollary}
We can contrast this with singular (co)homology. Here's an example:
\begin{example}[Barratt-Milnor]
A two-dimensional version $K$ of the Hawaiian earring, i.e., nested spheres all tangent to a point whose radii are going to zero. What they proved is that $H_q(K;\QQ)$ is uncountable for every $q>1$. But if you look at the \v{C}ech cohomology, stuff vanishes.
\end{example}
That's nice.

How about an even more special subcase? Suppose $M=\RR^n$. The result is called Alexander duality. This says:
\begin{theorem}[Alexander duality]
If $\emptyset\neq K\subseteq \RR^n$ be compact. Then $\cHH^{n-q}(K;R)\xrightarrow{\cong}\widetilde{H}_{q-1}(\RR^n-K;R)$
\end{theorem}
\begin{proof}
We have the LES of a pair, which gives an isomorphism $\partial:H_q(\RR^n,\RR^n-K;R)\xrightarrow{\cong}\widetilde{H}_{q-1}(\RR^n-K;R)$, so the composition $\partial\circ(-\cap[M]_K)$ is an isomorphism by Poincar\'e duality.
\end{proof}
For most purposes, this is the most useful duality theorem.
\begin{example}[Jordan curve theorem]
$q=1$ and $R=\Z$. Then this is saying that $\cHH^{n-1}(K)\xrightarrow{\cong}\widetilde{H}_0(\RR^n-K)$. But $\widetilde{H}_0(\RR^n-K)$ is free on $\#\pi_0(\RR^n-K)-1$ generators. If $n=2$, for example, and $K\cong S^1$, then $\cHH^{n-1}(K)=H^{n-1}(K)\cong H^{n-1}(S^1)$, so $H^1(S^1)\cong \widetilde{H}_0(\RR^2-K)$. Hence there are \emph{two} components in the complement of $K$. This could also be the topologist's sine curve as well. This is the Jordan curve theorem.
\end{example}
Consider the UCT, which states that there's a sexseq $0\to\Ext^1_\Z(H_{q-1}(X),\Z)\to H^q(X)\to\Hom(H_q(X),\Z)\to 0$ that splits, but not naturally. First, note that $\Hom(H_q(X),\Z)$ is always torsion-free. If I assume that $H_{q-1}(X)$ is finitely generated, then $\Ext^1_\Z(H_{q-1}(X),\Z)$ is a finite abelian group, but in particular it's torsion.

The UCT is making the decomposition of $H^q(X)$ into its torsion-free and torsion parts. I can divide by torsion, so that $H^q(X)/\mathrm{tors}\cong \Hom(H_q(X),\Z)$. But there's also an isomorphism $\Hom(H_q(X)/\mathrm{tors},\Z)\to \Hom(H_q(X),\Z)$ because $\Z$ is torsion-free. Therefore I get an isomorphism $\alpha:H^q(X)/\mathrm{tors}\to \Hom(H_q(X)/\mathrm{tors},\Z)$. I.e.:
\begin{equation*}
\xymatrix{
	0\ar[r] & \Ext^1_\Z(H_{q-1}(X),\Z)\ar[r] & H^{q}(X)\ar[r]\ar[d] & \Hom(H_q(X),\Z)\ar[r] & 0\\
 & & H^q(X)/\mathrm{tors}\ar[ur]^{\cong}\ar[r]^{\alpha} & \Hom(H_q(X)/\mathrm{tors}/\Z)\ar[u]^{\cong}
}
\end{equation*}
Or I could say it like this: the Kronecker pairing can be quotiented by torsion, and you get an induced map $H^q(X)/\mathrm{tors}\otimes H_q(X)/\mathrm{tors}\to\Z$ is a perfect pairing, which means that the adjoint map $H^q(X)/\mathrm{tors}\xrightarrow{\cong}\Hom(H_q(X)/\mathrm{tors},\Z)$. Let's combine this with Poincar\'e duality.

Let $X=M$ be a compact oriented $n$-manifold. Then $H^{n-q}(X)\xrightarrow{-\cap[M],\cong}H_q(M)$, and so we get a perfect pairing $H^q(X)/\mathrm{tors}\otimes H^{n-q}(X)/\mathrm{tors}\to\Z$. And what is that pairing? It's the cup product! We have:
\begin{equation*}
\xymatrix{
	H^q(M)\otimes H^{n-q}(M)\ar[r]\ar[d]_{1\otimes(-\cap [M])} & \Z\\
	H^q(M)\otimes H_q(M)\ar[ur]_{\langle,\rangle}
}
\end{equation*}
And, well:
\begin{equation*}
\langle a,b\cap [M]\rangle = \langle a\cup b,[M]\rangle
\end{equation*}
Thus the map $H^q(M)\otimes H^{n-q}(M)\to \Z$ is $a\otimes b\mapsto\langle a\cup b,[M]\rangle$, and it's a perfect pairing. This is a purely cohomological version, and is the most useful statement.
\begin{example}
Suppose $M=\CP^2=D^0\cup D^2\cup D^4$, and its homology is $\Z \, 0 \, \Z \, 0\, \Z$, and so its cohomology is the same. Let $a\in H^2(\CP^2)$. Then we have $H^2(\CP^2)\otimes H^2(\CP^2)\to \Z$, and so $a\cup a$ is a generator of $H^4(\CP^2)$, and hence specifies an orientation for $\CP^2$. The conclusion is that $H^\ast(\CP^2)=\Z[a]/(a^3)$ where $|a|=2$.

How about $\CP^3$? It just adds a $6$-cell, so its homology is $\Z \, 0 \, \Z \, 0\, \Z \, 0 \, \Z$, and so its cohomology is the same. But then $a^3=a\cup a\cup a$ is a generator of $H^6(\CP^2)$, and etc. Thus in general, we have:
$$H^\ast(\CP^n)=\Z[a]/(a^{n+1})$$
These things are finite CW-complexes, so you find:
\begin{equation}
H^\ast(\CP^\infty)=\Z[a]
\end{equation}
\end{example}
\begin{example}
Suppose I look at maps $f:S^m\to S^n$. One of the most interesting things is that there are lots of non null-homotopic maps $S^m\to S^n$ if $m>2$. For example, $\eta:S^3\to S^2$ that's the attaching map for the $4$-cell in $\CP^2$. This is called the Hopf fibration. It's essential. Why is it nullhomotopic? If $\eta$ was null homotopic, then $\CP^2\simeq S^2\wedge S^4$. That's compatible with the cohomology in each dimension, but not into the cohomology ring! There's a map $S^2\wedge S^4\to S^2$ that collapses $S^4$, and the generator in $H^\ast(S^2)$ has $a^2=0$, so $a^2=0$ in $H^\ast(S^2\wedge S^4)$. But this is not compatible with our computation that $H^\ast(\CP^2)=\Z[a]/(a^3)$ where $|a|=2$.
\end{example}

With coefficients in a field $k$, then the torsion is zero, so you find that if $M$ is compact $k$-oriented, then if the characteristic of $k=2$, there's no condition for $M$ to be oriented, and if the characteristic of $k$ is not $2$, then $M$ is $\Z$-oriented. Thus we get that $H^q(M;k)\otimes_k H^{n-q}(M;k)\to k$ is a perfect pairing.
\begin{example}
Exactly the same argument as for complex projective space shows that:
\begin{equation*}
H^\ast(\RP^n;\FF_2)=\FF_2[a]/(a^{n+1})
\end{equation*}
where $|a|=1$. So:
\begin{equation}
H^\ast(\RP^\infty;\FF_2)=\FF_2[a]
\end{equation}
where $|a|=1$.
\end{example}
I'll end with the following application.
\begin{theorem}
Suppose $f:\RR^{m+1}\supseteq S^m\to S^n\subseteq \RR^{n+1}$ that is equivariant with respect to the antipodal action, i.e., $f(-x)=-f(x)$. Then $m\leq n$.
\end{theorem}
So there are \emph{no} equivariant maps from $S^m\to S^n$ if $m>n$!
\begin{proof}
Suppose I have a map like that: the map on spheres induces a map $\overline{f}:\RP^m\to\RP^n$. We claim that $H_1(\overline{f})$ is an isomorphism. Let $\pi:S^n\to\RP^n$ denote the map. Let $\sigma:I\to S^m$ be defined via $\sigma(0)=v$ and $\sigma(1)=-v$. So this gives a $1$-cycle $\sigma:I\to S^m\to\RP^m$, and $H_1(\RP^n)=[\pi\sigma]$ is generated by this thing. When I map this thing to $\RP^n$, we send $\pi\sigma$ to a generator. What we've actually proved, therefore, is that $H_1(\RP^m)\cong H_1(\RP^n)$. This is also true with mod $2$ coefficients, i.e., $H_1(\overline{f},\FF_2)\neq 0$.

That means that $H^1(\overline{f};\FF_2)\neq 0$ by UCT. But what is this? This is a map $H^1(f;\FF_2):H^\ast(\RP^n;\FF_2)\to H^\ast(\RP^n;\FF_2)$, i.e., a map $\FF_2[a]/(a^{n+1})\to \FF_2[a]\to(a^{m+1})$. Thus $a\mapsto a$. There's not a lot of ways to do this if $m>n$. Thus what we've shown that $m\leq n$.
\end{proof}
This is the Borsuk-Ulam theorem from the '20s, I think. This is an example of how you can use the cohomology ring structure for projective space.

Please check the website for details about your finals. I will ask you to sign a form, to make sure that you don't share the questions or that you haven't heard the questions beforehand. I have a fixed set of questions that'll guide the conversation.