.%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%                                                         %%
%% Preamble of document                                    %%
%%                                                         %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Sets up the standard "article" document class
\documentclass[12pt]{article}

% Loads packages
% Note that I typically load all packages at once (as below).
% Sometimes you must load them in a certain order...
\usepackage{amsmath,
            amsfonts,
            amssymb,
            amsthm,
            enumerate,
            enumitem,
            geometry,
            mleftright,
            nicefrac,
            mathtools,
            xparse,
            ifthen,
            tikz,
            pgfplots,
            floatrow,
            multicol,
            caption,
            bbm,
            xurl}
            
% Set up PGF plots (used for graphing)
\pgfplotsset{compat=newest} 

% Fixes spacing caused by using "\left" or "\right"
\mleftright

% Sets the margins of the document.
% One can do this using plain TeX, but why?  
\geometry{margin=1in}
          
% Lines 45-48 set up my use of the cleveref package.
% This provides a "smarter" cross referencing style
\usepackage[colorlinks=true]{hyperref}
\usepackage[nosort,capitalise]{cleveref}
\newcommand{\creflastconjunction}{, and\nobreakspace}
\crefname{enumi}{item}{items}

% My personal command macros that I use.
% Note that "\displaySpace" is use to create the correct spacing for displayed math puctuation (and can be easily updated throughout the entire document by changing line 50).
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\E}{\mathbb{E}}
\renewcommand{\P}{\mathbb{P}}

% Creates my paired math delimiters for easy use.
\DeclarePairedDelimiter{\pr}{(}{)}
\DeclarePairedDelimiter{\br}{[}{]}
\DeclarePairedDelimiter{\cu}{\{}{\}}
\DeclarePairedDelimiter{\abs}{\lvert}{\rvert}
\DeclarePairedDelimiter{\norm}{\lVert}{\rVert}
\DeclarePairedDelimiter{\vt}{\langle}{\rangle}

% Creates desired header and footer for document.
% Note that it uses the adaptively created date macro.
\setlength{\headheight}{15pt}
\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyhf{}
\fancyhead[R]{}
\cfoot{}
\lfoot{\small{\textsc{Dissemination prohibited. \today}}}
\rfoot{\thepage}

% Updates equation and figure numbering.
\counterwithin{equation}{section}
\counterwithin{figure}{section}
		
% Defines theorem-like environments
\theoremstyle{definition}
\newtheorem{definition}{Definition}[section]
\newtheorem{theorem}[definition]{Theorem}
\newtheorem{problem}[definition]{Problem}
\newtheorem{remark}[definition]{Remark}
\newtheorem{assumption}[definition]{Assumption}
\newtheorem{setting}[definition]{Setting}
\newtheorem{corollary}[definition]{Corollary}
\newtheorem{proposition}[definition]{Proposition}
\newtheorem{lemma}[definition]{Lemma}
\renewcommand{\thetheorem}{\theenumi\arabic{theorem}}

% Helps create better arrows in figures
\usetikzlibrary{arrows.meta}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\title{Specifications for writing\\ a thesis or a scientific article}

\author{Joshua Lee Padgett}

\maketitle

\tableofcontents

\section{Preface}

%

The following document provides detailed specifications 
regarding the way in which a thesis or a scientific article should be written. 
In principle, all specifications are up for discussion.
If you intend to deviate from one of the specifications below, then please contact me 
\emph{in advance} before deviating from the proposed specifications 
to discuss with me whether I agree to the deviations that you propose. 
If you do not contact me regarding the proposed specifications, 
then I assume that you fully agree with the proposed specifications. 
Please let me know if you have any questions regarding this document.
Please also study carefully the \LaTeX\ code used to create this document!

\section{General comments}


\subsection{Logical precision}

\begin{enumerate}
\item
\emph{Each predicate} should be quantified by the use of a quantifier (often in written text instead of with a symbol) 
\emph{in front} of the predicate. The most common quantifiers are the \emph{universal} quantifier (text: ``for all,'' symbol: $ \forall $) 
and the \emph{existential} quantifier
(text: ``there exist(s),'' symbol: $ \exists $). 
The word ``For'' is \underline{not} a quantifier.

For example, please write:
note that for all $ x \in \R $ there exists a real number $ y \in \R $
such that for all $ t \in [0,\infty) $
it holds that
\begin{equation}
  e^x
  \leq 
  y \, e^t 
\end{equation}
instead of, for example:
% 
% 
% 
\begin{itemize}


 \item 
note that for all $ x \in \R $ there exists a real number $ y \in \R $ such that
\begin{equation}
\label{eq:pr1}
  e^x
  \leq 
  y \, e^t 
\end{equation}
for all $ t \in [0,\infty) $
(Problem: it is not entirely clear whether \cref{eq:pr1} means 
``$ \forall \, x \in \R \colon \exists \, y \in \R \colon \big( e^x \leq y \, e^t \ \forall \, t \in [0,\infty) \big) $''
or whether it means
``$ \forall \, x \in \R \colon \big( \big( \exists \, y \in \R \colon e^x \leq y \, e^t \big) \, \forall \, t \in [0,\infty) \big)$''.
Putting the quantifiers in front of the predicate avoids such issues.),


\item
note that for all $ x \in \R $ there exists a real number $ y \in \R $ such that
for $ t \in [0,\infty) \colon $
\begin{equation}
  e^x
  \leq 
  y \, e^t 
\end{equation}
(Problem: it is not clear what ``for $ t \in [0,\infty) $'' means. Does it mean 
``for all $ t \in [0,\infty) $ it holds that'' or does it mean 
``for some $ t \in [0,\infty) $ it holds that''?
``For some $ t \in [0,\infty) $ it holds that,'' in turn, is another way of saying that
``there exists a $ t \in [0,\infty) $ such that.''),
or

\item
note that for all $ x \in \R $ there exists a real number $ y \in \R $ such that
\begin{equation}
\label{eq:pr3}
  e^x
  \leq 
  y \, e^t 
\end{equation}
(Problem: $ t $ is not quantified in \cref{eq:pr3}; that is, \cref{eq:pr3} is not a statement 
but an $ 1 $-ary predicate. So, it does not make sense to say that we should note 
that \cref{eq:pr3} holds.).

\end{itemize}

Wikipedia (German)
\url{http://de.wikipedia.org/wiki/Pr%C3%A4dikatenlogik_erster_Stufe}
(access date: October 8, 2021):

``The first-order logic is a branch of mathematical logic. 
It is concerned with the structure of certain mathematical expressions and the logical inference that goes from such expressions to others. 
In doing so, it is possible to define both the language and the inference in a purely syntactic way, i.e., without reference to mathematical meanings. 
The interplay of purely syntactic considerations on the one hand and semantic considerations on the other hand leads to important findings that are important for all of mathematics, because this can be formulated using the Zermelo-Fraenkel set theory in the first-level predicate logic.''


\item
Please also use 
``Let $ n \in [0,\infty) $''
or
``Let $ n \in \{ 0, 1, 2, \dots \} $''
(depending on what you mean)
instead of 
``Let $ n \geq 0 $''
as it is not entirely clear whether 
``Let $ n \geq 0 $''
means 
``Let $ n \in \{ 0, 1, 2, \dots \} $''
or 
``Let $ n \in [0,\infty) $.''


\item 
We never use the word ``define'' and we never use the expression ``:='' in the article\slash thesis.
Instead we write, for example,
\begin{enumerate}[label=(\roman*)]
 \item let $ A $ be the set given by $ A = \{ 2, 4, 6, 8, \dots \} $,
 \item let $ A_r $, $ r \in \R $, be the sets which satisfy for all $ r \in \R $ 
 that $ A_r = [r,\infty) $,
 or 
 \item 
let $ f_r \colon \R \to \R $, $ r \in \R \backslash \{ 0 \} $, 
be the functions which satisfy for all $ r \in \R \backslash \{ 0 \} $, $ x \in \R $
that
\begin{equation}
  f_r( x ) = 
  \frac{ 1 }{ \sqrt{ 2 \pi r^2 } }
  \exp\left(
    - \frac{ x^2 }{ 2 r^2 } 
  \right)
  \text{.}
\end{equation}
\end{enumerate}

\end{enumerate}


\subsection{Writing style and scientific English}

\begin{enumerate}[label=(\roman*)]
\item 
The \LaTeX\ package \texttt{hyperref} must be included into the document.

\item 
Every time after the symbol $ \forall $ or the symbol $ \exists $ has appeared
there must appear the expression \verb+\,+ 
in the \LaTeX\ code.

\item 
The following 
\LaTeX\ phrases may not appear in your document:
\begin{enumerate}
\item 
``\verb+cf. +''
\item 
``\verb+i.i.d. +''
\end{enumerate}
Instead the following 
\LaTeX\ phrases may appear in your document:
\begin{enumerate}
\item 
``\verb+cf.\ +''
\item 
``\verb+i.i.d.\ +''
\end{enumerate}
Analogous comments apply to other abbreviations.

\item 
The following 
\LaTeX\ phrases may not appear in your document:
\begin{enumerate}
\item 
``\verb+$ d, m \in \N, x \in \R $+''
\item 
``\verb+$ d_1, d_2, \dots, d_k \in \N, x \in \R $+''
\end{enumerate}
Instead the following 
\LaTeX\ phrases may appear in your document:
\begin{enumerate}
\item 
``\verb+$ d, m \in \N $, $ x \in \R $+''
\item 
``\verb+$ d_1, d_2, \dots, d_k \in \N $, $ x \in \R $+''
\end{enumerate}
Analogous comments apply to similar expressions.

\item 
Immediately after the phrase ``e.g.,''
immediately after the phrase ``for example,''
and immediately after the phrase ``for instance'' 
must follow a comma.

\item 
One should never write that a function ``is increasing.''
Instead one should write that a function ``is strictly increasing'' or ``non-decreasing'' (depending on the correct content). 
The analogous comment applies to the phrase that a function ``is decreasing.''

\item 
The expressions 
$ \subset $,
$ \R^{ + } $, 
$ \R_{ \geq 0 } $,
or 
$ \R^{ * } $
may nowhere appear in the document. 
Instead the expressions 
$ \subseteq $, $ \subsetneq $, 
$ (0,\infty) $,  
$ [0,\infty) $,
or 
$ \R \backslash \{ 0 \} $
can be used in the document.

\item 
Immediately after every 
result (by which we mean a theorem, a lemma, a proposition, or a corollary) 
of the document (by which we mean a scientific article or a thesis) 
\begin{enumerate}[label=(\roman*)]
\item 
we add a proof which starts with
``Proof of $ \dots $'' and which ends, e.g., with 
``The proof of $ \dots $ is thus complete.'' or 
\item
we add a reference to another article\slash thesis 
for the proof of the result in front of the result.
\end{enumerate}
See the proof of \cref{prop:prop1} for an example.

 \item 
 Typically, every sentence, 
 except for the last one (and sometimes also the first one) 
 in the proof of a theorem, a lemma, 
 a proposition, or a corollary
 starts with something like
 \begin{enumerate}
  \item This implies/shows/proves/ensures/assures/yields/demonstrates that $ \dots $
  \item This, equation~(...), inequality~(...), estimate~(...), Theorem~..., Corollary~...,
  Lemma~..., and Lemma~... imply/show/prove/ensure/assure/yield/demonstrate that $ \dots $
  \item Combining $ \dots $ (with $ \dots $) (hence/therefore) implies/shows/$ \dots $ that $ \dots $
  \item Theorem~..., Corollary~..., Corollary~..., and Proposition~... (hence/therefore) imply/show/$ \dots $
  that $ \dots $
 \end{enumerate}
 Typically, the last sentence in the proof reads 
 as ``The proof of $ \dots $ is thus complete.''.
 Throughout the entire article/thesis the words ``thus'' typically appears
 only in the last sentence of a proof.
 A proof starts often with
 ``Throughout this proof let $ \dots $,
 let $ \dots $,
 $ \dots $,
 and let $ \dots $. Next observe/note that $ \dots $''.

\item 
Just before every display (\LaTeX\: \verb+\begin{equation} ... \end{equation}+) 
in the document should appear the phrase ``that'', the phrase ``and'', the phrase ``Then'', or the phrase ``be the set given by''
without any punctuation mark.


For example, we often write 

This implies for all $ \dots $ that \\
\verb+\begin{equation} ... \end{equation}+

or 

This implies that for all $ \dots $ it holds that\\ 
\verb+\begin{equation} ... \end{equation}+

in a proof.



\item 
\label{item:and_that}
The phrase ``and that'' may not appear in any result 
nor in any proof.


\item
Typically, we do not write a long text between the results.
One sentence relating the result with other results/notions/concepts in the document 
as well as with other results/notions/concepts in the literature is often sufficient.
We may add more explanatory sentences (written text) in the introduction and in the beginning of the 
section/chapter.

\item 
The document should start with an abstract and, thereafter, a table of contents.


\item 
 Typically, all assumptions in a setting are presented in exactly one sentence.
 In particular, we typically formulate a setting as follows:
 
 Throughout this article the following setting is (frequently\footnote{depending upon whether the setting
 is always or only sometimes used throughout the document}) used. Assume/let $ \dots $, assume/let $ \dots $,
 assume/let $ \dots $, $ \dots $, and assume/let $ \dots $.

\item
Every display has exactly one number on the right hand side. In particular, we never use
\verb+\begin{equation*} ... \end{equation*}+
or
\verb+$$ ... $$+.\\
We always use\\
\verb+\begin{equation} ... \end{equation}+
\\
or\\
\verb+\begin{equation} \begin{split} ... \end{split} \end{equation}+
\\
instead; see \cref{eq:young}--\cref{eq:Lyapunov_end} for a few examples.

\item 
We never use 
\begin{assumption}
Throughout this article/thesis assume that $ T p < 1 $,
that for all $ x \in \R $ it holds that
\begin{equation}
  f( x ) \leq 2 x \text{,}
\end{equation}
and that for all $ y \in [0,\infty) $ it holds that
\begin{equation}
  g( y ) \leq 2 y \text{.}
\end{equation}
\end{assumption}
but we may use one or more settings 
(see \cref{item:and_that} above and see \cref{sec:setting}
and \cref{prop:prop3} for an example).


 \item 
 Often the structure of a thesis/research article is as follows:
 \begin{enumerate}
  \item Introduction
  \begin{enumerate}
  \item[1.1] Notation
  \item[1.2] Setting
  \end{enumerate}
  \item First section for results $ \dots $
  \item Second section for results $ \dots $
  \item $ \dots $
 \end{enumerate}




\item 
Serial comma (also called Oxford comma):
We write 
``$ A $, $ B $, and $ C $'' 
instead of ``$ A $, $ B $ and $ C $''.
Similarly we write 
``$ A $, $ B $, or $ C $'' 
instead of
``$ A $, $ B $ or $ C $.'' 
\begin{enumerate}[label=(\alph*)]
 \item 
However, please note that the phrase 
``let $ f \colon \R \to \R $, and $ g \colon \R \to \R $ be functions which satisfy''
may not appear in your document.
\item 
Instead the phrase 
``let $ f \colon \R \to \R $ and $ g \colon \R \to \R $ be functions which satisfy''
may appear in your document.
\end{enumerate}


\item 
There should be no significant \LaTeX\ badboxes in the document.

\item 
You are welcome to use the axiom schema of specification 
to specify a set. More specifically, if $ S $ is a set and $ P $ 
is a predicate with one single argument, then you can use 
the set 
\begin{equation}
\label{eq:set_with_colon_command}
  \left\{ x \in S \colon P(x) \right\} 
\end{equation}
according to the axiom schema of specification in your document. 
In contrast, the expression 
\begin{equation}
  \left\{ x \in S \, | \, P(x) \right\} 
\end{equation}
may not appear in your document. Moreover, the expression
\begin{equation}
\label{eq:set_without_colon_command}
  \left\{ x \in S : P(x) \right\} 
\end{equation}
may also not appear in your document. 
As usual please check the employed \LaTeX\ code for \cref{eq:set_with_colon_command}
and \cref{eq:set_without_colon_command}, respectively.


\item 
\label{item:cases}
You are welcome to employ to the 
\LaTeX\ 
command \verb+\begin{cases} ... \end{cases}+ 
to create a distinction of cases within a display. 
If you employ the 
\LaTeX\ 
command \verb+\begin{cases} ... \end{cases}+, 
then it must employ the following structure:
\begin{verbatim}
\begin{cases}
Value in the first case
&
\colon
First case
\\
Value in the second case
&
\colon
Second case
\\
Value in the third case
&
\colon
Third case
\\
...
\\
Value in the n-th case
&
\colon
n-th case
\end{cases}
\end{verbatim}
For example, you may use the following phrase
``let $ a \in \R $, 
let $ \operatorname{sgn} \colon \R \to \R $ be the function which 
satisfies for all $ x \in \R $ that
\begin{equation}
\label{eq:sign_with_colon}
  \operatorname{sgn}(x)
  = 
  \begin{cases}
    1
  &
    \colon x \geq 0
  \\
    - 1
  &
    \colon x < 0 
    \text{,}
  \end{cases}
\end{equation}
and let $ \dots $'' 
to introduce a real number $ a \in \R $ 
and to specify the sign function (a sign function). 
The phrase 
``let $ \operatorname{sgn} \colon \R \to \R $ be the function that satisfies''
may not appear in your document. 


\item
The phrase ``let $ a \in \R $, 
let $ \operatorname{sgn} : \R \to \R $ be the function which 
satisfies'' 
may not appear in your document. 
The phrase ``let $ a \in \R $, 
let $ sgn \colon \R \to \R $ be the function which 
satisfies'' 
may also not appear in your document. 
Please note the differences 
between 
$ \operatorname{sgn} \colon \R \to \R $ 
(cf.\ \cref{eq:sign_with_colon} in \cref{item:cases} above), 
$ \operatorname{sgn} : \R \to \R $ (see above),
and 
$ sgn \colon \R \to \R $ (see above) 
and as usual please also study the \LaTeX\ codes used to create the above phrases.


\item 
The phrase ``(c.f.\ \cref{eq:sign_with_colon})'' may not appear in your document.
The phrase ``(cf. \cref{eq:sign_with_colon}'' may also not appear in your document.
In contrast, the phrase ``(cf.\ \cref{eq:sign_with_colon})'' 
and the phrase ``(see \cref{eq:sign_with_colon})'' 
may appear in your document. 
As usual please also study the \LaTeX\ codes used to create the above phrases.


\item 
The phrase 
``let $ \left\lVert \cdot \right\lVert_{ \R^d } \colon \R^d \to [0,\infty) $ be the Euclidean norm on $ \R^d $'' 
may appear in your document. 
\begin{enumerate}[label=(\alph*)]
\item
The phrase 
``let $ \left\lVert \cdot \right\rVert_{ \R^d } \colon \R^d \to [0,\infty) $ be the euclidean norm on $ \R^d $'' 
may not appear in your document. 
Analogously, one writes ``is H\"{o}lder continuous'' instead of is ``h\"{o}lder continuous.''

\item
The phrase 
``let $ \| \cdot \|_{ \R^d } \colon \R^d \to [0,\infty) $ be the Euclidean norm on $ \R^d $'' 
may also not appear in your document. 

\end{enumerate}
As usual please also study the \LaTeX\ codes used to create the above phrases.


\item 
The phrase ``function'' and the phrase ``functions'' may appear in your document. 
In contrast, each of the following phrases may not appear in your document:
\begin{enumerate}[label=(\alph*)]
\item
mapping
\item 
Mapping
\item 
map 
\item 
Map
\item 
Function 
\item 
Functions
\end{enumerate}
The following phrases may appear in your document:
\begin{enumerate}[label=(\alph*)]

\item
let $ f \colon \R \to \R $ be the function which satisfies for all

\item
let $ f \colon \R \to \R $ satisfy for all

\item 
let $ g_r \colon \R \to \R $, $ r \in (0,\infty) $, be the functions which satisfy for all 
$ r \in (0,\infty) $, $ x \in \R $ that $ g_r( x ) = x + r $

\item 
let $ g_r \colon \R \to \R $, $ r \in (0,\infty) $, satisfy for all 
$ r \in (0,\infty) $, $ x \in \R $ that $ g_r( x ) = x + r $

\item 
let $ f \colon \R^2 \to \R $ and $ g \colon \R \to \R $ be functions
\item 
let $ f = ( f(x,y) )_{ (x,y) \in \R^2 } \colon \R^2 \to \R $ and $ g \colon \R \to \R $ be functions
\item 
let $ f = ( f_x(y) )_{ (x,y) \in \R^2 } \colon \R^2 \to \R $ and $ g \colon \R \to \R $ be functions
\item 
let $ g_r \in C( \R, \R ) $, $ r \in (0,\infty) $, satisfy
\item 
let $ a_n \in \R $, $ n \in \N $, satisfy for all $ n \in \N $ that 
$ a_n = n $
\item 
let $ ( a_n )_{ n \in \N } \subseteq \R $ satisfy for all $ n \in \N $ that
$ a_n = n $
\end{enumerate}
In contrast, the following phrases may not appear in your document:
\begin{enumerate}[label=(\alph*)]
\item
let $ f : \R \to \R $ be the function which satisfies for all
\item 
let $ g_r \colon \R \to \R $, $ r \in (0,\infty) $ be the functions which satisfy for all 
$ r \in (0,\infty) $, $ x \in \R $ that $ g_r( x ) = x + r $
\item 
let $ g_r \colon \R \to \R $, $ r \in (0,\infty) $, be the functions which satisfy for all 
$ x \in \R $ that $ g_r( x ) = x + r $
\item 
let $ f \colon \R^2 \to \R $, $ g \colon \R \to \R $ be functions
\item 
let $ f \colon \R^2 \to \R $, $ g \colon \R \to \R $, be functions

\item 
let $ g_r \in C( \R, \R ) $, $ r \in (0,\infty) $ satisfy

\item 
Let $ g_r \in C( \R, \R ) $ satisfy

\item 
Let $ a \in \R $ and let $ g_r \in C( \R, \R ) $ satisfy

\item 
let $ g_r \in C( \R, \R ), r \in (0,\infty) $, satisfy

\item 
let $ g_r \in C( \R, \R ), r \in (0,\infty), $ satisfy

\item 
let $ a_n \in \R $, $ n \in \N $, satisfy that 
$ a_n = n $

\item 
let $ ( a_n )_{ n \in \N } \subseteq \R $ satisfy that
$ a_n = n $

\item 
let $ ( a_n )_{ n \in \N } \subseteq \R $, $ n \in \N $, satisfy for all $ n \in \N $ that
$ a_n = n $


\item 
let $ ( a_n )_{ n \in \N } \subseteq \R $, $ n \in \N $ satisfy for all $ n \in \N $ that
$ a_n = n $

\end{enumerate}
As usual please also study the \LaTeX\ codes used to create the above phrases.
\begin{enumerate}[label=(\Alph*)]
\item 
\label{item:with_functions}
In your document you may use the following phrases:
% 
% 
\begin{enumerate}[label=(\alph*)]
% 
\item
let $ f \colon \R \to \R $ be the function which satisfies for all

\item 
let $ g_r \colon \R \to \R $, $ r \in (0,\infty) $, be the functions which satisfy for all 
$ r \in (0,\infty) $, $ x \in \R $ that $ g_r( x ) = x + r $

\item 
let $ g_r \in C( \R, \R ) $, $ r \in (0,\infty) $, satisfy

\end{enumerate}
% 
% 
\item
\label{item:without_functions}
Instead of the phrases in 
\cref{item:with_functions} 
you may also use the following phrases:
\begin{enumerate}[label=(\alph*)]
\item
let $ f \colon \R \to \R $ satisfy for all

\item 
let $ g_r \colon \R \to \R $, $ r \in (0,\infty) $, satisfy for all 
$ r \in (0,\infty) $, $ x \in \R $ that $ g_r( x ) = x + r $

\item
let $ g_r \in C( \R, \R ) $, $ r \in (0,\infty) $, satisfy
\end{enumerate}
\end{enumerate}
You may, however, not mix between 
\cref{item:with_functions} 
and 
\cref{item:without_functions}.
Please use either the phrases in
\cref{item:with_functions} 
or the phrases in 
\cref{item:without_functions}.


\item 
Except in the introduction and the text between the presented results, 
the phrase ``satisfying''
% , the phrase ``and for'',  
and the phrase ``fulfilling'' may not appear in your document.

\end{enumerate}


\section{Precise specifications}


\subsection{Structure of the document}

A document (by which we mean a scientific article or a thesis) 
consists of exactly the following parts (in this order):
\begin{enumerate}
\item 
Title of the document (Title)
\item
Name of the authors 
\item 
Affiliations of the authors
\item 
Abstract of the document (Abstract)
\item 
Table of contents of the document
\item 
Main text of the document (Main text)
\item 
List of references.
\end{enumerate}
The main text contains of the following \LaTeX\ sections:
\begin{enumerate}
\item 
Introduction
\item 
Name of the 2nd section 
\item 
Name of the 3rd section 
\item 
\dots
\end{enumerate}

\subsection{\LaTeX\ header}

The \LaTeX\ header should not import packages which are not used in the document. 
The following \LaTeX\ code provides an illustrative example.
\begin{verbatim}
\documentclass[a4paper,12pt]{article}
\usepackage{amsthm,
            amsmath,
            amssymb,
            bbm,
            enumerate,
            geometry,
            nicefrac,
            hyperref
            }

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{remark}{Remark}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}

\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\E}{\mathbb{E}}
\renewcommand{\P}{\mathbb{P}}
\newcommand{\tr}{\operatorname{trace}}

\title{Title of the document}

\author{First author$^1$, Second author$^2$, and Third Author$^{3}$
\bigskip
\\
\small{$^1$Department of Mathematics, Some University,
USA, e-mail: $ \dots @ \dots $}
\smallskip
\\
\small{$^2$Department of Mathematics, ETH Zurich,
Switzerland, e-mail: $ \dots @ \dots $}
\smallskip
\\
\small{$^3$Faculty of Mathematics, Bielefeld University,
Germany, e-mail: $ \dots @ \dots $}}

\begin{document}

\maketitle

\begin{abstract}
Abstract ...
\end{abstract}

\tableofcontents

\section{Introduction}

\bibliographystyle{acm}
\bibliography{bibfile}
\end{document}
\end{verbatim}


\subsection{Main text of the document}
% 
% 
% 
The main text consists of environments (\LaTeX\ environments) 
with free text in between. 
The free text does not follow precise specifications. 
In Section~1 there is typically a lot of free text. 
In all other sections there is typically only very little free text. 
The purpose of the free text in all other sections beside 
Section~1 is typically to link the presented environment
to other environments in the document as well as 
to link the presented environment to other findings in the 
scientific literature. 
% 
% 
% 
%
\subsection{\LaTeX\ environments}
% 
% 
% 
An environment is 
\begin{enumerate}
\item a remark, 
\item a setting,
\item a framework,
\item a definition, 
\item a result (by which we mean a theorem, a lemma, a proposition, or a corollary),
or 
\item the proof of a result.
\end{enumerate}
In \LaTeX, an environment is formulated by means of the 
commands:
\begin{verbatim}
\begin{type_of_the_environment}
\label{...} 
... 
\end{type_of_the_environment}
\end{verbatim} 
It is not allowed to use the word 
\emph{where}, the word \emph{from}, the word \emph{since}, the word \emph{as}, the word \emph{by}, the word \emph{because}, 
or the word \emph{using} 
in a \LaTeX\ environment.

% 
% 
\subsubsection{Remarks}
A remark does not necessarily need to follow precise specifications. All other environments follow 
precise specifications.
% 
% 
% 
\subsubsection{Settings and frameworks}
% 
% 
% 
A setting and a framework consists of exactly one sentence. 
This sentence has the following structure (illustrated here in the case of 
a setting environment):
\begin{setting}
Assume/let $ \dots $, assume/let $ \dots $, 
assume/let $ \dots $, 
$ \dots $,
assume/let $ \dots $,
and assume/let $ \dots $.
\end{setting}
A framework environment is sometimes useful to describe algorithms. 
In most cases the document under consideration 
does not contain any framework but there might be several settings.
% 
% 
% 
\subsubsection{Definitions}
%
%
Every definition either consists 
of exactly one sentence 
or consists of exactly two sentences.
% 
% 
\paragraph{Definitions consisting of exactly one sentence}
%
%
If a definition consists of exactly one sentence, then 
the definition must follow the following structure: 
% 
% 
\begin{definition} 
We denote by $ \dots $. 
\end{definition}
%
% 
\paragraph{Definitions consisting of exactly two sentences}
%
%
If a definition consists of exactly two sentences, then 
the definition must follow the following structure: 
% 
% 
\begin{definition} 
Let $ \dots $, let $ \dots $, $ \dots $, and let $ \dots $ 
Then we denote by $ \dots $. 
\end{definition}
% 
%
\subsubsection{Results}
% 
% 
If a result consists of more than one sentence, then the second sentence of the result must start with the word ``Then.''
% 
\paragraph{Results consisting of exactly one sentence}
% A result consists of at least one sentence and at most two sentences. 
If a result consists of exactly one sentence, 
then the result must follow one of the following two structures (illustrated here in the case of 
a theorem environment):
\begin{enumerate}
\item 
First option for a result (theorem, lemma, proposition, or corollary) 
with exactly one sentence (illustrated here in the case of 
a theorem environment):
\begin{theorem}
It holds for all $ \dots $ that $ \dots $.
\end{theorem}
\item 
Second option for a result (theorem, lemma, proposition, or corollary) 
with exactly one sentence (illustrated here in the case of 
a theorem environment):
\begin{theorem}
There exist(s) $ \dots $ that $ \dots $.
\end{theorem}
\end{enumerate}
% 
% 
\paragraph{Results consisting of exactly two sentences}
% 
% 
In most cases a result consists of exactly two sentences. 
In this case the result (theorem, lemma, proposition, or corollary) 
must follow the following structure (illustrated here in the case of 
a theorem environment):
\begin{theorem}
\label{thm:example}
Assume/let $ \dots $, 
assume/let $ \dots $, 
assume/let $ \dots $, 
$ \dots $,
assume/let $ \dots $,
and assume/let $ \dots $.
Then $ \dots $.
\end{theorem}
The first sentence in \cref{thm:example} formulates
the employed hypotheses. If a setting is employed as one hypothesis, 
then one writes (where $ n \in \N = \{ 1 , 2 , 3, \dots \} $ is a natural number): 
\emph{Assume Setting~??} (1st hypothesis), 
\emph{assume/let $ \dots $} (2nd hypothesis), 
\emph{assume/let $ \dots $} (3rd hypothesis),
\emph{$ \dots $},
\emph{assume/let $ \dots $} ($ ( n - 1 ) $-th hypothesis),
\emph{and 
assume/let $ \dots $} ($ n $-th hypothesis).
% The second sentence of a result must start with the word ``Then''. 
In most cases the second sentence of a result starts 
\begin{enumerate}[label=(\roman*)]
\item 
with ``Then it holds for all $ \dots $'' or 
\item 
with ``Then there exist(s) $ \dots $.''
\end{enumerate}
% 
% 
% 
% 
\paragraph{Results consisting of more than two sentences}
% 
% 
If a result (theorem, lemma, proposition, or corollary) consists of more than two sentences, 
then the result must follow the following structure (illustrated here in the case of 
a theorem environment):
\begin{theorem}
Assume/let $ \dots $, 
assume/let $ \dots $, 
assume/let $ \dots $, 
$ \dots $,
assume/let $ \dots $,
and assume/let $ \dots $.
Then the following two/three/four/five/$ \dots $ statements are equivalent:
\begin{enumerate}[label=(\roman*)]
\item It holds $ \dots $/There exist(s) $ \dots $.
\item It holds $ \dots $/There exist(s) $ \dots $.
\item It holds $ \dots $/There exist(s) $ \dots $.
\item $ \dots $
\item $ \dots $.
\end{enumerate}

\end{theorem}
% 
% 
% 
\subsubsection{Proofs}
% 
% 
% 
A proof of a result should consist of \emph{the introductory paragraph of the proof} 
(see \cref{sec:beginning} below), 
of a finite possibly empty sequence of \emph{argumentation sentences} 
(see \cref{sec:argumentation} below), 
and of \emph{the closing sentence} (see \cref{sec:closing} below).
% 
% 
% 
\paragraph{The introductory paragraph of the proof}
\label{sec:beginning}
Every proof of a result consists of at least two sentences. 
The beginning of the proof starts 
\begin{enumerate}[label=(\roman*)]
\item 
with 
``Throughout this proof assume w.l.o.g.\ that $ \dots $ (1st hypothesis), 
assume w.l.o.g.\ that $ \dots $ (2nd hypothesis),
$ \dots $,
assume w.l.o.g.\ that $ \dots $ ($ k $-th hypothesis),
let $ \dots $ (1st set introduction),
let $ \dots $ (2nd set introduction),
$ \dots $,
let $ \dots $ ($ l - 1 ) $-th set introduction),
and 
let $ \dots $ ($ l $-th set introduction). Note/observe that $ \dots $''
(where $ k, l \in \N_0 = \N \cup \{ 0 \} = \{ 0, 1, 2, \dots \} $), 
\item 
with 
``Note/observe that $ \dots $,''
or 
\item 
with
``First, note/observe that $ \dots $.''
\end{enumerate}
% 
% 
% 
\paragraph{Argumentation sentences}
\label{sec:argumentation}
% 
% 
% 
% Subsequent to these formulations, the proof consists 
% of \emph{argumentation sentences} 
% and one \emph{closing sentence} after these argumentation sentences 
% (see Subsection~\ref{sec:closing} below). 
In each argumentation sentence we distinguish between 
\begin{enumerate}[label=\arabic*.]
\item 
a \emph{Type A1 argumentation sentence},
\item 
a \emph{Type A2 argumentation sentence},
\item 
a \emph{Type B1 argumentation sentence}, and
\item 
a \emph{Type B2 argumentation sentence}.
\end{enumerate}
% 
% 
An argumentation sentence which employs 
the previous argumentation sentence is  
a Type A1 argumentation sentence or 
a Type A2 argumentation sentence.
An argumentation sentence which does not employ the previous 
argumentation sentence 
is a Type B1 argumentation sentence 
or a Type B2 argumentation sentence. 
Each argumentation sentence may only use the following verbs:
\begin{enumerate}
\item 
note 
\item 
observe
\item 
imply/implies 
\item 
assure/assures
\item 
ensure/ensures
\item 
demonstrate/demonstrates
\item 
prove/proves
\item 
establish/establishes
\item 
show/shows
\item 
yield/yields
\item 
hold/holds
\item 
combine
\item 
obtain
\item 
satisfy/satisfies
\item 
exist/exists
\item 
be
\end{enumerate}
Besides the above verbs, no other verb may be used in an argumentation sentence. 
If an argumentation sentence contains the phrase ``combine,'' then 
this argumentation sentence must contain exactly one of 
the following phrases:
\begin{enumerate}
\item 
``In the next step we combine $ \dots $ to obtain that $ \dots $.''
\item
``Next, we combine $ \dots $ to obtain that $ \dots $.''
\end{enumerate}
The phrase ``combine'' may only appear in a Type B1 argumentation sentence.
% 
% 
% 
% 
\paragraph{Type A1 argumentation sentences} 
A Type A1 argumentation sentence 
must contain exactly one of the following phrases:
\begin{enumerate}
\item ``therefore''
\item ``Therefore, we obtain that $ \dots $''
\item ``hence''
\item ``Hence, we obtain that $ \dots $''
\item ``this''
\item ``This $ \dots $''
\end{enumerate}
% 
% 
% 
Each Type A1 argumentation sentence must use exactly one of the following structures:
\begin{enumerate}


\item 
Hence, we obtain that $ \dots $ 
(followed by possible quantification of a predicate)


\item 
Therefore, we obtain that $ \dots $
(followed by possible quantification of a predicate)


\item 
Combining $ S_1 $, $ S_2 $, $ \dots $, and $ S_n $ 
(where $ n \in \{ 2, 3, 4, \dots \} $)
hence\slash 
therefore
implies\slash 
ensures\slash 
assures\slash 
demonstrates\slash 
proves\slash 
establishes\slash
yields\slash
shows
(followed by possible\linebreak quantification of a predicate)


\item 
Combining 
$ S_1 $, $ S_2 $, $ \dots $, and $ S_n $ 
(where $ n \in \N = \{ 1, 2, 3, \dots \} $)
with 
$ T_1 $, $ T_2 $, $ \dots $, and $ T_k $ 
(where $ k \in \N = \{ 1, 2, 3, \dots \} $)
hence\slash 
therefore
implies\slash 
ensures\slash 
assures\slash 
demonstrates\slash 
proves\slash 
establishes\slash
yields\slash
shows
(followed by possible quantification of a predicate)


\item 
Combining this, $ S_1 $, $ S_2 $, $ \dots $, and $ S_n $ 
(where $ n \in \N $)
implies\slash 
ensures\slash 
assures\slash 
demon\-strates\slash 
proves\slash 
establishes\slash
yields\slash
shows
(followed by possible quantification of a predicate)


\item 
Combining 
this, $ S_1 $, $ S_2 $, $ \dots $, and $ S_n $ 
(where $ n \in \N_0 = \N \cup \{0\} $)
with 
$ T_1 $, $ T_2 $, $ \dots $, and $ T_k $ 
(where $ k \in \N$)
implies\slash 
ensures\slash 
assures\slash 
demonstrates\slash 
proves\slash 
establishes\slash
yields\slash
shows
(followed by possible quantification of a predicate)


% \item 
% Combining 
% this, $ S_1 $, $ S_2 $, $ \dots $, and $ S_n $ 
% (where $ n \in \N_0 $)
% with 
% $ T_1 $, $ T_2 $, $ \dots $, and $ T_k $ 
% (where $ k \in \N$) allows us to choose $ \dots $


\item 
This 
implies\slash 
ensures\slash 
assures\slash 
demonstrates\slash 
proves\slash 
establishes\slash
yields\slash
shows
that
(followed by possible quantification of a predicate)


\item 
This, $ S_1 $, $ S_2 $, $ \dots $, and $ S_n $ 
(where $ n \in \N $)
imply\slash
ensure\slash
assure\slash
demonstrate\slash
prove\slash
establish\slash
yield\slash
show
% that
(followed by possible quantification of a predicate)


% \item 
% $ S_1 $, $ S_2 $, $ \dots $, $ S_n $ (where $ n \in \N $), 
% this, $ T_1 $, $ T_2 $, $ \dots $, and $ T_k $ (where $ k \in \N_0 $) 
% imply/ensure/assure/demonstrate/prove/establish 
% that
% (possible quantification of a predicate)


\item 
$ S_1 $, $ S_2 $, $ \dots $, and $ S_n $ (where $ n \in \{ 2, 3, 4, \ldots \} $) 
hence\slash
therefore 
imply\slash
ensure\slash
assure\slash
demonstrate\slash
prove\slash
establish\slash
yield\slash
show
(followed by possible quantification of a predicate)
(For example: Lemma~1 and Theorem~3 hence prove that for all $ x \in \R $ it holds that $ f(x) = y $.)


\item 
$ S_1 $ hence\slash therefore 
implies\slash 
ensures\slash 
assures\slash 
demonstrates\slash 
proves\slash 
establishes\slash
yields\slash\linebreak
shows
(followed by possible quantification of a predicate)
(For example: Lemma~1 hence proves that for all $ x \in \R $ it holds that $ f(x) = y $.)


\end{enumerate}
% 
%
The phrase ``This'' may only appear in a Type A1 argumentation sentence.
The phrase ``this'' may only appear in a Type A1 argumentation sentence.
% 
% 
\paragraph{Type A2 argumentation sentences} 
A Type A2 argumentation sentence 
must follow the following structure: 
``Observe/Note that $ \dots $ 
imply\slash
implies\slash
ensure\slash
ensures\slash
demonstrate\slash
demonstrates\slash
show\slash
shows\slash
prove\slash
proves\slash
yield\slash
yields
% that 
$ \dots $.''
% 
% 
% 
\paragraph{Type B1 argumentation sentences} 
A Type B1 argumentation sentence 
must contain exactly one of the following phrases:
\begin{enumerate}
\item ``Moreover, note that $ \dots $''
\item ``Moreover, observe that $ \dots $''
\item ``Furthermore, note that $ \dots $''
\item ``Furthermore, observe that $ \dots $''
\item ``In addition, note that $ \dots $''
\item ``In addition, observe that $ \dots $''
\item ``Next, note that $ \dots $''
\item ``Next, observe that $ \dots $''
\item ``In the next step, we note that $ \dots $''
\item ``In the next step, we observe that $ \dots $''
\item ``In the next step, we combine $ \dots $ to obtain that $ \dots $''
\item ``Next, we combine $ \dots $ to obtain that $ \dots $''
\end{enumerate}
% 
% 
\paragraph{Type B2 argumentation sentences}
% 
% 
A Type B2 argumentation sentence 
must follow the following structure: 
``In the next step/Next, let $ \dots $ 
which satisfy/satisfies $ \dots $.''
% 
% 
\paragraph{The closing sentence of the proof}
\label{sec:closing}
% 
% 
The last sentence of the proof reads
\begin{enumerate}
\item 
as ``The proof of $ \dots $ is thus complete.'' or 
\item 
as ``This completes the proof of $ \dots $.''

\end{enumerate}

%\begin{comment}

\newpage
 
\section{An illustrative example}

We now provide an illustrative example of how these writing styles may be used in a scientific article/thesis.
Note that in an actual article/thesis, we would utilize \verb+\section{...}+ rather than \verb+\subsection{...}+.

\subsection{Introduction}

Here should be an introduction.

\subsubsection{Notation}

Throughout this article/thesis we use the following notation.
For sets $ A $ and $ B $ we denote by
$ \mathbb{M}( A, B) $
the set of all functions from $ A $ to $ B $.
Moreover, for measurable spaces
$ (A, \mathcal{A} ) $
and
$ (B, \mathcal{B} ) $
we denote by
$
  \mathcal{M}( \mathcal{A}, \mathcal{B} )
$
the set of all 
$ \mathcal{A} $/$ \mathcal{B} $-measurable spaces.


\subsubsection{Setting}
\label{sec:setting}


Throughout this article/thesis the following setting is frequently used.
Let $ T \in (0,\infty) $, $m \in \N$,
let 
$ ( \Omega , \mathcal{F}, \P ) $
be a probability space with a normal filtration
$ ( \mathcal{F}_t )_{ t \in [0,T] } $,
and let
$
  W \colon [0,T] \times \Omega \to \R^m
$
be a standard $ ( \mathcal{F}_t )_{ t \in [0,T] } $-Brownian motion.



\subsection{Results}


\begin{proposition}[Young's inequality]
\label{prop:prop1}
Let $ n \in \N $, $ a_1, \dots, a_n \in \R $,
$ p_1, \dots, p_n \in (1,\infty) $
satisfy
$
  \frac{ 1 }{ p_1 } + \dots + \frac{ 1 }{ p_n } = 1
$.
Then it holds that
\begin{equation}
\label{eq:young}
  a_1 \cdot a_2 \cdot \ldots \cdot a_n 
\leq 
  \frac{ 
    \left\lvert a_1 \right\rvert^{ p_1 } 
  }{ p_1 } 
  + 
  \ldots
  +
  \frac{ 
    \left\lvert a_n \right\rvert^{ p_n } 
  }{ p_n } 
  \text{.}
\end{equation}
\end{proposition}


\begin{proof}[Proof of \cref{prop:prop1}]
Throughout this proof without loss of generality assume that $ a_1 \cdot a_2 \cdot \ldots \cdot a_n \neq 0 $
and let 
$ x_1, \dots, x_n \in (0,\infty) $
satisfy
$ 
  x_1 
  = 
  \ln\left( 
    \left\lvert a_1 \right\rvert^{ p_1 } 
  \right)
$,
$ 
  x_2
  = 
  \ln\left( 
    \left\lvert a_2 \right\rvert^{ p_2 } 
  \right)
$,
$ \ldots $,
$ 
  x_n
  = 
  \ln\left( 
    \left\lvert a_n \right\rvert^{ p_n } 
  \right)
$.
This ensures that for all $ k \in \{ 1, 2, \dots, n \} $
it holds that
$ a_k \neq 0 $.
Next, note that the convexity of the function 
$
  ( 0, \infty ) \ni x \mapsto e^x \in (0,\infty)
$
and the assumption that 
$
  \frac{ 1 }{ p_1 } + \dots + \frac{ 1 }{ p_n } = 1
$
ensure that
\begin{equation}
\begin{split}
a_1 \cdot a_2 \cdot \ldots \cdot a_n
& \leq
  \left\lvert
    a_1 \cdot a_2 \cdot \ldots \cdot a_n
  \right\rvert
\\ & =
  \exp\left(
    \frac{ 
      \ln\left(
        \left\lvert a_1 \right\rvert^{ p_1 }
      \right)
    }{ p_1 }
  \right)
  \cdot
  \exp\left(
    \frac{ 
      \ln\left(
        \left\lvert a_2 \right\rvert^{ p_2 }
      \right)
    }{ p_2 }
  \right)
  \cdot
  \ldots 
  \cdot
  \exp\left(
    \frac{ 
      \ln\left(
        \left\lvert a_n \right\rvert^{ p_n }
      \right)
    }{ p_n }
  \right)
\\ & =  
  \exp\left(
    \frac{ 
      x_1
    }{ p_1 }
  \right)
  \cdot
  \exp\left(
    \frac{ 
      x_2
    }{ p_2 }
  \right)
  \cdot
  \ldots 
  \cdot
  \exp\left(
    \frac{ 
      x_n
    }{ p_n }
  \right)
=
  \exp\left(
    \frac{ x_1 }{ p_1 }
    +
    \frac{ x_2 }{ p_2 }
    +
    \ldots
    +
    \frac{ x_n }{ p_n }
  \right)
\\ & \leq
  \frac{ \exp( x_1 ) }{ p_1 }
  +
  \frac{ \exp( x_1 ) }{ p_1 }
  +
  \ldots
  +
  \frac{ \exp( x_1 ) }{ p_1 }
  =
  \frac{ 
    \left\vert a_1 \right\rvert^{ p_1 } 
  }{ p_1 } 
  + 
  \ldots
  +
  \frac{ 
    \left\lvert a_n \right\rvert^{ p_n } 
  }{ p_n } 
  \text{.}
\end{split}
\end{equation}
The proof of \cref{prop:prop1}
is thus complete.
\end{proof}


More results on stochastic differential equations can, e.g., be found 
in Kloeden and Platen \cite{kp92}.

\begin{proposition}[Lyapunov-type functions]
\label{prop:prop2}
Let $ c, T \in [0,\infty) $, 
$ d, m \in \N $,
let $ D \subseteq \R^d $ be an open set,
let 
$ 
  \mu \in 
  \mathcal{M}( \mathcal{B}( D ) , \mathcal{B}( \R^d ) ) 
$,
$ 
  \sigma \in 
  \mathcal{M}( \mathcal{B}( D ) , \mathcal{B}( \R^{ d \times m } ) ) 
$,
$ V \in C^2( D, [0,\infty) $
satisfy that for all $ x \in D $ it holds that
\begin{equation}
  V'(x) \mu( x ) 
  +
  \sum_{ k = 1 }^m
  V''( x )\left( 
    \sigma_k( x ) 
    ,
    \sigma_k( x )
  \right)
\leq 
  c \, V( x )
  \text{,}
\end{equation}
let $ ( \Omega, \mathcal{F}, \P ) $ be a probability space
with a normal filtration $ ( \mathcal{F}_t )_{ t \in [0,T] } $,
let
$
  W \colon [0,T] \times \Omega \to \R^m
$
be a standard $ ( \mathcal{F}_t )_{ t \in [0,T] } $-Brownian motion,
and let $ X \colon [0,T] \times \Omega \to D $ be an
$ ( \mathcal{F}_t )_{ t \in [0,T] } $-adapted stochastic process
with continuous sample paths satisfying that for all 
$ t \in [0,T] $ it holds $ \P $-a.s.\ that
\begin{equation}
  X_t = 
  X_0 
  +
  \int_0^t
  \mu( X_s ) \, {\rm d}s
  +
  \int_0^t
  \sigma( X_s ) \, {\rm d}W_s
  \text{.}
\end{equation}
Then for all $ t \in [0,T] $
it holds that
\begin{equation}
  \E\big[ V( X_t ) \big]
  \leq 
  e^{ c t }
  \,
  \E\big[ V( X_0 ) \big]
  \text{.}
\end{equation}
\end{proposition}

\begin{proposition}[Lyapunov-type functions revisited]
\label{prop:prop3}
Assume the setting outlined in \cref{sec:setting},
let $ c \in [0,\infty) $, 
$ d\in \N $,
let $ D \subseteq \R^d $ be an open set,
let 
$ 
  \mu \in 
  \mathcal{M}( \mathcal{B}( D ) , \mathcal{B}( \R^d ) ) 
$,
$ 
  \sigma \in 
  \mathcal{M}( \mathcal{B}( D ) , \mathcal{B}( \R^{ d \times m } ) ) 
$,
$ V \in C^2( D, [0,\infty) ) $
satisfy that for all $ x \in D $ it holds that
\begin{equation}
  V'(x) \mu( x ) 
  +
  \sum_{ k = 1 }^m
  V''( x )\left( 
    \sigma_k( x ) 
    ,
    \sigma_k( x )
  \right)
\leq 
  c \, V( x )
  \text{,}
\end{equation}
and let $ X \colon [0,T] \times \Omega \to D $ be an
$ ( \mathcal{F}_t )_{ t \in [0,T] } $-adapted stochastic process
with continuous sample paths satisfying that for all 
$ t \in [0,T] $ it holds $ \P $-a.s.\ that
\begin{equation}
  X_t = 
  X_0 
  +
  \int_0^t
  \mu( X_s ) \, {\rm d}s
  +
  \int_0^t
  \sigma( X_s ) \, {\rm d}W_s
  \text{.}
\end{equation}
Then it holds for all $ t \in [0,T] $ that
\begin{equation}
\label{eq:Lyapunov_end}
  \E\big[ V( X_t ) \big]
  \leq 
  e^{ c t }
  \,
  \E\big[ V( X_0 ) \big]
  \text{.}
\end{equation}
\end{proposition}



\begin{theorem}
\label{thm:sum}
Assume the setting outlined in \cref{sec:setting}
and let $ \alpha \in \R $.
Then 
\begin{equation}
  \lim_{ N \to \infty }
  \left(
    N^{ \alpha }
    \,
    \E\big[ \lvert W_{ T / N } \rvert \big]
  \right)
  =
  \begin{cases}
    \infty
  &
    \colon
    \alpha \geq \nicefrac{ 1 }{ 2 }
  \\
    \E\big[ \lvert W_{ T } \rvert \big]
  &
    \colon
    \alpha = \nicefrac{ 1 }{ 2 }
  \\
    0
  &
    \colon
    \alpha < \nicefrac{ 1 }{ 2 }
  \end{cases}
  \text{.}
\end{equation}
\end{theorem}

\bibliographystyle{acm}
\bibliography{bibfile}

%\end{comment}

\mathfrak{Y}


\end{document}
 
  