start adding comments v1

chapters/chap02/chap02.tex

@@ -10,15 +10,14 @@
 \chapter{Theoretical Background   12}
 \label{chap:background}
 
-This chapter introduces the theoretical knowledge that forms the foundation of
-the work presented in this thesis. In~\Cref{sec:domain}
-and~\Cref{sec:differentialEq}, we talk about differential equations and the
-underlying theory. In these Sections both the explanations and the approach are
-strongly based on the book on analysis by Rudin~\cite{Rudin2007} and the book
-about ordinary differential equations by Tenenbaum and
-Pollard~\cite{Tenenbaum1985}. Subsequently, we employ this knowledge to examine
-various pandemic models in~\Cref{sec:epidemModel}.
-Finally, we address the topic of neural networks with a focus on the multilayer
+This chapter introduces the theoretical foundations for the work presented in
+this thesis. In~\Cref{sec:domain} and~\Cref{sec:differentialEq}, we describe
+differential equations and the underlying theory. In these Sections both the
+explanations and the approach are based on a book on analysis by
+Rudin~\cite{Rudin2007} and a book about ordinary differential equations by
+Tenenbaum and Pollard~\cite{Tenenbaum1985}. Subsequently, we employ this
+knowledge to examine various pandemic models in~\Cref{sec:epidemModel}. Finally,
+we address the topic of neural networks with a focus on the multilayer
 perceptron in~\Cref{sec:mlp} and physics informed neural networks
 in~\Cref{sec:pinn}.
 
@@ -27,16 +26,16 @@ in~\Cref{sec:pinn}.
 \section{Mathematical Modelling using Functions   1}
 \label{sec:domain}
 
-To model a physical problem using mathematical tools, it is necessary to define
-a set of fundamental numbers or quantities upon which the subsequent
-calculations will be based. These sets may represent, for instance, a specific
-time interval or a distance. The term \emph{domain} describes these fundamental
-sets of numbers or quantities~\cite{Rudin2007}. A \emph{variable} is a changing
-entity living in a certain domain. In this thesis, we will focus on domains of
-real numbers in $\mathbb{R}$.\\
+To model a physical problem mathematically, it is necessary to define a set of
+fundamental numbers\todo{meeting question 1} or quantities upon which the subsequent calculations will be
+based. These sets may represent, for instance, a specific time interval or a
+distance. The term \emph{domain} describes these fundamental sets of numbers or
+quantities~\cite{Rudin2007}. A \emph{variable} is a changing entity living in a
+certain domain. In this thesis, we will focus on domains of real numbers in
+$\mathbb{R}$.\\
 
-The mapping between variables enables the modeling of the process and depicts
-the semantics. We use functions in order to facilitate this mapping. Let
+The mapping between variables enables the modeling of a physical process and may
+depict semantics. We use functions in order to facilitate this mapping. Let
 $A, B\subset\mathbb{R}$ be to subsets of the real numbers, then we define a
 function as the mapping
 \begin{equation}
@@ -46,52 +45,55 @@ In other words, the function $f$ maps elements $x\in A$ to values
 $f(x)\in B$. $A$ is the \emph{domain} of $f$, while $B$ is the \emph{codomain}
 of $f$. Functions are capable of representing the state of a system as a value
 based on an input value from their domain. One illustrative example is a
-function that maps a time point to the distance covered since a starting point.
+function that maps a time step to the distance covered since a starting point.
 In this case, time serves as the domain, while the distance is the codomain.
 
 % -------------------------------------------------------------------
 
-\section{Basics of Differential Equations   1}
+\section{Mathematical Modelling using Differential Equations   1}
 \label{sec:differentialEq}
 
-Often, the change of a system is more interesting than its current state.
-Functions are able to give us the latter, but only passively give information
-about the change of a system. The objective is to determine an effective method
-for calculating the change of a function across its domain. Let $f$ be a
-function and $[a, b]\subset \mathbb{R}$ an interval of real numbers, the
-expression
+Often, the behavior of a variable or a quantity across a domain is more
+interesting than its current state. Functions are able to give us the latter, \todo{meeting question 2}
+but only passively give information about the change of a system. The objective
+is to determine an effective method for calculating the change of a function
+across its domain. Let $f$ be a function and $[a, b]\subset \mathbb{R}$ an
+interval of real numbers. The expression
 \begin{equation}
   m = \frac{f(b) - f(a)}{a-b}
 \end{equation}
 gives the average rate of change. While the average rate of change is useful in
-many cases, the momentary rate of change is more accurate. To calculate this,
+many cases, the momentary rate of change is more accurate. To calculate this, \todo{look up in Rudin - cite (wordly)}
 we need to narrow down, the interval to an infinitesimal. For each $x\in[a, b]$
 we calculate
 \begin{equation} \label{eqn:differential}
   \frac{df}{dx} = \lim_{t\to x} \frac{f(t) - f(x)}{t-x},
 \end{equation}
-if it exists. $\frac{df}{dx}$ is the \emph{derivative}, or
-\emph{differential equation}, it returns the momentary rate of change of $f$ for
-each value $x$ of $f$'s domain. Repeating this process on $\frac{df}{dx}$ yields
-$\frac{d^2f}{dx^2}$, which is the function that calculates the rate of change of
-the rate of change and is called the second order derivative. Iterating this $n$
-times results in $\frac{d^nf}{dx^n}$, the derivative of the $n$'th order.
-Another method for obtaining a differential equation is to create it from the
-semantics of a problem. This method is useful if no basic function exists for a
-system. Differential equations find application in several areas such as
-engineering, physics, economics, epidemiology, and beyond.\\
-
-\todo{Here insert definition of differential equations (take from books)}
+if it exists. As the Tenenbaum and Pollard~\cite{Tenenbaum1985} define,
+$\nicefrac{df}{dx}$ is the \emph{derivative}, which is ``the rate of change of a
+variable with respect to another''. The relation between a variable and its
+derivative is modeled in a \emph{differential equation}. The derivative of
+$\nicefrac{df}{dx}$ yields $\nicefrac{d^2f}{dx^2}$, which is the function that
+calculates the rate of change of the rate of change and is called the
+\emph{second order derivative}. Iterating this $n$ times results in
+$\nicefrac{d^nf}{dx^n}$, the derivative of the $n$'th order. A method for
+obtaining a differential equation is to derive it from the semantics of a
+problem. For example, in physics a differential equation can be derived from the
+law of the conservation of energy~\cite{Demtroeder2021}. Differential equations \todo{is this good?}
+find application in several areas such as engineering \eg, the Chua's
+circuit~\cite{Matsumoto1984}, physics with, \eg, the Schrödinger
+equation~\cite{Schroedinger1926}, economics, \eg, Black-Scholes
+equation~\cite{Oksendal2000}, epidemiology, and beyond.\\
 
 In the context of functions, it is possible to have multiple domains, meaning
 that function has more than one parameter. To illustrate, consider a function
-operating in two-dimensional space, wherein each parameter represents one axis
-or one that, employs with time and locations as inputs. The term
-\emph{partial differential equations} (\emph{PDE}'s) describes differential
-equations of such functions, which require a derivative for each of their
-domains. In contrast, \emph{ordinary differential equations} (\emph{ODE}'s) are
-the single derivatives for a function having only one domain. In this thesis, we
-only need ODE's.\\
+operating in two-dimensional space, wherein each parameter represents one axis.
+Another example would be a function, that maps its inputs of a location variable
+and a time variable on a height. The term \emph{partial differential equations}
+(\emph{PDE}'s) describes differential equations of such functions, which contain
+partial derivatives with respect to each individual domain. In contrast, \emph{ordinary differential
+  equations} (\emph{ODE}'s) are the single derivatives for a function having only
+one domain~\cite{Tenenbaum1985}. In this thesis, we restrict ourselves to ODE's.\\
 
 A \emph{system of differential equations} is the name for a set of differential
 equations. The derivatives in a system of differential equations each have their
@@ -115,34 +117,36 @@ we can rewrite the~\Cref{eq:newtonSecLaw} to
 \begin{equation}
   F=ma=m\frac{d^2s}{dt^2}.
 \end{equation}\\
-This explanation of differential equations focuses on the aspects deemed crucial
-for this thesis and is not intended to be a complete explanation of the subject.
-To gain a better understanding of it, we recommend the books mentioned
-above~\cite{Rudin2007,Tenenbaum1985}. In the following section we
-describe the application of these principles in epidemiological models.
+
+To conclude, note that this explanation of differential equations focuses on the
+aspects deemed crucial for this thesis and is not intended to be a complete
+explanation of the subject. To gain a better understanding of it, we recommend
+the books mentioned above~\cite{Rudin2007,Tenenbaum1985}. In the following
+section we describe the application of these principles in epidemiological
+models.
 
 % -------------------------------------------------------------------
 
 \section{Epidemiological Models   4}
 \label{sec:epidemModel}
 
-Pandemics, like \emph{COVID-19}, which has resulted in a significant
-number of fatalities. The question arises: How should we fight a pandemic
-correctly? Also, it is essential to study whether the employed countermeasures
-efficacious in combating the pandemic. Given the unfavorable public response to
-measures such as lockdowns, it is imperative to investigate that their efficacy
-remains commensurate with the costs incurred to those affected. In the event
-that alternative and novel technologies were in use, such as the mRNA vaccines
-in the context of COVID-19, it is needful to test the effect and find the
-optimal variant. In order to shed light on the aforementioned events we need to
-develop a method to quantize the pandemic along with its course of
-progression.\\
+Pandemics, like \emph{COVID-19}, which have resulted in a significant
+number of fatalities. Hence, the question arises: How should we analyze a \todo{Better?}
+pandemic effectively? It is essential to study whether the employed
+countermeasures are efficacious in combating the pandemic. Given the unfavorable
+public response to measures such as lockdowns, it is imperative to investigate
+that their efficacy remains commensurate with the costs incurred to those
+affected. In the event that alternative and novel technologies were in use, such
+as the mRNA vaccines in the context of COVID-19, it is needful to test the
+effect and find the optimal variant. In order to shed light on the
+aforementioned events, we need a method to quantify the pandemic along with its
+course of progression.\\
 
 The real world is a highly complex system, which presents a significant
-challenge attempting to describe it fully in a model. Therefore, the model must
-reduce the complexity while retaining the essential information. Furthermore, it
-must address the issue of limited data availability. For instance, during
-COVID-19 institutions such as the Robert Koch Institute
+challenge attempting to describe it fully in a mathematical model. Therefore,
+the model must reduce the complexity while retaining the essential information.
+Furthermore, it must address the issue of limited data availability. For
+instance, during COVID-19 institutions such as the Robert Koch Institute
 (RKI)\footnote[1]{\url{https://www.rki.de/EN/Home/homepage_node.html}} were only
 able to collect data on infections and mortality cases. Consequently, we require
 a model that employs an abstraction of the real world to illustrate the events
@@ -162,12 +166,12 @@ of SIR models.\\
 
 The SIR model is capable of illustrating diseases, which are transferred through
 contact or proximity of an individual carrying the illness and a healthy
-individual. This is possible due to the distinction between infected beings
+individual. This is possible due to the distinction between infected individuals
 who are carriers of the disease and the part of the population, which is
 susceptible to infection. In the model, the mentioned groups are capable to
 change, \eg,  healthy individuals becoming infected.  The model assumes the
 size $N$ of the population remains constant throughout the duration of the
-pandemic. The population $N$ comprises three distinct groups: the
+pandemic. The population $N$ comprises three distinct compartments: the
 \emph{susceptible} group $S$, the \emph{infectious} group $I$ and the
 \emph{removed} group $R$ (hence SIR model). Let $\mathcal{T} = [t_0, t_f]\subseteq
   \mathbb{R}_{\geq0}$ be the time span of the pandemic, then,
@@ -180,9 +184,9 @@ $t\in\mathcal{T}$. For $S$, $I$, $R$ and $N$ applies:
   N = S + I + R.
 \end{equation}
 The model makes another assumption by stating that recovered people are immune
-to the illness and infectious individual can not infect them. The individuals in
-the $R$ group are either recovered or deceased, and thus unable to transmit or
-carry the disease.
+to the illness and infectious individuals can not infect them. The individuals
+in the $R$ group are either recovered or deceased, and thus unable to transmit
+or carry the disease.
 \begin{figure}[h]
   \centering
   \includegraphics[scale=0.87]{sir_graph.pdf}
@@ -207,11 +211,12 @@ McKendrick~\cite{1927} propose the following set of differential equations:
     \frac{dR}{dt} &= \alpha I.
   \end{split}
 \end{equation}
-This, according to Edelstein-Keshet, is based on the following assumption:
+This set of differential equations, is based on the following assumption:
 ``The rate of transmission of a microparasitic disease is proportional to the
 rate of encounter of susceptible and infective individuals modelled by the
-product ($\beta S I$)''~\cite{EdelsteinKeshet2005}. The system shows the change
-in size of the groups per time unit due to infections, recoveries, and deaths.\\
+product ($\beta S I$)'', according to Edelstein-Keshet~\cite{EdelsteinKeshet2005}.
+The system shows the change in size of the groups per time unit due to
+infections, recoveries, and deaths.\\
 
 The term $\beta SI$ describes the rate of encounters of susceptible and infected
 individuals. This term is dependent on the size of $S$ and $I$, thus Anderson
@@ -223,8 +228,8 @@ and May~\cite{Anderson1991} propose a modified model:
     \frac{dR}{dt} &= \alpha I.
   \end{split}
 \end{equation}
-In which $\beta SI$ gets normalized by $N$, which is more correct in a
-real world aspect~\cite{Anderson1991}.\\
+In~\Cref{eq:modSIR} $\beta SI$ gets normalized by $N$, which is more correct in
+a real world aspect~\cite{Anderson1991}.\\
 
 The initial phase of a pandemic is characterized by the infection of a small
 number of individuals, while the majority of the population remains susceptible.
@@ -291,7 +296,9 @@ emerged.\\
       \end{subfigure}
     }
   \end{picture}
-  \caption{Synthetic data, using~\Cref{eq:modSIR} and $N=7.9\cdot 10^6$, $I_0=10$ with different sets of parameters.}
+  \caption{Synthetic data, using~\Cref{eq:modSIR} and $N=7.9\cdot 10^6$, $I_0=10$ with different sets of parameters.
+    We visualize the case with the reference parameters in (a). In (b) and (c) we keep $\alpha$ constant, while varying
+    the value of $\beta$. In contrast, (d) and (e) have varying values of $\alpha$.}
   \label{fig:synth_sir}
 \end{figure}
 
@@ -302,47 +309,46 @@ that the disease is unable to keep pace with the rate of recovery, resulting
 in its demise before it can exert a significant influence on the population. In
 contrast, an early and high peak means that the disease is rapidly transmitted
 through the population, with a significant proportion of individuals having been
-infected.~\Cref{fig:sir_model} illustrates the impact of modifying either
+infected.~\Cref{fig:sir_model} illustrates this effect by varying the values of
 $\beta$ or $\alpha$ while simulating  a pandemic using a model such
 as~\Cref{eq:modSIR}. It is evident that both the transmission rate $\beta$
 and the recovery rate $\alpha$ influence the height and time of the peak of $I$.
 When the number of infections exceeds the number of recoveries, the peak of $I$
 will occur early and will be high. On the other hand, if recoveries occur at a
-faster rate than new infections the peak will occur later and will be low. This
-means, that it is crucial to know both $\beta$ and $\alpha$ to be able to
-simulate a pandemic using the SIR model.\\
+faster rate than new infections the peak will occur later and will be low. Thus,
+it is crucial to know both $\beta$ and $\alpha$, as these parameters
+characterize how the pandemic evolves.\\
 
 The SIR model makes a number of assumptions that are intended to reduce the
 model's overall complexity while simultaneously increasing its divergence from
 actual reality. One such assumption is that the size of the population, $N$,
-remains constant. This depiction is not an accurate representation of the actual
-relations observed in the real world, as the size of a population is subject to
-a number of factors that can contribute to change. The population is increased
-by the occurrence of births and decreased by the occurrence of deaths. There are
-different reasons for mortality, including the natural aging process or the
-development of other diseases. Other examples are the absence of the possibility
-for individuals to be susceptible again, after having recovered, or the
-possibility for the transition rates to change due to new variants or the
-implementation of new countermeasures. We address this latter option in the
-next~\Cref{sec:pandemicModel:rsir}.
+remains constant, as the daily change is negligible to the total population.
+This depiction is not an accurate representation of the actual relations
+observed in the real world, as the size of a population is subject to a number
+of factors that can contribute to change. The population is increased by the
+occurrence of births and decreased by the occurrence of deaths. Other examples
+are the impossibility for individuals to be susceptible again, after having
+recovered, or the possibility for the transition rates to change due to new
+variants or the implementation of new countermeasures. We address this latter
+option in the next~\Cref{sec:pandemicModel:rsir}.
 
 % -------------------------------------------------------------------
 
 \subsection{Reduced SIR Model and the Reproduction Number   1}
 \label{sec:pandemicModel:rsir}
-The~\Cref{sec:pandemicModel:sir} presents the classical SIR model. The model
-comprises two parameters $\beta$ and $\alpha$, which describe the course of a
-pandemic over its duration. This is beneficial when examining the overall
+The~\Cref{sec:pandemicModel:sir} presents the classical SIR model. This model
+contains two scalar parameters $\beta$ and $\alpha$, which describe the course
+of a pandemic over its duration. This is beneficial when examining the overall
 pandemic; however, in the real world, disease behavior is dynamic, and the
-values of the parameters $\beta$ and $\alpha$ change at each time point. The
-reason for this is due to events such as the implementation of countermeasures
-that reduce the contact between the infectious and susceptible individuals, the
-emergence of a new variant of the disease that increases its infectivity or
-deadliness, or the administration of a vaccination that provides previously
-susceptible individuals with immunity without ever being infectious. To address
-this Millevoi \etal~\cite{Millevoi2023} introduce a model that simultaneously
-reduces the size of the system of differential equations and solves the problem
-of time scaling at hand.\\
+values of the parameters $\beta$ and $\alpha$ change throughout the course of
+the disease. The reason for this is due to events such as the implementation of
+countermeasures that reduce the contact between the infectious and susceptible
+individuals, the emergence of a new variant of the disease that increases its
+infectivity or deadliness, or the administration of a vaccination that provides
+previously susceptible individuals with immunity without ever being infected. \todo{sai correction -> is this point not already included?}
+To address this Millevoi \etal~\cite{Millevoi2023} introduce a model that \todo{are there older sources}
+simultaneously reduces the size of the system of differential equations and
+solves the problem of time scaling at hand.\\
 
 First, they alter the definition of $\beta$ and $\alpha$ to be dependent on the time interval
 $\mathcal{T} = [t_0, t_f]\subseteq \mathbb{R}_{\geq0}$,
@@ -356,7 +362,7 @@ $t_0$, the \emph{reproduction number},
   \RO = \beta(t_0)D(t_0) = \frac{\beta(t_0)}{\alpha(t_0)},
 \end{equation}
 represents the number of susceptible individuals, that one infectious individual
-infects at the onset of the pandemic.In light of the effects of $\beta$ and
+infects at the onset of the pandemic. In light of the effects of $\beta$ and
 $\alpha$ (see~\Cref{sec:pandemicModel:sir}), $\RO > 1$ indicates that the
 pandemic is emerging. In this scenario $\alpha$ is relatively low due to the
 limited number of infections resulting from $I(t_0) << S(t_0)$. When $\RO < 1$,

header.tex

@@ -44,5 +44,5 @@
 
 \usepackage{cleveref}
 \usepackage{todonotes}
-
+\usepackage{nicefrac}
 

thesis.bbl

@@ -12,6 +12,15 @@
 \newblock \emph{Infectious diseases of humans : dynamics and control}.
 \newblock Oxford University Press, 1991
 
+\bibitem[Dem21]{Demtroeder2021}
+\textsc{Demtröder}, Wolfgang:
+\newblock \emph{Lehrbuch}. Bd.~1: {\emph{Experimentalphysik 1}}.
+\newblock 9. Auflage.
+\newblock Berlin : Springer Spektrum, 2021. --
+\newblock ISBN 978--3--662--62727--3. --
+\newblock Auf dem Umschlag: Mit über 2,5 h Lösungsvideos zu ausgewählten
+  Aufgaben
+
 \bibitem[EK05]{EdelsteinKeshet2005}
 \textsc{Edelstein-Keshet}, Leah:
 \newblock \emph{Mathematical Models in Biology}.
@@ -51,6 +60,15 @@
 \newblock \url{http://dx.doi.org/10.48550/ARXIV.PHYSICS/9705023}. --
 \newblock DOI 10.48550/ARXIV.PHYSICS/9705023
 
+\bibitem[Mat84]{Matsumoto1984}
+\textsc{Matsumoto}, T.:
+\newblock A chaotic attractor from Chua’s circuit.
+\newblock {In: }\emph{IEEE Transactions on Circuits and Systems} 31 (1984),
+  Dezember, Nr. 12, S. 1055--1058.
+\newblock \url{http://dx.doi.org/10.1109/tcs.1984.1085459}. --
+\newblock DOI 10.1109/tcs.1984.1085459. --
+\newblock ISSN 0098--4094
+
 \bibitem[MP72]{Minsky1972}
 \textsc{Minsky}, Marvin ; \textsc{Papert}, Seymour~A.:
 \newblock \emph{Perceptrons}.
@@ -68,6 +86,15 @@
 \newblock \url{http://dx.doi.org/10.48550/ARXIV.2311.09944}. --
 \newblock DOI 10.48550/ARXIV.2311.09944
 
+\bibitem[Oks00]{Oksendal2000}
+\textsc{Oksendal}, Bernt:
+\newblock \emph{Stochastic Differential Equations}.
+\newblock 5th ed.
+\newblock Berlin, Heidelberg : Springer Berlin / Heidelberg, 2000 (Universitext
+  Ser.). --
+\newblock ISBN 3--540--63720--6. --
+\newblock Description based on publisher supplied metadata and other sources.
+
 \bibitem[Ros58]{Rosenblatt1958}
 \textsc{Rosenblatt}, F.:
 \newblock The perceptron: A probabilistic model for information storage and
@@ -88,6 +115,15 @@
 \newblock \emph{Analysis}.
 \newblock Oldenbourg Wissenschaftsverlag GmbH, 2007
 
+\bibitem[Sch26]{Schroedinger1926}
+\textsc{Schrödinger}, E.:
+\newblock An Undulatory Theory of the Mechanics of Atoms and Molecules.
+\newblock {In: }\emph{Physical Review} 28 (1926), Dezember, Nr. 6, S.
+  1049--1070.
+\newblock \url{http://dx.doi.org/10.1103/physrev.28.1049}. --
+\newblock DOI 10.1103/physrev.28.1049. --
+\newblock ISSN 0031--899X
+
 \bibitem[SRS21]{Shaier2021}
 \textsc{Shaier}, Sagi ; \textsc{Raissi}, Maziar  ; \textsc{Seshaiyer},
   Padmanabhan:

thesis.bib

@@ -168,4 +168,63 @@
   publisher = {arXiv},
 }
 
+@Book{Demtroeder2021,
+  author    = {Demtröder, Wolfgang},
+  publisher = {Springer Spektrum},
+  title     = {Experimentalphysik 1},
+  year      = {2021},
+  address   = {Berlin},
+  edition   = {9. Auflage},
+  isbn      = {978-3-662-62727-3},
+  note      = {Auf dem Umschlag: Mit über 2,5 h Lösungsvideos zu ausgewählten Aufgaben},
+  series    = {Lehrbuch},
+  volume    = {1},
+  pagetotal = {507},
+  ppn_gvk   = {176791136X},
+  subtitle  = {Wolfgang},
+}
+
+@Article{Schroedinger1926,
+  author    = {Schrödinger, E.},
+  journal   = {Physical Review},
+  title     = {An Undulatory Theory of the Mechanics of Atoms and Molecules},
+  year      = {1926},
+  issn      = {0031-899X},
+  month     = dec,
+  number    = {6},
+  pages     = {1049--1070},
+  volume    = {28},
+  doi       = {10.1103/physrev.28.1049},
+  publisher = {American Physical Society (APS)},
+}
+
+@Book{Oksendal2000,
+  author    = {Oksendal, Bernt},
+  publisher = {Springer Berlin / Heidelberg},
+  title     = {Stochastic Differential Equations},
+  year      = {2000},
+  address   = {Berlin, Heidelberg},
+  edition   = {5th ed.},
+  isbn      = {3-540-63720-6},
+  note      = {Description based on publisher supplied metadata and other sources.},
+  series    = {Universitext Ser.},
+  pagetotal = {1332},
+  ppn_gvk   = {1763063550},
+  subtitle  = {An Introduction with Applications},
+}
+
+@Article{Matsumoto1984,
+  author    = {Matsumoto, T.},
+  journal   = {IEEE Transactions on Circuits and Systems},
+  title     = {A chaotic attractor from Chua’s circuit},
+  year      = {1984},
+  issn      = {0098-4094},
+  month     = dec,
+  number    = {12},
+  pages     = {1055--1058},
+  volume    = {31},
+  doi       = {10.1109/tcs.1984.1085459},
+  publisher = {Institute of Electrical and Electronics Engineers (IEEE)},
+}
+
 @Comment{jabref-meta: databaseType:bibtex;}