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\author{Herman Kruis and Michiel Meeuwissen}
\date{2006-10-09}
\title{Aiming capabilities of gum chewing semi-social urban people}
\begin{document}
\maketitle
\tableofcontents
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\section{Introduction}
Nowadays a lot of people are used to chew gum while they are walking through town. After a certain
period of being chewed on, chewing gum looses its taste and people tend to get rid of it. Not
seldom, it ends up on the streets, where it forms a huge problem for city-cleaners all around the
world\cite{bbc,italy,singapore,china}. Some chewers though, have the decency to try to aim it in a
pit. This can be noticed by the high density of chewing gums around pits, evidently the result of
`misses'.

In this work we describe the distribution of chewing gums around several urban pits. Using this we
analyze the aiming capabilities of urban people, which proves to be rather limited.

\section{Methods}
The chewing gum density around urban pits was measured around 2 pits in Utrecht. They were selected
to be on a relatively open space, with pedestrians coming from several directions. These pits had a
clearly observable increased concentration of gum around them.

We divided the space around each pit into circular areas of 18, 30, 60, 90 and 150 cm radius, and
counted the number of sticked gums in these 4 regions (the region from 0 to 18 contains the pit, can
cannot be counted). From that the gum-density as a function of the distance of the pit can be
calculated.

The numbers were counted independently by two persons (the authors) and averaged. This to get a
feeling of the accuracy of the decision about when a gum should be counted or not, or to be noticed
at all. The results usually differed a few per cent.

Because the number of hits could not be counted directly - these gums did disappear in the sewer
system - the central area of the distribution was left out of the measurement. We will be able to
extrapolate this number from the rest of the distribution.

The distribution is supposed to be gaussian, dependent on two variables $N$ and $\sigma$ which we
will find by an optimization based on the measurements. From $N$ and $sigma$ then follows the
number of gums in the sewer. This can be expressed by the formula:
\begin{equation}
   \rho(r) = \rho'(r) + \rho_0 = \frac{N}{2 \pi \sigma ^ 2} e ^ { - \frac{r^2}{2 \sigma ^2}} + \rho_0
\label{rho}
\end{equation}
where $\rho(r)$ is the density of gums per square meter at distance $r$ from the pit. $N$ is the
total number of gums which were aimed at the pit. $\rho_0$ is the `background' density of gums,
caused by the anti-social individuals, who do not try to dispose their gum in a pit.

$\rho_0$ was separately measured on a similar location near to the pit, sufficiently far away to
assume that no `misses' land there.

The number of gums between $r_j$ and $r_{j+1}$ would according to formula~(\ref{rho}) be:
\begin{eqnarray}
  N_{j} & = &  \rho(R_j) \cdot \pi (r_{j+1} ^ 2 - r_j ^2)
          =   \int_{r_j}^{r_{j+1}} \rho(r) 2 \pi r dr \nonumber \\
        & = & \int_{r_j}^{r_{j+1}} \left( \frac{N}{2 \pi \sigma ^ 2} e ^ { - \frac{r^2}{2 \sigma ^2}} +  \rho_0 \right) 2 \pi r dr \nonumber \\
        & = &
              N \left( e ^ { - \frac{r_j^2}{2 \sigma ^2}} -  e ^ { - \frac{r_{j+1}^2}{2 \sigma ^2}}\right) +
              \rho_0  \cdot \pi (r_{j+1} ^ 2 - r_j ^2)
  \label{Nj}
\end{eqnarray}
Where we define $R_j$ to be the distance to the pit where the gum density $\rho(r)$ takes its average
value of the area defined by the two distances $r_j$ and $r_{j+1}$. The number of gums in this area
we call $N_j$, which we want to fit to the counted number $n_j$. This means that we are minimizing
as function of $\sigma$ and $N$ this expression:
\begin{equation}
\Sigma_{j} \left( N_j - n_j \right) ^ 2
\end{equation}
This optimalisation will be done by a Marquardt-Levenberg algorithm\cite{numrec} (using the
implementation of GNUPlot\cite{gnuplot}) to fit $N_j(R_j)$ to $n_j(R_j)$. The interesting thing is
that $R_j$ is also a function of $\sigma$ (but independent of $N$). This can be derived from
formula~\ref{Nj}.

For the fitting process this does not matter much, it only means that not only the shape and height
of the gaussian fit-function are changed but, while doing so, also the $x$-values of the fitted data
are varied (but in a {\em coupled} manner).

So we need an expression for $R_j$ (following from \ref{Nj}):
\begin{equation}
 R_j (\sigma) =  \sigma \cdot \sqrt{ 2 \cdot \ln \left( \frac {r_{j+1} ^ 2 - r_j ^ 2} {2 \cdot \sigma^2 \cdot \left( e ^ { - \frac{r_j^2}{2 \sigma ^2}} -  e ^ { - \frac{r_{j+1}^2}{2 \sigma ^2}}\right)} \right) }
\end{equation}



\section{Results}
The first measurement was done at the open space near the corner of the Amsterdamsestraatweg and
that other street in the centre of Utrecht. This is a square (30cm x 30cm) pit with 10 slits in
it. The results are presented graphicly in figure \ref{pit1} and figure \ref{pit2}.
\begin{figure}[p]
\begin{center}
 \htmlimage{thumbnail=0.5}
 \includegraphics[height=8cm]{put1}
\caption{Density distribution of the gums around the pit near the , together with a gaussian fit.}
\label{pit1}
\end{center}
\end{figure}

\begin{figure}[p]
\begin{center}
 \htmlimage{thumbnail=0.5}
 \includegraphics[height=8cm]{put2}
\caption{Density distribution of the gums around the pit near the , together with a gaussian fit.}
\label{pit2}
\end{center}
\end{figure}

The results fo the gaussian fits are\\
\begin{tabular}{c|c|c}
  pit &  $N$ (total number of gums) & $\sigma$ (width of the distribution) (m)\\
\hline
%% zonder x-punt functie:
%%aantal          = 161.39           +/- 3.053        (1.892%)
%%sigma           = 0.350299         +/- 0.00459      (1.31%)

%aantal1         = 159.685          +/- 1.605        (1.005%)
%sigma1          = 0.340551         +/- 0.002312     (0.679%)

 pit 1 & $159.7 \pm 1.6$ &  $0.340 \pm 0.002$ \\

%%aantal          = 174.483          +/- 20.75        (11.89%)
%%sigma           = 0.307809         +/- 0.02545      (8.267%)
%aantal2         = 176.553          +/- 18.06        (10.23%)
%sigma2          = 0.302867         +/- 0.02086      (6.889%)

 pit 2 & $177 \pm 18$      & $0.303 \pm 0.021$
\end{tabular}

We can now calculate how many aimed gums hit the central region (of 18 cm radius) (from eq. \ref{rho}) :
\begin{equation}
\int_0^{0.18m} 2 \pi r \rho'(r) dr=
\int_0^{0.18m} 2 \pi r \frac{N}{2 \pi \sigma ^ 2} e ^ { - \frac{r^2}{2 \sigma ^2}} dr =
 N \left( 1 - e ^ { - \frac{0.18^2}{2 \sigma ^2}} \right)
\end{equation}
Applying this formula for pit 1 gives a number of $20.8$ (13 \%) gums. And for pit 2 it gives a
number of $29.0$ (16 \%) gums. In reality even more gums would have been a `miss' because the pit
entrance is actually smaller, because it is not circular (but square) and for about $50\%$ blocked by
bars.

From the figures \ref{pit1} and \ref{pit2} it can be seen that 1.5 meters from the pit there are
virtually no misses any more, $\rho'(r > 1.5) \approx 0$. Therefore the total counted number in the
1.5 meter area (which did not include this central region) plus the calculated number of central
hits minus the `background' density should also give approximately the total number of aimed gums
$N$ at the pit.

For pit 1 the measured number of gums between $0.18 m$ and $1.5 m$ distance is: $217.5$. The number
of accidental drops in this region $\pi (r_2^2 - r_1^2) \rho_0 = \pi \cdot (1.5^2 - 0.18^2) \cdot
11.5 = 80.1$. So the total number of misses outside the central area is $217.5 - 80.1 = 137.4$, plus
the number of hits in the central area = $137.4 + 20.0 = 157.4$ should be equal to $161.4$, which
sounds reasonable.

For pit 2 these numbers are $194.1$ and $177 \pm 18$.

If we are interested only in the relative number of hits then we do not need the number $N$, and we
can also take the results of the two measurements together (assuming that the success ratio of the
gum aimers is a constant). The weighted average for $\sigma_{\mbox{pit 1}}$ and $\sigma_{\mbox{pit
  2}}$  is $\overline\sigma = 0.30$.

\section{Conclusions}

The most interesting number if of course the value of $\sigma$, which is the width of the gaussian
radial density distribution. This is a measure for the preciseness of aim of the subjects, which
are the urban semi-social people, who spit their gums at pits. We found 2 values for $\sigma$ for
two different targets. These values are very comparable, but probably not the same, since they
differ more then twice the estimated standard deviation. This could be caused by the slightly
varying demographic circumstances in the neighborhood of the two pits.

When averaged anyway we find that considerably less than $15\%$ of the targeted gums actually hits
its goal: disappearance in the sewer system, which is a regrettably low figure. This leads us to
conclude that these semi-social aiming individuals can as well not even try, or promote themselves
to decent civilians. They can also choose the solution based on swallowing.

\begin{thebibliography}{}
\bibitem{bbc} \mylink{bbc}{http://news.bbc.co.uk/1/hi/england/2701515.stm}
\bibitem{italy} \mylink{italy}{http://www.barganews.com/people/chewing_gum/}
\bibitem{singapore} \mylink{Singapore}{http://www.sfgate.com/examiner/bondage/BOND-23960.html}
\bibitem{china} \mylink{China}{http://news.bbc.co.uk/2/hi/asia-pacific/2786235.stm}
\bibitem{solution_england} \mylink{Englang}{http://www.ananova.com/news/story/sm_656602.html?menu=news.quirkies}
\bibitem{laser} \mylink{http://www.nesta.org.uk/ourawardees/profiles/2932/print.htm}{http://www.nesta.org.uk/ourawardees/profiles/2932/print.htm}
\bibitem{origin} \mylink{http://www.tridentgum.com/consumer/html/x13.htmlxsg}{http://www.tridentgum.com/consumer/html/x13.htmlxsg}
\bibitem{origin2} \mylink{http://www.wrigley.com/wrigley/kids/gumstory.pdf}{http://www.wrigley.com/wrigley/kids/gumstory.pdf}
\bibitem{numrec} Press, {\em et al.} -- Numerical Recipes in C, second edition -- Cambridge University
  press, 1988.
\bibitem{gnuplot}Thomas Williams, Colin Kelley, {\em et al.} -- Version 3.7 patchlevel 3 --  Copyright(C) 1986 - 1993, 1998 - 2002.


\end{thebibliography}

\end{document}

