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\begin{document}
\section*{Quiz 7}
\rule{\textwidth}{0.5ex}
{\bf Quiz7 Problem 1}. {\bf Heat Transfer} and the Mean Value Property.

Consider the cross section of a long rectangular dam on a river, represented
in the figure. 

{
\includegraphics[scale=0.6]{figures/rectangular-dam.png}
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\begin{minipage}[b]{0.5\textwidth}
The boundaries of the dam are subject to three factors:
the temperature in degrees Celsius of the air (20),  the water (25), and
the ground at its base (30). 

~\relax
\end{minipage}
}

An analysis of the heat transfer from the three sources will be done from the
equilibrium temperature, which is found by the Mean Value Property below.

{
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\begin{center}{\bf The Mean Value Property}\\[3ex] \end{center}

If a plate is at thermal equilibrium, and $C$ is a circle with center $P$
contained in
the plate, then the temperature at $P$ is the average
value of the temperature function over $C$.

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\end{minipage}
}

A version of the Mean Value Property says that the temperature at center $P$
of circle $C$ is the average of the temperatures at four equally-spaced points on $C$.
We construct a grid as in the figure below, 
label the unknown temperatures at interior grid points as $x_1,x_2,x_3,x_4$,
then use the property to obtain four equations.


{
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\begin{center}
{\bf Four-Point} \\
{\bf Temperature Averages} 
\end{center}


\begin{center}\large\rm
$x_1 = \dd\frac14(20+25+x_2+x_3)$ \\[1ex]
$x_2 = \dd\frac14(20+20+x_1+x_4)$ \\[1ex]
$x_3 = \dd\frac14(25+30+x_1+x_4)$ \\[1ex]
$x_4 = \dd\frac14(20+30+x_2+x_3)$ 
\end{center}


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}

{\bf (a)}  Solve the equations for the four temperatures 
$x_1=23.125, x_2=21.875, x_3=25.625, x_4=24.375$. Use technology.


{\bf (b)} Replace the temperatures $20, 25, 30$ by $20, 16, 12$ and
re-compute the four temperatures.

{\bf (c)} Using the temperatures from part (b), subdivide the grid to make it $6\times 6$. Assign 
$25$ unknown temperatures at the interior grid points. Find the equations and
solve using technology.



{\bf References}. EPH Chapters 12, 13, used for Partial Differential Equations 3150.

\vfill\newpage
\rule{\textwidth}{0.5ex}

{\bf Quiz7 Problem 2}. Archeology and the Dot Product.

\begin{quote}\sf
Archeologist
Sir Flinders Petrie collected and analyzed pottery fragments from
900 Egyptian graves. He deduced from the data an historical ordering of
the 900 sites. Petrie's ideas will be illustrated for $4$ sites and $3$
pottery types. The matrix rows below represent sites 1, 2, 3, 4 and the
matrix columns represent pottery types 1, 2, 3. A matrix entry is 1 if
the site has that pottery type and 0 if not. This is an {\bf incidence
matrix}.
$$
A=\left(\begin{array}{ccc}
0 & 1 & 0 \\
0 & 1 & 1\\
1 &0 & 1\\
1 & 0 & 0
\end{array}\right)
$$
{\bf Petrie Matrix}. It is an incidence matrix in which the ones in each
column appear together, like the matirx above. 

{\bf Counting Pottery Types}. The dot product of  row 2 and row 3 is
$$
\rowvector{0,1,1}\cdot\rowvector{1,0,1} = 0*1 + 1*0 + 1*1 = 1
$$
which means sites 2 and 3 have one pottery type in common. {\bf Please
pause on this arithmetic}, until you agree that the products
$0*1,1*0,1*1$ add to the number of pottery types in common.

Sites with pottery in common are expected to be historically close in
time. Because pottery types evolve, old types cease production when
newly created pottery types begin production, which gives meaning to the
clustered ones in the columns of $A$. 
\end{quote}

{\large\rm {\bf The Problem}. Find a sequence of row swaps which starts with
the incidence matrix
$$C=\left(\begin{array}{rrr}
1& 1& 1 \\
1& 1& 0 \\
0& 0& 1 \\
0& 1& 1 \\
1& 1& 1 \\
0& 1& 1 \\
1& 1& 0 
\end{array}\right)$$
and ends with a Petrie matrix $A$. Express the swaps as elementary matrices
$E_1,E_2,\ldots$ and write $A$ as the product of elementary matrices times $C$.
There is not a unique set of swaps, because of duplicate rows.

You may use Kendall's ideas with Robinson matrices or the simple
definition of a Petrie matrix, given above. In any case, after finding
Petrie matrix $A$, compute Robinson's matrix $R=AA^T$ and apply the ideas
in sample quiz 7, to double-check the answer.
}

{\bf References}. Edwards-Penney Sections 3.4, 3.5, 3.6. 
David Kendall's 1969 work, \href{https://projecteuclid.org/euclid.pjm/1102983306}{Incidence matrices, interval graphs and seriation in archeology}.
W.\ S.\ Robinson, the method for chronologically ordering archeological
deposits, in the April 1951 issue of
\href{http://www.jstor.org/stable/276978}{American Antiquity}.
The incidence matrix $C$ was invented from an archeology exercise on
\href{http://www.saa.org/publicftp/public/primarydocuments/Seriation.pdf}{Seriation}
of 7 Maryland sites based on 3 types of historical ceramics.

%Geographical Types of Ceramic
%Location Delftware Creamware Pearlware
%Site a Tidewater 50 % 40 % 10 %
%Site b Coastal Edge 90 % 10 % 0 %
%Site c Mountains 0 % 0 % 100 %
%Site d Mountains 0 % 40 % 60 %
%Site e Piedmont 10 % 60 % 30 %
%Site f Mountains 0 % 20 % 80 %
%Site g Tidewater 70 % 30 % 0 %

\end{document}
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{\bf Quiz7 Problem 3}. {\bf Traffic Flow}\newline

\end{document}
