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\begin{document}
\myheader \mytitle
\hr
\sectiontitle{Mathematical Modeling}
\hr
\usefirefox
\hr
\visual{Rating}{../../../../CommonInformation/Lessons/rating.png}
\section*{Rating} %one of
% Everyone: contains no mathematics.
Student: contains scenes of mild algebra or calculus that may require
guidance.
% Mathematicallly Mature: may contain mathematics beyond calculus with proofs.
% Mathematicians Only: prolonged scenes of intense rigor.
\hr
\visual{Section Starter Question}{../../../../CommonInformation/Lessons/question_mark.png}
\section*{Section Starter Question}
Do you believe in the ideal gas law? Does it make sense to ``believe
in'' an equation? What do we really mean when we say we ``believe in''
an equation?
\hr
\visual{Key Concepts}{../../../../CommonInformation/Lessons/keyconcepts.png}
\section*{Key Concepts}
\begin{enumerate}
\item
``All mathematical models are wrong, but some mathematical
models are useful.''
\cite{box87}
\item
If the modeling assumptions are satisfied, proper mathematical
models should predict well given conditions corresponding to the
assumptions.
\item
Knowing that all mathematical models are wrong, the real
question becomes how wrong the model must be before it becomes
unacceptable.
\item
When observed outcomes deviate unacceptably from predicted
behavior in honest scientific or engineering work, we must alter
our assumptions, re-derive the quantitative relationships,
perhaps with more sophisticated mathematics or introducing more
quantities and begin the cycle of modeling again.
\end{enumerate}
\hr
\visual{Vocabulary}{../../../../CommonInformation/Lessons/vocabulary.png}
\section*{Vocabulary}
\begin{enumerate}
\item
A \defn{mathematical model} is a mathematical structure (often
an equation) expressing a relationship among a limited number of
quantifiable elements from the ``real world'' or some isolated
portion of it.
\end{enumerate}
\hr
\visual{Mathematical Ideas}{../../../../CommonInformation/Lessons/mathematicalideas.png}
\section*{Mathematical Ideas}
\subsection*{Mathematical Modeling}
Remember the following proverb: \emph{All mathematical models are
wrong, but some mathematical models are useful.}
\cite{box87}%
\index{mathematical model}
Mathematical modeling involves two equally important activities:
\begin{itemize}
\item
building a mathematical structure, a model, based on hypotheses
about relations among the quantities that describe the real
world situation, and then deriving new relations;%
\item
evaluating the model, comparing the new relations with the real
world and making predictions from the model.%
\end{itemize}
Good mathematical modeling explains the hypotheses, the development of
the model and its solutions, and then supports the findings by comparing
them mathematically with the actual circumstances. A successful model
must allow a user to consider the effects of different hypotheses.
Successful modeling requires a balance between so much complexity that
making predictions from the model may be intractable and so little
complexity that the predictions are unrealistic and useless. Complex
models often give more precise, but not necessarily more \emph{accurate},
answers and so can fool the modeler into believing that the model is
better at prediction that it actually is. On the other hand, simple
models may be useful for understanding, but are probably too blunt to
make useful predictions. Nate Silver quotes economist Arnold Zellner
advising to ``Keep it sophisticatedly simple''.
\cite[page 225]{silver12}
At a more detailed level, mathematical modeling involves \( 4 \)
successive phases in the cycle of modeling:
\begin{enumerate}
\item
A real world situation,%
\label{enum:mathmodel:cycle1}
\item
a mathematical model,%
\label{enum:mathmodel:cycle2}
\item
a new relation among the quantities,%
\label{enum:mathmodel:cycle3}
\item
predictions and verifications.%
\label{enum:mathmodel:cycle4}
\end{enumerate}
Consider the diagram in Figure~%
\ref{fig:mathmodel:cycle} which illustrates the cycle of modeling.
Connecting phase~%
\ref{enum:mathmodel:cycle1} to~%
\ref{enum:mathmodel:cycle2} in the more detailed cycle builds the
mathematical structure and connecting phase~%
\ref{enum:mathmodel:cycle3} to phase~%
\ref{enum:mathmodel:cycle4} evaluates the model.%
\index{cycle of modeling}
\begin{figure}
\centering
% \includegraphics{mathmodel1.png}
\begin{asy}
size(5inches);
real myfontsize = 10;
real mylineskip = 1.2*myfontsize;
pen mypen = fontsize(myfontsize, mylineskip);
defaultpen(mypen);
draw( box( (-2, 1), (-1, 2) ), linewidth(2pt) );
label(minipage("\sloppy Real world \\ situation", width=0.75inch), (-1.5,1.5));
label('1', (-2,2), 2*SE);
draw( box( ( 1, 1), ( 2, 2) ), linewidth(2pt) );
label(minipage("\sloppy Mathematical \\ model", width=0.75), (1.2,1.5));
label('2', ( 1,2), 2*SE);
draw( box( ( 1,-2), ( 2,-1) ), linewidth(2pt) );
label(minipage("\sloppy New \\ relation \\ among quantities", width=0.75inch), (1.5,-1.5));
label('3', ( 1,-1), 2*SE);
draw( box( (-2,-2), (-1,-1) ), linewidth(2pt) );
label(minipage("\sloppy Prediction\\ and \\ verification", width=0.75inch), (-1.5,-1.5));
label('4', (-2,-1), 2*SE);
draw( L=minipage("Identify a limited
number of quantities and the relations among them", width =2inch),
(-1,1.5)--(1,1.5), Arrow);
draw( L=minipage("\sloppy Mathematical solution", width=1inch), ( 1.5, 1 )--( 1.5,-1 ), Arrow);
draw( L="Evaluation of the new relation", ( 1, -1.5)--(-1 ,-1.5), Arrow);
draw( L="Comparison and refinement", (-1.5,-1 )--(-1.5, 1 ), Arrow);
\end{asy}
\caption{The cycle of modeling.%
\label{fig:mathmodel:cycle}}
\end{figure}
\subsubsection*{Modeling: Connecting Phase 1 to Phase 2}
A good description of the model will begin with an organized and
complete description of important factors and observations. The
description will often use data gathered from observations of the
problem. It will also include the statement of scientific laws and
relations that apply to the important factors. From there, the model
must summarize and condense the observations into a small set of
hypotheses that capture the essence of the observations. The small set
of hypotheses is a restatement of the problem, changing the problem from
a descriptive, even colloquial, question into a precise formulation that
moves the question from the general to the specific. Here the modeler
demonstrates a clear link between the listed assumptions and the
building of the model.
The hypotheses translate into a mathematical structure that becomes the
heart of the mathematical model. Many mathematical models, particularly
those from physics and engineering, become a single equation but
mathematical models need not be a single concise equation. Mathematical
models may be a regression relation, either a linear regression, an
exponential regression or a polynomial regression. The choice of
regression model should explicitly follow from the hypotheses since the
growth rate is an important consequence of the observations. The
mathematical model may be a linear or nonlinear optimization model,
consisting of an objective function and a set of constraints. Again the
choice of linear or nonlinear functions for the objective and
constraints should explicitly follow from the nature of the factors and
the observations. For dynamic situations, the observations often
involve some quantity and its rates of change. The hypotheses express
some connection between these quantities and the mathematical model then
becomes a differential equation, either linear or nonlinear depending on
the explicit details of the scientific laws relating the factors
considered. For discretely sampled data instead of continuous time
expressions the model may become a difference equation. If an important
feature of the observations and factors is noise or randomness, then the
model may be a probability distribution or a stochastic process. The
classical models from science and engineering usually take one of these
equation-like forms but not all mathematical models need to follow this
format. Models may be a connectivity graph, or a group of
transformations.%
\index{mathematical models!equations}
If the number of variables is more than a few, or the relations are too
complicated to write in a concise mathematical expression then the model
can be a computer program. Programs written in either high-level
languages such as C, FORTRAN or Basic and very-high-level languages such
as R, Octave, MATLAB, Scientific Python, Perl Data Language or a
computer algebra system are mathematical models. Spreadsheets combining
the data and the calculations are a popular and efficient way to
construct a mathematical model. The collection of calculations in the
spreadsheet express the laws connecting the factors that
the data in the rows and columns of the spreadsheet represent. Some
mathematical models use more elaborate software
specifically designed for modeling. Some software allows the user to
graphically describe the connections among factors to create and alter a
model.%
\index{spreadsheets}
Although this set of examples of mathematical models varies in
theoretical sophistication and the equipment used, the core of each is
to connect the data and the relations into a mechanism that allows the
user to vary elements of the model. Creating a model, whether a single
equation, a complicated mathematical structure, a quick spreadsheet, or
a large program is the essence of the first step connecting the phases
labeled 1 and 2 in Figure~%
\ref{fig:mathmodel:cycle}.
First models need not be sophisticated or detailed. For beginning
analysis ``back of the envelope calculations'' and ``dimensional
analysis'' will be as effective as spending time setting up an elaborate
model or solving equations with advanced mathematics. Unit analysis to
check consistency and outcomes of relations is important to check the
harmony of the modeling assumptions. A good model pays attention to
units, the quantities should be sensible and match. Even more
important, a non-dimensionalized model reveals significant
relationships, and major influences. Unit analysis is an important part
of modeling, and goes far beyond simple checking to ``make sure units
cancel.''%
\index{mathematical model!unit analysis}
\cite{lin74, logan87}
\subsubsection*{Mathematical Solution: Connecting Phase 2 to Phase 3}
Once the modelers create the model, then they should derive some new
relations among the important quantities selected to describe the real
world situation. This is the step connecting the phases labeled 2 and 3
in the diagram. If the model is an equation, for instance the Ideal Gas
Law, then one can solve the equation for one of the variables in terms
of the others. In the Ideal Gas Law, solving for one of the gas
parameters is easy. A regression model may need no additional
mathematical solution, although it might be useful to find auxiliary
quantities such as rates of growth or maxima or minima. For an
optimization problem the solution is the set of optima or the rates of
change of optima with respect to the constraints. If the model is a
differential equation or a difference equation, then the solution may
have some mathematical substance. For instance, for a ballistics
problem, the model may be a differential equation and the solution by
calculus methods yields the equation of motion. For a problem with
randomness, the derivation may find the mean or the variance. For a
connectivity graph, some interesting quantities are the number of cycles,
components or the diameter of the graph. If the model is a computer
program, then this step usually involves running the program to obtain
the output.
It is easy for students to focus most attention on the solution stage of the
process, since the methods are the core of the typical
mathematical curriculum. This step usually requires no interpretation,
and the model dictates the methods to use. This step is often
the easiest in the sense that it is the clearest on how to proceed,
although the mathematical procedures may be daunting.
\subsubsection*{Testing and Sensitivity: Connecting Phase 3 to Phase 4}
Once this step is done, the model is ready for testing and sensitivity
analysis. This is the step that connects the phases labeled 3 and 4.
At the least, the modelers should try to verify, even with common sense,
the results of the solution. Typically for a mathematical model, the
previous step allows the modelers to produce some important or valuable
quantity of the model. Modelers compare the results of the model with
standard or common inputs with known quantities for the data or
statement of the problem. This may be as easy as substituting into the
derived equation, regression expression, or equation of motion. When
running a computer model or program, this may involve sequences of
program runs and related analysis. With any model, the results will
probably not be exactly the same as the known data so interpretation or
error analysis will be necessary. The interpretation requires judgment
on the relative magnitudes of the quantities produced in light of the
confidence in the exactness or applicability of the hypotheses.
Another important activity at this stage in the modeling process is
the sensitivity analysis. The modelers should choose some critical
feature of the model and then vary the parameter value that quantifies
that feature. The results should be carefully compared to the real
world and to the predicted values. If the results do not vary
substantially, then perhaps the feature or parameter is not as
critical as believed. This is important new information for the
model. On the other hand, if a predicted or modeled value varies
substantially in comparison to the parameter as it is slightly varied,
then the accuracy of measurement of the critical parameter assumes new
importance. In sensitivity analysis, just as in all modeling, we
measure ``varying substantially'' with significant digits, relative
magnitudes, and rates of change. Here is another area where
expressing parameters in dimensionless groups is important
\cite{logan87}. In some areas of applied mathematics such as linear
optimization and statistics, a side effect of the solution method is
that it automatically produces sensitivity parameters. In linear
optimization, these are sometimes called the shadow prices and these
additional solution values should be used whenever possible.%
\index{sensitivity analysis}
\subsubsection*{Interpretation and Refinement: Connecting Phase 4 to
Phase 1}
Finally the modelers must take the results from the previous steps and
use them to refine the interpretation and understanding of the real
world situation. In the diagram the interpretation step
connects the phases labeled 4 and 1, completing the cycle of modeling.
For example, if the situation is modeling motion, then examining results
may show that the predicted motion is faster than measured, or that the
object does not travel as far as the model predicts. Then it may be
that the model does not include the effects of friction, and so friction
should be incorporated into a new model. At this step, the modeler has
to be open and honest in assessing the strengths and weaknesses of the
model. It also requires an improved understanding of the real world
situation to include the correct new elements and hypotheses to correct
the discrepancies in the results. A good model can be useful even when
it fails. When a model fails to match the reality of the situation then
the modeler must understand how it is wrong, and what to do when it is
wrong, and minimizing the cost when it is wrong.
The step between stages 4 and 1 may suggest new processes, or
experimental conditions to alter the model. If the problem suggests
changes then the modeler should implement those changes should and
test them in another
cycle in the modeling process.
\subsubsection*{Summary} A good summary of the modeling process is that
it is an intense and structured application of ``critical thinking''.
Sophistication of mathematical techniques is not always necessary, the
mathematics connecting steps 2 and 3 or potentially steps 3 and 4 may
only be arithmetic. The key to good modeling is the critical thinking
that occurs between steps 1 and 2, steps 3 and 4, and 4 and 1. If a
model does not fit into this paradigm, it probably does not meet the
criteria for a good model.
Good mathematical modeling, like good critical thinking, does not
arise automatically or naturally. Scientists, engineers,
mathematicians and students must practice and develop the craft of
creating, solving, using, and interpreting a mathematical model. The
structured approach to modeling helps distinguish the distinct steps,
each requiring separate intellectual skills. It also provides a
framework for developing and explaining a mathematical model.
\subsection*{A classification of mathematical models}
The goal of mathematical modeling is to represent a real-world situation
in a way that is a close approximation, even ideally exact, with a
mathematical structure that is still solvable. This represents the
interplay between the mathematics occurring between steps 1 and 2 and
steps 2 and 3. Then we can classify models based on this interplay:
\begin{itemize}
\item
exact models with exact solutions;
\item
exact models with approximate solutions;
\item
approximate models with exact solutions;
\item
approximate models with approximate solutions.
\end{itemize}
Exact models are uncommon because only rarely do we know physical
representations well enough to claim to represent situations exactly.
Exact models typically occur in discrete situations where we
can enumerate all cases and represent them mathematically.
\begin{example}
The average value of the 1-cent, 5-cent, 10-cent and 25-cent coins
in Paula's purse is 20 cents (the average value is the total money
value divided by the number of coins.) If she had one more 25-cent
coin, the average value would be 21 cents. What coins does Paula
have in her purse?
The mathematical model is to let \( V \) be the total value of the
coins and \( c \) be the number of coins. Write two equations in
two unknowns
\begin{align*}
V/c &= 20 \\
(V+25)/(c+1) &= 21.
\end{align*}
The solution is \( 3 \) 25-cent coins and \( 1 \) 5-cent coin.
Because of the discrete nature and fixed value of the coins, and the
simplicity of the equations, an exact model has an exact solution.
\end{example}
Approximate models are much more typical.
\begin{example}
Modeling the motion of a pendulum is a classic problem in applied
mathematics. Galileo first modeled the pendulum mathematically and
mathematicians and physicists still investigate more sophisticated
models. Most books on
applied mathematics and differential equations consider some form
of the
pendulum model, for example
\cite{lin74}. An overview of modeling the pendulum appears in
\cite{sangwin11}.%
\index{pendulum}
\index{Galileo Galilei}
\begin{figure}
\centering
% \includegraphics{pendulum3.png}
\begin{asy}
size(5inches);
real myfontsize = 10;
real mylineskip = 1.2*myfontsize;
pen mypen = fontsize(myfontsize, mylineskip);
defaultpen(mypen);
//parameters of the diagram
real basethick=0.4;
real basewidth=4.0;
real baseshift=-1.0;
real stemheight=15.0;
real crossbarlength=7.0;
real crossbarposition=14.0;
// base, stem, and cross-bar
filldraw(shift(baseshift,0)*box((0,0), (basewidth, basethick)), gray); // base
draw(shift(baseshift,0)*box((0,0), (basewidth,basethick)), linewidth(2pt)); // base outline
draw(shift(0,basethick)*box((0,0), (basethick, 15)), linewidth(2pt)); // vertical stem
// cross-bar
filldraw(shift(baseshift,crossbarposition)*box((0,0), (crossbarlength, basethick)), white);
// cross-bar outline
draw(shift(baseshift,crossbarposition)*box((0,0), (crossbarlength, basethick)), linewidth(2pt));
// spring drawing sub-routines
pair coilpoint(real lambda, real r, real t)
{
return (2.0*lambda*t+r*cos(t),r*sin(t));
}
guide coil(guide g=nullpath, real lambda, real r, real a, real b, int n)
{
real width=(b-a)/n;
for(int i=0; i <= n; ++i) {
real t=a+width*i;
g=g..coilpoint(lambda,r,t);
}
return g;
}
guide spring(real x) {
real r=8;
real t1=-pi;
real t2=20*pi;
real lambda=(t2-t1+x)/(t2-t1);
pair b=coilpoint(lambda,r,t1);
pair c=coilpoint(lambda,r,t2);
pair a=b-20;
pair d=c+20;
guide spr=(a--b--coil(lambda,r,t1,t2,100)--c--d);
return spr;
}
// spring parameters
real lengthPend = point(spring(1), size(spring(1))).x - point(spring(1), 0).x;
real scaledLengthPend = 7.0/lengthPend; // length of spring is about same as crossbar
real pendAngle = -70;
real x0Pend = point( rotate(pendAngle)*scale(scaledLengthPend)*spring(1), 0).x;
real y0Pend = point( rotate(pendAngle)*scale(scaledLengthPend)*spring(1), 0).y;
real xnPend = point( scale(scaledLengthPend)*spring(1), size(spring(1))).x;
real ynPend = point( scale(scaledLengthPend)*spring(1), size(spring(1))).y;
// coordinates of pivot point P
real pivotPointx = baseshift+crossbarlength;
real pivotPointy = crossbarposition;
// draw spring
draw(shift(pivotPointx-x0Pend, pivotPointy-y0Pend)*rotate(pendAngle)*
scale(scaledLengthPend)*spring(1), linewidth(2));
//draw bob
path bob = shift(xnPend, ynPend)*( (0,-1/3)--(1,-1/2)--(1,1/2)--(0, 1/3)--cycle );
draw(shift(pivotPointx-x0Pend, pivotPointy-y0Pend)*rotate(pendAngle)*bob, linewidth(2));
//label end points
label("$P$", (pivotPointx, pivotPointy-basethick) , W);
label("$B$", shift(pivotPointx-x0Pend, pivotPointy-y0Pend)*rotate(pendAngle)*(xnPend,ynPend), 3*W);
// draw measuring lines and labels
// angle theta
real l = 4.0;
real xadj = 0.4;
real yadj = 3.0;
path angle0 = (0,0)--(0,-l);
path complAnglePend = rotate(90+pendAngle)*angle0;
draw( shift(pivotPointx,pivotPointy)*angle0);
draw( shift(pivotPointx,pivotPointy)*complAnglePend);
label("$\theta$", (pivotPointx+xadj, pivotPointy-yadj));
// space axes
path ax = (0,0)--(1,0);
path ay = (0,0)--(0,1);
path az = (0,0)--(0.5,0.5);
Label labx = Label("$x$",position=EndPoint);
Label laby = Label("$y$",position=EndPoint);
Label labz = Label("$z$",position=EndPoint);
draw(shift(pivotPointx,(1/3)*stemheight)*ax, L=labx, arrow=Arrow(TeXHead));
draw(shift(pivotPointx,(1/3)*stemheight)*ay, L=laby, arrow=Arrow(TeXHead));
draw(shift(pivotPointx,(1/3)*stemheight)*az, L=labz, arrow=Arrow(TeXHead));
// torsion arrow
path torsion=shift(xnPend+2.0,ynPend)*scale(1/2,1)*arc( (0,0), 1, -240, 60);
draw(shift(pivotPointx-x0Pend, pivotPointy-y0Pend)*rotate(pendAngle)*torsion,
arrow=Arrows(TeXHead));
label("torsion", shift(pivotPointx-x0Pend, pivotPointy-y0Pend)*rotate(pendAngle)*
(xnPend+2.0,ynPend));
// extension
path extension = shift(20*N)*( point(spring(1), 0) -- point(spring(1), size(spring(1))) );
pair mid_extension = ( (1/2)*(point(spring(1), size(spring(1))).x),
point(spring(1), size(spring(1))).y);
draw(shift(pivotPointx-x0Pend, pivotPointy-y0Pend)*rotate(pendAngle)*
scale(scaledLengthPend)*extension, Arrows(TeXHead), bar=Bars);
label("extension", shift(pivotPointx-x0Pend, pivotPointy-y0Pend)*
rotate(pendAngle)*scale(scaledLengthPend)*mid_extension, 8*E);
\end{asy}
\caption{Schematic diagram of a pendulum.%
\label{fig:mathmodel:pendulum}}
\end{figure}
As a physically inclusive model of the pendulum, consider the
general elastic pendulum shown in Figure~%
\ref{fig:mathmodel:pendulum}. The pendulum is a heavy bob \( B \)
attached to
a pivot point \( P \) by an elastic spring. The pendulum has at
least 5 degrees of freedom about its rest position. The pendulum
can swing in the \( x \)-\( y \)-\( z \) spatial dimensions. The
pendulum can rotate torsionally around the axis of the spring \( PB \).
Finally, the pendulum can lengthen and shorten along the axis \( PB \).
Although we usually think of a pendulum as the short rigid rod in a
clock, some pendula move in all these degrees of freedom, for
example weights dangling from a huge construction crane.
We usually simplify the modeling with assumptions:
\begin{enumerate}
\item
The pendulum is a rigid rod so that it does not lengthen or
shorten, and it does not rotate torsionally.
\item
All the mass is concentrated in the bob \( B \), that is,
the pendulum rod is massless.
\item
The motion occurs only in one plane, so the pendulum has
only one degree of freedom, measured by the angle \( \theta \).
\item
There is no air resistance or friction in the bearing at \(
P \).
\item
The only forces on the pendulum are at the contact point \(
P \) and the force of gravity due to the earth.
\item
The pendulum is small relative to the earth so that gravity
acts in the ``down'' direction only and is constant.
\end{enumerate}
Then with standard physical modeling with Newton's laws of motion,
we obtain the differential equation for the pendulum angle \( \theta
\) with respect to vertical,
\[
ml \theta'' + m g \sin \theta = 0, \qquad \theta(0) = \theta_0,
\theta'(0) = v_0.
\] Note the application of the math modeling process moving from
phase 1 to phase 2 with the identification of important variables,
the hypotheses that simplify, and the application of physical laws.
It is clear that already the differential equation describing the
pendulum is an approximate model.
Nevertheless, this nonlinear differential equation cannot be solved
in terms of elementary functions. It can be solved in terms of a
class of higher transcendental functions known as elliptic
integrals, but in practical terms that is equivalent to not being
able to solve the equation analytically. There are two
alternatives, either to solve this equation numerically or to reduce
the equation further to a simpler form.
For small angles \( \sin\theta \approx \theta \), so if the pendulum
does not have large oscillations the model becomes
\[
ml \theta'' + m g \theta = 0, \qquad \theta(0) = \theta_0,
\theta'(0) = v_0.
\] This differential equation is analytically solvable with standard
methods yielding
\[
\theta(t) = A \cos(\omega t + \phi)
\] for appropriate constants \( A \), \( \omega \), and \( \phi \)
derived from the constants of the model. This is definitely an
exact solution to an approximate model. We use the approximation of \(
\sin \theta \) by \( \theta \) specifically so that the
differential equation is explicitly solvable in elementary
functions. The solution to the differential equation is the process
that leads from step 2 to step 3 in the modeling process. The
analysis necessary to measure the accuracy of the exact solution
compared to the physical motion is difficult. The measurement of
the accuracy would be the process of moving from step 2 to step 3 in
the cycle of modeling.
For a clock maker the accuracy of the solution is important. A day
has \( 86{,}400 \) seconds, so an error on the order of even 1 part in
10,000 for a physical pendulum oscillating once per second
accumulates unacceptably. This error concerned physicists such as
C. Huygens%
\index{Huygens, C.}
who created advanced pendula with a period independent of the
amplitude. This illustrates step 3 to step 4 and the next modeling
cycle.
Models of oscillation, including the pendulum, have been a central
paradigm in applied mathematics for centuries. Here we use the
pendulum model to illustrate the modeling process leading to an
approximate model with an exact solution.
\end{example}
\subsection*{An example from physical chemistry}
This section illustrates the cycle of mathematical modeling with a
simple example from physical chemistry. This simple example provides a
useful analogy to the role of mathematical modeling in mathematical
finance. The historical order of discovery is slightly modified to
illustrate the idealized modeling cycle. Scientific progress rarely
proceeds in simple order.
Scientists observed that diverse gases such as air, water vapor,
hydrogen, and carbon dioxide all behave predictably and similarly. After
many observations, scientists derived empirical relations such as
Boyle's law,%
\index{Boyle's Law}
and the law of Charles and Gay-Lussac about the gas. These laws express
relations among the volume \( V \), the pressure \( P \), the amount \(
n \), and the temperature \( T \) of the gas.
In classical theoretical physics, we can define an \emph{ideal gas}%
\index{ideal gas}
by making the following assumptions
\cite{halliday66}:
\begin{enumerate}
\item
A gas consists of particles called molecules which have mass,
but essentially have no volume, so the molecules occupy a
negligibly small fraction of the volume occupied by the gas.
\item
The molecules can move in any direction with any speed.
\item
The number of molecules is large.
\item
No appreciable forces act on the molecules except during a
collision.
\item
The collisions between molecules and the container are elastic,
and of negligible duration so both kinetic energy and momentum
are conserved.
\item
All molecules in the gas are identical.
\end{enumerate}
From this limited set of assumptions about theoretical entities called
molecules physicists derive the equation of state for an ideal gas using
the 4 quantifiable elements of volume, pressure, amount, and
temperature. The \emph{equation of state}%
\index{equation of state}
or \emph{ideal gas law}%
\index{ideal gas law}
is
\[
PV = n R T.
\] where \( R \) is a measured constant, called the universal gas
constant. The ideal gas law is a simple algebraic equation relating the
4 quantifiable elements describing a gas. The equation of state or
ideal gas law predicts very well the properties of gases under the wide
range of pressures, temperatures, masses and volumes commonly
experienced in everyday life. The ideal gas law predicts with accuracy
necessary for safety engineering the pressure and temperature in car
tires and commercial gas cylinders. This level of prediction works even
for gases we know do not satisfy the assumptions, such as air, which
chemistry tells us is composed of several kinds of molecules which have
volume and do not experience completely elastic collisions because of
intermolecular forces. We know the mathematical model is wrong, but it
is still useful.
Nevertheless, scientists soon discovered that the assumptions of an
ideal gas predict that the difference in the constant-volume specific
heat and the constant-pressure specific heat of gases should be the same
for all gases, a prediction that scientists observe to be false. The
simple ideal gas theory works well for monatomic gases, such as helium,
but does not predict so well for more complex gases. This scientific
observation now requires additional assumptions, specifically about the
shape of the molecules in the gas. The derivation of the relationship
for the observables in a gas is now more complex, requiring more
mathematical techniques.
Moreover, under extreme conditions of low temperatures or high
pressures, scientists observe new behaviors of gases. The gases
condense into liquids, pressure on the walls drops and the gases no
longer behave according to the relationship predicted by the ideal gas
law. We cannot neglect these deviations from ideal behavior in accurate
scientific or engineering work. We now have to admit that under these
extreme circumstances we can no longer ignore the size of the molecules,
which do occupy some appreciable volume. We also must admit that we
cannot ignore
intermolecular forces. The two effects just
described can be incorporated into a modified equation of state proposed
by J.D. van der Waals%
\index{van der Waals, J.D.}
in 1873. Van der Waals' equation of state%
\index{equation of
state!van der Waals}
is:
\[
\left( P + \frac{na}{V^2} \right) (V- b) = RT.
\] The added constants \( a \) and \( b \) represent the new elements of
intermolecular attraction and volume effects respectively. If \( a \)
and \( b \) are small because we are considering a monatomic gas under
ordinary conditions, the Van der Waals equation of state can be well
approximated by the ideal gas law. Otherwise we must use this more
complicated relation for engineering our needs with gases.
Physicists now realize that because of complicated intermolecular
forces, a real gas cannot be rigorously described by any simple equation
of state. We can honestly say that the assumptions of the ideal gas are
not correct, yet are sometimes useful. Likewise, the predictions of the
van der Waals equation of state describe quite accurately the behavior
of carbon dioxide gas in appropriate conditions. Yet for very low
temperatures, carbon dioxide deviates from even these modified
predictions because we know that the van der Waals model of the gas is
wrong. Even this improved mathematical model is wrong, but it still is
useful.
\subsection*{An example from mathematical finance}
For modeling mathematical finance, we make a limited number of idealized
assumptions about securities markets. We start from empirical
observations of economists about supply and demand and the role of
prices as a quantifiable element relating them. We ideally assume that
\begin{enumerate}
\item
The market has a very large number of identical, rational
traders.
\item
All traders always have complete information about all assets
they are trading.
\item
Prices vary randomly with some continuous probability
distribution.
\item
Trading transactions take negligible time.
\item
Trading transactions can be in any amounts.
\end{enumerate}
These assumptions are similar to the assumptions about an ideal gas.
From the assumptions we can make some standard economic arguments to
derive some interesting relationships about option prices. These
relationships can help us manage risk, and speculate intelligently in
typical markets. However, caution is necessary. In discussing the
economic collapse of 2008-2009, blamed in part on the overuse or even
abuse of mathematical models of risk, Valencia \cite{valencia10} says
``Trying ever harder to capture risk in mathematical formulae can be
counterproductive if such a degree of accuracy is intrinsically
unobtainable.'' If the dollar amounts get very large (so that
rationality no longer holds!), or only the market has only a few
traders, or sharp jumps in prices occur, or the trades come too
rapidly for information to spread effectively, we must proceed with
caution. The observed financial outcomes may deviate from predicted
ideal behavior in accurate scientific or economic work, or financial
engineering.
We must then alter our assumptions, re-derive the quantitative
relationships, perhaps with more sophisticated mathematics or
introducing more quantities and begin the cycle of modeling again.
\subsection*{Sources}
Some of the ideas about mathematical modeling are adapted from the
article by Valencia
\cite{valencia10} and the book \emph{When Genius Failed} by Roger
Lowenstein. Silver
\cite{silver12} is an excellent popular survey of models and their
predictions, especially large models developed from big data sets. The
classification of mathematical models and solutions is adapted from
Sangwin,~%
\cite{sangwin11}. The example of an exact model with an exact solution
is adapted from the 2009 American Mathematics Competition 8. Glenn
Ledder reviewed this section and suggested several of the problems.
\nocite{lowenstein00}
\hr
\visual{Problems to Work}{../../../../CommonInformation/Lessons/solveproblems.png}
\section*{Problems to Work for Understanding}
\begin{enumerate}
\item
How many jelly beans fill a cubical box \( 10 \) cm on a side?
\begin{enumerate}
\item
Find an upper bound for number of jelly beans in the
box. Create and solve a mathematical model for this
situation, enumerating explicitly the simplifying
assumptions that you make to create and solve the model.
\item
Find an lower bound for number of jelly beans in the
box. Create and solve a mathematical model for this
situation, enumerating explicitly the simplifying
assumptions that you make to create and solve the model.
\item
Can you use these bounds to find a better estimate for
the number of jelly beans in the box?
\item
Suppose the jelly beans are in a 1-liter jar instead of
a cubical box. (Note that a cubical box \( 10 \) cm on
a side has a volume of 1 liter.) What would change about
your estimates? What would stay the same? How does the
container being a jar change the problem?
\end{enumerate}
A good way to do this problem is to divide into small groups,
and have each group work the problem separately. Then gather
all the groups and compare answers.
\item
How many ping-pong balls fit into the room where you
are reading this? Create and solve a mathematical model for
this situation, enumerating explicitly the simplifying
assumptions that you make to create and solve the model.
Compare this problem to the previous jelly bean problem. How is
it different and how is it similar? How is it easier and how is
it more difficult?
\item
How many flat toothpicks would fit on the surface of a sheet of
poster board?
Create and solve a mathematical model for this situation,
enumerating explicitly the simplifying assumptions that you make
to create and solve the model.
\item
If your life earnings were doled out to you at a certain rate
per hour for every hour of your life, how much is your time
worth?
Create and solve a mathematical model for this situation,
enumerating explicitly the simplifying assumptions that you make
to create and solve the model.
\item
A ladder stands with one end on the floor and the other against
a wall. The ladder slides along the floor and down the wall. A
cat is sitting at the middle of the ladder. What curve does the
cat trace out as the ladder slides?
Create and solve a mathematical model for this situation,
enumerating explicitly the simplifying assumptions that you make
to create and solve the model. How is this problem similar to,
and different from, the previous problems about jelly beans in
box, ping-pong balls in a room, toothpicks on a poster board,
and life-earnings?
\item
In the differential equations text by Edwards and Penney
\cite{edwardspenney}, the authors describe the process of
mathematical modeling with the diagram in Figure~%
\ref{fig:mathmodel:edwardspenney} and the following list
\begin{figure}[h]
\centering
%% \includegraphics{edwardspenney}
\begin{asy}
size(5inches);
real myfontsize = 10;
real mylineskip = 1.2*myfontsize;
pen mypen = fontsize(myfontsize, mylineskip);
defaultpen(mypen);
object A1 = draw(minipage("Real world situtation", width=0.80inch), box, xmargin=0.1inch, (0,4*sqrt(3)));
object A2 = draw(minipage("Mathematical model", width=0.80inch), box, xmargin=0.1inch, (-4,0) );
object A3 = draw(minipage("Mathematical results", width=0.80inch), box, xmargin=0.1inch, (4,0) );
object B1 = draw(minipage("Formulation", width=0.80inch), ellipse, xmargin=0.1inch, (-2,2*sqrt(3)));
object B2 = draw(minipage("Mathematical Analysis", width=0.80inch), ellipse, xmargin=0.1inch, (0,0));
object B3 = draw(minipage("Interpretation", width=0.80inch), ellipse, xmargin=0.1inch, (2,2*sqrt(3)));
add(new void(frame f, transform t) {
draw(f,point(A1,SSW,t)--point(B1,N,t), Arrow);
});
add(new void(frame f, transform t) {
draw(f,point(B1,SSW,t)--point(A2,N,t), Arrow);
});
add(new void(frame f, transform t) {
draw(f,point(A2,E,t)--point(B2,W,t), Arrow);
});
add(new void(frame f, transform t) {
draw(f,point(B2,E,t)--point(A3,W,t), Arrow);
});
add(new void(frame f, transform t) {
draw(f,point(A3,NNE,t)--point(B3,SSE,t), Arrow);
});
add(new void(frame f, transform t) {
draw(f,point(B3,N,t)--point(A1,SSE,t), Arrow);
});
\end{asy}
\caption{The process of mathematical modeling according to
Edwards and Penney.}%
\label{fig:mathmodel:edwardspenney}
\end{figure}
\begin{enumerate}
\item
The formulation of a real world problem in mathematical
terms; that is, the construction of a mathematical
model.
\item
The analysis or solution of the resulting mathematical
problem.
\item
The interpretation of the mathematical results in the
context of the original real-world situation -- for
example answering the question originally posed.
\end{enumerate}
Compare and contrast this description of mathematical modeling
with the cycle of mathematical modeling of this section,
explicitly noting similarities and differences.
\item
Glenn Ledder defines the process of mathematical modeling as
\begin{quotation}
``A mathematical model is a mathematical construction based
on a real setting and created in the hope that its
mathematical behavior will resemble the real behavior enough
to be useful.''
\end{quotation}
He uses the diagram in Figure~%
\ref{fig:mathmodel:glennledder} as a conceptual model of the
modeling process.
\begin{figure}[h]
\centering
%\includegraphics{glennledder}
\begin{asy}
size(5inches);
real myfontsize = 10;
real mylineskip = 1.2*myfontsize;
pen mypen = fontsize(myfontsize, mylineskip);
defaultpen(mypen);
draw( box( (-2, 0), (-1, 1) ), linewidth(2pt) );
label(minipage("\sloppy Real world \\ situation", width=0.75inch), (-1.5,0.5));
draw( box( (0, 0), (1, 1) ), linewidth(2pt) );
label(minipage("\sloppy Conceptual \\ model", width=0.75inch), (0.5,0.5));
draw( box( (2, 0), (3, 1) ), linewidth(2pt) );
label(minipage("\sloppy Mathematical \\ model", width=0.75inch), (2.5,0.5));
real eps=(1/10);
draw( L=Label(minipage("\sloppy approximation", width=1inch),align=N+eps*E),
(-1, 0.67)--(0,0.67), Arrow);
draw( L=Label(minipage("\sloppy validation", width=1inch),align=0.5*eps*E+eps*S),
(-1, 0.33)--(0,0.33), BeginArrow);
draw( L=Label(minipage("\sloppy derivation", width=1inch), align=N+3*eps*E),
(1, 0.67)--(2,0.67), Arrow);
draw( L=Label(minipage("\sloppy analysis", width=1inch), align=0.5*eps*E+eps*S),
(1, 0.33)--(2,0.33), BeginArrow);
\end{asy}
\caption{The process of mathematical modeling according to
Glenn Ledder.}%
\label{fig:mathmodel:glennledder}
\end{figure}
Compare and contrast this description of mathematical modeling
with the cycle of mathematical modeling of this section,
explicitly noting similarities and differences.
\end{enumerate}
\hr
\visual{Books}{../../../../CommonInformation/Lessons/books.png}
\section*{Reading Suggestion:}
\bibliography{../../../../CommonInformation/bibliography}
% \begin{enumerate}
% \item
% \emph{Physics Parts I and II}, by D. Halliday and R. Resnick, J.
% Wiley and Sons, New York, 1966.
% \item
% \emph{When Genius Failed: The Rise and Fall of Long-Term
% Capital Management} by Roger Lowenstein, Random House, New York,
% 2000. (HG 4930 L69 2000)
% \end{enumerate}
\hr
\visual{Links}{../../../../CommonInformation/Lessons/chainlink.png}
\section*{Outside Readings and Links:}
\begin{enumerate}
\item
\link{http://www.youtube.com/watch?v=IqiZ0xso-F0}{Duke
University Modeling Contest Team} Accessed August 29, 2009
\end{enumerate}
\hr
\mydisclaim \myfooter
Last modified: \flastmod
\end{document}
readability grades:
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ARI: 12.7
Coleman-Liau: 14.7
Flesch Index: 42.2
Fog Index: 16.0
Lix: 50.4 = school year 9
SMOG-Grading: 14.0
sentence info:
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60 paragraphs, average length 4.2 sentences
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sentence beginnings:
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File name : mathmodel.tex
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READABILITY INDICES
Fog : 16.9548
Flesch : 29.3092
Flesch-Kincaid : 13.7047