Steven R. Dunbar
Department of Mathematics
203 Avery Hall
Lincoln, NE 68588-0130
http://www.math.unl.edu
Voice: 402-472-3731
Fax: 402-472-8466

Stochastic Processes and

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Mathematical Modeling

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Note: These pages are prepared with MathJax. MathJax is an open source JavaScript display engine for mathematics that works in all browsers. See http://mathjax.org for details on supported browsers, accessibility, copy-and-paste, and other features.

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### Rating

Student: contains scenes of mild algebra or calculus that may require guidance.

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### 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?

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### Key Concepts

1. “All mathematical models are wrong, but some mathematical models are useful.” [1]
2. If the modeling assumptions are satisﬁed, proper mathematical models should predict well given conditions corresponding to the assumptions.
3. Knowing that all mathematical models are wrong, the real question becomes how wrong the model must be before it becomes unacceptable.
4. 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.

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### Vocabulary

1. A 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.

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### Mathematical Ideas

#### Mathematical Modeling

Remember the following proverb: All mathematical models are wrong, but some mathematical models are useful. [1]

Mathematical modeling involves two equally important activities:

• 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;
• evaluating the model, comparing the new relations with the real world and making predictions from the model.

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 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”. [8, page 225]

At a more detailed level, mathematical modeling involves $4$ successive phases in the cycle of modeling:

1. A real world situation,
2. a mathematical model,
3. a new relation among the quantities,
4. predictions and veriﬁcations.

Consider the diagram in Figure 1 which illustrates the cycle of modeling. Connecting phase 1 to 2 in the more detailed cycle builds the mathematical structure and connecting phase 3 to phase 4 evaluates the model.

##### 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.

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.

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 1.

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.” [45]

##### 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.

##### 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 [5]. 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.

##### 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.

##### 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.

#### 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:

• exact models with exact solutions;
• exact models with approximate solutions;
• approximate models with exact solutions;
• approximate models with approximate solutions.

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.

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

$\begin{array}{llll}\hfill V∕c& =20\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \left(V+25\right)∕\left(c+1\right)& =21.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

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.

Approximate models are much more typical.

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 [4]. An overview of modeling the pendulum appears in [7].

As a physically inclusive model of the pendulum, consider the general elastic pendulum shown in Figure 2. 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:

1. The pendulum is a rigid rod so that it does not lengthen or shorten, and it does not rotate torsionally.
2. All the mass is concentrated in the bob $B$, that is, the pendulum rod is massless.
3. The motion occurs only in one plane, so the pendulum has only one degree of freedom, measured by the angle $𝜃$.
4. There is no air resistance or friction in the bearing at $P$.
5. The only forces on the pendulum are at the contact point $P$ and the force of gravity due to the earth.
6. The pendulum is small relative to the earth so that gravity acts in the “down” direction only and is constant.

Then with standard physical modeling with Newton’s laws of motion, we obtain the diﬀerential equation for the pendulum angle $𝜃$ with respect to vertical,

$ml{𝜃}^{″}+mgsin𝜃=0,\phantom{\rule{2em}{0ex}}𝜃\left(0\right)={𝜃}_{0},{𝜃}^{\prime }\left(0\right)={v}_{0}.$

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 $sin𝜃\approx 𝜃$, so if the pendulum does not have large oscillations the model becomes

$ml{𝜃}^{″}+mg𝜃=0,\phantom{\rule{2em}{0ex}}𝜃\left(0\right)={𝜃}_{0},{𝜃}^{\prime }\left(0\right)={v}_{0}.$

This diﬀerential equation is analytically solvable with standard methods yielding

$𝜃\left(t\right)=Acos\left(\omega t+\varphi \right)$

for appropriate constants $A$, $\omega$, and $\varphi$ derived from the constants of the model. This is deﬁnitely an exact solution to an approximate model. We use the approximation of $sin𝜃$ by $𝜃$ 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 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.

#### 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, 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 ideal gas by making the following assumptions [3]:

1. 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.
2. The molecules can move in any direction with any speed.
3. The number of molecules is large.
4. No appreciable forces act on the molecules except during a collision.
5. The collisions between molecules and the container are elastic, and of negligible duration so both kinetic energy and momentum are conserved.
6. All molecules in the gas are identical.

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 equation of state or ideal gas law is

$PV=nRT.$

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 in 1873. Van der Waals’ equation of state is:

$\left(P+\frac{na}{{V}^{2}}\right)\left(V-b\right)=RT.$

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.

#### 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

1. The market has a very large number of identical, rational traders.
3. Prices vary randomly with some continuous probability distribution.
4. Trading transactions take negligible time.
5. Trading transactions can be in any amounts.

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 [9] 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.

#### Sources

Some of the ideas about mathematical modeling are adapted from the article by Valencia [9] and the book When Genius Failed by Roger Lowenstein. Silver [8] 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, [7]. 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.

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### Problems to Work for Understanding

1. How many jelly beans ﬁll a cubical box $10$ cm on a side?
1. 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.
2. 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.
3. Can you use these bounds to ﬁnd a better estimate for the number of jelly beans in the box?
4. 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?

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.

2. 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?

3. 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.

4. 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.

5. 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?

6. In the diﬀerential equations text by Edwards and Penney [2], the authors describe the process of mathematical modeling with the diagram in Figure 3 and the following list

1. The formulation of a real world problem in mathematical terms; that is, the construction of a mathematical model.
2. The analysis or solution of the resulting mathematical problem.
3. The interpretation of the mathematical results in the context of the original real-world situation – for example answering the question originally posed.

Compare and contrast this description of mathematical modeling with the cycle of mathematical modeling of this section, explicitly noting similarities and diﬀerences.

7. Glenn Ledder deﬁnes the process of mathematical modeling as

“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.”

He uses the diagram in Figure 4 as a conceptual model of the modeling process.

Compare and contrast this description of mathematical modeling with the cycle of mathematical modeling of this section, explicitly noting similarities and diﬀerences.

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### References

[1]   George E. P. Box and Norman R. Draper. Empirical Model-Building and Response Surfaces. Wiley, 1987.

[2]   C. Henry Edwards and David E. Penney. Diﬀernential Equations. Pearson Prentice Hall, 4th edition edition, 2008.

[3]   D. Halliday and R. Resnick. Physics. J. Wiley and Sons, 1966. QC21 .R42 1966.

[4]   C.C. Lin and Lee A. Segel. Mathematics applied to deterministic problems in the natural sciences. Macmillan, 1974. QA37.2 .L55.

[5]   J. David Logan. Applied Mathematics: A contemporary approach. John Wiley and Sons, 1987. QA37.2 .L64 1987.

[6]   Roger Lowenstein. When Genius Failed: The Rise and Fall of Long-Term Capital Management. Random House, 2000. HG 4930 L69 2000.

[7]   Chris Sangwin. Modelling the journey from elementary word problems to mathematical research. Notices of the American Mathematical Society, 58(10):1436–1445, November 2011.

[8]   Nate Silver. The Signal and the Noise. The Penguin Press, 2012. popular history.

[9]   Matthew Valencia. The gods strike back. The Economist, pages 3–5, February 2010. popular history.

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1. Duke University Modeling Contest Team. Accessed August 29, 2009

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