42 CHAPTER 1. BACKGROUND IDEAS
computer program. Programs written in either high-level languages such as
C, FORTRAN or Basic and very-high-level languages such as MATLAB 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 ex-
press the laws connecting the factors which are represented by the data in
the rows and columns of the spreadsheet. Some mathematical models may
be expressed by using more elaborate software specifically designed for mod-
eling. Some software allows the user to describe the connections between
factors graphically 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 boxes labeled 1 and 2 above.
First models need not be sophisticated or detailed. For beginning anal-
ysis “back of the envelope calculations” and “dimensional analysis” will be
as effective as spending time setting up an elaborate model or solving equa-
tions 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-dimensionalizized 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.” [34, 32]
Mathematical Solution
Once the modelers have created 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 boxes 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 quite easy.
A regression equation model may require almost no 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