2.2 Mathematical models

This power of abstraction, this ability to leave out some or many details about a concrete problem in order to use the “distillate” for a multitude of similar problems, is one of the most striking characteristics of our mind. A concrete, “real physical” problem need not even be there: We can make some useful considerations without a specific application in mind, as we have seen with the intersection problem. Mathematics and logic are essential tools in this process of abstraction. They are powerful languages to express and represent real phenomena. We could even define mathematics as the science of finding and representing patterns and structures of concrete or abstract problems. They are these patterns that are expressed in a mathematical model.

So, what is a mathematical model? Informally, it is a list of alternatives fulfilling certain criteria from which we choose one, often the “best”. Consider buying a house: we have a number of houses from which we choose one.

Formally, a mathematical model consists of symbols such as operators (arithmetic, Boolean, functions), data (the known entities), variables (the unknown entities), and constraints (the conditions). On a most basic level, we can define a mathematical model as a n-dimensional state space (or a simply as a set) fulfilling certain properties or conditions. Consider the problem of buying a house: the “state space” are all houses on the planet (or the houses in a certain region), the “variable” is the unknown house to choose, the “data” is our budget, etc. The “conditions or constraints” is our taste: “it must have at least 5 windows and 2 balconies”, (and here is a Boolean operator).

In mathematics, we use symbols for these concepts:

Y  = {x ∈ X  | R (x)}

Y is called the mathematical model (which is a set), x is a vector of variables, X is a vector space of dimension n (often n or n, but not necessarily), R are the constraints, that is, a function which maps every element x in X into true or false: R : X →{true,false}. By solving a model we means the find one or all xo X such that R(xo) is true. The problem is said infeasible if no such xo exists, in other words if the set Y is empty, otherwise it is said the be feasible.

Consider our house-buying problem: X is the set of all houses: house1, house2, house3, ..., etc. X is one-dimensional in this case, such a list of all houses. x is one unknown house out of all houses. R(x) is the required condition of each house x, it is true or false for every house: house1=true [R(x) = true,x = house1] (yes, it has at least 5 windows and 2 balconies), R(house2=false ((no, it is not true that it has at least 5 windows and 2 balconies), R(house3)=false, etc. Y is the list of all houses that have a true-property: Y = {house1,}, the list of eligible houses.

Let us give several examples (please go carefully through these examples to understand the concepts well).

Example 1:

x ∈ ℤ+ , R (x) =  (x =  1)

The state space consists of all positive numbers (+). x is a singleton variable which must be a positive integer, and its value is 1 by definition of R, that is, we choose exactly one number which value is true, namely 1. Hence Y = {1}, the model is feasible and has exactly one solution, namely x = 1. (It is as if we have only one house with at least 5 windows and 2 balconies, that the unique one we choose!)

Example 2:

x ∈ ℤ+ , R (x) de=f (x = 1 ∧ x = 2)

x is a singleton variable which must be a positive integer, and its value is 1 and at the same time 2 by definition of R. Hence Y = {} (the set Y is empty), the model is infeasible, because there is no x which value is 1 and 2 at the same time. This is a contradiction. (It is as if no house fulfills our requirements.)

Example 3:

x ∈ ℝ2 , R (x ) = (|x1| ≤ 1 ∧  |x2| ≤ 1 )

The state space is 2-dimensional (every tuple of two real numbers is a possible choice). x is a 2-dimensional vector variable (x = (x1,x2)) with the components x1 and x2, they must be real numbers each, and its absolute values must be smaller or equal 1 by definition of R. Hence, the model is given by Y = {(x1,x2) | - 1 x1,x2 1}, the model is feasible, and has an infinity number of solutions: every point in a 2-dimensional Euclidean space included in the 2 × 2 unit square is a solution: so: (-0.5, 0.3), (0, 0), (0.1, 0.2), (0.4, 1), etc. are solutions. (Keeping our analogy with the choosing a house problem: We choose a house (x1) together with a garage (x2.)

Example 4:

      2        def                       2
x ∈ ℝ  , R (x) =  (x2 = x1 + 2 ∧  x2 = x1 - 1)

The state space is again 2-dimensional. x is a 2-dimensional vector variable x = (x1,x2) with the components x1 and x2 that must be real numbers, and they fulfill the two conditions defined by R. The solutions are:

       1 + √13--5 + √13--    1 - √13-- 5 - √13-
Y =  {(--------,---------) , (-------, --------)}
          2         2            2        2
                                            =  {(2.3,4.3 ) , (- 1.3,0.7)}

The model is feasible, and has exactly two solutions. To find the solutions, we need to solve a quadratic equation. Geometrically, the problem can be drawn in a 2-dimensional space (see Figure 5), the two solutions are where the line meets the parable.


Figure 5: Geometric Solution

Some mathematical models are easy to solve (the examples so far), others are very difficult. For example the last conjecture of Fermat is a difficult model:

Example 5:

x = (a,b,c,n) ∈ ℤ4 , R(x ) = (an + bn = cn  ∧ n >  2)

This model is infeasible, that is, it has no solution. However, this 3-hundred year old conjecture has been proven by the mathematician A. Wiles only at 1997, after 9 years of intensive research.

Interestingly, a generalization by Euler, namely:

x = (a,b,c,d,n ) ∈ ℤ5 , R (x) = (an +  bn + cn = dn )

has solutions. But it was not until 1960 to find one (others were found later on):

      4        4        4          4         4
95899  + 27519  +  27519  + 414560  = 422481

Even worse, it is easy to formulate models that cannot be solved because the computer takes far too much time to find a solution, for other models one can prove that they are unsolvable by any method.