MATHEMATICAL MODELLING
Through life, the behavior of different types of phenomena, common life situations and problems associated with how things have been studied and generalized, noting in each of them, some similarity in their development and / or resolution, as always or almost always have a systematic behavior, this behavior is usually simulated through a mathematical approach or model that takes into account most of the variables that occur in the problem or phenomenon. These approaches mathematical models are developed from observations, empiricism and theoretical analysis.
Now let´s see, what is, and how works a mathematical model.
A mathematical model can be broadly defined as a formulation or equation that expresses the essential features of a physical system or process in mathematical terms. In a very general sense, it can be represented as a functional relationship of the form:
Where the dependent variable is a characteristic that usually reflects the behavior or state of the system; the independent variable are usually dimensions, such as time and space, along which the system`s behavior is begin determined; the parameters are reflective of the system`s properties or composition; and the forcing functions are external influences acting upon the system.
The modeling diagram
The nodes of the following diagram represent information to be collected, sorted, evaluated, and organized.
The edges of the diagram represent activities of two-way communication between the nodes and the corresponding sources of information.
A model can represent it using:
- Relations
- Functions
- Relations
A relationship is the allocation of items in a set A with objects of another set B
Ordered pairs
F={ (1, 2), (3, 2), (5,4), (7,4), (7,10), (11,12), (13,14) y (15,16) }
Matching Rule
C= {(x , y) | "y" es el doble de "x" y "x" å R}* o y = 2x
* It reads "the set C consists of the pairs (x, y) where y is twice the value" x "and also" x "is an element of the set of real numbers.
Charts
The ordered pairs are points on the plane, the coordinate x represents the domain components and elements of contradominio are represented by the y coordinate.
2. Features
A function is a relationship that meets a property: for any object in set A, can be assigned only one object of the set B.
Call, domain set A and contradominio set B.
Ordered Pairs
Matching rule
Charts
Equations
Ordered Pairs
An ordered pair (A, B) represents an element of a function, which belongs to domain A and B at contradominio.
There are times when a couple can find orderly indicated on the form (x, f (x)), this is to note that the second variable f (x) depends directly on the variable x.
F={ (1, 2), (3, 4), (5,6), (7,8), (9,10), (11,12), (13,14) y (15,16) }
Matching Rule
A matching rule tells us the yardstick by which couples choose the domain and contradominio elements.
This criterion can be given as an extension (indicating the conditions which must meet the elements) or can be given through an equation.
C = ((x, y) | y is twice the "x" and "x" å R) * y = 2x
* It reads "the set C consists of the pairs (x, y) where y is twice the value" x "and also" x "is an element of the set of real numbers.
Charts
A lot of functions with domain and contradominio to all real numbers, so it can be represented in the Cartesian plane.
The ordered pairs are points on the plane, the coordinate x represents the domain components and elements of contradominio are represented by the y coordinate.
Note that, as our graph represents a function to each element x corresponds to a single element of y.
Some applications of mathematical models
Biology
-Protein folding
-Humane genome project
-Population dynamics
-Morphogenesis
-Evolutionary pedigrees
-Spreading of infectious diseases (AIDS)
-Animal and plant breeding (genetic variability)
Chemical engineering
-Chemical equilibrium
-Planning of production units
Chemistry
-Chemical reaction dynamics
-Molecular modeling
-Electronic structure calculations
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