mathematical model

(noun)

An abstract mathematical representation of a process, device, or concept; it uses a number of variables to represent inputs, outputs, internal states, and sets of equations and inequalities to describe their interaction.

Related Terms

Examples of mathematical model in the following topics:

  • Linear Mathematical Models

    • A mathematical model is a description of a system using mathematical concepts and language.
    • Mathematical models are used not only in the natural sciences and  engineering disciplines, but also in the social sciences. 
    • Many everyday activities require the use of mathematical models, perhaps unconsciously.
    • One difficulty with mathematical models lies in translating the real world application into an accurate mathematical representation.
    • Here is the distance versus time graphic model of the two trains:

  • Applications and Mathematical Models

    • Systems of linear equations are common in science and mathematics, including Physics, Chemistry and maximization/minimization and constraint problems.

  • Population Growth

    • Population can fluctuate positively or negatively and can be modeled using an exponential function.
    • Population growth can be modeled by an exponential equation.
    • If the current rates of births and deaths hold, the world population growth can be modeled using an exponential function.
    • The graph below shows an exponential model for the growth of the world population.
    • The graph has the general shape of an exponential curve though it is not exact as is the case usually when we deal with real data as opposed to purely mathematical constructs.

  • Sequences of Mathematical Statements

    • Sequences of statements are logical, ordered groups of statements that are important for mathematical induction.
    • In mathematics, a sequence is an ordered list of objects, or elements.
    • In mathematics, a "sequence of statements" refers to the progression of logical implications of one statement.
    • Sequences of statements are necessary for mathematical induction.
    • Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true for all natural numbers.

  • Linear Equations and Their Applications

    • There is in fact a field of mathematics known as linear algebra, in which linear equations in up to an infinite number of variables are studied.
    • Using similar models we can solve equations pertaining to distance, speed, and time (Distance=Speed*Time); density (Density=Mass/Volume); and any other relationship in which all variables are first order.

  • Applications of the Parabola

    • This is the exact mathematical relationship we know as a parabola.
    • As in all cases in the physical world, using the equation of a parabola to model a projectile's trajectory is an approximation.

  • Exponential Decay

    • Exponential rate of change can be modeled algebraically by the following formula:

  • Fitting a Curve

    • Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints.
    • Linear regression is an approach to modeling the linear relationship between a dependent variable, $y$ and an independent variable, $x$.
    • The simplest and perhaps most common linear regression model is the ordinary least squares approximation.
    • Indeed, trying to fit linear models to data that is quadratic, cubic, or anything non-linear, or data with many outliers or errors can result in bad approximations.
    • Model a set of data points as a line using the least squares approximation

  • Introduction to Variables

    • Variables are used in mathematics to denote arbitrary or unknown numbers.
    • In elementary mathematics, a variable is an alphabetic character representing a number, called the value of the variable, that is arbitrary, not fully specified, or unknown.
    • Variables may describe mathematical relationships between quantities that vary.
    • Variables may describe some mathematical properties.
    • It is common that many variables appear in the same mathematical formula, and they may play different roles.

  • Radical Functions

    • These assumptions can be used to build mental models for topics that would otherwise be impossible to understand.
    • However, they are not restricted to roots, and may also appear in other mathematical constants (e.g. π, e, φ, etc.).