mathematical model

Examples of mathematical model in the following topics:

• Essential Functions for Mathematical Modeling

  • A mathematical model is a description of a system using mathematical concepts and language.
  • The process of developing a mathematical model is termed mathematical modeling.
  • Mathematical models can take many forms, including but not limited to dynamical systems, statistical models, differential equations, or game theoretic models.
  • In general, mathematical models may include logical models, as far as logic is taken as a part of mathematics.
  • Even many everyday activities carried out without a thought are uses of mathematical models.

• Models Using Differential Equations

  • Differential equations can be used to model a variety of physical systems.
  • Differential equations are very important in the mathematical modeling of physical systems.
  • In biology and economics, differential equations are used to model the behavior of complex systems.
  • Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena.
  • Give examples of systems that can be modeled with differential equations

• Four Ways to Represent a Function

  • Verbal: When modeling a process mathematically, one often first develops a verbal description of the problem.
  • Graphical: This involves modeling a function in a dimensional overlay.

• Optimization

  • Mathematical optimization is the selection of a best element (with regard to some criteria) from some set of available alternatives.
  • Mathematical optimization is the selection of a best element (with regard to some criteria) from some set of available alternatives.
  • Also, agents are often modeled as being risk-averse, thereby preferring to avoid risk.
  • Asset prices are also modeled using optimization theory, though the underlying mathematics relies on optimizing stochastic processes rather than on static optimization.
  • Operations research also uses stochastic modeling and simulation to support improved decision-making.

• Functions of Several Variables

  • Functions with independent variables corresponding to each of the degrees of freedom are often used to model these systems, and multivariable calculus provides tools for characterizing the system dynamics.
  • Multivariable calculus is used in many fields of natural and social science and engineering to model and study high-dimensional systems that exhibit deterministic behavior.
  • Non-deterministic, or stochastic, systems can be studied using a different kind of mathematics, such as stochastic calculus.

• Series

  • An example is the famous series from Zeno's dichotomy and its mathematical representation:
  • Unlike finite summations, infinite series need tools from mathematical analysis, specifically the notion of limits, to be fully understood and manipulated.
  • In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, and finance.

• Exponential Growth and Decay

  • Exponential growth occurs when the growth rate of the value of a mathematical function is proportional to the function's current value.
  • Exponential growth occurs when the growth rate of the value of a mathematical function is proportional to the function's current value.

• Logistic Equations and Population Grown

  • A logistic equation is a differential equation which can be used to model population growth.
  • In the equation, the early, unimpeded growth rate is modeled by the first term $rP$.
  • This antagonistic effect is called the bottleneck, and is modeled by the value of the parameter $K$.
  • It can be used to model population growth because of the limiting effect scarcity has on the growth rate.
  • Describe shape of the logistic function and its use for modeling population growth

• Exponential and Logarithmic Functions

  • The exponential function is widely used in physics, chemistry, engineering, mathematical biology, economics and mathematics.
  • The natural logarithm has the constant e ($\approx 2.718$) as its base; its use is widespread in pure mathematics, especially calculus.

• Summing an Infinite Series

  • Unlike finite summations, infinite series need tools from mathematical analysis, and specifically the notion of limits, to be fully understood and manipulated.
  • In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, and finance.