mathematical model

Calculus

(noun)

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

Related Terms

Algebra

(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:

  • 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.

  • 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:

  • Quantitative vs. Qualitative Research

    • Both quantitative and qualitative models seek to explain patterns in behavior, but the former is mathematical and the latter is more descriptive.
    • Quantitative Research is defined as the systematic empirical investigation of social phenomena via statistical, mathematical or computational techniques.
    • Its objective is to develop and employ mathematical models, theories and/or hypotheses pertaining to phenomena.
    • Qualitative Research is the examination, analysis and interpretation of observations for the purpose of discovering underlying meanings and patterns of relationships, including classifications of types of phenomena and entities, in a manner that does not involve 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

  • The Thomson Model

    • Thomson, who discovered the electron in 1897, proposed the plum pudding model of the atom in 1904 before the discovery of the atomic nucleus in order to include the electron in the atomic model.
    • In this model the atom was also sometimes described to have a "cloud" of positive charge.
    • His conclusions led him to propose the Rutherford model of the atom.
    • A schematic presentation of the plum pudding model of the atom; in Thomson's mathematical model the "corpuscles" (in modern language, electrons) were arranged non-randomly, in rotating rings.
    • Intro to the History of Atomic Theory - The Thomson Model

  • Blending Content with Pedagogy

    • Mathematics is social, students should be given the opportunity to construct, gather, discover, and create collectively their own mathematics.
    • Promote similar and different ideas about the same mathematical concepts.
    • Intuition is the starting place for modeling mathematics, try to privilege it not being the only place that students rely upon in problem solving situations.
    • This should help you to enable students in actively doing real mathematics.
    • Recursively elaborate mathematics.

  • Modeling Ecosystem Dynamics

    • Conceptual models describe ecosystem structure, while analytical and simulation models use algorithms to predict ecosystem dynamics.
    • Analytical models generally work best when dealing with relatively-simple, linear systems; specifically, those that can be accurately described by a set of mathematical equations whose behavior is well known.
    • They are mathematically complex models that are good at predicting components of ecosystems such as food chains.
    • Like analytical models, simulation models use complex algorithms to predict ecosystem dynamics.
    • They are generally considered more ecologically realistic, while analytic models are valued for their mathematical elegance and explanatory power.

  • Economic Models

    • Most models use mathematical techniques in order to investigate, theorize, and fit theories into economic situations.
    • Economists use models in order to study and portray situations.
    • However, creating a model does have two basic steps: 1) generate the model, and 2) checking the model for accuracy - also known as diagnostics.
    • In most cases, economic models use mathematical or quantitative analysis.
    • Some economic models also use qualitative analysis.

  • Studying Ecosystem Dynamics

    • Many different models are used to study ecosystem dynamics, including holistic, experimental, conceptual, analytical, and simulation models.
    • Three basic types of ecosystem modeling are routinely used in research and ecosystem management: conceptual models, analytical models, and simulation models.
    • Analytical and simulation models are mathematical methods of describing ecosystems that are capable of predicting the effects of potential environmental changes without direct experimentation, although with limitations in accuracy.
    • An analytical model is created using simple mathematical formulas to predict the effects of environmental disturbances on ecosystem structure and dynamics.
    • A simulation model is created using complex computer algorithms to holistically model ecosystems and to predict the effects of environmental disturbances on ecosystem structure and dynamics.

  • 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.