All Topics  
Ansatz

 

   Email Print
   Bookmark   Link






 

Ansatz



 
 
Ansatz (Ger.
German language

German is a West Germanic languages, thus related to and classified alongside English language and Dutch language. It is one of the world's world language and the most widely spoken mother tongue in the European Union....
, "onset"; today, "setup, starting point"; plural: ansätze) is a German noun with several meanings in the English language. The fact that the word Ansatz is found in the English language today suggests that it has been carried by those who have used it frequently,, such as mathematicians and physicists. There was considerable exchange between the English- and German-speaking worlds in the first half of the twentieth century.






Discussion
Ask a question about 'Ansatz'
Start a new discussion about 'Ansatz'
Answer questions from other users
Full Discussion Forum



Encyclopedia


Ansatz (Ger.
German language

German is a West Germanic languages, thus related to and classified alongside English language and Dutch language. It is one of the world's world language and the most widely spoken mother tongue in the European Union....
, "onset"; today, "setup, starting point"; plural: ansätze) is a German noun with several meanings in the English language. The fact that the word Ansatz is found in the English language today suggests that it has been carried by those who have used it frequently,, such as mathematicians and physicists. There was considerable exchange between the English- and German-speaking worlds in the first half of the twentieth century. Just before, and immediately after the Second World War, mathematicians and physicists of German-language background emigrated to the United States and took up positions in industry and academia. The term "Ansatz" is used in describing solution methods for differential equations.

Definition

More formally, in physics and mathematics, an ansatz is an educated guess that is verified later by its results. An ansatz is the establishment of the starting equation(s) describing a mathematical or physical problem. It can take into consideration boundary conditions. After an ansatz has been established, the equations are solved for the general function of interest. Typically, a word problem
Word problem (mathematics education)

In mathematics education, the term word problem is often used to refer to any mathematical exercise where significant background information on the problem is presented as text rather than in mathematical notation....
 starts by writing down the ansatz with subsequent refinement leading to the solution of the problem.

Examples

Given a set
Set

A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics....
 of experimental data that looks to be cluster
Cluster

A cluster is a small group or bunch of something.In science:* Cluster , a small group of atoms or molecules*In astrophysics:** Star cluster, groups of stars which are gravitationally bound...
ed about a line, a linear ansatz could be made to find the parameter
Parameter

In mathematics, statistics, and the mathematical sciences, a parameter is a quantity that defines certain characteristics of systems or function s....
s of the line by a least squares
Least squares

The method of least squares or ordinary least squares is used to solve overdetermined systems. Least squares is often applied in statistical contexts, particularly regression analysis....
 curve fit. Variational approximation
Approximation

An approximation is an Accuracy and precision representation of something that is still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as Function , shapes, and physical laws....
 methods use ansätze and then fit the parameters.

Another example could be the mass, energy, and entropy balance equations that, considered simultaneous for purposes of the elementary operations of linear algebra, are the ansatz to most basic problems of thermodynamics.

Another example of ansatz is to suppose the solutions of a homogeneous linear differential equation
Differential equation

A differential equation is a mathematics equation for an unknown function of one or several variable that relates the values of the function itself and its derivatives of various orders....
 and difference equation to have, respectively, exponential and power form. More generally one can guess a particular solution of a system of equations and test such an ansatz by direct substitution of the solution in the system of equations.

See also

  • Bethe ansatz
    Bethe ansatz

    In physics, the Bethe Ansatz is a method for finding the exact solutions of certain one-dimensional quantum many-body models. It was invented by Hans Bethe in 1931 to find the exact eigenvalues and eigenvectors of the one-dimensional antiferromagnetic Heisenberg model Hamiltonian....
  • Ersatz
    Ersatz

    Ersatz is a German language word literally meaning substitute or replacement. Although it is used as an adjective in English language, Ersatz can function in German as a noun on its own, or as a part in compound nouns such as Ersatzteile or Ersatzspieler ....
  • Guesstimate
    Guesstimate

    Guesstimate is a portmanteau of the words guess and estimate, first used by American statisticians in 1934 or 1935. It is defined as an estimate made without adequate or complete information, or, more strongly, as an estimate arrived at by guesswork or conjecture....