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Watt's curve, which arose in the context of early work on the steam engine, is a sextic in two variables.. One method of solving the cubic equation involves transforming variables to obtain a sextic equation having terms only of degrees 6, 3, and 0, which can be solved as a quadratic equation in the cube of the variable.
Let be the -valued Itô diffusion solving the stochastic differential equation = + (). The infinitesimal generator of is defined by its action on compactly-supported (twice differentiable with continuous second derivative) functions : as [2]
The polynomial () (+) is a cubic polynomial: after multiplying out and collecting terms of the same degree, it becomes + +, with highest exponent 3.. The polynomial (+ +) + (+ + +) is a quintic polynomial: upon combining like terms, the two terms of degree 8 cancel, leaving + + + +, with highest exponent 5.
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The generator is used in evolution equations such as the Kolmogorov backward equation, which describes the evolution of statistics of the process; its L 2 Hermitian adjoint is used in evolution equations such as the Fokker–Planck equation, also known as Kolmogorov forward equation, which describes the evolution of the probability density ...
In linear algebra, a Moore matrix, introduced by E. H. Moore (), is a matrix defined over a finite field.When it is a square matrix its determinant is called a Moore determinant (this is unrelated to the Moore determinant of a quaternionic Hermitian matrix).
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The generator is used in the formulation of Kolmogorov's backward equation. Intuitively, this equation tells us how the expected value of any suitably smooth statistic of X evolves in time: it must solve a certain partial differential equation in which time t and the initial position x are the independent variables.