Machine Improvisation builds upon a long musical tradition of statistical modeling that began with Hiller and Isaacson's Illiac Suite for String Quartet ( 1957 ) and Xenakis ' uses of Markov chains and stochastic processes.

Ordinary differential equations appear in the movement of heavenly bodies ( planets, stars and galaxies ); optimization occurs in portfolio management ; numerical linear algebra is important for data analysis ; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.

Andrey Markov introduced the notion of Markov chains ( 1906 ), which played an important role in stochastic processes theory and its applications.

Pseudorandom number generators are widely used in such applications as computer modeling ( e. g., Markov chains ), statistics, experimental design, etc.

There have also been other algorithms based on Markov chains.

Markov chains have many applications as statistical models of real-world processes.

Many other examples of Markov chains exist.

Markov chains are often described by a directed graph, where the edges are labeled by the probabilities of going from one state to the other states.

* Time-homogeneous Markov chains ( or stationary Markov chains ) are processes where

* Examples of Markov chains

Specific examples of mathematics, statistics, and physics applied to music composition are the use of the statistical mechanics of gases in Pithoprakta, statistical distribution of points on a plane in Diamorphoses, minimal constraints in Achorripsis, the normal distribution in ST / 10 and Atrées, Markov chains in Analogiques, game theory in Duel and Stratégie, group theory in Nomos Alpha ( for Siegfried Palm ), set theory in Herma and Eonta, and Brownian motion in N ' Shima.

This page contains examples of Markov chains in action.

Often, random walks are assumed to be Markov chains or Markov processes, but other, more complicated walks are also of interest.

Markov chains form a common context for applications in probability theory.

Specific examples of mathematics, statistics, and physics applied to music composition are the use of the statistical mechanics of gases in Pithoprakta, statistical distribution of points on a plane in Diamorphoses, minimal constraints in Achorripsis, the normal distribution in ST / 10 and Atrées, Markov chains in Analogiques, game theory in Duel and Stratégie, group theory in Nomos Alpha ( for Siegfried Palm ), set theory in Herma and Eonta, and Brownian motion in N ' Shima.

* Elementary probability theory and Markov chains

** Examples of Markov chains

* Nearly completely decomposable, a property of some Markov chains

Prominent examples of stochastic algorithms are Markov chains and various uses of Gaussian distributions.