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A Markov chain with two states, A and E. In probability, a discrete-time Markov chain (DTMC) is a sequence of random variables, known as a stochastic process, in which the value of the next variable depends only on the value of the current variable, and not any variables in the past.
In this case, the variable x of the logistic map is the number of individuals of an organism divided by the maximum population size, so the possible values of x are limited to 0 ≤ x ≤ 1. For this reason, the behavior of the logistic map is often discussed by limiting the range of the variable to the interval [0, 1]. [Hirsch,Smale & Devaney 1]
The Ising model (or Lenz–Ising model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics.The model consists of discrete variables that represent magnetic dipole moments of atomic "spins" that can be in one of two states (+1 or −1).
Define a discrete-time Markov chain Y n to describe the nth jump of the process and variables S 1, S 2, S 3, ... to describe holding times in each of the states where S i follows the exponential distribution with rate parameter −q Y i Y i.
A typical finite-dimensional mixture model is a hierarchical model consisting of the following components: . N random variables that are observed, each distributed according to a mixture of K components, with the components belonging to the same parametric family of distributions (e.g., all normal, all Zipfian, etc.) but with different parameters
The internal state variables are the smallest possible subset of system variables that can represent the entire state of the system at any given time. [13] The minimum number of state variables required to represent a given system, , is usually equal to the order of the system's defining differential equation, but not necessarily.
The difference in these decay behaviors, where correlations between microscopic random variables become zero versus non-zero at large distances, is one way of defining short- versus long-range order. In statistical mechanics, the correlation function is a measure of the order in a system, as characterized by a mathematical correlation function ...
This explains the duality in Darcy's law as a governing equation and a defining equation for absolute permeability. The non-linearity of the material derivative in balance equations in general, and the complexities of Cauchy's momentum equation and Navier-Stokes equation makes the basic equations in classical mechanics exposed to establishing ...