Markov chain in the context of Lyapunov function

An HI region or H I region (read H one) is a cloud in the interstellar medium composed of neutral atomic hydrogen (HI), in addition to the local abundance of helium and other elements. (H is the chemical symbol for hydrogen, and "I" is the Roman numeral. It is customary in astronomy to use the Roman numeral I for neutral atoms, II for singly-ionized—HII is H in other sciences—III for doubly-ionized, e.g. OIII is O, etc.) These regions do not emit detectable visible light (except in spectral lines from elements other than hydrogen) but are observed by the 21-cm (1,420 MHz) region spectral line. This line has a very low transition probability, so it requires large amounts of hydrogen gas for it to be seen. At ionization fronts, where HI regions collide with expanding ionized gas (such as an H II region), the latter glows brighter than it otherwise would. The degree of ionization in an HI region is very small at around 10 (i.e. one particle in 10,000). At typical interstellar pressures in galaxies like the Milky Way, HI regions are most stable at temperatures of either below 100 K or above several thousand K; gas between these temperatures heats or cools very quickly to reach one of the stable temperature regimes. Within one of these phases, the gas is usually considered isothermal, except near an expanding H II region. Near an expanding H II region is a dense HI region, separated from the undisturbed HI region by a shock front and from the H II region by an ionization front.

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt to find approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences like economics, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in medicine and biology.

Before modern computers, numerical methods often relied on hand interpolation formulas, using data from large printed tables. Since the mid-20th century, computers calculate the required functions instead, but many of the same formulas continue to be used in software algorithms.

In statistics, Markov chain Monte Carlo (MCMC) is a class of algorithms used to draw samples from a probability distribution. Given a probability distribution, one can construct a Markov chain whose elements' distribution approximates it – that is, the Markov chain's equilibrium distribution matches the target distribution. The more steps that are included, the more closely the distribution of the sample matches the actual desired distribution.

Markov chain Monte Carlo methods are used to study probability distributions that are too complex or too high dimensional to study with analytic techniques alone. Various algorithms exist for constructing such Markov chains, including the Metropolis–Hastings algorithm.

Numerical analysis is the study of algorithms for the problems of continuous mathematics. These algorithms involve real or complex variables (in contrast to discrete mathematics), and typically use numerical approximation in addition to symbolic manipulation.

Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences like economics, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in medicine and biology.