In mathematics, a stochastic matrix (also termed probability matrix, transition matrix, substitution matrix, or Markov matrix ) is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability. It has found use throughout a wide variety of scientific fields, including probability theory, statistics, mathematical finance and linear algebra, as well as computer science and population genetics. The stochastic matrix was first developed by Andrey Markov at the beginning of the 20th century. There are several different...

In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all elements are random variables. Many important properties of physical systems can be represented mathematically as matrix problems. For example, the thermal conductivity of a lattice can be computed from the dynamical matrix of the particle-particle interactions within the lattice. == Applications == === Physics === In nuclear physics, random matrices were introduced by Eugene Wigner to model the nuclei of heavy atoms. He postulated that the spacings between the...

In mathematics, a K3 surface is a complex or algebraic smooth minimal complete surface that is regular and has trivial canonical bundle. In the Enriques–Kodaira classification of surfaces they form one of the 4 classes of surfaces of Kodaira dimension 0. Together with two-dimensional complex tori, they are the Calabi–Yau manifolds of dimension two. Most complex K3 surfaces are not algebraic. This means that they cannot be embedded in any projective space as a surface defined by polynomial equations. == Definition == There are many equivalent properties that can be used to characterize a K3 surface...