In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space V with inner product is a linear map A that is its own adjoint. If V is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of A is a Hermitian matrix, i.e., equal to its conjugate transpose A∗. By the finite-dimensional spectral theorem, V has an orthonormal basis such that the matrix of A relative to this basis is a diagonal matrix with entries in the real numbers. In this article, we consider generalizations of this concept to operators on Hilbert spaces of arbitrary dimension.
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi polynomials.
In mathematics, the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues. More generally, the spectral radius of a bounded linear operator is the supremum of the absolute values of the elements of its spectrum. The spectral radius is often denoted by ρ(·).
The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. The populations change through time according to the pair of equations:
Population dynamics is the type of mathematics used to model and study the size and age composition of populations as dynamical systems.
In the theory of grand unification of particle physics, and, in particular, in theories of neutrino masses and neutrino oscillation, the seesaw mechanism is a generic model used to understand the relative sizes of observed neutrino masses, of the order of eV, compared to those of quarks and charged leptons, which are millions of times heavier. The name of the seesaw mechanism was given by Tsutomu Yanagida in a Tokyo conference in 1981.
The effective population size (Ne) is the number of individuals that an idealised population would need to have in order for some specified quantity of interest to be the same as in the real population. Idealised populations are based on unrealistic but convenient simplifications such as random mating, simultaneous birth of each new generation, constant population size, and equal numbers of children per parent. In some simple scenarios, the effective population size is the number of breeding individuals in the population. However, for most quantities of interest and most real populations, the census population size N of a real population is usually larger than the effective population size Ne. The same population may have multiple effective population sizes, for different properties of interest, including for different genetic loci.
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.
In probability theory the hypoexponential distribution or the generalized Erlang distribution is a continuous distribution, that has found use in the same fields as the Erlang distribution, such as queueing theory, teletraffic engineering and more generally in stochastic processes. It is called the hypoexponetial distribution as it has a coefficient of variation less than one, compared to the hyper-exponential distribution which has coefficient of variation greater than one and the exponential distribution which has coefficient of variation of one.
In queueing theory, a discipline within the mathematical theory of probability, a Jackson network is a class of queueing network where the equilibrium distribution is particularly simple to compute as the network has a product-form solution. It was the first significant development in the theory of networks of queues, and generalising and applying the ideas of the theorem to search for similar product-form solutions in other networks has been the subject of much research, including ideas used in the development of the Internet. The networks were first identified by James R. Jackson and his paper was re-printed in the journal Management Science’s ‘Ten Most Influential Titles of Management Sciences First Fifty Years.’
Covariance matrix adaptation evolution strategy (CMA-ES) is a particular kind of strategy for numerical optimization. Evolution strategies (ES) are stochastic, derivative-free methods for numerical optimization of non-linear or non-convex continuous optimization problems. They belong to the class of evolutionary algorithms and evolutionary computation. An evolutionary algorithm is broadly based on the principle of biological evolution, namely the repeated interplay of variation and selection: in each generation (iteration) new individuals are generated by variation, usually in a stochastic way, of the current parental individuals. Then, some individuals are selected to become the parents in the next generation based on their fitness or objective function value . Like this, over the generation sequence, individuals with better and better -values are generated.
In population biology and demography, generation time is the average time between two consecutive generations in the lineages of a population. In human populations, generation time typically ranges from 22 to 33 years. Historians sometimes use this to date events, by converting generations into years to obtain rough estimates of time.
In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and the class of compound Poisson–gamma distributions which have positive mass at zero, but are otherwise continuous. Tweedie distributions are a special case of exponential dispersion models and are often used as distributions for generalized linear models.
In probability theory and statistics, the Conway–Maxwell–Poisson distribution is a discrete probability distribution named after Richard W. Conway, William L. Maxwell, and Siméon Denis Poisson that generalizes the Poisson distribution by adding a parameter to model overdispersion and underdispersion. It is a member of the exponential family, has the Poisson distribution and geometric distribution as special cases and the Bernoulli distribution as a limiting case.
In statistics and machine learning, lasso is a regression analysis method that performs both variable selection and regularization in order to enhance the prediction accuracy and interpretability of the resulting statistical model. It was originally introduced in geophysics, and later by Robert Tibshirani, who coined the term.
In the study of age-structured population growth, probably one of the most important equations is the Euler–Lotka equation. Based on the age demographic of females in the population and female births, this equation allows for an estimation of how a population is growing.
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. It is named after French mathematician Siméon Denis Poisson. The Poisson distribution can also be used for the number of events in other specified interval types such as distance, area or volume.
In the mathematical theory of random matrices, the Marchenko–Pastur distribution, or Marchenko–Pastur law, describes the asymptotic behavior of singular values of large rectangular random matrices. The theorem is named after Ukrainian mathematicians Vladimir Marchenko and Leonid Pastur who proved this result in 1967.
The Fokas method, or unified transform, is an algorithmic procedure for analysing boundary value problems for linear partial differential equations and for an important class of nonlinear PDEs belonging to the so-called integrable systems. It is named after Greek mathematician Athanassios S. Fokas.
