Numerical analysisNumerical 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 at finding 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, medicine, business and even the arts.
Asymptotic safety in quantum gravityAsymptotic safety (sometimes also referred to as nonperturbative renormalizability) is a concept in quantum field theory which aims at finding a consistent and predictive quantum theory of the gravitational field. Its key ingredient is a nontrivial fixed point of the theory's renormalization group flow which controls the behavior of the coupling constants in the ultraviolet (UV) regime and renders physical quantities safe from divergences.
Permutation matrixIn mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. Each such matrix, say P, represents a permutation of m elements and, when used to multiply another matrix, say A, results in permuting the rows (when pre-multiplying, to form PA) or columns (when post-multiplying, to form AP) of the matrix A.
Mlpackmlpack is a machine learning software library for C++, built on top of the Armadillo library and the ensmallen numerical optimization library. mlpack has an emphasis on scalability, speed, and ease-of-use. Its aim is to make machine learning possible for novice users by means of a simple, consistent API, while simultaneously exploiting C++ language features to provide maximum performance and maximum flexibility for expert users. Its intended target users are scientists and engineers.
Topological conjugacyIn mathematics, two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy, and related-but-distinct of flows, are important in the study of iterated functions and more generally dynamical systems, since, if the dynamics of one iterative function can be determined, then that for a topologically conjugate function follows trivially. To illustrate this directly: suppose that and are iterated functions, and there exists a homeomorphism such that so that and are topologically conjugate.
Pivotal quantityIn statistics, a pivotal quantity or pivot is a function of observations and unobservable parameters such that the function's probability distribution does not depend on the unknown parameters (including nuisance parameters). A pivot quantity need not be a statistic—the function and its value can depend on the parameters of the model, but its distribution must not. If it is a statistic, then it is known as an ancillary statistic. More formally, let be a random sample from a distribution that depends on a parameter (or vector of parameters) .
Conservative systemIn mathematics, a conservative system is a dynamical system which stands in contrast to a dissipative system. Roughly speaking, such systems have no friction or other mechanism to dissipate the dynamics, and thus, their phase space does not shrink over time. Precisely speaking, they are those dynamical systems that have a null wandering set: under time evolution, no portion of the phase space ever "wanders away", never to be returned to or revisited. Alternately, conservative systems are those to which the Poincaré recurrence theorem applies.
T-statisticIn statistics, the t-statistic is the ratio of the departure of the estimated value of a parameter from its hypothesized value to its standard error. It is used in hypothesis testing via Student's t-test. The t-statistic is used in a t-test to determine whether to support or reject the null hypothesis. It is very similar to the z-score but with the difference that t-statistic is used when the sample size is small or the population standard deviation is unknown.
Cauchy's integral formulaIn mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration": complex differentiation, like integration, behaves well under uniform limits – a result that does not hold in real analysis.
Quadratic classifierIn statistics, a quadratic classifier is a statistical classifier that uses a quadratic decision surface to separate measurements of two or more classes of objects or events. It is a more general version of the linear classifier. Statistical classification considers a set of vectors of observations x of an object or event, each of which has a known type y. This set is referred to as the training set. The problem is then to determine, for a given new observation vector, what the best class should be.
