Mathematics and statistics

Rape statistics

Rape statistics are counts and rates of rape drawn from police-recorded offenses and population surveys; differing definitions, such as the FBI's 2013 revision, make figures hard to compare.

Rare event sampling

Rare event sampling is a family of Monte Carlo techniques, chiefly importance sampling and splitting, that estimate very low-probability events far more efficiently than naive simulation.

Rare event simulation

Rare event simulation is a class of Monte Carlo methods, including importance sampling and splitting, that estimates very low-probability events far more efficiently than naive sampling.

Rarefaction (statistics)

Rarefaction is a statistical method that standardizes species richness by subsampling individuals or samples to a common count, introduced by Howard Sanders in 1968 for marine benthic ecology.

Rate–distortion theory

Rate–distortion theory, also called distortion-rate theory, is the branch of information theory that gives the minimum rate for representing a source when a specified distortion is tolerated.

Ratio

A ratio is a comparison of two quantities expressing their relative size, written as a to b, a:b, or a/b; a 2:1 ratio means the first is twice the second.

Ratio distribution

A ratio distribution, also called a quotient distribution, is the probability distribution of a random variable formed as the quotient of two random variables, Z = X/Y.

Ratio estimation

Ratio estimation is a survey sampling method that estimates a population total or mean by scaling the sample ratio of two means by a known auxiliary total.

Ratio estimator

The ratio estimator is a survey-sampling estimator of a population total or mean that uses a known auxiliary total, such as cultivated area, to gain precision.

Ratio test

The ratio test, also known as d'Alembert's ratio test, is a convergence test for infinite series, giving absolute convergence when the limit of consecutive-term ratios is below 1.

Rational function

A rational function, also called a fractional-rational function, is a function expressible as the ratio of two polynomials; such ratios form the field of rational functions F(x) in abstract algebra.

Rational interpolation approximates a function by a ratio of two polynomials forced through given data points, preferred over a single polynomial when the function has poles or nearly singular behavior.

Rational mapping

A rational mapping, or rational map, in algebraic geometry is a map between algebraic varieties given by rational functions, defined away from a closed subset.

Rational number

A rational number is a number that can be written as a fraction p/q of two integers, with q not zero; every integer and every terminating or repeating decimal is one.

Rational point

A rational point is a solution of an algebraic variety's defining equations whose coordinates are all rational numbers; finding them is the central problem of arithmetic geometry.

Rational reconstruction

Rational reconstruction is a computer algebra problem: recovering the rational number n/d from its residue a modulo m under size bounds, first solved by Paul S. Wang in 1981.

Rational representation

A rational representation of an algebraic group G is a linear representation on a finite-dimensional vector space given by a regular homomorphism G → GL(V).

Rational root theorem

The rational root theorem states that any rational root p/q of a polynomial with integer coefficients has p dividing the constant term and q dividing the leading coefficient.

Rational surface

A rational surface is a two-dimensional algebraic variety defined over an algebraically closed field and birationally isomorphic to the projective plane, the simplest class in the Enriques–Kodaira classification of algebraic surfaces.

Raven paradox

The raven paradox, also called Hempel's paradox, is a problem in confirmation theory posed by Carl Gustav Hempel in 1945, showing that observing a white shoe confirms that all ravens are black.

Ray class field

A ray class field is a finite abelian extension of a number field attached to a modulus, with Galois group the ray class group via the Artin map.

Rayleigh distribution

The Rayleigh distribution is a continuous probability distribution for the magnitude of a 2D vector with independent zero-mean Gaussian components, derived by Lord Rayleigh in 1880.

Reaction–diffusion system

A reaction–diffusion system is a set of coupled partial differential equations describing substances spreading by diffusion while reacting, the standard mathematical model of spontaneous pattern formation.

Real analysis

Real analysis, also called real variable analysis, is the rigorous branch of mathematics underlying calculus, built on the completeness of the real numbers and covering limits, continuity, differentiation, and integration.

Real closed field

A real closed field is an ordered field where every non-negative element is a square and every odd-degree polynomial has a root, introduced by Artin and Schreier in the 1920s.

Real number

A real number is an element of the real number system, written ℝ, which contains the rational numbers and fills the gaps between them with irrationals such as √2 and π.

Real projective plane

The real projective plane is the space of lines through the origin in three-dimensional real space, a closed non-orientable surface that cannot be embedded in ordinary space.

Real rank zero

Real rank zero is a property of C-algebras introduced by Brown and Pedersen in 1991, meaning self-adjoint elements can be approximated by ones with finite spectrum.

Realizability

Realizability is a family of semantic techniques in mathematical logic, introduced by Stephen Cole Kleene in 1945, where a number or program counts as evidence for a statement.