Mathematics and statistics

Representation theory of the Lorentz group

The representation theory of the Lorentz group studies how Lorentz transformations act linearly on vector spaces, dividing into non-unitary finite-dimensional representations for classical fields and unitary representations for particles.

Representation theory of the symmetric group

The representation theory of the symmetric group (S_n) studies how permutations act on vector spaces; its irreducible representations, the Specht modules, are indexed by integer partitions of n.

Reproducing kernel Hilbert space

A reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which evaluation at any point is continuous, a condition equivalent to a reproducing kernel existing.

Reproducing kernel method

The reproducing kernel method (RKM) is a numerical technique that solves differential and integral equations by representing the solution as a series in a reproducing kernel Hilbert space.

Reptile search algorithm

The reptile search algorithm (RSA), also called reptile search, is a gradient-free metaheuristic mimicking crocodile encircling and hunting, introduced by Laith Abualigah and colleagues in 2021.

Repunit

A repunit is a whole number whose decimal digits are all 1, such as 11 or 111, with the name coined by Albert H. Beiler in 1966.

Resampling (statistics)

Resampling in statistics is a family of methods that repeatedly draw samples from observed data to estimate a statistic's uncertainty, including the bootstrap, jackknife, cross-validation, and permutation tests.

Resampling schemes in particle filters

Resampling schemes in particle filters are randomized procedures that discard low-weight particles, duplicate high-weight ones, and reset weights to 1/N, spanning multinomial, stratified, systematic, and residual variants.

Reshetikhin–Turaev invariant

The Reshetikhin–Turaev invariant (RT-invariant) is a family of invariants of compact oriented 3-manifolds and framed links, built in 1991 from quantum group representations as a rigorous version of Witten's Chern–Simons invariants.

Residual power series method

The residual power series method (RPSM) is a semi-analytical technique that builds power series approximations to differential equations, fixing each coefficient by setting residual derivatives to zero.

Residual spectrum

The residual spectrum of a linear operator T is the set of λ for which T − λI is injective but its range is not dense.

Residual sum of squares

The residual sum of squares (RSS), also called the sum of squared residuals, totals the squared gaps between observed data and a regression model's predictions; ordinary least squares minimizes it.

Residual time

Residual time, or forward recurrence time, is the time from a fixed instant in a renewal process until the next renewal, used in reliability and queueing.

Residuated lattice

A residuated lattice is an algebraic structure combining a lattice with a monoid whose multiplication admits residuals, introduced by Ward and Dilworth in the late 1930s.

Residue (complex analysis)

The complex residue of a function at an isolated singularity is the coefficient of the (z − z₀)⁻¹ term in its Laurent series, the key quantity for contour integration.

Residue number system

A residue number system (RNS) is a non-positional way of representing integers as remainders with respect to coprime moduli, so addition, subtraction, and multiplication run in parallel without carries.

Residue theorem

The residue theorem (Cauchy's residue theorem) is a result of complex analysis stating that the integral of a function around a closed contour equals 2πi times the sum of its residues at enclosed singularities.

Resolution of singularities

Resolution of singularities replaces the singular points of an algebraic variety by a proper birational map from a nonsingular variety, built from blow-ups; Hironaka proved existence in characteristic zero in 1964.

Resolvable design

A resolvable design is a block design whose blocks partition into parallel classes, each dividing the point set exactly once, introduced by Raj Chandra Bose in 1942.

Resolvent (Galois theory)

In Galois theory, a resolvent (Galois resolvent) is an auxiliary equation whose roots let one find the roots of a given equation, an idea Lagrange introduced in 1770.

Resolvent operator

The resolvent operator of a linear operator A is the bounded inverse R(λ, A) = (λI − A)⁻¹, defined for λ where the inverse exists; it generalizes matrix invertibility in spectral theory.

Response surface design

A response surface design is a design of experiments method that fits a quadratic model to input variables and responses, so a process can be optimized.

Response surface methodology

Response surface methodology (RSM), also called the Box-Wilson method, is a collection of statistical techniques for designing experiments and optimizing process responses, introduced by Box and Wilson in 1951.

Response-adaptive randomization

Response-adaptive randomization is a clinical trial design in which the probabilities for assigning each new patient change as outcome data accumulate, shifting allocation toward treatments performing better.

Restricted isometry property

The restricted isometry property (RIP) is a condition on a matrix stating that it approximately preserves the length of every sparse vector, proposed by Candès and Tao for compressed sensing.

Restricted maximum likelihood

Restricted maximum likelihood (REML) is a statistical method that estimates variance components in linear mixed models while avoiding the downward bias of ordinary maximum likelihood.

Restricted three-body problem

The restricted three-body problem studies a mass so small it exerts no gravitational force, moving under the attraction of two bodies whose motion is unaffected by it.

Restriction theorem (Fourier analysis)

A restriction theorem in Fourier analysis bounds the restriction of a function's Fourier transform to a curved hypersurface, beginning with Elias Stein's work in the 1960s.

Resultant

The resultant, also called the polynomial resultant or eliminant, is an expression in the coefficients of two polynomials that vanishes exactly when they share a root.