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Riemann sum
The Riemann sum, also called the rectangle method, approximates a function's integral by summing signed rectangle areas over subintervals; introduced by Bernhard Riemann in 1854, its limit defines the Riemann integral.
Riemann surface
A Riemann surface is a connected one-dimensional complex manifold, a surface locally like the complex plane with holomorphic coordinate changes, introduced by Bernhard Riemann in his 1851 Göttingen doctoral thesis to make multi-valued analytic functions single-valued.
Riemann theta function
The Riemann theta function is a first-order theta function associated with the Jacobi variety of an algebraic curve or compact Riemann surface, introduced by Bernhard Riemann in 1857.
Riemann xi function
The Riemann xi function ξ(s) is the completed, symmetrized form of the Riemann zeta function introduced by Bernhard Riemann in 1859, whose zeros are exactly the nontrivial zeros of the zeta function.
Riemann zeta function
The Riemann zeta function, also called the Riemann ζ-function or Euler–Riemann zeta function, is a function of a complex variable linked to prime numbers through Euler's product formula.
Riemann–Hilbert correspondence
The Riemann–Hilbert correspondence is a 1984 theorem of Masaki Kashiwara and Zogman Mebkhout stating that regular holonomic D-modules on a complex manifold are equivalent to constructible sheaves.
Riemann–Liouville fractional derivative
The Riemann–Liouville fractional derivative is an operator extending differentiation to non-integer orders, formulated by Liouville in 1832 and Riemann in 1876 and used in anomalous diffusion and fractional-order control.
Riemann–Liouville fractional integral
The Riemann–Liouville fractional integral is an operator generalizing repeated integration to non-integer order via a weighted integral, the basic building block of fractional calculus and its Riemann–Liouville and Caputo derivatives.
Riemann–Roch theorem
The Riemann–Roch theorem is a result in algebraic geometry computing the dimension of spaces of functions on algebraic curves, proved by Riemann and Roch in the 1850s and 1860s.
Riemannian geometry
Riemannian geometry, named for Bernhard Riemann, is the branch of differential geometry studying smooth manifolds with a metric defining length, distance, angles, and curvature. It arose from Riemann's 1854 Göttingen lecture.
Riemannian manifold
A Riemannian manifold is a smooth manifold equipped with a smoothly varying inner product on each tangent space, allowing lengths, angles, distances, and curvature to be defined.
Riemannian metric
A Riemannian metric is a smooth assignment of an inner product to each tangent space of a manifold, making lengths, angles, and distances meaningful on curved spaces.
Riemannian optimization
Riemannian optimization minimizes or maximizes an objective function over a smooth Riemannian manifold, using Riemannian gradients, retractions, and vector transports; typical spaces include Stiefel, Grassmann, and fixed-rank matrices.
Riesz representation theorem
The Riesz representation theorem, also called the Fréchet–Riesz theorem, is a functional analysis result stating that every continuous linear functional on a Hilbert space is an inner product with a unique vector.
Riesz space
A Riesz space, also called a vector lattice, is a real vector space with a partial order compatible with the vector operations in which every pair of elements has a supremum and infimum.
Riesz transform
The Riesz transform is a singular integral operator of harmonic analysis, defined on Euclidean space as a first derivative divided by the square root of the Laplacian.
Right angle
A right angle, also called a ninety-degree angle, is an angle of 90 degrees, a quarter of a full rotation; lines meeting at one are perpendicular.
Right triangle
A right triangle, also called a right-angled triangle, is a triangle with one 90-degree angle; its sides obey the Pythagorean theorem, and its ratios define sine, cosine, and tangent.
Rigid analytic space
A rigid analytic space, the subject of rigid analytic geometry, is an analytic space over a non-Archimedean field, introduced by John Tate in the early 1960s.
Rigid designator
A rigid designator is an expression, such as a proper name, that names the same object in every possible world, a concept introduced by Saul Kripke.
Rigidity matroid
The rigidity matroid of a graph encodes which edge sets are rigid, defined by linear independence in the rigidity matrix; for generic frameworks it depends only on the graph.
Rigour
Rigour, or rigor, in mathematics is the virtue of a proof that establishes its conclusion through acceptable deductive steps, with no reliance on intuition or unchecked assumptions.
Rijndael S-box
The Rijndael S-box, also known as the AES S-box, is the substitution table used in the Advanced Encryption Standard, the block cipher NIST selected in 2000.
Ring (mathematics)
In mathematics, a ring is a set with addition and multiplication forming an Abelian group under addition and linked by distributive laws; the integers are a familiar example.
Ring homomorphism
A ring homomorphism, also called a ring morphism, is a function between rings that preserves addition and multiplication, with kernel and image governed by the isomorphism theorems.
Ring of integers
The ring of integers of a number field K is the subring of algebraic integers in K, generalizing the ordinary integers and central to algebraic number theory.
Ring theory
Ring theory is the branch of algebra that studies rings, sets with addition and multiplication obeying basic axioms, with commutative and noncommutative branches and examples like the integers.
Ringel's conjecture
Ringel's conjecture is a 1963 graph theory conjecture stating that every tree with n edges decomposes the complete graph K(2n+1); it was proved for all sufficiently large n in 2021.
Risch algorithm
The Risch algorithm is a decision procedure, formulated by Robert H. Risch in 1969 and 1970, that determines whether an elementary function has an elementary antiderivative and finds it in closed form.
Risk difference
The risk difference is an epidemiological and biostatistical measure defined as the difference in the probability of an outcome between two groups, quantifying the absolute effect of an exposure or treatment.