All articles starting with R
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Riemann curvature tensor
The Riemann curvature tensor, also called the Riemann-Christoffel tensor, is a four-index tensor measuring the intrinsic curvature of a manifold through path-dependent parallel transport; it is central to general relativity.
Riemann hypothesis
The Riemann hypothesis, also called the Riemann conjecture, is an unproved 1859 conjecture stating that all nontrivial zeros of the Riemann zeta function have real part 1/2.
Riemann integral
The Riemann integral, also called Riemann integration, defines the integral of a bounded function over a finite interval through lower and upper sums; Bernhard Riemann introduced it in 1854.
Riemann mapping theorem
The Riemann mapping theorem states that any simply connected proper plane domain can be conformally mapped onto the open unit disk, formulated by Bernhard Riemann in 1851.
Riemann problem
The Riemann problem is an initial value problem for a hyperbolic partial differential equation with piecewise constant data, whose jump breaks up into waves; it underlies Godunov-type numerical methods.
Riemann solver
A Riemann solver is a numerical method that computes fluxes in finite-volume schemes for hyperbolic conservation laws by solving the Riemann problem at each cell interface.
Riemann sphere
The Riemann sphere, also called the extended complex plane, is the complex plane with one point at infinity added, modeled as a sphere by stereographic projection.
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.
Riemenschneider (Mercurian crater)
Riemenschneider is a 183 km IAU-approved impact crater on Mercury at 52.75°S, 99.95°W, named in 1979 for the German sculptor Tilman Riemenschneider (c. 1460–1531), in the H-12 Michelangelo quadrangle.
Rien van Genuchten
Rien van Genuchten, born Martinus Th. van Genuchten, is a Dutch-born soil physicist whose 1980 equation for unsaturated soil hydraulics won him the 2023 Wolf Prize in Agriculture.
Ries impact event
The Ries impact event was an asteroid collision in southern Germany about 14.8 million years ago that produced the Nördlinger Ries, a 26-km impact crater in the Middle Miocene.
Ries UNESCO Global Geopark
Ries UNESCO Global Geopark, designated 2022 in Bavaria and Baden-Württemberg, Germany, protects the Nördlinger Ries, a 25 km asteroid impact crater about 15 million years old, Europe's best-preserved meteorite crater.
Rieske protein
Rieske protein (Rieske iron-sulfur protein, RIP1/UQCRFS1): an iron-sulfur protein with a rare His2-Cys2-ligated [2Fe-2S] cluster, discovered 1964, serving as the high-potential (+280 mV) electron carrier and proton gate of cytochrome bc1 and b6f complexes.
Riesling
Riesling, also known as White Riesling or Rhine Riesling, is a white wine grape of German origin, prized for high acidity and wines from bone dry to sweet.
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.
Rietdijk–Putnam argument
The Rietdijk–Putnam argument is a philosophical argument that the relativity of simultaneity in special relativity entails an eternalist block universe, advanced independently in 1966 and 1967.