Fr.: scalaire de Ricci
The simplest curvature invariant for a → Riemannian manifold. It is derived from the → Ricci tensor Rμν ≡ Rαμαν by contracting indices. Taking the trace of the Ricci tensor gives the Ricci scalar: R ≡ Rμνgμnu; = Rμν = Rαμαμ. Also called → scalar curvature.
Fr.: tenseur de Ricci
A → rank 2, → symmetric tensor Rμν that is a contraction of the → Riemann curvature tensor Rλμνλ. More specifically, Rμν ≡ Σ (λ) Rλμνκ = Rλμνκ. Closely related to the Ricci tensor is the → Einstein tensor, which plays an important role in the theory of → general relativity.
Named after the Italian mathematician Gregorio Ricci-Curbastro (1853-1925); → tensor.
M.E., from O.E. rice "wealthy, powerful" (cf. Du. rijk, Ger. reich "rich"), from PIE base *reg- "move in a straight line," hence, "to direct, rule" (cf. Mod.Pers./Mid.Pers. râst "right, straight;" O.Pers. rāsta- "straight, true," rās- "to be right, straight, true;" Av. rāz- "to direct, put in line, set," razan- "order;" Skt. raj- "to direct, stretch," rjuyant- "walking straight;" Gk. orektos "stretched out;" L. regere "to lead straight, guide, rule," p.p. rectus "right, straight;" Ger. recht; E. right).
Por "full, much, very, too much" (Mid.Pers. purr "full;" O.Pers. paru- "much, many;" Av. parav-, pauru-, pouru-, from par- "to fill;" PIE base *pelu- "full," from *pel- "to be full;" cf. Skt. puru- "much, abundant;" Gk. polus "many," plethos "great number, multitude;" O.E. full); pordâr, literally "having much possession," from por + dâr "having, possessor," from dâštan "to have, to possess," → property.
Fr.: amas riche
A → galaxy cluster with a particularly large number of galaxies.
Fr.: cascade de Richarson
Same as → energy cascade
Named after L. F. Richardson (1922), Weather Prediction by Numerical Process (Cambridge Univ. Press); → cascade.
Fr.: critère de Richardson
Named after the British meteorologist Lewis Fry Richardson (1881-1953), who first arrived in 1920 to the dimensionless ratio now called → Richardson number. The first formal proof of the criterion, however, came four decades later for → incompressible flows (Miles, J. W. 1961, J. Fluid Mech., 10, 496; Howard, L. N., 1961, J. Fluid Mech., 10, 509). Its extension to → compressible flows was demonstrated subsequently (Chimonas 1970, J. Fluid Mech., 43, 833); → criterion.
Fr.: nombre de Richardson
A dimensionless number which is used according to the → Richardson criterion to describe the condition for the → stability of a flow in the presence of vertical density stratification. If the → shear flow is characterized by linear variation of velocity and density, with velocities and densities ranging from U1 to U2 and ρ1 to ρ2 (ρ2>ρ1), respectively, over a depth H, then the Richardson number is expressed as: Ri = (ρ2 - ρ1) gH / ρ0 (U1 - U2)2. If Ri < 0.25, somewhere in the flow turbulence is likely to occur. For Ri > 0.25 the flow is stable.
The property of being very abundant.
Fr.: classe de richesse
A classification of → galaxy clusters into six groups (0 to 5), as in the → Abell catalog. It depends on the number of galaxies in a given cluster that lie within a → magnitude range m3 to m3+2, where m3 is the magnitude of the 3rd brightest member of the cluster. The first group contains 30-49 galaxies and the last group more than 299 galaxies.
Fr.: énigme, devinette
1) A question or statement so framed as to exercise one's ingenuity in answering it
or discovering its meaning; conundrum.
M.E. redel, redels, from O.E. rædels "riddle; counsel; conjecture; imagination;" cf. O.Fr. riedsal "riddle," O.Sax. radisli, M.Du. raetsel, Du. raadsel, O.H.G. radisle, Ger. Rätsel "riddle."
Kervas "riddle, puzzle" [Dehxodâ], Kurd. karvâs "riddle," of unknown origin.
Fr.: faîte, dorsale
A long, narrow elevation of the Earth's surface, generally sharp crested with steep sides, either independently or as part of a larger mountain or hill. See also: → submarine ridge, → wrinkle ridge, → mid-Atlantic ridge.
M.E. rigge; O.E. hrycg "spine, back of a man or beast" (cf. O.Fris. hregg, Du. rug, O.H.G. hrukki, Ger. Rücken "the back").
Ruk, from dialectal Tabari ruk "mountain, ridge;" cf. (Dehxodâ) raš "hill."
Riemann curvature tensor
tânsor-e xamidegi-ye Riemann
Fr.: tenseur de courbure de Riemann
A 4th → rank tensor that characterizes the deviation of the geometry of space from the Euclidean type. The curvature tensor Rλμνκ is defined through the → Christoffel symbols Γλμν as follows: Rλμνκ = (∂Γλμκ)/(∂xν) - (∂Γλμν)/(∂xκ) + ΓημκΓλην - ΓημνΓληκ.
Fr.: problème de Riemann
The combination of a → partial differential equation and a → piecewise constant → initial condition. The Riemann problem is a basic tool in a number of numerical methods for wave propagation problems. The canonical form of the Riemann problem is: ∂u/∂t + ∂f(u)/∂x = 0, x ∈ R, t > 0, u(x,0) = ul if x < 0, and u(x,0) = ur if x > 0 .
Fr.: géométrie de Riemann
Same as → Riemannian geometry.
Of or pertaining to Georg Friedrich Bernhard Riemann (1826-1866) or his mathematics findings. → Riemannian geometry, → Riemannian manifold, → Riemannian metric, → Riemann problem, → Riemann curvature tensor.
After the German mathematician Georg Friedrich Bernhard Riemann (1826-1866), the inventor of the elliptic form of → non-Euclidean geometry, who made important contributions to analysis and differential geometry, some of them paving the way for the later development of → general relativity.
Fr.: géométrie riemannienne
A → non-Euclidean geometry in which there are no → parallel lines, and the sum of the → angles of a → triangle is always greater than 180°. Riemannian figures can be thought of as figures constructed on a curved surface. The geometry is called elliptic because the section formed by a plane that cuts the curved surface is an ellipse.
Fr.: variété riemannienne
Fr.: métrique riemannienne
A positive-definite inner product, (.,.)x, on Tx(M), the tangent space to a manifold M at x, for each x ∈ M, which varies continually with x (Douglas N. Clark, Dictionary of Analysis, Calculus, and Differential Equations).
Fr.: rift; fissure, crevasse
1) General: a gap or break made by splitting, cleaving, etc.; fissure; cleft.
M.E., from a Scandinavian source; cf. Dan., Norw. rift "a cleft."
1) Câk "fissure, rupture, cleft, crack," related to caqidan "to drive a nail, strike together, fix a spike;" caxidan "to quarrel, strive;" diatectal Shughni cuγ, Roshani coγ "breaking, tearing, ripping up;" Malâyeri ceqidan "cracking of a cold porcelain bowl when receiving hot water;" Yazghulami cok-, coxt "to pound, to stamp;" Wakhi cuk-, cukt "to strike;" Ossetic cædyn "to shake, strike iron;" Lori cok "a mark fixed to separate two grounds;" Sogd. cax "to battel, fight;" Mid.Pers. (prefixed) pacên "copy" (see also Cheung 2007). Gosal-darré, from gosal, → fault, + darré, → valley.
Fr.: fossé d'effondrement, ~ tectonique