Riemannn.
/ˈriːmən/, U.S. /ˈrimən/ Mathematics.
1.
a. Riemann surface n. (also Riemann's surface) a complex manifold of complex dimension one; esp. a branched covering of the complex plane, which can be used in plotting multi-valued functions.
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the world > relative properties > number > geometry > surface > [noun]
superficec1400
superficies1530
surface1604
superficie1702
wave-surface1833
developable1847
quartic1854
scroll1862
conicoid1863
regulus1874
Riemann surface1876
tetrahedroid1889
construct1902
skew1902
trend surface1956
1876 O. J. Lodge in London, Edinb., & Dublin Philos. Mag. 5th Ser. 1 386 We have expressed the resistance of a Riemann's surface of any number of leaves, but all connected at one branch-point O.
1913 S. E. Rasor tr. H. Burkhardt Theory of Functions of Complex Variable v. 333 These values are so arranged on the two sheets of our two-sheeted Riemann's surface that only one of the values belongs to each point of the surface.
1932 A. Huxley Brave New World iv. 73 Two thousand Beta-Minus mixed doubles were playing Riemann-surface tennis.
1985 H. A. Priestley Introd. Complex Anal. vi. 101 The Riemann surface for the logarithm is modelled by a car park with infinitely many floors each of infinite extent, with a ramp joining each floor to the next.
2004 Nature 14 Oct. 741/2 Penrose does not shy from introducing Riemann surfaces, holomorphicity and hyperfunctions.
b. Riemann sphere n. the simplest kind of Riemann surface, consisting of the plane of complex numbers with an additional point at infinity and used in various branches of mathematics and physics.
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1893 J. Harkness & F. Morley Treat. Theory Functions vi. 236 The process of conversion of a thin flat surface with two sheets, connected by p holes, into a double Riemann sphere, can be illustrated in a similar manner.
1963 G. F. Simmons Introd. Topol. & Mod. Anal. vii. 163 When S is identified in this manner with the extended complex plane, it is usually called the Riemann sphere.
2004 B. Zwiebach First Course String Theory xxii. 499 The world-sheet of a free closed string is a Riemann sphere..with two punctures.
2.
a. Riemann space n. a generalized Euclidean space of n dimensions in which the definition of distance involves a tensor.
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the world > relative properties > number > geometry > geometric space > [noun] > non-Euclidean
hyperspace1867
curvature1873
Riemann space1894
non-Euclidean space1939
1894 F. Klein Lect. Math. i. 7 There is no difficulty in conceiving a four-dimensional Riemann space corresponding to an equation f(x, y, z) = 0.
1920 Bull. Amer. Math. Soc. 26 370 Every closed orientable n-dimensional manifold can be represented on an n-dimensional hypersphere as a Riemann space.
1982 A. Pais Subtle is Lord xvii. 339 The equations Rλμνρ = 0 are necessary and sufficient conditions for a Riemann space to be everywhere flat (pseudo-Euclidean).
2008 Jrnl. Geom. & Physics 58 129 Biomembranes should be understood from the viewpoint of Riemann space.
b. Riemann geometry n. (also Riemann's geometry) Riemannian geometry.
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the world > relative properties > number > geometry > [noun] > branches of
planimetrya1393
conic?a1560
helicosophy1570
stereometry1570
spheric1660
planometry1669
mensuration1704
polygonometry1791
analytical geometry1802
isoperimetry1811
analytic geometry1817
algebraic geometry1821
coordinate geometry1837
non-Euclidean geometry1872
differential geometry1877
pangeometry1878
projective geometry1878
metageometry1890
Riemann geometry1895
variable geometry1957
1895 Nature 25 July 309/1 Riemann's geometry is characterised by the proposition: If, in a plane, two straight lines which intersect in a point A also intersect in a second point B, all straight lines passing through A will also cut the line A B a second time.
1953 M. Kline Math. in Western Culture xxvi. 425 The axioms of Riemann's geometry correctly describe facts about the sphere.
1974 R. M. Pirsig Zen & Art of Motorcycle Maintenance (1976) iii. xxii. 257 Is Euclidian geometry true or is Riemann geometry true? He answered, The question has no meaning.
1998 Physics Lett. A 244 222/1 Nowadays space–time has a weak relict local anisotropy and therefore it is not unlikely that it is described by Finsler geometry..rather than by Riemann geometry.
c. Riemann curvature n. a mathematical generalization of the concept of curvature to spaces of more than three dimensions.
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1905 H. Maschke in H. J. Rogers Congr. Arts & Sci. I. 523 The fundamental rôle which the Gaussian curvature plays in the case n = 2 is here represented by an invariant expression of ds2 which..might be regarded as a generalization of the Gaussian curvature, namely the Riemann curvature of the hyperspace.
1979 E. Nagel Struct. Sci. ix. 245 As in the case of two-dimensional surfaces, the Riemann curvature of three-dimensional continua may be positive, negative, or zero.
2001 E. R. Harrison Cosmology (ed. 2) xii. 230 All components of the Riemann curvature are determined when we take into account distant as well as local matter.
d. Riemann tensor n. (also Riemann's tensor) = Riemann–Christoffel tensor n.
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the world > relative properties > number > mathematical number or quantity > tensor > [noun]
Riemann–Christoffel tensor1916
tensor1916
Riemann tensor1922
Ricci tensor1923
Riemann curvature tensor1923
1922 Proc. National Acad. Sci. U.S.A. 8 24 Rpq,rs is the Riemann tensor of the first kind.
1973 S. W. Hawking & G. F. R. Ellis Large Scale Structure of Space–Time ii. 42 Split the Riemann tensor into a part represented by the Ricci tensor and a part represented by the Weyl tensor.
2007 Chaos, Solitons & Fractals 32 1246 The Riemann tensor tells us all what we need to know about the intrinsic curvature and thus gravity... Riemann's tensor displays various symmetries.
e. Riemann curvature tensor n. = Riemann–Christoffel tensor n.
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the world > relative properties > number > mathematical number or quantity > tensor > [noun]
Riemann–Christoffel tensor1916
tensor1916
Riemann tensor1922
Ricci tensor1923
Riemann curvature tensor1923
1923 G. D. Birkhoff Relativity & Mod. Physics vii. 127 This tensor, which is known as the Riemannian or ‘curvature’ tensor, is of fundamental importance.]
1923 Science 28 Dec. 540/1 In this chapter he [sc. Birkhoff in the work cited previously] uses them [sc. geodesic coordinates] to determine the Riemann curvature tensor.
1989 R. Penrose Emperor's New Mind v. 210 The full mathematical expression for the curvature of four-dimensional space–time..is given by something called the Riemann curvature tensor. This is a somewhat complicated object, needing twenty real numbers at each point in order to specify it.
3.
a. Riemann integral n. (also Riemann's integral) a definite integral calculated by dividing an area under a curve of a real function into a number of rectangular intervals of equal width, summing the areas of these rectangles, and then taking the limit of the sum as the width of the intervals tends to zero.
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the world > relative properties > number > calculus > [noun] > integral calculus > integration or integrability > integral
fluent1706
integral1728
gamma function1834
surface integral1867
Riemann integral1894
Cauchy's integral1898
Lebesgue integral1905
Stieltjes integral1914
convolution1934
1894 Mathematische Annalen 45 118 (heading) On the theory of Riemann's Integrals.
1925 Ann. Math. 25 173 The theorem on Riemann integrals is used to establish some facts concerning the convergence of the Riemann sum.
1993 E. Hairer et al. Solving Ordinary Differential Equations (ed. 2) i. 38 We introduce a third subdivision..which is a refinement of both subdivisions (just as is usually done in proving the existence of Riemann's integral).
2007 T. W. Epps Pricing Derivative Securities (ed. 2) ii. 32 Unlike the Riemann integral, however, Stieltjes integrals over open and closed sets may differ.
b. Riemann integration n. the operation of finding the Riemann integral of a function.
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the world > relative properties > number > calculus > [noun] > integral calculus
integral calculus1728
Riemann integration1908
1908 Proc. Cambr. Philos. Soc. 14 529 A systematic development of the theory of Riemann integration.
1939 I. S. Sokolnikoff Advanced Calculus iv. 99 Begin the study of Riemann integration by presenting a reasonably careful definition of the definite integral based on the intuitive concept of the area under the curve.
2005 Amer. Math. Monthly 112 794 A coherent understanding of all of the various facets of Riemann integration.
c. Riemann integrable adj. of a function: capable of undergoing Riemann integration.
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the world > relative properties > number > calculus > [adjective] > of integration
integrating1654
integrable1728
integral1728
subregular1887
Riemann integrable1918
1918 Bull. Amer. Math. Soc. 24 122 If f(x) is Riemann integrable, it is also Young integrable.
1972 A. G. Howson Handbk. Terms Algebra & Anal. xxvii. 136 If f is bounded on I then it is Riemann integrable..on I if and only if it is continuous almost everywhere..on I.
2009 Jrnl. Number Theory 129 690 Any Riemann integrable function f of bounded variation δ(f).
d. Riemann sum n. a sum of the form Σf(yi)(xi − xi − 1), summed over all the points i of a set, which can be used to approximate the area under a curve or the value of an integral.
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1925 Ann. Math. 25 173 The theorem on Riemann integrals is used to establish some facts concerning the convergence of the Riemann sum.
1948 Proc. Symp. Large-scale Digital Calculating Machinery 1947 181 This sum can be looked upon as a Riemann sum and can, by neglecting errors of higher order, be replaced by the integral.
2000 R. C. Mittelhammer et al. Econometric Found. App. 720 The Riemann sum partitions the integral into a finite number of pieces to approximate the corresponding areas under the graph.
4. Riemann zeta function n. (also Riemann's zeta function) an analytic function ζ of the complex variable z equal to 1−z + 2−z + 3−z + … when the real part of z is greater than 1. Also written Riemann ζ function.
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the world > relative properties > number > algebra > [noun] > expression > function
function1758
exponential1784
potential function1828
syzygy1850
permutant1852
Green function1863
theta-function1871
Greenian1876
Gudermannian1876
discriminoid1877
Weierstrassian function1878
gradient1887
beta function1888
distribution function1889
Riemann zeta function1899
Airy integral1903
Poisson bracket1904
Stirling approximation1908
functional1915
metric1921
Fourier transform1923
recursive function1934
utility function1934
Airy function1939
transfer function1948
objective function1949
restriction1949
multifunction1954
restriction mapping1956
scalar function1956
Langevin function1960
mass function1961
1899 Messenger of Math. 29 114 If ζ(s, a, ω) be the extended Riemann ζ function.
1931 Q. Jrnl. Math. 2 161 The theory of the Riemann zeta-function.
1992 Sci. Amer. Jan. 84/2 The calculation of the time delay leads straight into what is probably the most enigmatic object in mathematics, Riemann's zeta function.
2005 D. G. Wells Prime Numbers 242 The Riemann zeta function also shows up in the theory of Brownian motion and in the study of diffusion and percolation processes.
5. Riemann hypothesis n. (also Riemann's hypothesis) the hypothesis that all the values of the complex variable z with a positive real part for which the Riemann zeta function ζ(z) = 0 have the real part equal to ½.Whether or not the hypothesis is true is one of the famous unsolved problems of mathematics.
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the world > relative properties > number > mathematics > [noun] > mathematical enquiry > proposition > specific conjecture
Goldbach1902
Riemann hypothesis1918
Waring's problem1920
Poincaré conjecture1938
Church's theorem1939
Church's thesis1954
1918 Nature 12 Dec. 300/1 The Riemann hypothesis and the expression of a number as the sum of a stated number of primes.
1924 Proc. Cambr. Philos. Soc. 22 296 We assume Riemann's hypothesis.
1941 D. H. Lehmer Guide Tables Theory Numbers 9 These tables were computed with the hope of shedding some light on the still unsolved problem of the order of magnitude of M(x), a problem intimately connected with the Riemann hypothesis.
2007 Independent 23 Feb. 20/1 A US$1m..prize for the first mathematician to find a pattern in primes—a problem known as the Riemann hypothesis.
This entry has been updated (OED Third Edition, June 2010; most recently modified version published online March 2022).
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