Riemannn.

Brit.

/ˈriːmən/, U.S.

/ˈrimən/

Origin: From a proper name. Etymon: proper name Riemann.

Etymology: < the name of Georg Friedrich Bernhard Riemann (1826–66), German mathematician.

Mathematics.

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.

ΘΚΠ

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.

ΚΠ

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.

a. Riemann space n. a generalized Euclidean space of n dimensions in which the definition of distance involves a tensor.

ΘΚΠ

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 R^λ_μνρ = 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.

ΘΚΠ

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.

ΚΠ

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.

ΘΚΠ

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 R_pq,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.

ΘΚΠ

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.

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.

ΘΚΠ

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.

ΘΚΠ

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.

ΘΚΠ

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(y_i)(x_i − x_{i − 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.

ΚΠ

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.

ΘΚΠ

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.

ΘΚΠ

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).

单词	riemann
释义	Riemannn. Brit. /ˈriːmən/, U.S. /ˈrimən/ Origin: From a proper name. Etymon: proper name Riemann. Etymology: < the name of Georg Friedrich Bernhard Riemann (1826–66), German mathematician. 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. ΘΚΠ 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. ΚΠ 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. ΘΚΠ 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 R^λ_μνρ = 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. ΘΚΠ 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. ΚΠ 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. ΘΚΠ 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 R_pq,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. ΘΚΠ 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. ΘΚΠ 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. ΘΚΠ 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. ΘΚΠ 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(y_i)(x_i − x_{i − 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. ΚΠ 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. ΘΚΠ 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. ΘΚΠ 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). < n.1876
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