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A manifold with an analytic atlas. The structure of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a0123201.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a0123202.png" /> over a complete non-discretely normed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a0123203.png" /> on a topological space is defined by specifying an analytic atlas over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a0123204.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a0123205.png" />, i.e. a collection of charts (cf. [[Chart|Chart]]) with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a0123206.png" /> covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a0123207.png" />, any two charts of which are analytically related. Two atlases are said to define the same structure if their union is an analytic atlas. The sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a0123208.png" /> of germs of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a0123209.png" />-valued analytic functions is defined on an analytic manifold. The class of ringed spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232010.png" /> which results in this way is identical with the class of smooth analytic spaces over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232011.png" />.
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A manifold with an analytic atlas. The structure of an $n$-dimensional manifold $M$ over a complete non-discretely normed field $k$ on a topological space is defined by specifying an analytic atlas over $k$ on $M$, i.e. a collection of charts (cf. [[Chart|Chart]]) with values in $k^n$ covering $M$, any two charts of which are analytically related. Two atlases are said to define the same structure if their union is an analytic atlas. The sheaf $\mathcal O$ of germs of $k$-valued analytic functions is defined on an analytic manifold. The class of ringed spaces $(M,\mathcal O)$ which results in this way is identical with the class of smooth analytic spaces over $k$.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232012.png" /> is the field of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232013.png" /> one speaks of real-analytic manifolds; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232014.png" /> is the field of complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232015.png" />, of complex-analytic or simply complex manifolds; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232016.png" /> is the field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232017.png" />-adic numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232018.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232020.png" />-adic analytic manifolds. Examples of analytic manifolds include the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232021.png" />-dimensional Euclidean spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232022.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232023.png" />-dimensional projective spaces over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232024.png" />, the affine and projective algebraic varieties without singular points over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232025.png" />, and Lie groups and their homogeneous spaces.
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If $k$ is the field of real numbers $\mathbf R$ one speaks of real-analytic manifolds; if $k$ is the field of complex numbers $\mathbf C$, of complex-analytic or simply complex manifolds; if $k$ is the field of $p$-adic numbers $\mathbf Q_p$, of $p$-adic analytic manifolds. Examples of analytic manifolds include the $n$-dimensional Euclidean spaces $k^n$, the $n$-dimensional projective spaces over $k$, the affine and projective algebraic varieties without singular points over $k$, and Lie groups and their homogeneous spaces.
  
The concept of an analytic manifold goes back to B. Riemann and F. Klein, but was precisely formulated for the first time by H. Weyl [[#References|[4]]] for the case of Riemann surfaces, i.e. one-dimensional complex manifolds. At present (the 1970's) it is natural to regard analytic manifolds as a special case of analytic spaces (cf. [[Analytic space|Analytic space]]), which may be roughly described as "varieties with singular points" . The concept of an analytic space was introduced in the 1950's and became the principal subject of the theory of analytic functions; many fundamental results obtained for analytic manifolds could be successfully applied to the non-smooth case. For an account of the general properties of analytic manifolds over an arbitrary field see [[#References|[3]]].
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The concept of an analytic manifold goes back to B. Riemann and F. Klein, but was precisely formulated for the first time by H. Weyl [[#References|[4]]] for the case of Riemann surfaces, i.e. one-dimensional complex manifolds. At present (the 1970's) it is natural to regard analytic manifolds as a special case of analytic spaces (cf. [[Analytic space|Analytic space]]), which may be roughly described as "varieties with singular points" . The concept of an analytic space was introduced in the 1950's and became the principal subject of the theory of analytic functions; many fundamental results obtained for analytic manifolds could be successfully applied to the non-smooth case. For an account of the general properties of analytic manifolds over an arbitrary field see [[#References|[3]]].
  
There is a close relationship between the theories of real-analytic and differentiable manifolds (cf. [[Differentiable manifold|Differentiable manifold]]), and also between the theories of real-analytic and complex-analytic manifolds, Clearly, the natural structure of a manifold of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232026.png" /> is defined on each real-analytic manifold. It was shown by H. Whitney in 1936 that the converse proposition is also true: It is possible to define on any paracompact manifold of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232027.png" /> an analytic structure over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232028.png" /> which induces the initial smooth structure. It follows from Grauert's theorem on the imbeddability of a paracompact analytic manifold over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232029.png" /> in a Euclidean space that this analytic structure is unambiguously defined up to an isomorphism (not necessarily the identity) [[#References|[2]]].
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There is a close relationship between the theories of real-analytic and differentiable manifolds (cf. [[Differentiable manifold|Differentiable manifold]]), and also between the theories of real-analytic and complex-analytic manifolds, Clearly, the natural structure of a manifold of class $C^\infty$ is defined on each real-analytic manifold. It was shown by H. Whitney in 1936 that the converse proposition is also true: It is possible to define on any paracompact manifold of class $C^\infty$ an analytic structure over $\mathbf R$ which induces the initial smooth structure. It follows from Grauert's theorem on the imbeddability of a paracompact analytic manifold over $\mathbf R$ in a Euclidean space that this analytic structure is unambiguously defined up to an isomorphism (not necessarily the identity) [[#References|[2]]].
  
A natural structure of a real-analytic manifold (of double dimension) is defined on all complex manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232030.png" />. The answer to the converse problem — viz. whether a complex structure on a given real-analytic manifold exists and whether it is unique — has been given in the simplest cases only. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232031.png" /> is a connected two-dimensional real-analytic manifold, then a necessary and sufficient condition for the existence of a complex structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232032.png" /> is paracompactness and orientability, while the problem of classification of these structures is identical with the classical moduli problem for Riemann surfaces (cf. [[Moduli of a Riemann surface|Moduli of a Riemann surface]]). There is a classification of compact analytic surfaces (i.e. of two-dimensional complex manifolds, cf. [[Analytic surface|Analytic surface]]), which gives a partial answer to the above problem for four-dimensional real-analytic manifolds. On the other hand it is possible, using topological methods, to identify classes of real manifolds that do not permit almost-complex or, a fortiori, complex structures. Such manifolds include the spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232033.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232034.png" />. A description of complex structures which are sufficiently near to a given complex structure is given by the theory of deformations of analytic structures (cf. [[Deformation|Deformation]]), in which an important role is played by Banach analytic manifolds — infinite-dimensional analogues of analytic manifolds.
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A natural structure of a real-analytic manifold (of double dimension) is defined on all complex manifolds $M$. The answer to the converse problem — viz. whether a complex structure on a given real-analytic manifold exists and whether it is unique — has been given in the simplest cases only. Thus, if $M$ is a connected two-dimensional real-analytic manifold, then a necessary and sufficient condition for the existence of a complex structure on $M$ is paracompactness and orientability, while the problem of classification of these structures is identical with the classical moduli problem for Riemann surfaces (cf. [[Moduli of a Riemann surface|Moduli of a Riemann surface]]). There is a classification of compact analytic surfaces (i.e. of two-dimensional complex manifolds, cf. [[Analytic surface|Analytic surface]]), which gives a partial answer to the above problem for four-dimensional real-analytic manifolds. On the other hand it is possible, using topological methods, to identify classes of real manifolds that do not permit almost-complex or, a fortiori, complex structures. Such manifolds include the spheres $S^{2k}$ for $k\neq1,3$. A description of complex structures which are sufficiently near to a given complex structure is given by the theory of deformations of analytic structures (cf. [[Deformation|Deformation]]), in which an important role is played by Banach analytic manifolds — infinite-dimensional analogues of analytic manifolds.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki,   "Elements of mathematics. Differentiable and analytic manifolds" , Addison-Wesley (1966) (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R. Narasimhan,   "Analysis on real and complex manifolds" , Springer (1971)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J.-P. Serre,   "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> H. Weyl,   "Die Idee der Riemannschen Fläche" , Teubner (1955)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Differentiable and analytic manifolds" , Addison-Wesley (1966) (Translated from French) {{MR|0205211}} {{MR|0205210}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R. Narasimhan, "Analysis on real and complex manifolds" , Springer (1971) {{MR|0832683}} {{MR|0346855}} {{MR|0251745}} {{ZBL|0583.58001}} {{ZBL|0188.25803}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) {{MR|0218496}} {{ZBL|0132.27803}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> H. Weyl, "Die Idee der Riemannschen Fläche" , Teubner (1955) {{MR|0069903}} {{ZBL|0068.06001}} </TD></TR></table>
  
  
  
 
====Comments====
 
====Comments====
A much related basic problem in complex analysis is the question whether there are any complex structures on projective space besides the usual one (and inducing the same topology). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232035.png" /> this is very classical (all Riemann surfaces of genus zero are isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232036.png" />). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232037.png" />, uniqueness of the complex structure follows from combined work of F. Hirzebruch, K. Kodaira [[#References|[a4]]], and S.T. Yau [[#References|[a5]]]. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232038.png" /> one has that a compact manifold that is bimeromorphically equivalent to a [[Kähler manifold|Kähler manifold]] and that is also topologically <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232039.png" /> is analytically isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232040.png" /> [[#References|[a6]]].
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A much related basic problem in complex analysis is the question whether there are any complex structures on projective space besides the usual one (and inducing the same topology). For $n=1$ this is very classical (all Riemann surfaces of genus zero are isomorphic to $P_\mathbf C^1$). For $n=2$, uniqueness of the complex structure follows from combined work of F. Hirzebruch, K. Kodaira [[#References|[a4]]], and S.T. Yau [[#References|[a5]]]. For $n=3$ one has that a compact manifold that is bimeromorphically equivalent to a [[Kähler manifold|Kähler manifold]] and that is also topologically $P_\mathbf C^3$ is analytically isomorphic to $P_\mathbf C^3$ [[#References|[a6]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Whitney,   "Complex analytic varieties" , Addison-Wesley (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W. Barth,   C. Peters,   A. van der Ven,   "Compact complex surfaces" , Springer (1984)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> R.O. Wells jr.,   "Differential analysis on complex manifolds" , Springer (1980)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> F. Hirzebruch,   K. Kodaira,   "On the complex projective spaces" ''J. Math. Pures Appl.'' , '''36''' (1957) pp. 201–216</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> S.-T. Yau,   "Calabi's conjecture and some new results in algebraic geometry" ''Proc. Nat. Acad. Sci. USA'' , '''74''' (1977) pp. 1798–1799</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> T. Peternell,   "A rigidity theorem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012320/a01232041.png" />"  ''Manuscripta Math.'' , '''50''' (1985) pp. 397–428</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Whitney, "Complex analytic varieties" , Addison-Wesley (1972) {{MR|0387634}} {{ZBL|0265.32008}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W. Barth, C. Peters, A. van der Ven, "Compact complex surfaces" , Springer (1984) {{MR|0749574}} {{ZBL|0718.14023}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) {{MR|0608414}} {{ZBL|0435.32004}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> F. Hirzebruch, K. Kodaira, "On the complex projective spaces" ''J. Math. Pures Appl.'' , '''36''' (1957) pp. 201–216 {{MR|0092195}} {{ZBL|0090.38601}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> S.-T. Yau, "Calabi's conjecture and some new results in algebraic geometry" ''Proc. Nat. Acad. Sci. USA'' , '''74''' (1977) pp. 1798–1799</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> T. Peternell, "A rigidity theorem for $P_3(\mathbf C)$" ''Manuscripta Math.'' , '''50''' (1985) pp. 397–428 {{MR|0784150}} {{ZBL|0573.32027}} </TD></TR></table>

Latest revision as of 13:31, 25 April 2014

A manifold with an analytic atlas. The structure of an $n$-dimensional manifold $M$ over a complete non-discretely normed field $k$ on a topological space is defined by specifying an analytic atlas over $k$ on $M$, i.e. a collection of charts (cf. Chart) with values in $k^n$ covering $M$, any two charts of which are analytically related. Two atlases are said to define the same structure if their union is an analytic atlas. The sheaf $\mathcal O$ of germs of $k$-valued analytic functions is defined on an analytic manifold. The class of ringed spaces $(M,\mathcal O)$ which results in this way is identical with the class of smooth analytic spaces over $k$.

If $k$ is the field of real numbers $\mathbf R$ one speaks of real-analytic manifolds; if $k$ is the field of complex numbers $\mathbf C$, of complex-analytic or simply complex manifolds; if $k$ is the field of $p$-adic numbers $\mathbf Q_p$, of $p$-adic analytic manifolds. Examples of analytic manifolds include the $n$-dimensional Euclidean spaces $k^n$, the $n$-dimensional projective spaces over $k$, the affine and projective algebraic varieties without singular points over $k$, and Lie groups and their homogeneous spaces.

The concept of an analytic manifold goes back to B. Riemann and F. Klein, but was precisely formulated for the first time by H. Weyl [4] for the case of Riemann surfaces, i.e. one-dimensional complex manifolds. At present (the 1970's) it is natural to regard analytic manifolds as a special case of analytic spaces (cf. Analytic space), which may be roughly described as "varieties with singular points" . The concept of an analytic space was introduced in the 1950's and became the principal subject of the theory of analytic functions; many fundamental results obtained for analytic manifolds could be successfully applied to the non-smooth case. For an account of the general properties of analytic manifolds over an arbitrary field see [3].

There is a close relationship between the theories of real-analytic and differentiable manifolds (cf. Differentiable manifold), and also between the theories of real-analytic and complex-analytic manifolds, Clearly, the natural structure of a manifold of class $C^\infty$ is defined on each real-analytic manifold. It was shown by H. Whitney in 1936 that the converse proposition is also true: It is possible to define on any paracompact manifold of class $C^\infty$ an analytic structure over $\mathbf R$ which induces the initial smooth structure. It follows from Grauert's theorem on the imbeddability of a paracompact analytic manifold over $\mathbf R$ in a Euclidean space that this analytic structure is unambiguously defined up to an isomorphism (not necessarily the identity) [2].

A natural structure of a real-analytic manifold (of double dimension) is defined on all complex manifolds $M$. The answer to the converse problem — viz. whether a complex structure on a given real-analytic manifold exists and whether it is unique — has been given in the simplest cases only. Thus, if $M$ is a connected two-dimensional real-analytic manifold, then a necessary and sufficient condition for the existence of a complex structure on $M$ is paracompactness and orientability, while the problem of classification of these structures is identical with the classical moduli problem for Riemann surfaces (cf. Moduli of a Riemann surface). There is a classification of compact analytic surfaces (i.e. of two-dimensional complex manifolds, cf. Analytic surface), which gives a partial answer to the above problem for four-dimensional real-analytic manifolds. On the other hand it is possible, using topological methods, to identify classes of real manifolds that do not permit almost-complex or, a fortiori, complex structures. Such manifolds include the spheres $S^{2k}$ for $k\neq1,3$. A description of complex structures which are sufficiently near to a given complex structure is given by the theory of deformations of analytic structures (cf. Deformation), in which an important role is played by Banach analytic manifolds — infinite-dimensional analogues of analytic manifolds.

References

[1] N. Bourbaki, "Elements of mathematics. Differentiable and analytic manifolds" , Addison-Wesley (1966) (Translated from French) MR0205211 MR0205210
[2] R. Narasimhan, "Analysis on real and complex manifolds" , Springer (1971) MR0832683 MR0346855 MR0251745 Zbl 0583.58001 Zbl 0188.25803
[3] J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) MR0218496 Zbl 0132.27803
[4] H. Weyl, "Die Idee der Riemannschen Fläche" , Teubner (1955) MR0069903 Zbl 0068.06001


Comments

A much related basic problem in complex analysis is the question whether there are any complex structures on projective space besides the usual one (and inducing the same topology). For $n=1$ this is very classical (all Riemann surfaces of genus zero are isomorphic to $P_\mathbf C^1$). For $n=2$, uniqueness of the complex structure follows from combined work of F. Hirzebruch, K. Kodaira [a4], and S.T. Yau [a5]. For $n=3$ one has that a compact manifold that is bimeromorphically equivalent to a Kähler manifold and that is also topologically $P_\mathbf C^3$ is analytically isomorphic to $P_\mathbf C^3$ [a6].

References

[a1] H. Whitney, "Complex analytic varieties" , Addison-Wesley (1972) MR0387634 Zbl 0265.32008
[a2] W. Barth, C. Peters, A. van der Ven, "Compact complex surfaces" , Springer (1984) MR0749574 Zbl 0718.14023
[a3] R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) MR0608414 Zbl 0435.32004
[a4] F. Hirzebruch, K. Kodaira, "On the complex projective spaces" J. Math. Pures Appl. , 36 (1957) pp. 201–216 MR0092195 Zbl 0090.38601
[a5] S.-T. Yau, "Calabi's conjecture and some new results in algebraic geometry" Proc. Nat. Acad. Sci. USA , 74 (1977) pp. 1798–1799
[a6] T. Peternell, "A rigidity theorem for $P_3(\mathbf C)$" Manuscripta Math. , 50 (1985) pp. 397–428 MR0784150 Zbl 0573.32027
How to Cite This Entry:
Analytic manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Analytic_manifold&oldid=16602
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article