...d if and only if it is isomorphic to a smooth algebraic subvariety of some complex projective space (Kodaira's projective imbedding theorem).
...D> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , '''1''' , Wiley (1978) {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR></tab
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...rojective line $\mathbf P^1$. A complete singular curve $X$ is rational if and only if its geometric genus $g$ is zero, that is, when there are no regular
When $k$ is the field $\mathbf C$ of complex numbers, the (only) non-singular complete rational curve $X$ is the Riemann
1 KB (191 words) - 10:10, 2 November 2014
One of the principal objects of study in algebraic geometry. The
modern definition of an algebraic variety as a reduced
5 KB (706 words) - 13:35, 17 March 2023
A generalization of the concept of a compact complex algebraic variety. A separated variety $X$ is called complete if for any variety $Y$
...tilde X$ (Nagata's theorem). A generalization of the concept of a complete algebraic variety to the relative case is that of a [[proper morphism]] of schemes.
1 KB (231 words) - 19:12, 24 November 2023
...{i\in I} a_i m_i = 0$. Thus there arises the module of syzygies, the chain complex of syzygies, etc. See also [[Hilbert syzygy theorem]].
...als and the theory of regular algebras and regular sequences, cf. [[Koszul complex]]; [[Depth of a module]].
906 bytes (150 words) - 16:48, 23 November 2023
...is the following: A number $\alpha$ is a root of the polynomial $f(x)$ if and only if $f(x)$ is divisible by the binomial $x-\alpha$ without remainder.
...orem is called after E. Bezout [[#References|[1]]], who studied systems of algebraic equations of higher degrees.
2 KB (268 words) - 15:02, 14 February 2020
...e point, also called a node or crunode; the cusp, or spinode; the tacnode; and the ramphoid cusp.
The terms "crunode" and "spinode" are seldom used nowadays (2000).
1 KB (186 words) - 13:06, 23 July 2014
...proof uses only techniques from [[Algebraic topology|algebraic topology]] and so could be considered elementary. They introduced the transfer into fibre
...TD> <TD valign="top"> D.G. Quillen, "Some remarks on etale homotopy theory and a conjecture of Adams" ''Topology'' , '''7''' (1968) pp. 111–116 {{MR|022
2 KB (351 words) - 14:26, 19 April 2014
...f this complex is said to be the $n$-th [[singular homology]] group of $X$ and is denoted by $H_n(X)$.
The concept of the cohomology of a cochain complex is defined in a dual manner.
2 KB (265 words) - 21:51, 22 October 2016
...connected (for example, if $G$ is a Lie group), then $G^0$ is open in $G$ and $G/G^0$ is discrete.
...th the irreducible components. For every polynomial homomorphism $\phi$ of algebraic groups one has $\phi(G^0)=\phi(G)^0$. If $G$ is defined over a field, then
3 KB (419 words) - 11:16, 18 October 2014
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A numerical invariant of a one-dimensional [[Algebraic variety|algebraic variety]] defined over a field $ k $.
3 KB (483 words) - 06:28, 31 March 2023
...implex of sufficiently high dimension. The dimension of a finite geometric complex is the largest dimension of its constituent simplices.
...upper bound of the dimensions of its simplices. For an infinite geometric complex, the topology of the polyhedron induced by its imbedding into the ambient H
3 KB (419 words) - 17:16, 7 February 2011
A [[characteristic class]] of a complex bundle $\zeta$, equal to
...iplicative sequence]] corresponding to the [[power series]] $t/(1-e^{-t})$ and $c_i$ are the [[Chern class]]es.
919 bytes (125 words) - 07:42, 18 March 2023
...l plane of the torus, and the boundaries of the circles lying on the torus and obtained by its rotation from the given circle are said to be meridians of
is the radius of the axial circle, and $ \epsilon = a/l $
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$#C+1 = 25 : ~/encyclopedia/old_files/data/P075/P.0705170 Projective algebraic set
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3 KB (364 words) - 08:09, 13 July 2022
...tructions, which renders the algebraic space a natural object of algebraic geometry.
...f schemes becomes identical with a complete subcategory of the category of algebraic spaces.
4 KB (591 words) - 20:16, 9 October 2017
...ncyclopedia/old_files/data/I052/I.0502010 Intersection index (in algebraic geometry)
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2 KB (289 words) - 06:42, 29 December 2021
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A numerical birational invariant of a two-dimensional algebraic variety defined over an algebraically closed field $ k $.
2 KB (319 words) - 14:59, 30 March 2023
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An imbedding of an algebraic variety $ X $
5 KB (747 words) - 06:29, 30 May 2020
The classical polylogarithms are special functions of a complex variable $z$, defined by the series
for $z$ a complex number inside the unit disk and $n$ an integer at least equal to $1$. These functions satisfy the different
2 KB (237 words) - 19:07, 1 May 2023
A generalization of the concept of a non-singular [[Algebraic variety|algebraic variety]]. A [[Scheme|scheme]] $ X $
with its algebraic closure $ \overline{k} $
3 KB (540 words) - 11:38, 12 October 2023
$#C+1 = 1 : ~/encyclopedia/old_files/data/A010/A.0100450 Abstract algebraic geometry
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8 KB (1,165 words) - 16:08, 1 April 2020
...re on analytically hyperbolic varieties|range of conjectures]] linking the geometry of varieties to their Diophantine properties.
...ariski closure]] of the union of the images of all non-trivial maps from [[algebraic group]]s into $V$. Lang conjectured that the complement of the special set
2 KB (277 words) - 18:16, 2 October 2023
algebraic subvarieties of $A^n$. If $X$ is an affine algebraic variety
(see [[Affine algebraic set|Affine algebraic set]]) in $A^n$, the [[induced topology]] on $X$ is also known as the Zaris
2 KB (304 words) - 20:50, 15 March 2023
.../encyclopedia/old_files/data/A012/A.0102450 Analytic surface (in algebraic geometry)
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8 KB (1,169 words) - 18:47, 5 April 2020
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...are topologically equivalent to a circle in the Euclidean plane, while the complex projective straight line is topologically equivalent to a two-dimensional s
2 KB (335 words) - 08:23, 6 June 2020
...index ([[signature]]) $\sigma(M)$ of a compact [[Kähler manifold]] $M$ of complex dimension $2n$ can be computed by the formula
...or]]). This was proved by W.V.D. Hodge [[#References|[2]]] for the de Rham complex
3 KB (413 words) - 08:52, 23 November 2023
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...ut the importance of this invariant in the theory of the classification of algebraic varieties.
5 KB (777 words) - 22:14, 5 June 2020
$#C+1 = 27 : ~/encyclopedia/old_files/data/A011/A.0101720 Algebraic varieties, arithmetic of,
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5 KB (793 words) - 16:10, 1 April 2020
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...f a complex Kähler manifold, and whose geometry is strongly related to the geometry of this manifold.
6 KB (953 words) - 12:29, 29 December 2021
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is the field of complex numbers, the analytic bundle is said to be holomorphic). The number $ n $
6 KB (951 words) - 06:45, 22 February 2022
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of an [[Algebraic variety|algebraic variety]] $ X $,
8 KB (1,076 words) - 07:58, 21 January 2024
...sisting of a single variety implies resolution of the singularities of $V$ and can be obtained from the local uniformization theorem in dimension $\leq3$.
...">[3]</TD> <TD valign="top"> W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , '''3''' , Cambridge Univ. Press (1954) {{MR|0061846}} {{ZBL|0055.38705}}
3 KB (440 words) - 14:25, 1 May 2014
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...this concept reduces to the concept of a regular mapping (a submersion) of complex manifolds. A finitely-represented (local) morphism of schemes $ f: X \ri
3 KB (548 words) - 06:26, 21 April 2024
...cal problem of the rationality or uni-rationality of the moduli variety of algebraic curves of genus $g$.
...an algebraically closed field $k$ has the structure of a quasi-projective algebraic variety $M_g$ (see [[#References|[3]]]–[[#References|[5]]]).
5 KB (777 words) - 22:00, 7 July 2014
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as a compact algebraic variety with the aid of Grassmann coordinates (cf. [[Exterior algebra|Exter
7 KB (1,104 words) - 19:42, 5 June 2020
A (real or complex) number having the property that any number does not change if zero is adde
...y be zero. Such elements are called [[zero divisor]]s. Fields, skew-fields and [[integral domain]]s do not have zero divisors.
3 KB (471 words) - 21:06, 3 January 2015
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A [[Kähler metric|Kähler metric]] on a [[Complex manifold|complex manifold]] (or orbifold) whose [[Ricci tensor|Ricci tensor]] $\operatorname
6 KB (832 words) - 16:09, 27 January 2024
...orem]]: Let $M$ and $N$ be two [[Differentiable manifold|$C^r$ manifolds]] and $f:M\to N$ a surjective $C^r$ map. If $r> \max \{0, \dim M - \dim N\}$, th
...also used. It refers to a "general" set of conditions which is consistent and is also such that every sentence from a specified set of sentences is decid
4 KB (704 words) - 11:07, 6 September 2013
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over the field of real or complex numbers is a mapping $ x \rightarrow x ^ {*} $
4 KB (550 words) - 22:13, 5 June 2020
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and $ N $
6 KB (927 words) - 22:11, 5 June 2020
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A [[Hermitian metric|Hermitian metric]] on a [[Complex manifold|complex manifold]] whose fundamental form $ \omega $
4 KB (543 words) - 22:15, 5 June 2020
of sections; and
...sheaves. If $\cF$ and $\cS$ are coherent, then so are $\cF\otimes_\cO \cS$ and ${\rm Hom}_\cO(\cF,\cS)$
3 KB (497 words) - 13:34, 6 January 2022
...(over a field $k$) all irreducible components of which have dimension $n$, and which is defined by a homogeneous ideal $I$ in the ring $k[T_0,\dots,T_N]$,
...$X$ is a non-singular connected variety, and $k=\mathbf C$ is the field of complex numbers, then
2 KB (416 words) - 12:41, 19 August 2014
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of integers, the ring of integers of some algebraic number field, the field $ \mathbf R $
4 KB (512 words) - 10:59, 29 May 2020
A complex (sometimes, real) number that is a root of a polynomial
...ynomial $x^2+1$, while $2^{1/n}$, where $n$ is any positive integer, is an algebraic number of degree $n$, being a root of the irreducible polynomial $x^n-2$.
10 KB (1,645 words) - 17:08, 14 February 2020
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sheeted covering surface over the Riemann sphere, extended over all finite and infinitely-distant branch points of $ S $.
1 KB (165 words) - 06:29, 30 May 2020
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A semi-algebraic set in $ \mathbf R ^ {n} $ (or $ k ^ {n} $,
4 KB (573 words) - 18:05, 26 January 2022
...encyclopedia/old_files/data/I050/I.0500320 Implicit function (in algebraic geometry)
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5 KB (768 words) - 22:11, 5 June 2020
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''of an algebraic variety''
3 KB (388 words) - 17:12, 7 February 2021
$#C+1 = 57 : ~/encyclopedia/old_files/data/C024/C.0204160 Complex space,
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5 KB (691 words) - 17:46, 4 June 2020
The divisor class group under algebraic equivalence on a non-singular projective variety.
...ent group $D(X)/D_{\mathrm{a}}(X)$ is called the Néron–Severi group of $X$ and is denoted by $\mathrm{NS}(X)$. The Néron–Severi theorem asserts that th
4 KB (687 words) - 05:47, 15 April 2023
geometry, which reduces the problem of the existence of rational
points on an algebraic variety over a global field to the analogous
3 KB (543 words) - 15:48, 17 February 2012
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...ements points, straight lines, planes, circles, spheres, conics, quadrics, and multi-dimensional analogues of these.
8 KB (1,191 words) - 06:39, 9 April 2023
An [[Algebraic variety|algebraic variety]] which is minimal relative to the existence of birational morphism
...gularities]]) for varieties of arbitrary dimension in characteristic 0 for and for varieties of dimension $n\leq3$ in characteristic $p>5$.
7 KB (1,005 words) - 21:13, 21 November 2018
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of a finite [[CW-complex|CW-complex]] (cf. also [[Cellular space|Cellular space]]) $ X $
11 KB (1,584 words) - 11:51, 8 April 2023
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on a compact [[Complex manifold|complex manifold]] $ X $.
5 KB (719 words) - 22:14, 5 June 2020
[[Cohomology ring|cohomology ring]] of a complex flag manifold $G/P$ (cf. also
[[Linear algebraic group|linear algebraic group]] and $P$ is a
3 KB (371 words) - 18:45, 30 March 2012
...ordinates in a Cartesian coordinate system satisfy an [[Algebraic equation|algebraic equation]] of degree two:
...in the general equation (*), one may transform it by parallel translation and rotation in the coordinate system to one of the 17 canonical forms given be
19 KB (2,446 words) - 17:35, 31 March 2012
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...or finite-dimensional orientable topological manifolds [[#References|[1]]] and for finite cell complexes (see [[#References|[2]]], [[#References|[3]]]). T
10 KB (1,491 words) - 19:04, 26 March 2023
A pair $(X,\xi)$, where $V$ is a complex smooth variety (cf.
[[Algebraic variety|Algebraic variety]]) over an algebraically closed
4 KB (644 words) - 13:06, 17 April 2023
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dimensional manifold) and for which the identity imbedding $ i: N \rightarrow M $
7 KB (987 words) - 08:24, 6 June 2020
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...rmulas in algebraic geometry are the Plücker formulas for a planar reduced and irreducible curve $ Z \subset \mathbf C P ^ {2} $,
5 KB (681 words) - 08:06, 6 June 2020
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and a decomposition $ H _ {\mathbf C } = \oplus _ {p + q = n } H ^ {p,q} $
9 KB (1,281 words) - 12:14, 12 January 2021
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onto the complex plane $ \mathbf C $
7 KB (1,010 words) - 19:41, 5 June 2020
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A [[Homogeneous complex manifold|homogeneous complex manifold]] that is isomorphic to a bounded domain in $ \mathbf C ^ {n} $
5 KB (656 words) - 03:11, 4 January 2022
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...non-singular variety. More precisely, a resolution of singularities of an algebraic variety $ X $
7 KB (965 words) - 08:11, 6 June 2020
[[Algebraic group|algebraic group]] that is a complete
[[Algebraic variety|algebraic variety]]. The completeness condition
8 KB (1,216 words) - 20:39, 5 March 2012
...ive; the property of points to be on one line (collinearity), the order of algebraic curves, etc., are such properties.
...ted belong to the improper line completing this plane. All improper points and lines belong to the improper plane. By completing affine space to projectiv
7 KB (1,119 words) - 13:21, 12 December 2013
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is normal. In a neighbourhood of such a point the space has a reduced and irreducible model. Every simple (non-singular) point is normal. The simples
8 KB (1,143 words) - 16:48, 19 February 2022
''of a complete smooth algebraic variety $X$ over an algebraically
property: For any algebraic family of divisors $D$ on $X$ with base
6 KB (948 words) - 11:41, 31 January 2022
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in a complex space $ X $
7 KB (990 words) - 19:38, 5 June 2020
...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...schemes over $\mathbf{Z}$ with Hermitian complex geometry on their set of complex points. The goal is to provide a geometric framework for the study of Dioph
8 KB (1,219 words) - 21:00, 13 July 2020
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and $ T _ {Y} $
4 KB (554 words) - 06:02, 18 April 2023
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An unbounded domain in the $ ( n + m ) $-dimensional complex affine space of the form
4 KB (618 words) - 07:11, 30 December 2021
...ctive algebraic varieties without singular points over $k$, and Lie groups and their homogeneous spaces.
...ar 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 fund
7 KB (1,028 words) - 13:31, 25 April 2014
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...cally irreducible; for such a scheme the concepts of a connected component and an irreducible component are the same. The set of singular points of a Noet
5 KB (706 words) - 15:10, 7 June 2020
...ertion about the various objects in algebraic geometry (cohomology spaces, algebraic varieties, schemes, fibrations, etc.), stating that these objects depend on
...rst circle of finiteness theorems relates to cohomology spaces of coherent algebraic sheaves. The fundamental theorem states that these spaces are finite-dimens
11 KB (1,549 words) - 20:08, 17 December 2019
The theory of eliminating unknowns from systems of algebraic equations. More precisely, suppose one is given a system of equations
...that of finding conditions on the coefficients of a system of homogeneous algebraic equations ensuring that this system has a non-zero solution.
8 KB (1,363 words) - 16:04, 13 January 2021
$#C+1 = 183 : ~/encyclopedia/old_files/data/A011/A.0101490 Algebraic function
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20 KB (3,036 words) - 07:17, 15 June 2022
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...zed to be related to problems of [[Algebraic geometry|algebraic geometry]] and [[Representation theory|representation theory]], besides [[Spectral theory|
8 KB (1,099 words) - 08:01, 20 February 2023
$#C+1 = 126 : ~/encyclopedia/old_files/data/A011/A.0101460 Algebraic cycle
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12 KB (1,874 words) - 06:08, 17 April 2024
...ls representing geometric conditions used to solve problems in enumerative geometry. This originated in work of M. Chasles
{{Cite|Ch}} on conics and was systematized and used to great effect by H. Schubert in
8 KB (1,263 words) - 08:49, 30 March 2012
...s'''; they correspond to invertible sheaves of $ \mathcal{O}_{X} $-modules and are closely connected with [[Divisor|divisors]] on $ X $; the set of line b
...bundle $ \lambda^{n} E $ is said to be the '''determinant bundle'''. To an algebraic vector bundle $ E $, one can associate the projective bundle $ \mathbf{P}(E
14 KB (2,169 words) - 08:12, 14 December 2016
non-zero elements is not empty and forms a group under multiplication
[[Associative rings and algebras|Associative rings and algebras]]). A
6 KB (929 words) - 00:28, 18 May 2013
...encyclopedia/old_files/data/E037/E.0307070 Extension theorems (in analytic geometry)
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7 KB (1,033 words) - 19:38, 5 June 2020
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and some finite set of numbers (real or complex) by means of a finite number of arithmetical operations. A rational functio
8 KB (1,257 words) - 03:49, 4 March 2022
$#C+1 = 43 : ~/encyclopedia/old_files/data/G043/G.0403740 General\AAhtype algebraic surface,
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7 KB (986 words) - 19:41, 5 June 2020
...K$. In particular, $\operatorname{Max}T_n$ is the polydisc of unit radius, and, more generally, for arbitrary $A$ the space $\operatorname{Max}A$ is an an
...coverings by admissible sets. Morphisms of spaces with a rigid structure, and also the concept of the rigid structure induced on a subspace, are defined
6 KB (892 words) - 20:07, 22 December 2018
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An algebraic or analytic complete non-singular surface $ X $
15 KB (2,046 words) - 05:46, 13 June 2022
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...he lines that belongs to this surface are called (rectilinear) generators, and every curve that intersects all the generators is called a directrix. If $
12 KB (1,684 words) - 08:12, 6 June 2020
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on an algebraic or analytic variety $ X $
10 KB (1,385 words) - 03:10, 2 March 2022
...h of algebra whose main study is derived functors on various categories of algebraic objects (modules over a given ring, sheaves, etc.).
...is done in analytic geometry (with the difference that the transition from geometry to algebra in homology theory is irreversible).
12 KB (1,885 words) - 23:48, 23 April 2017
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is a [[Complex (in homological algebra)|complex (in homological algebra)]] $ C $:
4 KB (611 words) - 01:36, 5 March 2022
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...bra, algebraic geometry and algebraic $K$-theory; see [[#References|[a5]]] and [[#References|[a6]]].
8 KB (1,239 words) - 17:01, 1 July 2020
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be the set of all such non-splitting curves and let $ \rho $
8 KB (1,242 words) - 07:04, 6 May 2022
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is surjective, and let $ {\mathcal J} = \mathop{\rm Ker} f ^ { \# } $.
5 KB (799 words) - 08:03, 6 June 2020
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...etric is induced by the [[Fubini–Study metric|Fubini–Study metric]] on the complex projective space. Similarly, every submanifold in the affine space $ \mat
12 KB (1,750 words) - 19:20, 27 October 2023
The ''Chow ring'' of a non-singular quasi-projective algebraic variety
...of rational equivalence classes of algebraic cycles (cf. [[Algebraic cycle|Algebraic cycle]]) on that variety. Multiplication in this ring is defined in terms o
4 KB (714 words) - 21:54, 24 April 2012
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...nalogue of a [[Hyper-elliptic curve|hyper-elliptic curve]]. A non-singular algebraic projective surface $ X $
5 KB (749 words) - 07:33, 22 December 2023
...that is, an analytic sheaf locally isomorphic to the sheaf $\mathcal O^p$) and also the sheaf of ideals of an analytic set $Y\subset X$, that is, the shea
...\Gamma(X,\mathcal G)$. If $\mathcal F$ is a coherent analytic sheaf on $X$ and $M$ is a submodule of $\mathcal F_x$, $x\in X$, then the submodule $\{s\in\
4 KB (597 words) - 07:18, 22 August 2014
A theory studying continuous families of objects in algebraic geometry.
be a class of objects in algebraic geometry (varieties, schemes, vector bundles, etc.) on which an equivalence relation
16 KB (2,402 words) - 11:49, 16 December 2019
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''on an algebraic variety''
12 KB (1,730 words) - 22:13, 5 June 2020
of algebraic varieties over the field $ \mathbf C $
of complex numbers the cohomology spaces $ H ^{*} (X _{s} ) $
12 KB (1,699 words) - 09:49, 20 December 2019
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...unction|elliptic function]] of one complex variable to the case of several complex variables. A function $ f(z) $
11 KB (1,602 words) - 16:08, 1 April 2020
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...the concept of an [[Analytic manifold|analytic manifold]]. A local model (and, at the same time, the most important example) of an analytic space over a
22 KB (3,277 words) - 01:53, 19 January 2022
...algebraic geometry. In the sequel, an algebraic curve means an irreducible algebraic curve over an algebraically closed field.
The simplest and clearest concept is that of a plane affine algebraic curve. This is a set of points in an affine plane $ A _{k} ^{2} $
22 KB (3,307 words) - 17:02, 17 December 2019
...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...conjugates), obtained in 1976 by A.M. Odlyzko. See also [[Algebraic number|Algebraic number]]; [[Number field|Number field]].
6 KB (866 words) - 19:51, 24 November 2023
* For the field of [[complex number]]s, $u(\mathbb{C}) = 1$.
* The function field of an [[algebraic curve]] over an [[algebraically closed field]] has $u \le 2$; this follows
3 KB (488 words) - 20:22, 15 November 2023
...l closed subgroups of a real or $ p $ -adic analytic group are analytic, and each continuous homomorphism of such groups is analytic (Cartan's theorems,
...omplex and $ p $ -adic analytic groups are called, respectively, complex and $ p $ -adic Lie groups.
6 KB (902 words) - 17:45, 12 December 2019
In algebraic geometry, the term ''divisor'' is used as a generalization of the concept of a divis
...$. The image $\phi(a)\in D_0$ of an element $a\in A^*$ is denoted by $(a)$ and is called the principal divisor of the element $a$. One says that $a\in A^*
16 KB (2,805 words) - 02:18, 6 January 2022
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over a [[Complex space|complex space]] $ X $
7 KB (1,004 words) - 08:07, 6 June 2020
''of one
complex variable in a domain <img align="absmiddle" border="0" src="https://www.enc
...//www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m0634607.png" />,
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
44 KB (5,974 words) - 22:47, 29 November 2014
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A projective algebraic variety defined by a homogeneous equation $ F _ {3} ( x _ {0}, \dots, x _
10 KB (1,376 words) - 11:12, 26 March 2023
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...References|[1]]], who showed that the rationality of the [[zeta-function]] and $ L $-
5 KB (732 words) - 08:12, 21 January 2024
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For non-singular complex projective algebraic varieties there are a number of (co)homological properties, such as Poincar
7 KB (1,043 words) - 22:13, 5 June 2020
...-linear or smooth (diffeomorphisms); imbeddings of one object into another and also immersions (local imbeddings). (cf. [[Homeomorphism]]; [[Diffeomorphis
A very important notion in algebraic topology is that of a [[deformation]]. The principal types of deformations
19 KB (2,788 words) - 09:43, 28 October 2023
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...topological $ K $-theory, and it is characterized by specific algebraic and topological methods of investigation, which are called methods of $ K $-t
17 KB (2,459 words) - 07:32, 26 February 2022
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...isciplines, such as theoretical programming, graph theory, systems theory, and mathematical linguistics.
8 KB (1,265 words) - 17:31, 5 June 2020
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a closed algebraic subvariety of an $ N $-
7 KB (1,079 words) - 16:43, 4 June 2020
...eory of algebraic surfaces one of the most interesting fields of algebraic geometry.
...valent surfaces, and its representations were considered to be imbedded in complex three-dimensional projective space $ \mathbf C P ^ {3} $,
26 KB (3,736 words) - 13:08, 8 February 2020
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...[Differentiable manifold|differentiable manifold]] (or [[Algebraic variety|algebraic variety]]) $ M $,
20 KB (3,056 words) - 08:25, 6 June 2020
A ''linear algebraic group'' is
[[Algebraic group|algebraic group]] that is isomorphic to an algebraic
14 KB (2,165 words) - 22:05, 5 March 2012
[[Algebraic curve|algebraic curve]] of genus 1. The theory of elliptic
curves is the source of a large part of contemporary algebraic
19 KB (3,251 words) - 20:37, 19 September 2017
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...atics in which one studies algebraic operations (cf. [[Algebraic operation|Algebraic operation]]).
17 KB (2,478 words) - 16:09, 1 April 2020
...e consists of a topological space $X$ (the underlying space of the scheme) and a
[[Algebraic variety|algebraic variety]]. For the history of the concept of a scheme, see
12 KB (2,191 words) - 13:20, 24 November 2013
...(schemes, algebraic spaces, etc., cf. [[Scheme|Scheme]]; [[Algebraic space|Algebraic space]]).
...g into a proof as for the formulation of such proofs in their most obvious and most general form.
29 KB (4,414 words) - 17:20, 17 December 2019
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...odic function|double-periodic function]] that is meromorphic in the finite complex $ z $-plane. An elliptic function has the following basic properties.
9 KB (1,292 words) - 19:08, 20 January 2022
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of the complex plane at which $ f ( z) $
9 KB (1,400 words) - 11:09, 26 March 2023
...groups has numerous applications in algebraic geometry, class field theory and cobordism theory.
and field of residues $ k $ ,
17 KB (2,537 words) - 22:38, 15 December 2019
[[Classical group|Classical group]]). In the case when $E=K^{2m}$ and the matrix of $\Phi$ with respect to the canonical basis $\{e_i\}$ of $E$ h
...c group is called the symplectic group of $2m$ variables over the ring $K$ and is denoted by $\def\Sp{ {\rm Sp}}\Sp(m,K)$ or $\Sp_{2m}(K)$. The matrix of
6 KB (1,078 words) - 14:22, 3 November 2013
...frak{M}$ (see [[Functor]]) and a distinguished object $I$ are given on it, and if it admits natural isomorphisms
...1) the natural isomorphisms $\alpha, \lambda, \rho, \kappa$ are coherent; and 2) every functor
3 KB (412 words) - 20:13, 22 December 2017
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for all $g \in G$ and $m , m ^ { \prime } \in M$, where $( g , m ) \rightarrow \square ^ { g } m$
9 KB (1,326 words) - 16:58, 1 July 2020
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...a–Viehweg vanishing theorem states that for a nef and big line bundle on a complex projective manifold $X$,
8 KB (1,253 words) - 15:30, 1 July 2020
...roups of a semi-simple splittable Lie algebra, of a symmetric space, of an algebraic group, etc.
be a connected affine [[Algebraic group|algebraic group]] defined over an algebraically closed field $ k $ .
9 KB (1,394 words) - 16:55, 17 December 2019
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...{ 0 }$ is a [[Polynomial|polynomial]] in $X$ with coefficients in $\bf Z$ and there exists an $x \in \widetilde{\mathbf{Z}}$ such that $f ( x )$ is a uni
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be a scheme and let $ X _ {et} $
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and $ q $
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In the classification of algebraic surfaces, Abelian surfaces are exactly the smooth complete surfaces $ A $
18 KB (2,511 words) - 06:25, 26 March 2023
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induced by two anti-commutative complex structures $ J _ {1} , J _ {2} $
8 KB (1,179 words) - 15:10, 19 January 2021
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The geometry of the Yang–Mills equations (cf. [[Yang–Mills field|Yang–Mills field]
7 KB (1,126 words) - 17:45, 1 July 2020
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A smooth projective [[Algebraic surface|algebraic surface]] $ X $
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A [[Cohomology|cohomology]] theory with values in a [[Sheaf|sheaf]] and with supports contained in a given subset. Let $ X $
10 KB (1,477 words) - 22:17, 5 June 2020
are called the points of the homogeneous space, and the group $ G $
and the subgroup $ H $
20 KB (2,996 words) - 08:42, 16 December 2019
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...ce]]) over an Abelian variety. It was shown by A. Weil [[#References|[1]]] and, in one special case, by F. Châtelet, that for an arbitrary Abelian variet
7 KB (1,109 words) - 16:59, 1 July 2020
...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...nn $\overline { \partial }$-problem]]), the prescribed curvature equation, and global analysis, [[#References|[a1]]], [[#References|[a7]]], [[#References|
10 KB (1,540 words) - 17:43, 1 July 2020
...$f_1, \ldots, f_k$, then the curve $L$ defined by (1) is called reducible and is the union of the curves $L_1, \ldots, L_k$ (the components of $L$) defin
...cible curve. Two irreducible plane real algebraic curves, one of order $n$ and the other of order $m$, intersect in at most $mn$ points (Bezout's theorem)
11 KB (1,916 words) - 00:44, 12 August 2019
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...ication, is a canonically determined projective normal [[Algebraic variety|algebraic variety]] $V ^ { * }$, defined over $\mathbf{C}$, in which $V$ is Zariski-o
13 KB (1,983 words) - 17:00, 1 July 2020
For example, the models of the theory of the complex field $ ( \mathbf C ,\ +,\ \cdot ,\ 0,\ 1) $
...r each model of the theory (cf. also [[Stable and unstable theories|Stable and unstable theories]]).
11 KB (1,671 words) - 11:38, 22 December 2019
A maximal torus of a linear algebraic group $ G $
is an algebraic subgroup of $ G $
9 KB (1,402 words) - 11:29, 17 December 2019
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The complex $ ( \Omega ^ {*} ( M) , d ) $
27 KB (4,062 words) - 01:31, 7 May 2022
...up|compact group]] into the group of continuous linear automorphisms of a (complex) Banach space that is continuous with respect to the strong operator topolo
be a Banach space and let $ \phi : \ G \rightarrow \mathop{\rm GL}\nolimits (V) $
5 KB (643 words) - 00:34, 12 June 2021
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''algebraic integral''
10 KB (1,594 words) - 06:20, 17 April 2024
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...axially-symmetric rigid body, with centre of mass on the axis of symmetry; and iii) the so-called Kowalewski top, which is a solid body rotating about a f
13 KB (1,808 words) - 22:15, 5 June 2020
[[Brauer group|Brauer group]] of $K$ onto the group of all complex
$A$ with generators $a,b$ and defining relations $a^n=x$, $b^n=y$, $ba=\epsilon ab$, where
8 KB (1,182 words) - 22:00, 5 March 2012
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...t (cf. [[Associative rings and algebras|Associative rings and algebras]]), and let $ A $
16 KB (2,248 words) - 22:15, 5 June 2020
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...ery powerful tool in many applications, particularly communication science and coding theory.
12 KB (1,769 words) - 08:31, 26 March 2023
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be an $ A $-module of finite type and $ \mathfrak a $
4 KB (548 words) - 06:47, 16 June 2022
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i) in general, they are non-commutative in dimensions $1$ and $2$; and
13 KB (1,937 words) - 13:10, 24 December 2020
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are called lines and I is the incidence relation. A subspace of $ \pi $
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and $ \delta $
8 KB (1,166 words) - 19:43, 5 June 2020
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...{SL} ( 2 , \mathbf{Q} )$-equivariant topology that is the given one on $H$ and makes $\mathbf{P} ^ { 1 } ( \mathbf{Q} )$ discrete; a deleted neighbourhood
10 KB (1,509 words) - 16:57, 1 July 2020
...beta$ is transcendental if $\alpha \ne 0,1$ is algebraic and $\beta$ is an algebraic irrational (the so-called Hilbert's seventh problem); A. Baker [[#Reference
...ry general statements of $\mathbf{Q}$-linear independence on commutative [[algebraic group]]s defined over $\mathbf{Q}$.
7 KB (1,017 words) - 17:25, 12 November 2023
...technology and natural science, the accumulation of quantitative relations and spatial forms studied in mathematics is continuously expanding; so this gen
...with very simple aspects of economic life. The first problems of mechanics and physics were already in this collection of fundamental mathematical ideas.
22 KB (3,256 words) - 21:54, 30 March 2012
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...undle]]; [[Tangent bundle|Tangent bundle]]). For a comprehensive treatment and lists of references, see [[#References|[a8]]], [[#References|[a9]]]. See al
12 KB (1,916 words) - 16:55, 1 July 2020
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...and formulation are given, together with a counterpart of the theorem for algebraic transformation groups, called the étale slice theorem.
11 KB (1,819 words) - 15:30, 1 July 2020
...f invariants arose from a number of problems in number theory, algebra and geometry. C.F. Gauss in his studies on the theory of binary quadratic forms posed th
and $ \alpha $ ,
22 KB (3,406 words) - 07:08, 6 May 2022
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...ty of $C$, see [[#References|[a1]]]. For an interplay between the geometry and the algebra, see [[#References|[a2]]].
16 KB (2,510 words) - 17:43, 1 July 2020
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The branch of algebra studying the properties of commutative rings and objects relating to them (ideals, modules, valuations, etc., cf. [[Ideal|Id
16 KB (2,400 words) - 17:45, 4 June 2020
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...rable difficulties and is carried out differently in different branches of geometry.
19 KB (2,971 words) - 19:30, 26 March 2023
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and a function $ \widehat{a} $
3 KB (474 words) - 19:41, 5 June 2020
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and of Abelian integrals (cf. [[Abelian integral|Abelian integral]]) of the fir
7 KB (963 words) - 14:55, 7 June 2020
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A series of functions used in the representation of automorphic forms and functions (cf. [[Automorphic form|Automorphic form]]; [[Automorphic functio
5 KB (778 words) - 08:25, 6 June 2020
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...e unit sphere in $E$ (in case $E$ has finite dimension, $G$ is semi-simple and also has finite dimension — the induced [[Unitary representation|unitary
10 KB (1,477 words) - 06:10, 26 March 2024
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and b) for any $ \alpha , \beta \in A $
17 KB (2,329 words) - 01:38, 30 December 2021
$#C+1 = 88 : ~/encyclopedia/old_files/data/D032/D.0302630 Diophantine geometry,
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24 KB (3,602 words) - 11:48, 26 March 2023
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and an $ U $-isomorphism of fibre bundles $ \phi : \pi ^ {-1} ( U) \right
11 KB (1,695 words) - 20:37, 23 December 2023
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of an algebraic or Kähler variety $ X $
6 KB (967 words) - 08:26, 6 June 2020
...of transformations]] $ \Gamma $ of analytic transformations of a given complex manifold $ M $ :
...defined on a bounded connected domain $ D $ of the $ n $-dimensional complex space $ \mathbf C ^{n} $ that are invariant under a discrete group $ \
17 KB (2,502 words) - 06:16, 12 July 2022
and $ F _{0} $;
and is called the extension of the field $ F _{0} $
13 KB (1,981 words) - 20:02, 17 December 2019
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...the phrase "magnetic monopole" usually denotes a Yang–Mills potential $A$ and Higgs field $\phi$ whose equations of motion are determined by the Yang–M
11 KB (1,671 words) - 16:59, 1 July 2020
...s considered. Henceforth, as a rule, real Lie groups are considered (every complex Lie group is naturally endowed with the structure of a real Lie group by me
...$ \mathbf R $ of real numbers (see also [[Linear group|Linear group]]) and its subgroups, closed in the natural Euclidean topology (J. von Neumann, 19
26 KB (3,980 words) - 17:06, 13 June 2020
...vering as a whole (in particular, the number of elements, the multiplicity and the combinatorial properties) reflect essential global characteristics of t
...nd where a finite set of these vertices constitutes an abstract simplex if and only if the intersection of the corresponding elements of $\gamma$ is not e
10 KB (1,606 words) - 07:49, 17 May 2017
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...her dimension is due to A. Grothendieck [[#References|[a3]]]. For a proper and smooth morphism $ f: X \rightarrow S $
11 KB (1,657 words) - 11:55, 12 January 2021
...nnected with solving inequalities in integers — Diophantine inequalities — and also with solving equations in integers (cf. [[Diophantine equations|Diopha
...fmath.org/legacyimages/d/d032/d032600/d03260013.png" />-adic number fields and Diophantine approximations in the field of power series).
54 KB (7,359 words) - 18:32, 31 March 2017
...these theorems usually become invalid, and the Grothendieck group $K_0(R)$ and the Whitehead group $K_1(R)$ are, in a certain sense, a measure of their de
...jects may be studied with the aid of the homotopy theory of vector bundles and of topological
14 KB (2,405 words) - 22:14, 10 January 2015
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'' $ \theta $-function, of one complex variable''
14 KB (1,941 words) - 05:01, 23 February 2022
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pairs of non-intersecting discs and by identifying each pair of boundary circles with boundaries of a bent cyli
20 KB (3,075 words) - 08:26, 6 June 2020
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...oduct theorem [[#References|[1]]]: For any given sequence of points in the complex plane $ \mathbf C $,
22 KB (3,120 words) - 01:51, 24 February 2022
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is an integer and $ F $
4 KB (621 words) - 08:26, 6 June 2020
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...l manifold with boundary $\partial T ( h ) = \partial F \times S ^ { 1 }$, and the union
7 KB (1,078 words) - 15:30, 1 July 2020
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A complete [[algebraic curve]] $ X _ {\widetilde \Gamma } $
8 KB (1,144 words) - 05:57, 13 February 2024
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...ction|residue of an analytic function]] of one complex variable to several complex variables. Let $ X $
11 KB (1,662 words) - 08:11, 6 June 2020
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Consider all tensors with even covariant and contravariant valencies at a point of the space $ A _ {n} $(
7 KB (1,010 words) - 10:59, 29 May 2020
$#C+1 = 269 : ~/encyclopedia/old_files/data/P075/P.0705650 Pseudo\AAhconvex and pseudo\AAhconcave
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19 KB (2,892 words) - 07:31, 11 May 2024
...finite mathematics is sometimes used as a synonym for discrete mathematics and discrete analysis. In what follows the term "discrete analysis" is understo
...thematics are used to study continuous models and vice versa — the methods and formulations of problems of classical mathematics are employed in the study
9 KB (1,264 words) - 16:56, 15 April 2012
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...cteristic class]] defined for complex vector bundles. A Chern class of the complex vector bundle $ \xi $
13 KB (1,910 words) - 19:40, 7 January 2024
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...the notion of a quantum group (cf. also [[Quantum groups|Quantum groups]]) and a [[Super-group|super-group]], in which the role of Bose–Fermi statistics
21 KB (3,130 words) - 17:42, 1 July 2020
...arrow\lVert x\rVert$ from a [[vector space]] $X$ over the field of real or complex numbers into the real numbers, subject to the conditions:
# $\lVert x\rVert\geq 0$, and $\lVert x\rVert=0$ for $x=0$ only;
8 KB (1,284 words) - 21:49, 6 June 2016
$#C+1 = 83 : ~/encyclopedia/old_files/data/F040/F.0400960 Foundations of geometry
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25 KB (3,631 words) - 19:39, 5 June 2020
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One of the [[Interpolation|interpolation]] problems in the complex domain.
23 KB (3,450 words) - 17:45, 1 July 2020
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and therefore surfaces with $ H = 0 $
29 KB (4,187 words) - 08:00, 6 June 2020
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.... Such a $\xi $ is called the characteristic vector field on $\mathcal{S}$ and its properties can be used as an alternative characterization of a Sasakian
17 KB (2,475 words) - 12:23, 12 December 2020
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Let $G$ be a [[Group|group]] of order $v$, and let $D$ be a $k$-subset of $G$, such that the list of quotients $d e ^ { -
9 KB (1,331 words) - 19:36, 13 February 2024
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are arbitrary real numbers and $ x = ( x _ {1}, \dots, x _ {n} ) $.
14 KB (2,052 words) - 10:49, 21 March 2022
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and an oriented, locally flat, submanifold of it, $ k ^ {n} $,
13 KB (1,934 words) - 19:26, 17 January 2024
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...oncepts in number theory. For example, the concept of an algebraic integer and a divisor first arose in the study of cyclotomic fields.
12 KB (1,769 words) - 11:19, 26 March 2023
...and physical studies of self-similarity. Although it is geometrically more complex than the classic [[Cantor set|Cantor set]], it is still one of the simplest
The Sierpiński pre-gaskets (left) and the Sierpiński gasket (right)
26 KB (3,837 words) - 19:21, 17 March 2024
...$ has the form $\lambda_iI+N_i$ where $I$ is the identity mapping on $V_i$ and each $N_i$ is nilpotent (see e.g. [[#References|[a1]]]).
An equivalent and perhaps more familiar description of $f$ is through the [[Matrix|matrix]] $
6 KB (984 words) - 06:48, 16 March 2024
...number]]s and over the field $R$ of [[real number]]s; they are said to be complex, respectively real, vector spaces.
...on]]s, spaces of [[summable function]]s, spaces of [[analytic function]]s, and spaces of [[Function of bounded variation|functions of bounded variation]].
14 KB (2,558 words) - 11:28, 21 June 2016
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...is also a very active field of contemporary research, both for small scale and large scale problems.
10 KB (1,478 words) - 17:02, 1 July 2020
$#C+1 = 85 : ~/encyclopedia/old_files/data/D032/D.0302180 Differential geometry of manifolds
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30 KB (4,323 words) - 19:35, 5 June 2020
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...teraction in the Weinberg–Salam theory of electrically weak interactions), and finally, the gluon field (the carrier of the strong interaction) are descri
14 KB (2,063 words) - 02:45, 18 July 2022
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...non-degenerate systems of equations and also as various sets of geometric and other objects allowing local parametrization (see below); for example, the
30 KB (4,462 words) - 07:59, 6 June 2020
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...the Henselization of a local ring essentially of finite type over a field, and an excellent Dedekind ring all have the Artin approximation property. The f
16 KB (2,447 words) - 00:27, 15 February 2024
If $A$ is a commutative [[Noetherian ring]] and $A[X_1,\ldots,X_n]$ is the [[ring of polynomials]] in $X_1,\ldots,X_n$ with
and only for them. This theorem, which was obtained by D. Hilbert in 1892, was
18 KB (2,720 words) - 19:17, 19 December 2019
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...matrix-functions holomorphic inside (respectively, outside) $\mathcal{T}$, and $\delta = \operatorname { diag } ( z ^ { k _ { i } } )$ is a diagonal matri
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...homotopies coming from subcollections or subcomposites of the $n$ mappings and these various homotopies at different levels are related by coherence condi
9 KB (1,449 words) - 00:50, 15 February 2024
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...nti-holomorphic involution, cf. also [[Riemann surface|Riemann surface]]), and with the theory of multi-dimensional systems. The theory of operator vessel
24 KB (3,136 words) - 20:00, 24 November 2023
''of two locally convex spaces $E_1$ and $E_2$''
...om $E_1 \times E_2$ into $F$. Then the topological tensor product of $E_1$ and $E_2$ (with respect to $T(F)$) is the (unique) locally convex space $E_1 \t
5 KB (896 words) - 23:16, 10 April 2019
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is a metric space and the transformations from $ \Gamma $
15 KB (2,190 words) - 11:51, 26 March 2023
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''of an analytic function $ w = f( z) $ of a complex variable $ z $''
34 KB (4,972 words) - 13:13, 6 January 2022
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...with the problem of classification of all possible pairwise non-isomorphic complex structures on a given differentiable real manifold. The fundamental idea (w
41 KB (5,916 words) - 11:24, 26 March 2023
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...e|Incidence]]) between them. Configurations can be both located in a place and in space.
27 KB (3,830 words) - 09:57, 26 March 2023
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...o account both the geometry of the domain on which the function is defined and the behaviour of the function while approaching the boundary, so that the s
21 KB (3,060 words) - 20:29, 10 January 2021
Galois group of an extension $K/k$ and suppose
defined by the complex $(C^n,d)$, where
17 KB (2,785 words) - 22:37, 23 December 2014
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...below. Let $K$ be a number field. Fix a [[Prime number|prime number]] $p$ and consider a set $\mathcal{S}$ of square-free ideals $L$ in $\mathcal{O} _ {
19 KB (2,901 words) - 17:41, 25 November 2023
...od for studying many problems in contemporary algebra, geometry, topology, and analysis.
...orphism. The space $\cF$ together with the stalk-wise algebraic operations and the projection $p$ is called the sheaf of Abelian groups (rings, etc.) over
26 KB (4,342 words) - 15:06, 15 July 2014
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'' $ f( z) $ of one complex variable at a finite isolated [[Singular point|singular point]] $ a $ of
16 KB (2,407 words) - 07:56, 11 January 2022
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and transforms them into vector-valued functions $ v = ( v _ {1}, \dots, v _
25 KB (3,768 words) - 09:07, 14 June 2022
The notion of Orientation is a formalization and far-reaching generalization of the concept of direction on a curve. The ori
[[Poincaré complex|Poincaré complex]], etc.). The modern view of orientation is given in
18 KB (2,980 words) - 17:05, 13 June 2020
...n as the [[Continuum hypothesis|Continuum Hypothesis]]. Solved by K. Gödel and P.J. Cohen in the (unexpected) sense that the continuum hypothesis is indep
...y of mathematics associated with this problem, see [[Hilbert 2nd problem]] and [[Hilbert program]].
29 KB (4,109 words) - 19:54, 18 March 2018
''Associative rings and algebras'' are
rings and algebras with an associative multiplication, i.e., sets with
11 KB (1,726 words) - 20:09, 15 December 2020
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...ebra concerned with braids, the groups formed by their equivalence classes and various generalizations of these groups [[#References|[1]]].
24 KB (3,637 words) - 08:41, 26 March 2023
and $ b $
and $ b $
20 KB (3,011 words) - 09:59, 26 March 2023
The most known examples are the tangent and cotangent bundle of a smooth manifold. The [[covering]]s are also a special
...es [[fibre space]] or ''fibered space'' are used. </ref> and the ''base'', and $F$ yet another topological space called the ''(generic) fiber'', such that
32 KB (5,299 words) - 12:27, 12 December 2020
...), differential geometry (semi-groups of partial transformations), and the algebraic theory of automata (semi-groups of automata).
...the general theory and in some applications. Let $X$ be an arbitrary set, and let an operation on the set $F_X$ of all finite sequences of elements of $X
17 KB (2,435 words) - 09:18, 2 April 2023
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if and only if $ X = \emptyset $.
38 KB (5,928 words) - 19:35, 5 June 2020
...Riemann's zeta-function, its generalizations and analogues. Zeta-functions and their generalizations in the form of $L$-functions (cf. [[Dirichlet L-funct
which converges absolutely and uniformly in any bounded domain of the complex $s$-plane for which $\sigma\geq1+\delta$, $\delta>0$. If $\sigma>1$, a vali
45 KB (7,251 words) - 02:20, 29 June 2022
'''On a ''general'' complex quintic threefold, there are only finitely many smooth rational curves of f
...{P}^1</math> and rescaling yields a <math>(5d+1)</math>-dimensional smooth and irreducible space. Fix a choice of ''smooth'' rational curve <math>C</math>
14 KB (2,200 words) - 18:27, 23 October 2017
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..., to the theory of groups, as well as to spinor and twistor calculus. More and more often computers are being used to do calculations (including symbolic
18 KB (2,719 words) - 19:42, 5 June 2020
$#C+1 = 92 : ~/encyclopedia/old_files/data/R082/R.0802480 Rings and algebras
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21 KB (3,225 words) - 09:25, 13 July 2022
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...operators acting on the [[Hilbert space|Hilbert space]] of square summable complex sequences.
14 KB (2,126 words) - 16:45, 1 July 2020
$#C+1 = 247 : ~/encyclopedia/old_files/data/T092/T.0902810 Tight and taut immersions
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24 KB (3,546 words) - 19:41, 1 May 2023
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...a20]]] (cf. also [[Fermat last theorem|Fermat last theorem]]). For details and generalizations of Iwasawa theory, see [[#References|[a10]]], [[#References
19 KB (2,876 words) - 05:38, 15 February 2024
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...oth a bracket and an associative multiplication defined on the same space, and to impose a "Leibniz rule" relating both operations, stating that the adj
9 KB (1,452 words) - 06:17, 15 February 2024
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...cise definition; for simplicity, one may think of a polyhedron): The chain complex $\underline{\underline{C}} ( \tilde { K } )$ encodes the relators of the pr
31 KB (4,667 words) - 17:46, 1 July 2020
...rs, with an involution $x \rightarrow x^*$, $x \in A$, such that the norm and the involution are connected by the relation $\norm{x^* x} = \norm{x}^2$ f
1) The algebra $C_0(X)$ of continuous complex-valued functions on a locally compact Hausdorff space $X$ which tend towar
15 KB (2,316 words) - 21:05, 20 April 2012
...are the foundations of the theory developed in the 19th century by S. Lie and his school (see ).
and suppose that in local coordinates in a neighbourhood of the set $ \{ e \}
8 KB (1,148 words) - 10:21, 16 December 2019
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...|[a5]]], [[#References|[a3]]], [[#References|[a6]]], [[#References|[a16]]] and the original articles cited therein.
18 KB (2,722 words) - 17:47, 1 July 2020