$#C+1 = 261 : ~/encyclopedia/old_files/data/A110/A.1100040 Abelian surface
An [[Abelian variety|Abelian variety]] of dimension two, i.e. a complete connected group variety of dime
18 KB (2,511 words) - 06:25, 26 March 2023
and an Abelian group $ G $.
see [[De Rham theorem|de Rham theorem]]).
16 KB (2,386 words) - 16:47, 20 January 2024
...lpotent as an abstract group (cf. [[Nilpotent group|Nilpotent group]]). An Abelian Lie group is nilpotent. If $ F = \{ V _{i} \} $ is a [[Flag|flag]] in a
...p version of Engel's theorem admits the following strengthening (Kolchin's theorem): If $ G $ is a subgroup of $ \mathop{\rm GL}\nolimits (V) $ , where
5 KB (803 words) - 18:12, 12 December 2019
The Mittag-Leffler theorem on expansion of a meromorphic function (see , ) is one of the basic theorem
The Mittag-Leffler theorem implies that any given meromorphic function $f(z)$ in $\mathbb{C}$ with pol
6 KB (980 words) - 18:47, 24 May 2017
...also [[Graph automorphism|Graph automorphism]]). This is know as Frucht's theorem. In 1949, Frucht [[#References|[a3]]] extended this result by showing that
...ese results see [[#References|[a4]]]. For an alternative proof of Frucht's theorem see [[#References|[a5]]].
3 KB (387 words) - 09:29, 19 January 2021
...oduct|Direct product]]) of $R$-groups, are $R$-groups. The following local theorem is valid for the class of $R$-groups: If all finitely-generated subgroups o
1 KB (254 words) - 11:47, 29 June 2014
Theorem 1) was later strengthened; namely, it was proved that the condition $ a _
...should mean. In [[#References|[a1]]], p. 195, is written: " a theorem is Abelian if it says something about an average of a sequence from a hypothesis about
11 KB (1,603 words) - 10:19, 7 May 2021
...athbf{Z}$, then this is true only under the condition that $L/M$ is a free Abelian group [[#References|[2]]]. The finitely-generated subalgebras of a free Lie
...belong to $L(X)$ are given by the Specht–Wever theorem and the Friedrichs theorem, respectively. The first one says that a homogeneous element $a$ of degree
3 KB (564 words) - 19:53, 15 March 2023
...uction associated with special radical subcategories; it first appeared in Abelian categories in the description of the so-called Grothendieck categories in t
be an [[Abelian category|Abelian category]]. A full subcategory $ {\mathfrak A ^ \prime } $
10 KB (1,375 words) - 22:17, 5 June 2020
...n a number of areas of analysis. If $\{x_k\}$ is a sequence in a Hausdorff Abelian [[Topological group|topological group]] $(G,\tau)$, then $\{x_k\}$ is $\tau
...rem|Banach–Steinhaus theorem]] and the [[Mazur–Orlicz theorem|Mazur–Orlicz theorem]] on the joint continuity of separately continuous bilinear operators are p
3 KB (545 words) - 12:08, 3 August 2014
...e of this generalized limit by using the [[Hahn–Banach
theorem|Hahn–Banach
theorem]]. Today (1996), Banach limits are studied via the notion of amenability.
...real numbers are amenable (left and right). M.M. Day has proved that every
Abelian semi-group is left and right amenable. On the other hand, <img align="absmi
10 KB (1,395 words) - 06:44, 9 October 2016
...[[Topological field|Topological field]]) satisfying the implicit function theorem, see [[#References|[a3]]].
...a1]</TD> <TD valign="top"> F.-V. Kuhlmann, "Valuation theory of fields, abelian groups and modules" , ''Algebra, Logic and Applications'' , Gordon&Brea
3 KB (405 words) - 19:33, 28 April 2014
...C ^ { * } ( G )$ and called the full $C ^ { * }$-algebra of $G$. If $G$ is Abelian and $\hat { C }$ its dual group, then $C ^ { * } ( G )$ is isometrically is
This Banach algebra is called the Fourier–Stieltjes algebra of $G$. If $G$ is Abelian, then $B ( G )$ is isometrically isomorphic to the Banach algebra of all bo
7 KB (1,059 words) - 15:30, 1 July 2020
...} )$, where $\hat { C }$ is the dual group of $G$. For $G$ not necessarily Abelian, $A _ { 2 } ( G )$ is precisely the [[Fourier-algebra(2)|Fourier algebra]]
...1]]]) a kind of "non-commutative harmonic analysis on G" , where (for $G$ Abelian) $A _ { p } ( G )$ replaces $L _ { \text{C} } ^ { 1 } ( \hat { G } )$ and $
11 KB (1,698 words) - 07:42, 27 January 2024
...nces|[1]]] and, in one special case, by F. Châtelet, that for an arbitrary Abelian variety $ A $
...trary orders [[#References|[4]]], [[#References|[5]]]. According to Lang's theorem, $ { \mathop{\rm WC} } ( A, k) = 0 $
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is only a [[Complex manifold|complex manifold]], but the Baily–Borel theorem [[#References|[a2]]] endows it with a canonical structure of a quasi-projec
...type), or Abelian motives with additional structure (Shimura varieties of Abelian type) [[#References|[a4]]], [[#References|[a6]]] (cf. also [[Moduli theory|
6 KB (890 words) - 19:42, 20 February 2021
...s , s ) ) )$. The main result of the theory is the arithmetic Riemann–Roch theorem, which computes the behaviour of the Chern character under direct image [[#
...nal points of $X$ is contained in the union of finitely many translates of Abelian proper subvarieties of $A$.
8 KB (1,219 words) - 21:00, 13 July 2020
...d relative homotopy groups as crossed modules, thus giving non-trivial non-Abelian information and often determining the $2$-type of a space. Some of the expl
...]]] for the presentation. This module should be thought of as giving a non-Abelian form of syzygies (cf. also [[Syzygy|Syzygy]]), and as the start of a free c
9 KB (1,326 words) - 16:58, 1 July 2020
the Hodge conjecture is equivalent to the [[Lefschetz theorem]] on cohomology of type $ ( 1, 1) $.
is a simple five-dimensional Abelian variety (see [[#References|[6]]]).
6 KB (935 words) - 09:01, 21 January 2024
...Maxwell equations|Maxwell equations]] (in vacuum). The quantization of non-Abelian gauge theories is still in its infancy.
...ohomology class. An analogous formula in dimension two is Gauss' classical theorem expressing the [[Euler characteristic|Euler characteristic]] as the integra
7 KB (1,013 words) - 16:45, 1 July 2020