...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 $
7 KB (1,109 words) - 16:59, 1 July 2020
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
theorem).
Other examples of group schemes are Abelian (group) varieties {{Cite|Mu}}.
5 KB (831 words) - 21:59, 5 March 2012
L. Stickelberger proved the following theorem: For $ r \geq 1 $,
Stickelberger's theorem implies that $ S $
7 KB (1,035 words) - 05:58, 19 March 2022
...m_{i+j=n} \binom{m_1}{i} \binom{m_2}{j}$ follows by the binomial expansion theorem from $(X+Y)^{m_1+m_2} = (X+Y)^{m_1} (X+Y)^{m_2}$.
.... A pre-$\lambda$-ring structure on $\Lambda(R)$ defines a homomorphism of Abelian groups $\lambda_t : R \rightarrow \Lambda(R)$, $\lambda_t(x) = \lambda^0(x)
10 KB (1,721 words) - 07:44, 23 March 2016
...operator whose norm does not exceed one) on a Hilbert space (von Neumann's theorem). This result is closely connected with the existence of a unitary power di
.... In the case when $\mathfrak A$ is the group algebra of a locally compact Abelian group, spectral sets are also called sets of harmonic synthesis.
2 KB (295 words) - 15:46, 29 December 2018
===Siegel's theorem on Dirichlet L-functions===
...lass number of a quadratic field of discriminant $-D$, it follows from the theorem that
5 KB (784 words) - 20:40, 18 October 2014
and an Abelian group $ G $.
By adding $ r $-dimensional chains as linear forms one obtains the Abelian group $ C _ {r} ( K, G) $
6 KB (897 words) - 08:54, 25 April 2022
but this is inessential.) Sheaves of Abelian groups, rings and other structures can be defined similarly.
Giraud's little theorem). Categories equivalent to one of the form $ Sh ( C, \tau ) $
8 KB (1,216 words) - 18:08, 14 November 2023
...cohomology theory" (but not the designation) while studying "generalized Abelian integrals" (now called "Eichler integrals" ; see below).
...idered here, a suitable version of the [[Riemann–Roch theorem|Riemann–Roch theorem]] shows that $C ^ { + } ( \Gamma , k , \mathbf{v} )$ has finite dimension o
13 KB (1,993 words) - 07:12, 15 February 2024
In formulation 1), the Lévy–Cramér theorem admits a generalization to the convolution of two signed measures with rest
...mér theorem to random variables in Euclidean spaces and in locally compact Abelian groups.
4 KB (647 words) - 19:21, 24 March 2023
...er of a group|Character of a group]]). Indeed, if $G$ is a locally compact Abelian group, the Fourier–Stieltjes transform of a finite measure $\mu$ on $\hat
...-definite functions on $G$. This definition is still valid when $G$ is not Abelian.
14 KB (2,163 words) - 19:56, 8 February 2024