...presented to cover cases when an arbitrary locally compact [[Abelian group|Abelian group]] $G$ is represented by invertible operators $\{ U _ { t } \} _ { t \
...$\mathcal{I} \neq L ^ { 1 } ( G )$, i.e., if $x \neq 0$, by the Tauberian theorem (cf. also [[Tauberian theorems|Tauberian theorems]]). This hull is called t
14 KB (2,151 words) - 17:43, 1 July 2020
...nk, $ t _ \lambda = t ( H _ \lambda ( M) ) $ (the torsion rank of an Abelian group $ A $
...a Morse function for which the Morse inequalities are equalities (Smale's theorem, see [[#References|[2]]]). In particular, on any closed manifold that is ho
6 KB (845 words) - 01:33, 5 March 2022
be a free Abelian group with basis $ \{ e _ \omega \} = \{ e _ {i _ {1} \dots i _ {k} }
The cited theorem asserts that the homomorphism
8 KB (1,150 words) - 18:42, 13 January 2024
The theorem is also associated with the names of E.T. Whittaker, K. Ogura, V.A. Kotel'n
...ver, both situations are covered by the so-called "approximate sampling" theorem, which is valid for not necessarily band-limited signals. It is due to J.R.
9 KB (1,428 words) - 07:01, 19 March 2024
...ed on the [[Cauchy integral theorem|Cauchy integral theorem]]. The residue theorem is fundamental in this theory. Let $ f( z) $
in a neighbourhood of the point at infinity. The residue theorem implies the theorem on the total sum of residues: If $ f( z) $
16 KB (2,407 words) - 07:56, 11 January 2022
...s with $ \mathop{\rm End}\nolimits (V) $ ( "Burnside theoremBurnside's theorem" ). Every normal subgroup of a completely reducible linear group is complet
...explicit function of $ n $ (see also [[Lie–Kolchin theorem|Lie–Kolchin theorem]]); in particular, the commutator subgroup of $ H $ is a unipotent group
16 KB (2,362 words) - 18:01, 12 December 2019
...inear forms it is possible to introduce an addition, converting it into an Abelian group — the Witt group of $ k $(
cf. [[Witt theorem|Witt theorem]]), and also to the case of symmetric bilinear forms associated with quadra
8 KB (1,152 words) - 08:29, 6 June 2020
Then there is the following characterization theorem for the Witt vectors. There is a unique functor $ W : \ \mathbf{Ring} \r
be the Abelian group $ 1 + t A [[t]] $
17 KB (2,502 words) - 17:25, 22 December 2019
...theorem]], which was first obtained for Lie groups. A consequence of this theorem is the fact that every compact group admits a complete system of finite-dim
...here is the following fundamental structure theorem: Every locally compact Abelian group $ G $
14 KB (2,197 words) - 16:40, 31 March 2020
ramified at exactly four points; and also a one-dimensional Abelian
$X$ with the structure of an Abelian group that is compatible with the
19 KB (3,251 words) - 20:37, 19 September 2017
an Abelian group (or module over some ring) $ H _ {q} ( X, A) $
...(the uniqueness theorem). In the category of all polyhedra, the uniqueness theorem holds when the requirement is added that the homology (cohomology) of a uni
5 KB (779 words) - 19:51, 16 January 2024
''Hurwitz formula, Hurwitz theorem''
...n="top">[5]</TD> <TD valign="top"> S. Lang, "Introduction to algebraic and Abelian functions" , Addison-Wesley (1972) {{MR|0327780}} {{ZBL|0255.14001}} </TD><
4 KB (523 words) - 08:11, 6 June 2020
...quivalent to studying modules over the group algebra $KG$. Thus, Maschke's theorem is formulated in the language of group algebras as follows: If $G$ is a fin
...f an ordered group is imbeddable in a skew-field (the Mal'tsev–von Neumann theorem). It is believed that this is also true for any right-ordered group.
5 KB (872 words) - 22:00, 29 April 2012
...nected with the name of W. Burnside, who noted in 1897 that all simple non-Abelian groups which were known at that time were of even order [[#References|[1]]]
is an elementary Abelian group of order $ 2 ^ {d} $;
15 KB (2,211 words) - 08:50, 26 March 2023
Cayley's theorem).
is a simple non-Abelian group. Hölder's theorem: For $ n \neq 2, 6 $,
7 KB (1,003 words) - 08:24, 6 June 2020
form an Abelian group of order $ m $
form an Abelian group with respect to multiplication. The unit of this group is the class
19 KB (3,006 words) - 06:05, 26 May 2024
...ht modification of the same argument yields the so-called six exponentials theorem [[#References|[a2]]], [[#References|[a4]]]: If $x_1,x_2$ are two complex nu
...as results of algebraic independence on values of exponential, elliptic or Abelian functions; more generally it applies to the arithmetic study of commutative
5 KB (726 words) - 09:23, 20 December 2014
...cessarily free, J.R. Stallings [[#References|[a13]]] made use of Grushko's theorem [[#References|[a6]]], which asserts that if a [[Group|group]] $G$ is genera
In attempting to generalize the above Stallings' theorem to pairs of relative cohomological dimension one, C.T.C. Wall [[#References
9 KB (1,405 words) - 15:30, 1 July 2020
field), any finite extension is separable (the theorem on the
$\Gal(K/P)=H$. The main theorem in Galois theory states that these
11 KB (1,965 words) - 04:47, 16 January 2022
the Birkhoff–Frink theorem). Any lattice can be imbedded in the lattice $ \mathop{\rm Sub} A $
...c. The classic example of lattice definability is given by the first basic theorem of projective geometry (see [[#References|[1]]]), where vector spaces over
7 KB (1,037 words) - 08:24, 6 June 2020