...are the three more extensive problems. The unidirectional word problem for Abelian semi-groups is decidable and, hence, so are the three subproblems described
...ility|Unsolvability]]; [[Gödel incompleteness theorem|Gödel incompleteness theorem]]. It was shown already by K. Gödel that the existence of undecidable prop
7 KB (1,138 words) - 13:14, 10 April 2014
...ative multiplication and addition exist; a near-ring is a (not necessarily Abelian) group with respect to addition, and the right distributive property
...$, the equality $A=M_S(\Gamma)$ holds (an analogue of the Wedderburn–Artin theorem). For every $\nu=0,1,2$, the Jacobson radical $J_\nu(A)$ of type $\nu$ can
5 KB (767 words) - 11:23, 27 October 2014
...s were introduced by J. Tate in order to study degenerations of curves and Abelian varieties over $ K $.
cf. also [[Weierstrass theorem|Weierstrass theorem]]); affinoid algebras are Noetherian rings, and even excellent rings if the
8 KB (1,161 words) - 08:25, 6 June 2020
...inable in an $o$-minimal expansion of $\mathbf{R}$" . This is a finiteness theorem, and van den Dries aims to explain the other finiteness phenomena in real a
2) An [[ordered group]] is $o$-minimal if and only if it is divisible Abelian.
5 KB (799 words) - 09:27, 26 November 2016
$#C+1 = 102 : ~/encyclopedia/old_files/data/D110/D.1100020 De Finetti theorem
...ent is De Finetti's theorem. Thus, an equivalent statement of De Finetti's theorem is that the extremal points of the convex set of exchangeable probability m
13 KB (1,888 words) - 11:23, 26 March 2023
...t})$. A theorem of Morita says that if $\mathcal{C}$ and $\mathcal{D}$ are Abelian full subcategories with $A \in \mathcal{C}$ and $B \in \mathcal{D}$, then a
3 KB (499 words) - 20:17, 26 November 2016
is a finite extension, then, according to the Chebotarev density theorem, for any automorphism $ \sigma \in \mathop{\rm Gal} ( K/k) $
For an Abelian extension $ K/k $,
4 KB (569 words) - 05:59, 11 October 2023
==Riesz decomposition theorem for super- or subharmonic functions.==
...nd [[Riesz theorem(2)|Riesz theorem]] (where it is simply called the Riesz theorem), [[#References|[a12]]], [[#References|[a20]]]. See also [[#References|[a8]
8 KB (1,180 words) - 17:00, 1 July 2020
$#C+1 = 96 : ~/encyclopedia/old_files/data/R081/R.0801980 Riemann\ANDRoch theorem
A theorem expressing the [[Euler characteristic|Euler characteristic]] $ \chi ( {\m
10 KB (1,385 words) - 03:10, 2 March 2022
...eory of compact groups, and all the results of that theory (the Peter–Weyl theorem, the theory of characters, orthogonality relations, etc.) are valid (and si
...ion of every irreducible representation is a divisor of the index of every Abelian normal subgroup of $ G $(
10 KB (1,488 words) - 19:39, 5 June 2020
Although the above theorem divides the Baumslag–Solitar groups into three classes: those that are re
...numbers. Thus, such groups are meta-Abelian (cf. [[Meta-Abelian group|Meta-Abelian group]]) and have strong structural properties; in particular, they do not
18 KB (2,803 words) - 16:46, 1 July 2020
...almost-periodic functions on a group depends essentially on the mean-value theorem (cf. [[#References|[5]]], [[#References|[8]]]). A linear functional $ M _
Theorem 1 (the Parseval equality). For an almost-periodic function $ f (x) $
12 KB (1,716 words) - 11:05, 10 May 2020
...from the category of pointed topological spaces into the category of (non-Abelian) groups. For any path $ \phi $
Poincaré's theorem).
6 KB (829 words) - 05:44, 13 April 2023
are coherent (the finiteness theorem). A similar fact holds for étale cohomology. In particular, if $ X $
the comparison theorem). 3) If $ X $
9 KB (1,267 words) - 08:08, 6 June 2020
...h unique root extraction. The following groups are orderable: torsion-free Abelian groups, torsion-free nilpotent groups, free groups, and free solvable group
...rderable groups is itself orderable. For orderable groups there is a local theorem (see [[Mal'tsev local theorems|Mal'tsev local theorems]]). A totally ordere
3 KB (495 words) - 19:18, 17 August 2014
...divisors corresponding to $ D $ (see [[Riemann–Roch theorem|Riemann–Roch theorem]]). If $ D $
is a free Abelian group of rank $ \rho $,
10 KB (1,443 words) - 15:43, 1 March 2022
is an Abelian group, then a transgression in $ E $
...fibre bundles. An important role is played here by the Borel transgression theorem: If $ A $
3 KB (536 words) - 09:05, 8 April 2023
...lian extension of $\textbf{Q}$, or, equivalently (by the [[Kronecker–Weber theorem]]), the maximal cyclotomic extension of $\textbf{Q}$.
...y, inverse problem of|Galois theory, inverse problem of]]). By the Iwasawa theorem [[#References|[a7]]], p. 567, (see also [[#References|[a1]]], Cor. 24.2), a
11 KB (1,686 words) - 18:54, 6 February 2021
even an Abelian group). By definition, if $ x = [ u ] $
even in the category of Abelian groups).
33 KB (4,910 words) - 10:04, 15 December 2019
...umbers without an obvious analogue for algebraic numbers is related to the theorem on unique factorization of rational integers $ n $ in prime factors: $$
...complicated. The question arises: What becomes of the unique factorization theorem, does it have a meaning at all in algebraic number fields?
28 KB (4,440 words) - 22:00, 11 December 2019