User:Richard Pinch/sandbox-WP
Baer–Specker group
An example of an infinite Abelian group which is a building block in the structure theory of such groups.
Definition
The Baer-Specker group is the group B = ZN of all integer sequences with componentwise addition, that is, the direct product of countably many copies of Z.
Properties
Reinhold Baer proved in 1937 that this group is not free abelian; Specker proved in 1950 that every countable subgroup of B is free abelian.
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Descendant subgroup
A subgroup of a group for which there is an descending series starting from the subgroup and ending at the group, such that every term in the series is a normal subgroup of its predecessor.
The series may be infinite. If the series is finite, then the subgroup is subnormal.
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Essential subgroup
A subgroup that determines much of the structure of its containing group. The concept may be generalized to essential submodules.
Definition
A subgroup \(S\) of a (typically abelian) group \(G\) is said to be essential if whenever H is a non-trivial subgroup of G, the intersection of S and H is non-trivial: here "non-trivial" means "containing an element other than the identity".
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Lambert summation
In mathematical analysis, Lambert summation is a summability method for a class of divergent series.
Definition
A series \(\sum a_n\) is Lambert summable to A, written \(\sum a_n = A (\mathrm{L})\), if
\[\lim_{r \rightarrow 1-} (1-r) \sum_{n=1}^\infty \frac{n a_n r^n}{1-r^n} = A . \, \]
If a series is convergent to A then it is Lambert summable to A (an Abelian theorem).
Examples
- \(\sum_{n=0}^\infty \frac{\mu(n)}{n} = 0 (\mathrm{L})\), where μ is the Möbius function. Hence if this series converges at all, it converges to zero.
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Pinch point
A pinch point or cuspidal point is a type of singular point on an algebraic surface. It is one of the three types of ordinary singularity of a surface.
The equation for the surface near a pinch point may be put in the form
\[ f(u,v,w) = u^2 - vw^2 + [4] \, \]
where [4] denotes terms of degree 4 or more.
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Residual property
In the mathematical field of group theory, a group is residually X (where X is some property of groups) if it "can be recovered from groups with property X".
Formally, a group G is residually X if for every non-trivial element g there is a homomorphism h from G to a group with property X such that \(h(g)\neq e\).
More categorically, a group is residually X if it embeds into its pro-X completion (see profinite group, pro-p group), that is, the inverse limit of \(\phi\colon G \to H\) where H is a group with property X.
Examples
Important examples include:
- Residually finite
- Residually nilpotent
- Residually solvable
- Residually free
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Stably free module
A module which is close to being free.
Definition
A module M over a ring R is stably free if there exist free modules F and G over R such that
\[ M \oplus F = G . \, \]
Properties
- A projective module is stably free if and only if it possesses a finite free resolution.
See also
References
Richard Pinch/sandbox-WP. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-WP&oldid=30236