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(Start article: Descendant subgroup)
(Start article: Lambert summation)
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==References==
 
==References==
 
* {{cite book | author=Phillip A. Griffith | title=Infinite Abelian group theory | series=Chicago Lectures in Mathematics | publisher=University of Chicago Press | year=1970 | isbn=0-226-30870-7 | page=19}}
 
* {{cite book | author=Phillip A. Griffith | title=Infinite Abelian group theory | series=Chicago Lectures in Mathematics | publisher=University of Chicago Press | year=1970 | isbn=0-226-30870-7 | page=19}}
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=Lambert summation=
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In [[mathematical analysis]], '''Lambert summation''' is a summability method for a class of [[divergent series]].
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==Definition==
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A series <math>\sum a_n</math> is ''Lambert summable'' to ''A'', written <math>\sum a_n = A (\mathrm{L})</math>, if
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:<math>\lim_{r \rightarrow 1-} (1-r) \sum_{n=1}^\infty \frac{n a_n r^n}{1-r^n} = A . \, </math>
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If a series is convergent to ''A'' then it is Lambert summable to ''A'' (an [[Abelian theorem]]). 
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==Examples==
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* <math>\sum_{n=0}^\infty \frac{\mu(n)}{n} = 0 (\mathrm{L})</math>, where &mu; is the [[Möbius function]].  Hence if this series converges at all, it converges to zero.
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==See also==
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* [[Lambert series]]
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* [[Abelian and tauberian theorems]]
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==References==
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* {{cite book | author=Jacob Korevaar | title=Tauberian theory. A century of developments | series=Grundlehren der Mathematischen Wissenschaften | volume=329 | publisher=[[Springer-Verlag]] | year=2004 | isbn=3-540-21058-X | pages=18 }}
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*{{cite book | author=Hugh L. Montgomery | authorlink=Hugh Montgomery (mathematician) | coauthors=[[Robert Charles Vaughan (mathematician)|Robert C. Vaughan]] | title=Multiplicative number theory I. Classical theory | series=Cambridge tracts in advanced mathematics | volume=97 | year=2007 | isbn=0-521-84903-9 | pages=159-160}}
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*{{cite journal | author=Norbert Wiener | authorlink=Norbert Wiener | title=Tauberian theorems | journal=Ann. Of Math. | year=1932 | volume=33 | pages=1–100 | doi=10.2307/1968102 }}
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=Pinch point=
 
=Pinch point=
 
A '''pinch point''' or '''cuspidal point''' is a type of [[Singular point of an algebraic variety|singular point]] on an [[algebraic surface]].  It is one of the three types of ordinary singularity of a surface.   
 
A '''pinch point''' or '''cuspidal point''' is a type of [[Singular point of an algebraic variety|singular point]] on an [[algebraic surface]].  It is one of the three types of ordinary singularity of a surface.   

Revision as of 18:28, 25 August 2013



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.

See also

References

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.

See also

References

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".

References

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.

See also

References


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.

References

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:

References

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

How to Cite This Entry:
Richard Pinch/sandbox-WP. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-WP&oldid=30237