Difference between revisions of "User:Richard Pinch/sandbox-WP"
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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= | ||
| + | In [[mathematical analysis]], '''Lambert summation''' is a summability method for a class of [[divergent series]]. | ||
| + | |||
| + | ==Definition== | ||
| + | A series <math>\sum a_n</math> is ''Lambert summable'' to ''A'', written <math>\sum a_n = A (\mathrm{L})</math>, if | ||
| + | |||
| + | :<math>\lim_{r \rightarrow 1-} (1-r) \sum_{n=1}^\infty \frac{n a_n r^n}{1-r^n} = A . \, </math> | ||
| + | |||
| + | If a series is convergent to ''A'' then it is Lambert summable to ''A'' (an [[Abelian theorem]]). | ||
| + | |||
| + | ==Examples== | ||
| + | |||
| + | * <math>\sum_{n=0}^\infty \frac{\mu(n)}{n} = 0 (\mathrm{L})</math>, where μ is the [[Möbius function]]. Hence if this series converges at all, it converges to zero. | ||
| + | |||
| + | ==See also== | ||
| + | * [[Lambert series]] | ||
| + | * [[Abelian and tauberian theorems]] | ||
| + | |||
| + | ==References== | ||
| + | * {{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 }} | ||
| + | *{{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}} | ||
| + | *{{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:
- Residually finite
- Residually nilpotent
- Residually solvable
- Residually free
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
Richard Pinch/sandbox-WP. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-WP&oldid=30237