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In [[mathematics]], '''flat convergence''' is a notion for convergence of submanifolds of Euclidean space.   It was first introduced by [[Hassler Whitney]] in 1957, and then extended to "integral currents" by [[Herbert Federer|Federer]] and Fleming in 1960.  It forms a fundamental part of the field of [[geometric measure theory]]. The notion was applied to find solutions to [[Plateau's problem]].   In 2001 the notion of an integral current was extended to arbitrary metric spaces by [[Luigi Ambrosio|Ambrosio]] and Kirchheim.
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A family with young children may need a 20 year or a 25 year term policy to protect the family through to the end of the college years but because of cost they use the 10 year term policy until they can afford the more costly policies. It stays activated until the end of the term given that you pay the premiums duly.
 
==Integral currents==
A ''k''-dimensional current ''T'' is a multilinear real-valued operator on smooth ''k''-formsFor example, given a [[Lipschitz continuity|Lipschitz map]] from a [[manifold]] into [[Euclidean space]], ''F'': ''N''<sup>''k''</sup> → '''R'''<sup>''N''</sup>, one has an integral current ''T''(''&omega;'') defined by integrating the [[pullback]] of the differential ''k''-form, ''&omega;'', over ''N''.   Currents have a notion of boundary <math>\partial{T} </math>  
(which is the usual boundary when ''N'' is a manifold with boundary) and a notion of mass, ''M''(''T''), (which is the volume of the image of&nbsp;''N'')An [[integer rectifiable current]] is defined as a countable sum of currents formed in this respect.   An [[integral current]] is an integer rectifiable current whose boundary has finite mass.   It is a deep theorem of
Federer-Fleming that the boundary is then also an integral current.
 
==Flat norm and flat distance==
The flat norm |''T''| of a ''k''-dimensional integral current ''T'' is the infimum of ''M''(''A'')&nbsp;+&nbsp;''M''(''B''), where the infimum is taken over all integral currents ''A'' and ''B'' such that <math>T=A + \partial B</math>.
 
The flat distance between two integral currents is then ''d''<sub>''F''</sub>(''T'',''S'') =&nbsp;|''T''&nbsp;&minus;&nbsp;''S''|.
 
==Compactness theorem ==
Federer-Fleming proved that if one has a sequence of integral currents <math>T_i</math> whose supports lie in a compact set ''K'' with a uniform upper bound on <math>M(T_i) + M(\partial T_i)</math>, then a subsequence converges in the flat sense to an integral current.  
 
This theorem was applied to study sequences of submanifolds of fixed boundary whose volume approached the infimum over all volumes of submanifolds with the given boundary.   It produced a candidate weak solution to [[Plateau's problem]].
 
==References==
* {{citation
| last = Federer
| first = Herbert
| authorlink = Herbert Federer
| title = Geometric measure theory
| publisher = Springer-Verlag New York Inc.
| location = New York
| year = 1969
| pages = xiv+676
| isbn = 978-3-540-60656-7
| mr= 0257325
| series = series Die Grundlehren der mathematischen Wissenschaften
| volume = Band 153 }}
*{{citation|first=H. |last=Federer|title=Colloquium lectures on geometric measure theory|journal= Bull. Amer. Math. Soc. |volume= 84 |issue= 3 |year=1978|pages= 291–338
|url=http://www.ams.org/bull/1978-84-03/S0002-9904-1978-14462-0/|doi=10.1090/S0002-9904-1978-14462-0}}
 
* {{citation
| last = Morgan
| first = Frank
| authorlink = Frank Morgan (mathematician)
| title = Geometric measure theory: A beginner's guide
| edition = Fourth
| publisher = Academic Press Inc.
| location = San Diego, CA
| year = 2009
| pages = viii+249
| isbn = 978-0-12-374444-9
| mr=2455580
}}
*{{springer|id=G/g130040|first=T.C. |last=O'Neil}}
 
* {{citation
|last=Ambrosio|first=Luigi|coauthors=Kirchheim, Bernd|title=Currents in Metric Spaces|journal=Acta Mathematica|year=2000|volume=185|pages=1–80}}
 
[[Category:Metric geometry]]
[[Category:Riemannian geometry]]
[[Category:Convergence (mathematics)]]

Revision as of 19:10, 13 February 2014

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Because the cost of renewing a term life insurance policy will be higher than the original premiums it is generally a good idea to purchase a policy with the longest term available, provided that it fits within your budget. After the term is up, you can usually keep the policy, albeit usually at very high (and annually increasing) rates. The truth is this that they are certainly on the wrong footings. The number of days listed in the policy as elimination time, is the number of days that must pass, before the insurance starts covering your long- term -care expenses. If you therefore have a policy for $100,000 the life insurance company will pay $200,000. A family with young children may need a 20 year or a 25 year term policy to protect the family through to the end of the college years but because of cost they use the 10 year term policy until they can afford the more costly policies. It stays activated until the end of the term given that you pay the premiums duly.