FACTOID # 20: Statistically, Delaware bears more cost of the US Military than any other state.

 Home Encyclopedia Statistics States A-Z Flags Maps FAQ About

 WHAT'S NEW RELATED ARTICLES People who viewed "Overspill" also viewed:

SEARCH ALL

Search encyclopedia, statistics and forums:

(* = Graphable)

Encyclopedia > Overspill

In mathematics, particularly in non-standard analysis, overspill is a widely used proof technique. It is based on the fact that N is not an internal subset of the nonstandard integers *N. Indeed, by applying the induction principle and transfer principle we get the following general principle Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Interactive Mathematics Miscellany and Puzzles â€” A collection of articles on various math topics, with interactive Java... Non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of the form 1/n, for n a natural... In mathematical logic, in particular in model theory and non-standard analysis, an internal set is a set that is a member of a model. ... In mathematics particularly in non-standard analysis, the transfer principle is a rule which transforms assertions about standard sets, mappings etc. ...

for any internal subset A of *N, if

1 is an element of A and
for every element n of A, n+1 also belongs to A
then
A= *N

Instantiating this general principle with N, it would follow N=*N which we know not to be the case.

This principle has a number of extremely useful consequences:

• The set of standard hyperreals is not internal.
• The set of bounded hyperreals is not internal.
• The set of infinitesimal hyperreals is not internal.

In particular:

• If an internal set contains all infinitesimal non-negative hyperreals, it contains a positive non-infinitesimal (or appreciable) hyperreal.
• If an internal set contains N it contains an unbounded element of *N.

We can use these facts to prove equivalence of the following two conditions for an internal hyperreal-valued function f defined on *R.

$forall epsilon >!!!> 0, exists delta >!!!> 0, |h| leq delta implies |f(x+h) - f(x)| leq epsilon$

and

$forall h cong 0, |f(x+h) - f(x)| cong 0$

The proof that the second fact implies the first uses overspill, since given a non-infinitesimal positive ε

$forall mbox{ positive } delta cong 0, (|h| leq delta implies |f(x+h) - f(x)| < epsilon)$

By overspill a positive appreciable δ with the requisite properties exists.

These equivalent conditions express the property known in non-standard analysis as S-continuity of f at x. S-continuity is referred to as an external property, since its extension (e.g. the set of pairs (f, x) such that f is S-continuous at x) is not an internal set. In metaphysics, extension is the property of taking up space; see Extension (metaphysics). ...

Results from FactBites:

 Overspill - Wikipedia, the free encyclopedia (268 words) In mathematics, particularly in non-standard analysis, overspill is a widely used proof technique. It is based on the fact that N is not an internal subset of the nonstandard integers *N. By overspill a positive appreciable δ with the requisite properties exists.
 Hampshire County Council (1551 words) The third application is for the formation of an overspill/ replacement student car park area which is intended to be provided on the northern campus near a cul-de-sac called Lower Brook Street. Regarding the application for the overspill car park to be constructed on the northern campus I note the comments expressed by the objector. On balance, therefore, I consider the applications for the two-storey extension, construction of an overspill car park area and the relocation of the four temporary classroom units to be acceptable and planning permission should be granted.
More results at FactBites »

Share your thoughts, questions and commentary here