Overspill

Overspill

In non-standard analysis, a branch of mathematics, overspill (referred to as overflow by Goldblatt (1998, p. 129)) is a widely used proof technique. It is based on the fact that the set of standard natural numbers N is not an internal subset of the internal set *N of hypernatural numbers.

By applying the induction principle for the standard integers N and the transfer principle we get the principle of internal induction:

For any internal subset A of *N, if

  1. 1 is an element of A, and
  2. for every element n of A, n + 1 also belongs to A,

then

A = *N

If N were an internal set, then instantiating the internal induction principle with N, it would follow N = *N which is known not to be the case.

The overspill principle has a number of 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.

Example

These facts can be used to prove the equivalence of the following two conditions for an internal hyperreal-valued function ƒ defined on *R.

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

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)| < \varepsilon).\,

Applying overspill, we obtain a positive appreciable δ with the requisite properties.

These equivalent conditions express the property known in non-standard analysis as S-continuity of ƒ at x. S-continuity is referred to as an external property, since its extension (e.g. the set of pairs (ƒ, x) such that ƒ is S-continuous at x) is not an internal set.

References

  • Robert Goldblatt (1998). Lectures on the hyperreals. An introduction to nonstandard analysis. Springer.

Wikimedia Foundation. 2010.

Игры ⚽ Поможем решить контрольную работу

Look at other dictionaries:

  • Overspill — bezeichnet in der Rundfunktechnik eine Versorgung eines Bereiches mit Signalen, der eigentlich nicht innerhalb des eigenen Versorgungsgebietes liegt. Overspill entsteht dadurch, dass sich Radiosignale gleichmäßig von der Sendestation ausbreiten,… …   Deutsch Wikipedia

  • overspill — ► NOUN Brit. ▪ a surplus population moving from an overcrowded area to live elsewhere …   English terms dictionary

  • overspill — [[t]o͟ʊvə(r)spɪl[/t]] 1) N UNCOUNT: also a N, oft N n Overspill is used to refer to people who live near a city because there is no room in the city itself. [BRIT] ...new towns built to absorb overspill from nearby cities. ...overspill council… …   English dictionary

  • overspill — 1. noun That which spills over. New towns were to accommodate overspill from established cities. 2. verb To spill over, to overflow, to spill out of. Further, Coventrys …   Wiktionary

  • overspill — o|ver|spill [ˈəuvəˌspıl US ˈouvər ] n [singular,U] BrE people who move out of a big city because there are too many people living there, and go to live in new houses outside the city ▪ an overspill of workers from London …   Dictionary of contemporary English

  • overspill — noun (U) BrE people who move out of a big city because there are too many people living there, and go to live in new houses outside the city: a new town built to accommodate London s overspill …   Longman dictionary of contemporary English

  • overspill — UK [ˈəʊvə(r)ˌspɪl] / US [ˈoʊvərˌspɪl] noun [uncountable] a) the people who begin to live and work in places just outside a crowded city, making it bigger b) used generally about people or things that cannot fit into a crowded place an overspill… …   English dictionary

  • overspill — verb (i) /oʊvəˈspɪl/ (say ohvuh spil) (overspilt or overspilled, overspilling) 1. to spill over. –noun /ˈoʊvəspɪl/ (say ohvuhspil) 2. that which spills out. 3. excess or surplus population: new towns are planned to take Sydney s overspill. {over… …  

  • Overspill parking — Cars parked on the sidewalk in Moscow Overspill parking is the parking of vehicles beyond the main area provided for the purpose. It can occur because provided parking spaces are insufficient for demand or considered unsatisfactory for some… …   Wikipedia

  • Overspill estate — The Darnhill estate near Heywood, Greater Manchester was originally built by Manchester Corporation between 1947 and the 1960s An overspill estate is a housing estate planned and built for the rehousing of people from decaying inner city areas[1 …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”