overspill

American Heritage Dictionary:

o·ver·spill

(ō'vər-spĭl') pronunciation

intr.v., -spilled, or -spilt (-spĭlt), -spill·ing, -spills.
To spill over.

n. (ō'vər-spĭl')

The act of spilling over.
Something that spills over: an overspill of milk.
Chiefly British. Movement of people from overcrowded cities to less populated areas.

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Oxford Dictionary of Geography:

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The population which is dispersed from large cities to relieve congestion and overcrowding, and, possibly, unemployment. It occurs with redevelopment in the city where new building is at much lower densities so that some people—the overspill—cannot be housed in the city.

Rhymes:

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Wikipedia on Answers.com:

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This article is about mathematics. For housing estates, see overspill estate.

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 is an element of A, and
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.

v d e Infinitesimals

History	Adequality Infinitesimal calculus Leibniz's notation Integral sign Criticism of non-standard analysis The Analyst The Method of Mechanical Theorems Cavalieri's principle

Related branches of mathematics	Non-standard analysis Non-standard calculus Internal set theory Synthetic differential geometry Constructive non-standard analysis

Formalizations of infinitesimal quantities	Differentials Hyperreal numbers Dual numbers Surreal numbers

Individual concepts	Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set Law of Continuity Overspill Microcontinuity

Scientists	Gottfried Leibniz Abraham Robinson Pierre de Fermat Augustin-Louis Cauchy

Infinitesimals in physics and engineering	Infinitesimal strain theory Infinitesimal oscillations of a pendulum

Textbooks	Analyse des Infiniment Petits Elementary Calculus

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

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Translations:

Overspill

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Dansk (Danish)
n. - overskud, befolkningsoverskud
v. intr. - løbe over

Nederlands (Dutch)
het gemorste, overbevolking (die wegtrekt)

Français (French)
n. - excédent de population
v. intr. - avoir un excédent de population, déborder de

Deutsch (German)
n. - Bevölkerungsüberschuß, Übergelaufenes
v. - überlaufen

Ελληνική (Greek)
n. - πλεονάζων πληθυσμός (εξαπλούμενος σε περίχωρα), υπερπληθυσμός, πλεονάζουσα ποσότητα υγρού που ξεχειλίζει από δοχείο
v. - ξεχειλίζω

Italiano (Italian)
sovrappopolazione

Português (Portuguese)
n. - transbordamento (m), excedente populacional, emigrantes (m)

Русский (Russian)
перенаселенность, переливание через край

Español (Spanish)
n. - exceso de población, traslado de población, derrame
v. intr. - derramar

Svenska (Swedish)
n. - befolkningsöverskott
v. - svämma över

中文（简体）(Chinese (Simplified))
溢落的东西, 过剩人口, 溢出

中文（繁體）(Chinese (Traditional))
n. - 溢落的東西, 過剩人口
v. intr. - 溢出

한국어 (Korean)
n. - 넘침, 넘친 물, 과잉인구
v. intr. - 넘치다

日本語 (Japanese)
n. - あふれ出し, 剰余
v. - あふれる

العربيه (Arabic)
‏(الاسم) انتقال السكان الى مناطق أقل ازدحاما (فعل) ينتقل الى مكان أقل ازدحاما‏

עברית (Hebrew)
n. - ‮כמות עודפת, הכמות שגלשה, האוכלוסיה העודפת המהגרת למקום אחר (בריטניה)‬
v. intr. - ‮גלש אל מעבר ל-‬

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		American Heritage Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved. Read more
		Oxford Dictionary of Geography. A Dictionary of Geography. Copyright © Susan Mayhew 1992, 1997, 2004. All rights reserved. Read more
		Rhymes. Oxford University Press. © 2006, 2007 All rights reserved. Read more
		Wikipedia on Answers.com. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article Overspill. Read more
		Translations. Copyright © 2007, WizCom Technologies Ltd. All rights reserved. Read more

	satellite town
	Banbury (city, England)
	Basingstoke (city, England)