overspill

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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Geography Dictionary: overspill

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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.

WordNet: overspill

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Note: click on a word meaning below to see its connections and related words.

The noun has 2 meanings:

Meaning #1: too much population
Synonym: overpopulation

Meaning #2: the occurrence of surplus liquid (as water) exceeding the limit or capacity
Synonyms: overflow, runoff

Wikipedia: Overspill

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

In non-standard analysis, a branch of mathematics, overspill 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 hyperintegers.

Indeed, 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 we know not to be the case.

The overspill 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.

Example

We can use these facts to prove 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,

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

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 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.

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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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Copyrights:

		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
		Geography Dictionary. A Dictionary of Geography. Copyright © Susan Mayhew 1992, 1997, 2004. All rights reserved. Read more
		WordNet. WordNet 1.7.1 Copyright © 2001 by Princeton University. All rights reserved. Read more
		Wikipedia. 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