Copyright © Philip M. Parker, INSEAD. Terms of Use.
Definition: Equaliser |
EqualiserNoun1. Electronic equipment that reduces frequency distortion. 2. An equivalent counterbalancing weight. Source: WordNet 1.7.1 Copyright © 2001 by Princeton University. All rights reserved. |
| Domain | Definition |
Computing | The element connected in the forward path or in an auxiliary feedback path of a feedback control system and whose transfer function is such that the overall performances of the feedback control system are improved. Source: European Union. (references) |
Electrical Engineering | A network which corrects a circuit's transmission-frequency characteristics to allow it to transmit selected frequencies in a uniform manner. Source: European Union. (references) |
| The connection made between points on a winding to minimise any undesirable potential difference between these points. Source: European Union. (references) | |
| The apparatus designed to compensate over a certain frequency range the amplitude/frequency distortion or the phase/frequency distortion introduced by lines or equipment. Source: European Union. (references) | |
| A device to reduce distortion in a system by introducing networks that compensate for the particular type of distortion over the required frequency band. Source: European Union. (references) | |
Source: compiled by the editor from various references; see credits. | |
(From Wikipedia, the free Encyclopedia)
This article is about equalisers in mathematics. For equalizers in audio signal processing, see Equalization.In mathematics, an equaliser, or equalizer, is a set of arguments where two or more functions have equal values. An equaliser is the solution set of an equation. In certain contexts, a difference kernel is the equaliser of exactly two functions.
Definitions
Let X and Y be sets. Let f and g be functions, both from X to Y. Then the equaliser of f and g is the set of elements x of X such that f(x) equals g(x) in Y. Symbolically:
The equaliser may be denoted Eq(f,g) or a variation on that theme (such as with lowercase letters "eq"). In informal contexts, the notation {f = g} is common.
The definition above used two functions f and g, but there is no need to restrict to only two functions, or even to only finitely many functions. In general, if F is a set of functions from X to Y, then the equaliser of the members of F is the set of elements x of X such that, given any two members f and g of F, f(x) equals g(x) in Y. Symbolically:
This equaliser may be denoted Eq(F), or Eq(f,g,h,...) if F is the set {f,g,h,...}. In the latter case, one may also find {f = g = h = ···} in informal contexts.
As a degenerate case of the general definition, let F be a singleton {f}. Since f(x) always equals itself, the equaliser must be the entire domain X. As an even more degenerate case, let F be the empty set {}. Then the equaliser is again the entire domain X, since the universal quantification in the definition is vacuously true.
Difference kernels
A binary equaliser (that is, an equaliser of just two functions) is also called a difference kernel. This may also be denoted DiffKer(f,g), Ker(f,g), or Ker(f - g). The last notation shows where this terminology comes from, and why it is most common in the context of abstract algebra: The difference kernel of f and g is simply the kernel of the difference f - g. Conversely, the kernel of a single function f can be reconstructed as the difference kernel Eq(f,0), where 0 is the constant function with value zero.
Of course, all of this presumes an algebraic context where the kernel of a function is its preimage under zero; that is not true in all situations. However, the terminology "difference kernel" has no other meaning.
In category theory
Equalisers can be defined by a universal property, which allows the notion to be generalised from the category of sets to arbitrary categories.
In the general context, X and Y are objects, while f and g are morphisms from X to Y. These objects and morphisms form a diagram in the category in question, and the equaliser is simply the limit of that diagram.
In more explicit terms, the equaliser consists of an object E and a morphism eq from E to X satisfying f·eq = g·eq (where "·" denotes composition of morphisms); and such that, given any other object O and morphism m from O to X, if f·m = g·m, then there exists a unique morphism u from O to E such that eq·u = m.
There should be a picture here.
In any universal algebraic category, including the categories where difference kernels are used, as well as the category of sets itself, the object E can always be taken to be the ordinary notion of equaliser, and the morphism eq can in that case be taken to be the inclusion function of E as a subset of X.
The generalisation of this to more than two morphisms is straightforward; simply use a larger diagram with more morphisms in it. The degenerate case of only one morphism is also straightforward; then eq can be any isomorphism from an object E to X.
The degenerate case of no morphisms at all might be confusing at first; it may seem that the diagram in question consists of only the objects X and Y, with no morphsims. The limit of that diagram is the product of X and Y, which doesn't agree with the set-theoretic definition above. However, the correct interpretation is that the diagram is based on X and includes Y only because Y is the codomain of a morphism in the diagram. Then if there are no morphisms involved, then the diagram consists of X alone, so the limit is again any isomorphism between E and X.
It can be proved that any equaliser in any category is a monomorphism. If the converse holds in a given category, then that category is said to be regular (in the sense of monomorphisms). More generally, a regular monomorphism in any category is any morphism m that is an equaliser of some set of morphisms. Some authorities require (more strictly) that m be a binary equaliser, that is an equaliser of exactly two morphisms. However, if the category in question is complete, then both definitions agree.
The notion of difference kernel also makes sense in a category-theoretic context. The terminology "difference kernel" is common throughout category theory for any binary equaliser. In the case of a preadditive category (a category enriched over the category of Abelian groups), the term "difference kernel" may be interpreted literally, since subtraction of morphisms makes sense. That is, Eq(f,g) = Ker(f - g), where Ker denotes the category-theoretic kernel.
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Equaliser."
Synonyms: EqualiserSynonyms: balance (n), counterbalance (n), counterpoise (n), counterweight (n), equalizer (n). (additional references) |
| Synonyms by domain: compensating element (computing, electrical engineering), equalizer (electrical engineering). |
Crosswords: Equaliser |
| Specialty definitions using "equaliser": Bode equaliser, Bode equalizer. (references) |
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| "Equaliser 05" by Nicholas Sales Commentary: "Equaliser." |
Source: photographs selected by the editor, with permission from the photographers. |
| "Equaliser" is generally used as a noun (singular) -- approximately 100.00% of the time. "Equaliser" is used about 290 times out of a sample of 100 million words spoken or written in English. Its rank is based on over 700,000 words used in the English language. Some parts-of-speech are not covered due to the samples used by the British National Corpus. (note: percents less than one-hundredth of one percent have been omitted) |
| Parts of Speech | Percent | Usage per 100 Million Words | Rank in English |
| Noun (singular) | 100% | 290 | 17,096 |
Source: compiled by the editor from several corpora; see credits.
Expression using "equaliser": Bode equaliser. Additional references. | |
| Source: compiled by the editor from various references; see credits. |
| The following statistics estimate the number of searches per day across the major English-language search engines as identified by various trade publications. Hyperlinks lead to commercial use of the expression at Amazon.com. |
| Expression | Frequency per Day |
equaliser | 2 |
| Source: compiled by the editor from various references; see credits. | |
| Language | Translations for "equaliser"; alternative meanings/domain in parentheses. | |
Chinese | 调平器 (Equalizer). (various references) | |
Danish | equalizer (equalizer), udligningsudstyr (equalizer), udligningsforbindelse (equalizer), udligningsenhed (equalizer), regenerator (equalizer, repeater, telephone amplifier, telephone repeater), modforvraenger (equalizer), modforvrænger (equalizer), kompenseringsenhed (compensating element, equalizer), kompenseringselement (compensating element, equalizer), kompenseringsdel (compensating element, equalizer), genskabelsesudstyr (equalizer), genskabelsesenhed (equalizer). (various references) | |
Dutch | evenwichtsverbinding (equalizer), equipotentiaalverbinding (equalizer), equalizer (equalizer), egalisator (equalizer), egalisatienetwerk (equalizer), effeningselement (equalizer), vereffeningscircuit (equalizer), vereffenaar (equalizer), compenserend element (compensating element, equalizer), balans (balance, balance sheet, equilibrium, fly, scales). (various references) | |
Finnish | tasoitusyhdistys (equalizer), tasain (balancing network, equalizer, impedance simulating network, piloted counterbore), taajuuskorjain (equalizer), korjauspiiri (equalizer), korjain (equalizer), kompensointielin (compensating element, equalizer). (various references) | |
French | correcteur d'affaiblissement (equalizer), correcteur (equalizer), connexion équipotentielle (equalizer), compensateur de phase (equalizer), équilibreur (equalizer), égaliseur (equalizer), égalisateur (equalizer). (various references) | |
German | Equalizer (equalizer), Entzerrer (equalizer, regenerative repeater), Pegelregler (equalizer, level control), Korrektfilter (equalizer), Kompensationsglied (compensating element, equalizer), Ausgleichsverbinder (equalizer). (various references) | |
Greek | ισοφάριση, εξισωτής (equalizer), εξισορροπητής (equalizer). (various references) | |
Italian | equalizzatore (equalizer). (various references) | |
Pig Latin | equaliseray.(various references) | |
Portuguese | equilibrador (equalizer, hillside, self-levelling), ligação equipotencial (equalizer), igualizador (compensating element, equalizer), igualador (sway-beam), compensador (compensatory, sway-beam). (various references) | |
Russian | эквалайзер. (various references) | |
Spanish | equilibrador (balance, balance poising tool, balancering network, equalizer, impedance simulating network), ecualizador (compensating element, equalizer), regulador de nivel (equalizer), igualador (equalizer), corrector (amender, calipers, callipers, copyreader, proofreader, reader, reviser), conexión equipotencial (equalizer). (various references) | |
Swedish | equalizer (equalizer), utjämningsförbindning (equalizer), utjämnare (equalizer, leveller), tonkontroll (equalizer), kompenseringslänk (compensating element, equalizer), kompenseringsenhet (equalizer), dämpningsutjämnare (equalizer). (various references) | |
Turkish | ekolayzer. (various references) | |
| Source: compiled by the editor from various translation references. | ||
Derivations | |
Words beginning with "equaliser": equalisers. (additional references) | |
| Source: compiled by the editor, based on several corpora (additional references). | |
Scrabble® Enable2K-Verified Anagrams | |
| Words within the letters "a-e-e-i-l-q-r-s-u" | |
-1 letter: equalise, queasier, reliques, squealer. | |
-2 letters: esquire, leisure, quaeres, queries, realise, relique, sequela. | |
-3 letters: aeries, ariels, easier, equals, larees, leaser, quaere, quails, queers, quires, reales, relies, resail, resale, reseal, reseau, resile, risque, sailer, saurel, sealer, sequel, serail, serial, square, squeal, squire, urease, urials. | |
-4 letters: aerie, aisle, ariel, arils, arise, arles, aurei, aures, auris, earls, easel. | |
| Words containing the letters "a-e-e-i-l-q-r-s-u" | |
+1 letter: equalisers, equalizers. | |
+2 letters: quarterlies, reliquaries, requalifies. | |
+3 letters: equilibrates, prequalifies. | |
+4 letters: harlequinades. | |
+5 letters: disequilibrate, superqualities, tranquilnesses. | |
| Source: compiled by the editor from various references; see credits. SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro. | |
| 1. Definition 2. Synonyms 3. Crosswords 4. Images: Digital Art | 5. Usage Frequency 6. Expressions 7. Expressions: Internet 8. Translations: Modern | 9. Derivations 10. Anagrams 11. Bibliography |
Copyright © Philip M. Parker, INSEAD. Terms of Use.
