different between inverse vs adjoint

English

Etymology

Recorded since 1440, from Latin inversus, the past participle of invertere (“to invert”), itself from in- (“in, on”) + vertere (“to turn”).

Pronunciation

• (General American) IPA^(key): /?n?v?s/, /??nv?s/

Adjective

inverse (not comparable)

1. Opposite in effect, nature or order.
2. Reverse, opposite in order.
3. (botany) Inverted; having a position or mode of attachment the reverse of that which is usual.
4. (mathematics) Having the properties of an inverse; said with reference to any two operations, which, when both are performed in succession upon any quantity, reproduce that quantity.

5. (geometry) That has the property of being an inverse (the result of a circle inversion of a given point or geometrical figure); that is constructed by circle inversion.

6. (category theory, of a category) Whose every element has an inverse (morphism which is both a left inverse and a right inverse).

Derived terms

• inverse function
• inverse image
• inverse spelling
• inversely
• inverse video

Related terms

• inversion
• inversive
• reverse

Translations

Noun

inverse (plural inverses)

1. An inverted state: a state in which something has been turned (properly) upside down or (loosely) inside out or backwards.

   Cowgirl is the inverse of missionary.

   321 is the inverse of 123.

2. The result of an inversion, particularly:
   1. The reverse of any procedure or process.

      Uninstalling is the inverse of installation.

   2. (mathematics) A ratio etc. in which the antecedents and consequents are switched.

      The inverse of a:b is b:a.

   3. (geometry) The result of a circle inversion; the set of all such points; the curve described by such a set.

      The inverse P‘ of a point P is the point on a ray from the center O through P such that OP × OP‘ = r² or the set of all such points.

   4. (logic) The non-truth-preserving proposition constructed by negating both the premise and conclusion of an initially given proposition.

      "Anything that isn't a dog doesn't go to heaven" is the inverse of "All dogs go to heaven." More generally, ${\displaystyle \lnot {\mathsf {p}}\to \lnot {\mathsf {q}}}$ is the inverse of ${\displaystyle {\mathsf {p}}\to {\mathsf {q}}}$ and is equivalent to the converse proposition ${\displaystyle {\mathsf {q}}\to {\mathsf {p}}}$.

      • 1896, James Welton, A Manual of Logic, 2nd ed., Bk iii, Ch. iii, §102:

        Inversion is the inferring, from a given proposition, another proposition whose subject is the contradictory of the subject of the original proposition. The given proposition is called the Invertend, that which is inferred from it is termed the Inverse... The rule for Inversion is: Convert either the Obverted Converse or the Obverted Contrapositive.

3. (mathematics) A second element which negates a first; in a binary operation, the element for which the binary operation—when applied to both it and an initially given element—yields the operation's identity element, specifically:
   1. (addition) The negative of a given number.

      The additive inverse of ${\displaystyle x}$ is ${\displaystyle -x}$, as ${\displaystyle x-x=0}$, as ${\displaystyle 0}$ is the additive identity element.

   2. (multiplication) One divided by a given number.

      The multiplicative inverse of ${\displaystyle x}$ is ${\displaystyle x^{-1}}$, as ${\displaystyle x\times x^{-1}=1}$, as ${\displaystyle 1}$ the multiplicative identity element.

   3. (functions) A second function which, when combined with the initially given function, yields as its output any term inputted into the first function.

      The compositional inverse of a function ${\displaystyle f}$ is ${\displaystyle f^{-1}}$, as ${\displaystyle f\ f^{-1}={\mathit {I}}}$, as ${\displaystyle {\mathit {I}}}$ is the identity function. That is, ${\displaystyle \forall x,f(f^{-1}(x))={\mathit {I}}(x)=x}$.

4. (category theory) A morphism which is both a left inverse and a right inverse.
5. (card games) The winning of the coup in a game of rouge et noir by a card of a color different from that first dealt; the area of the table reserved for bets upon such an outcome.
   • 1850, Henry George Bohn, The Hand-book of Games, p. 343:

     If the player... be determined to try his luck on the inverse, he must place his money on a yellow circle, or rather a collection of circles, situated at the extremity of the table.

   • 1950, Lawrence Hawkins Dawson, Hoyle's Games Modernized, 20th ed., p. 291:

     The tailleur never mentions the words ‘Black’ or ‘Inverse’, but always says that Red wins or Red loses, and that the colour wins or the colour loses.

6. (linguistics, Kiowa-Tanoan) A grammatical number marking that indicates the opposite grammatical number (or numbers) of the default number specification of noun class.

Synonyms

• (addition): additive inverse
• (multiplication): multiplicative inverse
• (composition): compositional inverse
• (geometry): inverse point, inverse curve

Translations

See also

• (logic): obverse, converse, contraposition

Verb

inverse (third-person singular simple present inverses, present participle inversing, simple past and past participle inversed)

1. (surveying) To compute the bearing and distance between two points.

Antonyms

• compute (a point).

Anagrams

• Severin, enviers, inserve, veiners, venires, versine

Danish

Adjective

inverse

1. plural and definite singular attributive of invers

Dutch

Pronunciation

• Hyphenation: in?ver?se

Noun

inverse m or f (plural inversen)

1. inverse

Adjective

inverse

1. Inflected form of invers

Anagrams

• viseren

French

Pronunciation

• IPA^(key): /??.v??s/

Etymology 1

From Latin inversus.

Adjective

inverse (plural inverses)

1. inverse, the other way round

Derived terms

• barre inverse

• inversement

Noun

inverse m (plural inverses)

1. the inverse, the contrary

   Synonyms: contraire, envers

Derived terms

Etymology 2

Verb

inverse

1. first-person singular present indicative of inverser
2. third-person singular present indicative of inverser
3. first-person singular present subjunctive of inverser
4. third-person singular present subjunctive of inverser
5. second-person singular imperative of inverser

Further reading

• “inverse” in Trésor de la langue française informatisé (The Digitized Treasury of the French Language).

Anagrams

• enivres, enivrés
• reviens
• Séverin
• vernies

German

Pronunciation

Adjective

inverse

1. inflection of invers:
   1. strong/mixed nominative/accusative feminine singular
   2. strong nominative/accusative plural
   3. weak nominative all-gender singular
   4. weak accusative feminine/neuter singular

Italian

Adjective

inverse

1. feminine plural of inverso

Anagrams

• svenire

Latin

Participle

inverse

1. vocative masculine singular of inversus

English

Etymology

From French adjoindre (“to join”), from late 19th C; see also adjoin. Doublet of adjunct.

In the case of category theory (which brings together concepts from numerous fields), the term is often confounded with adjunct and the relationship is called an adjunction. The origin of any particular usage may therefore be uncertain.

Pronunciation

• IPA^(key): /?æd?.??nt/

Adjective

adjoint (not comparable)

1. (mathematics) Used in certain contexts, in each case involving a pair of transformations, one of which is, or is analogous to, conjugation (either inner automorphism or complex conjugation).
2. (mathematics, category theory, of a functor) That is related to another functor by an adjunction.
3. (geometry, of one curve to another curve) Having a relationship of the nature of an adjoint (adjoint curve); sharing multiple points with.
   • 1933, H. F. Baker, Principles of Geometry, 2010, Volume 5, page 103,

     The sets A + A₀, B + B₀, together, form the complete intersection, with f = 0, of a composite adjoint curve of order m + k, consisting of the adjoint curve of order m through A + B, together with the non-adjoint curve ? = 0; and the set B + B₀ consists of p points, and lies on i + j adjoint ?-curves of f = 0.

   • 1963, Julian Lowell Coolidge, A History of Geometrical Methods, page 205,

     As we have stated before, a curve ${\displaystyle f'}$ is adjoint to a curve ${\displaystyle f}$ if it have at least the multiplicity ${\displaystyle r_{i}-1}$ at each point where ${\displaystyle f}$ has the multiplicity ${\displaystyle r_{i}}$. A first polar ${\displaystyle \sum _{i}y_{i}\left(\partial f/\partial x_{i}\right)=0}$ is an example of an adjoint curve.

   • 2016, Eugene Wachspress, Rational Bases and Generalized Barycentrics: Applications to Finite Elements and Graphics, page 216,

     This imposes n(n - 3)/2 conditions on the n-gon adjoint curve.

Usage notes

The adjoint operator, or Hermitian transpose, of an operator generalises the concept of transpose conjugate of a matrix. (See Hermitian adjoint on Wikipedia.Wikipedia )

In the case of an adjoint representation of a Lie group, the representation in question describes the group's elements as linear transformations of its Lie algebra, itself considered as a vector space. The representation is obtained by differentiating ("linearising") the group action of conjugation (i.e., differentiating the function x ? gxg^-1 for each element g).

The adjoint representation of a Lie algebra is the differential of the adjoint representation of a Lie group at the identity element of the group.

In relation to functors in category theory (and therefore in numerous fields of mathematics), the term is often synonymous with adjunct and the functors are said to be related by an adjunction. Functors may be left or right adjoint (adjunct).

Synonyms

• (mathematics): adjunct (in certain contexts)

Derived terms

Related terms

• adjunction (noun)
• coadjoint
• self-adjoint
• sub-adjoint

Translations

Noun

adjoint (plural adjoints)

1. (mathematics) The transpose of the cofactor matrix of a given square matrix.
2. (mathematics, linear algebra, of a matrix) Transpose conjugate.
3. (mathematics, mathematical analysis, of an operator) Hermitian conjugate.
4. (mathematics, category theory) A functor related to another functor by an adjunction.
5. (geometry, algebraic geometry) A curve A such that any point of a given curve C of multiplicity r has multiplicity at least r–1 on A. Sometimes the multiple points of C are required to be ordinary, and if this condition is not satisfied the term sub-adjoint is used.
6. An assistant to someone who holds a position in the military or civil service.
7. An assistant mayor of a French commune.

Derived terms

Translations

References

• Adjoint on Wolfram MathWorld

French

Pronunciation

• IPA^(key): /ad.?w??/

Noun

adjoint m (plural adjoints)

1. deputy, assistant
2. (linguistics) adjunct

Verb

adjoint m (feminine singular adjointe, masculine plural adjoints, feminine plural adjointes)

1. past participle of adjoindre

Further reading

• “adjoint” in Trésor de la langue française informatisé (The Digitized Treasury of the French Language).