adjoint

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.

adj

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.
   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 ω = 0; and the set B + B₀ consists of p points, and lies on i + j adjoint φ-curves of f = 0. 1933, H. F. Baker, Principles of Geometry, volume 5, published 2010, page 103
   As we have stated before, a curve f' is adjoint to a curve f if it have at least the multiplicity r_i-1 at each point where f has the multiplicity r_i. A first polar ∑ᵢy_i(∂f/∂x_i)=0 is an example of an adjoint curve. 1963, Julian Lowell Coolidge, A History of Geometrical Methods, page 205
   This imposes n(n - 3)/2 conditions on the n-gon adjoint curve. 2016, Eugene Wachspress, Rational Bases and Generalized Barycentrics: Applications to Finite Elements and Graphics, page 216

noun

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.
   . Nominated the citizens Fayolles, captain of infantry, and Favery of the Engineers to be my adjoints, and dispatched the letters of nomination to the Minister at War, so now I am fairly afloat. 1798, Theobald Wolfe Tone, The Writings of Theobald Wolfe Tone 1763-98, Volume 3
   Even as it is, by the reckless aggregation of machine-workers and their adjoints in the great cities and the manufacturing districts, it has kept down life amongst us, and keeps it down to a miserably low standard; so low that any standpoint for improvement is hard to think of even. 1884, William Morris, Art and Socialism
   The chef and his adjoints are seen by this code as part of the military, subject to military jurisdiction despite their obligation to execute the orders of the administrative council and of the justice of the peace. 1975, Elizabeth L. Saxe, Sidney Wilfred Mintz, Working Papers in Haitian Society and Culture, page 60
7. An assistant mayor of a French commune.
   A vigorous, aggressive, and ambitious mayor will not rely overmuch upon his adjoints, but a mayor of the more passive type will lean heavily upon them. 1929, Chester Collins Maxey, Urban Democracy, page 89
   Mayors and their adjoints were often simple cultivateurs like the majority of the community. 1989, Alan Forrest, Conscripts and Deserters