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A conjugate acid, within the Brønsted–Lowry acid–base theory, is a chemical compound formed when an acid gives a proton (H +) to a base—in other words, it is a base with a hydrogen ion added to it, as it loses a hydrogen ion in the reverse reaction.
In mathematics, especially group theory, two elements and of a group are conjugate if there is an element in the group such that =. This is an equivalence relation whose equivalence classes are called conjugacy classes .
In mathematics, in particular field theory, the conjugate elements or algebraic conjugates of an algebraic element α, over a field extension L/K, are the roots of the minimal polynomial p K,α (x) of α over K. Conjugate elements are commonly called conjugates in contexts where this is not ambiguous.
In Bayesian probability theory, if, given a likelihood function (), the posterior distribution is in the same probability distribution family as the prior probability distribution (), the prior and posterior are then called conjugate distributions with respect to that likelihood function and the prior is called a conjugate prior for the likelihood function ().
[2] [3] The basic concept of this theory is that when an acid and a base react with each other, the acid forms its conjugate base, and the base forms its conjugate acid by exchange of a proton (the hydrogen cation, or H +). This theory generalises the Arrhenius theory.
Conjugate variables are pairs of variables mathematically defined in such a way that they become Fourier transform duals, [1] [2] or more generally are related through Pontryagin duality. The duality relations lead naturally to an uncertainty relation—in physics called the Heisenberg uncertainty principle —between them.
Saying that conjugation of x by a leaves x unchanged is equivalent to saying that a and x commute: a − 1 x a = x x a = a x . {\displaystyle a^{-1}xa=x\iff xa=ax.} Therefore the existence and number of inner automorphisms that are not the identity mapping is a kind of measure of the failure of the commutative law in the group (or ring).
Conjugate transpose, the complex conjugate of the transpose of a matrix; Harmonic conjugate in complex analysis; Conjugate (graph theory), an alternative term for a line graph, i.e. a graph representing the edge adjacencies of another graph; In group theory, various notions are called conjugation: Inner automorphism, a type of conjugation ...