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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.
These forces and their associated displacements are called conjugate variables. [1] For example, consider the p V {\displaystyle pV} conjugate pair. The pressure p {\displaystyle p} acts as a generalized force: Pressure differences force a change in volume d V {\displaystyle \mathrm {d} V} , and their product is the energy lost by the system ...
Conjugation (biochemistry), covalently linking a biomolecule with another molecule; Conjugate (acid-base theory), a system describing a conjugate acid-base pair; Conjugated system, a system of atoms covalently bonded with alternating single and multiple bonds; Conjugate variables (thermodynamics), pairs of variables that always change ...
A conjugate prior is an algebraic convenience, giving a closed-form expression for the posterior; otherwise, numerical integration may be necessary. Further, conjugate priors may give intuition by more transparently showing how a likelihood function updates a prior distribution.
Geometric representation (Argand diagram) of and its conjugate ¯ in the complex plane.The complex conjugate is found by reflecting across the real axis.. In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.
According to the correspondence principle, in certain limits the quantum equations of states must approach Hamilton's equations of motion.The latter state the following relation between the generalized coordinate q (e.g. position) and the generalized momentum p: {˙ = = {,}; ˙ = = {,}.
where x i and y i are conjugate pairs, and the y i are the natural variables of the potential Φ. From the chain rule it follows that: = {} where {y i ≠ j} is the set of all natural variables of Φ except y j that are held as constants. This yields expressions for various thermodynamic parameters in terms of the derivatives of the potentials ...
In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation , Fenchel transformation , or Fenchel conjugate (after Adrien-Marie Legendre and Werner Fenchel ).