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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 ...
Conjugate variables are pairs of thermodynamic concepts, with the first being akin to a "force" applied to some thermodynamic system, the second being akin to the resulting "displacement", and the product of the two equaling the amount of energy transferred. The common conjugate variables are: Pressure-volume (the mechanical parameters);
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.
The set of natural variables for each of the above four thermodynamic potentials is formed from a combination of the T, S, p, V variables, excluding any pairs of conjugate variables; there is no natural variable set for a potential including the T-S or p-V variables together as conjugate variables for energy.
Conjugate (square roots), the change of sign of a square root in an expression; Conjugate element (field theory), a generalization of the preceding conjugations to roots of a polynomial of any degree; Conjugate transpose, the complex conjugate of the transpose of a matrix; Harmonic conjugate in complex analysis
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 ).
Time itself cannot be part of a conjugate pair because time is just a coordinate. Uncertainty in time itself or spread in time makes no sense since time marches forward for everything in a predictable way and is not variable in a single reference frame (time can be variable from one reference frame to the next, but that is an entirely different ...