Furthermore, any probabilistic prediction system should be derived, ideally, from the basic physical principle for predicting probability, that is, the

Liouville equation (Yano and Ouchtar 2017), although its practical use may appear difficult (see "Data assimilation" section).

Equation (1) has a widespread adoption in many physical branches, such as conserved current of

Liouville equation, two-dimensional quantum gravity gauge field, and conformal field theory [7-13].

Saarloos[3] shown that the density function (mass, momentum and energy fields) obeys a

Liouville equation for hydrodynamics ideal fluid.

Tarasov in [36] based on the

Liouville equation obtained the fractional analogues of the classical kinetic and transport equations.

For the spheroidal solutions, there is a singularity in the appended Sturm-

Liouville equation Eq.

one can obtain the quantum

Liouville equation for the Wigner distribution [40]

In classical mechanics, there are no equations that describe the evolution of the probability densities P(r, t) (or P(p,t), where p is momentum); only joint probability densities, [W.sub.cl](p,r,t), can be expressed by the

Liouville equation. Therefore, corresponding quantum equations for P(r, t), as used in (1), and P(p, t) cannot exist.

Sturm

Liouville equation has been extensively studied in both continuous and discrete cases [7, 20, 29, 33, 36].

(For [epsilon] = 0 this is usually called the

Liouville equation.) Suppose that [[mu].sub.0] is an invariant measure for (1.1) and the density [[rho].sub.0] of [[mu].sub.0] is a [C.sup.2] function.

One of the retrieved results is the

Liouville Equation, an equation Books24x7 has made "live." The user can then click on "Get MathML" to download the live equation and manipulate it.

where A is the normalization constant related to [eta] whose value is calculated with (14), whereas [chi](y) is obtained through the

Liouville equation. This latter gives