In this section, we are going to study the

conserved quantities of the solutions derived in Section 3.

The potential topics of this special issue include symmetries, differential equations, and applications; optimal control; equivalence transformations and classical and non-classical symmetries; reduction techniques and solutions and linearization;

conserved quantities in natural phenomena; completely integrable equations in mathematical physics; recursion operators, infinite hierarchy of symmetries, and/or conservation laws; equations admitting weak soliton solutions; models for air pollution and underground pollution; mathematical methods for extended thermodynamics; numerical techniques for problems arising in the modeling of physical process; ad hoc methods for solutions.

After reviewing Lagrangian formalism of classical mechanics, the graduate text develops the field theoretical technique in general relativity to construct

conserved quantities for isolated astronomical systems, a theory of cosmological perturbations, and three approaches for constructing conservations laws.

The two

conserved quantities that equation (1.1) possess are given by

In particular,

conserved quantities (i.e., momentum and scalar concentration) are treated on the same foot, leading to unified dimensionless governing equations and solutions.

One standard approach to constructing

conserved quantities is to restrict to localized waveforms, specifically such that [phi] [right arrow] [[phi].sub.[+ or -]] "sufficiently fast" as x [right arrow] [+ or -][infinity], where [[phi].sub.[+ or -]] are constants.

A working hypothesis is that all observers are made up of long lasting quasi-localized packets of fields that determine discrete state machines and these are distinguished by localized collections of mass, charge and other

conserved quantities.

In general, the boundary conditions will determine which conservation law is to be applied to obtain

conserved quantities of the BVP of (28).

In the present paper we give an elementary proof of the correspondence between

conserved quantities and stationary Bernoulli distributions in surjective one-dimensional cellular automata.

The normal coordinates n and b obtained from explicit integration of the one-forms [eta] and [beta] represent the global

conserved quantities. They appear arbitrarily in the Riccati equation and hence in the Poisson structures.

Section 2 provides the necessary background information on the invariance and

conserved quantities of dynamical system and especially the Noether's theorem.