This paper proves the existence and uniqueness of a time-invariant measure for the 2D Navier-Stokes equations on the sphere under a random kick-force and a time-periodic deterministic force. Several examples of determ...This paper proves the existence and uniqueness of a time-invariant measure for the 2D Navier-Stokes equations on the sphere under a random kick-force and a time-periodic deterministic force. Several examples of deterministic force satisfying the necessary conditions for a unique invariant measure to exist are given. The support of the measure is examined and given explicitly for several cases.展开更多
We expand previously established results concerning the uniform representability of classical and relativistic gravitational field equations by means of velocity-field divergence equations by demonstrating that conser...We expand previously established results concerning the uniform representability of classical and relativistic gravitational field equations by means of velocity-field divergence equations by demonstrating that conservation equations for (probability) density functions give rise to velocity-field divergence equations the solutions of which generate—by way of superposition—the totality of solutions of various well-known classical and quantum-mechanical wave equations.展开更多
Using an alternative representation of the Ricci tensor, we argue that the theory of gravitation can be easily developed in such a way that the formal description of gravity in the transition from classical Newtonian ...Using an alternative representation of the Ricci tensor, we argue that the theory of gravitation can be easily developed in such a way that the formal description of gravity in the transition from classical Newtonian physics to general relativity remains essentially unchanged. That is to say, we show how arguments concerning the plausible conceptual compatibility of Newtonian and general-relativistic models of gravity can be replaced by a demonstration of their actual formal identity. More specifically, we find that both the classical Newtonian and the general relativistic field equations are equivalent to a velocity-field divergence equation of the form v [div (v)] + div (v,v) = -4πρ where the term div (v,v) is defined to be the trace of the square of the Jacobian derivative matrix of v (or of its general-relativistic analogue).展开更多
文摘This paper proves the existence and uniqueness of a time-invariant measure for the 2D Navier-Stokes equations on the sphere under a random kick-force and a time-periodic deterministic force. Several examples of deterministic force satisfying the necessary conditions for a unique invariant measure to exist are given. The support of the measure is examined and given explicitly for several cases.
文摘We expand previously established results concerning the uniform representability of classical and relativistic gravitational field equations by means of velocity-field divergence equations by demonstrating that conservation equations for (probability) density functions give rise to velocity-field divergence equations the solutions of which generate—by way of superposition—the totality of solutions of various well-known classical and quantum-mechanical wave equations.
文摘Using an alternative representation of the Ricci tensor, we argue that the theory of gravitation can be easily developed in such a way that the formal description of gravity in the transition from classical Newtonian physics to general relativity remains essentially unchanged. That is to say, we show how arguments concerning the plausible conceptual compatibility of Newtonian and general-relativistic models of gravity can be replaced by a demonstration of their actual formal identity. More specifically, we find that both the classical Newtonian and the general relativistic field equations are equivalent to a velocity-field divergence equation of the form v [div (v)] + div (v,v) = -4πρ where the term div (v,v) is defined to be the trace of the square of the Jacobian derivative matrix of v (or of its general-relativistic analogue).
