Mathematical concepts and their physical foundation in the nonstandard analysis theory of turbulence

Main mathematical concepts and their physical foundation in the nonstandard analysis theory of turbulence are presented and discussed. The underlying fact is that there does not exist the absolute zero fluid-volume. Therefore, the physical object corresponding to the absolute point is just the uniform fluid-particle. The fluid-particle, in general, corresponds to the monad. The uniform fluid-particle corresponds to the uniform monad, while the nonuniform fluid-particle to the nonuniform monad. There are two kinds of the differentiations, one is based on the absolute point, and the other based on the monad. The former is adopted in the Navier-Stokes equations, and the latter in the fundamental equations presented in this paper for the nonstandard analysis theory of turbulence. The continuity of fluid is elucidated by virtue of the concepts of the fluid-particle and fluid-particle at a lower level. Furthermore, the characters of the continuity in two cases, i.e. in the standard and nonstandard analyses, are presented in this paper. And the difference in discretization between the Navier-Stokes equations and the fundamental equations given herein is also pointed out.

Canards Flying on Bifurcation

There exists a property “structural stability” for “4-dimensional canards” which is a singular-limit solution in a slow-fast system with a bifurcation parameter. It means that the system includes the possibility to have some critical values on the bifurcation parameter. Corresponding to these values, the pseudo-singular point, which is a singular point in the time-scaled-reduced system should be changed to another one. Then, the canards may fly to another pseudo-singular point, if possible. Can the canards fly? The structural stability gives the possibility for the canards flying. The precise reasons why happen are described in this paper.

On Relative Stability in 4-Dimensional Duck Solution

This paper gives the existence of a relatively stable duck solution in a slow-fast system in R2＋2 with an invariant manifold. The slow-fast system in R2＋： has a 2-dimensional slow vector field and a 2-dimensional fast vector field. The fast vector field restricts a feasible region of the slow vector field strictly. In the case of the slow-fast system in R2＋1 , that is, the fast vector field is l-dimension, it is classified according to its sign, because it gives only negative（-）, positive（＋） or zero sign. Then it is attractive, repulsive or stationary. On the other hand, in R2~2 , the fast vector field has combinatorial cases. It causes a complex state to analyze the system. First, we introduce a local model near the pseudo singular point on which we classify the fast vector field attractive（-,-）, attractive-repulsive（-,＋） or repulsive（＋,＋）, simply as possible. We prove the existence of a 4-dimensional duck solution in the local model. Secondarily, we assume that the slow-fast system has an invariant manifold near the pseudo singular point. When the invariant manifold has a homoclinic point near the pseudo singular point, we show that the slow-fast sytem has a 4-dimensional duck solution having a relatively stable region.

ON THE EXISTENCE OF SOLUTIONS TO D＇－VALUED STOCHASTIC DIFFERENTIAL EQUATIONS INVOLVING EVOLUTION DRIFT

In this paper, nonstandard analysis is employed to present an existence theory of -valued stochastic differential equations involving evolution drift. And (C0, 1)-evolution systems are also defined and investigated on dual multi-Hilbertian spaces.

Structural Stability in 4-Dimensional Canards

Let us consider higher dimensional canards in a sow-fast system R2+2 with a bifurcation parameter. Then, the slow manifold sometimes shows various aspects due to the bifurcation. Introducing a key notion “symmetry” to the slow-fast system, it becomes clear when the pseudo singular point obtains the structural stability or not. It should be treated with a general case. Then, it will also be given about the sufficient conditions for the existence of the center manifold under being “symmetry”. The higher dimensional canards in the sow-fast system are deeply related to Hilbert’s 16th problem. Furthermore, computer simulations are done for the systems having Brownian motions. As a result, the rigidity for the system is confirmed.

A Nonstandard Lévy-Khintchine Formula and Lévy Processes

We use methods from nonstandard analysis to obtain a short and simple derivation of the Levy-Khintchine formula via an explicit construction of certain laws of the infinitesimal increments. Consequently, any arbitrary Levy process is representable as the standard part of a hyperfinite sum of infinitesimal increments.

A Reexamination on the Problem of Advanced Response

In the theory of signal analysis, the existence of the advanced response, whose appearance is earlier than the input, violates the law of causation. It is believed that the presence of advanced response is due to unreasonable setting of the characteristic of the system. Hereof, by means of nonstandard analysis, it is concluded that the advanced response arises from the course of Fourier analysis other than the transmission process of the signal.