Functions and Analysis

1206 Submissions

[3] viXra:1206.0086 [pdf] replaced on 2014-06-09 10:32:56

The Navier-Stokes Equations And Turbulence

Authors: Bertrand Wong
Comments: 2 Pages.

The Navier-Stokes differential equations describe the motion of fluids which are incompressible. The three-dimensional Navier-Stokes equations misbehave very badly although they are relatively simple-looking. The solutions could wind up being extremely unstable even with nice, smooth, reasonably harmless initial conditions. A mathematical understanding of the outrageous behaviour of these equations would dramatically alter the field of fluid mechanics. This paper describes why the three-dimensional Navier-Stokes equations are not solvable, i.e., the equations cannot be used to model turbulence, which is a three-dimensional phenomenon.
Category: Functions and Analysis

[2] viXra:1206.0017 [pdf] submitted on 2012-06-05 10:52:56

Random Consensus in Nonlinear Systems Under Fixed Topology

Authors: Radha F. Gupta, Poom Kumam
Comments: 9 Pages.

This paper investigates the consensus problem in almost sure sense for uncertain multi-agent systems with noises and fixed topology. By combining the tools of stochastic analysis, algebraic graph theory, and matrix theory, we analyze the convergence of a class of distributed stochastic type non-linear protocols. Numerical examples are given to illustrate the results.
Category: Functions and Analysis

[1] viXra:1206.0005 [pdf] submitted on 2012-06-02 21:56:55

Fractional Geometric Calculus: Toward A Unified Mathematical Language for Physics and Engineering

Authors: Xiong Wang
Comments: 6 Pages.

This paper discuss the longstanding problems of fractional calculus such as too many definitions while lacking physical or geometrical meanings, and try to extend fractional calculus to any dimension. First, some different definitions of fractional derivatives, such as the Riemann-Liouville derivative, the Caputo derivative, Kolwankar's local derivative and Jumarie's modified Riemann-Liouville derivative, are discussed and conclude that the very reason for introducing fractional derivative is to study nondifferentiable functions. Then, a concise and essentially local definition of fractional derivative for one dimension function is introduced and its geometrical interpretation is given. Based on this simple definition, the fractional calculus is extended to any dimension and the \emph{Fractional Geometric Calculus} is proposed. Geometric algebra provided an powerful mathematical framework in which the most advanced concepts modern physic, such as quantum mechanics, relativity, electromagnetism, etc., can be expressed in this framework graciously. At the other hand, recent developments in nonlinear science and complex system suggest that scaling, fractal structures, and nondifferentiable functions occur much more naturally and abundantly in formulations of physical theories. In this paper, the extended framework namely the Fractional Geometric Calculus is proposed naturally, which aims to give a unifying language for mathematics, physics and science of complexity of the 21st century.
Category: Functions and Analysis