[7] **viXra:1305.0174 [pdf]**
*submitted on 2013-05-28 22:01:07*

**Authors:** Jin He, Xiaoli Yang

**Comments:** 17 Pages. 2 Figures. Two theorems of rational structure are proved,

Rational structure in two dimension means that not only there exists an orthogonal net of curves in the plane but also, for each curve, the stellar density on one side of the curve is in constant ratio to the density on the other side of the curve. Such a curve is called a proportion curve or a Darwin curve. Such a distribution of matter is called a rational structure. Spiral galaxies are blended with dust and gas. Their longer wavelength (e.g. infrared) images present mainly the stellar distribution, which is called the naked galaxies. Jin He found many evidences that galaxies are rational stellar distribution. We list a few examples. Firstly, galaxy components (disks and bars) can be fitted with rational structure. Secondly, spiral arms can be fitted with Darwin curves. Thirdly, rational structure dictates New Universal Gravity which explains constant rotation curves simply and elegantly. This article presents the systematic theory of rational structure, its general solution and geometric meaning. A preliminary application to spiral galaxies is also discussed.

**Category:** Functions and Analysis

[6] **viXra:1305.0156 [pdf]**
*submitted on 2013-05-27 04:09:38*

**Authors:** Claude Michael Cassano

**Comments:** 5 Pages.

A formula for Particular solutions to any Linear Second Order Inhomogeneous Ordinary Diffrential equations is presented. For second order ODEs these make the methods of undetermined coefficients and variation of parameters obsolete.

**Category:** Functions and Analysis

[5] **viXra:1305.0147 [pdf]**
*submitted on 2013-05-23 19:33:33*

**Authors:** Jin He, Xiaoli Yang

**Comments:** 14 Pages. 1 figure. Solve the beautiful equation and reveal the secret of galaxies

We have not found the general solution to the Creator's equation system. However, we have outlined the strategy for determining the solution. Firstly, we should study the stretch equation which is the first order linear and homogeneous partial differential equation, and find its all stretches which correspond to the given vector field (i.e., the gradient of the logarithmic stellar density). Our solution G(x,y), however, must be simultaneously the modulus of some analytic complex function. It is called the modulus stretch. Secondly, among all possible modulus stretches, we find the right solution (i.e., the orthogonal net of curves) which satisfies the Creator's standard equation.

**Category:** Functions and Analysis

[4] **viXra:1305.0094 [pdf]**
*submitted on 2013-05-15 22:17:49*

**Authors:** Jin He, Xiaoli Yang

**Comments:** 11 Pages. 1 Figure. Nobel prizes come from the solutions of the Creator's equation.

Is the sum of rational structures also a rational structure? It is called the Creator's big question for humans. Numerical calculation suggests that it is approximately rational for the fitted parameter values of barred spiral galaxies. However, we need mathematical justification. The authors present the Creator's equation system without composite functions, the equation system being the necessary and sufficient condition for rational structure. However, we have not found its general solution. Please help us find the general solution.

**Category:** Functions and Analysis

[3] **viXra:1305.0082 [pdf]**
*submitted on 2013-05-14 00:58:36*

**Authors:** Yaqub Azari

**Comments:** 9 Pages.

This paper presents the local radial basis function based on finite difference (LRBF-FD) for the sine-Gordon equation. Advantages of the proposed method are that this method is mesh free unlike finite difference (FD) and finite element (FE) methods, and its coefficient matrix is sparse and well-conditioned as compared with the global RBF collocation method (GRBF). Numerical results show that the LRBF-FD method has good accuracy as compared with GRBF.

**Category:** Functions and Analysis

[2] **viXra:1305.0052 [pdf]**
*replaced on 2013-05-15 22:21:50*

**Authors:** Jin He, Xiaoli Yang

**Comments:** 8 Pages. 1 Figure. The authors are very old and are not experts in mathematics. Please help us humans to resolve the question.

Is the sum of rational structures also a rational structure? It is called the Creator's big question for humans. Numerical calculation suggests that it is approximately rational for the fitted parameter values of barred spiral galaxies. However, we need mathematical justification. The authors are very old and are not experts in mathematics. Please help us humans to resolve the question.

**Category:** Functions and Analysis

[1] **viXra:1305.0040 [pdf]**
*submitted on 2013-05-07 04:43:29*

**Authors:** Nasser Almismari

**Comments:** 12 Pages.

This paper is to find by proof the first derive of known tetration functions, fixed base iterated functions b^^x , general case for b^^f(x) and variable base with variable height iterated function x^^x. although the case of b^^x is already known by using the base change method but its derive function f(x) is still depend on the derive of f(x-1) which gives a shortcoming derivation. However, in the coming proofs, the resulted derivative functions are proved by applying differentiation elementary concepts step by step up to the final first derive ,but an unknown limit and a non-elementary product part of the resulted derivative function still needs study, Although I included approximation method for numerical solutions.

**Category:** Functions and Analysis