[6] **viXra:1909.0658 [pdf]**
*submitted on 2019-09-30 21:12:36*

**Authors:** Saburou Saitoh

**Comments:** 3 Pages. I think many and many mathematicians will be able to enjoy the result, indeed, over mathematicians. It will be a general interest.

In this short note, we will consider the value of the function $\exp {(ax)}/f(a)$ at $a=0$ for $f(a)=0$. This case appears for the construction of the special solution of some differential operator $f(D)$ for the polynomial case of $D$ with constant coefficients. We would like to show the power of the new method of the division by zero calculus, simply and typically.

**Category:** Functions and Analysis

[5] **viXra:1909.0572 [pdf]**
*submitted on 2019-09-26 16:06:21*

**Authors:** Bertrand Wong

**Comments:** 4 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. [The paper is published in an international journal.]

**Category:** Functions and Analysis

[4] **viXra:1909.0366 [pdf]**
*submitted on 2019-09-17 19:20:58*

**Authors:** Teo Banica

**Comments:** 200 Pages.

The unitary group U_N has a free analogue U_N^+, and the study of the closed subgroups G\subset U_N^+ is a problem of general interest. We review here the general theory of U_N^+ and its subgroups, with all the needed preliminaries included. We discuss as well a number of more advanced topics, selected for their beauty, and potential importance.

**Category:** Functions and Analysis

[3] **viXra:1909.0294 [pdf]**
*replaced on 2019-09-16 03:56:42*

**Authors:** Johan Aspegren

**Comments:** 5 Pages.

In this article we will prove a version of the Beppo Levi`s lemma for the complex valued functions. This achieved by making a more stronger asumption that is assumed in Beppo Levi`s lemma. We will assume that the sum of measurable functions that is absolutely convergent almost everywhere is integrable. We will prove that it implies the asumptions of the Beppo Levi lemma, if we consider functions that are non-negative. It can be argued that our version is more suitable to applications, and we will prove a new probability law. We will show that with our asumptions in probability theory it follows that the expected value is countable additive. Moreover, it follows that in strong law of large numbers we don`t need to make any asumptions on distributions and the mean of the sample will convergence almost surely to the mean of the expected values.

**Category:** Functions and Analysis

[2] **viXra:1909.0200 [pdf]**
*submitted on 2019-09-09 16:24:17*

**Authors:** Saburou Saitoh

**Comments:** 6 Pages. A very surprising and mysterious property at the point at infinity.

Okumura's Disc Series Can Beyond the Crucial Point of D\"aumler-Puha's Horn Torus Models for the Riemann Sphere

**Category:** Functions and Analysis

[1] **viXra:1909.0189 [pdf]**
*submitted on 2019-09-10 05:20:53*

**Authors:** Kouider Mohammed Ridha

**Comments:** 3 Pages.

The most basic problem in numerical analysis (methods) is the root finding problem. In this paper we are interesting in represent numerical method which is the Modified Bisection Algorithm(MBA) referred to by Tanakan, (2013, [9]) for finding the multi-roots of a function. Hence in this study we programming the MBA for multi-roots with the R software.

**Category:** Functions and Analysis