# Functions and Analysis

## 1909 Submissions

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

### A General Form of the Beppo Levi`s Lemma

**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*

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

**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*

### Modified Bisection Algorithm for Multiple Roots of Nonlinear Equation with the R Software

**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