[9] **viXra:1708.0419 [pdf]**
*submitted on 2017-08-28 08:22:53*

**Authors:** Edgar Valdebenito

**Comments:** 14 Pages.

This note presents some results on pi.

**Category:** General Mathematics

[8] **viXra:1708.0418 [pdf]**
*submitted on 2017-08-28 08:28:05*

**Authors:** Edgar Valdebenito

**Comments:** 3 Pages.

In this note we give some double integrals involving pi^2 and pi^3.

**Category:** General Mathematics

[7] **viXra:1708.0416 [pdf]**
*submitted on 2017-08-28 08:44:11*

**Authors:** Edgar Valdebenito

**Comments:** 2 Pages.

This note presents some infinite products involving pi.

**Category:** General Mathematics

[6] **viXra:1708.0415 [pdf]**
*submitted on 2017-08-28 08:47:32*

**Authors:** Edgar Valdebenito

**Comments:** 13 Pages.

This note presents fifteen integrals and ten fractals.

**Category:** General Mathematics

[5] **viXra:1708.0330 [pdf]**
*submitted on 2017-08-24 13:34:48*

**Authors:** David Njeru Kathuri

**Comments:** 52 Pages.

Burden of proof is the obligation on somebody presenting a new idea (a claim) to provide evidence to support its truth (a warrant). Once evidence has been presented, it is up to any opposing "side" to prove the evidence presented is not adequate.
The document is about solving the Division by Zero problem. This is done via scientifically analyzing what quantitative division really entails -After a scientific analysis, it yields a principal/law that states that division of any countable quantity cannot occur without movement. This principle helps in the solving of the Division by Zero problem.
It falls under Mathematical Analysis category and tries to distinguish the scope of limits involving zero and its inverse, that is, divisions by zero.

**Category:** General Mathematics

[4] **viXra:1708.0240 [pdf]**
*replaced on 2017-10-01 12:50:17*

**Authors:** Koji KOBAYASHI

**Comments:** 5 Pages.

This paper describes about complexity of NP problems by using “Effective circuit” independency, and apply SAT problem.
Inputs of circuit family that compute P problem have some explicit symmetry that indicated circuit structure. To clarify this explict symmetry, we define “Effective circuit” as partial circuit which are necessary to compute target inputs. Effective circuit set divide problem to some symmetric partial problems.
The other hand, inputs of NTM that compute NP problem have extra implicit symmetry that indicated nondeterministic transition functions. To clarify this implicit symmetry, we define special DTM “Concrete DTM”which index i correspond to selection of nondeterministic transition functions. That is, NTM split many different asymmetry DTM and compute all DTM in same time.
Consider concrete DTM and effective circuit set, circuit family [SAT] that solve SAT problem have to include all effective circuit set [CVPi] that correspond to concrete DTM as Circuit Value Problem. [CVPi] have unique gate and [SAT] must include all [CVPi]. Number of [CVPi] is over polynomial size of input. Therefore, [SAT] is over polynomial size.

**Category:** General Mathematics

[3] **viXra:1708.0224 [pdf]**
*submitted on 2017-08-18 20:48:18*

**Authors:** Claude Michael Cassano

**Comments:** 2 Pages.

Elementary ordinary differential equations texts often present factorization of ordinary differential equations with constant coefficients by linear operators. The theorem, here, demonstrates that any Second Order Homogeneous Linear Ordinary Differential Equation with differentiable coefficients may be factored via two linear differential operators, by way of the reduction of order formula. Since the reduction of order formula applies at any order, this theorem may be generalized to any order.

**Category:** General Mathematics

[2] **viXra:1708.0205 [pdf]**
*submitted on 2017-08-17 05:20:01*

**Authors:** Mohsen Soori, Behrooz Arezoo, Mohsen Habibi

**Comments:** 14 Pages.

Accuracy of produced parts in machining process is influenced by many errors such as tool deflection as well as geometrical deviations of machine tool structure. To increase accuracy
and productivity in part manufacturing, the errors are modelled by using mathematical concepts. This paper presents an application for the virtual machining systems to analyse accuracy in modelling of tool deflection error. A virtual machining system is used to create actual parts in
virtual environments. Then, a comparison for different methods of tool deflection error such as
cantilever beam model of the cutting tool, Finite Element Method (FEM) of the cutting tool and
workpiece and geometrical model of the cutting tool effects on the workpiece is presented to
show accuracy and reliability of the methods in prediction of milled surfaces. So, capabilities and difficulties of the methods in the error modelling are presented to increase accuracy and
efficiency in part manufacturing.

**Category:** General Mathematics

[1] **viXra:1708.0102 [pdf]**
*submitted on 2017-08-09 13:30:28*

**Authors:** Edgar Valdebenito

**Comments:** 15 Pages.

This note presents some results related with fractals,polynomials and number pi.

**Category:** General Mathematics