[4] **viXra:1708.0240 [pdf]**
*replaced on 2017-08-21 12:33:00*

**Authors:** Koji KOBAYASHI

**Comments:** 5 Pages.

This paper describes about complexity of NP problems by using
minimal circuit, and divide class P and NP.
Inputs of uniform circuit family that compute P problem have some symme-
try that indicated curcit structure. To clarify this symmetry, we define “Min-
imal circuit” as subgraph of circuit which are necessary to compute subset of
inputs. Minimal circuit divide problem to some symmetric partial problems.
The other hand, inputs of NTM that compute NP problem have extra im-
plicit symmetry that indicated nondeterministic transition functions. To clar-
ify this implicit symmetry, we define special DTM “Concrete DTM Di”which
index i correspond to selection of nondeterministic transition functions. That
is, NTM split many different asymmetry DTM Di and compute all Di in same
time.
Consider Di and minimal circuit family, uniform circuit family N that solve
NP problem have to include minimal circuit family that correspond to Di.
These minimal circuit family have unique circuit gate and N must include
these minimal circuit family and gates. Number of such minimal circuit is
over polynomial size of input. Therefore, N is over polynomial size, and P is
not NP.

**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