# Algebra

## 1610 Submissions

[3] **viXra:1610.0353 [pdf]**
*submitted on 2016-10-29 08:05:38*

### Duplex Fraction Method To Compute The Determinant Of A 4 × 4 Matrix

**Authors:** Reza Farhadian

**Comments:** 4 Pages.

In this paper, We present a new method to compute the determinant of a 4 × 4 matrix, that is very simplest than previous methods in this subject. This method is obtained by a new definition of fraction and also by using the Dodgson’s condensation method and Salihu’s method.

**Category:** Algebra

[2] **viXra:1610.0178 [pdf]**
*submitted on 2016-10-16 13:16:55*

### MOD Natural Neutrosophic Semirings

**Authors:** W. B. Vasantha Kandasamy, K. Ilanthenral, Florentin Smarandach

**Comments:** 262 Pages.

In this book for the first time authors describe and develop the new notion of MOD natural neutrosophic semirings using Z^I_n, C_I(Zn), _I, _I, _I and _I.
Several interesting properties about this structure is derived. Using these MOD natural neutrosophic semirings MOD natural
neutrosophic matrix semirings and MOD natural neutrosophic polynomial semirings and defined and described. Special elements of these structures are analysed. When MOD intervals [0, n) and MOD natural neutrosophic intervals [0, n) are used we see the MOD semirings do not in general satisfy the associative laws and the distributive laws leading to the definition of pseudo semirings of infinite order. These are also introduced in this book. We also define and develop MOD subset pseudo semiring and MOD subset natural neutrosophic pseudo semirings. This study is innovative and interesting by providing a large class of MOD pseudo semirings. Special elements in them are analysed.
Using these MOD subset matrix pseudo semirings and MOD subset polynomial pseudo semirings and developed.
They enjoy very many special features. Several problems are suggested and these notions will certainly attract semiring theorists.

**Category:** Algebra

[1] **viXra:1610.0118 [pdf]**
*submitted on 2016-10-11 13:57:41*

### Symmetric Normal Matrices and the Marcus-de Oliveira Determinantal Conjecture

**Authors:** Ameet Sharma

**Comments:** 15 Pages.

We present notes on the Marcus-de Oliveira conjecture. The conjecture concerns the region in the complex plane covered by the determinants of the sums of two normal matrices with prescribed eigenvalues. Call this region ∆U . Let ∆O be the restriction of ∆U to determinants of sums of symmetric normal matrices. In this paper, we conjecture that ∆O has the same boundary as ∆U. We prove the conjecture for the cases: 1) at least one of the two matrices has just one eigenvalue, 2) at least one of the two matrices has distinct eigenvalues. The implication of this theorem is that proving the Marcus-de Oliveira conjecture for symmetric normal matrices would prove it for the general case. This paper builds on work in [1].

**Category:** Algebra