[7] **viXra:1605.0306 [pdf]**
*submitted on 2016-05-30 15:53:54*

**Authors:** Bin Wang

**Comments:** 25 Pages. This is the second of three papers, all of which are posted on this site.

This paper includes two main chapters, \S 2 and \S3. Each deals with one type of algebraic Poincar\'e duality (APD) on
linear spaces originated from algebraic cycles. Two types of APD confirm the following conjectures:
(1) the Griffiths' conjecture on the incidence equivalence versus Abel-Jacobi equivalence.
(2) the standard conjectures including the ``D" conjecture over $\mathbb C$.

**Category:** Algebra

[6] **viXra:1605.0305 [pdf]**
*submitted on 2016-05-30 15:57:17*

**Authors:** Bin Wang

**Comments:** 27 Pages. This is the last of three papers, all of which are posted on this site.

This is the sequel of three papers. In this paper we apply algebraic Poincar\'e duality to
the maximal sub Hodge structures to show
(1) Generalized Hodge conjecture of level 1 is correct.
(2) The generalized Hodge conjecture of level 0, i.e. the usual Hodge conjecture, is correct.

**Category:** Algebra

[5] **viXra:1605.0302 [pdf]**
*submitted on 2016-05-30 15:49:46*

**Authors:** Bin Wang

**Comments:** 20 Pages. This is the first of three papers, all of which are posted on this site.

We discuss a structure that exists in many problems on smooth projective varieties over the field of complex numbers,
and name it as ``Algebraic Poincar\'e duality" or ``APD" for abbreviation.
In particular, over the complex numbers with singular cohomology, it is a solution to
(1) Griffiths' conjecture on the incidence equivalence versus Abel-Jacobi equivalence,
(2) Generalized Hodge conjecture of level 1,
(3) Generalized Hodge conjecture of level 0, i.e. the usual Hodge conjecture,
(4) The standard conjectures,
(5) Grothendieck's ``D" conjecture.
However it is not the goal of this paper to show APD implies these conjectures. In this paper we'll build the foundation for the structure by introducing the APD in its simplest form over the complex numbers.

**Category:** Algebra

[4] **viXra:1605.0139 [pdf]**
*submitted on 2016-05-13 12:00:15*

**Authors:** José de Jesús Camacho Medina

**Comments:** 13 Pages.

In the following document shows a particular form of simplify the root of a sum as the sum of roots, through an algebraic expression entitled: "Camacho Identity".

**Category:** Algebra

[3] **viXra:1605.0022 [pdf]**
*submitted on 2016-05-03 01:16:32*

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

**Comments:** 257 Pages.

In this book authors for the first time define a special type of fixed points using MOD rectangular matrices as operators. In this case the special fixed points or limit cycles are pairs which
is arrived after a finite number of iterations. Such study is both new and innovative for it can find lots of applications in mathematical modeling.

**Category:** Algebra

[2] **viXra:1605.0021 [pdf]**
*submitted on 2016-05-03 01:17:45*

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

**Comments:** 202 Pages.

In this book authors for the first time introduce a special type of fixed points using MOD square matrix operators. These special type of fixed points are different from the usual classical
fixed points.

**Category:** Algebra

[1] **viXra:1605.0013 [pdf]**
*submitted on 2016-05-02 05:34:51*

**Authors:** Yakov A. Iosilevskii

**Comments:** 34 Pages. An additional category is "Mathematics: Set Theory and Logic"

There are two presently common onamastic (onomatological) methods of logographically naming and thus concisely describing an algebraic system; both methods are often used simultaneously. According to one method, an algebraic system is equivocally denoted by an atomic logographic symbol that originally denotes a certain underlying set of elements, which is regarded as the principal one, while all other objects of the algebraic system, properly named, are kept in mind and are regarded as implicit properties of that set or of its separate elements. That is to say, according to this method, an algebraic system is its principal underlying set of elements together with all its properties, which are implied and are not mentioned explicitly. According to the other method, an algebraic system is regarded as an ordered multiple, whose coordinates properly denote the defining objects of the algebraic system, and consequently the ordered multiple name is equivocally used as a proper name of the algebraic system. Thus, in this case, the togetherness of all constituents of the algebraic system is expressed by the pertinent ordered multiple name in terms of its coordinate names. In my recent article available at http://viXra.org/abs/1604.0124¸ I have demonstrated that both above onomastic methods are inconsistent. Therefore, in that article and also in my earlier article appearing at http://arxiv.org/abs/1510.00328, I suggested and used another onomastic method of logographically naming the pertinent algebraic systems, namely that employing, as a name of an algebraic system, a complex logographic name the union of all explicit constituent sets of the system, namely, the underlying sets of elements, the surjective binary composition functions, and the bijective singulary inversion functions; a function is a set (class) of ordered pairs. In the present article, the latter onomastic method is substantiated and generalized in two respects. First, the set of explicit constituent sets of an algebraic system is now extended to include the injective choice, or selection, functions of all additive and multiplicative identity elements of the algebraic system, belonging to its underlying sets, so that all those elements are now mentioned by the logographic name of the system. A general definition of an algebraic system is elaborated in such a way so as to make the new onomastic method universally applicable to any algebraic system.

**Category:** Algebra