[3] **viXra:1511.0267 [pdf]**
*submitted on 2015-11-27 16:20:45*

**Authors:** Józef Radomański

**Comments:** 17 Pages.

Paravectors just like integers have a ring structure. By introducing an integrated product we get geometric properties which make paravectors similar to vectors. The concepts of parallelism, perpendicularity and the angle are conceptually similar to vector counterparts, known from the Euclidean geometry. Paravectors meet the idea
of parallelogram law, Pythagorean theorem and many other properties well-known to everyone from school.

**Category:** Algebra

[2] **viXra:1511.0034 [pdf]**
*submitted on 2015-11-03 11:48:25*

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

**Comments:** 215 Pages.

Study of MOD planes happens to a very recent one. In this book, systematically algebraic structures on MOD planes like, MOD semigroups, MOD groups and MOD rings of different types are defined and studied. Such study is innovative for a large four quadrant planes are made into a small MOD planes. Several distinct features enjoyed by these MOD planes are defined, developed and described.

**Category:** Algebra

[1] **viXra:1511.0032 [pdf]**
*submitted on 2015-11-03 11:52:39*

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

**Comments:** 205 Pages.

In this book the notion of MOD functions are defined on MOD planes. This new concept of MOD functions behaves in a very different way. Even very simple functions like y = nx has several zeros in MOD planes where as they are nice single line graphs with only (0, 0) as the only zero. Further polynomials in MOD planes do not in general follows the usual or classical laws of differentiation or integration.

**Category:** Algebra