[17] **viXra:1306.0234 [pdf]**
*submitted on 2013-06-29 19:34:04*

**Authors:** John Frederick Sweeney

**Comments:** 6 Pages.

In math, especially within the Lie Algebra groups, there exist a small group of "exceptional isomorphisms" or "accidental isomorphisms." Following the lead of Swiss psychologist Carl Jung, the author does not accept the existence of "exceptional" or "accidental;" instead, these are merely phenomena which heretofore have not been explained satisfactorily by mathematical theory. However, articulation of the Qi Men Dun Jia Cosmic Board Model in a recent and forthcoming paper has helped to explain several of the exceptional isomorphisms.

**Category:** Algebra

[16] **viXra:1306.0178 [pdf]**
*submitted on 2013-06-21 04:43:19*

**Authors:** Eckhard Hitzer

**Comments:** 8 Pages. 3 figures, 1 table. Support Website for the Linear Algebra Lectures of the University of the Air Japan 2004-2008.

Axioms for Geometric Algebra R_{p,q} - Definitions using quadratic form, basic multiplication rules. Grade r subspaces, geometric algebra R_2, complex numbers, reflections and rotations, 2-dim. point groups, geometric algebra R_3 and quaternions.

**Category:** Algebra

[15] **viXra:1306.0177 [pdf]**
*submitted on 2013-06-21 04:46:56*

**Authors:** Eckhard Hitzer

**Comments:** 3 Pages. Support Website for the Linear Algebra Lectures of the University of the Air Japan 2004-2008.

Definition, Adjoint and inverse linear mappings, References.

**Category:** Algebra

[14] **viXra:1306.0176 [pdf]**
*submitted on 2013-06-21 04:49:53*

**Authors:** Eckhard Hitzer

**Comments:** 2 Pages. Support Website for the Linear Algebra Lectures of the University of the Air Japan 2004-2008.

We describe Gram-Schmidt orthogonalization in W.K. Clifford's geometric algebra.

**Category:** Algebra

[13] **viXra:1306.0175 [pdf]**
*submitted on 2013-06-21 04:54:51*

**Authors:** Eckhard Hitzer

**Comments:** 11 Pages. Support Website for the Linear Algebra Lectures of the University of the Air Japan 2004-2008.

The aim of this work is to show how the geometric product of multivectors is defined in
general, extending the basic geometric product of vectors given by Clifford. An alternative
definition of Clifford geometric algebra, that guarantees existence as quotient algebra of
the tensor algebra was given by Chevalley in 1954.[2] We further treat the scalar product, the outer product, the cross product in three dimensions, linear dependence and independence, as well as right- and left contractions.
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**Category:** Algebra

[12] **viXra:1306.0174 [pdf]**
*submitted on 2013-06-21 04:59:20*

**Authors:** Eckhard Hitzer

**Comments:** 6 Pages. Support Website for the Linear Algebra Lectures of the University of the Air Japan 2004-2008.

Definition of geometric algebra with quadratic form - examples of quadratic forms and associated geometric algebras - geometric algebras with degenerate quadratic forms - new interpretation of the geometric algebra of the Minkowski plane - generalizing to the geometric mother algebra with p=q=n.

**Category:** Algebra

[11] **viXra:1306.0173 [pdf]**
*submitted on 2013-06-21 05:03:08*

**Authors:** Eckhard Hitzer

**Comments:** 4 Pages. Support Website for the Linear Algebra Lectures of the University of the Air Japan 2004-2008.

The previous Japanese emperor is said to have asked this question. Today many students and scientists still ask it, but the traditional canon of mathematics at school and university needs to be widened for the answer.

**Category:** Algebra

[10] **viXra:1306.0150 [pdf]**
*submitted on 2013-06-19 03:03:40*

**Authors:** Eckhard Hitzer, Christian Perwass

**Comments:** 2 Pages. 1 figure, 1 table. Bulletin of the Society for Science on Form, 21(1), pp. 55,56 (2006).

This paper focuses on the symmetries of crystal space lattices. All two dimensional
(2D) and three dimensional (3D) point groups of 2D and 3D crystal cells are exclusively described
by vectors (two in 2D, three in 3D for one particular cell) taken from the physical cells. Geometric
multiplication of these vectors completely generates all symmetries, including reflections,
rotations, inversions, rotary-reflections and rotary-inversions. We then extend this treatment to 3D
space groups by including translations, glide reflections and screw rotations. We focus on the
monoclinic case as an example. A software demonstration shows the spacegroup visualizer.
**Keywords:** Crystal lattice, space group, geometric algebra, OpenGL, interactive software.

**Category:** Algebra

[9] **viXra:1306.0125 [pdf]**
*submitted on 2013-06-17 02:20:49*

**Authors:** Eckhard Hitzer, Luca Redaelli

**Comments:** 22 Pages. 27 figures. Advances in Applied Clifford Algebras, 13(2), pp. 157-181 (2003). DOI: 10.1007/s00006-003-0013-x .

Conventional illustrations of the rich elementary relations and physical applications of geometric algebra are helpful,
but restricted in communicating full generality and time dependence. The main restrictions are one special perspective
in each graph and the static character of such illustrations. Several attempts have been made to overcome such
restrictions. But up till now very little animated and fully interactive, free, instant access, online material is available.
This report presents therefore a set of over 90 newly developed (freely online accessible [1]) JAVA
applets. These applets range from the elementary concepts of vector, bivector, outer product and rotations to triangle
relationships, oscillations and polarized waves. A special group of 21 applets illustrates three geometrically different
approaches to the representation of conics; and even more ways to describe ellipses. Next Clifford's famous circle chain
theorem is illustrated. Finally geometric applications important for crystallography and structural mechanics give a
glimpse of the vast potential for applied mathematics. The interactive geometry software Cinderella [2]
was used for creating these applets. The interactive features of many of the applets invite the user to freely explore
by a few mouse clicks as many different special cases and perspectives as he likes. This is of great help in "visualizing"
geometry encoded by geometric algebra.

**Category:** Algebra

[8] **viXra:1306.0113 [pdf]**
*submitted on 2013-06-17 04:25:16*

**Authors:** Eckhard Hitzer

**Comments:** 21 Pages. in T. M. Karade (ed.), Proc. of the Nat. Symp. on Math. Sc., 1-5 March, 2001, Nagpur, India, Einst. Foundation Int. 1, pp. 1-26 (2001).

This paper first reviews how anti-symmetric matrices in two dimensions yield
imaginary eigenvalues and complex eigenvectors. It is shown how this carries on to
rotations by means of the Cayley transformation. Then the necessary tools from real
geometric algebra are introduced and a real geometric interpretation is given to the
eigenvalues and eigenvectors. The latter are seen to be two component eigenspinors
which can be further reduced to underlying vector duplets. The eigenvalues are
interpreted as rotors, which rotate the underlying vector duplets. The second part of
this paper extends and generalizes the treatment to three dimensions. The final part
shows how all entities and relations can be obtained in a constructive way, purely
assuming the geometric algebras of 2-space and 3-space.

**Category:** Algebra

[7] **viXra:1306.0112 [pdf]**
*submitted on 2013-06-17 04:30:00*

**Authors:** Eckhard Hitzer

**Comments:** 16 Pages. Mem. Fac. Eng. Fukui Univ. 49(2), pp. 283-298 (2001).

This paper briefly reviews the conventional method of obtaining the canonical form of an antisymmetric (skew-symmetric, alternating) matrix. Conventionally a vector space over the complex field has to be introduced. After a short introduction to the universal mathematical "language" Geometric Calculus, its fundamentals, i.e. its "grammar" Geometric Algebra (Clifford Algebra) is explained. This lays the groundwork for its real geometric and coordinate free application in order to obtain the canonical form of an antisymmetric matrix in terms of a bivector, which is isomorphic to the conventional canonical form. Then concrete applications to two, three and four dimensional antisymmetric square matrices follow. Because of the physical importance of the Minkowski metric, the canonical form of an antisymmetric matrix with respect to the Minkowski metric is derived as well. A final application to electromagnetic fields concludes the work.
Keywords: Geometric Calculus, Geometric Algebra, Clifford Algebra, antisymmetric (alternating, skewsymmetric) matrix, Real Geometry

**Category:** Algebra

[6] **viXra:1306.0038 [pdf]**
*submitted on 2013-06-06 10:12:21*

**Authors:** H. J. Siamwalla, A.S.Muktibodh

**Comments:** 7 Pages.

In this paper we prove some results towards classifying Smarandache groupoids
which are in Z*(n) and not in Z(n) when n is even and n is odd.

**Category:** Algebra

[5] **viXra:1306.0036 [pdf]**
*submitted on 2013-06-06 10:15:44*

**Authors:** Arun S. Muktibodh

**Comments:** 7 Pages.

In [4] we have introduced Smarandache quasigroups which are Smarandache
non-associative structures. A quasigroup is a groupoid whose composition table is a Latin
square. There are squares in the Latin squares which seem to be of importance to study the
structure of Latin Squares. We consider the particular type of squares properly contained in
the Latin squares which themselves contain a Latin square. Such Latin squares are termed
as Smarandache Mukti-Squares or SMS. Extension of some SMS to Latin squares is also
considered.

**Category:** Algebra

[4] **viXra:1306.0035 [pdf]**
*submitted on 2013-06-06 10:17:09*

**Authors:** Arun S. Muktibodh

**Comments:** 6 Pages.

In this paper, we have introduced Smarandache quasigroups which are Smarandache non-
associative structures. W.B.Kandasamy [2] has studied groupoid ring and loop ring. We have de¯ned
Smarandache quasigroup rings which are again non-associative structures having two binary operations.
Substructures of quasigroup rings are also studied.

**Category:** Algebra

[3] **viXra:1306.0034 [pdf]**
*submitted on 2013-06-06 10:18:29*

**Authors:** Arun S. Muktibodh

**Comments:** 7 Pages.

In this paper, we have introduced Smarandache quasigroups which are Smaran-
dache non-associative structures. W.B.Kandasamy [2] has studied Smarandache groupoids
and Smarandache semigroups etc. Substructure of Smarandache quasigroups are also studied.

**Category:** Algebra

[2] **viXra:1306.0033 [pdf]**
*submitted on 2013-06-06 10:20:01*

**Authors:** Arun S. Muktibodh

**Comments:** 3 Pages.

G. Pilz [1] has dened near-rings and semi-near-rings. In this paper we introduce
the concepts of quasi-near ring and semiquasi-near ring. We have also dened Smarandache
semiquasi-near-ring. Some examples are constructed. We have posed some open problems.

**Category:** Algebra

[1] **viXra:1306.0019 [pdf]**
*submitted on 2013-06-05 00:29:16*

**Authors:** S. Kalimuthu

**Comments:** 4 Pages. NA

The history of matrices goes back to ancient times! But the term "matrix" was not applied to the concept until 1850."Matrix" is the Latin word for womb, and it retains that sense in English. It can also mean more generally any place in which something is formed or produced. The orgins of mathematical matrices lie with the study of systems of simultaneous linear equations. An important Chinese text from between 300 BC and AD 200, Nine Chapters of the Mathematical Art (Chiu Chang Suan Shu), gives the first known example of the use of matrix methods to solve simultaneous equations. Since their first appearance in ancient China, matrices have remained important mathematical tools. Today, they are used not simply for solving systems of simultaneous linear equations, but also for describing the quantum mechanics of atomic structure, designing computer game graphics, analyzing relationships, and even plotting complicated dance steps! The elevation of the matrix from mere tool to important mathematical theory owes a lot to the work of female mathematician Olga Taussky Todd (1906-1995), who began by using matrices to analyze vibrations on airplanes during World War II and became the torchbearer for matrix theory. In this work, by applying the fundamental concepts of matrices, the author attempts to study the fifth Euclidean postulate problem and Godel’s incompleteness theorems.

**Category:** Algebra