[4] **viXra:1003.0229 [pdf]**
*submitted on 7 Mar 2010*

**Authors:** Linfan Mao

**Comments:** 275 pages

A Smarandache multi-space is a union of n different spaces equipped with some
different structures for an integer n ≥ 2, which can be used both for discrete or
connected spaces, particularly for geometries and spacetimes in theoretical physics.

**Category:** Combinatorics and Graph Theory

[3] **viXra:1003.0227 [pdf]**
*replaced on 26 Jun 2011*

**Authors:** Linfan Mao

**Comments:** 399 pages.

Automorphism groups survey similarities on mathematical systems, which appear nearly
in all mathematical branches, such as those of algebra, combinatorics, geometry, ... and
theoretical physics, theoretical chemistry, etc. In geometry, configurations with high
symmetry born symmetrical patterns, a kind of beautiful pictures in aesthetics. Naturally,
automorphism groups enable one to distinguish systems by similarity. More automorphisms
imply more symmetries of that system. This fact has established the fundamental
role of automorphism groups in modern sciences. So it is important for graduate students
knowing automorphism groups with applications.

**Category:** Combinatorics and Graph Theory

[2] **viXra:1003.0223 [pdf]**
*submitted on 7 Mar 2010*

**Authors:** Linfan Mao

**Comments:** 215 pages

SMARANDACHE GEOMETRIES
&
MAP THEORY WITH APPLICATIONS

**Category:** Combinatorics and Graph Theory

[1] **viXra:1003.0221 [pdf]**
*replaced on 27 Jun 2011*

**Authors:** Linfan Mao

**Comments:** 502 pages.

Accompanied with humanity into the 21st century, a highlight trend for developing
a science is its overlap and hybrid, and harmoniously with other sciences, which
enables one to handle complex systems in the WORLD. This is also for developing
mathematics. As a powerful tool for dealing with relations among objectives,
combinatorics, including combinatorial theory and graph theory mushroomed in last
century. Its related with algebra, probability theory and geometry has made it to an
important subject in mathematics and interesting results emerged in large number
without metrics. Today, the time is come for applying combinatorial technique to
other mathematics and other sciences besides just to find combinatorial behavior
for objectives. That is the motivation of this book, i.e., to survey mathematics and
fields by combinatorial principle.

**Category:** Combinatorics and Graph Theory