Automorphism Groups Of Maps Surfaces And Smarandache Geometries Second Edition Graduate Text Book In Mathematics

A combinatorial map is a connected topological graph cellularly embedded in a surface. This monograph concentrates on the automorphism group of a map, which is related to the automorphism groups of a Klein surface and a Smarandache manifold, also applied to the enumeration of unrooted maps on orientable and non-orientable surfaces. A number of results for the automorphism groups of maps, Klein surfaces and Smarandache manifolds and the enumeration of unrooted maps underlying a graph on orientable and non-orientable surfaces are discovered. An elementary classification for the closed s-manifolds is found. Open problems related to the combinatorial maps with the differential geometry, Riemann geometry and Smarandache geometries are also presented in this monograph for the further applications of the combinatorial maps to the classical mathematics.

The Mathematical Combinatorics (International Book Series) is a fully refereed international book series, quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences.

The International J. Mathematical Combinatorics is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly, which publishes original research papers and survey articles in all aspects of mathematical combinatorics, Smarandache multi-spaces, Smarandache geometries, non-Euclidean geometry, topology and their applications to other sciences.

A thing is complex, and hybrid with other things sometimes. Then, what is the reality of a thing? The reality of a thing is its state of existed, exists, or will exist in the world, independent on the understanding of human beings, which implies that the reality holds on by human beings maybe local or gradual, not the reality of a thing. Hence, to hold on the reality of things is the main objective of science in the history of human development.

Topics in detail to be covered are: Smarandache multi-spaces with applications to other sciences, such as those of algebraic multi-systems, multi-metric spaces; Smarandache geometries; Differential Geometry; Geometry on manifolds; Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration; Low Dimensional Topology; Differential Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations with Manifold Topology; Applications of Smarandache multi-spaces to theoretical physics; Applications of Combinatorics to mathematics and theoretical physics; Mathematical theory on gravitational fields; Mathematical theory on parallel universes; Other applications of Smarandache multi-space and combinatorics.

Mathematical science is the human recognition on the evolution laws of things that we can understand with the principle of logical consistency by mathematics, i.e., mathematical reality. So, is the mathematical reality equal to the reality of thing? The answer is not because there always exists contradiction between things in the eyes of human, which is only a local or conditional conclusion. Such a situation enables us to extend the mathematics further by combinatorics for the reality of thing from the local reality and then, to get a combinatorial reality of thing. This is the combinatorial conjecture for mathematical science, i.e., CC conjecture that I put forward in my postdoctoral report for Chinese Acade- my of Sciences in 2005, namely any mathematical science can be reconstructed from or made by combinatorialization. After discovering its relation with Smarandache multi-spaces, it is then be applied to generalize mathematics over 1-dimensional topological graphs, namely the mathematical combinatorics that I promoted on science internationally for more than 20 years. This paper surveys how I proposed this conjecture from combinatorial topology, how to use it for characterizing the non-uniform groups or contradictory systems and furthermore, why I introduce the continuity ow GL as a mathematical element, i.e., vectors in Banach space over topological graphs with operations and then, how to apply it to generalize a few of important conclusions in functional analysis for providing the human recognition on the reality of things, including the subdivision of substance into elementary particles or quarks in theoretical physics with a mathematical supporting.