多面形理論

內(nèi)容概要

This monograph is for a unified theory of surfaces, embeddings and maps all considered as polyhedra via the joint tree modal which was initiated from the author's articles in the seventies of last century and has been basically developed in recent decades. Complete invariants for each classification are topologically, combinatorially or isomorphically extracted. A number of counting polynomials including handle and crosscap polynomials are presented. In particular, an appendix serves as the exhaustive counting super maps (rooted and nonrooted) including these polynomials with under graphs of small size for the reader's digests.    Although the book is mainly for researchers in mathematics, theoretical physics, chemistry, biology and some others related, the basic part in each chapter can also be chosen for graduates and college teachers as references.

書籍目錄

PrefaceChapter Ⅰ  Preliminaries  Ⅰ.1  Sets and mappings  Ⅰ.2  Partitions and permutations  Ⅰ.3  Group actions  Ⅰ.4  Networks  Ⅰ.5  NotesChapter Ⅱ  Surfaces  Ⅱ.1  Polyhedra  Ⅱ.2  Elementary equⅣalence  Ⅱ.3  Polyhegons  Ⅱ.4  Orientability  Ⅱ.5  Classification  Ⅱ.6  NotesChapter Ⅲ  Embeddings of Graphs  Ⅲ.1  Geometric consideration  Ⅲ.2  Surface closed curve axiom  Ⅲ.3  Distinction  Ⅲ.4  Joint tree model  Ⅲ.5  Combinatorial properties  Ⅲ.6  NotesChapter Ⅳ  Mathematical Maps  Ⅳ.1  Basic permutations  Ⅳ.2  Conjugate axiom  Ⅳ.3  TransitⅣity  Ⅳ.4  Included angles  Ⅳ.5  NotesChapter Ⅴ  Duality on Surfaces  Ⅴ.1  Dual partition of edges  Ⅴ.5  NotesChapter Ⅵ  Invariants on Basic Class  Ⅵ.1  Orientability  Ⅵ.2  Euler characteristic  Ⅵ.3  Basic equⅣalence  Ⅵ.4  Orientable maps  Ⅵ.5  Nonorientable maps  Ⅵ.6  NotesChapter Ⅶ  Asymmetrization  Ⅶ.1  Isomorphisms  Ⅶ.2  Recognition  Ⅶ.3  Upper bound of group order  Ⅶ.4  Determination of the group  Ⅶ.5  Rootings  Ⅶ.6  NotesChapter Ⅷ  Asymmetrized Census  Ⅷ.1  Orientable equation  Ⅷ.2  Planar maps...  Ⅷ.3  Nonorientable equation  Ⅷ.4  Gross equation  Ⅷ.5  The number of maps  Ⅷ.6  NotesChapter Ⅸ  Petal Bundles  Ⅸ.1  Orientable petal bundles  Ⅸ.2  Planar pedal bundles  Ⅸ.3  Nonorientable pedal bundles  Ⅸ.4  The number of pedal bundles  Ⅸ.5  NotesChapter Ⅹ Super Maps of Genus ZeroChapter Ⅺ Symmetric CensusChapter Ⅻ Cycle Oriented MapsChapter ⅫⅠ Census by GenusChapter ⅩⅣ Classic ApplicationsAppendix Ⅰ Embeddings and maps of Small Size Distributed by GenusAppendix Ⅱ Orientable Forms of Surfaces and Their Nonorientable Genus PolynomialsBibliographySubject IndexAuthor Index

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