This unique book gives a comprehensive account of new mathematical tools useful to solve polygon problems. In the 20th and 21st centuries, many problems inmathematics, theoretical physics and theoretical chemistry - and more recently in molecular biology and bio-informatics - can be expressed as counting problems, in which specified graphs or shapes are counted. One special class of shapes is that of polygons. These are closed, connected paths in space. We usually sketch them in two dimensions, but they can exist in any dimension. The typical questions asked include “how many are there of a given perimeter?”, “how big is the average polygon of given perimeter?”, and corresponding questions about the area or volume enclosed. For example, “how many enclosing a given area?” and “how large is an average polygon of given area?” Simple though they seems, these questions are extraordinarily difficult to answer. Only book devoted to polygons First published account of a class of ultra-fast counting algorithms New experimental mathematics techniques to conjecture exact solutions Accessible accounts of new mathematical tools to solve polygon problems Powerful tools such as SLE and conformal invariance from mathematical physics INDICE: 1. History of polygon models, polyominoes and polyhedra. 2. Lattice polygons and related objects. 3. Exactly solved models. 4. Why are so many problems unsolved? 5. The anisotropic generating function of self-avoiding polygons is not D-finite. 6. Polygons and the Lace Expansions. 7. Exact enumerations. 8. Series analysis. 9. Monte Carlo methods for lattice polygons. 10. Effect of confinement: polygonsin strips, slabs and rectangles. 11. Limit distributions and scaling functions. 12. Interacting lattice polygons. 13. Fully packedloop models on finite geometries. 14. Conformal field theory applied to loop models. 15. Stochastic Löwner evolution and the scaling limit of critical models. 16. Series data and growth constant, amplitude and exponent estimates. References. Index.
- ISBN: 978-1-4020-9926-7
- Editorial: Springer
- Encuadernacion: Cartoné
- Páginas: 520
- Fecha Publicación: 01/04/2009
- Nº Volúmenes: 1
- Idioma: Inglés