Polygons, Polyominoes and Polycubes | Buch

Cover: 9781402099267 | Polygons, Polyominoes and Polycubes | A. J. Guttmann | Buch | xix

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Beschreibung

The problem of counting the number of self-avoiding polygons on a square grid, - therbytheirperimeterortheirenclosedarea,is aproblemthatis soeasytostate that, at ?rst sight, it seems surprising that it hasn¿t been solved. It is however perhaps the simplest member of a large class of such problems that have resisted all attempts at their exact solution. These are all problems that are easy to state and look as if they should be solvable. They include percolation, in its various forms, the Ising model of ferromagnetism, polyomino enumeration, Potts models and many others. These models are of intrinsic interest to mathematicians and mathematical physicists, but can also be applied to many other areas, including economics, the social sciences, the biological sciences and even to traf?c models. It is the widespread applicab- ity of these models to interesting phenomena that makes them so deserving of our attention. Here however we restrict our attention to the mathematical aspects. Here we are concerned with collecting together most of what is known about polygons, and the closely related problems of polyominoes. We describe what is known, taking care to distinguish between what has been proved, and what is c- tainlytrue,but has notbeenproved. Theearlierchaptersfocusonwhatis knownand on why the problems have not been solved, culminating in a proof of unsolvability, in a certain sense. The next chapters describe a range of numerical and theoretical methods and tools for extracting as much information about the problem as possible, in some cases permittingexactconjecturesto be made.

Zusammenfassung

The only book devoted to polygons

Presents a class of ultra-fast counting algorithms

New experimental mathematics techniques to conjecture exact solutions

Powerful mathematical tools to solve polygon problems

Inhaltsverzeichnis

History and Introduction to Polygon Models and Polyominoes.- Lattice Polygons and Related Objects.- Exactly Solved Models.- Why Are So Many Problems Unsolved?.- The Anisotropic Generating Function of Self-Avoiding Polygons is not D-Finite.- Polygons and the Lace Expansion.- Exact Enumerations.- Series Analysis.- Monte Carlo Methods for Lattice Polygons.- Effect of Confinement: Polygons in Strips, Slabs and Rectangles.- Limit Distributions and Scaling Functions.- Interacting Lattice Polygons.- Fully Packed Loop Models on Finite Geometries.- Conformal Field Theory Applied to Loop Models.- Stochastic Lowner Evolution and the Scaling Limit of Critical Models.- Appendix: Series Data and Growth Constant, Amplitude and Exponent Estimates.

Details

Erscheinungsjahr:	2009
Fachbereich:	Geometrie
Genre:	Importe , Mathematik
Rubrik:	Naturwissenschaften & Technik
Medium:	Buch
Inhalt:	xix 490 S.
ISBN-13:	9781402099267
ISBN-10:	1402099266
Sprache:	Englisch
Herstellernummer:	12465118
Einband:	Gebunden
Autor:	Guttmann, A. J.
Redaktion:	Guttmann, A. J.
Herausgeber:	A J Guttmann
Hersteller:	Springer Netherland Springer Netherlands
Verantwortliche Person für die EU:	Springer Verlag GmbH, Tiergartenstr. 17, D-69121 Heidelberg, juergen.hartmann@springer.com
Maße:	241 x 160 x 33 mm
Von/Mit:	A. J. Guttmann
Erscheinungsdatum:	18.05.2009
Gewicht:	0,928 kg

Artikel-ID: 101681531