Polygons, Polyominoes and Polycubes

Buch | Softcover

490 Seiten

2016 | Softcover Reprint of the Original 1st 2009 ed.
Springer (Verlag)
978-94-017-7712-4 (ISBN)

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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.

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.

Erscheinungsdatum	16.07.2016
Reihe/Serie	Lecture Notes in Physics ; 775
Zusatzinfo	XIX, 490 p.
Verlagsort	Dordrecht
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Informatik ► Theorie / Studium ► Algorithmen
	Mathematik / Informatik ► Mathematik ► Algebra
	Mathematik / Informatik ► Mathematik ► Analysis
	Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
	Mathematik / Informatik ► Mathematik ► Graphentheorie
	Naturwissenschaften ► Biologie ► Evolution
	Naturwissenschaften ► Chemie
	Naturwissenschaften ► Physik / Astronomie ► Allgemeines / Lexika
	Naturwissenschaften ► Physik / Astronomie ► Theoretische Physik
	Naturwissenschaften ► Physik / Astronomie ► Thermodynamik
ISBN-10	94-017-7712-8 / 9401777128
ISBN-13	978-94-017-7712-4 / 9789401777124
Zustand	Neuware