Polygons Explained
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Polygons, Polyominoes and Polycubes
Author: A. J. Guttmann
language: en
Publisher: Springer Science & Business Media
Release Date: 2009-05-18
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.
Introduction to Computer Graphics
A basic understanding of the key techniques in computer graphics can open the door to this exciting field and its many applications, including for video games and for augmented and virtual reality. This easy-to-follow textbook and reference introduces the fundamental concepts of computer graphics, integrating both technical background and theory with practical examples and applications throughout. Thoroughly revised and updated, this new edition continues to present a user-friendly approach to creating images and animations, complementing the expanded coverage of topics with usage of example programs and exercises. Topics and features: Contains pedagogical tools, including easy-to-understand example programs and end-of-chapter exercises Presents a practical guide to basic computer graphics programming using the Open Graphics Library (OpenGL) and the widely used Java programming language Includes new and expanded content on the OpenGL graphics pipelines, shader programming, drawing basic objects using the OpenGL, three-dimensional modelling, quaternions, rasterisation, antialiasing and more Supplies complete Java project examples as supplementary material This reader-friendly textbook is an essential tool for second-year undergraduate students and above, providing clear and concise explanations of the basic concepts of computer graphics. It will enable readers to immediately implement these concepts using the OpenGL and Java (with only elementary knowledge of the programming language). Prof. Dr.-Ing. Karsten Lehn works at the Faculty of Information Technology at Fachhochschule Dortmund, University of Applied Sciences and Arts. Prof. Dr. Merijam Gotzes is teaching at Hamm-Lippstadt University of Applied Sciences. Prof. Dr. Frank Klawonn is head of the Data Analysis and Pattern Recognition Laboratory at the Ostfalia University of Applied Sciences and heads the Biostatistics Research Group at the Helmholtz Centre for Infection Research.
Introduction to Structural Analysis
This Book Deals With The Subject Of Structural Analysis Of Statically Determinate Structures Prescribed For The Degree And Diploma Courses Of Various Indian Universities And Polytechnics. It Is Useful As Well For The Students Appearing In Gate, Amie And Various Other Competitive Examinations Like That For Central And State Engineering Services. It Is A Valuable Guide For The Practising Engineers And Other Professionals.The Scope Of The Material Presented In This Book Is Sufficiently Broad To Include All The Basic Principles And Procedures Of Structural Analysis Needed For A Fresh Engineering Student. It Is Also Sufficiently Complete For One To Become Familiar With The Principles Of Mechanics And Proficient In The Use Of The Fundamentals Involved In Structural Analysis Of Simple Determinate Structures. The Book Is Written In Easy To Understand English With Clarity Of Expression And Continuity Of Ideas. The Chapters Have Been Arranged Systematically And The Subject Matter Developed Step By Step From The Very Fundamentals To A Fully Advanced Stage. In Each Chapter, The Design Significance Of Various Concepts And Their Subsequent Applications In Field Problems Have Been Highlighted.The Theory Has Been Profusely Illustrated Through Well Designed Examples Throughout The Book. Several Numerical Problems For Practice Have Also Been Included.