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Random Sequential Packing Of Cubes - Yoshiaki Itoh

Random Sequential Packing Of Cubes

Hardcover

Published: 26th January 2011
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In this volume very simplified models are introduced to understand the random sequential packing models mathematically. The 1-dimensional model is sometimes called the Parking Problem, which is known by the pioneering works by Flory (1939), Renyi (1958), Dvoretzky and Robbins (1962). To obtain a 1-dimensional packing density, distribution of the minimum of gaps, etc., the classical analysis has to be studied. The packing density of the general multi-dimensional random sequential packing of cubes (hypercubes) makes a well-known unsolved problem. The experimental analysis is usually applied to the problem. This book introduces simplified multi-dimensional models of cubes and torus, which keep the character of the original general model, and introduces a combinatorial analysis for combinatorial modelings.

Prefacep. vii
Introductionp. 1
The Flory modelp. 9
One-dimensional discrete random sequential packingp. 9
Application of generating functionp. 10
Number of gapsp. 13
Minimum of gapsp. 17
Packing on circle and numerical studyp. 20
Appendix: Complex Analysisp. 21
Random interval packingp. 23
The probabilistic setup of the problemp. 24
The solution of the delay differential equation using Laplace transformp. 26
The computation of the limitp. 28
Packing on circle and the speed of convergencep. 31
Appendix: The Laplace transformp. 36
On the minimum of gaps generated by 1-dimensional random packingp. 39
Main properties of Pr(L(x) ≥ h)p. 39
Laplace transform of Pr(L(x) ≥ h)p. 42
Numerical calculations for a (h)p. 45
Asymptotic analysis for a(h)p. 49
Renewal equation techniquep. 50
Approximation for small 1 - hp. 53
Approximation for small hp. 56
Maximum of gapsp. 63
Appendix: Renewal equationsp. 64
Integral equation method for the 1-dimensional random packingp. 69
Estimating M(x) - CRxp. 69
The variance and the central limit theoremp. 76
Random sequential bisection and its associated binary treep. 83
Random sequential bisectionp. 84
Binary search treep. 86
Expected number of nodes at the d-th levelp. 91
Exponential distribution and uniform distributionp. 92
Asymptotic size of the associated treep. 93
Asymptotic shape of the associated treep. 94
More on the associated treep. 97
The unified Kakutani Rényi modelp. 99
The limit random packing densityp. 100
Expectation and variance of number of cars for l = 0p. 102
The central limit theoremp. 106
Almost sure convergence resultsp. 109
The limit distribution of a randomly chosen gapp. 112
Parking cars with spin but no lengthp. 123
Integral equationsp. 124
Existence of the limit packing densityp. 126
Laplace transform and explicitly solvable casesp. 130
General solution methodsp. 131
The power series solutionp. 135
Numerical computationsp. 139
Random sequential packing simulationsp. 145
Sequential random packing and the covering problemp. 145
Random packing of spheresp. 148
Random packing of cubesp. 151
Random sequential coding by Hamming distancep. 153
Frequency of getting Golay code by a random sequential packingp. 157
Discrete cube packings in the cubep. 161
Setting of a goalp. 161
Reduction to another problemp. 162
Proof of the theoremp. 167
Discrete cube packings in the torusp. 171
Introductionp. 171
Algorithm for generating cube packingsp. 173
Non-extensible cube packingsp. 176
The second momentp. 182
Appendix: Crystallographic groupsp. 185
Continuous random cube packings in cube and torusp. 189
Introductionp. 189
Combinatorial cube packingsp. 191
Discrete random cube packings of the cubep. 199
Combinatorial torus cube packings and lamination constructionp. 203
Properties of non-extensible cube packingsp. 210
Combinatorial Enumerationp. 219
The isomorphism and automorphism problemsp. 220
Sequential exhaustive enumerationp. 224
The homomorphism principlep. 225
Bibliographyp. 227
Indexp. 237
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9789814307833
ISBN-10: 9814307831
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 256
Published: 26th January 2011
Publisher: World Scientific Publishing Co Pte Ltd
Country of Publication: SG
Dimensions (cm): 22.86 x 15.24  x 2.03
Weight (kg): 0.48