Random Sequential Packing Of CubesIn 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. |
Contents
| 1 | |
2 The Flory model | 9 |
3 Random interval packing | 23 |
4 On the minimum of gaps generated by 1dimensional random packing | 39 |
5 Integral equation method for the 1dimensional random packing | 69 |
6 Random sequential bisection and its associated binary tree | 83 |
7 The unified Kakutani Renyi model | 99 |
8 Parking cars with spin but no length | 123 |
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2-cubes algorithm asymptotic binary search tree binary tree cars central limit theorem Chapter combinatorial cube packing combinatorial torus cube compute consider convergence coordinate cube K4 cube packing CP cube tiling define differential equation dimension discrete cube packing Dutour Sikiric enumeration estimate exists expected number exponential extensible external nodes finite formula function gaps Golay code graph Hamming bound Hamming distance Hence implies integral equations interval splitting isomorphism Laplace transform Lemma length limit packing density matrix me(x method metric space non-extensible cube packings number of cubes obtained with strictly p(CP packing problem parameters parking possible Proof prove r-packing random cube packing random sequential packing random variable Renewal Theory Rényi's result satisfies sequential bisection sequential random packing solution space strictly positive probability t)dt Theorem torus cube packings u)du upper bound vector vertices Wallpaper groups