In a sequential packing problem, random objects are uniformly and independently selected from some space. A selected object is either packed or rejected, depending on the distance between it and the nearest object which has been previously packed. A saturated packing is said to exist when it is no longer possible to pack any additional selections. The random packing density is the average proportion of the space which is occupied by the packed objects at saturation. Results concerning the time of the first rejection in a packing sequence are given in Chapter 1. The accuracy of some common approximation formulas is investigated for several settings. The problems considered may be thought of as generalizations of the classical birthday problem. Exact results concerning random packing densities are generally known only for some packing sequences in one-dimensional spaces. In Chapter 2, the packing densities of various computer generated codes are examined. These stochastically constructed codes provide a convenient way to study packing in multidimensional spaces. Asymptotic approximation formulas are given for the packing densities which arise from several different coding schemes. In Chapter 3 the distribution of the number of random selections needed to achieve a saturated packing is considered. In each of the settings examined, the results are compared with analogous results from an associated random covering problem.

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./a

This book provides an in-depth discussion of the theory of finite packings and coverings by convex bodies.

This book covers methods of Mathematical Morphology to model and simulate random sets and functions (scalar and multivariate). The introduced models concern many physical situations in heterogeneous media, where a probabilistic approach is required, like fracture statistics of materials, scaling up of permeability in porous media, electron microscopy images (including multispectral images), rough surfaces, multi-component composites, biological tissues, textures for image coding and synthesis. The common feature of these random structures is their domain of definition in n dimensions, requiring more general models than standard Stochastic Processes.The main topics of the book cover an introduction to the theory of random sets, random space tessellations, Boolean random sets and functions, space-time random sets and functions (Dead Leaves, Sequential Alternate models, Reaction-Diffusion), prediction of effective properties of random media, and probabilistic fracture theories.

This contributed volume provides readers with an overview of the most recent developments in the mathematical fields related to fractals, including both original research contributions, as well as surveys from many of the leading experts on modern fractal theory and applications. It is an outgrowth of the Conference of Fractals and Related Fields III, that was held on September 19-25, 2015 in île de Porquerolles, France. Chapters cover fields related to fractals such as harmonic analysis, multifractal analysis, geometric measure theory, ergodic theory and dynamical systems, probability theory, number theory, wavelets, potential theory, partial differential equations, fractal tilings, combinatorics, and signal and image processing. The book is aimed at pure and applied mathematicians in these areas, as well as other researchers interested in discovering the fractal domain.

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