The dispersion of carbon black agglomerates suspended in polydimethyl siloxane (PDMS) or polybutadiene (PBD) liquids has been studied. Agglomerates comprised of either a low-structure or a high-structure carbonblack were subjected to simple shear flow. Two characteristic length scales that affect the dispersion process are identified. One length scale (Lp) is a measure of the ease with which fl...

In this paper the packing density of various layered permutations is calculated, thus solving some problems suggested by Albert, Atkinson, Handley, Holton & Stromquist [Electron. J. Combin. 9 (2002), #R5]. Specifically, the density is found for layered permutations of type [m1, . . . ,mr] when log(r +1) ≤ min{mi}. It is also shown how to derive good estimates for the packing density of permutat...

In the classic circle packing problem, one asks whether a given set of circles can be packed into the unit square. This problem is known to be NP-hard. In this thesis, we present a new sufficient condition using only the circles’ combined area: It is possible to pack any circle instance with a combined area of up to ≈53.90% of the square’s area. This area condition is tight, in the sense that f...

In 1984, S. K. Stein and his co-authors posed a problem concerning simple three-dimensional shapes, known as semicrosses, or tripods. By definition, a tripod is formed by a corner and the three adjacent edges of an integer cube. How densely can one fill the space with nonoverlapping tripods of a given size? In particular, is it possible to fill a constant fraction of the space as the tripod siz...

Let Ak = {0 = a1 < a2 < ... < ak} and B = {0 = b1 < b2 < ... < bn ...} be sets of k integers and infinitely many integers, respectively. Suppose B has asymptotic density x t d(B) x. If, for every integer n _> 0, there is at most one representation n a^ + bj , then we say that Ak has a packing complement of density j> x. Given Ak and x9 there is no known algorithm for determining whether or not ...

Preface In this thesis we give a new approach to the classical problems of finite and infinite packings and lattice packings of convex bodies. This approach is based on the introduction of parameterized densities δ ρ (K, C), ρ ∈ R >0. The parameterized density of a finite packing set C of a convex body K with respect to the parameter ρ is defined by δ ρ (K, C) = #(C)V (K) V (conv(C) + ρK) , whe...

The diffraction pattern of molten metals shows several intensity-maxima, which can be interpreted in a similar way as the Debye-Scherrer pattern of crystalline powders. For this purpose, however, it is necessary to introduce liquid-like distortions. Such an approach has been given by the concept of paracrystallinity [1] , which contains the well known equations of crystallography [2] as limitin...

The high packing density of residues in proteins ought to be manifested in some order; to date this packing order has not been thoroughly characterized. The packing regularity in proteins is important because the internal organization of proteins can have a dominant effect on functional dynamics, and it can aid in the design, simulation and evaluation of structures. Packing metrics could also i...

We introduce a parameter space for periodic point sets, given as a union of m translates of a point lattice. In it we investigate the behavior of the sphere packing density function and derive sufficient conditions for local optimality. Using these criteria we prove that perfect, strongly eutactic lattices cannot be locally improved to yield a denser periodic sphere packing. This in particular ...

This paper formalizes the local density inequality approach to getting upper bounds for sphere packing densities in Rn. This approach was first suggested by L. Fejes-Tóth in 1954 as a method to prove the Kepler conjecture that the densest packing of unit spheres in R has density π √ 18 , which is attained by the “cannonball packing.” Local density inequalities give upper bounds for the sphere p...