Asymptotic enumeration of 2-covers and line graphs

In this paper we find asymptotic enumerations for the number of line graphs on $n$-labelled vertices and for different types of related combinatorial objects called 2-covers. We find that the number of 2-covers, $s_n$, and proper 2-covers, $t_n$, on $[n]$ both have asymptotic growth $$s_n\sim t_n\sim B_{2n}2^{-n}\exp(-\frac12\log(2n/\log n))= B_{2n}2^{-n}\sqrt{\frac{\log n}{2n}},$$ where $B_{2n}$ is the $2n$th Bell number, while the number of restricted 2-covers, $u_n$, restricted, proper 2-covers on $[n]$, $v_n$, and line graphs $l_n$, all have growth $$u_n\sim v_n\sim l_n\sim B_{2n}2^{-n}n^{-1/2}\exp(-[\frac12\log(2n/\log n)]^2).$$ In our proofs we use probabilistic arguments for the unrestricted types of 2-covers and and generating function methods for the restricted types of 2-covers and line graphs

Asymptotics for incidence matrix classes

We define {\em incidence matrices} to be zero-one matrices with no zero rows or columns. A classification of incidence matrices is considered for which conditions of symmetry by transposition, having no repeated rows/columns, or identification by permutation of rows/columns are imposed. We find asymptotics and relationships for the number of matrices with $n$ ones in these classes as $n\to\infty$.Comment: updated and slightly expanded versio

Asymptotic enumeration of incidence matrices

We discuss the problem of counting {\em incidence matrices}, i.e. zero-one matrices with no zero rows or columns. Using different approaches we give three different proofs for the leading asymptotics for the number of matrices with $n$ ones as $n\to\infty$. We also give refined results for the asymptotic number of $i\times j$ incidence matrices with $n$ ones.Comment: jpconf style files. Presented at the conference "Counting Complexity: An international workshop on statistical mechanics and combinatorics." In celebration of Prof. Tony Guttmann's 60th birthda

A note on higher-dimensional magic matrices

We provide exact and asymptotic formulae for the number of unrestricted, respectively indecomposable, $d$-dimensional matrices where the sum of all matrix entries with one coordinate fixed equals 2.Comment: AmS-LaTeX, 9 page

Solid Freeform Fabrication of Artificial Human Teeth

In this paper, we describe a solid freeform fabrication procedure for human dental restoration via porcelain slurry micro-extrusion. Based on submicron-sized dental porcelain powder obtained via ball milling process, a porcelain slurry formulation has been developed. The formulation developed allows the porcelain slurry to show a pseudoplastic behavior and moderate viscosity, which permits the slurry to re-shape to form a near rectangular cross section. A well-controlled cross-section geometry of the extrudate is important for micro-extrusion to obtain uniform 2-D planes and for the addition of the sequential layers to form a 3-D object. Human teeth are restored by this method directly from CAD digital models. After sintering, shrinkage of the artificial teeth is uniform in all directions. Microstructure of the sintered teeth is identical to that made via traditional dental restoration processes.Mechanical Engineerin

Laser Densification of Extruded Dental Porcelain Bodies in Multi-Material Laser Densification (MMLD) Process

In this study commercial dental porcelain powder was deposited via slurry extrusion and laser densified to fabricate dental restorations in a Multi-Material Laser Densification (MMLD) process. The processing conditions for laser densification of single lines and closed rings were investigated in order to avoid warping and cracking. Multi-layer rings were also investigated to study the dependence of bonding between layers on the laser densification conditions. The laser densified rings showed no warping, and good bonding between layers could be achieved when the laser densification condition was selected properly. The mechanism to achieve porcelain rings without warping and cracking is discussed. The understanding developed will pave the way for fabricating a physical dental restoration unit.Mechanical Engineerin