Fubini Numbers and Polynomials of Graphs

In this paper, we introduce the Fubini number and Fubini polynomial of a graph in connection with the enumeration of ordered independent partitions of its set of vertices. We prove several properties of them, and study how these notions cover other variants of Fubini numbers and polynomials for special graphs.

A Facile and Rapid Route to Self-Digitization of Samples into a High Density Microwell Array for Digital Bioassays

ABSTRACTDigital bioassays are powerful methods to detect rare analytes from complex mixtures and study the temporal processes of individual entities within biological systems. In digital bioassays, a crucial first step is the discretization of samples into a large number of identical independent partitions. Here, we developed a rapid and facile sample partitioning method for versatile digital bioassays. This method is based on a detachable self-digitization (DSD) chip which couples a reversible assembly configuration and a predegassing-based self-pumping mechanism to achieve an easy, fast and large-scale sample partitioning. The DSD chip consists of a channel layer used for loading sample and a microwell layer used for holding the sample partitions. Benefitting from its detachability, the chip avoids a lengthy oil flushing process used to remove the excess sample in loading channels and can compartmentalize a sample into more than 100,000 wells of picoliter volume with densities up to 14,000 wells/cm2 in less than 30 s. We also demonstrated the utility of the proposed method by applying it to digital PCR and digital microbial assays.

A SUPER VOXEL-BASED RIEMANNIAN GRAPH FOR MULTI SCALE SEGMENTATION OF LIDAR POINT CLOUDS

Automatically segmenting LiDAR points into respective independent partitions has become a topic of great importance in photogrammetry, remote sensing and computer vision. In this paper, we cast the problem of point cloud segmentation as a graph optimization problem by constructing a Riemannian graph. The scale space of the observed scene is explored by an octree-based over-segmentation with different depths. The over-segmentation produces many super voxels which restrict the structure of the scene and will be used as nodes of the graph. The Kruskal coordinates are used to compute edge weights that are proportional to the geodesic distance between nodes. Then we compute the edge-weight matrix in which the elements reflect the sectional curvatures associated with the geodesic paths between super voxel nodes on the scene surface. The final segmentation results are generated by clustering similar super voxels and cutting off the weak edges in the graph. The performance of this method was evaluated on LiDAR point clouds for both indoor and outdoor scenes. Additionally, extensive comparisons to state of the art techniques show that our algorithm outperforms on many metrics.

Chromaticity of complete 4-partite graphs with certain star or matching deleted

Let P(G,?) be the chromatic polynomial of a graph G. Two graphs G and H are said to be chromatically equivalent, denoted G ~ H, if P(G,?) = P(H,?). We write [G] = {H |H ~ G}. If [G] = {G}, then G is said to be chromatically unique. In this paper, we first characterize certain complete 4-partite graphs G accordingly to the number of 5-independent partitions of G. Using these results, we investigate the chromaticity of G with certain star or matching deleted. As a by-product, we obtain new families of chromatically unique complete 4-partite graphs with certain star or matching deleted.

On the number of independent partitions

In [3], Shelah defined the cardinals κn(T) and , for each theory T and n < ω. κn(T) is the least cardinal κ without a sequence (pi)i<κ of complete n-types such that pi is a forking extension of pj for all i < j < κ. It is essential in computing the stability spectrum of a stable theory. On the other hand is called the number of independent partitions of T. (See Definition 1.2 below.) Unfortunately this invariant has not been investigated deeply. In the author's opinion, this unfortunate situation of is partially due to the fact that its definition is complicated in expression. In this paper, we shall give equivalents of which can be easily handled.In §1 we shall state the definitions of κn(T) and . Some basic properties of forking will be stated in this section. We shall also show that if = ∞ then T has the independence property.In §2 we shall give some conditions on κ, n, and T which are equivalent to the statement . (See Theorem 2.1 below.) We shall show that does not depend on n. We introduce the cardinal ı(T), which is essential in computing the number of types over a set which is independent over some set, and show that ı(T) is closely related to . (See Theorems 2.5 and 2.6 below.) The author expects the reader will discover the importance of via these theorems.Some of our results are motivated by exercises and questions in [3, Chapter III, §7]. The author wishes to express his heartfelt thanks to the referee for a number of helpful suggestions.