Zero forcing is an iterative coloring process on a graph that has been widely used in such different areas as the modelling of propagation phenomena networks and study minimum rank problems matrices graphs. This paper deals with zero hypergraphs. (Representing network by hypergraph allows us to account for its community structure more general interactions between subsets nodes.) We consider two...

The study of classes of models of a finite diagram was initiated by S. Shelah in 1969. A diagram D is a set of types over the empty set, and the class of models of the diagram D consists of the models of T which omit all the types not in D. In this work, we introduce a natural dependence relation on the subsets of the models for the No-stable case which share many of the formal properties of fo...

In search environments where large document collections are partitioned into smaller subsets (shards), processing the query against only the relevant shards improves search efficiency. The problem of ranking the shards based on their estimated relevance to the query has been studied extensively. However, a related important task of identifying how many of the top ranked relevant shards should b...

Let 5 be a family of subsets of an n-set, considered as a subposet of the Boolean algebra B.. Adjoin a minimum 0 and maximum i if necessary to form @. Let ~(95) denote the value of the Mdbius function p(6,i) in &. We compute the maximum value of Ip( as 9 ranges over three types of families in B,: lower order ideals, intervals of rank levels, and arbitrary rank-selections. The maxima are obtaine...

1. (A result of Bourbaki on finite geometries, from Răzvan) Let X be a finite set, and let F be a family of distinct proper subsets of X. Suppose that for every pair of distinct elements in X, there is a unique member of F which contains both elements. Prove that |F| ≥ |X|. Solution: Let X = [n] and F = {A1, . . . , Am}. We need to show that n ≤ m. Define the m × n incidence matrix A over R by ...

February 1, 2017 Version 3.0.0 Date 2016-09-16 Title Rank Product method for identifying differentially expressed genes with application in meta-analysis Author Francesco Del Carratore , Andris Janckevics Fangxin Hong , Ben Wittner , Rainer Breitling <r.breitling...

Classification of harmonic and semiharmonic graphs according to their cyclomatic number became of interest recently. All finite harmonic graphs with up to four independent cycles, as well as all finite semiharmonic graphs with at most one cycle were determined. Here, we determine all finite semiharmonic bicyclic graphs. Besides that, we present several methods to construct semiharmonic graphs f...

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In this note, all chromatic equivalence classes for 2-connected 3-chromatic graphs with five triangles and cyclomatic number six are described. New families of chromatically unique graphs of order n are presented for each n ≥ 8. This is a generalization of a result stated in [5]. Moreover, a proof for the conjecture posed in [5] is given.