A maximum weighted independent set of a permutation graph is a maximum subset of noncrossing chords in a matching diagram (i.e., a set of chords with end-points on two horizontal lines). The problem of nding, among all noncrossing subsets of with density at most k, one with maximum size is considered, where the density of a subset is the maximum number of chords crossing a vertical line and k i...

Let A, B, be finite subsets of an abelian group, and let G ⊂ A × B be such that #A, #B, #{a + b : (a, b) ∈ G} ≤ N. We consider the question of estimating the quantity #{a − b : (a, b) ∈ G}. In [2] Bourgain obtained the bound of N 2− 1 13 , and applied this to the Kakeya conjecture. We improve Bourgain's estimate to N

Let A, B, be finite subsets of an abelian group, and let G ⊂ A × B be such that #A, #B, #{a + b : (a, b) ∈ G} ≤ N. We consider the question of estimating the quantity #{a − b : (a, b) ∈ G}. In [2] Bourgain obtained the bound of N 2− 1 13 , and applied this to the Kakeya conjecture. We improve Bourgain's estimate to N

It is known that if N is finitely colored, then some color class is piecewise syndetic. (See Definition 1.1 below for a definition of piecewise syndetic.) We generalize this result by considering finite colorings of the set of all finite subsets of N . The monochromatic objects obtained are “d-copies” of arbitrary finite forests and arbitrary infinite forests of finite height. Van der Waerden’s...

There are two main approximations of mining big data in memory. One is to partition a big dataset to several subsets, so as to mine each subset in memory. By this way, global patterns can be obtained by synthesizing all local patterns discovered from these subsets. Another is the statistical sampling method. This indicates that data partitioning should be an important strategy for mining big da...

INRIA In this paper a general class of tree algorithms is analyzed. It is shown that, by using an appropriate probabilistic representation of the quantities of interest, the asymptotic behavior of these algorithms can be obtained quite easily without resorting to the usual complex analysis techniques. This approach gives a unified probabilistic treatment of these questions. It simplifies and ex...

We survey results concerning the maximum size of a family F of subsets of an n-element set such that a certain configuration is avoided. When F avoids a chain of size two, this is just Sperner’s Theorem. Here we give bounds on how large F can be such that no four distinct sets A, B, C, D ∈ F satisfy A ⊂ B, C ⊂ B, C ⊂ D. In this case, the maximum size satisfies ( n b 2 c ) ( 1 + 1 n + Ω ( 1 n ))...

These numbers will be regarded as distinct although two or more may be equal in value. Let «,■ (t=l, 2, • • • , k) be any fixed set of k positive integers whose sum is n, let C= Yli-i w»' and let R = n\/C. We shall consider partitions P* = {Bf, B2*, ■ ■ ■ , Bk*\ of the set B into disjoint subsets Bf, each B* containing n,elements of B (* = 1, 2, • • • , k). By a partition here is meant an order...

Background: The appendix is considered as part of the gut-associated lymphoid tissue; however, lymphocyte subsets in this tissue are not fully defined. Objective: To investigate and compare the function and phenotype of lymphocyte subsets in peripheral blood and appendix of patients with normal and inflamed appendix tissues. Methods: Peripheral blood samples and appendiceal mononuclear cells we...

We examine the binary codes C2(Ai + I) from matrices Ai + I where Ai is an adjacency matrix of a uniform subset graph Γ(n, 3, i) of 3-subsets of a set of size n with adjacency defined by subsets meeting in i elements of Ω, where 0 ≤ i ≤ 2. Most of the main parameters are obtained; the hulls, the duals, and other subcodes of the C2(Ai + I) are also examined. We obtain partial PD-sets for some of...