A set A of positive integers is relatively prime to n if gcd(A∪{n}) = 1. Given positive integers l ≤ m ≤ n, let Φ([l,m], n) denote the number of nonempty subsets of {l, l +1, . . . ,m} which are relatively prime to n and let Φk([l,m], n) denote the number of such subsets of cardinality k. In this paper we give formulas for these functions for the case l = 1. Intermediate consequences include id...

Let X = {1,. . ., n) and Í" = {1,..., k), k < n. Let C(n, k) be the subsets of X which intersect Y, ordered by inclusion. Lih showed that C(n, k) has the Sperner property. Here it is shown that C(n, k) has several stronger properties. A nested chain decomposition is constructed for C(n, k) by bracketing. C(n, k) is shown to have the LYM property. A more general class of collections of subsets i...

The definition of n-width of a bounded subset A in a normed linear space X is based on the existence of n-dimensional subspaces. Although the concept of an n-dimensional subspace is not available for metric trees, in this paper, using the properties of convex and compact subsets, we present a notion of n-widths for a metric tree, called Tn-widths. Later we discuss properties of Tn-widths, and s...

Background: Atherosclerosis is a chronic inflammatory disease affecting large and medium arteries. CD4+ T cells are known to play a role in the progression of the disease. CD4+CD25+Foxp3+ natural Treg (nTreg) cells seem to have a protective role in the disease and their reduction in acute coronary syndrome is recently shown. Objective: To investigate the frequency of nTreg subsets in the pe...

Different approaches have been proposed for feature selection to obtain suitable features subset among all features. These methods search feature space for feature subsets which satisfies some criteria or optimizes several objective functions. The objective functions are divided into two main groups: filter and wrapper methods.  In filter methods, features subsets are selected due to some measu...

Let  $left{a_alpharight}_{alphain I}$ be a bounded net in a Banach algebra $A$ and $varphi$ a nonzero multiplicative linear functional on $A$. In this paper, we deal with the problem of when $|aa_alpha-varphi(a)a_alpha|to0$ uniformly for all $a$ in weakly compact subsets of $A$. We show that Banach algebras associated to locally compact groups such  as Segal algebras and $L^1$-algebras are resp...

Background: Lymphocyte subsets enumeration is considered prominent in the management of primary and acquired immunodeficiency disorders. Because of local variations due to race, age, gender, and environmental conditions on lymphocyte subsets, and to improve the accuracy of interpretation of laboratory findings, reference intervals must be determined in every population. Objective: To establish ...

The Kneser graph K (n; k) has as vertices the k-subsets of f1;2;:::;ng. Two vertices are adjacent if the corresponding k-subsets are disjoint. It was recently proved by the rst author 2] that Kneser graphs have Hamilton cycles for n 3k. In this note, we give a short proof for the case when k divides n. x 1. Preliminaries. Suppose that n k 1 are integers and let n] := f1; 2; :::; ng. We denote t...