Assume that G = (V, E) is an undirected graph with vertex set V and edge set E. The ball Br(v) denotes the vertices within graphical distance r from v. A subset C ⊆ V is called an (r,≤ l)-locating-dominating code of type B if the sets Ir(F ) = ⋃ v∈F (Br(v)∩C) are distinct for all subsets F ⊆ V \C with at most l vertices. A subset C ⊆ V is an (r,≤ l)-locatingdominating code of type A if sets Ir(...

Let M to be a matroid defined on a finite set E and L ⊂ E. L is locked in M if M |L and M∗|(E\L) are 2-connected, and min{r(L), r∗(E\L)} ≥ 2. In this paper, we prove that the nontrivial facets of the bases polytope of M are described by the locked subsets. We deduce that finding the maximum–weight basis ofM is a polynomial time problem for matroids with a polynomial number of locked subsets. Th...

This paper is the continuation of formalization of [4]. The definitions of meet and join of subsets of relational structures are introduced. The properties of these notions are proved. [6] provide the notation and terminology for this paper. 1. PRELIMINARIES One can prove the following propositions: (1) Let L be a relational structure, X be a set, and a be an element of L. If a ∈ X and sup X ex...

Abstract The space whose subsets we analyse with respect to lineability is $$L_0(\Omega ,P)^{\mathbb {N}}$$ L 0 ( Ω , P ) N </mml:math...

It has recently become clear that distinct subsets of CD8 T cells, analogous to their CD4 counterparts, exist in rodents and humans. To examine functional differences between human CD8 T-cell subsets, we generated Tc1, Tc2, and Tc0 T-cell clones from the peripheral blood of healthy individuals. The majority of CD8 T-cell clones generated displayed a classic Tc1 phenotype, but 10% to 20% secrete...

This work is motivated by Nathanson’s recent paper on relatively prime sets and a phi function for subsets of {1, 2, 3, . . . , n}. We establish enumeration formulas for the number of relatively prime subsets and the number of relatively prime subsets of cardinality k of {1, 2, 3, . . . , n} under various constraints. Further, we show how this work links up with the study of multicompositions. ...

Let n ≥ k ≥ l ≥ 2 be integers, and let F be a family of k-element subsets of an n-element set. Suppose that l divides the size of the intersection of any two (not necessarily distinct) members inF . We prove that the size ofF is at most (⌊n/l⌋ k/l ) provided n is sufficiently large for fixed k and l.