For an Euclidean space Rn, let Ln denote the modal logic of chequered subsets of Rn. For every n ≥ 1, we characterize Ln using the more familiar Kripke semantics, thus implying that each Ln is a tabular logic over the well-known modal system Grz of Grzegorczyk. We show that the logics Ln form a decreasing chain converging to the logic L∞ of chequered subsets of R∞. As a result, we obtain that L...

While some probiotics have shown beneficial effects on preventing or treating colitis development, others have shown no effects. In this study, we have assessed the immunomodulating effects of two probiotic strains, Lactobacillus rhamnosus (L. rhamnosus) and Bifidobacterium breve (B. breve) on T cell polarization in vitro, using human peripheral blood mononuclear cells (PBMC), and in vivo, usin...

The aim of the present paper is to prove that the family of all closed nonempty subsets of a complete probabilistic metric space L is complete with respect to the probabilistic Pompeiu-Hausdorff metric H . The same is true for the families of all closed bounded, respectively compact, nonempty subsets of L. If L is a complete random normed space in the sense of Šerstnev, then the family of all n...

For two finite sets of integers A and B their additive energy E(A, B) is the number of solutions to a + b = a + b, where a, a ∈ A and b, b ∈ B. Given finite sets A, B ⊆ Z with additive energy E(A, B) = |A||B| + E, we investigate the sizes of largest subsets A ⊆ A and B ⊆ B with all |A||B| sums a + b, a ∈ A, b ∈ B, being different (we call such subsets A, B co-Sidon). In particular, for |A| = |B...

Heterogeneity amongst dendritic cell (DC) subsets leads to a spectrum of immune response capacity against pathogens. Several DC subsets in spleen have been described which differ in terms of phenotype and function. We have previously reported a distinct population of CD11c(lo)CD11b(hi)MHC-II(-)CD8(-) dendritic-like "L-DC" in murine spleen, which can also be generated in splenic stromal longterm...

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...