Dynamics of $\mathscr B$-free systems generated by Behrend sets. I

Generated topology on infinite sets by ultrafilters

Let $X$ be an infinite set, equipped with a topology $tau$. In this paper we studied the relationship between $tau$, and ultrafilters on $X$. We can discovered, among other thing, some relations of the Robinson's compactness theorem, continuity and the separation axioms. It is important also, aspects of communication between mathematical concepts.

Simplest Rules Characterizing Classes Generated by δ-Free Sets

Behrend-Type Constructions for Sets of Linear Equations

A linear equation on k unknowns is called a (k, h)-equation if it is of the form ∑k i=1 aixi = 0, with ai ∈ {−h, . . . , h} and ∑ ai = 0. For a (k, h)-equation E, let rE(n) denote the size of the largest subset of the first n integers with no solution of E (besides certain trivial solutions). Several special cases of this general problem, such as Sidon’s equation and sets without threeterm arit...

Linear adiabatic dynamics generated by operators with continuous spectrum. I

We are interested in the asymptotic behavior of the solution to the Cauchy problem for the linear evolution equation iε∂tψ = A(t)ψ, A(t) = A0 + V (t), ψ(0) = ψ0, in the limit ε → 0. A case of special interest is when the operator A(t) has continuous spectrum and the initial data ψ0 is, in particular, an improper eigenfunction of the continuous spectrum of A(0). Under suitable assumptions on A(t...

On b-I-Open Sets and b-I-Continuous Functions

One of the important and basic topics in the theory of classical point set topology and several branches of mathematics, which have been researched by many authors, is continuity of functions. This concept has been extended to the setting of I-continuity of functions. Janković and Hamlett [1, 2] introduced the notion of I-open sets in topological spaces. Abd El-Monsef et al. [3] further investi...