Long-time behavior is one of the most fundamental properties in dynamical systems. The limit behaviors flows on surfaces are captured by Poincaré-Bendixson theorem using ω-limit sets. This paper demonstrates that positive and negative long-time not independent. In fact, we show dependence between sets α-limit points surfaces, which partially generalizes theorem. Applying dependency result to so...

A Kakeya set is a subset of R that contains a unit line segment in every direction. We introduce a technique called ‘cut-and-move’ designed to study Kakeya sets and use this to make several simple observations in support of the famous Kakeya conjecture. For example, we give a very simple proof that the lower box-counting dimension of any Kakeya set is at least d/2. We also examine the generic v...

We study separable metric spaces with few types of countable dense sets. We present a structure theorem for locally compact spaces having precisely n types of countable dense sets: such a space contains a subset S of size at most n−1 such that S is invariant under all homeomorphisms of X and X \ S is countable dense homogeneous. We prove that every Borel space having fewer than c types of count...

Given a timed regular expression and a dense-time Boolean signal we compute the set of all matches of the expression in the signal, that is, the set of all segments of the signal that satisfy the regular expression. The set of matches is viewed as a set of points in a two-dimensional space with each point indicating the beginning and end of a matching segment on the real time axis. Our procedur...

The paper is an attempt to represent a study of limit points, boundary points, exterior points, border, interior points and closure points in the common generalized topological space. This paper takes a look at the possibilities of an extended topological space and it also considers the new characterizations of dense set.

In this paper, an inverse nodal problem for a second-order differential equation having a chemical potential on a finite interval is investigated. First, we estimate the nodal points and nodal lengths of differential operator. Then, we show that the potential can be uniquely determined by a dense set of nodes of the eigenfunctions.

This work is motivated by the problem of finding locally compact group topologies for piecewise full groups (a.k.a.~ topological groups). We determine that any in compact-open topology on self-homeomorphisms Cantor set must be uniformly discrete, a precise sense we introduce here. Uniformly discrete are particular countable, finite, residually finite and topology. The resulting form subclass am...

In this note we show that for every measurable function on $\mathbb{R}^n$ the set of points where blowup exists and is not constant $(n-1)$-rectifiable. particular, $u\in L^1_{loc}(\mathbb{R}^n)$ jump $J_u$