Abstract

It is well known that any Vitali set on the real line $\Bbb{R}$ does not possess the Baire property. In this article we prove the following: Let $S$ be a Vitali set, $S_r$ be the image of $S$ under the translation of $\Bbb {R}$ by a rational number $r$ and $\Cal F = \{S_r: r \text{ is rational}\}$. Then for each non-empty proper subfamily $\Cal F'$ of $\Cal F$ the union $\bigcup \Cal F'$ does not possess the Baire property.

Keywords: Vitali set; Baire property.

MSC: 03E15, 03E20

Abstract

In this work, we apply a modified box-counting method to estimate the fractal dimension $D$ of a chaotic attractor $E$ generated by a two-dimensional mapping. The obtained numerical results show that the computed value of the capacity dimension $(d_{cap})$ tends to a limit value when the number of points $(n=card(E))$ increases. The function which fits the points $(n,D(n))$ has a sigmoidal form, and its expression characterizes the capacity dimension of chaotic attractors related to different discrete dynamical systems.

Keywords: Dynamical system; chaotic attractor; fractal set; capacity dimension; information dimension.

MSC: 37D45, 37L30, 65D10, 65Y20, 28A80.

Abstract

System of delayed differential equations is used to model a pair of FitzHugh-Nagumo excitable systems with time-delayed fast threshold modulation coupling. The Hopf bifurcation of the stationary solution, due to coupling is completely described. The critical time delays, that include indirect and direct Hopf bifurcations, and conditions on the parameters for such bifurcations are found. It is shown that there is a domain for values of time lags and coupling strength where instability of the equilibrium introduced by coupling can disappear due to interaction delay.

Keywords: Hopf bifurcation; delayed differential equations.

MSC: 34K18, 37N25

Abstract

Characterizations of $\beta$-connectedness and $\Cal S$-connectedness of topological spaces are investigated. Further results concerning preservation of these connectedness-like properties under surjections are obtained. The paper completes our previous study [Z. Duszyński, {\it On some concepts of weak connectedness of topological spaces}, Acta Math. Hungar. {\bf 110} (2006), 81--90].

Keywords: $\alpha$-open; semi-open; b-open; b-connected; $\beta$-connected; semi-connected.

MSC: 54C08, 54B05

Abstract

In this paper, we establish the existence of bounded continuous solutions over any measurable subset of $\R^{n}$ of some nonlinear integral equations. Our method is based on fixed point theorems.

Keywords: Hammerstein integral equations; fixed point theorems.

MSC: 45N05, 47J05

Abstract

As two separated concepts connectedness and homogeneity are generalized by slight homogeneity. Sum theorems and product theorems regarding slight homogeneous spaces are obtained. Many results concerning slight homogeneous components are given. Also, SCDH spaces generalize CDH spaces. It is proved in an extremally disconnected SCDH space that, all slightly homogenous components are clopen and SCDH subspaces. Many other results and many examples and counter-examples concerning slight homogeneity and SCDH spaces are obtained.

Keywords: Clopen sets; slight continuity; extremally disconnected spaces, homogeneity; countable dense homogeneity; homogeneous components.

MSC: 54A05, 54B05, 54B10, 54C08

Abstract

Without directly involving the role of points, we introduce and study the notions of fuzzy $\lambda$-Hausdorff spaces and fuzzy $\mu$-compact spaces. A characterization of a map $f$ from a fuzzy $\lambda$-Hausdorff space $X$ to a fuzzy $\mu$-compact space $Y$, where $\lambda=f^{-1}(\mu)$, to be fuzzy $\lambda$-continuous is obtained, which puts such a characterization for the continuity of $f$ in ordinary topological setting, for fuzzy topological spaces. These notions and results have been formulated for intuitionistic fuzzy topological spaces also.

Keywords: Fuzzy topology; intuitionistic fuzzy topology; fuzzy $\lambda$-Hausdorff space; fuzzy $\mu$-compact space; intuitionistic fuzzy $A$-Hausdorff space; intuitionistic fuzzy $A$-compact space.

MSC: 54A05