Let $C$ be a Cantor subset of the real line. For a real number $t$, let $C+t$ be the translate of $C$ by $t$. We say two real numbers $s,t$ are translation equivalent, if the intersection of $C$ and $C+s$ is a translate of the intersection of $C$ and $C+t$. We consider a class of Cantor sets determined by similarities with one fixed positive contraction ratio. For this class of Cantor set, we show that an "initial segment" of the intersection of $C$ and $C+t$ is a self-similar set with contraction ratios that are powers of the contraction ratio used to describe $C$ as a selfsimilar set if and only if $t$ is translation equivalent to a rational number. Many of our results are new even for the middle thirds Cantor set.

In this paper, we show that the space of three-interval scaling functions with the induced metric of $L^2(\mathbb R)$ consists of three pathcomponents each of which is contractible and hence, the first fundamental group of these spaces is zero. One method to construct simple scaling sets for $L^2(\mathbb R)$ and $H^2(\mathbb R)$ is described. Further, we obtain a characterization of a method to provide simple scaling sets for higher dimensions with the help of lower dimensional simple scaling sets and discuss scaling sets, wavelet sets and multiwavelet sets for a reducing subspace of $L^2(\mathbb R^n)$. The contractibility of simple scaling sets for different subspaces are also discussed.

The authors use fixed point index properties to prove existence of positive solutions to the abstract Hammerstein equation $u=LFu$ where $L:E\rightarrow E$ is a compact linear operator, $F:K\rightarrow K$ is a continuous and bounded mapping, $E$ is a Banach space, and $K$ is a cone in $E$. The results obtained are used to prove existence results for positive solutions to two point boundary value problems associated with differential equations.

In this paper, we establish sharp maximal function estimates for parameterized Littlewood-Paley operators and area integrals with kernel satisfying the logarithmic type Lipschitz condition. As applications, the weighted $L^p$ bounds and Morrey boundedness of these operators can be obtained.

We consider the problem of electromagnetic wave propagation in homogeneous dielectric dispersive waveguides $\Pi=\mathcal{D}\times\mathbb{R}$ where $\mathcal{D}$ is a bounded domain in $\mathbb{R}^{2},$ produced by non-uniformly moving sources of the form \begin{equation} \mathbf{j(x,}t)=\mathbf{A}(t)\delta(\mathbf{x-x}_{0}(t)) \tag{0.1} \end{equation} where $\mathbf{j(x,}t)$ is the current density, $\mathbf{A}(t)$ is a vector amplitude, $\mathbf{x=x}_{0}(t)$ is a trajectory of the source.

We consider the propagation of $TE$ and $TM$ waves in the waveguide $\Pi,$ produced by the source (0.1). As example we study the propagation of electromagnetic waves in a waveguides filled by a cold, non magnetized plasma.