Volume 36, Issue 3

In this paper we study the harmonic functions and the Dirichlet eigenfunctions of the Hata set, and their restrictions to the interval [0,1]$[0,1]$, its main edge. We prove that these restrictions of the harmonic functions are singular, i.e. monotone and with zero derivatives almost everywhere, and provide numerical evidence that this is also the case for the Dirichlet eigenfunctions.

For subsets of ℝ+$\mathbb{R}^{+}$ we consider the local right upper porosity and the local right lower porosity as elements of a cluster set of all porosity numbers. The use of a scaling function μ:ℕ→ℝ+${\mu:\mathbb{N}\to\mathbb{R}^{+}}$ provides an extension of the concept of porosity numbers on subsets of ℕ${\mathbb{N}}$. The main results describe interconnections between porosity numbers of a set, features of the scaling functions, and the geometry of so-called pretangent spaces to this set.

We investigate semi-orthogonal wavelet frames on local fields of positive characteristic and provide a characterization of frame wavelets by means of some basic equations in the frequency domain. The theory of frame multiresolution analysis recently proposed by Shah [20] on local fields is used to establish equivalent conditions for a finite number of functions ψ1,ψ2,…,ψL${\psi_{1},\psi_{2},\ldots,\psi_{L}}$ in L2⁢(K)${L^{2}(K)}$ to generate a semi-orthogonal wavelet frame for L2⁢(K)${L^{2}(K)}$.

We consider the perturbed Hammerstein equation y⁢(t)=γ1⁢(t)⁢H1⁢(φ⁢(y))+γ2⁢(t)⁢H2⁢(ψ⁢(y))+λ⁢∫01G⁢(t,s)⁢f⁢(s,y⁢(s))⁢𝑑s,$y(t)=\gamma_{1}(t)H_{1}(\varphi(y))+\gamma_{2}(t)H_{2}(\psi(y))+\lambda\int_{0% }^{1}G(t,s)f(s,y(s))\,ds,$ where φ and ψ are linear functionals, which can be represented as Stieltjes integrals with signed measures, and λ>0${\lambda>0}$ is a parameter. We demonstrate that by imposing some mild conditions on H1${H_{1}}$ and H2${H_{2}}$, together with an assumption that the functionals decompose in a particular way, it follows that this problem can have multiple positive solutions even if no conditions at all are imposed on f except for a positivity assumption and continuity. Applications are, in particular, given to boundary value problems with nonlocal boundary conditions, and one of these existence results allows H1${H_{1}}$ to be affine and H2${H_{2}}$ to be linear.

We determine all pairs (f,g)${(f,g)}$ of meromorphic functions that share four pairs of values (aν,bν)${(a_{\nu},b_{\nu})}$, 1≤ν≤4${1\leq\nu\leq 4}$, and a fifth pair (a5,b5)${(a_{5},b_{5})}$ under some natural constraint.

Many properties of gamma functions are known. In this paper, we extend similar properties to incomplete gamma functions. In particular, it is known that Γ⁢(1-a)⁢Γ⁢(a)=π⁢csc⁡(π⁢a)${\Gamma(1-a)\Gamma(a)=\pi\csc(\pi a)}$. In this paper we find similar results for products of incomplete gamma functions. Since the error functions are a special case of the incomplete gamma functions, we extend these results to these functions. Also, we prove an interesting relation between the error functions and the Laplace transform.

Many Dirichlet series are either continuable to the whole complex plane or admit half-planes as their maximal domain of meromorphic continuation. Here we prove that this need not always be true.

In this paper, we prove some normality criteria for families of zero-free meromorphic functions. The results of this paper are generalizations of some problems studied by Chang [2] and Dethloff–Tan–Thin [5].