## Abstract

We study the properties of the weighted space H^{k}
_{2α}(Ω) and weighted set W^{k}
_{2α}(Ω, δ)for boundary value problem with singularity.

We study the properties of the weighted space H^{k}
_{2α}(Ω) and weighted set W^{k}
_{2α}(Ω, δ)for boundary value problem with singularity.

In p. 219 of R.K. Guy's Unsolved Problems in Number Theory, 3rd edn., Springer, New York, 2004, we are asked to prove that the Diophantine equation x^{n} + y^{n} = n!z^{n} has no integer solutions with n ∈ N_{+} and n > 2. But, contrary to this expectation, we show that for n = 3, this equation has in finitely many primitive integer solutions, i.e. the solutions satisfying the condition gcd(x, y, z) = 1.

In this paper, we first give several properties with respect to subgroups of self-similar groups in the sense of iterate function system (IFS). We then prove that some subgroups of p-adic numbers ℚ_{p} are strong self-similar in the sense of IFS.

Let ℌ be an in finite-dimensional complex Hilbert space and A be a standard operator algebra on ℌ which is closed under the adjoint operation. It is shown that every nonlinear *-Lie higher derivation D = {δ_{n}}gn∈N of A is automatically an additive higher derivation on A. Moreover, D = {δ_{n}}gn∈N is an inner *-higher derivation.

The survey collects many recent advances on area Nevanlinna type classes and related spaces of analytic functions in the unit disk concern- ing zero sets and factorization representations of these classes and discusses approaches, used in proofs of these results.

In this article, we first present some integral inequalities for Gauss-Jacobi type quadrature formula involving generalized relative semi-(r; m; h)-preinvex mappings. And then, a new identity concerning twice differentiable mappings defined on m-invex set is derived. By using the notion of generalized relative semi-(r; m; h)-preinvexity and the obtained identity as an auxiliary result, some new estimates with respect to Hermite-Hadamard, Ostrowski and Simpson type inequalities via fractional integrals are established. It is pointed out that some new special cases can be deduced from main results of the article.

In this paper, defining new interesting classes, λ-pseudo bi-starlike functions with respect to symmetrical points and λ-pseudo bi-convex functions with respect to symmetrical points in the open unit disk U, we obtain upper bounds for the initial coefficients of functions belonging to these new classes.

In this paper we introduce and study the Stancu type generalization of the integral type operators defined in (1.1). First, we obtain the moments of the operators and then prove the Voronovskaja type asymptotic theorem and basic convergence theorem. Next, the rate of convergence and weighted approximation for the above operators are discussed. Then, weighted L_{p}-approximation and pointwise estimates are studied. Further, we study the A-statistical convergence of these operators. Lastly, we give better estimations of the above operators using King type approach.

In this paper, we introduce a new extended generalized Burr III family of distributions in the so- called T-Burr III {Y} family by using the quantile functions of a few popular distributions. We derive the general mathematical properties of this extended family including explicit expressions for the quantile function, Shannon entropy, moments and mean deviations. Three new Burr III sub-families are then investigated, and four new extended Burr III models are discussed. The density of Burr III extended distributions can be symmetric, left-skewed, right-skewed or reversed-J shaped, and the hazard rate shapes can be increasing, decreasing, bathtub and upside-down bathtub. The potentiality of the newly generated distributions is demonstrated through applications to censored and complete data sets.

We present an existence theorem for at least one continuous solution for a coupled system of nonlinear functional (delay) integral equations of Urysohn-Stieltjes type.