Volume 31, Issue 4 -

A celebrated result of S. Bernstein [3] states that every solution of the minimal surface equation over the entire plane ℝ 2 has to be an affine linear function. Since the paper of Bernstein appeared in 1927, many different proofs and generalizations of this beautiful theorem were given, namely to higher dimensions and to more general equations, for a careful account we refer to the paper by Simon [6] and to the monograph by Dierkes, Hildebrandt, and Tromba [4, Chap. 3]. In his paper Simon [5] posed the question whether the equation (1+ u x 2 ) u xx +2 u x u y u xy +(1+ u y 2 ) u yy  = 0          (1.1) has the Bernstein property i.e. whether every C 2 -solution defined on all of ℝ 2 necessarily has to be affine. We here show by a very simple argument that this is not the case.

In this work, we develop two families of optimal fourth order iterative methods requiring three evaluations. During each step, methods need evaluation of two derivatives and one function. According to the Kung–Traub conjecture an optimal iterative method without memory based on 3 evaluations could achieve an optimal convergence order of 4 . The proposed iterative methods are especially befitting for finding zeros of functions whose derivative is easy to evaluate. For example, polynomial functions and functions defined via integrals.

Given a two-dimensional semi-Riemannian metric with positive Gaussian curvature, we construct local isometric embeddings into Minkowski space ℝ 2,1 . Using reasonable geometric initial conditions, we control the number of solutions and deduce symmetry properties of the solutions.

We study the uniqueness of entire functions sharing a non-zero value with linear differential polynomials generated by them. We improve some existing results.

This paper studies the uniqueness of meromorphic functions that share one small function CM with their k -th derivative. The result in this paper is generalization for the results in [1] and [2].

A finite difference scheme for finding viscosity solutions of specifically stated problems for Hamilton–Jacobi equations is proposed. Solutions of such problems satisfy two differential inequalities involved into the definition of viscosity solutions. The specifics is that the inequalities must hold only in the region where solutions are less than a given function or greater than another given function. From the point of view of optimal control theory and the theory of differential games, these results can be applied to finding value functions in problems with nonterminal payoff functionals and state constraints.

The purpose of this study is to extend the notions of statistically convergence, strongly p -summability, statistical limit point and statistical cluster point to functions defined on discrete countable amenable semigroups.