In the present article, a modification of Jakimovski-Leviatan operators is presented which reproduce constant and e–x functions. We prove uniform convergence order of a quantitative estimate for the modified operators. We also give a quantitative Voronovskya type theorem.

Even though the fundamental main structure of cellular automata (CA) is a discrete special model, the global behaviors at many iterative times and on big scales could be a close, nearly a continuous, model system. CA theory is a very rich and useful phenomena of dynamical model that focuses on the local information being relayed to the neighboring cells to produce CA global behaviors. The mathematical points of the basic model imply the computable values of the mathematical structure of CA. After modeling the CA structure, an important problem is to be able to move forwards and backwards on CA to understand their behaviors in more elegant ways. A possible case is when CA is to be a reversible one. In this paper, we investigate the structure and the reversibility of two-dimensional (2D) finite, linear, triangular von Neumann CA with null boundary case. It is considered on ternary field [Formula: see text] (i.e. 3-state). We obtain their transition rule matrices for each special case. For given special triangular information (transition) rule matrices, we prove which triangular linear 2D von Neumann CAs are reversible or not. It is known that the reversibility cases of 2D CA are generally a much challenged problem. In the present study, the reversibility problem of 2D triangular, linear von Neumann CA with null boundary is resolved completely over ternary field. As far as we know, there is no structure and reversibility study of von Neumann 2D linear CA on triangular lattice in the literature. Due to the main CA structures being sufficiently simple to investigate in mathematical ways, and also very complex to obtain in chaotic systems, it is believed that the present construction can be applied to many areas related to these CA using any other transition rules.

We investigate main theoretical aspects of two-dimensional linear-hybrid cellular automata with periodic boundary condition over the Galois field GF(2). We focus on the characterization of two-dimensional hybrid linear cellular automata by way of a special algorithm. Here we set up a relation between reversibility of cellular automata and characterization of two-dimensional hybrid linear cellular automata with a special boundary conditions, i.e. periodic case. The determination of the characterization problem of special type of cellular automaton is studied by means of the matrix algebra theory. It is believed that this type of cellular automata could find many different applications in special case situations, e.g. image processing area, textile design, video processing, DNA research, etc., in the near future.

In this study, we introduce a new kind of nonlinear Bernstein-Chlodowsky operators based on q-integers. Firstly, we define the nonlinear q?Bernstein-Chlodowsky operators of max-product kind. Then, we give an error estimation for the q?Bernstein Chlodowsky operators of max-product kind by using a suitable generalizition of the Shisha-Mond Theorem. There follows an upper estimates of the approximation error for some subclasses of functions.