Abstract

We prove two Grüss-type inequalities for positive linear operator approximation, i.e., inequalities explaining the non-multiplicativity of such mappings. Instead of the least concave majorant of the first order modulus of continuity, we employ second order moduli of smoothness and show in the case of the classical Bernstein operators that in certain cases this leads to better results than those obtained earlier.

Keywords: Grüss-type inequality; positive linear operators; Bernstein operator; variation-diminishing Schoenberg-splines; second order modulus of smoothness.

MSC: 41A36, 41A17, 41A25, 39B62

Abstract

A new technique is evolved to give operator representations of hypergeometric functions and certain polynomials.

Keywords: Operational representation; polynomials; generalized hypergeometric functions.

MSC: 33C45, 33C20, 47F05

Abstract

In [Comput.~Math.~Appl. 41 (2001), 135--147], A.A. Ungar employs the Möbius gyrovector spaces for the introduction of the hyperbolic trigonometry. This A.A. Ungar's work, plays a major role in translating some theorems in Euclidean geometry to corresponding theorems in hyperbolic geometry. In this paper we present (i)~the hyperbolic Breusch's lemma, (ii)~ the hyperbolic Urquhart's theorem, and (iii)~ the hyperbolic Steiner-Lehmus theorem in the Poincaré ball model of hyperbolic geometry by employing results from A.A. Ungar's work.

Keywords: Möbius transformation; Gyrogroups; Hyperbolic geometry; Gyrovector spaces and hyperbolic trigonometry.

MSC: 51B10, 51M10, 30F45, 20N05

Abstract

Generalized hypersubstitutions are mappings from the set of all fundamental operations into the set of all terms of the same language, which do not necessarily preserve the arities. Strong hyperidentities are identities which are closed under generalized hypersubstitutions and a strongly solid variety is a variety for which each of its identities is a strong hyperidentity. In this paper we determine the greatest $Reg_{G}$-strongly solid variety of commutative semigroups.

Keywords: Generalized hypersubstitution; regular generalized hypersubstitution; regular strongly solid variety; commutative semigroup.

MSC: 20M07, 08B15, 08B25

Abstract

In the present paper we prove a unique common fixed point theorem for four weakly compatible self maps in non-Archimedean Menger PM-spaces without using the notion of continuity. Our result generalizes and extends the results of Khan and Sumitra [M.A. Khan, Sumitra, A common fixed point theorem in non-Archimedean Menger PM-space, Novi Sad J. Math. 39 (1) (2009), 81--87] and others.

Keywords: Non-Archimedean Menger PM-spaces; weakly compatible maps; common fixed point.

MSC: 47H10, 54H25

Abstract

Newton's method to find the zero of a function in one variable uses the ratio of the function and derivative values, but it does not use the information provided by these quantities separately. It is a natural question to ask what a method would look like that does take into account these values instead of just their ratio. We answer that question in the case of a polynomial with all real zeros, the result being a method that is somewhat reminiscent of Laguerre's method.

Keywords: Newton; Laguerre; polynomial; zero; root.

MSC: 65H04

Abstract

A generalization of the total graph of a ring is presented. Various properties are proved and some relations to the total graph of a ring as well as to the zero-divisor graph are established.

Keywords: Total graph; module.

MSC: 13C99, 05C25