Volume 49, Issue 4

We find values of superposition of Marczewski-Burstin operation and operation of finding Bernstein sets. Several counterexamples of Marczewski-Burstin systems are given.

In this paper, we consider Gauthier’s generalized convolution and then define its binomial analogue as well as alternating binomial analogue. We formulate these convolutions and give some applications of them.

The aim of this paper is to show that if F is a differential field and y is a PII transcendent such that tr.deg.F 〈y〉 = 2, then every constant in F〈y〉 is in F. We also show that in this case, F〈y〉 is not contained in any strongly normal extension.

Some Ostrowski type inequalities for functions whose second derivatives in absolute value at certain powers are s-convex in the second sense are established. Two mistakes in a recently published paper are pointed out and corrected.

We determine the solutions f : S → H of the following functional equation f(xy) + f(σ(y)x) = 2f(x); x; y ∈ S; and the solutions f 1 ; f 2 ; f 3 : M → H of the functional equation f 1 (xy) + f 2 (σ(y)x) = 2f 31 (x); x; y ∈ M; where S is a semigroup, M is a monoid, H is an abelian group 2-torsion free, and σ is an involutive automorphism.

We present some new formulas for interpolations of functions by polynomials. We apply them to some special functions.

In this paper we characterize the compact, invertible and Fredholm multiplication operators on Cesàro sequence spaces.

We give a probabilistic generalization of the theory of generalized metric spaces [2]. Then, we prove a fixed point theorem for a self-mapping of a probabilistic generalized metric space, satisfying the very general nonlinear contraction condition without the assumption that the space is Hausdorff.

In the paper, some aspects of the convergence of series of dependent Gaussian sequences problem are solved. The necessary and sufficient conditions for the convergence of series of centered dependent indicators are obtained. Some strong convergence results for weighted sums of Gaussian functionals are discussed.

Lieb’s extension theorem holds for generalized p + q ∈ [0; 1] and Ando convexity theorem holds for q - r > 1. In this paper, we give a complete characterization for concavity or convexity of Lieb well known theorem in the case where p + q ≥ 1 or p+q ≤ 0. We also characterize some auxiliary results including Ando theorem for q-r ≤ 1.