Volume 12, Issue 2

In this paper we use the version of the Covering Property Axiom, which has been formulated by Ciesielski and Pawlikowski and holds in the iterated perfect set model, to study the relations between different kinds of ultrafilters on ω and ℚ. In particular, we will give a full account for the logical relations between the properties of being a selective ultrafilter, a P -point, a Q -point, and an ω 1 -OK point.

In this paper, we prove a best approximation theorem in generalized convex spaces. As an application, we derive a result on the existence of a maximal element and a coincidence point theorem in generalized convex spaces. The results of this paper generalize some known results in the literature.

We look at analogues of the σ -algebras of events occurring up to a time and the events which are strictly prior to a time of the classical (commutative) theory. In the second case, we define the p-times and investigate the order structure of time projections associated with these times in an abstract set up.

The Apollonian metric is a generalization of the hyperbolic metric to arbitrary open sets in Euclidean spaces. In this article we show that the Apollonian metric is comparable to the j G metric in the set G if and only if its complement is unbounded and thick in the sense of Väisälä, Vuorinen and Wallin [ Thick sets and quasisymmetric maps , Nagoya Math. J. 135 (1994), 121–148]. These conditions are also equivalent to the following: there exists L > 1 such that all Euclidean L -bilipschitz mappings are Apollonian bilipschitz with uniformly bounded constant.

A function satisfies the condition Q I (x) (resp. , ) at a point if for each real ε > 0 and for each set U ∋ x belonging to Euclidean topology in (resp. to the strong density topology [to the ordinary density topology]) there is an open set O such that O ∩ U ≠ ∅ and These notions are some analogies of the quasicontinuity or the approximate quasicontinuity. In this article we compare these notions with the classical notion of the quasicontinuity.

It is shown that the random transition count is complete for Markov chains with a fixed length and a fixed initial state, for some subsets of the set of all transition probabilities.

In this paper, we study the problem of uniqueness on meromorphic functions involving in differential polynomials and obtain some results which extend and improve the theorems of M. Fang and W. Hong et al.

The purpose of this paper is to study the weak and strong convergence of an new implicit iteration process to a common fixed point for a finite family of asymptotically nonexpansive mappings in Banach spaces. The results presented in this paper extend and improve important known results in [Bauschke, J. Math. Anal. Appl. 202: 150–159, 1996], [Chang, Cho, Nonlinear Anal. Appl. 1: 369–382, 2003], [Goebel, Kirk, Proc. Amer. Math. Soc. 35: 171–174, 1972]–[Reich, J. Math. Anal. Appl. 75: 287–292, 1980], [Sun, J. Math. Anal. Appl. 286: 351–358, 2003]–[Zhou, Chang, Numer. Funct. Anal. Optim. 23: 911–921, 2002] and others.

It is proved that there exists a Sierpiński-Zygmund function, which is measurable with respect to a certain invariant extension of the Lebesgue measure on the real line ℝ.

We are going to prove an extension theorem on a functional equation for special means studied by Domsta and Matkowski.