. In this paper, we aim to introduce two new families of analytic and bi-univalent functions associated with the Attiya’s operator, which is defined by the Hadamard product of a generalized Mittag-Leffler function and analytic functions on the open unit disk. Then we estimate the second and third coefficients of the Taylor-Maclaurin series expansions of functions belonging to these families. Also, we investigate Fekete-Szeg¨o problem for these families. Some relevant connections of certain special cases of the main results with those in several earlier works are also pointed out. Two naturally-arisen problems are given for further investigation.
In this paper, we deal with a problem to determine the type of automorphisms of the unit disc in C in terms of intrinsic geometry. We will characterize the hyperbolicity and parabolicity of automorphism by the distance function of the Poincaré metric.
The purpose of this thesis is to analyze the effect on formation of computational abilities and dispositions to the first grade of elementary school students by applying of communicative teaching and learning activity. As a result of analysis, we could get some suggestive points and also we could make sure that computational abilities and dispositions of the first grade students are formed by applying of communicative teaching and learning activity. However, help and control of teacher have to be with it.
For a positive integer n, let μd(n) be the number of multiplicative d-dimensional partitions of n ∏ i=1 pi, where pi denotes the ith prime. The number of multiplicative partitions of a square free number with n prime factors is the Bell number μ1(n) = Bn. By the definition of the function μd(n), it can be seen that for all positive integers n, μ1(n) = Tn(1) = Bn, where Tn(x) is the nth Touchard (or exponential ) polynomial. We show that, for a positive n, μ2(n) = 2Tn(1/2). We also conjecture that for all m, μ3(m) ≤ 3Tm(1/3).
. We establish existence and uniqueness of Green functions for flow velocity of stationary Stokes systems, under a continuity assumption of weak solutions to the system, in a bounded domain such that the divergence equation is solvable there. We also obtain pointwise bounds of the Green functions.
. In this paper, we introduce the new concept of a cone metric-like space and consider some fixed point theorems for generalized contrac- tive mappings under suitable conditions in cone metric-like spaces. Our results generalize and unify the several main results of [1, 2, 9].
. We discuss the condition that if ab = 0 for elements a,b in a ring R then aIb = 0 for some essential ideal I of R . A ring with such condition is called IEIP . We prove that a ring R is IEIP if and only if D n ( R ) is IEIP for every n ≥ 2, where D n ( R ) is the ring of n by n upper triangular matrices over R whose diagonals are equal. We construct an IEIP ring that is not Abelian and show that a well-known Abelian ring is not IEIP, noting that rings with the insertion-of-factors-property are Abelian.
. The purpose of this paper is to investigate various kinds of degeneracy of maximal surfaces in L 4 in view of the generalized Gauss map.
. The classical limit theorems like strong law of large numbers, central limit theorems and law of iterated logarithms are fundamental the- ories in probability and statistics. These limit theorems are proved under additivity of probabilities and expectations. In this paper, we investigate strong law of large numbers under sub-linear expectation which generalize the classical ones. We give strong law of large numbers under sub-linear expectation with respect to the partial sums and some conditions similar to Petrov’s. It is an extension of the classical Chung type strong law of large numbers of Jardas et al.’s result. As an application, we obtain Chung’s strong law of large number and Marcinkiewicz’s strong law of large number for independent and identically distributed random variables under the sub-linear expectation. Here the sub-linear expectation and its related capacity are not additive.
Huang et al. proved that the n by n upper triangular matrix ring over a domain is weakly reversible-over-center by using the property of regular matrices. In this article we provide a concrete proof which is able to be available in the related study of centers. Next we extend an example of weakly reversible-over-center, which was argued by Huang et al., to the general case. Throughout this note every ring is an associative ring with identity unless otherwise stated. Let R be a ring. We denote the center and the set of all idempotents of R by Z(R) and I(R), respectively. Denote the n by n (n ≥ 2) full (resp., upper triangular) matrix ring over R by Matn(R) (resp., Tn(R)). In denotes the identity matrix of both Matn(R) and Tn(R). Write Dn(R) = {(aij) ∈ Tn(R) | a11 = · · · = ann}. Use Eij for the matrix with (i, j)-entry 1 and zeros elsewhere. The following definitions are due to the literature. An element u of R is right regular if ur = 0 implies r = 0 for r ∈ R. Similarly, left regular elements can be defined. An element is regular if it is both left and right regular (and hence not a zero divisor). R is called Abelian if I(R) ⊆ Z(R), and R is called reduced if N(R) = 0. Reduced rings are easily shown to be Abelian. R is said to be directly finite if ab = 1 for a, b ∈ R implies ba = 1. Abelian rings are easily shown to be directly finite. 1. Weakly reversible-over-center rings Following Choi et al. [2], a ring R is called reversible-over-center if ab ∈ Z(R) for a, b ∈ R implies ba ∈ Z(R). Reduced rings are reversible-overcenter and reversible-over-center rings are Abelian by [2, Theorem 1.1] and [2, Proposition 1.3(1)], respectively. In this article, we consider a generalization of reversible-over-center rings, concentrating upon the nonzero case of ab ∈ Z(R). Following [3], a ring R is called weakly reversible-over-center if 0 6= ab ∈ Z(R) for a, b ∈ R implies ba ∈ Z(R). Every reversible-over-center ring is clearly Received April 16, 2020; Accepted May 23, 2020. 2010 Mathematics Subject Classification. 16U80, 16U70, 16S50, 16U10.
