On graded Ie-prime submodules of graded modules over graded commutative rings

Let G be a group with identity e. Let R be a G-graded commutative ring with identity and M a graded R-module. We introduce the concept of graded Ie-prime submodule as a generalization of a graded prime submodule for I =?g?G Ig a fixed graded ideal of R. We give a number of results concerning this class of graded submodules and their homogeneous components. A proper graded submodule N of M is said to be a graded Ie-prime submodule of M if whenever rg ? h(R) and mh ? h(M) with rgmh ? N ? IeN, then either rg ? (N :R M) or mh ? N.

On a conjecture of Seneta on regular variation of truncated moments

We prove that h?(x) = ??x0 y??1F?(y)dy is regularly varying with index ? [0, ?) if and only if V?(x) = ?[0,x] y?dF(y) is regularly varying with the same index, where ? > 0, F(x) is a distribution function of a nonnegative random variable, and F?(x) = 1?F(x). This contains at ? = 0, ?= 1 a result of Rogozin [8] on relative stability, and at ? = 0, ? = 2 a new, equivalent characterization of the domain of attraction of the normal law. For ? = 0 and ? > 0 our result implies a recent conjecture by Seneta [9].

A theory of variations via P-statistical convergence

We introduce some notions of variation using the statistical convergence with respect to power series method. By the use of the notions of variation, we prove criterions that can be used to verify convergence without using limit value. Also, some results that give relations between P-statistical variations are studied.

Linear combinations of polynomials with three-term recurrence

We study the zero distribution of the sum of the first n polynomials satisfying a three-term recurrence whose coefficients are linear polynomials. We also extend this sum to a linear combination, whose coefficients are powers of az + b for a, b ? R, of Chebyshev polynomials. In particular, we find necessary and sufficient conditions on a, b such that this linear combination is hyperbolic.

A relation between porosity convergence and pretangent spaces

The convergence of porosity is one of the relatively new concept in Mathematical analysis. It is completely structurally different from the other convergence concepts. Here we give a relation between porosity convergence and pretangent spaces.

Construction of metrics on the set of elliptic curves over a finite field

We consider metrics on the set of elliptic curves in short Weierstrass form over a finite field of characteristic greater than three. The metrics have been first found by Mishra and Gupta (2008). Vetro (2011) constructs other metrics which are independent on the choice of a generator of the multiplicative group of the underlying finite field, whereas the metrics found by Mishra and Gupta, are dependent on the choice of a generator of the multiplicative group of the underlying finite field. Hakuta (2015, 2018) constructs metrics on the set of non-supersingular elliptic curves in shortWeierstrass form over a finite field of characteristic two and three, respectively. The aim of this paper is to point out that the metric found by Mishra and Gupta is in fact not a metric. We also construct new metrics which are slightly modified versions of the metric found by Mishra and Gupta.

On starlike functions associated with cardioid domain

Analytic functions are characterized by the geometry of their image domains. That?s why, geometry of image domain is of substantial importance to have a comprehensive study of analytic functions. To introduce and study new geometrical structures as image domain and to define their subsequent analytic functions is an ongoing part of research in geometric function theory. We introduced a new domain named as cardioid domain and defined the corresponding analytic function, see [14]. Here we further study the cardioid domain, to define and study starlike functions associated with cardioid domain.

Continued fractions and diophantine equations in positive characteristic

We exhibit explicitly the continued fraction expansion of some algebraic power series over a finite field. We also discuss some Diophantine equations on the ring of polynomials, which are intimately related to these power series.

Weighted generalization of some inequalities for double integrals

We give some weighted double integral inequalities of Hermite-Hadamard type for co-ordinated convex functions in a rectangle from R2. The inequalities obtained provide generalizations of some result given in earlier works.

On 2-microlocal Morrey type Besov and Triebel-Lizorkin spaces with variable exponents

We introduce 2-microlocal Morrey type Besov and Triebel-Lizorkin spaces with variable exponents and give some characterizations of these spaces by so-called Peetre?s maximal functions. The atomic and molecular decompositions of these spaces are obtained. Finally, using molecular decomposition and the property of local means, we get the wavelet characterizations of these spaces.