Turkish Journal of Analysis and Number Theory

3

Authors: Manvendra Tamba

Let G(n) denote the number of partitions of n into distinct parts which are of the form 2m, 3m, 5m, 6m-3, 8m-3, 9m-3 or 11m-3 with parts of the form 2m, 3m, 6m-3, or 11m-3 being even in number minus the number of them with parts of the form 2m, 3m, 6m-3, or 11m-3 being odd in number. In this paper, we prove that G(n) assumes all integral values and does so infinitely often.

3

Authors: Sanying Shi

Suppose that E is an algebraic number field over the rational field Let a(n) be the number of integral ideals in E with norm n and Δ(x) denote the remainder term in the asymptotic formula of the l-th integral power sum of a(n). In this paper the bound of the average behavior of Δ(x) is given. This result constitutes an improvement upon that of Lü and Wang for the error terms in mean value.

4

Authors: Kwara Nantomah

This paper presents some inequalities concerning certain ratios of the classical Euler’s Gamma function. The results generalized some recent results.

6

Authors: ERHAN SET, MUHARREM TOMAR, SELAHATTIN MADEN

In this study, s-convex stochastic processes in the second sense are presented and some well-known results concerning s-convex functions are extended to s-convex stochastic processes in the second sense. Also, we investigate relation between s-convex stochastic processes in the second sense and convex stochastic processes.

10

Authors: Ranjit Biswas

In this paper the author introduces a new kind of numbers called by ‘Compound Numbers’. A region R may or may not have imaginary object. A region even may have more than one imaginary objects too. Corresponding to an imaginary object (if exists) of a region R, we get compound objects for the region R. Imaginary objects and compound objects of a region R are not members of R and so they are called imaginary with respect to the region R only (i.e. it is a local characteristics property with respect to the region concerned), as they could be core members of another region. Every region has its own set of imaginary numbers (if exist). As a particular instance, the compound objects of the set of real numbers are the complex numbers (of existing concept). In this paper the author discovers imaginary objects of the region C (the set of complex numbers). The compound objects of C are called by ‘compound numbers’. Collection of all compound numbers is denoted by the set E. This work just reports the birth of compound numbers, not further details at this stage. It is claimed that “Theory of Numbers” will get a new direction by the birth of compound numbers. A new “Theory of Objects’ and the classical “Theory of Numbers” as a special case of it were also studied in [16]. In this paper we say that every complete region has its own ‘Theory of Numbers’, where the classical ‘theory of numbers’ is just a special instance corresponding to a particular complete region RR. Consequently, we also introduce a new field called by “Object Geometry” of a complete region, being a generalization of our classical geometry of the existing style, from elementary to the higher level.