In this paper we perform a further investigation for r-gcd-sum and r-lcm-sum functions. By making use of the properties generalization Euler’s φ-function, Abel’s identity elementary arguments, derive asymptotic formulas average function, function their generalizations. Moreover, also study sums reciprocals r-gcd r-lcm.

Lost circulation refers to loss of part or whole drilling fluid into the formation while operataion. It wastes costly mud and time spent on rig operations. In this study, investigation a natural plant called, Prosopis Farcta (PF), has been done show its effectiveness as lost control material. A series experimental tests have conducted in order discover PF an LCM. Moreover, different particle si...

Let R be an integral domain in which every two nonzero elements have a greatest common divisor. Let (an)n>1 be a sequence of nonzero elements in R. We prove that gcd(an, am) = agcd(n,m) for all n,m > 1 if and only if an = ∏

We study the distribution of generalized gcd and lcm functions on average. The function, denoted by [Formula: see text], is greatest text]th power divisor common to text] text]. Likewise, smallest multiple derive asymptotic formulas for average order arithmetic, geometric, harmonic means Additionally, we also deduce with error terms over a set lattice points, thereby generalizing some previous ...

the production of colony-stimulating factors (csf) is delicately controlled through a complex network of humoral and environmental factors. we have studied some of the mechanisms which regulate the production of csf as compared to general protein synthesis in the lung tissue ill vitro. when lung tissue from mice was cultured for various times in serum free medium, the first detectable level of ...

The multipotent hemopoietic stem cell has fastidious growth requirements in vitro. Traditionally, phytohemagglutinin-stimulated leukocyte conditioned medium has been used to supply the undefined growth factors required for culture of the human multipotent hemopoietic progenitor. We describe the use of medium conditioned by the bladder carcinoma cell line, 5637, to replace PHA-LCM in CFU-GEMM cu...