In this paper, we define the neutrosophic valued (and generalized or G) metric spaces for the first time. Besides, we newly determine a mathematical model for clustering the neutrosophic big data sets using G-metric. Furthermore, relative weighted neutrosophic-valued distance and weighted cohesion measure, is defined for neutrosophic big data set. We offer a very practical method for data analysis of neutrosophic big data although neutrosophic data type (neutrosophic big data) are in massive and detailed form when compared with other data types.

The purpose of this paper is to find the quantities and surfaces of a line congruence via examining it in the dual space and to represent the results more appropriately for computational approximations. For this, we take mainly two-dual parameter motion on the dual unit sphere (DUS) so, we get a line congruence corresponding this motion by a new method. Thus, the equations of the developable surfaces, the principal surfaces, the focal surfaces and the center surface of the line congruence are found by coordinate functions. The results are illustrated by examples.

American scientific publishing group

By utilizing the Darboux frames, along with a regular surface whose parametric curves are lines of curvature, we analyzed the normal line congruence which preserves the asymptotic curves between its focal surfaces. This allows deriving systems of partial differential equations through which the problem of determining the director surface and the corresponding normal line congruence could be solved. Moreover, a necessary and sufficient condition that the focal surfaces of the normal line congruence are degenerates into curves is derived. As a result, the middle focal surface of the normal line congruence is presented as a new surface interrogation tool.