Based on the fundamental theories of null curves in Minkowski 3-space, Darboux mate a curve are defined which can be regarded as kind extension for Bertrand and Mannheim 3-space. The relationships pairs explored their expression forms presented explicitly.

The concept of a pair curves, called as Bertrand partner was introduced by in 1850. curves have been studied widely the literature from past to present. In this study, we take into account trajectories according Positional Adapted Frame (PAF) for particles moving 3-dimensional Euclidean space. Some characterizations are given these with aid PAF elements. Then, obtain some special cases trajecto...

In this work, first, we express some characterizations of helices and ccr curves in the Euclidean 4-space. Thereafter, relations among Frenet-Serret invariants of Bertrand curve of a helix are presented. Moreover, in the same space, some new characterizations of involute of a helix are presented.

one of the most practical routine tests for convergence of a positive series makes use of the ratio test. if this test fails, we can use rabbe's test. when rabbe's test fails the next sharper criteria which may sometimes be used is the bertrand's test. if this test fails, we can use a generalization of bertrand's test and such tests can be continued innitely. for simplicity...