Uysal, M.; Özkan, E.; Soykan, Y.

In this paper, we introduce a new application of Blaise and Blaise-Lucas numbers. We define quaternions of the Blaise and Blaise-Lucas numbers. We calculate some important theorems; Cassini, Catalan, Vajda and d’Ocagne identities for quaternions of the Blaise and Blaise-Lucas numbers. We give the concepts of recurrence relation, characteristic equation, roots of the characteristic equation, Binet formula and generating function for these numbers. Finally, we give matrices representation of the Blaise and Blaise-Lucas Quaternions.

Üregen, R.N.; Koç, S.

Let M be a nonzero unital module over a commutative ring R with a nonzero identity. A proper submodule N of M is said to be a weakly n-submodule if whenever 0 ≠ am ∈ N for some a ∈ R and m ∈ M, then a ∈√ann(M) or m ∈ N. We examine the relations between weakly n-submodules and classical submodules such as prime, weakly prime, weakly primary and r-submodules. Also, we characterize modules over which every nonzero submodule is secondary by using weakly n-submodules.

Akkuş, Hakan; Özkan, Engin

In this study, we define the k-Woodall, k-Woodall-Lucas, and Modified k-Woodall sequences, and some terms of these sequences are given. Then we obtain the Binet formulas, generating functions, sum formulas, etc. Also, the relations among the terms of the kWoodall, k-Woodall-Lucas, modified k-Woodall, Woodall, Woodall-Lucas, Modified Woodall, Cullen, Cullen-Lucas, and Modified Cullen sequences are presented. In addition, we calculate Vajda’s identity, D’ocagne identity, etc. Finally, we associate the terms of these sequences with matrices.