The purpose of the paper is to present a new generalization of the dual Fibonacci quaternions called the generalized dual bi-periodic Fibonacci quaternions. This new generalization allow us to state several number of dual quaternion sequences in a unique sequence. Furthermore, we give the generating function, the Binet formula, the norm value and some basic properties of these dual quaternions.

In this paper, two new Hamilton operators are defined and the algebra of dual split quaternions is developed using these operators. It is shown that finite screw motions in Minkowski 3-space can be expressed by dual-number r (3×3) matrices in dual Lorentzian space. Moreover, by means of Hamilton operators, screw motion is obtained in 3-dimensional Minkowski space R31.

In this paper, we introduced $m$-extension dual complex Fibonacci $p$-numbers. We established the properties of $m$-extension dual complex Fibonacci $p$-numbers. They are connected to complex Fibonacci numbers, complex Fibonacci $p$-numbers and dual complex Fibonacci $p$-numbers.

Quantum calculus, which arises in the mathematical fields of combinatorics and special functions as well as in a number of areas, involving the study of fractals and multi−, fractal measures, and expressions for the entropy of chaotic dynamical systems, has attracted the attention of many researchers in recent years. In this paper, by virtue of some useful notations from q−, calculus, we define the q−, Fibonacci dual bicomplex numbers and q−, Lucas dual bicomplex numbers with a different perspective. Afterwards, we give the Binet formulas, binomial sums, exponential generating functions, Catalan identities, Cassini identities, d’, Ocagne identities and some algebraic properties for the q−, Fibonacci dual bicomplex numbers and q−, Lucas dual bicomplex numbers.

In this paper, a new interconnection network structure called hierarchical extended Fibonacci cubes interconnection networks HEFC1(n) is proposed, and its properties are evaluated. A set of the HEFC1(n)s constructed by the proposed method is a two level conventional hierarchical network. For each n (dimension of extended Fibonacci cube-EFC1(n)), a HEFC1(n)   interconnection network can be constructed. All EFC1(n) for n>4 are recursively constructible, hence HEFC1(n)s for n>4 are also recursively constructible with some extra EFC1(n-1)s and EFC1(n-2)s. Furthermore, since a HEFC1(n) has a hierarchically structured character and the feature of uniformity, a wide variety of inter-cluster connections are possible. The comparisons of HEFC1(n)s with some of the traditional cubic and hierarchical cubic networks are presented in this study. The routing in HEFC1(n+2)s is as easy as routing in HCN(n,n) and the scalability of HEFC1(n+2) is better than the scalability of HCN(n,n) and H(2n). H(2n) and HCN(n,n) are symmetric interconnection networks, however, HEFC1(n+2) is not a symmetric interconnection network and HEFC1(n+2)s have a recurrent structure as H(2n) and HCN(n,n). The cost of HEFC1(n+2) is better than the costs of HCN(n,n) and H(2n).The edge connectivity of HEFC1(n+2) is worse than the edge connectivity of H(2n) and HCN(n,n). This makes the VLSI design of HEFC1(n+2) simpler than the VLSI designs of H(2n) and HCN(n,n).

The objective of this study was to develop a new optimal parallel algorithm for matrix multiplication which could run on a Fibonacci Hypercube structure. Most of the popular algorithms for parallel matrix multiplication can not run on Fibonacci Hypercube structure, therefore giving a method that can be run on all structures especially Fibonacci Hypercube structure is necessary for parallel matrix multiplication. For creating this method, a new model for matrix multiplication with an algorithm for data distribution on Fibonacci Hypercube structure was provided. Other than this, another optimized algorithm was designed on Mesh structure. By running the algorithms on a simulative parallel system and giving the results in graphical mode, it has been found that these two algorithms have optimized value in parallel matrix multiplication and they are more efficient than the previous algorithms.

We study two foremost Mahonian statistics, the major index and the inversion number for a class of binary words called restricted Fibonacci words. The language of restricted Fibonacci words satisfies recurrences which allow for the calculation of the generating functions in two different ways. These yield identities involving the $q$-binomial coefficients and provide non-standard $q$-analogues of the Fibonacci numbers. The major index generating function for restricted Fibonacci words turns out to be a $q$-power multiple of the inversion generating function.

The aim of this paper is to introduce the hyperbolic generalized $k$-Horadam quaternions and octonions and investigate their algebraic properties. We present some properties and identities of these quaternions and octonions for generalized $k$-Horadam numbers. Moreover, we give some determinants related to the hyperbolic generalized $k$-Horadam quaternions and octonions. Finally, we evaluate its determinants through the Chebyshev polynomials of the second kind and give an illustrative example as well.

The aim of this paper is to introduce the cyclic-Fibonacci hybrid sequence and give some properties. By taking into account the cyclic-Fibonacci hybrid sequence modulo m, the method will be given to determine the period lengths of this sequence according to the different m values. In the final part of this paper, we study the cyclic-Fibonacci hybrid sequence in groups and then we calculate the cyclic-Fibonacci hybrid lengths of polyhedral groups (2, 2, 2), (2, n, 2) and (n, 2, 2) as applications of the results produced.