It is known that the hybrid numbers are generalizations of complex, hyperbolic and dual numbers. Recently, they have attracted the attention of many scientists. At this paper, we provide the Euler’s and De Moivre’s formulas for the 4×4 matrices associated with hybrid numbers by using trigonometric identities. Also, we give the roots of the matrices of hybrid numbers. Moreover, we give some illustrative examples to support the main formulas.
In the literature, Holditch theorem was obtained under periodic rotation and translation motions in [H. Holditch, Geometrical theorem, Q. J. Pure Appl. Math. 2 (1858) 38] or periodic shear and translation motions in [O. Röschel, Der satz von Holditch in der isotropen ebene. Abh. Braunschweig. Wiss. Ges. 36 (1984) 27–32]. In this paper, by introducing the projection of a vector onto a plane, scalar area, area vector of a surface, we investigate Holditch theorem under periodic rotation, translation and shear motions. We give two interpretations for the Holditch type theorem in Galilean space.
In this paper, we obtain Euler’s and De Moivre’s formulas for the [Formula: see text] matrix representation of Pauli quaternions. Moreover, we provide De Moivre’s formula for the light-like Pauli quaternions. Additionally, we give the [Formula: see text] roots of the matrix representation of Pauli quaternions. Moreover, we exemplify some of the results with illustrative examples to support the main formulas.
The Pell numbers, named after the English diplomat and mathematician John Pell, are studied by many authors. At this work, by inspiring the definition harmonic numbers, we define harmonic Pell numbers. Moreover, we construct one type of symmetric matrix family whose elements are harmonic Pell numbers and its Hadamard exponential matrix. We investigate some linear algebraic properties and obtain inequalities by using matrix norms. Furthermore, some summation identities for harmonic Pell numbers are obtained. Finally, we give a MATLAB-R2016a code which writes the matrix with harmonic Pell entries and calculates some norms and bounds for the Hadamard exponential matrix.
The aim of this study is to investigate the oil prices, which have crucial impact of an economy, using new ratios called Nickel ratios instead of the golden ratios on technical analysis. The Nickel ratios are developed considering Nickel Fibonacci sequence. This study is the first to use Nickel ratios in technical analysis in economics and finance. In this study, graphs comprising of weekly, daily, $4-$hour and $30-$minute periods are analyzed using Nickel ratios in Fibonacci retracement, fan, arcs and time zones applications, and the results are compared with the golden ratio obtained from the Fibonacci number sequence. In addition, the support and resistance points obtained from Nickel ratios have more significant levels than the golden ratio. The retracement, fan, arcs and time zones graphs with weekly, daily, four hourly and half-hourly data based on the golden and Nickel ratios show that the levels regarding the Nickel ratios confirm more significant points in comparison with the levels regarding the golden ratios. Finally, more efficient results are observed when the ratios of golden and Nickel are considered together.
This paper focuses on a specially constructed matrix whose entries are harmonic Fibonacci numbers and considers its Hadamard exponential matrix. A lot of admiring algebraic properties are presented for both of them. Some of them are determinant, inverse in usual and in the Hadamard sense, permanents, some norms, etc. Additionally, a MATLAB-R2016a code is given to facilitate the calculations and to further enrich the content.
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