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Öğe Algebraic Construction for Dual Quaternions with GCN(Bitlis Eren Üniversitesi Rektörlüğü, 2022) Şentürk, Gülsüm Yeliz; Gürses, Nurten; Yüce, SalimIn this paper, we explain how dual quaternion theory can extend to dual quaternions with generalized complex number (GCN) components. More specifically, we algebraically examine this new type dual quaternion and give several matrix representations both as a dual quaternion and asa GCN.Öğe A comprehensive survey of dual-generalized complex Fibonacci and Lucas numbers(Yıldız Teknik Üniversitesi, 2022) Gürses, Nurten; Şentürk, Gülsüm Yeliz; Yüce, SalimThis paper aims to develop dual-generalized complex Fibonacci and Lucas numbers and obtain recurrence relations. Fibonacci and Lucas’s approach to dual-generalized complex numbers contains dual-complex, hyper-dual and dual-hyperbolic situations as special cases and allows general contributions to the literature for all real number p. For this purpose, Binet’s formulas along with Tagiuri’s, Hornsberger’s, D’Ocagne’s, Cassini’s and Catalan’s identities, are calculated for dual-generalized complex Fibonacci and Lucas numbers. Finally, the results are given, and the special cases for this unification are classified.Öğe Construction of dual-generalized complex Fibonacci and Lucas quaternions(VASYL STEFANYK PRECARPATHIAN NATL UNIV, VUL SHEVCHENKA 57, IVANO-FRANKIVSK 76018, UKRAINE, 2022) Şentürk, Gülsüm Yeliz; Gürses, Nurten; Yüce, SalimThe aim of this paper is to construct dual-generalized complex Fibonacci and Lucas quaternions. It examines the properties both as dual-generalized complex number and as quaternion. Additionally, general recurrence relations, Binet’s formulas, Tagiuri’s (or Vajda’s like), Honsberger’s, d’Ocagne’s, Cassini’s and Catalan’s identities are obtained. A series of matrix representations of these special quaternions is introduced. Finally, the multiplication of dual-generalized complex Fibonacci and Lucas quaternions are also expressed as their different matrix representations.Öğe Dual quaternion theory over HGC numbers(Taru Publications, 2024) Şentürk, Gülsüm Yeliz; Gürses, NurtenKnowing the applications of quaternions in various fields, such as robotics, navigation, computer visualization and animation, in this study, we give the theory of dual quaternions considering Hyperbolic-Generalized Complex (HGC) numbers as coefficients via generalized complex and hyperbolic numbers. We account for how HGC number theory can extend dual quaternions to HGC dual quaternions. Some related theoretical results with HGC Fibonacci/Lucas numbers are established, including their dual quaternions. Given HGC Fibonacci/Lucas numbers, their special matrix correspondences have been identified and these are carried out to HGC Fibonacci/Lucas dual quaternions. Furthermore, we provide a more accurate way to quickly calculate HGC Fibonacci numbers and associate this with HGC generalized Fibonacci numbers. For implementation, we produce an algorithm in Maple. Lastly, we put the theory into practice.Öğe EXAMINATION OF SETS OF POINTS AND NON-NULL LINES UNDER THE TWO-PARAMETER LORENTZIAN MOTION(Editura Bibliotheca-Bibliotheca Publ House, 2020) Şentürk, Gülsüm Yeliz; Yüce, SalimIn this paper, firstly, we interested in finding the relationships among the densities of sets of collinear points, among the densities of non-collinear points and among the densities of sets of intersecting non-null lines in Lorentzian plane. Furthermore, we concerned with the density formulas of sets of points and the sets of non-null lines under the two-parameter planar Lorentzian motion.Öğe Examination of Sets of Points and Non-Null Lines Under the Two-Parameter Lorentzian Motion(EDITURA BIBLIOTHECA-BIBLIOTHECA PUBL HOUSE, STRADA NICOLAE RADIAN KB2-3, TARGOVISTE, 130082, ROMANIA, 2020) Şentürk, Gülsüm Yeliz; Yüce, SalimIn this paper, firstly, we interested in finding the relationships among the densities of sets of collinear points, among the densities of non-collinear points and among the densities of sets of intersecting non-null lines in Lorentzian plane. Furthermore, we concerned with the density formulas of sets of points and the sets of non-null lines under the two-parameter planar Lorentzian motion.Öğe Fundamental properties of extended Horadam numbers(Bulgarian Acad Science, 2021) Şentürk, Gülsüm Yeliz; Gürses, Nurten; Yüce, SalimIn this paper, the extended Horadam numbers are introduced by using dual-generalized complex, hyperbolic-generalized complex and complex-generalized complex numbers. Then, generating function, Binet's formula, D'Ocagne's, Catalan's and Cassini's identities are given. Moreover, specialy matrix representations of the extended Horadam numbers are investigated. In conclusion, the results and classification of the special cases are introduced.Öğe Investigating Generalized Quaternions with Dual-Generalized Complex Numbers(INST MATHEMATICS, AS CR, ZITNA 25, PRAHA 1 115 67, CZECH REPUBLIC, 2023) Gürses, Nurten; Şentürk, Gülsüm Yeliz; Yüce, SalimWe aim to introduce generalized quaternions with dual-generalized complex number coefficients for all real values ?, ? and p. Furthermore, the algebraic structures, properties and matrix forms are expressed as generalized quaternions and dual-generalized complex numbers. Finally, based on their matrix representations, the multiplication of these quaternions is restated and numerical examples are given.Öğe New Insight Into Quaternions and Their Matrices(Ankara Üniversitesi Fen Fakültesi, 2023) Şentürk, Gülsüm Yeliz; Gürses, Nurten; Yüce, SalimThis paper aims to bring together quaternions and generalized complex numbers. Generalized quaternions with generalized complex number components are expressed and their algebraic structures are examined. Several matrix representations and computational results are introduced. An alternative approach for a generalized quaternion matrix with elliptic number entries has been developed as a crucial part.Öğe On Ruled Non-Degenerate Surfaces with Darboux Frame in Minkowski 3-Space(TURKIC WORLD MATHEMATICAL SOC, Z KHALILOV STR, 23, BAKU, AZ 1148, AZERBAIJAN, 2020) Şentürk, Gülsüm Yeliz; Yüce, SalimIn this paper, ruled non-degenerate surfaces with respect to Darboux frame are studied. Characterization of them which are related to the geodesic torsion, the normal curvature and the geodesic curvature with respect to Darboux frame are examined. Furthermore, some special cases of non-null rulings are demonstrated according to Frenet frame {T, N, B} with Darboux frame {T, g, n}. Finally, the integral invariants of these surfaces are examined.Öğe ON RULED NON-DEGENERATE SURFACES WITH DARBOUX FRAME IN MINKOWSKI 3-SPACE(Turkic World Mathematical Soc, 2020) Şentürk, Gülsüm Yeliz; Yüce, SalimIn this paper, ruled non-degenerate surfaces with respect to Darboux frame are studied. Characterization of them which are related to the geodesic torsion, the normal curvature and the geodesic curvature with respect to Darboux frame are examined. Furthermore, some special cases of non-null rulings are demonstrated according to Frenet frame {T, N, B} with Darboux frame {T, g, n}. Finally, the integral invariants of these surfaces are examined.Öğe A Study on Dual-Generalized Complex and Hyperbolic-Generalized Complex Numbers(Gazi Üniversitesi, 2021) Gürses, Nurten; Şentürk, Gülsüm Yeliz; Yüce, SalimThis work is intended to introduce the theories of dual-generalized complex and hyperbolicgeneralized complex numbers. The algebraic properties of these numbers are taken into consideration. Besides, dual-generalized complex and hyperbolic-generalized complex valued functions are defined and different matrix representations of these numbers are examined. Moreover, a remarkable classification are given for special cases and the set of complex-generalized complex numbers are mentioned.
