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Öğe An extended framework for bihyperbolic generalized Tribonacci numbers(Ankara University, 2024) Gürses, Nurten; İşbilir, ZehraThe aim of this article is to identify and analyze a new type special number system which is called bihyperbolic generalized Tribonacci numbers (BGTN for short). For this purpose, we give both classical and several new properties such as; recurrence relation, Binet formula, generating function, exponential generating function, summation formulae, matrix formula, and special determinant equations of BGTN . Also, the system of BGTN is quite a big family and includes several type special cases with respect to initial values and $r,~ s, ~t$ values, we give the subfamilies and special cases of it. In addition to these, we construct some numerical algorithms including recurrence relation and special two types determinant equations related to calculating the terms of this new type special number system. Then, we examine several properties by taking two special cases and including some illustrative numerical examples.Öğe Characterization of Tzitzeica Curves Using Positional Adapted Frame(Mehmet Zeki SARIKAYA, 2022) Özen, Kahraman Esen; İşbilir, Zehra; Tosun, MuratIn this study, Tzitz\'eica curves are taken into consideration in the Euclidean 3-space by using the Positional Adapted Frame (PAF). Such curves are characterized according to PAF elements. Also, some results are obtained on spherical Tzitz\'eica curves. The results obtained in this study are new contributions to the field. It is expected that these results will be useful in various application areas of differential geometry and applied mathematics in the future.Öğe Generalized osculating-type ruled surfaces of singular curves(Wiley, 2023) Yazıcı, Bahar Doğan; İşbilir, Zehra; Tosun, MuratIn this study, we introduce generalized osculating-type ruled surfaces of special singular curves. We give some theories and results about the geometric structure of the surface. In addition, the singular point classes of the surface are examined, and the conditions for being a cross-cap surface are expressed. Generalized osculating-type ruled surface is considered as a framed surface and its basic invariants are found and some results are given. Finally, we give some examples and figures to support the theories.Öğe MANNHEIM PARTNER P-TRAJECTORIES IN THE EUCLIDEAN 3-SPACE E-3(Honam Mathematical Soc, 2022) İşbilir, Zehra; Özen, Kahraman Esen; Tosun, MuratMannheim introduced the concept of a pair of curves, called as Mannheim partner curves, in 1878. Until now, Mannheim partner curves have been studied widely in the literature. In this study, we take into account of this concept according to Positional Adapted Frame (PAF) for the particles moving in the 3-dimensional Euclidean space. We introduce a new type special trajectory pairs which are called Mannheim partner P-trajectories in the Euclidean 3-space. The relationships between the PAF elements of this pair are investigated. Also, the relations between the Serret-Frenet basis vectors of Mannheim partner P-trajectories are given. Afterwards, we obtain the necessary conditions for one of these trajectories to be an osculating curve and for other to be a rectifying curve. Moreover, we provide an example including an illustrative figure.Öğe On the combined Jacobsthal-Padovan generalized quaternions(Tbilisi Centre Math Sci, 2022) Gürses, Nurten; İşbilir, ZehraIn this article, we examine the combined Jacobsthal-Padovan (CJP) generalized quaternions with four special cases: Jacobsthal-Padovan, Jacobsthal-Perrin, adjusted Jacobsthal-Padovan and modified Jacobsthal-Padovan generalized quaternions. Then, recurrence relation, generating function, Binet-like formula and exponential generating function of these quaternions are examined. In addition to this, some new properties, special determinant equations, matrix formulas and summation formulas are discussed.Öğe Padovan, Perrin and Pell-Padovan Dual Quaternions(2023) İşbilir, Zehra; Gürses, NurtenIn this present study, we intend to determine the Padovan, Perrin and Pell-Padovan dual quaternions with nonnegative and negative subscripts. In line with this purpose, we construct some new properties such as; special determinant equalities, new recurrence relations, matrix formulas, Binet-like formulas, generating functions, exponential generating functions, summation formulas, and binomial properties for these special dual quaternions.Öğe Spinor representation of framed Mannheim curves(Scientific and Technological Research Council Turkey, 2022) Doğan Yazıcı, Bahar; İşbilir, Zehra; Tosun, MuratIn this paper, we obtain spinor with two complex components representations of Mannheim curves of framed curves. Firstly, we give the spinor formulas of the frame corresponding to framed Mannheim curve. Later, we obtain the spinor formulas of the frame corresponding to framed Mannheim partner curve. Moreover, we explain the relationships between spinors corresponding to framed Mannheim pairs and their geometric interpretations. Finally, we present some geometrical results of spinor representations of framed Mannheim curves.Öğe The spinor representations of framed Bertrand curves(Univ Nis, Fac Sci Math, 2023) İşbilir, Zehra; Yazıcı, Bahar Doğan; Tosun, MuratIn this study, we intend to examine the framed Bertrand curves in three-dimensional Euclidean space E3 by using the spinors, which have a fundamental place and importance in different disciplines from mathematics to physics. For this purpose, we investigate the spinor representations of framed Bertrand mates in E3. Additionally, we present some geometric results and interpretations. Then, we construct numerical examples with illustrative figures in order to support the given materials.
