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    Bertrand partner P-Trajectories in the Euclidean 3-space e3
    (Ankara Univ, Fac Sci, 2023) Isbilir, Zehra; Ozen, Kahraman Esen; Tosun, Murat
    The concept of a pair of curves, called as Bertrand partner curves, was introduced by Bertrand in 1850. Bertrand partner curves have been stud-ied widely in the literature from past to present. In this study, we take into account of the concept of Bertrand partner trajectories according to Positional Adapted Frame (PAF) for the particles moving in 3-dimensional Euclidean space. Some characterizations are given for these trajectories with the aid of the PAF elements. Then, we obtain some special cases of these trajectories. Moreover, we provide a numerical example.
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    Examination of generalized Tribonacci dual quaternions
    (Univ Tartu Press, 2023) Isbilir, Zehra; Guerses, Nurten
    This manuscript deals with introducing and discussing of a new type dual quaternions which are named generalized Tribonacci dual quaternions (GTDQ, for short). For this purpose, several new properties, such as Binet formula, generating function, exponential generating function, matrix formula, and determinant equations, are established. In addition to these, some numerical algorithms are constructed. In the last part, some special cases of the family of the GTDQ are examined regarding r, s, t values and initial values considering concluded results.
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    Generalized Rectifying Ruled Surfaces of Special Singular Curves
    (Ovidius Univ Press, 2023) Isbilir, Zehra; Yazici, Bahar Dogan; Tosun, Murat
    In this study, generalized rectifying ruled surfaces of Frenet-type framed base curves in the three-dimensional Euclidean space are introduced. These surfaces are a generalization of not only the tangent and binormal surfaces of Frenet-type framed base curves, but also the tangent and binormal surfaces of regular curves. Additionally, we present some geometric characterizations and properties of these surfaces. Then, the singular point classes of the surface are scrutinized and the conditions for being a cross-cap surface are stated. Moreover, generalized rectifying surfaces are examined as framed surfaces by using the framed surface theory, and we investigate the basic invariants and curvatures of them. Then, several illustrative examples with figures are given to support the theorems and results.
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    Mannheim partner trajectories related to pafors
    (Univ Nis, 2024) Isbilir, Zehra; Ozen, Kahraman Esen; Tosun, Murat
    In this study, we consider the concept of Mannheim partner trajectories related to the Positional Adapted Frame on Regular Surfaces (PAFORS) for the particles moving on the different regular surfaces in Euclidean 3 -space. We give the relations between the PAFORS elements of these aforementioned trajectories. Also, we obtain the relations between Darboux basis vectors of these trajectories. Furthermore, some special cases of these trajectories are written.
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    A New Insight on Rectifying-Type Curves in Euclidean 4-Space
    (Int Electronic Journal Geometry, 2023) Isbilir, Zehra; Tosun, Murat
    In this study, our purpose is to determine the generalized rectifying-type curves with Frenet-type frame in Myller configuration for Euclidean 4-space E4. Also, some characterizations of them are given. We construct some correlations between curvatures and invariants of generalized rectifyingtype curves. Additionally, we obtain an illustrative example with respect to the rectifying-type curves with Frenet-type frame in Myller configuration for Euclidean 4-space E4.
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    On generalized osculating-type curves in Myller configuration
    (Ovidius Univ Press, 2024) Isbilir, Zehra; Tosun, Murat
    In this study, we examine osculating-type curves with Frenet-type frame in Myller configuration for Euclidean 3-space E-3. We present the necessary characterizations for a curve to be an osculating-type curve. Characterizations originating from the natural structure of Myller configuration are a generalization of osculating curves according to the Frenet frame. Also, we introduce some new results that are not valid for osculating curves. Then, we give an illustrative numerical example supported by a figure.
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    On special curves in Lie groups with Myller configuration
    (Wiley, 2024) Isbilir, Zehra; Dogan Yazici, Bahar; Tosun, Murat
    In this study, we determine a new type comprehensive frame, which is called the generalized Frenet-type frame in three-dimensional Lie groups with Myller configurations, and it includes several special and classical type frames for Euclidean 3-space and three-dimensional Lie groups. After constructing this new comprehensive frame, we obtain derivative formulas with the help of the Lie curvature. In addition, we define some special type curves. The geometry of versor fields along a curve with Frenet-type frame in three-dimensional Lie groups with Myller configurations is a generalization of the usual theory of curves. Since this particular relationship, the osculating-type and rectifying-type curves with Frenet-type frame in three-dimensional Lie groups with Myller configurations include some special cases for osculating and rectifying curves in different spaces.
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    Padovan and Perrin generalized quaternions
    (Wiley, 2021) Isbilir, Zehra; Gurses, Nurten
    In this study, we investigate the Padovan (or Cordonnier) and Perrin generalized quaternions. We obtain the new identities for these special quaternions related to matrix forms. We also introduce Binet-like formulae, generating functions, several summation, and binomial properties concerning these quaternions.
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    Pell-Padovan generalized quaternions
    (Bulgarian Acad Science, 2021) Isbilir, Zehra; Gurses, Nurten
    The aim of this article is to introduce Pell-Padovan generalized quaternions. It also derives new properties associated with these and takes into account negative indices. Additionally, it presents generating function, Binet-like formula, Simson formula, matrix representations, and several summation properties.
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    Pentanacci and Pentanacci-Lucas hybrid numbers
    (Taylor & Francis Ltd, 2021) Isbilir, Zehra; Gurses, Nurten
    The aim of this study is to introduce the Pentanacci and Pentanacci-Lucas hybrid numbers. Then, some properties with respect to these special numbers and relations between them are obtained. In addition, matrix formulations, generating functions, Binet-like formulas and some summation formulas of them are examined.
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    Spinor representations of framed curves in the three-dimensional lie groups
    (Taru Publications, 2023) Isbilir, Zehra; Yazici, Bahar Dogan; Tosun, Murat
    In this study, we determine the spinor representations of special singular curves (framed curves) in the three-dimensional Lie groups with a biinvariant metric. Also, we construct spinor framed equations for some special cases and obtain the relations between the spinor representations of the general frame and adapted frame along the framed curves in the three-dimensional Lie groups. Then, we give some geometric properties and results with respect to them.
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    Spinor Representations of Positional Adapted Frame in the Euclidean 3-Space
    (Int Electronic Journal Geometry, 2023) Isbilir, Zehra; Ozen, Kahraman Esen; Guner, Mehmet
    The main goal of this paper is to study together the spinors, which have a major place in several disciplines from mathematics to physics, and Positional Adapted Frame (PAF) which is a new frame that attracts the attention of many researchers. In accordance with this purpose, we introduce the spinor representations for the trajectories endowed with PAF in the Euclidean 3-space E3, and construct the spinor equations of PAF vectors. Then, we find the relations between spinor representations of PAF and Serret-Frenet frame. Also we give some results and present some geometric interpretations with respect to this relationship. Moreover, we present an illustrative numerical example in order to support the given theorems and results.