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Öğe A new approach to Leonardo number sequences with the dual vector and dual angle representation(Amer Inst Mathematical Sciences-Aims, 2024) Babadağ, Faik; Atasoy, AliIn this paper, we introduce dual numbers with components including Leonardo number sequences. This novel approach facilitates our understanding of dual numbers and properties of Leonardo sequences. We also investigate fundamental properties and identities associated with Leonardo number sequences, such as Binet's formula and Catalan's, Cassini's and D'ocagne's identities. Furthermore, we also introduce a dual vector with components including Leonardo number sequences and dual angles. This extension not only deepens our understanding of dual numbers, it also highlights the interconnectedness between numerical sequences and geometric concepts. In the future it would be valuable to replicate a similar exploration and development of our findings on dual numbers with Leonardo number sequences.Öğe Obtaining triplet from quaternions(Ramazan Yaman, 2021) Atasoy, Ali; Yaylı, YusufIn this study, we obtain triplets from quaternions. First, we obtain triplets from real quaternions. Then, as an application of this, we obtain dual triplets from the dual quaternions. Quaternions, in many areas, it allows ease in calculations and geometric representation. Quaternions are four dimensions. The triplets are in three dimensions. When we express quaternions with triplets, our study is conducted even easier. Quaternions are very important in the display of rotational movements. Dual quaternions are important in the expression of screw movements. Reducing movements from four dimensions to three dimensions makes our study easier. This simplicity is achieved by obtaining triplets from quaternions.Öğe A new approach to curve couples with Bishop frame(Ankara Univ, Fac Sci, 2024) Babadağ, Faik; Atasoy, Ali. This paper presents a detailed study of a new generation of the Bishop frame with components including three orthogonal unit vectors, which are tangent vector, normal vector and binormal vector. It is a frame field described on a curve in Euclidean space, which is an alternative to the Frenet frame. It is useful for curves for which the second derivative is not available. Moreover, the conditions which the Bishop frame of one curve coincides with the Bishop frame of another curve are defined. It would be valuable to replicate similar approaches in the Bishop frame of one curve coincides with the Bishop frame of another curve.Öğe Al1070/TiB2 Kompozitlerin Tornalanmasında TiB2 Takviye Miktarının Esas Kesme Kuvveti Ve Yüzey Pürüzlülüğüne Etkisi(2020) Pul, MuharremBu çalışmada Al1070 matrisli TiB2takviyeli kompozitin işlenebilirliği incelenmiştir. İlk aşamada ağırlıkça %2, %4 ve %8 TiB2 takviyeli Al1070 kompozitler üretilmiştir. Daha sonra; kuru kesme şartlarında, 1 mm sabit kesme derinliğinde, 100, 200, 300 m/dak kesme hızlarında, 0,10- 0,15 - 0,25 mm/dev ilerleme değerlerinde ve sementit karbür (SK), kaplamalı sementit karbür (KSK) ve kübik bor nitrür (KBN) takımlar ile tornada işlenebilirlik deneyleri yapılmıştır. Deneylerde kesme kuvvetleri ve yüzey pürüzlülük değerleri kaydedilmiştir. Sonuçta, kesme hızının artmasıyla kesme kuvvetleri ve yüzey pürüzlülük değerleri azalmış, TiB2 takviye miktarındaki artışla hem kesme kuvvetleri hem de yüzey pürüzlülük değerleri artmıştır. En düşük esas kesme kuvveti % 2 TiB2 takviyeli kompozitte, KSK takım ile 0,10 mm/dev ilerleme ve 300 m/dak kesme hızında 74 N olarak elde edilmiştir. En düşük yüzey pürüzlülüğü ise yine % 2 TiB2 takviyeli kompozit numuneden, SK takım ile 0,10 mm/dev ilerleme ve 100 m/dak kesme hızında 0,63 µm olarak ölçülmüştür. Tüm numunelerde ilerleme miktarlarının artışı ile yüzey pürüzlülük değerleri ve esas kesme kuvvetleri artış göstermiştir. SK takımlardan düşük ilerleme ve kesme hızı değerlerinde daha düşük yüzey pürüzlülük değerleri elde edilirken, yüksek ilerleme ve kesme hızı değerlerinde ise KBN takımlar daha iyi performans göstermiştir. Genel olarak, işlenebilirlik yönünden en olumlu sonuçlar SK takımlar ile elde edilmiştir.Öğe A New Polar Representation for Split and Dual Split Quaternions(Springer Basel Ag, 2017) Atasoy, Ali; Ata, Erhan; Yayli, Yusuf; Kemer, YaseminWe present a new different polar representation of split and dual split quaternions inspired by the Cayley-Dickson representation. In this new polar form representation, a split quaternion is represented by a pair of complex numbers, and a dual split quaternion is represented by a pair of dual complex numbers as in the Cayley-Dickson form. Here, in a split quaternion these two complex numbers are a complex modulus and a complex argument while in a dual split quaternion two dual complex numbers are a dual complex modulus and a dual complex argument. The modulus and argument are calculated from an arbitrary split quaternion in Cayley-Dickson form. Also, the dual modulus and dual argument are calculated from an arbitrary dual split quaternion in Cayley-Dickson form. By the help of polar representation for a dual split quaternion, we show that a Lorentzian screw operator can be written as product of two Lorentzian screw operators. One of these operators is in the two-dimensional space produced by 1 and i vectors. The other is in the three-dimensional space generated by 1, j and k vectors. Thus, an operator in a four-dimensional space is expressed by means of two operators in two and three-dimensional spaces. Here, vector 1 is in the intersection of these spaces.
