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This 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 . 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.

Document Information Language:English Accession Number: WOS:000522768900024, In 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., The first author has been partially supported by TUBITAK (2211-Domestic PhD Scholarship) The Scientific and Technological Research Council of Turkey.

In 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., This work was supported by Research Fund of the Yildiz Technical University, Project no. FDK-2018-3320. G. Y. Senturk has been partially supported by TUBITAK (2211-Domestic Ph.D. Scholarship), The Scientific and Technological Research Council of Turkey.

Bu tez 7 bölümden oluşmaktadır. Birinci bölüm, tezin giriş kısmına ayrılmış olup bu bölümde literatür özeti, tezin amacı ve bulgular verilmiştir. İkinci bölümde, ilk olarak tezin bütünü için temel kavram olan Öklid ve Lorentz düzlemlerinde nokta ve doğru kümelerinin yoğunluğu tanımlanmıştır. Öklid düzleminde 2-parametreli hareketler tanımlanarak kutup ekseni ve hareketli koordinat sistemleri verilmiştir. Üçüncü bölümde ise hareketin kutup ekseni ve kutup eksen yoğunluk değişmezliği esas alınarak W. Blaschke and H. R. Müller'in [1,2] bakış açısı ile Öklid düzleminde 2-parametreli hareketler geometrik olarak ayrıntılı incelenmiştir. Böylece, Öklid düzleminde 2-parametreli hareketler ile ilgili W. Blaschke ve H. R. Müller'in [1] Almanca kitabında olup H. R. Müller'in [2] kitabında olmayan konular ortak bir notasyon ve anlatım tarzı ile incelenmiştir.Dördüncü ve daha sonraki temel bölümler tezin orijinal kısmını oluşturmaktadır.Dördüncü bölümde, W. Blaschke and H. R. Müller'in [1,2] Öklid düzlemindeki bakış açısı ile Lorentz düzleminde 2-parametreli hareket incelendi. İlk olarak, Lorentz düzlemindeki 1-parametreli hareketin hiperbolik ifadesi yardımı ile 2-parametreli Lorentzian hareketi tanımlandı. Sonrasında, 2-parametreli Lorentzian hareketler altında hareketli koordinat sistemleri, hareketin non-null kutup ekseni, non-null kutup eksen dönüşümünün yoğunluk değişmezliği, sıfır yoğunluklu non-null kutup eksen dönüşümü, eğri elementi çizen noktaları ve hareketin destek fonksiyonu araştırıldı.Beşinci bölümde, Lorentz düzleminde nokta ve non-null doğru kümelerinin yoğunlukları ile ilgili özel durumlar incelendi.Altıncı bölümde, Lorentz düzleminde 2-parametreli hareket altında nokta ve non-null doğru kümelerinin yoğunlukları tanımlandı; nokta ve non-null doğru kümelerinin yoğunlukları ile ilgili özel durumlar verildi.Yedinci bölüm olan son bölümde, orijinal bölümlerde elde edilen sonuçlardan bahsedildi ve gelecek çalışmalar için önerilerde bulunuldu. This thesis consists of seven chapters.The first chapter is devoted to the introduction part that includes literature review, objective of the thesis and hypothesis. In the second chapter, firstly, the densities of the set of points and lines are defined in Euclidean and Lorentzian planes as the basic concepts for the whole thesis. The polar axis and the moving coordinate systems are given by defining 2-parameter planar motions in Euclidean plane. In the third chapter, 2-parameter motions in Euclidean plane are examined geometrically by taking into account W. Blaschke and H. R. Müller [1,2] viewpoint using the polar axes of the motion and the density invariance of the polar axes. Thus, the topics related to 2-parameter motions in the Euclidean plane in W. Blaschke and H. R. Müller's German book [1] but not in H. R. Müller's [2] book are investigated by using common notations and expressions.The original parts of this thesis are composed of the fourth chapter and the following chapters.In the fourth chapter, 2-parameter Lorentzian motions are investigated by using the viewpoint of W. Blaschke and H. R. Müller [1,2] in the Euclidean plane. Firstly, 2-parameter planar Lorentzian motions are defined by using the hyperbolic expression of 1-parameter motion in the Lorentzian plane. And then, during 2-parameter planar Lorentzian motions the moving coordinate systems, non-null polar axes, the density invariance of non-null polar axis transformation, zero-density of non-null polar axes, the curve plotter points and the support function of the motion are examined.In the fifth chapter, special cases of the densities of sets points and non-null lines in the Lorentzian plane are investigated.In the sixth chapter, the density formulas of sets of points and non-null lines are defined; special cases of the densities of sets points and non-null lines are given under 2-parameter planar Lorentzian motion.In the seventh chapter, which is the last chapter of thesis, the original obtained results are mentioned and some suggestions are made for future studies. 149

This 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 complexgeneralized complex numbers are mentioned.