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Öğe AN OVERVIEW TO ANALYTICITY OF DUAL FUNCTIONS(Ankara Univ, Fac Sci, 2022) Durmaz, Olgun; Kec, Osman; Aktas, BusraIn this paper, the analyticity conditions of dual functions are clearly examined and the properties of the concept derivative are given in detail. Then, using the dual order relation, the dual analytic regions of dual analytic functions are constructed such that a collection of these regions forms a basis on Dn. Finally, the equivalent of the inverse function theorem in dual space is given by a theorem and proved.Öğe The derivative and tangent operators of a motion in Lorentzian space(World Scientific Publ Co Pte Ltd, 2017) Durmaz, Olgun; Aktas, Busra; Gundogan, HalitIn this paper, by using Lorentzian matrix multiplication, L-Tangent operator is obtained in Lorentzian space. The L-Tangent operators related with planar, spherical and spatial motion are computed via special matrix groups. L-Tangent operators are related to vectors. Some illustrative examples for applications of L-Tangent operators are also presented.Öğe Dual uzayın temel yapıları ve diferensiyel geometrisi(Kırıkkale Üniversitesi, 2022) Durmaz, Olgun; Gündoğan, HalitBu tez dört bölümden oluşmaktadır. İlk bölüm giriş kısmına ayrılmıştır. Bu bölümde, çalışmanın konusu ile ilgili literatürde yer alan bilgilere yer verilmiştir. Ayrıca, bu kısımda çalışmanın amacı ve önemi belirtilmiştir. İkinci bölümde, sonraki bölümlerde kullanılacak olan temel tanım ve teoremlere yer verilmiştir. Üçüncü bölümün ilk kısmında, ?² üzerinde tanımlı olan sözlük sıralama bağıntısının yardımıyla tanımlanan dual sıralama bağıntısının özellikleri detaylı bir şekilde incelenmiştir. Bu bölümün ikinci kısmında, dual analitik fonksiyonların tanımına yer verilmiş ve bu tanım ışığında dual analitik fonksiyonların özellikleri bir akış içerisinde birkaç örnekle de desteklenerek ele alınmıştır. Dördüncü bölümün ilk kısmında, diferensiyel geometride önemli bir yere sahip olan yöne göre türev, vektör alanı, türev dönüşümü ve lie çarpımı tanımları dual uzayda ele alınmıştır. Daha sonra, dual parametreye sahip D-modül eğrisinin tanımına yer verilmiş ve bu eğrinin özellikleri verilen teoremlerle ayrıntılı bir şekilde incelenmiştir. Bu bölümün son kısmında, dual kovaryant türevin tanımı yapılmış ve özellikleri araştırılmıştır.Öğe İndirgenmiş riemann metriği ve riemann manifoldları(Kırıkkale Üniversitesi, 2015) Durmaz, Olgun; Gündoğan, HalitBu çalışma üç bölümden oluşmaktadır. Birinci Bölümde diğer bölümler için gerekli olan temel kavramlara ayrılmıştır. İkinci Bölümde Riemann manifoldları tanıtılmıştır. Bir Riemann manifoldu üzerindeki eğri- nin yay uzunluğu, bir tanjant vektörün uzunluğu ve iki tanjant vektörün iç çarpımı gibi bazı özellikler incelenmiş ve sonra örnekler verilmiştir. Son olarak, Üçüncü Bö- lümde Öklid uzayında gömülü yüzey üzerindeki Riemann yapı incelenmiş ve bu yü- zey üzerinde Riemann metriğinin katsayıları tanıtılmıştır. Sonra da Pseudo-Riemann metriği, Riemann manifoldunun izometrileri ve Hacim elementi çalışılmıştır.Öğe New Approaches On Dual Space(UNIV NIS, 2020) Durmaz, Olgun; Aktas, Busra; Gundogan, HalitIn this paper, we have explained how to define the basic concepts of differential geometry on Dual space. To support this, dual tangent vectors that have (p) over bar as dual point of application have been defined. Then, the dual analytic functions defined by Dimentberg have been examined in detail, and by using the derivative of the these functions, dual directional derivatives and dual tangent maps have been introduced.Öğe On Constraint Manifolds of Lorentz Sphere(OVIDIUS UNIV PRESS, 2020) Aktas, Busra; Durmaz, Olgun; Gundogan, Hal'tThe expression of the structure equation of a mechanism is significant to present the last position of the mechanism. Moreover, in order to attain the constraint manifold of a chain, we need to constitute the structure equation. In this paper, we determine the structure equations and the constraint manifolds of a spherical open-chain in the Lorentz space. The structure equations of spherical open chain with reference to the causal character of the first link are obtained. Later, the constraint manifolds of the mechanism are determined by means of these equations. The geometric constructions corresponding to these manifolds are studied.Öğe On constraint manifolds of spatial closed chains in Lorentzian space(World Scientific Publ Co Pte Ltd, 2024) Aktas, Busra; Durmaz, Olgun; Aydin, OeznurThe objective of this paper is to explore the unexplored algebraic representations of the constraint manifolds associated with 4C and 6C spatial closed chains in Lorentzian space. Initially, we establish the structure equations of spatial closed chains by utilizing the structure equations for spatial open chains in Lorentzian space. Subsequently, we employ these structure equations to derive the algebraic expressions defining the constraint manifolds of 4C and 6C spatial closed chains in Lorentzian space, considering factors such as the causal character of the first link and the rotational axis of the cylindrical joint. This study will finalize all algebraic representations of constraint manifolds for spacelike and timelike mechanisms in Lorentzian space.Öğe On Interpretation of Hyperbolic Angle(KYUNGPOOK NATL UNIV, DEPT MATHEMATICS, 2020) Aktas, Busra; Gundogan, Halit; Durmaz, OlgunMinkowski spaces have long been investigated with respect to certain properties and substructues such as hyperbolic curves, hyperbolic angles and hyperbolic arc length. In 2009, based on these properties, Chung et al. [3] defined the basic concepts of special relativity, and thus; they interpreted the geometry of the Minkowski spaces. Then, in 2017, E. Nesovic [6] showed the geometric meaning of pseudo angles by interpreting the angle among the unit timelike, spacelike and null vectors on the Minkowski plane. In this study, we show that hyperbolic angle depends on time, t. Moreover, using this fact, we investigate the angles between the unit timelike and spacelike vectors.Öğe On Parallelizable Spheres in Semi Euclidean Space(SOUTHEAST ASIAN MATHEMATICAL SOC-SEAMS, 2020) Durmaz, Olgun; Aktas, Busra; Gundogan, HalitIn Euclidean space, there exist four theorems which show that S-n sphere is not parallelizable for n not equal 1, 3, 7. While three of them are shown by using Bott theorem, the last one is shown by using Hurwitz-Radon numbers. In this paper, a theorem and the proof of this theorem about parallelization of spheres in semi-Euclidean space is given. It is presented that some spheres are parallelizable with respect to specific number systems.Öğe On The Basic Structures Of Dual Space(UNIV NIS, 2020) Aktas, Busra; Durmaz, Olgun; Gundogan, HalitTopology studies the properties of spaces that are invariant under any continuous deformation. Topology is needed to examine the properties of the space. Fundamentally, the most basic structure required to do math in the space is topology. There exists little information on the expression of the basis and topology on dual space. The main point of the research is to explain how to define the basis and topology on dual space D-n. Then, we will study the geometric constructions corresponding to the open balls in D and D-2, respectively.Öğe Rodrigues parameters on dual hyperbolic unit sphere H20(Taylor & Francis Ltd, 2018) Aktas, Busra; Durmaz, Olgun; Gundogan, HalitRodrigues parameters depend on the tangent of the half rotation angle in Euclidean space but in Dual space, dual Rodrigues parameters contain both rotation angle and distance corresponding the shortest distance between the straight lines in R-3. In this paper, we give Cayley's formula for the dual hyperbolic spherical motion and explain 3x3 type L-Dual skew symmetric matrices by using properties of this formula. Then, we obtain Rodrigues parameters of dual Hyperbolic unit sphere and show that Rodrigues parameters contain the hyperbolic rotation angle which is being between timelike lines and distance which is the minimal Lorentzian distance between the timelike lines of R-1(3)Öğe Structure equations and constraint manifolds on Lorentz plane(Wiley, 2019) Durmaz, Olgun; Aktas, Busra; Gundogan, HalitCalculating the structure equation of a chain is important to represent the position of the end link on the chain. Furthermore, the structure equation helps to determine the constraint manifold of the chain. The constraint manifold satisfies to make geometric interpretations about the form that is obtained. What is more, the constraint forced on the positions of the end link by the rest of the chain is represented by the manifold. In Lorentz space, the structure equations change according to the causal characters of the first link. In this paper, we attain the structure equations of a planar open chain in terms of the causal character of the first link in this space. Later, the constraint manifolds of the chain by using these equations are given. Some geometric comments about these manifolds are explained.Öğe The inequalities on dual numbers and their topological structures(Tubitak Scientific & Technological Research Council Turkey, 2023) Aktaş, Büşra; Durmaz, Olgun; Gündoğan, HalitInequalities are frequently used in various fields of mathematics to prove theorems. The existence of inequalities contributes significantly to the foundations of such branches. In this paper, we study the properties of order relations in the system of dual numbers, which is inspired by order relations defined on real numbers. Besides, some special inequalities that are used in various fields of mathematics, such as Cauchy-Schwarz, Minkowski, and Chebyshev are studied in this framework. An example is also provided to validate our research findings.
