# Salim YüceYildiz Technical University · Department of Mathematics

Salim Yüce

Professor

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99

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Introduction

## Publications

Publications (99)

We aim to introduce generalized quaternions with dual-generalized complex number coefficients for all real values α, β and p. Furthermore, the algebraic structures, properties and matrix forms are expressed as generalized quaternions and dual-generalized complex numbers. Finally, based on their matrix representations, the multiplication of these qu...

In this paper, we explain how dual quaternion theory can extend to dual quaternions with generalized complex number (GCN) components. More specifically, we algebraically examine this new type dual quaternion and give several matrix representations both as a dual quaternion and as a GCN.

In this paper, we explain how dual quaternion theory can extend to dual quaternions with generalized complex number (GCN) components. More specifically, we algebraically examine this new type dual quaternion and give several matrix representations both as a dual quaternion and as a GCN .

This paper aims to bring together Vietoris' sequence, which is a rational sequence, and hybrid numbers, and then examine some properties of the hybrid numbers with Vietoris' number coefficients. Some relations between this hybrid number and its norm, the recurrence relations, the generating function and Binet's formula are also calculated.

This study has identified new dual quaternion sequence with Vietoris' components , which is a rational sequence. Additionally, some classical expressions related to the conjugate and its norm have been examined. Finally, the recurrence relations, several well-known identities have been discussed.

In this paper, the extended Horadam numbers are introduced by using dual-generalized complex, hyperbolic-generalized complex and complex-generalized complex numbers. Then, generating function, Binet's formula, D'Ocagne's, Catalan's and Cassini's identities are given. Moreover, special matrix representations of the extended Horadam numbers are inves...

Our main interest in this paper is to explore dual-generalized complex (DGC) Oresme sequence extension. We present two new types of Oresme numbers. We investigate special linear recurrence relations and summation properties for DGC Oresme numbers of type-1. Furthermore, we describe the recurrence relation of DGC Oresme numbers of type-1 in matrix f...

Composite materials are frequently used in the construction of rail, tunnels, and pipelines as well as in the construction of aircraft, ships, and chemical pipelines. When such structural elements are formed from new-generation composites, such as CNT-reinforced composites, and their interaction with the ground, there is a need to renew the dynamic...

Our main interest in this paper is to explore dual-generalized complex (DGC) Oresme sequence extension. We investigate special linear recurrence relations and sums statements for DGC Oresme numbers. Furthermore, we describe recurrence relation of DGC Oresme numbers in matrix form. We also discuss the theory using doubling approach to DGC Oresme seq...

The paper aims to introduce generalized quaternions with dual-generalized complex number coefficient. Furthermore, the algebraic structures, properties and matrix forms are expressed as generalized quaternion and dual-generalized complex number. Finally, based on their matrix representations, the multiplication of these quaternions are restated. Co...

In this study, during the 1-parameter closed homothetic motion, the Holditch-Type Theorems are presented for the polar moments of inertia of the closed orbit curves of three non-collinear points.

Our main interest in this paper is to explore dual-generalized complex (DGC) Oresme sequence extension, and to what extent. We investigate the special linear recurrence relations, sums and multiplication statements for DGC Oresme numbers. Furthermore, we describe recurrence relations of DGC Oresme numbers in matrix form. We also generalize the theo...

In this study, we focused on n-dimensional quaternionic space Hn. To create the module structure, first part is devoted to define a metric depending on the product order relation of Rn. The set of Hn has been rewritten with a different representation of n-vectors. Using this notation, formulations corresponding to the basic operations in Hn are obt...

In this study, firstly, real, complex and quaternion combinations of octonionic matrices are defined. In terms of these defined combinations, basic operations of octonionic matrices are given. Later, algebraic structures of octonionic matrix set are obtained. In addition, the modulus structures of the octonionic matrix set on the real, complex and...

The aim of this paper is to bring together quaternions and generalized complex numbers. Generalized quaternions with generalized complex number components are expressed and their algebraic structures are examined. Several matrix representations and computational results are introduced. As a crucial part, alternative approach for generalized quatern...

The aim of this paper is to bring together quaternions and generalized complex numbers. Generalized quaternions with generalized complex number components are expressed and theiralgebraic structures are examined. Several matrix representations and computational results are introduced. As a crucial part, alternative approach for generalized quaterni...

This work is intended to introduce the theories of dual-generalized complex and hyperbolic-generalized 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 ex...

In this paper, the extended Horadam numbers are introduced by using dual-generalized complex, hyperbolic-generalized complex and complex-generalized complex numbers. Then, generating function, Binet's formula, D'Ocagne's, Catalan's and Cassini's identities are given. Moreover, special matrix representations of these Horadam numbers are investigated...

4- dimensional Pseudo Galilean Geometry

Geometri, Cebir ve Analiz alanlarının buluştuğu, matematiğin güzel ve nispeten temel bir
hesaplaması olan ve sayıların matematik düzeyinde geniş bir görünümünü sunan bu kitap,
matematiğin tüm alanlarındaki lisans öğrencilerinin yanında üniversitelerin tüm bölümlerindeki
öğrenciler tarafından okunabilir. Ayrıca, bu kitap temel konuların çekici ve ol...

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 aim of this paper is to construct dual-generalized complex Fibonacci and Lucas quaternions. Moreover, general recurrence relations, Binet's formulas, D'ocagne's, Cassini's and Catalan's identities for dual-generalized complex Fibonacci and Lucas quaternions are obtained. The matrix representations of dual-generalized complex Fibonacci and Lucas...

The aim of this paper is to develop dual-generalized complex Fibonacci and Lucas numbers. The general recurrence relations of dual-generalized complex Fibonacci and Lucas numbers are obtained. Also, Binet's formulas, D'Ocagne's, Cassini's and Catalan's identities are calculated for these type of numbers.

This work is intended to introduce the theories of dual-generalized complex and hyperbolic-generalized complex numbers. We define dual-generalized complex and hyperbolic-generalized complex valued functions and examine different matrix representations of these numbers. By using similar thought, the theories can be given for the set of complex-gener...

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,...

In this paper, firstly, our purpose is to give the relationship among the densities of the sets of collinear points, the relationship among the densities of the sets of noncollinear points, and the relationship among the densities of the sets of the intersecting lines in Euclidean plane, respectively. In addition to that, we define the density form...

Metallic ratio is a root of the simple quadratic equation x2 = kx + 1 for k is any positive integer which is the characteristic equation of the recurrence relation of k‐Fibonacci (k‐Lucas) numbers. This paper is about the metallic ratio in . We define k‐Fibonacci and k‐Lucas numbers in , and we show that metallic ratio can be calculated in if and o...

In this paper, using Darboux frame {T, g, n} of ruled surface phi(s,v), the evolute offsets phi* (s, v) with Darboux frame {T* ,g*, n*} of phi(s,v) are defined. The striction curve, distribution parameter and orthogonal trajectory of phi* (s, v) are investigated by using the Darboux frame {T, g, n} . The distribution parameters of ruled surfaces ph...

Bu çalışmada, ilk olarak, reel kuaterniyon
matrislerin kümesinin reel matris halkası
üzerinde boyutlu bir modül olduğu
ve kompleks matris halkası üzerinde boyutlu bir modül olduğu
gösterilmiştir. Ayrıca, reel kuaterniyon matrislerin bazı yeni özellikleri
tanımlanmıştır. Daha sonra, reel kuaterniyon matrislerin matris temsilleri
Matlab uygulamaları...

In this paper, by taking into account the beginning of the hypersurface theory in Euclidean space $E^4$, a practical method for the matrix of the Weingarten map (or the shape operator) of an oriented hypersurface $M^3$ in $E^4$ is obtained. By taking this efficient method, it is possible to study of the hypersurface theory in $E^4$ which is analog...

After a brief review of the different types of quaternions, we develop a new perspective for dual quaternions with dividing two parts. Due to this new perspective, we will define the isotropic and nonisotropic dual quaternions. Then we will also give the basic algebraic concepts about the dual quaternions. Moreover, we define isotropic dual quatern...

In this paper, using Darboux frame {T, g, n} of ruled surface phi(s,v), Mannheim offsets phi (s, v) with Darboux frame {T, g, n} of phi(s,v) are defined. The striction curve, distribution parameter and orthogonal trajectory of phi (s, v) are investigated by using the Darboux frame {T, g, n} . The distribution parameters of ruled surfaces phi(T),phi...

The aim of this study is to view the role of Bézier curves in both the Euclidean plane E 2 and Euclidean space E 3 with the help of the fundamental algorithm which is commonly used in Computer Science and Applied Mathematics and without this algorithm. The Serret-Frenet elements of non-unit speed curves in the Euclidean plane E 2 and Euclidean spac...

We examined the moving coordinate systems, the polar axes, the density invariance of the polar axis transformation, and the curve plotter points and the support function of the two-parameter planar Lorentzian motion. Furthermore, we were concerned with the determination of the motion using the polar axes and analyzed the motion when the density of...

In the literature, Holditch theorem was obtained under periodic rotation and translation motions in [H. Holditch, Geometrical theorem, Q. J. Pure Appl. Math. 2 (1858) 38] or periodic shear and translation motions in [O. Röschel, Der satz von Holditch in der isotropen ebene. Abh. Braunschweig. Wiss. Ges. 36 (1984) 27–32]. In this paper, by introduci...

In this study, using Darboux frame \({\left\{ \mathbf{T},\mathbf{g},\mathbf{n}
\right\}}\) of ruled surface \({\varphi(s,v)}\), Bertrand offset \({\varphi^*(s,v) }\) of ruled surface \({\varphi(s,v)}\) is identified. Characteristic properties of Bertrand mates of ruled surfaces as a striction curve, distribution parameter and orthogonal trajectorie...

The purpose of this paper is to obtain De Moivre’s and Euler’s formulas for the matrices of octonions, and to find the powers and the roots of these matrices. A simple method is provided to find the powers of \(8\times 8\,\) matrices with the help of De Moivre’s formula obtained from the matrices of octonions. We also show that any powers of these...

In this study, a brief summary about quaternions and quaternionic curves are firstly presented. Also, the definition of focal curve is given. The focal curve of a smooth curve consists of the centers of its osculating hypersphere. By using this definition and the quaternionic osculating hyperspheres of these curves, the quaternionic focal curves in...

The Jacobsthal quaternions defined by Szynal-Liana and Wloch [35]. In this paper, we defined some properties of Jacobsthal quaternions. Also, we investigated the relations between the Jacobsthal quaternions which connected with Jacobsthal and Jacobsthal-Lucas numbers. Furthermore, we gave the Binet formulas and Cassini identities for these quaterni...

The one parameter planar hyperbolic homothetic motion was introduced in Ersoy and Akyigit (Adv Appl Clifford Algebras 21:297–313, 2011). We give a formula for higher order accelerations and poles under this motion. In the case of the homothetic rate h≡ 1 we obtain the higher order accelerations and poles under one parameter planar hyperbolic motion...

In this study, we determine TN-Smarandache curves whose posi-
tion vector is composed by Frenet frame vectors of another regular curve in
Minkowski 3-space R31
. Then, we present some characterisations of Smaran-
dache curves and calculate Frenet invariants of these curves. Moreover, we
classify TN;TB;NB and TNB-Smarandache curves of a regular curv...

Dual number coefficient octonion (DNCO) is one of the kind of octonion, it has 16 components with an additional dual unit (Formula presented.). Starting with DNCO algebra, we develop the generalized electromagnetic field equations of dyons regarding the DNCOS spaces, which has two octonionic space-times namely the octonionic internal space-time and...

Dual Fibonacci quaternions are first defined by [5]. In fact, these quaternions must be called as dual coefficient Fibonacci quaternions. In this paper, dual Fibonacci quaternions are redefined by using the dual quaternions given in [16]. Since the generalization of the complex numbers is the real quaternions, the generalization of the dual numbers...

In this study, the osculating curves in Euclidean space E3 and E4, well known in differential geometry, are studied through the instrumentality of quaternions. We inoculate sundry delineations for quaternionic osculating curves in the Euclidean space E3, then we portray the quaternionic osculating curve in E4 as a quaternionic curve whose position...

In this article, we consider a base curve, a rolling curve and a roulette on complex plane. We investigate the third one when any two of the base curve, the rolling curve, and a roulette are known. We obtain Euler Savary’s formula, which gives the relation between the curvatures of these three curves.

In this paper, we defined the generalized dual Fibonacci quaternions. Also, we investigated the relations between different generalized dual Fibonacci quaternions. Furthermore, we gave the Binet’s formulas and Cassini identities for these quaternions.

In this paper, we introduce one-parameter homothetic motions in the generalized complex number plane (\({\mathfrak{p}}\)-complex plane)
$$\mathbb{C}_{J}=\left\{x+Jy:\,\,\, x,y \in \mathbb{R},\quad J^2=\mathfrak{p},\quad \mathfrak{p} \in \{-1,0,1\} \right\} \subset \mathbb{C}_\mathfrak{p}$$where
$$\mathbb{C}_\mathfrak{p}=\{x+Jy:\,\,\, x,y \in \mathb...

In this study, we have given new characteristics results about the pitch and angle
of the pitch which are the integral invariants of the involute evolute
offsets of ruled surface with geodesic Frenet frame.

In this study, the ruled surface with Darboux frame is defined. Then, the ruled surfaces
characteristic properties which are related to the geodesic curvature, the normal curvature and
the geodesic torsion are investigated. The relation between the Darboux frame and the Frenet
frame on the ruled surface is presented. Moreover, some theorems about t...

In this study, we firstly give the basic notations of the generalized complex number plane (\({\mathfrak{p}}\)-complex plane) \({\mathbb{C}_{\mathfrak{p}}}\). Then, we introduce the one-parameter planar motions in \({\mathfrak{p}}\)-complex plane \({\mathbb{C}_{J}}\) such that \({\mathbb{C}_{J} \subset \mathbb{C}_{\mathfrak{p}}}\). These motions co...

In this paper, we introduce one-parameter homothetic motions in the generalized complex number plane (p-complex plane) CJ = x + Jy : x, y ∈ R, J 2 = p, p ∈ {−1, 0, 1} ⊂ Cp where Cp = {x + Jy : x, y ∈ R, J 2 = p} such that −∞ < p < ∞. The velocities, accelerations and pole points of the motion are analysed. Moreover, three generalized complex number...

In this work, a base curve, a rolling curve and a roulette in generalized complex number plane (p-complex plane) CJ = x + Jy : x, y ∈ R, J 2 = p, p ∈ {−1, 0, 1} ⊂ Cp where Cp = {x + Jy : x, y ∈ R, J 2 = p} such that −∞ < p < ∞, is taken into consideration and Euler Savary's formula which gives the relation between the curvatures of these three curv...

In this paper, firstly, we define principal direction curve and binormal direction curve of a given Frenet curve by using integral curves of the Frenet vector field. Besides, we introduce W-direction curve and W-rectifying curve of a Frenet curve in 3-dimensional Galilean space G3 by using the unit Darboux vector field W of the Frenet curve and giv...

In this study, we describe octonionic inclined curves and harmonic curvatures for the octonionic curves. We give a characterizations for an octonionic curve to be an octonionic inclined curve. And finally, we obtain some characterisations for the octonionic inclined curves in terms of the harmonic curvatures.

In this study, we treate a subject is named as spatial real octonionic
curves (SROC) in R7 and real octonionic curves (ROC) in R8. First of
all, we single out the spatial real octonionic curves (SROC)' Serret Frenet
equations and curvatures in R7. Thereafter, Serret Frenet equations for
the real octonionic curves (ROC) in R8 are calculated by doing...

In this study, we have approached the issue of how to determine prop-
erties of curve for the special octonionic curves (octonionic involute evolute
curves, octonionic Bertrand curves, octonionic Smarandache curves and
octonionic Mannheim curves) by means of real octonions in Euclidean 8-
space. Firstly, the title of the special octonionic curves h...

In this study, We identify the instantaneous Pfaffian vector of the closed motion which define along the striction curve of the ruled surface $\varphi^*$ which Mannheim offset of closed ruled surface $\varphi$ in $\mathbb{E}^3$. Using this vector, we get the Steiner rotation vector and the Steiner translation vector of this motion and give new char...

We expressed the higher-order velocities, accelerations, and poles under the one-parameter planar hyperbolic motions and their inverse motions. The higher-order accelerations and poles are also presented by considering the rotation angle as a parameter of the motion and its inverse motion.
1. Introduction
First of all, we need to define the set of...

Determination of the viewpoint of academic staff and graduate students in teaching geometry, which is an essential component of mathematics, has great importance. From this point of view, the aim of this study is to investigate views of academic staff and graduate students on methods such as technology, material, projects, teaching of arithmetic, a...

As it is well known to all , the real quaternion division algebra H is isomorphic to a 4×4 real matrix algebra. But the real split octonion algebra O_{S} can not be isomorphic to any matrix algebras over the real number field, because O_{S} is a nonassociative algebra over R. The split octonions O_{S} are an enlargement of the quaternions (or the s...

Müller, H. R. [2], on the Euclidean plane E2, introduced the one-parameter planar motion and obtained the relation between absolute, relative, sliding velocities (and accelerations). Also, Müller, H. R. [6] provided the relation between the velocities (in the sense of Complex) under the one-parameter motion on the Complex plane ℂ = {x + iy|x, y ∈ℝ,...

In this study, we specify some special Smarandache curves with reference to Darboux frame in Euclidean 3-space We dispose certain particularizations and outcomes about Smarandache curves. However we defray an example about our study

The aim of this paper is to obtain De Moivre's and Euler's formulas for matrices of octonion, and to find powers and roots of these matrices. Moreover, we give a new method to find powers of 8×8 matrices by the help of De Moivre's formulas obtaining from matrices of octonion, and show that all powers of matrices in relation with this matrices can b...

In this paper, we take into account the opinion of involute-evolute curves
which lie on fully surfaces and by taking into account the Darboux frames of
them we illustrate these curves as special involute-evolute partner D-curves in
E3. Besides, we find the relations between the normal curvatures, the geodesic
curvatures and the geodesic torsions of...

In this study, we determine some special Smarandache curves in E1 3.We give
some characterizations consequences of Smarandache curves.

In this study, we determine some special Smarandache curves according to
Darboux frame in E3. We give some characterizations and consequences of
Smarandache curves.

In this paper, we consider split quaternions and split quaternion matrices. Firstly, we give some properties of split quaternions. After that we investigate split quaternion matrices using properties of complex matrices. Then we define the complex adjoint matrix of split quaternion matrices and we describe some of their properties. Furthermore, we...

The Steiner formula and the Holditch Theorem for one-parameter closed planar Euclidean motions [1, 7] were expressed by H.R.
Müller [9] under the one-parameter closed planar motions in the complex sense.
In this paper, in analogy with complex motions as given by Müller [9], the Steiner formula and the mixture area formula are
obtained under one pa...

In this article, we give the area formula of the closed projection curve of a closed space curve in Lorentzian 3-space L3. For the 1-parameter closed Lorentzian space motion in L3, we obtain a Holditch Theorem taking into account the Lorentzian matrix multiplication for the closed space curves by using their othogonal projections onto the Euclidean...

In this paper, after presenting a summary of the one-parameter planar homothetic motion on the complex plane given by N. Kuruoğlu et al. [Int. J. Appl. Math. 6, No. 4, 439–447 (2001; Zbl 1034.70001)], high-order velocities, accelerations and poles are analyzed under the one-parameter homothetic motions in the complex plane. Also, high-order velocit...

. In the Lorentzian plane, we give Cauchy-length formulas to the envelope of a family of lines. Using these, we prove the length
of the enveloping trajectories of non-null lines under the planar Lorentzian motions and give the Holditch-type theorems for
the length of the enveloping trajectories. Furthermore, Holditch-type theorem for the orbit area...

A generalization of the theory of involute-evolute curves is presented for ruled surfaces based on line geometry. Using lines instead of points, two ruled surfaces which are offset in the sense of involute-evolute are defined. Moreover, the results are clarified using computer-aided examples.

Müller [3], in the Euclidean plane \({{\mathbb{E}}}^2\), introduced the one parameter planar motions and obtained the relation between absolute, relative, sliding velocities (and accelerations). Also, Müller [11] provided the relation between the velocities (in the sense of Complex) under the one parameter motions in the Complex plane \({\mathbb{C}...

The polar moment of inertia of a closed orbit curve of a point under one-parameter closed planar motions was studied by H. R. Müller, [1]. We study the polar moment of inertia of the enveloping curve of a line under one-parameter closed homothetic motions of planar kinematics.

The present paper is concerned with the generalization of the Holditch Theorem under one- parameter homothetic motion on Lorentzian planes. In this paper, for the homothetic Lorentzian motion, we expressed the Steiner formula. Furthermore, we present the Holditch-Type Theorems.

A. Tutar and N. Kuruoǧlu [1] had given the following theorem as a generalization of the classical Holditch Theorem [2]: During the closed planar homothetic motions with the period T, if the chord AB of fixed lenght a + b is moved around once on an oval k0, then a point X ∈ AB̄ (a = AX̄, b = BX̄) describes a closed path k0(X) and the "Holditch Ring"...

In this study, we first compute the polar moment of inertia of orbit curves under planar Lorentzian motions and then give the following theorems for the Lorentzian circles: When endpoints of a line segment AB with length a +b move on Lorentzian circle (its total rotation angle is ÃŽÂ´) with the polar moment of inertia T, a point X which is collinea...

In this paper, we present the Steiner formula for one-parameter open planar homothetic motions. Using this area formula, the generalization of the Holditch Theorem given by W. Blaschke and H. R. Muller (4, p.142) is expressed during one-parameter open planar homothetic motions. Furthermore, we obtain another formula for the swept surface area.

In this paper, we present the Steiner formula for one-parameter open planar homothetic motions. Using this area formula, the generalization of the Holditch Theorem given by W. Blaschke and H. R. Müller [4, p.142] is expressed during one-parameter open planar homothetic motions. Furthermore, we obtain another formula for the swept surface area.

In this paper, under the one-parameter closed planar homothetic motion, a generalization of Holditch Theorem is obtained by using two different line segments (with fixed lengths) whose endpoints move along two different closed curves.

In this paper we give some general results for the polar inertia momentums, given by H. R. Müller [Abh. Braunschw. Wiss. Ges. 29, 115–119 (1978; Zbl 0485.53009)], of the closed orbit curves obtained during the one-parameter closed planar motions.

Holditch-type theorems for the length of the enveloping trajectories are given under the one-parameter closed planar homothetic motions.

W. Blaschke and H. R. Müller [4, p. 142] have given the following theorem as a generalization of the classic Holditch Theorem: Let E/E′ be a 1-parameter closed planar Euclidean motion with the rotation number ν and the period T. Under the motion E/E′, let two points A = (0, 0), B = (a + b, 0) ∈ E trace the curves k
A, k
B ⊂ E′ and let F
A, F
B be t...

It is clearly that, If f : E3 → E3 is a homothety and φ is a surface in three dimensional Euclidean space E3, f(φ) = φ̄ is also a surface in E3. In this paper, the surface φ has been taken a ruled surface, specially. It was shown that image surface f(φ) = φ̄ has been a ruled surface, too. Furthermore, It was investigated whether some properties rel...