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The purpose of this study was to characterize the growth of one student’s mathematical understanding and use of different representations about a geometric transformation, dilation. We accomplished this purpose by using the Pirie-Kieren model jointly with the Semiotic Representation Theory as a lens. Elif, a 10th- grade student, was purposefully chosen as the case for this study because of the growth of mathematical understanding about dilation she exhibited over time. Elif participated in task-based interviews before, during and after participating in a variety of transformation lessons where she used multiple representations, including physical and virtual manipulatives. The results revealed that Elif was able to progress in her mathematical understanding from informal levels to the formal levels in the Pirie-Kieren model as she performed treatments and conversions, movements involving different registers of representations. The results also showed numerous examples of Elif’s mathematical understanding based on folding back activities, complementary aspects of acting and expressing, and interventions. Using the two theories together provides a powerful and holistic approach to a deeper understanding of mathematical learning by characterizing and articulating the growth of mathematical understanding and the way of mathematical thinking.

This paper presents a dual method to study motion of a robot end-effector by using the curvature theory of a dual curve which corresponds to a ruled surface generated by a line fixed in the end-effector. By using dual method, translational and angular differential properties of motion such as velocity and acceleration are determined without redundant parameters. These properties are important information in robot trajectory planning. As a practical example, motion of a robot end-effector in which a line fixed in the end-effector generates a surface of hyperbolic paraboloid is investigated.

Bu calismada, Lorentziyen uzayda hareket eden bir robot uc-islevcinin hareketi, robot uc-islevciye sabitlenmis bir dogru tarafindan olusturulan spacelike regle yuzeye dual uzayda karsilik gelen dualLorentziyen birim kuresel timelike egrilerin egrilik teorisi kullanilarak incelenmistir. Bu inceleme ile robot yorunge planlamasinda onemli rol oynayan robot uc-islevcinin zamana bagli lineer ve acisal hizi ile lineer ve acisal ivmesi belirlenmistir.

In this paper, using the classifications of timelike and spacelike ruled surfaces, we define and study the Mannheim offsets of spacelike ruled surfaces in Minkowski 3-space. We give the conditions for offset surfaces to be developable. © 2017, Shiraz University.

Classification is one of the most researched issues in Machine Learning. In this study, the Lorentzian Support Vector Machine (LSVM) method is proposed that performs classification in Lorentzian space. This proposed new classifier forms a hyperplane separating the classes based on the Lorentzian metric and maximize margins between nearest points to the hyperplane according to the Lorentzian distance. Thus, samples from different classes are classified in Lorentzian space. Also, for the purpose of increasing the classification accuracy, a pre-preprocessing is applied. Experimental results taken from LSVT, SONAR, TELESCOPE and WISCONSIN data sets validate the proposed LSVM method.

In this paper, using the classifications of timelike and spacelike ruled surfaces, we study the Mannheim offsets of timelike ruled surfaces in the Minkowski 3-space. First, we define the Mannheim offsets of a timelike ruled surface by considering the Lorentzian casual character of the offset surface. We obtain that the Lorentzian casual character of the Mannheim offset of a timelike ruled surface may be timelike or spacelike. Furthermore, we give characterizations for developable Mannheim offsets of a timelike ruled surface.

In this study, we obtain the basic equations of a point trajectory in Lorentzian spatial motion. We derive the invariants of Lorentzian axodes and their kinematic meaning. The instantaneous properties of a point trajectory are discussed and the Euler-Savary analogue equations of a point trajectory are set up in Lorentzian spatial motion.

In this study, we give the dual characterizations of Mannheim offsets of ruled surfaces in terms of their integral invariants and obtain a new characterization of the Mannheim offsets of developable surface, i.e., we show that the striction lines of developable Mannheim offset surfaces are Mannheim partner curves. Furthermore, we obtain the relationships between the area of projections of spherical images for Mannheim offsets of ruled surfaces and their integral invariants.

In this paper, we consider the idea of Bertrand partner curves for curves lying on surfaces and by considering the Darboux frames of surface curves, we call these curves as Bertrand partner D ‐curves and give the characterizations for these curves by means of the geodesic curvatures, the normal curvatures and the geodesic torsions of these associated curves. Bu çalışmada, Bertrand partner eğrileri düşüncesi yüzey üzerinde yatan eğriler için ele alınmış ve yüzey eğrilerinin Darboux çatıları dikkate alınarak bu eğriler Bertrand partner D ‐eğrileri olarak adlandırılmıştır. Bu eğrilerin karakterizasyonları, bağlantılı eğrilerin geodezik eğriliklerine, normal eğriliklerine ve geodezik burulmalarına göre verilmiştir.

Planar and spherical elliptic motions had been introduced by Karger and Novak [A. Karger, J. Novak, Space Kinematics and Lie Groups, Gordon and Breach Science Publishers, New York, London, 1985]. Also, a dual spherical elliptic motion has been defined by Yapar [Z. Yapar, Spatial conchoidal and elliptic motions, Mechanism and Machine Theory 27 (1) (1992) 75–91]. In this paper, we define the elliptic motion on a dual hyperbolic unit sphere H ∼ 0 2 in the dual Lorentzian space D 1 3 with dual signature (+, +, −) and carry the results to the Lorentzian line space IR 1 3 by means of Study’s mapping.

Smarandache Curves of a Spacelike Curve lying on Unit dual Lorentzian Sphere In this paper, we give Darboux approximation for dual Smarandache curves of spacelike curve on unit dual Lorentzian sphere . Firstly, we define the types of dual Smarandache curves of a spacelike curve lying on dual Lorentzian sphere . Then, we obtain the relationships between the dual curvatures of spacelike curve and its dual Smarandache curves. Finally, we give an example for Smarandache curves of a spacelike curve on unit dual Lorentzian sphere .

In this study, we propose a new algorithm which works in Lorentzian space with a similar sense in the k-NN method. We exploit the distance metric of Lorentzian space in classification problem. It is a special metric which may give a zero distance for far points. To take best benefit from structural and other properties of the Lorentzian space, a special projection over the data sets is applied. By this projection, basic geometrical operations are used; namely translation (shifting), compression and rotation. Our new algorithm does classification according to the nearest neighbor in Lorentzian space. The usability and validity of the proposed classification method is tested by some public data sets such as WHOLE, VERTEBRAL, RELAX, ECOLI. The results are compared with results of well-known classical classification methods such as kNN, LDA, SVM and Bayes. As a result, our proposed algorithm produces more successful results.

In this paper, we give Darboux approximation for dual Smarandache curves of time like curve on unit dual Lorentzian sphere. Firstly, we define the four types of dual Smarandache curves of a timelike curve lying on dual Lorentzian sphere.

In this paper we study the motion of a robot end-effector by using the curvature theory of a dual unit hyperbolic spherical curve which corresponds to a timelike ruled surface with timelike ruling generated by a line fixed in the end-effector. In this way, the linear and angular differential properties of the motion of a robot end-effector such as velocities and accelerations which are important information in robot trajectory planning are determined. Moreover, the motion of a robot end-effector which moves on the surface of a right circular hyperboloid of one sheet is examined as a practical example.

In this paper, we give Darboux approximation for dual Smarandache curves of spacelike curve on dual unit hyperbolic sphere (H) over tilde (2)(0). Firstly, we define the four types of dual Smarandache curves of a dual hyperbolic curve (alpha) over tilde (s). Then, we obtain the relationships between the dual curvatures of dual hyperbolic curve (alpha) over tilde (s) and its dual Smarandache curves. Finally, we give an example for Smarandache curves of a spacelike curve on dual unit hyperbolic sphere (H) over tilde (2)(0).

In this work, we proved the Cosine Rule II for a spherical triangle on the dual unit sphere

In this paper Disteli diagram is introduced for hyperbolic spatial motion by considering the E. Study mapping of timelike lines. Disteli’s diagram of pure rotations is obtained on a circle of Lorentzian space and also given for the general case. The distribution parameters of the axodes are obtained in Disteli’s diagram.

In this paper, we introduce the dual geodesic trihedron (dual Darboux frame) of a timelike ruled surface. By the aid of the E. Study Mapping, we consider timelike ruled surfaces as dual hyperbolic spherical curves and define the Mannheim offsets of timelike ruled surfaces by means of dual Darboux frame. We obtain the relationships between invariants of Mannheim timelike surface offsets. Furthermore, we give the conditions for these surface offsets to be developable.

Lorentzian geometry is a subject of mathematics and has famous applications in physics, especially in relativity theory. This geometry has interesting features, e. g. one axis has a negative sign in metric definition (time axis). In this study, we try to apply Lorentzian geometry for feature extraction and dimensionality reduction. We use a Lorentzian Manifold (LM) for face recognition and reduce the dimensionality in this new feature space. We compare results with different feature extraction methods; Principal Component Analysis (PCA), Linear Discriminant Analysis (LDA) and Locality Preserving Projection (LPP). Our experiments show that the best feature extraction method is LM and it produces the best face recognition rates. It is also powerful in dimensionality reduction.

In this paper, depending on the Darboux instantaneous rotation vector of a solid perpendicular trihedron in the Minkowski 3-space [...]

In this study, a fundamental relation, as a base for the geometry of the time-like surfaces, among the Darboux vectors of an arbitrary time-like curve (c) on a time-like surface and the parameter curves (c1) and (c2) in the Minkowski 3-space R13 was founded.

In this paper, we define dual geodesic trihedron(dual Darboux frame) of a spacelike ruled surface. Then, we study Mannheim offsets of spacelike ruled surfaces in dual Lorentzian space by considering the E. Study Mapping. We represent spacelike ruled surfaces by dual Lorentzian unit spherical curves and define Mannheim offsets of the spacelike ruled surfaces by means of dual Darboux frame. We obtain relationships between the invariants of Mannheim spacelike offset surfaces and offset angle, offset distance. Furthermore, we give conditions for these surface offsets to be developable., 13 pages. arXiv admin note: substantial text overlap with arXiv:1108.6076

In this paper, we give the characterizations for Mannheim offsets of timelike ruled surfaces with spacelike rulings in dual Lorentzian space [Formula: see text]. We obtain the relations between terms of their integral invariants and also we give new characterization for the Mannheim offsets of developable timelike ruled surface. Moreover, we find relations between the area of projections of spherical images for Mannheim offsets of timelike ruled surfaces and their integral invariants.

In this work, we state and prove the sine, cosine I, cosine II, sine-cosine and cotangent rules for spherical triangles on the hyperbolic unit sphere H02 in the Lorentzian space R13.

The formulae for a time-like surface, given by Enneper, O. Bonnet, Euler and Liouville, were obtained by nsing the hyperbolic angle between time-like curves on the surface

In this study, the E. Study mapping was dermed for the space-like and time-like lines in the Minkowski 3-space R13. Hence, there is one to one correspondence between directed space-like (resp., time-like) lines of R13 and ordered pair of vectors (a,a0.) such that < a,a> = 1 (resp., < a,a> = -1) and = 0.

In this paper, we give some characterizations of Mannheim partner curves in the Minkowski 3-space E-1(3). Firstly, we classify these curves in E-1(3). Next, we give some relationships characterizing these curves and we show that the Mannheim theorem is not valid for the Mannheim partner curves in E-1(3). Moreover, by considering the spherical indicatrix of the Frenet vectors of those curves, we obtain some new relationships between the curvatures and torsions of the Mannheim partner curves in E-1(3).

In this study we give the Frenet frames and Frenet invariants of timelike ruled surfaces. We show that a timelike ruled surface and its directing cone have the same base of Frenet frame. We define instantaneous rotation vectors of the Frenet frames of timelike ruled surfaces. Also we prove the Chasles Theorem for timelike ruled surfaces.

Bu araştırmanın amacı bir grup ortaöğretim matematik öğretmen adayının kendi programlarında verilen geometri derslerinden ne ölçüde yararlandıkları ve bu dersler ile ortaöğretimde kendileri öğretmen olduklarında okutacak oldukları dersler arasındaki benzerlik ve farklılıkların ortaya çıkarılmasıdır. Katılımcılar Türkiye'deki bir büyükşehir devlet üniversitesinin Ortaöğretim Fen ve Matematik Alanları Eğitimi Bölümü Matematik Öğretmenliği programında lisans eğitimi alan beş öğretmen adayından oluşmaktadır. Verilerin toplanması amacıyla katılımcılara 1985-2006 yılları arasında Öğrenci Seçme Sınavında (ÖSS) çıkmış sorulardan oluşan çoktan seçmeli bir test uygulanmış, daha sonra birebir görüşmeler yapılarak bu soruları çözerken, işlenen derslerden ne ölçüde yararlandıkları konusunda görüşleri alınmıştır. Veriler gömülü teori kapsamındaki tekniklerden biri olan açık kodlama yöntemi ile analiz edilmiştir. Bu çerçevede öğretmen adaylarıyla yapılan görüşmeler bulguların güvenilirliği açısından, araştırmacı ve aynı üniversitede yüksek lisans yapan bir arkadaşı tarafından cümle cümle kodlanarak benzer veriler bir araya getirilmiştir. Bundan sonra yapılan görüşmelerden elde edilen veriler benzerlik ve farklılıklarına göre boyutları ile birlikte gruplanmış ve bunların sonucunda kategorilere ulaşılmıştır. Bu kategoriler de birbirleriyle ve kendi aralarında karşılaştırılarak aşağıdaki bulgulara ulaşılmıştır: 1. Öğretmen adayları üniversitede verilen derslerin fazla akademik olduğunu ve uygulamaya gereken önemin verilmediğini düşünmektedir. 2. Öğretmen adayları üniversitede okutulan geometri dersleri ile ortaöğretimde öğretmen olunca işleyecekleri dersler arasında yeterli ölçüde bağlantı kuramamaktadır. 3. Öğretmen adayları ortaöğretim geometrisinde yer alan temel kavramların nasıl öğretileceği konusunda eğitim almak istemektedir. Son olarak, bu bulgular yardımıyla eğitim süresince karşılaşılan zorlukların giderilmesi için bazı öneriler sunulmuştur. Bu önerilerden ön plana çıkan birkaç tanesi şu şekildedir: 1. Eğitim Fakültelerinin Matematik Öğretmenliği bölümünde öğrencilere verilen geometri derslerinin içerik olarak gözden geçirilmesi tavsiye edilebilir. 2. Geometri derslerinin çok ağır ve akademik olmaktan çıkarılıp uygulamaya yönelik derslere ağırlık verilmesi tavsiye edilir. 3. Öğretmen adaylarına Ortaöğretim geometrisindeki bazı temel kavramların öğretimi ile ilgili bir ders verilebilir.