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Öğe BETCHOV-DA RIOS EQUATION BY NULL CARTAN, PSEUDO NULL AND PARTIALLY NULL CURVE IN MINKOWSKI SPACETIME(Korean Mathematical Soc, 2023) Erdogdu, Melek; Li, Yanlin; Yavuz, AyseThe aim of this paper is to investigate Betchov-Da Rios equation by using null Cartan, pseudo null and partially null curve in Minkowski spacetime. Time derivative formulas of frame of s parameter null Cartan, pseudo null and partially null curve are examined, respectively. By using the obtained derivative formulas, new results are given about the solution of Betchov-Da Rios equation. The differential geomet-ric properties of these solutions are obtained with respect to Lorentzian causal character of s parameter curve. For a solution of Betchov-Da Rios equation, it is seen that null Cartan s parameter curves are space curves in three-dimensional Minkowski space. Then all points of the soliton sur-face are flat points of the surface for null Cartan and partially null curve. Thus, it is seen from the results obtained that there is no surface corresponding to the solution of Betchov-Da Rios equation by using the pseudo null s parameter curve.Öğe Congruence of degenerate surface along pseudo null curve and Landau-Lifshitz equation(Elsevier, 2022) Yavuz, Ayse; Erdogdu, MelekThis paper is devoted to the geometry of pseudo null curve in terms of anholonomic coordinates in Minkowski space. Firstly, extended Frenet formulas for pseudo null curves are deeply discussed. Then binormal congruence of degenerate surfaces containing the s - lines and n - lines are investigated with the condition mu(b) = 0. This condition represents the necessary and sufficient condition for the existence of a one-parameter family of surfaces containing the s - lines and n - lines. Moreover, normal congruence of surfaces containing the s - lines and b - lines are examined with the condition mu(n) = 0. Again, the condition represents the necessary and sufficient condition for the existence of a one-parameter family of surfaces containing the s - lines and b - lines. Considering the compatibility conditions, Gauss-Mainardi-Codazzi equations are obtained for this binormal and congruence of surfaces, respectively. Finally, some relations are given for constructing the moving pseudo null curve with using the integrable Landau-Lifshitz equation. (C) 2022 Elsevier B.V. All rights reserved.Öğe CONSTRUCTION OF BINORMAL MOTION AND CHARACTERIZATION OF CURVES ON SURFACE BY SYSTEM OF DIFFERENTIAL EQUATIONS FOR POSITION VECTOR(Editura Bibliotheca-Bibliotheca Publ House, 2021) Yavuz, AyseThe main purpose of this paper is to investigate unit speed curve with constant geodesic, normal curvature and geodesic torsion of curve on a surface in the Euclidean 3-space. In accordance with this scope, the position vector of a curve is stated by a linear combination of its Darboux Frame with differentiable functions. Some special results have been obtained within the scope of this position curve and differentiable functions. As a physical application of obtained results, differential geometric properties of a surface with binormal motion of a given curve are given with the obtained characterization of the curve.Öğe A Different Approach by System of Differential Equations for the Characterization Position Vector of Spacelike Curves(Univ Punjab, Dept Mathematics, 2021) Yavuz, Ayse; Erdogdu, MelekThe purpose of this study is to obtain a characterization of unit speed spacelike curve with constant curvature and torsion in the Minkowski 3-space. According to this purpose, the position vector of a spacelike curve is expressed by a linear combination of its Serret Frenet Frame with differentiable functions. Since a spacelike curve has different kinds of frames, then we investigate the curve with respect to the Lorentzian casual characterizations of the frame. Hence we examine the results in three different cases including different subcases. Moreover, we illustrate some examples for each case.Öğe DIFFERENTIAL GEOMETRIC ASPECTS OF NONLINEAR SCHRODINGER EQUATION(Ankara Univ, Fac Sci, 2021) Erdogdu, Melek; Yavuz, AyseThe main scope of this paper is to examine the smoke ring (or vortex flament) equation which can be viewed as a dynamical system on the space curve in E-3: The differential geometric properties the soliton surface associated with Nonlinear Schrodinger (NLS) equation, which is called NLS surface or Hasimoto surface, are investigated by using Darboux frame. Moreover, Gaussian and mean curvature of Hasimoto surface are found in terms of Darboux curvatures k(n), k(g) and tau(g). Then, we give a different proof of that the s-parameter curves of NLS surface are the geodesics of the soliton surface. As applications we examine two NLS surfaces with Darboux Frame.Öğe A Dynamical Approach to Position Vector of Timelike Curve by Vectorial Momentum, Torque and Tangential Dual Curve(Springernature, 2022) Yavuz, AyseIn this study, the position vector of a timelike curve p is stated by a linear combination of its Serret Frenet frame with differentiable functions. The definition of tangential dual curve of the curve p is stated by using these differentiable functions. Moreover, tangential torque curve of timelike curve p is defined and investigated. New dynamically and physical results are stated depending on the torque of the timelike curve p and the direction of the tangent vector component of the curve. Then, the position vector of a timelike W curve is again stated by differentiable functions. Therefore, solutions of differential equation of the position vector of timelike W curve with two different types depending on the values of curvature and torsion of timelike curve are obtained. By using the differentiable functions obtained as a result of these solutions, tangential dual and torque curve of the timelike W curve are obtained. Depending on the tangential dual and torque curve of the timelike W curve, results are given for two different cases separately.Öğe Fracture Resistance of CAD/CAM Monolithic Zirconia Crowns Supported by Titanium and Ti-Base Abutments: The Effect of Chewing Simulation and Thermocyclic Aging(Quintessence Publishing Co Inc, 2023) Yavuz, Ayse; Buyukerkmen, Emine BeguemPurpose: To evaluate the effect of chewing simulation and thermocyclic aging on the fracture resistance of CAD/CAM monolithic zirconia crowns supported by titanium and Ti-base abutments. Materials and Methods: Two implant abutment groups-titanium (Ti) and titanium base (Ti-base; Medentika)-were used. A total of 40 mandibular first molar CAD/CAM monolithic zirconia crowns (Vita YZ T) were fabricated, then cemented onto the abutments with Panavia V5. Each abutment group was divided into two subgroups (n = 10). The Ti and Ti-base groups were subjected to a single load until fracture, and the Ti/CT and Ti-base/CT groups (CT: chewing simulation and thermocyclic aging) underwent chewing simulation (1.2 x 10(6) cycles x 50 N load, 1.4 Hz) and thermocylic aging (3,911 cycles/5 degrees C to 55 degrees C). The fracture resistances of the crowns were tested with a universal testing machine (1 mm/minute). Shapiro-Wilk and one-way ANOVA test were used for statistical analysis (P =.05). Results: The survival rates after chewing simulation and thermocyclic aging were 100% for both CT groups. The fracture resistance values (mean +/- SD) of the groups were as follows: Ti = 1,718.18 +/- 331.06 N, Ti-base = 1,713.53 +/- 233.24 N, Ti/CT = 1,664.82 +/- 188.62 N, and Ti-base/CT = 1,551.28 +/- 344.79 N. According to one-way ANOVA test results, there was no statistically significant difference between the four groups (P =.526). Conclusion: CAD/CAM monolithic zirconia crowns supported by Ti-base or titanium abutments were found to have sufficient fracture resistance in the treatment of an absent single posterior tooth. However, more in vitro and clinical studies are required to evaluate the long-term performance of Ti-base abutments and CAD/CAM zirconia crowns.Öğe A GEOMETRIC APPROACH TO TIMELIKE FLOWS IN TERMS OF ANHOLONOMIC COORDINATES(Honam Mathematical Soc, 2022) Yavuz, Ayse; Erdogdu, MelekThis paper is devoted to the geometry of vector fields and timelike flows in terms of anholonomic coordinates in three dimensional Lorentzian space. We discuss eight parameters which are related by three partial differential equations. Then, it is seen that the curl of tangent vector field does not include any component in the direction of principal normal vector field. This implies the existence of a surface which contains both s - lines and b - lines. Moreover, we examine a normal congruence of timelike surfaces containing the s - lines and b - lines. Considering the compatibility conditions, we obtain the Gauss-Mainardi-Codazzi equations for this normal congruence of timelike surfaces in the case of the abnormality of normal vector field is zero. Intrinsic geometric properties of these normal congruence of timelike surfaces are obtained. We have dealt with important results on these geometric properties.Öğe Harmonic curvatures of the strip in Minkowski space(World Scientific Publ Co Pte Ltd, 2018) Kaya, Filiz Ertem; Yavuz, AyseThis study aimed to give definitions and relations between strip theory and harmonic curvatures of the strip in Minkowski space. Previously, the same was done in Euclidean Space (see [F. Ertem Kaya, Y. Yayli and H. H. Hacisalihoglu, A characterization of cylindrical helix strip, Commun. Fac. Sci. Univ. Ank. Ser. A1 59(2) (2010) 37-51]). The present paper gives for the first time a generic characterization of the harmonic curvatures of the strip, helix strip and inclined strip in Minkowski space.Öğe Non-lightlike Bertrand W curves: A new approach by system of differential equations for position vector(Amer Inst Mathematical Sciences-Aims, 2020) Yavuz, Ayse; Erdogdu, MelekIn this study, the characterization of position vectors belonging to non-lightlike Bertrand W curve mate with constant curvature are obtained depending on differentiable functions. The position vector of Bertrand W curve is stated by a linear combination of its Frenet frame with differentiable functions. There exist also different cases for the curve depending on the value of curvature and torsion. The relationships between Frenet apparatuas of these curves are stated separately for each case. Finally, the timelike and spacelike Bertrand curve mate visualized of given curves as examples, separately.Öğe NONNULL SOLITON SURFACE ASSOCIATED WITH THE BETCHOV-DA RIOS EQUATION(Pergamon-Elsevier Science Ltd, 2022) Li, Yanlin; Erdogdu, Melek; Yavuz, AyseThe aim of this paper is to investigate the nonnull soliton surfaces associated with Betchov-Da Rios equation in Minkowski space-time. The differential geometric properties of these kind of nonnull soliton surfaces are examined with respect to the Lorentzian casual characterizations. Moreover, the linear maps of Weingarten type are obtained which are defined on tangent spaces of these soliton surfaces. Some new results are obtained by means of two geometric invariants ?? and h which are generated by linear maps of Weingarten type. Then, the mean curvature vector field and Gaussian curvature of the nonnull soliton surface are obtained. Finally, it is shown that this kind of soliton surface consists of flat points as a numerical example.Öğe On Backlund transformation and motion of null Cartan curves(World Scientific Publ Co Pte Ltd, 2022) Erdogdu, Melek; Yavuz, AyseThe main scope of this paper is to examine null Cartan curves especially the ones with constant torsion. In accordance with this scope, the position vector of a null Cartan curve is stated by a linear combination of the vector fields of its pseudo-orthogonal frame with differentiable functions. However, the most important difference that distinguishes this study from the other studies is that the Bertrand curve couples (timelike, spacelike or null) of null Cartan curves are also examined. Consequently, it is seen that all kinds of Bertrand couples of a given null Cartan curve with constant curvature functions have also constant curvature functions. This result is the most valuable result of the study, but allows us to introduce a transformation on null Cartan curves. Then, it is proved that aforesaid transformation is a Backlund transformation which is well recognized in modern physics. Moreover, motion of an inextensible null Cartan curve is investigated. By considering time evolution of null Cartan curve, the angular momentum vector is examined. And three different situations are given depending on the character of the angular momentum vector Omega. In the case of tau(t) = 0, we discuss the solution of the system which is obtained by compatibility conditions. Finally, we provide the relation between torsion of the curve and the velocity vector components of the moving curve C.Öğe On differential analysis of spacelike flows on normal congruence of surfaces(Amer Inst Mathematical Sciences-Aims, 2022) Erdogdu, Melek; Yavuz, AyseThe present paper examines the differential analysis of flows on normal congruence of spacelike curves with spacelike normal vector in terms of anholonomic coordinates in three dimensional Lorentzian space. Eight parameters, which are related by three partial differential equations, are discussed. Then, it is seen that the curl of tangent vector field does not include any component with principal normal direction. Thus there exists a surface which contains both s-lines and b - lines. Also, we examine a normal congruence of surfaces containing the s - lines and b - lines. By compatibility conditions, Gauss-Mainardi-Codazzi equations are obtained for this normal congruence of surface. Intrinsic geometric properties of this normal congruence of surfaces arc given.Öğe On normal congruence of surfaces and position vector of optical fiber by electromagnetic wave vectors(Elsevier Gmbh, 2022) Yavuz, AyseThe present paper gives an extraordinary view of the normal congruence of surfaces including the s- lines and b- linesin terms of electromagnetic wave vectors in ordinary space. Frenet- Serret frame of given a space curve are described in E3 in terms of anholonomic coordinates which includes eight parameters. Using the expression the Frenet frames and electromagnetic wave vectors on the curve with a linear transformation in terms of each other, the changes of <(t)over right arrow >, <(E)over right arrow > and <(B)over right arrow > between any two points in the tangential and binormal direction along with the curved path sigma = sigma(s, n, b) are obtained in terms of geometric phase beta, respectively. Moreover, the solution of the systems of differential equations of optical fiber with position vector is obtained. Intrinsic geometric properties of this normal congruence of surfaces are obtained in terms of electromagnetic wave vectors. The conditions under which electromagnetic and magnetic vectors satisfy Maxwell's equations given electric charge and current densities are investigated. Finally, an application is stated to investigate a normal congruence of surfaces by using electromagnetic wave vectors. Also, illustrations of polarization and magnetic field vector of EM wave are given.Öğe RULED SURFACES WITH CONSTANT SLOPE RULING ACCORDING TO DARBOUX FRAME IN MINKOWSKI SPACE(Etamaths Publ, 2020) Yavuz, Ayse; Yayli, YusufIn this study, three different types of ruled surfaces are defined. The generating lines of these ruled surfaces are given by points on a curve X in Minkowski Space, while the position vector of X have constant slope with respect to the planes (t, y), (t, n), (n, y). It is observed that the Lorentzian casual characters of the ruled surfaces with constant slope can be timelike or spacelike. Furthermore, striction lines of these surfaces are obtained and investigated under various special cases. Finally, new investigations are obtained on the base curve of these types of ruled surfaces.Öğe Some New Properties of Surfaces Generated by Null Cartan Curves(Int Electronic Journal Geometry, 2022) Yavuz, Ayse; Erdogdu, MelekIn this paper, some special types of surfaces with null Cartan base curve are investigated. The generating lines of the surfaces are chosen as a linear combination of Cartan frame fields with non-constant differentiable functions. Firstly, the surfaces whose generating lines have the same direction of Cartan frame fields B, N and T are examined respectively. As a special case, Gaussian and mean curvatures of one parameter family of Bertrand curves of a given null Cartan curve and the singular points of this type of surface are stated. Furthermore, an example is also stated to explain the obtained results. Then, the surfaces with null Cartan base curve are investigated where generating lines lie on the planes spanned by {N, B}, {T, B} and {T, N}, respectively. Finally, some differential geometric properties of these surface are given mainly in three different cases.Öğe SURFACES WITH VANISHING ABNORMALITY OF NORMAL DIRECTION IN MINKOWSKI SPACE(Editura Bibliotheca-Bibliotheca Publ House, 2022) Li, Yanlin; Yavuz, Ayse; Erdogdu, MelekThis paper is investigated geometry of vector fields along spacelike curve with timelike normal vector by using anholonomic coordinates. Derivative formulas of Frenet Serret frame of the curve are stated which includes eight parameters. Surfaces with vanishing abnormality of normal direction in Minkowski space are examined. Intrinsic geometric properties of these spacelike surfaces are investigated. Finally, the relations between spacelike surfaces with vanishing abnormality of normal direction and NLS, Heisenberg spin equation are investigated as applications.
