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Artykuły w czasopismach na temat „Bertrand curve”

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Data publikacji: 3 czerwca 2025
Data aktualizacji: 31 lipca 2025

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1

Öztekin, Handan. "Special Bertrand Curves in 4D Galilean Space." Mathematical Problems in Engineering 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/318458.

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The generalization of Bertrand curves in Galilean 4-space is introduced and the characterization of the generalized Bertrand curves is obtained. Furthermore, it is proved that no special curve is a classical Bertrand curve in Galilean 4-space such that the notion of classical Bertrand curve is definite only in three-dimensional spaces.
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2

Zhang, Chunxiao, and Donghe Pei. "Generalized Bertrand Curves in Minkowski 3-Space." Mathematics 8, no. 12 (2020): 2199. http://dx.doi.org/10.3390/math8122199.

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We define a generalized lightlike Bertrand curve pair and a generalized non-lightlike Bertrand curve pair, discuss their properties and prove the necessary and sufficient condition of a curve which is a generalized lightlike or a generalized non-lightlike Bertrand curve. Moreover, we study the relationship between slant helices and generalized Bertrand curves.
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3

Erdem, Hatice Altın, and Kazım İlarslan. "Spacelike Bertrand curves in Minkowski 3-space revisited." Analele Universitatii "Ovidius" Constanta - Seria Matematica 31, no. 3 (2023): 87–109. https://doi.org/10.2478/auom-2023-0033.

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Abstract In the geometry of curves in 𝔼3, if the principal normal vector field of a given space curve ϕ with non-zero curvatures is the principal normal vector field of another space curve ϕ *, then the curve ϕ is called a Bertrand curve and ϕ * is called Bertrand partner of ϕ. These curves have been studied in di erent space over a long period of time and found wide application in di erent areas. Therefore, we have a great knowledge of geometric properties of these curves. In this paper, revested results for spacelike Bertrand curves with non-null normal vectors will be given with the previou
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4

Li, Yanlin, Osman Keçilioğlu, and Kazım İlarslan. "Generalized Bertrand Curve Pairs in Euclidean Four-Dimensional Space." Axioms 14, no. 4 (2025): 253. https://doi.org/10.3390/axioms14040253.

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In this study, the existence of Bertrand curves (in the classical sense, i.e., curves with a common principal normal vector field) in four-dimensional Euclidean space is demonstrated using a novel approach. The necessary conditions for a regular curve to be a Bertrand curve pair are obtained. Furthermore, the relationship between Bertrand curves and Combescure-related curves (pairs of curves with parallel Frenet vectors) is established, and several geometric properties are derived. Additionally, examples are constructed for both Bertrand curve pairs and Combescure-related curve pairs, and thei
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5

Tamta, Stuti, and Ram Shankar Gupta. "New Parametrization of Bertrand Partner D-curves in $\mathbb{E}^{3}$." Boletim da Sociedade Paranaense de Matemática 42 (April 19, 2024): 1–11. http://dx.doi.org/10.5269/bspm.63309.

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We define and study a new parametrization of a Bertrand partner D-curve \{$\alpha$,$\alpha^{*}$\} in Euclidean 3-space by not taking Darboux frame element $g^{*}$ of Bertrand partner D-curve $\alpha^{*}$ parallel to $\overrightarrow{\alpha \alpha^{*}}$. We obtain a necessary and sufficient condition for a curve to be such type of Bertrand D-curves. Also, we obtain a characterization of a new parametrization of asymptotic Bertrand D-curves and provide an example.
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6

Aksoyak, Ferdag Kahraman, Ismail Gok, and Kazim Ilarslan. "Generalized Null Bertrand Curves In Minkowski Space-Time." Annals of the Alexandru Ioan Cuza University - Mathematics 60, no. 2 (2014): 489–502. http://dx.doi.org/10.2478/aicu-2013-0031.

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Abstract Çöken and ÇIFTCI proved that a null Cartan curve in Minkowski space-time E41 is a null Bertrand curve if and only if k2 is nonzero constant and k3 is zero. That is, the null curve with non-zero curvature k2 is not a Bertrand curve in Minkowski space-time E41. So, in this paper we defined a new type of Bertrand curve in Minkowski space-time E41 for a null curve with non-zero curvature k3 by using the similar idea of generalized Bertrand curve given by Matsuda and Yorozu and we called it a null (1, 3)-Bertrand curve. Also, we proved that if a null curve with non-zero curvatures in Minko
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7

ÖNDER, Mehmet. "Direction curves of generalized Bertrand curves and involute-evolute curves in $E^{4}$." Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics 71, no. 2 (2022): 326–38. http://dx.doi.org/10.31801/cfsuasmas.950707.

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In this study, we define (1,3)-Bertrand-direction curve and (1,3)-Bertrand-donor curve in the 4-dimensional Euclidean space $E^{4}$. We introduce necessary and sufficient conditions for a special Frenet curve to have a (1,3)-Bertrand-direction curve. We introduce the relations between Frenet vectors and curvatures of these direction curves. Furthermore, we investigate whether (1,3)-evolute-donor curves in $E^{4}$ exist and show that there is no (1,3)-evolute-donor curve in $E^{4}$ .
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8

Kızıltuğ, Sezai, Mehmet Onder, and Omer Tarakçi. "Bertrand and Mannheim partner -curves on parallel surfaces." Boletim da Sociedade Paranaense de Matemática 35, no. 2 (2017): 159–69. http://dx.doi.org/10.5269/bspm.v35i2.24309.

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In this paper we study Bertrand and Mannheim partner -curves on parallel surface. Using the definition of parallel surfaces, first we find images of two curves lying on two different surfaces and satisfying the conditions to be Bertrand partner -curve or Mannheim partner -curve. Then we obtain relationships between Bertrand and Mannheim partner -curves and their image curves.
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9

Erdem, H. A., A. Uçum, K. İlarslan, and Ç. Camcı. "New approach to timelike Bertrand curves in 3-dimensional Minkowski space." Carpathian Mathematical Publications 15, no. 2 (2023): 482–94. http://dx.doi.org/10.15330/cmp.15.2.482-494.

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In the theory of curves in Euclidean $3$-space, it is well known that a curve $\beta $ is said to be a Bertrand curve if for another curve $\beta^{\star}$ there exists a one-to-one correspondence between $\beta $ and $\beta^{\star}$ such that both curves have common principal normal line. These curves have been studied in different spaces over a long period of time and found wide application in different areas. In this article, the conditions for a timelike curve to be Bertrand curve are obtained by using a new approach in contrast to the well-known classical approach for Bertrand curves in Mi
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10

DOĞAN YAZICI, Bahar, Osman Zeki OKUYUCU, and Murat TOSUN. "On special singular curve couples of framed curves in 3D Lie groups." Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics 72, no. 3 (2023): 710–20. http://dx.doi.org/10.31801/cfsuasmas.1197154.

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In this paper, we introduce Bertrand and Mannheim curves of framed curves, which are a special singular curve in 3D Lie groups. We explain the conditions for framed curves to be Bertrand curves and Mannheim curves in 3D Lie groups. We give relationships between framed curvatures and Lie curvatures of Bertrand and Mannheim curves of framed curves. In addition, we obtain the characterization of Bertrand and Mannheim curves according to the various frames of framed curves in 3D Lie groups.
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11

Aydemir, İsmail, and Fırat Yerlikaya. "New Representations of Spherical Indicatricies of Bertrand Curves in Minkowski 3-Space." Geometry 2015 (January 28, 2015): 1–5. http://dx.doi.org/10.1155/2015/509058.

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We obtained a new representation for timelike Bertrand curves and their Bertrand mate in 3-dimensional Minkowski space. By using this representation, we expressed new representations of spherical indicatricies of Bertrand curves and computed their curvatures and torsions. Furthermore in case the indicatricies of a Bertrand curve are slant helices, we investigated some new characteristic features of these curves.
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12

Şenyurt, Süleyman, Yasin Altun, and Ceyda Cevahir. "Smarandache curves for spherical indicatrix of the Bertrand curves pair." Boletim da Sociedade Paranaense de Matemática 38, no. 2 (2018): 27–39. http://dx.doi.org/10.5269/bspm.v38i2.33899.

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In this paper, we investigate special Smarandache curves with regard to Sabban frame for Bertrand partner curve spherical indicatrix. Some results have been obtained. These results were expressed depending on the Bertrand curve. Besides, we are given examples of our results.
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13

Kızıltuğ, Sezai, and Yusuf Yaylı. "Bertrand Curves of AW(k)-Type in the Equiform Geometry of the Galilean Space." Abstract and Applied Analysis 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/402360.

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We consider curves of AW(k)-type (1≤k≤3) in the equiform geometry of the Galilean spaceG3. We give curvature conditions of curves of AW(k)-type. Furthermore, we investigate Bertrand curves in the equiform geometry ofG3. We have shown that Bertrand curve in the equiform geometry ofG3is a circular helix. Besides, considering AW(k)-type curves, we show that there are Bertrand curves of weak AW(2)-type and AW(3)-type. But, there are no such Bertrand curves of weak AW(3)-type and AW(2)-type.
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14

KAHRAMAN AKSOYAK, Ferdağ. "Quaternionic Bertrand curves according to type 2-quaternionic frame in $\mathbb{R}^{4}$." Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics 71, no. 2 (2022): 395–406. http://dx.doi.org/10.31801/cfsuasmas.991631.

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In this paper, we give some characterization of quaternionic Bertrand curves whose the torsion is non-zero but bitorsion is zero in $\mathbb{R}^{4}$ according to Type 2-Quaternionic Frame. One of the most important points in working on quaternionic curves is that given a curve in $\mathbb{R}^{4}$, the curve in $\mathbb{R}^{3}$ associated with this curve is determined individually. So, we obtain some relationships between quaternionic Bertrand curve $\alpha^{(4)}$ in $\mathbb{R}^{4}$ and its associated spatial quaternionic curve $\alpha$ in $\mathbb{R}^{3}$. Also, we support some theorems in th
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15

Alo, Jeta, and Leyla Zeren Akgün. "Bertrand Curves in $n$-Dimensional Riemann-Otsuki Space." Journal of New Theory, no. 50 (March 28, 2025): 38–47. https://doi.org/10.53570/jnt.1630419.

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In this paper, we extend the classic properties of Bertrand curves in Euclidean 3-space to an $n$-dimensional Riemann-Otsuki space. We introduce the concept of infinitesimal deformations of curves within this space, and by applying the Frenet formulas concerning the contravariant component of the covariant derivative, we derive conditions under which a given deformation of a curve corresponds to a Bertrand curve in this $n$-dimensional space.
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16

Hanif, Muhammad, Hou Hua, and Emilija Nesovic. "On involutes of order k of a null Cartan curve in Minkowski spaces." Filomat 33, no. 8 (2019): 2295–305. http://dx.doi.org/10.2298/fil1908295h.

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In this paper, we define an involute and an evolving involute of order k of a null Cartan curve in Minkowski space En1 for n ? 3 and 1 ? k ? n-1. In relation to that, we prove that if a null Cartan helix has a null Cartan involute of order 1 or 2, then it is Bertrand null Cartan curve and its involute is its Bertrand mate curve. In particular, we show that Bertrand mate curve of Bertrand null Cartan curve can also be a non-null curve and find the relationship between the Cartan frame of a null Cartan curve and the Frenet or the Cartan frame of its non-null or null Cartan involute of order 1 ?
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17

Tanriöver, Necmettin. "Some properties of Bertrand curves in Lorentzian n-space 𝕃n". International Journal of Geometric Methods in Modern Physics 13, № 05 (2016): 1650064. http://dx.doi.org/10.1142/s021988781650064x.

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In this paper, Bertrand curves in [Formula: see text]-dimensional Lorentz space [Formula: see text] are defined and some of their properties are determined. Various relationships and characterizations are found between higher order curvatures and their derivatives for Bertrand curve pair. In addition, some relationships are obtained between these curves and general helix, harmonic curvature.
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18

Alluhaibi, Nadia, and Rashad A. Abdel-Baky. "Surface Pencil Couple with Bertrand Couple as Joint Principal Curves in Galilean 3-Space." Axioms 12, no. 11 (2023): 1022. http://dx.doi.org/10.3390/axioms12111022.

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A principal curve on a surface plays a paramount role in reasonable implementations. A curve on a surface is a principal curve if its tangents are principal directions. Using the Serret–Frenet frame, the surface pencil couple can be expressed as linear combinations of the components of the local frames in Galilean 3-space G3. With these parametric representations, a family of surfaces using principal curves (curvature lines) are constructed, and the necessary and sufficient condition for the given Bertrand couple to be the principal curves on these surfaces are derived in our approach. Moreove
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19

Senyurt, Suleyman, Abdussamet Çaliskan, and Unzile Çelik. "Smarandache curves of Bertrand curves pair according to Frenet frame." Boletim da Sociedade Paranaense de Matemática 39, no. 5 (2021): 163–73. http://dx.doi.org/10.5269/bspm.41546.

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In this paper, the curvature and the torsion of Smarandache curves obtained by the vectors of the Bertrand partner curve are calculated. These values are expressed depending upon the curve. Besides, we illustrate example with our main results.
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20

Qian, Jinhua, Mingyu Sun, Pei Yin, and Young-Ho Kim. "Null Darboux Curve Pairs in Minkowski 3-Space." Axioms 10, no. 3 (2021): 142. http://dx.doi.org/10.3390/axioms10030142.

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Based on the fundamental theories of null curves in Minkowski 3-space, the null Darboux mate curves of a null curve are defined which can be regarded as a kind of extension for Bertrand curves and Mannheim curves in Minkowski 3-space. The relationships of null Darboux curve pairs are explored and their expression forms are presented explicitly.
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21

Elsharkawy, Ayman, Yusra Tashkandy, Walid Emam, Clemente Cesarano, and Noha Elsharkawy. "On Some Quasi-Curves in Galilean Three-Space." Axioms 12, no. 9 (2023): 823. http://dx.doi.org/10.3390/axioms12090823.

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In this paper, the quasi-frame and quasi-formulas are introduced in Galilean three-space. In addition, the quasi-Bertrand and the quasi-Mannheim curves are studied. It is proven that the angle between the tangents of two quasi-Bertrand or quasi-Mannhiem curves is not constant. Furthermore, the quasi-involute is studied. Moreover, we prove that there is no quasi-evolute curve in Galilean three-space. Also, we introduce quasi-Smarandache curves in Galilean three-space. Finally, we demonstrate an illustrated example to present our findings.
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22

Alo, Jeta. "Null Hybrid Curves and Some Characterizations of Null Hybrid Bertrand Curves." Symmetry 17, no. 2 (2025): 312. https://doi.org/10.3390/sym17020312.

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In this paper, we investigate null curves in R24, the four-dimensional Minkowski space of index 2, utilizing the concept of hybrid numbers. Hybrid and spatial hybrid-valued functions of a single variable describe a curve in R24. We first derive Frenet formulas for a null curve in R23, the three-dimensional Minkowski space of index 2, by means of spatial hybrid numbers. Next, we apply the Frenet formulas for the associated null spatial hybrid curve corresponding to a null hybrid curve in order to derive the Frenet formulas for this curve in R24. This approach is simpler and more efficient than
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23

Çelik, Oğuzhan. "A Generalization of Curve Mates: Normal Mate of a Curve." Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi 17, no. 2 (2024): 338–52. http://dx.doi.org/10.18185/erzifbed.1396745.

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This a paper, a new curve pair is defined that generalizes some pairs of curves well known as Mannheim and Bertrand curve pairs. A normal curve pair is defined in such a way that a vector u obtained by overlapping the normal planes of the G and G* curves makes the same angle as the binormals of these curves. The relationship between torsions and curvatures of curve pairs was analyzed. Moreover, The unit quaternion q corresponding to the rotation matrix between the Frenet vectors of the curves was defined. In the conclusion, it is expressed express which famous pairs of curves will be obtained
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24

ECHABBI, Nidal, and Amina OUAZZANI CHAHDI. "Some Associated Curves of Normal Indicatrix of a Regular Curve." Journal of Mathematics Research 12, no. 1 (2020): 84. http://dx.doi.org/10.5539/jmr.v12n1p84.

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In this paper, we consider integral curves of a vector field generated by Frenet vectors of normal indicatrix of a given curve in Euclidean 3-space. We define some new associated curves such as evolute direction curves, Bertrand direction curves and Mannheim directon curves of the normal indicatrix of a regular curve, respectively. We also found the relationships between curvatures of these curves. By using these associated curves, we give a new approach to construct slant helices and C- slant helices. Finally, we present some examples.
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ECHABBI, Nidal, and Amina OUAZZANI CHAHDI. "Some Associated Curves of Normal Indicatrix of a Regular Curve." Journal of Mathematics Research 12, no. 1 (2020): 92. http://dx.doi.org/10.5539/jmr.v12n1p92.

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In this paper, we consider integral curves of a vector field generated by Frenet vectors of normal indicatrix of a given curve in Euclidean 3-space. We define some new associated curves such as evolute direction curves, Bertrand direction curves and Mannheim directon curves of the normal indicatrix of a regular curve, respectively. We also found the relationships between curvatures of these curves. By using these associated curves, we give a new approach to construct slant helices and C- slant helices. Finally, we present some examples.
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26

Alghanemi, Azeb, and Meraj Ali Khan. "Position Vectors of the Natural Mate and Conjugate of a Space Curve." Advances in Mathematical Physics 2023 (May 8, 2023): 1–5. http://dx.doi.org/10.1155/2023/7565988.

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The concept of the natural mate and the conjugate curves associated to a smooth curve in Euclidian 3-space were introduced initially by Dashmukh and others. In this paper, we give some extra results that add more properties of the natural mate and the conjugate curves associated with a smooth space curve in E 3 . The position vectors of the natural mate and the conjugate of a given smooth curve are investigated. Also, using the position vector of the natural mate, the necessary and sufficient condition for a smooth given curve to be a Bertrand curve is introduced. Moreover, a new characterizat
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27

Çakmak, Ali. "New Type Direction Curves in 3-Dimensional Compact Lie Group." Symmetry 11, no. 3 (2019): 387. http://dx.doi.org/10.3390/sym11030387.

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In this paper, new types of associated curves, which are defined as rectifying-direction, osculating-direction, and normal-direction, in a three-dimensional Lie group G are achieved by using the general definition of the associated curve, and some characterizations for these curves are obtained. Additionally, connections between the new types of associated curves and the curves, such as helices, general helices, Bertrand, and Mannheim, are given.
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28

Koçak, Zehra Nur, and Emel Karaca. "On special ruled surface pairs in fractional calculus." Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 74, no. 2 (2025): 267–76. https://doi.org/10.31801/cfsuasmas.1500845.

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In this paper, we extend the theory of special fractional curve pairs (i.e., F-Bertrand, FMannheim, and F-involute-evolute curve pairs) to fractional ruled surfaces with the perspective of fractional calculus. Next, we characterize two fractional ruled surfaces, offset in the senses of F-Bertrand, F-Mannheim, and F-involute-evolute. Moreover, considering the chain rules in fractional calculus, some significant theorems are proved, and the developability conditions are examined by calculating the distribution parameters. Finally, we give examples to verify the results.
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29

ŞENYURT, Süleyman, Davut CANLI, and Kebire Hilal AYVACI. "Associated curves from a different point of view in $E^3$." Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics 71, no. 3 (2022): 826–45. http://dx.doi.org/10.31801/cfsuasmas.1026359.

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In this paper, tangent, principal normal and binormal wise associated curves are defined such that each of these vectors of any given curve lies on the osculating, normal and rectifying plane of its partner, respectively. For each associated curve, a new moving frame and the corresponding curvatures are formulated in terms of Frenet frame vectors. In addition to this, the possible solutions for distance functions between the curve and its associated mate are discussed. In particular, it is seen that the involute curves belong to the family of tangent associated curves in general and the Bertra
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30

Öztürk, İskender, Hasan Çakır, and Mustafa Özdemir. "Osculating Mate of a Curve in Minkowski 3-Space." Axioms 14, no. 6 (2025): 468. https://doi.org/10.3390/axioms14060468.

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In this paper, we introduce and develop the concept of osculating curve pairs in the three-dimensional Minkowski space. By defining a vector lying in the intersection of osculating planes of two non-lightlike curves, we characterize osculating mates based on their Frenet frames. We then derive the transformation matrix between these frames and investigate the curvature and torsion relations under varying causal characterizations of the curves—timelike and spacelike. Furthermore, we determine the conditions under which these generalized osculating pairs reduce to well-known curve pairs such as
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31

Schneider, Baruch. "Some Notes on the Poincaré-Bertrand Formula." Journal of Applied Mathematics 2012 (2012): 1–10. http://dx.doi.org/10.1155/2012/969685.

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32

Alluhaibi, Nadia, and Rashad A. Abdel-Baky. "A Surface Family with a Common Asymptotic Null Curve in Minkowski 3-Space E 1 3." Mathematical Problems in Engineering 2021 (December 23, 2021): 1–8. http://dx.doi.org/10.1155/2021/3901527.

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This approach is on constructing a surface family with a common asymptotic null curve. It has provided the necessary and sufficient condition for the curve to be an asymptotic null curve and extended the study to ruled and developable surfaces. Subsequently, the study has examined the Bertrand offsets of a surface family with a common asymptotic null curve. Lastly, we support the results of this approach by some examples.
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33

López, Rafael, Željka Milin Šipuš, Ljiljana Primorac Gajčić, and Ivana Protrka. "Harmonic evolutes of B-scrolls with constant mean curvature in Lorentz–Minkowski space." International Journal of Geometric Methods in Modern Physics 16, no. 05 (2019): 1950076. http://dx.doi.org/10.1142/s0219887819500762.

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In this paper, we study harmonic evolutes of [Formula: see text]-scrolls, that is, of ruled surfaces in Lorentz–Minkowski space having no Euclidean counterparts. Contrary to Euclidean space where harmonic evolutes of surfaces are surfaces again, harmonic evolutes of [Formula: see text]-scrolls turn out to be curves. In particular, we show that the harmonic evolute of a [Formula: see text]-scroll of constant mean curvature together with its base curve forms a null Bertrand pair. This allows us to characterize [Formula: see text]-scrolls of constant mean curvature and reconstruct them from a giv
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34

Da Silva, Luiz C. B. "Characterization of spherical and plane curves using rotation minimizing frames." Boletim da Sociedade Paranaense de Matemática 41 (December 26, 2022): 1–6. http://dx.doi.org/10.5269/bspm.49075.

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In this work, we study plane and spherical curves in Euclidean and Lorentz-Minkowski 3-spaces by employing rotation minimizing (RM) frames. By conveniently writing the curvature and torsion for a curve on a sphere, we show how to find the angle between the principal normal and an RM vector field for spherical curves. Later, we characterize plane and spherical curves as curves whose position vector lies, up to a translation, on a moving plane spanned by their unit tangent and an RM vector field. Finally, as an application, we characterize Bertrand curves as curves whose so-called natural mates
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35

Mofarreh, Fatemah. "Timelike Surface Couple with Bertrand Couple as Joint Geodesic Curves in Minkowski 3-Space." Symmetry 16, no. 6 (2024): 732. http://dx.doi.org/10.3390/sym16060732.

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A curve on a surface is a geodesic curve if its principal normal vector is anywhere aligned with the surface normal. Using the Serret–Frenet frame, a timelike surface couple (TLSC) with the symmetry of a Bertrand couple (BC) can be specified in terms of linear combinations of the components of the local frames in Minkowski 3-space E13. With these parametric representations, the necessary and sufficient conditions for the specified BC are derived to be the geodesic curves defining these surfaces. Afterward, the definition of a TL ruled surface (RS) is also provided. Furthermore, the application
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CEAYIR, HASIM, and FIDAN JABRAILZADE. "NOTES ON LIFTING OF BERTRAND CURVE ON TANGENT SPACE TR3." Poincare Journal of Analysis and Applications 05, no. 2.1 (2018): 57–63. http://dx.doi.org/10.46753/pjaa.2018.v05i02(i).002.

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37

Etro, Federico, and Lorenza Rossi. "New-Keynesian Phillips curve with Bertrand competition and endogenous entry." Journal of Economic Dynamics and Control 51 (February 2015): 318–40. http://dx.doi.org/10.1016/j.jedc.2014.10.009.

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38

Schneider, Baruch, and Ömer Kavaklıoğlu. "Poincaré–Bertrand formula on a piecewise Liapunov curve in two-dimensional." Applied Mathematics and Computation 202, no. 2 (2008): 814–19. http://dx.doi.org/10.1016/j.amc.2008.03.026.

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39

CEYLAN, Ayşe Yilmaz, and Abdullah Aziz ERGİN. "BERTRAND MATE OF A BIHARMONIC CURVE IN CARTAN-VRANCEANU 3-DIMENSIONAL SPACE." International Electronic Journal of Geometry 8, no. 1 (2015): 45–52. http://dx.doi.org/10.36890/iejg.592796.

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40

Askar, Sameh. "Complex Investigations of a Piecewise-Smooth Remanufacturing Bertrand Duopoly Game." Mathematics 9, no. 20 (2021): 2558. http://dx.doi.org/10.3390/math9202558.

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This paper considers a Bertrand competition between two firms whose decision variables are derived from a quadratic utility function. The first firm produces new products with their own prices while the second firm re-manufactures returned products and sells them in the market at prices that may be less than or equal to the price of the first firm. Dynamically, this competition is constructed on which boundedly rational firms apply a gradient adjustment mechanism to update their prices in each period. According to this mechanism and the nature of the competition, a two-dimensional piecewise sm
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41

TUINSTRA, JAN. "A PRICE ADJUSTMENT PROCESS IN A MODEL OF MONOPOLISTIC COMPETITION." International Game Theory Review 06, no. 03 (2004): 417–42. http://dx.doi.org/10.1142/s0219198904000289.

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We consider a price adjustment process in a model of monopolistic competition. Firms have incomplete information about the demand structure. When they set a price they observe the amount they can sell at that price and they observe the slope of the true demand curve at that price. With this information they estimate a linear demand curve. Given this estimate of the demand curve they set a new optimal price. We investigate the dynamical properties of this learning process. We find that, if the cross-price effects and the curvature of the demand curve are small, prices converge to the Bertrand-N
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42

Barroero, Fabrizio. "CM RELATIONS IN FIBERED POWERS OF ELLIPTIC FAMILIES." Journal of the Institute of Mathematics of Jussieu 18, no. 5 (2017): 941–56. http://dx.doi.org/10.1017/s1474748017000287.

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Let $E_{\unicode[STIX]{x1D706}}$ be the Legendre family of elliptic curves. Given $n$ points $P_{1},\ldots ,P_{n}\in E_{\unicode[STIX]{x1D706}}(\overline{\mathbb{Q}(\unicode[STIX]{x1D706})})$, linearly independent over $\mathbb{Z}$, we prove that there are at most finitely many complex numbers $\unicode[STIX]{x1D706}_{0}$ such that $E_{\unicode[STIX]{x1D706}_{0}}$ has complex multiplication and $P_{1}(\unicode[STIX]{x1D706}_{0}),\ldots ,P_{n}(\unicode[STIX]{x1D706}_{0})$ are linearly dependent over End$(E_{\unicode[STIX]{x1D706}_{0}})$. This implies a positive answer to a question of Bertrand
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43

ERDEM, ESRA, and MÜNEVVER YILDIRIM YILMAZ. "SPECIAL CURVES ACCORDING TO TYPE-2 QUATERNIONIC FRAME IN R4." Journal of Science and Arts 25, no. 1 (2025): 133–44. https://doi.org/10.46939/j.sci.arts-25.1-a12.

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Quaternions, which were defined by William Rowan Hamilton in 1843, are a number system in four-dimensional space and are analogous to complex numbers. However, quaternion multiplication is not commutative, distinguishing them from complex numbers. Quaternions are special mathematical tools used in computer science, robotics, and many other mathematical sciences. From this point of view, they also get attention in differential geometry. In particular, their characterizations given by the Serret-Frenet apparatus are challenging. For this reason, Bharathi and Nagaraj obtained Serret-Frenet formul
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44

Çelik, Serkan, Hacı Bayram Karadağ, and Hatice Kuşak Samancı. "The Conchoidal Twisted Surfaces Constructed by Anti-Symmetric Rotation Matrix in Euclidean 3-Space." Symmetry 15, no. 6 (2023): 1191. http://dx.doi.org/10.3390/sym15061191.

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A twisted surface is a type of mathematical surface that has a nontrivial topology, meaning that it cannot be smoothly deformed into a flat surface without tearing or cutting. Twisted surfaces are often described as having a twisted or Möbius-like structure, which gives them their name. Twisted surfaces have many interesting mathematical properties and applications, and are studied in fields such as topology, geometry, and physics. In this study, a conchoidal twisted surface is formed by the synchronized anti-symmetric rotation matrix of a planar conchoidal curve in its support plane and this
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45

Zhao, Liuwei. "Dynamic Analysis and Chaos Control of Bertrand Triopoly Based on Differentiated Products and Heterogeneous Expectations." Discrete Dynamics in Nature and Society 2020 (July 11, 2020): 1–17. http://dx.doi.org/10.1155/2020/2012680.

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Price competition has become a universal commercial phenomenon nowadays. This paper considers a dynamic Bertrand price game model, in which enterprises have heterogeneous expectations. By the stability theory of the dynamic behavior of the Bertrand price game model, the instability of the boundary equilibrium point and the stability condition of the internal equilibrium point are obtained. Furthermore, bifurcation diagram, basin of attraction, and critical curve are introduced to investigate the dynamic behavior of this game. Numerical analysis shows that the change of model parameters in a dy
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46

Takahashi, Masatomo, and Haiou Yu. "Bertrand and Mannheim Curves of Spherical Framed Curves in a Three-Dimensional Sphere." Mathematics 10, no. 8 (2022): 1292. http://dx.doi.org/10.3390/math10081292.

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We investigated differential geometries of Bertrand curves and Mannheim curves in a three-dimensional sphere. We clarify the conditions for regular spherical curves to become Bertrand and Mannheim curves. Then, we concentrate on Bertrand and Mannheim curves of singular spherical curves. As singular spherical curves, we considered spherical framed curves. We define Bertrand and Mannheim curves of spherical framed curves. We give conditions for spherical framed curves to become Bertrand and Mannheim curves.
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47

Wu, Linlin, Anjie Zhou, Kaixin Yao, and Donghe Pei. "Generalized Bertrand Curves of Non-Light-like Framed Curves in Lorentz–Minkowski 3-Space." Mathematics 12, no. 16 (2024): 2593. http://dx.doi.org/10.3390/math12162593.

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In this paper, we define the generalized Bertrand curves of non-light-like framed curves in Lorentz–Minkowski 3-space; their study is essential for understanding many classical and modern physics problems. Here, we consider two non-light-like framed curves as generalized Bertrand pairs. Our generalized Bertrand pairs can include Bertrand pairs with either singularities or not, and also include Mannheim pairs with singularities. In addition, we discuss their properties and prove the necessary and sufficient conditions for two non-light-like framed curves to be generalized Bertrand pairs.
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48

Tasdemir, Mert, Elif Canfes, and Banu Uzun. "On Caputo fractional Bertrand Curves in E3 and E31." Filomat 38, no. 5 (2024): 1681–702. http://dx.doi.org/10.2298/fil2405681t.

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In this article, we examine Bertrand curves in E3 and E31 by using the Caputo fractional derivative which we call ?-Bertrand Curves. First, we consider ?-Bertrand curves in E3 and we give a characterization of them. Then, we study ?-Bertrand curves in E31 and we prove the necessary and sufficient condition for a ?-Bertrand curves in E31 by considering time like, space like and null curves. We also give the related examples by using Python.
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49

Al-Jedani, Awatif, and Rashad A. Abdel-Baky. "Surface Family with Bertrand Curves as Joint Asymptotic Curves in 3D Galilean Space." Mathematics 11, no. 19 (2023): 4100. http://dx.doi.org/10.3390/math11194100.

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The primary objective of this work is to discuss a surface family with the similarity of Bertrand curves in 3D Galilean space. Subsequently, by applying the Serret–Frenet frame, we estimate the sufficient and necessary statuses of a surface family with Bertrand curves as joint asymptotic curves. The dilation to ruled surfaces is also summarized. Meanwhile, the epitomes are illustrated to provide an explanation of the theoretical results.
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50

Li, Yanlin, Ali Uçum, Kazım İlarslan, and Çetin Camcı. "A New Class of Bertrand Curves in Euclidean 4-Space." Symmetry 14, no. 6 (2022): 1191. http://dx.doi.org/10.3390/sym14061191.

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Bertrand curves are a pair of curves that have a common principal normal vector at any point and are related to symmetry properties. In the present paper, we define the notion of 1,3-V Bertrand curves in Euclidean 4-space. Then we find the necessary and sufficient conditions for curves in Euclidean 4-space to be 1,3-V Bertrand curves. Some related examples are given.
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