scholarly journals A study on null cartan curve in Minkowski 3-space

2020 ◽  
Vol 5 (1) ◽  
pp. 413-424
Author(s):  
Muhammad Abubakar Isah ◽  
Mihriban Alyamaç Külahcı
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Bishop Frame ◽  
Minkowski Spaces ◽  
Necessary And Sufficient

AbstractNull cartan curves have been studied by some geometers in both Euclidean and Minkowski spaces, but some special characters of the curves are not considered. In this paper, we study weak AW (k) – type and AW (k) – type null cartan curve in Minkowski 3-space E_1^3 . We define helix according to Bishop frame in E_1^3 . Furthermore, the necessary and sufficient conditions for the helices in Minkowski 3-space are obtained.

scholarly journals Special Curves According to Bishop Frame in Minkowski 3-Space

2020 ◽  
Vol 5 (1) ◽  
pp. 237-248
Author(s):  
Muhammad Abubakar Isah ◽  
Mihriban Alyamaç Külahçı
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Bishop Frame ◽  
Null Curve ◽  
Minkowski Spaces ◽  
Slant Helix ◽  
Null Curves ◽  
Necessary And Sufficient

AbstractPseudo null curves were studied by some geometers in both Euclidean and Minkowski spaces, but some special characters of the curve are not considered. In this paper, we study weak AW (k) – type and AW (k) – type pseudo null curve in Minkowski 3-space [E_1^3 . We define helix and slant helix according to Bishop frame in [E_1^3 . Furthermore, the necessary and sufficient conditions for the slant helix and helix in Minkowski 3-space are obtained.

scholarly journals Characteristic Properties of Type-2 Smarandache Ruled Surfaces According to the Type-2 Bishop Frame in E 3

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Ibrahim Al-Dayel ◽  
E. M. Solouma
Keyword(s):  
Sufficient Conditions ◽  
Special Kind ◽  
Necessary And Sufficient Conditions ◽  
Ruled Surfaces ◽  
Minimal Amount ◽  
Bishop Frame ◽  
Necessary And Sufficient

In this paper, we define and investigate a special kind of ruled surfaces called type-2 Smarandache ruled surfaces related to the type-2 Bishop frame in E 3 . From this point and depending on the type-2 Bishop curvature, we provide the necessary and sufficient conditions that allow these surfaces to be developable in a minimal amount of time. Furthermore, an example is given to clear the results.

scholarly journals Timelike Sweeping Surfaces According to Type-2 Bishop Frame in Minkowski 3–space

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Ruled Surfaces ◽  
Developable Surface ◽  
Bishop Frame ◽  
Tangential Surface ◽  
Necessary And Sufficient

In this paper, we express timelike sweeping surfaces using rotation minimizing frames in Minkowski 3–Space E3 1 . Necessary and sufficient conditions for timelike sweeping surfaces to be developable ruled surfaces are derived. Using these, we analyze the conditions when the resulting timelike developable surface is a cylinder, cone or tangential surface.

scholarly journals Surfaces family with common smarandache asymptotic curve according to bishop frame in euclidean 3-space

2016 ◽  
Vol 34 (1) ◽  
pp. 187-202 ◽  
Author(s):  
Gulnur Saffak Atalay ◽  
Emin Kasap
Keyword(s):  
Linear Combination ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Bishop Frame ◽  
The Family ◽  
Necessary And Sufficient

In this paper, we analyzed the problem of constructing a family of surfaces from a given some special Smarandache curves in  Euclidean 3-space. Using the Bishop frame of the curve in Euclidean 3-space, we express the family of surfaces as a linear combination of the components of this frame, and derive the necessary and sufficient conditions for coefficients to satisfy both the asymptotic and isoparametric requirements. Finally, examples are given to show the family of surfaces with common Smarandache asymptotic curve.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

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