scholarly journals NC -Smarandache Ruled Surface and NW -Smarandache Ruled Surface according to Alternative Moving Frame in E 3

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Soukaina Ouarab
Keyword(s):  
Sufficient Conditions ◽  
Ruled Surface ◽  
Necessary And Sufficient Conditions ◽  
Moving Frame ◽  
Necessary And Sufficient

In this study, original definitions of NC − Smarandache ruled surface and NW − Smarandache ruled surface according to the alternative moving frame are introduced in E 3 . The main results of the study are presented in theorems that give necessary and sufficient conditions for those special surfaces to be developable and minimal. Finally, an example with illustrations is presented.

scholarly journals A Note on Parametric Surfaces in Minkowski 3-Space

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Esra Betul Koc Ozturk
Keyword(s):  
Linear Combination ◽  
Sufficient Conditions ◽  
Ruled Surface ◽  
Necessary And Sufficient Conditions ◽  
Frenet Frame ◽  
Parametric Surface ◽  
Parametric Surfaces ◽  
Null Curve ◽  
Necessary And Sufficient

With the help of the Frenet frame of a given pseudo null curve, a family of parametric surfaces is expressed as a linear combination of this frame. The necessary and sufficient conditions are examined for that curve to be an isoparametric and asymptotic on the parametric surface. It is shown that there is not any cylindrical and developable ruled surface as a parametric surface. Also, some interesting examples are illustrated about these surfaces.

scholarly journals Smarandache Ruled Surfaces according to Frenet-Serret Frame of a Regular Curve in E 3

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Soukaina Ouarab
Keyword(s):  
Sufficient Conditions ◽  
Ruled Surface ◽  
Necessary And Sufficient Conditions ◽  
Ruled Surfaces ◽  
Regular Curve ◽  
Necessary And Sufficient

In this paper, we introduce original definitions of Smarandache ruled surfaces according to Frenet-Serret frame of a curve in E 3 . It concerns TN-Smarandache ruled surface, TB-Smarandache ruled surface, and NB-Smarandache ruled surface. We investigate theorems that give necessary and sufficient conditions for those special ruled surfaces to be developable and minimal. Furthermore, we present examples with illustrations.

scholarly journals Smarandache Ruled Surfaces according to Darboux Frame in E 3

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Soukaina Ouarab
Keyword(s):  
Architectural Design ◽  
Sufficient Conditions ◽  
Surface Modeling ◽  
Ruled Surface ◽  
Necessary And Sufficient Conditions ◽  
Ruled Surfaces ◽  
Regular Surface ◽  
New Approach ◽  
Darboux Frame ◽  
Necessary And Sufficient

This paper presents a new approach of constructing special ruled surfaces and aims to study their developability and minimalist conditions. Our concept opens opportunities for application in engineering, surface modeling, and architectural design. The principle of our study is to introduce the original definitions of Smarandache ruled surfaces according to Darboux frame of a curve lying on an arbitrary regular surface in E 3 . It concerns T g -Smarandache ruled surface, T n -smarandache ruled surface, and g n -Smarandache ruled surface. New theorems giving necessary and sufficient conditions for those surfaces to be developable and minimal are investigated. Finally, an example with illustrations is presented.

scholarly journals Sweeping surface of center curve on surface in Euclidean 3-space E³

2021 ◽  
Vol 20 ◽  
pp. 235-243
Author(s):  
Rashad A. Abdel-Baky ◽  
Fatemah Mofarreh
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Ruled Surfaces ◽  
Moving Frame ◽  
Regular Surface ◽  
Geometrical Properties ◽  
Darboux Frame ◽  
Curve On Surface ◽  
Necessary And Sufficient

For the curve on the regular surface, there is moving frame with this thatis named Darboux frame. Sweeping surfaces through the curve associated with Darboux frame are introduced and their geometrical properties are investigated. Moreover, we obtain the necessary and sufficient conditions of this kind of surfaces to be developable ruled surfaces. Finally, an example to illustrate the application of the results is introduced.

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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