A Note on Parametric Surfaces in Minkowski 3-Space

Frenet Frame ◽

Parametric Surface ◽

Parametric Surfaces ◽

Null Curve ◽

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.

On the Construction of a Surface Family with Common Geodesic in Galilean Space G3

Open Physics ◽

10.1515/phys-2016-0041 ◽

2016 ◽

Vol 14 (1) ◽

pp. 360-363 ◽

Cited By ~ 3

Author(s):

Zühal Küçükarslan Yüzbaşı ◽

Mehmet Bektaş

Keyword(s):

Sufficient Conditions ◽

Parametric Representation ◽

Geodesic Curve ◽

Frenet Frame ◽

Parametric Surfaces ◽

Galilean Space ◽

AbstractIn this paper, we investigate the parametric representation for a family of surfaces through a given geodesic curve G3. We provide necessary and sufficient conditions for this curve to be an isogeodesic curve on the parametric surfaces using Frenet frame in Galilean space. Also, for the sake of visualizing of this study, we plot an example for this surfaces family.

Hypersurface Family with a Common Isoasymptotic Curve

Geometry ◽

10.1155/2014/623408 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6 ◽

Cited By ~ 1

Author(s):

Ergin Bayram ◽

Emin Kasap

Keyword(s):

Linear Combination ◽

Sufficient Conditions ◽

Frenet Frame ◽

Necessary And Sufficient ◽

The Given

We handle the problem of finding a hypersurface family from a given asymptotic curve in R4. Using the Frenet frame of the given asymptotic curve, we express the hypersurface as a linear combination of this frame and analyze the necessary and sufficient conditions for that curve to be asymptotic. We illustrate this method by presenting some examples.

Surfaces family with common Smarandache asymptotic curve

Boletim da Sociedade Paranaense de Matemática ◽

10.5269/bspm.v34i1.24392 ◽

2016 ◽

Vol 34 (1) ◽

pp. 9-20

Author(s):

Gulnur Saffak Atalay ◽

Emin Kasap

Keyword(s):

Linear Combination ◽

Sufficient Conditions ◽

Frenet Frame ◽

The Family ◽

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 Frenet 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 curve.

Special Curves According to Bishop Frame in Minkowski 3-Space

Applied Mathematics and Nonlinear Sciences ◽

10.2478/amns.2020.1.00021 ◽

2020 ◽

Vol 5 (1) ◽

pp. 237-248

Author(s):

Muhammad Abubakar Isah ◽

Mihriban Alyamaç Külahçı

Keyword(s):

Sufficient Conditions ◽

Bishop Frame ◽

Null Curve ◽

Minkowski Spaces ◽

Slant Helix ◽

Null Curves ◽

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.

Injectivity of linear combinations in $\mathcal{B}(\mathcal{H})$

Electronic Journal of Linear Algebra ◽

10.13001/ela.2021.5049 ◽

2021 ◽

Vol 37 ◽

pp. 359-369

Author(s):

Marko Kostadinov

Keyword(s):

Linear Combination ◽

Necessary Conditions ◽

Sufficient Conditions ◽

Linear Combinations ◽

Special Cases ◽

Sufficient And Necessary Conditions ◽

Alpha Beta ◽

The aim of this paper is to provide sufficient and necessary conditions under which the linear combination $\alpha A + \beta B$, for given operators $A,B \in {\cal B}({\cal H})$ and $\alpha, \beta \in \mathbb{C}\setminus \lbrace 0 \rbrace$, is injective. Using these results, necessary and sufficient conditions for left (right) invertibility are given. Some special cases will be studied as well.

Linear combinations of polynomials with three-term recurrence

Publications de l Institut Mathematique ◽

10.2298/pim2124029t ◽

2021 ◽

Vol 110 (124) ◽

pp. 29-40

Author(s):

Khang Tran ◽

Maverick Zhang

Keyword(s):

Linear Combination ◽

Chebyshev Polynomials ◽

Sufficient Conditions ◽

Linear Combinations ◽

Zero Distribution ◽

Necessary And Sufficient ◽

Linear Polynomials

We study the zero distribution of the sum of the first n polynomials satisfying a three-term recurrence whose coefficients are linear polynomials. We also extend this sum to a linear combination, whose coefficients are powers of az + b for a, b ? R, of Chebyshev polynomials. In particular, we find necessary and sufficient conditions on a, b such that this linear combination is hyperbolic.

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

Abstract and Applied Analysis ◽

10.1155/2021/5526536 ◽

2021 ◽

Vol 2021 ◽

pp. 1-8

Author(s):

Soukaina Ouarab

Keyword(s):

Sufficient Conditions ◽

Ruled Surface ◽

Ruled Surfaces ◽

Regular Curve ◽

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.

On a Certain Subclass of Analytic Functions Involving Integral Operator Defined by Polylogarithm Function

Mathematics ◽

10.3390/math7010066 ◽

2019 ◽

Vol 7 (1) ◽

pp. 66

Author(s):

Danyal Soybaş ◽

Santosh B. Joshi ◽

Haridas Pawar

Keyword(s):

Integral Operator ◽

Linear Combination ◽

Analytic Functions ◽

Sufficient Conditions ◽

Distortion Theorem ◽

Partial Sums ◽

Polylogarithm Function ◽

In the present paper, we have introduced a new subclass of analytic functions involving integral operator defined by polylogarithm function. Necessary and sufficient conditions are obtained for this class. Further distortion theorem, linear combination and results on partial sums are investigated.

Smarandache Ruled Surfaces according to Darboux Frame in E 3

Journal of Mathematics ◽

10.1155/2021/9912624 ◽

2021 ◽

Vol 2021 ◽

pp. 1-10

Author(s):

Soukaina Ouarab

Keyword(s):

Architectural Design ◽

Sufficient Conditions ◽

Surface Modeling ◽

Ruled Surface ◽

Ruled Surfaces ◽

Regular Surface ◽

New Approach ◽

Darboux Frame ◽

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.

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

Journal of Mathematics ◽

10.1155/2021/9951434 ◽

2021 ◽

Vol 2021 ◽

pp. 1-6

Author(s):

Soukaina Ouarab

Keyword(s):

Sufficient Conditions ◽

Ruled Surface ◽

Moving Frame ◽

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.