arXiv subjects

Hassan Al-Zoubi

Publications and source records attributed to Hassan Al-Zoubi.

At least 19 recordsLinked to original sources

arXiv2025

Surfaces of coordinate finite II-type

In this article, we study the class of surfaces of revolution in the 3-dimensional Euclidean space $E^{3}$ with nonvanishing Gauss curvature whose position vector $\boldsymbol{x}$ satisfies the condition $Δ^{II}\boldsymbol{x}=A\boldsymbol{x}$, where $A$ is a square matrix of order 3 and $Δ^{II}$ denotes the Laplace operator of the second fundamental form $II$ of the surface. We show that a surface of revolution satisfying the preceding relation is a catenoid or part of a sphere.

math.GM↗

arXiv2023

Characterization of quadric surfaces in terms of coordinate finite type Gauss map

In this article, we introduce an important class of surfaces, namely, quadrics in the Euclidean 3-space $\mathbb{E}^{3}$. We prove that planes, spheres and circular cylinders are the only quadric surfaces whose Gauss map $\boldsymbol{G}$ satisfies a relation of the form $Δ^{I}\boldsymbol{G}= M \boldsymbol{G}$, where $M$ is a square matrix of order 3 and $Δ^{I}$ is the Laplace-Beltrami operator corresponding to the first fundamental form $I$ of the surface.

math.GM↗

arXiv2022

Surfaces of coordinate finite type in the Lorentz-Minkowski 3-space

In this article, we study the class of surfaces of revolution in the 3-dimensional Lorentz-Minkowski space with nonvanishing Gauss curvature whose position vector x satisfies the condition ΔIIIx = Ax, where A is a square matrix of order 3 and ΔIII denotes the Laplace operator of the second fundamental form III of the surface. We show that such surfaces are either minimal or pseudospheres of a real or imaginary radius.

math.GM↗

arXiv2022

Ruled and quadric surfaces of finite Chen-type

In this paper, we study ruled surfaces and quadrics in the 3-dimensional Euclidean space which are of finite $III$-type, that is, they are of finite type, in the sense of B.-Y. Chen, with respect to the third fundamental form. We show that helicoids and spheres are the only ruled and quadric surfaces of finite $III$-type, respectively.

math.DG↗

arXiv2022

Tubular Surfaces Whose Gauss Map N Satisfies $Δ^{II}N = ΛN$

In this paper, we consider tubes in the Euclidean 3-space whose Gauss map N is of coordinate finite II-type, i.e., the position vector N satisfies the relation $Δ^{II}N = ΛN$, where $Δ^{II}$ is the Laplace operator with respect to the second fundamental form I of the surface and $Λ$ is a square matrix of order 3. We show that circular cylinders are the only class of surfaces mentioned above of coordinate finite I-type Gauss map.

math.GM↗

arXiv2021

Some properties of surfaces of finite III-type

In this paper, we firstly investigate some relations regarding the first and the second Laplace operators corresponding to the third fundamental form III of a surface in the Euclidean space E3. Besides, we introduce the finite Chen type surfaces of revolution with nonvanishing Gauss curvature with respect to the third fundamental form. We present a special case of this family of surfaces of revolution in E3, namely, surfaces of revolution with R is constant, where R denotes the sum of the radii of the principal curvature of a surface.

math.DG↗

arXiv2020

Tubes of finite $II$-type Gauss map

In this paper, we continue the classification of finite type Gauss map surfaces in the 3-dimensional Euclidean space $\mathbb{E}^{3}$. We present an important family of surfaces, namely, tubes in $\mathbb{E}^{3}$. We show that the Gauss map of a tube is of an infinite type corresponding to the second fundamental form.

math.GM↗

arXiv2020

Quadric surfaces of coordinate finite type Gauss map

We study quadric surfaces in the 3-dimensional Euclidean space which are of coordinate finite type Gauss map with respect to the second fundamental form $II$, i.e., their Gauss map vector $\boldsymbol{n}$ satisfies the relation $Δ^{II}\boldsymbol{n}=\varLambda \boldsymbol{n}$, where $Δ^{II}$ denotes the Laplace operator of the second fundamental form $II$ of the surface and $\varLambda$ is a square matrix of order 3. We show that helicoids and spheres are the only class of surfaces mentioned above satisfying $Δ^{II}\boldsymbol{n}=\varLambda \boldsymbol{n}$.

math.GM↗

arXiv2019

Linear Optimization of Polynomials and Rational Functions over Boxes

In this paper, we investigate the problem of finding tight linear lower bounding functions for multivariate polynomials over boxes. These functions are obtained by the expansion of polynomials into Bernstein form and using the linear least squares function. Convergence properties of the given polynomials to their lower bounds are shown with respect to raising the degree, width of the box and subdivision. Subsequently, we provide a new method for constructing an affine lower bounding function for a multivariate rational function based on the Bernstein control points, the convex hull of a non-positive polynomial $s$ and degree elevation. Numerical comparisons with the well known Bernstein constant lower bounding function are finally given.

math.OC↗

arXiv2019

Surfaces of revolution of finite III-type

In this paper, we consider surfaces of revolution in the 3-dimensional Euclidean space E3 with nonvanishing Gauss curvature. We introduce the finite Chen type surfaces concerning the third fundamental form of the surface. We present a special case of this class of surfaces of revolution in E3, namely, surfaces of revolution where the sum of the radii of the principal curvature R is constant.

math.GM↗

arXiv2019

Optimization and Positivity Certificates of Rational Functions using Bernstein Form

Rational functions of total degree $l$ in n variables have a representation in the Bernstein form defined over $n$ dimensional simplex. The range of a rational function is bounded by the smallest and the largest rational Bernstein coefficients over a simplex. Convergence properties of the bounds to the range are reviewed. Algebraic identities certifying the positivity of a given rational function over a simplex are given. Subsequently, a bound established in this work does not depend on the given dimension.

math.OC↗

arXiv2019

Tubes of coordinate finite type Gauss map in the Euclidean 3-space

In this paper, we consider tubes in the Euclidean 3-space whose Gauss map n is of coordinate finite I-type, i.e., the position vector n satisfies the relation ΔIn = Λn, where ΔI is the Laplace operator with respect to the first fundamental form I of the surface and Λ is a square matrix of order 3. We show that circular cylinders are the only class of surfaces mentioned above of coordinate finite I-type Gauss map.

math.DG↗

arXiv2019

Tubes of finite Chen-type

In this paper, we consider surfaces in the 3-dimensional Euclidean space E3 which are of finite III-type, that is, they are of finite type, in the sense of B.-Y. Chen, corresponding to the third fundamental form. We present an important family of surfaces, namely, tubes in E3 .We show that tubes are of infinite III-type.

math.GM↗

arXiv2019

Ruled surfaces of finite type with respect to the second fundamental form

In this article, we consider surfaces in the 3-dimensional Euclidean space E3 without parabolic points which are of finite II-type, that is, they are of finite type, in the sense of B.-Y. Chen, corresponding to the second fundamental form. We study an important family of surfaces, namely, ruled surfaces in E3. We show that ruled surfaces are of infinite II-type.

math.GM↗

arXiv2019

Translation surfaces of coordinate finite type

We consider translation surfaces in the 3-dimensional Euclidean space which are of coordinate finite type with respect to the third fundamental form $III$, i.e. their position vector $x$ satisfies the relation $Δ^{III}x = Λx$, where $Λ$ is a square matrix of order 3. We show that Sherk's minimal surface is the only translation surface satisfying $Δ^{III}x = Λx$.

math.GM↗

arXiv2019

Anchor rings of finite type Gauss map in the Euclidean 3-space

In this article, we continue the classification of finite type Gauss map surfaces in the Euclidean 3-space E3 with respect to the first fundamental form by studying a subclass of tubes, namely the anchor rings. We show that anchor rings are of infinite type Gauss map.

math.DG↗

arXiv2016

Ruled and quadric surfaces in the 3-dimensional Euclidean space satisfying $Δ^{III}\boldsymbol{x} = \varLambda \boldsymbol{x}$

We consider ruled and quadric surfaces in the 3-dimensional Euclidean space which are of coordinate finite type with respect to the third fundamental form $III$, i.e., their position vector $\boldsymbol{x}$ satisfies the relation $Δ^{III}\boldsymbol{x}=\varLambda \boldsymbol{x}$ where $\varLambda $ is a square matrix of order 3. We show that helicoids and spheres are the only surfaces in $E^3$ satisfying the preceding relation.

math.DG↗

arXiv2015

On Surfaces of finite Chen-type

We investigate some relations concerning the first and the second Beltrami operators corresponding to the fundamental forms I, II, III of a surface in the three-dimensional Euclidean space and we study surfaces which are of finite type in the sense of B.-Y. Chen with respect to the fundamental forms II and III.

math.DG↗