Quadrics of finite chen type regarding the second fundamental form

Al-Zoubi, Hassan

Indian Journal of Mathematics

Quadrics of finite chen type regarding the second fundamental form

Author(s): Hassan Al-Zoubi Hassan Al-Zoubi Department of Mathematics,
Al-Zaytoonah University of Jordan,
P.O. Box 130, Amman, Jordan 11733,
Jordan. dr.hassanz@zuj.edu.jo

Vol. 67, No. 1 (2025) • pp. 25–38

Published 27-04-2026 Article History Received 15-05-2025 Revised 26-02-2026 Published 27-04-2026

Add to Mendeley

Twitter / X LinkedIn Facebook

Export Citation to RIS Export Citation to BibTeX Export Citation to EndNote

Abstract

This paper focuses on the study of quadric surfaces in 3-dimensional Euclidean space that are of finite II-type, a concept introduced by B.-Y. Chen regarding the 2nd fundamental form II. It was shown that the only ones of the class of quadrics in 3-space according to the definition of finite Chen type are the spheres.

Keywords

Quadrics in 3-space, Surfaces of finite Chen-type , Second Beltrami operator.

2020 Mathematics Subject Classification

53A45, 53A05.

More from the issue

HIV/AIDS modeling of sexual and gender-based violence in conflict-affected populations with therapeutic intervention: global stability and sensitivity analysis

Author: Abdelkadir Muzey Mohammed, Habtu Alemayehu Atsbaha, Yohannes Yirga Kefela, Woldegebriel Assefa Woldegerima, Kiros Tedla Gebrehiwot

Armed conflicts intensify HIV/AIDS transmission by increasing sexual and gender-based violence (SGBV) and disrupting healthcare systems, particularly in low-income regions such as Sub-Saharan Africa. These intersecting factors substantially accelerate the progression of existing infections and increase the incidence of new cases, yet the combined impact of conflict-related SGBV and biomedical interventions remains insufficiently quantified. To address this, we developed a deterministic compartmental model to investigate HIV transmission dynamics, explicitly incorporating rape-related infections, post-exposure prophylaxis (PEP), and antiretroviral therapy (ART), and evaluated their interactions across varying conflict intensity scenarios. The model assessed disease-free and endemic equilibria, computed the basic reproduction number ($\mathcal{R}_0$), and conducted global stability and sensitivity analyses. Numerical simulations examined the influence of SGBV prevalence, treatment coverage, and escalating conflict on HIV transmission. Results demonstrated that HIV transmission escalates sharply with increasing SGBV and conflict intensity. Timely initiation of PEP and ART substantially reduces new infections, with PEP exerting the greatest impact among rape-exposed populations. Combined PEP and ART interventions produced synergistic reductions in $\mathcal{R}_0$, whereas treatment interruptions facilitated persistent endemicity. These findings underscore effective HIV control in conflict-affected populations requires integrated strategies that concurrently reduce SGBV exposure and maintain treatment access. The model provides a quantitative framework for epidemic preparedness, evidence-based policy development, and survivor-centered interventions, emphasizing the urgent need for targeted public health strategies in conflict-impacted regions.

View Details →

On a generalization of $D3$-modules

Author: Papa Cheikhou Diop, Modou Seye, Sanjeev Kumar Maurya, Mamadou Barry

$ R $ is supposed to be an associative ring with identity and $ M $ be a unital left $R$-module. In this paper, we present the concept of a $D31$-module that generalizes the concept of a $D3$-module introduced in [14] and studied in [25] . A module $M$ is termed as $ D31 $-module if, whenever $ W $ and $ X $ are summands of $ M $ with $ W + X = M $ and $ W $ is cosingular, we have $ W \cap X $ is a direct summand of $ M $. We explore fundamental properties of these modules and establish that the class of rings $ R $ such that every $ D31 $-module is also a $ D3 $-module coincides precisely with the class of COSP-rings. Additionally, we examine the connections between $ D31 $-modules and other related module classes.

View Details →

A grid-based method for solving the initial-boundary value problem for a multidimensional third-order hyperbolic partial differential equation

Author: M. KH. Beshtokov

The initial boundary value problem for a multidimensional partial differential equation of the third order of hyperbolic type, which serves as a mathematical model of the movement of moisture and salts in soils, is studied. For an approximate solution of the problem, the original equation is reduced to an integro-differential equation with a small parameter. It is shown that when the small parameter tends to zero, the solution of the resulting modified problem converges to the solution of the original problem. A. A. Samarsky's locally one-dimensional difference scheme is constructed. An a priori estimate is obtained using the method of energy inequalities, from which the uniqueness and stability of the solution of the scheme follow, the convergence of the solution of the locally one-dimensional difference scheme to the solution of the modified differential problem is proved.

View Details →

ISO $C2$ and ISO $C3$-Modules

Author: Surya Prakash, Ajim Uddin Ansari, Anurag Shukla

In this paper, we introduce and study the concept of iso $C2$-modules and iso $C3$-modules as generalizations of the direct injective modules. We prove that $M$ is an iso $C3$-module if for any two isosimple submodules $A, B$ of $M$ with $A\cong B, B\subseteq ^\oplus M$ and $A\cap B = 0$ implies that $A\subseteq ^ \oplus M$. Also, if $M$ is an iso $C3$- module then $Im f\subseteq ^\oplus B$, whenever $M = A\oplus B$ with $A, B$ isosimple and $f: A\rightarrow B$ be an $R$-homomorphism. Furthermore, we provide a number of properties and characterizations of these new classes of modules.

View Details →

Symplectic Newton–Cotes product Integration methods for fractional Hamiltonian systems

Author: Shruti Tiwari

This work presents a novel class of composite product-integration closed Newton--Cotes (CNC) integrators specifically designed for Hamiltonian systems governed by Caputo fractional derivatives of order $1 < \alpha < 2$. The proposed methodology establishes explicit full-history convolution weights while constructing a discrete variational principle within an extended phase space, ultimately yielding a symplectic mapping. Through rigorous analysis, we demonstrate global error bounds of $O(h^{2-\alpha})$ for CNC-2 and $O(h^{4-\alpha})$ for CNC-4 under conventional regularity assumptions. The linear stability characteristics are analyzed via the generating function of convolution weights, while long-term energy drift behavior is quantified. Additionally, efficient implementation strategies are developed that achieve $O(1)$ per-step computational complexity through systematic history truncation and sum-of-exponentials compression techniques. Comprehensive numerical experiments validate the theoretical predictions and demonstrate superior performance relative to existing fractional integrators.

View Details →

Approximation of zeros of nonlinear real functions using a variant of the false position method based on quadratic polynomials

Author: Y. Castillo, S. Correa, R. Ipanaqué, C. Iman, J. Farfán

We present a quadratic variant of the classical False Position Method (VQFPM) for approximating the zeros of nonlinear real functions. The method is based on second-degree Lagrange interpolation using three dynamic points within each iteration interval, which significantly improves the convergence rate without requiring derivatives. We establish the theoretical foundations of the method and derive rigorous bounds for its convergence error. Through numerical experiments implemented in the open-source software Maxima, we demonstrate that VQFPM achieves high precision with fewer iterations than traditional methods, including Newton and Secant. These results highlight VQFPM as a robust and efficient alternative for derivative-free root finding.

View Details →

Allahabad Mathematical Society

Quadrics of finite chen type regarding the second fundamental form

Abstract

Keywords

2020 Mathematics Subject Classification

Full Text / PDF

References