The Sombor index (\(SO\)) and the modified Sombor index (\(^mSO\)) are two closely related vertex-degree-based graph invariants. Both were introduced in the 2020s, and have already found a variety of chemical, physicochemical, and network-theoretical applications. In this paper, we examine the product \(SO \cdot {^mSO}\) and determine its main properties. It is found that the structure-dependence of this product is fully different from that of either \(SO\) or \(^mSO\). Lower and upper bounds for \(SO \cdot {^mSO}\) are established and the extremal graphs are characterized. For connected graphs, the minimum value of the product \(SO \cdot {^mSO}\) is the square of the number of edges. In the case of trees, the maximum value pertains to a special type of eclipsed sun graph, trees with a single branching point.

This present paper introduces two new subclasses of p-valent functions. The coefficient bounds and Fekete-Szego inequalities for the functions in these classes are also obtained.

This paper deals with the maximum number of limit cycles bifurcating from the degenerate centre
\[ \dot{x}=-y(3x^2+y^2),\: \dot{y}=x(x^2-y^2), \]
when we perturb it inside a class of all homogeneous polynomial differential systems of degree \(5\). Using averaging theory of second order, we show that, at most, five limit cycles are produced from the periodic orbits surrounding the degenerate centre under quintic perturbation. In addition, we provide six examples that give rise to exactly \(5, 4, 3, 2, 1\) and \(0\) limit cycles.

In this paper, we work on expanding the Jensen \((\Gamma_{1},\Gamma_{2})\)-function inequalities by relying on the general Jensen \((\eta,\lambda)\)-functional equation with \(3k\)-variables on the complex Banach space. That is the main result of this.

In this paper, we concentrate on norms of derivations implemented by self-adjoint operators. We determine the upper and lower norm estimates of derivations implemented by self-adjoint operators.The results show that the knowledge of self-adjoint governs the quantum chemical system in which the eigenvalue and eigenvector of a self-adjoint operator represents the ground state energy and the ground state wave function of the system respectively.

This manuscript proposed high-precision methods for obtaining solutions for nonlinear models. The method uses the Newton method as its predictor and an iterative function that involves the perturbed Newton method with the quotient of two power series as the corrector function. The theoretical analysis of convergence indicates that the methods class is of convergence order four, requiring three functions evaluation per cycle. The computation performance comparison with some existing methods shows that the developed methods class has perfect precision.

The purpose of this study is to present a generalized class of estimators using the three-stage Optional Randomized Response Technique (ORRT) in the presence of non-response and measurement errors on a sensitive study variable. The proposed estimator makes use of dual auxiliary information. The expression for the bias and mean square error of the proposed estimator are derived using Taylor series expansion. The proposed estimator’s applicability is proven using real data sets. A numerical study is used to compare the efficiency of the proposed estimator with adapted estimators of the finite population mean. The suggested estimator performs better than adapted ordinary, ratio, and exponential ratio-type estimators in the presence of both non-response and measurement errors. The efficiency of the proposed estimator of population mean declines as the inverse sampling rate, non-response rate, and sensitivity level of the survey question increase.

Second-order macroscopic vehicular traffic flow models are categorized under two broad headings based on the direction of their characteristics. Faster-than-vehicle waves are often called isotropic models vis-\'{a}-vis anisotropic models with slower-than-vehicle characteristic speed. The dispute on the supremacy among these families of models is the motivation for this paper. This paper compares and contrasts six distinctive second-order macroscopic models using a numerical simulation and analysis. Three models are characterized by faster-than-vehicle waves with their corresponding anisotropic counterparts. Simulation results on the formation of deceleration waves and the dissolution of acceleration fans are presented to graphically compare the wave profiles of the selected isotropic and anisotropic traffic models. Observably, these opposing models can all characterize these physical traffic phenomena to the same degree. Thus, faster characteristic speed conceptualization of second-order macroscopic equations does not tantamount to model failure but rather lies in the explanation of this property.

In this paper, the comparative study of Fadugba-Falodun Scheme (FFS) and the Improved Euler’s Method (IEM) is presented. IEM and FFS have been used successfully for the solution of second order ordinary differential equation. FFS is a numerical method recently proposed by means of an interpolating function involving a transcendental function of exponential type. In order to discuss the efficiency and accuracy of the two methods, an illustrative example has been presented in the context of the Exact Solution (ES) and the absolute relative errors computed at each mesh point of the integration interval under consideration. The numerical results show that there is no significant difference between the FFS and ES, unlike its counterpart IEM. Hence, FFS is a good numerical method for the solution of the second order initial value problem in ordinary differential equations. All calculations have been carried out via MATLAB (R2014a) in double precision.

This paper is devoted to a study of the numerical solution of the thermoelastic model describing the contact problem between the body and a rigid foundation that is thermally conducting. The linear thermoelastic constitutive law describes the behavior of the material. The contact is frictionless and described with Signorini’s condition and a thermal contact condition when the heat exchange coefficient depends on the contact pressure. We aim to present a detailed description of the numerical modeling of the problem. on this end, we use a penalty method to approximate the constraints. Finally, we provide numerical simulations to study a two-dimensional example and compare the penalty problem with the Lagrangian one.