In this paper, we make use of the concept of fractional q-calculus to introduce two new classes of biBazilevic functions involving q-Ruscheweyh differential operator that are subordinate to Gegenbauer polynomials and q-analogue of hyperbolic tangent functions. This study explores the characteristics and behaviors of these functions, offering estimates for the modulus of the initial Taylor series coefficients a2 and a3 within this specific class and their various subclasses. Additionally, this study delves into the classical Fekete-Szeg¨o functional problem concerning functions f that are part of our newly defined class and several of their subclasses.

Our article focuses on the study of quaternions topic introduced by Hamilton. Quaternions are a generalization of complex numbers and have multiple applications in mathematical physics. Another application of quaternions is robotics because what generalizes the imaginary axis is the family i, j, k modeling Euler angles and rotations in space. The first part of the article we recall the different definitions of how the algebra of quaternions is well constructed. The main results are given in the third part and concern: spatial quaternionics rectifying-direction (sqRD) curves and and spatial quaternionic rectifying-donor (sqRDnr) curves. We study a new tip of unit speed associated curves in E 3 , which is also used in robotic systems and kinematics, like a spatial quaternionic rectifying-direction curve and spatial quaternionic rectifying-donor curve. Then, we achieve qualification for the curves. Moreover, we present applications of spatial quaternionic rectifying-direction to some specific curves like helix, slant helix, Salkowski and anti-Salkowski curves or rectifying curves. In addition, we establish different theorems which generalize the results obtained on the quaternionic curves in Q. Then, we give some examples are finally discussed. Consequently, Our paper is centered around theoretical analysis in geometry rather than experimental investigations.

Let λ and μ be monotonically increasing strictly positive sequences, i.e., the speeds of convergence. Earlier the notions of boundedness, convergence, and zero-convergence with the speed are known. In this paper, the notions of paranormed boundedness, convergence, and zero-convergence with speed have been defined. The matrix transforms from the sets of bounded, convergent, and zero-convergent sequences with the speed λ into the sets of paranormed bounded, paranormed convergent, and paranormed zero-convergent sequences with the speed μ are studied.

In this article, the focus is on exploring planar piecewise smooth quadratic systems, a significant class of dynamical systems that exhibit changes in behavior under different conditions but with smooth transitions between these states. The main objective is to introduce a second-order averaged method designed specifically to identify limit cycles, repeating trajectories in a system's phase space indicative of periodic behavior. This innovative method not only allows for the detection of these cycles but also quantifies their number, providing a deeper understanding of the system's long-term behavior. The paper highlights its applicability by demonstrating the maximum number of limit cycles that can exist in two distinct systems, offering valuable insights into the dynamics of such systems and contributing to the broader field of mathematical modeling and analysis.

This paper investigated the stability of the dynamical behavior of the susceptible (S), infectious (I) and recovered (R) (SIR) disease epidemic model with intracellular time delay that is unable to stabilize the unstable interior non-hyperbolic equilibrium. The study employed characteristics and bifurcation methods for investigating conditions of stability and instability of the SIR disease epidemic model using the dimensionless threshold reproduction value 𝑅0 for the disease-free equilibrium (DFE) point and the endemic equilibrium point. The study confirms that disease-free equilibrium (DFE) point and the endemic equilibrium point cannot coexist simultaneously. The paper equally investigated the local stability analysis of the reduced nonlinear SIR disease epidemic delay model when at least one of the characteristic roots has zero real parts while every other eigenvalue(s) has negative real parts. The result of the analysis of the model showed that the conditions for Hopf bifurcation obtained from the behavior of the systems are sufficient but not necessary since the model is unable to stabilize the unstable interior non-hyperbolic equilibrium. Specifically, the direction of Hopf bifurcation, the stability behavior and the period of the bifurcating periodic solutions of the interior nonhyperbolic equilibrium of the infectious disease model were explicitly determined using methods of the normal form concept (NFC) and the center manifold theorem (CMT) to investigate the transformed reduced operator differential equation (OpDE). The contribution of this paper is based on applications to assess the effectiveness of different control strategies of parameter values for stability properties of infectious disease models and can be found useful to bio-mathematicians, ecologists, biologists and public health workers for decision-making. Finally, a numerical example to verify the analytical finding was performed using the MATLAB software.

The significance of Multistep Methods with constant coefficients and their application in addressing various Natural Science issues is universally acknowledged. Dahlquist conducted foundational research on these methods. Building on this, this text outlines certain developments in these theories and their use in solving Ordinary Differential, Volterra Integral, and Volterra Integro-Differential Equations. Advanced (forward-jumping) methods are examined, with a comparison made between the outcomes of these methods and those established by Dahlquist. Additionally, the study focuses on advanced second derivative multistep methods, demonstrating that the stable variants of these advanced methods yield greater accuracy. Furthermore, the research identifies the maximum achievable degree for the advanced methods. The constructed methods have been utilized to tackle model problems, and the resulting findings are presented here for illustration.

In this paper, we discuss a one-parameter generalization of Jacobsthal numbers that preserves the recurrence relation with arbitrary initial conditions. We introduce generalized Jacobsthal-Lucas-like numbers, which are simple associations of generalized Jacobsthal numbers. Consequently, we give some new and well-known identities. Furthermore, we propose integral representations of these numbers associated with generalized Jacobsthal and Jacobsthal-Lucas-like numbers. Our results not only generalize the integral representations of the Jacobsthal and Jacobsthal-Lucas numbers but also apply to all one-parameter generalizations of Jacobsthal numbers.

Quadric surfaces of finite type are a class of three-dimensional surfaces in geometry that are defined by second-degree equations in three variables, which are an essential part of the study of conic sections, and they exhibit a wide range of interesting geometric properties and real-world applications. This paper explores the intriguing domain of quadric surfaces, particularly emphasizing those of finite type. This will start by defining the ideas of the second Laplace-Beltrami operators, involving a surface's second fundamental form (II) in the Euclidean space E3. Then, we characterize the coordinate finite type quadrics involving the second fundamental form.

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This paper aims to introduce new classes of square mean (ω, c)-periodic limit-type stochastic processes. The main structural characterizations and some qualitative properties of the introduced classes are established, as well as their invariance under the actions of convolution product and the composition principle.

Network Marketing is a business tool that builds a network of business partners or distributors by directly selling products and services through a word-of-mouth marketing strategy. NM graph is a graphical representation of the network marketing industry. Just like a graph consists of vertices and edges, network marketing consists of distributors/ business partners which represent the vertices of the NM graph, and edges determine the genealogy of the network. In this paper, we explore the structure of an NM graph by graph parameters and see how the graph reflects various properties of a network of people. It is seen that an NM graph is a rooted binary tree. The measure of the influence of a network and the measure of profitability of a network is determined. It is seen that the objective to maximize profit by allocating different stakes to different persons is a typical case of an assignment problem. The genealogy of a network can be determined by studying the adjacency matrix of the NM digraph. Graph parameter like eccentricity represents the strongest person in the network earning maximum profitability. The domination number acts as a measure of the stability of an NM graph.

In this note, we propose a new proof of existence of 𝐿∞ strong trace of entropy solution for multidimensional degenerate parabolic-hyperbolic equation in a bounded domain Ω ⊂ IRℓ reached by 𝐿1 convergence. The proof is based on using of concept of quasi solution.

In this paper, new vector fields are defined along a curve lying on an orientable hypersurface with nonvanishing extended Darboux curvatures of the second kind, and some new planes and curves are introduced using these vector fields in Euclidean 4-space. It is also shown that in 4-dimensional space, these new planes play the same role that the Darboux vector W = 𝜏𝑔T − 𝜅𝑛V + 𝜅𝑔N plays in Euclidean 3-space. Besides, developable and non-developable ruled hypersurfaces associated with these new vector fields are defined.

In this paper, we delve into the fascinating realm of quadric surfaces, with a specific focus on those of finite type. We first define relations regarding the first and the second Laplace operators corresponding to the second fundamental form II of a surface in the Euclidean space E 3 . We focus on quadric surfaces from two sides, on one side, we study quadric surfaces of the first kind whose Gauss map N satisfies a relation of the form ΔΙΙn = AN, where A is a square matrix of order 3 and ∆ is the second Laplace operator. On the other side, we study quadric surfaces of the second kind with the same property.

Transformation such as integral transform is needed to obtain the exact solutions for linear ordinary differential equations (ODEs) with constant coefficients of higher orders. MAHA transformation exact solution of ODEs is simpler and easier than the previous with two parameters. The major steps of this transform are applying the MAHA transform on the given equation followed by taking the inverse transform. The general steps in numerical solutions involve defining the ODE as a function, defining initial conditions and the range of the independent variable, using an appropriate ODE solver function, and calling the solver and plotting the solution using a programming language. The exact and analytical solutions are validated. Both methods are easy and simple to be deployed in computing scientific applications such as nuclear physics and medical applications.

The aim of this research to study the approximation of functions in the space- 𝐿𝑝 by the “algebraic polynomial” in terms of the” average modulus” of the k-order also, we will estimate the degree of the (O-S- A), (means one – sided approximation) in term of averaged modulus where all the results which number is eleven we need to prove the main theorem that (the degree of best (O-S- A) of 𝑓 by trigonometric polynomials of order 𝑛 in 𝐿 𝑝 (𝑋 ), (𝐸̃𝑛 (𝑓)𝑝) ) is less than or equal to (Averaged modulus of smoothness of 𝑓 of order- 𝑘, (𝜏𝑘 (𝑓 , 1 𝑛 ) 𝑝 ) ) have been proven, It has also been proven the converse theorem for the main theorem in this research.

In this note, we investigate about discrete entropy solution of scalar conservation law. We establish uniqueness of finite volume approximate solution to scalar conservation laws with a non Lipschitz flux function in the whole space. Our arguments are based on properties of moduli of continuity of the components of the numerical flux.

A new method for solving the singular one-dimensional boundary value problem with a strong degeneracy is proposed in this paper. In the case of the strong degeneration of the differential equation, the boundary condition is set only at one end of the interval. This method is based on the use of the polynomial and non-polynomial Lagrangian and Hermitian type local splines and the variational method. The use of splines of Hermitian type with the first level is convenient if it is needed to obtain simultaneously a solution and the first derivative of the solution at the grid nodes. Next, it is possible to construct the solution between the grid nodes using the same spline approximation formulas. The non-polynomial splines help us to construct a more accurate solution. The results of solving a one-dimensional boundary value problem with strong degeneracy are presented in this paper.

The paper deals with the G 2 continuity for planar curves. The G 2 continuity is considered as a superior quality of curvature, which is often sought after in high-precision designs and industrial applications. It ensures a perfectly smooth transition between different parts of a surface or curve, which can improve the functionality, aesthetics, and durability of the finished object. This article describes an algorithm to achieve a G 2 junction between two sets of data –point, tangent, curvature–. The junction is based on a rational Bézier curve defined by control mass points. The control mass points generalize those of classical Bézier curves defined with weighted points with no negative weights. It is necessary as vectors and points with negative weights are coming out while applying homographic parameter change on a curve segment or converting any polynomial function into a rational Bézier representation. Here, from two sets of data –point, tangent and curvature–, a Bézier curve of degree n is built. This curve is described by control mass points. In most situations, the best degree for G 2 connection of those two sets equals 5.

In this paper, we define Riemann-Liouville generalized fractional integral. Moreover, we obtained some significant inequalities for Riemann-Liouville generalized fractional integrals.

The continuation of general Dirichlet series to meromorphic functions in the complex plane remains an outstanding problem. It has been completely solved only for Dirichlet L-series. A sufficient condition for the general case exists, however it is impossible to verify that it is fulfilled. We solve this problem here for another class of general Dirichlet series, namely those series which are obtained from infinite Blaschke products by a particular change of variable. This is a source of examples of general Dirichlet series with infinitely many poles. An interesting new case is now revealed, in which the singular points of the extended function form a continuum. We take a closer look at the case of Dirichlet series with natural boundary and give examples of such series. Some figures illustrate the theory.

The delta-Birnbaum-Saunders distribution is a relatively novel concept that combines the Birnbaum-Saunders with the binomial distributions. As a result, datasets containing both positive and zero values conform well with this distribution, making it particularly intriguing. Additionally, coefficients of variation are among the important statistics for comparing the dispersion of data. Therefore, there is an interest in proposing methods for constructing confidence intervals, which play a crucial role in statistical inference for the difference between two coefficients of variation in delta-Birnbaum-Saunders distributions. There are four methods proposed: the method of variance estimates recovery, the bootstrap confidence interval, the generalized confidence interval based on the variance stabilized transformation, and the generalized confidence interval based on the Wilson score method. All the methods are compared in terms of performance using coverage probability and average width through Monte Carlo simulations. The simulation results show that the bootstrap confidence interval performs similarly to the method of variance estimate recovery, except in cases where the shape parameter is large. In addition, it is shown that the generalized confidence interval based on variance stabilized transformation and the generalized confidence interval based on the Wilson score method yield similar results and demonstrate the highest efficiency compared to other methods. Finally, two datasets are used to illustrate the application of the proposed confidence intervals.

In this paper, a random predator-prey system is established, based on which cooperative hunting and regional switching are considered. Firstly, the existence and uniqueness of the global positive solution of the model are proved. Secondly, the sufficient conditions for extinction and stationary distribution are obtained by using Lyapunov function. Finally, numerical simulation is used to demonstrate the correctness of the conclusion.