On graded Jgr -classical 2-absorbing submodules of graded modules over graded commutative rings

Let G be an abelian group with identity e e . Let R be a G-graded commutative ring with identity 1, and M M be a graded R-module. In this paper, we introduce the concept of graded J g r {J}_{gr} -classical 2-absorbing submodule as a generalization of a graded classical 2-absorbing submodule. We give some results concerning of these classes of graded submodules. A proper graded submodule C C of M M is called a graded J g r {J}_{gr} -classical 2-absorbing submodule of M M , if whenever r g , s h , t i ∈ h ( R ) {r}_{g},{s}_{h},{t}_{i}\in h\left(R) and x j ∈ h ( M ) {x}_{j}\in h\left(M) with r g s h t i x j ∈ C {r}_{g}{s}_{h}{t}_{i}{x}_{j}\in C , then either r g s h x j ∈ C + J g r ( M ) {r}_{g}{s}_{h}{x}_{j}\in C+{J}_{gr}\left(M) or r g t i x j ∈ C + J g r ( M ) {r}_{g}{t}_{i}{x}_{j}\in C+{J}_{gr}\left(M) or s h t i x j ∈ C + J g r ( M ) , {s}_{h}{t}_{i}{x}_{j}\in C+{J}_{gr}\left(M), where J g r ( M ) {J}_{gr}\left(M) is the graded Jacobson radical.

Global existence and dynamic structure of solutions for damped wave equation involving the fractional Laplacian

We consider strong damped wave equation involving the fractional Laplacian with nonlinear source. The results of global solution under necessary conditions on the critical exponent are established. The existence is proved by using the Galerkin approximations combined with the potential well theory. Moreover, we showed new decay estimates of global solution.

A statistical study of COVID-19 pandemic in Egypt

The spread of the COVID-19 started in Wuhan on December 31, 2019, and a powerful outbreak of the disease occurred there. According to the latest data, more than 165 million cases of COVID-19 infection have been detected in the world (last update May 19, 2021). In this paper, we propose a statistical study of COVID-19 pandemic in Egypt. This study will help us to understand and study the evolution of this pandemic. Moreover, documenting of accurate data and taken policies in Egypt can help other countries to deal with this epidemic, and it will also be useful in the event that other similar viruses emerge in the future. We will apply a widely used model in order to predict the number of COVID-19 cases in the coming period, which is the autoregressive integrated moving average (ARIMA) model. This model depicts the present behaviour of variables through linear relationship with their past values. The expected results will enable us to provide appropriate advice to decision-makers in Egypt on how to deal with this epidemic.

Two strongly convergent self-adaptive iterative schemes for solving pseudo-monotone equilibrium problems with applications

The aim of this paper is to propose two new modified extragradient methods to solve the pseudo-monotone equilibrium problem in a real Hilbert space with the Lipschitz-type condition. The iterative schemes use a new step size rule that is updated on each iteration based on the value of previous iterations. By using mild conditions on a bi-function, two strong convergence theorems are established. The applications of proposed results are studied to solve variational inequalities and fixed point problems in the setting of real Hilbert spaces. Many numerical experiments have been provided in order to show the algorithmic performance of the proposed methods and compare them with the existing ones.

New class of operators where the distance between the identity operator and the generalized Jordan ∗-derivation range is maximal

A new class of operators, larger than ∗ \ast -finite operators, named generalized ∗ \ast -finite operators and noted by Gℱ ∗ ( ℋ ) {{\mathcal{G {\mathcal F} }}}^{\ast }\left({\mathcal{ {\mathcal H} }}) is introduced, where: Gℱ ∗ ( ℋ ) = { ( A , B ) ∈ ℬ ( ℋ ) × ℬ ( ℋ ) : ∥ T A − B T ∗ − λ I ∥ ≥ ∣ λ ∣ , ∀ λ ∈ C , ∀ T ∈ ℬ ( ℋ ) } . {{\mathcal{G {\mathcal F} }}}^{\ast }\left({\mathcal{ {\mathcal H} }})=\{(A,B)\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }})\times {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}):\parallel TA-B{T}^{\ast }-\lambda I\parallel \ge | \lambda | ,\hspace{0.33em}\forall \lambda \in {\mathbb{C}},\hspace{0.33em}\forall T\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }})\}. Basic properties are given. Some examples are also presented.

A new iteration method for the solution of third-order BVP via Green's function

In this study, a new iterative method for third-order boundary value problems based on embedding Green’s function is introduced. The existence and uniqueness theorems are established, and necessary conditions are derived for convergence. The accuracy, efficiency and applicability of the results are demonstrated by comparing with the exact results and existing methods. The results of this paper extend and generalize the corresponding results in the literature.

Some results on generalized finite operators and range kernel orthogonality in Hilbert spaces

Let ℋ {\mathcal{ {\mathcal H} }} be a complex Hilbert space and ℬ ( ℋ ) {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}) denotes the algebra of all bounded linear operators acting on ℋ {\mathcal{ {\mathcal H} }} . In this paper, we present some new pairs of generalized finite operators. More precisely, new pairs of operators ( A , B ) ∈ ℬ ( ℋ ) × ℬ ( ℋ ) \left(A,B)\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }})\times {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}) satisfying: ∥ A X − X B − I ∥ ≥ 1 , for all X ∈ ℬ ( ℋ ) . \parallel AX-XB-I\parallel \ge 1,\hspace{1.0em}\hspace{0.1em}\text{for all}\hspace{0.1em}\hspace{0.33em}X\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}). An example under which the class of such operators is not invariant under similarity orbit is given. Range kernel orthogonality of generalized derivation is also studied.

A revised model for the effect of nanoparticle mass flux on the thermal instability of a nanofluid layer

A revised model of the nanoparticle mass flux is introduced and used to study the thermal instability of the Rayleigh-Benard problem for a horizontal layer of nanofluid heated from below. The motion of nanoparticles is characterized by the effects of thermophoresis and Brownian diffusion. The nanofluid layer is confined between two rigid boundaries. Both boundaries are assumed to be impenetrable to nanoparticles with their distribution being determined from a conservation condition. The material properties of the nanofluid are allowed to depend on the local volume fraction of nanoparticles and are modelled by non-constant constitutive expressions developed by Kanafer and Vafai based on experimental data. The results show that the profile of the nanoparticle volume fraction is of exponential type in the steady-state solution. The resulting equations of the problem constitute an eigenvalue problem which is solved using the Chebyshev tau method. The critical values of the thermal Rayleigh number are calculated for several values of the parameters of the problem. Moreover, the critical eigenvalues obtained were real-valued, which indicates that the mode of instability is via a stationary mode.

Solving two-dimensional nonlinear fuzzy Volterra integral equations by homotopy analysis method

The purpose of the paper is to find an approximate solution of the two-dimensional nonlinear fuzzy Volterra integral equation, as homotopy analysis method (HAM) is applied. Studied equation is converted to a nonlinear system of Volterra integral equations in a crisp case. Using HAM we find approximate solution of this system and hence obtain an approximation for the fuzzy solution of the nonlinear fuzzy Volterra integral equation. The convergence of the proposed method is proved. An error estimate between the exact and the approximate solution is found. The validity and applicability of the HAM are illustrated by a numerical example.

On almost e-ℐ-continuous functions

This work is concerned with a new class of functions called almost e e - ℐ {\mathcal{ {\mathcal I} }} -continuous functions containing the class of almost e e -continuous functions. This notion is stronger than almost δ β ℐ \delta {\beta }_{{\mathcal{ {\mathcal I} }}} -continuous functions and is weaker than both almost e e -continuous functions and e e - ℐ {\mathcal{ {\mathcal I} }} -continuous functions. Relationships between this new class and other classes of functions are investigated and some characterizations of this new class of functions are studied.