On Hermitian Solutions of the Generalized Quaternion Matrix Equation A X B + C X D = E

The paper deals with the matrix equation A X B + C X D = E over the generalized quaternions. By the tools of the real representation of a generalized quaternion matrix, Kronecker product as well as vec-operator, the paper derives the necessary and sufficient conditions for the existence of a Hermitian solution and gives the explicit general expression of the solution when it is solvable and provides a numerical example to test our results. The paper proposes a unificated algebraic technique for finding Hermitian solutions to the mentioned matrix equation over the generalized quaternions, which includes many important quaternion algebras, such as the Hamilton quaternions and the split quaternions.

Fractional Propagation of Short Light Pulses in Monomode Optical Fibers: Comparison of Beta Derivative and Truncated M- Fractional Derivative

The present study is dedicated to the computation and analysis of solitonic structures of a nonlinear Sasa-Satsuma equation that comes in handy to understand the propagation of short light pulses in the monomode fiber optics with the aid of Beta Derivative and Truncated M- fractional derivative. We employ new direct algebraic technique for nonlinear Sasa-Satsuma equation to derive novel soliton solutions. A variety of soliton solutions are retrieved in trigonometric, hyperbolic, exponential, rational forms. The vast majority of obtained solutions represent the lead of this method on other techniques. The prime advantage of considered technique over the other techniques is that it provides more diverse solutions with some free parameters. Moreover, the fractional behavior of the obtained solutions is analyzed thoroughly by using two and three dimensional graphs. Which shows that for lower fractional orders i.e $\beta=0.1$, the magnitude of truncated M-fractional derivative is greater whereas for increasing fractional orders i.e $\beta=0.7$ and $\beta=0.99$, magnitude remains same for both definitions except for a phase shift in some spatial domain that eventually vanishes and two curves coincide.

Analytical methods: Nonlinear longitudinal wave equation in a magnetoelectro-elastic circular rod, foam drainage and modified Degasperis–Procesi models arising in nonlinear water wave models

We form the analytical solitary wave solutions with the execution of generalized direct algebraic technique on three well-known nonlinear wave models, namely, called foam drainage, longitudinal magnetoelectro-elastic circular rod and modified Degasperis–Procesi equations. The derived solutions are hyperbolic functions in which some are plotted graphically on meticulous values to the parameters which provides the basic knowledge to understand physical significant of these three wave models. The obtained solutions show the efficiency and precision of our scheme. These derived new results prove that our novel technique is awfully effective and can be productive as a instrument solving for sundry other nonlinear evolution equations.

A Novel Construction of Perfect Strict Avalanche Criterion S-box using Simple Irreducible Polynomials

An irreducible polynomial is one of the main components in building an S-box with an algebraic technique approach. The selection of the precise irreducible polynomial will determine the quality of the S-box produced. One method for determining good S-box quality is strict avalanche criterion will be perfect if it has a value of 0.5. Unfortunately, in previous studies, the strict avalanche criterion value of the S-box produced still did not reach perfect value. In this paper, we will discuss S-box construction using selected irreducible polynomials. This selection is based on the number of elements of the least amount of irreducible polynomials that make it easier to construct S-box construction. There are 17 irreducible polynomials that meet these criteria. The strict avalanche criterion test results show that the irreducible polynomial p17(x) =x8 + x7 + x6 + x + 1 is the best with a perfect SAC value of 0.5. One indicator that a robust S-box is an ideal strict avalanche criterion value of 0.5

Cramer’s rule over quaternions and split quaternions: A unified algebraic approach in quaternionic and split quaternionic mechanics

This paper aims to present, in a unified manner, Cramer’s rule which are valid on both the algebras of quaternions and split quaternions. This paper, introduces a concept of v-quaternion, studies Cramer’s rule for the system of v-quaternionic linear equations by means of a complex matrix representation of v-quaternion matrices, and gives an algebraic technique for solving the system of v-quaternionic linear equations. This paper also gives a unification of algebraic techniques for Cramer’s rule in quaternionic and split quaternionic mechanics.

Optical solitons of the paraxial wave dynamical model in kerr media and its applications in nonlinear optics

The solitons and other solutions illustrate nondiffractive and nondispersive spatio-temporal localized packets of wave propagating in the media of optical Kerr. In this paper, solitons, elliptic function and other solutions of dimensionless time-dependent paraxial wave model are constructed via employing three mathematical techniques, namely, the improved simple equation technique, [Formula: see text]-expansion technique and modified extended direct algebraic technique. These wave solutions have key applications and help to understand the physical phenomena of this wave model. By giving appropriate parameter values, different types of solitons structures can be depicted graphically. Several precise solutions and computations have proved the straightforwardness, consistency and power of the these techniques.

Upper bounds for Ricci curvatures for submanifolds in Bochner-Kaehler manifolds

Chen established the relationship between the Ricci curvature and the squared norm of meancurvature vector for submanifolds of Riemannian space form with arbitrary codimension knownas Chen-Ricci inequality. Deng improved the inequality for Lagrangian submanifolds in complexspace form by using algebraic technique. In this paper, we establish the same inequalitiesfor different submanifolds of Bochner-Kaehler manifolds. Moreover, we obtain improvedChen-Ricci inequality for Kaehlerian slant submanifolds of Bochner-Kaehler manifolds.