ON CAPABLE GROUPS OF ORDER p4

A group $H$ is said to be capable, if there exists another group$G$ such that $\frac{G}{Z(G)}~\cong~H$, where $Z(G)$ denotes thecenter of $G$. In a recent paper \cite{2}, the authorsconsidered the problem of capability of five   non-abelian $p-$groups of order $p^4$ into account. In this paper, we continue this paper by considering three other groups of order $p^4$.  It is proved that the group $$H_6=\langle x, y, z \mid x^{p^2}=y^p=z^p= 1, yx=x^{p+1}y, zx=xyz, yz=zy\rangle$$ is not capable. Moreover, if $p > 3$ is  prime and $d \not\equiv 0, 1 \ (mod \ p)$ then the following groups are not capable:\\{\tiny $H_7^1=\langle x, y, z \mid x^{9} = y^3 = 1, z^3 = x^{3}, yx = x^{4}y, zx = xyz, zy = yz \rangle$,\\$H_7^2= \langle x, y, z \mid x^{p^2} = y^p = z^p = 1, yx = x^{p+1}y, zx = x^{p+1}yz, zy = x^pyz \rangle,$ \\$H_8^1=\langle x, y, z \mid x^{9} = y^3 = 1, z^3 = x^{-3}, yx = x^{4}y, zx = xyz, zy = yz \rangle$,\\$H_8^2=\langle x, y, z \mid x^{p^2} = y^p = z^p = 1, yx = x^{p+1}y, zx = x^{dp+1}yz, zy = x^{dp}yz \rangle$.

A note on “Numerical solutions of fuzzy differential equations by extended Runge–Kutta-like formulae of order 4”

In this note we show that the example presented in a recent paper by Ghazanfari et al. is incorrect. Namely, the “exact solution” suggested by the authors is not solution of the given fuzzy differential equation (FDE). Indeed, the authors have proposed an exact solution which is independent from the initial condition. So, we obtain the correct exact solution using the characterization theorem proposed by Bede et al. under Seikkala differentiability. Also, some details are given for the mentioned example

Nearest interval-valued approximation of interval-valued fuzzy numbers

In this paper, we proposed a new interval-valued approximation of interval-valued fuzzy numbers, which is the best one with respect to a certain measure of distance between interval-valued fuzzy numbers. Also, a set of criteria for interval-valued approximation operators is suggested

An improved Runge-Kutta method for solving fuzzy differential equations under generalized differentiability

In this paper, a new Runge-Kutta method be presented which has the fifth order local truncation error with lower function evaluation in comparison with classical one's. Also we use the generalized derivative instead of Seikkala's derivative to illustrate the efficiency of this derivative. The method's applicability is illustrated by solving a linear first order fuzzy differential equation

On a numerical solution for fuzzy fractional differential equation using an operational matrix method

In the current manuscript we suggest an approach to obtain the solutions of the fuzzy fractional differential equations (FFDEs). We found the operational matrix within the modified Laguerre functions. In this way the investigated equations are turned into a set of algebraic equations. We provide examples to illustrate both accuracy and simplicity of the suggested approach

A fractional multistep method for solving a class of linear fractional differential equations under uncertainty

The objective of this research has been devoted to solve linear fuzzy fractional differential equations (FFDEs) of the Caputo sense. The basic idea is to develop a fractional linear multistep method for solving linear FFDEs under fuzzy fractional generalized differentiability. For safely illustrating the advantages and potential of the presented method, a comparison with the fractional Euler method has to be analyzed in depth. We are interested in using a simple method to obtain gripping results

Toward the existence and uniqueness of solutions for fractional integro-differential equations under uncertainty

The main contribution of the current paper is to obtain new results on the existence and uniqueness of the solution of fractional integro-differential equations under uncertainty with nonlocal conditions. For this purpose, we have used two basic tools, the contraction mapping principle and Krasnoselskii's fixed-point theorem. Indeed, we have considered the original problem involving fuzzy Caputo differentiability, together with fuzzy nonlinear condition

An eigenvalue-eigenvector method for solving a system of fractional differential equations with uncertainty

A new method is proposed for solving systems of fuzzy fractional differential equations (SFFDEs) with fuzzy initial conditions involving fuzzy Caputo differentiability. For this purpose, three cases are introduced based on the eigenvalue-eigenvector approach; then it is shown that the solution of system of fuzzy fractional differential equations is vector of fuzzy-valued functions. Then the method is validated by solving several examples

The behavior of logistic equation with alley effect in fuzzy environment: fuzzy differential equation approach

In this paper a fuzzy logistic equation with alley effect is introduced by considering some parameter as fuzzy numbers. Due to presence of the fuzzy number the corresponding differential equation in logistic equation model with alley effect becomes fuzzy differential equation. Considering generalized Hukuhara derivative approach the fuzzy logistic equation converted to system of two crisp differential equations. We obtain the conditions of stability criterion for different cases. Different numerical examples are given to support our work

A high-order nonlinear Schrödinger equation with the weak non-local nonlinearity and its optical solitons

Abstract The present paper explores a high-order nonlinear Schrodinger equation in a non-Kerr law media with the weak non-local nonlinearity describing solitons' propagation through nonlinear optical fibers. To this end, the real and imaginary parts of the model are firstly extracted using a wave variable transformation. The modified Kudryashov method and symbolic computations are then adopted to successfully retrieve optical solitons of the model. The results presented in the current study demonstrate the great performance of the modified Kudryashov method in handling high-order nonlinear Schrodinger equations