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19 pages, 473 KiB

Open AccessArticle

Converging Cylindrical Symmetric Shock Waves in a Real Medium with a Magnetic Field

by Munesh Devi, Rajan Arora, Mustafa M. Rahman and Mohd Junaid Siddiqui

Symmetry 2019, 11(9), 1177; https://0-doi-org.brum.beds.ac.uk/10.3390/sym11091177 - 17 Sep 2019

Cited by 7 | Viewed by 2373

The topic “converging shock waves” is quite useful in Inertial Confinement Fusion (ICF). Most of the earlier studies have assumed that the medium of propagation is ideal. However, due to very high temperature at the axis of convergence, the effect of medium on [...] Read more.

The topic “converging shock waves” is quite useful in Inertial Confinement Fusion (ICF). Most of the earlier studies have assumed that the medium of propagation is ideal. However, due to very high temperature at the axis of convergence, the effect of medium on shock waves should be taken in account. We have considered a problem of propagation of cylindrical shock waves in real medium. Magnetic field has been assumed in axial direction. It has been assumed that electrical resistance is zero. The problem can be represented by a system of hyperbolic Partial Differential Equations (PDEs) with jump conditions at the shock as the boundary conditions. The Lie group theoretic method has been used to find solutions to the problem. Lie’s symmetric method is quite useful as it reduces one-dimensional flow represented by a system of hyperbolic PDEs to a system of Ordinary Differential Equations (ODEs) by means of a similarity variable. Infinitesimal generators of Lie’s group transformation have been obtained by invariant conditions of the governing and boundary conditions. These generators involves arbitrary constants that give rise to different possible cases. One of the cases has been discussed in detail by writing reduced system of ODEs in matrix form. Cramer’s rule has been used to find the solution of system in matrix form. The results are presented in terms of figures for different values of parameters. The effect of non-ideal medium on the flow has been studied. Guderley’s rule is used to compute similarity exponents for cylindrical shock waves, in gasdynamics and in magnetogasdynamics (ideal medium), in order to set up a comparison with the published work. The computed values are very close to the values in published articles. Full article

(This article belongs to the Special Issue Matrices and Symmetry)

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9 pages, 237 KiB

Open AccessArticle

An Efficient Algorithm for Nontrivial Eigenvectors in Max-Plus Algebra

by Mubasher Umer, Umar Hayat and Fazal Abbas

Symmetry 2019, 11(6), 738; https://0-doi-org.brum.beds.ac.uk/10.3390/sym11060738 - 30 May 2019

Cited by 5 | Viewed by 2658

Abstract

The eigenproblem for matrices in max-plus algebra describes the steady state of the system, and therefore it has been intensively studied by many authors. In this paper, we propose an algorithm to compute the eigenvalue and the corresponding eigenvectors of a square matrix [...] Read more.

The eigenproblem for matrices in max-plus algebra describes the steady state of the system, and therefore it has been intensively studied by many authors. In this paper, we propose an algorithm to compute the eigenvalue and the corresponding eigenvectors of a square matrix in an iterative way. The algorithm is extended to compute the nontrivial eigenvectors for Latin squares in max-plus algebra. Full article

(This article belongs to the Special Issue Matrices and Symmetry)

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10 pages, 256 KiB

Open AccessArticle

Rank Equalities Related to the Generalized Inverses A^{‖(B₁,C₁)}, D^{‖(B₂,C₂)} of Two Matrices A and D

by Wenjie Wang, Sanzhang Xu and Julio Benítez

Symmetry 2019, 11(4), 539; https://0-doi-org.brum.beds.ac.uk/10.3390/sym11040539 - 15 Apr 2019

Cited by 2 | Viewed by 2031

Abstract

Let A be an

n \times n

complex matrix. The

(B, C)

-inverse

A^{∥ (B, C)}

of A was introduced by Drazin in 2012. For given matrices A and B, several rank equalities related to [...] Read more.

Let A be an

n \times n

complex matrix. The

(B, C)

-inverse

A^{∥ (B, C)}

of A was introduced by Drazin in 2012. For given matrices A and B, several rank equalities related to

A^{∥ (B_{1}, C_{1})}

and

B^{∥ (B_{2}, C_{2})}

of A and B are presented. As applications, several rank equalities related to the inverse along an element, the Moore-Penrose inverse, the Drazin inverse, the group inverse and the core inverse are obtained. Full article

(This article belongs to the Special Issue Matrices and Symmetry)

9 pages, 258 KiB

Open AccessArticle

Some Results on the Symmetric Representation of the Generalized Drazin Inverse in a Banach Algebra

by Yonghui Qin, Xiaoji Liu and Julio Benítez

Symmetry 2019, 11(1), 105; https://0-doi-org.brum.beds.ac.uk/10.3390/sym11010105 - 17 Jan 2019

Cited by 4 | Viewed by 2501

Abstract

Based on the conditions

a b^{2} = 0

and

b^{π} (a b) \in A^{d}

, we derive that

{(a b)}^{n}

,

{(b a)}^{n}

, and

a b + b a

are all [...] Read more.

Based on the conditions

a b^{2} = 0

and

b^{π} (a b) \in A^{d}

, we derive that

{(a b)}^{n}

,

{(b a)}^{n}

, and

a b + b a

are all generalized Drazin invertible in a Banach algebra

A

, where

n \in N

and a and b are elements of

A

. By using these results, some results on the symmetry representations for the generalized Drazin inverse of

a b + b a

are given. We also consider that additive properties for the generalized Drazin inverse of the sum

a + b

. Full article

(This article belongs to the Special Issue Matrices and Symmetry)

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