CENTER OF THE DIAGONAL MATRIX RING WITH QUATERNION ENTRIES

It is now known (3) that if is a regular rank ring, then the rank function can be extended to the matrix ring in such a way that R(a) = R(a ꕕ n) ; here, a is an arbitrary element of is the n × n diagonal matrix with a for each entry on the diagonal, and R denotes rank in and also in . It is also known (2) that every regular rank ring has a rankmetric completion which is again a regular rank ring.

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On the Generalized Distance Energy of Graphs

Mathematics ◽

10.3390/math8010017 ◽

2019 ◽

Vol 8 (1) ◽

pp. 17 ◽

Cited By ~ 4

Author(s):

Abdollah Alhevaz ◽

Maryam Baghipur ◽

Hilal A. Ganie ◽

Yilun Shang

Keyword(s):

Lower Bounds ◽

Complete Graph ◽

Connected Graph ◽

Distance Matrix ◽

Wiener Index ◽

Diagonal Matrix ◽

Upper And Lower Bounds ◽

Star Graph ◽

Extremal Graphs ◽

Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

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The normalized distance Laplacian

Special Matrices ◽

10.1515/spma-2020-0114 ◽

2021 ◽

Vol 9 (1) ◽

pp. 1-18

Author(s):

Carolyn Reinhart

Keyword(s):

Spectral Radius ◽

Characteristic Polynomial ◽

Adjacency Matrix ◽

Connected Graph ◽

Distance Matrix ◽

Laplacian Matrix ◽

Diagonal Matrix ◽

The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

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Some Criteria for Hermite Rings and Elementary Divisor Rings

Canadian Journal of Mathematics ◽

10.4153/cjm-1974-131-8 ◽

1974 ◽

Vol 26 (6) ◽

pp. 1380-1383 ◽

Cited By ~ 3

Author(s):

Thomas S. Shores ◽

Roger Wiegand

Keyword(s):

Triangular Matrix ◽

Elementary Divisor ◽

Diagonal Matrix ◽

Finitely Generated ◽

Elementary Divisor Ring ◽

Hermite Ring

Recall that a ring R (all rings considered are commutative with unit) is an elementary divisor ring (respectively, a Hermite ring) provided every matrix over R is equivalent to a diagonal matrix (respectively, a triangular matrix). Thus, every elementary divisor ring is Hermite, and it is easily seen that a Hermite ring is Bezout, that is, finitely generated ideals are principal. Examples have been given [4] to show that neither implication is reversible.

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INVESTIGATION COMPARISON ON THE GATE SPEED OF THREE-LEVEL AND TWO-LEVEL QUANTUM NOT GATES WITH ASYMMETRIC POTENTIAL OF rf-SQUID

Modern Physics Letters B ◽

10.1142/s0217984907013699 ◽

2007 ◽

Vol 21 (19) ◽

pp. 1261-1270 ◽

Cited By ~ 3

Author(s):

YING-HUA JI ◽

JU-JU HU ◽

SHI-HUA CAI

Keyword(s):

Microwave Pulse ◽

Diagonal Matrix ◽

Matrix Elements ◽

The Asymmetry ◽

Rf Squid ◽

Double Well Potential

We investigate the relation between the speed of quantum NOT gate and the asymmetry or detuning of the potential in system of the interaction of a two-level rf-SQUID qubit with a classical microwave pulse. The rf-SQUID is characterized by an asymmetric double well potential that gives rise to diagonal matrix elements. Then in resonance, we compare the gate speeds for three-level and two-level quantum NOT gates. We show that in general, a three-level gate is much faster than the conventional two-level gate.

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Buckling of Double Layer Grid Edge Members

International Journal of Space Structures ◽

10.1177/026635119701200203 ◽

1997 ◽

Vol 12 (2) ◽

pp. 81-87

Author(s):

Erling Murtha-Smith ◽

Thuyen P. Nguyen

Keyword(s):

Double Layer ◽

External Load ◽

Stiffness Matrix ◽

Diagonal Matrix ◽

Moment Of Inertia ◽

Effective Length ◽

Stiffness Coefficient ◽

Stability Functions

Stability equations are developed for edge joints for Double Layer Grids. Translations are neglected and rotations at each joint are related. Hence, the stiffness matrix reduces to a diagonal matrix of unit bandwidth so each joint becomes an independent substructure. Instability of an edge joint occurs when the minimum principal stiffness coefficient of the joint goes to zero. Using stability functions and the regular geometric relationships of DLG topology, the buckling forces in the members and hence the external load on the system are determined. A simple example in which the members were all of the same length, material and moment of inertia, gives effective length factors for the edge members of between 0.77 to 0.81.

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Wear Model for Piston Ring and Cylinder Bore System

ASME 2005 Internal Combustion Engine Division Fall Technical Conference (ICEF2005) ◽

10.1115/icef2005-1340 ◽

2005 ◽

Cited By ~ 1

Author(s):

Yunchao Qiu ◽

Qian Zou ◽

Gary C. Barber ◽

Harold E. McCormick ◽

Dequan Zou ◽

...

Keyword(s):

Piston Ring ◽

Thickness Distribution ◽

Matrix Ring ◽

Experimental Investigations ◽

General Tendency ◽

Wear Model ◽

Wear Process ◽

The Matrix ◽

Surface Profiles ◽

Cylinder Bore

A new wear model for piston ring and cylinder bore system has been developed to predict wear process with high accuracy and efficiency. It will save time and cost compared with experimental investigations. Surfaces of ring and bore were divided into small domains and assigned to corresponding elements in two-dimensional matrix. Fast Fourier Transform (FFT) and Conjugate Gradient Method (CGM) were applied to obtain pressure distribution on the computing domain. The pressure and film thickness distribution were provided by a previously developed ring/bore lubrication module. By changing the wear coefficients of the ring and bore with accumulated cycles, wear was calculated point by point in the matrix. Ring and bore surface profiles were modified when wear occurred. The results of ring and bore wear after 1 cycle, 10 cycles and 2 hours at 3600 rpm were calculated. They coincided well with the general tendency of wear in a ring and bore system.

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Broadband EM radiation amplification by means of a monochromatically driven two-level system

Modern Physics Letters B ◽

10.1142/s0217984917500270 ◽

2017 ◽

Vol 31 (04) ◽

pp. 1750027 ◽

Cited By ~ 2

Author(s):

Andrey V. Soldatov

Keyword(s):

Dipole Moment ◽

Quantum System ◽

High Frequency ◽

Laser Field ◽

Diagonal Matrix ◽

Matrix Elements ◽

Level System ◽

Dipole Moment Operator ◽

Moment Operator ◽

Radiation Amplification

It is shown that a two-level quantum system possessing dipole moment operator with permanent non-equal diagonal matrix elements and driven by external semiclassical monochromatic high-frequency electromagnetic (EM) (laser) field can amplify EM radiation waves of much lower frequency.

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