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Fiber Bragg Gratings for Microwave Photonics Applications
Published in Chi H. Lee, Microwave Photonics, 2017
A solution to the problem is to use only a single linearly CFBG [114]. A schematic diagram to show the Fourier transform optical pulse shaping system using only a single linearly CFBG is illustrated in Figure 4.29a. An input ultrashort optical pulse is first temporally stretched by the linearly CFBG, and then completely compressed by the same linearly CFBG by directing the dispersed optical pulse into the FBG from the opposite direction. Therefore, a perfect dispersion cancellation is obtained. At the same time, the linearly CFBG also acts as an optical spectral shaper which is designed to have a user-defined spectral reflection response according to the target temporal waveform. The impulse response of the entire system is equal to the Fourier transform of the square of the power reflectivity function of the linearly CFBG. For example, to generate a triangular waveform, the linearly CFBG should have a reflection response corresponding to a sinc function, as shown in Figure 4.29a. The key device in the system is the linearly CFBG, which should be designed to have a strict linear group delay response and a user-defined reflection magnitude response. A simple and effective method to synthesize the required linearly CFBG is to perform an accurate mapping of the grating reflection response to the refractive index apodization, as have been discussed in Section 4.2.2. It is different from the design based on the Born approximation; the method here can make the grating have a high reflectivity, leading to improved energy efficiency. Such a linearly CFBG was fabricated. As can be seen from Figure 4.29b, the spectral response matches well with the desired spectral response. A linear group delay response is also achieved.
Matrix Groups
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
A linear group of degreen is a subgroup of GL(V) (where dim V = n). (A subgroup of GL(n, F) is a linear group of degree n, since an n × n matrix can be viewed as a linear operator acting on Fn by matrix multiplication.)
Matrices and Linear Algebra
Published in William S. Levine, Control System Fundamentals, 2019
The invertible matrices in Rn×n, along with the operation of matrix multiplication, form a group, the general linear group, denoted by GL(R, n); In is the identity element of the group.
Computational study of bioconvection rheological nanofluid flow containing gyrotactic microorganisms: a model for bioengineering nanofluid fuel cells
Published in International Journal of Modelling and Simulation, 2023
Adebowale Martins Obalalu, Sulyman Olakunle Salawu, Olalekan Adebayo Olayemi, Christopher Bode Odetunde, Akintayo Oladimeji Akindele
The following is the summary of the literature review. The literature review on bioconvection microorganisms, thermo-bioconvectional transport, and oxytactic microorganisms is presented with a health literature background.Different cases of non-Newtonian fluids were studied to gain a better understanding of the literature that discusses the power-law fluid model.The literature regarding the linear group of transformations and numerical computation is presented.The significance of thermophoresis and Brownian motion on the fluid was discussed at the end of the literature.
A high-order Lie groups scheme for solving the recovery of external force in nonlinear system
Published in Inverse Problems in Science and Engineering, 2018
Yu-Chen Liu, Yung-Wei Chen, Yu-Ting Wang, Jiang-Ren Chang
The general linear group is a Lie group set, whose manifold is an open subset of the linear space of all n × n non-singular matrices. Therefore, is also an n × n-dimensional manifold. The general linear group has a uniquely real Lie algebra . We consider a one-parameter subgroup , of the general linear group , which is a curve passing through the group identity at τ = 0,
Diagonal Born–Oppenheimer correction for coupled-cluster wave-functions
Published in Molecular Physics, 2018
Therefore, the change of normalisation by different amounts at every geometry as in (12) can be viewed as a kind of gauge transformation [12,33,36]. Here, it arises from the fact that the adiabatic separation ansatz (7) is invariant to arbitrary geometry-dependent normalisation changing transformations [33] (from now onwards, we will refer to them simply as gauge transformations). This shows that the symmetry is a result of the decision to separate the electronic and nuclear degrees of freedom and to consider the electronic motion first. The gauge group which includes all possible gauge transformations at a geometry is the general linear group GL(1).