Radiative transfer solutions for coupled atmosphere ocean systems using the matrix operator technique

2012 ◽  
Vol 113 (7) ◽  
pp. 536-548 ◽  
Author(s):  
André Hollstein ◽  
Jürgen Fischer
Keyword(s):  
Radiative Transfer ◽  
Matrix Operator ◽  
Operator Technique ◽  
The Matrix

An Analytical Model Predicting Fixed Index Assembly Machine Performance

1972 ◽  
Vol 94 (2) ◽  
pp. 419-424
Author(s):  
W. B. Heginbotham ◽  
G. E. Rippon
Keyword(s):  
Mathematical Model ◽  
Analytical Model ◽  
Digital Computer ◽  
Matrix Operator ◽  
Operator Technique ◽  
Machine Performance ◽  
Simplified Method ◽  
The Matrix ◽  
Operating Methods

A simplified method of analyzing the performance of fixed index assembly machines using the matrix operator technique is described. This method can be used widely for exploring the effects of operating methods for assembly machines, and an example is quoted where each workhead has a different reliability. By using a digital computer, an exact mathematical model can be built which will enable studies to be made about machine performance when all the head parameters and method of operation are different for each individual head.

scholarly journals Symmetric Determinants and the Cayley and Capelli Operator

1948 ◽  
Vol 8 (2) ◽  
pp. 76-86 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Invariant Theory ◽  
Differential Operators ◽  
Matrix Operator ◽  
Linear Combinations ◽  
Classical Invariant Theory ◽  
Independent Variables ◽  
The Matrix ◽  
Classical Invariant ◽  
Initial Discovery ◽  
Special Case

The result obtained by Lars Gårding, who uses the Cayley operator upon a symmetric matrix, is of considerable interest. The operator Ω = |∂/∂xij|, which is obtained on replacing the n2 elements of a determinant |xij by their corresponding differential operators and forming the corresponding n-rowed determinant, is fundamental in the classical invariant theory. After the initial discovery in 1845 by Cayley further progress was made forty years later by Capelli who considered the minors and linear combinations (polarized forms) of minors of the same order belonging to the whole determinant Ω: but in all this investigation the n2 elements xij were regarded as independent variables. The apparently special case, undertaken by Gårding when xij = xji and the matrix [xij] is symmetric, is essentially a new departure: and it is significant to have learnt from Professor A. C. Aitken in March this year 1946, that he too was finding the symmetrical matrix operator [∂/∂xij] of importance and has already written on the matter.

Matrix Operator Theory of Radiative Transfer 1: Rayleigh Scattering

1973 ◽  
Vol 12 (2) ◽  
pp. 314 ◽  
Author(s):  
Gilbert N. Plass ◽  
George W. Kattawar ◽  
Frances E. Catchings
Keyword(s):  
Radiative Transfer ◽  
Operator Theory ◽  
Rayleigh Scattering ◽  
Matrix Operator

scholarly journals Boundary null-controllability of two coupled parabolic equations : simultaneous condensation of eigenvalues and eigenfunctions.

2020 ◽  
Author(s):  
EL Hadji SAMB
Keyword(s):  
Parabolic Equations ◽  
Riesz Basis ◽  
Jordan Block ◽  
Null Controllability ◽  
Matrix Operator ◽  
Eigenvalues And Eigenfunctions ◽  
Minimal Time ◽  
The Matrix ◽  
New Phenomena

Let the matrix operator $L=D\partial_{xx}+q(x)A_0 $, with  $D=diag(1,\nu)$, $\nu\neq 1$, $q\in L^{\infty}(0,\pi)$, and $A_0$ is a Jordan block of order $1$. We analyze the boundary null controllability  for the system $y_{t}-Ly=0$. When $\sqrt{\nu} \notin \mathbb{Q}_{+}^*$ and  $q$ is constant, $q=1$ for instance, there exists a family of root vectors of $(L^*,\mathcal{D}(L^*))$ forming a Riesz basis of $L^{2}(0,\pi;\mathbb{R}^2 )$. Moreover in  \cite{JFA14} the authors show the existence of a minimal time of control depending on condensation of eigenvalues of $(L^*,\mathcal{D}(L^*))$, that is to say the existence of $T_0(\nu)$ such that the system is null controllable at time $T > T_0(\nu)$ and not null controllable at time  $T < T_0(\nu)$. In the same paper, the authors prove that for all $\tau \in [0, +\infty]$, there exists $\nu \in ]0, +\infty[$ such that $T_0(\nu)=\tau$. When $q$ depends on $x$, the property of Riesz basis is no more guaranteed. This leads to a new phenomena: simultaneous condensation of eigenvalues and eigenfunctions. This condensation affects the time of null controllability.

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