Sylvester theorem

If the kinetic matrix is constant, eq. (12.127) can be solved either analytically by using the Sylvester theorem of the Laplace transformation, or numerically. Finally, the relation between the intrinsic time scale and the laboratory time scale t can be determined from t = u ( )] A constant kinetic matrix is sufficient for... 

The inverse problem in this case is formulated as recovery of the unknown coefficient 7 of the elliptic operator from the known values of the field p(r, u>) in some domain or in the boundary of observations. In a number of brilliant mathematical papers the corresponding uniqueness theorems for this mathematical inverse problem have been formulated and proved The key result is that the unknown coefficient 7 (r) of an elliptic differential operator can be determined uniquely from the boundary measurements of the field, if 7 (r) is a real-analytical function, or a piecewise real-analytical function. In other words, from the physical point of view we assume that 7 (r) is a smooth function in an entire domain, or a piecewise smooth function. Note that this result corresponds well to Wcidelt s and Gusarov s uniqueness theorems for the magnetotelluric inverse problem. I would refer the readers for more details to the papers by Calderon (1980), Kohn and Vogelius (1984, 1985), Sylvester and Uhlmann, (1987), and Isakov (1993). 

Sylvester, J., and G. Uhlmann, 1987, Global uniqueness theorem for an inverse boundary value problem Ann. Math., 125, 153-169. 

Step 5 [3] is calculated using Sylvester s theorem. First, the eigenvalues Sq/ from Eqs. 8.3.45... 

Application of continuous-lag Markov chains requires the knowledge of lag-dependent transition probabilities. Through Sylvester s theorem (Agterberg, 1974 Carle and Fogg, 1997) these can be derived from ... 

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