Kramer s rule

Making use of Kramer s rule, we should identify (derive ) the system characteristic equation ... 

We apply Kramer s rule to find C2 just as we had with C,. The solution has the same characteristic polynomial in (10-22). The transfer functions ... 

Equation 11.16 forms a system of M linear nonhomogeneous equations for the M unknowns a L=, m and can be solved by using any of the standard methods (e.g., Kramer s rule). In matrix notation, the solution is... 

Then the unknown variables x, can be obtained by the rule of Cramer (sometimes also addressed as Kramer s rule). The ith component of the vector x, x, can be obtained as... 

Solving a system of n equations analytically is generally cumbersome and one may have to use Kramer s rule or analyze an inverse of H instead, see Bernstein and Federgruen (2000) for an example. The only way to avoid this... 

The examination of the role of the two-electron term in the Hamiltonian shows that elements of the many-electron basis with occupied Kramers pairs of spin orbitals generally will have a larger energy than others and that the ground state configuration conforms with Hund s rule. Kramers pairs are related to the Racah seniority approximate quantum number. The rotationally invariant geminal creator... 

Ito [51] obtains different expressions for the Kramers-Moyal coefficients in which the spurious drift term is absent. However use of Ito coefficients involves new rules for calculus and so Stratonovich s method will be used here since it is also in agreement with the original method of Brown [8] and is the correct definition to use in the case of a physical noise which always has a finite correlation time [58] (see B.2). 

One such link between semiempirical theory and experiment that appeared about that time was the development of calculational methods for optical rotatory dispersion. Moffitt s theoretical work with Kronig—Kramers transforms coupled with Djerassi s experimental data on steroids gave rise to rules for the prediction of the sign of optical rotation. Computer calculations with semiempirical methods played a role. i Wavefunctions of at least an approximate sort were needed for the dipole and dipole velocity matrix elements of the theory. 

Hansen compares bis observations with Sommer i eld s term values and the transition probabilities calculated by Kramers from the Correspondence Principle (Fig. 1). The predicted pattern in zero field is very simple since the selection rule Afc = 1 prohibits the transitions a9 c and 6. These components should appear weakly, however, in consequence of the Stark effect, in an electric field of 100 volt/cm. Hansen brings evidence to show that his fields were certainly less than this. 

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