By T S Moss
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Extra info for Optical properties of semi-conductors (Semi-conductor monographs)
77); the , (i"1,2, n) within the brackets G are n algebraically independent (`denominatora) invariants and , ( "0,2, k) with "1 are ? algebraically dependent (`numeratora or auxiliary) invariants which form the basis of the module of invariant polynomials. erent modules. For example the obvious equivalence P%"P[ , , , ]E(1, ,2, ) 2 L I "P[ , ,2, ]E(1, ,2, )(1, ) . (79) L I L enables one to change the system of denominator invariants (to go from to ) with simultaneous L L doubling of the number of numerator invariants.
To get the symbolic information about the set of syzygies we can multiply both numerator and denominator of the Molien function (108) by (1! ) to get the expression with six denominator invariants of degree two. 3 M G" . (1! ) (111) Naturally, the numbers of invariant polynomials in each degree do not alter upon such a transformation but we have new symbolic information. There are six syzygies of the "rst kind, eight syzygies of the second kind and three syzygies of the third kind. erent kinds (sometimes this is even impossible without independent explicit calculation of syzygies as shown for example by Stanley (1996, Chapter I-11)).
15. Contour plot of the level set of the invariant function in Eq. (118) on the orbifold. 1/10. erent types of regular levels (b, d). Fig. 16. Contour plot of the level set of the invariant function in Eq. (118) with / & on the orbifold. The function possesses stationary points outside of critical orbits. erent types of regular levels (b, d, f ). The non-critical stationary point located at the touching point of level c and the boundary. O group which have the opposite values of the numerator invariant .
Optical properties of semi-conductors (Semi-conductor monographs) by T S Moss