Gauss inequality

The above inequality, which was first derived by Ahlrichs, provides an upper bound for the magnitudes of the [ssjss] integrals in terms of a product of two quantities that can be readily evaluated. Unfortunately, the inequality [22] does not hold for other types of ERIs, and therefore it can be rigorously used only for the Gaussian lobe functions. Despite this limitation, the estimate [22] was successfully applied in prescreening of all ERIs by Cremer and Gauss. Another nonrigorous estimate was derived and used by Almlof et al. ... 

By Gauss theorem and (3.68) we obtain entropy inequality in the local form called the Clausius-Duhem inequality... 

Lastly, it is also possible to use the method that exploits the null space of the constraints. Once again in this case, all the active bounds must first be removed from the problem. Only then it is possible to use either LQ factorization or a stable Gauss factorization of all the equality and active inequality constraints. This gives the KKT conditions for an unconstrained problem, as has already been demonstrated for equality constraints. 

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