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On the Principle of Exchange of Stabilities in Rayleigh-Bénard Convection

The problem of Rayleigh--Benard convection with internal heat sources and a variable gravity field is treated. For the case of stress-free boundary conditions, it is proved that the principle of exchange of stabilities holds as long as the product of gravity field and the integral of the heat sourcesis nonnegative throughout the layer. The proof is based on the idea of a positive operator, and uses the positivity properties of Green's function.