This analysis leads to a natural boundary condition at the bottom, which is a special case of a generalized entropy condition for the type of partial differential equation under consideration.
It is often used in information theory and natural language processing as a special case of relative entropy approach similar to the averaged Kullback-Leibler divergence, satisfying the condition of symmetry in the entire range of values.
It is a special case of the generalized entropy theory, similar to a Shannon index, where T = maximum possible entropy of the data − observed entropy of the data.
Applying Bernoulli's equation in the case of an ideal gas with constant entropy gives: frac{1}{2}{u}^2+frac{gamma}{gamma hbox 1}kern0.5em frac{p}{rho} = frac{1}{2}{u}_{mathrm{ex}}^2+frac{gamma}{gamma hbox 1}kern0.5em frac{p_{mathrm{ex}}}{rho_{ex}} (5).
We conclude our contribution with the formulation of this theorem for the case of logical entropy.
Let us illustrate the case of maximum entropy binary state with numerical examples.
In the case of maximum entropy state, can be decoded first with arbitrary low probability of error if satisfies (15) for and.
In the case of maximum entropy channel state, we obtain the capacity region for binary noiseless MAC with one informed encoder.
Moreover, it is clear that, by using the same line of reasoning we can prove the existence of local extrema in many other objective functionals as, for example, for the case of Renyi entropy which was already proposed and studied for ICA[6, 7].
In the case of the entropy (mathcal {F} varrho )=int f varrho )) with (f(t)=tlog t) then everything works fine because, again, the corresponding term is linear: begin{aligned} varrho nabla left( frac{delta mathcal {F}}{delta varrho } varrho )right) =varrho frac{nabla varrho }{varrho }=nabla varrho.
