Past work shows that one can associate a notion of Shannon entropy to Dirichlet polynomial, regarded as an empirical distribution. Indeed, be extracted from any d:Dir by two-step process, where the first step is rig homomorphism out Dir, *set* polynomials, with structure given standard addition and multiplication. In this short note, we show upgraded *functor*, when replace set polynomials *cat...

It is pointed out that the case for Shannon entropy and von Neumann entropy, as measures of uncertainty in quantum mechanics, is not as bleak as suggested in quant-ph/0006087. The main argument of the latter is based on one particular interpretation of Shannon’s H-function (related to consecutive measurements), and is shown explicitly to fail for other physical interpretations. Further, it is s...

In this paper the differentiable structure of the Shannon wavelets is defined and the projection of a linear differential operators is given for any order. As application, the wavelet solution of a heat propagation problem is computed and the contribution of the different scale components is explicitly shown.

Claude Shannon died on 24 February 2001 of the after-effects of Alzheimer disease. With him, one of the greatest scientific minds of the century, and even of all times, disappears. His work exercised a deep influence, although often misknown, in the communication techniques hence in the world where we are living, as well as in the thoughts of the XX century. I shall try, after having evoked the...

Chain rules for Tsallis type entropies are proven, as including the famous pseudoadditivity in nonextensive statistics as a special case. They give important relations between Tsallis conditional entropy and Tsallis joint entropy. Moreover the subadditivity of Tsallis type entropies is shown for the correlated two random variables in the case of q ≥ 1 with the help of the generalized Shannon in...

We establish the optimal quantization problem for probabilities under constrained Rényi-α-entropy of the quantizers. We determine the optimal quantizers and the optimal quan-tization error of one-dimensional uniform distributions including the known special cases α = 0 (restricted codebook size) and α = 1 (restricted Shannon entropy).

We compare the elementary theories of Shannon information and Kolmogorov complexity, the extent to which they have a common purpose, and where they are fundamentally different. We discuss and relate the basic notions of both theories: Shannon entropy versus Kolmogorov complexity, the relation of both to universal coding, Shannon mutual information versus Kolmogorov (‘algorithmic’) mutual inform...

We provide a new result that links two crucial entropy notions: Shannon Entropy H1 and collision entropy H2. Our formula gives the worst possible amount of collision entropy in a probability distribution, when its Shannon Entropy is fixed. Our results and techniques used in the proof immediately imply many quantitatively tight separations between Shannon and smooth Renyi entropy, which were pre...