By Guy Jumarie
Every idea is a throw of cube. Stephane Mallarme This booklet is the final certainly one of a trilogy which studies part of our examine paintings over approximately thirty years (we discard our non-conventional ends up in computerized regulate thought and purposes at the one hand, and fuzzy units at the other), and its major keyword phrases are info conception, Entropy, greatest Entropy precept, Linguistics, Thermodynamics, Quantum Mechanics, Fractals, Fractional Brownian movement, Stochastic Differential Equations of Order n, Stochastic optimum keep an eye on, laptop imaginative and prescient. Our obsession has been continually a similar: Shannon's details idea may still play a uncomplicated function within the foundations of sciences, yet topic to the situation that or not it's definitely generalized to permit us to house difficulties which aren't unavoidably relating to conversation engineering. With this target in brain, questions are of maximum value: (i) How will we introduce that means or value of knowledge in Shannon's details conception? (ii) How do we outline and/or degree the quantity of knowledge keen on a kind or a development with no utilizing a probabilistic scheme? it truly is compulsory to discover appropriate solutions to those difficulties if we wish to observe Shannon's concept to technological know-how with a few probability of good fortune. for example, its use in biology has been very disappointing, for the very cause that the that means of knowledge is there of uncomplicated value, and isn't taken with this approach.
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Additional resources for Maximum Entropy, Information Without Probability and Complex Fractals: Classical and Quantum Approach
2) to define it appears to be unnecessarily complicated. However, we shall consider instances where X and X' do not play such symmetrical roles. We assume that X is still a random variable of which the entropy H(X) is defined in the usual way. We assume further that we have some plausible definitions of H(X,X') but none for H(X'). 2) will be used to define H(X'). 3). ' 1- S J9! ). ). In the following we shall generalize to the n-dimensional case. Therefore, we consider a function f: 9t n ~ 9t n , x ~ f(x), but no change in notation will result.
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49) and 1 1- s IR ,(Y,X) := - - I n J. 50) In the present reminder, for the sake of brevity, we have formally generalized the definitions related to discrete probabilities, but it is important to point out that the entropy of continuous probabilities is not the limiting form of the entropy of discrete probabilities when the discretizing span (of approximation) tends to zero, and it is exactly this remark which led us to introduce the following concept of total entropy. 8. 1. Total Shannon entropy Preliminary notations.