Mutual relationship between Entropies

Our paper analyzes some new lines to advance on an evolving concept, the so-called Entropy. We need to model this measure by adequate conditions, departing from vague pieces of information. For this, it will be very necessary to analyze the relationship between some such measures, which may be of di¤erent types, with their very interesting applications, as Graph Entropy, Metric Entropy, Algorithmic Entropy, Quantum Entropy, and Topological Entropy. Keywords: Measure Theory, Fuzzy Measures, Symmetry, Entropy. Mathematics Subject Classi…cation: 68R10, 68R05, 05C78, 78M35. 1. Introduction The study of di¤erent concepts of Entropy will be very interesting now, and not only on Physics, but also on Information Theory [27] and other Mathematical Sciences, considered in its more general vision [6, 44]. Also may be a very useful tool on Biocomputing, for instance, or in many others, as studying Environmental Sciences. Because, among other interpretations, with important practical consequences, the law of Entropy means that energy cannot be fully recycled. Many quotations are made until now referring to the content and signi…cance of this fuzzy measure. Between them, we pick up "Gain in Entropy always means loss of Information, and nothing more" (G. N. Lewis). "Information is just known Entropy. Entropy is just unknown Information" (M. P. Frank, Physical Limits of Computing). Mutual Information and Relative Entropy , also called Kullback-Leibler divergence, among other related concepts [6, 11-14, 17], have been very useful in Learning Systems, both on supervised and on unsupervised cases. Our paper attempt to analyze the mutual relationship between the distinct types of entropies, as the Quantum Entropy, also called Von Neumann Entropy; 1AMO Advanced Modeling and Optimization. ISSN: 1841-4311