Entropic Forms and Related Algebras

Abstract

Starting from a very general trace-form entropy, we introduce a pair of algebraic structures endowed by a generalized sum and a generalized product. These algebras form, respectively, two Abelian fields in the realm of the complex numbers isomorphic each other. We specify our results to several entropic forms related to distributions recurrently observed in social, economical, biological and physical systems including the stretched exponential, the power-law and the interpolating Bosons-Fermions distributions. Some potential applications in the study of complex systems are advanced.

20 pages, no figures

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• Theoretical foundations and mathematical formalism of the power-law tailed statistical distributions
• The k-statistics approach to epidemiology
• New power-law tailed distributions emerging in $\kappa$-statistics
• Composition law of $\kappa$-entropy for statistically independent systems
• An overview of generalized entropic forms
• k-Deformed Fourier Transform
• Relativistic Roots of $\kappa$-Entropy
• Generalized Ensemble Theory with Non-extensive Statistics
• Solutions for a q-generalized Schrödinger equation of entangled interacting particles
• Advantages of $q$-logarithm representation over $q$-exponential representation from the sense of scale and shift on nonlinear systems
• Nonlinear Kinetics on Lattices based on the Kinetic Interaction Principle
• $\alpha$-divergence derived as the generalized rate function in a power-law system
• The ${\tau }_q$-Fourier transform: covariance and uniqueness