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Entropy

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The first law of thermodynamics is quantitatively formulated via an equation relating the internal energy of a system, the heat exchanged by it, and the work done on it. A quantitative formulation of the second law of thermodynamics leads to defining a state function, the entropy.
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Salt particles that have dissolved in water never spontaneously come back together in solution to reform solid particles. Moreover, a gas that has expanded in a vacuum remains dispersed and never spontaneously reassembles. The unidirectional nature of these phenomena is the result of a thermodynamic state function called entropy (S). Entropy is the measure of the extent to which the energy is dispersed throughout a system, or in other words, it is proportional to the degree of disorder of a...

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Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section

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The Rényi Entropies Operate in Positive Semifields.

Francisco J Valverde-Albacete¹, Carmen Peláez-Moreno¹

¹Department of Signal Theory and Communications, Universidad Carlos III de Madrid, 28911 Leganés, Spain.

Entropy (Basel, Switzerland)

|December 3, 2020

Summary

This summary is machine-generated.

Rényi entropies function within a positive semifield structure, not a non-linear semiring. This framework explains the success of tropical algebra and computational intelligence methods in information processing.

Keywords:

Pap’s g-calculus artificial intelligence computational intelligence idempotent semifields machine learning non-Newtonian calculus positive commutative semifields shifted Rényi entropy

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Area of Science:

Information Theory
Algebraic Structures
Computational Intelligence

Background:

Rényi entropies are fundamental in quantifying information.
Existing frameworks may not fully capture the underlying mathematical structure of Rényi entropies.
Computational intelligence applications often utilize algebraic procedures with unclear theoretical underpinnings.

Purpose of the Study:

To demonstrate that Rényi entropies operate within a positive semifield structure.
To connect Rényi's postulates to Pap's g-calculus and its transformations.
To elucidate the algebraic basis for the success of certain computational intelligence algorithms.

Main Methods:

Analysis of Rényi's postulates to derive Pap's g-calculus.
Transformation of standard arithmetic operations (product to sum, sum to power-emphasized sum) using Rényi's information function.
Identification of the resulting structure as a positive semifield.

Main Results:

Rényi's postulates naturally lead to Pap's g-calculus.
The transformation results in a positive semifield, where standard product becomes sum and standard sum becomes power-emphasized sum.
This construction unifies idempotent analysis, tropical algebra, and other structures.

Conclusions:

Rényi entropies are best understood as operating in a positive semifield.
The positive semifield structure provides a theoretical foundation for the efficacy of tropical algebra and related methods in computational intelligence.
This framework offers insights into the broad applicability of information processing techniques in diverse computational fields.