The entropy of fuzzy dynamical systems and generators
TL;DR: The entropy and conditional entropy of stochastical complete repartitions are defined and the Kolmogorov-Sinaj theorem on generators is proved for the fuzzy case.
About: This article is published in Fuzzy Sets and Systems.The article was published on 1992-06-25. It has received 57 citations till now. The article focuses on the topics: Joint quantum entropy & Maximum entropy probability distribution.
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01 Jan 1997
TL;DR: In this paper, a generalization of the model expounded in the previous chapter is presented. But the model is restricted to the family of all measurable functions f: (Ω, ℒ) → [0, 1] with two binary operations (denote them by ⊕ and ⊙), a unary operation * and two fixed elements 0 Ω, 1Ω where ==================>>\s.
Abstract: In this chapter we shall study a generalization of the model expounded in the previous chapter. We introduced a basic example, at the beginning of Section 8.1. We considered there the family of all measurable functions f: (Ω, ℒ) → [0, 1] with two binary operations (denote them by ⊕ and ⊙), a unary operation * and two fixed elements 0Ω, 1Ω where
$$ f \oplus g = (f + g) \wedge 1$$
$$ f \odot g = (f + g - 1) \vee 0$$
$${f^*} = 1 - f]$$
160 citations
TL;DR: It is shown that the class of duality fitting implicators I is much richer than the residuals of the dualityfitting conjunctors in the study of Mesiar et al, and also shows that the I-fuzzy partitions have a ''constant-wise'' structure.
55 citations
01 Jan 2002
TL;DR: In this paper, a survey of non-commutative generalization of Boolean algebraic probability theory can be found, which can be thought of as a generalisation of classical Boolean algebraal probability theory.
Abstract: This chapter surveys the MV-algebraic generalization of notions and results of Boolean algebraic probability theory. MV-algebras are a noncommutative generalization of Boolean algebras. In the approach of Birkhoff and von Neumann, properties of a quantum system measure are identified with self-adjoint idempotent elements in the algebra of continuous Hilbert operators on the Hilbert space. No Hilbert space can be canonically assigned to a system with infinitely many degrees of freedom, such as those naturally occurring in quantum statistical mechanics and in quantum field theory. The MV-algebraic probability theory developed in this survey can be thought of as a noncommutative generalization of classical Boolean algebraic probability theory. An MV-algebra is said to be semisimple if for each nonzero element, there is a homomorphism. The weak σ-distributivity of the underlying group structure of any Riesz space is a necessary and sufficient condition for every valued measure to be extendable from the subalgebra of a set to its enveloping σ-algebra. The MV-algebraic central limit theorem is also elaborated in the chapter.
41 citations
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Book•
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01 Jan 1977
TL;DR: In this paper, the authors discuss the relationship between Riemann-Stieltjes and Lebesgue Integrals, and the Lp spaces, and apply the Heine-Borel Theorem for the convergence of Fourier coefficients.
Abstract: Preface to the Second Edition Preface to the First Edition Authors Preliminaries Points and Sets in Rn Rn as a Metric Space Open and Closed Sets in Rn, and Special Sets Compact Sets and the Heine-Borel Theorem Functions Continuous Functions and Transformations The Riemann Integral Exercises Functions of Bounded Variation and the Riemann-Stieltjes Integral Functions of Bounded Variation Rectifiable Curves The Riemann-Stieltjes Integral Further Results about Riemann-Stieltjes Integrals Exercises Lebesgue Measure and Outer Measure Lebesgue Outer Measure and the Cantor Set Lebesgue Measurable Sets Two Properties of Lebesgue Measure Characterizations of Measurability Lipschitz Transformations of Rn A Nonmeasurable Set Exercises Lebesgue Measurable Functions Elementary Properties of Measurable Functions Semicontinuous Functions Properties of Measurable Functions and Theorems of Egorov and Lusin Convergence in Measure Exercises The Lebesgue Integral Definition of the Integral of a Nonnegative Function Properties of the Integral The Integral of an Arbitrary Measurable f Relation between Riemann-Stieltjes and Lebesgue Integrals, and the Lp Spaces, 0 Riemann and Lebesgue Integrals Exercises Repeated Integration Fubini's Theorem Tonelli's Theorem Applications of Fubini's Theorem Exercises Differentiation The Indefinite Integral Lebesgue's Differentiation Theorem Vitali Covering Lemma Differentiation of Monotone Functions Absolutely Continuous and Singular Functions Convex Functions The Differential in Rn Exercises Lp Classes Definition of Lp Holder's Inequality and Minkowski's Inequality Classes l p Banach and Metric Space Properties The Space L2 and Orthogonality Fourier Series and Parseval's Formula Hilbert Spaces Exercises Approximations of the Identity and Maximal Functions Convolutions Approximations of the Identity The Hardy-Littlewood Maximal Function The Marcinkiewicz Integral Exercises Abstract Integration Additive Set Functions and Measures Measurable Functions and Integration Absolutely Continuous and Singular Set Functions and Measures The Dual Space of Lp Relative Differentiation of Measures Exercises Outer Measure and Measure Constructing Measures from Outer Measures Metric Outer Measures Lebesgue-Stieltjes Measure Hausdorff Measure Caratheodory-Hahn Extension Theorem Exercises A Few Facts from Harmonic Analysis Trigonometric Fourier Series Theorems about Fourier Coefficients Convergence of S[f] and SP[f] Divergence of Fourier Series Summability of Sequences and Series Summability of S[f] and SP[f] by the Method of the Arithmetic Mean Summability of S[f] by Abel Means Existence of f P Properties of f P for f Lp, 1 Application of Conjugate Functions to Partial Sums of S[f] Exercises The Fourier Transform The Fourier Transform on L1 The Fourier Transform on L2 The Hilbert Transform on L2 The Fourier Transform on Lp, 1 2 Exercises Fractional Integration Subrepresentation Formulas and Fractional Integrals L1, L1 Poincare Estimates and the Subrepresentation Formula Holder Classes Norm Estimates for Ialpha Exponential Integrability of Ialphaf Bounded Mean Oscillation Exercises Weak Derivatives and Poincare-Sobolev Estimates Weak Derivatives Approximation by Smooth Functions and Sobolev Spaces Poincare-Sobolev Estimates Exercises Notations Index
981 citations
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TL;DR: In this paper, Caratheodory Herausgegegeben von P Finsler, A Rosenthal and R Steuerwald (Lehrbucher und Monographien aus dem Gebiete der Exakten Wissenschaften Mathematische Reihe, Band 10) Pp 337 (Basel und Stuttgart: Birkhauser Verlag, 1956) 3850 francs; 3850 DM
Abstract: Mass und Integral und ihre Algebraisierung Von Prof C Caratheodory Herausgegeben von P Finsler, A Rosenthal und R Steuerwald (Lehrbucher und Monographien aus dem Gebiete der Exakten Wissenschaften Mathematische Reihe, Band 10) Pp 337 (Basel und Stuttgart: Birkhauser Verlag, 1956) 3850 francs; 3850 DM
480 citations
Book•
01 Jan 1975
TL;DR: In this paper, the authors propose a measure-preserving transformation with pure point spectrum and topological entropy, and show that it is invariant to spectral invariants and isomorphism.
Abstract: Preliminaries.- Measure-preserving transformations.- Isomorphism and spectral invariants.- Measure-preserving transformations with pure point spectrum.- Entropy.- Topological dynamics.- Topological entropy.
160 citations