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Approach to Entropy and Co-Entropy in Measurable and Probability Spaces (Invited Lecture — Extended Abstract)

100%

Cattaneo G.

Annals of Computer Science and Information Systems

2023

tom Vol. 35

63--64

In this talk, we discuss an abstract approach to entropy in countable measurable and induced probability spaces. We consider applications and interpretations of this approach in the context of Rough Set Theory, Fuzzy Set Algebras, as well as Conservative Classical Logics and Quantum Computing. The talk consists of three parts. The first part considers entropy as associated with measure distributions understood as sequences of non-negative values. The second part does the same for partitions. Finally, the third part refers to the aforementioned applications.

The converse of the Hölder inequality and its generalizations

100%

Matkowski J.

nr 2

171-182

Let (Ω,Σ,μ) be a measure space with two sets A,B ∈ Σ such that 0 < μ (A) < 1 < μ (B) < ∞ and suppose that ϕ and ψ are arbitrary bijections of [0,∞) such that ϕ(0) = ψ(0) = 0. The main result says that if $ʃ_Ω xydμ ≤ ϕ^{-1} (ʃ_{Ω} ϕ∘x dμ) ψ^{-1} (ʃ_{Ω} ψ∘x dμ)$ for all μ-integrable nonnegative step functions x,y then ϕ and ψ must be conjugate power functions. If the measure space (Ω,Σ,μ) has one of the following properties: (a) μ (A) ≤ 1 for every A ∈ Σ of finite measure; (b) μ (A) ≥ 1 for every A ∈ Σ of positive measure, then there exist some broad classes of nonpower bijections ϕ and ψ such that the above inequality holds true. A general inequality which contains integral Hölder and Minkowski inequalities as very special cases is also given.