We revisit the notion of Kolmogorov-Sinai entropy for classical dynamical systems in terms of an algebraic formalism. This is the starting point for deening the entropy for general non-commutative systems. Hereby typical quantum tools are introduced in the statistical description of classical dynamical systems. We illustrate the power of these techniques by providing a simple, self-contained pr...

We shall try to exhibit a relation between black hole entropy and topological entropy using the famous Baum-Connes conjecture for foliated manifolds which are particular examples of noncommutative spaces. Our argument is qualitative and it is based on the microscopic origin of the Beckenstein-Hawking area-entropy formula for black holes, provided by superstring theory, in the more general nonco...

The quantum entropy is usually defined using von Neumann's formula, which measures lack of information and vanishes for pure states. In contrast, we obtain a formula for the entropy of a pure state as it is measured from thermodynamic experiments, solely from the self-entanglement of the wave function, and find strong numerical evidence that the two are in agreement for nonintegrable systems, b...

Abstract. Estimating the entropy of a sample set is required in solving numerous learning scenarios involving information theoretic optimization criteria. A number of entropy estimators are available in the literature; however, these require a batch of samples to operate on in order to yield an estimate. In this paper, we derive a recursive formula to estimate Renyi’s quadratic entropy on-line,...

In the present paper, we introduce a natural extension of AKMtopological entropy for noncompact spaces and prove a variational principle which states that the topological entropy, the supremum of the measure theoretical entropies and the minimum of the metric theoretical entropies always coincide. We apply the variational principle to show that the topological entropy of automorphisms of simply...

In this paper, we propose a new entropy measure with geometrical interpretation of intuitionistic fuzzy sets. Compared with the entropy measure provided by Szmidt and Kacprzyk, the new entropy formula in this paper can measure both fuzziness and intuitionism for intuitionistic fuzzy sets. According to the relationship between entropy and similarity measure, we construct a new similarity measure...

Entropy is a measurement of the degree of uncertainty. Meanentropy method can be used for modeling the choice among uncertain outcomes. In this paper, we consider the portfolio selection problem under the assumption that security returns are characterized by type-2 fuzzy variables. Since the expectation and entropy of type-2 fuzzy variables haven’t been well defined, type-2 fuzzy variables need...

Toeplitz matrices have applications to different problems of statistical mechanics. Recently it was used for calculation of entanglement entropy in exactly solvable models including spin chains. We use Fisher-Hartwig formula to calculate entanglement entropy [as well as Rényi entropy] of large block of spins in the ground state of XY spin chain. In the end of the paper we announce our recent re...

Complexity is one of the most important issues of any production planning. The increase in complexity of production planning can cause inconsistency between a production plan and an actual outcome. The complexity generally can be divided in two categories, the static complexity and the dynamic complexity, which can be computed using the ant ropy formula. The formula considers the probability of...

We give a “limit-free formula” simplifying the computation of the topological entropy for topological automorphisms of totally disconnected locally compact groups. This result allows us to extend to our setting several basic properties of the topological entropy known for compact groups.