Entropy Of Partitions Research Articles

Spectroscopic studies of the quercetin/rutin-nonionic surfactant interactions

• Interactions between Ru with TX114 and Q with T80 are strong. • Interactions is mostly based on hydrogen bonds and hydrophobic forces. • For Ru with TX114 there are the highest partition coefficient. The behaviour of quercetin (Q) and rutin (Ru) in the aqueous solution of two nonionic surfactants, Triton X-114 (TX114) and Tween 80 (T80) was investigated using the spectroscopic techniques. The object of investigations were the interactions between the given flavonoid and the surfactant in a wide range of surfactant concentrations and the temperature range from 293 to 313 K. The obtained UV–Visible and emission spectra were the basis for determination of the binding constant, partition coefficient, Gibbs free energy of binding as well as partition, enthalpy and entropy of binding and partition. It was proved that the interactions between Ru with TX114 and Q with T80 are strong and become weaker with the increasing temperature. The nature of interactions between the molecules is mostly based on the hydrogen bonds and hydrophobic forces.

Entropy of Dynamical Systems on Interval-Valued Intuitionistic Fuzzy Sets

In this work, we introduce the concepts of Shannon entropy and conditional entropy of experiments in the interval-valued intuitionistic fuzzy case, and study the basic properties of the information measures. Subsequently, by means of the suggested notion of entropy of partitions, we define the entropy of a dynamical system on interval-valued intuitionistic fuzzy sets (IVIF). A version of the Kolmogorov–Sinai theorem on generators for dynamical systems on the IVIF is proved. It is shown that this entropy is an invariant under isomorphisms of interval-valued intuitionistic fuzzy dynamical systems; thus, we obtain a tool for distinguishing some non-isomorphic interval-valued intuitionistic fuzzy dynamical systems. The proposed measure can be used as a measure of information of experiment whose outcomes are interval-valued intuitionistic fuzzy events.

On Relative Metric Mean Dimension with Potential and Variational Principles

In this article, we introduce a notion of relative mean metric dimension with potential for a factor map \(\pi : (X,d, T)\rightarrow (Y, S)\) between two topological dynamical systems. To link it with ergodic theory, we establish four variational principles in terms of metric entropy of partitions, Shapira’s entropy, Katok’s entropy and Brin–Katok local entropy respectively. Some results on local entropy with respect to a fixed open cover are obtained in the relative case. We also answer an open question raised by Shi (On variational principles for metric mean dimension, 2021. arXiv:2101.02610) partially for a very well-partitionable compact metric space, and in general we obtain a variational inequality involving box dimension of the space. Corresponding inner variational principles given an invariant measure of (Y, S) are also investigated.

On the Entropy of Partitions in Hyper Effect Algebra

On the Entropy of Partitions in Hyper Effect Algebra

The Shannon–McMillan–Breiman theorem beyond amenable groups

We introduce a new isomorphism-invariant notion of entropy for measure-preserving actions of arbitrary countable groups on probability spaces, which we call orbital Rokhlin entropy. It employs Danilenko’s orbital approach to entropy of a partition, and it is motivated by Seward’s recent generalization of Rokhlin’s characterization of entropy from amenable to general groups. A key ingredient in our approach is the use of an auxiliary probability-measure-preserving hyperfinite equivalence relation. Under the assumption of ergodicity of the auxiliary equivalence relation, our main result is a Shannon–McMillan–Breiman pointwise almost sure convergence theorem for the orbital entropy of partitions in measure-preserving group actions, the first such convergence result going beyond the realm of amenable groups. As a special case, we obtain a Shannon–McMillan–Breiman theorem for all strongly mixing actions of any countable group. Furthermore, we compare orbital Rokhlin entropy to Rokhlin entropy, and using an important recent result of Seward, we show that they coincide for free ergodic actions of any countable group. Finally, we consider actions of non-abelian free groups and demonstrate the geometric significance of the entropy equipartition property implied by the Shannon–McMillan–Breiman theorem. We show that the orbital entropy of a partition is the limit of the information functions of the sequence of partitions arising from refining any given finite partition along almost every horoball in the group.

Application of protic ammonium-based ionic liquids with carboxylate anions for phenol extraction from aqueous solution and their cytotoxicity on human cells

In this work, the removal of phenol from aqueous solution was investigated by screening three ionic liquids (ILs) having one common cation, trioctylammonium (TOA), combined with three different anions, namely 3,4-dimethylbenzoate (IL1), 4-tert-butylbenzoate (IL2), and 4-phenylbutanoate (IL3), at room temperature. The results indicate that [TOA][4-phenylbutanoate] (IL3) gave the best efficiency and, hence, was selected for the optimization of process parameters: pH, contact time, concentration, phase ratio, and temperature. The optimum conditions were found to be pH 5, contact time of 5 min, phase ratio (VIL/Vw) of 1:5, initial concentration of phenol of 3000 ppm, and temperature of 298 K. The extraction thermodynamics were studied, and the enthalpy, entropy, and free energy of partition were evaluated. The modes of interaction of these ILs with phenol were modeled using the Conductor-Like Screening Model for Real Solvent (COSMO-RS). Additionally, the cytotoxicity of these ILs toward human cells was investigated. The results indicate that they possess less toxicity than the more commonly used 1-butyl-1-methylpyrrolidinium bis-(trifluoromethylsulfonyl)imide. It has been concluded from the experimental results that ammonium-based ILs with carboxylate anions are efficient for extraction of phenol from polluted water.

Shannon’s entropy of partitions determined by hierarchical clustering trees in asymmetry and dimension identification

In the multivariate statistics community, it is commonly acknowledged that among the hierarchical clustering tree (HCT) procedures, the single linkage rule for inter-cluster distance, tends to produce trees which are significantly more asymmetric than those obtained using other rules such as complete linkage, for instance. We consider the use of Shannon’s entropy of the partitions determined by HCTs as a measure of the asymmetry of the clustering trees. On a different direction, our simulations show an unexpected relationship between Shannon’s entropy of partitions and dimension of the data. Based on this observation a procedure for intrinsic dimension identification based on entropy of partitions is proposed and studied. A theoretical result is established for the dimension identification method stating that, locally, for continuous data on a d-dimensional manifold, the entropy of partitions behaves as if the local data were uniformly sampled from the unit ball of Evaluation on simulated examples shows that the method proposed compares favorably with other procedures for dimension identification available in the literature.

Logical Entropy of Partitions in Hyperproduct MV-algebras

In this paper, we introduce the algebraic structure hyperproduct MV-algebras which is a generalization of product MV-algebras. In addition, we study the logical entropy and the logical conditional entropy of partitions in a hyperproduct MV-algebra and provide their properties.

Rényi Entropy and Rényi Divergence in Sequential Effect Algebra

The aim of this study is to extend the results concerning the Shannon entropy and Kullback–Leibler divergence in sequential effect algebra to the case of Rényi entropy and Rényi divergence. For this purpose, the Rényi entropy of finite partitions in sequential effect algebra and its conditional version are proposed and the basic properties of these entropy measures are derived. In addition, the notion of Rényi divergence of a partition in sequential effect algebra is introduced and the basic properties of this quantity are studied. In particular, it is proved that the Kullback–Leibler divergence and Shannon’s entropy of partitions in a given sequential effect algebra can be obtained as limits of their Rényi divergence and Rényi entropy respectively. Finally, to illustrate the results, some numerical examples are presented.

Entanglement entropy of random partitioning

We study the entanglement entropy of random partitions in one- and two-dimensional critical fermionic systems. In an infinite system we consider a finite, connected (hypercubic) domain of linear extent L, the points of which with probability p belong to the subsystem. The leading contribution to the average entanglement entropy is found to scale with the volume as a(p)LD, where a(p) is a non-universal function, to which there is a logarithmic correction term, b(p)LD−1 ln L. In 1D the prefactor is given by b(p)=c/3f(p),where c is the central charge of the model and f(p) is a universal function. In 2D the prefactor has a different functional form of p below and above the percolation threshold.Graphical abstract

Unifying Entropies of Quantum Logic Based on Rényi Entropies

The aim of this study it to unify entropies of quantum logic. For this purpose, Renyi entropy of partition of quantum logic corresponding to a state is presented and formal properties of a proper definition of an entropy are detected. In addition, an appropriate definition of conditional Renyi entropy of quantum logic having the expected properties such as monotonicity is proposed. Finally, Renyi entropy of quantum dynamical system is mentioned and it is proved that it is invariant under isomorphism.

Probing Rényi entanglement entropy via randomized measurements.

Entanglement is a key feature of many-body quantum systems. Measuring the entropy of different partitions of a quantum system provides a way to probe its entanglement structure. Here, we present and experimentally demonstrate a protocol for measuring the second-order Rényi entropy based on statistical correlations between randomized measurements. Our experiments, carried out with a trapped-ion quantum simulator with partition sizes of up to 10 qubits, prove the overall coherent character of the system dynamics and reveal the growth of entanglement between its parts, in both the absence and presence of disorder. Our protocol represents a universal tool for probing and characterizing engineered quantum systems in the laboratory, which is applicable to arbitrary quantum states of up to several tens of qubits.

Rényi entropy of fuzzy dynamical systems

Abstract The present paper is devoted to the study of Renyi entropy in the fuzzy case. We define the Renyi entropy of a fuzzy partition and its conditional version and derive basic properties the suggested entropy measures. In particular, it was shown that the Renyi entropy of a fuzzy partition is monotonically decreasing. Consequently, using the proposed concept of Renyi entropy, the notion of Renyi entropy of a fuzzy dynamical system is introduced. Finally, it is proved that the Renyi entropy of a fuzzy dynamical system is invariant under isomorphism of fuzzy dynamical systems.

R-norm entropy and R-norm divergence in product MV-algebras

The aim of the paper is to extend the results concerning the Shannon entropy and Kullback–Leibler divergence in product MV-algebras to the case of R-norm entropy and R-norm divergence. We define the R-norm entropy of finite partitions in product MV-algebras and its conditional version and derive the basic properties of these entropy measures. In addition, we introduce the concept of R-norm divergence in product MV-algebras and we prove basic properties of this quantity. In particular, it is proven that the Kullback–Leibler divergence and Shannon’s entropy of partitions in a given product MV-algebra can be obtained as the limits of their R-norm divergence and R-norm entropy, respectively.

Tsallis Entropy of Partitions in Quantum Logics

The purpose of the present paper is to study the Tsallis entropy Sq,t(P) of a partition P, and the Tsallis conditional entropy Sq,t(P|Q) of finite partitions P,Q in a quantum logic (L,t), where L is a quantum logic and t is a Bayessian state. We extend some results of Furuichi (J. Math. Phys. 47: 023302, 2006) to the partitions in (L,t). We study the concavity and subadditivity properties for Tsallis entropy of partitions in (L,t). We show that the subadditivity property in the case of q 1,Sq,t(P|Q) ≤ Sq,t(P), but in the case of $q<1, S_{q,t} (P|Q) \nleq S_{q,t} (P)$ , in general. We also study the chain rules for the information measures in (L,t); and present a numerical example to illustrate the results. We study Tsallis entropy and Tsallis conditional entropy of t-independent partitions. Finally, we show that, Tsallis entropy and Tsallis conditional entropy of partitions of a couple (L,t), are invariant under the relation of $\circeq _{t}$ .

Tsallis Entropy of Fuzzy Dynamical Systems

This article deals with the mathematical modeling of Tsallis entropy in fuzzy dynamical systems. At first, the concepts of Tsallis entropy and Tsallis conditional entropy of order where is a positive real number not equal to 1, of fuzzy partitions are introduced and their mathematical behavior is described. As an important result, we showed that the Tsallis entropy of fuzzy partitions of order satisfies the property of sub-additivity. This property permits the definition of the Tsallis entropy of order of a fuzzy dynamical system. It was shown that Tsallis entropy is an invariant under isomorphisms of fuzzy dynamical systems; thus, we acquired a tool for distinguishing some non-isomorphic fuzzy dynamical systems. Finally, we formulated a version of the Kolmogorov–Sinai theorem on generators for the case of the Tsallis entropy of a fuzzy dynamical system. The obtained results extend the results provided by Markechová and Riečan in Entropy, 2016, 18, 157, which are particularized to the case of logical entropy.

Rényi Entropy and Rényi Divergence in Product MV-Algebras.

This article deals with new concepts in a product MV-algebra, namely, with the concepts of Rényi entropy and Rényi divergence. We define the Rényi entropy of order q of a partition in a product MV-algebra and its conditional version and we study their properties. It is shown that the proposed concepts are consistent, in the case of the limit of q going to 1, with the Shannon entropy of partitions in a product MV-algebra defined and studied by Petrovičová (Soft Comput. 2000, 4, 41–44). Moreover, we introduce and study the notion of Rényi divergence in a product MV-algebra. It is proven that the Kullback–Leibler divergence of states on a given product MV-algebra introduced by Markechová and Riečan in (Entropy 2017, 19, 267) can be obtained as the limit of their Rényi divergence. In addition, the relationship between the Rényi entropy and the Rényi divergence as well as the relationship between the Rényi divergence and Kullback–Leibler divergence in a product MV-algebra are examined.

The entropy of hyperfuzzy sets

This paper deals with studying the entropy of dynamical systems on hyperfuzzy sets. At first we define the entropy of hyperfuzzy partitions and conditional entropy. Then, some basic properties of these notions are established. Furthermore, we introduce the notion of a dynamical systems on hyperfuzzy sets and the entropy of it. Finally, we define a generator of this dynamical system and prove Kolmogorov-Sinai’s Theorem is also true for our mentioned generator.

Logical entropy on effect algebras with the Riesz decomposition property

In this study, the notion of logical entropy is generalized for the case when the considered probability space is an effect algebra with the Riesz decomposition property. We define the logical entropy and conditional logical entropy of finite partitions in an effect algebra with the Riesz decomposition property and prove the basic properties of these measures. Furthermore, we introduce the concepts of logical cross entropy and logical divergence and discuss their desirable properties.

Observational Modeling of Logical Entropy of Dynamical Systems with local approach

In this paper, we introduce a new kind of the logical entropy through a local relative approach. The notions of local relative logical entropy and local relative conditional logical entropy from an observer's viewpoint on local relative probability measure space are introduced and some of their ergodic properties are studied. Some properties of the local relative logical entropy of independent partitions are investigated and the concavity property for the local relative logical entropy has been proved. We show that, the basic properties of Shannon entropy of partitions on probability measure spaces, are established for the case of the local relative logical entropy. So the suggested measures can be used besides of the Shannon entropy of partitions . Using the concept of the local relative logical entropy of partitions, we define the local relative logical entropy of a dynamical system and present some of its properties. Finally, it is shown that the local relative logical entropy of dynamical systems is invariant under isomorphism. So the notion of local relative logical entropy of dynamical systems can be a new tool for distinction of non-isomorphic relative dynamical systems.