Multiple Entropy Measures for Multi-particle Pure Quantum State

We propose to use a set of averaged entropies, the multiple entropy measures (MEMS), to partiallyquantify quantum entanglement of multipartite quantum state.The MEMS is vector-like with m = [N/2] components:[S_1, S_2,..., S_m], and the i-th component S_i is the geometric mean of i-qubits partial entropy of the system.The S_imeasures how strong an arbitrary i qubits from the system are correlated with the rest of the system.It satisfies theconditions for a good entanglement measure.We have analyzed the entanglement properties of the GHZ-state, theW-states, and cluster-states under MEMS.

ENTROPY MEASUREMENT OF INTERVAL GREY SEQUENCE AND ITS APPLICATION

The uncertainty measurement method for grey information theory and the metric formula are established, and its application in decision-making is researched. The entropy measurement of grey sequence based on the limited interval grey number sequence is different from the Shannon probability entropy. The measurement formula of grey number and its properties are studied, such as the invariance, the applicable conditions, and the grey entropy of union and intersection of two grey numbers, and so on. Finally, the algorithm for interval grey sequence and an example are given to show the effectiveness of the method.

On some new fuzzy entropy measure of Pythagorean fuzzy sets for decision-making based on an extended TOPSIS approach

Fuzzy entropy measures are valuable tools in decision-making when dealing with uncertain or imprecise information.There exist many entropy measures for Pythagorean Fuzzy Sets(PFS)in the literature that fail to deal with the problem of providing reasonable or consistent results to the decision-makers.To deal with the shortcomings of the existing measures,this paper proposes a robust fuzzy entropy measure for PFS to facilitate decision-making under uncertainty.The usefulness of the measure is illustrated through an illustration of decision-making in a supplier selection problem and compared with existing fuzzy entropy measures.The Technique for Order Performance by Similarity to Ideal Solution(TOPSIS)approach is also explored to solve the decision-making problem.The results demonstrate that the proposed measure can effectively capture the degree of uncertainty in the decision-making process,leading to more accurate decision outcomes by providing a reliable and robust ranking of alternatives.

Applying the Shearlet-Based Complexity Measure for Analyzing Mass Transfer in Continuous-Flow Microchannels

Continuous-flow microchannels are widely employed for synthesizing various materials,including nanoparticles,polymers,and metal-organic frameworks(MOFs),to name a few.Microsystem technology allows precise control over reaction parameters,resulting in purer,more uniform,and structurally stable products due to more effective mass transfer manipulation.However,continuous-flow synthesis processes may be accompanied by the emergence of spatial convective structures initiating convective flows.On the one hand,convection can accelerate reactions by intensifying mass transfer.On the other hand,it may lead to non-uniformity in the final product or defects,especially in MOF microcrystal synthesis.The ability to distinguish regions of convective and diffusive mass transfer may be the key to performing higher-quality reactions and obtaining purer products.In this study,we investigate,for the first time,the possibility of using the information complexity measure as a criterion for assessing the intensity of mass transfer in microchannels,considering both spatial and temporal non-uniformities of liquid’s distributions resulting from convection formation.We calculate the complexity using shearlet transform based on a local approach.In contrast to existing methods for calculating complexity,the shearlet transform based approach provides a more detailed representation of local heterogeneities.Our analysis involves experimental images illustrating the mixing process of two non-reactive liquids in a Y-type continuous-flow microchannel under conditions of double-diffusive convection formation.The obtained complexity fields characterize the mixing process and structure formation,revealing variations in mass transfer intensity along the microchannel.We compare the results with cases of liquid mixing via a pure diffusive mechanism.Upon analysis,it was revealed that the complexity measure exhibits sensitivity to variations in the type of mass transfer,establishing its feasibility as an indirect criterion for assessing mass transfer intensity.The method presented can extend beyond flow analysis,finding application in the controlling of microstructures of various materials(porosity,for instance)or surface defects in metals,optical systems and other materials that hold significant relevance in materials science and engineering.

Possibilistic entropy-based measure of importance in fault tree analysis

With respect to the subjective factors and nonlinear characteristics inherent in the important identification of fault tree analysis （FTA）, a new important measure of FTA is proposed based on possibilistic information entropy. After investigating possibilistic information semantics, measure-theoretic terms, and entropy-like models, a two-dimensional framework has been constructed by combining both the set theory and the measure theory. By adopting the possibilistic assumption in place of the probabilistic one, an axiomatic index of importance is defined in the possibility space and then the modelling principles are presented. An example of the fault tree is thus provided, along with the concordance analysis and other discussions. The more conservative numerical results of importance rankings, which involve the more choices can be viewed as “soft” fault identification under a certain expected value. In the end, extension to evidence space and further research perspectives are discussed.

Shannon Entropy as a Measurement of the Information in a Multiconfiguration Dirac-Fock Wavefunction

Discrete Shannon entropy is applied to describe the information in a multiconfiguration Dirac Fock wavefunction. The dependence of Shannon entropy is shown as enlarging the configuration space and it can reach saturation when there are enough configuration state wavefunctions to obtain the convergent energy levels; that is, the calculation procedure in multiconfiguration Dirae Fock method is an entropy saturation process. At the same accuracy level, the basis sets for the smallest entropy are best able to describe the energy state. Additionally, a connection between the sudden change of Shannon information entropies and energy level crossings along with isoelectronic sequence can be set up, which is helpful to find the energy level crossings of interest in interpreting and foreseeing the inversion scheme of energy levels for an x-ray laser.

Minimum Cross Fuzzy Entropy Problem, The Existence of Its Solution and Generalized Minimum Cross Fuzzy Entropy Problems

In the present study we have formulated a Minimum Cross Fuzzy Entropy Problem （Minx（F）EntP） and proposed sufficient conditions for existence of its solution. Mentioned problem can be formulated as follows. In the set of membership functions satisfying the given moment constraints generated by given moment functions it is required to choose the membership function that is closest to a priori membership function in the sense of cross fuzzy entropy measure. The existence of solution of formulated problem is proved by virtue of concavity property of cross fuzzy entropy measure, the implicit function theorem and Lagrange multipliers method. Moreover, Generalized Cross Fuzzy Entropy Optimization Methods in the form of MinMinx（F）EntM and MaxMinx（F）EntM are suggested on the basis of primary phase of minimizing cross fuzzy entropy measure for fixed moment vector function and on the definition of the special functional with Minx（F）Ent values of cross fuzzy entropy measure. Next phase for obtaining mentioned distributions consists of optimization of defined functional with respect to moment vector functions. Distributions obtained by mentioned methods are defined as （MinMinx（F）Ent）m and （MaxMinx（F）Ent）m distributions.

Generalized Maximum Fuzzy Entropy Methods with Applications on Wind Speed Data

This study is connected with new Generalized Maximum Fuzzy Entropy Methods （GMax（F）EntM） in the form of MinMax（F）EntM and MaxMax（F）EntM belonging to us. These methods are based on primary maximizing Max（F）Ent measure for fixed moment vector function in order to obtain the special functional with maximum values of Max（F）Ent measure and secondary optimization of mentioned functional with respect to moment vector functions. Distributions, in other words sets of successive values of estimated membership function closest to （furthest from） the given membership function in the sense of Max（F）Ent measure, obtained by mentioned methods are defined as （MinMax（F）Ent）m which is closest to a given membership function and （MaxMax（F）Ent）m which is furthest from a given membership function. The aim of this study consists of applying MinMax（F）EntM and MaxMax（F）EntM on given wind speed data. Obtained results are realized by using MATLAB programme. The performances of distributions （MinMax（F）En0m and （MaxMax（F）Ent）m generated by using Generalized Maximum Fuzzy Entropy Methods are established by Chi-Square, Root Mean Square Error criterias and Max（F）Ent measure.

Oscillatory Shannon entropy in the process of equilibration of nonequilibrium crystalline systems

We present a study of the equilibration process of some nonequilibrium crystalline systems by means of molecular dynamics simulation technique. The nonequilibrium conditions are achieved in the systems by randomly defining velocity components of the constituent atoms. The calculated Shannon entropy from the probability distribution of the kinetic energy among the atoms at different instants during the process of equilibration shows oscillation as the system relaxes towards equilibrium. Fourier transformations of these oscillating Shannon entropies reveal the existence of Debye frequency of the concerned system.

Emergency Decision-Making Based on q-Rung Orthopair Fuzzy Rough Aggregation Information

With the frequent occurrences of emergency events,emergency decision making(EDM)plays an increasingly significant role in coping with such situations and has become an important and challenging research area in recent times.It is essential for decision makers to make reliable and reasonable emergency decisions within a short span of time,since inappropriate decisions may result in enormous economic losses and social disorder.To handle emergency effectively and quickly,this paper proposes a new EDM method based on the novel concept of q-rung orthopair fuzzy rough(q-ROPR)set.A novel list of q-ROFR aggregation information,detailed description of the fundamental characteristics of the developed aggregation operators and the q-ROFR entropy measure that determine the unknown weight information of decision makers as well as the criteria weights are specified.Further an algorithm is given to tackle the uncertain scenario in emergency to give reliable and reasonable emergency decisions.By using proposed list of q-ROFR aggregation information all emergency alternatives are ranked to get the optimal one.Besides this,the q-ROFR entropy measure method is used to determine criteria and experts’weights objectively in the EDM process.Finally,through an illustrative example of COVID-19 analysis is compared with existing EDM methods.The results verify the effectiveness and practicability of the proposed methodology.

Towards Improving Predictive Statistical Learning Model Accuracy by Enhancing Learning Technique

The accuracy of the statistical learning model depends on the learning technique used which in turn depends on the dataset’s values.In most research studies,the existence of missing values(MVs)is a vital problem.In addition,any dataset with MVs cannot be used for further analysis or with any data driven tool especially when the percentage of MVs are high.In this paper,the authors propose a novel algorithm for dealing with MVs depending on the feature selec-tion(FS)of similarity classifier with fuzzy entropy measure.The proposed algo-rithm imputes MVs in cumulative order.The candidate feature to be manipulated is selected using similarity classifier with Parkash’s fuzzy entropy measure.The predictive model to predict MVs within the candidate feature is the Bayesian Ridge Regression(BRR)technique.Furthermore,any imputed features will be incorporated within the BRR equation to impute the MVs in the next chosen incomplete feature.The proposed algorithm was compared against some practical state-of-the-art imputation methods by conducting an experiment on four medical datasets which were gathered from several databases repository with MVs gener-ated from the three missingness mechanisms.The evaluation metrics of mean abso-lute error(MAE),root mean square error(RMSE)and coefficient of determination(R2 score)were used to measure the performance.The results exhibited that perfor-mance vary depending on the size of the dataset,amount of MVs and the missing-ness mechanism type.Moreover,compared to other methods,the results showed that the proposed method gives better accuracy and less error in most cases.

A purely data driven method for European option valuation

An alternative option pricing method is proposed based on a random walk market model. The minimal entropy martingale measure which adopts no arbitrage opportunity in the market, is deduced for this market model and is used as the pricing measure to evaluate European call options by a Monte Carlo simulation method. The proposed method is a purely data driven valuation method without any distributional assumption about the price process of underlying asset. The performance of the proposed method is compared with the canonical valuation method and the historical volatility-based Black-Scholes method in an artificial Black-Scholes world. The simulation results show that the proposed method has merits, and is valuable to financial engineering.

ON A LEMMA OF BOWEN

We are concerned with the sets of quasi generic points in finite symbolic space. We estimate the sizes of the sets by the Billingsley dimension defined by Gibbs measures. A dimension formula of such set is given, which generalizes Bowen＇s result. An application is given to the level sets of Birkhoff average.

Trivial and Simple Spectrum for SL(d,R) Cocycles with Free Base and Fiber Dynamics

Let ACD（M, SL（d,R）） denote the pairs （f, A） so that f∈ A C Diff^1（M） is a C^1-Anosov transitive diffeomorphisms and A is an SL（d,R） cocycle dominated with respect to f. We prove that open and densely in ACD（M, SL（d,R））, in appropriate topologies, the pair （f,A） has simple spectrum with respect to the unique maximal entropy measure μf. Then, we prove prevalence of trivial spectrum near the dynamical cocycle of an area-preserving map and also for generic cocycles in AUtLeb（M） × LP（M, SL（d, R））.