Anomaly-Resistant Decentralized State Estimation Under Minimum Error Entropy With Fiducial Points for Wide-Area Power Systems

This paper investigates the anomaly-resistant decentralized state estimation(SE) problem for a class of wide-area power systems which are divided into several non-overlapping areas connected through transmission lines. Two classes of measurements(i.e., local measurements and edge measurements) are obtained, respectively, from the individual area and the transmission lines. A decentralized state estimator, whose performance is resistant against measurement with anomalies, is designed based on the minimum error entropy with fiducial points(MEEF) criterion. Specifically, 1) An augmented model, which incorporates the local prediction and local measurement, is developed by resorting to the unscented transformation approach and the statistical linearization approach;2) Using the augmented model, an MEEF-based cost function is designed that reflects the local prediction errors of the state and the measurement;and 3) The local estimate is first obtained by minimizing the MEEF-based cost function through a fixed-point iteration and then updated by using the edge measuring information. Finally, simulation experiments with three scenarios are carried out on the IEEE 14-bus system to illustrate the validity of the proposed anomaly-resistant decentralized SE scheme.

Entropy squeezing for a V-type three-level atom interacting with a single-mode field and passing through the amplitude damping channel with weak measurement

The entropy squeezing of a V-type three-level atom interacting with a single-mode field and passing through the amplitude damping channel is investigated in detail. Our results show that when coupled to the single-mode field, the atom in appropriate initial states can not only generate obvious entropy squeezing but also keep in the optimal squeezing state,while passing through the amplitude damping channel, the atom can generate entropy squeezing under the control of the weak measurement. Besides, it is proved again that as a measurement method for atomic squeezing, the entropy squeezing is precise and effective. Therefore our work is instructive for experiments in preparing three-level system information resource with ultra-low quantum noise.

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.

Aggregating metasearch engine results based on maximal entropy OWA operator

The maximal entropy ordered weighted averaging （ME-OWA） operator is used to aggregate metasearch engine results, and its newly analytical solution is also applied. Within the current context of the OWA operator, the methods for aggregating metasearch engine results are divided into two kinds. One has a unique solution, and the other has multiple solutions. The proposed method not only has crisp weights, but also provides multiple aggregation results for decision makers to choose from. In order to prove the application of the ME-OWA operator method, under the context of aggregating metasearch engine results, an example is given, which shows the results obtained by the ME-OWA operator method and the minimax linear programming （ minimax-LP ） method. Comparison between these two methods are also made. The results show that the ME-OWA operator has nearly the same aggregation results as those of the minimax-LP method.

Integrating unascertained measurement and information entropy theory to assess blastability of rock mass

Due to the complex features of rock mass blastability assessment systems, an evaluation model of rock mass blastability was established on the basis of the unascertained measurement （UM） theory and the actual characteristics of the project. Considering a comprehensive range of intact rock properties and discontinuous structures of rock mass, twelve main factors influencing the evaluation blastability of rock mass were taken into account in the UM model, and the blastability evaluation index system of rock mass was constructed. The unascertained evaluation indices corresponding to the selected factors for the actual situation were solved both qualitatively and quantitatively. Then, the UM function of each evaluation index was obtained based on the initial data for the analysis of the blastability of six rock mass at a highway improvement cutting site in North Wales. The index weights of the factors were calculated by entropy theory, and credible degree identification （CDI） criteria were established according to the UM theory. The results of rock mass blastability evaluation were obtained by the CDI criteria. The results show that the UM model assessment results agree well with the actual records, and are consistent with those of the fuzzy sets evaluation method. Meanwhile, the unascertained superiority degree of rock mass blastability of samples S1-$6 which can be calculated by scoring criteria are 3.428 5, 3.453 3, 4.058 7, 3.675 9, 3.516 7 and 3.289 7, respectively. Furthermore, the proposed method can take into account large amount of uncertain information in blastability evaluation, which can provide an effective, credible and feasible way for estimating the blastability of rock mass. Engineering practices show that it can complete the blastability assessment systematically and scientifically without any assumption by the proposed model, which can be applied to practical engineering.

ENDPOINT ESTIMATE FOR MAXIMAL COMMUTATORS WITH NON-DOUBLING MEASUES

The authors establish the weak type endpoint estimate for the maximal commutators generated by Calderon-Zygmund singular integrals and Orlicz type functions with non-doubling measures.

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.

A Knowledge Measure with Parameter of Intuitionistic Fuzzy Sets

A new knowledge measure with parameter of intuitionistic fuzzy sets (IFSs) is presented based on the membership degree and the non-membership degree of IFSs, which complies with the extended form of Szmidt-Kacprzyk axioms for intuitionistic fuzzy entropy. And a sufficient and necessary condition of order property in the Szmidt-Kacprzyk axioms is discussed. Additionally, some numerical examples are given to illustrate the applications of the proposed knowledge measure and some conventional entropies and knowledge measures of IFSs. The experimental results show that the results of the parametric model proposed in this paper are more accurate than those of most of the classic models.

ESTIMATES FOR THE MAXIMAL SINGULAR INTEGRALS WITHOUT DOUBLING CONDITION

It is shown that the maximal singular integral operator with kernels satisfying Ho rmander＇s condition is of weak type （1,1） and L^p （1〈p〈∞） bounded without assuming that the underlying measure p is doubling. Under stronger smoothness conditions,such estimates can be obtained by using a Cotlar＇s inequality. This inequality is not applicable here and it is noticeable that the Cotlar＇s inequality maybe fails under Hormander＇s condition.

Coherence measures based on sandwiched Rényi relative entropy

Coherence is a fundamental ingredient for quantum physics and a key resource for quantum information theory.Baumgratz,Cramer and Plenio established a rigorous framework(BCP framework)for quantifying coherence[Baumgratz T,Cramer M and Plenio M B Phys.Rev.Lett.113140401(2014)].In the present paper,under the BCP framework we provide two classes of coherence measures based on the sandwiched Rényi relative entropy.We also prove that we cannot get a new coherence measure f(C(·))by a function f acting on a given coherence measure C.

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.

A Novel Method for Prediction of Protein Domain Using Distance-Based Maximal Entropy

Detecting the boundaries of protein domains is an important and challenging task in both experimental and computational structural biology. In this paper, a promising method for detecting the domain structure of a protein from sequence information alone is presented. The method is based on analyzing multiple sequence alignments derived from a database search. Multiple measures are defined to quantify the domain information content of each position along the sequence. Then they are combined into a single predictor using support vector machine. What is more important, the domain detection is first taken as an imbal- anced data learning problem. A novel undersampling method is proposed on distance-based maximal entropy in the feature space of Support Vector Machine （SVM）. The overall precision is about 80%. Simulation results demonstrate that the method can help not only in predicting the complete 3D structure of a protein but also in the machine learning system on general im- balanced datasets.

Quantitative comparison of similarity measure and entropy for fuzzy sets

Comparison and data analysis with the similarity measures and entropy for fuzzy sets were carried out. The distance proportional value between the fuzzy set and the corresponding crisp set was considered by the fuzzy entropy. The relation between the similarity measure and the entropy for fuzzy set was also analyzed. The fuzzy entropy was reformulated as the dissimilarity measure. Furthermore, crisp set having the minimum uncertainty with respect to the corresponding fuzzy set was also proposed. Finally, derivation of a similarity measure from entropy with the help of total information property was derived. A simple example shows the relation between similarity measure and fuzzy entropy, in which the summation of similarity measure and fuzzy entropy represents a constant value.

An Entropy Measurement Method of Quantum Information System under Uncertain Environment

Under uncertain environment, it is very difficult to measure the entropy of quantum information system, because there is no effective method to model the randomness. First, different from the traditional classic uncertainty, a quantum uncertain model is proposed to simulate a quantum information system under uncertain environment, and to simplify the entropy measurement of quantum information system. Second, different from the classic random seed under uncertain environment which is often called as pseudo-random number, here the quantum random is employed to provide us a true random model for the entropy of quantum information system. Third, the complex interaction and entangling activity of uncertain factors of quantum information is modeled as quantum binary instead of classic binary, which can help us to evaluate the entropy of uncertain environment, to analyze the entropy divergence in quantum information system. This work presents a non-classic risk factor measurement for quantum information system and a helpful entropy measurement.

Entropy and Similarity Measure for T2SVNSs and Its Application

The objective of this paper is to present a new approach for solving the multicriteria group decision-making(MCGDM)problems in type-2 single valued neutrosophic set(T2SVNS)environment.Firstly,we give the concepts SVNS,T2SVNS and tangent similarity measure with T2SVN information.Secondly,we define a new entropy function for determining unknown attribute weights.In addition,a MCGDM method is developed based on entropy and tangent similarity measure of T2SVNSs.Finally,an illustrative example and comparative analysis are given to confirm the rationality and feasibility of the proposed method.

Non-commitment Entropy： A Noval Modalityfor Uncertainty Measurement

Three-way decision rule is the extension of traditional two-way decision. In the real environment, a decision maker is not easy tomake choice between acceptance and rejection for the uncertainly or incomplete information. In this case, people used to choosethree-way decision for the uncertain and high risky decision with extra but necessary cost. Meanwhile some general uncertaintymeasures are proposed by generalizing Shannon＇s entropy. The theory of information entropy makes the uncertainty measuresmore accuracy in boundary of three-way decision. In this paper, we propose several types of non-commitment entropy by usingthe relation of the ＇third＇ decision-non-commitment, and employ the proposed model to evaluate the significance of theattributes for classification as well.

Entropy of Measurements of Electric and Non-eleCtric Systems Fluctuating Parameters

Fluctuations of the measured parameters of the investigated electric and non-electric systems are considered on the thermodynamics basis.We have defined that entropy of the investigated system in relation to environment depends on the correlation between measuring durationТmeas of the system’s fluctuation parameter and the system’s relaxation timeτ.Decrement of the energy spectrum of fluctuations with the decrease of frequency is explained byТmeas>τ.WhenТmeas≈Nmaxτ(where Nmax is the number of possible ways to create the balanced state of the system),entropy of investigated system tends to zero in relation to the environment,and the measurement error caused by fluctuations becomes the minimal.

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.

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.

Entropy at the Level of Individual Particles: Analysis of Maxwell’s Agent with a Hidden-Variable Theory

Problem: Maxwell’s Agent (MA) is a thought experiment about whether the second Law is violated at smaller scales. This is a complex problem because the scale dependencies are unclear for perfect gas assumptions, quantum coherence, thermalisation, and contextual measurement. Purpose: The MA is explored from a non-local hidden-variable (NLHV) perspective. Approach: The Cordus theory, a specific NLHV solution, was applied at macroscopic to fundamental scales. Physical realism requires the Agent be included in the analysis. Findings: The primary function is sorting, i.e. a one-time separation of species by some attribute. The thermodynamic MA situation is merely a special case for reducing disorder (entropy). A one-time extraction of energy is possible. This requires input energy, hence the device only has thermodynamic leverage and is not a perpetual motion device. Inefficiencies arise from thermalisation causing short mean free path of Brownian motion, perfect gases having minimal interaction with the gate, ambiguity about spatial location arising from quantum superposition, contextual measurement interfering with the particle velocity, and bremsstrahlung hysteresis losses occurring when the Agent operates. Implications: Entropy is a group property at the bulk level, not a characteristic of the individual particle, and can be reversed at an energy cost at the particle level. Originality: The explanation spans multiple levels from macroscopic down to fundamental, which is unusual. Achieving an explanation from the NLHV sector is novel. The theory accommodates superposition, irreversibility, entropy, contextual measurement, coherence-discord transition, and Brownian motion.