Performance Evaluation of Public Bus Transportation by Using DEA Models and Shannon’s Entropy:An Example From a Company in a Large City of China

The purpose of this paper is to assess the operational efficiency of a public bus transportation via a case study from a company in a large city of China by using data envelopment analysis(DEA)model and Shannon’s entropy.This company operates 37 main routes on the backbone roads.Thus,it plays a significant role in public transportation in the city.According to bus industry norms,an efficiency evaluation index system is constructed from the perspective of both company operations and passenger demands.For passenger satisfaction,passenger waiting time and passenger-crowding degree are considered,and they are undesirable indicators.To describe such indicators,a superefficient DEA model is constructed.With this model,by using actual data,efficiency is evaluated for each bus route.Results show that the DEA model with Shannon’s entropy being combined achieves more reasonable results.Also,sensitivity analysis is presented.Therefore,the results are meaningful for the company to improve its operations and management.

The Use of GIS, Remote Sensing and Shannon’s Entropy Statistical Techniques to Analyze and Monitor the Spatial and Temporal Patterns of Urbanization and Sprawl in Zarqa City, Jordan

The aim of this study is to understand and quantify the urban growth and trend in Zarqa city during the period 1990 to 2014 and to produce land use and cover map for the studied area through the use of the GIS and remote sensing techniques with Shannon’s Entropy statistical method. For this purpose, three Landsat images were used for land use classification by using supervised maximum likelihood classification techniques to extract and assess the changes of urban lands. The results indicated that the urban areas in Zarqa city increased by 22.15% in the period from 1990 to 2005 and 14.86% from 2005 to 2014, with a rate of expansion of 0.96 and by 1.31 km2/ year for the two time periods respectively. The entropy value increased from 1.20 in the first period to 1.38 in the second, while the entropy value for the NE, NW, SE and SW zones showed high values, which confirmed that urban expansion and sprawling had existed in the past twenty four years in the study area. Urban expansion and sprawl cause different impacts on the natural, economic, and aesthetic aspects of the city which lead and guide government officials and planners to understand and monitor current growth and visualize future growth.

Modeling and Mapping of Urban Sprawl Pattern in Cairo Using Multi-Temporal Landsat Images, and Shannon’s Entropy

Cairo city, being the Egypt’s industrial and cultural center, has a problem of rapid urban sprawl. The city has an extremely high population density which is continuously increasing through informal settlements that grow by sprawling due to migration from the Nile Delta villages and the high population growth rates. The present study attempts to understand, detect and quantify the spatial pattern of Cairo’s urban sprawl using Shannon’s entropy and multi-temporal Landsat TM and ETM images acquired for the period from 1984 to 2013. Supervised classification was applied to extract the built-up areas and to measure the changes in the urban land-use class among the city wards. Shannon’s entropy was applied to model the city’s urban sprawl, trend and spatial change. The entropy values for the city’s electoral wards were modeled and used in an interpolation function to create an entropy surface (index) for each acquired temporal image. Such index indicates the spatial pattern of the urban sprawl and provides a visual comparison of the entropy phenomenon in such wards. Results indicate that Shannon’s entropy index increased from (1.4615) in year 1984 to (2.1023) in year 2013, indicating more dispersed urban growth, a sign of urban sprawl. The maximum entropy values are found in the eastern wards namely El Nozha, Awal Nasr District, Thany Nasr-District, El Salam, El Marg and El Bassatein. A regression analysis was carried for the population growth rate and the built-up areas. Findings help in understanding the sprawl patterns and dynamics among Cairo’s electoral wards and provide a visual comparison. The applied methodology provides explanations and facilitates tracing and measuring the urban sprawl which is needed by decision makers and city planners of mega cities.

Re-Formulation of Mean King’s Problem Using Shannon’s Entropy

Mean King’s problem is formulated as a retrodiction problem among noncommutative observables. In this paper, we reformulate Mean King’s problem using Shannon’s entropy as a first step of introducing quantum uncertainty relation with delayed classical information. As a result, we give informational and statistical meanings to the estimation on Mean King problem. As its application, we give an alternative proof of nonexistence of solutions of Mean King’s problem for qubit system without using entanglement.

基于Shannon熵与声发射信号的CBN砂轮性能监测方法研究

为了在磨削加工过程中能够有效判别CBN(Cubic Boron Nitride)砂轮的磨削性能,提出了一种基于Shannon熵理论与声发射信号的CBN砂轮性能监测方法。首先,利用声发射传感器采集CBN砂轮磨削加工过程中的声发射信号,基于最大信息熵对CBN砂轮磨削加工过程中的声发射信号进行概率密度估计,获得磨削加工过程中声发射信号的最大熵概率密度分布。然后,通过分析研究CBN砂轮在修整过后循环磨削以及不同直径剩余磨削时的声发射信号特征,根据交叉熵原理分析CBN砂轮不同磨削性能时声发射信号最大熵概率密度分布,并通过设定交叉熵阈值来辨别磨削加工过程中CBN砂轮的磨削性能。最后,为验证该方法的实用性,在某工厂CBN砂轮磨削产品生产线上进行大量实验研究,结果表明,该方法对CBN砂轮磨损状态及CBN砂轮剩余寿命进行有效监测,验证了该方法监测CBN砂轮在磨削加工过程中磨削性能的有效性。

基于D-S证据理论改进AHP-熵权的流域洪涝灾害评估研究

考虑致灾因子危险性、孕灾环境敏感性以及承灾体易损性,选取指标构建小清河流域洪涝灾害风险评估指标体系,提出一种基于D-S证据理论的改进AHP-熵权法计算指标权重,求取洪涝灾害风险指数,运用自然断点分级法确定洪涝灾害风险等级,分析小清河流域洪涝灾害风险空间分布情况。结果表明:小清河流域洪涝灾害风险总体上表现出南低北高的趋势,其中高风险区和较高风险区分别占流域面积的8.7%和14.3%,主要分布在小清河干流以及主要支流两岸。所得评估结果同“利奇马”台风发生期间实际洪灾风险分布情况一致,对比证明基于D-S证据理论的改进AHP-熵权法优于AHP和熵权法,可为小清河流域防洪减灾决策提供依据。

Nonrelativistic Shannon information entropy for Kratzer potential

The Shannon information entropy is investigated within the nonrelativistic framework. The Kratzer potential is con- sidered as the interaction and the problem is solved in a quasi-exact analytical manner to discuss the ground and first excited states. Some interesting features of the information entropy densities as well as the probability densities are demonstrated. The Bialynicki-Birula-Mycielski inequality is also tested and found to hold for these cases.

水泵水轮机S特性区能量损失及流动特性研究

为了调节电网的稳定性,抽水蓄能电站需要频繁启停和变换工况运行,导致水泵水轮机容易进入S特性区,机组振动增加,并网失败。本文以模型水泵水轮机为研究对象,采用熵产理论详细分析了S特性区不同工况下的能量损失规律,明确了熵产率分布与内部流动结构的关系。结果表明:S特性区内近飞逸工况总熵产最大,约为设计工况的5.1倍,脉动熵产占据的比例接近80%,随着流量的减小,转轮熵产占比逐渐降低,活动导叶和尾水管的熵产占比增加。小流量工况转轮进口靠近下环位置首先出现了明显的漩涡,导致了活动导叶出口和转轮进口的高熵产区,随着流量进一步减小,漩涡逐渐向上冠转移,并且切向速度增大,在转轮进口形成挡水环,阻碍水流进入转轮,在无叶区内出现了环状分布的高熵产区。反水泵工况,水流在低压边与逆时针旋转的叶片撞击,导致水流很难进入叶片内部,形成了大尺度的回流涡结构;双列叶栅内充满大量涡结构,导致活动导叶吸力面的熵产率增大,并且向固定导叶传播。

RAPD Markers and Genetic Information Entropy in Environmental Monitoring: A Case Study with Wild Mushrooms

Mushrooms have a remarkable scientific value due to their nutritional, medicinal properties and industrial applications in enzyme production, so that effort in the maintenance of native wild mushroom varieties is increasing. The present study focuses on the use of Random Amplified Polymorphic DNA (RAPD) markers for biodiversity measure of wild mushroom species of the Northwest mountainous region of Greece. Data mining of similarity matrices from RAPD analysis was used to extract measurable entropy parameters for mushroom biodiversity monitoring based on Shannon’s information entropy. Shannon information index provides an easy assessment of the entropy of the genetic information of the germplasm per mushroom species while the total equitability index (EH) = 0.871 offers an overall estimation of the genetic variation evenness of all species in the population of the studied mushrooms. Application of RAPDs with parallel entropy analysis is an easily applicable and low-cost valuable technology in environmental monitoring, using genetic information of wild mushroom species as an indicator that can lead to future actions in biodiversity maintenance and germplasm protection. The provided methodology can serve as a pilot procedure enriched with other environmental factors to monitor and protect wild mushroom communities native to the Greek countryside or in any part of the world and provide comparable results about biodiversity from different regions using common entropy indices.

On Clausius, Boltzmann and Shannon Notions of Entropy

Discrete dynamical systems are given by the pair (X,f) where X is a compact metric space and f: X→X is a continuous map. During years, a long list of results have appeared to precise and understand what is the complexity of the systems. Among them, one of the most popular is that of topological entropy. In modern applications, other conditions on X and f have been considered. For example, X can be non-compact or f can be discontinuous (only in a finite number of points and with bounded jumps on the values of f or even non-bounded jumps). Such systems are interesting from theoretical point of view in Topological Dynamics and appear frequently in applied sciences such as Electronics and Control Theory. In this paper, we are reviewing the origins of the notion of entropy and studying some developing of it leading to modern notions of entropies. At the same time, we will incorporate some mathematical foundations of such old and new ideas until the appearance of Shannon entropy. To this end, we start with the introduction for the first time of the notion of entropy in thermodynamics by R. Clausius and its evolution by L. Boltzmann until the appearing in the twenty century of Shannon and Kolmogorov-Sinai entropies and the subsequent topological entropy. In turn, such notions have evolved to other recent situations where it is necessary to give some extended versions of them adapted to new problems. Of special interest is to appreciate the connexions of the notions of entropy from Boltzmann and Shannon. Since this history is long, we will not deal with the Kolmogorov-Sinai entropy or with topological entropy and modern approaches.

Effects of Shannon entropy and electric field on polaron in RbCl triangular quantum dot

In this paper, the time evolution of the quantum mechanical state of a polaron is examined using the Pekar type variational method on the condition of the electric-LO-phonon strong-coupling and polar angle in RbC1 triangular quantum dot. We obtain the eigenenergies, and the eigenfunctions of the ground state, and the first excited state respectively. This system in a quantum dot can be treated as a two-level quantum system qubit and the numerical calculations are performed. The effects of Shannon entropy and electric field on the polaron in the RbC1 triangular quantum dot are also studied.

Shannon information entropies for position-dependent mass Schrdinger problem with a hyperbolic well

The Shannon information entropy for the Schrodinger equation with a nonuniform solitonic mass is evaluated for a hyperbolic-type potential. The number of nodes of the wave functions in the transformed space z are broken when recovered to original space x. The position Sx and momentum S p information entropies for six low-lying states are calculated. We notice that the Sx decreases with the increasing mass barrier width a and becomes negative beyond a particular width a,while the Sp first increases with a and then decreases with it. The negative Sx exists for the probability densities that are highly localized. We find that the probability density ρ（x） for n = 1, 3, 5 are greater than 1 at position x = 0. Some interesting features of the information entropy densities ρs（x） and ρs（p） are demonstrated. The Bialynicki-Birula-Mycielski（BBM）inequality is also tested for these states and found to hold.

Shannon information entropies for rectangular multiple quantum well systems with constant total lengths

We first study the Shannon information entropies of constant total length multiple quantum well systems and then explore the effects of the number of wells and confining potential depth on position and momentum information entropy density as well as the corresponding Shannon entropy.We find that for small full width at half maximum（FWHM） of the position entropy density,the FWHM of the momentum entropy density is large and vice versa.By increasing the confined potential depth,the FWHM of the position entropy density decreases while the FWHM of the momentum entropy density increases.By increasing the potential depth,the frequency of the position entropy density oscillation within the quantum barrier decreases while that of the position entropy density oscillation within the quantum well increases.By increasing the number of wells,the frequency of the position entropy density oscillation decreases inside the barriers while it increases inside the quantum well.As an example,we might localize the ground state as well as the position entropy densities of the1 st,2 nd,and 6 th excited states for a four-well quantum system.Also,we verify the Bialynicki–Birula–Mycieslki（BBM）inequality.

Dense Fractal Networks, Trends, Noises and Switches in Homeostasis Regulation of Shannon Entropy for Chromosomes’ Activity in Living Cells for Medical Diagnostics

We analyze correlations and patterns of oxidative activity of 3D DNA at DNA fluorescence in complete sets of chromosomes in neutrophils of peripheral blood. Fluorescence of DNA is registered by method of flow cytometry with nanometer spatial resolution. Experimental data present fluorescence of many ten thousands of cells, from different parts of body in each population, in various blood samples. Data is presented in histograms as frequency distributions of flashes in the dependence on their intensity. Normalized frequency distribution of information in these histograms is used as probabilistic measure for definition of Shannon entropy. Data analysis shows that for this measure of Shannon entropy common sum of entropy, i.e. total entropy E, for any histogram is invariant and has identical trends of changes all values of E (r) = lnr at reduction of rank r of histogram. This invariance reflects informational homeostasis of chromosomes activity inside cells in multi-scale networks of entropy, for varied ranks r. Shannon entropy in multi-scale DNA networks has much more dense packing of correlations than in “small world” networks. As the rule, networks of entropy differ by the mix of normal D 2 and abnormal D > 2 fractal dimensions for varied ranks r, the new types of fractal patterns and hinges for various topology (fractal dimension) at different states of health. We show that all distributions of information entropy are divided on three classes, which associated in diagnostics with a good health or dominants of autoimmune or inflammatory diseases. This classification based on switching of stability at transcritical bifurcation in homeostasis regulation. We defined many ways for homeostasis regulation, coincidences and switching patterns in branching sequences, the averages of Hölder for deviations of entropy from homeostasis at different states of health, with various saturation levels the noises of entropy at activity of all chromosomes in support regulation of homeostasis.

基于最大信息系数法和邓熵的D-S证据理论改进

物联网中大量传感器采集的数据存在不确定性,针对D-S证据理论在处理冲突证据时融合决策结果与事实相悖的问题,文章提出一种新的基于改进D-S证据理论的多传感器数据融合算法,首先使用最大信息系数法计算证据间的可信度;然后结合信息熵对证据的不确定度进行分析,以确定新的权重;最后使用Dempster组合规则得到融合结果。算例分析表明,文章所提方法能有效融合冲突证据,较经典算法有较高的基本概率分配。将所提方法用于传感器数据处理,不仅能降低数据中存在的不确定性,还能有效处理D-S理论中的冲突问题,从而得到较为准确的融合结果。

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.

Invariance and Noises of Shannon Entropy for Information on Oxidative Activity of DNA in All Living Cells for Medical Diagnostics

We analyze oxidative activity of DNA due to fluorescence of chromosomes inside cells, using flow cytometry method with nanometer spatial resolution. Statistics of fluorescence is presented in histogram as frequency distributions of flashes in the dependence on their intensity and in distributions of Shannon entropy, which was defined on the base of normalized distribution of information in original histogram for frequency of flashes. We show that overall sum of entropy, i.e. total entropy E , for any histogram is invariant and has identical trends of changes all values of E(r) = lnr at reduction of histogram’ rank r. This invariance reflects informational homeostasis of chromosomes activity in multi-scale networks of entropy inside all cells in various samples of blood for DNA inside neutrophils, lymphocytes, inside all leukocytes of human and inside chicken erythrocytes for various dyes, colors and various excitations of fluorescence. Informational homeostasis of oxidative activity of 3D DNA in the full set of chromosomes inside living cells exists for any Shannon-Weaver index of biodiversity of cells, at any state of health different beings. Regulation perturbations in information activity DNA provides informational adaptability and vitality of cells at homeostasis support. Noises of entropy, during regulation of informational homeostasis, depend on the states of health in real time. The main structural reconstructions of chromosomal correlations, corresponding to self-regulation of homeostasis, occur in the most large-scale networks of entropy, for rank r<32. We show that stability of homeostasis is supported by activity of all 46 chromosomes inside cells. Patterns, hidden switching and branching in sequences of averages of H?lder and central moments for noises in regulation of homeostasis define new opportunities in diagnostics of health and immunity. All people and all aerobic beings have one overall homeostatic level for countdown of information activity of DNA inside cells. We noted very bad and dangerous properties of artificial cells with other levels of informational homeostasis for all aerobic beings in foods, medical treatment and in biotechnologies.

基于改进D-S证据理论的相控阵雷达作战效能评估

针对相控阵雷达作战效能评估中战场环境等不确定因素的影响,采用改进Dempster-Shafer(D-S)证据理论对其进行作战效能评估方法的优化。首先,通过构建相控阵雷达作战效能评估指标体系,明确评估指标并赋值权。其次,引入Kulsinski差异和模糊熵方法,对传统的D-S证据理论进行改进,减少指标在融合过程中的融合权重不确定性的影响。最后,通过改进D-S证据理论对评估指标进行数据融合,建立评估准确性较高的相控阵雷达作战效能评估模型。通过算例验证分析,所提方法能够有效减少融合过程中的不确定性影响,可实现相控阵雷达作战效能评估的目的,提高作战效能评估精度。

基于云D-S证据理论的极端天气下电力系统应急能力评判

以提高应对极端天气下电力系统突发事件能力为目标,将电力系统应急能力作为研究对象,提出基于云模型和证据理论融合的评估算法。采用层次分析法和熵值法相结合的方法确定应急指标权重,运用云模型做为基础对电力系统应急能力进行评估,以解决评价指标模糊性和随机性的问题,并引入D-S证据理论对指标进行综合评估,构建极端天气下电力系统的应急能力评估模型。通过案例证明,该方法可以有效准确地定量和定性评估电力网系统的应急能力,为提高电力网系统的应急管理能力提供科学依据。

非平衡群体基因变异测量的Shannon信息量方法

在Ｓｈａｎｎｏｎ信息量的基础上，对非平衡群体建立了群体基因型相对信息量Ｓ＇（Ｇ），纯合体相对信息量Ｓ＇Ｊ（Ｇ）、杂合体相对信息量Ｓ＇Ｈ（Ｇ）的概念，并赋予它们以遗传学意义．与基因一致度Ｊ和基因多样度Ｄ进行了理论比较，结果表明，二者在数量规律上有很好的一致性，但又是相对独立的指标体系，且各相对信息量还有新的内涵．Ｓ＇（Ｇ）既能表征基因变异，又能反映基因型水平上的遗传变异，Ｓ＇Ｊ（Ｇ）主要反映纯合体的遗传变异，Ｓ＇Ｈ（Ｇ）主要反映杂合体的遗传变异．各相对信息量既可反映群体的遗传变异程度，又能比较不同位点间的遗传变异程度．