ALBERT

Description: Entropy, Vol. 20, Pages 550: Information Geometry of Nonlinear Stochastic Systems Entropy doi: 10.3390/e20080550 Authors: Rainer Hollerbach Donovan Dimanche Eun-jin Kim We elucidate the effect of different deterministic nonlinear forces on geometric structure of stochastic processes by investigating the transient relaxation of initial PDFs of a stochastic variable x under forces proportional to -xn (n=3,5,7) and different strength D of δ-correlated stochastic noise. We identify the three main stages consisting of nondiffusive evolution, quasi-linear Gaussian evolution and settling into stationary PDFs. The strength of stochastic noise is shown to play a crucial role in determining these timescales as well as the peak amplitude and width of PDFs. From time-evolution of PDFs, we compute the rate of information change for a given initial PDF and uniquely determine the information length L(t) as a function of time that represents the number of different statistical states that a system evolves through in time. We identify a robust geodesic (where the information changes at a constant rate) in the initial stage, and map out geometric structure of an attractor as L(t→∞)∝μm, where μ is the position of an initial Gaussian PDF. The scaling exponent m increases with n, and also varies with D (although to a lesser extent). Our results highlight ubiquitous power-laws and multi-scalings of information geometry due to nonlinear interaction.

Description: Entropy, Vol. 20, Pages 553: Hierarchical Structure of Generalized Thermodynamic and Informational Entropy Entropy doi: 10.3390/e20080553 Authors: Pierfrancesco Palazzo The present research aimed at discussing the thermodynamic and informational aspects of entropy concept to propose a unitary perspective of its definitions as an inherent property of any system in any state. The dualism and the relation between physical nature of information and the informational content of physical states of matter and phenomena play a fundamental role in the description of multi-scale systems characterized by hierarchical configurations. A method is proposed to generalize thermodynamic and informational entropy property and characterize the hierarchical structure of its canonical definition at macroscopic and microscopic levels of a system described in the domain of classical and quantum physics. The conceptual schema is based on dualisms and symmetries inherent to the geometric and kinematic configurations and interactions occurring in many-particle and few-particle thermodynamic systems. The hierarchical configuration of particles and sub-particles, representing the constitutive elements of physical systems, breaks down into levels characterized by particle masses subdivision, implying positions and velocities degrees of freedom multiplication. This hierarchy accommodates the allocation of phenomena and processes from higher to lower levels in the respect of the equipartition theorem of energy. However, the opposite and reversible process, from lower to higher level, is impossible by virtue of the Second Law, expressed as impossibility of Perpetual Motion Machine of the Second Kind (PMM2) remaining valid at all hierarchical levels, and the non-existence of Maxwell’s demon. Based on the generalized definition of entropy property, the hierarchical structure of entropy contribution and production balance, determined by degrees of freedom and constraints of systems configuration, is established. Moreover, as a consequence of the Second Law, the non-equipartition theorem of entropy is enunciated, which would be complementary to the equipartition theorem of energy derived from the First Law.

Description: Entropy, Vol. 20, Pages 552: The Gibbs Paradox Entropy doi: 10.3390/e20080552 Authors: Simon Saunders The Gibbs Paradox is essentially a set of open questions as to how sameness of gases or fluids (or masses, more generally) are to be treated in thermodynamics and statistical mechanics. They have a variety of answers, some restricted to quantum theory (there is no classical solution), some to classical theory (the quantum case is different). The solution offered here applies to both in equal measure, and is based on the concept of particle indistinguishability (in the classical case, Gibbs’ notion of ‘generic phase’). Correctly understood, it is the elimination of sequence position as a labelling device, where sequences enter at the level of the tensor (or Cartesian) product of one-particle state spaces. In both cases it amounts to passing to the quotient space under permutations. ‘Distinguishability’, in the sense in which it is usually used in classical statistical mechanics, is a mathematically convenient, but physically muddled, fiction.

Description: Entropy, Vol. 20, Pages 549: Entropy-Based Feature Extraction for Electromagnetic Discharges Classification in High-Voltage Power Generation Entropy doi: 10.3390/e20080549 Authors: Imene Mitiche Gordon Morison Alan Nesbitt Brian G. Stewart Philip Boreham This work exploits four entropy measures known as Sample, Permutation, Weighted Permutation, and Dispersion Entropy to extract relevant information from Electromagnetic Interference (EMI) discharge signals that are useful in fault diagnosis of High-Voltage (HV) equipment. Multi-class classification algorithms are used to classify or distinguish between various discharge sources such as Partial Discharges (PD), Exciter, Arcing, micro Sparking and Random Noise. The signals were measured and recorded on different sites followed by EMI expert’s data analysis in order to identify and label the discharge source type contained within the signal. The classification was performed both within each site and across all sites. The system performs well for both cases with extremely high classification accuracy within site. This work demonstrates the ability to extract relevant entropy-based features from EMI discharge sources from time-resolved signals requiring minimal computation making the system ideal for a potential application to online condition monitoring based on EMI.

Description: Symmetry, Vol. 10, Pages 300: Computing Metric Dimension and Metric Basis of 2D Lattice of Alpha-Boron Nanotubes Symmetry doi: 10.3390/sym10080300 Authors: Zafar Hussain Mobeen Munir Maqbool Chaudhary Shin Min Kang Concepts of resolving set and metric basis has enjoyed a lot of success because of multi-purpose applications both in computer and mathematical sciences. For a connected graph G(V,E) a subset W of V(G) is a resolving set for G if every two vertices of G have distinct representations with respect to W. A resolving set of minimum cardinality is called a metric basis for graph G and this minimum cardinality is known as metric dimension of G. Boron nanotubes with different lattice structures, radii and chirality’s have attracted attention due to their transport properties, electronic structure and structural stability. In the present article, we compute the metric dimension and metric basis of 2D lattices of alpha-boron nanotubes.

Description: Symmetry, Vol. 10, Pages 299: Game-Theoretic Solutions for Data Offloading in Next Generation Networks Symmetry doi: 10.3390/sym10080299 Authors: Muhammad Asif Shafi Ullah Khan Rashid Ahmad Dhananjay Singh In recent years, global mobile data traffic has seen an unprecedented increase. This is due to worldwide usage of smart devices, availability of fast internet connections, and the popularity of social media. The Mobile Network Operators (MNOs) are, therefore, facing problems in handling this huge traffic flow. Each type of traffic, including real-time video, audio, and text has its own Quality of Services (QoS) requirements which, if not met, may cause a sufficient loss of profit. Offloading of these traffics can be made more efficient so that values of QoS parameters are enhanced. In this work, we propose an incentive-based game-theoretic frame work for downloading data. The download of each type of data will get an incentive determined by the two-stage Stackelberg game. We model the communication among single Mobile Base Station (MBS) and multiple Access Points (APs) in a crowded metropolitan environment. The leader offers an economic incentive based on the traffic type and followers respond to the incentive and offload traffic accordingly. The model optimizes strategies of both the MBS and APs in order to make the best use of their utilities. For the analysis, we have used a combination of analytical and experimental methods. The numerical outcome characterized a direct process of the best possible offloading ratio and legalized the efficiency of the proposed game. Optimal incentives and optimal offloading was the achievement of our proposed game-theoretic approach. We have implemented the model in MATLAB, and the experimental results show a maximum payoff was achieved and the proposed scheme achieved Nash Equilibria.

Description: Entropy, Vol. 20, Pages 555: Quantum Games with Unawareness Entropy doi: 10.3390/e20080555 Authors: Piotr Frąckiewicz Games with unawareness model strategic situations in which players’ perceptions about the game are limited. They take into account the fact that the players may be unaware of some of the strategies available to them or their opponents as well as the players may have a restricted view about the number of players participating in the game. The aim of the paper is to introduce this notion into theory of quantum games. We focus on games in strategic form and Eisert–Wilkens–Lewenstein type quantum games. It is shown that limiting a player’s perception in the game enriches the structure of the quantum game substantially and allows the players to obtain results that are unattainable when the game is played in a quantum way by means of previously used methods.

Description: Entropy, Vol. 20, Pages 560: Ensemble Estimation of Information Divergence † Entropy doi: 10.3390/e20080560 Authors: Kevin R. Moon Kumar Sricharan Kristjan Greenewald Alfred O. Hero Recent work has focused on the problem of nonparametric estimation of information divergence functionals between two continuous random variables. Many existing approaches require either restrictive assumptions about the density support set or difficult calculations at the support set boundary which must be known a priori. The mean squared error (MSE) convergence rate of a leave-one-out kernel density plug-in divergence functional estimator for general bounded density support sets is derived where knowledge of the support boundary, and therefore, the boundary correction is not required. The theory of optimally weighted ensemble estimation is generalized to derive a divergence estimator that achieves the parametric rate when the densities are sufficiently smooth. Guidelines for the tuning parameter selection and the asymptotic distribution of this estimator are provided. Based on the theory, an empirical estimator of Rényi-α divergence is proposed that greatly outperforms the standard kernel density plug-in estimator in terms of mean squared error, especially in high dimensions. The estimator is shown to be robust to the choice of tuning parameters. We show extensive simulation results that verify the theoretical results of our paper. Finally, we apply the proposed estimator to estimate the bounds on the Bayes error rate of a cell classification problem.

Description: Entropy, Vol. 20, Pages 556: Multivariate Multiscale Complexity Analysis of Self-Reproducing Chaotic Systems Entropy doi: 10.3390/e20080556 Authors: Shaobo He Chunbiao Li Kehui Sun Sajad Jafari Designing a chaotic system with infinitely many attractors is a hot topic. In this paper, multiscale multivariate permutation entropy (MMPE) and multiscale multivariate Lempel–Ziv complexity (MMLZC) are employed to analyze the complexity of those self-reproducing chaotic systems with one-directional and two-directional infinitely many chaotic attractors. The analysis results show that complexity of this class of chaotic systems is determined by the initial conditions. Meanwhile, the values of MMPE are independent of the scale factor, which is different from the algorithm of MMLZC. The analysis proposed here is helpful as a reference for the application of the self-reproducing systems.

Description: Entropy, Vol. 20, Pages 564: A New Fractional-Order Chaotic System with Different Families of Hidden and Self-Excited Attractors Entropy doi: 10.3390/e20080564 Authors: Jesus Munoz-Pacheco Ernesto Zambrano-Serrano Christos Volos Sajad Jafari Jacques Kengne Karthikeyan Rajagopal In this work, a new fractional-order chaotic system with a single parameter and four nonlinearities is introduced. One striking feature is that by varying the system parameter, the fractional-order system generates several complex dynamics: self-excited attractors, hidden attractors, and the coexistence of hidden attractors. In the family of self-excited chaotic attractors, the system has four spiral-saddle-type equilibrium points, or two nonhyperbolic equilibria. Besides, for a certain value of the parameter, a fractional-order no-equilibrium system is obtained. This no-equilibrium system presents a hidden chaotic attractor with a `hurricane’-like shape in the phase space. Multistability is also observed, since a hidden chaotic attractor coexists with a periodic one. The chaos generation in the new fractional-order system is demonstrated by the Lyapunov exponents method and equilibrium stability. Moreover, the complexity of the self-excited and hidden chaotic attractors is analyzed by computing their spectral entropy and Brownian-like motions. Finally, a pseudo-random number generator is designed using the hidden dynamics.

Description: Entropy, Vol. 20, Pages 561: Statistical Significance of Earth’s Electric and Magnetic Field Variations Preceding Earthquakes in Greece and Japan Revisited Entropy doi: 10.3390/e20080561 Authors: Nicholas Sarlis By analyzing the seismicity in a new time domain, termed natural time, we recently found that the change of the entropy under time reversal (Physica A2018, 506, 625–634) and the relevant complexity measures (Entropy2018, 20, 477) exhibit pronounced variations before the occurrence of the M8.2 earthquake in Mexico on 7 September 2017. Here, the statistical significance of precursory phenomena associated with other physical properties and in particular the anomalous variations observed in the Earth’s electric and magnetic fields before earthquakes in different regions of the world and in particular in Greece since 1980s and Japan during 2001–2010 are revisited (the latter, i.e., the magnetic field variations are alternatively termed ultra low frequency (ULF) seismo-magnetic phenomena). Along these lines we employ modern statistical tools like the event coincidence analysis and the receiver operating characteristics technique. We find that these precursory variations are far beyond chance and in addition their lead times fully agree with the experimental findings in Greece since the 1980s.

Description: Entropy, Vol. 20, Pages 563: A New Underwater Acoustic Signal Denoising Technique Based on CEEMDAN, Mutual Information, Permutation Entropy, and Wavelet Threshold Denoising Entropy doi: 10.3390/e20080563 Authors: Yuxing Li Yaan Li Xiao Chen Jing Yu Hong Yang Long Wang Owing to the complexity of the ocean background noise, underwater acoustic signal denoising is one of the hotspot problems in the field of underwater acoustic signal processing. In this paper, we propose a new technique for underwater acoustic signal denoising based on complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN), mutual information (MI), permutation entropy (PE), and wavelet threshold denoising. CEEMDAN is an improved algorithm of empirical mode decomposition (EMD) and ensemble EMD (EEMD). First, CEEMDAN is employed to decompose noisy signals into many intrinsic mode functions (IMFs). IMFs can be divided into three parts: noise IMFs, noise-dominant IMFs, and real IMFs. Then, the noise IMFs can be identified on the basis of MIs of adjacent IMFs; the other two parts of IMFs can be distinguished based on the values of PE. Finally, noise IMFs were removed, and wavelet threshold denoising is applied to noise-dominant IMFs; we can obtain the final denoised signal by combining real IMFs and denoised noise-dominant IMFs. Simulation experiments were conducted by using simulated data, chaotic signals, and real underwater acoustic signals; the proposed denoising technique performs better than other existing denoising techniques, which is beneficial to the feature extraction of underwater acoustic signal.

Description: Entropy, Vol. 20, Pages 567: Nonlocality in Bell’s Theorem, in Bohm’s Theory, and in Many Interacting Worlds Theorising Entropy doi: 10.3390/e20080567 Authors: Mojtaba Ghadimi Michael J. W. Hall Howard M. Wiseman “Locality” is a fraught word, even within the restricted context of Bell’s theorem. As one of us has argued elsewhere, that is partly because Bell himself used the word with different meanings at different stages in his career. The original, weaker, meaning for locality was in his 1964 theorem: that the choice of setting by one party could never affect the outcome of a measurement performed by a distant second party. The epitome of a quantum theory violating this weak notion of locality (and hence exhibiting a strong form of nonlocality) is Bohmian mechanics. Recently, a new approach to quantum mechanics, inspired by Bohmian mechanics, has been proposed: Many Interacting Worlds. While it is conceptually clear how the interaction between worlds can enable this strong nonlocality, technical problems in the theory have thus far prevented a proof by simulation. Here we report significant progress in tackling one of the most basic difficulties that needs to be overcome: correctly modelling wavefunctions with nodes.

Description: Symmetry, Vol. 10, Pages 315: Long Time Behavior and Global Dynamics of Simplified Von Karman Plate Without Rotational Inertia Driven by White Noise Symmetry doi: 10.3390/sym10080315 Authors: Huatao Chen Dengqing Cao Jingfei Jiang Xiaoming Fan Without the assumption that the coefficient of weak damping is large enough, the existence of the global random attractors for simplified Von Karman plate without rotational inertia driven by either additive white noise or multiplicative white noise are proved. Instead of the classical splitting method, the techniques to verify the asymptotic compactness rely on stabilization estimation of the system. Furthermore, a clear relationship between in-plane components of the external force that act on the edge of the plate and the expectation of radius of the global random attractors can be obtained from the theoretical results. Based on the relationship between global random attractor and random probability invariant measure, the global dynamics of the plates are analyzed numerically. With increasing the in-plane components of the external force that act on the edge of the plate, global D-bifurcation, secondary global D-bifurcation and complex local dynamical behavior occur in motion of the system. Moreover, increasing the intensity of white noise leads to the dynamical behavior becoming simple. The results on global dynamics reveal that random snap-through which seems to be a complex dynamics intuitively is essentially a simple dynamical behavior.

Description: Symmetry, Vol. 10, Pages 314: Systematic Review of Decision Making Algorithms in Extended Neutrosophic Sets Symmetry doi: 10.3390/sym10080314 Authors: Mohsin Khan Le Hoang Son Mumtaz Ali Hoang Thi Minh Chau Nguyen Thi Nhu Na Florentin Smarandache The Neutrosophic set (NS) has grasped concentration by its ability for handling indeterminate, uncertain, incomplete, and inconsistent information encountered in daily life. Recently, there have been various extensions of the NS, such as single valued neutrosophic sets (SVNSs), Interval neutrosophic sets (INSs), bipolar neutrosophic sets (BNSs), Refined Neutrosophic Sets (RNSs), and triangular fuzzy number neutrosophic set (TFNNs). This paper contains an extended overview of the concept of NS as well as several instances and extensions of this model that have been introduced in the last decade, and have had a significant impact in literature. Theoretical and mathematical properties of NS and their counterparts are discussed in this paper as well. Neutrosophic-set-driven decision making algorithms are also overviewed in detail.

Description: Symmetry, Vol. 10, Pages 311: On p-Adic Fermionic Integrals of q-Bernstein Polynomials Associated with q-Euler Numbers and Polynomials † Symmetry doi: 10.3390/sym10080311 Authors: Lee-Chae Jang Taekyun Kim Dae San Kim Dmitry Victorovich Dolgy We study a q-analogue of Euler numbers and polynomials naturally arising from the p-adic fermionic integrals on Zp and investigate some properties for these numbers and polynomials. Then we will consider p-adic fermionic integrals on Zp of the two variable q-Bernstein polynomials, recently introduced by Kim, and demonstrate that they can be written in terms of the q-analogues of Euler numbers. Further, from such p-adic integrals we will derive some identities for the q-analogues of Euler numbers.

Description: Symmetry, Vol. 10, Pages 312: Connecting Electroweak Symmetry Breaking and Flavor: A Light Dilaton D and a Sequential Heavy Quark Doublet Q Symmetry doi: 10.3390/sym10080312 Authors: Wei-Shu Hou The 125 GeV boson is quite consistent with the Higgs boson of the Standard Model (SM), but there is a challenge from Anderson as to whether this particle is in the Lagrangian. As Large Hadron Collider (LHC) Run 2 enters its final year of running, we ought to reflect and make sure we have gotten everything right. The ATLAS and CMS combined Run 1 analysis claimed a measurement of 5.4σ vector boson fusion (VBF) production which is consistent with SM, which seemingly refutes Anderson. However, to verify the source of electroweak symmetry breaking (EWSB), we caution that VBF measurement is too important for us to be imprudent in any way, and gluon–gluon fusion (ggF) with similar tag jets must be simultaneous measured, which should be achievable in LHC Run 2. The point is to truly test the dilaton possibility—the pseudo-Goldstone boson of scale invariance violation. We illustrate EWSB by dynamical mass generation of a sequential quark doublet (Q) via its ultrastrong Yukawa coupling and argue how this might be consistent with a 125 GeV dilaton, D. The ultraheavy 2mQ≳4–5 TeV scale explains the absence of New Physics so far, while the mass generation mechanism shields us from the UV theory for the strong Yukawa coupling. Collider and flavor physics implications are briefly touched upon. Current Run 2 analyses show correlations between the ggF and VBF measurements, but the newly observed tt¯H production at LHC poses a challenge.

Description: Symmetry, Vol. 10, Pages 305: Evaluation of a Third-Party Logistics (3PL) Provider Using a Rough SWARA–WASPAS Model Based on a New Rough Dombi Agregator Symmetry doi: 10.3390/sym10080305 Authors: Siniša Sremac Željko Stević Dragan Pamučar Miloš Arsić Bojan Matić For companies active in various sectors, the implementation of transport services and other logistics activities has become one of the key factors of efficiency in the total supply chain. Logistics outsourcing is becoming more and more important, and there is an increasing number of third party logistics providers. In this paper, logistics providers were evaluated using the Rough SWARA (Step-Wise Weight Assessment Ratio Analysis) and Rough WASPAS (Weighted Aggregated Sum Product Assessment) models. The significance of the eight criteria on the basis of which evaluation was carried out was determined using the Rough SWARA method. In order to allow for a more precise consensus in group decision-making, the Rough Dombi aggregator was developed in order to determine the initial rough matrix of multi-criteria decision-making. A total of 10 logistics providers dealing with the transport of dangerous goods for chemical industry companies were evaluated using the Rough WASPAS approach. The obtained results demonstrate that the first logistics provider is also the best one, a conclusion confirmed by a sensitivity analysis comprised of three parts. In the first part, parameter ρ was altered through 10 scenarios in which only alternatives four and five change their ranks. In the second part of the sensitivity analysis, a calculation was performed using the following approaches: Rough SAW (Simple Additive Weighting), Rough EDAS (Evaluation Based on Distance from Average Solution), Rough MABAC (MultiAttributive Border Approximation Area Comparison), and Rough TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution). They showed a high correlation of ranks determined by applying Spearman’s correlation coefficient in the third part of the sensitivity analysis.

Description: Symmetry, Vol. 10, Pages 318: Detectability Improved Tamper Detection Scheme for Absolute Moment Block Truncation Coding Compressed Images Symmetry doi: 10.3390/sym10080318 Authors: Wien Hong Xiaoyu Zhou Der-Chyuan Lou Xiaoqin Huang Cancan Peng Since digital media is gaining popularity nowadays, people are more concerned about its integrity protection and authentication since tampered media may result in unexpected problems. Considering a better media protection technique, this paper proposes an efficient tamper detection scheme for absolute moment block truncation coding (AMBTC) compressed images. In AMBTC, each image block is represented by two quantization levels (QLs) and a bitmap. Requiring insignificant computation cost, it attracts not only a wide range of application developers, but also a variety of studies to investigate the authentication of its codes. While the existing methods protect the AMBTC codes to a large extent, the leakage of some unprotected codes may be insensitive to intentional tampering. The proposed method fully protects the AMBTC codes by embedding authentication codes (ACs) into QLs. Meanwhile, the most significant bits of QLs are symmetrically perturbed to generate the candidates of ACs. The ACs that cause the minimum distortion are embedded into the least significant bits of QLs to minimize the distortion. When compared with prior works, the experimental results reveal that the proposed method offers a significant sensitivity-of-tamper property while providing a comparable image quality.

Description: Symmetry, Vol. 10, Pages 319: CRCM: A New Combined Data Gathering and Energy Charging Model for WRSN Symmetry doi: 10.3390/sym10080319 Authors: Yuhou Wang Ying Dong Shiyuan Li Hao Wu Mengyao Cui With the development of wireless sensor networks (WSNs), the problem about how to increase the lifecycle of the WSNs is always a hot discussion point, and some researchers have devoted to the ‘energy saving’ to decrease the energy consumption of the sensor nodes by different algorithms. However, the fundamental technique is ‘energy acquiring’ for the battery which can solve the limited capacity problem. In this paper, we study the data gathering and energy charging by a mobile charger (MC) at the same time that most energy consumption can be saved by short communication distance. We have named this as the recharging model-combined recharging and collecting data model on-demand (CRCM). Firstly, the hexagon-based (HB) algorithm is proposed to sort all sensor nodes in the region to make data collecting and energy charging work at the same time. Then we consider both residual energy and geographic position (REGP) of the sensor node to calculate the priority of each cluster. Thirdly, the dynamic mobile charger (DMC) algorithm is proposed to calculate the number of MCs to make sure no sensor node will die in each charging queue. Finally, the simulations show that our REGP algorithm is better than Earliest Deadline First (EDF) and Nearest-Job-Next with Preemption (NJNP), and the DMC plays well when the number of sensor nodes increase.

Description: Symmetry, Vol. 10, Pages 323: Temperature-Dependent s± ↔ s++ Transitions in the Multiband Model for Fe-Based Superconductors with Impurities Symmetry doi: 10.3390/sym10080323 Authors: V. A. Shestakov M. M. Korshunov O. V. Dolgov We study the dependence of the superconducting gaps on both the disorder and the temperature within the two-band model for iron-based materials. In the clean limit, the system is in the s± state with sign-changing gaps. Scattering by nonmagnetic impurities leads to the change of the sign of the smaller gap, resulting in a transition from the s± to the s++ state with the sign-preserving gaps. We show here that the transition is temperature-dependent. Thus, there is a line of s±→s++ transition in the temperature–disorder phase diagram. There is a narrow range of impurity scattering rates, where the disorder-induced s±→s++ transition occurs at low temperatures, but then the low-temperature s++ state transforms back to the s± state at higher temperatures. With increasing impurity scattering rate, the temperature of such s++→s± transition shifts to the critical temperature Tc, and only the s++ state is left for higher amounts of disorder.

Description: Entropy, Vol. 20, Pages 455: A Novel Fault Diagnosis Method of Rolling Bearings Based on AFEWT-KDEMI Entropy doi: 10.3390/e20060455 Authors: Mingtao Ge Jie Wang Fangfang Zhang Ke Bai Xiangyang Ren According to the dynamic characteristics of the rolling bearing vibration signal and the distribution characteristics of its noise, a fault identification method based on the adaptive filtering empirical wavelet transform (AFEWT) and kernel density estimation mutual information (KDEMI) classifier is proposed. First, we use AFEWT to extract the feature of the rolling bearing vibration signal. The hypothesis test of the Gaussian distribution is carried out for the sub-modes that are obtained by the twice decomposition of EWT, and Gaussian noise is filtered out according to the test results. In this way, we can overcome the noise interference and avoid the mode selection problem when we extract the feature of the signal. Then we combine the advantages of kernel density estimation (KDE) and mutual information (MI) and put forward a KDEMI classifier. The mutual information of the probability density combining the unknown signal feature vector and the probability density of the known type signal is calculated. The type of the unknown signal is determined via the value of the mutual information, so as to achieve the purpose of fault identification of the rolling bearing. In order to verify the effectiveness of AFEWT in feature extraction, we extract signal features using three methods, AFEWT, EWT, and EMD, and then use the same classifier to identify fault signals. Experimental results show that the fault signal has the highest recognition rate by using AFEWT for feature extraction. At the same time, in order to verify the performance of the AFEWT-KDEMI method, we compare two classical fault signal identification methods, SVM and BP neural network, with the AFEWT-KDEMI method. Through experimental analysis, we found that the AFEWT-KDEMI method is more stable and effective.

Description: Entropy, Vol. 20, Pages 458: Vacuum Landscaping: Cause of Nonlocal Influences without Signaling Entropy doi: 10.3390/e20060458 Authors: Gerhard Grössing Siegfried Fussy Johannes Mesa Pascasio Herbert Schwabl In the quest for an understanding of nonlocality with respect to an appropriate ontology, we propose a “cosmological solution”. We assume that from the beginning of the universe each point in space has been the location of a scalar field representing a zero-point vacuum energy that nonlocally vibrates at a vast range of different frequencies across the whole universe. A quantum, then, is a nonequilibrium steady state in the form of a “bouncer” coupled resonantly to one of those (particle type dependent) frequencies, in remote analogy to the bouncing oil drops on an oscillating oil bath as in Couder’s experiments. A major difference to the latter analogy is given by the nonlocal nature of the vacuum oscillations. We show with the examples of double- and n-slit interference that the assumed nonlocality of the distribution functions alone suffices to derive the de Broglie–Bohm guiding equation for N particles with otherwise purely classical means. In our model, no influences from configuration space are required, as everything can be described in 3-space. Importantly, the setting up of an experimental arrangement limits and shapes the forward and osmotic contributions and is described as vacuum landscaping.

Description: Symmetry, Vol. 10, Pages 210: An Improved Whale Optimization Algorithm Based on Different Searching Paths and Perceptual Disturbance Symmetry doi: 10.3390/sym10060210 Authors: Wei-zhen Sun Jie-sheng Wang Xian Wei Whale optimization algorithm (WOA) is a swarm intelligence optimization algorithm inspired by humpback whale hunting behavior. WOA has many similarities with other swarm intelligence algorithms (PSO, GWO, etc.). WOA’s unique search mechanism enables it to have a strong global search capability while taking into account the strong global search capabilities. In this work, considering the the deficiency of WOA in local search mechanism, combined with the optimization methods of other group intelligent algorithms, perceptual perturbation mechanism is introduced, which makes the agent perform more detailed searches near the local extreme point. At the same time, since the WOA uses a logarithmic spiral curve, the agent cannot fully search all the spaces within its search range, even though the introduction of the perturbation mechanism may still lead to the algorithm falling into a local optimum. Therefore, the equal pitch Archimedes spiral curve is chosen to replace the classic logarithmic spiral curve. In order to fully verify the effect of the search path on the performance of the algorithm, several other spiral curves have been chosen for experimental comparison. By utilizing the 23 benchmark test functions, the simulation results show that WOA (PDWOA) with perceptual perturbation significantly outperforms the standard WOA. Then, based on the PDWOA, the effect of the search path on the performance of the algorithm has been verified. The simulation results show that the equal pitch of the Archimedean spiral curve is best.

Description: Symmetry, Vol. 10, Pages 201: Topological Characterization of the Symmetrical Structure of Bismuth Tri-Iodide Symmetry doi: 10.3390/sym10060201 Authors: Muhammad Imran Muhammad Arfan Ali Sarfaraz Ahmad Muhammad Kamran Siddiqui Abdul Qudair Baig The bismuth tri-iodide ( B i I 3 ) is an inorganic compound. It is the result of the response of bismuth and iodine, which has inspired enthusiasm for subjective inorganic investigation. The topological indices are the numerical invariants of the molecular graph that portray its topology and are normally graph invariants. In 1975, Randic presented, in a bond-added substance, a topological index as a descriptor for portraying subatomic branching. In this paper, we investigate the precious stone structure of bismuth tri-iodide chain and sheet. Moreover, exact formulas of degree-based added-substance topological indices principally the first, second, and hyper Zagreb indices, the general Randic index, the geometric-arithmetic index, the fourth atom-bond connectivity index, and the fifth geometric arithmetic index of the subatomic graph of bismuth tri-iodide for both chain and sheet structures are determined.

Description: Entropy, Vol. 20, Pages 436: Information Geometry of κ-Exponential Families: Dually-Flat, Hessian and Legendre Structures Entropy doi: 10.3390/e20060436 Authors: Antonio M. Scarfone Hiroshi Matsuzoe Tatsuaki Wada In this paper, we present a review of recent developments on the κ -deformed statistical mechanics in the framework of the information geometry. Three different geometric structures are introduced in the κ -formalism which are obtained starting from three, not equivalent, divergence functions, corresponding to the κ -deformed version of Kullback–Leibler, “Kerridge” and Brègman divergences. The first statistical manifold derived from the κ -Kullback–Leibler divergence form an invariant geometry with a positive curvature that vanishes in the κ → 0 limit. The other two statistical manifolds are related to each other by means of a scaling transform and are both dually-flat. They have a dualistic Hessian structure endowed by a deformed Fisher metric and an affine connection that are consistent with a statistical scalar product based on the κ -escort expectation. These flat geometries admit dual potentials corresponding to the thermodynamic Massieu and entropy functions that induce a Legendre structure of κ -thermodynamics in the picture of the information geometry.

Description: Entropy, Vol. 20, Pages 440: On a Common Misconception Regarding the de Broglie–Bohm Theory Entropy doi: 10.3390/e20060440 Authors: Oliver Passon We discuss a common misconception regarding the de Broglie–Bohm (dBB) theory; namely, that it not only assigns a position to each quantum object but also contains the momenta as “hidden variables”. Sometimes this alleged property of the theory is even used to argue that the dBB theory is inconsistent with quantum theory. We explain why this claim is unfounded and show in particular how this misconception veils the true novelty of the dBB theory.

Description: Entropy, Vol. 20, Pages 438: Analysis of Cell Signal Transduction Based on Kullback–Leibler Divergence: Channel Capacity and Conservation of Its Production Rate during Cascade Entropy doi: 10.3390/e20060438 Authors: Tatsuaki Tsuruyama Kullback–Leibler divergence (KLD) is a type of extended mutual entropy, which is used as a measure of information gain when transferring from a prior distribution to a posterior distribution. In this study, KLD is applied to the thermodynamic analysis of cell signal transduction cascade and serves an alternative to mutual entropy. When KLD is minimized, the divergence is given by the ratio of the prior selection probability of the signaling molecule to the posterior selection probability. Moreover, the information gain during the entire channel is shown to be adequately described by average KLD production rate. Thus, this approach provides a framework for the quantitative analysis of signal transduction. Moreover, the proposed approach can identify an effective cascade for a signaling network.

Description: Entropy, Vol. 20, Pages 437: Complexity Analysis of Global Temperature Time Series Entropy doi: 10.3390/e20060437 Authors: António M. Lopes J. A. Tenreiro Machado Climate has complex dynamics due to the plethora of phenomena underlying its evolution. These characteristics pose challenges to conducting solid quantitative analysis and reaching assertive conclusions. In this paper, the global temperature time series (TTS) is viewed as a manifestation of the climate evolution, and its complexity is calculated by means of four different indices, namely the Lempel–Ziv complexity, sample entropy, signal harmonics power ratio, and fractal dimension. In the first phase, the monthly mean TTS is pre-processed by means of empirical mode decomposition, and the TTS trend is calculated. In the second phase, the complexity of the detrended signals is estimated. The four indices capture distinct features of the TTS dynamics in a 4-dim space. Hierarchical clustering is adopted for dimensional reduction and visualization in the 2-dim space. The results show that TTS complexity exhibits space-time variability, suggesting the presence of distinct climate forcing processes in both dimensions. Numerical examples with real-world data demonstrate the effectiveness of the approach.

Description: Entropy, Vol. 20, Pages 439: Factoid Question Answering with Distant Supervision Entropy doi: 10.3390/e20060439 Authors: Hongzhi Zhang Xiao Liang Guangluan Xu Kun Fu Feng Li Tinglei Huang Automatic question answering (QA), which can greatly facilitate the access to information, is an important task in artificial intelligence. Recent years have witnessed the development of QA methods based on deep learning. However, a great amount of data is needed to train deep neural networks, and it is laborious to annotate training data for factoid QA of new domains or languages. In this paper, a distantly supervised method is proposed to automatically generate QA pairs. Additional efforts are paid to let the generated questions reflect the query interests and expression styles of users by exploring the community QA. Specifically, the generated questions are selected according to the estimated probabilities they are asked. Diverse paraphrases of questions are mined from community QA data, considering that the model trained on monotonous synthetic questions is very sensitive to variants of question expressions. Experimental results show that the model solely trained on generated data via the distant supervision and mined paraphrases could answer real-world questions with the accuracy of 49.34%. When limited annotated training data is available, significant improvements could be achieved by incorporating the generated data. An improvement of 1.35 absolute points is still observed on WebQA, a dataset with large-scale annotated training samples.

Description: Entropy, Vol. 20, Pages 434: Complexity Analysis of Carbon Market Using the Modified Multi-Scale Entropy Entropy doi: 10.3390/e20060434 Authors: Jiuli Yin Cui Su Yongfen Zhang Xinghua Fan Carbon markets provide a market-based way to reduce climate pollution. Subject to general market regulations, the major existing emission trading markets present complex characteristics. This paper analyzes the complexity of carbon market by using the multi-scale entropy. Pilot carbon markets in China are taken as the example. Moving average is adopted to extract the scales due to the short length of the data set. Results show a low-level complexity inferring that China’s pilot carbon markets are quite immature in lack of market efficiency. However, the complexity varies in different time scales. China’s carbon markets (except for the Chongqing pilot) are more complex in the short period than in the long term. Furthermore, complexity level in most pilot markets increases as the markets developed, showing an improvement in market efficiency. All these results demonstrate that an effective carbon market is required for the full function of emission trading.

Description: Symmetry, Vol. 10, Pages 218: Gauss Map and Its Applications on Ruled Submanifolds in Minkowski Space Symmetry doi: 10.3390/sym10060218 Authors: Sun Jung Young Kim We study ruled submanifolds in Minkowski space in regard to the Gauss map satisfying some partial differential equation. As a generalization of usual cylinders, cones and null scrolls in a three-dimensional Minkowski space, a cylinder over a space curve, a product manifold of a right cone and a k-plane, a product manifold of a hyperbolic cone and a k-plane which look like kinds of cylinders over cones in 3-space, and the generalized B-scroll kind in Minkowski space are characterized with the partial differential equation regarding the Gauss map, where k is a positive integer.

Description: Symmetry, Vol. 10, Pages 217: Decision-Making via Neutrosophic Support Soft Topological Spaces Symmetry doi: 10.3390/sym10060217 Authors: Parimala Mani Karthika Muthusamy Saeid Jafari Florentin Smarandache Udhayakumar Ramalingam The concept of interval neutrosophic sets has been studied and the introduction of a new kind of set in topological spaces called the interval valued neutrosophic support soft set has been suggested. We study some of its basic properties. The main purpose of this paper is to give the optimum solution to decision-making in real life problems the using interval valued neutrosophic support soft set.

Description: Entropy, Vol. 20, Pages 462: On Bohmian Mechanics, Particle Creation, and Relativistic Space-Time: Happy 100th Birthday, David Bohm! Entropy doi: 10.3390/e20060462 Authors: Roderich Tumulka The biggest and most lasting among David Bohm’s (1917–1992) many achievements is to have proposed a picture of reality that explains the empirical rules of quantum mechanics. This picture, known as pilot wave theory or Bohmian mechanics among other names, is still the simplest and most convincing explanation available. According to this theory, electrons are point particles in the literal sense and move along trajectories governed by Bohm’s equation of motion. In this paper, I describe some more recent developments and extensions of Bohmian mechanics, concerning in particular relativistic space-time and particle creation and annihilation.

Description: Entropy, Vol. 20, Pages 465: Ontological Clarity via Canonical Presentation: Electromagnetism and the Aharonov–Bohm Effect Entropy doi: 10.3390/e20060465 Authors: Tim Maudlin Quantum physics demands some radical revision of our fundamental beliefs about physical reality. We know that because there are certain verified physical phenomena—two-slit interference, the disappearance of interference upon monitoring, violations of Bell’s inequality—that have no classical analogs. But the exact nature of that revision has been under dispute since the foundation of quantum theory. I offer a method of clarifying what the commitments of a clearly formulated physical theory are, and apply it to a discussion of some options available to account for another non-classical phenomenon: the Aharonov–Bohm effect.

Description: Entropy, Vol. 20, Pages 464: Non-Quadratic Distances in Model Assessment Entropy doi: 10.3390/e20060464 Authors: Marianthi Markatou Yang Chen One natural way to measure model adequacy is by using statistical distances as loss functions. A related fundamental question is how to construct loss functions that are scientifically and statistically meaningful. In this paper, we investigate non-quadratic distances and their role in assessing the adequacy of a model and/or ability to perform model selection. We first present the definition of a statistical distance and its associated properties. Three popular distances, total variation, the mixture index of fit and the Kullback-Leibler distance, are studied in detail, with the aim of understanding their properties and potential interpretations that can offer insight into their performance as measures of model misspecification. A small simulation study exemplifies the performance of these measures and their application to different scientific fields is briefly discussed.

Description: Symmetry, Vol. 10, Pages 214: Intelligent Prognostics of Degradation Trajectories for Rotating Machinery Based on Asymmetric Penalty Sparse Decomposition Model Symmetry doi: 10.3390/sym10060214 Authors: Qing Li Steven Y. Liang The ability to accurately track the degradation trajectories of rotating machinery components is arguably one of the challenging problems in prognostics and health management (PHM). In this paper, an intelligent prediction approach based on asymmetric penalty sparse decomposition (APSD) algorithm combined with wavelet neural network (WNN) and autoregressive moving average-recursive least squares algorithm (ARMA-RLS) is proposed for degradation prognostics of rotating machinery, taking the accelerated life test of rolling bearings as an example. Specifically, the health indicators time series (e.g., peak-to-peak value and Kurtosis) is firstly decomposed into low frequency component (LFC) and high frequency component (HFC) using the APSD algorithm; meanwhile, the resulting non-convex regularization problem can be efficiently solved using the majorization-minimization (MM) method. In particular, the HFC part corresponds to the stable change around the zero line of health indicators which most extensively occurs; in contrast, the LFC part is essentially related to the evolutionary trend of health indicators. Furthermore, the nonparametric-based method, i.e., WNN, and parametric-based method, i.e., ARMA-RLS, are respectively introduced to predict the LFC and HFC that focus on abrupt degradation regions (e.g., last 100 points). Lastly, the final predicted data could be correspondingly obtained by integrating the predicted LFC and predicted HFC. The proposed methodology is tested using degradation health indicator time series from four rolling bearings. The proposed approach performed favorably when compared to some state-of-the-art benchmarks such as WNN and largest Lyapunov (LLyap) methods.

Description: Symmetry, Vol. 10, Pages 222: Bayer Image Demosaicking Using Eight-Directional Weights Based on the Gradient of Color Difference Symmetry doi: 10.3390/sym10060222 Authors: Yizheng Liu Chengyou Wang Hongming Zhao Jiayang Song Shiyue Chen In this paper, we propose a new demosaicking algorithm which uses eight-directional weights based on the gradient of color difference (EWGCD) for Bayer image demosaicking. To obtain the interpolation of green (G) pixels, the eight-directional G pixel values are first estimated in red (R)/blue (B) pixels. This estimate is used to calculate the color difference in R/B pixels of the Bayer image in diagonal directions. However, in horizontal and vertical directions, the new estimated G pixels are defined to obtain the color difference. The eight-directional weights of estimated G pixels can be obtained by considering the gradient of the color difference and the gradient of the RGB pixels of the Bayer image. Therefore, the eight-directional weighted values and the first estimated G pixel values are combined to obtain the full G image. Compared with six similar algorithms using the same eighteen McMaster images, the results of the experiment demonstrate that the proposed algorithm has a better performance not only in the subjective visual measurement but also in the assessments of peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) index measurement.

Description: Entropy, Vol. 20, Pages 466: The Gibbs Paradox and Particle Individuality Entropy doi: 10.3390/e20060466 Authors: Dennis Dieks A consensus seems to have developed that the Gibbs paradox in classical thermodynamics (the discontinuous drop in the entropy of mixing when the mixed gases become equal to each other) is unmysterious: in any actual situation, two gases can be separated or not, and the associated harmless discontinuity from “yes” to “no” is responsible for the discontinuity. By contrast, the Gibbs paradox in statistical physics continues to attract attention. Here, the problem is that standard calculations in statistical mechanics predict a non-vanishing value of the entropy of mixing even when two gases of the same kind are mixed, in conflict with thermodynamic predictions. This version of the Gibbs paradox is often seen as a sign that there is something fundamentally wrong with either the traditional expression S=klnW or with the way W is calculated. It is the aim of this article to review the situation from the orthodox (as opposed to information theoretic) standpoint. We demonstrate how the standard formalism is not only fully capable of dealing with the paradox, but also provides an intuitively clear picture of the relevant physical mechanisms. In particular, we pay attention to the explanatory relevance of the existence of particle trajectories in the classical context. We also discuss how the paradox survives the transition to quantum mechanics, in spite of the symmetrization postulates.

Description: Entropy, Vol. 20, Pages 470: Noise Enhanced Signal Detection of Variable Detectors under Certain Constraints Entropy doi: 10.3390/e20060470 Authors: Ting Yang Shujun Liu Wenguo Liu Jishun Guo Pin Wang In this paper, a noise enhanced binary hypothesis-testing problem was studied for a variable detector under certain constraints in which the detection probability can be increased and the false-alarm probability can be decreased simultaneously. According to the constraints, three alternative cases are proposed, the first two cases concerned minimization of the false-alarm probability and maximization of the detection probability without deterioration of one by the other, respectively, and the third case was achieved by a randomization of two optimal noise enhanced solutions obtained in the first two limit cases. Furthermore, the noise enhanced solutions that satisfy the three cases were determined whether randomization between different detectors was allowed or not. In addition, the practicality of the third case was proven from the perspective of Bayes risk. Finally, numerous examples and conclusions are presented.

Description: Entropy, Vol. 20, Pages 483: Non-Commutative Worlds and Classical Constraints Entropy doi: 10.3390/e20070483 Authors: Louis Kauffman This paper reviews results about discrete physics and non-commutative worlds and explores further the structure and consequences of constraints linking classical calculus and discrete calculus formulated via commutators. In particular, we review how the formalism of generalized non-commutative electromagnetism follows from a first order constraint and how, via the Kilmister equation, relationships with general relativity follow from a second order constraint. It is remarkable that a second order constraint, based on interlacing the commutative and non-commutative worlds, leads to an equivalent tensor equation at the pole of geodesic coordinates for general relativity.

Description: Entropy, Vol. 20, Pages 481: Relating Vertex and Global Graph Entropy in Randomly Generated Graphs Entropy doi: 10.3390/e20070481 Authors: Philip Tee George Parisis Luc Berthouze Ian Wakeman Combinatoric measures of entropy capture the complexity of a graph but rely upon the calculation of its independent sets, or collections of non-adjacent vertices. This decomposition of the vertex set is a known NP-Complete problem and for most real world graphs is an inaccessible calculation. Recent work by Dehmer et al. and Tee et al. identified a number of vertex level measures that do not suffer from this pathological computational complexity, but that can be shown to be effective at quantifying graph complexity. In this paper, we consider whether these local measures are fundamentally equivalent to global entropy measures. Specifically, we investigate the existence of a correlation between vertex level and global measures of entropy for a narrow subset of random graphs. We use the greedy algorithm approximation for calculating the chromatic information and therefore Körner entropy. We are able to demonstrate strong correlation for this subset of graphs and outline how this may arise theoretically.

Description: Entropy, Vol. 20, Pages 485: Symmetric Logarithmic Derivative of Fermionic Gaussian States Entropy doi: 10.3390/e20070485 Authors: Angelo Carollo Bernardo Spagnolo Davide Valenti In this article, we derive a closed form expression for the symmetric logarithmic derivative of Fermionic Gaussian states. This provides a direct way of computing the quantum Fisher Information for Fermionic Gaussian states. Applications range from quantum Metrology with thermal states to non-equilibrium steady states with Fermionic many-body systems.

Description: Entropy, Vol. 20, Pages 489: A Moment-Based Maximum Entropy Model for Fitting Higher-Order Interactions in Neural Data Entropy doi: 10.3390/e20070489 Authors: N. Alex Cayco-Gajic Joel Zylberberg Eric Shea-Brown Correlations in neural activity have been demonstrated to have profound consequences for sensory encoding. To understand how neural populations represent stimulus information, it is therefore necessary to model how pairwise and higher-order spiking correlations between neurons contribute to the collective structure of population-wide spiking patterns. Maximum entropy models are an increasingly popular method for capturing collective neural activity by including successively higher-order interaction terms. However, incorporating higher-order interactions in these models is difficult in practice due to two factors. First, the number of parameters exponentially increases as higher orders are added. Second, because triplet (and higher) spiking events occur infrequently, estimates of higher-order statistics may be contaminated by sampling noise. To address this, we extend previous work on the Reliable Interaction class of models to develop a normalized variant that adaptively identifies the specific pairwise and higher-order moments that can be estimated from a given dataset for a specified confidence level. The resulting “Reliable Moment” model is able to capture cortical-like distributions of population spiking patterns. Finally, we show that, compared with the Reliable Interaction model, the Reliable Moment model infers fewer strong spurious higher-order interactions and is better able to predict the frequencies of previously unobserved spiking patterns.

Description: Symmetry, Vol. 10, Pages 231: Observers and Their Notion of Spacetime beyond Special Relativity Symmetry doi: 10.3390/sym10070231 Authors: José Manuel Carmona José Luis Cortés José Javier Relancio It is plausible that quantum gravity effects may lead us to a description of Nature beyond the framework of special relativity. In this case, either the relativity principle is broken or it is maintained. These two scenarios (a violation or a deformation of special relativity) are very different, both conceptually and phenomenologically. We discuss some of their implications on the description of events for different observers and the notion of spacetime.

Description: Symmetry, Vol. 10, Pages 233: Symmetries of Differential Equations in Cosmology Symmetry doi: 10.3390/sym10070233 Authors: Michael Tsamparlis Andronikos Paliathanasis The purpose of the current article is to present a brief albeit accurate presentation of the main tools used in the study of symmetries of Lagrange equations for holonomic systems and subsequently to show how these tools are applied in the major models of modern cosmology in order to derive exact solutions and deal with the problem of dark matter/energy. The key role in this approach are the first integrals of the field equations. We start with the Lie point symmetries and the first integrals defined by them, that is, the Hojman integrals. Subsequently, we discuss the Noether point symmetries and the well-known method for deriving the Noether integrals. By means of the Inverse Noether Theorem, we show that, to every Hojman quadratic first integral, it is possible to associate a Noether symmetry whose Noether integral is the original Hojman integral. It is emphasized that the point transformation generating this Noether symmetry need not coincide with the point transformation defining the Lie symmetry which produces the Hojman integral. We discuss the close connection between the Lie point and the Noether point symmetries with the collineations of the metric defined by the kinetic energy of the Lagrangian. In particular, the generators of Noether point symmetries are elements of the homothetic algebra of that metric. The key point in the current study of cosmological models is the introduction of the mini superspace, which is the space that is defined by the physical variables of the model, which is not the spacetime where the model evolves. The metric in the mini superspace is found from the kinematic part of the Lagrangian and we call it the kinetic metric. The rest part of the Lagrangian is the effective potential. We consider coordinate transformations of the original mini superspace metric in order to bring it to a form where we know its collineations, that is, the Killing vectors, the homothetic vector, etc. Then, we write the field equations of the cosmological model and we use the connection of these equations with the collineations of the mini superspace metric to compute the first integrals and subsequently to obtain analytic solutions for various allowable potentials and finally draw conclusions about the problem of dark energy. We consider the ΛCDM cosmological model, the scalar field cosmology, the Brans–Dicke cosmology, the f(R) gravity, the two scalar fields cosmology with interacting scalar fields and the Galilean cosmology. In each case, we present the relevant results in the form of tables for easy reference. Finally, we discuss briefly the higher order symmetries (the contact symmetries) and show how they are applied in the cases of scalar field cosmology and in the f(R) gravity.

Description: Entropy, Vol. 20, Pages 498: Assessing Information Transmission in Data Transformations with the Channel Multivariate Entropy Triangle Entropy doi: 10.3390/e20070498 Authors: Francisco J. Valverde-Albacete Carmen Peláez-Moreno Data transformation, e.g., feature transformation and selection, is an integral part of any machine learning procedure. In this paper, we introduce an information-theoretic model and tools to assess the quality of data transformations in machine learning tasks. In an unsupervised fashion, we analyze the transformation of a discrete, multivariate source of information X¯ into a discrete, multivariate sink of information Y¯ related by a distribution PX¯Y¯. The first contribution is a decomposition of the maximal potential entropy of (X¯,Y¯), which we call a balance equation, into its (a) non-transferable, (b) transferable, but not transferred, and (c) transferred parts. Such balance equations can be represented in (de Finetti) entropy diagrams, our second set of contributions. The most important of these, the aggregate channel multivariate entropy triangle, is a visual exploratory tool to assess the effectiveness of multivariate data transformations in transferring information from input to output variables. We also show how these decomposition and balance equations also apply to the entropies of X¯ and Y¯, respectively, and generate entropy triangles for them. As an example, we present the application of these tools to the assessment of information transfer efficiency for Principal Component Analysis and Independent Component Analysis as unsupervised feature transformation and selection procedures in supervised classification tasks.

Description: Symmetry, Vol. 10, Pages 245: A Forecasting Model Based on Multi-Valued Neutrosophic Sets and Two-Factor, Third-Order Fuzzy Fluctuation Logical Relationships Symmetry doi: 10.3390/sym10070245 Authors: Hongjun Guan Jie He Aiwu Zhao Zongli Dai Shuang Guan Making predictions according to historical values has long been regarded as common practice by many researchers. However, forecasting solely based on historical values could lead to inevitable over-complexity and uncertainty due to the uncertainties inside, and the random influence outside, of the data. Consequently, finding the inherent rules and patterns of a time series by eliminating disturbances without losing important details has long been a research hotspot. In this paper, we propose a novel forecasting model based on multi-valued neutrosophic sets to find fluctuation rules and patterns of a time series. The contributions of the proposed model are: (1) using a multi-valued neutrosophic set (MVNS) to describe the fluctuation patterns of a time series, the model could represent the fluctuation trend of up, equal, and down with degrees of truth, indeterminacy, and falsity which significantly preserve details of the historical values; (2) measuring the similarities of different fluctuation patterns by the Hamming distance could avoid the confusion caused by incomplete information from limited samples; and (3) introducing another related time series as a secondary factor to avoid warp and deviation in inferring inherent rules of historical values, which could lead to more comprehensive rules for further forecasting. To evaluate the performance of the model, we explored the Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) as the major factor we forecast, and the Dow Jones Index as the secondary factor to facilitate the predicting of the TAIEX. To show the universality of the model, we applied the proposed model to forecast the Shanghai Stock Exchange Composite Index (SHSECI) as well.

Description: Symmetry, Vol. 10, Pages 244: Computing Zagreb Indices and Zagreb Polynomials for Symmetrical Nanotubes Symmetry doi: 10.3390/sym10070244 Authors: Zehui Shao Muhammad Kamran Siddiqui Mehwish Hussain Muhammad Topological indices are numbers related to sub-atomic graphs to allow quantitative structure-movement/property/danger connections. These topological indices correspond to some specific physico-concoction properties such as breaking point, security, strain vitality of chemical compounds. The idea of topological indices were set up in compound graph hypothesis in view of vertex degrees. These indices are valuable in the investigation of mitigating exercises of specific Nanotubes and compound systems. In this paper, we discuss Zagreb types of indices and Zagreb polynomials for a few Nanotubes covered by cycles.

Description: Entropy, Vol. 20, Pages 568: Symbolic Entropy Analysis and Its Applications Entropy doi: 10.3390/e20080568 Authors: Raúl Alcaraz This editorial explains the scope of the special issue and provides a thematic introduction to the contributed papers.

Description: Entropy, Vol. 20, Pages 569: Random Finite Set Based Parameter Estimation Algorithm for Identifying Stochastic Systems Entropy doi: 10.3390/e20080569 Authors: Peng Wang Ge Li Yong Peng Rusheng Ju Parameter estimation is one of the key technologies for system identification. The Bayesian parameter estimation algorithms are very important for identifying stochastic systems. In this paper, a random finite set based algorithm is proposed to overcome the disadvantages of the existing Bayesian parameter estimation algorithms. It can estimate the unknown parameters of the stochastic system which consists of a varying number of constituent elements by using the measurements disturbed by false detections, missed detections and noises. The models used for parameter estimation are constructed by using random finite set. Based on the proposed system model and measurement model, the key principles and formula derivation of the proposed algorithm are detailed. Then, the implementation of the algorithm is presented by using sequential Monte Carlo based Probability Hypothesis Density (PHD) filter and simulated tempering based importance sampling. Finally, the experiments of systematic errors estimation of multiple sensors are provided to prove the main advantages of the proposed algorithm. The sensitivity analysis is carried out to further study the mechanism of the algorithm. The experimental results verify the superiority of the proposed algorithm.

Description: Symmetry, Vol. 10, Pages 302: Fermion Scattering in a CPT-Even Lorentz Violation Quantum Electrodynamics Symmetry doi: 10.3390/sym10080302 Authors: Frederico Santos Manoel Ferreira In this work, we reassess two known processes of Quantum Electrodynamics involving electrons and muons. The photon propagator is modified by a CPT-even Lorentz-violating (LV) tensor, while fermion lines and the vertex interaction are not altered. Using the Feynman rules, the associated cross sections for unpolarized scatterings are evaluated, revealing the usual energy dependence and Lorentz-violating contributions that induce space anisotropy. A possible route to constraining the LV coefficients is presented and the results properly commented.

Description: Symmetry, Vol. 10, Pages 301: Evaluating Water Use Efficiency in China’s Western Provinces Based on a Slacks-Based Measure (SBM)-Undesirable Window Model and a Malmquist Productivity Index Symmetry doi: 10.3390/sym10080301 Authors: Yuyan Luo Lu Yin Yong Qin Zhong Wang Yanfeng Gong This paper evaluated the water use efficiency in 12 Chinese western provinces from 2005 to 2015. Based on data availability and the index selection rationality and the slacks-based measure (SBM)-undesirable Window analysis model and the Malmquist productivity index, the water resource inputs and outputs were measured to analyze water use efficiency. Total investment in fixed assets for the whole society and total water, made up of total agricultural water, total industrial water, and total domestic water, were used as the input indexes, and regional GDP and waste water discharge were the output indexes, with the waste water discharge being regarded as an undesirable output. The data from different years and different provinces in the same period, and data from the same year and the same provinces in different periods were compared in order to derive the water resource efficiency and technical changes over time and space. It was found that the total water resource factor productivity in the 12 provinces grew slowly in the study period, that water resource technical progress positively affected water use efficiency, and that a lack of technical efficiency restricted water use efficiency growth. Several suggestions are given to optimize water use efficiency in the 12 provinces. The research findings and suggestions provide valuable reference for studies in related fields.

Description: Entropy, Vol. 20, Pages 570: Analysis of Solar Irradiation Time Series Complexity and Predictability by Combining Kolmogorov Measures and Hamming Distance for La Reunion (France) Entropy doi: 10.3390/e20080570 Authors: Dragutin T. Mihailović Miloud Bessafi Sara Marković Ilija Arsenić Slavica Malinović-Milićević Patrick Jeanty Mathieu Delsaut Jean-Pierre Chabriat Nusret Drešković Anja Mihailović Analysis of daily solar irradiation variability and predictability in space and time is important for energy resources planning, development, and management. The natural variability of solar irradiation is being complicated by atmospheric conditions (in particular cloudiness) and orography, which introduce additional complexity into the phenomenological records. To address this question for daily solar irradiation data recorded during the years 2013, 2014 and 2015 at 11 stations measuring solar irradiance on La Reunion French tropical Indian Ocean Island, we use a set of novel quantitative tools: Kolmogorov complexity (KC) with its derivative associated measures and Hamming distance (HAM) and their combination to assess complexity and corresponding predictability. We find that all half-day (from sunrise to sunset) solar irradiation series exhibit high complexity. However, all of them can be classified into three groups strongly influenced by trade winds that circulate in a “flow around” regime: the windward side (trade winds slow down), the leeward side (diurnal thermally-induced circulations dominate) and the coast parallel to trade winds (winds are accelerated due to Venturi effect). We introduce Kolmogorov time (KT) that quantifies the time span beyond which randomness significantly influences predictability.

Description: Symmetry, Vol. 10, Pages 304: No-Reference Image Blur Assessment Based on Response Function of Singular Values Symmetry doi: 10.3390/sym10080304 Authors: Shanqing Zhang Pengcheng Li Xianghua Xu Li Li Ching-Chun Chang Blur is an important factor affecting the image quality. This paper presents an efficient no-reference (NR) image blur assessment method based on a response function of singular values. For an image, the grayscale image is computed to the acquire spatial information. In the meantime, the gradient map is computed to acquire the shape information, and the saliency map can be obtained by using scale-invariant feature transform (SIFT). Then, the grayscale image, the gradient map, and the saliency map are divided into blocks of the same size. The blocks of the gradient map are converted into discrete cosine transform (DCT) coefficients, from which the response function of singular values (RFSV) are generated. The sum of the RFSV are then utilized to characterize the image blur. The variance of the grayscale image and the DCT domain entropy of the gradient map are used to reduce the impact of the image content. The SIFT-dependent weights are calculated in the saliency map, which are assigned to the image blocks. Finally, the blur score is the normalized sum of the RFSV. Extensive experiments are conducted on four synthetic databases and two real blur databases. The experimental results indicate that the blur scores produced by our method are highly correlated with the subjective evaluations. Furthermore, the proposed method is superior to six state-of-the-art methods.

Description: Entropy, Vol. 20, Pages 583: Information Perspective to Probabilistic Modeling: Boltzmann Machines versus Born Machines Entropy doi: 10.3390/e20080583 Authors: Song Cheng Jing Chen Lei Wang We compare and contrast the statistical physics and quantum physics inspired approaches for unsupervised generative modeling of classical data. The two approaches represent probabilities of observed data using energy-based models and quantum states, respectively. Classical and quantum information patterns of the target datasets therefore provide principled guidelines for structural design and learning in these two approaches. Taking the Restricted Boltzmann Machines (RBM) as an example, we analyze the information theoretical bounds of the two approaches. We also estimate the classical mutual information of the standard MNIST datasets and the quantum Rényi entropy of corresponding Matrix Product States (MPS) representations. Both information measures are much smaller compared to their theoretical upper bound and exhibit similar patterns, which imply a common inductive bias of low information complexity. By comparing the performance of RBM with various architectures on the standard MNIST datasets, we found that the RBM with local sparse connection exhibit high learning efficiency, which supports the application of tensor network states in machine learning problems.

Description: Symmetry, Vol. 10, Pages 327: Different Forms of Triangular Neutrosophic Numbers, De-Neutrosophication Techniques, and their Applications Symmetry doi: 10.3390/sym10080327 Authors: Avishek Chakraborty Sankar Prasad Mondal Ali Ahmadian Norazak Senu Shariful Alam Soheil Salahshour In this paper, we introduce the concept of neutrosophic number from different viewpoints. We define different types of linear and non-linear generalized triangular neutrosophic numbers which are very important for uncertainty theory. We introduced the de-neutrosophication concept for neutrosophic number for triangular neutrosophic numbers. This concept helps us to convert a neutrosophic number into a crisp number. The concepts are followed by two application, namely in imprecise project evaluation review technique and route selection problem.

Description: Entropy, Vol. 20, Pages 588: On a Matrix Inequality Related to the Distillability Problem Entropy doi: 10.3390/e20080588 Authors: Yi Shen Lin Chen We investigate the distillability problem in quantum information in Cd⊗Cd. One case of the problem has been reduced to proving a matrix inequality when d=4. We investigate the inequality for three families of non-normal matrices. We prove the inequality for the first two families with d=4 and for the third family with d≥5. We also present a sufficient condition for the fulfillment of the inequality with d=4.

Description: Symmetry, Vol. 10, Pages 328: Thermodynamic Response of Beams on Winkler Foundation Irradiated by Moving Laser Pulses Symmetry doi: 10.3390/sym10080328 Authors: Yuxin Sun Shoubin Liu Zhangheng Rao Yuhang Li Jialing Yang In this paper, the exact analytical solutions are developed for the thermodynamic behavior of an Euler-Bernoulli beam resting on an elastic foundation and exposed to a time decaying laser pulse that scans over the beam with a uniform velocity. The governing equations, namely the heat conduction equation and the vibration equation are solved using the Green’s function approach. The temporal and special distributions of temperature, deflection, strain, and the energy absorbed by the elastic foundation are calculated. The effects of the laser motion speed, the modulus of elastic foundation reaction, and the laser pulse duration time are studied in detail.

Description: Symmetry, Vol. 10, Pages 331: Applications of Neutrosophic Bipolar Fuzzy Sets in HOPE Foundation for Planning to Build a Children Hospital with Different Types of Similarity Measures Symmetry doi: 10.3390/sym10080331 Authors: Raja Muhammad Hashim Muhammad Gulistan Florentin Smarandache In this paper we provide an application of neutrosophic bipolar fuzzy sets in daily life’s problem related with HOPE foundation that is planning to build a children hospital, which is the main theme of this paper. For it we first develop the theory of neutrosophic bipolar fuzzy sets which is a generalization of bipolar fuzzy sets. After giving the definition we introduce some basic operation of neutrosophic bipolar fuzzy sets and focus on weighted aggregation operators in terms of neutrosophic bipolar fuzzy sets. We define neutrosophic bipolar fuzzy weighted averaging ( N B FWA ) and neutrosophic bipolar fuzzy ordered weighted averaging ( N B FOWA ) operators. Next we introduce different kinds of similarity measures of neutrosophic bipolar fuzzy sets. Finally as an application we give an algorithm for the multiple attribute decision making problems under the neutrosophic bipolar fuzzy environment by using the different kinds of neutrosophic bipolar fuzzy weighted/fuzzy ordered weighted aggregation operators with a numerical example related with HOPE foundation.

Description: Symmetry, Vol. 10, Pages 332: On the Distinguishing Number of Functigraphs Symmetry doi: 10.3390/sym10080332 Authors: Muhammad Fazil Muhammad Murtaza Zafar Ullah Usman Ali Imran Javaid Let G 1 and G 2 be disjoint copies of a graph G and g : V ( G 1 ) → V ( G 2 ) be a function. A functigraph F G consists of the vertex set V ( G 1 ) ∪ V ( G 2 ) and the edge set E ( G 1 ) ∪ E ( G 2 ) ∪ { u v : g ( u ) = v } . In this paper, we extend the study of distinguishing numbers of a graph to its functigraph. We discuss the behavior of distinguishing number in passing from G to F G and find its sharp lower and upper bounds. We also discuss the distinguishing number of functigraphs of complete graphs and join graphs.

Description: Entropy, Vol. 20, Pages 597: Work Availability and Exergy Analysis Entropy doi: 10.3390/e20080597 Authors: Pouria Ahmadi Behnaz Rezaie n/a

Description: Entropy, Vol. 20, Pages 598: Entropy Applications in Environmental and Water Engineering Entropy doi: 10.3390/e20080598 Authors: Huijuan Cui Bellie Sivakumar Vijay P. Singh n/a

Description: Symmetry, Vol. 10, Pages 333: A Multi-Level Privacy-Preserving Approach to Hierarchical Data Based on Fuzzy Set Theory Symmetry doi: 10.3390/sym10080333 Authors: Jinyan Wang Guoqing Cai Chen Liu Jingli Wu Xianxian Li Nowadays, more and more applications are dependent on storage and management of semi-structured information. For scientific research and knowledge-based decision-making, such data often needs to be published, e.g., medical data is released to implement a computer-assisted clinical decision support system. Since this data contains individuals’ privacy, they must be appropriately anonymized before to be released. However, the existing anonymization method based on l-diversity for hierarchical data may cause a serious similarity attack, and cannot protect data privacy very well. In this paper, we utilize fuzzy sets to divide levels for sensitive numerical and categorical attribute values uniformly (a categorical attribute value can be converted into a numerical attribute value according to its frequency of occurrences), and then transform the value levels to sensitivity levels. The privacy model ( α l e v h , k)-anonymity for hierarchical data with multi-level sensitivity is proposed. Furthermore, we design a privacy-preserving approach to achieve this privacy model. Experiment results demonstrate that our approach is obviously superior to existing anonymous approach in hierarchical data in terms of utility and security.

Description: Symmetry, Vol. 10, Pages 335: Finite Element Study of a Threaded Fastening: The Case of Surgical Screws in Bone Symmetry doi: 10.3390/sym10080335 Authors: J. A. López-Campos A. Segade E. Casarejos J. R. Fernández J. A. Vilán P. Izquierdo This paper studies the stress state of a threaded fastening by using Finite Element (FE) models, applied to surgical screws in cortical bone. There is a general interest in studying the stress states induced in the different elements of a joint caused by the thread contact. Analytical models were an initial approach, and later FE models allowed detailed studies of the complex phenomena related to these joints. Different studies have evaluated standard threaded joints in machinery and structures, being the thread symmetric. However, surgical screws employ asymmetric thread geometry, selected to improve the stress level generated in the bone. Despite the interest and widespread use, there is scarce documentation on the actual effect of this thread type. In this work, we discuss the results provided by FE models with detailed descriptions of the contacts comparing differences caused by the materials of the joint, the thread geometry and the thread’s three-dimensional helical effects. The complex contacts at the threaded surfaces cause intense demand on computational resources that often limits the studies including these joints. We analyze the results provided by one commercial software package to simplify the threaded joints. The comparison with detailed FE models allows a definition of the level of uncertainty and possible limitations of this type of simplifications, and helps in making suitable choices for complex applications.

Description: Symmetry, Vol. 10, Pages 339: A Three-Dimensional Constrained Ordered Weighted Averaging Aggregation Problem with Lower Bounded Variables Symmetry doi: 10.3390/sym10080339 Authors: Yi-Fang Chen Hui-Chin Tang We consider the constrained ordered weighted averaging (OWA) aggregation problem with a single constraint and lower bounded variables. For the three-dimensional constrained OWA aggregation problem with lower bounded variables, we present four types of solution(x1',x2',x3') depending on the number of zero elements. According to the computerized experiment we perform, the lower bounds can affect the solution(x1',x2',x3') types, thereby affecting the optimal solution of the three-dimensional constrained OWA aggregation problem with lower bounded variables.

Description: Entropy, Vol. 20, Pages 613: Time-Dependent Probability Density Functions and Attractor Structure in Self-Organised Shear Flows Entropy doi: 10.3390/e20080613 Authors: Quentin Jacquet Eun-jin Kim Rainer Hollerbach We report the time-evolution of Probability Density Functions (PDFs) in a toy model of self-organised shear flows, where the formation of shear flows is induced by a finite memory time of a stochastic forcing, manifested by the emergence of a bimodal PDF with the two peaks representing non-zero mean values of a shear flow. Using theoretical analyses of limiting cases, as well as numerical solutions of the full Fokker–Planck equation, we present a thorough parameter study of PDFs for different values of the correlation time and amplitude of stochastic forcing. From time-dependent PDFs, we calculate the information length ( L ), which is the total number of statistically different states that a system passes through in time and utilise it to understand the information geometry associated with the formation of bimodal or unimodal PDFs. We identify the difference between the relaxation and build-up of the shear gradient in view of information change and discuss the total information length ( L ∞ = L ( t → ∞ ) ) which maps out the underlying attractor structures, highlighting a unique property of L ∞ which depends on the trajectory/history of a PDF’s evolution.

Description: Entropy, Vol. 20, Pages 612: Permutation Entropy Based on Non-Uniform Embedding Entropy doi: 10.3390/e20080612 Authors: Mei Tao Kristina Poskuviene Nizar Faisal Alkayem Maosen Cao Minvydas Ragulskis A novel visualization scheme for permutation entropy is presented in this paper. The proposed scheme is based on non-uniform attractor embedding of the investigated time series. A single digital image of permutation entropy is produced by averaging all possible plain projections of the permutation entropy measure in the multi-dimensional delay coordinate space. Computational experiments with artificially-generated and real-world time series are used to demonstrate the advantages of the proposed visualization scheme.

Description: Entropy, Vol. 20, Pages 611: Multi-Fault Diagnosis of Gearbox Based on Improved Multipoint Optimal Minimum Entropy Deconvolution Entropy doi: 10.3390/e20080611 Authors: Fuhe Yang Xingquan Shen Zhijian Wang Under complicated conditions, the extraction of a multi-fault in gearboxes is difficult to achieve. Due to improper selection of methods, leakage diagnosis or misdiagnosis will usually occur. Ensemble Empirical Mode Decomposition (EEMD) often causes energy leakage due to improper selection of white noise during signal decomposition. Considering that only a single fault cycle can be extracted when MOMED (Multipoint Optimal Minimum Entropy Deconvolution) is used, it is necessary to perform the sub-band processing of the compound fault signal. This paper presents an adaptive gearbox multi-fault-feature extraction method based on Improved MOMED (IMOMED). Firstly, EEMD decomposes the signal adaptively and selects the intrinsic mode functions with strong correlation with the original signal to perform FFT (Fast Fourier transform); considering the mode-mixing phenomenon of EEMD, reconstruct the intrinsic mode functions with the same timescale, and obtain several intrinsic mode functions of the same scale to improve the entropy of fault features. There is a lot of white noise in the original signal, and EEMD can improve the signal-to-noise ratio of the original signal. Finally, through the setting of different noise-reduction intervals to extract fault features through MOMED. The proposed method is compared with EEMD and VMD (Variational Mode Decomposition) to verify its feasibility.

Description: Entropy, Vol. 20, Pages 610: SU(2) Decomposition for the Quantum Information Dynamics in 2d-Partite Two-Level Quantum Systems Entropy doi: 10.3390/e20080610 Authors: Francisco Delgado The gate array version of quantum computation uses logical gates adopting convenient forms for computational algorithms based on the algorithms classical computation. Two-level quantum systems are the basic elements connecting the binary nature of classical computation with the settlement of quantum processing. Despite this, their design depends on specific quantum systems and the physical interactions involved, thus complicating the dynamics analysis. Predictable and controllable manipulation should be addressed in order to control the quantum states in terms of the physical control parameters. Resources are restricted to limitations imposed by the physical settlement. This work presents a formalism to decompose the quantum information dynamics in S U ( 2 2 d ) for 2 d -partite two-level systems into 2 2 d − 1 S U ( 2 ) quantum subsystems. It generates an easier and more direct physical implementation of quantum processing developments for qubits. Easy and traditional operations proposed by quantum computation are recovered for larger and more complex systems. Alternating the parameters of local and non-local interactions, the procedure states a universal exchange semantics on the basis of generalized Bell states. Although the main procedure could still be settled on other interaction architectures by the proper selection of the basis as natural grammar, the procedure can be understood as a momentary splitting of the 2 d information channels into 2 2 d − 1 pairs of 2 level quantum information subsystems. Additionally, it is a settlement of the quantum information manipulation that is free of the restrictions imposed by the underlying physical system. Thus, the motivation of decomposition is to set control procedures easily in order to generate large entangled states and to design specialized dedicated quantum gates. They are potential applications that properly bypass the general induced superposition generated by physical dynamics.

Description: Entropy, Vol. 20, Pages 616: Dynamic Clustering and Coordinated User Scheduling for Cooperative Interference Cancellation on Ultra-High Density Distributed Antenna Systems Entropy doi: 10.3390/e20080616 Authors: Kazuki Maruta This paper proposes dynamic clustering and user scheduling for previously conceived inter-cluster interference cancellation scheme on ultra-high density distributed antenna system (UHD-DAS). UHD-DAS is composed of one central unit (CU) and densely deployed remote radio units (RUs) serving as small cell access points. It can enhance spatial spectral efficiency by alleviating traffic load imposed per radio unit; however, intenser small cell deployment revives the inter-cell interference (ICI) problem. Cell clustering, cooperation of multiple RUs, can mitigate ICI partially, whereas inter-cluster interference (ICLI) still limits its possible capacity. Simplified ICLI cancellation based on localized RU cooperation was previously proposed to mitigate interference globally. The resolved issue is that it required frequency reuse distance to fully obtain its interference cancellation ability. This paper introduces dynamic clustering with coordinated user scheduling to ensure reuse distance without extra frequency reuse. Joint dynamic clustering and ICLI cancellation can effectively work and almost reaches ideal performance as full cooperative spatial multiplexing transmission.

Description: Symmetry, Vol. 10, Pages 349: Optimizing the High-Level Maintenance Planning Problem of the Electric Multiple Unit Train Using a Modified Particle Swarm Optimization Algorithm Symmetry doi: 10.3390/sym10080349 Authors: Jianping Wu Boliang Lin Hui Wang Xuhui Zhang Zhongkai Wang Jiaxi Wang Electric multiple unit (EMU) trains’ high-level maintenance planning is a discrete problem in mathematics. The high-level maintenance process of the EMU trains consumes plenty of time. When the process is undertaken during peak periods of the passenger flow, the transportation demand may not be fully satisfied due to the insufficient supply of trains. In contrast, if the process is undergone in advance, extra costs will be incurred. Based on the practical requirements of high-level maintenance, a 0–1 programming model is proposed. To simplify the description of the model, candidate sets of delivery dates, i.e., time windows, are generated according to the historical data and maintenance regulations. The constraints of the model include maintenance regulations, the passenger transportation demand, and capacities of workshop. The objective function is to minimize the mileage losses of all EMU trains. Moreover, a modified particle swarm algorithm is developed for solving the problem. Finally, a real-world case study of Shanghai Railway is conducted to demonstrate the proposed method. Computational results indicate that the (approximate) optimal solution can be obtained successfully by our method and the proposed method significantly reduces the solution time to 500 s.

Description: Symmetry, Vol. 10, Pages 348: Sharp Bounds on the Higher Order Schwarzian Derivatives for Janowski Classes Symmetry doi: 10.3390/sym10080348 Authors: Nak Eun Cho Virendra Kumar V. Ravichandran Higher order Schwarzian derivatives for normalized univalent functions were first considered by Schippers, and those of convex functions were considered by Dorff and Szynal. In the present investigation, higher order Schwarzian derivatives for the Janowski star-like and convex functions are considered, and sharp bounds for the first three consecutive derivatives are investigated. The results obtained in this paper generalize several existing results in this direction.

Description: Entropy, Vol. 20, Pages 621: The Maximum Entropy Method in Ultrasonic Non-Destructive Testing—Increasing the Resolution, Image Noise Reduction and Echo Acquisition Rate Entropy doi: 10.3390/e20080621 Authors: Evgeny Gennadievich Bazulin The use of linear methods, for example, the Combined Synthetic Aperture Focusing Technique (C–SAFT), does not allow one to obtain images with high resolution and low noise, especially structural noise in all cases. Non-linear methods should improve the quality of the reconstructed image. Several examples of the application of the maximum entropy (ME) method for ultrasonic echo processing in order to reconstruct the image of reflectors with Rayleigh super-resolution and a high signal-to-noise ratio are considered in the article. The use of the complex phase-shifted Barker code signal as a probe pulse and the compression of measured echoes by the ME method made it possible to increase the signal-to-noise ratio by more than 20 dB for the image of a flat-bottom hole with a diameter of 1 mm in a model experiment. A modification of the ME method for restoring the reflector image by the time-of-flight diffraction (TOFD) method is considered, taking into account the change of the echo signal shape, depending on the depth of the reflector. Using the ME method, 2.5D-images of models of dangling cracks in a pipeline with a diameter of 800 mm were obtained, which make it possible to determine their dimensions. In the object with structural noise, using the ME method, it was possible to increase the signal-to-noise ratio of the reflector image by more than 12 dB. To accelerate the acquisition of echoes in the dual scan mode, it is proposed to use code division multiple access (CDMA) technology based on simultaneous emission by all elements of the array of pseudo-orthogonal signals. The model experiment showed the effectiveness of applying the ME method.

Description: Entropy, Vol. 20, Pages 619: A Classical Interpretation of the Scrooge Distribution Entropy doi: 10.3390/e20080619 Authors: William K. Wootters The Scrooge distribution is a probability distribution over the set of pure states of a quantum system. Specifically, it is the distribution that, upon measurement, gives up the least information about the identity of the pure state compared with all other distributions that have the same density matrix. The Scrooge distribution has normally been regarded as a purely quantum mechanical concept with no natural classical interpretation. In this paper, we offer a classical interpretation of the Scrooge distribution viewed as a probability distribution over the probability simplex. We begin by considering a real-amplitude version of the Scrooge distribution for which we find that there is a non-trivial but natural classical interpretation. The transition to the complex-amplitude case requires a step that is not particularly natural but that may shed light on the relation between quantum mechanics and classical probability theory.

Description: Symmetry, Vol. 10, Pages 345: Neutrosophic Duplets of {Zpn,×} and {Zpq,×} and Their Properties Symmetry doi: 10.3390/sym10080345 Authors: Vasantha Kandasamy W.B. Ilanthenral Kandasamy Florentin Smarandache The notions of neutrosophy, neutrosophic algebraic structures, neutrosophic duplet and neutrosophic triplet were introduced by Florentin Smarandache. In this paper, the neutrosophic duplets of Z p n , Z p q and Z p 1 p 2 … p n are studied. In the case of Z p n and Z p q , the complete characterization of neutrosophic duplets are given. In the case of Z p 1 … p n , only the neutrosophic duplets associated with p i s are provided; i = 1 , 2 , … , n . Some open problems related to neutrosophic duplets are proposed.

Description: Entropy, Vol. 20, Pages 633: Fractal Structure and Non-Extensive Statistics Entropy doi: 10.3390/e20090633 Authors: Airton Deppman Tobias Frederico Eugenio Megías Debora P. Menezes The role played by non-extensive thermodynamics in physical systems has been under intense debate for the last decades. With many applications in several areas, the Tsallis statistics have been discussed in detail in many works and triggered an interesting discussion on the most deep meaning of entropy and its role in complex systems. Some possible mechanisms that could give rise to non-extensive statistics have been formulated over the last several years, in particular a fractal structure in thermodynamic functions was recently proposed as a possible origin for non-extensive statistics in physical systems. In the present work, we investigate the properties of such fractal thermodynamical system and propose a diagrammatic method for calculations of relevant quantities related to such a system. It is shown that a system with the fractal structure described here presents temperature fluctuation following an Euler Gamma Function, in accordance with previous works that provided evidence of the connections between those fluctuations and Tsallis statistics. Finally, the scale invariance of the fractal thermodynamical system is discussed in terms of the Callan–Symanzik equation.

Description: Entropy, Vol. 20, Pages 634: An Entropy-Based Approach for Measuring Factor Contributions in Factor Analysis Models Entropy doi: 10.3390/e20090634 Authors: Nobuoki Eshima Minoru Tabata Claudio Giovanni Borroni In factor analysis, factor contributions of latent variables are assessed conventionally by the sums of the squared factor loadings related to the variables. First, the present paper considers issues in the conventional method. Second, an alternative entropy-based approach for measuring factor contributions is proposed. The method measures the contribution of the common factor vector to the manifest variable vector and decomposes it into contributions of factors. A numerical example is also provided to demonstrate the present approach.

Description: Entropy, Vol. 20, Pages 631: Entropic Equilibria Selection of Stationary Extrema in Finite Populations Entropy doi: 10.3390/e20090631 Authors: Marc Harper Dashiell Fryer We propose the entropy of random Markov trajectories originating and terminating at the same state as a measure of the stability of a state of a Markov process. These entropies can be computed in terms of the entropy rates and stationary distributions of Markov processes. We apply this definition of stability to local maxima and minima of the stationary distribution of the Moran process with mutation and show that variations in population size, mutation rate, and strength of selection all affect the stability of the stationary extrema.

Description: Entropy, Vol. 20, Pages 635: Detailed Fluctuation Theorems: A Unifying Perspective Entropy doi: 10.3390/e20090635 Authors: Riccardo Rao Massimiliano Esposito We present a general method to identify an arbitrary number of fluctuating quantities which satisfy a detailed fluctuation theorem for all times within the framework of time-inhomogeneous Markovian jump processes. In doing so, we provide a unified perspective on many fluctuation theorems derived in the literature. By complementing the stochastic dynamics with a thermodynamic structure (i.e., using stochastic thermodynamics), we also express these fluctuating quantities in terms of physical observables.

Description: Symmetry, Vol. 10, Pages 361: Porous Scaffold Design Based on Minimal Surfaces: Development and Assessment of Variable Architectures Symmetry doi: 10.3390/sym10090361 Authors: Rita Ambu Anna Eva Morabito In tissue engineering, biocompatible porous scaffolds that try to mimic the features and function of the bone are of great relevance. In this paper, an effective method for the design of 3D porous scaffolds is applied to the modelling of structures with variable architectures. These structures are of interest since they are more similar to the stochastic configuration of real bone with respect to architectures made of a unit cell replicated in three orthogonal directions, which are usually considered for this kind of applications. This property configures them as, potentially, more suitable to satisfy simultaneously the biological requirements and those relative to the mechanical strength. The procedure implemented is based on the implicit surface modelling method and the use of a triply periodic minimal surface (TPMS), specifically, the Schwarz’s Primitive (P) minimal surface, whose geometry was considered for the development of scaffolds with different configurations. The representative structures modelled were numerically analysed by means of finite element analysis (FEA), considering them made of a biocompatible titanium alloy. The architectures considered were thus assessed in terms of the relationship between the geometrical configuration and the mechanical response to compression loading.

Description: Symmetry, Vol. 10, Pages 366: The (A)symmetry between the Exterior and Interior of a Schwarzschild Black Hole Symmetry doi: 10.3390/sym10090366 Authors: Pawel Gusin Andy Augousti Filip Formalik Andrzej Radosz A black hole in a Schwarzschild spacetime is considered. A transformation is proposed that describes the relationship between the coordinate systems exterior and interior to an event horizon. The application of this transformation permits considerations of the (a)symmetry of a range of phenomena taking place on both sides of the event horizon. The paper investigates two distinct problems of a uniformly accelerated particle. In one of these, although the equations of motion are the same in the regions on both sides, the solutions turn out to be very different. This manifests the differences of the properties of these two ranges.

Description: Symmetry, Vol. 10, Pages 365: On Connection between Second-Degree Exterior and Symmetric Derivations of Kähler Modules Symmetry doi: 10.3390/sym10090365 Authors: Hamiyet Merkepçi Mathematical physics looks for ways to apply mathematical ideas to problems in physics. In differential forms, the tensor form is first defined, and the definitions of exterior and symmetric differential forms are made accordingly. For instance, M is an R-module, M ⊗ R M the tensor product of M with itself and H a submodule of M ⊗ R M generated by x ⊗ y − y ⊗ x , where x , y in M. Then, ∨ 2 ( M ) = M ⊗ R M / H is called the second symmetric power of M. A role of the exterior differential forms in field theory is related to the conservation laws for physical fields, etc. In this study, I present a new approach to emphasize the properties of second exterior and symmetric derivations on Kahler modules, and I find a connection between them. I constitute exact sequences of ∨ 2 ( Ω 1 ( S ) ) and Λ 2 ( Ω 1 ( S ) ) , and I describe and prove a new isomorphism in the following: Let S be an affine algebra presented by R / I , where R = k [ x 1 , … x s ] is a polynomial algebra and I = ( f 1 , … f m ) an ideal of R. Then, I have J 1 Ω 1 ( S ) ≃ Ω 1 ( S ) ⊕ ∨ 2 ( Ω 1 ( S ) ) ⊕ Λ 2 ( Ω 1 ( S ) .

Description: Entropy, Vol. 20, Pages 644: Model Error, Information Barriers, State Estimation and Prediction in Complex Multiscale Systems Entropy doi: 10.3390/e20090644 Authors: Andrew Majda Nan Chen Complex multiscale systems are ubiquitous in many areas. This research expository article discusses the development and applications of a recent information-theoretic framework as well as novel reduced-order nonlinear modeling strategies for understanding and predicting complex multiscale systems. The topics include the basic mathematical properties and qualitative features of complex multiscale systems, statistical prediction and uncertainty quantification, state estimation or data assimilation, and coping with the inevitable model errors in approximating such complex systems. Here, the information-theoretic framework is applied to rigorously quantify the model fidelity, model sensitivity and information barriers arising from different approximation strategies. It also succeeds in assessing the skill of filtering and predicting complex dynamical systems and overcomes the shortcomings in traditional path-wise measurements such as the failure in measuring extreme events. In addition, information theory is incorporated into a systematic data-driven nonlinear stochastic modeling framework that allows effective predictions of nonlinear intermittent time series. Finally, new efficient reduced-order nonlinear modeling strategies combined with information theory for model calibration provide skillful predictions of intermittent extreme events in spatially-extended complex dynamical systems. The contents here include the general mathematical theories, effective numerical procedures, instructive qualitative models, and concrete models from climate, atmosphere and ocean science.

Description: Entropy, Vol. 20, Pages 643: Random Spacing between Metal Tree Electrodeposits in Linear DLA Arrays Entropy doi: 10.3390/e20090643 Authors: Jaad Tannous Lina Anouti Rabih Sultan When we examine the random growth of trees along a linear alley in a rural area, we wonder what governs the location of those trees, and hence the distance between adjacent ones. The same question arises when we observe the growth of metal electro-deposition trees along a linear cathode in a rectangular film of solution. We carry out different sets of experiments wherein zinc trees are grown by electrolysis from a linear graphite cathode in a 2D film of zinc sulfate solution toward a thick zinc metal anode. We measure the distance between adjacent trees, calculate the average for each set, and correlate the latter with probability and entropy. We also obtain a computational image of the grown trees as a function of parameters such as the cell size, number of particles, and sticking probability. The dependence of average distance on concentration is studied and assessed.

Description: Entropy, Vol. 20, Pages 656: “The Heisenberg Method”: Geometry, Algebra, and Probability in Quantum Theory Entropy doi: 10.3390/e20090656 Authors: Arkady Plotnitsky The article reconsiders quantum theory in terms of the following principle, which can be symbolically represented as QUANTUMNESS → PROBABILITY → ALGEBRA and will be referred to as the QPA principle. The principle states that the quantumness of physical phenomena, that is, the specific character of physical phenomena known as quantum, implies that our predictions concerning them are irreducibly probabilistic, even in dealing with quantum phenomena resulting from the elementary individual quantum behavior (such as that of elementary particles), which in turn implies that our theories concerning these phenomena are fundamentally algebraic, in contrast to more geometrical classical or relativistic theories, although these theories, too, have an algebraic component to them. It follows that one needs to find an algebraic scheme able make these predictions in a given quantum regime. Heisenberg was first to accomplish this in the case of quantum mechanics, as matrix mechanics, whose matrix character testified to his algebraic method, as Einstein characterized it. The article explores the implications of the Heisenberg method and of the QPA principle for quantum theory, and for the relationships between mathematics and physics there, from a nonrealist or, in terms of this article, “reality-without-realism” or RWR perspective, defining the RWR principle, thus joined to the QPA principle.

Description: Entropy, Vol. 20, Pages 648: The Effects of Mo and Nb on the Microstructures and Properties of CrFeCoNi(Nb,Mo) Alloys Entropy doi: 10.3390/e20090648 Authors: Chun-Huei Tsau Meng-Chi Tsai The effects of niobium and molybdenum additions on the microstructures, hardness and corrosion behaviors of CrFeCoNi(Nb,Mo) alloys were investigated. All of the CrFeCoNi(Nb,Mo) alloys displayed dendritic microstructures. The dendrites of CrFeCoNiNb and CrFeCoNiNb0.5Mo0.5 alloys were a hexagonal close packing (HCP) phase and the interdendrites were a eutectic structure of HCP and face-centered cubic (FCC) phases. Additionally, the dendrites of CrFeCoNiMo alloys were a simple cubic (SC) phase and the interdendrites were a eutectic structure of SC and FCC phases. The volume fraction of dendrites and interdendrites in these alloys were calculated. The influences of the volume fraction of dendrite in the alloys on the overall hardness were also discussed. The CrFeCoNiNb alloy had the larger volume fraction of dendrite and thus had the highest hardness among these alloys. The CrFeCoNi(Nb,Mo) alloys also showed better corrosion resistances in 1 M H2SO4 and 1 M NaCl solutions by comparing with commercial 304 stainless steel. The CrFeCoNiNb0.5Mo0.5 alloy possessed the best corrosion resistances in these solutions among the CrFeCoNi(Nb,Mo) alloys.

Description: Entropy, Vol. 20, Pages 646: Tensile Behavior and Evolution of the Phases in the Al10Co25Cr8Fe15Ni36Ti6 Compositionally Complex/High Entropy Alloy Entropy doi: 10.3390/e20090646 Authors: Anna Maria Manzoni Sebastian Haas Haneen Daoud Uwe Glatzel Christiane Förster Nelia Wanderka Compositionally complex alloys, or high entropy alloys, are good candidates for applications at higher temperatures in gas turbines. After their introduction, the equiatomic Al17Co17Cr17Cu17Fe17Ni17 (at.%) served as a starting material and a long optimization road finally led to the recently optimized Al10Co25Cr8Fe15Ni36Ti6 (at.%) alloy, which shows promising mechanical properties. Investigations of the as-cast state and after different heat treatments focus on the evolution of the microstructure and provide an overview of some mechanical properties. The dendritic solidification provides two phases in the dendritic cores and two different ones in the interdendritic regions. Three of the four phases remain after heat treatments. Homogenization and subsequent annealing produce a γ-γ’ based microstructure, similar to Ni-based superalloys. The γ phase is Co-Cr-Fe rich and the γ’ phase is Al-Ni-Ti rich. The understanding of the mechanical behavior of the investigated alloy is supported and enhanced by the study of the different phases and their nanohardness measurements. The observations are compared with mechanical and microstructural data from commercial Ni-based superalloys, Co-based alloys, and Co-Ni-based alloys at the desired application temperature of ~800 °C.

Description: Symmetry, Vol. 10, Pages 369: A Comprehensive Comparative Analysis of the Basic Theory of the Short Term Bus Passenger Flow Prediction Symmetry doi: 10.3390/sym10090369 Authors: Huawei Zhai Licheng Cui Yu Nie Xiaowei Xu Weishi Zhang In order to meet the real-time public travel demands, the bus operators need to adjust the timetables in time. Therefore, it is necessary to predict the variations of the short-term passenger flow. Under the help of the advanced public transportation systems, a large amount of real-time data about passenger flow is collected from the automatic passenger counters, automatic fare collection systems, etc. Using these data, different kinds of methods are proposed to predict future variations of the short-term bus passenger flow. Based on the properties and background knowledge, these methods are classified into three categories: linear, nonlinear and combined methods. Their performances are evaluated in detail in the major aspects of the prediction accuracy, the complexity of training data structure and modeling process. For comparison, some long-term prediction methods are also analyzed simply. At last, it points that, with the help of automatic technology, a large amount of data about passenger flow will be collected, and using the big data technology to speed up the data preprocessing and modeling process may be one of the directions worthy of study in the future.

Description: Entropy, Vol. 20, Pages 667: Application of Mutual Information-Sample Entropy Based MED-ICEEMDAN De-Noising Scheme for Weak Fault Diagnosis of Hoist Bearing Entropy doi: 10.3390/e20090667 Authors: Fen Yang Ziming Kou Juan Wu Tengyu Li In this paper, a novel weak fault features extraction scheme is proposed to extract weak fault features in head sheave bearings of floor-type multi-rope friction mine hoists in strong noise environments. A mutual information-based sample entropy (MI-SE) is proposed to select the effective intrinsic mode function (IMF). The numerical simulation presented in this paper has demonstrated that the improved complete ensemble empirical mode decomposition with adaptive noise (ICEEMDAN) has a poor performance on weak signals processing under a strong noise background, and fault features cannot be identified clearly. The de-noised signal is decomposed into several IMFs by the ICEEMDAN method, with the help of the minimum entropy deconvolution (MED), which works as a pre-filter to increase the kurtosis value by about 3.2 times. The envelope spectrum of the effective IMF selected by the MI-SE method shows almost all fault features clearly. An analogous experiment system was built to verify the feasibility of the proposed scheme, whose results have also shown that the proposed hybrid scheme has better performance compared with ICEEMDAN or MED on the weak fault features extraction under a strong noise background. This paper provides a novel method to diagnose the weak faults of the slow speed and heavy load rolling bearings in a strong noise environment.

Description: Symmetry, Vol. 10, Pages 372: Decaying Dark Energy in Light of the Latest Cosmological Dataset Symmetry doi: 10.3390/sym10090372 Authors: Ivan de Martino Decaying Dark Energy models modify the background evolution of the most common observables, such as the Hubble function, the luminosity distance and the Cosmic Microwave Background temperature–redshift scaling relation. We use the most recent observationally-determined datasets, including Supernovae Type Ia and Gamma Ray Bursts data, along with H ( z ) and Cosmic Microwave Background temperature versus z data and the reduced Cosmic Microwave Background parameters, to improve the previous constraints on these models. We perform a Monte Carlo Markov Chain analysis to constrain the parameter space, on the basis of two distinct methods. In view of the first method, the Hubble constant and the matter density are left to vary freely. In this case, our results are compatible with previous analyses associated with decaying Dark Energy models, as well as with the most recent description of the cosmological background. In view of the second method, we set the Hubble constant and the matter density to their best fit values obtained by the Planck satellite, reducing the parameter space to two dimensions, and improving the existent constraints on the model’s parameters. Our results suggest that the accelerated expansion of the Universe is well described by the cosmological constant, and we argue that forthcoming observations will play a determinant role to constrain/rule out decaying Dark Energy.

Description: Symmetry, Vol. 10, Pages 370: Multimedia Data Modelling Using Multidimensional Recurrent Neural Networks Symmetry doi: 10.3390/sym10090370 Authors: Zhen He Shaobing Gao Liang Xiao Daxue Liu Hangen He Modelling the multimedia data such as text, images, or videos usually involves the analysis, prediction, or reconstruction of them. The recurrent neural network (RNN) is a powerful machine learning approach to modelling these data in a recursive way. As a variant, the long short-term memory (LSTM) extends the RNN with the ability to remember information for longer. Whilst one can increase the capacity of LSTM by widening or adding layers, additional parameters and runtime are usually required, which could make learning harder. We therefore propose a Tensor LSTM where the hidden states are tensorised as multidimensional arrays (tensors) and updated through a cross-layer convolution. As parameters are spatially shared within the tensor, we can efficiently widen the model without extra parameters by increasing the tensorised size; as deep computations of each time step are absorbed by temporal computations of the time series, we can implicitly deepen the model with little extra runtime by delaying the output. We show by experiments that our model is well-suited for various multimedia data modelling tasks, including text generation, text calculation, image classification, and video prediction.

Description: Entropy, Vol. 20, Pages 661: Non-Orthogonal eMBB-URLLC Radio Access for Cloud Radio Access Networks with Analog Fronthauling Entropy doi: 10.3390/e20090661 Authors: Andrea Matera Rahif Kassab Osvaldo Simeone Umberto Spagnolini This paper considers the coexistence of Ultra Reliable Low Latency Communications (URLLC) and enhanced Mobile BroadBand (eMBB) services in the uplink of Cloud Radio Access Network (C-RAN) architecture based on the relaying of radio signals over analog fronthaul links. While Orthogonal Multiple Access (OMA) to the radio resources enables the isolation and the separate design of different 5G services, Non-Orthogonal Multiple Access (NOMA) can enhance the system performance by sharing wireless and fronthaul resources. This paper provides an information-theoretic perspective in the performance of URLLC and eMBB traffic under both OMA and NOMA. The analysis focuses on standard cellular models with additive Gaussian noise links and a finite inter-cell interference span, and it accounts for different decoding strategies such as puncturing, Treating Interference as Noise (TIN) and Successive Interference Cancellation (SIC). Numerical results demonstrate that, for the considered analog fronthauling C-RAN architecture, NOMA achieves higher eMBB rates with respect to OMA, while guaranteeing reliable low-rate URLLC communication with minimal access latency. Moreover, NOMA under SIC is seen to achieve the best performance, while, unlike the case with digital capacity-constrained fronthaul links, TIN always outperforms puncturing.

Description: Entropy, Vol. 20, Pages 665: Assessing Time Series Reversibility through Permutation Patterns Entropy doi: 10.3390/e20090665 Authors: Massimiliano Zanin Alejandro Rodríguez-González Ernestina Ruiz David Papo Time irreversibility, i.e., the lack of invariance of the statistical properties of a system under time reversal, is a fundamental property of all systems operating out of equilibrium. Time reversal symmetry is associated with important statistical and physical properties and is related to the predictability of the system generating the time series. Over the past fifteen years, various methods to quantify time irreversibility in time series have been proposed, but these can be computationally expensive. Here, we propose a new method, based on permutation entropy, which is essentially parameter-free, temporally local, yields straightforward statistical tests, and has fast convergence properties. We apply this method to the study of financial time series, showing that stocks and indices present a rich irreversibility dynamics. We illustrate the comparative methodological advantages of our method with respect to a recently proposed method based on visibility graphs, and discuss the implications of our results for financial data analysis and interpretation.

Description: Entropy, Vol. 20, Pages 666: Chart for Thermoelectric Systems Operation Based on a Ternary Diagram for Bithermal Systems Entropy doi: 10.3390/e20090666 Authors: Julien Ramousse Christophe Goupil Thermoelectric system’s operation needs careful attention to ensure optimal power conversion depending on the application aims. As a ternary diagram of bithermal systems allows a synthetic graphical analysis of the performance attainable by any work-heat conversion system, thermoelectric systems operation is plotted as a parametric curve function of the operating conditions (electric current and reservoirs’ temperature), based on the standard model of Ioffe. The threshold of each operating mode (heat engine, heat pump, thermal dissipation, and forced thermal transfer), along with the optimal efficiencies and powers of the heat pump and heat engine modes, are characterized graphically and analytically as a function of the material properties and the operating conditions. The sensibility of the performance aims (maximum efficiency vs. maximum power) with the operating conditions is, thus, highlighted. In addition, the specific contributions of each phenomenon involved in the semiconductor (reversible Seebeck effect, irreversible heat leakage by conduction and irreversible thermal dissipation by Joule effect) are discussed in terms of entropy generation. Finally, the impact of the exo-irreversibilities on the performance is analyzed by taking the external thermal resistances into account.

Description: Entropy, Vol. 20, Pages 663: Cross-Sectoral Information Transfer in the Chinese Stock Market around Its Crash in 2015 Entropy doi: 10.3390/e20090663 Authors: Xudong Wang Xiaofeng Hui This paper applies effective transfer entropy to research the information transfer in the Chinese stock market around its crash in 2015. According to the market states, the entire period is divided into four sub-phases: the tranquil, bull, crash, and post-crash periods. Kernel density estimation is used to calculate the effective transfer entropy. Then, the information transfer network is constructed. Nodes’ centralities and the directed maximum spanning trees of the networks are analyzed. The results show that, in the tranquil period, the information transfer is weak in the market. In the bull period, the strength and scope of the information transfer increases. The utility sector outputs a great deal of information and is the hub node for the information flow. In the crash period, the information transfer grows further. The market efficiency in this period is worse than that in the other three sub-periods. The information technology sector is the biggest information source, while the consumer staples sector receives the most information. The interactions of the sectors become more direct. In the post-crash period, information transfer declines but is still stronger than the tranquil time. The financial sector receives the largest amount of information and is the pivot node.

Description: Symmetry, Vol. 10, Pages 384: A New Evaluation for Solving the Fully Fuzzy Data Envelopment Analysis with Z-Numbers Symmetry doi: 10.3390/sym10090384 Authors: Ali Namakin Seyyed Esmaeil Najafi Mohammad Fallah Mehrdad Javadi There are numerous models for solving the efficiency evaluation in data envelopment analysis (DEA) with fuzzy input and output data. However, because of the limitation of those strategies, they cannot be implemented for solving fully fuzzy DEA (FFDEA). Furthermore, in real-world problems with imprecise data, fuzziness is not sufficient to consider, and the reliability of the information is also very vital. To overcome these flaws, this paper presented a new method for solving the fully fuzzy DEA model where all parameters are Z-numbers. The new approach is primarily based on crisp linear programming and has a simple structure. Moreover, it is proved that the only existing method to solve FFDEA with Z-numbers is not valid. An example is also presented to illustrate the efficiency of our proposed method and provide an explanation for the content of the paper.

Description: Symmetry, Vol. 10, Pages 380: The Weighted Arithmetic Mean–Geometric Mean Inequality is Equivalent to the Hölder Inequality Symmetry doi: 10.3390/sym10090380 Authors: Yongtao Li Xian-Ming Gu Jianxing Zhao In the current note, we investigate the mathematical relations among the weighted arithmetic mean–geometric mean (AM–GM) inequality, the Hölder inequality and the weighted power-mean inequality. Meanwhile, the proofs of mathematical equivalence among the weighted AM–GM inequality, the weighted power-mean inequality and the Hölder inequality are fully achieved. The new results are more generalized than those of previous studies.

Description: Symmetry, Vol. 10, Pages 381: Time-Dependent Behavior of a Circular Symmetrical Tunnel Supported with Rockbolts Symmetry doi: 10.3390/sym10090381 Authors: Wei Han Gang Wang Chuanzheng Liu Hengjie Luan Ke Wang Under the effect of initial stress and excavation disturbance, there exists interaction between rock mass and rockbolt in deeply buried tunnels. In order to fully explore the mechanism of rock mass supported with rockbolts, this article studied the time-dependent behavior of the rock mass supported with discretely mechanically or frictionally coupled (DMFC) rockbolts. The interaction model elastic solutions under distributed force model were analyzed, then the viscoelastic analytical solutions were conducted to describe the rheological properties of the coupling model, and the solutions were acquired by setting the constitutive models of the rockbolt and rock mass in terms of a one-dimensional Kelvin model and a three-dimensional Burgers model based on material properties and dimension. Several examples were performed and the influence of initial stress σ0, the viscosity parameters η1 and η2 of the three-dimensional Burgers model as well as the pre-tension T0 on reinforcement effect were analyzed. According to the proposed model, the smaller η2 is or the larger the pre-tension T0 is, the more effective the support effect. However, when the pre-tension is too large, the support effect is no longer significantly enhanced. In addition, the early reinforcement effect is controlled by the first creep stage in the Burgers model while the ultimate support effect is mainly influenced by the viscosity coefficient of the second creep stage in the Burgers model. This research can provide an important theoretical reference to guide the parameter design of rockbolt reinforcement engineering in a circular symmetrical tunnel.

Description: Entropy, Vol. 20, Pages 676: Trailing-Edge Flap Control for Mitigating Rotor Power Fluctuations of a Large-Scale Offshore Floating Wind Turbine under the Turbulent Wind Condition Entropy doi: 10.3390/e20090676 Authors: Bofeng Xu Junheng Feng Tongguang Wang Yue Yuan Zhenzhou Zhao Wei Zhong A trailing-edge flap control strategy for mitigating rotor power fluctuations of a 5 MW offshore floating wind turbine is developed under turbulent wind inflow. The wind shear must be considered because of the large rotor diameter. The trailing-edge flap control strategy is based on the turbulent wind speed, the blade azimuth angle, and the platform motions. The rotor power is predicted using the free vortex wake method, coupled with the control strategy. The effect of the trailing-edge flap control on the rotor power is determined by a comparison with the rotor power of a turbine without a trailing-edge flap control. The optimal values of the three control factors are obtained. The results show that the trailing-edge flap control strategy is effective for improving the stability of the output rotor power of the floating wind turbine under the turbulent wind condition.