Entropy, Volume 20, Issue 12 (December 2018) – 100 articles

We apply the stochastic thermodynamics formalism to describe the dynamics of systems of complex Langevin and Fokker-Planck equations. We provide in particular a simple and general recipe to calculate thermodynamical currents, dissipated and propagating heat for networks of nonlinear oscillators. By using the Hodge decomposition of thermodynamical forces and fluxes, we derive a formula for entropy production that generalises the notion of non-potential forces and makes transparent the breaking of detailed balance and of time reversal symmetry for states arbitrarily far from equilibrium. Our formalism is then applied to describe the off-equilibrium thermodynamics of a few examples, notably a continuum ferromagnet, a network of classical spin-oscillators and the Frenkel-Kontorova model of nano friction. Full article

One type of fractal dimension definition is based on the generalized entropy function. Both entropy and fractal dimensions can be employed to characterize complex spatial systems such as cities and regions. Despite the inherent connection between entropy and fractal dimensions, they have different application scopes and directions in urban studies. This paper focuses on exploring how to convert entropy measurements into fractal dimensions for the spatial analysis of scale-free urban phenomena using the ideas from scaling. Urban systems proved to be random prefractal and multifractal systems. The spatial entropy of fractal cities bears two properties. One is the scale dependence: the entropy values of urban systems always depend on the linear scales of spatial measurement. The other is entropy conservation: different fractal parts bear the same entropy value. Thus, entropy cannot reflect the simple rules of urban processes and the spatial heterogeneity of urban patterns. If we convert the generalized entropies into multifractal spectrums, the problems of scale dependence and entropy homogeneity can be solved to a degree for urban spatial analysis. Especially, the geographical analyses of urban evolution can be simplified. This study may be helpful for students in describing and explaining the spatial complexity of urban evolution. Full article

Detecting and classifying ships based on radiated noise provide practical guidelines for the reduction of underwater noise footprint of shipping. In this paper, the detection and classification are implemented by auditory inspired convolutional neural networks trained from raw underwater acoustic signal. The proposed model includes three parts. The first part is performed by a multi-scale 1D time convolutional layer initialized by auditory filter banks. Signals are decomposed into frequency components by convolution operation. In the second part, the decomposed signals are converted into frequency domain by permute layer and energy pooling layer to form frequency distribution in auditory cortex. Then, 2D frequency convolutional layers are applied to discover spectro-temporal patterns, as well as preserve locality and reduce spectral variations in ship noise. In the third part, the whole model is optimized with an objective function of classification to obtain appropriate auditory filters and feature representations that are correlative with ship categories. The optimization reflects the plasticity of auditory system. Experiments on five ship types and background noise show that the proposed approach achieved an overall classification accuracy of 79.2%, which improved by 6% compared to conventional approaches. Auditory filter banks were adaptive in shape to improve accuracy of classification. Full article

Experimental investigations of nano-scale spatio-temporal effects that occur on the friction surface under extreme tribological stimuli, in combination with thermodynamic modeling of the self-organization process, are presented in this paper. The study was performed on adaptive PVD (physical vapor deposited) coatings represented by the TiAlCrSiYN/TiAlCrN nano-multilayer PVD coating. A detailed analysis of the worn surface was conducted using scanning electron microscopy and energy dispersive spectroscopy (SEM/EDS), transmission electron microscopy (TEM), X-ray photoelectron spectroscopy (XPS), and Auger electron spectroscopy (AES) methods. It was demonstrated that the coating studied exhibits a very fast adaptive response to the extreme external stimuli through the formation of an increased amount of protective surface tribo-films at the very beginning of the running-in stage of wear. Analysis performed on the friction surface indicates that all of the tribo-film formation processes occur in the nanoscopic scale. The tribo-films form as thermal barrier tribo-ceramics with a complex composition and very low thermal conductivity under high operating temperatures, thus demonstrating reduced friction which results in low cutting forces and wear values. This process presents an opportunity for the surface layer to attain a strong non-equilibrium state. This leads to the stabilization of the exchanging interactions between the tool and environment at a low wear level. This effect is the consequence of the synergistic behavior of complex matter represented by the dynamically formed nano-scale tribo-film layer. Full article

Thermostability is a protein property that impacts many types of studies, including protein activity enhancement, protein structure determination, and drug development. However, most computational tools designed to predict protein thermostability require tertiary structure data as input. The few tools that are dependent only on the primary structure of a protein to predict its thermostability have one or more of the following problems: a slow execution speed, an inability to make large-scale mutation predictions, and the absence of temperature and pH as input parameters. Therefore, we developed a computational tool, named KStable, that is sequence-based, computationally rapid, and includes temperature and pH values to predict changes in the thermostability of a protein upon the introduction of a mutation at a single site. KStable was trained using basis features and minimal redundancy–maximal relevance (mRMR) features, and 58 classifiers were subsequently tested. To find the representative features, a regular-mRMR method was developed. When KStable was evaluated with an independent test set, it achieved an accuracy of 0.708. Full article

Consider a uniquely ergodic $C^{*}$ -dynamical system based on a unital *-endomorphism $\Phi$ of a $C^{*}$ -algebra. We prove the uniform convergence of Cesaro averages $\frac{1}{n}{\sum }_{k=0}^{n-1}{\lambda }^{-n}\Phi \left(a\right)$ for all values $\lambda$ in the unit circle, which are not eigenvalues corresponding to “measurable non-continuous” eigenfunctions. This result generalizes an analogous one, known in commutative ergodic theory, which turns out to be a combination of the Wiener–Wintner theorem and the uniformly convergent ergodic theorem of Krylov and Bogolioubov. Full article

The purpose of this reply is to provide a discussion and closure for the comment paper by Dr. Bormashenko on the present authors’ article, which discussed the application of the principle of least action in reversible thermodynamic processes and cycles. Dr. Bormashenko’s questions and misunderstandings are responded to, and the differences between the present authors’ work and Lucia’s are also presented. Full article

As it is well known both atmospheric and mantle convection are very complex phenomena. The dynamical description of these processes is a very difficult task involving complicated 2-D or 3-D mathematical models. However, a first approximation to these phenomena can be by means of simplified thermodynamic models where the restriction imposed by the laws of thermodynamics play an important role. An example of this approach is the model proposed by Gordon and Zarmi in 1989 to emulate the convective cells of the atmospheric air by using finite-time thermodynamics (FTT). In the present article we use the FTT Gordon-Zarmi model to coarsely describe the convection in the Earth’s mantle. Our results permit the existence of two layers of convective cells along the mantle. Besides the model reasonably reproduce the temperatures of the main discontinuities in the mantle, such as the 410 km-discontinuity, the Repetti transition zone and the so-called D-Layer. Full article

Graph kernels are of vital importance in the field of graph comparison and classification. However, how to compare and evaluate graph kernels and how to choose an optimal kernel for a practical classification problem remain open problems. In this paper, a comprehensive evaluation framework of graph kernels is proposed for unattributed graph classification. According to the kernel design methods, the whole graph kernel family can be categorized in five different dimensions, and then several representative graph kernels are chosen from these categories to perform the evaluation. With plenty of real-world and synthetic datasets, kernels are compared by many criteria such as classification accuracy, F1 score, runtime cost, scalability and applicability. Finally, quantitative conclusions are discussed based on the analyses of the extensive experimental results. The main contribution of this paper is that a comprehensive evaluation framework of graph kernels is proposed, which is significant for graph-classification applications and the future kernel research. Full article

In 1923, Hadamard encountered a class of integrals with strong singularities when using a particular Green’s function to solve the cylindrical wave equation. He ignored the infinite parts of such integrals after integrating by parts. Such an idea is very practical and useful in many physical models, e.g., the crack problems of both planar and three-dimensional elasticities. In this paper, we present the rectangular and trapezoidal formulas to approximate the Hadamard derivative by the idea of the finite part integral. Then, we apply the proposed numerical methods to the differential equation with the Hadamard derivative. Finally, several numerical examples are displayed to show the effectiveness of the basic idea and technique. Full article

The processing and analyzing of multimedia data has become a popular research topic due to the evolution of deep learning. Deep learning has played an important role in addressing many challenging problems, such as computer vision, image recognition, and image detection, which can be useful in many real-world applications. In this study, we analyzed visual features of images to detect advertising images from scanned images of various magazines. The aim is to identify key features of advertising images and to apply them to real-world application. The proposed work will eventually help improve marketing strategies, which requires the classification of advertising images from magazines. We employed convolutional neural networks to classify scanned images as either advertisements or non-advertisements (i.e., articles). The results show that the proposed approach outperforms other classifiers and the related work in terms of accuracy. Full article

As the complementary concept of intuitionistic fuzzy entropy, the knowledge measure of Atanassov’s intuitionistic fuzzy sets (AIFSs) has attracted more attention and is still an open topic. The amount of knowledge is important to evaluate intuitionistic fuzzy information. An entropy-based knowledge measure for AIFSs is defined in this paper to quantify the knowledge amount conveyed by AIFSs. An intuitive analysis on the properties of the knowledge amount in AIFSs is put forward to facilitate the introduction of axiomatic definition of the knowledge measure. Then we propose a new knowledge measure based on the entropy-based divergence measure with respect for the difference between the membership degree, the non-membership degree, and the hesitancy degree. The properties of the new knowledge measure are investigated in a mathematical viewpoint. Several examples are applied to illustrate the performance of the new knowledge measure. Comparison with several existing entropy and knowledge measures indicates that the proposed knowledge has a greater ability in discriminating different AIFSs and it is robust in quantifying the knowledge amount of different AIFSs. Lastly, the new knowledge measure is applied to the problem of multiple attribute decision making (MADM) in an intuitionistic fuzzy environment. Two models are presented to determine attribute weights in the cases that information on attribute weights is partially known and completely unknown. After obtaining attribute weights, we develop a new method to solve intuitionistic fuzzy MADM problems. An example is employed to show the effectiveness of the new MADM method. Full article

The goal of this comment note is to express my concerns about the recent paper by Tian Zhao et al. (Entropy 2018, 20, 542). It is foreseen that this comment will stimulate a fruitful discussion of the issues involved. The principle of the least thermodynamic action is applicable for the analysis of the Carnot cycle using the entropy (not heat) generation extrema theorem. The transversality conditions of the variational problem provide the rectangular shape of the ST diagram for the Carnot cycle. Full article

A new type of multi-baffle-type heat sink is proposed in this paper. The heat-transfer coefficient and pressure drop penalty of the employed six heat sink models are numerically investigated under five different inlet velocities. It is shown that Model 6 (M6) has excellent heat transfer performance as its heat-transfer coefficient reaches a value of 1758.59 W/m²K with a pressure drop of 2.96 × 10⁴ Pa, and the temperature difference between the maximum and the minimum temperature of the heating surface is 51.7 K. The results showed that the coolant for M6 is distributed evenly to each channel at the maximal degree. The phenomena of the maldistribution of temperature is effectively improved. Moreover, the thermal resistance and thermal enhancement factor for the six models is also examined. M6 possesses the lowest total thermal resistance and largest thermal enhancement factor compared to the other five models. Furthermore, an experimental platform is set up to verify the simulation results obtained for M6. The simulated heat-transfer coefficient and pressure drop values agree well with the experimental results. Full article

The fundamental equations of various disciplines often seem to share the same basic structure. Natural selection increases information in the same way that Bayesian updating increases information. Thermodynamics and the forms of common probability distributions express maximum increase in entropy, which appears mathematically as loss of information. Physical mechanics follows paths of change that maximize Fisher information. The information expressions typically have analogous interpretations as the Newtonian balance between force and acceleration, representing a partition between the direct causes of change and the opposing changes in the frame of reference. This web of vague analogies hints at a deeper common mathematical structure. I suggest that the Price equation expresses that underlying universal structure. The abstract Price equation describes dynamics as the change between two sets. One component of dynamics expresses the change in the frequency of things, holding constant the values associated with things. The other component of dynamics expresses the change in the values of things, holding constant the frequency of things. The separation of frequency from value generalizes Shannon’s separation of the frequency of symbols from the meaning of symbols in information theory. The Price equation’s generalized separation of frequency and value reveals a few simple invariances that define universal geometric aspects of change. For example, the conservation of total frequency, although a trivial invariance by itself, creates a powerful constraint on the geometry of change. That constraint plus a few others seem to explain the common structural forms of the equations in different disciplines. From that abstract perspective, interpretations such as selection, information, entropy, force, acceleration, and physical work arise from the same underlying geometry expressed by the Price equation. Full article

In this paper, a hierarchical prior model based on the Haar transformation and an appropriate Bayesian computational method for X-ray CT reconstruction are presented. Given the piece-wise continuous property of the object, a multilevel Haar transformation is used to associate a sparse representation for the object. The sparse structure is enforced via a generalized Student-t distribution ( $\mathcal{S}{t}_g$ ), expressed as the marginal of a normal-inverse Gamma distribution. The proposed model and corresponding algorithm are designed to adapt to specific 3D data sizes and to be used in both medical and industrial Non-Destructive Testing (NDT) applications. In the proposed Bayesian method, a hierarchical structured prior model is proposed, and the parameters are iteratively estimated. The initialization of the iterative algorithm uses the parameters of the prior distributions. A novel strategy for the initialization is presented and proven experimentally. We compare the proposed method with two state-of-the-art approaches, showing that our method has better reconstruction performance when fewer projections are considered and when projections are acquired from limited angles. Full article

The paper presents possible applications of entropy measures in analysis of biosignals recorded during head up tilt testing (HUTT) in patients with suspected vasovagal syndrome. The study group comprised 80 patients who developed syncope during HUTT (57 in the passive phase of the test (HUTT(+) group) and 23 who had negative result of passive phase and developed syncope after provocation with nitroglycerine (HUTT(−) group)). The paper focuses on assessment of monitored signals’ complexity (heart rate expressed as R-R intervals (RRI), blood pressure (sBP, dBP) and stroke volume (SV)) using various types of entropy measures (Sample Entropy (SE), Fuzzy Entropy (FE), Shannon Entropy (Sh), Conditional Entropy (CE), Permutation Entropy (PE)). Assessment of the complexity of signals in supine position indicated presence of significant differences between HUTT(+) versus HUTT(−) patients only for Conditional Entropy (CE(RRI)). Values of CE(RRI) higher than 0.7 indicate likelihood of a positive result of HUTT already at the passive phase. During tilting, in the pre-syncope phase, significant differences were found for: (SE(sBP), SE(dBP), FE(RRI), FE(sBP), FE(dBP), FE(SV), Sh(sBP), Sh(SV), CE(sBP), CE(dBP)). HUTT(+) patients demonstrated significant changes in signals’ complexity more frequently than HUTT(−) patients. When comparing entropy measurements done in the supine position with those during tilting, SV assessed in HUTT(+) patients was the only parameter for which all tested measures of entropy (SE(SV), FE(SV), Sh(SV), CE(SV), PE(SV)) showed significant differences. Full article

Contemporary non-representationalist interpretations of the quantum state (especially QBism, neo-Copenhagen views, and the relational interpretation) maintain that quantum states codify observer-relative information. This paper provides an extensive defense of such views, while emphasizing the advantages of, specifically, the relational interpretation. The argument proceeds in three steps: (1) I present a classical example (which exemplifies the spirit of the relational interpretation) to illustrate why some of the most persistent charges against non-representationalism have been misguided. (2) The special focus is placed on dynamical evolution. Non-representationalists often motivate their views by interpreting the collapse postulate as the quantum mechanical analogue of Bayesian probability updating. However, it is not clear whether one can also interpret the Schrödinger equation as a form of rational opinion updating. Using results due to Hughes & van Fraassen as well as Lisi, I argue that unitary evolution has a counterpart in classical probability theory: in both cases (quantum and classical) probabilities relative to a non-participating observer evolve according to an entropy maximizing principle (and can be interpreted as rational opinion updating). (3) Relying on a thought-experiment by Frauchiger and Renner, I discuss the differences between quantum and classical probability models. Full article

An image encryption algorithm is presented in this paper based on a chaotic map. Different from traditional methods based on the permutation-diffusion structure, the keystream here depends on both secret keys and the pre-processed image. In particular, in the permutation stage, a middle parameter is designed to revise the outputs of the chaotic map, yielding a temporal delay phenomena. Then, diffusion operation is applied after a group of random numbers is inserted into the permuted image. Therefore, the gray distribution can be changed and is different from that of the plain-image. This insertion acts as a one-time pad. Moreover, the keystream for the diffusion operation is designed to be influenced by secret keys assigned in the permutation stage. As a result, the two stages are mixed together to strengthen entirety. Experimental tests also suggest that our algorithm, permutation– insertion–diffusion (PID), performs better when expecting secure communications for images. Full article

We derive the work cost of using generalized thermal baths from the physical equivalence of quantum mechanics under unitary transformations. We demonstrate our method by considering a qubit extracting work from a single bath to amplify a cavity field. There, we find that only half of the work investment is converted into useful output, the rest being wasted as heat. These findings establish the method as a promising tool for studying quantum resources within the framework of classical thermodynamics. Full article

It is demonstrated how to construct a Galois connection between two related systems with entropy. The construction, called the Landauer’s connection, describes coupling between two systems with entropy. It is straightforward and transfers changes in one system to the other one, preserving ordering structure induced by entropy. The Landauer’s connection simplifies the description of the classical Landauer’s principle for computational systems. Categorification and generalization of the Landauer’s principle opens the area of modeling of various systems in presence of entropy in abstract terms. Full article

In this paper, based on the background of channel cooperation and service of the supply chain, this paper constructs a Nash game model and a Stackeberg game model in the multi-channel supply chain considering an online-to-store channel (OSC). Based on maximizing the profits and the bounded rationality expectation rule (BRE), this paper builds a dynamic game model, respectively, and analyzes the stability of the equilibrium points by mathematical analysis and explores the influences of parameters on stability domain and entropy of the system by using bifurcation diagram, the entropy diagram, the largest Lyapunov exponent and the chaotic attractor etc. Besides, the influences of service level and profit distribution rate on system’s profit are discussed. The theoretical results show that the greater the service level and profit distribution rate are, the smaller the stability domain of the system is; the system will go into chaotic state and the system’s entropy will increase when operators adjust her/his price decision quickly; when the manufacturer or the retailer keeps service level in the appropriate value which is conducive to maximizing her/his profit; the manufacturer should carefully set the service level of OSC to ensure the system’s profit; in Nash game model, the stability of the system weakens than that in Stackelberg game model. Furthermore, this paper puts forward some suggestions to help the manufacturer and retailer in multi-channel supply chain to do the better decision. Full article

This paper combines Bayesian networks (BN) and information theory to model the likelihood of severe loss of separation (LOS) near accidents, which are considered mid-air collision (MAC) precursors. BN is used to analyze LOS contributing factors and the multi-dependent relationship of causal factors, while Information Theory is used to identify the LOS precursors that provide the most information. The combination of the two techniques allows us to use data on LOS causes and precursors to define warning scenarios that could forecast a major LOS with severity A or a near accident, and consequently the likelihood of a MAC. The methodology is illustrated with a case study that encompasses the analysis of LOS that have taken place within the Spanish airspace during a period of four years. Full article

The minimum error entropy principle (MEE) is an alternative of the classical least squares for its robustness to non-Gaussian noise. This paper studies the gradient descent algorithm for MEE with a semi-supervised approach and distributed method, and shows that using the additional information of unlabeled data can enhance the learning ability of the distributed MEE algorithm. Our result proves that the mean squared error of the distributed gradient descent MEE algorithm can be minimax optimal for regression if the number of local machines increases polynomially as the total datasize. Full article

The effects of atomic size difference on the microstructure and mechanical properties of single face-centered cubic (FCC) phase high-entropy alloys are studied. Single FCC phase high-entropy alloys, namely, CoCrFeMnNi, Al_0.2CoCrFeMnNi, and Al_0.3CoCrCu_0.3FeNi, display good workability. The recrystallization and grain growth rates are compared during annealing. Adding Al with 0.2 molar ratio into CoCrFeMnNi retains the single FCC phase. Its atomic size difference increases from 1.18% to 2.77%, and the activation energy of grain growth becomes larger than that of CoCrFeMnNi. The as-homogenized state of Al_0.3CoCrCu_0.3FeNi high-entropy alloy becomes a single FCC structure. Its atomic size difference is 3.65%, and the grain growth activation energy is the largest among these three kinds of single-phase high-entropy alloys. At ambient temperature, the mechanical properties of Al_0.3CoCrCu_0.3FeNi are better than those of CoCrFeMnNi because of high lattice distortion and high solid solution hardening. Full article

We consider the likelihood ratio test of a simple null hypothesis (with density $f_0$ ) against a simple alternative hypothesis (with density $g_0$ ) in the situation that observations $X_i$ are mismeasured due to the presence of measurement errors. Thus instead of $X_i$ for $i=1,\dots ,n,$ we observe $Z_i=X_i+\sqrt{\delta }{V}_i$ with unobservable parameter $\delta$ and unobservable random variable $V_i$ . When we ignore the presence of measurement errors and perform the original test, the probability of type I error becomes different from the nominal value, but the test is still the most powerful among all tests on the modified level. Further, we derive the minimax test of some families of misspecified hypotheses and alternatives. The test exploits the concept of pseudo-capacities elaborated by Huber and Strassen (1973) and Buja (1986). A numerical experiment illustrates the principles and performance of the novel test. Full article

The elastic properties of seventy different compositions were calculated to optimize the composition of a V–Mo–Nb–Ta–W system. A new model called maximum entropy approach (MaxEnt) was adopted. The influence of each element was discussed. Molybdenum (Mo) and tungsten (W) are key elements for the maintenance of elastic properties. The V–Mo–Nb–Ta–W system has relatively high values of C₄₄, bulk modulus (B), shear modulus (G), and Young’s modulus (E), with high concentrations of Mo + W. Element W is brittle and has high density. Thus, low-density Mo can substitute part of W. Vanadium (V) has low density and plays an important role in decreasing the brittleness of the V–Mo–Nb–Ta–W system. Niobium (Nb) and tantalum (Ta) have relatively small influence on elastic properties. Furthermore, the calculated results can be used as a general guidance for the selection of a V–Mo–Nb–Ta–W system. Full article

As a non-invasive diagnostic tool, Magnetic Resonance Imaging (MRI) has been widely used in the field of brain imaging. The classification of MRI brain image conditions poses challenges both technically and clinically, as MRI is primarily used for soft tissue anatomy and can generate large amounts of detailed information about the brain conditions of a subject. To classify benign and malignant MRI brain images, we propose a new method. Discrete wavelet transform (DWT) is used to extract wavelet coefficients from MRI images. Then, Tsallis entropy with DNA genetic algorithm (DNA-GA) optimization parameters (called DNAGA-TE) was used to obtain entropy characteristics from DWT coefficients. At last, DNA-GA optimized support vector machine (called DNAGA-KSVM) with radial basis function (RBF) kernel, is applied as a classifier. In our experimental procedure, we use two kinds of images to validate the availability and effectiveness of the algorithm. One kind of data is the Simulated Brain Database and another kind of image is real MRI images which downloaded from Harvard Medical School website. Experimental results demonstrate that our method (DNAGA-TE+KSVM) obtained better classification accuracy. Full article

Fault prediction is an important research area that aids software development and the maintenance process. It is a field that has been continuously improving its approaches in order to reduce the fault resolution time and effort. With an aim to contribute towards building new approaches for fault prediction, this paper proposes Entropy Churn Metrics (ECM) based on History Complexity Metrics (HCM) and Churn of Source Code Metrics (CHU). The study also compares performance of ECM with that of HCM. The performance of both these metrics is compared for 14 subsystems of 5different software projects: Android, Eclipse, Apache Http Server, Eclipse C/C++ Development Tooling (CDT), and Mozilla Firefox. The study also analyses the software subsystems on three parameters: (i) distribution of faults, (ii) subsystem size, and (iii) programming language, to determine which characteristics of software systems make HCM or ECM more preferred over others. Full article