Open AccessArticle
Group Sparse Precoding for Cloud-RAN with Multiple User Antennas
Entropy 2018, 20(2), 144; doi:10.3390/e20020144 -
Abstract
Cloud radio access network (C-RAN) has become a promising network architecture to support the massive data traffic in the next generation cellular networks. In a C-RAN, a massive number of low-cost remote antenna ports (RAPs) are connected to a single baseband unit (BBU)
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Cloud radio access network (C-RAN) has become a promising network architecture to support the massive data traffic in the next generation cellular networks. In a C-RAN, a massive number of low-cost remote antenna ports (RAPs) are connected to a single baseband unit (BBU) pool via high-speed low-latency fronthaul links, which enables efficient resource allocation and interference management. As the RAPs are geographically distributed, group sparse beamforming schemes attract extensive studies, where a subset of RAPs is assigned to be active and a high spectral efficiency can be achieved. However, most studies assume that each user is equipped with a single antenna. How to design the group sparse precoder for the multiple antenna users remains little understood, as it requires the joint optimization of the mutual coupling transmit and receive beamformers. This paper formulates an optimal joint RAP selection and precoding design problem in a C-RAN with multiple antennas at each user. Specifically, we assume a fixed transmit power constraint for each RAP, and investigate the optimal tradeoff between the sum rate and the number of active RAPs. Motivated by the compressive sensing theory, this paper formulates the group sparse precoding problem by inducing the 0-norm as a penalty and then uses the reweighted 1 heuristic to find a solution. By adopting the idea of block diagonalization precoding, the problem can be formulated as a convex optimization, and an efficient algorithm is proposed based on its Lagrangian dual. Simulation results verify that our proposed algorithm can achieve almost the same sum rate as that obtained from an exhaustive search. Full article
Open AccessArticle
Stochastic Dynamics of a Time-Delayed Ecosystem Driven by Poisson White Noise Excitation
Entropy 2018, 20(2), 143; doi:10.3390/e20020143 -
Abstract
We investigate the stochastic dynamics of a prey-predator type ecosystem with time delay and the discrete random environmental fluctuations. In this model, the delay effect is represented by a time delay parameter and the effect of the environmental randomness is modeled as Poisson
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We investigate the stochastic dynamics of a prey-predator type ecosystem with time delay and the discrete random environmental fluctuations. In this model, the delay effect is represented by a time delay parameter and the effect of the environmental randomness is modeled as Poisson white noise. The stochastic averaging method and the perturbation method are applied to calculate the approximate stationary probability density functions for both predator and prey populations. The influences of system parameters and the Poisson white noises are investigated in detail based on the approximate stationary probability density functions. It is found that, increasing time delay parameter as well as the mean arrival rate and the variance of the amplitude of the Poisson white noise will enhance the fluctuations of the prey and predator population. While the larger value of self-competition parameter will reduce the fluctuation of the system. Furthermore, the results from Monte Carlo simulation are also obtained to show the effectiveness of the results from averaging method. Full article
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Open AccessArticle
Coarse-Graining Approaches in Univariate Multiscale Sample and Dispersion Entropy
Entropy 2018, 20(2), 138; doi:10.3390/e20020138 -
Abstract
The evaluation of complexity in univariate signals has attracted considerable attention in recent years. This is often done using the framework of Multiscale Entropy, which entails two basic steps: coarse-graining to consider multiple temporal scales, and evaluation of irregularity for each of those
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The evaluation of complexity in univariate signals has attracted considerable attention in recent years. This is often done using the framework of Multiscale Entropy, which entails two basic steps: coarse-graining to consider multiple temporal scales, and evaluation of irregularity for each of those scales with entropy estimators. Recent developments in the field have proposed modifications to this approach to facilitate the analysis of short-time series. However, the role of the downsampling in the classical coarse-graining process and its relationships with alternative filtering techniques has not been systematically explored yet. Here, we assess the impact of coarse-graining in multiscale entropy estimations based on both Sample Entropy and Dispersion Entropy. We compare the classical moving average approach with low-pass Butterworth filtering, both with and without downsampling, and empirical mode decomposition in Intrinsic Multiscale Entropy, in selected synthetic data and two real physiological datasets. The results show that when the sampling frequency is low or high, downsampling respectively decreases or increases the entropy values. Our results suggest that, when dealing with long signals and relatively low levels of noise, the refine composite method makes little difference in the quality of the entropy estimation at the expense of considerable additional computational cost. It is also found that downsampling within the coarse-graining procedure may not be required to quantify the complexity of signals, especially for short ones. Overall, we expect these results to contribute to the ongoing discussion about the development of stable, fast and robust-to-noise multiscale entropy techniques suited for either short or long recordings. Full article
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Open AccessArticle
Lagrangian Function on the Finite State Space Statistical Bundle
Entropy 2018, 20(2), 139; doi:10.3390/e20020139 -
Abstract
The statistical bundle is the set of couples (Q,W) of a probability density Q and a random variable W such that Full article
Open AccessArticle
A Chemo-Mechanical Model of Diffusion in Reactive Systems
Entropy 2018, 20(2), 140; doi:10.3390/e20020140 -
Abstract
The functional properties of multi-component materials are often determined by a rearrangement of their different phases and by chemical reactions of their components. In this contribution, a material model is presented which enables computational simulations and structural optimization of solid multi-component systems. Typical
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The functional properties of multi-component materials are often determined by a rearrangement of their different phases and by chemical reactions of their components. In this contribution, a material model is presented which enables computational simulations and structural optimization of solid multi-component systems. Typical Systems of this kind are anodes in batteries, reactive polymer blends and propellants. The physical processes which are assumed to contribute to the microstructural evolution are: (i) particle exchange and mechanical deformation; (ii) spinodal decomposition and phase coarsening; (iii) chemical reactions between the components; and (iv) energetic forces associated with the elastic field of the solid. To illustrate the capability of the deduced coupled field model, three-dimensional Non-Uniform Rational Basis Spline (NURBS) based finite element simulations of such multi-component structures are presented. Full article
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Open AccessArticle
Finding a Hadamard Matrix by Simulated Quantum Annealing
Entropy 2018, 20(2), 141; doi:10.3390/e20020141 (registering DOI) -
Abstract
Hard problems have recently become an important issue in computing. Various methods, including a heuristic approach that is inspired by physical phenomena, are being explored. In this paper, we propose the use of simulated quantum annealing (SQA) to find a Hadamard matrix, which
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Hard problems have recently become an important issue in computing. Various methods, including a heuristic approach that is inspired by physical phenomena, are being explored. In this paper, we propose the use of simulated quantum annealing (SQA) to find a Hadamard matrix, which is itself a hard problem. We reformulate the problem as an energy minimization of spin vectors connected by a complete graph. The computation is conducted based on a path-integral Monte-Carlo (PIMC) SQA of the spin vector system, with an applied transverse magnetic field whose strength is decreased over time. In the numerical experiments, the proposed method is employed to find low-order Hadamard matrices, including the ones that cannot be constructed trivially by the Sylvester method. The scaling property of the method and the measurement of residual energy after a sufficiently large number of iterations show that SQA outperforms simulated annealing (SA) in solving this hard problem. Full article
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Open AccessArticle
A Simple and Adaptive Dispersion Regression Model for Count Data
Entropy 2018, 20(2), 142; doi:10.3390/e20020142 -
Abstract
Regression for count data is widely performed by models such as Poisson, negative binomial (NB) and zero-inflated regression. A challenge often faced by practitioners is the selection of the right model to take into account dispersion, which typically occurs in count datasets. It
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Regression for count data is widely performed by models such as Poisson, negative binomial (NB) and zero-inflated regression. A challenge often faced by practitioners is the selection of the right model to take into account dispersion, which typically occurs in count datasets. It is highly desirable to have a unified model that can automatically adapt to the underlying dispersion and that can be easily implemented in practice. In this paper, a discrete Weibull regression model is shown to be able to adapt in a simple way to different types of dispersions relative to Poisson regression: overdispersion, underdispersion and covariate-specific dispersion. Maximum likelihood can be used for efficient parameter estimation. The description of the model, parameter inference and model diagnostics is accompanied by simulated and real data analyses. Full article
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Open AccessArticle
Engine Load Effects on the Energy and Exergy Performance of a Medium Cycle/Organic Rankine Cycle for Exhaust Waste Heat Recovery
Entropy 2018, 20(2), 137; doi:10.3390/e20020137 -
Abstract
The Organic Rankine Cycle (ORC) has been proved a promising technique to exploit waste heat from Internal Combustion Engines (ICEs). Waste heat recovery systems have usually been designed based on engine rated working conditions, while engines often operate under part load conditions. Hence,
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The Organic Rankine Cycle (ORC) has been proved a promising technique to exploit waste heat from Internal Combustion Engines (ICEs). Waste heat recovery systems have usually been designed based on engine rated working conditions, while engines often operate under part load conditions. Hence, it is quite important to analyze the off-design performance of ORC systems under different engine loads. This paper presents an off-design Medium Cycle/Organic Rankine Cycle (MC/ORC) system model by interconnecting the component models, which allows the prediction of system off-design behavior. The sliding pressure control method is applied to balance the variation of system parameters and evaporating pressure is chosen as the operational variable. The effect of operational variable and engine load on system performance is analyzed from the aspects of energy and exergy. The results show that with the drop of engine load, the MC/ORC system can always effectively recover waste heat, whereas the maximum net power output, thermal efficiency and exergy efficiency decrease linearly. Considering the contributions of components to total exergy destruction, the proportions of the gas-oil exchanger and turbine increase, while the proportions of the evaporator and condenser decrease with the drop of engine load. Full article
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Open AccessArticle
Uncertainty Relation Based on Wigner–Yanase–Dyson Skew Information with Quantum Memory
Entropy 2018, 20(2), 132; doi:10.3390/e20020132 -
Abstract
We present uncertainty relations based on Wigner–Yanase–Dyson skew information with quantum memory. Uncertainty inequalities both in product and summation forms are derived. It is shown that the lower bounds contain two terms: one characterizes the degree of compatibility of two measurements, and the
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We present uncertainty relations based on Wigner–Yanase–Dyson skew information with quantum memory. Uncertainty inequalities both in product and summation forms are derived. It is shown that the lower bounds contain two terms: one characterizes the degree of compatibility of two measurements, and the other is the quantum correlation between the measured system and the quantum memory. Detailed examples are given for product, separable and entangled states. Full article
Open AccessArticle
Applying Time-Dependent Attributes to Represent Demand in Road Mass Transit Systems
Entropy 2018, 20(2), 133; doi:10.3390/e20020133 -
Abstract
The development of efficient mass transit systems that provide quality of service is a major challenge for modern societies. To meet this challenge, it is essential to understand user demand. This article proposes using new time-dependent attributes to represent demand, attributes that differ
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The development of efficient mass transit systems that provide quality of service is a major challenge for modern societies. To meet this challenge, it is essential to understand user demand. This article proposes using new time-dependent attributes to represent demand, attributes that differ from those that have traditionally been used in the design and planning of this type of transit system. Data mining was used to obtain these new attributes; they were created using clustering techniques, and their quality evaluated with the Shannon entropy function and with neural networks. The methodology was implemented on an intercity public transport company and the results demonstrate that the attributes obtained offer a more precise understanding of demand and enable predictions to be made with acceptable precision. Full article
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Open AccessArticle
Investigating the Configurations in Cross-Shareholding: A Joint Copula-Entropy Approach
Entropy 2018, 20(2), 134; doi:10.3390/e20020134 -
Abstract
The complex nature of the interlacement of economic actors is quite evident at the level of the Stock market, where any company may actually interact with the other companies buying and selling their shares. In this respect, the companies populating a Stock market,
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The complex nature of the interlacement of economic actors is quite evident at the level of the Stock market, where any company may actually interact with the other companies buying and selling their shares. In this respect, the companies populating a Stock market, along with their connections, can be effectively modeled through a directed network, where the nodes represent the companies, and the links indicate the ownership. This paper deals with this theme and discusses the concentration of a market. A cross-shareholding matrix is considered, along with two key factors: the node out-degree distribution which represents the diversification of investments in terms of the number of involved companies, and the node in-degree distribution which reports the integration of a company due to the sales of its own shares to other companies. While diversification is widely explored in the literature, integration is most present in literature on contagions. This paper captures such quantities of interest in the two frameworks and studies the stochastic dependence of diversification and integration through a copula approach. We adopt entropies as measures for assessing the concentration in the market. The main question is to assess the dependence structure leading to a better description of the data or to market polarization (minimal entropy) or market fairness (maximal entropy). In so doing, we derive information on the way in which the in- and out-degrees should be connected in order to shape the market. The question is of interest to regulators bodies, as witnessed by specific alert threshold published on the US mergers guidelines for limiting the possibility of acquisitions and the prevalence of a single company on the market. Indeed, all countries and the EU have also rules or guidelines in order to limit concentrations, in a country or across borders, respectively. The calibration of copulas and model parameters on the basis of real data serves as an illustrative application of the theoretical proposal. Full article
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Open AccessArticle
Complexity of Simple, Switched and Skipped Chaotic Maps in Finite Precision
Entropy 2018, 20(2), 135; doi:10.3390/e20020135 -
Abstract
In this paper we investigate the degradation of the statistic properties of chaotic maps as consequence of their implementation in a digital media such as Digital Signal Processors (DSP), Field Programmable Gate Arrays (FPGA) or Application-Specific Integrated Circuits (ASIC). In these systems, binary
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In this paper we investigate the degradation of the statistic properties of chaotic maps as consequence of their implementation in a digital media such as Digital Signal Processors (DSP), Field Programmable Gate Arrays (FPGA) or Application-Specific Integrated Circuits (ASIC). In these systems, binary floating- and fixed-point are the numerical representations available. Fixed-point representation is preferred over floating-point when speed, low power and/or small circuit area are necessary. Then, in this paper we compare the degradation of fixed-point binary precision version of chaotic maps with the one obtained by using floating point 754-IEEE standard, to evaluate the feasibility of their FPGA implementation. The specific period that every fixed-point precision produces was investigated in previous reports. Statistical characteristics are also relevant, it has been recently shown that it is convenient to describe the statistical characteristic using both, causal and non-causal quantifiers. In this paper we complement the period analysis by characterizing the behavior of these maps from an statistical point of view using cuantifiers from information theory. Here, rather than reproducing an exact replica of the real system, the aim is to meet certain conditions related to the statistics of systems. Full article
Open AccessArticle
Robustification of a One-Dimensional Generic Sigmoidal Chaotic Map with Application of True Random Bit Generation
Entropy 2018, 20(2), 136; doi:10.3390/e20020136 -
Abstract
The search for generation approaches to robust chaos has received considerable attention due to potential applications in cryptography or secure communications. This paper is of interest regarding a 1-D sigmoidal chaotic map, which has never been distinctly investigated. This paper introduces a generic
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The search for generation approaches to robust chaos has received considerable attention due to potential applications in cryptography or secure communications. This paper is of interest regarding a 1-D sigmoidal chaotic map, which has never been distinctly investigated. This paper introduces a generic form of the sigmoidal chaotic map with three terms, i.e., xn+1 = ∓AfNL(Bxn) ± Cxn ± D, where A, B, C, and D are real constants. The unification of modified sigmoid and hyperbolic tangent (tanh) functions reveals the existence of a “unified sigmoidal chaotic map” generically fulfilling the three terms, with robust chaos partially appearing in some parameter ranges. A simplified generic form, i.e., xn+1 = ∓fNL(Bxn) ± Cxn, through various S-shaped functions, has recently led to the possibility of linearization using (i) hardtanh and (ii) signum functions. This study finds a linearized sigmoidal chaotic map that potentially offers robust chaos over an entire range of parameters. Chaos dynamics are described in terms of chaotic waveforms, histogram, cobweb plots, fixed point, Jacobian, and a bifurcation structure diagram based on Lyapunov exponents. As a practical example, a true random bit generator using the linearized sigmoidal chaotic map is demonstrated. The resulting output is evaluated using the NIST SP800-22 test suite and TestU01. Full article
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Open AccessArticle
On a Dynamical Approach to Some Prime Number Sequences
Entropy 2018, 20(2), 131; doi:10.3390/e20020131 -
Abstract
We show how the cross-disciplinary transfer of techniques from dynamical systems theory to number theory can be a fruitful avenue for research. We illustrate this idea by exploring from a nonlinear and symbolic dynamics viewpoint certain patterns emerging in some residue sequences generated
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We show how the cross-disciplinary transfer of techniques from dynamical systems theory to number theory can be a fruitful avenue for research. We illustrate this idea by exploring from a nonlinear and symbolic dynamics viewpoint certain patterns emerging in some residue sequences generated from the prime number sequence. We show that the sequence formed by the residues of the primes modulo k are maximally chaotic and, while lacking forbidden patterns, unexpectedly display a non-trivial spectrum of Renyi entropies which suggest that every block of size m>1, while admissible, occurs with different probability. This non-uniform distribution of blocks for m>1 contrasts Dirichlet’s theorem that guarantees equiprobability for m=1. We then explore in a similar fashion the sequence of prime gap residues. We numerically find that this sequence is again chaotic (positivity of Kolmogorov–Sinai entropy), however chaos is weaker as forbidden patterns emerge for every block of size m>1. We relate the onset of these forbidden patterns with the divisibility properties of integers, and estimate the densities of gap block residues via Hardy–Littlewood k-tuple conjecture. We use this estimation to argue that the amount of admissible blocks is non-uniformly distributed, what supports the fact that the spectrum of Renyi entropies is again non-trivial in this case. We complete our analysis by applying the chaos game to these symbolic sequences, and comparing the Iterated Function System (IFS) attractors found for the experimental sequences with appropriate null models. Full article
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Open AccessArticle
Exergoeconomic Assessment of Solar Absorption and Absorption–Compression Hybrid Refrigeration in Building Cooling
Entropy 2018, 20(2), 130; doi:10.3390/e20020130 -
Abstract
The paper mainly deals with the match of solar refrigeration, i.e., solar/natural gas-driven absorption chiller (SNGDAC), solar vapor compression–absorption integrated refrigeration system with parallel configuration (SVCAIRSPC), and solar absorption-subcooled compression hybrid cooling system (SASCHCS), and building cooling based on the exergoeconomics. Three types
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The paper mainly deals with the match of solar refrigeration, i.e., solar/natural gas-driven absorption chiller (SNGDAC), solar vapor compression–absorption integrated refrigeration system with parallel configuration (SVCAIRSPC), and solar absorption-subcooled compression hybrid cooling system (SASCHCS), and building cooling based on the exergoeconomics. Three types of building cooling are considered: Type 1 is the single-story building, type 2 includes the two-story and three-story buildings, and type 3 is the multi-story buildings. Besides this, two Chinese cities, Guangzhou and Turpan, are taken into account as well. The product cost flow rate is employed as the primary decision variable. The result exhibits that SNGDAC is considered as a suitable solution for type 1 buildings in Turpan, owing to its negligible natural gas consumption and lowest product cost flow rate. SVCAIRSPC is more applicable for type 2 buildings in Turpan because of its higher actual cooling capacity of absorption subsystem and lower fuel and product cost flow rate. Additionally, SASCHCS shows the most extensive cost-effectiveness, namely, its exergy destruction and product cost flow rate are both the lowest when used in all types of buildings in Guangzhou or type 3 buildings in Turpan. This paper is helpful to promote the application of solar cooling. Full article
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Open AccessArticle
On Points Focusing Entropy
Entropy 2018, 20(2), 128; doi:10.3390/e20020128 -
Abstract
In the paper, we consider local aspects of the entropy of nonautonomous dynamical systems. For this purpose, we introduce the notion of a (asymptotical) focal entropy point. The notion of entropy appeared as a result of practical needs concerning thermodynamics and the problem
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In the paper, we consider local aspects of the entropy of nonautonomous dynamical systems. For this purpose, we introduce the notion of a (asymptotical) focal entropy point. The notion of entropy appeared as a result of practical needs concerning thermodynamics and the problem of information flow, and it is connected with the complexity of a system. The definition adopted in the paper specifies the notions that express the complexity of a system around certain points (the complexity of the system is the same as its complexity around these points), and moreover, the complexity of a system around such points does not depend on the behavior of the system in other parts of its domain. Any periodic system “acting” in the closed unit interval has an asymptotical focal entropy point, which justifies wide interest in these issues. In the paper, we examine the problems of the distortions of a system and the approximation of an autonomous system by a nonautonomous one, in the context of having a (asymptotical) focal entropy point. It is shown that even a slight modification of a system may lead to the arising of the respective focal entropy points. Full article
Open AccessArticle
Logical Divergence, Logical Entropy, and Logical Mutual Information in Product MV-Algebras
Entropy 2018, 20(2), 129; doi:10.3390/e20020129 -
Abstract
In the paper we propose, using the logical entropy function, a new kind of entropy in product MV-algebras, namely the logical entropy and its conditional version. Fundamental characteristics of these quantities have been shown and subsequently, the results regarding the logical entropy have
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In the paper we propose, using the logical entropy function, a new kind of entropy in product MV-algebras, namely the logical entropy and its conditional version. Fundamental characteristics of these quantities have been shown and subsequently, the results regarding the logical entropy have been used to define the logical mutual information of experiments in the studied case. In addition, we define the logical cross entropy and logical divergence for the examined situation and prove basic properties of the suggested quantities. To illustrate the results, we provide several numerical examples. Full article
Open AccessArticle
Mesoscopic Moment Equations for Heat Conduction: Characteristic Features and Slow–Fast Mode Decomposition
Entropy 2018, 20(2), 126; doi:10.3390/e20020126 -
Abstract
In this work, we derive different systems of mesoscopic moment equations for the heat-conduction problem and analyze the basic features that they must hold. We discuss two- and three-equation systems, showing that the resulting mesoscopic equation from two-equation systems is of the telegraphist’s
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In this work, we derive different systems of mesoscopic moment equations for the heat-conduction problem and analyze the basic features that they must hold. We discuss two- and three-equation systems, showing that the resulting mesoscopic equation from two-equation systems is of the telegraphist’s type and complies with the Cattaneo equation in the Extended Irreversible Thermodynamics Framework. The solution of the proposed systems is analyzed, and it is shown that it accounts for two modes: a slow diffusive mode, and a fast advective mode. This latter additional mode makes them suitable for heat transfer phenomena on fast time-scales, such as high-frequency pulses and heat transfer in small-scale devices. We finally show that, if proper initial conditions are provided, the advective mode disappears, and the solution of the system tends asymptotically to the transient solution of the classical parabolic heat-conduction equation. Full article
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Open AccessArticle
Application of Multiscale Entropy in Assessing Plantar Skin Blood Flow Dynamics in Diabetics with Peripheral Neuropathy
Entropy 2018, 20(2), 127; doi:10.3390/e20020127 -
Abstract
Diabetic foot ulcer (DFU) is a common complication of diabetes mellitus, while tissue ischemia caused by impaired vasodilatory response to plantar pressure is thought to be a major factor of the development of DFUs, which has been assessed using various measures of skin
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Diabetic foot ulcer (DFU) is a common complication of diabetes mellitus, while tissue ischemia caused by impaired vasodilatory response to plantar pressure is thought to be a major factor of the development of DFUs, which has been assessed using various measures of skin blood flow (SBF) in the time or frequency domain. These measures, however, are incapable of characterizing nonlinear dynamics of SBF, which is an indicator of pathologic alterations of microcirculation in the diabetic foot. This study recruited 18 type 2 diabetics with peripheral neuropathy and eight healthy controls. SBF at the first metatarsal head in response to locally applied pressure and heating was measured using laser Doppler flowmetry. A multiscale entropy algorithm was utilized to quantify the regularity degree of the SBF responses. The results showed that during reactive hyperemia and thermally induced biphasic response, the regularity degree of SBF in diabetics underwent only small changes compared to baseline and significantly differed from that in controls at multiple scales (p < 0.05). On the other hand, the transition of regularity degree of SBF in diabetics distinctively differed from that in controls (p < 0.05). These findings indicated that multiscale entropy could provide a more comprehensive assessment of impaired microvascular reactivity in the diabetic foot compared to other entropy measures based on only a single scale, which strengthens the use of plantar SBF dynamics to assess the risk for DFU. Full article
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Open AccessArticle
Dissolution or Growth of a Liquid Drop via Phase-Field Ternary Mixture Model Based on the Non-Random, Two-Liquid Equation
Entropy 2018, 20(2), 125; doi:10.3390/e20020125 -
Abstract
We simulate the diffusion-driven dissolution or growth of a single-component liquid drop embedded in a continuous phase of a binary liquid. Our theoretical approach follows a diffuse-interface model of partially miscible ternary liquid mixtures that incorporates the non-random, two-liquid (NRTL) equation as a
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We simulate the diffusion-driven dissolution or growth of a single-component liquid drop embedded in a continuous phase of a binary liquid. Our theoretical approach follows a diffuse-interface model of partially miscible ternary liquid mixtures that incorporates the non-random, two-liquid (NRTL) equation as a submodel for the enthalpic (so-called excess) component of the Gibbs energy of mixing, while its nonlocal part is represented based on a square-gradient (Cahn-Hilliard-type modeling) assumption. The governing equations for this phase-field ternary mixture model are simulated in 2D, showing that, for a single-component drop embedded in a continuous phase of a binary liquid (which is highly miscible with either one component of the continuous phase but is essentially immiscible with the other), the size of the drop can either shrink to zero or reach a stationary value, depending on whether the global composition of the mixture is within the one-phase region or the unstable range of the phase diagram. Full article
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