Entropy

24 pages, 654 KiB

Open AccessArticle

Macroscopic Internal Variables and Mesoscopic Theory: A Comparison Considering Liquid Crystals

by Christina Papenfuss and Wolfgang Muschik

Entropy 2018, 20(1), 81; https://doi.org/10.3390/e20010081 - 22 Jan 2018

Cited by 4 | Viewed by 4216

Internal and mesoscopic variables differ fundamentally from each other: both are state space variables, but mesoscopic variables are additionally equipped with a distribution function introducing a statistical item into consideration which is missing in connection with internal variables. Thus, the alignment tensor of [...] Read more.

Internal and mesoscopic variables differ fundamentally from each other: both are state space variables, but mesoscopic variables are additionally equipped with a distribution function introducing a statistical item into consideration which is missing in connection with internal variables. Thus, the alignment tensor of the liquid crystal theory can be introduced as an internal variable or as one generated by a mesoscopic background using the microscopic director as a mesoscopic variable. Because the mesoscopic variable is part of the state space, the corresponding balance equations change into mesoscopic balances, and additionally an evolution equation of the mesoscopic distribution function appears. The flexibility of the mesoscopic concept is not only demonstrated for liquid crystals, but is also discussed for dipolar media and flexible fibers. Full article

(This article belongs to the Special Issue Phenomenological Thermodynamics of Irreversible Processes)

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7 pages, 243 KiB

Open AccessArticle

Entanglement Entropy of the Spin-1 Condensates at Zero Temperature

by Zhibing Li, Yimin Liu, Wei Zheng and Chengguang Bao

Entropy 2018, 20(1), 80; https://doi.org/10.3390/e20010080 - 22 Jan 2018

Cited by 1 | Viewed by 3780

Abstract

For spin-1 condensates, the spatial degrees of freedom can be considered as being frozen at temperature zero, while the spin-degrees of freedom remain free. Under this condition, the entanglement entropy has been derived exactly with an analytical form. The entanglement entropy is found [...] Read more.

For spin-1 condensates, the spatial degrees of freedom can be considered as being frozen at temperature zero, while the spin-degrees of freedom remain free. Under this condition, the entanglement entropy has been derived exactly with an analytical form. The entanglement entropy is found to decrease monotonically with the increase of the magnetic polarization as expected. However, for the ground state in polar phase, an extremely steep fall of the entropy is found when the polarization emerges from zero. Then the fall becomes a gentle descent after the polarization exceeds a turning point. Full article

(This article belongs to the Special Issue Residual Entropy and Nonequilibrium States)

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(color online) The entanglement entropy <math display="inline"> <semantics> <msubsup> <mi>S</mi> <mrow> <mi>e</mi> <mi>e</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </semantics> </math> of the g.s. in polar phase against <math display="inline"> <semantics> <mrow> <mi>M</mi> <mo>/</mo> <mi>N</mi> </mrow> </semantics> </math>. The solid, dash, and dash-dot lines have N = 102, 103, and 104, respectively. Full article ">Figure 2
(color online) The entanglement entropy <math display="inline"> <semantics> <msubsup> <mi>S</mi> <mrow> <mi>e</mi> <mi>e</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </semantics> </math> of the g.s. in polar phase against <math display="inline"> <semantics> <mrow> <mi>M</mi> <mo>/</mo> <mi>N</mi> </mrow> </semantics> </math>. The solid, dash, and dash-dot lines have N = 102, 103, and 104, respectively. Full article ">Figure 3
(color online) The entanglement entropy <math display="inline"> <semantics> <msubsup> <mi>S</mi> <mrow> <mi>e</mi> <mi>e</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </semantics> </math> and <math display="inline"> <semantics> <msubsup> <mi>S</mi> <mrow> <mi>e</mi> <mi>e</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </semantics> </math> of the g.s. in ferromagnetic phase against <math display="inline"> <semantics> <mrow> <mi>M</mi> <mo>/</mo> <mi>N</mi> </mrow> </semantics> </math>. Curves for N = 102, 103, and 104 are plotted. Dependence on N is not observable. Full article ">

14 pages, 1207 KiB

Open AccessArticle

A Newly Secure Solution to MIMOME OFDM-Based SWIPT Frameworks: A Two-Stage Stackelberg Game for a Multi-User Strategy

by Makan Zamanipour

Entropy 2018, 20(1), 79; https://doi.org/10.3390/e20010079 - 22 Jan 2018

Viewed by 4595

Abstract

The paper technically proposes a newly secure scheme for simultaneous wireless power and information transfer (SWIPT) frameworks. We take into account an orthogonal frequency division multiplexing (OFDM)-based game which is in relation to a multi-input multi-output multi-antenna Eavesdropper (MIMOME) strategy. The transceiver is [...] Read more.

The paper technically proposes a newly secure scheme for simultaneous wireless power and information transfer (SWIPT) frameworks. We take into account an orthogonal frequency division multiplexing (OFDM)-based game which is in relation to a multi-input multi-output multi-antenna Eavesdropper (MIMOME) strategy. The transceiver is generally able to witness the case imperfect channel state information (ICSI) at the transmitter side. Transferring power and information are conducted via orthogonally provided sub-carriers. We propose a two-step Stackelberg game to optimise the Utility Functions of both power and information parts. The price for the first stage (in connection with information) is the total power of the other sub-carriers over which the energy is supported. In this stage, the sum secrecy rate should be essentially maximised. The second level of the proposed Stackelberg game is in association with the energy part. In this stage, the price essentially is the total power of the other sub-carriers over which the information is transferred. In this stage, additionally, the total power transferred is fundamentally maximised. Subsequently, the optimally and near-optimally mathematical solutions are derived, for some special cases such as ICSI one. Finally, the simulations validate our scheme as well, authenticating our contribution’s tightness and efficiency. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

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MIMOME OFDM-based Transceiver in downlink. Full article ">Figure 2
Secrecy capacity vs. iterations’ number. Full article ">Figure 3
Secrecy capacity vs. the interception channel’s norm. Full article ">Figure 4
Secrecy capacity vs. <math display="inline"> <semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>(</mo> <mi>d</mi> <mi>B</mi> <mo>)</mo> </mrow> </semantics> </math>. (a) Complete <math display="inline"> <semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>(</mo> <mi>d</mi> <mi>B</mi> <mo>)</mo> </mrow> </semantics> </math> regime; (b) Low <math display="inline"> <semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>(</mo> <mi>d</mi> <mi>B</mi> <mo>)</mo> </mrow> </semantics> </math> regime. Full article ">Figure 5
Cumulative distribution function vs. the secrecy capacity. Full article ">Figure 6
Goodput vs. <math display="inline"> <semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>(</mo> <mi>d</mi> <mi>B</mi> <mo>)</mo> </mrow> </semantics> </math>. Full article ">

8 pages, 653 KiB

Open AccessArticle

Anomalous Advection-Dispersion Equations within General Fractional-Order Derivatives: Models and Series Solutions

by Xin Liang, Yu-Gui Yang, Feng Gao, Xiao-Jun Yang and Yi Xue

Entropy 2018, 20(1), 78; https://doi.org/10.3390/e20010078 - 22 Jan 2018

Cited by 6 | Viewed by 5174

Abstract

In this paper, an anomalous advection-dispersion model involving a new general Liouville–Caputo fractional-order derivative is addressed for the first time. The series solutions of the general fractional advection-dispersion equations are obtained with the aid of the Laplace transform. The results are given to [...] Read more.

In this paper, an anomalous advection-dispersion model involving a new general Liouville–Caputo fractional-order derivative is addressed for the first time. The series solutions of the general fractional advection-dispersion equations are obtained with the aid of the Laplace transform. The results are given to demonstrate the efficiency of the proposed formulations to describe the anomalous advection dispersion processes. Full article

(This article belongs to the Special Issue Wavelets, Fractals and Information Theory III)

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15 pages, 3113 KiB

Open AccessArticle

Using Entropy in Web Usage Data Preprocessing

by Michal Munk and Lubomir Benko

Entropy 2018, 20(1), 67; https://doi.org/10.3390/e20010067 - 22 Jan 2018

Cited by 5 | Viewed by 5085

Abstract

The paper is focused on an examination of the use of entropy in the field of web usage mining. Entropy creates an alternative possibility of determining the ratio of auxiliary pages in the session identification using the Reference Length method. The experiment was [...] Read more.

The paper is focused on an examination of the use of entropy in the field of web usage mining. Entropy creates an alternative possibility of determining the ratio of auxiliary pages in the session identification using the Reference Length method. The experiment was conducted on two different web portals. The first log file was obtained from a course of virtual learning environment web portal. The second log file was received from the web portal with anonymous access. A comparison of the results of entropy estimation of the ratio of auxiliary pages and a sitemap estimation of the ratio of auxiliary pages showed that in the case of sitemap abundance, entropy could be a full-valued substitution for the estimate of the ratio of auxiliary pages. Full article

(This article belongs to the Special Issue Entropy-based Data Mining)

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18 pages, 18367 KiB

Open AccessArticle

Adaptive Diagnosis for Rotating Machineries Using Information Geometrical Kernel-ELM Based on VMD-SVD

by Zhipeng Wang, Limin Jia and Yong Qin

Entropy 2018, 20(1), 73; https://doi.org/10.3390/e20010073 - 21 Jan 2018

Cited by 29 | Viewed by 4915

Abstract

Rotating machineries often work under severe and variable operation conditions, which brings challenges to fault diagnosis. To deal with this challenge, this paper discusses the concept of adaptive diagnosis, which means to diagnose faults under variable operation conditions with self-adaptively and little prior [...] Read more.

Rotating machineries often work under severe and variable operation conditions, which brings challenges to fault diagnosis. To deal with this challenge, this paper discusses the concept of adaptive diagnosis, which means to diagnose faults under variable operation conditions with self-adaptively and little prior knowledge or human intervention. To this end, a novel algorithm is proposed, information geometrical extreme learning machine with kernel (IG-KELM). From the perspective of information geometry, the structure and Riemannian metric of Kernel-ELM is specified. Based on the geometrical structure, an IG-based conformal transformation is created to improve the generalization ability and self-adaptability of KELM. The proposed IG-KELM, in conjunction with variation mode decomposition (VMD) and singular value decomposition (SVD) is utilized for adaptive diagnosis: (1) VMD, as a new self-adaptive signal processing algorithm is used to decompose the raw signals into several intrinsic mode functions (IMFs). (2) SVD is used to extract the intrinsic characteristics from the matrix constructed with IMFs. (3) IG-KELM is used to diagnose faults under variable conditions self-adaptively with no requirement of prior knowledge or human intervention. Finally, the proposed method was applied on fault diagnosis of a bearing and hydraulic pump. The results show that the proposed method outperforms the conventional method by up to 7.25% and 7.78% respectively, in percentages of accuracy. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

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19 pages, 1072 KiB

Open AccessFeature PaperArticle

Characterizing Normal and Pathological Gait through Permutation Entropy

by Massimiliano Zanin, David Gómez-Andrés, Irene Pulido-Valdeolivas, Juan Andrés Martín-Gonzalo, Javier López-López, Samuel Ignacio Pascual-Pascual and Estrella Rausell

Entropy 2018, 20(1), 77; https://doi.org/10.3390/e20010077 - 19 Jan 2018

Cited by 17 | Viewed by 5750

Abstract

Cerebral palsy is a physical impairment stemming from a brain lesion at perinatal time, most of the time resulting in gait abnormalities: the first cause of severe disability in childhood. Gait study, and instrumental gait analysis in particular, has been receiving increasing attention [...] Read more.

Cerebral palsy is a physical impairment stemming from a brain lesion at perinatal time, most of the time resulting in gait abnormalities: the first cause of severe disability in childhood. Gait study, and instrumental gait analysis in particular, has been receiving increasing attention in the last few years, for being the complex result of the interactions between different brain motor areas and thus a proxy in the understanding of the underlying neural dynamics. Yet, and in spite of its importance, little is still known about how the brain adapts to cerebral palsy and to its impaired gait and, consequently, about the best strategies for mitigating the disability. In this contribution, we present the hitherto first analysis of joint kinematics data using permutation entropy, comparing cerebral palsy children with a set of matched control subjects. We find a significant increase in the permutation entropy for the former group, thus indicating a more complex and erratic neural control of joints and a non-trivial relationship between the permutation entropy and the gait speed. We further show how this information theory measure can be used to train a data mining model able to forecast the child’s condition. We finally discuss the relevance of these results in clinical applications and specifically in the design of personalized medicine interventions. Full article

(This article belongs to the Special Issue Information Theory Applied to Physiological Signals)

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Probability distributions of the Permutation Entropy (PE), as calculated in control subjects and Cerebral Palsy (CP) patients, the latter including aggregated and disaggregated (w.r.t. the Gross Motor Function Classification System (GMFCS) scale) results. Rows, from top to bottom, respectively correspond to pelvis, hip, knee, ankle and forefoot; columns, from left to right, to the abduction-adduction, sagittal and rotational axes. Full article ">Figure 2
Forest plots showing the beta coefficients of linear mixed models comparing gait PE values according to the patient’s GMFCS level. Squares represent the mean value of each beta coefficient and horizontal lines the corresponding <math display="inline"> <semantics> <mrow> <mn>95</mn> <mo>%</mo> </mrow> </semantics> </math> bias-corrected and accelerated bootstrap intervals. See main text and <a href="#sec4dot3-entropy-20-00077" class="html-sec">Section 4.3</a> for details. Full article ">Figure 3
Permutation entropy as a function of the normalized walking speed, for control subjects (black dots) and CP patients (red dots). Each panel corresponds to the same joint/axis as in <a href="#entropy-20-00077-f001" class="html-fig">Figure 1</a>. Full article ">Figure 4
Forest plots showing the beta coefficients of linear mixed models comparing gait PE values according to normalized walking speed (left panel), condition (using healthy as the reference, central panel) and the interaction of normalized walking speed with condition (right panel). The magnitudes of the effects are indicated on the X axis. Squares represent the mean values of each beta coefficient and horizontal lines the corresponding <math display="inline"> <semantics> <mrow> <mn>95</mn> <mo>%</mo> </mrow> </semantics> </math> bias-corrected and accelerated bootstrap intervals. See <a href="#sec4dot3-entropy-20-00077" class="html-sec">Section 4.3</a> for further details. Full article ">Figure 5
Forest plots showing the beta coefficients of linear mixed models that measure the effect of PE on the Gait Deviation Index (GDI, left), the Global Profile Score (GPS, center) and elements of the Movement Analysis Profile (MAP, right). The magnitudes of the effects are indicated in the X axis. Squares represent the mean values of each beta coefficient and horizontal lines the corresponding <math display="inline"> <semantics> <mrow> <mn>95</mn> <mo>%</mo> </mrow> </semantics> </math> bias-corrected and accelerated bootstrap intervals. See <a href="#sec4dot3-entropy-20-00077" class="html-sec">Section 4.3</a> for further details. Full article ">Figure 6
(Top) Multi-scale PE, for control subjects (black lines) and CP patients (red lines), as a function of the down-sampling <math display="inline"> <semantics> <mi>υ</mi> </semantics> </math>; see <a href="#sec4dot2dot2-entropy-20-00077" class="html-sec">Section 4.2.2</a> for details. Each panel corresponds to the same joint/axis as in <a href="#entropy-20-00077-f001" class="html-fig">Figure 1</a>. (Bottom) <math display="inline"> <semantics> <mo>Δ</mo> </semantics> </math>MSE for all joint/axis pairs; see Equation (<a href="#FD1-entropy-20-00077" class="html-disp-formula">1</a>). Full article ">Figure 7
Classifying patients according to their gait entropy. The left panel depicts the Receiver Operating Characteristic (ROC) curve (blue solid line), obtained through a random forest model; the dashed grey line represents the result obtained by a random classification. The right panel depicts the drop in the Area Under the Curve (AUC) when individual features (joint/axis pairs) are deleted from the dataset; the higher the value, the more important is the considered feature. Full article ">Figure 8
Random forest classification of the patients’ GMFCS stage according to the PE of the joint time series. The left panel shows the importance of the individual features. The higher the value in the X axis of the left panel, the more important the corresponding feature is for an accurate classification. Importance is estimated according to the increase in the classification error when this feature is randomly permuted. The right panels show the adjusted class probability for healthy, GMFCS I, GMFCS II, GMFCS III and GMFCS IV stages according to an RF classification. In the X axis, values of PE of hip flexion (upper plot) and ankle flexion (lower plot) are shown. Different values presented in the split are shown by a small mark in the axis. The left Y axis of the panel indicates the adjusted class probability, while the right one shows the proportion of cycles of the different classes. Full article ">

20 pages, 901 KiB

Open AccessArticle

Maximum Entropy Expectation-Maximization Algorithm for Fitting Latent-Variable Graphical Models to Multivariate Time Series

by Saïd Maanan, Bogdan Dumitrescu and Ciprian Doru Giurcăneanu

Entropy 2018, 20(1), 76; https://doi.org/10.3390/e20010076 - 19 Jan 2018

Cited by 8 | Viewed by 4451

Abstract

This work is focused on latent-variable graphical models for multivariate time series. We show how an algorithm which was originally used for finding zeros in the inverse of the covariance matrix can be generalized such that to identify the sparsity pattern of the [...] Read more.

This work is focused on latent-variable graphical models for multivariate time series. We show how an algorithm which was originally used for finding zeros in the inverse of the covariance matrix can be generalized such that to identify the sparsity pattern of the inverse of spectral density matrix. When applied to a given time series, the algorithm produces a set of candidate models. Various information theoretic (IT) criteria are employed for deciding the winner. A novel IT criterion, which is tailored to our model selection problem, is introduced. Some options for reducing the computational burden are proposed and tested via numerical examples. We conduct an empirical study in which the algorithm is compared with the state-of-the-art. The results are good, and the major advantage is that the subjective choices made by the user are less important than in the case of other methods. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

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24 pages, 4590 KiB

Open AccessArticle

Finite-Time Thermodynamic Modeling and a Comparative Performance Analysis for Irreversible Otto, Miller and Atkinson Cycles

by Jinxing Zhao and Fangchang Xu

Entropy 2018, 20(1), 75; https://doi.org/10.3390/e20010075 - 19 Jan 2018

Cited by 15 | Viewed by 4723

Abstract

Finite-time thermodynamic models for an Otto cycle, an Atkinson cycle, an over-expansion Miller cycle (M1), an LIVC Miller cycle through late intake valve closure (M2) and an LIVC Miller cycle with constant compression ratio (M3) have been established. The models for the two [...] Read more.

Finite-time thermodynamic models for an Otto cycle, an Atkinson cycle, an over-expansion Miller cycle (M1), an LIVC Miller cycle through late intake valve closure (M2) and an LIVC Miller cycle with constant compression ratio (M3) have been established. The models for the two LIVC Miller cycles are first developed; and the heat-transfer and friction losses are considered with the effects of real engine parameters. A comparative analysis for the energy losses and performances has been conducted. The optimum compression-ratio ranges for the efficiency and effective power are different. The comparative results of cycle performances are influenced together by the ratios of the energy losses and the cycle types. The Atkinson cycle has the maximum peak power and efficiency, but the minimum power density; and the M1 cycle can achieve the optimum comprehensive performances. The less net fuel amount and the high peak cylinder pressure (M3 cycle) have a significantly adverse effect on the loss ratios of the heat-transfer and friction of the M2 and M3 cycles; and the effective power and energy efficiency are always lower than the M1 and Atkinson cycles. When greatly reducing the weights of the heat-transfer and friction, the M3 cycle has significant advantage in the energy efficiency. The results obtained can provide guidance for selecting the cycle type and optimizing the performances of a real engine. Full article

(This article belongs to the Section Thermodynamics)

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15 pages, 6335 KiB

Open AccessReview

Constructal Optimizations for Heat and Mass Transfers Based on the Entransy Dissipation Extremum Principle, Performed at the Naval University of Engineering: A Review

by Lingen Chen, Qinghua Xiao and Huijun Feng

Entropy 2018, 20(1), 74; https://doi.org/10.3390/e20010074 - 19 Jan 2018

Cited by 40 | Viewed by 4253

Abstract

Combining entransy theory with constructal theory, this mini-review paper summarizes the constructal optimization work of heat conduction, convective heat transfer, and mass transfer problems during the authors’ working time in the Naval University of Engineering. The entransy dissipation extremum principle (EDEP) is applied [...] Read more.

Combining entransy theory with constructal theory, this mini-review paper summarizes the constructal optimization work of heat conduction, convective heat transfer, and mass transfer problems during the authors’ working time in the Naval University of Engineering. The entransy dissipation extremum principle (EDEP) is applied in constructal optimizations, and this paper is divided into three parts. The first part is constructal entransy dissipation rate minimizations of heat conduction and finned cooling problems. It includes constructal optimization for a “volume-to-point” heat-conduction assembly with a tapered element, constructal optimizations for “disc-to-point” heat-conduction assemblies with the premise of an optimized last-order construct and without this premise, and constructal optimizations for four kinds of fin assemblies: T-, Y-, umbrella-, and tree-shaped fins. The second part is constructal entransy dissipation rate minimizations of cooling channel and steam generator problems. It includes constructal optimizations for heat generating volumes with tree-shaped and parallel channels, constructal optimization for heat generating volume cooled by forced convection, and constructal optimization for a steam generator. The third part is constructal entransy dissipation rate minimizations of mass transfer problems. It includes constructal optimizations for “volume-to-point” rectangular assemblies with constant and tapered channels, and constructal optimizations for “disc-to-point” assemblies with the premise of an optimized last-order construct and without this premise. The results of the three parts show that the mean heat transfer temperature differences of the heat conduction assemblies are not always decreased when their internal complexity increases. The average heat transfer rate of the steam generator obtained by entransy dissipation rate maximization is increased by 58.7% compared with that obtained by heat transfer rate maximization. Compared with the rectangular mass transfer assembly with a constant high permeability pathway (HPP), the maximum pressure drops of the element and first-order assembly with tapered HPPs are decreased by 6% and 11%, respectively. The global transfer performances of the transfer bodies are improved after optimizations, and new design guidelines derived by EDEP, which are different from the conventional optimization objectives, are provided. Full article

(This article belongs to the Section Thermodynamics)

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12 pages, 266 KiB

Open AccessArticle

Non-Gaussian Closed Form Solutions for Geometric Average Asian Options in the Framework of Non-Extensive Statistical Mechanics

by Pan Zhao, Benda Zhou and Jixia Wang

Entropy 2018, 20(1), 71; https://doi.org/10.3390/e20010071 - 18 Jan 2018

Cited by 7 | Viewed by 3661

Abstract

In this paper we consider pricing problems of the geometric average Asian options under a non-Gaussian model, in which the underlying stock price is driven by a process based on non-extensive statistical mechanics. The model can describe the peak and fat tail characteristics [...] Read more.

In this paper we consider pricing problems of the geometric average Asian options under a non-Gaussian model, in which the underlying stock price is driven by a process based on non-extensive statistical mechanics. The model can describe the peak and fat tail characteristics of returns. Thus, the description of underlying asset price and the pricing of options are more accurate. Moreover, using the martingale method, we obtain closed form solutions for geometric average Asian options. Furthermore, the numerical analysis shows that the model can avoid underestimating risks relative to the Black-Scholes model. Full article

(This article belongs to the Special Issue Nonadditive Entropies and Complex Systems)

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Call option price versus strike price. The dashed curve is for the Black-Scholes model. The solid curve is for our model. Full article ">Figure 2
Put option price versus strike price. The dashed curve is for the Black-Scholes model. The solid curve is for our model. Full article ">Figure 3
Call option price versus strike price, for different values of q. The dashed curve is for <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics> </math>. The Solid Curve is for <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>1.4</mn> </mrow> </semantics> </math>. Full article ">Figure 4
Put option price versus strike price, for different values of q. The dashed curve is for <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics> </math>. The Solid Curve is for <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>1.4</mn> </mrow> </semantics> </math>. Full article ">

19 pages, 8006 KiB

Open AccessArticle

An Operation Reduction Using Fast Computation of an Iteration-Based Simulation Method with Microsimulation-Semi-Symbolic Analysis

by Vladimir Mladenovic, Danijela Milosevic, Miroslav Lutovac, Yigang Cen and Matjaz Debevc

Entropy 2018, 20(1), 62; https://doi.org/10.3390/e20010062 - 18 Jan 2018

Cited by 4 | Viewed by 4414

Abstract

This paper presents a method for shortening the computation time and reducing the number of math operations required in complex calculations for the analysis, simulation, and design of processes and systems. The method is suitable for education and engineering applications. The efficacy of [...] Read more.

This paper presents a method for shortening the computation time and reducing the number of math operations required in complex calculations for the analysis, simulation, and design of processes and systems. The method is suitable for education and engineering applications. The efficacy of the method is illustrated with a case study of a complex wireless communication system. The computer algebra system (CAS) was applied to formulate hypotheses and define the joint probability density function of a certain modulation technique. This innovative method was used to prepare microsimulation-semi-symbolic analyses to fully specify the wireless system. The development of an iteration-based simulation method that provides closed form solutions is presented. Previously, expressions were solved using time-consuming numerical methods. Students can apply this method for performance analysis and to understand data transfer processes. Engineers and researchers may use the method to gain insight into the impact of the parameters necessary to properly transmit and detect information, unlike traditional numerical methods. This research contributes to this field by improving the ability to obtain closed form solutions of the probability density function, outage probability, and considerably improves time efficiency with shortened computation time and reducing the number of calculation operations. Full article

(This article belongs to the Special Issue Symbolic Entropy Analysis and Its Applications)

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13 pages, 2725 KiB

Open AccessArticle

Energetic and Exergetic Analysis of a Transcritical N₂O Refrigeration Cycle with an Expander

by Ze Zhang, Yu Hou and Francis A. Kulacki

Entropy 2018, 20(1), 31; https://doi.org/10.3390/e20010031 - 18 Jan 2018

Cited by 6 | Viewed by 4895

Abstract

Comparative energy and exergy investigations are reported for a transcritical N₂O refrigeration cycle with a throttling valve or with an expander when the gas cooler exit temperature varies from 30 to 55 °C and the evaporating temperature varies from −40 to [...] Read more.

Comparative energy and exergy investigations are reported for a transcritical N₂O refrigeration cycle with a throttling valve or with an expander when the gas cooler exit temperature varies from 30 to 55 °C and the evaporating temperature varies from −40 to 10 °C. The system performance is also compared with that of similar cycles using CO₂. Results show that the N₂O expander cycle exhibits a larger maximum cooling coefficient of performance (COP) and lower optimum discharge pressure than that of the CO₂ expander cycle and N₂O throttling valve cycle. It is found that in the N₂O throttling valve cycle, the irreversibility of the throttling valve is maximum and the exergy losses of the gas cooler and compressor are ordered second and third, respectively. In the N₂O expander cycle, the largest exergy loss occurs in the gas cooler, followed by the compressor and the expander. Compared with the CO₂ expander cycle and N₂O throttling valve cycle, the N₂O expander cycle has the smallest component-specific exergy loss and the highest exergy efficiency at the same operating conditions and at the optimum discharge pressure. It is also proven that the maximum COP and the maximum exergy efficiency cannot be obtained at the same time for the investigated cycles. Full article

(This article belongs to the Special Issue Phenomenological Thermodynamics of Irreversible Processes)

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18 pages, 334 KiB

Open AccessArticle

The Fractality of Polar and Reed–Muller Codes

by Bernhard C. Geiger

Entropy 2018, 20(1), 70; https://doi.org/10.3390/e20010070 - 17 Jan 2018

Cited by 4 | Viewed by 3790

Abstract

The generator matrices of polar codes and Reed–Muller codes are submatrices of the Kronecker product of a lower-triangular binary square matrix. For polar codes, the submatrix is generated by selecting rows according to their Bhattacharyya parameter, which is related to the error probability [...] Read more.

The generator matrices of polar codes and Reed–Muller codes are submatrices of the Kronecker product of a lower-triangular binary square matrix. For polar codes, the submatrix is generated by selecting rows according to their Bhattacharyya parameter, which is related to the error probability of sequential decoding. For Reed–Muller codes, the submatrix is generated by selecting rows according to their Hamming weight. In this work, we investigate the properties of the index sets selecting those rows, in the limit as the blocklength tends to infinity. We compute the Lebesgue measure and the Hausdorff dimension of these sets. We furthermore show that these sets are finely structured and self-similar in a well-defined sense, i.e., they have properties that are common to fractals. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

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15 pages, 7227 KiB

Open AccessArticle

An Entropic Model for the Assessment of Streamwise Velocity Dip in Wide Open Channels

by Domenica Mirauda, Marilena Pannone and Annamaria De Vincenzo

Entropy 2018, 20(1), 69; https://doi.org/10.3390/e20010069 - 17 Jan 2018

Cited by 20 | Viewed by 4148

Abstract

The three-dimensional structure of river flow and the presence of secondary currents, mainly near walls, often cause the maximum cross-sectional velocity to occur below the free surface, which is known as the “dip” phenomenon. The present study proposes a theoretical model derived from [...] Read more.

The three-dimensional structure of river flow and the presence of secondary currents, mainly near walls, often cause the maximum cross-sectional velocity to occur below the free surface, which is known as the “dip” phenomenon. The present study proposes a theoretical model derived from the entropy theory to predict the velocity dip position along with the corresponding velocity value. Field data, collected at three ungauged sections located along the Alzette river in the Grand Duchy of Luxembourg and at three gauged sections located along three large rivers in Basilicata (southern Italy), were used to test its validity. The results show that the model is in good agreement with the experimental measurements and, when compared with other models documented in the literature, yields the least percentage error. Full article

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26 pages, 505 KiB

Open AccessArticle

Multi-Attribute Decision-Making Based on Bonferroni Mean Operators under Cubic Intuitionistic Fuzzy Set Environment

by Gagandeep Kaur and Harish Garg

Entropy 2018, 20(1), 65; https://doi.org/10.3390/e20010065 - 17 Jan 2018

Cited by 112 | Viewed by 5915

Abstract

Cubic intuitionistic fuzzy (CIF) set is the hybrid set which can contain much more information to express an interval-valued intuitionistic fuzzy set and an intuitionistic fuzzy set simultaneously for handling the uncertainties in the data. Unfortunately, there has been no research on the [...] Read more.

Cubic intuitionistic fuzzy (CIF) set is the hybrid set which can contain much more information to express an interval-valued intuitionistic fuzzy set and an intuitionistic fuzzy set simultaneously for handling the uncertainties in the data. Unfortunately, there has been no research on the aggregation operators on CIF sets so far. Since an aggregation operator is an important mathematical tool in decision-making problems, the present paper proposes some new Bonferroni mean and weighted Bonferroni mean averaging operators between the cubic intuitionistic fuzzy numbers for aggregating the different preferences of the decision-maker. Then, we develop a decision-making method based on the proposed operators under the cubic intuitionistic fuzzy environment and illustrated with a numerical example. Finally, a comparison analysis between the proposed and the existing approaches have been performed to illustrate the applicability and feasibility of the developed decision-making method. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

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10 pages, 904 KiB

Open AccessArticle

Changes in the Complexity of Heart Rate Variability with Exercise Training Measured by Multiscale Entropy-Based Measurements

by Frederico Sassoli Fazan, Fernanda Brognara, Rubens Fazan Junior, Luiz Otavio Murta Junior and Luiz Eduardo Virgilio Silva

Entropy 2018, 20(1), 47; https://doi.org/10.3390/e20010047 - 17 Jan 2018

Cited by 18 | Viewed by 5982

Abstract

Quantifying complexity from heart rate variability (HRV) series is a challenging task, and multiscale entropy (MSE), along with its variants, has been demonstrated to be one of the most robust approaches to achieve this goal. Although physical training is known to be beneficial, [...] Read more.

Quantifying complexity from heart rate variability (HRV) series is a challenging task, and multiscale entropy (MSE), along with its variants, has been demonstrated to be one of the most robust approaches to achieve this goal. Although physical training is known to be beneficial, there is little information about the long-term complexity changes induced by the physical conditioning. The present study aimed to quantify the changes in physiological complexity elicited by physical training through multiscale entropy-based complexity measurements. Rats were subject to a protocol of medium intensity training (

n = 13

) or a sedentary protocol (

n = 12

). One-hour HRV series were obtained from all conscious rats five days after the experimental protocol. We estimated MSE, multiscale dispersion entropy (MDE) and multiscale SDiff

_{q}

from HRV series. Multiscale SDiff

_{q}

is a recent approach that accounts for entropy differences between a given time series and its shuffled dynamics. From SDiff

_{q}

, three attributes (q-attributes) were derived, namely SDiff

_{q_{m a x}}

,

q_{m a x}

and

q_{z e r o}

. MSE, MDE and multiscale q-attributes presented similar profiles, except for SDiff

_{q_{m a x}}

.

q_{m a x}

showed significant differences between trained and sedentary groups on Time Scales 6 to 20. Results suggest that physical training increases the system complexity and that multiscale q-attributes provide valuable information about the physiological complexity. Full article

(This article belongs to the Special Issue Evaluation of Systems’ Irregularity and Complexity: Sample Entropy, Its Derivatives, and Their Applications across Scales and Disciplines)

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15 pages, 276 KiB

Open AccessArticle

Analytical Solutions for Multi-Time Scale Fractional Stochastic Differential Equations Driven by Fractional Brownian Motion and Their Applications

by Xiao-Li Ding and Juan J. Nieto

Entropy 2018, 20(1), 63; https://doi.org/10.3390/e20010063 - 16 Jan 2018

Cited by 19 | Viewed by 4600

Abstract

In this paper, we investigate analytical solutions of multi-time scale fractional stochastic differential equations driven by fractional Brownian motions. We firstly decompose homogeneous multi-time scale fractional stochastic differential equations driven by fractional Brownian motions into independent differential subequations, and give their analytical solutions. [...] Read more.

In this paper, we investigate analytical solutions of multi-time scale fractional stochastic differential equations driven by fractional Brownian motions. We firstly decompose homogeneous multi-time scale fractional stochastic differential equations driven by fractional Brownian motions into independent differential subequations, and give their analytical solutions. Then, we use the variation of constant parameters to obtain the solutions of nonhomogeneous multi-time scale fractional stochastic differential equations driven by fractional Brownian motions. Finally, we give three examples to demonstrate the applicability of our obtained results. Full article

(This article belongs to the Special Issue Entropy in Dynamic Systems)

22 pages, 184 KiB

Open AccessEditorial

Acknowledgement to Reviewers of Entropy in 2017

by Entropy Editorial Office

Entropy 2018, 20(1), 66; https://doi.org/10.3390/e20010066 - 15 Jan 2018

Viewed by 5126

Abstract

Peer review is an essential part in the publication process, ensuring that Entropy maintains high quality standards for its published papers.[...] Full article

24 pages, 1686 KiB

Open AccessArticle

Joint Content Recommendation and Delivery in Mobile Wireless Networks with Outage Management

by Yaodong Li, Lingyu Chen, Haibin Shi, Xuemin Hong and Jianghong Shi

Entropy 2018, 20(1), 64; https://doi.org/10.3390/e20010064 - 15 Jan 2018

Cited by 5 | Viewed by 4101

Abstract

Personalized content retrieval service has become a major information service that consumes a large portion of mobile Internet traffic. Joint content recommendation and delivery is a promising design philosophy that could effectively improve the overall user experience with personalized content retrieval services. Existing [...] Read more.

Personalized content retrieval service has become a major information service that consumes a large portion of mobile Internet traffic. Joint content recommendation and delivery is a promising design philosophy that could effectively improve the overall user experience with personalized content retrieval services. Existing research mostly focused on a push-type design paradigm called proactive caching, which, however, has multiple inherent drawbacks such as high device cost and low energy efficiency. This paper proposes a novel, interactive joint content recommendation and delivery system as an alternative to overcome the drawbacks of proactive caching systems. We present several optimal and heuristic algorithms for the proposed system and analyze the system performance in terms of user interest and transmission outage probability. Some theoretical performance bounds of the system are also derived. The effectiveness of the proposed system and algorithms is validated by simulation results. Full article

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15 pages, 464 KiB

Open AccessEditor’s ChoiceArticle

Low Computational Cost for Sample Entropy

by George Manis, Md Aktaruzzaman and Roberto Sassi

Entropy 2018, 20(1), 61; https://doi.org/10.3390/e20010061 - 13 Jan 2018

Cited by 51 | Viewed by 6608

Abstract

Sample Entropy is the most popular definition of entropy and is widely used as a measure of the regularity/complexity of a time series. On the other hand, it is a computationally expensive method which may require a large amount of time when used [...] Read more.

Sample Entropy is the most popular definition of entropy and is widely used as a measure of the regularity/complexity of a time series. On the other hand, it is a computationally expensive method which may require a large amount of time when used in long series or with a large number of signals. The computationally intensive part is the similarity check between points in m dimensional space. In this paper, we propose new algorithms or extend already proposed ones, aiming to compute Sample Entropy quickly. All algorithms return exactly the same value for Sample Entropy, and no approximation techniques are used. We compare and evaluate them using cardiac inter-beat (RR) time series. We investigate three algorithms. The first one is an extension of the

k d

-trees algorithm, customized for Sample Entropy. The second one is an extension of an algorithm initially proposed for Approximate Entropy, again customized for Sample Entropy, but also improved to present even faster results. The last one is a completely new algorithm, presenting the fastest execution times for specific values of m, r, time series length, and signal characteristics. These algorithms are compared with the straightforward implementation, directly resulting from the definition of Sample Entropy, in order to give a clear image of the speedups achieved. All algorithms assume the classical approach to the metric, in which the maximum norm is used. The key idea of the two last suggested algorithms is to avoid unnecessary comparisons by detecting them early. We use the term unnecessary to refer to those comparisons for which we know a priori that they will fail at the similarity check. The number of avoided comparisons is proved to be very large, resulting in an analogous large reduction of execution time, making them the fastest algorithms available today for the computation of Sample Entropy. Full article

(This article belongs to the Special Issue Evaluation of Systems’ Irregularity and Complexity: Sample Entropy, Its Derivatives, and Their Applications across Scales and Disciplines)

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24 pages, 2986 KiB

Open AccessArticle

k-Same-Net: k-Anonymity with Generative Deep Neural Networks for Face Deidentification

by Blaž Meden, Žiga Emeršič, Vitomir Štruc and Peter Peer

Entropy 2018, 20(1), 60; https://doi.org/10.3390/e20010060 - 13 Jan 2018

Cited by 56 | Viewed by 10287

Abstract

Image and video data are today being shared between government entities and other relevant stakeholders on a regular basis and require careful handling of the personal information contained therein. A popular approach to ensure privacy protection in such data is the use of [...] Read more.

Image and video data are today being shared between government entities and other relevant stakeholders on a regular basis and require careful handling of the personal information contained therein. A popular approach to ensure privacy protection in such data is the use of deidentification techniques, which aim at concealing the identity of individuals in the imagery while still preserving certain aspects of the data after deidentification. In this work, we propose a novel approach towards face deidentification, called k-Same-Net, which combines recent Generative Neural Networks (GNNs) with the well-known k-Anonymitymechanism and provides formal guarantees regarding privacy protection on a closed set of identities. Our GNN is able to generate synthetic surrogate face images for deidentification by seamlessly combining features of identities used to train the GNN model. Furthermore, it allows us to control the image-generation process with a small set of appearance-related parameters that can be used to alter specific aspects (e.g., facial expressions, age, gender) of the synthesized surrogate images. We demonstrate the feasibility of k-Same-Net in comprehensive experiments on the XM2VTS and CK+ datasets. We evaluate the efficacy of the proposed approach through reidentification experiments with recent recognition models and compare our results with competing deidentification techniques from the literature. We also present facial expression recognition experiments to demonstrate the utility-preservation capabilities of k-Same-Net. Our experimental results suggest that k-Same-Net is a viable option for facial deidentification that exhibits several desirable characteristics when compared to existing solutions in this area. Full article

(This article belongs to the Special Issue Selected Papers from IWOBI—Entropy-Based Applied Signal Processing)

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18 pages, 1520 KiB

Open AccessArticle

Exergetic Analysis, Optimization and Comparison of LNG Cold Exergy Recovery Systems for Transportation

by Paweł Dorosz, Paweł Wojcieszak and Ziemowit Malecha

Entropy 2018, 20(1), 59; https://doi.org/10.3390/e20010059 - 13 Jan 2018

Cited by 33 | Viewed by 7597

Abstract

LNG (Liquefied Natural Gas) shares in the global energy market is steadily increasing. One possible application of LNG is as a fuel for transportation. Stricter air pollution regulations and emission controls have made the natural gas a promising alternative to liquid petroleum fuels, [...] Read more.

LNG (Liquefied Natural Gas) shares in the global energy market is steadily increasing. One possible application of LNG is as a fuel for transportation. Stricter air pollution regulations and emission controls have made the natural gas a promising alternative to liquid petroleum fuels, especially in the case of heavy transport. However, in most LNG-fueled vehicles, the physical exergy of LNG is destroyed in the regasification process. This paper investigates possible LNG exergy recovery systems for transportation. The analyses focus on “cold energy” recovery systems as the enthalpy of LNG, which may be used as cooling power in air conditioning or refrigeration. Moreover, four exergy recovery systems that use LNG as a low temperature heat sink to produce electric power are analyzed. This includes single-stage and two-stage direct expansion systems, an ORC (Organic Rankine Cycle) system, and a combined system (ORC + direct expansion). The optimization of the above-mentioned LNG power cycles and exergy analyses are also discussed, with the identification of exergy loss in all components. The analyzed systems achieved exergetic efficiencies in the range of

20 %

to

36 %

, which corresponds to a net work in the range of 214 to 380 kJ/kg

_{L N G}

. Full article

(This article belongs to the Special Issue Work Availability and Exergy Analysis)

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21 pages, 1581 KiB

Open AccessArticle

Biological Networks Entropies: Examples in Neural Memory Networks, Genetic Regulation Networks and Social Epidemic Networks

by Jacques Demongeot, Mariem Jelassi, Hana Hazgui, Slimane Ben Miled, Narjes Bellamine Ben Saoud and Carla Taramasco

Entropy 2018, 20(1), 36; https://doi.org/10.3390/e20010036 - 13 Jan 2018

Cited by 6 | Viewed by 5564

Abstract

Networks used in biological applications at different scales (molecule, cell and population) are of different types: neuronal, genetic, and social, but they share the same dynamical concepts, in their continuous differential versions (e.g., non-linear Wilson-Cowan system) as well as in their discrete Boolean [...] Read more.

Networks used in biological applications at different scales (molecule, cell and population) are of different types: neuronal, genetic, and social, but they share the same dynamical concepts, in their continuous differential versions (e.g., non-linear Wilson-Cowan system) as well as in their discrete Boolean versions (e.g., non-linear Hopfield system); in both cases, the notion of interaction graph G(J) associated to its Jacobian matrix J, and also the concepts of frustrated nodes, positive or negative circuits of G(J), kinetic energy, entropy, attractors, structural stability, etc., are relevant and useful for studying the dynamics and the robustness of these systems. We will give some general results available for both continuous and discrete biological networks, and then study some specific applications of three new notions of entropy: (i) attractor entropy, (ii) isochronal entropy and (iii) entropy centrality; in three domains: a neural network involved in the memory evocation, a genetic network responsible of the iron control and a social network accounting for the obesity spread in high school environment. Full article

(This article belongs to the Section Statistical Physics)

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24 pages, 4193 KiB

Open AccessEditor’s ChoiceArticle

Transfer Entropy as a Tool for Hydrodynamic Model Validation

by Alicia Sendrowski, Kazi Sadid, Ehab Meselhe, Wayne Wagner, David Mohrig and Paola Passalacqua

Entropy 2018, 20(1), 58; https://doi.org/10.3390/e20010058 - 12 Jan 2018

Cited by 19 | Viewed by 5921

Abstract

The validation of numerical models is an important component of modeling to ensure reliability of model outputs under prescribed conditions. In river deltas, robust validation of models is paramount given that models are used to forecast land change and to track water, solid, [...] Read more.

The validation of numerical models is an important component of modeling to ensure reliability of model outputs under prescribed conditions. In river deltas, robust validation of models is paramount given that models are used to forecast land change and to track water, solid, and solute transport through the deltaic network. We propose using transfer entropy (TE) to validate model results. TE quantifies the information transferred between variables in terms of strength, timescale, and direction. Using water level data collected in the distributary channels and inter-channel islands of Wax Lake Delta, Louisiana, USA, along with modeled water level data generated for the same locations using Delft3D, we assess how well couplings between external drivers (river discharge, tides, wind) and modeled water levels reproduce the observed data couplings. We perform this operation through time using ten-day windows. Modeled and observed couplings compare well; their differences reflect the spatial parameterization of wind and roughness in the model, which prevents the model from capturing high frequency fluctuations of water level. The model captures couplings better in channels than on islands, suggesting that mechanisms of channel-island connectivity are not fully represented in the model. Overall, TE serves as an additional validation tool to quantify the couplings of the system of interest at multiple spatial and temporal scales. Full article

(This article belongs to the Special Issue Transfer Entropy II)

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15 pages, 664 KiB

Open AccessArticle

Granger Causality and Jensen–Shannon Divergence to Determine Dominant Atrial Area in Atrial Fibrillation

by Raquel Cervigón, Francisco Castells, José Manuel Gómez-Pulido, Julián Pérez-Villacastín and Javier Moreno

Entropy 2018, 20(1), 57; https://doi.org/10.3390/e20010057 - 12 Jan 2018

Cited by 8 | Viewed by 5471

Abstract

Atrial fibrillation (AF) is already the most commonly occurring arrhythmia. Catheter pulmonary vein ablation has emerged as a treatment that is able to make the arrhythmia disappear; nevertheless, recurrence to arrhythmia is very frequent. In this study, it is proposed to perform an [...] Read more.

Atrial fibrillation (AF) is already the most commonly occurring arrhythmia. Catheter pulmonary vein ablation has emerged as a treatment that is able to make the arrhythmia disappear; nevertheless, recurrence to arrhythmia is very frequent. In this study, it is proposed to perform an analysis of the electrical signals recorded from bipolar catheters at three locations, pulmonary veins and the right and left atria, before to and during the ablation procedure. Principal Component Analysis (PCA) was applied to reduce data dimension and Granger causality and divergence techniques were applied to analyse connectivity along the atria, in three main regions: pulmonary veins, left atrium (LA) and right atrium (RA). The results showed that, before the procedure, patients with recurrence in the arrhythmia had greater connectivity between atrial areas. Moreover, during the ablation procedure, in patients with recurrence in the arrhythmial both atria were more connected than in patients that maintained sinus rhythms. These results can be helpful for procedures designing to end AF. Full article

(This article belongs to the Special Issue Entropy in Signal Analysis)

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Figure 1
Diagrammatic representation illustrating Orbiter and Lasso catheters’ distribution of electric poles along both atria and a pulmonary vein. The lower right figure represents the anatomic relation of the four cardiac heart chambers, showing atria and ventricles and one pulmonary vein. Full article ">Figure 2
Four original RA electrogram signals (top) and pre-processing signals using the Botteron pre-processing chain (bottom). Full article ">Figure 3
Five Gaussian-shaped pulses with increasing delays (a) and PC decomposition (b). Full article ">Figure 4
Variance content of the first component vs. delay between activations. Full article ">Figure 5
G-Causality in relation to atrial area in AF recurrent patients (red) and patients that maintain sinus rhythm (blue). Full article ">Figure 6
Jensen–Shannon divergence in relation to atrial area in recurrent AF patients (red) and patients that maintain sinus rhythm (blue). Full article ">Figure 7
Granger causality in relation to all the phases in recurrent AF patients (red) and patients that maintain sinus rhythm (blue). Full article ">Figure 8
Jensen–Shannon Divergence between LA and RA along the for phases in recurrent AF patients (red) and patients in sinus rhythm (blue). Full article ">

17 pages, 296 KiB

Open AccessArticle

Entropy of Iterated Function Systems and Their Relations with Black Holes and Bohr-Like Black Holes Entropies

by Christian Corda, Mehdi FatehiNia, MohammadReza Molaei and Yamin Sayyari

Entropy 2018, 20(1), 56; https://doi.org/10.3390/e20010056 - 12 Jan 2018

Cited by 15 | Viewed by 4460

Abstract

In this paper we consider the metric entropies of the maps of an iterated function system deduced from a black hole which are known the Bekenstein–Hawking entropies and its subleading corrections. More precisely, we consider the recent model of a Bohr-like black hole [...] Read more.

In this paper we consider the metric entropies of the maps of an iterated function system deduced from a black hole which are known the Bekenstein–Hawking entropies and its subleading corrections. More precisely, we consider the recent model of a Bohr-like black hole that has been recently analysed in some papers in the literature, obtaining the intriguing result that the metric entropies of a black hole are created by the metric entropies of the functions, created by the black hole principal quantum numbers, i.e., by the black hole quantum levels. We present a new type of topological entropy for general iterated function systems based on a new kind of the inverse of covers. Then the notion of metric entropy for an Iterated Function System (

I F S

) is considered, and we prove that these definitions for topological entropy of IFS’s are equivalent. It is shown that this kind of topological entropy keeps some properties which are hold by the classic definition of topological entropy for a continuous map. We also consider average entropy as another type of topological entropy for an

I F S

which is based on the topological entropies of its elements and it is also an invariant object under topological conjugacy. The relation between Axiom A and the average entropy is investigated. Full article

(This article belongs to the Special Issue Entropic Aspects Arising from Geometric Descriptions of Physical Phenomena)

20 pages, 29827 KiB

Open AccessArticle

A Sequential Algorithm for Signal Segmentation

by Paulo Hubert, Linilson Padovese and Julio Michael Stern

Entropy 2018, 20(1), 55; https://doi.org/10.3390/e20010055 - 12 Jan 2018

Cited by 10 | Viewed by 7095

Abstract

The problem of event detection in general noisy signals arises in many applications; usually, either a functional form of the event is available, or a previous annotated sample with instances of the event that can be used to train a classification algorithm. There [...] Read more.

The problem of event detection in general noisy signals arises in many applications; usually, either a functional form of the event is available, or a previous annotated sample with instances of the event that can be used to train a classification algorithm. There are situations, however, where neither functional forms nor annotated samples are available; then, it is necessary to apply other strategies to separate and characterize events. In this work, we analyze 15-min samples of an acoustic signal, and are interested in separating sections, or segments, of the signal which are likely to contain significant events. For that, we apply a sequential algorithm with the only assumption that an event alters the energy of the signal. The algorithm is entirely based on Bayesian methods. Full article

(This article belongs to the Special Issue MaxEnt 2017 - The 37th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)

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16 pages, 2660 KiB

Open AccessArticle

Synchronization in Fractional-Order Complex-Valued Delayed Neural Networks

by Weiwei Zhang, Jinde Cao, Dingyuan Chen and Fuad E. Alsaadi

Entropy 2018, 20(1), 54; https://doi.org/10.3390/e20010054 - 12 Jan 2018

Cited by 30 | Viewed by 4234

Abstract

This paper discusses the synchronization of fractional order complex valued neural networks (FOCVNN) at the presence of time delay. Synchronization criterions are achieved through the employment of a linear feedback control and comparison theorem of fractional order linear systems with delay. Feasibility and [...] Read more.

This paper discusses the synchronization of fractional order complex valued neural networks (FOCVNN) at the presence of time delay. Synchronization criterions are achieved through the employment of a linear feedback control and comparison theorem of fractional order linear systems with delay. Feasibility and effectiveness of the proposed system are validated through numerical simulations. Full article

(This article belongs to the Special Issue Research Frontier in Chaos Theory and Complex Networks)

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14 pages, 6244 KiB

Open AccessArticle

Chaotic Dynamics of the Fractional-Love Model with an External Environment

by Linyun Huang and Youngchul Bae

Entropy 2018, 20(1), 53; https://doi.org/10.3390/e20010053 - 12 Jan 2018

Cited by 22 | Viewed by 4520

Abstract

Based on the fractional order of nonlinear system for love model with a periodic function as an external environment, we analyze the characteristics of the chaotic dynamic. We analyze the relationship between the chaotic dynamic of the fractional order love model with an [...] Read more.

Based on the fractional order of nonlinear system for love model with a periodic function as an external environment, we analyze the characteristics of the chaotic dynamic. We analyze the relationship between the chaotic dynamic of the fractional order love model with an external environment and the value of fractional order (α, β) when the parameters are fixed. Meanwhile, we also study the relationship between the chaotic dynamic of the fractional order love model with an external environment and the parameters (a, b, c, d) when the fractional order of the system is fixed. When the parameters of fractional order love model are fixed, the fractional order (α, β) of fractional order love model system exhibit segmented chaotic states with the different fractional orders of the system. When the fractional order (α = β) of the system is fixed, the system shows the periodic state and the chaotic state as the parameter is changing as a result. Full article

(This article belongs to the Special Issue Theoretical Aspect of Nonlinear Statistical Physics)

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