Entropy Curves Research Articles

Quantitative Fault Diagnosis of Planetary Gearboxes Based on Improved Symbolic Dynamic Entropy

To realize a quantitative diagnosis of faults in the planetary gearboxes of wind turbines by processing the complex frequency signals of the planetary gear boxes and avoiding the aliasing problem of the resulting frequencies, this paper proposes a diagnosis method based on improved variational mode decomposition (IVMD) and average multi-scale double symbolic dynamic entropy (AMDSDE). Moreover, an IVMD algorithm based on multi-scale permutation entropy is introduced to reduce noise interference and realize signal demodulation. Considering the effects of complex transfer paths and the correlation between current and adjacent state modes, AMDSDE is proposed. Each fault size is obtained based on the entropy curve, and the AMDSDE of unknown faults is calculated. To verify the accuracy of the proposed method, simulations and experimental signals are processed. The quantitative diagnosis of the planetary gearboxes of wind turbines is realized, providing a reliable basis for evaluating the health status of planetary gearboxes.

End of the world brane meets TT¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ T\\overline{T} $$\\end{document}

End of the world branes in AdS have been recently used to study problems deeply connected to quantum gravity, such as black hole evaporation and holographic cosmology. With non-critical tension and Neumann boundary condition, the end of the world brane often represents part of the degrees of freedom in AdS gravity and geometrically it is only part of the entire boundary. On the other hand, holographic TT¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ T\\overline{T} $$\\end{document} deformation can also give a boundary as a cutoff surface for AdS gravity. In this paper we consider AdS gravity with both the end of the world boundary and the cutoff boundary. Using partial reduction we obtain a brane world gravity glued to a TT¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ T\\overline{T} $$\\end{document} deformed bath. We compute both entanglement entropy and Page curve, and find agreement between the holographic results and island formula results.

A two-dimensional harmonic oscillator confined in a circle in the presence of a constant electric field: an informational approach

In this work, we study an electron subjected to a harmonic oscillator potential confined in a circle of radius r0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r_0$$\\end{document} and in the presence of a constant electric field. We obtain energies and eigenfunctions for three different confinement radii as a function of the electric field strength. We have used the linear variational method by constructing the trial function as a linear combination of two-dimensional confined harmonic oscillator wave functions. We calculate the radial standard deviation as a measure of the dispersion of the probability density. We also computed the Shannon entropy and Fisher information, in configuration and momentum spaces, as localization-delocalization measures for three different confinement radii and as a function of the electric field strength. We find that Shannon entropy and Fisher information are more reliable than variance in determining electron location. The behaviour of Shannon entropy and Fisher information curves is shown to depend on each specific state under study.Graphical abstract

Wavelet entropy and complexity–entropy curves approach for energy commodity price predictability amid the transition to alternative energy sources

In recent years, energy commodities have emerged as pivotal and widely debated subjects, driven by their profound influence on the global economy and their intricate interconnections. Moreover, the challenges stemming from the predictability of energy commodity prices have become a prominent and intensifying focus of discussion. To this aim, in this paper, we employ wavelet analysis with an entropy approach to investigate and evaluate fluctuations, low-frequency events, and rare events in energy commodity prices and the consequent predictability of such time series. In particular, wavelet analysis can differentiate high-frequency from low-frequency movements in a time series, and entropy is a valuable mathematical tool used to evaluate the state of disorder, randomness, or uncertainty in a time series. Specifically, we employ Rényi Entropy instead of the Shannon entropy because it allows for enhanced consideration of low-frequency events and spikes in a time series. Therefore, to analyze the predictability of a series, we use Wavelet Rényi Entropy (WRE) and the Rényi complexity–entropy curves by combining the wavelet transform with Rényi entropy. Finally, we apply our analysis to real financial data, including energy commodities and indices that can describe the transition to alternative energy resources.

Binary AdS black holes coupled to a bath in Type IIB

We construct Type IIB string theory setups which, via double holography, realize two gravitational systems in separate AdS spaces which interact with each other and with a non-gravitational bath. We employ top-down string theory solutions with concrete field theory duals in the form of 4d \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}$$\\end{document} = 4 SYM BCFTs and a first-principles notion of double holography. The setups are used to realize pairs of ‘near’ and ‘far’ black holes from the perspective of the bath, which exchange Hawking radiation with each other and radiate into the bath. We identify three phases for the entropy in the bath characterized as no island, partial island and full island, and discuss the entropy curves. The setups differ from the black hole binaries observed in gravitational wave experiments but may capture certain aspects.

On the atypical solutions of the symmetric binary perceptron

We study the random binary symmetric perceptron problem, focusing on the behavior of rare high-margin solutions. While most solutions are isolated, we demonstrate that these rare solutions are part of clusters of extensive entropy, heuristically corresponding to non-trivial fixed points of an approximate message-passing algorithm. We enumerate these clusters via a local entropy, defined as a Franz–Parisi potential, which we rigorously evaluate using the first and second moment methods in the limit of a small constraint density α (corresponding to vanishing margin κ ) under a certain assumption on the concentration of the entropy. This examination unveils several intriguing phenomena: (i) we demonstrate that these clusters have an entropic barrier in the sense that the entropy as a function of the distance from the reference high-margin solution is non-monotone when κ⩽1.429−α/log⁡α , while it is monotone otherwise, and that they have an energetic barrier in the sense that there are no solutions at an intermediate distance from the reference solution when κ⩽1.239−α/log⁡α . The critical scaling of the margin κ in −α/log⁡α corresponds to the one obtained from the earlier work of Gamarnik et al (2022 (arXiv:2203.15667)) for the overlap-gap property, a phenomenon known to present a barrier to certain efficient algorithms. (ii) We establish using the replica method that the complexity (the logarithm of the number of clusters of such solutions) versus entropy (the logarithm of the number of solutions in the clusters) curves are partly non-concave and correspond to very large values of the Parisi parameter, with the equilibrium being reached when the Parisi parameter diverges.

Unitary (semi)causal quantum-circuit representation of black hole evaporation

A general structure of unitary evolution (evaporation) of the black hole, respecting causality imposed by the event horizon (semicausality), has been derived and presented in the language of quantum circuits. The resulting consequences for the evolution of the corresponding entanglement entropy and the entropy curve have been determined. As an illustration of the general scheme, two families of qubit toy models have been discussed: tensor product models and controlled non-product models.

Hysteresis-free reversible magnetocaloric effect and critical behavior of Ni39.56Fe7.27Co2.97Mn40.65Sn9.55 full heusler alloy

Utilizing the magnetocaloric effect (MCE), magnetic refrigeration technology offers an advantage over traditional gas-based refrigerators, which produce harmful gases. Herein, we study the MCE and critical magnetic behavior of Fe and Co substituted Ni39.56Fe7.27Co2.97Mn40.65Sn9.55 Heusler alloy near the paramagnetic (PM) to ferromagnetic (FM) transition temperature. With Fm3̅m space group, the present sample crystallizes in cubic L21 structure. Arrott plots and hysteresis-free continuous phase transition confirm a second-order transition from FM to PM. Again, this second-order transition is verified by collapsing scaled entropy curves for various fields into a single universal curve. The critical exponent values obtained using several analytical techniques, like the Kouvel-Fisher method, modified Arrott plot, and critical isotherm, correspond to the mean-field theory universality class having long-range ordering. The above findings suggest our sample to be a suitable magnetic refrigerant by overcoming the difficulty of obtaining a reversible and high value of magnetic entropy change along with substantial relative cooling power across the second-order phase transition.

Assessing the impact of gait speed on gait stability using multi-scale entropy fused with plantar pressure signals.

Introduction: Walking speed can affect gait stability and increase the risk of falling. Methods: In this study, we design a device to measure the distribution of the plantar pressure to investigate the impact of the walking speed on the stability of the human gait and movements of the body. We fused the entropy acquired at multiple scales with signals of the plantar pressure to evaluate the effects of the walking speed on the stability of the human gait. We simultaneously collected data on the motion-induced pressure from eight plantar regions to obtain the fused regional pressure. To verify their accuracy, we obtained data on the plantar pressure during walking by using the force table of the Qualisys system. We then extracted the peak points and intervals of the human stride from pressure signals fused over three regions, and analyzed the mechanics of their regional fusion by using the regional amplitude-pressure ratio to obtain the distribution of the plantar pressure at an asynchronous walking speed. Furthermore, we introduced multi-scale entropy to quantify the complexity of the gait and evaluate its stability at different walking speeds. Results: The results of experiments showed that increasing the speed from 2 to 6km/h decreased the stability of the gait, with a 26.7% increase in the amplitude of pressure in the region of the forefoot. The hindfoot and forefoot regions were subjected to the minimal pressure at a speed of 2km/h, while the most consistent stress was observed in regions of the forefoot, midfoot, and hindfoot. Moreover, the curve of entropy at a speed of 2km/h exhibited a slow decline at a small scale and high stability at a large scale. Discussion: The multi-scale entropy increased the variation in the stability of the synchronous velocity of walking compared with the sample entropy and the analysis of regional fusion mechanics. Multi-scale entropy can thus be used to qualitatively assess the relationship between the speed and stability of the gait, and to identify the most stable gait speed that can ensure gait stability and posture control.

A new approach of image contrast enhancement based on entropy curve

A new approach of image contrast enhancement based on entropy curve

Mathematical diversity of parts for a continuous distribution

The current paper is part of a series exploring how to link diversity measures (e.g., Gini-Simpson index, Shannon entropy, Hill numbers) to a distribution’s original shape and to compare parts of a distribution, in terms of diversity, with the whole. This linkage is crucial to understanding the exact relationship between the density of an original probability distribution, denoted by p(x), and the diversity D in non-uniform distributions, both within parts of a distribution and the whole. Empirically, our results are an important advance since we can compare various parts of a distribution, noting that systems found in contemporary data often have unequal distributions that possess multiple diversity types and have unknown and changing frequencies at different scales (e.g. income, economic complexity ratings, rankings, etc.). To date, we have proven our results for discrete distributions. Our focus here is continuous distributions. In both instances, we do so by linking case-based entropy, a diversity approach we developed, to a probability distribution’s shape for continuous distributions. This allows us to demonstrate that the original probability distribution g 1, the case-based entropy curve g 2, and the slope of diversity g 3 (c (a,x) versus the c(a,x)*lnA(a,x) curve) are one-to-one (or injective). Put simply, a change in the probability distribution, g 1, leads to variations in the curves for g 2 and g 3. Consequently, any alteration in the permutation of the initial probability distribution, which results in a different form, will distinctly define the graphs g 2 and g3 . By demonstrating the injective property of our method for continuous distributions, we introduce a unique technique to gauge the level of uniformity as indicated by D/c. Furthermore, we present a distinct method to calculate D/c for different forms of the original continuous distribution, enabling comparison of various distributions and their components.

Generalized weighted permutation entropy analysis of satellite hot-pixel time series in Brazilian biomes

In this work we analyze hot-pixel satellite data, proxy for natural and anthropogenic vegetation and forest fires, using recently proposed method Generalized weighted permutation entropy (GWPE). The current analysis of data from 2002 to 2022 for six Brazilian biomes sheds new light on differences among the biomes in terms of ordinal pattern entropy, for both small and large fluctuations. It is found that for all biomes large fluctuations in fire temporal series show higher complexity level than small fluctuations, small fluctuations being more predictable as indicated by lower entropy values. Moreover, for fire sensitive biomes Amazon and Atlantic Forest the three dimensional complexity entropy curves in the complexity-entropy-scale causality box (CESCB) are rather similar (indicating similar dynamics for both small and large fluctuations), as well as for fire dependent biomes Pantanal and Pampa, while fire independent Caatinga biome shows different, distinct behavior. Finally, the similarity in lower parts of the CESCB curves (corresponding to small fluctuations) between Amazon and Cerrado reflects the changes of fire dynamics due to human activities.

Enhanced magnetocaloric effect accompanying successive magnetic transitions in TbMn2Si2-xGex compounds

The lack of magnetic refrigeration (MR) materials with high magnetocaloric effect (MCE) and large relative cooling power (RCP) in the temperature range required for hydrogen liquefaction (20 K–77 K) is a bottleneck for practical applications of MR cooling systems. The present investigation of TbMn2Si2-xGex compounds (x = 0.1, 0.2) by variable temperature neutron and synchrotron X-ray diffraction, magnetization and heat capacity measurements, establish that substitution of Si with Ge in TbMn2Si2 leads to a significant enlargement of the unit cell and modification of the magnetic properties. Two consecutive ferromagnetic first-order transitions occur below 77 K with the third transition from paramagnetism to a collinear antiferromagnetic state being determined around 500 K. The resultant plateau-like MCE with large RCP below 77 K in these designed compounds offers scope for application for hydrogen liquefaction. Detailed neutron investigation confirm that four magnetic states exist within the temperature range 5 K to 500 K, with two successive first-order magnetic transitions below 77 K responsible for the large MCE. Our specific heat studies provide evidence of strong contributions from the nuclear specific heat and the corresponding nuclear specific heat coefficients of A = 430 ± 50 mJ mol−1 K and A = 418 ± 60 mJ mol−1 K have been determined for TbMn2Si2-xGex with x = 0.1 and x = 0.2, respectively. The overlapping entropy curves near these successive transitions lead to a plateau-like magnetothermal effect as well as a large reversible MCE for both samples (e.g. ΔSMmax = 14.0 J/kg K and ΔTmax = 7.6 K; RCP = 379 J/kg for TbMn2Si1.9Ge0.1 for an applied field of 5 T) indicating that the material can operate over a wide temperature range – particularly for hydrogen liquefaction.

Hydrological complexity analysis using multiscale entropy: Methodological explorations and insights

Multiscale entropy is a popular and valuable methodology for analyzing hydrological complexity. However, some errors and limitations remain in its use and interpretation. This study identified three key issues, incorrect mapping of entropy values with frequency signals, an insufficient time scale range, and an unclear relationship between features in the multiscale entropy curve and hydrological interpretation. To address these issues, we focused on the sediment concentration and runoff in the Yanhe Watershed and conducted in-depth explorations using refined composite multiscale entropy (RCMFE) and three other methods, power spectrum density analysis, wavelet transform, and surrogate analysis.The results indicate that the entropy at each scale is influenced by signals with time scales equal to or greater than its own scale and that its variation is controlled by the overall signal dispersion. The RCMFE method demonstrated suitability for the complexity analysis of sediment and runoff sequences with short-length data. To effectively capture hydrological complexity, attention to the scale range issue is crucial, ensuring that the richness of complex features at larger scales in multiscale entropy curves is not overlooked. Interpretation of the local minima and maxima in the multiscale entropy curve is highlighted. Local minima signify high periodic patterns and predictability, whereas the first peak, typically occurring at approximately 2–3 months, is a valuable indicator of the overall complexity of the hydrological series. Linear characteristics were found to contribute significantly to the hydrological complexity, with stronger linear correlations resulting in an overall increase, lower local minima, and increased volatility in the multiscale entropy curve. The notable reduction in the hydrological complexity of the Yanhe Watershed calls for ecological sustainability and sustainable soil and water conservation efforts.

A quantitative information measure applied to texture perception attributes during mastication.

We have calculated an entropy or information measure of previously reported experimentally determined temporal dominance of sensations (TDS) data of texture attributes for two sets of emulsion filled gels throughout the mastication cycle. The samples were emulsion filled gels and two-layered emulsion filled gels. We find that the entropy measure follows an average curve, which is different for each set. The specifics of the entropy curve may serve as a fingerprint for the perception of a specific food sample.

Itinerant ferromagnetism and room temperature giant magnetocaloric effect in (Mn0.6Fe0.4) (Ni0.5Co0.5) Si shape memory alloy

In this article, the feasibility of obtaining a magneto-structural coupling over a wide working temperature range of about 127 K (from 206.5 K to 333.5 K) with negligible thermal hysteresis in combination with the structural, microstructural, magnetic, and magnetocaloric properties of a 50 % Co-doped (at the Ni site) polycrystalline (Mn0.6Fe0.4) (Ni0.5Co0.5) Si (MFNCS-0.5) magnetic shape memory alloy (MSMA) has been thoroughly investigated. Along with leaf-like microstructure, at room temperature, the system shows a mixed phase of TiNiSi type low-temperature high-magnetic (ferromagnetic) orthorhombic martensite and Ni2In type high-temperature low-magnetic (paramagnetic) hexagonal austenite, with the dominance of the second one. To estimate the magnetocaloric effect (MCE) across the phase transition, the isothermal magnetic measurement technique has been employed. The merging of heating and cooling iso-field thermo-magnetization (M-T) curves, the presence of negligible thermal hysteresis in the MH curves, universal magnetic entropy curves, a proposed phenomenological model (across the Curie temperature), and the positive slopes of the Arrott plot (Banerjee criterion) show that the first-order structural and magnetic transition are coupled along with a second-order ferromagnetic to paramagnetic (FM → PM) phase transition (SOPT). By extrapolating the Arrott plot to the H/M = 0 axis the spontaneous magnetization value very close to 0 K, Ms(0), is found to be 71.92 emu/g (∼1.83 µB/f.u.). The linear dependency of MS2T on T2 below SOPT and the RWR (Rhodes–Wolhfarth ratio) = 4.29 > 1 indicate the long-range itinerant ferromagnetism in this alloy. A giant magnetic entropy change (ΔSm) of 25.77 J kg−1 K−1 and a refrigerant capacity (RC) of 472.83 J kg−1 are obtained at 6 T field. This observed magnetic refrigerant's key characteristics, including a giant magnetocaloric effect (MCE) near room temperature (RT), negligible thermal hysteresis loss, a wide operating temperature range of 127 K, and the use of cost-effective, non-toxic constituent elements, make it a suitable and very promising member for applications in solid-state refrigeration technology near RT.

A new approach of contrast enhancement for Medical Images based on entropy curve

X-ray, MRI, and CT scan images etc, are widely used for the detection of various prevalent diseases. The efficient identification of the disease depends very much on the quality of the image. Sometimes due to environmental conditions such as insufficient and nonuniform illumination, shacking and movement of objects or the imaging device, and some other fault in the device, the resultant image may have low contrast and the presence of noise or artifacts. In such situation, a contrast enhancement algorithm that can efficiently improve the contrast without increasing noise and artifacts can play an important role. In this article, we propose a simple yet efficient method for these types of images by using the image’s entropy curve and homomorphic filtering. It enhances the contrast of medical images without increasing the content of noise and artifacts. Also, the performance of the proposed method is evaluated by a complete qualitative and quantitative analysis.

Power unit Burr-XII distribution: Statistical inference with applications

This article introduces a new statistical distribution called the power unit Burr-XII distribution, which is an extension of the Burr XII distribution obtained through a power transform. This paper investigates several statistical properties of this new model, including mode, moments, moment generating and characteristic functions, entropy, and Bonferroni and Lorenz curves. In addition, various classical estimators for the parameters of the power unit Burr-XII distribution are evaluated, including maximum likelihood estimates, ordinary least squares estimates, maximum product of interval estimates, Anderson–Darling, right and left tail Anderson–Darling, and Cramer–von Mises estimators. This study provides tables and graphs to illustrate the performance of these methods under different scenarios and sample sizes. The results of the proposed distribution in different fields of application are shown using the mortality rates of COVID-19 patients in Spain and flood data from 20 observations. As a result of both applications, it was observed that the proposed distribution gave better results compared to other well-known distributions, such as the beta distribution, transformed gamma distribution, log-weighted power distribution, transmuted power distribution, Kumaraswamy distribution, Topp–Leone distribution, exponentiated reduce Kies distribution, and exponentiated Topp–Leone distribution.

On the comparison of diversity of parts of a distribution

The literature on diversity measures, regardless of the metric used (e.g., Gini-Simpson index, Shannon entropy) has a notable gap: not much has been done to connect these measures back to the shape of the original distribution, or to use them to compare the diversity of parts of a given distribution and their relationship to the diversity of the whole distribution. As such, the precise quantification of the relationship between the probability of each type p i and the diversity D in non-uniform distributions, both among parts of a distribution as well as the whole, remains unresolved. This is particularly true for Hill numbers, despite their usefulness as ‘effective numbers’. This gap is problematic as most real-world systems (e.g., income distributions, economic complexity indices, rankings, ecological systems) have unequal distributions, varying frequencies, and comprise multiple diversity types with unknown frequencies that can change. To address this issue, we connect case-based entropy, an approach to diversity we developed, to the shape of a probability distribution; allowing us to show that the original probability distribution g 1, the case-based entropy curve g 2 and the c {1,k} versus the curve g 3, which we call the slope of diversity, are one-to-one (or injective), i.e., a different probability distribution g 1 gives a different curve for g 2 and g 3. Hence, a different permutation of the original probability distribution g 1(that leads to a different shape) will uniquely determine the graphs g 2 and g 3. By proving the injective nature of our approach, we will have established a unique way to measure the degree of uniformity of parts as measured by D P /c P for a given part P of the original probability distribution, and also have shown a unique way to compute the D P /c P for various shapes of the original distribution and (in terms of comparison) for different curves.

Formation of trapped meta-stable liquid–vapour interfaces in polar liquids in presence of excess gas-like molecules: Anomalous heat capacities and emergence of microscopic bubbles

We have elucidated the wetting behaviour of a polar-liquid in contact of an attractive planar surface. The formation of meta-stable liquid column-like structures, separated by vapour phase molecules, forming liquid–vapor interfaces, has been investigated in confined planar geometry. We have used lattice based Monte Carlo simulation for both canonical and grand-canonical systems. The presence of excess gas-like molecules has been reported to alter partial wetting conditions due to excess pressure in canonical ensemble. The system is driven towards formation of microscopic liquid–vapor interfaces, and the distinct liquid–vapor interface forming bridge-like structures disappear at high temperatures. The existence of meta-stable interfaces and meso- and nano-scale liquid structures under confinement appears to be a direct manifestation of the restricted movement of the molecules. We conclude that associated irreversibility due to stacking of atoms in extremely narrow geometries results into formation of meta-stable equilibrium structures. The negative heat capacities reduce at lower temperatures. Further it also depends on the extent of vapor phase as well as presence of gas-like molecules. Adsorption isotherm for liquid molecules adsorbed at the first layer of the attractive surface confirms that the negative heat capacities are rooted into the formation of trapped meta-stable states due to complete wetting of the surface due to adsorption of liquid molecules at first (few) layers of the surface. Due to the inaccessibility of the microstates, the system is trapped into a meta-stable or quasi-equilibrium state. The final structure is an outcome of the reconciliation between molecular interactions, interfacial energies and the extent of physical confinement. The micrographs and density profiles recorded at different temperatures T = 1.0, 1.2, 1.4 and 1.6, phase-coexistence, specific heat capacity and configurational entropy curves confirm for the same. The computational model for our case study fully explores the origin of the interfaces and successfully accounts definite reasons for the associated negative heat capacities, which has been riddling scientist for many decades.