Cauchy Model Research Articles

Energy absorption characteristics and auxetic effect of novel elliptic-arc re-entrant honeycomb structures

The novel elliptical-arc re-entrant honeycomb (ERH) is designed by substituting the straight inclined struts within the re-entrant honeycomb with elliptical-arc struts. To assess its performance effectively, the study established the 3D equivalent Cauchy model (3D-ECM) and 2D equivalent Kirchhoff–Love model (2D-EKM) using the variational asymptotic method. The equivalent properties derived from the unit-cell constitutive model were integrated into the equivalent models for macroscopic analysis. Through 3D printing experiments and numerical simulations, the model’s accuracy in predicting the compression behaviors and auxetic effects of various uni- and multi-cellular ERHs under uniaxial compression, as well as the three-point bending behaviors of ERH panels were confirmed. In addition, this model substantially simplifies the modeling process, leading to a 8-fold increase in computational efficiency. Parametric analyses demonstrated that the ERH structure can uphold a beneficial auxetic effect while achieving lightweight and high strength characteristics when the axial ratio of the ellipse equals 1.25. Furthermore, ERH structures outperform arc-shaped re-entrant and regular re-entrant honeycombs in energy absorption and specific energy absorption capacity. These findings offer valuable insights for the preliminary design and optimization of ERH structures.

Physical properties of La:ZnO thin films prepared at different thicknesses using spray pyrolysis technique

Herein we investigate the impact of film thickness on the physical properties of Lanthanum (La) doped ZnO thin films. The films were fabricated using the spray pyrolysis technique with a consistent La content of 5 weight (wt) % in the initial solution. X-ray diffraction analysis indicated the presence of a hexagonal ZnO phase with preferred orientation along the (002) direction and no other phases were detected. The crystallite sizes were calculated using the Halder-Wagner equation, with a maximum size of 16.1 nm observed for a film thickness of 106 nm. Field-emission scanning electron microscopy (FE-SEM) images revealed the formation of a continuous film with an average grain size that increased as the thickness of the film increased. The grain size ranged from 74.5 to 136 nm as the film thickness varied from 106 to 426 nm. Films with lower thicknesses up to 196 nm exhibited two band gaps at approximately 3.2 and 4 eV, while films with higher thicknesses displayed a single band gap around 3.2 eV. The refractive index dispersion for all films was modeled using the Cauchy model, with parameters showing high dependence on the thickness values.The refractive index at high frequency, as calculated using the Cauchy model, was observed to decrease with increasing film thickness, ranging from 1.87 at 106nm to 1.63 at 426nm. Similar values were obtained by fitting the optical refractive index data with the Wemple-DiDomenico relation. Additionally, the UV sensing performance of the films was evaluated against UV light of a single wavelength (365 nm) at applied voltages of 10, 20, and 30V. The rise and decay times were measured, with the film thickness of 426 nm exhibiting the shortest rise and decay times at a specific applied voltage.

Theoretical Model for Determining the Thickness and Optical Constants of Transparent Conducting Oxide Thin Films From the Measured Reflectance Spectra

ABSTRACTThickness and optical constants of transparent conducting oxide (TCO) of metal oxide thin films have been commonly estimated by invoking theoretical models based on measured optical reflectance or transmittance. One of such widely employed models is reflectance‐based Kushev model. However, the relatively higher value of the absorption coefficient at the absorption edge of TCOs of metal oxide thin films results in a large deviation of their thickness values estimated by this model. To address the underlying problems, in the present work, we have improved the Kushev model. Specifically, for minimizing the error in the estimated thickness values from the proposed model, we have employed t test for discarding outliers of a set of thickness values of a given thin film, which were calculated from nearby extrema of a reflectance curve. Then the thickness calculated from model was validated by comparing the value of ZnO thin transparent films extracted from ellipsometric measurement. Our model shows excellent agreement with the thickness values of ZnO films from ellipsometric measurement with only 1.2% deviation. Furthermore, observed values were also compared with the values from already established Cauchy model. The perfectly matching of the simulated reflectance spectra from the present model with the measured reflectance and excellent agreement of thickness values derived from the ellipsometric measurements with the values extracted from the model may infer us that the present model is suitable for fairly accurately estimating film thicknesses of TCO thin films.

Thermal distribution and viscous heating of electromagnetic radiative Eyring–Powell fluid with slippery wall conditions

The research examines the intricate between Eyring-Powell microfluidic heat transfer, electromagnetic radiation and viscous heating in a Riga slippery device as applied in heat exchangers, cooling systems, biomedical devices, energy generation, polymer processing, and others. The viscoelastic property of the fluid is characterized by the Eyring-Powell Cauchy fluid model with shear-thickening and shear-thinning phenomena that are useful in several heat transport processes and thermal management. The developed governing model is taken from the constitutive relations and conservation principles and solved via an adaptive partition weighted residual method. The essential fluid term sensitivities are systematically investigated on the flow and thermal distribution characteristics. An appropriate validation and comparison of results is done and found to agree quantitatively, this confirmed the correctness of the presented outcomes. The results reveal the significant impact of the thermofluidic terms interaction on the viscous flowing fluid and thermal behaviour. As seen, slip conditions momentously increase the flow rate gradients for about 1.7% to 2.4% close to the boundary wall and consequently raise the microfluidic thermal propagation rates with 3.7% non-Newtonian fluid and thickness of the boundary layer. Moreover, the thermal gradient and distribution complexities are modulated within the fluid regime due to radiation and Lorentz force.

Innovative synthesis, structural characteristics, linear and nonlinear optical properties, and optoelectric parameters of newly developed A2ZnGeO4 (A = K, Li) thin films.

The synthesis of high-quality thin films through spin coating deposition on meticulously cleaned glass substrates is presented. Optical band gaps E g of both samples using the Kubelka-Munk function are determined. The data analysis uncovers the presence of optical allowed direct transition for A2ZnGeO4 (A = K, Li). Spectroscopic ellipsometry measurements on A2ZnGeO4 (A = K, Li) thin films and an analysis of their optical properties using the Cauchy model are presented. Furthermore, the increase of the thickness of the thin film results in improvements in their optoelectrical parameters, such as electrical conductivity, optical mobility, and optical conductivity. Using the Kubodera and Kobayashi comparative model, the third order nonlinear susceptibility (χ (3)) was estimated based on the compounds' high linear absorption of the generated third harmonic wavelength (355 nm). This paper presents remarkable NLO results that reveal potential uses in optoelectronics and photonics.

Remaining useful life prediction of rolling bearings based on performance evaluation and multifractional generalized Cauchy model with adaptive drift

The proposed Remaining Useful Life (RUL) prediction framework utilizes several advanced techniques to accurately estimate the remaining service life of rolling bearings. The framework includes early failure assessment, adaptive failure threshold (FT) determination, and a multifractional generalized Cauchy model (MfGC). The early failure assessment is enabled by establishing early FTs and health indicator (HI) curves generated by the Mahalanobis distance cumulative sum (MD-CUSUM) technique. The proposed dynamic fault threshold update method uses the BOX-COX transformation and Chebyshev inequality to determine confidence intervals for evaluating the fault threshold time. The multifractional nature of the MfGC process is characterized by independent, time-varying Hurst indices and fractional dimensions, and the long-range dependence (LRD) characteristics and stochasticity of the process are explained by the diffusion terms generated from the MfGC differential time series. The MfGC model with adaptive drift is constructed for various degenerate trajectories, and a method for estimating the model’s parameters is proposed. The effectiveness of the proposed RUL prediction method is demonstrated using the XJTU-SY bearing dataset.

3D equivalent Cauchy model for serrated re-entrant auxetic honeycombs based on variational asymptotic method

The presence of serrated ligaments in the re-entrant auxetic honeycomb (abbreviated as serrated RAH) effectively enhances the stiffness while maintaining auxeticity in the re-entrant direction. To investigate its auxetic characteristics, a 3D equivalent Cauchy model (3D-ECM) was established using the variational asymptotic method. The unique aspect is that the engineering constants are obtained by homogenizing the representative unit-cell and then utilized in the 3D-ECM for global analysis. Subsequently, the resulting global responses are fed into the recovery relationships for localized field analysis. The accuracy of the 3D-ECM and the recovered local field distributions were confirmed by comparing them with the results obtained from the three-dimensional FE model (3D-FEM) and the experimental data collected from the 3D-printed specimen. The effects of the geometric parameters on the negative Poisson’s ratio and engineering constants were in-depth discussed. Specifically, an optimal value of 0.5 for the re-entrant ratio and d-value (corresponding to the re-entrant angle of 90 degree) yield maximum auxeticity and comparatively higher elastic moduli. In addition, the 3D-ECM greatly enhanced the computational efficiency without compromising the accuracy in predicting global behaviors and local field distributions. The unit-cell tailorability in the proposed model offers valuable guidance for optimizing the design of serrated RAHs.

Spatial heterogeneity of long-range dependence and self-similarity of global sea surface chlorophyll concentration with their environmental impact factors analysis

Understanding the long-range dependence and self-similarity of global sea surface chlorophyll concentration (SSCC) will enrich its characteristics description and analysis with global change patterns. The satellite SSCC products were collected from the European Space Agency during the period from 29 July 1998 to 31 December2020. After resampling the SSCC products into the spatial resolution of 1°, the missing values were interpolated by Bayesian maximum entropy with mean absolute error of cross validation equaling to 0.1295 mg/m3. Generalized Cauchy model was employed to quantitatively determine the long-range dependence and self-similarity of SSCC at a global scale by using the Hurst parameter and fractal dimension. Good fitted results were achieved with an averaged R2 of 0.9141 and a standard deviation of 0.0518 across the 32,281 spatial locations of the entire ocean; the averaged values of Hurst parameter and fractal dimension were 0.8667 and 1.2506, respectively, suggesting strong long-range dependence and weak self-similarity of SSCC in the entire oceans. Univariate and multivariate generalized addictive models (GAM) were introduced to depict the influence of sea surface height anomaly, sea surface salinity, sea surface temperature and sea surface wind on the Hurst parameter and fractal dimension of SSCC; and smaller mean absolute error were achieved for the GAM of Hurst parameter than that of fractal dimension. Sea surface height anomaly showed the strongest influence for the Hurst parameter than the other three factors, and sea surface wind depicted similar influence; the sea surface temperature owned opposite influence on Hurst parameter compared to sea surface salinity.

Space–Time Variations in the Long-Range Dependence of Sea Surface Chlorophyll in the East China Sea and the South China Sea

Gaining insights into the space–time variations in the long-range dependence of sea surface chlorophyll is crucial for the early detection of environmental issues in oceans. To this end, 12 locations were selected along the Yangtze River and Pearl River estuaries, varying in distances from the Chinese coastline. Daily satellite-observed sea surface chlorophyll concentration data at these 12 locations were collected from the Copernicus Marine Service website, spanning from December 1997 to November 2023. The main objective of the current study is to introduce a multi-fractional generalized Cauchy model for calculating the values of Hurst exponents and quantitatively assessing the long-range dependence strength of sea surface chlorophyll at different spatial locations and time instants during the study period. Furthermore, ANOVA was utilized to detect the differences of calculated Hurst exponent values among the locations during various months and seasons. From a spatial perspective, the findings reveal a significantly stronger long-range dependence of sea surface chlorophyll in offshore regions compared to nearshore areas, with Hurst exponent values > 0.5 versus <0.5. It is noteworthy that the values of Hurst exponents at each location exhibit significant differences during various seasons, from a temporal perspective. Specifically, the long-range dependence of sea surface chlorophyll in summer in the nearshore region is weaker than in other seasons, whereas that in the offshore region is stronger than in other seasons. The study concludes that long-range dependence is inversely related to the distance from the coastline, and anthropogenic activity plays a dominant role in shaping the long-range dependence of sea surface chlorophyll in the coastal regions of China.

Plasma-Enhanced Atomic Layer Deposition of Lithium Nickel Oxide Thin Film Model Systems

The main bottleneck in the development of the next generation of Li-ion batteries with higher energy densities, as demanded for long-range electrical mobility, is the lack of understanding and control of the reactions at the electrode-electrolyte interfaces. Reactions at the cathode side cause rapid capacity loss during cycling and represent a major issue for batteries with high-voltage cathodes, which are crucial in reaching the required energy densities. Promising candidates for use in next generation of batteries are Ni-rich Li(Ni1-x-yMnxCoy)O2 (NMC) cathodes, which show high capacity, high potential, and contribute to address the geopolitical issue of decreasing the Co content in the cathode. Ni-rich NMCs exhibit similar electrochemical properties as pure LiNiO2 and therefore LiNiO2 thin films can be used as model systems to gain fundamental understanding of the interfacial processes that hinder the large-scale implementation of Ni-rich NMCs. Atomic layer deposition (ALD) is a suitable technique to fabricate these model systems, as it gives precise control over film thickness, uniformity, chemical composition, and crystal orientation.Previous work on LiNiO2 by ALD is not extensively reported in literature. Moreover, films grown by a supercycle process, in which two ALD subcycles are alternated, suffered from a low Ni at.% and a large C content because of precursor ligand residues, and films fabricated using a complex multilayered approach showed severe Si contamination from the Li precursor [1]. In this contribution we report on ALD lithium nickel oxide (LNO) thin films with stoichiometries close to LiNiO2 and low levels of contaminants.The ALD process is based on lithium hexamethyldisilazide (LiHMDS) and nickel bis(N,N’-di-tert-butylacetamidinate) (Ni(tBu-MeAMD)2) as precursors and O2 plasma as co-reactant. 1 ALD cycle of Li2O is combined with x cycles of NiO using a supercycle approach. In this contribution we explore the relation between the Ni/Li cycle ratio x and material properties of the LNO films.The growth of the LNO films is monitored by means of in-situ spectroscopic ellipsometry. The ellipsometric data of films with a high Ni/Li cycle ratio are fitted with a general oscillator model used for NiO films, while a Cauchy model was adopted in case of a low Ni/Li cycle ratio and Li2O films. The growth per supercycle (GPSC) of LNO shows an intermixing effect, as it does not follow the linear trend calculated from the growth per cycle (GPC) values of the individual Li2O and NiO ALD processes, which are 1.0 Å/cycle and 0.34 Å/cycle respectively.XPS surface measurements show that the Ni/Li atomic ratio is tuned from 0 to 10 as function of the Ni/Li cycle ratio x. Low levels of Si and N contamination, below 4% and 2% respectively, are observed. A shift of the main peak in the Ni 2p3/2 spectrum to higher binding energies indicates a shift in oxidation state from Ni2+ to Ni3+ as decreases [2]. According to surface measurements, the LNO film with x=5 shows a stoichiometry close to LiNiO2, but depth profile measurements indicate that the Ni at.% is significantly higher in the bulk than on the surface of the films, most certainly attributed to the formation of a Li2CO3 surface layer upon air exposure. Li2CO3 is also present in the bulk of the LNO films, but the bulk C at.% stays below 4%.These results show that we can use ALD to fabricate a high-voltage cathode model system, which we plan to characterize in-situ and post-mortem in combination with both liquid and solid electrolytes to study the cathode-electrolyte interface.[1] Maximov et al., Energies, 13 (2020) 2345[2] Fu et al., Applied Surface Science 441 (2018) 1048 This work is part of the ‘BatteryNL – Next Generation Batteries based on Understanding Materials Interfaces’ project (with project number NWA.1389.20.089) of the NWA research programme ‘Research on Routes by Consortia (ORC)’ funded by the Dutch Research Council (NWO). Figure:(a) The GPSC of LNO ALD process shows an intermixing effect when comparing the experimental results with the expected GPSC based on the two processes (b) Ni 2p3/2 XPS spectrum for different Ni/Li cycle ratios x. A shift to higher binding energies is observed with decreasing x, indicating a shift from Ni2+ to Ni3+. (c) XPS surface measurements of LNO films show low levels of contaminants and a Ni/Li atomic ratio varying with the Ni/Li cycle ratio. (d) Depth profile of LNO with x=5 shows an increased Ni at.% and a low C at.% in the bulk of the film. Figure 1

Spectroscopic Ellipsometry: Advancements, Applications and Future Prospects in Optical Characterization

Spectroscopic ellipsometry (SE), a non-invasive optical technique, is a powerful tool for characterizing surfaces, interfaces, and thin films. By analyzing the change in the polarization state of light upon reflection or transmission through a sample, ellipsometry provides essential parameters such as thin film thickness (t) and optical constants (n, k). This review article discusses the principles of ellipsometry, including the measurement of key values ∆ and Ψ, and the complex quantity ρ. The article also presents the Fresnel equations for s and p polarizations and the importance of oblique angles of incidence in ellipsometry. Data analysis in ellipsometry is explored, including the determination of bandgap and data referencing the electrical properties of materials. The article emphasizes the importance of choosing the appropriate models to fit ellipsometric data accurately, with examples of the Cauchy and Lorentz models. Additionally, the Kramers–Kronig relations are introduced, illustrating the connection between real and imaginary components of optical constants. The review underscores the significance of ellipsometry as a non-destructive and versatile technique for material characterization across a wide range of applications.

The Optical Properties of Thin Film Alloys of ZnO, TiO2 and ZrO2 with Al2O3 Synthesised Using Atomic Layer Deposition

In this work, the results of ellipsometric studies of thin films of broadband oxides (ZnO, TiO2, ZrO2) and broadband oxides doped with Al2O3 (Al2O3–ZnO, Al2O3–TiO2, Al2O3–ZrO2) are presented. All layers have been produced using the atomic layer deposition method. Ellipsometric studies were performed in the wavelength range of 193–1690 nm. Sellmeier and Cauchy models were used to describe the optical properties of the tested layers. Dispersion dependencies of refractive indices were determined for thin layers of broadband oxides on silicon substrates, and then for layers of Al2O3 admixture. The EDX investigations enabled estimation of the composition of the alloys. The Bruggeman effective medium approximation (EMA) model was used to determine the theoretical dependencies of the dispersion refractive indices of the studied alloys. The refractive index values determined using the Bruggeman EMA model are in good agreement with the values determined from the ellipsometric measurements. The doping of thin layers of ZnO, ZrO2 and TiO2 with Al2O3 enables the creation of anti-reflective layers and filters with a specific refractive index.

A Family of Truncated Positive Distributions

In this paper, a new family of continuous distributions with positive support is introduced. This family is generated by a truncation of the family of univariate symmetrical distributions. In this new family of distributions, general properties, such as moments, asymmetry and kurtosis coefficients, are derived. Particular cases of interest based on the normal, logistic, Laplace and Cauchy models are discussed in depth. The estimation of parameters is carried out by applying moments and maximum likelihood methods. Also, a simulation study was conducted to illustrate the good performance of estimators. An application to the Survival Times (in days) of Guinea Pigs dataset is included, where the special cases of distributions in this family are fitted. The option which provides the best fit is ultimately chosen. An R package, called “tpn”, has been implemented, which includes the relevant cases of interest in this family.

Epitaxial growth of nano-texturized NiO films on MgO (001) substrates by a reactive magnetron sputtering method

Epitaxial films with nanostructures are of great significance for some special applications. In this work, nano-texturized NiO films are epitaxially grown on MgO (001) substrates by a reactive magnetron sputtering method. The evolution in morphology is studied as a function of deposition time, and the influences of substrate temperature, oxygen partial pressure, and the substrate treatment before deposition of NiO on the formation of nano-texturized NiO films are explored. It is found that the non-symmetrically epitaxial growth of nano-texturized NiO films can be linked with the surface damages of MgO (001) substrate, which leads to the coalescence of adjacent NiO grains preferentially occurs along the direction of 〈100〉 and 〈010〉, but they do not coalesce simultaneously following the other directions. In addition, the optical property of nano-texturized NiO films is studied using the Cauchy model.

Single oscillator model assessments and dielectric loss of non-crystalline brilliant green films, and characterization of brilliant green/p-Si photodetectors

Herein, spin-coated brilliant green films (from 95 to 237 nm) were characterized by an X-ray diffractometer (XRD). The extracted cell parameters of the analyzed monoclinic unit cell are a = 14.9650 Ǻ, b = 11.7708 Ǻ, c = 10.6658 Ǻ, α=90.00 Ǻ, β=134.76 Ǻ and γ=90.00 Ǻ, and space group P2. The extracted characteristic XPS peaks for brilliant green thin films at 283.82, 531.49, 398.44 and 168 eV are assigned to C 1s, O 1s, N1 s and S 2p, respectively. All the fabricated brilliant green films exhibit noncrystalline nature with indirect bandgaps. The recorded transmittance and reflectance are used to calculate the absorption coefficient. The refractive index of the brilliant green thin films is estimated by mathematical relations, depending on the extracted optical energy gap. In the non-absorbing region, the absorption edges are overlapped for all films at >1600 nm. The rate of optical energy dissipation revealed two triggering peaks in the visible region at 1.85 and 2.87 eV and 4.1 and 5.3 eV in the ultraviolet region with a higher absorption coefficient. The Cauchy and Sellimeier models analyzed and simulated observed dispersion data. The estimated values of the ideality factor, =3.41, the reverse saturation current, Io=1.06×10−7A and =0.64 eV. The junction resistance, Rj, demonstrated reasonable rectification with low series resistance, Rs =795 Ω, and shunt resistance, Rsh =0.21 MΩ. The photodetector parameters of the brilliant green/p-Si interfaces are characterized, such as the SNR and LDR ratios, and show a linear, stable response with the photon energy ranging from 7 to 32 mW/cm2. The evaluated detectivity has values of order 1011 Jones with no detect value distortion.

Robust color image watermarking algorithm based on synchronization correction with multi-layer perceptron and Cauchy distribution model

A robust color image watermarking algorithm based on the multi-layer Perceptron (MLP) and Cauchy distribution model is proposed. The algorithm has great performance against geometric attacks. Firstly, the central region of the Y-color channel is extracted, and the watermark is embedded into the low-frequency component by dual-tree complex wavelet transform (DTCWT). A synchronization correction technique based on the MLP model is used prior to the watermark extraction. When the MLP model is trained, the position and scale parameters of the Cauchy distribution are employed to generate the image feature vector after first calculating the quaternion discrete cosine transform (QDCT) of the original color host image. The new 3D hyperchaotic mapping is used during the algorithmic process to increase security, which is employed for scrambling, encryption, and embedding of watermarks, respectively. The experimental results demonstrate that the proposed algorithm has good imperceptibility, accurate model prediction, and excellent robustness against common image processing attacks and geometric attacks.

Plekhanov Russian University of Economics; JSC «Mechanical Plant»

Lean production is the dominant direction of transformations taking place in industrial enterprises. The benefits of lean manufacturing come from streamlining resource flows, increasing productivity, eliminating waste, and reducing delays. This article discusses the problem of optimizing current assets based on the Cauchy model. The subject of the study is the change in the system of current assets management, taking into account their composition, structure, cyclicality in terms of lean production. The purpose of this article is to consider the constructed model for managing changes in current assets. The methodological basis of this study is an interdisciplinary approach. The findings of the study can be used both to develop the assessment and optimization of current assets at enterprises that switch to lean production and create a "smart factory".

Limitations of the Wasserstein MDE for univariate data

Minimum Kolmogorov and Wasserstein distance estimates, hat{theta }_{MKD} and hat{theta }_{MWD}, respectively, of model parameter, theta (in Theta ), are empirically compared, obtained assuming the model is intractable. For the Cauchy and Lognormal models, simulations indicate both estimates have expected values nearly theta , but hat{theta }_{MKD} has in all repetitions of the experiments smaller SD than hat{theta }_{MWD}, and hat{theta }_{MKD}’s relative efficiency with respect to hat{theta }_{MWD} improves as the sample size, n, increases. The minimum expected Kolmogorov distance estimate, hat{theta }_{MEKD}, has eventually bias and SD both smaller than the corresponding Wasserstein estimate, hat{theta }_{MEWD}, and hat{theta }_{MEKD}’s relative efficiency improves as n increases. These results hold also for stable models with stability index alpha =.5 and alpha =1.1. For the Uniform and the Normal models the estimates have similar performance. The disturbing empirical findings for hat{theta }_{MWD} are due to the unboudedness and non-robustness of the Wasserstein distance and the heavy tails of the underlying univariate models.Theoretical confirmation is provided for stable models with 1<alpha <2, which have finite first moment. Similar results are expected to hold for multivariate heavy tail models. Combined with existing results in the literature, the findings do not support the use of Wasserstein distance in statistical inference, especially for intractable and Black Box models with unverifiable heavy tails.

An Adaptive Generalized Cauchy Model for Remaining Useful Life Prediction of Wind Turbine Gearboxes with Long-Range Dependence

Remaining useful life (RUL) prediction is important for wind turbine operation and maintenance. The degradation process of gearboxes in wind turbines is a slowly and randomly changing process with long-range dependence (LRD). The degradation trend of the gearbox is constantly changing, and a single drift coefficient is not accurate enough to describe the degradation trend. This paper proposes an original adaptive generalized Cauchy (GC) model with LRD and randomness to predict the RUL of wind turbine gearboxes. The LRD is explained jointly by the fractal dimension and the Hurst exponent, and the randomness is explained by the diffusion term driven by the GC difference time sequence. The estimated value of the unknown parameter of adaptive GC model is deduced, and the specific expression of the RUL estimation is deduced. The adaptability is manifested in the time-varying drift coefficient of the GC model: by continuously updating the drift coefficient to adapt to the change in the degradation trend, the adaptive GC model offers high accuracy in the prediction of the degradation trend. The performance of the proposed model is analyzed using real wind turbine gearbox data.

Bivariate covariance functions of Pólya type

We provide sufficient conditions of Pólya type which guarantee the positive definiteness of an isotropic 2 × 2-matrix-valued function in R and R3. Several isotropic bivariate covariance models have been proposed in literature, where all components of the covariance matrix are of the same parametric family, such as the bivariate Matérn model. Based on the Pólya type conditions, we introduce two novel bivariate parametric covariance models of this class, the powered exponential (or stable) covariance model and the generalized Cauchy covariance model. Both models allow for flexible smoothness, variance, scale, and cross-correlation parameters. The smoothness parameters are in (0,1]. Additionally, the bivariate generalized Cauchy model allows for distinct long range parameters. We also show that the univariate spherical model can be generalized to the bivariate case within the above class only in a trivial way.