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Exponential, logarithmic and hyperbolic functions and an introduction to complex functions
Published in Alan Jeffrey, Mathematics, 2004
In Chapter 2 we used the term ‘a real-valued function of a real variable’ to mean any rule that associates with each real number from the domain of definition of the function a unique real number from the range of that function. Symbolically, if D denotes the set of points in the domain of a function ƒ, and R denotes the set of points in the range of ƒ, this relationship or mapping is given by R=f(D).
Degradation modeling based on the gamma process with random initial degradation level and random threshold
Published in Quality Technology & Quantitative Management, 2023
Luis Alberto Rodríguez-Picón, Luis Carlos Méndez-González, Víctor Hugo Flores-Ochoa, Iván JC Pérez Olguín, Vicente García
where is defined on a real line as a function of the real variable , and . The CFs of and can be obtained with (4) and are defined as and . If the CF of can be obtained as (Rodriguez-Picon et al., 2019), then the PDF of can be obtained by considering the inverse Fourier transform (IFT) of as,
Modified Grey Wolf Randomized Optimization in Dementia Classification Using MRI Images
Published in IETE Journal of Research, 2022
N. Bharanidharan, R. Harikumar
The PCA technique is used to extract the small number of uncorrelated samples called principal components to represent a set containing a large number of uncorrelated samples. PCA technique is based on orthogonal transformation to find the principal components [15]. Usually, DFA is used to compute the statistical self-affinity of a signal in time series analysis. In DFA implementation, the variation from the trend is called fluctuation is computed initially and this fluctuation is measured over different window sizes [16]. HT is a linear operator that transforms the function of a real variable into another function of a real variable. HT incorporates the transformation in convolution with the Cauchy kernel [17,18]. K-means clustering is an iterative algorithm that splits the data points into K clusters based on the distance of data points from the mean [19].
Identification, tracking and warning of vortex induced vibration using k-means clustering method
Published in Structure and Infrastructure Engineering, 2022
Min He, Peng Liang, Yang Zhang, Yang Wang, Kang-di Wang
The Hilbert transform (Kress, 1989) is a specific linear operator that takes a function of a real variable and produces another function of a real variable by doing a convolution with as: where denotes the original signal; Since the in-field monitoring data is discrete, the Hilbert transform of the discrete signal can be obtained using inverse discrete-time Fourier transform as: where denotes the inverse discrete Fourier transform of denote the discrete Fourier transform of denotes the sign function. The relationship between the discrete Fourier transform of and can be described as: