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Modeling the Diffusion of Chemical Contamination in Soil with Non-Conventional Differential Operators
Published in Abdon Atangana, Mathematical Analysis of Groundwater Flow Models, 2022
A fractal is a continuous pattern that repeats itself at various scales. Fractals are used to investigate anomalous systems occurring in nature. They are one of the latest developments in statistics and inhabit scaling and self-similar properties. Scaling is the ability of objects to increase or decrease in size by a scale factor that is the same in all dimensions. Self-similarity is the capacity of an object to duplicate itself. Fractal derivatives are of a self-similar nature and repeat themselves at various scales in space and time. Fractals are believed to have links to fractional derivatives, Levy statistics, Brownian motion, and empirical power-law scaling (Chen, 2005). The fractal derivative can be used describe motion in turbulence, fractal flow, and viscos-elastic behavior (Chen et al., 2017). Chen (2017) replaces the inter-order time derivative with a fractal derivative using the power law to describe the anomalous diffusion in heterogeneous media. The following equation gives the fractal derivative: ∂u∂tα=limt1→tut1−utt1α−tαα>0
Pavement curling and warping analysis using wavelet techniques
Published in International Journal of Pavement Engineering, 2021
Shuo Yang, Ahmad Alhasan, Yang Zhang, Halil Ceylan, Sunghwan Kim
CWT decomposition produces a coefficient at each point to measure the similarity of the mother wavelet to the profile at the current scale factor and translation factor . The reconstruction of a certain layer (scale) was performed by setting the coefficients at all the other layers to zero, then applying inverse wavelet transforms (see Equation (4)) to the newly formed data set; the mean elevations of the reconstructed layers thus remain identical to the original profiles. New profiles were then constructed by cumulatively subtracting the layers from the original profile in ascending order. This established a low-pass filter in the CWT analysis and in a manner comparable to the previous approach utilised in DWT. For example, the new profile at layer 6 () based on Morlet decomposition is defined by subtracting the sum of layers 1–6 from the original profile, meaning that profile features associated with pseudo-frequencies >3 circle/m were filtered out. Twenty-five newly constructed profiles were produced and their curling and filters profiles were subsequently obtained as well for further analysis.
Semi-supervised machine learning of optical in-situ monitoring data for anomaly detection in laser powder bed fusion
Published in Virtual and Physical Prototyping, 2023
Ngoc Vu Nguyen, Allen Jun Wee Hum, Truong Do, Tuan Tran
Data augmentation is applied on the training dataset to enhance the robustness of the ML model. The training images are augmented with random reflection, random translation between pixels and pixels, random rotation with the angle between and , and random scaling with scale factor varying from to . To optimise memory usage, augmented images are randomly created in Matlab during the training process without saving them to memory.
Drought risk analysis based on multivariate copula function in Henan Province, China
Published in Geomatics, Natural Hazards and Risk, 2023
Yunliang Wen, Liwei Zhou, Ling Kang, Hao Chen, Jinlei Guo
Proposed by McKee in 1993, the SPI (Belayneh et al. 2014; Li et al. 2020) describes the severity of drought with the probability of precipitation. This index applies to droughts on multiple time scales. Suppose precipitation x obeys gamma distribution. Then, the probability density of x, i.e. the SPI, can be described as: where is the shape factor; is the scale factor; is the gamma function. These parameters can be computed through maximum likelihood estimation (MLE):