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Applied Analysis
Published in Nirdosh Bhatnagar, Introduction to Wavelet Transforms, 2020
The number e is called Euler’s number, after the mathematician Leonhard Euler (1707–1783). The series expansion of ex≜exp(x) is also valid if x is a complex number. Substituting x = iθ in the series expansion of ex, yields the well-known Euler identity. ()eiθ=cosθ+isinθ
Single Degree‐of‐Freedom Damped Vibration
Published in Haym Benaroya, Mark Nagurka, Seon Han, Mechanical Vibration, 2017
Haym Benaroya, Mark Nagurka, Seon Han
In 1748 Euler made ideas of Johann Bernoulli more precise in defining a function. He stated that mathematical analysis was the study of functions, and based the calculus and the theory of elementary functions. Euler discovered the equation eix=cosx+icosx, $$ e^{{ix}} = {\text{~cos~}}x + i{\text{~cos~}}x, $$
Special Functions
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
∫exdx=dexdx=ex. $$ \mathop \smallint \limits_{{}}^{{}} e^{x} dx = \frac{{de^{x} }}{{dx}} = e^{x} . $$
Optimization of empirical typhoon model considering the difference of radius between pressure gradient and wind speed distributions
Published in Coastal Engineering Journal, 2022
Masaya Toyoda, Nobuhito Mori, Jun Yoshino
Here, e is the base of natural logarithms. This equation assumes the gradient–wind equilibrium and considers the surface friction and moving speed of the TC. Therefore, it is inferred that the distribution is not suitable at the time of landfall. Figure 1 shows an example of the cross-sectional atmospheric pressure distribution for different values of B. The horizontal axis represents the distance from the center, and the vertical axis represents the atmospheric pressure. A large difference can be observed in the pressure distribution depending on the setting of B. In the case of a small B value (< 1.0), the pressure gradient until the typhoon approaches becomes gradual, and near the center of the typhoon, the pressure gradient becomes steep. In the case of a large B value (> 1.0), the opposite tendency is observed.
Evaporative Cooling Heat Transfer of Water From Hierarchically Porous Aluminum Coating
Published in Heat Transfer Engineering, 2018
Nabeel Fathi, Jinsub Kim, Seongchul Jun, Ryan M. King, Miguel Amaya, Seung M. You
Here, e refers to Euler number which is the base of the natural logarithm. The average pore radius R was obtained from the wicking test using hexane which is known to have a 0° contact angle with most metals. And, the dynamic contact angle θd was also obtained as approximately 55° regardless of the particle size from the successive wicking tests using water. The detailed procedure to obtain R and θd was presented in Fathi et al.[19]. The wicking heights with time calculated from Eq. (4) are also depicted in Figure 8 according to the particle size. It exhibits good agreement with the experimental data at the largerst particle size of 114 μm. The average difference between calculated heights and experimental data was 7.0%. However, at the smaller particle sizes of 70 μm and 27 μm, the average differences increased up to 13.1% and 11.5%, respectively. The differences are considered to result from the assumption that the porous structures are regarded as straight capillary tubes in Eq. (4). However, the calculated wicking height with time agrees with the overall trend obtained using the vertically erected coated sample of Al-HPC in Figure 8.
Data cleansing for energy-saving: a case of Cyber-Physical Machine Tools health monitoring system
Published in International Journal of Production Research, 2018
Changyi Deng, Ruifeng Guo, Chao Liu, Ray Y. Zhong, Xun Xu
where SP is the size of the neighbouring space, indicating the highest correlation of neighbouring nodes. R is the global correlation of the current neighbourhood space (0 < R ≤ 1). e is a mathematical constant. Δ indicates the amount of change in the measured value, which is the difference between the current measured value and the previous period value. α indicates the fluctuation adjustment parameter (α > 0). The value of α depends on the reliability of the data to be obtained. To ensure the reliability of the data, set as the lower bound of the elastic space. To prevent the elastic space from being too large, set as the upper bound of the elastic space. When the elastic space exceeds the upper bound , the data from the sensor is identified as unreliable data by the local node.