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Sequences, limits and continuity
Published in Alan Jeffrey, Mathematics, 2004
The fundamental mathematical constant e = 2.718281…, can be defined in various different ways. The choice of definition used to introduce e in this chapter makes use of a special form of limit that can also be used to define the exponential function ex that plays such an important role throughout mathematics. A different, but equivalent, definition of e that is useful for the numerical computation of the exponential function and for determining its differentiability properties is given later in Chapter 5. The exponential function and the related hyperbolic functions form topics for study in Chapter 6, in which the inverse of the exponential function, called the natural logarithmic function ln x, is introduced.
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.
Research on the correlation mechanism between eye-tracking data and aesthetic ratings in product aesthetic evaluation
Published in Journal of Engineering Design, 2023
Yong Wang, Fanghao Song, Yan Liu, Yaying Li, Xiaomin Ma, Weihao Wang
Based on the analysis results in Table 7, we construct an evaluation model for the product aesthetic correlation mechanism to predict products belonging to high, medium or low visual aesthetics categories. First, we defined the constants G1, G2 and G3 in the logistic regression equation as low, medium and high visual aesthetics, respectively. In addition, time to first fixation, fixation before, first fixation duration, total fixation duration, fixation count, visit count, and total visit duration are defined as Y1, Y2, Y3, Y4, Y5, Y6 and Y7, respectively. The B value of the intercept in Table 4 (−0.107) and the B value of the gaze time (0.251) were selected as the regression coefficient of the intercept and the regression coefficient of Y4, respectively. In addition, the B value of the number of gaze points (−1.748) and the B value of the number of gaze times (2.773) were chosen as the regression coefficient of Y5 and the regression coefficient of Y6, respectively. Furthermore, the B value of the visit time (−0.968) was selected as the regression coefficient of Y7, and these sets of regression coefficients with the test data are in accordance with the principle of linear summation to establish Eq. (1). Equations (2) and (3) are developed accordingly (Geng et al. 2007). Finally, Equations (4), (5) and (6) are constructed to predict the probability of low, medium and high visual aesthetics of products, respectively. e is as a mathematical constant and is the base of the natural logarithmic function.