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Exponential functions
Published in John Bird, Engineering Mathematics, 2017
Logarithms having a base of ‘e’ are called hyperbolic, Napierian or natural logarithms and the Napierian logarithm of x is written as logex $ \log _{e}x $ , or more commonly as lnx $ \ln x $ . Logarithms were invented by JohnNapier*, a Scotsman (1550–1617).
Exponential functions
Published in John Bird, Basic Engineering Mathematics, 2017
Logarithms having a base of e are called hyperbolic, Napierian or natural logarithms and the Napierian logarithm of x is written as loge $ \log _{e} $ x or, more commonly, as ln x. Logarithms were invented by John Napier∗ $ ^{*} $
Automatic generation control of an interconnected power system
Published in International Journal of Ambient Energy, 2022
Ali M. Ali, Mohamed S. Saad, Adel A. El-Amari, Mohammed A. Moustafa Hassan
is the inertia weight factor and t is the iteration number. is calculated with respect to the initial and final values of with the same way as and ln is the napierian logarithm. is determined according to the fitness value of and at iteration t. , are acceleration coefficients and initial values of inertia weight factor, respectively with i = 1 or 2. is the mean value of the best positions according to all particles at iteration t, as explained in Abbas et al. (2017).
Process capability index for Poisson regression profile based on the S pmk index
Published in Quality Engineering, 2019
Vasileios Alevizakos, Christos Koukouvinos, Philippe Castagliola
Let be a p × n matrix of explanatory variables where . The Poisson regression model assumes that the dependent variable of observation, yi, is modeled as a Poisson random variable with mean λi where each λi is a function of (link function). In Poisson regression profile, the link function is the Napierian logarithm function and the model is set as: where is the parameter vector of the model in which . It is usual to set in order for the coefficient β1 to be the intercept of the model. Alternatively, Eq. [1] can be written as:
Rheological behaviour of a La-based bulk metallic glass (BMG) used in 3D printing
Published in Philosophical Magazine, 2021
At low-stress level (ασ < 0.8), In the high-stress level (ασ > 1.2), The value of α can be calculated from the equation α = β/n, at constant temperature, n and β are the slope of the curves of ln-lnσ and ln-σ, respectively. Take the napierian logarithm of both sides of Equations (2) and (3) for linear fitting. As shown in Figure 8(a,b), when the temperature is 487 K, n = 0.823, β = 0.0174, so α = β/n = 0.021.