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Special Functions
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
For positive values of z the logarithm is a monotonic function, as is the exponential for real z. Any monotonic function has a single-valued inverse function; the natural logarithm is the inverse of the exponential. If x = ey, then y=lnx $ y = {\text{ln }}x $ , and x=elnx. $ x = e^{{ {\text{ln }}x}} . $ The same inverse relations hold for bases other than e. That is, if u = aw, then
Combustion with Multiple Flames under High Strain Rates
Published in Combustion Science and Technology, 2021
The solution for a simple counterflow diffusion flame is given in Figure 15. is shown as a monotonic function of . The functions and of have linear segments except for curved portions where the chemical reaction is occurring. Similar principles hold for the behavior in space with three flames as shown in Figure 16. Of course, now, there are three reaction zones and four line segments connecting them in high cases. In the multiple-flame case, merging of flames is seen as decreases, reducing the number of line segments. For all configurations, the reaction zones broaden as decreases, increasing the curvature of the lines in the plots.
Frequency and time domain comparison of selective polynomial filters with corrected phase characteristics
Published in International Journal of Electronics, 2019
Miona Andrejević Stošović, Dragan Topisirović, Vančo Litovski
Since we are looking for a critical monotonic amplitude characteristic, En−1(x2) is to be chosen so that it enables critical monotonicity. The first derivative of a critical-monotonic function never changes its sign when it has a maximal number of zeros. The first condition for that is En−1(x2) to be a full square, i.e. to be expressed as a square of another polynomial i.e. , where Vn−1(x) is to be an odd or an even polynomial. The second condition asks for all the zeros of Vn−1(x) to be real and to be located in the interval {0, 1}. To that end, Vn−1(x) was expressed as a sum of orthogonal polynomials, with the interval of orthogonality defined by the normalized passband of the filter, i.e. ω ∈ {0, 1}.
The short-run and long-run equilibria for commuting with autonomous vehicles
Published in Transportmetrica B: Transport Dynamics, 2022
Fangni Zhang, Wei Liu, Gabriel Lodewijks, S. Travis Waller
A particular δ function can be either monotonic or non-monotonic. We introduce two particular functions to represent each and to illustrate the evolution of the dynamical system established in Section 3.2 as well as the stability of long-run equilibria. The exponential function is designated as the monotonic form, which is an adopted version from the well-known Newell model (Newell 1961). The non-monotonic function is represented by a polynomial function In the following sections, we will present numerical results with the two functions, respectively. It would be interesting to identify and calibrate the ‘true’ functions for δ, which is beyond the scope of this study.