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Basics of Discrete Volterra Series
Published in Andrzej Borys, Nonlinear Aspects of Telecommunications, 2018
where ln means a logarithm with the base equal to e. Note that inequality 1.263d determines the number of components in the truncated Volterra series that are necessary to guarantee the truncation error does not to exceed the value of er. Consider now other results3 of Sandberg published in 1992 that can be seen as a continuation of the considerations presented in Reference 17 for continuous-time systems. The paper3 develops further ideas given in Reference 17 with respect to the accuracy of truncations often made in practical applications using the associated Volterra series. The problem of a uniform approximation with doubly finite Volterra series is formulated and solved in Reference 3. We now present the results of Sandberg's paper3. In particular, the results regarding nonlinear systems, which can be modeled by the following system equations, w = A(x) + C (z) y = D(x) + B(z) z = N (w) (1.264a) (1.264b) (1.264c)
Organic and Polymeric Photonic Band Gap Materials and Devices
Published in Sam-Shajing Sun, Larry R. Dalton, Introduction to Organic Electronic and Optoelectronic Materials and Devices, 2016
In practice, most of the photo-curable monomers utilized are multifunctional acrylate derivatives by virtue of their fast curing nature and chemical versatility. More importantly, multifunctional acrylates serve as matrix binders to contain the dispersion of nematic liquid crystals. Nematic liquid crystals (cyanobiphenyl derivatives) are generally used as a second component for the purpose of electrical switching. In other words, the orientation of the liquid crystal director may be altered upon the application of the external electric field. In this way, the direction of the polarized light may be controlled at ease. Table 26.1 illustrates the chemical structures of a typical eutectic liquid crystal E7 (Merck Co.) and di-, tri-, tetra-, penta-, and hexa-acrylate monomers with varying functionality from 2 to 6. The E7 is a eutectic mixture of different LC constituents containing four to eight carbons in the alkyl side groups; the reason to use the above LC mixture is that the nematic region of liquid crystal can be expanded relative to the single neat LC constituent, which is needed for improving electro-optical switching characteristics. Table 26.2 lists the refractive indices of some monomers and nematic liquid crystal commonly found in photolithography. Note that the liquid crystal molecule is represented by the ordinary and extraordinary indices due to the anisotropic nature of the liquid crystal mesogen, hereafter designated as no and ne, respectively. That is to say, the light polarized along the liquid crystal molecule axis propagates at a different velocity than the light polarized perpendicular to the liquid crystal molecular axis.
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.