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Interaction of Radiation with Plasmas
Published in R A Cairns, A D R Phelps, P Osborne, Generation and Application of High Power Microwaves, 2020
Once the dielectric tensor is known, we obtain a set of three homogeneous linear equations for the components of the electric field. The condition that these have a non-trivial solution gives a relation connecting k and ⍵ the dispersion relation for the waves. The imaginary part in the Z function means that the solution of the dispersion relation for real ⍵ gives rise to a complex value of κ in which the imaginary part results in damping of the wave. There is significant damping only if the argument of the Z function is small. Typically ⍵ ≫ k||υth, where υth is the thermal velocity ms/2Ts and so there is only significant damping if ⍵ is close to nΩs for some value of n, so making the corresponding value of ξn small. For n = 0 this is Landau damping, and for n nonzero it is cyclotron damping. In an inhomogeneous magnetic field cyclotron damping is localised around layers where the cyclotron frequency is a multiple of the wave frequency. Landau or cyclotron damping is essential to most schemes for radiofrequency heating of plasma. In the next section we shall explore further the physical origin of these effects.
Probabilistic approach and risk analysis
Published in Krystiaim W. Pilarczyk, Dikes Aimd Revetments, 2017
Harne KT. Kuijper, Johannes (Han) K. Vrijling
The function is depending of the point where it will be linearised. The mean value and the standard deviation of the linear Z-function are: μz≈Z(x→*)+∑i=1n∂Z∂xi⋅(μxi−xi*)σz=∑i=1n(∂Z∂xi⋅σxi)2
Technology of Intelligent Systems
Published in James A. Momoh, Mohamed E. El-Hawary, Electric Systems, Dynamics, and Stability with Artificial Intelligence Applications, 2018
James A. Momoh, Mohamed E. El-Hawary
The triangular membership function is the most frequently used function and the most practical, but other shapes are also used. One is the trapezoid which contains more information than the triangle. A fuzzy set can also be represented by a quadratic equation (involving squares, n2, or numbers to the second power) to produce a continuous curve. Three additional shapes which are named for their appearance are: the S-function, the PI-function, and the Z-function.
Mining large-gradient subsidence monitoring using D-InSAR optimized by GNSS
Published in The Imaging Science Journal, 2021
Haodi Fan, Xugang Lian, Wenfu Yang, Linlin Ge, Haifeng Hu, Zheyuan Du
To facilitate the calculation, some small abrupt changes in the settlement curve are discarded and smoothed. The D-InSAR subsidence curve obtained is shown in Figure 3(a). Taking the maximum subsidence point of GNSS and the maximum subsidence point on the left and right sides of D-InSAR as the demarcation point, three straight lines perpendicular to the horizontal axis were made. The curve can be divided into four segments: a, b, c, and d. The horizontal axis can be regarded as the t axis, the D-InSAR subsidence curve is the z function, and the GNSS subsidence curve is the y function. When observing segments a and b in Figure 3(b), it is obvious that the y values are not unique under the same z values.
Development of Damage Patterns and Fragility Curves in Brick-nogging Buildings from the Thabeikkyin Earthquake, Myanmar, 2012
Published in Journal of Earthquake Engineering, 2018
Zin Zin Nwe, Nan Pawt Sai Awar, Aye Mya Cho, Kyaw Moe Aung, Maki Koyama, Junji Kiyono
Membership functions are set for each questionnaire item category. The relationship between probability distribution of each item category and seismic intensity is classified into three types. They are Z (monotonically decreasing Z-shape), Π (Bell shape), and S (monotonically increasing S-shape) functions as shown in Fig. 7. The Z function means that the probability decreases as the seismic intensity increases, the Π function is a unimodal probability distribution, and the S function means that the probability increases as the seismic intensity increases. These functions are represented by the following equations:
Study of Electron Anomalous Heat Diffusion Affected by Mode Number of Perturbed Magnetic Island in Tokamak Plasmas
Published in Fusion Science and Technology, 2022
Hong Gao, Zewen Shao, Muzhi Tan
Besides that, the ratio between the parallel and the perpendicular heat diffusivities, i.e., the g(z) function, is an important factor. It is found that g(z), i.e., , dominates how fast a perturbed island that contributes to radial enhanced radial heat diffusivity drops from maximum values at the rational surface to zero under a single perturbed magnetic island and weak stochastic magnetic field situations and