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Inelastic scattering and non-equilibrium transport
Published in DAVID K. FERRY, Semiconductor Transport, 2016
The first of equations (4.65) is exactly the relax at ion-time approximation (3.51). However, the relaxation-rime approximation has not been assumed for this equation. Rather, the relaxation time approximation arises by the disregard of the second equation in (4.65). In equilibrium, it is usually assumed that the distribution function is the equilibrium function, which means that the second equation may be disregarded. However, when inelastic and anisotropic scatterers are present, this is not the case, no matter how small the electric field. Thus the second equation, with replaced by the first equation, gives a differential equation for the zero-order distribution function. The term f0{E) is often called the energy distribution function.
Nonlinear Equations
Published in Jeffery J. Leader, Numerical Analysis and Scientific Computation, 2022
The root-finding problem is to find a solution x∗ of f(x)=0. (There may be many solutions, but we are only seeking any one of them.) The solution is called a root of the equation or a zero of the function f. For nearly all methods it's an implicit assumption that the zeroes of the function f are isolated, that is, that there is an open interval that contains that zero but no other zeroes of the function. See Problem 14 for a case of a non-isolated zero.
Propagation II:Mathematical Models (Part One)
Published in Paul C. Etter, Underwater Acoustic Modeling and Simulation, 2017
The solution of the Bessel equation (Equation 4.24) for outgoing waves is given by the zero-order Hankel function of the first kind: () S=H0(1)(k0r)
On the structural description of random fields
Published in Waves in Random and Complex Media, 2022
In the approximation of external force F given by its correlation function, the scale of focusing is defined as: The behavior of trajectories in the vicinity of the zero of the function F averaged over this scale is determined by the matrix Fij which enters the following equation: By representing the solution of the system (20) in the form (17), we are able to produce the dispersion equation with respect to the square of the instant frequency, P2. Note that this equation has the same structure as (18). The overall structure of the solution is also similar to those described above, but the trajectories have an oscillatory behavior. Its temporary scales are determined by the properties of tensor Fij. In the two-dimensional case we have: The solution of this equation describes both the rotationally-oscillating regime and the regime of the compression/stretching.
Experimental validation of the diffusion function model for accuracy-enhanced thermoreflectometry
Published in Quantitative InfraRed Thermography Journal, 2021
Benjamin Javaudin, Rémi Gilblas, Thierry Sentenac, Yannick Le Maoult
The Fresnel function, , gives the specular reflectance of the micro-facet oriented by an angle . The local incident and reflection angle is then . This function includes the spectral dependence of the BRDF with the complex index of refraction () associated with the material. In Equation (5), assuming that the angle stays close to zero, the Fresnel function, , is approximated by the normal Fresnel factor as follows:
Closed-form solutions of singular KYP lemma: strongly passive systems, and fast lossless trajectories
Published in International Journal of Control, 2020
Chayan Bhawal, Debasattam Pal, Madhu N. Belur
Definition 5.2 means that lossless trajectories of are those for which the rate of change of stored energy is equal to the power supplied. It is crucial to note here that Definition 5.2 does not preclude the possibility of multiplication of Dirac delta impulse δ and its derivatives with themselves. We treat Equation (23) only formally here. By this we mean that Equation (23) is said to hold if and only if the expression is zero as a function for , and each of the coefficients of the monomials in the quadratic expression involving symbols in the expression is zero.