Heaviside’s step function

Here, kq = k /s where e is the dielectric constant of the bulk solvent and is the Debye length 9 stands for the Heaviside unit step function, which reflects here the distance of closest approach, I, of ions to the electrode [I is calculated from the edge of the metal skeleton). 

Here h(x) is the Heaviside step function with h(x > 0) = 1 and h(x > 0) = 0 (not to be confused with Planck s constant). The limit a(J.. . ) indicates that the sunnnation is restricted to channel potentials witir a given set of good quantum numbers (J.. . ). 

For PF, the F function requires another type of special mathematical representation. For this, however, consider a sudden change in a property of the fluid flowing that is maintained (and not pulsed) (e.g., a sudden change from pure water to a salt solution). If the change occurs at the inlet at t = 0, it is not observed at the outlet until t = t. For the exit stream, F(t) = 0 from ( = 0 to t = t, since the fraction of the exit stream of age less than ( is 0 for t < f in other words, the exit stream is pure water. For t > t, F(t) = 1, since all the exit stream (composed of the salt solution) is of age less than t. This behavior is represented by the unit step function S(t - b) (sometimes called the Heaviside unit function), and is illustrated in Figure 13.7, in which the arbitrary constant b = t. With this change, the unit step function is... 

The surface integral in Eq. (5.70) may be expressed as a volume integral if we include the delta function S(S(q)) in the integrand such that we only get contributions when the coordinates satisfy Eq. (5.72). We also need the flux of system points across the surface. Both of these may be introduced by considering the flux function F(p, q) given in terms of the Heaviside step function as... 

We also need a function that shows whether a given trajectory, starting on the reactant side of the dividing surface, ends up at the product side at time t —> oo and thereby contributes to the formation of products. The Heaviside step function in Eq. (5.50) may also be used to specify whether the phase-space point of a system is at the dividing surface (S(q(t)) - 0), on the product side (say S(q(t)) > 0), or on the reactant side (S(q(t)) < 0). We then define the function P(p,q) according to... 

The Kubo relation (25) of section 2.1 is obtained as the Fourier transform of (70). The term linear in V is the retarded two-time Green s function, first introduced in this context by Bogoliubov and Tyablikov [30]. The identification with Green s functions stems from the presence of the Heaviside step function that in part were introduced to allow integration over the full time interval and whose time derivative gives a Dirac delta function. For instance, t — to)U(tfo) is a solution of the inhomogeneous equation [31]... 

The Heaviside step function 0(s+As) in Eq.(19) indicates that the FO VP rate is nonzero... 

We next define the Dirac delta function S(x) as the derivative of the Heaviside step function ... 

An iteration method is used the solve the above the equation by marching in the pseudotime, Xp. The color function is defined as a mollified Heaviside step function based on s. The value of varies from 1 to 0 in moving from the interior to exterior with a sharp gradient over a thin interface region. 

Another approach is to extend the classical contact theory for indenters on elastic half-spaces developed by Hertz [77] and Huber [78] to the case of layered materials. An example of such an approach is ref. [79], in which the authors modify the Hertz/Huber analysis by considering the coating material properties as a function of indentation depth. Mathematically, the authors treat the transition from coating to substrate as a discontinuity in Young s modulus and Poisson s ratio represented by a Heaviside step function, and re-derive the appropriate Hertzian equations. The results match FEA calculations well. 

Joint distribution of BB and AB pairs is shown in Fig. 6.44. The distribution of similar mobile particles B at long times in the asymmetric case practically is the same as in the symmetric case (when X = Xb). The behaviour of Xb (r, t) is determined by the Coulomb repulsion of B s for which the non-equilibrium screening effect does not take place. In its turn, some deviation for the joint dissimilar functions Y(r, t) seen in Fig. 6.44 for the symmetric and asymmetric cases is a direct consequence of different screening effects in the latter case the effective recombination radius increases in time which results in an increase of the Y(r,t) gradient at r = ro at long times this correlation function itself strives for the Heaviside step-like form. 

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