Heaviside step function

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 functions of a single variable (e.g., energy, momentum or time) the projector Prz)(x) is simply 0(a ), the Heaviside step function, or a combination thereof. When also replacing x, k by the variables , t, the Fourier transform in Eq. (5) is given by... 

According to the results obtained for the first derivative, Equations 2.12 and 2.13, the second derivative, i.e., the hardness, is zero when evaluated from the left or from the right, and it is not defined for integer number of electrons. However, Ayers [25] has shown that if one makes use of the Heaviside step function... 

For an isotropic tortuosity in three dimensions, = 1/3. 0 is the Heaviside step-function that accounts for the existence of a percolation fhreshold, A in the water-permeating network. A can be obtained from the specific law of swelling established for the membrane under consideration, as discussed in Section 6.7.4. The mean square radius of pores that contribute to the water flow in a local volume element with water content A is... 

Sign and Heaviside Step Functions We may define the sign function... 

For successive overrelaxation, we understand Eq. (26) to incorporate the use of o(k+1 values in place of o k) values in the convolution product as soon as they are formed for preceding x values. This adaptation can be explicitly displayed by the appropriate use of the Heaviside step function in a modified version of Eq. (26). The method of Van Cittert is a special case of simultaneous relaxation in which C = 1. 

Here H is the Heaviside step function. Physical realizability now implies positivity for dp. With this modification, only negative values of 6p contribute to the integral. Unfortunately, 0 is no longer differentiable. The restriction on O expressed in Eq. (45) inhibits derivation of the least-squares normal equations. Instead, Howard considered the Heaviside step as a limit ... 

As an example of the use of this method, consider a reaction between two species, as usual with a boundary condition at infinity where p - 1. Let the reactive sink term be given by the Heaviside step function / — u(R — [rj). The solution for the rate coefficient has already been given (Chap. 8, Sect. 2.4) as... 

Derive also, 6 being the Heaviside step function,... 

Here p is a defect creation rate per unit time and volume, called also dose rate, f r) is their initial distribution function over relative distances, normalised according to f f(r)dr= 1, o(r) the AB pair recombination rate. For the annihilation mechanism o(r) = cro0(ro - r), 0 is the Heaviside step-function (Section 3.1). 

To construct the functional for inhomogeneous fluids and account for the correlations, the coarse-grained densities rather than the local densities should be used. For convenience, we use the Heaviside step function to estimate the weighted densities. In this work, the attraction... 

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... 

---