Step discontinuity

W.P. Huang (Ed.), Methods for modelling and simulation of guided-wave optoelectronic devices waves and interactions (PIERS 11, EMW Publishing, Cambridge, MA, 1995). P.G. Suchoski, Jr., and V. Ramaswamy, Exact numerical technique for the analysis of step discontinuities and tapers in optical dielectric waveguides , J.Opt.Soc.Am.A. 3, 194-203 (1986). 

W. Biehlig, et al, Light propagation in a planar dielectric slab waveguide with step discontinuities. Optical and Quantum Electronics 18, 219-228 (1986). 

The Quattro system allows recording of signals in real time, corresponding to channel function. However, the higher throughput limits the complexity of the protocols that can be used, as well as constrains the number of compound additions to a two-step discontinuous clamp process, as with the first generation system. 

Glass et al. [7] reported that MacCormack s method [22], which is a second-order accurate explicit scheme, can handle these moving discontinuities quite well and is valid for the HHC problems. Since the hyperbolic problems considered here have step discontinuities at the thermal wave front, MacCormack s prediction-correction scheme is used in the present study. When MacCormack s method is applied to Eqs. (2.7) and (2.8), the following finite difference formulation results ... 

On the basis of analysis of experimental facts Ehrenfest has introduced the following classification of phase transitions a conversion is called the phase transition of nth order if successive derivatives of a thermodynamic function U up to and including (n — 1) are continuous functions, whereas the nth derivative has a step discontinuity at the transition point the... 

A function does not have to be analytic, or even continuous, in order to be represented by a Fourier series. It is only necessary that the function be integrable. As mentioned in Chapter 5, an integrable function can have step discontinuities, as long as the step in the function is finite. At a step discontinuity, a Fourier series will converge to a value halfway between the value just to the right of the discontinuity and the value just to the left of the discontinuity. 

Lansing, N. F., Analytical Prediction of Ion Chamber Response to Step Discontinuity in MTR Pouer, ORNL eFr-50-2-32, February 8, 1950. 

The kinetics of melt-crystallization of PLA has been analyzed by a number of research groups [14, 35, 39, 71-75]. Isothermal bulk crystallization rates were determined in a wide temperature range from 70 to 165 C [71,72]. The maximum crystaUization rate is observed around 100 C, and the most peculiar behavior is a discontinuity in the phase change kinetics around 110-120°C, an example of which is shown in Figure 5.10. Figure 5.10a reports the half-time of crystallization of PLA as function of the isothermal crystallization temperature. The data set shows a broad minimum around 108 C and a step/discontinuity around 116-118°C, indicated by the arrow. The sudden variation in crystallization rate... 

Kasche V (2001) Physico-chemical boundaries in the continuous and one-step discontinuous... 

Example 11.2. Show that the series of the previous example equals zero at x = —L, x = L, and x = 3L, the value halfway between the two values of the function at the step discontinuity. 

The continuity properties of Eq. (33-1) follow by reductio ad absurdum. Suppose P is not continuous, so that somewhere it has a step, or jump in value. The first derivatives will then contain a singularity described by the Dirac delta function, and the second derivatives will contain the derivative of the delta function. Since the profile (x,y) contains, at most, a step discontinuity, it is clear that Eq. (33-1) cannot be satisfied. Consequently T must be continuous. A similar argument is used to prove that the first derivatives are also continuous. It is not necessary that the second derivatives be continuous. For example, if n(x,y) is a step profile, then at least one of the second derivatives of F must be discontinuous. Finally, since is constructed from solutions of the scalar wave equation, it must have the same continuity properties as F. 

The transmission coefficient of Eq. (35-35) for tunneling rays assumes a continuous profile. If we introduce a step discontinuity at the interface x = p, in Fig. 35-4(a), and assume that the fields on either side remain evanescent, it may be shown that the effect of the discontinuity is to multiply the transmission coefficient by an extra factor jTf, so that [11]... 

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