Equal-time approximation

Fig. 6 Comparison between the standard evolution operator, the Green s function and the covariant evolution operator for single-photon exchange in the equal-time approximation. The solid lines between heavy dots represent electron propagators and xh free lines electron creation and absorption operators...

Connection to Bethe-Salpeter Equation 5.1 Equal-Time Approximation... 

We have presented a relativistically covariant many-body perturbation procedure, based upon the CEO and the GO. This represents a unification of the many-body perturbation theory and quantum electrodynamics. Applied to all orders, the procedure leads in the equal-time approximation to the BSE in the effective-potential form. By relaxing this restriction, the procedure is consistent with the full BSE. The new procedure will be of importance in cases where QED effects beyond first order in combination with high-order electron correlation are significant. 

When this holds, the kinetic equations reduce to single exponentials. Chipperfield6 demonstrates that approximate adherence to Eq. (4-25) suffices to fit 20 absorbancetime pairs spaced at equal times over the first 75 percent of the reaction with correlation coefficients better than 0.999. 

Third, it is often useful to assume that the concentration of one or more of the intermediate species is not changing very rapidly with time (i.e., that one has a quasistationary state situation). This approximation is also known as the Bodenstein steady-state approximation for intermediates. It implies that the rates of production and consumption of intermediate species are nearly equal. This approximation is particularly good when the intermediates are highly reactive. 

The duration of the reflected pressure depends on the dimensions of the reflecting surface, up to a maximum time approximately equal to the positive phase duration of the incident blast wave. This upper limit corresponds to the total reflection of the entire blast wave without any diffraction around the edges of the reflecting surface. Further details of the duration are provided in Section 3,5.1,... 

In many early experiments, hysteresis was observed for highly hydrophobic compounds such as PCBs (79, 80). Since the time to reach equilibrium can be quite long for strongly hydrophobic compounds, a solute may have never reached equilibrium during the sorption isotherm experiment. Consequently, Kj would be underestimated, which leads to the discrepancy between the sorption and desorption coefficients that was attributed to hysteresis. The case for hysteresis being an artifact is supported by recent data for tetrachlorobenzene (log K = 4.7), illustrating that sorption and desorption require approximately two days to reach equilibrium with approximately equal time constants (78). Finally, the diffusion model is consistent with the observation that the extent of hysteresis was inversely related to particle size (81). 

The antiproliferative effects of bisindole alkaloids are well established. In general, these compounds are extremely toxic to a wide variety of cells in culture. The cytotoxic effects require a minimal exposure time approximately equal to the doubling time of the cells under test, usually 12 to 72 hr, at a drug concentration of 10 nM to 1 xM in order to achieve an ED, 8,9). Cell types that are characterized by short doubling times (e.g., lymphocytes) tend to be more sensitive than those cell types that divide more... 

The first term on the right-hand side of eqn. (11) decays away and, after a time approximately equal to 5t, the second term alone will remain. Note that this is a sine wave of the same frequency as the forcing function, but that its amplitude is reduced and its phase is shifted. This second term is called the frequency response of the system such responses are often characterised by observing how the amplitude ratio and phase lag between the input and output sine waves vary as a function of the input frequency, k. To recover the system RTD from frequency response data is more complex tnan with step or impulse tests, but nonetheless is possible. Gibilaro et al. [22] have described a short-cut route which enables low-order system moments to be determined from frequency response tests, these in turn approximately defining the system transfer function G(s) [see eqn. (A.5), Appendix 1]. From G(s), the RTD can be determined as in eqn. (8). 

Considering equal time steps (At) Dee and Shuler propose that the amount of virus bound during the first time step could be approximated by ... 

Figures 4.34a,b demonstrate the emission lines of titanite, which according to their spectral positions may be confidently connected with Nd " ". The luminescence spectrum in the 860-940 nm spectral range, corresponding to the transition, contains six peaks at 860, 878, 888, 906, 930 and 942 nm, while around 1,089 nm corresponding to F3/2- fn/2 transition it contains five peaks at 1,047,1,071,1,089,1,115 and 1,131 nm. The decay time of IR luminescence of Nd " equal to approximately 30 ps in titanite is evidently the shortest one in the known systems activated by Nd ". The typical radiative lifetime of this level depends on the properties of the solid matrix and varies from approximately 100 ps to 600 ps (Kaminskii 1996). To explain the fast decay time of Nd " in titanite, the energy level quenching by the host matrix may be considered.

The etching process may be characterized by an amount of light energy received by a sample for the optimal etching time. For some semiconductor materials this amount is equal to approximately 1 J/cm2. The minimum operating illumination intensity, at which the rate of photoetching exceeds noticeably the rate of dark corrosion, is 10"4 W/cm2. 

The sampling schedule was chosen in order to accumulate early data rapidly since we expected the largest changes in the leach rate at the beginning of the experiment. Also, when plotted on log paper, this sequence maintained approximately equal time separation between data points. 

Doubling time approximately equals 70 percent divided by the percent growth rate. For example, if money in the bank grows at a rate of 7 percent per year, then the doubling time equals about 70%/7% = 10 years. In other words, at a rate of 7 percent, a 50 investment will double into 100 in 1 o years. Calculate the population... 

Some care is needed in deriving a short-time approximation for qF since introduction of eqn. (35a) would make the bracketed term zero. Evidently, the second-order term in the series expansion of exp (l21) erfc (ltv2) should be taken into account. Since this term equals l2t [22], eqn. (36) reduces for ltv2 <0.1 to the simple form... 

To account for absorption effects, an additional measurement is required. With the internal standard method, the profile for the internal standard must be measured for both calibration standards and samples. The rather small quantity of analyte present in the sample requires the addition of a similar quantity of internal standard (a, 200 ug) necessitating the measurement of the standards profile under essentially the same conditions as that of the analyte. Consequently, the internal standard method requires an additional measurement time approximately equal to that of the analyte. Under conditions noted above, this would amount to approximately thirty minutes total analysis time per sample. 

The heavy oily liquid is separated from the rest of the solution and is placed in another glass-stoppered flask. Water is added dropwise to the oily emulsion until it becomes clear. To this solution is added a volume of ethanol equal to approximately one-third the volume of the solution itself. After the mixture has been shaken and allowed to stand for a few minutes, two immiscible phases again result. The heavy oily liquid is again drawn off, and further extractions with water and with ethanol, respectively, are made (about 10 times ) until the dropwise addition of water causes a white flocculent precipitate to form. After the solution has stood several minutes, it is filtered through dry fine-textured filter paper (S S 589 or Whatman 42) and placed in a desiccator. 

The extrudate must possess sufficient mechanical strength when wet, yet it must be brittle enough to be broken down to short lengths in the spheronizer, but not to be so friable that it disintegrates completely. To achieve a narrow size distribution of spheres, the extrudate is ideally reduced to cylindrical rods of uniform length equal to approximately one and a half times their diameter. ... 

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