Gaussian distribution propagation

GRBs emit photons in pulses containing photons with a combination of different wavelengths, whose sources are believed to be ultrarelativistic shocks with Lorentz factor y = (9(100) [28]. Let us consider a wavepacket of photons emitted with a Gaussian distribution in x at the time t = 0. One has to find out how such a pulse would be modified at the observation point at a subsequent time t, because of the propagation through the spacetime foam, as a result of the refractive index. This is similar to the motion of a wavepacket in a conventional dispersive medium. The Gaussian wavepacket may be expressed at t 0 as the real part of... 

A second theory proposes that the propagating radical exists with a Gaussian distribution of dihedral angles, centered about a single preferred conformation. 

Of primary importance are the estimates of the errors of the parameters which may be used directly in interpreting or evaluating the parameters or in the propagation of errors upon the calculation of subsequent quantities. The actual parameters should be the most probable values if the errors are truly random, following a Gaussian distribution. From the experimental data and the associated errors, an experimental apparatus (procedure) can be pre-evaluated in regards to the desired accuracy of the results. 

As long as the photodissociation reaction is fairly direct, the time-dependent formulation is fruitful and provides insight into both the process itself and the relationship of the final-state distributions to the absorption spectrum features. Moreover, solution of the time-dependent Schrodinger equation is feasible for these short-time evolutions, and total and partial cross sections may be calculated numerically.5 Finally, in those cases where the wavepacket remains well localized during the entire photodissociation process, a semi-classical gaussian wavepacket propagation will yield accurate results for the various physical quantities of interest.6... 

This heterogeneity may, however, also be expressed in terms of polymer lifetimes instead of propagation rates. By assuming a Gaussian distribution of chain lifetimes Mussa (35) has been able to derive molecular weight distributions of the type observed experimentally. This treatment appears to imply a chain termination which has the form of an error function round a mean value. On the whole it seems that a variable propagation rate is the more likely. 

Computer simulation was also performed on another model. Thus, O Donnell et al. and Iwasaki et al. independently illustrated that the 13-line, 9-line, and 5-line spectra can be simulated by assuming a statistical distribution of conformational angles about the most probable conformation of the propagating radical. When the conformation with 0 = 65° and 0, = 55° is assumed as the most stable, the 13-line spectrum shown in Fig. 15 a is obtained- When a Gaussian distribution of conformational angles is assumed about this conformation, the 13-line spectrum changes to a 9-line one, with an intensity distribution very sensitive to the half-width of the Gaussian... 

OR is derived from statistical propagation of error theory. Consider a set of J wavelengths, r, where the observed errors are independently and identically distributed along a Gaussian distribution with a mean of zero and a standard deviation of o. Given a regression vector, b, the magnitude of the projection r onto b is indicative of a sample quality or quantity, c. This quantity parameter is hence... 

Fig. 3.13 Amplitude of the electric field as a function of position along the propagation axis (y) in wave groups with Gaussian distributions of wavelengths. The widths (FWHM) of the distributions are 0.1% ( short dashes), 2% (long dashes) or 5% (solid curve) of the mean wavelength (X. The amplitudes are normalized at y = 0...

Cabaniss, S., Propagation of Uncertainty in Aqueous Equilibrium Calculations Non-Gaussian Output Distributions, Anal. Chem. 69, 1997, 3658-3664. 

Figure 5.1.7 shows the propagator of the motion measured for a clean and a biofilm impacted capillary [14,15] and the residence time distributions calculated for each from these velocity distributions. The clean capillary gives an experimental propagator equal to the theoretical velocity distribution convolved with a Gaussian diffusion curve [14], as shown in Figure 5.1.2. For the flow around the biofilm structure note the appearance of a high velocity tail indicating higher probability of large displacements relative to the clean capillary. The slow flow peak near zero displacement results from the protons trapped within the EPS gel matrix where the...

In conclusion, the ordinary central limit theorem (CLT) establishes that all the propagators sharing the same second moment (c2) yields in the time asymptotic limit the same gaussian pdf. What is the property that the propagators with a diverging second moment must have in common to produce the same pdf in the time asymptotic limit The answer to this question is equivalent to establishing the GCLT [45]. The answer to this important question rests on the anti-Fourier transform of the expression of Eq. (102), which turns out [46] to yield for v oc a distribution proportional to 1/ E,, with... 

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