Near-Gaussian

Chaimelling phenomena were studied before Rutherford backscattering was developed as a routine analytical tool. Chaimelling phenomena are also important in ion implantation, where the incident ions can be steered along the lattice planes and rows. Channelling leads to a deep penetration of the incident ions to deptlis below that found in the nonnal, near Gaussian, depth distributions characterized by non-chaimelled energetic ions. Even today, implanted chaimelled... 

Resolution The chromatographic separation of two components, A and B, under trace conditions with small feed injections can be characterized in terms of the resolution, R,. For nearly Gaussian peaks ... 

The globally optimal laser field for this example is presented in Fig. 2. The field is relatively simple with structure at early times, followed by a large peak with a nearly Gaussian profile. Note that the control formalism enforces no specific structure on the field a priori. That is, the form of the field is totally unconstrained during the allotted time interval, so simple solutions are not guaranteed. Also shown in Fig. 2 is the locally optimal... 

The Gibbs-Bogoliubov inequalities set bounds on A A of (AU)0 and (AU) which are easier a priori to estimate. These bounds are of considerable conceptual interest, but are rarely sufficiently tight to be helpful in practice. Equation (2.17) helps to explain why this is so. For distributions that are nearly Gaussian, the bounds are tight only if a is small enough. 

A different expansion relies on using Gram-Charlier polynomials, which are the products of Hermite polynomials and a Gaussian function [41] These polynomials are particularly suitable for describing near-Gaussian functions. Even and odd terms of the expansion describe symmetric and asymmetric deformations of the Gaussian, respectively. To ensure that P0(AU) remains positive for all values of AU, we take... 

Check. Use the Crooks relation (5.35) to check whether the forward and backward work distributions are consistent. Check for consistency of free energies obtained from different estimators. If the amount of dissipated work is large, caution may be necessary. If cumulant expressions are used, the work distributions should be nearly Gaussian, and the variances of the forward and backward perturbations should be of comparable size [as required by (5.35) for Gaussian work distributions]. Systematic errors from biased estimators should be taken into consideration. Statistical errors can be estimated, for instance, by performing a block analysis. 

These are the first two terms in a cumulant expansion [50]. We note here that the convergence of cumulant expansions is a subtle issue. Generally, if the statistics are nearly Gaussian, the cumulant expansion yields a good approximation. If the statistical distribution is not Gaussian, however, the cumulant expansion diverges with the inclusion of higher-order terms. See [29] and references therein for more discussion of this point. 

The behavior predicted by this equation is illustrated in Fig. 16-33 with N = 80. xF = (evtp/L)/[( 1 - e)(p K, + ,)] is the dimensionless duration of the feed step and is equal to the amount of solute fed to the column divided by tne sorption capacity. Thus, at xF = 1, the column has been supplied with an amount of solute equal to the stationary phase capacity. The graph shows the transition from a case where complete saturation of the bed occurs before elution (xF= 1) to incomplete saturation as xF is progressively reduced. The lower curves with xF < 0.4 are seen to be nearly Gaussian and centered at a dimensionless time xm — (1 — tf/2). Thus, as xF -> 0, the response curve approaches a Gaussian centered at t = 1. 

Figure 1.11. A non-premixed scalar PDF as a function of time for inert-scalar mixing. Note that at very short times the PDF is bi-modal since all molecular mixing occurs in thin diffusion layers between regions of pure fluid where = 0 or 1. On the other hand, for large times, the scalar PDF is nearly Gaussian.

In fully developed homogeneous turbulence,7 the one-point joint velocity PDF is nearly Gaussian (Pope 2000). A Gaussian joint PDF is uniquely defined by a vector of expected values (j, and a covariance matrix C ... 

The scalar fields appearing in Figs. 3.7 to 3.9 were taken from the same DNS database as the velocities shown in Figs. 2.1 to 2.3. The one-point joint velocity, composition PDF found from any of these examples will be nearly Gaussian, even though the temporal and/or spatial variations are distinctly different in each case.11 Due to the mean scalar... 

Almost simultaneously with the first attempt to determine molecular weight from equilibrium sedimentation, Rinde tried to widen this method to include determination of the molecular weight distribution (MWD) of a polydisperse system (3). Unfortunately, this attempt proved to be more complicated and did not result in establishment of a reliable routine. Since the appearance of Rinde s dissertation in 1928, many investigators have tried to determine MWD. Most of these efforts, however, did not provide a successful comprehensive technique (4-16). This objective has been accomplished only in a few cases under very limited conditions, such as in case of a Gaussian or near Gaussian MWD, in which only characterizing parameters had to be determined. Scholte (17, 18) determined MWD by performing an experimental procedure based on several equilibrium experiments. 

Nonlinear optical studies were carried out using a combination of nonlinear absorption, self-focusing and degenerate four wave mixing measurements. The measurements were made using a Quantel Nano-Pico system that permits operation at either 10 ns or 100 ps pulse lengths at 1064 nm. The 10 ns pulses were TEM mode and temporally smoothed to a near gaussian (H). 

The main source of Lorentzian broadening is dipole-dipole interaction between paramagnetic species, often termed spin-spin broadening . This will arise if solutions are too concentrated, and can in general be avoided by suitable dilution. Hence most observed lines are nearly Gaussian in shape. 

Sometimes a measurement involves a single piece of calibrated equipment with a known measurement uncertainty value o, and then confidence limits can be calculated just as with the coin tosses. Usually, however, we do not know o in advance it needs to be determined from the spread in the measurements themselves. For example, suppose we made 1000 measurements of some observable, such as the salt concentration C in a series of bottles labeled 100 mM NaCl. Further, let us assume that the deviations are all due to random errors in the preparation process. The distribution of all of the measurements (a histogram) would then look much like a Gaussian, centered around the ideal value. Figure 4.2 shows a realistic simulated data set. Note that with this many data points, the near-Gaussian nature of the distribution is apparent to the eye. 

Figure 11 Examples of transient anti-Stokes spectra with coherent SFG artifacts, obtained from methanol at 300 K with C-H stretch (3020 cm-1) pumping, (a) and (c) Experimental results at two times. The dashed curves represent the (nearly Gaussian) spectrum of the SFG artifact, (b) and (d) Recovered lineshapes with SFG contribution subtracted away. At early time (—1 ps) mainly the higher frequency C-FI stretch is seen. As times passes, energy is redistributed among the two C-FI stretch and the O-FI stretch transitions. (From L. K. Iwaki, unpublished.)...

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