Amplitudes and Distributions

sidechain atoms normally increase in their fluctuation amplitudes as one goes away from the mainchain exceptions can occur, for example, for polar sidechains that are hydrogen bonded, so that the middle portion of the sidechain has larger fluctuations than either end. Table II uses a lysozyme simulation to illustrate some of these results.191 

Root-Mean-Square Fluctuations of Lysozyme Atoms 

Of interest also are the results concerning deviations of the atomic fluctuations from simple isotropic and harmonic motion. As discussed in Chapt. XI, most X-ray refinements of proteins assume (out of necessity, because of the limited data set) that the motions are isotropic and harmonic. Simulations have shown that the fluctuations of protein atoms are highly anisotropic and for some atoms, strongly anharmonic. The anisotropy and anharmonicity of the atomic distribution functions in molecular dynamics simulations of proteins have been studied in considerable detail.193 197 To illustrate these aspects of the motions, we present some results for lysozyme196 and myoglobin.197 If Ux, Uy, and Uz are the fluctuations from the mean positions along the principal X, Y, and Z axes for the motion of a given atom and the mean-square fluctuations are 

The quantity A determines the amount by which the ratio of the fluctuation in the principal X direction to that of the average of the fluctuations in the other two directions (Y and Z) exceeds that of an isotropic distribution, for which Ai is zero. A second measure of the anisotropy is 

The fluctuations of a significant fraction of the protein atoms are found to be anharmonic i.e., the potentials of mean force for the atomic displacements deviate from the simple parabolic form that would be obtained at sufficiently low temperature. The third and fourth moments of the distribution 

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Figure 2. Correlation between DSC amplitude and distributed NMR T2 relaxation times measured during step-wise heating ofpork from 25 °C to 75 °C. Reported values are Jack-knife based significance levels presented as number of standard deviations away from zero (presenting non-significance).

ANS You are asking how sensitive is the inverse problem to a priori knowledge of the amplitude and distribution of potentials on the epicardium. From stability estimates of the inverse problem, derived with the help of Bob Grossman from Princeton, using an integral operator approach to the inverse problem, I can answer that the inverse problem is very sensitive to a priori information. Although the epicardial potentials are sensitive to noise on the torso, the solution of the inverse problem is even more sensitive to the a priori bounds that you impose on the epicardial potentials. 

Figure Bl.2.7. Time domain and frequency domain representations of several interferograms. (a) Single frequency, (b) two frequencies, one of which is 1.2 times greater than the other, (c) same as (b), except the high frequency component has only half the amplitude and (d) Gaussian distribution of frequencies.

If T2>T, then AS>0 since Cp and T are >0. From a molecular point of view, heating a solid increases the amplitudes and energy distributions of the vibrations of the molecules in the solid, resulting in increased disorder. 

Since a heterodyne receiver is an amplitude and phase detector, it could nicely be used to correlate optical signals received at various remote sites. The local oscillator can be a single laser distributed by optical fiber to the various sites or local lasers that can be synchronized "a posteriori" by reference to a common source (e.g. a bright star). 

Because the wavelength of the RF signal is of the order of the substrate dimensions (3 m at 100 MHz), it can be expected that uniform deposition is more difficult at these high frequencies [477]. In fact, a practical optimum frequency around 60-70 MHz is used [478, 479], which provides a good compromise between high deposition rate and attainability of uniform deposition. Further, the use of a distributed RF electrode network where all nodes have identical amplitude and phase improves the homogeneity of deposition [480]. 

In the individual compartments quasi-steady state is achieved depending on emissions, degradation rates and spatial distribution of DDT. According to the seasonality of the parameters affecting degradation rates, e.g. temperature and oxidant abundance, the compartmental burdens in steady state follow a seasonal cycle. As the sources and consequently most of the DDT mass is located in the northern hemisphere, the cycle is defined by the climate of that hemisphere. Times needed to to achieve quasi staty state in the compartments are equal in the AGG and SAT experiment, as well as amplitude and phase of the burden time series. Vegetation reaches quasi-steady state within 2-4 years, and atmosphere already within 2 years. These... 

The crystallinity of organic pigment powders makes X-ray diffraction analysis the single most important technique to determine crystal modifications. The reflexions that are recorded at various angles from the direction of the incident beam are a function of the unit cell dimensions and are expected to reflect the symmetry and the geometry of the crystal lattice. The intensity of the reflected beam, on the other hand, is largely controlled by the content of the unit cell in other words, since it is indicative of the structural amplitudes and parameters and the electron density distribution, it provides the basis for true structural determination [32],... 

Brod, F.P.R., Park, K.J. and de Almeida, R.G., Image analysis to obtain the vibration amplitude and the residence time distribution of a vibro-fluidized dryer. Food Bioprod. Proc., 82 (2004) 157-163. 

A cross-sectional view (yz) of the amplitude and phase of the focal field along the optical axis is depicted in Figure 9.1c and d, respectively. The phase distribution is plotted in its unwrapped representation and the linear phase due to propagation has been subtracted for clarity. It is evident that the phase before the focal plane and the phase after the focal plane (z = 0) are not the same. Along the optical axis, the phase... 

The results of two different optimisations of the production of charged states >11+ are presented in Fig. 2b. The dashed curve is the TOF distribution obtained when optimising 80 independent phases across the spectrum. By contrast with the Fourier Transform-limited pulse, ions up to 25+ are present in the TOF distribution The corresponding pulse shape (as determined from the autocorrelation in Fig. 2c) is a sequence of two pulses of equal amplitude and separated by 500 fs. To test the importance of the time delay between the two pulses, we performed restricted optimisations where a periodic phase was applied across the spectrum along with a quadratic term. In this case the period and amplitude of the oscillatory part... 

The terms that occur high in the series have a small amplitude and contribute little. Therefore, the zeros of the zeta function will be determined essentially by the first few terms associated with the least unstable periodic orbits. Contrary to systems with one degree of freedom, no factorization is possible so that the different periodic orbits have additive contributions that interfere. The distribution of zeros will therefore have the tendency to become irregular, contrary to classically integrable systems. 

As it is known [5], the intensity of the scattered light gives us an information about the system s disorder, e.g., presence therein of pores, impurities etc. Since macroscopically liquid is homogeneous, critical opalescence arises due to local microscopic inhomogeneities - an appearance of small domains with different local densities. In other words, liquid is ordered inside these domains but still disorded on the whole since domains are randomly distributed in size and space, they appear and disappear by chance. Fluctuations of the order parameter have large amplitude and involve a wide spectrum of the wavelengths (which results in the milk colour of the scattered light). 

The behaviour of the correlation functions shown in Fig. 8.5 corresponds to the regime of unstable focus whose phase portrait was earlier plotted in Fig. 8.1. For a given choice of the parameter k = 0.9 the correlation dynamics has a stationary solution. Since a complete set of equations for this model has no stationary solution, the concentration oscillations with increasing amplitude arise in its turn, they create the passive standing waves in the correlation dynamics. These latter are characterized by the monotonous behaviour of the correlations functions of similar and dissimilar particles. Since both the amplitude and oscillation period of concentrations increase in time, the standing waves do not reveal a periodical motion. There are two kinds of particle distributions distinctive for these standing waves. Figure 8.5 at t = 295 demonstrates the structure at the maximal concentration... 

More complicated case of standing waves emerges in the regime of chaotic oscillations. Here the equations for the correlation dynamics are able to describe auto-oscillations (for d — 3). However, a noise in concentrations changes stochastically the amplitude and period of the standing waves. It results finally in the correlation functions with non-monotonous behaviour. Despite the fact that the motion of both concentrations and of the correlation functions is aperiodic, the time evolution of the correlation functions reveals several distinctive distributions shown in Fig. 8.6. 

Pfr(r,q) is related to the framework vibration and represents the probability distribution of the individual distance of a hypothetical molecule with a fixed value for q. P(q) is the large amplitude probability distribution, which in a classical approximation may be expressed as... 

It is important to remember that this equation depends on the assumption that the quantizer is a fixed point, mid-tread converter with sufficient resolution so that the resulting quantization noise (enoise) is white. Furthermore, the input is assumed to be a full scale sinusoidal input. Clearly, few real world signals fit this description, however, it suffices for an upper bound. In reality, the RMS energy of the input is quite different due to the wide amplitude probability distribution function of real signals. One must also remember that the auditory system is not flat (see the chapter by Kates) and therefore SNR is at best an upper bound. 

Mean Frequencies/Amplitudes and Corresponding Density Distributions. Form of Rotational Absorption Band... 

The same arguments as were used in Section IX.B.3 express the mean values and the distributions over angular amplitudes and the RR frequencies in the form... 

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