Amplitude randomness

Fig. 5 Object (shaded) used in computer simulations Its diffraction image is the solid curve The data image (dashed curve) is the diffraction image plus 4% amplitude random noise All plotted points are spaced by one-half the Nyquist interval. Hence to resolve the central dip in the object would require superresolution.

Likewise, mechanoreceptors in the skin can be made more sensitive by applying a small-amplitude random signal (Figure 8.2.21). By itself, this electrical or mechanical noise is less than the sensor threshold, and not detectable. When added to a small primary signal, however, some of the noise pulses exceed the threshold, and the sensor fires. Thus, the magnitude of the adequate stimulus has been made smaller. 

In ZF-/tSR the longitudinal muon spin relaxation function G t) is directly deduced from the time-differential measurement of the forward/backward muon decay asymmetry, without any disturbance of the spin-glass system by an external field. (No depolarization of the muon spin means G = l, complete depolarization G =0.) The observed time evolution G (t) of muon-spin polarization reflects amplitudes, randomness, and fluctuations of local magnetic fields at muon sites in the specimen. There appear two essential problems in analyzing pSR experiments on spin glasses (i) One has to make model assumptions about the shape of G (t) (ii) Any relaxation slower than 10 s appears as a static component in pSR (lifetime of the muon is = 2.2 x 10 s). 

Fig. 11.34 Two variable-amplitude random-sequence load fluctuation investigated by Barsom [50]...

As the spins precess in the equatorial plane, they also undergo random relaxation processes that disturb their movement and prevent them from coming together fiilly realigned. The longer the time i between the pulses the more spins lose coherence and consequently the weaker the echo. The decay rate of the two-pulse echo amplitude is described by the phase memory time, which is the time span during which a spin can remember its position in the dephased pattern after the first MW pulse. Tyy is related to the homogeneous linewidth of the individual spin packets and is usually only a few microseconds, even at low temperatures. 

Figure C3.6.10 Defect-mediated turbulence in tire complex Ginzburg-Landau equation, (a) The phase, arg( ), as grey shades, (b) The amplitude [A], witli a similar color coding. In tire left panel topological defects can be identified as points around which one finds all shades of grey. Note tire apparently random spatial pattern of amplitudes.

The two sources of stochasticity are conceptually and computationally quite distinct. In (A) we do not know the exact equations of motion and we solve instead phenomenological equations. There is no systematic way in which we can approach the exact equations of motion. For example, rarely in the Langevin approach the friction and the random force are extracted from a microscopic model. This makes it necessary to use a rather arbitrary selection of parameters, such as the amplitude of the random force or the friction coefficient. On the other hand, the equations in (B) are based on atomic information and it is the solution that is approximate. For ejcample, to compute a trajectory we make the ad-hoc assumption of a Gaussian distribution of numerical errors. In the present article we also argue that because of practical reasons it is not possible to ignore the numerical errors, even in approach (A). 

Although 0-switching produces shortened pulses, typically 10-200 ns long, if we require pulses in the picosecond (10 s) or femtosecond (10 s) range the technique of mode locking may be used. This technique is applicable only to multimode operation of a laser and involves exciting many axial cavity modes but with the correct amplitude and phase relationship. The amplitudes and phases of the various modes are normally quite random. 

While the random fluctuations apparent are a function of the scaling factor for the traces, the two show different amplitudes. The top trace has a relatively small fluc tuation, while the bottom trace shows a larger one. 

Signals generated by high-speed maehinery are very eomplex in nature and are generated by several forees with a net effeet that masks the pure tones. The random portion of the signal, whieh is blended with the pure tones, is ealled noise. The ratio of the total amplitude (area under speetrum) to that of the noise is ealled the signal-to-noise (S/N) ratio. Sometimes this ratio is expressed in deeibels, or db, as follows ... 

Averaging is a teehnique to improve the S/N ratio. Two or more sueeessive speetra made up of both periodie and random (noise) signals are added together and then averaged. This eombination results in a speetrum with a periodie eomponent that is mueh the same as when viewed in the instantaneous signals but with random peaks of mueh less amplitude. This result oeeurs beeause the period peak stays at a fixed frequeney in the speetrum, while the noise peak is fluetuating in frequeney over the speetrum. 

Electrochemical noise This is a non-perturbation method and is defined as random low frequency low amplitude fluctuations either of the potential or current in a corroding system. Analysis of the corrosion potential noise can provide information relating to both the mechanism and kinetics of the cor-... 

For explicitness, let us assume that off-diagonal disorder is caused by chain twists, which randomly diminish the overlap between the n-orbitals of neighboring carbon atoms (see Fig. 3-7). The electron hopping amplitudes that depend both on the interatomic distances and on the relative orientation of the electronic orbitals on neighboring atoms can then be written in the form ... 

Here, the second term describes the change of the hopping amplitudes due to the displacement of the atoms parallel to the chain [cf. Eq. (3.5)] and the third term is a random contribution resulting from the conformational disorder (chain twists). While the lattice displacements u are dynamic variables, the fluctuations dl +t due to disorder are assumed to be frozen ( quenched disorder). 

Note that, while the random chain twists always decrease the hopping amplitudes (all ()/ , + are negative), // (a) can be both positive and negative, as it is the alternating part of the fluctuations. As in the FCM, we consider white noise disorder with a correlation function given by Eq. (3.22). This corresponds to independent random variations of the hopping amplitudes <5/ on different bonds. 

Randomness.—The word random is used frequently to describe erratic and apparently unpredictable variations of an observed quantity. The noise voltage measured at the terminals of a hot resistor, the amplitude of a radar signal that has been reflected from the surface of the sea, and the velocity measured at some point in a turbulent air flow are all examples of random or unpredictable phenomena. 

In astronomy, we are interested in the optical effects of the turbulence. A wave with complex amplitude U(x) = exp[ irefractive index, resulting in a random phase structure by the time it reaches the telescope pupil. If the turbulence is weak enough, the effect of the aberrations can be approximated by summing their phase along a path (the weak phase screen approximation), then the covariance of the complex amplitude at the telescope can be shown to be... 

Leadbetter AJ, Norris EK (1979) Molec Phys 38 669. There are different contributions which give rise to a broadening a of the molecular centre of mass distribution function f(z). The most important are the long-wave layer displacement thermal fluctuations and the individual motions of molecules having a random diffusive nature. The layer displacement amplitude depends on the magnitude of the elastic constants of smectics ... 

In the random phase approximation, the transition amplitude from state 0) to l) for any one electron operator O may be written as... 

When it is employed to specify an ensemble of random structures, in the sense mentioned above, the MaxEnt distribution of scatterers is the one which rules out the smallest number of structures, while at the same time reproducing the experimental observations for the structure factor amplitudes as expectation values over the ensemble. Thus, provided that the random scatterer model is adequate, deviations from the prior prejudice (see below) are enforced by the fit to the experimental data, while the MaxEnt principle ensures that no unwarranted detail is introduced. 

The calculations discussed in the previous section fit the noise-free amplitudes exactly. When the structure factor amplitudes are noisy, it is necessary to deal with the random error in the observations we want the probability distribution of random scatterers that is the most probable a posteriori, in view of the available observations and of the associated experimental error variances. 

The error-free likelihood gain, V,( /i Z2) gives the probability distribution for the structure factor amplitude as calculated from the random scatterer model (and from the model error estimates for any known substructure). To collect values of the likelihood gain from all values of R around Rohs, A, is weighted with P(R) ... 

At each stage during the structure determination process, the current structural model gives an estimate of the prediction variance Z2 to be associated with the calculated amplitude. The contribution of the random part of the structure to this prediction variance decreases while the structure determination proceeds, and uncertainty is removed by the fit to the observations. Rescaling of Z2 would be needed during the optimisation of the Bayesian score. 

Samples of the poly(dialkylphosphazenes) 1 and 2 displayed X-ray powder diffraction patterns characteristic of crystalline regions in the materials. The peaks in the diffraction pattern of 1 were of lower amplitude and greater angular breadth than those of 2. These data indicate that poly(diethylphosphazene) (2) is highly crystalline while poly(dimethyl-phosphazene) (1) is more amorphous with smaller crystalline zones. This high degree of crystallinity is probably responsible for the insolubility of 2 as noted above. All of the phenyl substituted polymers 3-6 were found to be quite amorphous in the X-ray diffraction studies, a result that is further evidence for an atactic structure of the poly(alkylphenylphosphazenes) 3 and 4 and for a random substitution pattern in the copolymers 5 and 6. 

The random coil amide I VCD pattern is exacdy the same shape, but smaller in amplitude and shifted in frequency from the pattern characteristic of poly-L-proline II (PLP II) which is a left-handed 3ihelix of trans peptides (Kobrinskaya et al., 1988 Dukor and Keiderling, 1991 Dukor et al., 1991 Dukor and Keiderling, 1996 Keiderling et al., 1999b). This... 

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