Frequency function, experimental

We are unaware of any previously proposed model, statistical or otherwise, which will explain the frequency functions experimentally observed. The following diseussion is an attempt to establish a consistent model. 

Our results indicate that dispersion coefficients obtained from fits of pointwise given frequency-dependent hyperpolarizabilities to low order polynomials can be strongly affected by the inclusion of high-order terms. A and B coefficients derived from a least square fit of experimental frequency-dependent hyperpolarizibility data to a quadratic function in ijf are therefore not strictly comparable to dispersion coefficients calculated by analytical differentiation or from fits to higher-order polynomials. Ab initio calculated dispersion curves should therefore be compared with the original frequency-dependent experimental data. 

In practice, PSD curves can be obtained directly from the experimental, digitized peak (fractogram), yfV. i) once it is converted to a function of particle diameter dj with the use of Eq. (1). The frequency function of particle size f is expressed as [3]... 

Mixing sequences for total through-bond correlation spectroscopy in solids (TOBSY) have been developed for fast MAS experiments. Possible sequences with the desired Hamiltonian (the homonuclear isotropic J interaction) have been identified using lowest order average Hamiltonian theory combined with numerical simulations as a function of the MAS frequency. An experimental TOBSY spectrum of a uniformly C-labelled decapeptide at 20 kHz MAS has been obtained using one of the new sequences. The spectrum allows to assign the resonances to the respective spin systems. 

While clever experimental design may make it possible to test predictions of single-cell models without recourse to measurements of f, direct experimental access to the frequency function is often desirable and sometimes necessary in order to evaluate single-cell control and kinetic properties using the strategy just outlined. 

Figure 9. Comparison of experimentally measured protein content frequency function (dots) for S. pombe, grown in a chemostat with D = 0.246 hr , with best fit simulations (lines) using the kinetic forms of Equation 2. Reproduced, with permission, from Ref. 25. Copyright 1981, John Wiley Sons, Inc.

As a test of the range of applicability of the kinetics determined in the steady-state measurements, the transient population balance equation has been solved, using the kinetics determined from steady state, to simulate the sequence of protein content frequency functions obtained in synchronous growth of this organism. The simulation results are in very good qualitative agreement with the experimental measurements of the corresponding quantities (28). 

Cox and Merz [C20] have made the remarkable experimental observation that the non-Newtonian shear viscosity function of flexible chain polymers has the same form as the complex viscosity-frequency function, i.e.. 

In Fig. 3 we show the stimulated emission frequencies observed in the forward direction as a function of laser frequency. The experimental points were obtained from many stimulated emission spectra and at various angles for each laser frequency. The solid lines indicate the expected positions of thp various TPS and ESR processes at zero-field intensity. 

The photoabsorption spectrum a(co) of a cluster measures the cross-section for electronic excitations induced by an external electromagnetic field oscillating at frequency co. Experimental measurements of a(co) of free clusters in a beam have been reported, most notably for size-selected alkali-metal clusters [4]. Data for size-selected silver aggregates are also available, both for free clusters and for clusters in a frozen argon matrix [94]. The experimental results for the very small species (dimers and trimers) display the variety of excitations that are characteristic of molecular spectra. Beyond these sizes, the spectra are dominated by collective modes, precursors of plasma excitations in the metal. This distinction provides a clear indication of which theoretical method is best suited to analyze the experimental data for the very small systems, standard chemical approaches are required (Cl, coupled clusters), whereas for larger aggregates the many-body perturbation methods developed by the solid-state community provide a computationally more appealing alternative. We briefly sketch two of these approaches, which can be adapted to a DFT framework (1) the random phase approximation (RPA) of Bohm and Pines [95] and the closely related time-dependent density functional theory (TD-DFT) [96], and (2) the GW method of Hedin and Lundqvist [97]. 

However, this approach is extremely sensitive to noise and can fail to produce good results for experimental data with noise. One successful approach was a Bayesian-based iterative method [6], which treats point spread functions as probability-frequency functions and applies Bayes s theorem [7],... 

Figure 16 shows typical proton spin-lattice relaxation dispersion data for polyethylene melts as an illustration of the three-component behavior of polymer melts. For comparison with model theories the chain-mode regime represented by component B is suited best and will be discussed in detail. It will be shown that the NMR relaxometry frequency window of typically 10 Hz< V <10 Hz (for proton resonance) almost exclusively probes the influence of chain modes represented by component B (compare Fig. 5). That is, the correlation function experimentally relevant for spin-lattice relaxation dispersion may be identified with component B according to...

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