Uni-modal distribution

Method of cumulants is suitable for characterizing a reasonably narrow uni modal distribution. When G(T, 0) is broad or can not be adequately represented by unimodal distribution, the method of cumulants fails. Indeed, two very different distributions may have the same first two moments, and thus can not be distinguished from each other based on the method of cumulants [46],... 

Harris s three-parameter equation is very flexible and can be closely fitted to most uni-modal distributions. As its mathematical form lends itself easily to further treatment it has been used in a relatively involved evaluation of results from scanning centrifugal sedimentation instruments for particle size analysis. ... 

For a uni-modal distribution centered around (x), it follows from the exact equation (3.76) that the approximate closed equation is ... 

In order to estimate how long a uni-modal distribution can survive in the migration process the approximate mean value and variance equations (4.53, 31) are solved, as an example, for parameters chosen as for the results of Fig. 4.8 a, b and for a reasonably rhosen initial variance. 

The equations (4.84, 85) and their solutions are based on the approximation (4.83) which is valid for a uni-modal probability distribution only. Contrary to this assumption, however, any initially uni-modal distribution can be expected to develop first into the doubled-peaked quasi-stationary solution Pqs (n) shown in Fig. 4.10 and then to go over finally to the exact stationary solution, that of an extinct population. Consequently deviations of the true time paths of (n), and o t) from those described by (4.84, 85) are to be expected. A calculation of the development of a model population with time with the exact master equation (4.63) and with the parameters A = 0.5, n = 0.2, and bi = 0.01 confirms this expectation (Figs. 4.11, 12). The Fig. 4.11 shows the exactly calculated change with time of a distribution which starts as normal distribution but soon develops into the form of the bimodal quasi-stationary distribution Pqs(n). In Fig. 4.12 and for the same model parameters the exact paths of the mean value (n)(and the variance a t) are compared with the paths obtained by solving the approximate equations (4.84, 85). 

The stress magnitude s history of three maker-particles which have different residence time is described in Fig. 7. The curve oscillates periodically due to the screw rotation. We focus on the highest value of the stress level as circled in Fig. 7 and its distribution is estimated by Eq. 4. The bimodal stress distribution is obtained in the case of the rotor at low rotational speed in Fig. 8. The height of the distribution curve corresponds to the flow rate and about half of the polymer melt does not overpass the wingtip at lOOrpm. On the contrary, the KB and rotor at high rotational speed shows uni-modal distribution and uniform stress induced mixing is expected. 

Figure 5. Left Frame Comparison between a measured post-Pinatubo bimodal size distribution (solid line) and that retrieved by the LUT from its best-fit extinction spectrum (i.e., at = 1.6) (dashed line). The fitting parameters of the measured bimodal and the LUT retrieved uni-modal are shown in the table. Right Frame Calculated extinction spectra for size distributions in the left frame The error bars on the spectrum calculated from the measured bimodal (open circles) are derived from the relative errors on coincident SAGE II and CLAES measure-...

Figure 6. Results of applying the LUT algorithm to synthetic extinction spectra calculated from measured pre- and post-Pinatubo size distributions obtained from Pueschel el al. [9], Goodman el al [10] and Deshler el al. [11,12]. R,/bimodal) is the effective radius of the measured bimodal size distribution, and R /uni modal) is the corresponding effective radius returned by the LUT. The dots are results obtained when a range of distribution widths are considered in the LUT calculations and the crosses are results obtained when at is restricted to the value that yields the best fit between calculated and measured extinction spectra. The solid and dashed curves are second order polynomial fits to the dots and crosses, respectively.

Unfolded state a is characterized by a wide uni-modal and asymmetrical distribution of R, with a slow rise and a sharp drop. The distribution becomes narrower and more symmetrical when the averaging time window is larger than the characteristic time scale of conformational dynamics in this unfolded state. This indicates a correlation between structure and dynamics, where conformations that correspond to shorter R move on faster time scales. 

Unfolded state (3 is characterized by a wide uni-modal and symmetrical distribution of the end-to-end distance. The distribution narrows down in a uniform manner with increasing Tw Thus, in the case of unfolded state / , one does not observe a correlation between R and the time scale of conformational dynamics. This behavior is similar to that previously observed in the case of a free-jointed homopolymer model. [54]... 

Doki, N. Kubota, N. Yokota, M. Kimura, S. Sasaki, S. Production of sodium chloride crystals of uni-modal size distribution by batch dilution crystallization. J. Chem. Eng. Japan 2002, 35 (11), 1099-1104. 

In a number of cases, the initial bond-length distribution was clearly not uni-modal, e.g. Figure A.2a. Where possible, such distributions were resolved into their unimodal components (as in Figure A.2c) on chemical or structural criteria. The case illustrated in Figure A.2, for Cu-Cl bonds, is one of the most spectacular examples, owing to the dramatic consequences of changes in oxidation state and coordination number and of Jahn-Teller effects on the structures of copper complexes. 

The correlation between the experimental results and the calculated values for both the uni-modal and bi-model distribution is good, see Figure 8. For both, low and... 

Hgurc 2 indicates different types of distribution found in emulsions uni-modal of the log normal type produced by turbulent homogeneous stirring narrowly shaped or highly polydi.spersed bimodal emulsions resulting from a mi, -ture of iw o emulsions, which by the way might be intentionally made to attain a low viscosity. 

Some Rheovibron viscoelasticity results have been reported for bimodal PDMS networks. Measurements are first carried out on uni-modal networks consisting of the chains used in combination in the bimodal networks. One of the important observations was the dependence of crystallinity on the network chain-length distribution. 

The non-stationary migration process will now be examined. Starting from a probability distribution P(x, y t = 0) with one sharp peak only, it seems reasonable to assume that the distribution remains uni-modal over a considerable period of time. As long as this assumption is correct the approximate but... 

The equations (4.80, 81) are not closed. However, as long as the accompanying distribution remains essentially uni-modal, the approximations... 

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