Unimodal distributions

The mode of distribution is simply the value of the most frequent size present. A distribution exhibiting a single maximum is referred to as a unimodal distribution. When two or more maxima are present, the distribution is caUed bimodal, trimodal, and so on. The mode representing a particle population may have different values depending on whether the measurement is carried out on the basis of particle length, surface area, mass, or volume, or whether the data are represented ia terms of the diameter or log (diameter). 

The last example brings out very clearly that knowledge of only the mean and variance of a distribution is often not sufficient to tell us much about the shape of the probability density function. In order to partially alleviate this difficulty, one sometimes tries to specify additional parameters or attributes of the distribution. One of the most important of these is the notion of the modality of the distribution, which is defined to be the number of distinct maxima of the probability density function. The usefulness of this concept is brought out by the observation that a unimodal distribution (such as the gaussian) will tend to have its area concentrated about the location of the maximum, thus guaranteeing that the mean and variance will be fairly reasdnable measures of the center and spread of the distribution. Conversely, if it is known that a distribution is multimodal (has more than one... 

The inset shows a unimodal distribution of relaxation times r = I 1 obtained by a CONTIN analysis. Besides CONTIN there is a number of alternative techniques [51] for the determination of the distribution from the correlation function. Detailed discussions of this topic have been given by Stock and Ray [52] and by Stepanek [50]. 

Most bimodal networks synthesized to date have been prepared from PDMS [88], One reason for this choice is the fact that the polymer is readily available with either hydroxyl or vinyl end groups, and the reactions these groups participate in are relatively free of complicating side reactions. These ideas can obviously be extended to higher modalities (trimodal, etc., eventually approaching an extremely broad, effectively-unimodal distribution) [102-104],... 

The use of unimodal meridional reflections for determination of the orientation distribution and desmearing (Sect. 9.6). 

The influence of finite size and imperfect orientation of the entities on the shape of the reflections. Separation of unimodal orientation distributions by means of Ruland s streak method, and assessment of the analytical shape of the orientation distribution (Sect. 9.7). 

Unimodal is a fiber-symmetrical orientation distribution g([Pg.211]

Split-Meridional Distribution. Figure 9.4a displays a case of a unimodal, non-meridional orientation distribution the most probable orientation of the structural entities does not coincide with the meridian. If the orientation distribution itself is broad, the split character of the distribution may be invisible. Then it is an apparently meridional distribution. 

If the smeared image of a point-reflection on the meridian is unimodal, the orientation distribution g(texture measurement through the reflection, I((p,s = c) contains the information sought-after [256]... 

For a unimodal equatorial reflection the treatment is more involved. If the distribution is narrow, it follows from Fig. 9.5 an approximation

approximative solution, in turn (Ruland cited by Thunemann [257], P- 28)... 

Weight average molecular weights of poly(HAMCL) with saturated or unsaturated pendent groups are relatively low, compared to Mw s of poly(HASCL), and in the range of 60,000 to 360,000 g mol as depicted in Table 2 [4,30,35,36]. Also for the poly(HAMCL) copolymers, the molecular weight distributions are unimodal. Their polydispersities are in the range of 1.6-2.4, which is narrower than the polydispersity of poly(3HB-co-3HV) copolymers, and close to the theoretical value of 2.0 for synthetic polycondensates such as chemically synthesized polyesters [54]. 

Pore size optimization is one area where developmental efforts have been focused. Unimodal pore (NiMo) catalysts were found highly active for asphaltene conversion from resids but a large formation of coke-like sediments. Meanwhile, a macroporous catalyst showed lower activity but almost no sediments. The decrease of pore size increases the molecular weight of the asphaltenes in the hydrocracked product. An effective catalyst for VR is that for which average pores size and pore size distribution, and active phase distribution have been optimized. Therefore, the pore size distribution must be wide and contain predominantly meso-pores, but along with some micro- and macro-pores. However, the asphaltene conversion phase has to be localized in the larger pores to avoid sediment formation [134],... 

To finish the discussion of MAXCOV, we want to note that the assumption of zero nuisance correlation is not the only assumption of MAXCOV. Our interpretation of a hitmax as an interval with the 50/50 mix of taxon and nontaxon members is based on the presumption that two unimodal distributions underlie the data. There are two components in this assumption. The first component is presuming that latent distributions are unimodal. In other words, any kind of distribution will work as long as it has only one maximum, which encompasses the majority of distributions one finds in statistics textbooks (e.g., normal, chi-square, gamma and many others). This is an extremely lenient assumption, but it does pose some constraint on the flexibility of the procedure. However, we do not consider this restriction particularly important, especially in the context of our state of knowledge regarding psychopathology. 

Either bimodal or unimodal distribution curves may result from environmental, dietary, and behavioral factors, the bimodal curve indicating... 

Unimodal distribution, 166-167 Universities, genetic research by, 10-15 University of Utah Genome Center, 147 University of Washington Genome Center, 147... 

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