Bimodal distribution approximation

A second important characteristic is the value otj. of the elongation at which rupture occurs. The corresponding values of r/rm show that rupture generally occurred at approximately 80-90% of maximum chain extensibility (12). These quantitative results on chain dimensions are very important but may not apply directly to other networks, in which the chains could have very different configurational characteristics and in which the chain length distribution would presumably be quite different from the very unusual bimodal distribution intentionally produced in the present networks. 

The basis of all performance criteria are prediction errors (residuals), yt - yh obtained from an independent test set, or by CV or bootstrap, or sometimes by less reliable methods. It is crucial to document from which data set and by which strategy the prediction errors have been obtained furthermore, a large number of prediction errors is desirable. Various measures can be derived from the residuals to characterize the prediction performance of a single model or a model type. If enough values are available, visualization of the error distribution gives a comprehensive picture. In many cases, the distribution is similar to a normal distribution and has a mean of approximately zero. Such distribution can well be described by a single parameter that measures the spread. Other distributions of the errors, for instance a bimodal distribution or a skewed distribution, may occur and can for instance be characterized by a tolerance interval. 

In a similar study, Allen and co-workers (1996) determined the particle size distribution for 15 PAHs with molecular weights ranging from 178 (e.g., phenan-threne) to 300 (coronene) and associated with urban aerosols in Boston, Massachusetts. As for BaP in the winter (Venkataraman and Friedlander, 1994b), PAHs with MW >228 were primarily present in the fine aerosol fraction (Dp < 2 /Am). A study of 6-ring, MW 302 PAH at the same site showed bimodal distributions, with most of the mass in the 0.3- to 1.0-/zm particle size size range a smaller fraction was in the ultrafine mode particles (0.09-0.14 /xm) (Allen et al., 1998). For PAHs with MW 178—202, the compounds were approximately evenly distributed between the fine and coarse (D > 2 /am) fractions. Polycyclic aromatic hydrocarbons in size-segregated aerosols col-... 

If the system suddenly and irreversibly leaks into the region around r=40, indicating a bottleneck, the function f(r) should be revised to flatten out both peaks of the bimodal distribution, and produce an approximately uniform distribution over the whole range r=20 to 40. Sampling this flattened-out ensemble serves two purposes ... 

Chloride ions are three- and four-coordinated. The neutron diffraction analyses of the amino acid hydrochlorides provided the data in Thble 11.1, and the more extensive data obtained from the X-ray analyses of hydrochlorides of nucleic acid constituents are given in Thble 11.2. The data indicate clearly that the chloride ion may be three- or four-coordinated. When three-coordinated, the bonds may be in planar or in pyramidal configuration, with no bimodal distribution. The ligands of four-coordinated chloride ions are only very approximately in tetrahedral con-... 

Small angle neutron scattering may be used in a somewhat similar way. It is especially sensitive to concentrations of hydrogen atoms. A study of the 3-25 nm range by this method showed a bimodal distribution of pores peaking at approximately 5 nm and 10 nm, but accounting for less than 2% of the total porosity (A16,P39). The pores were approximately spherical, and on heating the material at 105°, partial collapse of the pore structure was observed, with loss of the 10 nm peak. 

Figure 8 The bimodal distribution of pH in coal-mine drainage, where approximately half the discharges from bituminous and anthracite coal mines are acidic with a pH less than 5 (source http //pa.water.usgs.gov/projects/ amd/index.html).

These polymers can be blended either in solutions or in the melt. A blend of up to 2% of polymer 3 with polystyrene appeared clear, but turbid blends resulted from higher percentages of polymer 3. The Tg s of blended polymers containing up to 30% of polymer 3 did not change from that of pure polystyrene. Transmission electron microscope (TEM) analysis of the 5% blend showed that there are bimodal distributions of polymer 3 domains, a small amount of approximately 10 nm domains, and a large portion of approximately 1 l domains. 

It should be noted that our analysis does not preclude the possibility that there are Impurities in our sanq)les. Furthermore, the presence of even a trace amount of an unknown impurity in our sample may be the cause of the postulated aggregation process. Finally, the histogram procedure requires that segments with very small amplitudes be set equal to zero. Therefore, the region between the two histogram peaks may be taken as having a very small, but not necessarily zero, contribution to the total distribution. The observed bimodal distribution, in fact, could be a very broad unimodal distribution with peaks at the extremes and a central valley, or a more conq)licated multimodal distribution which our histogram approximation cannot accomodate. 

In spite of have been proposed many approximated solutions to Boltzmann equation (including the Grad s method of 13 moments, expansions of generalized polynomial, bimodal distributions functions), however the Chapman-Enskog is the most popular outline for generalize hydrodynamic equations starting from kinetics equations kind Boltzmann (James William, 1979 Cercignani, 1988). 

It was shown by Williams (49) and Fernando and Williams (84) that the measured stress intensity factor IQ for a mixed mode of failure can be approximated as a combination of contributions from the plane strain (A lc) and plane stress (/fic) regions using a ratio between specimen width B and the size of the plastic zone and assuming a bimodal distribution of K, for B > 2Ry ... 

Williams (49), Ward (79), and Jancar et al. (89) proposed an approximate model of mixed mode of fracture to account for the effect of finite specimen dimensions for Kc and G, respectively. The basic idea in both theories is a substitution of the actual distribution of fracture toughness across the cross-section by a simple bimodal distribution, assuming plane strain value in the center and plane stress value at the surface area of the specimen. Size of the plastic zone IR relative to the specimen width B gives the contribution of plane stress regions and is a measure of the displacement of the state of stress at the crack tip from the plane strain conditions. Note that this approach can be used only if the mode of failure does not change with the test conditions or material composition (i.e., it attains its brittle character). 

In Fig. 25.2, a comparison is made between the measured centerline fluctuations for a plume in grid turbulence with Eq. (25.18). It is to be noted that Eq. (25.18) exhibits a maximum value for when C = o/2. In Fig. 25.2, the k = 0 approximation appears to retain this feature. Another salient feature of Eq. (25.18) is that the distribution of t across a steady flow at a given downstream location starts as a near-source bimodal distribution and changes to a unimodal distribution after the centerline mean value reaches and falls... 

Wu [5] got the MWD from storage modulus G and stress relaxation modulus G(f) using approximations derived from the Doi-Edwards description of chain dynamics. Wu s method accurately predicted the MWD of polymers with narrow distribution. However, often, it led to a distorted shape of the MWD for the sample with bimodal distributions. Therefore, Tuminello [6-8] developed a theory based on a diluted assumption in 1986. His method rigorously applies only to linear polymers. Especially, it works better for linear polymers with PI < 3.5. According to his theory, the relative differential AIFVD of polymer can be determined well from dynamic modulus master curve. 

Some authors have pointed out that DFT based methods commonly indicate that carbon based materials have a bimodal distribution of pores, where pores of an approximate width of Inm are not present in the sample [2,4-6]. It has been suggested that this is due to artifacts in the model or in the method used for the inversion of the integral to obtain the PSD. It has been pointed out that surface energetic heterogeneity does not explain the appearance of this minimum in the PSD, which is due to packing effects [5]. In this work we examine whether the use of different models to describe the adsorbate, with different ways of packing, can explain the appearanee of the minimum in the PSD. 

The distribution may, of course, be constructed to any degree of accuracy if sufficient moments of sufficient accuracy are available. Unfortunately, in the case of bimodal distributions a very large number of moments may be necessary before even approximate bimodal character is achieved. 

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