Sampling zero-dimensional

The need for multiscale modeling of biological networks in zero-dimensional (well mixed) systems has been emphasized in Rao et al. (2002). The multiscale nature of stochastic simulation for well-mixed systems arises from separation of time scales, either disparity in rate constants or population sizes. In particular, the disparity in species concentrations is commonplace in biological networks. The disparity in population sizes of biological systems was in fact recognized early on by Stephanopoulos and Fredrickson (1981). This disparity in time scales creates slow and fast events. Conventional KMC samples only fast events and cannot reach long times. 

Gy refers to this sampling situation as zero-dimensional. As we saw in Chapter 2, he develops a model for the variance of the FE and bases it on statistical sampling theory, where all the sampling units are discrete and well defined. As long as all units are accessible., we can easily define and extract the units for the sample. Adhering to the principle of correct sampling is thus straightforward. 

Figure 5.17 shows a HREM image of a carbon allotrope having a zerodimensional (point) polymer of carbon observed at tire lower part of postshock sample I. The characteristic of the electron diffraction pattern of this type consists of two-dimensional asymmetric (hk) bands and symmetric (0001) reflections of tlie graphite structure and did not alter with tlie angle of tlie incident electron beam. In general, such patterns can be obtained when llie particle is a spherical shell, tlrat is, a zero-dimensional characteristic. Therefore,... 

With zero-dimensional lots, we can apply SRS. All other lots of bulk material are three-dimensional and thus impossible to sample correctly. By reducing the sampling dimension, we can reduce our overall sampling error. 

An example of analysis in which a more elaborate data analysis was performed is the analysis of stretched samples of poly(etherester) (PEE) (73,74). This material consists of hard and soft domains. By stretching this material the soft domains elongate and become needle shaped. The analysis performed here is in the form of projections which are, in this case, integral operators that map fimc-tions onto subspaces of the domain in which they are defined. The most well known of these projections is the one that maps the scattered intensity onto a zero-dimensional subspace, ie, a number or scalar. For instance,... 

When we examine the plots in Figure 56 we see that the PRESS decreases each time we add another factor to the basis space. When all of the factors are included, the PRESS drops all the way to zero. Thus, these fits cannot provide us with any information about the dimensionality of the data. The problem is that we are trying to use the same data for both the training and validation data. We lose the ability to assess the optimum rank for the basis space because we do not have independent validation samples that contain independent noise. So, the more factors we add, the better the calibration is able to model the particular noise in these samples. When we use all of the factors, we are able to model the noise completely. Thus, when we predict the concentrations for... 

Percolation theory describes [32] the random growth of molecular clusters on a d-dimensional lattice. It was suggested to possibly give a better description of gelation than the classical statistical methods (which in fact are equivalent to percolation on a Bethe lattice or Caley tree, Fig. 7a) since the mean-field assumptions (unlimited mobility and accessibility of all groups) are avoided [16,33]. In contrast, immobility of all clusters is implied, which is unrealistic because of the translational diffusion of small clusters. An important fundamental feature of percolation is the existence of a critical value pc of p (bond formation probability in random bond percolation) beyond which the probability of finding a percolating cluster, i.e. a cluster which spans the whole sample, is non-zero. 

After the N, C-labelled NBD peptide was obtained, the triple-resonance experiment developed to measure rNHC dipole-CSA CCR-rates [41] was performed as a 2-dimensional experiment. The spectra are obtained by applying a pulse sequence that was developed by Kay and co-workers [41] to measure /nhc with increased sensitivity. The experiment is one of the sensitive triple-resonance experiments, and is therefore suitable to test the feasibihty of CCR measurements and to optimize sample conditions. The pulse sequence is derived from an HNCOCA and the double and zero-quantum co-... 

The same sample was subsequently used to measure /nhch dipole-dipole CCR. In this case the pulse sequence proposed by Yang and Kay [43] was applied. The experiment is also based on an NH(CO)CA experiment. The zero and double quantum coherences result in two 2-dimensional (2D) datasets and the 2D spectra obtained (black and red cross peaks in Fig. 4) result after pairwise adding and subtracting the measured 2D datasets. The signals are detected at the frequency and split by the JcaHa coupling. The black and red cross peaks are shifted by Jnhhq . Also in this case, the CCR-rate can directly be obtained from the intensities of the individual peaks ... 

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