Random case

Figure 7.3 also shows that the proportion of alternating Mi units decreases and the fraction of longer sequences increases as rir2 increases. The 50 mol % entry in Table 7.5 shows that the distribution of sequence lengths gets flatter and broader for rir2 = 1, the random case. 

Quality of evidence-1, evidence from >1 properly randomized, controlled trials II, evidence from >1 well-designed clinical trials without randomization, case-controlled analytic studies, multiple time series, or dramatic results from uncontrolled experiments III, evidence from expert opinion, clinical experience, descriptive studies, or reports of expert committees. 

Figure 3 The collapse of the peptide Ace-Nle30-Nme under deeply quenched poor solvent conditions monitored by both radius of gyration (Panel A) and energy relaxation (Panel B). MC simulations were performed in dihedral space 81% of moves attempted to change angles, 9% sampled the w angles, and 10% the side chains. For the randomized case (solid line), all angles were uniformly sampled from the interval —180° to 180° each time. For the stepwise case (dashed line), dihedral angles were perturbed uniformly by a maximum of 10° for 4>/ / moves, 2° for w moves, and 30° for side-chain moves. In the mixed case (dash-dotted line), the stepwise protocol was modified to include nonlocal moves with fractions of 20% for 4>/ J/ moves, 10% for to moves, and 30% for side-chain moves. For each of the three cases, data from 20 independent runs were combined to yield the traces shown. CPU times are approximate, since stochastic variations in runtime were observed for the independent runs. Each run comprised of 3 x 107 steps. Error estimates are not shown in the interest of clarity, but indicated the results to be robust.

All statistical parameters can be determined by standard formulas valid not only for the random case. Of general validity is... 

In the random case when p = 1/6, f = 1. In the most correlated case when p = 1, f = 0. When p = 0 and the atom cannot jump backward to erase its previous jump, f = 3/2 and diffusion is enhanced relative to the random, uncorrelated case. 

The complete balance of the upturn by the polydispersity is only obtained for random branching processes. Often the reaction is impeded by serious constraints, or the primary chains before cross-linking are monodisperse. Then the resultant final molecular-weight distribution is narrower than in the random case, and the characteristic upturn as a result of branching, develops again. A strange coincidence in behavior is observed with star-molecules, where the rays are polydisperse, and with the ABC-type polycondensates. In both cases the particle-scattering factors can be expressed as ... 

The rest of the work consists in the cascade substitution for the construction of a path-weight generating function which is done as usual. Gordon and Parker have focussed their interest on the position of the gel point which became shifted from ac = 0.5000, in the random case, to ac = 0.5733, for the steric hindrance case. It will certainly be of interest to investigate whether... 

Note that for the random case, equivalent to the Flory-Stockmayer model (model 1 in Table 4), i=2/3 and 2=1/3. 

Although the values of K will depend on the nature of Q, Z, T, and v, there is, for each v, a special situation, the ideal random case, where the sorting of the substituents about the central atom follows the laws of random statistics. For this case, the Z and T substituents become arranged about the Q in a completely random fashion irrespective of other substituents which are attached to Q. The K values for the ideal random case may be derived mathematically. If it is assumed that the Z/T atom ratio is p/q (with p + q = 1), the probability P of having a central atom Q with i Z substituents and (v — i) T substituents in the v sites subject to redistribution is... 

Relative Distribution of Molecules for the Ideally Random Case for Two Exchanging Substituents on One Type of Central Moiety at the Overall Composition R—vl2... 

Fig. 1. Equilibrium distribution of molecules for the ideal random case in systems QZV vs QT as a function of the composition parameter R. A, v=2 B, v=3 C, v=4.

For the ideal random case, the AH term is zero, thus giving the following equation for the free energy ... 

One is generally interested in the enthalpies of scrambling reactions since these may be measured thermochemically and thus brought into relationship with equilibrium constants. Real scrambling equilibria, however, deviate more or less from the ideal random case. And, in order to be able to calculate enthalpies, it is assumed that for large values of the enthalpy, (JS)real (JS)rand. Accordingly, we may now estimate the enthalpy of a real scrambling reaction. 

In boron trihalide adducts and tetrahaloborate ions, halogen redistribution equilibria are reasonably close to this ideal random case when chlorine, bromine, and iodine are involved (27, 28, 80, 100, 112), as are equilibria in the uncomplexed heavier boron trihalides (111). 

Radicals of TPO (Fig. 12.16) participate in self- and cross-termination and second-order kinetics fit is probably a random case. 

Simulation experiments have shown a tighter bound of 1.25r for the random case. So the actual number of subsets used by the Subset Difference scheme is expected to be slightly lower than the 2r — 1 worst case result. 

When chain transfer agents are used to control polymer molecular weight the molecular weight distribution will tend to narrow toward the random case from the characteristics it would have in the absence of transfer agents. 

Large doses of vitamin A in children in poor socioeconomic conditions have often for logistic reasons been combined with other health-care interventions, such as immunization. A randomized, case-control study of 336 infants receiving either 33 000 micrograms RE/day (100 000 lU) of vitamin A or placebo simultaneously with measles vaccine showed a lower seroconversion to measles in the vitamin A group (62). 

In the random case, the number of contacts between empty sites is given, in analogy to Equation 2.A30, by the equation ... 

The nonrandom T factors in Equation 2.A33 are, of course, equal to one in the random case. These numbers must satisfy the following material balance equations ... 

In the random case (T, = roo= 1) this equation reduces to the familiar LF equation of state, Equation 2.A19. The second minimization condition. Equation 2.A37, leads to the equation ... 

When the variance model is heteroscedastic, the algorithm for bootstrapping the residuals will not be valid because the bootstrapped data set might not have the same variance model as the original data. In fact, more than likely, bootstrapping heteroscedastic residuals will lead to a homoscedastic model. Heteroscedas-ticity is not a problem for the random case because heteroscedasticity will be preserved after bootstrapping. In the heteroscedastic case, the modified residuals need to be corrected for their variance so that Eq. (A. 102) becomes... 

Statistical copolymers are those in which the monomer sequence follows a specific statistical law (e.g., Markovian statistics of order zero, one, two). Random copolymers are a special case of statistical copolymers in which the nature of a monomeric unit is independent of the nature of the adjacent unit (Bernoullian or zero-order Markovian statistics). They exhibit the structure shown in Figure 6.1. If A and B are the two monomers forming the copolymer, the nomenclature is poly (A-stat-B) for statistical copolymers and poly (A-ran-B) for the random case. It should be noted that sometimes the terms random and statistical are used indistinctly. The commercial examples of these copolymers include SAN poly (styrene-ran-acrylonitrile) [4] and poly (styrene-ran-methyl methacrylate) (MMA) [5]. 

To calculate the suitability indices of the seven sites, the weight of four factors should randomly be assigned first. Five random cases of the weights of the four factors were assumed as shown in Table 3.3. Table 3.4 illustrates the resultant suitability indices in each case after computation. Based on the suitability indices of the five cases, the seven sites could be classified into three groups, namely... 

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