Initial canonical value

Three AOAs in the CO3.0 chondrite Y-81020 that were studied for Al-Mg systematics showed Mg excesses corresponding to initial A1/ A1 ratios of (2.7 0.7) X 10 (2.9 1.5) X 10 and (3.2 1.5) X 10. Comparable excesses of Mg were not detected in the AOA-bearing chondrule (Itoh et al, 2002). The lower A1/ A1 ratios in AOAs compared to a canonical value of 5 X 10 found in most CAIs (e.g., MacPherson et al, 1995) suggest that either AOAs formed 0.1-0.5Ma after CAIs or that A1 was heterogeneously distributed in the CAI-forming region (Itoh et al, 2002). 

To first order, the larger data set now available extends and confirms the general assessments of MacPherson et al. (1995), albeit with some modifications and enhancements. The distribution of inferred initial Al/ Al in CAIs is bimodal (Figure 3(a)), with the dominant peak at the so-called canonical value of 4.5 X 10, and a second peak at dead aluminum (i.e., Al/ Al = 0). MacPherson et al. (1995) demonstrated that this pattern applied to all classes of carbonaceous chondrites, although the relative... 

The initial AF Al and °Be/ Be values have been measured in a variety of refractory phases from both CV and CM carbonaceous chondrites (Figure 7). Normal CAIs of both petrologic types A and B have inferred AF Al values that plot within error of the canonical solar system initial even for cases where the Al-Mg system is disturbed in anorthite, other phases in the inclusion plot near the canonical value (e.g., Sugiura et al., 2001). As noted above, initial Be/ Be ratios for normal CV CAIs also show no discrimination based on petrology and the total range covered is approximately a factor of 2, which is only marginally outside of experimental uncertainty. Thus, for normal CAIs it is difficult... 

The atoms of a molecule can be partitioned into equivalence classes on the basis of a single property or a set of properties. The value of each class for the property(s) will be called the initial canonical value. The set of properties could include the atomic number, predominant in the Cahn-Ingold-Prelog system if this is used to number the atoms of the molecule, or the number of ligand atoms, which is the property... 

Different schemes of canonicalization can be distinguished by the properties selected to establish the initial canonical value and whether the ligands of each atom are characterized by a tree of such initial canonical values or the sum of such values obtained by a successive summation process to be detailed later. 

Now let us examine the behaviour of the corresponding sum algorithm. Here we will use the same initial canonical value. The second canonical values are the sum of the ligands initial canonical values. In general the ith canonical value for an atom is the sum of the i - 1th canonical values of its ligands. In this way we obtain a seqond canonical value of 12 for both... 

The canonicalization procedure of Professor Ugi s group is a tree algorithm, in which the initial canonical value is composed of the atomic number and the coordination number of the atom. (If one of these two numbers makes the atom unique in the molecule then only the one number is used as the initial canonical value.)... 

If there are at least two nonterminal double bonds in the molecule the algorithm continues and calculates a "stereochemical canonical value", in which the initial canonical value is a number meaning cis or trans with respect to a double bond. If the atoms concerned are still equivalent with respect to the steroeochemical canonical value, then they are truly equivalent unless they are both chiral. This step is required to reveal the inequivalence of atoms 1 and 2 of the Figure below. 

Calculate the initial canonical values for all the atoms by copying the first 32 bits of each connection table row and shifting right two bits to eliminate the chirality information. 

Sort the rows of the connection table and the stereorelation table according to the precedence determined first by the initial canonical values and in case where the initial canonical values are equal, by the final canonical values. In this process prepare a translation table such that the ith entry of the table is the new number of old atom number i. 

In our last example we return to the issue of the possible damaging effects of the standard geometry constraints. Two long trajectories have been computed for a partially hydrated dodecamer DNA duplex of the previous example, first by using ICMD and second with Cartesian coordinate molecular dynamics without constraints [54]. Both trajectories started from the same initial conformation with RMSD of 2.6 A from the canonical B-DNA form. Figure 5 shows the time evolution of RMSD from the canonical A and B conformations. Each point in the figure corresponds to a 15 ps interval and shows an average RMSD value. We see that both trajectories approach the canonical B-DNA, while the RMSD... 

Fig. 3.4. Evolution of the weights i],(N) and the histograms fi(N) in a grand-canonical implementation of the multicanonical method for the Lennard-lones fluid al V 125. The temperature is T = 1.2 and the initial chemical potential is /./ = —3.7. The weights are updated after each 10-million-step interval, and the numbers indicate the iteration number. The second peak in the weights at large particle numbers indicates that the initial chemical potential is close to its value at coexistence...

Consider the example of condensed phase transitions between vibrational states, which have energies that are significantly drfferent compared with knT. The momentum on the initial surface before a hop and the final surface momentum after the hop are considerably drfferent for typical values of the initial momentum sampled Irom a canonical distribution. This causes the two branches of the combined trajectory to quickly diverge, and action for the combined trajectory to grow rapidly. The result is that the integrand converges very quickly as a function of x, particularly after the and Fj integrations have been performed. 

If the initial coordinates in array A correspond to the canonical basis, we set A(I,0) = A(0,J) = 0 for all I and J. Notice that the elements A(0,J) can be obtained from the values in A(I,0), thus we store redundant information. This redundancy, however, will be advantageous in the programs that call the module M10. 

Where T is the initial phase point of the system, L is the Liouville operator, y(tf)(F) is the canonical distribution function, and Bk(T) and k(T) are the values of the classical properties Bk and iLk when the system is in the classical state T. Much work has been done to determine how the quantum-mechanical functions approach the corresponding classical functions. 

Direct dynamics calculations were carried out with quasiclassical normalmode sampling from a canonical ensemble at 923 K (the experimental reaction temperature). Simulations initiated at the vicinity of TS for rearrangement of carbene 13 to 14 via oxirene 12, and 300 trajectories were obtained at DFT methods. The preliminary results reported in the manuscript showed that preferred formation of 15a over 15b by the ratio of 1.8 7.6 depends on the method used. The results were qualitatively consistent with the value of 2.5 deduced from the experiment. The non-unity ratio likely arises from the situation that two methyl groups in 14 are dynamically unequal on the carbene formation process. 

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