Cluster equality

FIGURE 6.15 Three different models for model-based clustering equal and diagonal covariance (left), diagonal covariance of different size (middle), and different covariance matrix for each cluster (right). 

Cluster and Path-Cluster MCIs A cluster is a starlike substructure with one central atom and three or more neighbor atoms to which it is connected. The order of a cluster equals the degree dz = (v)z of the central atom. The general MCI formula is... 

Each cluster is characterised by its size s, which is defined as the number of sites belonging to the cluster. A cluster of size s is called an s-cluster. The normalised cluster number ns is defined as the number of s-clusters per lattice site. An equivalent way to define ns(p) is by noting that the probability that an arbitrary lattice site belongs to an s-cluster equals ns-s. 

The two examples we just examined used the surface reaction as a reference for the study of clusters. Equally important are the implications of cluster chemistry to reactions in solution. Here, the chemistry of cluster ions has received much attention this is particularly important because reactions of charged reactants are often quite different in the gas phase and in solution. One canonical case is that of anion-neutral Sn2 reactions such as... 

The Nillson-Clemenger model [45] predicts that the ratio of the two axes R and of a deformed cluster equals the ratio of the resonance energies. The connection between the energetic splitting of the resonances and the deformation is given by... 

In order to determine the structures of total-energy minima we used the basin-hopping method. For the binary clusters the additional existence of so-called homotops increases the computational demands enormously. Homotops for, e.g., clusters are defined as clusters with the same size, composition and geometric arrangement, differing only in the way in which A and B-type atoms are arranged. Their number for the ApB cluster equals... 

Derivation of Eq. (9 a) We assert that very large but finite clusters (s > S ) have for p above the same internal structure as the gel, and that analogously to assumption (6d) the cluster radius Rs(p), divided by its value Rj(pc) at the gel point, is for large molecules near p a function of s/S only. In particular, in the cluster interior of these very large macromolecules the probability of a monomer to be part of the cluster equals the probability G of that monomer to be part of the infinite cluster. Hien, in d dimensions that mass s of the very large but finite cluster equals the product of the density G (p - Pc) and the cluster volume R. The above similarity assumption allows to apply this equality (apart from a constant factor) even at s = s ... 

For a general dimension d, the cluster size distribution fiinction n(R, x) is defined such that n(R, x)dR equals the number of clusters per unit volume with a radius between andi + dR. Assuming no nucleation of new clusters and no coalescence, n(R, x) satisfies a continuity equation... 

As noted above, one of the goals of NAMD 2 is to take advantage of clusters of symmetric multiprocessor workstations and other non-uniform memory access platforms. This can be achieved in the current design by allowing multiple compute objects to run concurrently on different processors via kernel-level threads. Because compute objects interact in a controlled manner with patches, access controls need only be applied to a small number of structures such as force and energy accumulators. A shared memory environment will therefore contribute almost no parallel overhead and generate communication equal to that of a single-processor node. 

Cussler studied diffusion in concentrated associating systems and has shown that, in associating systems, it is the size of diffusing clusters rather than diffusing solutes that controls diffusion. is a reference diffusion coefficient discussed hereafter is the activity of component A and iC is a constant. By assuming that could be predicted by Eq. (5-223) with P = 1, iC was found to be equal to 0.5 based on five binaiy systems and vahdated with a sixth binaiy mixture. The limitations of Eq. (5-225) using and K defined previously have not been explored, so caution is warranted. Gurkan showed that K shoiild actually be closer to 0.3 (rather than 0.5) and discussed the overall results. 

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