Cluster center

In applying Equation (19), the objective is to find a set of cluster centers that minimize the Euclidean distance between each data point and the cluster center,... 

The algorithm originally proposed by Moody and Darken (1989) uses >means clustering to determine the centers of the clusters. The hypersphere around each cluster center is then determined to ensure sufficient overlap between the clusters for a smooth fit by criteria such as the P-nearest neighbor heuristic,... 

Layer-by-layer Ru3 cluster-based multilayers were fabricated onto preorganized self-assembled monolayer gold electrode surfaces by Abe et al. [15], in which [Ru3(q3-0)( i-0Ac)6(4,4/-bpy)2(C0)] was utilized as the synthetic precursor. The stepwise connection of oxo-centered triruthenium cluster units onto the gold electrode surface is a feasible approach for construction of Ru3 cluster-based oligomers on a solid surface, in which the bridging ligand 4,4 -bipyridine appears to mediate weak cluster-cluster electronic interaction between the Ru3 cluster centers. 

Cleavage of C—H bonds is also well documented for mononuclear complexes (66-68, 147), and seems likely for dinuclear systems [(84, 386) also Section II,A, Eq. (7)] but apart from some cyclopropane systems (387), cleavage of C—C bonds has not been demonstrated at mononuclear centers. Equation (65) shows such cleavage at a cluster center (388) ... 

Table 2. Binding Energy Per Atom Eb, Distance D from Atoms to the Cluster Center, and Average Magnetic Moment Per Atom p for Octahedral Six-Atom Clusters. Data Collected from Zhang et al.107... 

Model calculations for the Cs suboxides in comparison with elemental Cs have shown that the decrease in the work function that corresponds to an increase in the Fermi level with respect to the vacuum level can be explained semi-quantitatively with the assumption of a void metal [65], The Coulomb repulsion of the conduction electrons by the cluster centers results in an electronic confinement and a raising of the Fermi energy due to a quantum size effect. 

Similarly, a measure of heterogeneity between two clusters can be based on the maximum, minimum, or average of all pairwise distances between the objects of the two clusters (compare complete, single, and average linkage), or on the pairwise distances between the cluster centers. The latter choice results in a measure of heterogeneity /i / between cluster j and l as... 

Variations of R with A suggest a two-step hydration process solvation and formation of disconnected water clusters centered on polar head groups, followed by the formation of a continuous hydrogen-bond network. At low A, Ri depends logarithmically on co, suggesting bidimensional diffusion of protons in the interfacial region between polymer and water. 

The denominator of the ratio can be Interpreted as the weighted scatter of the two classes about their respective centers, as measured on a line In the direction of the maximizing eigenvector z. The numerator measures the distance between the two cluster centers, projected upon the same line. 

Fig. 1. Thirty molecules (open and solid circles) are grouped into five clusters in a two-dimensional chemical space. In A, the original, unsealed variables are used, whereas the variables are scaled to have unit sample standard deviations in B. The lines connect each molecule to its cluster center. Solid circles denote the compounds chosen at random open circles are unselected compounds.

This simulated dataset is generated as follows. First, nine cluster centers are defined at different locations in a 2D space with coordinate values within [-3.0, 3.0]. Second, a random number (between 1 and 100) of points is generated around each cluster center within a distance of 0.5. Finally, additional points are generated, which are randomly distributed in the 2D space so that a total of 1000 points is obtained. This dataset simulates the situation where clusters of molecules exist in their descriptor space, and the number of members for each cluster is different, i.e., some regions are more densely populated than other regions. 

Spectroscopic studies indicate that the siroheme and 4Fe 4S cluster are exchange coupled in all oxidation states (Christner et al., 1984 Cline et al, 1985a,b). The structural basis of this coupling has been provided by a preliminary 3 A resolution X-ray structure of sulfite reductase in the oxidized state (McRee et al., 1986). The siroheme and the 4Fe 4S cluster are packed against each other and appear to share a common ligand (Fig. 18). The distance from the siroheme iron to the cluster center is... 

In the preceding sections of this article, a systematization of basic cluster reactions has been attempted based on the present status of knowledge. It can be seen that the number of reactions reported is still relatively small for certain reaction types. The organization of the material described here, which mostly follows reactivity patterns, may not be the best one when more facts are available. An alternative way of presentation is outlined below, based on a cluster-centered viewpoint. It brings out which areas might show future developments or deserve research activities. 

A hypsochromic shift in the emission band would be observed if the amount of excited-state distortion were lower in a rigid matrix than in fluid solution. Such an interpretation has been recently suggested for Cu4l4py4 and related copper clusters, and this constitutes an unusual case of molecules that exhibit rigidochromism but not solvatochromism in its emission from a cluster-centered excited state [115]. It is likely that both excited-state destabilization and distortion arise during the polymerization processes in the tungsten and rhenium organometallic systems. 

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