Disjoint clusters

The coupled-cluster (CC) method (Cizek and Paldus, 1971 Harris, 1977a,b Bartlett and Purvis, 1978) is an attempt to introduce interactions among electrons within clusters (predominantly pairs) as well as coupling among these clusters of electrons and to permit the wavefunction to contain all possible disjoint clusters. For example, we know, from the early work of Sinanoglu (1962) and others, that electron pair interactions are of utmost importance and that contributions of quadruply excited configurations to... 

A particular realization C of the Cayley tree for a certain placement of bonds is formed by disjoint clusters of beads, a cluster being a set of beads connected... 

Partitional clustering methods attempt to directly decompose a data set into a fixed number of disjoint clusters. The methods attempt to minimize a criterion function, typically based on minimizing the dissimilarity inside classes, while maximizing the dissimilarity between different classes. 

For each of these PLE, a Weibull interpolation was fitted due to lager intervals of the PLE step function in comparison to the univariate PLE. The reason for that is obvious because by splitting the data into 25 disjoint clusters, fewer failures are taken into account for each PLE, leading to lager intervals. Also the last fitted Weibull distribution per cluster was used to extrapolate the failure behavior for further failure predictions. As can be seen in Figure 2, up to three different shape parameters for Weibull distributions are used for the inter- and extrapolation of each PLE, so the changing failure behavior can be taken into account for prediction purposes. 

T introduces true n-particle correlation, and products like T T etc., arising out of the expansion eq.(3.2 ), generate simultaneous presence of k-particle amd m-particle correlations in a (k+m)-fold excited determinants etc. The truncation of T == T then corresponds to the pair—correlation model of Sinanoglu, while incorporating higher excited states with several disjoint pair excitations induced through the powers T - The amplitudes for T may be called linked or connected clusters for n electrons. The difficulty of a linear variation method such as Cl lies in its inability to realize the cluster expansion structure eq.(3.2),in a simple and practicable manner. 

If the data set X is comprised of two classes, the cluster structure of X may be described by two disjoint fuzzy sets >4j and A 2 whose union is X. Each fuzzy set corresponds to a class (or cluster) of samples in X. The disjointness condition of 1, 2 minimal separation condition for the... 

Note that commutation of cluster operators holds only when the occupied and virtual orbital spaces are disjoint, as is the case in spin-orbital or spin-restricted closed-shell theories. For spin-restricted open-shell approaches, where singly occupied orbitals contribute terms to both the occupied and virtual orbital subspaces, the commutation relations of cluster operators are significantly more complicated. See Ref. 36 for a discussion of this issue. 

This behavior of the mixtures of ionic liquids with molecular solvents was recently studied, experimentally and by molecular simulation, by Del Popolo et al. [45], for mixtures of [C4C1im][PF6] with naphthalene over the entire composition range. As in the previously discussed mixtures of ionic liquids with aromatics, naphthalene is able to cleave some interionic contacts, but not all of them. With compositions in which the ionic liquid is more dilute, the ionic network subsists in the shape of filaments of continuous cation-anion contacts in a medium composed mostly of the molecular fluid. If dilution is increased, then disjoint ionic clusters, down to ion pairs, will form. Some aromatic compounds are not sufficiently good solvents to the ionic liquids and cannot disrupt the ionic network, leading to an immiscibility gap (as is the case with benzene and toluene, for example, at mole fractions around 0.7-0.8 [46, 47],... 

Hierarchical clustering procedures iteratively partition the item set into disjointed subsets. There are top-down and bottom-up techniques. The top-down techniques partition can be into two or more subsets, and the number of subsets can be fixed or variable. The aim is to maximize the similarity of the items within the subset or to maximize the difference of the items between subsets. The bottom-up techniques work the other way around and build a hierarchy by assembling iteratively larger clusters from smaller clusters until the whole item set is contained in a single cluster. A popular hierarchical technique is nearest-neighbor clustering, a technique that works bottom up by iteratively joining two most similar clusters to a new cluster. 

The large number of TIs, and the fact that many of them are highly correlated, confounds the development of predictive models. Therefore, we attempted to reduce the number of TIs to a smaller set of relatively independent variables. Variable clustering " was used to divide the TIs into disjoint subsets (clusters) that are essentially unidimensional. These clusters form new variables which are the first principal component derived from the members of the cluster. From each cluster of indexes, a single index was selected. The index chosen was the one most correlated with the cluster variable. In some cases, a member of a cluster showed poor group membership relative to the other members of the cluster, i.e., the correlation of an index with the cluster variable was much lower than the other members. Any variable showing poor cluster membership was selected for further studies as well. A correlation of a TI with the cluster variable less than 0.7 was used as the definition of poor cluster membership. 

Eq. (6) shows that communication is a reflexive, symmetric and transitive relation. A relation that exhibits these properties is an equivalence relation (Kemeny Snell 1960). Equivalence relations have the ability to partition the universe Q upon which it is defined to disjointed partitions (Dartmouth College Writing Group. Cogan 1958). Each of these partitions defines a unique cluster of communicating literals, which is referred to as abstract equivalence classes. 

Here we have to use the operators E q and not the X q. The reason is that to ensure that the commutator expansions in coupled-cluster theory truncate, the wave operator must be expressed in terms of excitation operators between two disjoint one-particle spaces. This expression and the corresponding one for T2 form the basis for the Kramers-restricted CCSD (KRCCSD) method (Visscher et al. 1995). 

These experiments are performed as follows the image data-set is randomly divided into three completely disjoint sets of equal size. To avoid any kind of influence in the test results, different images of the same insect instance are placed in the same set. The first set is used as the clustering set to create the GMM clusters for each object class (as detailed before) the second set is used to train the 15 decision trees that comprise the final classifier (described earlier), while the third is used to measure the classification accuracy of the classifier. We use separate clustering and training sets to reduce over-fitting of the classifier to the training data. 

These equations show which excitation processes contribute at each excitation level. Thus, the quadruply excited configurations are generated by five distinct mechanisms, where, for instance, the disconnected term represents the independent interactions within two distinct pairs of electrons and the connected f4 terra describes the simultaneous interaction of four electrons. The disconnected terms represent interactions of product clusters within disjoint sets of electrons and vanish whenever two or more spin-orbital indices are identical. 

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