Recursion method structure

AUGMENTED SPACE RECURSION METHOD FOR THE CALCULATION OF ELECTRONIC STRUCTURE OF RANDOM ALLOYS... 

While the supercell approach works well for localized systems, it is typically necessary to consider a very large supercell. This results in a plane-wave basis replicating not only the relevant electronic states but also vacuum regions imposed by the supercell. A much more efficient method to implement for investigating the electronic structures of localized systems is to use real space methods such as the recursion methods [27] and the moments methods [28], These methods do not require symmetry and their cost grows linearly with the number of inequivalent atoms being considered. For these reasons, real space methods are very useful for a description of the electronic properties of complex systems, for which the usual k-space methods are either inapplicable or extremely costly. 

When the SLD profile of an interface is known, a matrix method or a recursion method can easily be used to calculate reflectivity curves. However, for pedagogical purposes the relationship between the reflectivity and the structure of the interface is better revealed by the analytical expressions derived with the help of the kinematic approximation. The kinematic approximation has been shown to describe the reflection of neutrons from stratified media very well when the reflectivity is significantly less than unity. When a film of SLD pi and thickness t, is sandwiched between two phases of identical SLD p, the expression for the reflectivity derived from a kinematic approach is [36]... 

The Structurally Recursive Method is then expanded, and a second, non-recursive algorithm fw the manipulator inertia matrix is derived from it A finite summation, which is a function of individual link inertia matrices and columns of the propriate Jacobian matrices, is defined fw each element of the joint space inertia matrix in the Inertia Projection Method. Further manipulation of this expression and application of the composite-rigid-body inertia concept [42] are used to obtain two additional algwithms, the Modified Composite-Rigid-Body Method and the Spatial Composite-Rigid-Body Method, also in the fourth section. These algorithms do make use of recursive expressions and are more computationally efficient. 

Four algorithms for computing the joint space inertia matrix of a manipulator are presented in this section. We begin with the most physically intuitive algorithm the Structurally Recursive Method. Development of the remaining three methods, namely, the Inertia Projection Method, the Modified Composite-Rigid-Body Method, arid the Spatial Composite-Rigid-Body Method, follows directly from the results of this tot intuitive derivation. 

To begin the development of the Structurally Recursive Method, a one-link manipulator, as shown in Figure 3.1, is examined. A free-body force equation may be written fix the single link as follows ... 

In the next analysis, we will examine the components of successive inotia matrices as defined by the algorithm given in Table 3.1. First, the expansion of the equations for the Structurally Recursive Method leads to an exjnession for H,j, the Tii X itj submatrix of H, in the form of a summation. Its terms involve projections of individual link inertias onto the preceding joint axis vectors, which... 

Extrapolating from these expanded versions of the equations for the Structurally Recursive Method, a general expression fw the (i,j) submatrix of the 7 T-link manipulator inertia matrix, Hat (or simply H), may be written as follows ... 

In this simple recursion, the operational space inertia matrix of the base member, Ao, is propagated across joint 1 by La > a new spatial articulated transformation which is very similar in form to the acceloation propagator of the previous section. The propagated matrix is combined with Ii, the spatial inertia of link 1 to form Ai, the operational space inertia matrix of the two-link partial chain comprised of links 0 and 1. Note the similarity between this recursive procedure and the structural recursion used to derive the Structurally Recursive Method (Method I) in Ch t 3. 

Linear Recursive Methods - Kekule Structure Counting. - A rather... 

Recursive estimation methods are routinely used in many applications where process measurements become available continuously and we wish to re-estimate or better update on-line the various process or controller parameters as the data become available. Let us consider the linear discrete-time model having the general structure ... 

Evaluation of protein sequence analysis methods based on the use of PSSMs in benchmarking experiments and in a number of test cases shows that these methods are capable of systematically detecting relationships between proteins that previously have been deemed tractable only at the structure-comparison level. Clearly, however, there is still a lot of room for improvement, as many automated procedures missed subtle connections that subsequendy have been revealed on a case-by-case basis, in part thanks to a careful choice of starting points for the PSSM construction. An exhaustive exploration of the sequence space by recursive iterative searching is likely to yield additional, on many occasions unexpected, links between proteins and, in particular, is expected to increase the rate of structure prediction. 

At the end of a split synthesis, because the beads have been pooled and mixed, the exact identity of a molecule on a given bead is unknown. Likewise, the identity and structure of compounds in wells is unknown. Split synthesis is not a spatially addressable method. Fortunately, the exact structure does not need to be known unless a compound shows activity in a screen. If active, the structure of the compound in the well will need to be elucidated through a process called deconvolution. Deconvolution is generally accomplished through one of two methods recursive deconvolution21 or binary encoding.22... 

In addition to cell-based partitioning, statistical partitioning methods are widely used for compound classification. One of the most popular approaches is recursive partitioning (Rusinko et al. 1999), a decision tree method, as illustrated in Figure 1.8. Recursive partitioning divides data sets along decision trees formed by sequences of molecular descriptors. At each node of the tree, a descriptor-based decision is made and the molecular data set is subdivided. For example, a chosen descriptor could simply detect the presence or absence of a structural fragment in a molecule. Alternatively, the... 

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