Recursive sampling

Recursive sampling has the virtue of generating unbiased populations of chain conformations. While it has been shown to be highly efficient, some of its effectiveness is lost at low temperatures and for systems that exhibit long-range interactions [10]. Its application to complicated molecular architectures may face, however, some practical challenges if efficiency and algorithmic simplicity are to be maintained. 

Recently, recursive sampling has been combined with configurational bias methods (see Section III.D). In the Pruned-enriched Rosenbluth method of Grassberger [11], the number of copies of a growing chain is either reduced (pruning) or increased (enrichment) based on the chain s... 

Grassberger and Hegger have developed a recursive sampling approach to generate a population of trial conformations of homopolymer chains of length N (e.g.,. starting from a... 

If the sampling time is 0.1 seconds, the values of the discrete-time state transition and control matrices AfT) and BfT) calculated in Example 9.8 may be used in the recursive solution. 

Pd4oCu4oP2o, Pd5oCu3oP2o, and Pd6oCu2oP20 alloys were measured by resonant ultrasound spectroscopy (RUS). In this technique, the spectrum of mechanical resonances for a parallelepiped sample is measured and compared with a theoretical spectrum calculated for a given set of elastic constants. The true set of elastic constants is calculated by a recursive regression method that matches the two spectra [28,29]. 

Equations 13.14 to 13.16 constitute the well known recursive least squares (RLS) algorithm. It is the simplest and most widely used recursive estimation method. It should be noted that it is computationally very efficient as it does not require a matrix inversion at each sampling interval. Several researchers have introduced a variable forgetting factor to allow a more precise estimation of 0 when the process is not "sensed" to change. 

In any recursive partitioning technique, power drops dramatically as we split our data into smaller and smaller groups so that, except in huge samples, the final model selected understates the actual number of useful predictors. By giving all significant predictors the opportunity to be used for splitting, RRP makes it easy to find a more complete set of relevant variables. 

Frequency Analysis. The Discrete Fourier Transform (and its fast implementation, the Fast Fourier Transform [Brigham, 1974]) (FFT) as well as its cousin, the Discrete Cosine Transform [Rao and Yip, 1990] (DCT) require block operations, as opposed to single sample inputs. The DFT can be described recursively, with the basis being the 2 point DFT calculated as follows ... 

Perhaps surprisingly, it will be shown in a later section that the above recursion is exact at the sample points in spite of the apparent crudeness of the finite difference approximation. 

To compare this with the waveguide description, we substitute the traveling-wave decomposition y(n, m) = y (n - m) + y (n + m) (which is exact in the ideal case at the sampling instants) into the right-hand side of the FDA recursion above and see how good is the approximation to the left-hand side y(n+ 1, m) = y + (n + 1 -m) + y (n + 1 + m). Doing this gives... 

Thus, to update the sampled string displacement, past values are needed for each point along the string at time instants n and n— 1. Then the above recursion can be carried out for time n + 1 by iterating over all m along the string. 

In recursive deconvolution, after the addition of a new building block, a sample of that batch of resin is set aside and cataloged. Furthermore, after the final step, the resin is not repooled, so the last step for all the library members remains known. The cataloged resins and knowledge of the last step facilitate deconvolution.21... 

Recursive deconvolution is perhaps best explained through a simple eight compound library prepared by three steps, addition of A or B, then / or 2, and finally a or b (Scheme 9.12). In total, such a library has four resin samples. The resin samples are cataloged from intermediate steps—two samples each from the A/B step and the 112 step. Assume the library gives one hit in a screen. The screener of the library knows only the last step leading to the hit. In this case, the hit is Alb. The screener would know only the last step that produced hit, so the hit would be b, with being the unknown building blocks from steps one and two.21... 

In a related strategy termed recursive deconvolution [16], samples of all sublibraries at each stage are retained. With this approach it is not necessary to repeat the entire synthesis at each stage of the deconvolution. One simply adds the previously determined preferred synthons to the reserved sublibraries. 

Based on current knowledge of the process and its disturbance characteristics, one may know or choose a reasonable difference equation structure for the controller algorithm. Starting with some assumed initial parameter values in the controller equation, the controller can be implemented on the process as shown. The control algorithm is coupled with an on-line recursive estimation algorithm which receives information on the inputs and outputs at each sampling interval and uses this to recursively estimate the optimal controller parameters on-line and to update the controller accordingly. The idea is to use the data collected from the on-line control manipulations to tune the controller directly. 

In considering fluids, a variety of approaches have been used to compute entropy and free energy a free-volume method,111-112 thermodynamic perturbation theory,113-116 thermodynamic integration,117-121 umbrella sampling,122-124 and a Monte Carlo recursion method.125-126 The entropy of association of two protein molecules in water has also been computed.127... 

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