Kernel, binary

It is often obtained from Eq. (1.71) when the kernel is assumed to relax much quicker than the solution to be found. Then it is nothing more than a low-density gas approximation to Eq. (1.71), valid when conditions (1.83) or (1.88) are met. For these conditions the differential theory is expected to be binary in collisions, and... 

Whatever the X, the binary kernel gives a convergent contribution18 to the evolution of 9 1(p1, t). This solves, in principle at least, the problem of the divergence of the Born development (Eqs. 56, 57). 

The formulation in terms of binary kernels presents another advantage indeed, in the series (71) the selection rules (63) never impose k = 0. The terms k = 0 of Eq. (71) are not reducible and give negligible contributions to d[Pg.344]

When n > 2, one can draw the reducible contributions made up of sequences of binary kernels and where states k = 0 between these kernels exist. Thus, the class associated with the skeleton of Fig. 3b contains a state k = 0 and contributes, not to Eq. (56), but to Eq. (70). In the following we shall need the relation which expresses Yg,- n) as the difference between ) and the ensemble of reducible contributions to (70) (of the type of Fig. 3b for n = 3, for example). It is necessary for us now to study systematically the points k = 0 of Eq. (70) so as to extract the reducible contributions. A study of the selection rules will permit us to solve this problem. We shall associate the appearance of the points k = 0 with the structure of the skeletons that we have introduced we shall see that the reduci-bility will be a dynamical translation of certain topological properties of the equilibrium clusters. 

Table 8 Mean Recalls at 5% for Similarity Searching (SS), Conventional Turbo Similarity Searching using Croup Fusion (TSS-CF) and Turbo Similarity Searching using Substructural Analysis (TSS-SSA) and Binary Kernel Discrimination (TSS-BKD)...

Harper G, Bradshaw J, Gittins JC, Green DVS, Leach AR. (2001) Prediction of Biological Activity for High-Throughput Screening Using Binary Kernel Discrimination. J. Chem. Inf. Comp. Set. 41 1295-1300. 

Chen B, Harrison RF, Pasupa K, Wilton DJ, WiUett P, Wood DJ, Delaney J, Lawson K, MuUier G. Evaluation of Binary Kernel Discrimination for Virtual Screening in Lead-Discovery Programmes. /. Chem. Inf Model, submitted for publication. 

However, MET is not a unique theory accounting for the higher-order concentration corrections. Similar results were obtained within the fully renormalized YLS (Yang-Lee-Shin) theory [132], which is also integrodiffer-ential and employs the kernels containing concentration corrections as compared to those in IET. It was shown in Ref. 41 that both these theories, MET and YLS, provide the correct asymptotic expressions for binary kinetics, but differ slightly in the nonlinear terms of the concentration expansion. There were also a number of other attempts to overcome the concentration limitations of the theory made by the same Korean group earlier (in superposition approximation [139,141,142]) and later [numerous multiparticle kernel (MPK) theories [51,126]]. 

Harper et al. have demonstrated a much better performance of probabilistic binary kernel discrimination method to screen large databases compared to... 

Many different methods can be applied to virtual screening, and such methods are described in other chapters of this book and/or in the Handbooks of Che-minformatics Here we discuss the methods based on a probabilistic approach. Unfortunately, there are many publications in which the probabilistic or statistical approach items are farfetched. The Binary Kernel Discrimination and the Bayesian Machine Learning Models are actually special... 

For example, when the kernel is [2] of a binary scheme, the kernel of its quaternary square is [4], giving the matrix... 

Consider first the binary case. Let the scheme be of the form anK where K is the kernel of k entries. 

The kernel by definition has no further factors of a, but it can be expressed as a polynomial in a. In fact, because the kernel of a binary scheme always has an odd number of entries, its symmetric form can be expressed as a polynomial in a2. 

In the analysis of artifacts above, we observed that the mask of every binary, uniform, stationary scheme can be expressed as the product of a number of a = (1 + z)/2yfz factors and a further factor called the kernel. Then we saw that the kernel itself can be expressed as the sum of a number of terms, each of which is just a constant times an even power of a. 

Harper, G., Bradshaw, J., Gittins, J.C., Green, D.V.S. and Leach, A.R. (2001) Prediction of biological activity for high-throughput screening using binary kernel discrimination./. Chem. Inf. Comput. Sci., 41, 1295-1300. 

As an example, consider a monodisperse population of particles characterized by mass as internal coordinate and moments m = mjt(O) = 1 with k = 0,..., 2N 1. This population of particles is continuously fed to a system wherein particles undergo aggregation and symmetric binary breakage with constant kernels. The equations describing the evolution of the moments are... 

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