Iterated permutation

Signs c denote inclusion arrows denote constructions. The diagram at the bottom commutes, i.e., iterated squaring and doubling is the special case of iterated permutations where the construction of claw-intractable permutation pairs on the factoring assumption is used. 

Construction 8.56. Let a weak claw-intractable family of permutation pairs be given (see Definition 8.27). The corresponding family of iterated permutations as bundling functions is defined by the following components, which are written with an asterisk to distinguish them from the components of the family of permutation pairs ... 

Theorem 8.57 (Iterated permutations as bundling functions). If a strong claw-intractable family of permutation pairs is given. Construction 8.56 defines a collision-intractable family of bundling functions. If the underlying family is weak, all properties except for the bundling property are still guaranteed. ... 

In the case of the threshold rules defined in this section, we must consider sequential iterations of deterministic rules. Also, the choice of spins that may change state is not random but is fixed by some random permutation of the sites on the lattice. Such rules may be shown to correspond to spin glasses in the zero-temperature limit. 

The selected SARs depend on the declared EAL level, respecting the asset and risk value, and on specific needs implied by security objectives. Operations on SFRs, (e.g., iteration, assignment, refinement, selection), and user defined SFRs are allowed too. At this stage SOF (Strength of Function) claims should be attached when permutation or probabilistic mechanisms are presumed. SOF may be basic, medium or high. 

To make ( (s) diagonally dominant, it is necessary to select a specific type of controller. Rosenbrock has proposed that K(s) K.qKh(s)Kr (s), i.e., a product of three controller matrices. Ka is a permutation matrix, which scales the elements in G(s)K(s) and makes some preliminary assignment of single loop connections, usually to assure integrity. This step can be used to make (G(s)K(s)diagonally dominant as s 0. K s) can be chosen to meet stability criteria. Finally the elements of Kc(s), a diagonal matrix, can be selected to improve performance of the system. The proper selection of and Kt>(s) are the most difficult parts of the design process, and this step should be considered iterative, especially for an inexperienced user. 

This potency does not account for the inhibition observed for the library Ai (34%, 65,341 compounds), and examination of the various iterations clearly shows other families of active compounds. Nevertheless, a relatively short process (6 iterations, 51 reactions, 37 deprotections, 1 chromatography) detected a reasonably active lead compound as a starting point for chemical optimization. The sublibrary populations were reduced from 65341 to 12 in only four iterations (sublibrary E2, where only 12 permutations of the four monomers in the structure were possible), the second being a control of the validity of the first selection. Deconvolution of a few other families of positives (for example sublibraries Bs, E2, E4 and F2) could have produced different lead structures while maintaining a relatively modest number of iterations and reactions. 

Lemma 8.6. If/o and fi are permutations on their domain D, the iterated functions B and B[Pg.221]

The construction principle is always the iteration of the given permutations or functions, as shown in Figures 8.2 and 8.3, i.e., the functions B and B(j constructed in Definition 8.5. The idea why such constructions are collision-intractable is shown in Figure 8.10 If one has a collision of the iterated function 5, one knows two paths of applications of /o and /j that end at the same value z. Hence, except for the case where one path is a part of the other, those paths must meet at some point, coming from different directions. At this point, one has a claw between and/j, ifj and/j are permutations. 

Lemma 8.55. (Finding claws in collisions). Whenever a claw-intractable family of permutation pairs is given (strong or weak, see Definitions 8.26 and 8.27), the corresponding algorithm claw from collision is defined as follows. (Remember that with the conventions from Definition 8.5, the iterated functions are called Bf. ) It works on inputs of the form K, x, x ) with K e All, where X = (b, y) and x = (b , y ) with b, b e 0, 1 " and y, y e Df. ... 

The construction of a collision-intractable family of bundling homomorphisms from iterated squaring and doubling is basically Construction 8.56, based on the claw-intractable families of permutation pairs from Construction 8.64. The new points are ... 

Implementing the Mantel statistic requires an iterative procedure that is as follows (1) permute the rows and columns of the similarity matrix S N.B. that either similarity matrix can be permuted with the same result), (2) compute either or and (3) repeat the first two steps. This process is carried out a number of times in order to estimate the sampling distribution of the Mantel statistic under the assumed (null) hypothesis that the similarities in S are not linearly correlated with the corresponding similarities in S. The Mantel statistic derived directly from the similarity matrices is then compared with its distribution derived under H. Based on the distribution, if the Mantel statistic is likely to have been obtained, then the null hypothesis is accepted. Otherwise the alternative hypothesis, H, that the similarities between the two matrices are correlated is accepted. 

Here, the matrices H and V are symmetric and positive definite matrices, which are each, after suitable permutation of indices, tridiagonal matrices. The matrix S is a non-negative diagonal matrix. Recalling that tridiagonal matrix equations are efficiently solved by the Gauss elimination method, we consider now the Peaceman-Rachford iterative method [27], a particular variant of the lAD methods, which is defined by... 

Algorithm 5.11, taken from [95], shows how the structure generator behind MOL-GEN 3.5 [19, 20] fills the bond matrix. The filling of the matrix blocks (steps (3) and (4)) is iterated with canonicity testing for matrix blocks (step (5)). Only permutations from the automorphism group of blocks 1,..., r - 1 calculated eeuUer have to... 

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