Abstract

The sheaf cohomology of topological shift for the block-rectangular matrixrepresentation of the hierarchical Markov Model is endowed with the analytical codification of white noise and of random effects. New analytical techniques for fragment sequencing are developed. The fragment sequencing is obtained after the topological Markov chain of the adjacency matrix of the corresponding undirected graph; the presence of white noise and that of random effects are comprehended. The paradigm consists in defining the hierarchical block rectangular matrices, from which the Topological Hidden Markov Models are issued (as clusters), with the aspects of Hidden Markov Models of ’multivariate Gaussian data’ with vanishing mean; the generalized covariance matrix is studied. The model is compared with the stochastic properties of the corresponding decomposition of approximation of experimental data. One of the previous results of the applications of the new method can be looked at in the analysis of the numerical simulation of the sequencing techniques: as an example, it is known that the Gojobori-Ichii-Nei model fails in reproducing the Jukes-Cantor scheme, while the Kimura matrix model succeeds in it. The difference is explained as the former model is not obtained from the Jukes-Cantor paradigm after application of the differential operators (for substituting the entries of the matrix), while the latter model is. In the present paper, the new analytical result is further accomplished, to calculate the maximal likelihood analytically in protein sequencing and in DNAsequencing, in modes in which the sequences of elements varies in time; the method solves analytically the phylogenetics computer programs. The analysed problem belongs to the nondeterministic polynomial time (NP) hard class of complexity.