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Calculating the entropy for complex systems is a significant problem in science and engineering problems. However, this calculation is usually computationally expensive when the entropy is computed directly. This paper introduces three classes of information sources that for all members of each class, the entropy value is the same. These classes are characterized according to special dynamics created by three kinds of self-mappings on Ω, and A, where Ω is a probability space and A is a finite set. An approximation of rank variables of the product of information sources is made, and it is proved that the topological entropy of the product of two information sources is equal to the summation of their topological entropies. [ABSTRACT FROM AUTHOR]

The extraction of predictive features from the complex high-dimensional multi-omic data is necessary for decoding and overcoming the therapeutic responses in systems pharmacology. Developing computational methods to reduce high-dimensional space of features in in vitro , in vivo and clinical data is essential to discover the evolution and mechanisms of the drug responses and drug resistance. In this paper, we have utilized the matrix factorization (MF) as a modality for high dimensionality reduction in systems pharmacology. In this respect, we have proposed three novel feature selection methods using the mathematical conception of a basis for features. We have applied these techniques as well as three other MF methods to analyze eight different gene expression datasets to investigate and compare their performance for feature selection. Our results show that these methods are capable of reducing the feature spaces and find predictive features in terms of phenotype determination. The three proposed techniques outperform the other methods used and can extract a 2-gene signature predictive of a tyrosine kinase inhibitor treatment response in the Cancer Cell Line Encyclopedia. [ABSTRACT FROM AUTHOR]

Explaining the maintenance of diverse species assemblages is a central goal of ecology and conservation. Recent coexistence mechanisms highlight the role of dispersal as a source of the differences that allow similar species to coexist. Here, we propose a new mechanism for species coexistence that is based on dispersal differences, and on the geometry of the habitat patch. In a finite habitat patch with complex boundaries, species with different dispersal abilities will arrange themselves in stable, concentric patterns of dominance. Species with superior competitive and dispersal abilities will dominate the interior of the patch, with inferior species at the periphery. We demonstrate and explain the mechanism on a simple one-dimensional domain, and then on two-dimensional habitat patches with realistic geometries. Finally, we use metrics from landscape ecology to demonstrate that habitat patches with more complex geometries can more easily support coexistence. The factors that underpin this new coexistence mechanism-different dispersal abilities and habitat patches with complex geometries-are common to many marine and terrestrial ecosystems, and it is therefore possible that the mechanism is a common factor supporting diverse species assemblages.