Hopf algebras and Markov chains: Two examples and a theory

The operation of squaring (coproduct followed by product) in a combinatorial Hopf algebra is shown to induce a Markov chain in natural bases. Chains constructed in this way include widely studied methods of card shuffling, a natural "rock-breaking" process, and Markov chains on simplicial complexes. Many of these chains can be explictly diagonalized using the primitive elements of the algebra and the combinatorics of the free Lie algebra. For card shuffling, this gives an explicit description of the eigenvectors. For rock-breaking, an explicit description of the quasi-stationary distribution and sharp rates to absorption follow.Comment: 51 pages, 17 figures. (Typographical errors corrected. Further fixes will only appear on the version on Amy Pang's website, the arXiv version will not be updated.

A Typology of Digital Sharing Business Models: A Design Science Research Approach

The digitally enabled sharing economy, also called the “digital sharing economy” (DSE), has changed patterns of consumption by introducing new choices and channels for provision and receipt of services. The DSE encompasses sharing systems whose business models may vary distinctly from platform to platform. Although business models in the context of the sharing economy have been studied so far, we have observed that the current literature does not provide an approach that covers all the possible business models (in the broadest sense of the term) that (potentially) exist within the scope of the DSE. The present paper, therefore, aims to propose a typology of business models in the DSE that covers a wide space of models – even those which may not involve “business” in the commercial sense. This is achieved through an iterative inductive process based on a design science research approach. The typology can assist in positioning the current and future sharing systems in the DSE by systematically classifying their business models. It is intended to serve as a guiding tool for the sustainability assessment of platforms from both resource and socio-economic perspectives. The present study can also enable researchers and practitioners to capture and systematically analyse digital sharing business models based on a structured, actionable approach

Comparative analyses imply that the enigmatic sigma factor 54 is a central controller of the bacterial exterior

Contains fulltext : 95738.pdf (publisher's version ) (Open Access)BACKGROUND: Sigma-54 is a central regulator in many pathogenic bacteria and has been linked to a multitude of cellular processes like nitrogen assimilation and important functional traits such as motility, virulence, and biofilm formation. Until now it has remained obscure whether these phenomena and the control by Sigma-54 share an underlying theme. RESULTS: We have uncovered the commonality by performing a range of comparative genome analyses. A) The presence of Sigma-54 and its associated activators was determined for all sequenced prokaryotes. We observed a phylum-dependent distribution that is suggestive of an evolutionary relationship between Sigma-54 and lipopolysaccharide and flagellar biosynthesis. B) All Sigma-54 activators were identified and annotated. The relation with phosphotransfer-mediated signaling (TCS and PTS) and the transport and assimilation of carboxylates and nitrogen containing metabolites was substantiated. C) The function annotations, that were represented within the genomic context of all genes encoding Sigma-54, its activators and its promoters, were analyzed for intra-phylum representation and inter-phylum conservation. Promoters were localized using a straightforward scoring strategy that was formulated to identify similar motifs. We found clear highly-represented and conserved genetic associations with genes that concern the transport and biosynthesis of the metabolic intermediates of exopolysaccharides, flagella, lipids, lipopolysaccharides, lipoproteins and peptidoglycan. CONCLUSION: Our analyses directly implicate Sigma-54 as a central player in the control over the processes that involve the physical interaction of an organism with its environment like in the colonization of a host (virulence) or the formation of biofilm

Cancer Biomarker Discovery: The Entropic Hallmark

Background: It is a commonly accepted belief that cancer cells modify their transcriptional state during the progression of the disease. We propose that the progression of cancer cells towards malignant phenotypes can be efficiently tracked using high-throughput technologies that follow the gradual changes observed in the gene expression profiles by employing Shannon's mathematical theory of communication. Methods based on Information Theory can then quantify the divergence of cancer cells' transcriptional profiles from those of normally appearing cells of the originating tissues. The relevance of the proposed methods can be evaluated using microarray datasets available in the public domain but the method is in principle applicable to other high-throughput methods. Methodology/Principal Findings: Using melanoma and prostate cancer datasets we illustrate how it is possible to employ Shannon Entropy and the Jensen-Shannon divergence to trace the transcriptional changes progression of the disease. We establish how the variations of these two measures correlate with established biomarkers of cancer progression. The Information Theory measures allow us to identify novel biomarkers for both progressive and relatively more sudden transcriptional changes leading to malignant phenotypes. At the same time, the methodology was able to validate a large number of genes and processes that seem to be implicated in the progression of melanoma and prostate cancer. Conclusions/Significance: We thus present a quantitative guiding rule, a new unifying hallmark of cancer: the cancer cell's transcriptome changes lead to measurable observed transitions of Normalized Shannon Entropy values (as measured by high-throughput technologies). At the same time, tumor cells increment their divergence from the normal tissue profile increasing their disorder via creation of states that we might not directly measure. This unifying hallmark allows, via the the Jensen-Shannon divergence, to identify the arrow of time of the processes from the gene expression profiles, and helps to map the phenotypical and molecular hallmarks of specific cancer subtypes. The deep mathematical basis of the approach allows us to suggest that this principle is, hopefully, of general applicability for other diseases