Symmetries, groups and groupoids for Systems of Systems

In this paper we propose an algebraic model of systems based on the concept of symmetry that can be instrumental in representing Systems of Systems two main characteristics, namely complexity and (hierarchical) emergence

The Evolution of Homochirality

It is possible to reconsider the origin of biological homochirality in a novel way by formally invoking the standard groupoid approach to stereochemistry in a thermodynamic context that generalizes Landau's spontaneous symmetry breaking arguments. On Earth, limited metabolic free energy may have served as a low temperature analog to 'freeze' the system in the lowest energy state, i.e., the set of simplest homochiral transitive groupoids representing reproductive chemistries. These engaged in a Darwinian competition until a single configuration survived. Subsequent path dependent evolutionary process locked in this initial condition, in spite of increases in available metabolic free energy. Astrobiological outcomes, given higher initial metabolic free energy densities, could well be considerably richer, for example, of mixed chirality. One result would be a complicated distribution of biological chirality across a statistically large sample of extraterrestrial stereochemisty, in marked contrast with published predictions of a racemic average

Non-Hausdorff Symmetries of C*-algebras

Symmetry groups or groupoids of C*-algebras associated to non-Hausdorff spaces are often non-Hausdorff as well. We describe such symmetries using crossed modules of groupoids. We define actions of crossed modules on C*-algebras and crossed products for such actions, and justify these definitions with some basic general results and examples.Comment: very minor changes. To appear in Math. An

Protein folding disorders: Toward a basic biological paradigm

Mechanistic 'physics' models of protein folding fail to account for the observed spectrum of protein folding and aggregation disorders, suggesting that a more appropriately biological paradigm will be needed for understanding the etiology, prevention, and treatment of these diseases

Bost-Connes-Marcolli systems for Shimura varieties. I. Definitions and formal analytic properties

We construct a quantum statistical mechanical system $(A,s)$ analogous to the systems constructed by Bost-Connes and Connes-Marcolli in the case of Shimura varieties. Along the way, we define a new Bost-Connes system for number fields which has the ``correct'' partition function and the ``correct'' symmetries. We give a formalism that applies to general Shimura data $(G,X)$. The object of this series of papers is to show that these systems exhibit phase transitions with spontaneous symmetry breaking, and to classify their KMS states, at least for low temperature.Comment: 37 pages, LaTe

On biological homochirality

Generalizing Landau's spontaneous symmetry breaking arguments using the standard groupoid approach to stereochemistry allows reconsideration of the origin of biological homochirality. On Earth, limited metabolic free energy density may have served as a low temperature analog to 'freeze' the system into the set of simplest homochiral transitive groupoids representing reproductive chemistries. These engaged in Darwinian competition until a single configuration survived. Subsequent path dependent evolutionary process licked in this initial condition. Astrobiological outcomes, in the presence of higher initial metabolic free energy densities, could well be considerably richer, perhaps of mixed chirality. One result would be a complicated distribution of biological chirality across a statistically large sample of extraterrestrial stereochemistry, in contrast with a recent prediction of a racemic average

Evolution as a language that speaks itself

The 'self-referential' character of evolutionary process noted by Goldenfeld and Woese (2010) can be restated through a 'language' model in which genes, gene expression, and environment are represented as interacting information sources. The larger, composite, source that characterizes the high probability evolutionary paths then becomes, in a real sense, a language that speaks itself. The approach represents a significant extension of nonequilibrium condensed matter formalism in which the asymptotic limit theorems of information theory beat back the mathematical thicket a full step, providing necessary, but not sufficient, conditions for punctuated evolutionary transitions that can themselves be expressed as highly structured large deviations having their own grammar and syntax