Free Rota-Baxter algebras and rooted trees

A Rota-Baxter algebra, also known as a Baxter algebra, is an algebra with a linear operator satisfying a relation, called the Rota-Baxter relation, that generalizes the integration by parts formula. Most of the studies on Rota-Baxter algebras have been for commutative algebras. Two constructions of free commutative Rota-Baxter algebras were obtained by Rota and Cartier in the 1970s and a third one by Keigher and one of the authors in the 1990s in terms of mixable shuffles. Recently, noncommutative Rota-Baxter algebras have appeared both in physics in connection with the work of Connes and Kreimer on renormalization in perturbative quantum field theory, and in mathematics related to the work of Loday and Ronco on dendriform dialgebras and trialgebras. This paper uses rooted trees and forests to give explicit constructions of free noncommutative Rota--Baxter algebras on modules and sets. This highlights the combinatorial nature of Rota--Baxter algebras and facilitates their further study. As an application, we obtain the unitarization of Rota-Baxter algebras.Comment: 23 page

Amino Acid Features of P1B-ATPase Heavy Metal Transporters Enabling Small Numbers of Organisms to Cope with Heavy Metal Pollution

Phytoremediation refers to the use of plants for extraction and detoxification of pollutants, providing a new and powerful weapon against a polluted environment. In some plants, such as Thlaspi spp, heavy metal ATPases are involved in overall metal ion homeostasis and hyperaccumulation. P1B-ATPases pump a wide range of cations, especially heavy metals, across membranes against their electrochemical gradients. Determination of the protein characteristics of P1B-ATPases in hyperaccumulator plants provides a new opportuntity for engineering of phytoremediating plants. In this study, using diverse weighting and modeling approaches, 2644 protein characteristics of primary, secondary, and tertiary structures of P1B-ATPases in hyperaccumulator and nonhyperaccumulator plants were extracted and compared to identify differences between proteins in hyperaccumulator and nonhyperaccumulator pumps. Although the protein characteristics were variable in their weighting, tree and rule induction models; glycine count, frequency of glutamine-valine, and valine-phenylalanine count were the most important attributes highlighted by 10, five, and four models, respectively. In addition, a precise model was built to discriminate P1B-ATPases in different organisms based on their structural protein features. Moreover, reliable models for prediction of the hyperaccumulating activity of unknown P1B-ATPase pumps were developed. Uncovering important structural features of hyperaccumulator pumps in this study has provided the knowledge required for future modification and engineering of these pumps by techniques such as site-directed mutagenesis

Mixable Shuffles, Quasi-shuffles and Hopf Algebras

The quasi-shuffle product and mixable shuffle product are both generalizations of the shuffle product and have both been studied quite extensively recently. We relate these two generalizations and realize quasi-shuffle product algebras as subalgebras of mixable shuffle product algebras. As an application, we obtain Hopf algebra structures in free Rota-Baxter algebras.Comment: 14 pages, no figure, references update

Design of integrated pitch axis for autopilot/autothrottle and integrated lateral axis for autopilot/yaw damper for NASA TSRV airplane using integral LQG methodology

Two designs are presented for control systems for the NASA Transport System Research Vehicle (TSRV) using integral Linear Quadratic Gaussian (LQG) methodology. The first is an integrated longitudinal autopilot/autothrottle design and the second design is an integrated lateral autopilot/yaw damper/sideslip controller design. It is shown that a systematic top-down approach to a complex design problem combined with proper application of modern control synthesis techniques yields a satisfactory solution in a reasonable period of time

Environmental exposures to cadmium and lead as potential causes of eye diseases

Humans are exposed to cadmium and lead in various regions of the world daily due to industrial development and climate change. Increasing numbers of preclinical and clinical studies indicate that heavy metals, such as cadmium and lead, play a role in the pathogenesis of eye diseases. Excessive exposure to heavy metals such as cadmium and lead can increase the risk of impaired vision. Therefore, it is essential to better characterize the role of these non-essential metals in disease etiology and progression. This article discusses the potential role of cadmium and lead in the development of age-related eye diseases, including age-related macular degeneration, cataracts, and glaucoma. Furthermore, we discuss how cadmium and lead affect ocular cells and provide an overview of putative pathological mechanisms associated with their propensity to damage the eye.</p