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

Topological representations of matroid maps

The Topological Representation Theorem for (oriented) matroids states that every (oriented) matroid can be realized as the intersection lattice of an arrangement of codimension one homotopy spheres on a homotopy sphere. In this paper, we use a construction of Engstr\"om to show that structure-preserving maps between matroids induce topological mappings between their representations; a result previously known only in the oriented case. Specifically, we show that weak maps induce continuous maps and that the process is a functor from the category of matroids with weak maps to the homotopy category of topological spaces. We also give a new and conceptual proof of a result regarding the Whitney numbers of the first kind of a matroid.Comment: Final version, 21 pages, 8 figures; Journal of Algebraic Combinatorics, 201

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

Phonons in thin GaAs quantum wires

Phonon frequencies and potentials for an array of thin rectangular GaAs wires embedded in AlAs are calculated within a microscopic scheme. The confined and interface character of optical modes are clearly evident from their dispersion and from the spatial profiles. Our results allow us to conclude that macroscopic models based on the dielectric continuum scheme are adequate to describe confined phonon profiles at wave vectors relevant to el-ph scattering, in contrast with approaches based on mechanical boundary conditions, which yield modes with the wrong symmetry sequence. The implications for electron-phonon scattering rates are discussed

Composition-Diamond lemma for λ\lambda-differential associative algebras with multiple operators

In this paper, we establish the Composition-Diamond lemma for $\lambda$-differential associative algebras over a field $K$ with multiple operators. As applications, we obtain Gr\"{o}bner-Shirshov bases of free $\lambda$-differential Rota-Baxter algebras. In particular, linear bases of free $\lambda$-differential Rota-Baxter algebras are obtained and consequently, the free $\lambda$-differential Rota-Baxter algebras are constructed by words

Relationship between Lower Limbs Performance and Spinal Alignment in Parkinson’s Disease Patients: An Observational Study with Cross Sectional Design

Parkinson’s disease (PD) is a progressive neurodegenerative disease determining spinal deformities and muscle rigidity, weakness and dystonia that can be related to a change in muscular output during sit-to-stand tasks (STS). Purpose: The aim of this study was to determine the impacts of spinal alignment on lower limbs performance during STS tasks in Parkinson’s disease (PD) patients and healthy controls. Methods: In total, 43 consecutive PD patients (“PD” Group, 25 males and 18 females; age 73.7 ± 7.1) and 42 people not affected by any type of neurological disease (“CON” Group, 22 males, 20 females; age 69.8 ± 6.0) participated in the observational study. The clinical assessment included: IPAQ (International Physical Activity Questionnaire), Hoehn Yahr score, plumbline distance from the spinous process of C7, kyphosis apex and the spinous process of L3 and S1. We used the Muscle Quality Index test (MQI) to assess muscle power output during STS in both groups. Results: The MQI test measurements of absolute and relative lower limb power was significantly lower in the PD group, in addition to a negative correlation with age and a positive correlation with PL-L3 in that group of patients. Conclusions: A final consideration regarding our results leads to the possibility that the preservation of lumbar lordosis may be one of the factors for maintaining efficient biomechanics of the lower limb muscles, with the preservation of the physiological contractile characteristics of these muscles being the objective for a multidisciplinary rehabilitation based on postural exercises of the spine and a program of training exercises for the lower limb muscles

Toward a more ecologically informed view of severe forest fires

We use the historical presence of high-severity fire patches in mixed-conifer forests of the western United States to make several points that we hope will encourage development of a more ecologically informed view of severe wildland fire effects. First, many plant and animal species use, and have sometimes evolved to depend on, severely burned forest conditions for their persistence. Second, evidence from fire history studies also suggests that a complex mosaic of severely burned conifer patches was common historically in the West. Third, to maintain ecological integrity in forests born of mixed-severity fire, land managers will have to accept some severe fire and maintain the integrity of its aftermath. Lastly, public education messages surrounding fire could be modified so that people better understand and support management designed to maintain ecologically appropriate sizes and distributions of severe fire and the complex early-seral forest conditions it creates