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

Phononic thermal conductivity in silicene: the role of vacancy defects and boundary scattering

We calculate the thermal conductivity of free-standing silicene using the phonon Boltzmann transport equation within the relaxation time approximation. In this calculation, we investigate the effects of sample size and different scattering mechanisms such as phonon-phonon, phonon-boundary, phonon-isotope and phonon-vacancy defect. Moreover, the role of different phonon modes is examined. We show that, in contrast to graphene, the dominant contribution to the thermal conductivity of silicene originates from the in-plane acoustic branches, which is about 70\% at room temperature and this contribution becomes larger by considering vacancy defects. Our results indicate that while the thermal conductivity of silicene is significantly suppressed by the vacancy defects, the effect of isotopes on the phononic transport is small. Our calculations demonstrate that by removing only one of every 400 silicon atoms, a substantial reduction of about 58\% in thermal conductivity is achieved. Furthermore, we find that the phonon-boundary scattering is important in defectless and small-size silicene samples, specially at low temperatures.Comment: 9 pages, 11 figure

Quantum Analogy of Poisson Geometry, Related Dendriform Algebras and Rota-Baxter Operators

We will introduce an associative (or quantum) version of Poisson structure tensors. This object is defined as an operator satisfying a "generalized" Rota-Baxter identity of weight zero. Such operators are called generalized Rota-Baxter operators. We will show that generalized Rota-Baxter operators are characterized by a cocycle condition so that Poisson structures are so. By analogy with twisted Poisson structures, we propose a new operator "twisted Rota-Baxter operators" which is a natural generalization of generalized Rota-Baxter operators. It is known that classical Rota-Baxter operators are closely related with dendriform algebras. We will show that twisted Rota-Baxter operators induce NS-algebras which is a twisted version of dendriform algebra. The twisted Poisson condition is considered as a Maurer-Cartan equation up to homotopy. We will show the twisted Rota-Baxter condition also is so. And we will study a Poisson-geometric reason, how the twisted Rota-Baxter condition arises.Comment: 18 pages. Final versio

Partial discharge behavior under operational and anomalous conditions in HVDC systems

Power cables undergo various types of overstressing conditions during their operation that influence the integrity of their insulation systems. This causes accelerated ageing and might lead to their premature failure in severe cases. This paper presents an investigation of the impacts of various dynamic electric fields produced by ripples, polarity reversal and transient switching impulses on partial discharge (PD) activity within solid dielectrics with the aim of considering such phenomena in high voltage direct current (HVDC) cable systems. Appropriate terminal voltages of a generic HVDC converter were reproduced - with different harmonic contaminations - and applied to the test samples. The effects of systematic operational polarity reversal and superimposed switching impulses with the possibility of transient polarity reversal were also studied in this investigation. The experimental results show that the PD is greatly affected by the dynamic changes of electric field represented by polarity reversal, ripples and switching. The findings of this study will assist in understanding the behaviour of PDs under HVDC conditions and would be of interest to asset managers considering the effects of such conditions on the insulation diagnostics

Generalized shuffles related to Nijenhuis and TD-algebras

Shuffle and quasi-shuffle products are well-known in the mathematics literature. They are intimately related to Loday's dendriform algebras, and were extensively used to give explicit constructions of free commutative Rota-Baxter algebras. In the literature there exist at least two other Rota-Baxter type algebras, namely, the Nijenhuis algebra and the so-called TD-algebra. The explicit construction of the free unital commutative Nijenhuis algebra uses a modified quasi-shuffle product, called the right-shift shuffle. We show that another modification of the quasi-shuffle product, the so-called left-shift shuffle, can be used to give an explicit construction of the free unital commutative TD-algebra. We explore some basic properties of TD-operators and show that the free unital commutative Nijenhuis algebra is a TD-algebra. We relate our construction to Loday's unital commutative dendriform trialgebras, including the involutive case. The concept of Rota-Baxter, Nijenhuis and TD-bialgebras is introduced at the end and we show that any commutative bialgebra provides such objects.Comment: 20 pages, typos corrected, accepted for publication in Communications in Algebr