Rota-Baxter Algebras and Dendriform Algebras

In this paper we study the adjoint functors between the category of Rota-Baxter algebras and the categories of dendriform dialgebras and trialgebras. In analogy to the well-known theory of the adjoint functor between the category of associative algebras and Lie algebras, we first give an explicit construction of free Rota-Baxter algebras and then apply it to obtain universal enveloping Rota-Baxter algebras of dendriform dialgebras and trialgebras. We further show that free dendriform dialgebras and trialgebras, as represented by binary planar trees and planar trees, are canonical subalgebras of free Rota-Baxter algebras.Comment: Typos corrected and the last section on analog of Poincare-Birkhoff-Witt theorem deleted for a gap in the proo

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

Commissioning of the new multi-layer integration prototype of the CALICE tile hadron calorimeter

The basic prototype of a tile hadron calorimeter (HCAL) for the International Linear Collider (ILC) has been realised and extensively tested. A major aspect of the proposed concept is the improvement of the jet energy resolution by measuring details of the shower development and combining them with the data of the tracking system (particle flow). The prototype utilises scintillating tiles that are read out by novel Silicon Photomultipliers (SiPMs) and takes into account all design aspects that are demanded by the intended operation at the ILC. Currently, a new 12 layer prototype with about 3400 detector channels is under development. Alternative architectures for the scintillating tiles with and without wavelength-shifting fibres and tiles with individual wrapping with reflector foil is tested as well as different types of SiPMs. The new prototype was used for the first time at the CERN Proton Synchrotron test facility in fall 2014. Additionally, detector modules for the CALICE scintillator-based Electromagnetic Calorimeter (Sc-ECAL), that follow the proposed HCAL electronics architecture, were part of this new prototype. A new multi-layer Data Acquisition System (DAQ) was used for the detector configuration and operation.Comment: 2014 IEEE Nuclear Science Symposium and Medical Imaging Conference and 21st Symposium on Room-Temperature Semiconductor X-ray and Gamma-ray Detectors (NSS/MIC 2014 / RTSD 2014

Dendriform Equations

We investigate solutions for a particular class of linear equations in dendriform algebras. Motivations as well as several applications are provided. The latter follow naturally from the intimate link between dendriform algebras and Rota-Baxter operators, e.g. the Riemann integral or Jackson's q-integral.Comment: improved versio

Interaction between high-level and low-level image analysis for semantic video object extraction

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