On the Representation Theory of an Algebra of Braids and Ties

We consider the algebra ${\cal E}_n(u)$ introduced by F. Aicardi and J. Juyumaya as an abstraction of the Yokonuma-Hecke algebra. We construct a tensor space representation for ${\cal E}_n(u)$ and show that this is faithful. We use it to give a basis for ${\cal E}_n(u)$ and to classify its irreducible representations.Comment: 24 pages. Final version. To appear in Journal of Algebraic Combinatorics

Approximate solutions of the determinantal assignment problem and distance problems

The paper introduces the formulation of an exact algebrogeometric problem, the study of the Determinantal Assignment Problem (DAP) in the set up of design, where approximate solutions of the algebraic problem are sought. Integral part of the solution of the Approximate DAP is the computation of distance of a multivector from the Grassmann variety of a projective space. We examine the special case of the calculation of the minimum distance of a multivector in ∧2(ℝ5) from the Grassmann variety G 2(ℝ5). This problem is closely related to the problem of decomposing the multivector and finding its best decomposable approximation. We establish the existence of the best decomposition in a closed form and link the problem of distance to the decomposition of multivectors. The uniqueness of this decomposition is then examined and several new alternative decompositions are presented that solve our minimization problem based on the structure of the problem

FT Raman and DFT Study on a Series of All- anti Oligothienoacenes End-Capped with Triisopropylsilyl Groups

Herein, we study the Π-conjugational properties of a homologous series of all -anti oligothienoacenes containing four to eight fused thiophene rings by means of FT Raman spectroscopy and DFT calculations. The theoretical analysis of the spectroscopic data provides evidence that selective enhancement of a very limited number of Raman scatterings is related to the occurrence in these oligothienoacenes of strong vibronic coupling between collective Ν(C[bouble bond]C) stretching modes in the 1600–1300 cm −1 region and the HOMO/LUMO frontier orbitals (HOMO=highest occupied molecular orbital; LUMO=lowest unoccupied molecular orbital). The correlation of the Raman spectroscopic data and theoretical results for these all -anti oligothienoacenes with those previously collected for a number of all -syn oligothienohelicenes gives further support to the expectation that cross-conjugation is dominant in heterohelicenes. Fully planar all -anti oligothienoacenes display linear Π conjugation which seemingly does not reach saturation with increasing number of annulated thiophene rings in the oligomeric chain at least up to the octamer.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/64547/1/3069_ftp.pd

Oxidation of End-Capped Pentathienoacenes and Characterization of Their Radical Cations

A detailed investigation of the optical and electrochemical properties of two pentathienoacene derivatives, 2,6-bis(trimethylsilyl)-Α-pentathienoacene ( TMS-T5-TMS ) and 2,6-bis(triisopropylsilyl)-Α-pentathienoacene ( TIPS-T5-TIPS ), as the neutral and oxidized species was performed in the temperature range of 80–300 14K. The experimental solution UV/Vis and solid-state Raman spectra were interpreted by using time-dependent DFT and DFT quantum chemical calculations at the B3LYP/6-31G** level. Bond lengths, HOMO–LUMO positions, and charge distribution were also predicted by computational methods for both the neutral and oxidized states of each thienoacene. As evidenced by ESR and spectroelectrochemical data, upon oxidation the pentathienoacene derivative with the less sterically hindering trimethylsilyl solubilizing groups, TMS-T5-TMS , undergoes Π 14dimerization to form [ TMS-T5-TMS ] 2 2+ . In contrast, TIPS-T5-TIPS , with the more bulky triisopropylsilyl solubilizing groups, was oxidized to the radical cation but dimerization was prevented due to steric interactions. These experimental observations are supported by DFT calculations, which were used to investigate [ TMS-T5-TMS ] 2 2+ and [ TIPS-T5-TIPS ] 2 2+ Π 14dimers in the solid state and in solution. The redox potentials and absorption peak locations corresponding to the radical cations and Π 14dimer of TMS-T5-TMS were identified experimentally.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/64441/1/chem_200900246_sm_miscellaneous_information.pd

Tensor spaces and exterior algebra

This book explains, as clearly as possible, tensors and such related topics as tensor products of vector spaces, tensor algebras, and exterior algebras. You will appreciate Yokonuma's lucid and methodical treatment of the subject. This book is useful in undergraduate and graduate courses in multilinear algebra. Tensor Spaces and Exterior Algebra begins with basic notions associated with tensors. To facilitate understanding of the definitions, Yokonuma often presents two or more different ways of describing one object. Next, the properties and applications of tensors are developed, including the classical definition of tensors and the description of relative tensors. Also discussed are the algebraic foundations of tensor calculus and applications of exterior algebra to determinants and to geometry. This book closes with an examination of algebraic systems with bilinear multiplication. In particular, Yokonuma discusses the theory of replicas of Chevalley and several properties of Lie algebras deduced from them