To some Hecke symmetries (i.e. Yang-Baxter braidings of Hecke type) we associate ”noncommutative varieties” called braided spheres. An example of such a variety is the Podles’ nonstandard quantum sphere. On any braided sphere we introduce and compute an ”equivariant” analogue of Connes’ noncommutative index. In contrast with the Connes’ construction our version of equivariant NC index is based ...

To some Hecke symmetries (i.e. Yang-Baxter braidings of Hecke type) we assign algebras called braided non-commutative spheres. For any such algebra, we introduce and compute a q-analog of the Chern-Connes index. Unlike the standard Chern-Connes index, ours is based on the so-called categorical trace specific for a braided category in which the algebra in question is represented. AMS Mathematics...

We give a simple necessary and sufficient condition for a Schubert variety X w to be smooth when w is a freely braided element of a simply laced Weyl group; such elements were introduced by the authors in a previous work. This generalizes in one direction a result of Fan concerning varieties indexed by short-braid avoiding elements. We also derive generating functions for the freely braided ele...

In this study, carbon nano fibers (CNFs) were mixed into epoxy resin through a magnetic stirrer and again mixed using ultra sonicator. Using hand layup technique, biaxial braided fiber composites were prepared with unfilled, 0.2, 0.5 and 1 wt% CNF. Tensile test and shear test was performed to identify the tensile strength and shear strength of the composites. Fractured surface of the tensile sp...

Braided bialgebras were defined by mimicking the definition of bialgebras in a braided category; see [Ta]. In this paper we are interested in those braided bialgebras that are connected as a coalgebra, and such that, up to multiplication by a certain scalar, their braiding restricted to the primitive part is a Hecke operator. To every braided bialgebra as above we associate a braided Lie algebr...

Braided differential operators ∂ are obtained by differentiating the addition law on the braided covector spaces introduced previously (such as the braided addition law on the quantum plane). These are affiliated to a Yang-Baxter matrix R. The quantum eigenfunctions expR(x|v) of the ∂ i (braided-plane waves) are introduced in the free case where the position components xi are totally noncommuti...

The helix geometry model of 3D braided composites has been presented, which truly reflects the braided manner and coincides with the actual configuration of the braided composites. The longitudinal tensile stress-strain relationships and the strength of 3D braided composites under the tension loading have been predicted by a finite multiphase element method (FMEM) based on the helix geometry mo...