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 ...

Abstract We use skein theory to compute the coefficients of certain power series considered by Habiro in his theory of sl2 invariants of integral homology 3-spheres. Habiro originally derived these formulas using the quantum group Uqsl2 . As an application, we give a formula for the colored Jones polynomial of twist knots, generalizing formulas of Habiro and Le for the trefoil and the figure ei...

The hydrogenases [FeFe] and Hmd feature at first sight rather different active sites. A closer inspection reveals striking similarities, which allow us to define swapped ligand spheres in such a way that the single active iron center of Hmd functions in a first-shell ligand environment resembling the reacting iron atom in [FeFe] hydrogenase and vice versa. These redesigned ligand environments c...

In this paper we consider   multivariate Lagrange mean-value interpolation problem, where interpolation parameters are integrals over spheres. We have   concentric spheres. Indeed, we consider the problem in three variables when it is not correct.  

Let M be a rational homology 3–sphere with |H1(M,Z)| = b. For any odd divisor c of b, we construct a unified invariant IM,c lying in a cyclotomic completion of a certain polynomial ring, which dominates Witten–Reshetikhin–Turaev SO(3) invariants of M at all roots of unity whose order r satisfies (r, b) = c. For c = 1, we recover the unified invariant constructed by Le and Beliakova–Le. If b = 1...

We study an exact asymptotic behavior of the Witten–Reshetikhin–Turaev invariant for the Brieskorn homology spheres Σ(p1, p2, p3) by use of properties of the modular form following a method proposed by R. Lawrence and D. Zagier. Key observation is that the invariant coincides with a limiting value of the Eichler integral of the modular form with weight 3/2. We show that the Casson invariant is ...

Effect of (a) particle size, (b) lattice shrinkage, and (c) vacancies on the upconversion luminescence in GdVO 4 :Yb 3+ /Er . The colored spheres represent different systems (left) with respective volumes being proportional to quantum yield.

The effects of using different counter electrode metal sulfides on the performances of solar cells made with CdS/CdSe/ZnS quantum dots co-sensitized onto hierarchical TiO2 spheres (HTSs) used as photo-electrode are reported. The HTS in the QDSSCs is composed of an assembly of numerous TiO2 spheres made by the solvolthermal method. The photoelectrical performance of HTS/CdS/CdSe/ZnS coupled to C...

We show that the unit area Liouville quantum gravity sphere can be constructed in two equivalent ways. The first, which was introduced by the authors and Duplantier in [DMS14], uses a Bessel excursion measure to produce a Gaussian free field variant on the cylinder. The second uses a correlated Brownian loop and a “mating of trees” to produce a Liouville quantum gravity sphere decorated by a sp...

The time-window for processing electron spin information (spintronics) in solid-state quantum electronic devices is determined by the spin-lattice and spin-spin relaxation times of electrons. Minimizing the effects of spin-orbit coupling and the local magnetic contributions of neighbouring atoms on spin-lattice and spin-spin relaxation times at room temperature remain substantial challenges to ...