We study random walks on the three-strand braid group B3, and in particular compute the drift, or average topological complexity of a random braid, as well as the probability of trivial entanglement. These results involve the study of magnetic random walks on hyperbolic graphs (hyperbolic Harper-Hofstadter problem), what enables to build a faithful representation of B3 as generalized magnetic t...

Abstract. We compute the asymptotical growth rate of a large family of Uq(sl2) 6j-symbols and we interpret our results in geometric terms by relating them to the volumes of suitable hyperbolic objects. We propose an extension of S. Gukov’s generalized volume conjecture to cover the case of hyperbolic links in S or #k . We prove this conjecture for the infinite family of universal hyperbo...

We name an indecomposable symmetrizable generalized Cartan matrix A and the corresponding Kac–Moody Lie algebra g(A) of the arithmetic type if for any β ∈ Q with (β|β) < 0 there exist n(β) ∈ N and an imaginary root α ∈ ∆ such that n(β)β ≡ α mod Ker (.|.) on Q. Here Q is the root lattice. This generalizes ”symmetrizable hyperbolic” type of Kac and Moody. We show that generalized Cartan matrices ...

The Huygens-Wilker type inequalities involving generalized trigonometric functions and generalized hyperbolic functions are established. The first and the second inequalities of Huygens and Wilker, for classes of functions under discussion, are also investigated.

A generalized multivariate problem due to V. M. Zolotarev is considered. Some related results on geometric random sums and (multivariate) stable distributions are extended a more general case of “anisotropic” summation where independent vectors with index having special distribution Anisotropic-geometric introduced. It demonstrated that these coordinate-wise scale mixtures elliptically contoure...