We construct the positive principal series representations for Uq(gR) where g is of simply-laced type, parametrized by Rr where r is the rank of g. In particular, the positivity of the operators and the transcendental relations between the generators of the modular double are shown. We define the modified quantum group $\mathbf{U}_{q\tilde{q}(g_R)$ of the modular double and show that the representation of both parts of the modular double commute with each other, there is an embedding into the q-tori polynomials, and the commutant is the Langlands dual. We write down explicitly the action for type An,Dn and give the details of calculations for type E6,E7 and E8.

In this paper, we introduce and study a triple hierarchical variational inequality (THVI) with constraints of minimization and equilibrium problems. More precisely, let Fix ( T ) be the fixed point set of a nonexpansive mapping, let Fix ( T ) be the solution set of a mixed equilibrium problem (MEP), and let <i></i> be the solution set of a minimization problem (MP) for a convex and continuously Frechet differential functional in Hilbert spaces. We want to find a solution Fix ( T ) of a variational inequality with a variational inequality constraint over the intersection of Fix ( T ) , Fix ( T ) , and <i></i>. We propose a hybrid iterative algorithm with regularization to compute approximate solutions of the THVI, and we present the convergence analysis of the proposed iterative algorithm.

When estimating a mean regression function and its derivatives, locally weighted least squares regression has proven to be a very attractive technique. The present paper focuses on the important issue of how to select the smoothing parameter or bandwidth. In the case of estimating curves with a complicated structure, a variable bandwidth is desirable. Furthermore, the bandwidth should be indicated by the data themselves. Recent developments in nonparametric smoothing techniques inspired us to propose such a datadriven bandwidth selection procedure, which can be used to select both constant and variable bandwidths. The idea is based on a residual squares criterion along with a good approximation of the bias and variance of the estimator. The procedure can be applied to select bandwidths not only for estimating the regression curve but also for estimating its derivatives. The resulting estimation

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