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In this paper a Wiener-type characterization is presented of those boundary points of a Denjoy domain where the Green function is Lipschitz continuous. This property is linked with the splitting of a Euclidean boundary point into two minimal Martin boundary points.

Abstract We investigate a system of N Brownian particles with the Coulomb interaction in any dimension \(d\ge 2\), and we assume that the initial data are independent and identically distributed with a common density \(\rho _0\) satisfying \(\int _{\mathbb {R}^{d}}\rho _0\ln \rho _0\,\hbox {d}x<\infty \) and \(\rho _0\in L^{\frac{2d}{d+2}} (\mathbb {R}^{d}) \cap L^1(\mathbb {R}^{d}, (1+|x|^2)\,\hbox {d}x)\). We prove that there exists a unique global strong solution for this interacting partsicle system and there is no collision among particles almost surely. For \(d=2\), we rigorously prove the propagation of chaos for this particle system globally in time without any cutoff in the following sense. When \(N\rightarrow \infty \), the empirical measure of the particle system converges in law to a probability measure and this measure possesses a density which is the unique weak solution to the mean-field Poisson–Nernst–Planck equation of single component.

We construct the positive principal series representations for Uq(gR) where g is of type Bn, Cn, F4 or G2, parametrized by Rn where n is the rank of g. We show that under the representations, the generators of the Langlands dual group Uq(LgR) are related to the generators of Uq(gR) by the transcendental relations. This gives a new and very simple analytic relation between the Langlands dual pair. We define the modified quantum group Uqq(gR)=Uq(gR) ⊗ Uq(LgR) of the modular double and show that the representations of both parts of the modular double commute with each other, and there is an embedding into the q-tori polynomials.

This paper is devoted to the study of the proximal point algorithm for solving monotone second-order cone complementarity problems. The proximal point algorithm is to generate a sequence by solving subproblems that are regularizations of the original problem. After given an appropriate criterion for approximate solutions of subproblems by adopting a merit function, the proximal point algorithm is verified to have global and superlinear convergence properties. For the purpose of solving the subproblems efficiently, we introduce a generalized Newton method and show that only one Newton step is eventually needed to obtain a desired approximate solution that approximately satisfies the appropriate criterion under mild conditions. Numerical comparisons are also made with the derivative-free descent method used by Pan and Chen (Optimization 59:11731197, 2010), which confirm the theoretical results