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Abstract We give a description of the Hallnäs–Ruijsenaars eigenfunctions of the 2-particle hyperbolic Ruijsenaars system as matrix coefficients for the order 4 element$$S\in SL(2,{\mathbb {Z}})$$ $S\in SL(2,Z)$acting on the Hilbert space ofGL(2) quantum Teichmüller theory on the punctured torus. TheGL(2) Macdonald polynomials are then obtained as special values of the analytic continuation of these matrix coefficients. The main tool used in the proof is the cluster structure on the moduli space of framedGL(2)-local systems on the punctured torus, and an$$SL(2,{\mathbb {Z}})$$ $SL(2,Z)$-equivariant embedding of theGL(2) spherical DAHA into the quantized coordinate ring of the corresponding cluster Poisson variety. 

The goal of this paper is to develop a methodology for the systematic analysis of asymptotic statistical properties of data-driven DRO formulations based on their corresponding non-DRO counterparts. We illustrate our approach in various settings, including both phidivergence and Wasserstein uncertainty sets. Different types of asymptotic behaviors are obtained depending on the rate at which the uncertainty radius decreases to zero as a function of the sample size and the geometry of the uncertainty sets. 

Abstract Practical engineering designs typically involve many load cases. For topology optimization with many deterministic load cases, a large number of linear systems of equations must be solved at each optimization step, leading to an enormous computational cost. To address this challenge, we propose a mirror descent stochastic approximation (MD-SA) framework with various step size strategies to solve topology optimization problems with many load cases. We reformulate the deterministic objective function and gradient into stochastic ones through randomization, derive the MD-SA update, and develop algorithmic strategies. The proposed MD-SA algorithm requires only low accuracy in the stochastic gradient and thus uses only a single sample per optimization step (i.e., the sample size is always one). As a result, we reduce the number of linear systems to solve per step from hundreds to one, which drastically reduces the total computational cost, while maintaining a similar design quality. For example, for one of the design problems, the total number of linear systems to solve and wall clock time are reduced by factors of 223 and 22, respectively.