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Let$M$be a non-compact connected Riemann surface of a finite type, and$R$⋐$M$be a relatively compact domain such that$H$_{1}($M$,$Z$)=$H$_{1}($R$,$Z$). Let $$\tilde R \to R$$ be a covering. We study the algebra$H$^{∞}($U$) of bounded holomorphic functions defined in certain subdomains $$U \subset \tilde R$$ . Our main result is a Forelli type theorem on projections in$H$^{∞}($D$).

Recent years have witnessed the surge of asynchronous parallel (async-parallel) iterative algorithms due to problems involving very large-scale data and a large number of decision variables. Because of asynchrony, the iterates are computed with outdated information, and the age of the outdated information, which we call delay, is the number of times it has been updated since its creation. Almost all recent works prove convergence under the assumption of a finite maximum delay and set their stepsize parameters accordingly. However, the maximum delay is practically unknown. This paper presents convergence analysis of an async-parallel method from a probabilistic viewpoint, and it allows for large unbounded delays. An explicit formula of stepsize that guarantees convergence is given depending on delays’ statistics. With p+1 identical processors, we empirically measured that delays closely follow the Poisson distribution with parameter p, matching our theoretical model, and thus, the stepsize can be set accordingly. Simulations on both convex and nonconvex optimization problems demonstrate the validness of our analysis and also show that the existing maximum-delay-induced stepsize is too conservative, often slows down the convergence of the algorithm.

Many optimization algorithms converge to stationary points. When the underlying problem is nonconvex, they may get trapped at local minimizers and occasionally stagnate near saddle points. We propose the Run-and-Inspect Method, which adds an “inspect” phase to existing algorithms that helps escape from non-global stationary points. The inspection samples a set of points in a radius R around the current point. When a sample point yields a sufficient decrease in the objective, we resume an existing algorithm from that point. If no sufficient decrease is found, the current point is called an approximate R-local minimizer. We show that an R-local minimizer is globally optimal, up to a specific error depending on R, if the objective function can be implicitly decomposed into a smooth convex function plus a restricted function that is possibly nonconvex, nonsmooth. Therefore, for such nonconvex objective functions, verifying global optimality is fundamentally easier. For high-dimensional problems, we introduce blockwise inspections to overcome the curse of dimensionality while still maintaining optimality bounds up to a factor equal to the number of blocks. Our method performs well on a set of artificial and realistic nonconvex problems by coupling with gradient descent, coordinate descent, EM, and prox-linear algorithms.

We construct a smooth Artin stack parameterizing the stable weighted curves of genus one with twisted fields and prove that it is isomorphic to the blowup stack of the moduli of genus one weighted curves studied by Hu and Li. This leads to a blowup-free construction of Vakil-Zinger's desingularization of the moduli of genus one stable maps to projective spaces. This construction provides the cornerstone of the theory of stacks with twisted fields, which is thoroughly studied in arXiv:2005.03384 and leads to a blowup-free resolution of the stable map moduli of genus two.