The chapter discusses the group actions on R³. The Smith conjecture has many equivalent forms, and each of these forms has various consequences and generalizations. The Smith conjecture is a structure theorem about symmetries of the product of a compact surface with an interval. Here the interval may be closed or open. The usual Smith conjecture is equivalent to proving the smooth <i>Z_n</i> actions on <i>S</i>² [0, 1] are conjugate to actions that preserve the product structure. This generalized Smith conjecture represents the belief that all the symmetries of the product of a compact surface with an interval actually arise from the symmetries of the surface extended trivially to the product structure. The chapter presents how to apply minimal surfaces to study the generalized Smith conjecture on <i>S</i>² <i>I</i>, where <i>I</i> is an open or closed interval. A geodesic version of the loop theorem and a new equivariant

In 1962 Sprind\v{z}uk proved Mahler's conjecture in both the real and complex cases. Baker gave a generalized result by using a modified version of Sprind\v{z}uk's method. Later, Baker and Schmidt derived the Hausdorff dimensions of sets which are defiend in terms of approximation by algebraic numbers of bounded degrees by using Baker's theorem. In this article we will prove two analogue theorems in the fields of formal power series over finite finite fields.

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Processing streaming data as they arrive is often necessary for high dimensional data analysis. In this paper, we analyze the convergence of a subspace online PCA iteration, as a followup of the recent work of Li, Wang, Liu, and Zhang [Math. Program., Ser. B, DOI 10.1007/s10107-017-1182-z] who considered the case for the most significant principal component only, i.e., a single vector. Under the sub-Gaussian assumption, we obtain a finite-sample error bound that closely matches the minimax information lower bound of Vu and Lei [Ann. Statist. 41:6 (2013), 2905-2947].