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We show that specializations of the 4d =2 superconformal index labeled by an integer N is given by TrN where  is the Kontsevich-Soibelman monodromy operator for BPS states on the Coulomb branch. We provide evidence that the states enumerated by these limits of the index lead to a family of 2d chiral algebras N. This generalizes the recent results for the N=−1 case which corresponds to the Schur limit of the superconformal index. We show that this specialization of the index leads to the same integrand as that of the elliptic genus of compactification of the superconformal theory on S2×T2 where we turn on 12N units of U(1)r flux on S2.

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The first author has associated in a natural way a profinite group to each irreducible subshift. The group in question was initially obtained as a maximal subgroup of a free profinite semigroup. In the case of minimal subshifts, the same group is shown in the present paper to also arise from geometric considerations involving the Rauzy graphs of the subshift. Indeed, the group is shown to be isomorphic to the inverse limit of the profinite completions of the fundamental groups of the Rauzy graphs of the subshift. A further result involving geometric arguments on Rauzy graphs is a criterion for freeness of the profinite group of a minimal subshift based on the Return Theorem of Berthé et al.

We consider image deblurring problem in the presence of impulsive noise. It is known that \emph{total variation} (TV) regularization with L1-norm penalized data fitting (TVL1 for short) works reasonably well only when the level of impulsive noise is relatively low. For high level impulsive noise, TVL1 works poorly. The reason is that all data, both corrupted and noise free, are equally penalized in data fitting, leading to insurmountable difficulty in balancing regularization and data fitting. In this paper, we propose to combine TV regularization with nonconvex \emph{smoothly clipped absolute deviation} (SCAD) penalty for data fitting (TVSCAD for short). Our motivation is simply that data fitting should be enforced only when an observed data is not severely corrupted, while for those data more likely to be severely corrupted, less or even null penalization should be enforced. A \emph{difference of convex functions} algorithm is adopted to solve the nonconvex TVSCAD model, resulting in solving a sequence of TVL1-equivalent problems, each of which can then be solved efficiently by the alternating direction method of multipliers. Theoretically, we establish global convergence to a critical point of the nonconvex objective function. The R-linear and at-least-sublinear convergence rate results are derived for the cases of anisotropic and isotropic TV, respectively. Numerically, experimental results are given to show that the TVSCAD approach improves those of the TVL1 significantly, especially for cases with high level impulsive noise, and is comparable with the recently proposed iteratively corrected TVL1 method [\textit{Inverse Problems}, 32 (2016), pp.~085004].