It is well known that every Hölder continuous function on the unit circle is the sum of two functions such that one of these functions extends holomorphically into the unit disc and the other extends holomorphically into the complement of the unit disc. We prove that an analogue of this holds for Hölder continuous functions on an annulus A which have zero averages on all circles contained in A which surround the hole.

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The SMS4 block cipher is part of the Chinese WAPI wireless standard. This paper describes the specification and offers a specification for a toy version called simplified SMS4 (S-SMS4). We explore algebraic attacks on SMS4 and S-SMS4 using Gröbner basis attacks on equation systems over GF(2) and GF(2^8), as well as attacks using a SAT solver derived from the GF(2) model. A comparison of SAT and Gröbner basis attacks is provided.

Fusion of images with same or different modalities has been conquering medical imaging field more rapidly due to the presence of highly accessible patients’ information in recent years. For example, cross platform non-rigid registration of CT with MRI images has found a significant role in different clinical application. In some instances labelling of anatomical features by medical experts are also involved to further improve the accuracy and authenticity of the registration. Being motivated by these, we propose a new algorithm to compute diffeomorphic hybrid multi-modality registration with large deformations. Our iterative scheme consists of mainly two steps. First, we obtain the optimal Beltrami coefficient corresponding to the diffeomorphic mapping that exactly superimposes the feature points. The second step detects the intensity difference in the framework of mutual information. A non-rigid deformation which minimizes the intensity difference is then obtained. Experiments have been carried out on both synthetic and real data. Results demonstrate the stability and efficacy of the proposed algorithm to obtain diffeomorphic image registration.

Let M be a closed surface diffeomorphic to S2 endowed with a Riemannian metric. Denote the diameter of M by d. We prove that for every x 2 M and every positive integer k there exist k distinct geodesic loops based at x of length  20kd.