$ {{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes
Tingting Wu Jian Gao Yun Gao Fang-Wei Fu
Advances in Mathematics of Communications 2018, 12(4): 641-657 doi: 10.3934/amc.2018038

This paper is concerned with ${{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes. These codes can be identified as submodules of the ring ${\mathbb{Z}}_{2}[x]/\langle x^r-1\rangle × {\mathbb{Z}}_{2}[x]/\langle x^s-1\rangle × {\mathbb{Z}}_{4}[x]/\langle x^t-1\rangle$. There are two major ingredients. First, we determine the generator polynomials and minimum generating sets of this kind of codes. Furthermore, we investigate their dual codes. We determine the structure of the dual of separable ${{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes completely. For the dual of non-separable ${{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes, we give their structural properties in a few special cases.

keywords: \begin{document}$ {{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$\end{document}-cyclic codes generator polynomials minimum generating sets dual code
Two-step methods for image zooming using duality strategies
Tingting Wu Yufei Yang Huichao Jing
Numerical Algebra, Control & Optimization 2014, 4(3): 209-225 doi: 10.3934/naco.2014.4.209
In this paper we propose two two-step methods for image zooming using duality strategies. In the first method, instead of smoothing the normal vector directly as did in the first step of the classical LOT model, we reconstruct the unit normal vector by means of Chambolle's dual formulation. Then, we adopt the split Bregman iteration to obtain the zoomed image in the second step. The second method is based on the TV-Stokes model. By smoothing the tangential vector and imposing the divergence free condition, we propose an image zooming method based on the TV-Stokes model using the dual formulation. Furthermore, we give the convergence analysis of the proposed algorithms. Numerical experiments show the efficiency of the proposed methods.
keywords: Two-step method TV-Stokes model split Bregman iteration image zooming. duality strategy

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