The internal space of a N = 4 supersymmetric model with WessZumino term has a connection with totally skew-symmetric torsion and holonomy in SP(<i>n</i>). We study the mathematical background of this type of connection. In particular, we relate it to classical Hermitian geometry, construct homogeneous as well as inhomogeneous examples, characterize it in terms of holomorphic data, develop its potential theory and reduction theory.

A review of the current state of multivariate public-key cryptosystems compares and contrasts the most promising multivariate schemes in digital signatures and public-key encryption as well as their security.

In this paper, we develop an energy stable fully discrete discontinuous Galekin (DG) finite element method for the thin film epitaxy problem. Based on the method of lines, we construct and prove the energy stability of the spatial semi-discrete DG scheme firstly. To avoid the strict time step restriction of the explicit time integration method, the first order convex splitting method is used to get an unconditionally stable fully discrete DG method, which is a linearly implicit scheme for this nonlinear problem. The energy stability of the fully discrete convex splitting DG scheme is also proved. To improve the temporal accuracy, spectral deferred correction (SDC) method is adapted to achieve the high order accuracy in both time and space. Combining with the convex splitting method, the SDC method can be linearly solvable, high order accurate and stable in our numerical tests. These advantages ensure that the resulting fully discrete DG scheme is efficient to perform the long time simulation of the thin film epitaxy model. Numerical experiments of the accuracy and long time simulation show the capability and efficiency of the method.

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The total variation (TV) model is attractive in that it is able to preserve sharp attributes in images. However, the restored images from TV-based methods do not usually stay in a given dynamic range, and hence projection is required to bring them back into the dynamic range for visual presentation or for storage in digital media. This will affect the accuracy of the restoration as the projected image will no longer be the minimizer of the given TV model. In this paper, we show that one can get much more accurate solutions by imposing box constraints on the TV models and solving the resulting constrained models. Our numerical results show that for some images where there are many pixels with values lying on the boundary of the dynamic range, the gain can be as great as 10.28 decibel in the peak signal-to-noise ratio. One traditional hindrance using the constrained model is that it is difficult to solve. However, in this paper, we propose using the alternating direction method of multipliers (ADMM) to solve the constrained models. This leads to a fast and convergent algorithm that is applicable for both Gaussian and impulse noise. Numerical results show that our ADMM algorithm is better than some state-of-the-art algorithms for unconstrained models in terms of both accuracy and robustness with respect to the regularization parameter.