We present a generalized learning with errors (LWE) problem, which is essentially a simple and direct extension of the original LWE problem to the case of matrices. Then we use this new version of LWE problem, which we call matrix LWE problem to build new cryptographic schemes, which include a new scalable key distribution scheme, a new key exchanges scheme and a new simple identity-based encryption scheme.

The construction of the Q-operator for twisted affine superalgebra Uq(C(2)(2)) is given. It is shown that the corresponding prefundamental representations give rise to evaluation modules some of which do not have a classical limit, which nevertheless appear to be a necessary part of fusion relations.

Nous savons que la méthode dite du balayage permet de démontrer le principe de$Dirichlet$dans le cas général.

In this paper, we develop an adaptive Generalized Multiscale Discontinuous Galerkin Method (GMs-DGM) for a class of high-contrast flow problems, and derive a-priori and a-posteriori error estimates for the method. Based on the a-posteriori error estimator, we develop an adaptive enrichment algorithm for our GMsDGM and prove its convergence. The adaptive enrichment algorithm gives an automatic way to enrich the approximation space in regions where the solution requires more basis functions, which are shown to perform well compared with a uniform enrichment. We also discuss an approach that adaptively selects multiscale basis functions by correlating the residual to multiscale basis functions (cf.[4]). The proposed error indicators are L2-based and can be inexpensively computed which makes our approach efficient. Numerical results are presented that demonstrate the robustness of the proposed error indicators.

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