Let G=(V,E) be a locally finite graph. Firstly, using calculus of variations, including a direct method of variation and the mountain-pass theory, we get sequences of solutions to several local equations on G (the Schrödinger equation, the mean field equation, and the Yamabe equation). Secondly, we derive uniform estimates for those local solution sequences. Finally, we obtain global solutions by extracting convergent sequence of solutions. Our method can be described as a variational method from local to global.

SMS4 is a 128-bit block cipher used in the WAPI standard for providing data confidentiality in wireless networks. In this paper we investigate and explain the origin of the S-Box employed by the cipher, show that an embedded cipher similar to BES can be obtained for SMS4 and demonstrate the fragility of the cipher design by giving variants that exhibit 2^{64} weak keys. We also show attacks on reduced round versions of the cipher. The best practical attack we found is an integral attack that works on 10 rounds out of 32 rounds with a complexity of 2^{18} operations; it can be extended to 13 rounds using round key guesses, resulting in a complexity of 2^{114} operations and a data complexity of 2^{16} chosen pairs.

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Let Ω⊂ℝ^{$n$}be an arbitrary open set. We characterize the space$W$^{1,1}_{loc}(Ω) using variants of the classical area and coarea formulas. We use these characterizations to obtain a norm approximation and trace theorems for functions in the space$W$^{1,1}(ℝ^{$n$}).

The aim of this paper is twofold. First we give an explicit construction of the in¯nitesimal deformations of the category Coh(X) of coherent sheaves on a smooth projective variety X. Secondly,we show that any Fourier-Mukai transform ©: Db(X) ! Db(Y ) extends to an equivalence between the derived categories of the deformed Abelian categories.