A Practical E-Payment Protocol Based on Quantum Multi-Proxy Blind Signature

A practical E-payment protocol is presented in this paper. It is based on quantum multi-proxy blind signature. Adopting the techniques of quantum key distribution, one-time pad and quantum multi-proxy blind signature, our E-payment system could protect user's anonymity as the traditional E-payment systems do, and also have unconditional security, which the classical E-payment systems can not provide. Furthermore, compared with the existing quantum E-payment systems, this practical system could not only support mobile E-payment transactions but also inter-bank transactions.     

Burgers方程的一类交替分组方法

对于Burgers方程给出了一组新的Saul'yev型非对称差分格式，并用这些差分格式构造了求解非线性Burgers方程的交替分组四点方法．该算法把剖分节点分成若干组，在每组上构造能够独立求解的差分方程．因此算法具有并行本性，能直接在并行计算机上使用．章还证明了所给算法线性绝对稳定．数值试验表明，该方法使用简便，稳定性好，有很好的精度。     

A generalized Fourier inversion Theorem

In this work we define operator-valued Fourier transforms for suitable integrable elements with respect to the Plancherel weight of a (not necessarily Abelian) locally compact group. Our main result is a generalized version of the Fourier inversion Theorem for strictly-unconditionally integrable Fourier transforms. Our results generalize and improve those previously obtained by Ruy Exel in the case of Abelian groups. Supported by CAPES, Brazil.     

Unconditional basis recursively generated in L p on compact sets

We construct an unconditional basis in the Banach space L ^p(Ω) for p 1 by using the refinement equation and the basic operation of translation and scale, where Ω is a compact subset in ℝⁿ. We also give an algorithm of how to construct an unconditional basis in L ^p(Ω p). At the end of this paper, we give the characterization of the functions in L ^p(Ω p) by using the wavelet coefficients.     

多项时间分数阶扩散方程各向异性线性三角元的高精度分析

在各向异性网格下,针对具有Caputo导数的二维多项时间分数阶扩散方程,给出了线性三角形元的高精度分析.首先,基于线性三角形元和改进的L1格式,建立了一个全离散逼近格式,并证明了其无条件稳定性;其次,利用有限元插值算子与Riesz投影算子之间的关系及相关的高精度结果,导出了超逼近性质.进而,借助于插值后处理技术得到了超收敛估计.值得指出的是,单独利用插值算子或Riesz投影都无法得到上述超逼近和超收敛结果.最后,利用数值算例验证了理论分析的正确性.此外,对一些常见的有限单元在该方程的数值逼近方面,作了进一步探讨.     

Superconvergence analysis of an energy stable scheme with three step backward differential formula-finite element method for nonlinear reaction–diffusion equation

A three step backward differential formula scheme is proposed for nonlinear reaction–diffusion equation and superconvergence results are studied with Galerkin finite element method unconditionally. Energy stability is testified for the constructed scheme with an artificial term. Splitting technique is utilized to get rid of the ratio between the time step size and the subdivision parameter . Temporal error estimate in H²-norm is derived, which leads to the boundedness of the solutions of the time-discrete equations. Unconditional spatial error estimate in L²-norm is deduced which help bound the numerical solutions in L^∞-norm. Superconvergent property of in H¹-norm with order is obtained by taking difference between two time levels of the error equations unconditionally. The global superconvergent property is deduced through the above results. Two numerical examples show the validity of the theoretical analysis.     

Orthogonal systems of spline wavelets as unconditional bases in Sobolev spaces

We exhibit the necessary range for which functions in the Sobolev spaces $L_p^s$ can be represented as an unconditional sum of orthonormal spline wavelet systems, such as the Battle–Lemarié wavelets. We also consider the natural extensions to Triebel–Lizorkin spaces. This builds upon, and is a generalization of, previous work of Seeger and Ullrich, where analogous results were established for the Haar wavelet system.     

Block preconditioners for energy stable schemes of magnetohydrodynamics equations

In this article, we develop the first and second order unconditionally energy stable schemes for magnetohydrodynamics (MHD) equations and design robust preconditioners for these schemes. Inspired by operator preconditioning ideas, appropriate parameter-dependent norms on function spaces are subtly defined to uniformly bound the bilinear and trilinear terms, which implies the uniform well-posedness of the schemes under the newly defined norms. Then robust block preconditioners are constructed using the Riesz operators. We prove that, if time step size $k\le C$, the proposed preconditioners are uniformly robust with respect to physical parameters and discrete parameters. Various numerical experiments, including energy stability tests, Kelvin–Helmholtz instability and magnetic driven cavity physical benchmark problems, are presented to manifest unconditional energy stability of the schemes and robustness of our preconditioners.