On a nonlinear multigrid algorithm with primal relaxation for the image total variation minimisation

Digital image restoration has drawn much attention in the recent years and a lot of research has been done on effective variational partial differential equation models and their theoretical studies. However there remains an urgent need to develop fast and robust iterative solvers, as the underlying problem sizes are large. This paper proposes a fast multigrid method using primal relaxations. The basic primal relaxation is known to get stuck at a ‘local’ non-stationary minimum of the solution, which is usually believed to be ‘non-smooth’. Our idea is to utilize coarse level corrections, overcoming the deadlock of a basic primal relaxation scheme. A further refinement is to allow non-regular coarse levels to correct the solution, which helps to improve the multilevel method. Numerical experiments on both 1D and 2D images are presented.     

Multipartite Entanglement Preparation and Quantum Communication with Atomic Ensembles

We propose an experimentally feasible scheme to generate Greenberger-Horne-Zeilinger type of maximal entanglement among three atomic ensembles, and show one of the applications, controlled secure direct communication. The scheme involves laser manipulation of atomic ensembles, quarter- and half-wave plates, beam splitters, polarizing beam splitters and single-photon detectors, which are within the reach of current experimental technology.     

Solutions in Sobolev spaces of vector refinement equations with a general dilation matrix

In this paper, we present a necessary and sufficient condition for the existence of solutions in a Sobolev space W_p^k(ℝ^s) (1≤p≤∞) to a vector refinement equation with a general dilation matrix. The criterion is constructive and can be implemented. Rate of convergence of vector cascade algorithms in a Sobolev space W_p^k(ℝ^s) will be investigated. When the dilation matrix is isotropic, a characterization will be given for the L_p (1≤p≤∞) critical smoothness exponent of a refinable function vector without the assumption of stability on the refinable function vector. As a consequence, we show that if a compactly supported function vector φ∈L_p(ℝ^s) (φ∈C(ℝ^s) when p=∞) satisfies a refinement equation with a finitely supported matrix mask, then all the components of φ must belong to a Lipschitz space Lip(ν,L_p(ℝ^s)) for some ν>0. This paper generalizes the results in R.Q. Jia, K.S. Lau and D.X. Zhou (J. Fourier Anal. Appl. 7 (2001) 143–167) in the univariate setting to the multivariate setting. Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 42C20, 41A25, 39B12. Research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC Canada) under Grant G121210654.     

用13CNMR研究聚甲基丙烯酸锡酯的立构序列分布

本文测定了自由基型溶液聚合的聚甲基丙烯酸三甲基锡酯(PTMTM),聚甲基丙烯酸三乙基锡酯(PTETM)和聚甲基丙烯酸三丁基锡酯(PTBTM)的¹³C-NMR谱,对其C-2谱扩展分峰,计算立构序列,从定量数据表明,聚合物样品的空间立构富于或稍富于间规立构,随取代基的体积增大而趋于无规分布,作者认为受空间位阻效应和分子内或分子间形成配位键的影响。     

Elevation of the glass transition temperature in flexible‐chain semicrystalline polymers

In the idealized two‐phase model of a semicrystalline polymer, the amorphous intercrystalline layers are considered to have the same properties as the fully‐amorphous polymer. In reality, these thin intercrystalline layers can be substantially influenced by the presence of the crystals, as individual polymer molecules traverse both crystalline and amorphous phases. In polymers with rigid backbone units, such as poly(etheretherketone), PEEK, previous work has shown this coupling to be particularly severe; the glass transition temperature (T_g) can be elevated by tens of degrees celsius, with the magnitude of the elevation correlating directly with the thinness of the amorphous layer. However, this connection has not been explored for flexible‐chain polymers, such as those formed from vinyl‐type monomers. Here, we examine T_g in both isotactic polystyrene (iPS) and syndiotactic polystyrene (sPS), crystallized under conditions that produce a range of amorphous layer thicknesses. T_g is indeed shown to be elevated relative to fully‐amorphous iPS and sPS, by an amount that correlates with the thinness of the amorphous layer; the magnitude of the effect is severalfold less than that in PEEK, consistent with the minimum lengths of polymer chain required to make a fold in the different cases. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 1198–1204, 2007     

Super-Wavelets and Decomposable Wavelet Frames

A wavelet frame is called decomposable whenever it is equivalent to a superwavelet frame of length greater than one. Decomposable wavelet frames are closely related to some problems on super-wavelets. In this article we first obtain some necessary or sufficient conditions for decomposable Parseval wavelet frames. As an application of these conditions, we prove that for each n > 1 there exists a Parseval wavelet frame which is m-decomposable for any 1 < m ≤ n, but not k-decomposable for any k > n. Moreover, there exists a super-wavelet whose components are non-decomposable. Similarly we also prove that for each n > 1, there exists a Parseval wavelet frame that can be extended to a super-wavelet of length m for any 1 < m ≤ n, but can not be extended to any super-wavelet of length k with k > n. The connection between decomposable Parseval wavelet frames and super-wavelets is investigated, and some necessary or sufficient conditions for extendable Parseval wavelet frames are given.     

Polynomial-time approximation schemes for packing and piercing fat objects

We consider two problems: given a collection of n fat objects in a fixed dimension, (1) ( packing) find the maximum subcollection of pairwise disjoint objects, and (2) ( piercing) find the minimum point set that intersects every object. Recently, Erlebach, Jansen, and Seidel gave a polynomial-time approximation scheme (PTAS) for the packing problem, based on a shifted hierarchical subdivision method. Using shifted quadtrees, we describe a similar algorithm for packing but with a smaller time bound. Erlebach et al.'s algorithm requires polynomial space. We describe a different algorithm, based on geometric separators, that requires only linear space. This algorithm can also be applied to piercing, yielding the first PTAS for that problem.     

Numerical simulation on an interaction of a vortex street with an elliptical leading edge using an unstructured grid

An algorithm for a time accurate incompressible Navier–Stokes solver on an unstructured grid is presented. The algorithm uses a second order, three‐point, backward difference formula for the physical time marching. For each time step, a divergence free flow field is obtained based on an artificial compressibility method. An implicit method with a local time step is used to accelerate the convergence for the pseudotime iteration. To validate the code, an unsteady laminar flow over a circular cylinder at a Reynolds number of 200 is calculated. The results are compared with available experimental and numerical data and good agreements are achieved. Using the developed unsteady code, an interaction of a Karman vortex street with an elliptical leading edge is simulated. The incident Karman vortex street is generated by a circular cylinder located upstream. A clustering to the path of the vortices is achieved easily due to flexibility of an unstructured grid. Details of the interaction mechanism are analysed by investigating evolutions of vortices. Characteristics of the interactions are compared for large‐ and small‐scale vortex streets. Different patterns of the interaction are observed for those two vortex streets and the observation is in agreement with experiment. Copyright © 2004 John Wiley & Sons, Ltd.     

Time resolved spectroscopic NMR imaging using hyperpolarized Xe

We have visualized the melting and dissolution processes of xenon (Xe) ice into different solvents using the methods of nuclear magnetic resonance (NMR) spectroscopy, imaging, and time resolved spectroscopic imaging by means of hyperpolarized ¹²⁹Xe. Starting from the initial condition of a hyperpolarized solid Xe layer frozen on top of an ethanol (ethanol/water) ice block we measured the Xe phase transitions as a function of time and temperature. In the pure ethanol sample, pieces of Xe ice first fall through the viscous ethanol to the bottom of the sample tube and then form a thin layer of liquid Xe/ethanol. The xenon atoms are trapped in this liquid layer up to room temperature and keep their magnetization over a time period of 11 min. In the ethanol/water mixture (80 vol%/20%), most of the polarized Xe liquid first stays on top of the ethanol/water ice block and then starts to penetrate into the pores and cracks of the ethanol/water ice block. In the final stage, nearly all the Xe polarization is in the gas phase above the liquid and trapped inside the pores. NMR spectra of homogeneous samples of pure ethanol containing thermally polarized Xe and the spectroscopic images of the melting process show that very high concentrations of hyperpolarized Xe (about half of the density of liquid Xe) can be stored or delivered in pure ethanol.