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The sparsity-based approaches have demonstrated promising performance in image processing. In this paper, for better preservation of the salient edge structures of images, we propose an ℓ 0 + ℓ 2-norm based analysis model， which requires solving a challenging non-separable ℓ 0-norm related minimization problem, and we also propose an inexact augmented Lagrangian method with proven convergence to a local minimum. Extensive experiments in image smoothing, including texture removal and context smoothing, show that our method achieves better visual results over various sparsity-based models and the CNN method. Also, experiments on sparse view CT reconstruction further validate the advantage of the proposed method.

We study scattering of higher spin closed string states at arbitrary mass levels from D-particle in the Regge regime. We extract the infinite ratios among high-energy amplitudes of different string states in the fixed angle regime from these Regge string scattering amplitudes. In this calculation, we have used an identity proved recently based on a signless Stirling number identity in combinatorial theory. The complete ratios calculated by this indirect method include a subset of ratios calculated previously by direct fixed angle calculation. Moreover, we discover that in spite of the non-factorizability of the closed string D-particle scattering amplitudes, the complete ratios derived for the fixed angle regime are found to be factorized. These ratios are consistent with the decoupling of high-energy zero norm states calculated previously., 19 pages v2. Eqs.(3.19) to (3.21) added. v3. 27 pages. example added in sec.III. final in Prog. Theo. Phys

We give an explicit example of an exactly solvable -symmetric Hamiltonian with unbroken symmetry which has one eigenfunction with zero norm. The set of its eigenfunctions is not complete in the corresponding Hilbert space and it is non-diagonalizable. In the case of a regular Sturm–Liouville problem any diagonalizable -symmetric Hamiltonian with unbroken symmetry has a complete set of positive -normalizable eigenfunctions. For non-diagonalizable Hamiltonians, a complete set of -normalizable functions is possible but the functions belonging to the root subspace corresponding to multiple zeros of the characteristic determinant are no longer eigenfunctions of the Hamiltonian.

We calculate the NS–R fermionic zero-norm states of the type II string spectrum. The massless and some possible massive zero-norm states are seen to be responsible for the space-time supersymmetry. The existence of other fermionic massive zero-norm states with higher spinor–tensor indices correspond to new enlarged boson–fermion symmetries of the theory at high energy. We also discuss the R–R charges and R–R zero-norm states and justify the idea that the perturbative string does not carry the massless R–R charges.

The structure of a relativistic field which contains "ghost" (zero norm) states is investigated. Restrictions are made for the case of free fields, which can be thought of as asymptotic parts of a more complicated theory. According to the interpretation the in- or out-going fields may be eliminated from the physical results. If explicit calculations are to be made in any more concrete theory, it is easier to handle dipole ghosts than exponential ghosts. (B.O.G.)