As a special shift-invariant spaces, spline subspaces yield many advantages so that there are many practical applications for signal or image processing. In this paper, we pay attention to the sampling and reconstruction problem in spline subspaces. We improve lower bound of sampling set conditions in spline subspaces. Based on the improved explicit lower bound, a improved explicit convergence ratio of reconstruction algorithm is obtained. The improved convergence ratio occupies faster convergence rate than old one. At the end, some numerical examples are shown to validate our results.
This paper discusses conditions under which the solution of linear system with minimal Schatten-p norm, 0 〈 p ≤ 1, is also the lowest-rank solution of this linear system. To study this problem, an important tool is the restricted isometry constant （RIC）. Some papers provided the upper bounds of RIC to guarantee that the nuclear-norm minimization stably recovers a low-rank matrix. For example, Fazel improved the upper bounds to δ4Ar 〈 0.558 and δ3rA 〈 0.4721, respectively. Recently, the upper bounds of RIC can be improved to δ2rA 〈 0.307. In fact, by using some methods, the upper bounds of RIC can be improved to δ2tA 〈 0.4931 and δrA 〈 0.309. In this paper, we focus on the lower bounds of RIC, we show that there exists linear maps A with δ2rA 〉1√2 or δrA 〉 1/3 for which nuclear norm recovery fail on some matrix with rank at most r. These results indicate that there is only a little limited room for improving the upper bounds for δ2rA and δrA.Furthermore, we also discuss the upper bound of restricted isometry constant associated with linear maps A for Schatten p （0 〈 p 〈 1） quasi norm minimization problem.
We consider efficient methods for the recovery of block sparse signals from underdetermined system of linear equations. We show that if the measurement matrix satisfies the block RIP with δ2s 〈 0.4931, then every block s-sparse signal can be recovered through the proposed mixed l2/ll-minimization approach in the noiseless case and is stably recovered in the presence of noise and mismodeling error. This improves the result of Eldar and Mishali （in IEEE Trans. Inform. Theory 55： 5302-5316, 2009）. We also give another sufficient condition on block RIP for such recovery method： 58 〈 0.307.
Biofuels are indeed clean, green, renewable and sustainable, which have a great effect on alleviating social energy crisis as they can be extracted from various kinds of biomass. According to the crucial enzymes involved in the synthesis of sucrose, starch and lipid, various combinations of crucial enzymes were used to retrieval in NCBI （the National Center for Biotechnology Information） database to screen out species with high similarities. Moreover, comprehensive evaluation formulas were designed to excavate potential biofuel plants, which considered several parameters including the homology index, availability rate, distribution area in China, price index and function index. Finally, tens of superior plant species, for example, Panicum virgatum, Zeal luxuriant and Brassica napus, are suggested to be the potentials in future biofuels productions. These plants have the characteristics of wide distribution, high yield, and low cost in production, and they are useful for environmental protection as well as sources of non-crop biomass fuels in China. The results provide a theoretical basis for further study and exploitation of biofuels.
Shubo ZhangJing LiuXiaobai JinDaniel Wai Tin ChanMiao He
