br Materials and methods br Results and discussion br

Materials and methods
Results and discussion
Conclusions Given that the activity of anti-plant virus compounds is related to host tobacco, virus, and compounds, we determined the effect of anti-CMV compound NNM on defense enzyme activity and chlorophyll index in tobacco, and we used RIB as the control agent. The results showed that the content of PAL, POD, SOD, and chlorophyll in tobacco leaves increased significantly after being sprayed with NNM for days 1–7. The best activity was achieved on day 5. Then, we selected the transcriptome sequencing study on the tobacco leaves of the NNM treatment group and the CK treatment group on day 5. GO enrichment results showed that the NNM treatment group can enhance the tobacco oxidation reduction process, metabolic process, and single-organism metabolic process. Differential gene KEGG enrichment results showed that the NNM treatment group can accelerate tobacco alpha-linolenic Hesperadin metabolism; alanine, aspartate, glutamate, and linoleic acid metabolism; and valine, leucine, and isoleucine degradation, and arginine, proline metabolism. Furthermore, q-PCR results showed that the defense genes PAL, POD, SOD, and PR-1a were up-regulated by NNM on day 5 in CMV-inoculated leaves. The results corresponded to recent studies that implied that NNM strengthens the defensive enzyme activities and promotes the systemic accumulation of pathogenesis-related proteins in TMV- and CMV-inoculated tobacco thereby producing antiviral activity (Han et al., 2014). The following are the supplementary data related to this article.
Introduction In recent years, orthogonal polynomials on the unit circle (OPUC) have been extensively studied, see [13] for a expository note and books [14], [15] for details. In the study of orthogonal polynomials on the real line (OPRL), Jacobi matrix representation is one of the key tools, hence people want to get the matrix realization of OPUC. While for OPUC case, different orthonormal bases corresponds to different matrix representation. In 2003, M. Cantero, L. Moral, L. Velázquez [6] gave the “right” basis and the corresponding matrix representation which is named after them, i.e. CMV matrix. Naturally, it is viewed as the unitary analog of Jacobi matrices. Since Jacobi matrices have been studied for more than one hundred years, there are fruitful results in this area and their relation with OPRL is clear. As CMV matrices are the unitary analog of Jacobi matrices, people expect the results for Jacobi matrices also hold for CMV matrices. Indeed, many of them have been carried out in Barry Simon\'s monographs [14], [15], while the formula in our paper is one of the exceptions. The importance of Anderson localization is due to the seminal work by the physicist P.W. Anderson [2], which is named after him and helped him get the 1977 physics Nobel prize. In physics, Anderson localization refers to the phenomena that disorder in the media will cause suppression of electron transport; while in mathematics, it means the corresponding operator has only pure point spectrum with exponentially decaying eigenfunctions. Mathematically rigorous studies of the Anderson Model and other models started in the 1970s and several powerful methods have been found to prove Anderson localization, such as multiscale analysis (MSA) introduced by J. Fröhlich and T. Spencer [8], fractional moments method (FMM) developed by M. Aizenman and S. Molchanov [1], etc. Recently, the method developed by J. Bourgain, M. Goldstein and W. Schlag [3], [4] for one-dimensional Schrödinger operators has been applied widely to other one-dimensional models [5], [17], [9], [10], [11], which motivated us to apply this method to CMV matrices.
Preliminaries
Proof of Theorems Actually, Theorem 1 and Theorem 2 are equivalent. In this section, we first prove the equivalency of them, and then give the proof of Theorem 1, since it needs some additional preparations.