We address an interesting question raised by Dos Santos Ferreira, Kenig and Salo [11] about regions Rg ⊂ C for which there can be uniform L 2n n+2 → L 2n n−2 resolvent estimates for ∆g + ζ, ζ ∈ Rg, where ∆g is the Laplace-Beltrami operator with metric g on a given compact boundaryless Riemannian manifold of dimension n ≥ 3. This is related to earlier work of Kenig, Ruiz and the third author [19] for the Euclidean Laplacian, in which case the region is the entire complex plane minus any disc centered at the origin. Presently, we show that for the round metric on the sphere, S n , the resolvent estimates in [11], involving a much smaller region, are essentially optimal. We do this by establishing sharp bounds based on the distance from ζ to the spectrum of ∆ S n . In the other direction, we also show that the bounds in [11] can be sharpened logarithmically for manifolds with nonpositive curvature, and by powers in the case of the torus, T n = R n /Z n , with the flat metric. The latter improves earlier bounds of Shen [22]. The work of [11] and [22] was based on Hadamard parametrices for (∆g + ζ) −1 . Ours is based on the related Hadamard parametrices for cos t −∆g, and it follows ideas in [26] of proving L p -multiplier estimates using small-time wave equation parametrices and the spectral projection estimates from [25]. This approach allows us to adapt arguments in Bérard [3] and Hlawka [15] to obtain the aforementioned improvements over [11] and [22]. Further improvements for the torus are obtained using recent techniques of the first author [5] and his work with Guth [7] based on the multilinear estimates of Bennett, Carbery and Tao [2]. Our approach also allows us to give a natural necessary condition for favorable resolvent estimates that is based on a measurement of the density of the spectrum of −∆g, and, moreover, a necessary and sufficient condition based on natural improved spectral projection estimates for shrinking intervals, as opposed to those in [25] for unit-length intervals. We show that the resolvent estimates are sensitive to clustering within the spectrum, which is not surprising given Sommerfeld's original conjecture [32] about these operators.

Vegetation restoration has been widely used in karst rocky desertification (KRD) areas of southwestern China, but the response of microbial community to revegetation has not been well characterized. We investigated the diversity, structure, and co-occurrence patterns of bacterial communities in soils of five vegetation types (grassland, shrubbery, secondary forest, pure plantation and mixed plantation) in KRD area using high-throughput sequencing of the 16S rRNA gene. Bray-Curtis dissimilarity analysis revealed that 15 bacterial community samples were clustered into five groups that corresponded very well to the five vegetation types. Shannon diversity was positively correlated with pH and Ca2+ content but negatively correlated with organic carbon, total nitrogen, and soil moisture. Redundancy analysis indicated that soil pH, Ca2+ content, organic carbon, total nitrogen, and soil moisture jointly influenced bacterial community structure. Co-occurrence network analysis revealed non-random assembly patterns of bacterial composition in the soils. Bryobacter, GR-WP33-30, and Rhizomicrobium were identified as keystone genera in co-occurrence network. These results indicate that diverse soil physicochemical properties and potential interactions among taxa during vegetation restoration may jointly affect the bacterial community structure in KRD regions.

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Camellia oleifera is a major tree species for producing edible oil. Its seed oil is well known for the high level of oleic acids; however, little is known regarding the molecular mechanism of lipid biosynthesis in C. oleifera. Here, we measured the oil contents and fatty acid (FA) compositions at four developmental stages and investigated the global gene expression profiles through transcriptomics sequencing. We identified differentially-expressed genes (DEGs) among the developmental stages and found that the distribution of numbers of DEGs was associated with the accumulation pattern of seed oil. Gene Ontology (GO) enrichment analysis revealed some critical biological processes related to oil accumulation, including lipid metabolism and phosphatidylcholine metabolism. Furthermore, we investigated the expression patterns of lipid biosynthesis genes. We showed that most of the genes were identified with single or multiple copies, and some had correlated profiles along oil accumulation. We proposed that the higher levels of stearoyl-ACP desaturases (SADs) coupled with lower activities of fatty acid desaturase 2 (FAD2) might be responsive to the boost of oleic acid at the late stage of C. oleifera seeds’ development. This work presents a comprehensive transcriptomics study of C. oleifera seeds and uncovers valuable DEGs that are associated with the seed oil accumulation.

Pecan (Carya illinoinensis) is an important tree nut throughout the world. The high concentration of flavonoid in its kernels makes it an excellent food with health benefits. However, the molecular basis of flavonoid biosynthesis in pecan remains unclear, which hinders quality breeding in this plant. Therefore, in order to find the crucial genes involved in flavonoid biosynthesis, the changes in flavonoid profiles and the transcriptomes of pecan kernels at four developmental stages (late water, gel, dough, and mature stages) were analyzed. As a result, the highest levels of total phenolic, condensed tannin, and flavan-3-ols were observed at the "late water stage". Catechin was the most abundant flavan-3-ol at different development stages. In total, 64 773 unigenes were obtained, and 46 924 (72.44%) unigenes were annotated. After differentially expressed gene (DEG) analysis, 12 750 unique DEGs were identified. Flavonoid-related DEGs of 36 structural genes and eight MYBs were obtained. The structural gene set contained three PALs, three CHSs, two CHIs, one F3H, two F3′Hs, two F3′5′Hs, one DFR, one ANS, two LARs, and two ANRs. The expression patterns of most of the structural genes were consistent with the changes in flavonoid profiles during kernel development. We believe that this RNA-Seq data set will provide valuable resources for unraveling the molecular mechanism of flavonoid metabolism in pecan and will significantly promote genetic studies and quality breeding in this plant.

Abstract. We study time decay estimates of the fourth-order Schrödinger operatorWe analyze the low energy and high energy behaviour of resolvent R(H; z), and then derive the Jensen-Kato dispersion decay estimate and local decay estimate for e −itH P ac under suitable spectrum assumptions of H. Based on Jensen-Kato type decay estimate and local decay estimate, we obtain the L 1 → L ∞ estimate of e −itH P ac in 3-dimension by Ginibre argument, and also establish the endpoint global Strichartz estimates of e −itH P ac for d ≥ 5. Furthermore, using the local decay estimate and the Georgescu-Larenas-Soffer conjugate operator method, we prove the Jensen-Kato type decay estimates for some functions of H.

Background As a perennial crop, oil-Camellia possesses a long domestication history and produces high-quality seed oil that is beneficial to human health. Camellia oleifera Abel. is a sister species to the tea plant, which is extensively cultivated for edible oil production. However, the molecular mechanism of the domestication of oil-Camellia is still limited due to the lack of sufficient genomic information. Results To elucidate the genetic and genomic basis of evolution and domestication, here we report a chromosome-scale reference genome of wild oil-Camellia (2.95 Gb), together with transcriptome sequencing data of 221 cultivars. The oil-Camellia genome, assembled by an integrative approach of multiple sequencing technologies, consists of a large proportion of repetitive elements (76.1%) and high heterozygosity (2.52%). We construct a genetic map of high-density corrected markers by sequencing the controlled-pollination hybrids. Genome-wide association studies reveal a subset of artificially selected genes that are involved in the oil biosynthesis and phytohormone pathways. Particularly, we identify the elite alleles of genes encoding sugar-dependent triacylglycerol lipase 1, β-ketoacyl-acyl carrier protein synthase III, and stearoyl-acyl carrier protein desaturases; these alleles play important roles in enhancing the yield and quality of seed oil during oil-Camellia domestication. Conclusions We generate a chromosome-scale reference genome for oil-Camellia plants and demonstrate that the artificial selection of elite alleles of genes involved in oil biosynthesis contributes to oil-Camellia domestication.

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