Twisted Courant algebroids and coisotropic Cartan geometries

In this paper, we show that associated to any coisotropic Cartan geometry there is a twisted Courant algebroid. This includes in particular parabolic geometries. Using this twisted Courant structure, we give some new results about the Cartan curvature and the Weyl structure of a parabolic geometry. As more direct applications, we have Lie 2-algebra and 3D AKSZ sigma model with background associated to any coisotropic Cartan geometry

Genome-wide comparison of microRNAs and their targeted transcripts among leaf, flower and fruit of sweet orange

BACKGROUND: In plants, microRNAs (miRNAs) regulate gene expression mainly at the post-transcriptional level. Previous studies have demonstrated that miRNA-mediated gene silencing pathways play vital roles in plant development. Here, we used a high-throughput sequencing approach to characterize the miRNAs and their targeted transcripts in the leaf, flower and fruit of sweet orange. RESULTS: A total of 183 known miRNAs and 38 novel miRNAs were identified. An in-house script was used to identify all potential secondary siRNAs derived from miRNA-targeted transcripts using sRNA and degradome sequencing data. Genome mapping revealed that these miRNAs were evenly distributed across the genome with several small clusters, and 69 pre-miRNAs were co-localized with simple sequence repeats (SSRs). Noticeably, the loop size of pre-miR396c was influenced by the repeat number of CUU unit. The expression pattern of miRNAs among different tissues and developmental stages were further investigated by both qRT-PCR and RNA gel blotting. Interestingly, Csi-miR164 was highly expressed in fruit ripening stage, and was validated to target a NAC transcription factor. This study depicts a global picture of miRNAs and their target genes in the genome of sweet orange, and focused on the comparison among leaf, flower and fruit tissues. CONCLUSIONS: This study provides a global view of miRNAs and their target genes in different tissue of sweet orange, and focused on the identification of miRNA involved in the regulation of fruit ripening. The results of this study lay a foundation for unraveling key regulators of orange fruit development and ripening on post-transcriptional level. ELECTRONIC SUPPLEMENTARY MATERIAL: The online version of this article (doi:10.1186/1471-2164-15-695) contains supplementary material, which is available to authorized users

The Pontryagin Class for Pre-Courant Algebroids

In this paper, we show that the Jacobiator $J$ of a pre-Courant algebroid is closed naturally. The corresponding equivalence class $[J^\flat]$ is defined as the Pontryagin class, which is the obstruction of a pre-Courant algebroid to be deformed into a Courant algebroid. We construct a Leibniz 2-algebra and a Lie 2-algebra associated to a pre-Courant algebroid and prove that these algebraic structures are isomorphic under deformations. Finally, we introduce the twisted action of a Lie algebra on a manifold to give more examples of pre-Courant algebroids, which include the Cartan geometry.Comment: 26 page

New type of solutions for the critical Lane-Emden system

In this paper, we consider the critical Lane-Emden system \begin{align*} \begin{cases} -\Delta u=K_1(y)v^p,\quad y\in \mathbb{R}^N,&\\ -\Delta v=K_2(y)u^q,\quad y\in \mathbb{R}^N,&\\ u,v>0, \end{cases} \end{align*} where $N\geq 5$, $p,q\in (1,\infty)$ with $\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}$, $K_1(y)$ and $K_2(y)$ are positive radial potentials. Under suitable conditions on $K_1(y)$ and $K_2(y)$, we construct a new family of solutions to this system, which are centred at points lying on the top and the bottom circles of a cylinder

Partial regularity of solutions to the 3D chemotaxis-Navier-Stokes equations at the first blow-up time

In this note, we investigate partial regularity of weak solutions of the three dimensional chemotaxis-Navier-Stokes equations, and obtain the $\frac53$-dimensional Hausdorff measure of the possible singular set is vanishing at the first blow-up time. The new ingredients are to establish certain type of local energy inequality and deal with the non-scaling invariant quantity of $n\ln n$, which seems to be the first description for the singular set of weak solutions of the chemotaxis-fluid model, which is motivated by Caffarelli-Kohn-Nirenberg's partial regularity theory \cite{CKN}

Capacity sharing, product differentiation and welfare

This article constructs a duopoly market with product differentiation and analyses profits, consumer surplus and social welfare under three conditions: (a) two enterprises have sufficient capacity; (b) one enterprise has insufficient capacity, and another enterprise has excess capacity that is not shared; and (c) one enterprise has insufficient capacity, and another enterprise has excess capacity and engages in capacity sharing. Through comparison, the implementation conditions for and effects of capacity sharing and the role of product differentiation are revealed. The results show that capacity sharing helps increase producer surplus and social welfare. Capacity constraints reduce social welfare but can be solved by capacity sharing. Capacity sharing can only be realised when both enterprises are profitable, and the charge for capacity sharing should not be too high or too low. Product differentiation has impacts on output, profit, consumer surplus and social welfare, and these impacts are restricted by the existence of capacity constraints and capacity sharing