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Alternative splicing (AS) significantly enriches the diversity of transcriptomes and proteomes, playing a pivotal role in the physiology and development of eukaryotic organisms. With the continuous advancement of high-throughput sequencing technologies, an increasing number of novel transcript isoforms, along with factors related to splicing and their associated functions, are being unveiled. In this review, we succinctly summarize and compare the different splicing mechanisms across prokaryotes and eukaryotes. Furthermore, we provide an extensive overview of the recent progress in various studies on AS covering different developmental stages in diverse plant species and in response to various abiotic stresses. Additionally, we discuss modern techniques for studying the functions and quantification of AS transcripts, as well as their protein products. By integrating genetic studies, quantitative methods, and high-throughput omics techniques, we can discover novel transcript isoforms and functional splicing factors, thereby enhancing our understanding of the roles of various splicing modes in different plant species.

 

The Degasperis-Procesi (DP) equation is an integrable Camassa-Holm-type model which is an asymptotic approximation for the unidirectional propagation of shallow water waves. This work establishes the orbital stability of localized smooth solitary waves to the DP equation on the real line, extending our previous work on their spectral stability [J. Math. Pures Appl. (9) 142 (2020), pp. 298–314]. The main difficulty stems from the fact that the natural energy space is a subspace of L 3 L^3 , but the translation symmetry for the DP equation gives rise to a conserved quantity equivalent to the L 2 L^2 -norm, resulting in L 3 L^3 higher-order nonlinear terms in the augmented Hamiltonian. But the usual coercivity estimate is in terms of L 2 L^2 norm for DP equation, which cannot be used to control the L 3 L^3 higher order term directly. The remedy is to observe that, given a sufficiently smooth initial condition satisfying some mild constraint, the L ∞ L^\infty orbital norm of the perturbation is bounded above by a function of its L 2 L^2 orbital norm, yielding the higher order control and the orbital stability in the L 2 ∩ L ∞ L^2\cap L^\infty space. 

We propose a novel combinatorial inference framework to conduct general uncertainty quantification in ranking problems. We consider the widely adopted Bradley-Terry-Luce (BTL) model, where each item is assigned a positive preference score that determines the Bernoulli distributions of pairwise comparisons’ outcomes. Our proposed method aims to infer general ranking properties of the BTL model. The general ranking properties include the “local” properties such as if an item is preferred over another and the “global” properties such as if an item is among the top K-ranked items. We further generalize our inferential framework to multiple testing problems where we control the false discovery rate (FDR) and apply the method to infer the top-K ranked items. We also derive the information-theoretic lower bound to justify the minimax optimality of the proposed method. We conduct extensive numerical studies using both synthetic and real data sets to back up our theory.