We survey finite element methods for approximating the time harmonic Maxwell equations. We concentrate on comparing error estimates for problems with spatially varying coefficients. For the conforming edge finite element methods, such estimates allow, at least, piecewise smooth coefficients. But for Discontinuous Galerkin (DG) methods, the state of the art of error analysis is less advanced (we consider three DG families of methods: Interior Penalty type, Hybridizable DG, and Trefftz type methods). Nevertheless, DG methods offer significant potential advantages compared to conforming methods.

HighlightCabZIP63, indirectly activated by CaWRKY40, positively modulates transcription of CabZIP63 and CaWRKY40, enhances the binding of CaWRKY40 to its target promoters, and, therefore, increases resistance to Ralstonia solanacearum and thermotolerance.

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The tripartite mitogen-activated protein kinase (MAPK) signaling cascades have been implicated in plant growth, development, and environment adaptation, but a comprehensive understanding of MAPK signaling at genome-wide level is limited in Capsicum annuum. Herein, genome-wide identification and transcriptional expression analysis of MAPK and MAPK kinase (MAPKK) were performed in pepper. A total of 19 pepper MAPK (CaMAPKs) genes and five MAPKK (CaMAPKKs) genes were identified. Phylogenetic analysis indicated that CaMAPKs and CaMAPKKs could be classified into four groups and each group contains similar exon-intron structures. However, significant divergences were also found. Notably, five members of the pepper MAPKK family were much less conserved than those found in Arabidopsis, and 9 Arabidopsis MAPKs did not have orthologs in pepper. Additionally, 7 MAPKs in Arabidopsis had either two or three orthologs in the pepper genome, and six pepper MAPKs and one MAPKK differing in sequence were found in three pepper varieties. Quantitative real-time RT-PCR analysis showed that the majority of MAPK and MAPKK genes were ubiquitously expressed and transcriptionally modified in pepper leaves after treatments with heat, salt, and Ralstonia solanacearum inoculation as well as exogenously applied salicylic acid, methyl jasmonate, ethephon, and abscisic acid. The MAPKK-MAPK interactome was tested by yeast two-hybrid assay, the results showed that one MAPKK might interact with multiple MAPKs, one MAPK might also interact with more than one MAPKKs, constituting MAPK signaling networks which may collaborate in transmitting upstream signals into appropriate downstream cellular responses and processes. These results will facilitate future functional characterization of MAPK cascades in pepper.

In the first part of this work, we analyzed an unconstrained Dirichlet boundary control problem for an elliptic convection diffusion PDE and proposed a new hybridizable discontinuous Galerkin (HDG) method to approximate the solution. For the case of a 2D convex polygonal domain, we also proved an optimal superlinear convergence rate for the control under certain assumptions on the domain and on the target state. In this work, we revisit the convergence analysis without these assumptions; in this case, the solution can have low regularity and we use a different analysis approach. We again prove an optimal convergence rate for the control, and present numerical results to illustrate the convergence theory.

We begin an investigation of hybridizable discontinuous Galerkin (HDG) methods for approximating the solution of Dirichlet boundary control problems governed by elliptic PDEs. These problems can involve atypical variational formulations, and often have solutions with low regularity on polyhedral domains. These issues can provide challenges for numerical methods and the associated numerical analysis. We propose an HDG method for a Dirichlet boundary control problem for the Poisson equation, and obtain optimal a priori error estimates for the control. Specifically, under certain assumptions, for a 2D convex polygonal domain we show the control converges at a superlinear rate. We present 2D and 3D numerical experiments to demonstrate our theoretical results.

CaWRKY40 is a positive regulator of pepper (Capsicum annum) response to Ralstonia solanacearum inoculation (RSI), but the underlying mechanism remains largely unknown. Here, we functionally characterize CaCDPK15 in the defense signaling mediated by CaWRKY40. Pathogen-responsive TGA, W, and ERE boxes were identified in the CaCDPK15 promoter (pCaCDPK15), and pCaCDPK15-driven GUS expression was significantly enhanced in response to RSI and exogenously applied salicylic acid, methyl jasmonate, abscisic acid, and ethephon. Virus-induced gene silencing (VIGS) of CaCDPK15 significantly increased the susceptibility of pepper to RSI and downregulated the immunity-associated markers CaNPR1, CaPR1, and CaDEF1. By contrast, transient CaCDPK15 overexpression significantly activated hypersensitive response associated cell death, upregulated the immunity-associated marker genes, upregulated CaWRKY40 expression, and enriched CaWRKY40 at the promoters of its targets genes. Although CaCDPK15 failed to interact with CaWRKY40, the direct binding of CaWRKY40 to pCaCDPK15 was detected by chromatin immunoprecipitation, which was significantly potentiated by RSI in pepper plants. These combined results suggest that RSI in pepper induces CaCDPK15 and indirectly activates downstream CaWRKY40, which in turn potentiates CaCDPK15 expression. This positive-feedback loop would amplify defense signaling against RSI and efficiently activate strong plant immunity.

We consider an unconstrained tangential Dirichlet boundary control problem for the Stokes equations with an L 2 penalty on the boundary control. The contribution of this paper is twofold. First, we obtain well-posedness and regularity results for the tangential Dirichlet control problem on a convex polygonal domain. The analysis contains new features not found in similar Dirichlet control problems for the Poisson equation; an interesting result is that the optimal control has higher local regularity on the individual edges of the domain compared to the global regularity on the entire boundary. Second, we propose and analyze a hybridizable discontinuous Galerkin (HDG) method to approximate the solution. For convex polygonal domains, our theoretical convergence rate for the control is optimal with respect to the global regularity on the entire boundary. We present numerical experiments to demonstrate the performance of the HDG method.AMS Subject Classification. 49J20 and 65N30.The dates will be set by the publisher.