Domain In R2 Research Articles

Free boundary Monge-Ampère equations

In this paper, we consider a class of Monge-Ampère equations in a free boundary domain of ℝ2 where the value of the unknown function is prescribed on the free boundary. From a variational point of view, these equations describe an optimal transport problem from an a priori undetermined source domain to a prescribed target domain. We prove the existence and uniqueness of a variational solution to these Monge-Ampère equations under a singularity condition on the density function on the source domain. Furthermore, we provide regularity results under some conditions on the prescribed domain.

PmMYB4, a Transcriptional Activator from Pinus massoniana, Regulates Secondary Cell Wall Formation and Lignin Biosynthesis

Wood formation originates in the biosynthesis of lignin and further leads to secondary cell wall (SCW) biosynthesis in woody plants. Masson pine (Pinus massoniana Lamb) is an economically important industrial timber tree, and its wood yield affects the stable development of the paper industry. However, the regulatory mechanisms of SCW formation in Masson pine are still unclear. In this study, we characterized PmMYB4, which is a Pinus massoniana MYB gene involved in SCW biosynthesis. The open reading frame (ORF) of PmMYB4 was 1473 bp, which encoded a 490 aa protein and contained two distinctive R2 and R3 MYB domains. It was shown to be a transcription factor, with the highest expression in semi-lignified stems. We overexpressed PmMYB4 in tobacco. The results indicated that PmMYB4 overexpression increased lignin deposition, SCW thickness, and the expression of genes involved in SCW formation. Further analysis indicated that PmMYB4 bound to AC-box motifs and might directly activate the promoters of genes (PmPAL and PmCCoAOMT) involved in SCW biosynthesis. In addition, PmMYB4-OE(over expression) transgenic lines had higher lignin and cellulose contents and gene expression than control plants, indicating that PmMYB4 regulates SCW mainly by targeting lignin biosynthetic genes. In summary, this study illustrated the MYB-induced SCW mechanism in Masson pine and will facilitate enhanced lignin and cellulose synthesis in genetically engineered trees.

Fused spatial point process intensity estimation with varying coefficients on complex constrained domains

The availability of large spatial data geocoded at accurate locations has fueled a growing interest in spatial modeling and analysis of point processes. The proposed research is motivated by the intensity estimation problem for spatial point patterns on complex domains in R2 (e.g., domains with irregular boundaries, sharp concavities, and/or interior holes due to geographic constraints) and linear networks, where many existing spatial point process models suffer from the problems of “leakage” and computation. We propose an efficient intensity estimation algorithm to estimate the spatially varying intensity function and to study the varying relationship between intensity and explanatory variables on complex domains. The method is built upon a graph regularization technique and hence can be flexibly applied to point patterns on complex domains such as regions with irregular boundaries and holes, or linear networks. An efficient proximal gradient optimization algorithm is proposed to solve the penalized composite likelihood problem for parameter estimations. Finally, we apply the method to study and visualize the intensity patterns of the accidents on the Western Australia road network, and the spatial variations in the effects of income, night light condition, and population density on the Toronto homicides occurrences.

Well-posedness of the Stokes-transport system in bounded domains and in the infinite strip

We consider the Stokes-transport system, a model for the evolution of an incompressible viscous fluid with inhomogeneous density. This equation was already known to be globally well-posed for any L1∩L∞ initial density with finite first moment in R3. We show that similar results hold on different domain types. We prove that the system is globally well-posed for L∞ initial data in bounded domains of R2 and R3 as well as in the infinite strip R×(0,1). These results contrast with the ill-posedness of a similar problem, the incompressible porous medium equation, for which uniqueness is known to fail for such a density regularity.

Loss of the R2R3 MYB Transcription Factor RsMYB1 Shapes Anthocyanin Biosynthesis and Accumulation in Raphanus sativus.

The red or purple color of radish (Raphanus sativus L.) taproots is due to anthocyanins, which have nutritional and aesthetic value, as well as antioxidant properties. Moreover, the varied patterns and levels of anthocyanin accumulation in radish roots make them an interesting system for studying the transcriptional regulation of anthocyanin biosynthesis. The R2R3 MYB transcription factor RsMYB1 is a key positive regulator of anthocyanin biosynthesis in radish. Here, we isolated an allele of RsMYB1, named RsMYB1Short, in radish cultivars with white taproots. The RsMYB1Short allele carried a 4 bp insertion in the first exon causing a frame-shift mutation of RsMYB1, generating a truncated protein with only a partial R2 domain at the N-terminus. Unlike RsMYB1Full, RsMYB1Short was localized to the nucleus and the cytoplasm and failed to interact with their cognate partner RsTT8. Transient expression of genomic or cDNA sequences for RsMYB1Short in radish cotyledons failed to induce anthocyanin accumulation, but that for RsMYB1Full activated it. Additionally, RsMYB1Short showed the lost ability to induce pigment accumulation and to enhance the transcript level of anthocyanin biosynthetic genes, while RsMYB1Full promoted both processes when co-expressed with RsTT8 in tobacco leaves. As the result of the transient assay, co-expressing RsTT8 and RsMYB1Full, but not RsMYB1Short, also enhanced the promoter activity of RsCHS and RsDFR. We designed a molecular marker for RsMYB1 genotyping, and revealed that the RsMYB1Short allele is common in white radish cultivars, underscoring the importance of variation at the RsMYB1 locus in anthocyanin biosynthesis in the radish taproot. Together, these results indicate that the nonsense mutation of RsMYB1 generated the truncated protein, RsMYB1Short, that had the loss of ability to regulate anthocyanin biosynthesis. Our findings highlight that the frame shift mutation of RsMYB1 plays a key role in anthocyanin biosynthesis in the radish taproot.

N-Amination Converts Amyloidogenic Tau Peptides into Soluble Antagonists of Cellular Seeding.

The spread of neurofibrillary tangles composed of tau protein aggregates is a hallmark of Alzheimer's and related neurodegenerative diseases. Early oligomerization of tau involves conformational reorganization into parallel β-sheet structures and supramolecular assembly into toxic fibrils. Despite the need for selective inhibitors of tau propagation, β-rich protein assemblies are inherently difficult to target with small molecules. Here, we describe a minimalist approach to mimic the aggregation-prone modules within tau. We carried out a backbone residue scan and show that amide N-amination completely abolishes the tendency of these peptides to self-aggregate, rendering them soluble mimics of ordered β-strands from the tau R2 and R3 domains. Several N-amino peptides (NAPs) inhibit tau fibril formation in vitro. We further demonstrate that NAPs 12 and 13 are effective at blocking the cellular seeding of endogenous tau by interacting with monomeric or fibrillar forms of extracellular tau. Peptidomimetic 12 is serum stable, non-toxic to neuronal cells, and selectivity inhibits the fibrilization of tau over Aβ42. Structural analysis of our lead NAPs shows considerable conformational constraint imposed by the N-amino groups. The described backbone N-amination approach provides a rational basis for the mimicry of other aggregation-prone peptides that drive pathogenic protein assembly.

Weak solutions to the time-fractional g-Navier–Stokes equations and optimal control

Abstract In this paper, we introduce the g-Navier–Stokes equations with time-fractional derivative of order α ∈ ( 0 , 1 ) {\alpha\in(0,1)} in domains of ℝ 2 {\mathbb{R}^{2}} . We then study the existence and uniqueness of weak solutions by means of the Galerkin approximation. Finally, an optimal control problem is considered and solved.

On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains: Clustering concentration layers

We consider the problemε2div(∇a(y)u)−V(y)u+up=0,u>0in Ω,∇a(y)u⋅ν=0on ∂Ω, where Ω is a bounded domain in R2 with smooth boundary, the exponent p is greater than 1, ε>0 is a small parameter, V is a uniformly positive smooth potential on Ω¯, and ν denotes the outward normal of ∂Ω. For two positive smooth functions a1(y),a2(y) on Ω¯, the operator ∇a(y) is given by∇a(y)u=(a1(y)∂u∂y1,a2(y)∂u∂y2).(1). Let Γ⊂Ω¯ be a smooth curve intersecting orthogonally with ∂Ω at exactly two points and dividing Ω into two parts. Moreover, Γ is a non-degenerate geodesic embedded in the Riemannian manifold R2 with metric V2σ(y)[a2(y)dy12+a1(y)dy22], where σ=p+1p−1−12. By assuming some additional constraints on the functions a(y), V(y) and the curves Γ, ∂Ω, we prove that there exists a sequence of ε such that the problem has solutions uε with clustering concentration layers directed along Γ, exponentially small in ε at any positive distance from it.(2). If Γ˜ is a simple closed smooth curve in Ω (not touching the boundary ∂Ω), which is also a non-degenerate geodesic embedded in the Riemannian manifold R2 with metric V2σ(y)[a2(y)dy12+a1(y)dy22], then a similar result of concentrated solutions is still true.

Sufficient and necessary conditions for ground state sign-changing solutions to the Schrödinger–Poisson system with cubic nonlinearity on bounded domains

In this paper we are interested in the existence of ground state sign-changing solutions to the following Schrödinger–Poisson system −Δu+ϕu=λu+μ|u|2uinΩ−Δϕ=u2inΩu=ϕ=0on∂Ω,where Ω is a bounded smooth domain of R3, μ>0, λ<λ1 and λ1 is the first eigenvalue of −Δ,H01(Ω). Using construction technique, we show that the above system possesses one ground state sign-changing solution if and only if μ>0, which improves the recent results by Khoutir (2021).

Genome-wide analysis of MYB transcription factors and their responses to salt stress in Casuarina equisetifolia

BackgroundMYB transcription factors are a kind of DNA binding protein that can specifically interact with the promoter region. Members of MYB TFs are widely involved in plant growth and development, secondary metabolism, stress response, and hormone signal transduction. However, there is no report of comprehensive bioinformatics analysis on the MYB family of Casuarina equisetifolia.ResultsIn this study, bioinformatics methods were used to screen out 182 MYB transcription factors from the Casuarina equisetifolia genome database, including 69 1R-MYB, 107 R2R3-MYB, 4 R1R2R3-MYB, and 2 4R-MYB. The C. equisetifolia R2R3-MYB genes were divided into 29 groups based on the phylogenetic topology and the classification of the MYB superfamily in Arabidopsis thaliana, while the remaining MYB genes (1R-MYB, R1R2R3-MYB, and 4R-MYB) was divided into 19 groups. Moreover, the conserved motif and gene structure analysis shown that the members of the CeqMYBs were divided into the same subgroups with mostly similar gene structures. In addition, many conserved amino acids in the R2 and R3 domains of CeqMYBs by WebLogo analysis, especially tryptophan residues (W), with 3 conserved W in R2 repeat and 2 conserved W in R3 repeat. Combining promoter and GO annotation analysis, speculated on the various biological functions of CeqMYBs, thus 32 MYB genes were selected to further explore its response to salt stress by using qPCR analysis technique. Most CeqMYB genes were differentially regulated following multiple salt treatments.ConclusionsSeven genes (CeqMYB164, CeqMYB4, CeqMYB53, CeqMYB32, CeqMYB114, CeqMYB71 and CeqMYB177) were assigned to the “response to salt stress” by GO annotation. Among them, the expression level of CeqMYB4 was up-regulated under various salt treatments, indicating CeqMYB4 might participated in the response to salt stress. Our results provide important information for the biological function of C. equisetifolia, as well as offer candidate genes for further study of salt stress mechanism.

A numerical method for two-dimensional Hammerstein integral equations

"In this paper we investigate a collocation method for the approximate solution of Hammerstein integral equations in two dimensions. As in [8], col- location is applied to a reformulation of the equation in a new unknown, thus reducing the computational cost and simplifying the implementation. We start with a special type of piecewise linear interpolation over triangles for a refor- mulation of the equation. This leads to a numerical integration scheme that can then be extended to any bounded domain in R2, which is used in collocation. We analyze and prove the convergence of the method and give error estimates. As the quadrature formula has a higher degree of precision than expected with linear interpolation, the resulting collocation method is superconvergent, thus requiring fewer iterations for a desired accuracy. We show the applicability of the proposed scheme on numerical examples and discuss future research ideas in this area."

An isoparametric mixed finite element method for approximating a class of fourth-order elliptic problem

A Ciarlet-Raviart type isoparametric mixed finite element method (MFEM) is constructed and analyzed for solving a class of fourth-order elliptic equation with Navier boundary condition defined on a curved domain in R2, and numerical quadrature is also considered in the scheme. The existence and uniqueness of the numerical solutions are proved under certain numerical quadrature. With the help of the special technically analysis, the optimal error estimates with H1 norm are obtained in Ωh, which yields better accuracy than using a convex polygonal domain to approximate the curved domain. For either constant coefficients or nonconstant coefficients problem, numerical examples are listed to confirm theoretical analysis respectively.

DkMYB14 is a bifunctional transcription factor that regulates the accumulation of proanthocyanidin in persimmon fruit

SUMMARYProanthocyanidins (PAs) are phenolic secondary metabolites that contribute to the protection of plant and human health. Persimmon (Diospyros kaki Thunb.) can accumulate abundant PAs in fruit, which cause a strong sensation of astringency. Proanthocyanidins can be classified into soluble and insoluble PAs; the former cause astringency but the latter do not. Soluble PAs can be converted into insoluble PAs upon interacting with acetaldehydes. We demonstrate here that DkMYB14, which regulates the accumulation of PA in persimmon fruit flesh, is a bifunctional transcription factor that acts as a repressor in PA biosynthesis but becomes an activator when involved in acetaldehyde biosynthesis. Interestingly, both functions contribute to the elimination of astringency by decreasing PA biosynthesis and promoting its insolubilization. We show that the amino acid Gly39 in the R2 domain and the ethylene response factor‐associated amphiphilic repression‐like motif in the C‐terminal of DkMYB14 are essential for the regulation of both PA and acetaldehyde synthesis. The repressive function of DkMYB14 was lost after the mutation of either motif, and all activities of DkMYB14 were eliminated following the mutation of both motifs. Our results demonstrate that DkMYB14 functions as both a transcriptional activator and a repressor, directly repressing biosynthesis of PA and promoting its insolubilization, resulting in non‐astringency in persimmon.

The Weyl formula for planar annuli

We study the zeros of cross-product of Bessel functions and obtain their approximations, based on which we reduce the eigenvalue counting problem for the Dirichlet Laplacian associated with a planar annulus to a lattice point counting problem associated with a special domain in R2. Unlike other lattice point problems, the one arisen naturally here has interesting features that lattice points under consideration are translated by various amounts and the curvature of the boundary is unbounded. By transforming this problem into a relatively standard form and using classical van der Corput's bounds, we obtain a two-term Weyl formula for the eigenvalue counting function for the planar annulus with a remainder of size O(μ2/3). If we additionally assume that certain tangent has rational slope, we obtain an improved remainder estimate of the same strength as Huxley's bound in the Gauss circle problem, namely O(μ131/208(log⁡μ)18627/8320). As a by-product of our lattice point counting results, we readily obtain this Huxley-type remainder estimate in the two-term Weyl formula for planar disks.

Convergence for a planar elliptic problem with large exponent Neumann data

We study positive solutions up of the nonlinear Neumann elliptic problem Δu=u in Ω, ∂u/∂ν=|u|p−1u on ∂Ω, where Ω is a bounded open smooth domain in R2. We investigate the asymptotic behavior of families of solutions up satisfying an energy bound condition when the exponent p is getting large. Inspired by the work of Davila-del Pino-Musso [8], we prove that up is developing m peaks xi∈∂Ω, in the sense upp/∫∂Ωupp approaches the sum of m Dirac masses at the boundary and we determine the localization of these concentration points.

On the Landau-Lifshitz-Bloch equation with spin torque effects

A mathematical model for current-induced magnetization dynamics in ferromagnets at elevated temperatures is considered. The model is represented by a classical Landau-Lifshitz-Bloch equation containing adiabatic and non-adiabatic torques. In the case of a ferromagnet confined to a bounded domain of R3, we prove existence of a weak solution by using Faedo-Galerkin approach and a compactness method. This paper extends the work done by Le et al. (2016) in the absence of adiabatic and non-adiabatic terms. Furthermore, uniqueness is proved in the case of dimension one and two. Finally, the limiting behaviour in large time is studied for the non-adiabatic case.

Generalized Laplacian decomposition of vector fields on fractal surfaces

We consider the behavior of generalized Laplacian vector fields on a Jordan domain of R3 with fractal boundary. Our approach is based on properties of the Teodorescu transform and suitable extension of the vector fields. Specifically, the present article addresses the decomposition problem of a Hölder continuous vector field on the boundary (also called reconstruction problem) into the sum of two generalized Laplacian vector fields in the domain and in the complement of its closure, respectively. In addition, conditions on a Hölder continuous vector field on the boundary to be the trace of a generalized Laplacian vector field in the domain are also established.

Sign-changing solutions for a Schrödinger–Kirchhoff–Poisson system with 4-sublinear growth nonlinearity

In this paper we consider the following Schrödinger–Kirchhoff–Poisson-type system { − ( a + b ∫ Ω | ∇ u | 2 d x ) Δ u + λ ϕ u = Q ( x ) | u | p − 2 u in Ω , − Δ ϕ = u 2 in Ω , u = ϕ = 0 on ∂ Ω , where Ω is a bounded smooth domain of R 3 , a &gt; 0 , b ≥ 0 are constants and λ is a positive parameter. Under suitable conditions on Q ( x ) and combining the method of invariant sets of descending flow, we establish the existence and multiplicity of sign-changing solutions to this problem for the case that 2 &lt; p &lt; 4 as λ sufficient small. Furthermore, for λ = 1 and the above assumptions on Q ( x ) , we obtain the same conclusions with 2 &lt; p &lt; 12 5 .

Compactness of extremals for critical singular Trudinger-Moser functionals

Let Ω be a smooth bounded domain in R2, and W01,2(Ω) be the usual Sobolev space. Assume 0∈Ω. For any 0<β<1, according to Csato-Roy [7] and Yang-Zhu [21], there exists some function uβ∈W01,2(Ω)∩Cloc1(Ω‾∖{0})∩C0(Ω‾) satisfying∫Ωe4π(1−β)uβ2|x|2βdx=supu∈W01,2(Ω),‖u‖W01,2(Ω)≤1⁡∫Ωe4π(1−β)u2|x|2βdx. In this paper, we concern the compactness of the function family (uβ)0<β<1. It is shown that up to a subsequence, uβ converges to some function u0 in C1(Ω‾) as β tends to 0. Furthermore, u0 is an extremal of the supremumsupu∈W01,2(Ω),‖u‖W01,2(Ω)≤1⁡∫Ωe4πu2dx.The geometric meaning reads as follows: If we write g0=dx12+dx22 and gβ=|x|−2β(dx12+dx22) as the Euclidean and conical metrics on Ω respectively, where x=(x1,x2)∈Ω, then as β→0, the extremal family (uβ)0<β<1 of the singular Trudinger-Moser functionalsJβ(u)=∫Ωe4π(1−β)u2dvgβ is compact under the constraint u∈W01,2(Ω,gβ) and ‖u‖W01,2(Ω,gβ)≤1.

Overexpression of PtrMYB121 Positively Regulates the Formation of Secondary Cell Wall in Arabidopsis thaliana

MYB transcription factors have a wide range of functions in plant growth, hormone signaling, salt, and drought tolerances. In this study, two homologous transcription factors, PtrMYB55 and PtrMYB121, were isolated and their functions were elucidated. Tissue expression analysis revealed that PtrMYB55 and PtrMYB121 had a similar expression pattern, which had the highest expression in stems. Their expression continuously increased with the growth of poplar, and the expression of PtrMYB121 was significantly upregulated in the process. The full length of PtrMYB121 was 1395 bp, and encoded protein contained 464 amino acids including conserved R2 and R3 MYB domains. We overexpressed PtrMYB121 in Arabidopsis thaliana, and the transgenic lines had the wider xylem as compared with wild-type Arabidopsis. The contents of cellulose and lignin were obviously higher than those in wild-type materials, but there was no significant change in hemicellulose. Quantitative real-time PCR demonstrated that the key enzyme genes regulating the synthesis of lignin and cellulose were significantly upregulated in the transgenic lines. Furthermore, the effector-reporter experiment confirmed that PtrMYB121 bound directly to the promoters of genes relating to the synthesis of lignin and cellulose. These results suggest that PtrMYB121 may positively regulate the formation of secondary cell wall by promoting the synthesis of lignin and cellulose.