Domain In R2 Research Articles

Robin–Dirichlet alternating iterative procedure for solving the Cauchy problem for Helmholtz equation in an unbounded domain

Abstract We consider the Cauchy problem for the Helmholtz equation with a domain in ℝ d {\mathbb{R}^{d}} , d ≥ 2 {d\geq 2} with N cylindrical outlets to infinity with bounded inclusions in ℝ d - 1 . {\mathbb{R}^{d-1}.} Cauchy data are prescribed on the boundary of the bounded domains and the aim is to find solution on the unbounded part of the boundary. In 1989, Kozlov and Maz’ya [14] proposed an alternating iterative method for solving Cauchy problems associated with elliptic, self-adjoint and positive-definite operators in bounded domains. Different variants of this method for solving Cauchy problems associated with Helmholtz-type operators exists. We consider the variant proposed by Berntsson, Kozlov, Mpinganzima and Turesson (2018) [4] for bounded domains and derive the necessary conditions for the convergence of the procedure in unbounded domains. For the numerical implementation, a finite difference method is used to solve the problem in a simple rectangular domain in ℝ 2 {\mathbb{R}^{2}} that represent a truncated infinite strip. The numerical results shows that by appropriate truncation of the domain and with appropriate choice of the Robin parameters μ 0 {\mu_{0}} and μ 1 {\mu_{1}} , the Robin–Dirichlet alternating iterative procedure is convergent.

Exploring membrane binding targets of tau oligomers on phase-separated lipid bilayers using multiscale MD simulations.

Self-aggregation of tau has been linked to the onset of Alzheimer's. Recent studies indicate that the disordered tau oligomers are more neurotoxic than the ordered tau fibrils in the intracellular neurofibrillary tangle. Details of tau oligomer interactions with the lipid nanodomains in phase-separated lipid bilayers (raft membranes), a model of neuronal membranes, are unknown. Using coarse-grained (CG) and all-atom (AA) MD simulations, a tau-K18 monomer containing the microtubule-binding domain (R1-R2-R3-R4) and its disordered oligomers were simulated. By mixing phosphatidylcholine, cholesterol, ganglioside (GM1), and phosphatidylserine (PS) lipids in water, raft membranes containing lipid nanodomains, liquid-ordered (Lo), liquid-disordered (Ld), and Lo/Ld boundary (Lod) on both lipid leaflets, as well as PS- or GM1-cluster on one lipid leaflet, were simulated. Lipid-binding kinetics and energetics of tau-K18 oligomers to these nanodomains were investigated. Within 20 μs, tau-K18 oligomers bound to the Lod domains in membranes without PS or GM1. In contrast, exclusive binding to PS- or GM1-clusters in membranes containing PS or GM1 occurred in less than 6 μs. After a CG to AA transformation and a 200 ns-AA simulation, disruptions of lipid chain order and beta-sheet formation on raft membranes containing membrane-bound tau-K18 oligomers were found. By mutating three hydrophobic residues to anionic residues at the R2 and R3 domain regions of tau-K18, the mutated tau-K18 oligomers bound much weaker to the PS- or GM1-cluster domains than the wild type did, suggesting the importance of electrostatic interactions in the tau-lipid interactions. Since PS and GM1 are found in the inner and outer leaflets of neuronal membranes, respectively, our results suggest that the PS- and GM1-clusters are leaflet-specific targets while the Lod domains are non-specific targets in membrane-associated tauopathies in Alzheimer's. (Supported by NSF [OAC-153159] and Welch Foundation [W-2057-2021-327].)

Spectral properties of the inhomogeneous Drude-Lorentz model with dissipation

We establish spectral enclosures and spectral approximation results for the inhomogeneous lossy Drude-Lorentz system with purely imaginary poles, in a possibly unbounded Lipschitz domain of R3. Under the assumption that the coefficients θe, θm of the material are asymptotically constant at infinity, we prove that spectral pollution due to domain truncation can lie only in the essential numerical range of a curlcurl0−f(ω) pencil.As an application, we consider a conducting metamaterial at the interface with the vacuum; we prove that the complex eigenvalues with non-trivial real part lie outside the set of spectral pollution. We believe this is the first result of enclosure of spectral pollution for the Drude-Lorentz model without assumptions of compactness on the resolvent of the underlying Maxwell operator.

Well-posedness and exponential stability for a Klein-Gordon system with locally distributed viscoelastic dampings in a past-history framework

Our main goal in the present work is to prove the well-posedness as well as the exponential stability for a strongly coupled Klein-Gordon system posed on a bounded domain of R2 with smooth boundary, subject to locally distributed viscoelastic effects driven by nonnegative functions a(x) and b(x) acting on Ω, where a=b=0 in a region A of Ω. Some difficulties imposed by the presence of integrals terms force us to consider a perturbed problem in order to obtain exponential decay rates. For this purpose, microlocal analysis tools combined with unique continuation property were used.

Strong attractors for weakly damped quintic wave equation in bounded domains

In this paper, we study the longtime dynamics for the weakly damped wave equation with quintic non-linearity in a bounded smooth domain of R3. Based on the Strichartz estimates for the case of bounded domains, we establish the existence of a strong global attractor in the phase space H2(Ω)∩H01(Ω)×H01(Ω). Moreover, the finite fractal dimension of the attractor is also shown with the help of the quasi-stable estimation.

Sobolev space theory and Hölder estimates for the stochastic partial differential equations on conic and polygonal domains

We establish existence, uniqueness, and Sobolev and Hölder regularity results for the stochastic partial differential equationdu=(∑i,j=1daijuxixj+f0+∑i=1dfxii)dt+∑k=1∞gkdwtk,t>0,x∈D given with non-zero initial data. Here {wtk:k=1,2,⋯} is a family of independent Wiener processes defined on a probability space (Ω,P), aij=aij(ω,t) are merely measurable functions on Ω×(0,∞), and D is either a polygonal domain in R2 or an arbitrary dimensional conic domain of the type(0.1)D(M):={x∈Rd:x|x|∈M},M⊊Sd−1,(d≥2) where M is an open subset of Sd−1 with C2 boundary. We measure the Sobolev and Hölder regularities of arbitrary order derivatives of the solution using a system of mixed weights consisting of appropriate powers of the distance to the vertices and of the distance to the boundary. The ranges of admissible powers of the distance to the vertices and to the boundary are sharp.

On Weights Which Admit Harmonic Bergman Kernel and Minimal Solutions of Laplace’s Equation

Abstract In this paper we consider spaces of weight square-integrable and harmonic functions L 2 H(Ω, µ). Weights µ for which there exists reproducing kernel of L 2 H(Ω, µ) are named ’admissible weights’ and such kernels are named ’harmonic Bergman kernels’. We prove that if only weight of integration is integrable in some negative power, then it is admissible. Next we construct a weight µ on the unit circle which is non-admissible and using Bell-Ligocka theorem we show that such weights exist for a large class of domains in ℝ2. Later we conclude from the classical result of reproducing kernel Hilbert spaces theory that if the set {f ∈ L 2 H(Ω, µ)|f(z) = c} for admissible weight µ is non-empty, then there is exactly one element with minimal norm. Such an element in this paper is called ’a minimal (z, c)-solution in weight µ of Laplace’s equation on Ω’ and upper estimates for it are given.

Characterizations of MYB Transcription Factors in Camellia oleifera Reveal the Key Regulators Involved in Oil Biosynthesis

MYB (myeloblastosis) transcription factors plays an important role in various physiological and biochemical processes in plants. However, little is known about the regulatory roles of MYB family genes underlying seed oil biosynthesis in Camellia oleifera. To identify potential regulators, we performed the genome-wide characterizations of the MYB family genes and their expression profiles in C. oleifera. A total of 186 CoMYB genes were identified, including 128 R2R3-type MYB genes that had conserved R2 and R3 domains. Phylogenetic analysis revealed the CoR2R3-MYBs formed 25 subgroups and possessed some highly conserved motifs outside the MYB DNA-binding domain. We investigated the promoter regions of CoR2R3-MYBs and revealed a series of cis-acting elements related to development, hormone response, and environmental stress response, suggesting a diversified regulatory mechanism of gene functions. In addition, we identified four tandem clusters containing eleven CoR2R3-MYBs, which indicated that tandem duplications played an important role in the expansion of the CoR2R3-MYB subfamily. Furthermore, we analyzed the global gene expression profiles at five stages during seed development and revealed seven CoR2R3-MYB genes that potentially regulated lipid metabolism and seed maturation in C. oleifera. These results provide new insights into understanding the function of the MYB genes and the genetic improvement of seed oil.

⊥-loss: A symmetric loss function for magnetic resonance imaging reconstruction and image registration with deep learning.

Convolutional neural networks (CNNs) are increasingly adopted in medical imaging, e.g., to reconstruct high-quality images from undersampled magnetic resonance imaging (MRI) acquisitions or estimate subject motion during an examination. MRI is naturally acquired in the complex domain C, obtaining magnitude and phase information in k-space. However, CNNs in complex regression tasks are almost exclusively trained to minimize the L2 loss or maximizing the magnitude structural similarity (SSIM), which are possibly not optimal as they do not take full advantage of the magnitude and phase information present in the complex domain. This work identifies that minimizing the L2 loss in the complex field has an asymmetry in the magnitude/phase loss landscape and is biased, underestimating the reconstructed magnitude. To resolve this, we propose a new loss function for regression in the complex domain called ⊥-loss, which adds a novel phase term to established magnitude loss functions, e.g., L2 or SSIM. We show ⊥-loss is symmetric in the magnitude/phase domain and has favourable properties when applied to regression in the complex domain. Specifically, we evaluate the ⊥+ℓ2-loss and ⊥+SSIM-loss for complex undersampled MR image reconstruction tasks and MR image registration tasks. We show that training a model to minimize the ⊥+ℓ2-loss outperforms models trained to minimize the L2 loss and results in similar performance compared to models trained to maximize the magnitude SSIM while offering high-quality phase reconstruction. Moreover, ⊥-loss is defined in Rn, and we apply the loss function to the R2 domain by learning 2D deformation vector fields for image registration. We show that a model trained to minimize the ⊥+ℓ2-loss outperforms models trained to minimize the end-point error loss.

Dirichlet Problem with L1(S) Boundary Values

Let D be a connected bounded domain in R2, S be its boundary, which is closed and C2-smooth. Consider the Dirichlet problem Δu=0inD,u|S=h, where h∈L1(S). The aim of this paper is to prove that the above problem has a solution for an arbitrary h∈L1(S), and this solution is unique. The result is new. The method of its proof is new. The definition of the L1(S)-boundary value of a harmonic in the D function is given. No embedding theorems are used. The history of the Dirichlet problem goes back to 1828. The result in this paper is, to the author’s knowledge, the first result in the 194 years of research (since 1828) that yields the existence and uniqueness of the solution to the Dirichlet problem with the boundary values in L1(S).

A posteriori error analysis for a distributed optimal control problem governed by the von Kármán equations

This article discusses the numerical analysis of the distributed optimal control problem governed by the von Kármán equations defined on a polygonal domain in ℝ2. The state and adjoint variables are discretised using the nonconforming Morley finite element method and the control is discretized using piecewise constant functions. A priori and a posteriori error estimates are derived for the state, adjoint and control variables. The a posteriori error estimates are shown to be efficient. Numerical results that confirm the theoretical estimates are presented.

The Pea R2R3-MYB Gene Family and Its Role in Anthocyanin Biosynthesis in Flowers.

Pea (Pisum sativum L.) is one of the most important legume crops in the world, and it has attracted great attention for its high nutritive values. Recently, the crop breeding program has been focused on the crop metabolic engineering (i.e., color, flavor, nutrition) to improve the quality of crop. As a major group of transcription factors forming the ternary MYB–bHLH–WD repeat protein (MBW) complex to regulate the anthocyanin biosynthesis pathway, members of R2R3-MYB gene family have always been the focus of research targets to improve the valuable metabolic product of crops. Until now, few report about the R2R3-MYB gene family of pea has been released. In this study, we identified 119 R2R3-MYB genes in the assembled pea genome (Version 1a), of which 111 were distributed across 14 chromosomes. Combining with the 126 R2R3-MYB protein sequences of Arabidopsis, we categorized 245 R2R3-MYB proteins into 36 subgroups according to sequence similarity and phylogenetic relationships. There was no member from subgroup 12, 15 and 29 existing in pea genome, whereas three novel subgroups were found in pea and named as N1-N3. Further analyses of conserved domains and Motifs, gene structures, and chromosomal locations showed that the typical R2 and R3 domains were present across all R2R3-MYB proteins, and Motif 1, 2, and 3 were identified in most members. Most of them had no more than two introns. Additionally, 119 pea R2R3-MYB genes did not experience large-scale duplication events. Finally, we concluded that several candidate genes may be responsible for the spatiotemporal accumulation of anthocyanins in pea petals. PsMYB116 was predominantly expressed in the dorsal petals to presumably activate the anthocyanin biosynthesis pathway, while PsMYB37 and PsMYB32 may positively regulates the anthocyanin accumulation in the lateral petals. This study not only provides a good reference to further characterize the diverse functions of R2R3-MYB genes but also helps researchers to understand the color formation of pea flowers.

Cloning and Functional Analysis of NtMYB9 in ‘Jinzhanyintai’ of Narcissus tazetta var. chinensis

Narcissus tazetta var. chinensis M.Roem. (Chinese Narcissus) is a traditional and famous flower in China, and its corona shows a transition from green to yellow with the opening of flowers, which is of great ornamental value. To investigate the molecular mechanism of the corona color-forming in the Chinese Narcissus, in this study, a MYB gene was screened from the transcriptome of the corona by bioinformatics analysis methods, named NtMYB9. The expression level of the NtMYB9 gene was significantly higher in the corona than in the tepal, filament, ovary and leaf, and reached the highest level at the fourth period of complete coloration of the corona. The full-length sequence of the NtMYB9 coding region was cloned using the corona cDNA as a template. Sequence analysis revealed that NtMYB9 protein contained R2 and R3 domains, phylogenetic tree analysis showed that the NtMYB9 protein was most closely related to flavonol activators. Subcellular localization showed that NtMYB9 was localized in the nucleus. The overexpression of NtMYB9 gene into tobacco leaves and the expression level of seven enzyme genes related to the flavonoid biosynthesis pathway were significantly up-regulated. The transient transformation of Petunia × hybrida (Petunia) and Phalaenopsis aphrodite (Phalaenopsis) significantly revealed that the NtMYB9 gene could turn flowers yellow. These results suggest that the NtMYB9 gene is involved in the positive regulation of flavonoid biosynthesis in the Chinese Narcissus and may promote the synthesis of flavonols. In conclusion, these findings provide a valuable resource for further studies on the regulatory mechanism of the flavonoid biosynthesis pathway, and they are also beneficial to the molecular breeding of Chinese Narcissus.

Establishment of a novel ELISA system for measuring periostin independently of formation of the IgA complex.

Periostin, a matricellular protein that modulates cell functions having various pathophysiological roles, has the potential to be a useful biomarker for various diseases. We recently found that periostin forms a complex with IgA in human serum, which may affect the periostin measurement. We investigated (1) whether the formation of the periostin-IgA complex affects the original periostin ELISA system, decreasing the values of serum periostin? (2) bow each domain of periostin affects periostin measurement by the original periostin ELISA system? (3) whether we can establish a novel ELISA system that is not affected by formation of the IgA complex? The periostin value at the reducing condition was significantly higher than that of the non-reducing condition, demonstrating that formation of the IgA complex affects periostin measurement. The monoclonal antibodies (mAbs) for periostin recognizing the EMI and R1 domains immunoprecipitated serum periostin in the reducing condition more than in the non-reducing condition, whereas the mAbs recognizing the R2 or R3 domain immunoprecipitated comparable amounts of serum periostin in the reducing and non-reducing conditions, suggesting the EMI and R1 domains contribute to formation of the complex with IgA. Using SS16A recognizing the R3 domain combined with SS17B recognizing the R4 domain, we established an ELISA system that was able to measure periostin independently of the IgA complex. We have established a novel ELISA system that measures periostin independently of the IgA complex. This system is promising in identifying periostin as a biomarker for various diseases.

Continuous Dependence for the Boussinesq Equations under Reaction Boundary Conditions in R2

In this paper, we studied the continuous dependence result for the Boussinesq equations. We considered the case where Ω was a bounded domain in R2. Temperatures T and C satisfied reaction boundary conditions. A first-order inequality for the differences of energy could be derived. An integration of this inequality produced a continuous dependence result. The result told us that the continuous dependence type stability was also valid for the Boussinesq coefficient λ of the Boussinesq equations with reaction boundary conditions.

Global well-posedness of 3-D nonhomogeneous incompressible MHD equations with bounded nonnegative density

In this paper, we consider a nonhomogeneous incompressible MHD equations in the torus T3 or a C2 simply connected bounded domain of R3. When the initial velocity and magnetic are in H1 and initial density is only bounded nonnegative, we first construct the existence of local in time solution for arbitrary large data and global in time for the model with some smallness conditions. Moreover, the uniqueness of the solution is also proved under some quite soft assumptions about its regularity through a Lagrangian approach. In particular, the initial vacuum is allowed.

Detection and computation of conservative kernels of models consisting of freeform curves and surfaces, using inequality constraints

We present an algorithm to compute a tight-as-needed conservative approximation of the kernel domain of freeform curves in R2 and freeform surfaces in R3. Inequality constraints to detect the interior of the kernel domain are formulated as multivariates, and solved with a subdivision-based approach to find the domains in R2 or R3 that satisfy the inequalities and are in the kernel. The convex hull of the computed domains is also included in the kernel, and adopted as the approximated kernel domain. We can apply the presented algorithm to detect the kernel domain of not only C1 continuous closed regular curves and surfaces, but also the kernel domains of multiple piecewise C1 continuous regular freeform curves and surfaces. Further, the presented algorithm can be applied to find the gamma-kernel as well as the kernel domain of open curves and surfaces, under some assumptions. We demonstrate our experimental result using various freeform curves and surfaces, and compare it with the kernel computation algorithm presented in Elber et al. (2006).

Homogalacturonan from squash: Characterization and tau-binding pattern of a sulfated derivative

A pectic polysaccharide (WAP) was isolated from squash and identified as a homogalacturonan with a molecular mass of 83.2 kDa by GPC, monosaccharide composition analysis, FT-IR and NMR spectra. Sulfation modification of WAP was carried out and a sulfated derivative (SWAP) was obtained with a substitution degree of 1.81. The NMR spectrum indicated that the sulfation modification mainly occurred at the C-2 and C-3 positions of galacturonan residues. The binding pattern of SWAP to tau K18 protein was observed in 2D 1H15N HSQC spectra of tau, which resembled the tau-heparin interaction, with R2 domain as the major binding region. These results suggest that SWAP has the potential to act as a heparin mimic to inhibit the transcellular spread of tau; thus natural polysaccharide from squash may be developed into therapies for AD and related tauopathies.

Multiplicity results for semilinear heterogeneous problems in exterior domains of [formula omitted] involving subcritical or critical nonlinearities à la Trudinger-Moser

Let Ω be an exterior domain in R2. We prove some multiplicity results, in the Beppo-Levi space, for solutions to the following boundary value problem−Δu=b(x)f(u)inΩ,u=0on∂Ω, where the nonnegative coefficient b(x) satisfies a suitable integrability assumption and the C1 nonlinearity f:R→R is superlinear at zero and infinity and has subcritical or critical growth at infinity in the sense of Trudinger-Moser.

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.