Compact Domain Research Articles

Density of algebraic points on Noetherian varieties

Let $${\Omega \subset \mathbb{R}^{n}}$$ be a relatively compact domain. A finite collection of real valued functions on $${{\Omega}}$$ is called a Noetherian chain if the partial derivatives of each function are expressible as polynomials in the functions. A Noetherian function is a polynomial combination of elements of a Noetherian chain. We introduce Noetherian parameters (degrees, size of the coefficients) which measure the complexity of a Noetherian chain. Our main result is an explicit form of the Pila–Wilkie theorem for sets defined using Noetherian equalities and inequalities: for any $${\varepsilon > 0}$$ , the number of points of height H in the transcendental part of the set is at most C· $${{H}^ \varepsilon}$$ where C can be explicitly estimated from the Noetherian parameters and $${\varepsilon}$$ . We show that many functions of interest in arithmetic geometry fall within the Noetherian class, including elliptic and abelian functions, modular functions and universal covers of compact Riemann surfaces, Jacobi theta functions, periods of algebraic integrals, and the uniformizing map of the Siegel modular variety $${\mathcal{A}_{g}}$$ . We thus effectivize the (geometric side of) Pila–Zannier strategy for unlikely intersections in various contexts.

Meshless local B-spline collocation method for heterogeneous heat conduction problems

Several numerical issues still pertain in the modeling of heterogeneous heat conduction, particularly from the viewpoints of material discontinuity and its handling and the presence of heat source. In this paper, a meshless local B-spline collocation method is presented for unsteady heat conduction problems of heterogeneous media. Unknown field variables are approximated by using B-spline basis functions within overlapped compact domains covering the geometry of materials. The present method is a truly meshless approach. The proposed approach is mainly coming with the following advantage that it is straightforward in dealing with discontinuity across the interface of heterogeneous materials. Treatment of discontinuity by using non-crossing interface compact domains allows material discontinuity to be handled geometrically without enforcing additional term/function at the interface. Several heat conduction problems in 2D and 3D heterogeneous media with arbitrary discontinuity shapes are considered. Attention is given for heterogeneous heat conduction problem accompanied by the presence of crack as well. The analysis is then completed by simulating effect of heat generation, in particular which produces high temperature rise inside a heterogeneous structure/component. Simulation results show that the proposed method is a simple and accurate numerical technique for solving unsteady heat conduction problems of heterogeneous media.

Reversing chromatin dynamics for development

Cell Differentiation Compacted chromatin regions, marked by trimethylation of histone H3 at position lysine 9 (H3K9me3), occur at highly repeated DNA sequences, helping to suppress recombination and gene expression. Because pluripotent cells contain low levels of H3K9me3 heterochromatin relative to differentiated cells, it has been thought that an increase in such heterochromatin helps to define cell differentiation. Nicetto et al. used two independent methods to examine compacted heterochromatic domains and found that H3K9me3 compaction increased at protein-coding genes during early mouse organogenesis. During differentiation, these domains open up to allow cell-specific expression. Loss of heterochromatin by genetic inactivation of the H3K9me3 methyltransferases caused ectopic expression of cell-inappropriate genes and tissue pathology. Science , this issue p. [294][1] [1]: /lookup/doi/10.1126/science.aau0583

Height estimates for constant mean curvature graphs in $$\mathrm {Nil}_3$$Nil3 and $${\widetilde{PSL_2}}({\mathbb {R}})$$PSL2~(R)

In this paper we obtain height estimates for compact, constant mean curvature vertical graphs in the homogeneous spaces $$\mathrm {Nil}_3$$ and $${\widetilde{PSL}}_2({\mathbb {R}})$$. As a straightforward consequence, we announce a structure-type result for complete graphs defined on relatively compact domains.

Size dependent buckling analysis of nano sandwich beams by two schemes

Within the framework of Timoshenko beam theory, the buckling of nano sandwich beams is developed. The material properties are assumed to vary arbitrarily in both axial and thickness directions. These types of beams are referred to as bi-directional functionally graded (BDFG) beams. Two types of nano sandwich beams with different material distribution patterns and immovable supports are considered. Since the size effects play a significant role in mechanical behavior of nanostructures, the small-scale effects are captured by Eringen’s nonlocal theory of elasticity. The governing equations are derived using the variational formulation. Symmetric smoothed particle hydrodynamics (SSPH) and the Galerkin method are adopted as numerical solution approaches. As a truly meshless method, the convergence of the SSPH technique mainly depends on the smoothing length value and distribution of particles in the compact support domain of the kernel function. The Revised Super Gauss Function is used as the kernel function and an optimum value for the smoothing length that bears the fastest convergence rate is obtained. The solution methods are verified through benchmark problems found in the literature. Numerical and illustrative results show that various parameters, including the aspect ratio, nonlocal parameter, gradient indexes, and cross-sectional types have significant effects on the buckling responses of BDFG nano sandwich beams.

H3K9me3-heterochromatin loss at protein-coding genes enables developmental lineage specification.

Gene silencing by chromatin compaction is integral to establishing and maintaining cell fates. Trimethylated histone 3 lysine 9 (H3K9me3)-marked heterochromatin is reduced in embryonic stem cells compared to differentiated cells. However, the establishment and dynamics of closed regions of chromatin at protein-coding genes, in embryologic development, remain elusive. We developed an antibody-independent method to isolate and map compacted heterochromatin from low-cell number samples. We discovered high levels of compacted heterochromatin, H3K9me3-decorated, at protein-coding genes in early, uncommitted cells at the germ-layer stage, undergoing profound rearrangements and reduction upon differentiation, concomitant with cell type-specific gene expression. Perturbation of the three H3K9me3-related methyltransferases revealed a pivotal role for H3K9me3 heterochromatin during lineage commitment at the onset of organogenesis and for lineage fidelity maintenance.

Phase singularities in complex arithmetic random waves

Complex arithmetic random waves are stationary Gaussian complex-valued solutions of the Helmholtz equation on the two-dimensional flat torus. We use Wiener-It\^o chaotic expansions in order to derive a complete characterization of the second order high-energy behaviour of the total number of phase singularities of these functions. Our main result is that, while such random quantities verify a universal law of large numbers, they also exhibit non-universal and non-central second order fluctuations that are dictated by the arithmetic nature of the underlying spectral measures. Such fluctuations are qualitatively consistent with the cancellation phenomena predicted by Berry (2002) in the case of complex random waves on compact planar domains. Our results extend to the complex setting recent pathbreaking findings by Rudnick and Wigman (2008), Krishnapur, Kurlberg and Wigman (2013) and Marinucci, Peccati, Rossi and Wigman (2016). The exact asymptotic characterization of the variance is based on a fine analysis of the Kac-Rice kernel around the origin, as well as on a novel use of combinatorial moment formulae for controlling long-range weak correlations.

C. elegans synMuv B proteins regulate spatial and temporal chromatin compaction during development.

Tissue-specific establishment of repressive chromatin through creation of compact chromatin domains during development is necessary to ensure proper gene expression and cell fate. Caenorhabditis elegans synMuv B proteins are important for the soma/germline fate decision and mutants demonstrate ectopic germline gene expression in somatic tissue, especially at high temperature. We show that C. elegans synMuv B proteins regulate developmental chromatin compaction and that the timing of chromatin compaction is temperature sensitive in both wild type and synMuv B mutants. Chromatin compaction in mutants is delayed into developmental time periods when zygotic gene expression is upregulated and demonstrates an anterior-to-posterior pattern. Loss of this patterned compaction coincides with the developmental time period of ectopic germline gene expression, which leads to a developmental arrest in synMuv B mutants. Finally, accelerated cell division rates at elevated temperature may contribute to a lack of coordination between expression of tissue specific transcription programs and chromatin compaction at high temperature. Thus, chromatin organization during development is regulated both spatially and temporally by synMuv B proteins to establish repressive chromatin in a tissue-specific manner to ensure proper gene expression.

High-Throughput Isolation of Soluble Protein Domains Using a Bipartite Split-GFP Complementation System.

The identification of soluble, folded domains of proteins is a recurring task in modern molecular biology. We detail a protocol for identifying compact soluble protein domains using a self-assembling two-part split-GFP comprised of a detector fragment (GFP β-strands 1 through 10, or GFP1-10) and a tagging fragment (GFP β-strand 11, or GFP11). The assay is performed in E. coli cells and in cell extracts. A selection step insures the protein fragments are in frame and contain no stop codons, while an inverse PCR is used to enrich protein fragment libraries containing a specific target sequence.

Cargo shuttling by electrochemical switching of core-shell microgels obtained by a facile one-shot polymerization.

Controlling and understanding the electrochemical properties of electroactive polymeric colloids is a highly topical but still a rather unexplored field of research. This is especially true when considering more complex particle architectures like stimuli-responsive microgels, which would entail different kinetic constraints for charge transport within one particle. We synthesize and electrochemically address dual stimuli responsive core-shell microgels, where the temperature-responsiveness modulates not only the internal structure, but also the microgel electroactivity both on an internal and on a global scale. In detail, a facile one-step precipitation polymerization results in architecturally advanced poly(N-isopropylacrylamide-co-vinylferrocene) P(NIPAM-co-VFc) microgels with a ferrocene (Fc)-enriched (collapsed/hard) core and a NIPAM-rich shell. While the remaining Fc units in the shell are electrochemically accessible, the electrochemical activity of Fc in the core is limited due to the restricted mobility of redox active sites and therefore restricted electron transfer in the compact core domain. Still, prolonged electrochemical action and/or chemical oxidation enable a reversible adjustment of the internal microgel structure from core-shell microgels with a dense core to completely oxidized microgels with a highly swollen core and a denser corona. The combination of thermo-sensitive and redox-responsive units being part of the network allows for efficient amplification of the redox response on the overall microgel dimension, which is mainly governed by the shell. Further, it allows for an electrochemical switching of polarity (hydrophilicity/hydrophobicity) of the microgel, enabling an electrochemically triggered uptake and release of active guest molecules. Hence, bactericidal drugs can be released to effectively kill bacteria. In addition, good biocompatibility of the microgels in cell tests suggests suitability of the new microgel system for future biomedical applications.

Optimal Transport Approximation of 2-Dimensional Measures

We propose a fast and scalable algorithm to project a given density on a set of structured measures defined over a compact 2D domain. The measures can be discrete or supported on curves, for instan...

Determination of Permittivity from the Modulus of the Electric Strength of a High-Frequency Electromagnetic Field

For an unmagnetized nonconducting medium, a system of electrodynamic equations corresponding to time-periodic oscillations is considered. For this system, we study the problem of determining the permittivity $$\varepsilon $$ from a given electric field magnitude in the electromagnetic field resulting from the interference of two fields produced by external point current sources. It is assumed that the permittivity differs from a given positive constant $${{\varepsilon }_{0}}$$ only inside a compact domain $${{\Omega }_{0}} \subset {{\mathbb{R}}^{3}}$$ and the electric field magnitude is given for all frequencies, starting at a fixed one $${{\omega }_{0}}$$ , on the boundary $$S$$ of a domain $$\Omega $$ surrounding $${{\Omega }_{0}}$$ . By analyzing an asymptotic expansion of the solution to a direct problem at high frequencies, it is shown that, with the use of given information, the original problem is reduced to the well-known inverse kinematic problem of determining the refraction coefficient inside $$\Omega $$ from the travel times of an electromagnetic wave propagating between boundary points of this domain. This reduction opens up the possibility of obtaining a constructive solution of the original problem.

Recovering two coefficients in an elliptic equation via phaseless information

For fixed $y \in \mathbb{R}^3$ , we consider the equation $L u+k^2u = - δ(x-y), \>x \in \mathbb{R}^3$ , where $L=\text{div}(n(x)^{-2}\nabla)+q(x)$ , $k >0$ is a frequency, $n(x)$ is a refraction index and $q(x)$ is a potential. Assuming that the refraction index $n(x)$ is different from $1$ only inside a bounded compact domain $Ω$ with a smooth boundary $S$ and the potential $q(x)$ vanishes outside of the same domain, we study an inverse problem of finding both coefficients inside $Ω$ from some given information on solutions of the elliptic equation. Namely, it is supposed that the point source located at point $y \in S$ is a variable parameter of the problem. Then for the solution $u(x,y,k)$ of the above equation satisfying the radiation condition, we assume to be given the following phaseless information $f(x,y,k)=|u(x,y,k)|^2$ for all $x,y \in S$ and for all $k≥ k_0>0$ , where $k_0$ is some constant. We prove that this phaseless information uniquely determines both coefficients $n(x)$ and $q(x)$ inside $Ω$ .

The propagation of active-passive interfaces in bacterial swarms

Propagating interfaces are ubiquitous in nature, underlying instabilities and pattern formation in biology and material science. Physical principles governing interface growth are well understood in passive settings; however, our understanding of interfaces in active systems is still in its infancy. Here, we study the evolution of an active-passive interface using a model active matter system, bacterial swarms. We use ultra-violet light exposure to create compact domains of passive bacteria within Serratia marcescens swarms, thereby creating interfaces separating motile and immotile cells. Post-exposure, the boundary re-shapes and erodes due to self-emergent collective flows. We demonstrate that the active-passive boundary acts as a diffuse interface with mechanical properties set by the flow. Intriguingly, interfacial velocity couples to local swarm speed and interface curvature, raising the possibility that an active analogue to classic Gibbs-Thomson-Stefan conditions may control this boundary propagation.

A Priori Estimates for Water Waves with Emerging Bottom

We study the beach problem for water waves. The case we consider is a compact fluid domain, where the free surface intersects the bottom along an edge, with a non-zero contact angle. Using elliptic estimates in domains with edges and a new equation on the Taylor coefficient, we establish a priori estimates for angles smaller than a dimensional constant. Local existence will be derived in a following paper.

Turán—Erőd Type Converse Markov Inequalities on General Convex Domains of the Plane in the Boundary Lq Norm

In 1939 P. Turan started to derive lower estimations on the norm of the derivatives of polynomials of (maximum) norm 1 on $$\mathbb{I}:= [-1,1]$$ (interval) and $$\mathbb{D}:= \{z \in \mathbb{C}: |z| \leq 1\}$$ (disk) under the normalization condition that the zeroes of the polynomial in question all lie in $$\mathbb{I} $$ or $$\mathbb{D} $$ , respectively. For the maximum norm he found that with n:= deg p tending to infinity, the precise growth order of the minimal possible derivative norm is √n for $$\mathbb{I} $$ and n for $$\mathbb{D} $$ . J. Erőd continued the work of Turan considering other domains. Finally, about a decade ago the growth of the minimal possible ∞-norm of the derivative was proved to be of order n for all compact convex domains. Although Turan himself gave comments about the above oscillation question in Lq norms, till recently results were known only for $$\mathbb{D} $$ and $$\mathbb{I} $$ . Recently, we have found order n lower estimations for several general classes of compact convex domains, and conjectured that even for arbitrary convex domains the growth order of this quantity should be n. Now we prove that in Lq norm the oscillation order is at least n/log n for all compact convex domains.

Low-cost compact multispectral spatial frequency domain imaging prototype for tissue characterization.

We present a low-cost, compact, and multispectral spatial frequency domain imaging prototype. Illumination components, including 9 LEDs (660 nm - 950 nm) placed on a custom-designed printed circuit board, linear and rotational motors, a printed sinusoidal pattern, and collimation and projection optics as well as the detection components are incorporated in a compact custom-designed 3D-printed probe. Reconstruction of absorption and reduced scattering coefficients is evaluated via imaging tissue mimicking phantoms and potentials of the probe for biological tissue imaging are evaluated via imaging human ovarian tissue ex vivo.

HP1 cooperates with CAF-1 to compact heterochromatic transgene repeats in mammalian cells

The nuclear organization of tightly condensed heterochromatin plays important roles in regulating gene transcription and genome integrity. Heterochromatic domains are usually present at chromosomal regions containing a large array of repeated DNA sequences. We previously showed that integration of a 1,000-copy tandem array of an inducible reporter gene into the genome of mammalian cells induces the formation of a highly compact heterochromatic domain enriched in heterochromatin protein 1 (HP1). It remains to be determined how these DNA repeats are packaged into a heterochromatic form and are silenced. Here, we show that HP1-mediated transgene condensation and silencing require the interaction with PxVxL motif-containing proteins. The chromatin assembly factor 1 (CAF-1) complex concentrates at the transgenic locus through the interaction of its PxVxL motif-containing p150 subunit with HP1. Knockdown of p150 relieves HP1-mediated transgene compaction and repression. When targeted to the transgenic locus, p150 mutants defective in binding HP1 cause transgene decondensation and activation. Taken together, these results suggest that HP1 cooperates with CAF-1 to compact transgene repeats. This study provides important insight into how heterochromatin is maintained at chromosomal regions with abundant DNA repeats.

Pattern evolution characterizes the mechanism and efficiency of CVD graphene growth

Growing large-area, high-quality monolayers is the holy grail of graphene synthesis. In this work, the efficiency of graphene growth and the quality of their continuous films are explored through the time evolution of individual domains and their surface coverage on the substrate. Our phase-field modeling results and experimental characterization clearly demonstrate the critical roles of the deposition flux, edge-reaction kinetics and the surface diffusion of active carbon sources in modulating the pattern evolution and rate of growth. We find that the contrast between the edge-kinetics-limited and surface-diffusion-limited regimes is remarkable, which can be characterized by the evolution of domain patterns and considered as an indicator of the growth regime. However, common features exist in these two regimes, showing that the growth rate scales with time as t2 in the early stage of growth and is regime-independent, which is explained by the coarsen profiles of carbon concentration for both the compact and dendritic domains. The rate decays rapidly in the final stage of growth due to the competition between neighboring domains on the limited carbon sources diffusing on the substrate, which is highly regime-sensitive and extremely low in the surface-diffusion-limited regime with narrow gaps between the domains to be filled. Based on these findings, synthesis strategies to improve the growth efficiency and film quality are discussed.

T4 DNA ligase structure reveals a prototypical ATP-dependent ligase with a unique mode of sliding clamp interaction.

DNA ligases play essential roles in DNA replication and repair. Bacteriophage T4 DNA ligase is the first ATP-dependent ligase enzyme to be discovered and is widely used in molecular biology, but its structure remained unknown. Our crystal structure of T4 DNA ligase bound to DNA shows a compact α-helical DNA-binding domain (DBD), nucleotidyl-transferase (NTase) domain, and OB-fold domain, which together fully encircle DNA. The DBD of T4 DNA ligase exhibits remarkable structural homology to the core DNA-binding helices of the larger DBDs from eukaryotic and archaeal DNA ligases, but it lacks additional structural components required for protein interactions. T4 DNA ligase instead has a flexible loop insertion within the NTase domain, which binds tightly to the T4 sliding clamp gp45 in a novel α-helical PIP-box conformation. Thus, T4 DNA ligase represents a prototype of the larger eukaryotic and archaeal DNA ligases, with a uniquely evolved mode of protein interaction that may be important for efficient DNA replication.