A Spectral Method for Elliptic Equations: The Neumann Problem

Let $\Omega$ be an open, simply connected, and bounded region in $\mathbb{R}^{d}$, $d\geq2$, and assume its boundary $\partial\Omega$ is smooth. Consider solving an elliptic partial differential equation $-\Delta u+\gamma u=f$ over $\Omega$ with a Neumann boundary condition. The problem is converted to an equivalent elliptic problem over the unit ball $B$, and then a spectral Galerkin method is used to create a convergent sequence of multivariate polynomials $u_{n}$ of degree $\leq n$ that is convergent to $u$. The transformation from $\Omega$ to $B$ requires a special analytical calculation for its implementation. With sufficiently smooth problem parameters, the method is shown to be rapidly convergent. For $u\in C^{\infty}(\overline{\Omega})$ and assuming $\partial\Omega$ is a $C^{\infty}$ boundary, the convergence of $\Vert u-u_{n}\Vert_{H^{1}}$ to zero is faster than any power of $1/n$. Numerical examples in $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$ show experimentally an exponential rate of convergence.Comment: 23 pages, 11 figure

A spectral method for elliptic equations: the Dirichlet problem

An elliptic partial differential equation Lu=f with a zero Dirichlet boundary condition is converted to an equivalent elliptic equation on the unit ball. A spectral Galerkin method is applied to the reformulated problem, using multivariate polynomials as the approximants. For a smooth boundary and smooth problem parameter functions, the method is proven to converge faster than any power of 1/n with n the degree of the approximate Galerkin solution. Examples in two and three variables are given as numerical illustrations. Empirically, the condition number of the associated linear system increases like O(N), with N the order of the linear system.Comment: This is latex with the standard article style, produced using Scientific Workplace in a portable format. The paper is 22 pages in length with 8 figure

Soft X-ray harmonic comb from relativistic electron spikes

We demonstrate a new high-order harmonic generation mechanism reaching the `water window' spectral region in experiments with multi-terawatt femtosecond lasers irradiating gas jets. A few hundred harmonic orders are resolved, giving uJ/sr pulses. Harmonics are collectively emitted by an oscillating electron spike formed at the joint of the boundaries of a cavity and bow wave created by a relativistically self-focusing laser in underdense plasma. The spike sharpness and stability are explained by catastrophe theory. The mechanism is corroborated by particle-in-cell simulations

X-ray harmonic comb from relativistic electron spikes

X-ray devices are far superior to optical ones for providing nanometre spatial and attosecond temporal resolutions. Such resolution is indispensable in biology, medicine, physics, material sciences, and their applications. A bright ultrafast coherent X-ray source is highly desirable, for example, for the diffractive imaging of individual large molecules, viruses, or cells. Here we demonstrate experimentally a new compact X-ray source involving high-order harmonics produced by a relativistic-irradiance femtosecond laser in a gas target. In our first implementation using a 9 Terawatt laser, coherent soft X-rays are emitted with a comb-like spectrum reaching the 'water window' range. The generation mechanism is robust being based on phenomena inherent in relativistic laser plasmas: self-focusing, nonlinear wave generation accompanied by electron density singularities, and collective radiation by a compact electric charge. The formation of singularities (electron density spikes) is described by the elegant mathematical catastrophe theory, which explains sudden changes in various complex systems, from physics to social sciences. The new X-ray source has advantageous scalings, as the maximum harmonic order is proportional to the cube of the laser amplitude enhanced by relativistic self-focusing in plasma. This allows straightforward extension of the coherent X-ray generation to the keV and tens of keV spectral regions. The implemented X-ray source is remarkably easily accessible: the requirements for the laser can be met in a university-scale laboratory, the gas jet is a replenishable debris-free target, and the harmonics emanate directly from the gas jet without additional devices. Our results open the way to a compact coherent ultrashort brilliant X-ray source with single shot and high-repetition rate capabilities, suitable for numerous applications and diagnostics in many research fields

DNA Methylation Analysis of Chromosome 21 Gene Promoters at Single Base Pair and Single Allele Resolution

Differential DNA methylation is an essential epigenetic signal for gene regulation, development, and disease processes. We mapped DNA methylation patterns of 190 gene promoter regions on chromosome 21 using bisulfite conversion and subclone sequencing in five human cell types. A total of 28,626 subclones were sequenced at high accuracy using (long-read) Sanger sequencing resulting in the measurement of the DNA methylation state of 580427 CpG sites. Our results show that average DNA methylation levels are distributed bimodally with enrichment of highly methylated and unmethylated sequences, both for amplicons and individual subclones, which represent single alleles from individual cells. Within CpG-rich sequences, DNA methylation was found to be anti-correlated with CpG dinucleotide density and GC content, and methylated CpGs are more likely to be flanked by AT-rich sequences. We observed over-representation of CpG sites in distances of 9, 18, and 27 bps in highly methylated amplicons. However, DNA sequence alone is not sufficient to predict an amplicon's DNA methylation status, since 43% of all amplicons are differentially methylated between the cell types studied here. DNA methylation in promoter regions is strongly correlated with the absence of gene expression and low levels of activating epigenetic marks like H3K4 methylation and H3K9 and K14 acetylation. Utilizing the single base pair and single allele resolution of our data, we found that i) amplicons from different parts of a CpG island frequently differ in their DNA methylation level, ii) methylation levels of individual cells in one tissue are very similar, and iii) methylation patterns follow a relaxed site-specific distribution. Furthermore, iv) we identified three cases of allele-specific DNA methylation on chromosome 21. Our data shed new light on the nature of methylation patterns in human cells, the sequence dependence of DNA methylation, and its function as epigenetic signal in gene regulation. Further, we illustrate genotype–epigenotype interactions by showing novel examples of allele-specific methylation

Isotope-Geochemical Evidence of the Nature of the Protoliths of Diamondiferous Rocks of the Kokchetav Subduction-Collision Zone (Northern Kazakhstan)

—The isotope-geochemical features of diamondiferous metamorphic rocks of the Kokchetav subduction–collision zone (KSCZ) show that both the basement rocks and the sediments of the Kokchetav massif were their protoliths. A whole-rock Sm–Nd isochron from the diamondiferous calc-silicate, garnet–pyroxene rocks and migmatized granite-gneisses of the western block of the KSCZ yielded an age of 1116 ± 14 Ma, while an age of 1.2–1.1 Ga was obtained by U–Pb dating of zircons from the granite-gneiss basement of the Kokchetav microcontinent. Based on these data, we assume that the protoliths of the calc-silicate, garnet–pyroxene rocks and the granite-gneisses of the KSCZ were the basement rocks sharing an initially single Nd source, which was not influenced by high- to ultrahigh-pressure metamorphism (~530 Ma). Therefore, their geochemical features are probably not directly related to ultrahigh-pressure metamorphism. The corresponding rock associations lack isotope-geochemical evidence of partial melting that would occur during ultrahigh-pressure metamorphism, which suggesting that they were metamorphosed under granulite-facies conditions. At the same time, the high-alumina diamondiferous rocks of the Barchi area (garnet–kyanite–mica schists and granofelses), which were depleted to different degrees in light rare-earth elements (REE) and K, have yielded a Sm–Nd whole-rock isochron age of 507 ± 10 Ma indicating partial melting of these rocks during their exhumation stage. The close ɛNd (1100) values of the basement rocks and garnet–kyanite–mica schist with geochemical characteristics arguing against its depletion during high-pressure metamorphism indicate that the basement rocks were a crustal source for high-alumina sediments. You do not currently have access to this article

Parental growth media of Siberian diamonds - relation to kimberlites

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Growth medium and carbon source of unusual rounded diamonds from alluvial placers of the North-East of Siberian Platform

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