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

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

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

High order harmonics from relativistic electron spikes

A new regime of relativistic high-order harmonic generation is discovered [Phys. Rev. Lett. 108, 135004 (2012)]. Multi-terawatt relativistic-irradiance (>1018 W/cm2) femtosecond (~30-50 fs) lasers focused to underdense (few×1019 cm-3) plasma formed in gas jet targets produce comb-like spectra with hundreds of even and odd harmonic orders reaching the photon energy of 360 eV, including the 'water window' spectral range. Harmonics are generated by either linearly or circularly polarized pulses from the J-KAREN (KPSI, JAEA) and Astra Gemini (CLF, RAL, UK) lasers. The photon number scalability has been demonstrated with a 120 TW laser producing 40 μJ/sr per harmonic at 120 eV. The experimental results are explained using particle-in-cell (PIC) simulations and catastrophe theory. A new mechanism of harmonic generation by sharp, structurally stable, oscillating electron spikes at the joint of boundaries of wake and bow waves excited by a laser pulse is introduced. In this paper detailed descriptions of the experiments, simulations and model are provided and new features are shown, including data obtained with a two-channel spectrograph, harmonic generation by circularly polarized laser pulses and angular distribution

Plume-lithosphere interaction, and the formation of fibrous diamonds

This work was financially supported though a JSPS international research fellowship PE 14721 (to MWB) and JSPS KAKENHI grant numbers JP 26287139 and JP15KK0150 (to HS). The work of DAZ and ALR was supported by Russian science foundation (16-17-10067). RB acknowledges funding from the NERC (NE/M000427/1). SM acknowledges funding from the NERC (NE/PO12167/1).Fluid inclusions in diamond provide otherwise inaccessible information on the origin and nature of carbonaceous fluid(s) in the mantle. Here we evaluate the role of subducted volatiles in diamond formation within the Siberian cratonic lithosphere. Specifically, we focus on the halogen (Cl, Br and I) and noble gas (He, Ne and Ar) geochemistry of fluids trapped within cubic, coated and cloudy fibrous diamonds from the Nyurbinskaya kimberlite, Siberia. Our data show Br/Cl and I/Cl ratios consistent with involvement of altered oceanic crust, suggesting subduction-derived fluids have infiltrated the Siberian lithosphere. 3He/4He ranging from 2 to 11 RA, indicates the addition of a primordial mantle component to the SCLM. Mantle plumes may therefore act as a trigger to re-mobilise subducted carbon-rich fluids from the sub-continental lithospheric mantle, and we argue this may be an essential process in the formation of fluid-rich diamonds, and kimberlitic magmatism.Publisher PDFPeer reviewe

On the Convergence of Kergin and Hakopian Interpolants at Leja Sequences for the Disk

We prove that Kergin interpolation polynomials and Hakopian interpolation polynomials at the points of a Leja sequence for the unit disk $D$ of a sufficiently smooth function $f$ in a neighbourhood of $D$ converge uniformly to $f$ on $D$. Moreover, when $f$ is $C^\infty$ on $D$, all the derivatives of the interpolation polynomials converge uniformly to the corresponding derivatives of $f$

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