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 $f$

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

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

The Density of Translates of Zonal Kernels on Compact Homogeneous Spaces

Compact manifolds embedded in Euclidean space which have a transitive group of linear isometries, like the spheres or the "flat" tori under rotations, admit a natural notion of a continuous self-adjoint zonal kernel function k(x; y), which generalizes the idea of a radial or distance dependent function on the spheres and tori. In connection with a study of quasi-interpolation on these spaces, we have reproved Research conducted at Leicester University with partial support from their Department of Mathematics and Computer Science, from the University of Washington and from an EPSRC Visiting Fellowship under grant GR/K79710 and extended results of Sun for the spheres, to characterize those kernels for which the span of the translates, P a n k(x; y n ), is dense in the continuous functions. The essence of the characterization is that the integral operator with kernel k(x; y) must be non-singular when restricted to any finite dimensional space of polynomial functions which is invari..

Fast evaluation of radial basis functions : methods for four-dimensional polyharmonic splines

As is now well known for some basic functions ϕ, hierarchical and fast multipole like methods can greatly reduce the storage and operation counts for fitting and evaluating radial basis functions. In particular for spline functions of the form [FORM] p a low degree polynomial and certain choices of ⏀, the cost of a single extra evaluation can be reduced from O(N) to O(log N), or even O(1), operations and the cost of a matrix-vector product (i.e., evaluation at all centres) can be decreased from O(N²) to O(N log N), or even O(N), operations. This paper develops the mathematics required by methods of these types for polyharmonic splines in R⁴. That is for splines s built from a basic function from the list ⏀(r) = r⁻² or ⏀(r) = r²n ln(r), n = 0, 1, .... We give appropriate far and near field expansions, together with corresponding error estimates, uniqueness theorems, and translation formulae. A significant new feature of the current work is the use of arguments based on the action of the group of non-zero quaternions, realised as 2 x 2 complex matrices [MATRICES] acting on C² = R⁴. Use of this perspective allows us to give a relatively efficient development of the relevant spherical harmonics and their properties

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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