The construction of Chebyshev approximations in the complex plane

Imperial Users onl

Polynomiography for the Polynomial Infinity Norm via Kalantari's Formula and Nonstandard Iterations

In this paper, an iteration process, referred to in short as MMP, will be considered. This iteration is related to finding the maximum modulus of a complex polynomial over a unit disc on the complex plane creating intriguing images. Kalantari calls these images polynomiographs independently from whether they are generated by the root finding or maximum modulus finding process applied to any polynomial. We show that the images can be easily modified using different MMP methods (pseudo-Newton, MMP-Householder, methods from the MMP-Basic, MMP-Parametric Basic or MMP-Euler-Schroder Families of Iterations) with various kinds of non-standard iterations. Such images are interesting from three points of views: scientific, educational and artistic. We present the results of experiments showing automatically generated non-trivial images obtained for different modifications of root finding MMP-methods. The colouring by iteration reveals the dynamic behaviour of the used root finding process and its speed of convergence. The results of the present paper extend Kalantari's recent results in finding the maximum modulus of a complex polynomial based on Newton's process with the Picard iteration to other MMP-processes with various non-standard iterations

The energy of waves in the photosphere and lower chromosphere: IV. Inversion results of Ca II H spectra

Most static 1D atmosphere models in the quiet Sun predict a rise of the gas temperature at chromospheric layers, but numerical simulations only yield an increase in the brightness temperature. We investigate the thermal structure in the solar chromosphere as derived from an LTE inversion of Ca II H spectra in QS and active regions. We investigate the temperature stratifications on differences between magnetic and field-free regions in the QS, and between QS and ARs. We determine the energy content of individual calcium bright grains (BGs). The rms temperature fluctuations are below 100 K in the photosphere and 200-300 K in the chromosphere. The average temperature stratification in the QS does not exhibit a clear chromospheric temperature rise, opposite to the AR case. We find an energy content of about 7*10E18 J for BGs that repeat with a cadence of about 160 secs. The precursors of BGs have a vertical extent of about 200 km and a horizontal extent of about 1 Mm. The comparison of observed with synthetic NLTE profiles confirms that the solar chromosphere in the QS oscillates between an atmosphere in radiative equilibrium and one with a moderate chromospheric temperature rise. Two-dimensional x-z temperature maps exhibit nearly horizontal canopy-like structures with a few Mm extent around photospheric magnetic field concentrations at a height of about 600 km. The large difference between QS regions and ARs, and the better match of AR and non-LTE reference spectra suggest that magnetic heating processes are more important than commonly assumed. The temperature fluctuations in QS derived by the LTE inversion do not suffice on average to maintain a stationary chromospheric temperature rise. The spatially and vertically resolved information on the temperature structure allows one to investigate in detail the topology and evolution of the thermal structure in the lower solar atmosphere.Comment: 16 pages, 16 figures + 1 page Appendix, accepted by A&

High dynamic range imaging with a single-mode pupil remapping system : a self-calibration algorithm for redundant interferometric arrays

The correction of the influence of phase corrugation in the pupil plane is a fundamental issue in achieving high dynamic range imaging. In this paper, we investigate an instrumental setup which consists in applying interferometric techniques on a single telescope, by filtering and dividing the pupil with an array of single-mode fibers. We developed a new algorithm, which makes use of the fact that we have a redundant interferometric array, to completely disentangle the astronomical object from the atmospheric perturbations (phase and scintillation). This self-calibrating algorithm can also be applied to any - diluted or not - redundant interferometric setup. On an 8 meter telescope observing at a wavelength of 630 nm, our simulations show that a single mode pupil remapping system could achieve, at a few resolution elements from the central star, a raw dynamic range up to 10^6; depending on the brightness of the source. The self calibration algorithm proved to be very efficient, allowing image reconstruction of faint sources (mag = 15) even though the signal-to-noise ratio of individual spatial frequencies are of the order of 0.1. We finally note that the instrument could be more sensitive by combining this setup with an adaptive optics system. The dynamic range would however be limited by the noise of the small, high frequency, displacements of the deformable mirror.Comment: 11 pages, 7 figures. Accepted for publication in MNRA

Parameterized Uniform Complexity in Numerics: from Smooth to Analytic, from NP-hard to Polytime

The synthesis of classical Computational Complexity Theory with Recursive Analysis provides a quantitative foundation to reliable numerics. Here the operators of maximization, integration, and solving ordinary differential equations are known to map (even high-order differentiable) polynomial-time computable functions to instances which are `hard' for classical complexity classes NP, #P, and CH; but, restricted to analytic functions, map polynomial-time computable ones to polynomial-time computable ones -- non-uniformly! We investigate the uniform parameterized complexity of the above operators in the setting of Weihrauch's TTE and its second-order extension due to Kawamura&Cook (2010). That is, we explore which (both continuous and discrete, first and second order) information and parameters on some given f is sufficient to obtain similar data on Max(f) and int(f); and within what running time, in terms of these parameters and the guaranteed output precision 2^(-n). It turns out that Gevrey's hierarchy of functions climbing from analytic to smooth corresponds to the computational complexity of maximization growing from polytime to NP-hard. Proof techniques involve mainly the Theory of (discrete) Computation, Hard Analysis, and Information-Based Complexity

Distributions of flux vacua

We give results for the distribution and number of flux vacua of various types, supersymmetric and nonsupersymmetric, in IIb string theory compactified on Calabi-Yau manifolds. We compare this with related problems such as counting attractor points.Comment: 43 pages, 7 figures. v2: improved discussion of finding vacua with discrete flux, references adde