10 research outputs found
Askey-Wilson Type Functions, With Bound States
The two linearly independent solutions of the three-term recurrence relation
of the associated Askey-Wilson polynomials, found by Ismail and Rahman in [22],
are slightly modified so as to make it transparent that these functions satisfy
a beautiful symmetry property. It essentially means that the geometric and the
spectral parameters are interchangeable in these functions. We call the
resulting functions the Askey-Wilson functions. Then, we show that by adding
bound states (with arbitrary weights) at specific points outside of the
continuous spectrum of some instances of the Askey-Wilson difference operator,
we can generate functions that satisfy a doubly infinite three-term recursion
relation and are also eigenfunctions of -difference operators of arbitrary
orders. Our result provides a discrete analogue of the solutions of the purely
differential version of the bispectral problem that were discovered in the
pioneering work [8] of Duistermaat and Gr\"unbaum.Comment: 42 pages, Section 3 moved to the end, minor correction
A ‘VISUAL’ APPROACH TO MENANDER - (A.K.) Petrides Menander, New Comedy and the Visual. Pp. xii + 322, ills. Cambridge: Cambridge University Press, 2014. Cased, £65, US$99. ISBN: 978-1-107-06843-8.
High-energy X-rays and gamma-rays from solar flares were discovered just over
fifty years ago. Since that time, the standard for the interpretation of
spatially integrated flare X-ray spectra at energies above several tens of keV
has been the collisional thick-target model. After the launch of the Reuven
Ramaty High Energy Solar Spectroscopic Imager (RHESSI) in early 2002, X-ray
spectra and images have been of sufficient quality to allow a greater focus on
the energetic electrons responsible for the X-ray emission, including their
origin and their interactions with the flare plasma and magnetic field. The
result has been new insights into the flaring process, as well as more
quantitative models for both electron acceleration and propagation, and for the
flare environment with which the electrons interact. In this article we review
our current understanding of electron acceleration, energy loss, and
propagation in flares. Implications of these new results for the collisional
thick-target model, for general flare models, and for future flare studies are
discussed.Comment: This is an article from a monograph on the physics of solar flares,
inspired by RHESSI observations. The individual articles are to appear in
Space Science Reviews (2011