Chunky and Equal-Spaced Polynomial Multiplication

Finding the product of two polynomials is an essential and basic problem in computer algebra. While most previous results have focused on the worst-case complexity, we instead employ the technique of adaptive analysis to give an improvement in many "easy" cases. We present two adaptive measures and methods for polynomial multiplication, and also show how to effectively combine them to gain both advantages. One useful feature of these algorithms is that they essentially provide a gradient between existing "sparse" and "dense" methods. We prove that these approaches provide significant improvements in many cases but in the worst case are still comparable to the fastest existing algorithms.Comment: 23 Pages, pdflatex, accepted to Journal of Symbolic Computation (JSC

Fermi surfaces and anomalous transport in quasicrystals

Fermi surfaces of several quasicrystalline approximants are calculated by means of ab-initio methods which enable direct comparison with dHvA experiments. A criterion for anomalous metallic transport is proposed and power-law temperature dependence of electronic conductivity is deduced from scaling analysis of the Kubo formula.Comment: 8 pages, 7 figures. to appear in Phys. Rev.

Multivariate sparse interpolation using randomized Kronecker substitutions

We present new techniques for reducing a multivariate sparse polynomial to a univariate polynomial. The reduction works similarly to the classical and widely-used Kronecker substitution, except that we choose the degrees randomly based on the number of nonzero terms in the multivariate polynomial, that is, its sparsity. The resulting univariate polynomial often has a significantly lower degree than the Kronecker substitution polynomial, at the expense of a small number of term collisions. As an application, we give a new algorithm for multivariate interpolation which uses these new techniques along with any existing univariate interpolation algorithm.Comment: 21 pages, 2 tables, 1 procedure. Accepted to ISSAC 201

The Galactic Disk Distribution of Planetary Nebulae with Warm Dust Emission Features: II

Can the distribution of warm-dust compositions in IR-bright galactic disk PNe be linked to the underlying stellar population? The PNe with warm dust emission represent a homogeneous population, which is presumably young and minimally affected by a possible dependence of PN lifetime on progenitor mass. The sample in paper I thus allows testing the predictions of single star evolution, through a comparison with synthetic distributions and under the assumption that tip-of-the-AGB and PN statistics are similar. We construct a schematic model for AGB evolution (adapted from Groenewegen & de Jong 1993), whose free-parameters are calibrated with the luminosity function (LF) of C stars in the LMC, the initial-final mass relation, and the range of PN compositions. The observed metallicity gradient and distribution of star forming regions with galactocentric radius (Bronfman et al. 2000) allow us to synthesise the galactic disk PN progenitor population. We find the fraction of O-rich PNe, f(O), is a tight constraint on AGB parameters. For our best model, a minimum PN progenitor mass Mmin=1Msun predicts that about 50% of all young PNe should be O-rich, compared to an observed fraction of 22%; thus Mmin=1.2Msun, at a 2sigma confidence level. By contrast, current AGB models for single stars can account neither for the continuous range of N enrichment (Leisy & Dennefeld 1996), nor for the observation that the majority of very C-rich PNe have Peimbert type I (paper I). f(O) is thus an observable much easier to model. The decrease in f(O) with galactocentric radius, as reported in paper I, is a strong property of the synthetic distribution, independent of Mmin. This trend reflects the sensitivity of the surface temperature of AGB stars and of the core mass at the first thermal pulse to the galactic metallicity gradient.Comment: accepted by MNRA

Parallel sparse interpolation using small primes

To interpolate a supersparse polynomial with integer coefficients, two alternative approaches are the Prony-based "big prime" technique, which acts over a single large finite field, or the more recently-proposed "small primes" technique, which reduces the unknown sparse polynomial to many low-degree dense polynomials. While the latter technique has not yet reached the same theoretical efficiency as Prony-based methods, it has an obvious potential for parallelization. We present a heuristic "small primes" interpolation algorithm and report on a low-level C implementation using FLINT and MPI.Comment: Accepted to PASCO 201

Integrating factors for second order ODEs

A systematic algorithm for building integrating factors of the form mu(x,y), mu(x,y') or mu(y,y') for second order ODEs is presented. The algorithm can determine the existence and explicit form of the integrating factors themselves without solving any differential equations, except for a linear ODE in one subcase of the mu(x,y) problem. Examples of ODEs not having point symmetries are shown to be solvable using this algorithm. The scheme was implemented in Maple, in the framework of the "ODEtools" package and its ODE-solver. A comparison between this implementation and other computer algebra ODE-solvers in tackling non-linear examples from Kamke's book is shown.Comment: 21 pages - original version submitted Nov/1997. Related Maple programs for finding integrating factors together with the ODEtools package (versions for MapleV R4 and MapleV R5) are available at http://lie.uwaterloo.ca/odetools.ht