Self-Similar Factor Approximants

The problem of reconstructing functions from their asymptotic expansions in powers of a small variable is addressed by deriving a novel type of approximants. The derivation is based on the self-similar approximation theory, which presents the passage from one approximant to another as the motion realized by a dynamical system with the property of group self-similarity. The derived approximants, because of their form, are named the self-similar factor approximants. These complement the obtained earlier self-similar exponential approximants and self-similar root approximants. The specific feature of the self-similar factor approximants is that their control functions, providing convergence of the computational algorithm, are completely defined from the accuracy-through-order conditions. These approximants contain the Pade approximants as a particular case, and in some limit they can be reduced to the self-similar exponential approximants previously introduced by two of us. It is proved that the self-similar factor approximants are able to reproduce exactly a wide class of functions which include a variety of transcendental functions. For other functions, not pertaining to this exactly reproducible class, the factor approximants provide very accurate approximations, whose accuracy surpasses significantly that of the most accurate Pade approximants. This is illustrated by a number of examples showing the generality and accuracy of the factor approximants even when conventional techniques meet serious difficulties.Comment: 22 pages + 11 ps figure

(Borel) convergence of the variationally improved mass expansion and the O(N) Gross-Neveu model mass gap

We reconsider in some detail a construction allowing (Borel) convergence of an alternative perturbative expansion, for specific physical quantities of asymptotically free models. The usual perturbative expansions (with an explicit mass dependence) are transmuted into expansions in 1/F, where $F \sim 1/g(m)$ for $m \gg \Lambda$ while $F \sim (m/\Lambda)^\alpha$ for m \lsim \Lambda, $\Lambda$ being the basic scale and $\alpha$ given by renormalization group coefficients. (Borel) convergence holds in a range of $F$ which corresponds to reach unambiguously the strong coupling infrared regime near $m\to 0$, which can define certain "non-perturbative" quantities, such as the mass gap, from a resummation of this alternative expansion. Convergence properties can be further improved, when combined with $\delta$ expansion (variationally improved perturbation) methods. We illustrate these results by re-evaluating, from purely perturbative informations, the O(N) Gross-Neveu model mass gap, known for arbitrary $N$ from exact S matrix results. Comparing different levels of approximations that can be defined within our framework, we find reasonable agreement with the exact result.Comment: 33 pp., RevTeX4, 6 eps figures. Minor typos, notation and wording corrections, 2 references added. To appear in Phys. Rev.

Applications of ICP magnetic sector multicollector mass spectrometry to basic energy research. Final report for period December 1st, 1993 - May 31st, 2000

The primary aims of this research were threefold: to develop and utilize the new technique of multiple collector inductively coupled plasma mass spectrometry and apply it to problems in the earth, ocean, and environmental sciences; to develop new chronometers and improve existing chronometers to allow the accurate determination of the ages of geological features and processes; and to study natural fluid-mediated mass transfer processes and source of components in the crust and the oceans. This technique has now become the preferred method for the determination of the isotopic compositions of a variety of elements in the periodic table. The prototype instrument was used to explore a vast array of isotopic systems and demonstrate applicability to problems as different as the origin of the solar system and smelting methods in the Bronze Age. Highlights of the program are briefly summarized under the following topics: tungsten isotopes and the early solar system; trace siderophile and chalcophile element geochemistry; hafnium isotopes and the early development of the continents; evolution of lead isotopic compositions of the oceans; the isotopic composition and residence time of Hf in seawater; the isotopic compositions of Sr, Hf, Pb, and Nd in dust; U-Th disequilibrium dating of carbonates and soils; in situ U-Th disequilibrium dating of opal