Mathematica program for extracting one-turn Lie generator map. application of TPSA
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Abstract
Abstract The Lie Algebra package LieMath, written in the Mathematica language, constructs the one-turn nonlinear map for a given lattice of optical elements. The method used is a BCH-based map concatenation. Truncated power series algebra (TPSA) techniques have been implemented to compute the Poisson bracket and extract the map faster than when one relies on the symbolic capabilities of Mathematica to operate on truncated multivariate Taylor series. In addition, this makes possible obtaining parameter-dependent numerical maps and optimization of nonlinear parameters. © 2008 Elsevier B.V. All rights reserved. PACS: 41.85.-p; 02.70.Wz Keywords: Beam optics, Lie algebra, Nonlinear maps 1. Introduction The Lie algebraic method allows us to create the nonlinear map of a beamline, either in Lie operator form, or as Taylor expansions of final coordinates in terms of initial coordinates. In a map concaten ation based on the Baker- Cambell-Hausdorff (BCH) expansion, the mo st computationally intensive part is calculation of the Poisson brackets (PB). Therefore, differential algebra libraries are needed to carry out fast computations w ith polynomials and vector functions of polynomials. To build the map, a symbolic computational system may be used such as Mathematica [1], [2], [3]. The most straightforward approach involves encoding the PB as an operator on multivariate polynomials produced by the truncated series expansion of the (piecewise constant) Hamiltonian. One then relies on the symbolic engine to perform the product and derivative needed for the PB. Using a symbolic system gives additional flexibility: easy switching between nume rical and analytical calculations and also getting analytical dependence on parameters [1],[5]. The main disadv antage is speed, which is slow for large lattices. In automatic differentiation, or Truncated power series algebra (TPSA) [6 8], the array of Taylor coefficients in the expansion of a multivariate function is computed not by symbolic differentiation, or by numerical
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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