Fast Numerical Solution of a Kind of Nonlinear Integral Equations—Dyson-Schwinger Equations for Quark Propagator in Hadron Physics
Bibliographic record
Abstract
Abstract The nonlinear integral equation has been widely studied and has become the heart of the matter in many scientific and engineering fields, such as seismology, optical fiber evolution, radio astronomy, and hadron physics with Quantum Chromodynamics. The Dyson-Schwinger Equations (DSEs) approach provides an essential nonperturbative approach to investigating the properties of hadrons and hot/dense quark matter. Mathematically, the Dyson-Schwinger Equations are a group of coupled nonlinear integral equations of quark propagators, gluon propagators, ghost propagators, and various vertices. On account of the non-linearity and singularity of the coupled equations, it is almost impossible to solve the DSEs analytically. One has to resort to the numerical solution of the equations, in which efficient fast algorithms are key points in practice. In this work, two improvements for numerically solving the nonlinear and singular integral equation for quark propagator in a vacuum are proposed. One is a modified interpolation method for unknown functions in the integral with high degrees of freedom. The other is the parallelization on CPUs with OpenMP in GCC Comparing the CPU times with different algorithms, our results indicate that our proposed methods can greatly improve the efficiency and reduce the computation time of the CPU.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".