Evaluation of quantum mechanical perturbative sums in terms of quadratic surds and their use in the approximation of ζ(3)/π<sup>3</sup>
Bibliographic record
Abstract
Abstract The first correction to the energy using Rayleigh–Schrödinger perturbation theory is calculated for the ground state of the one‐dimensional Hubbard model. An algebraic technique is developed which can be used to evaluate these perturbative sums such that the final result is expressed as a finite linear combination of the elements which occur in the individual terms of the sum. The first nontrivial correction to the energy for the ground state of the one‐dimensional Hubbard model in closed form is evaluated. A number theoretic application of this calculation will be given, which is related to the Euclidean construction of regular polygons inside the circle. It is shown that ζ(3) can be approximated by a product of nested radicals and rational constants, multiplied by π 3 , much in the same way that ζ(2) was expressed by Euler as a product of rational numbers and π 2 . It is discussed how this analysis can be continued to higher orders in perturbation theory such that ζ(2 m +1) is obtained as a multiple of π 2 m +1 and nested radicals. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 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)
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 teacher head, 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".