Bibliographic record
Abstract
Experimental computation with oscillatory integrals David H. Bailey ∗ Jonathan M. Borwein † June 26, 2009 Abstract A previous study by one of the present authors, together with D. Borwein and I. Leonard [8], studied the asymptotic behavior of the p-norm of the sinc function: sinc(x) = (sin x)/x and along the way looked at closed forms for integer values of p. In this study we address these integrals with the tools of experimental mathematics, namely by computing their numerical values to high precision, both as a challenge in itself, and also in an attempt to recognize the numerical values as closed-form constants. With this approach, we are able to reproduce several of the results of [8] and to find new results, both numeric and analytic, that go beyond the previous study. Introduction A previous work by one of the present authors, together with D. Borwein and I. Leonard [8], studied the behavior of the p-norm of the sinc function: sinc(x) = (sin x)/x. In particular, these authors considered the function I(p) defined by: I(p) p sin t t p dt Plots of I(p) over (0, 10) and (0, 100) are shown in Figures 1 and 2. In this study we wish to further explore this function, both numerically and analytically. Indeed, in [8] one finds proofs of the following composite result. Theorem 1 For all p > 1 one has I(p) > Moreover p→∞ 3π 2p 2 2p + 1 p 2p lim I(p) = and there are real constants c s such that I(p) p sin(x) x dx c s 2 p s=2 p s ∗ Lawrence Berkeley National Laboratory, Berkeley, CA 94720, dhbailey@lbl.gov. Supported in part by the Director, Office of Computational and Technology Research, Division of Mathematical, Information, and Computational Sciences of the U.S. Department of Energy, under contract number DE-AC02-05CH11231. † School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia jonathan.borwein@newcastle.edu.au and Faculty of Computer Science, Dalhousie University, Halifax, NS, B3H 2W5, Canada, jborwein@cs.dal.ca. Supported in part by ARC, NSERC and the Canada Research Chair Programme.
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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 | 0.000 | 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) | 0.001 | 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".