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Record W1531695429 · doi:10.1007/978-1-4612-0211-0_6

The Numerical Approach

2004· book-chapter· en· W1531695429 on OpenAlex

Why this work is in the frame

A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.

affAt least one author lists a Canadian institution in the pinned OpenAlex snapshot.

Bibliographic record

VenueBirkhäuser Boston eBooks · 2004
Typebook-chapter
Languageen
FieldMathematics
TopicNumerical methods for differential equations
Canadian institutionsUniversity of the Fraser ValleySimon Fraser University
Fundersnot available
KeywordsOdeNonlinear systemSolverOrdinary differential equationComputer sciencePartial differential equationSimple (philosophy)Numerical analysisApplied mathematicsFinite differenceCalculus (dental)Differential equationMathematicsMathematical analysisPhysicsProgramming language

Abstract

fetched live from OpenAlex

The combination of finite-difference approximations to the derivatives and the use of a high speed digital computer leads to a very powerful approach to solving the nonlinear ordinary and partial differential equations of physics. For many nonlinear systems, particularly those where the nonlinear terms are not small corrections to an otherwise linear behavior, the numerical route may be the best or only feasible way to travel. For the nonlinear ODEs encountered earlier in the text, the student has been allowed to use the Mathematica numerical ODE solver without any explanation provided of the principles on which it is based. In this chapter, we would like to partially fill that void by briefly describing how some of the common numerical schemes for solving nonlinear ODEs are derived. Our aim is to provide a simple conceptual framework that will make the reader more comfortable with the numerical approach while progressing through the rest of the topics that lie ahead. It should be emphasized that we are not attempting to explain the code which underlies Mathematica’s NDSolve command which is about 500 pages long.

Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.

Full frame distilled prediction

Teacher imitation

Not 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.

metaresearch head score (Codex)0.001
metaresearch head score (Gemma)0.000
Version: codex-gemma-dda1882f352aValidation status: machine_predicted_unvalidated
Candidate categoriesMeta-epidemiology (narrow)
Consensus categoriesnone
DomainCandidate signal: none · Consensus signal: none
Study designCandidate signal: Theoretical or conceptual · Consensus signal: Theoretical or conceptual
GenreCandidate signal: Other · Consensus signal: Other
Teacher disagreement score0.222
Threshold uncertainty score1.000

Codex and Gemma teacher scores by category

CategoryCodexGemma
Metaresearch0.0010.000
Meta-epidemiology (narrow)0.0010.000
Meta-epidemiology (broad)0.0010.001
Bibliometrics0.0000.000
Science and technology studies0.0010.000
Scholarly communication0.0000.000
Open science0.0010.000
Research integrity0.0010.001
Insufficient payload (model declined to judge)0.0000.000

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.

Opus teacher head0.101
GPT teacher head0.330
Teacher spread0.229 · how far apart the two teachers sit on this one work
Validation statusscore_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it