Bibliographic record
Abstract
Classical algebraIn this chapter we sketch the basic material -primarily algebra -needed in later chapters.As mentioned in the Introduction, the aspiration of this book isn't to 'Textbookhood'.There are plenty of good textbooks on the material of this chapter (e.g.[162]).What is harder to find are books that describe the ideas beneath and the context behind the various definitions, theorems and proofs.This book, and this chapter, aspire to that.What we lose in depth and detail, we hope to gain in breadth and conceptual content.The range of readers in mind is diverse, from mathematicians expert in other areas to physicists, and the chosen topics, examples and explanations try to reflect this range.Finite groups (Section 1.1) and lattices (Section 1.2.1) appear as elementary examples throughout the book.Lie algebras (Section 1.4), more than their nonlinear partners the Lie groups, are fundamental to us, especially through their representations (Section 1.5).Functional analysis (Section 1.3), category theory (Section 1.6) and algebraic number theory (Section 1.7) play only secondary roles.Section 1.2 provides some background geometry, but for proper treatments consult [113], [104], [527], [59], [478].Note the remarkable unity of algebra.Algebraists look at mathematics and science and see structure; they study form rather than content.The foundations of a new theory are laid by running through a fixed list of questions; only later, as the personality quirks of the new structure become clearer, does the theory become more individual.For instance, among the first questions asked are: What does 'finite' mean here?and What plays the role of a prime number?Mathematics (like any subject) evolves by asking questions, and though a good original question thunders like lightning at night, it is as rare as genius itself.See the beautiful book [504] for more of algebra presented in this style.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.076 | 0.021 |
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".