CHAPTER SIX: MANDELBROT POLYNOMIALS AND MATRICES
Bibliographic record
Abstract
6.1 A note to the student/readerThis chapter uses some ideas of the late Benoit Mandelbrot186 and some known facts about the Mandelbrot set, together with some of our own ideas, to help you to learn the following:1.A bit more Python programming, including more practice with the use of numerical libraries from NumPy, iteration and recursion; some new things, such as loop invariants, automatic differentiation, and symbolic computation; and a bit more about graphics.2.A bit more about polynomials: the cubic formula (we won't need the quartic formula, and even our use of the cubic formula is a bit contrived, but we think it's fun), the cost of numerical solution of polynomials (we will point you to the current numerical champion polynomial solver, MPSolve187), and the rather necessary to know notions of numerical stability and conditioning. Most of these topics are not taught as thoroughly as they should be in a first numerical analysis course, so this material should strengthen the results when you do encounter them.3.The surprising use of eigenvalues to find roots of polynomials. The first treatment of eigenvalues (typically in a linear algebra course, even though the problem is not, strictly speaking, linear) usually runs the other way around and defines the eigenvalues of a matrix in terms of the characteristic polynomial of the matrix. Indeed, we will teach you a concept not in the textbooks, namely the concept of a minimal-height companion matrix, and we will show you one such for the Mandelbrot polynomials.4.A bit more about dynamical systems, especially about iteration and composition.5.An excellent approximate formula for the largest-magnitude root of the Mandelbrot polynomial.6.An honest-to-goodness analytic solution to the Mandelbrot iteration (this is a very new result, published only in 2021), valid for all c outside the Mandelbrot set.7.That we don't know everything about Mandelbrot polynomials and matrices, and that you might be able to answer some open questions.186https://en.wikipedia.org/wiki/Benoit_Mandelbrot187https://en.wikipedia.org/wiki/MPSolve
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.004 | 0.004 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.005 |
| Insufficient payload (model declined to judge) | 0.054 | 0.022 |
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".