Bibliographic record
Abstract
A set of sufficient conditions which guarantee the existence of a point x ⋆ such that f (x ⋆ ) = x ⋆ is called a "fixed point theorem".Many such theorems are named after well-known mathematicians and economists.Fixed point theorems are among most useful ones in applied mathematics, especially in economics and game theory.Particularly important theorem in these areas is Kakutani's fixed point theorem which ensures existence of fixed point for point-to-set mappings, e.g., [2,3,4].John Nash developed and applied Kakutani's ideas to prove the existence of (what became known as) "Nash equilibrium" for finite games with mixed strategies for any number of players.This work earned him a Nobel Prize in Economics that he shared with two mathematicians.Nash's life was dramatized in the movie "Beautiful Mind" in 2001.In this paper, we approach the system f (x) = x differently.Instead of studying existence of its solutions our objective is to determine conditions which are both necessary and sufficient that an arbitrary point x ⋆ is a fixed point, i.e., that it satisfies f (x ⋆ ) = x ⋆ .The existence of solutions for continuous function f of the single variable is easy to establish using the Intermediate Value Theorem of Calculus.However, characterizing fixed points x ⋆ , i.e., providing answers to the question of finding both necessary and sufficient conditions for an arbitrary given x ⋆ to satisfy f (x ⋆ ) = x ⋆ , is not simple even for functions of the single variable.It is possible that constructive answers do not exist.Our objective is to find them.Our work may require some less familiar tools.One of these might be the "quadratic envelope characterization of zero-derivative point" recalled in the next section.The results are taken from the author's current Research project "Studying the Essence of Fixed Points".They are believed to be original.The author has received several feedbacks on the preliminary report and on parts of the project which can be seen on Internet [9].
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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 | 0.005 | 0.019 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.003 |
| Bibliometrics | 0.005 | 0.002 |
| Science and technology studies | 0.002 | 0.006 |
| Scholarly communication | 0.005 | 0.007 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.004 | 0.004 |
| Insufficient payload (model declined to judge) | 0.011 | 0.002 |
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".