Counting Zeros of <i>C</i> <sup>1</sup> Fredholm Maps of Index 1
Bibliographic record
Abstract
The degree defined by P. Benevieri and M. Furi (see ‘A simple notion of orientability for Fredholm maps of index zero between Banach manifolds and degree theory’, Ann. Sci. Math. Québec 22 (1998) 131–148) is used here to obtain some sharp lower bounds for the number of solutions of the λ-sections of the compact components of the set of non-trivial solutions of F(λ, x) = 0, where F is a C1 Fredholm map of index 1 such that F(λ, 0) = 0 for all λ ∈ R. These bounds are given in terms of the parity of the linearized Fredholm family D2F(·, 0). The parity is a local invariant measuring the change of the orientation of D2F(λ, 0) as λ crosses an interval. The set of eigenvalues of D2F(·, 0) is not assumed to be discrete. Therefore, even in the classical situation when F is a compact perturbation of the identity, the results presented here are completely new. 2000 Mathematics Subject Classification 47J15, 58C30 (primary), 47H11, 58C40 (secondary).
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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".