Some qualitative properties of solutions of higher-order lower semicontinus differential inclusions
Bibliographic record
Abstract
Let n, k ∈ N, T > 0, and F : [0, T ] × (R n ) k → 2 R n be a lower semicontinuos and bounded multifunction with nonempty closed values.We prove that there exists a bounded and upper semicontinuous multifunction G : R × (R n ) k → 2 R n with nonempty compact convex values such that every generalized solution u :) is a generalized solution to the differential inclusion u (k) ∈ F(t, u, u , . . ., u (k-1) ).As an application, we prove an existence and qualitative result for the generalized solutions of the Cauchy problem associated to the inclusion u (k) ∈ F(t, u, u , . . ., u (k-1) ).In particular, we prove that if F is lower semicontinuous and bounded with nonempty closed values, then the solution multifunction admits an upper semicontinuous multivalued selection with nonempty compact connected values.Finally, by applying the latter result, we prove an analogous existence and qualitative result for the generalized solutions of the Cauchy problem associated to the differential equation g(u (k) ) = f (t, u, u , . . ., u (k-1) ), where f is continuous.We only assume that g is continuous and locally nonconstant.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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 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".