A model of deformations of a discontinuous Stieltjes string with a nonlinear boundary condition
Bibliographic record
Abstract
Variational methods are used to study a model of the deformation of a discontinuous Stieltjes string (a chain of strings held together by springs) located along the segment [0, l].The model is described by the integro-differential equationpuwith derivatives with respect to the measure µ generated by a given strictly increasing function µ(x) on the segment [0, l], where the function u(x) determines the deformation of the string, p(x) characterizes the elasticity of the string, the functions Q(x) and F(x) describe the elastic response of the external environment and the external load, respectively.The integral x 0 ud[Q] is understood in the generalized sense according to Stieltjes.We are looking for solutions u(x) in the class of µ-absolutely continuous functions on [0, l], whose derivatives have bounded variation on [0, l].We assume that one of the boundary conditions is nonlinear and has the form -p(l -0)u µ (l -0)γu(l) ∈ N [-k,k] u(l), where N [-k,k] u(l) denotes the outward normal cone at the point u(l) to the segment [-k, k].This condition arises due to the presence of the limiter [-k, k] on the motion of the elastically fixed right end of the string (by a spring with elasticity γ) so that |u(l)| ≤ k.In this paper, necessary and sufficient conditions for the minimization of the energy functional of the Stieltjes string system are established, the critical loads at which the contact of the end of the string with the boundary points of the limiter occurs are determined, and the dependence of the solution on the length of the limiter is studied.
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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 | 0.001 |
| 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 | 0.003 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.003 | 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".