Clarification of Faber series and related applications to complex variable methods in two-dimensional elasticity
Bibliographic record
Abstract
Faber series are used extensively in the application of complex variable methods to two-dimensional elasticity theory, for example, in the mechanical analysis of composite materials where Faber series representations of complex potentials lead to convenient expressions for the corresponding displacement and stress distributions. In many cases, the use of the Faber series is combined with conformal mapping techniques which "transfer" a boundary value problem defined in the elastic body (physical plane) to a simpler problem posed in an imaginary plane characterized by the conformal mapping. In several instances in the literature, however, little attention has been paid to the domain of definition of the Faber series in the imaginary plane leading often to misunderstandings and erroneous conclusions regarding the concept and feasibility of the use of the Faber series. In this paper, we present a thorough and rigorous examination of the representation of the Faber series in both the physical (occupied by the material) and imaginary (defined by the conformal mapping) plane. In addition, we show that replacing a truncated Faber series by a truncated Taylor series does not induce any additional errors in the numerical analysis of the corresponding boundary value problem. We anticipate that the discussion in this paper will help clarify any existing misinterpretations regarding the application of the Faber series and help further extend their use to a range of problems in composite mechanics.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.010 | 0.024 |
| Meta-epidemiology (narrow) | 0.002 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.009 |
| Scholarly communication | 0.004 | 0.009 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.003 | 0.007 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
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".