Analytic Solution of Surface–Volume–Surface Electric Field Integral Equation on Dielectric Sphere and Analysis of Its Spectral Properties
Bibliographic record
Abstract
The exact solution of the surface–volume–surface electric field integral equation (SVS-EFIE) is presented for the problem of radiation in the vicinity of the homogeneous dielectric sphere. The solution is obtained via the Galerkin method of moments (MoM) utilizing rotational and irrotational complete sets of orthogonal vector spherical harmonics as basis and test functions according to the Helmholtz decomposition. In the case of radial or tangential electric dipole radiation, the electric field throughout the sphere evaluated via the analytic MoM solution of the SVS-EFIE is compared against the exact classical Mie series solution. The two are shown to agree to 12 digits of accuracy upon a sufficient number of basis/test functions taken in the MoM solution and the Mie series expansion. The exact solution confirms and validates the rigorous nature of the SVS-EFIE formulation. It also reveals the spectral properties of its individual operators, their products, and their linear combination, as well as the spectrum of the MoM impedance matrix. It is shown that, upon choosing basis and test functions in the Sobolev space$H_{div}^{1/2}(S)$and performing testing inner products evaluation in the space$H_{div}^{-1/2}(S)$, with$S$being the surface of the sphere, the MoM impedance matrix features bounded condition number with increasing order of discretization similar with analogous exact MoM solution of the surface EFIE (S-EFIE) on the perfectly electrically conducting (PEC) sphere.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".