Exact solution of surface-volume-surface electric field integral equation and analysis of its spectral properties
Bibliographic record
Abstract
Exact solution of the Surface-Volume-Surface Electric Field Integral Equation (SVS-EFIE) is presented for the problem of radiation in the vicinity of homogeneous dielectric sphere. The solution is obtained via Galerkin Method of Moments (MoM) utilizing rotational and irrotational complete sets of orthogonal vector spherical har- monics as basis and test functions according to the Helmholtz decomposition. In this work, two types of formulations derived from single-source integral equation rep- resentation namely, SVS-EFIE-J and SVS-EFIE-M have been analyzed and solved with the MoM discretization. Excitation in the form of radial and tangential electric dipole situated above the spherical scatterer and the electric field throughout the sphere evaluated via ana- lytic 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 suffi- cient 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 op- erators, 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 Sobolev space H1/2(S) for SVS-EFIE-J and L2(S) for SVS-EFIE-M, while perform- div ing the testing in the spaces H−1/2(S) and L2(S), respectively, S being the surface of the sphere, the MoM impedance matrix features bounded condition number simi- lar with analogous exact MoM solution of the surface EFIE on perfectly electrically conducting (PEC) sphere and the solution involving MoM of the surface EFIE on perfectly magnetic conductor (PMC). The selection of the proper functional spaces makes proposed SVS-EFIE formulation free of oversampling breakdown. Moreover, both the spectrum of the continuous integral operators of this new magnetic current based SVS-EFIE-M and the spectrum of its MoM impedance matrix are bounded at low frequencies for simply connected smooth dielectric scatterers.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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".