Bibliographic record
Abstract
The following is a summary description of some research results that were taught to students over many years and communicated by the author in IEEE Proceedings submission, in the book proposals, and poster sessions. A generalized transform, grouping Fourier and Laplace transform in the continuous-time domain and Fourier and z transform in the discrete-time domain is proposed. A generalized formalism is obtained for the continuous time domain and a similar one for the discrete time domain. The key to the existence of a generalized formalism and transform is based on a generalization of the formalism of generalized functions, and the Dirac-delta impulse in particular. A generalization of the Dirac-delta impulse is proposed, both in the continuous time domain and in the discrete time domain. The generalization of the impulse leads to a considerable extension of the domain of existence of Laplace and z transforms. Functions and sequences that have impulsive Fourier transforms and hitherto no Laplace or z transform are rendered within the region of convergence of these transforms. More importantly, a large class of functions and sequences that have no Fourier transform, such as two-sided infinite duration growing exponentials and exponentially diverging sinusoids, now have Laplace and z transforms. Two generalized impulses, namely, the Xi and Zeta impulses, defined on the complex transform plane are proposed. These generalized impulses may provide the missing link that bridges the gap between the theory of generalized functions as it applies to the Fourier transform and the more general Laplace and z transforms.
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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 | 0.002 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.003 | 0.002 |
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".