Generalisation of the Dirac-delta impulse extending Laplace and z transform domains
Bibliographic record
Abstract
A generalisation of the Dirac-delta impulse and its derivatives as two generalised distributions, namely, the xi and zeta impulses, and their derivatives, defined on the complex s-plane and z-plane of continuous-time and discrete-time functions, respectively, is proposed. The generalised impulses extend the existence of Laplace and z transforms to a large class of infinite duration two-sided functions, which hitherto had no transform or had only a Fourier transform in the form of distributions. The proposed generalised impulses are shown to bridge the gap between the theory of generalised functions and both the unilateral and bilateral Laplace and z transforms. The generalised impulses extend the existence of Laplace and z transforms to include both functions that have a Fourier transform as a distribution as well as exponentially rising infinite duration two-sided functions that have no Fourier transform. It is shown that a modulation theory can now be added to the properties of bilateral transforms. No such theorem has hitherto existed for these transforms. The proposed generalised impulses and the resulting extended Laplace and z transforms are shown to lead to new complex-plane operations, such as spatial convolution, and to simplify operations such as ordinary convolution, sampling and the solution of differential and difference equations. Bilateral Laplace and z transforms may receive greater attention now that these transforms can be applied to a new, large and basic class of functions, such as two-sided infinite duration exponentials and rising trigonometric and hyperbolic functions.
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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 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".