Bibliographic record
Abstract
Most of the physical processes arising in nature are modeled by differential equations, either ordinary (example: the spring/mass/damper system) or partial (example: heat diffusion). From the point of view of analog computability, the existence of an effective way to obtain solutions (either exact or approximate) of these systems is essential. A pioneering model of analog computation is the General Purpose Analog Computer (GPAC), introduced by Shannon [ Journal of Mathematical Physics 20 ( 1941 ), 337–354] as a model of the Differential Analyzer and improved by Pour-El [ Transactions of the American Mathematical Society 199 ( 1974 ), 1–28], Lipshitz and Rubel [ Proceedings of the American Mathematical Society 99(2) ( 1987 )], Costa and Graça [ Journal of Complexity 19(5) ( 2003 ), 644–664] and others. The GPAC is capable of manipulating real-valued data streams. Its power is known to be characterized by the class of differentially algebraic functions, which includes the solutions of initial value problems for ordinary differential equations. We address one of the limitations of this model, which is its fundamental inability to reason about functions of more than one independent variable (the ‘time’ variable), as noted by Rubel [ Advances in Applied Mathematics 14(1) ( 1993 ), 39–50]. In particular, the Shannon GPAC cannot be used to specify solutions of partial differential equations. We extend the class of data types using networks with channels which carry information on a general complete metric space X; here we take [Formula: see text], the class of continuous functions of one real (spatial) variable. We consider the original modules in Shannon’s construction (constants, adders, multipliers, integrators) and we add a differential module which has one input and one output. For input u, it outputs the spatial derivative [Formula: see text]. We then define an X-GPAC to be a network built with X-stream channels and the above-mentioned modules. This leads us to a framework in which the specifications of such analog systems are given by fixed points of certain operators on continuous data streams. Such a framework was considered by Tucker and Zucker [ Theoretical Computer Science 371 ( 2007 ), 115–146]. We study the properties of these analog systems and their associated operators, and present a characterization of the X-GPAC-generable functions which generalizes Shannon’s results.
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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 | 0.001 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.009 | 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".