Triple
T12138869
| Position | Surface form | Disambiguated ID | Type / Status |
|---|---|---|---|
| Subject | Pafnuty Chebyshev |
E289130
|
entity |
| Predicate | notableWork |
P4
|
FINISHED |
| Object |
Chebyshev rational approximation
Chebyshev rational approximation is a mathematical technique that uses rational functions to approximate other functions with near-optimal uniform accuracy over a given interval.
|
E968221
|
NE FINISHED |
How this triple was built (4 steps)
Every LLM step that produced this triple, in pipeline order — named-entity classification, the disambiguation choices (the exact options shown, with the pick highlighted), and the generated description. The batch + timestamp of each is in the Provenance table below.
NER
Named-entity recognition
gpt-5-mini
Instruction
Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).
Input
Phrase: Chebyshev rational approximation | Statement: [Pafnuty Chebyshev, notableWork, Chebyshev rational approximation]
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Chebyshev rational approximation Context triple: [Pafnuty Chebyshev, notableWork, Chebyshev rational approximation]
-
A.
Carathéodory–Fejér interpolation
Carathéodory–Fejér interpolation is a classical result in complex analysis and approximation theory that concerns constructing analytic functions, typically with bounded or positive real part, that match prescribed initial Taylor coefficients.
-
B.
Birkhoff interpolation
Birkhoff interpolation is a generalized form of polynomial interpolation that allows prescribing function and derivative values at selected points, not necessarily in a consecutive or complete pattern.
-
C.
Bernstein polynomials
Bernstein polynomials are a family of polynomials used in approximation theory that provide a constructive proof of the Weierstrass approximation theorem by uniformly approximating continuous functions on a closed interval.
-
D.
Runge phenomenon
The Runge phenomenon is a numerical analysis effect where high-degree polynomial interpolation, especially at equally spaced points, produces large oscillations and poor approximations near the interval endpoints.
-
E.
Gaussian quadrature rules
Gaussian quadrature rules are numerical integration methods that approximate definite integrals by optimally choosing evaluation points and weights to achieve exactness for polynomials up to a high degree.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NEDg
Description generation
gpt-5.1
Instruction
Generate a one-sentence description of the target entity. You are given a context triple in the form (subject, predicate, object), where the object is the target entity. # Instructions Use the triple to infer relevant information about the entity. Describe the entity based on what is most defining, well-known. Avoid repeating the information from the triple, unless really essential. # Response Format Return only the sentence: "Description: [one-sentence description of the target entity]"
Input
Entity: Chebyshev rational approximation Triple: [Pafnuty Chebyshev, notableWork, Chebyshev rational approximation]
Generated description
Chebyshev rational approximation is a mathematical technique that uses rational functions to approximate other functions with near-optimal uniform accuracy over a given interval.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Chebyshev rational approximation Target entity description: Chebyshev rational approximation is a mathematical technique that uses rational functions to approximate other functions with near-optimal uniform accuracy over a given interval.
-
A.
Carathéodory–Fejér interpolation
Carathéodory–Fejér interpolation is a classical result in complex analysis and approximation theory that concerns constructing analytic functions, typically with bounded or positive real part, that match prescribed initial Taylor coefficients.
-
B.
Birkhoff interpolation
Birkhoff interpolation is a generalized form of polynomial interpolation that allows prescribing function and derivative values at selected points, not necessarily in a consecutive or complete pattern.
-
C.
Bernstein polynomials
Bernstein polynomials are a family of polynomials used in approximation theory that provide a constructive proof of the Weierstrass approximation theorem by uniformly approximating continuous functions on a closed interval.
-
D.
Runge phenomenon
The Runge phenomenon is a numerical analysis effect where high-degree polynomial interpolation, especially at equally spaced points, produces large oscillations and poor approximations near the interval endpoints.
-
E.
Gaussian quadrature rules
Gaussian quadrature rules are numerical integration methods that approximate definite integrals by optimally choosing evaluation points and weights to achieve exactness for polynomials up to a high degree.
- F. None of above. chosen
Provenance (5 batches)
The batch behind each pipeline step, in order, with when it ran. Timestamps are batch-level — stages were processed in waves, so the object chain (NER → NED1 → NEDg → NED2) reads in order, but predicate / elicitation batches can sit in a different wave.
| Step | Stage | Batch ID | Status | When |
|---|---|---|---|---|
| creating | Elicitation | batch_69d6ab4b5e4c81909950b17151eb0951 |
completed | April 8, 2026, 7:23 p.m. |
| NER | Named-entity recognition | batch_69d9158eef48819083bdce283a363414 |
completed | April 10, 2026, 3:21 p.m. |
| NED1 | Entity disambiguation (via context triple) | batch_69f5f690ae408190966bb4fe8feaa7d2 |
completed | May 2, 2026, 1:05 p.m. |
| NEDg | Description generation | batch_69f6022ecf38819080f0eb6a3a815c5b |
completed | May 2, 2026, 1:54 p.m. |
| NED2 | Entity disambiguation (via description) | batch_69f606560934819092ba4d4fa162b799 |
completed | May 2, 2026, 2:12 p.m. |
Created at: April 8, 2026, 9:49 p.m.