Triple

T7871817
Position Surface form Disambiguated ID Type / Status
Subject Jacobi polynomials E182753 entity
Predicate generalizes P2372 FINISHED
Object Chebyshev polynomials of the first kind
Chebyshev polynomials of the first kind are a classical family of orthogonal polynomials on the interval [-1, 1] that play a central role in approximation theory, numerical analysis, and spectral methods.
E697760 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 polynomials of the first kind | Statement: [Jacobi polynomials, generalizes, Chebyshev polynomials of the first kind]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Chebyshev polynomials of the first kind
Context triple: [Jacobi polynomials, generalizes, Chebyshev polynomials of the first kind]
  • A. 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.
  • B. Jacobi polynomials
    Jacobi polynomials are a family of classical orthogonal polynomials depending on two parameters, widely used in approximation theory, numerical analysis, and solutions of differential equations.
  • C. Chebyshev functions
    Chebyshev functions are arithmetic functions in number theory that encode information about the distribution of prime numbers and play a key role in analytic approaches to the prime number theorem.
  • D. Bernoulli polynomials
    Bernoulli polynomials are a sequence of polynomials deeply connected to number theory and analysis, appearing in the study of special functions, series expansions, and the evaluation of sums of powers of integers.
  • E. Hermite
    Hermite is a French surname most famously associated with the 19th-century mathematician Charles Hermite, known for his contributions to number theory, algebra, and analysis.
  • 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 polynomials of the first kind
Triple: [Jacobi polynomials, generalizes, Chebyshev polynomials of the first kind]
Generated description
Chebyshev polynomials of the first kind are a classical family of orthogonal polynomials on the interval [-1, 1] that play a central role in approximation theory, numerical analysis, and spectral methods.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Chebyshev polynomials of the first kind
Target entity description: Chebyshev polynomials of the first kind are a classical family of orthogonal polynomials on the interval [-1, 1] that play a central role in approximation theory, numerical analysis, and spectral methods.
  • A. 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.
  • B. Jacobi polynomials
    Jacobi polynomials are a family of classical orthogonal polynomials depending on two parameters, widely used in approximation theory, numerical analysis, and solutions of differential equations.
  • C. Chebyshev functions
    Chebyshev functions are arithmetic functions in number theory that encode information about the distribution of prime numbers and play a key role in analytic approaches to the prime number theorem.
  • D. Bernoulli polynomials
    Bernoulli polynomials are a sequence of polynomials deeply connected to number theory and analysis, appearing in the study of special functions, series expansions, and the evaluation of sums of powers of integers.
  • E. Hermite
    Hermite is a French surname most famously associated with the 19th-century mathematician Charles Hermite, known for his contributions to number theory, algebra, and analysis.
  • 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_69ca82894d9081908a832bfce71a4714 completed March 30, 2026, 2:02 p.m.
NER Named-entity recognition batch_69cb39a5950481908399211c5dfe2569 completed March 31, 2026, 3:04 a.m.
NED1 Entity disambiguation (via context triple) batch_69cb5b6bc7248190adbf4377c52e16a9 completed March 31, 2026, 5:28 a.m.
NEDg Description generation batch_69cb5f1daac88190a162132bbd40fdc6 completed March 31, 2026, 5:43 a.m.
NED2 Entity disambiguation (via description) batch_69cb768ac1a48190bc6a59a64adf7144 completed March 31, 2026, 7:23 a.m.
Created at: March 30, 2026, 4:56 p.m.