Triple

T18480276
Position Surface form Disambiguated ID Type / Status
Subject Orthogonal Polynomials E451537 entity
Predicate associatedConcept P531 FINISHED
Object Favard's theorem — NE NERFINISHED

Named-entity recognition

Before disambiguation, gpt-5-mini classified whether the object phrase is a named entity — the step behind the object's NE type shown above.

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: Favard's theorem | Statement: [Orthogonal Polynomials, associatedConcept, Favard's theorem]

Disambiguation candidates (2 decisions)

The exact options the model was shown at each disambiguation step, with the option it chose highlighted — the evidence behind this triple's disambiguated ids.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07 
Target entity: Favard's theorem
Context triple: [Orthogonal Polynomials, associatedConcept, Favard's theorem]
  • 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. Chebyshev alternation theorem
    The Chebyshev alternation theorem is a fundamental result in approximation theory that characterizes the best uniform (minimax) polynomial approximation to a continuous function by the presence of alternating maximum errors at a finite set of points.
  • C. 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.
  • D. Bernstein inequalities
    Bernstein inequalities are fundamental results in approximation theory and probability that provide bounds on the derivatives or deviations of functions and random variables under certain smoothness or moment conditions.
  • E. Weierstrass approximation theorem
    The Weierstrass approximation theorem is a fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated by polynomials.
  • F. None of above. chosen
  • G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07 
Target entity: Favard's theorem
Target entity description: Favard's theorem is a fundamental result in the theory of orthogonal polynomials that characterizes such polynomial sequences precisely as those satisfying a three-term recurrence relation with appropriate coefficients.
  • 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. Chebyshev alternation theorem
    The Chebyshev alternation theorem is a fundamental result in approximation theory that characterizes the best uniform (minimax) polynomial approximation to a continuous function by the presence of alternating maximum errors at a finite set of points.
  • C. 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.
  • D. Bernstein inequalities
    Bernstein inequalities are fundamental results in approximation theory and probability that provide bounds on the derivatives or deviations of functions and random variables under certain smoothness or moment conditions.
  • E. Weierstrass approximation theorem
    The Weierstrass approximation theorem is a fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated by polynomials.
  • F. None of above. chosen

Provenance (2 batches)

Stage Batch ID Job type Status
creating batch_69d8d38465a0819099b9b42d2a662ac1 elicitation completed
NER batch_69e53066a7108190a50eda9b489c90ca ner completed
Created at: April 10, 2026, 11:35 a.m.