Triple

T18480387
Position Surface form Disambiguated ID Type / Status
Subject Szegő kernel E451540 entity
Predicate usedIn P98 FINISHED
Object Toeplitz operator theory
Toeplitz operator theory is a branch of functional analysis that studies linear operators with constant diagonals, connecting operator theory, complex analysis, and harmonic analysis.
E451536 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: Toeplitz operator theory | Statement: [Szegő kernel, usedIn, Toeplitz operator theory]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Toeplitz operator theory
Context triple: [Szegő kernel, usedIn, Toeplitz operator theory]
  • A. Hilbert–Schmidt operators
    Hilbert–Schmidt operators are a class of compact operators on Hilbert spaces characterized by having finite Hilbert–Schmidt norm, playing a central role in functional analysis and operator theory.
  • B. Toeplitz matrices
    Toeplitz matrices are structured matrices whose entries are constant along each diagonal, playing a central role in operator theory, numerical analysis, and signal processing.
  • C. Szegő kernel
    The Szegő kernel is a fundamental reproducing kernel in complex analysis and operator theory, associated with Hardy spaces on the boundary of a domain and central to the study of orthogonal polynomials and boundary behavior of analytic functions.
  • D. Calderón–Zygmund theory
    Calderón–Zygmund theory is a branch of harmonic analysis that studies singular integral operators and their boundedness properties on function spaces such as L^p.
  • E. Blaschke products
    Blaschke products are bounded analytic functions on the unit disk formed as (finite or infinite) products of Möbius transformations that map the disk to itself, playing a central role in complex analysis and function theory.
  • 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: Toeplitz operator theory
Triple: [Szegő kernel, usedIn, Toeplitz operator theory]
Generated description
Toeplitz operator theory is a branch of functional analysis that studies linear operators with constant diagonals, connecting operator theory, complex analysis, and harmonic analysis.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Toeplitz operator theory
Target entity description: Toeplitz operator theory is a branch of functional analysis that studies linear operators with constant diagonals, connecting operator theory, complex analysis, and harmonic analysis.
  • A. Hilbert–Schmidt operators
    Hilbert–Schmidt operators are a class of compact operators on Hilbert spaces characterized by having finite Hilbert–Schmidt norm, playing a central role in functional analysis and operator theory.
  • B. Toeplitz matrices chosen
    Toeplitz matrices are structured matrices whose entries are constant along each diagonal, playing a central role in operator theory, numerical analysis, and signal processing.
  • C. Szegő kernel
    The Szegő kernel is a fundamental reproducing kernel in complex analysis and operator theory, associated with Hardy spaces on the boundary of a domain and central to the study of orthogonal polynomials and boundary behavior of analytic functions.
  • D. Calderón–Zygmund theory
    Calderón–Zygmund theory is a branch of harmonic analysis that studies singular integral operators and their boundedness properties on function spaces such as L^p.
  • E. Blaschke products
    Blaschke products are bounded analytic functions on the unit disk formed as (finite or infinite) products of Möbius transformations that map the disk to itself, playing a central role in complex analysis and function theory.
  • F. None of above.

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_69d8d38465a0819099b9b42d2a662ac1 completed April 10, 2026, 10:40 a.m.
NER Named-entity recognition batch_69e53066a7108190a50eda9b489c90ca completed April 19, 2026, 7:43 p.m.
NED1 Entity disambiguation (via context triple) batch_6a04916189f48190ba0a3196dbcfe4e2 completed May 13, 2026, 2:57 p.m.
NEDg Description generation batch_6a0491db8c0081909bc90a13ccf17ca8 completed May 13, 2026, 2:59 p.m.
NED2 Entity disambiguation (via description) batch_6a0493148d348190b53330a239d8eeb7 completed May 13, 2026, 3:04 p.m.
Created at: April 10, 2026, 11:35 a.m.