Triple

T13686344
Position Surface form Disambiguated ID Type / Status
Subject Pierre-Louis Lions E328137 entity
Predicate notableWork P4 FINISHED
Object concentration-compactness principle
The concentration-compactness principle is a fundamental method in nonlinear analysis that overcomes loss-of-compactness issues in variational problems, especially those involving critical exponents.
 E1054397 NE FINISHED

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: concentration-compactness principle | Statement: [Pierre-Louis Lions, notableWork, concentration-compactness principle]

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: concentration-compactness principle
Context triple: [Pierre-Louis Lions, notableWork, concentration-compactness principle]
  • A. Rellich–Kondrachov compactness theorem
    The Rellich–Kondrachov compactness theorem is a fundamental result in functional analysis and the theory of Sobolev spaces that guarantees the compactness of certain embedding operators, playing a key role in the study of partial differential equations.
  • B. Cheeger–Gromov compactness theorem
    The Cheeger–Gromov compactness theorem is a fundamental result in Riemannian geometry that gives conditions under which a sequence of Riemannian manifolds has a subsequence converging (in the Gromov–Hausdorff or smooth sense) to a limit space.
  • C. Poincaré inequality
    The Poincaré inequality is a fundamental result in functional analysis and partial differential equations that bounds the average oscillation of a function by the size of its gradient, playing a key role in Sobolev space theory and the study of elliptic problems.
  • D. Sobolev inequality
    The Sobolev inequality is a fundamental result in functional analysis and partial differential equations that bounds the size of a function in certain Lebesgue spaces by the size of its derivatives, enabling key embedding and regularity properties.
  • E. Morawetz inequalities
    Morawetz inequalities are fundamental energy and decay estimates in the study of partial differential equations, especially wave and dispersive equations, that provide control over the long-time behavior of solutions.
  • 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: concentration-compactness principle
Target entity description: The concentration-compactness principle is a fundamental method in nonlinear analysis that overcomes loss-of-compactness issues in variational problems, especially those involving critical exponents.
  • A. Rellich–Kondrachov compactness theorem
    The Rellich–Kondrachov compactness theorem is a fundamental result in functional analysis and the theory of Sobolev spaces that guarantees the compactness of certain embedding operators, playing a key role in the study of partial differential equations.
  • B. Cheeger–Gromov compactness theorem
    The Cheeger–Gromov compactness theorem is a fundamental result in Riemannian geometry that gives conditions under which a sequence of Riemannian manifolds has a subsequence converging (in the Gromov–Hausdorff or smooth sense) to a limit space.
  • C. Poincaré inequality
    The Poincaré inequality is a fundamental result in functional analysis and partial differential equations that bounds the average oscillation of a function by the size of its gradient, playing a key role in Sobolev space theory and the study of elliptic problems.
  • D. Sobolev inequality
    The Sobolev inequality is a fundamental result in functional analysis and partial differential equations that bounds the size of a function in certain Lebesgue spaces by the size of its derivatives, enabling key embedding and regularity properties.
  • E. Morawetz inequalities
    Morawetz inequalities are fundamental energy and decay estimates in the study of partial differential equations, especially wave and dispersive equations, that provide control over the long-time behavior of solutions.
  • F. None of above. chosen

How the object was described

The object's one-sentence description was generated by prompting gpt-5.1 with the object name and this triple as context.

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: concentration-compactness principle
Triple: [Pierre-Louis Lions, notableWork, concentration-compactness principle]
Generated description
The concentration-compactness principle is a fundamental method in nonlinear analysis that overcomes loss-of-compactness issues in variational problems, especially those involving critical exponents.

Provenance (5 batches)

Stage Batch ID Job type Status
creating batch_69d8076f1fa8819094664a59b55010df elicitation completed
NER batch_69dbc670968881908e2b4fdf656c7285 ner completed
NED1 batch_69f7944981ec8190be5ff39b7c2c70ab ned_source_triple completed
NED2 batch_69f796e5c60c8190a19389bc4cdbd658 ned_description completed
NEDg batch_69f795e361c48190b37060312e7df181 nedg completed
Created at: April 9, 2026, 9:53 p.m.