Triple

T3884518
Position Surface form Disambiguated ID Type / Status
Subject Can one hear the shape of a drum? E92906 entity
Predicate mainConcept P533 FINISHED
Object Laplacian spectrum
The Laplacian spectrum is the collection of eigenvalues of the Laplace operator on a domain or manifold, encoding how functions vibrate or diffuse over it and serving as a key tool in spectral geometry and mathematical physics.
E394466 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: Laplacian spectrum | Statement: [Can one hear the shape of a drum?, mainConcept, Laplacian spectrum]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Laplacian spectrum
Context triple: [Can one hear the shape of a drum?, mainConcept, Laplacian spectrum]
  • A. Weyl law
    The Weyl law is a fundamental result in spectral theory that describes the asymptotic distribution of eigenvalues of the Laplacian (or similar operators) in terms of the volume of the underlying domain or manifold.
  • B. Laplace operator
    The Laplace operator is a second-order differential operator widely used in mathematics and physics to describe phenomena such as diffusion, heat flow, and wave propagation.
  • C. Selberg trace formula
    The Selberg trace formula is a fundamental result in analytic number theory and spectral theory that relates lengths of closed geodesics on a Riemannian manifold to the spectrum of its Laplace operator, serving as a non-abelian analogue of the Poisson summation formula.
  • D. Can one hear the shape of a drum?
    "Can one hear the shape of a drum?" is a famous 1966 paper by mathematician Mark Kac that explores whether the geometric shape of a domain can be uniquely determined from the spectrum of its Laplacian, encapsulated in the question of whether one can infer a drum’s shape from the sound it makes.
  • E. Laplace equation
    The Laplace equation is a fundamental second-order partial differential equation widely used in physics and engineering to describe steady-state phenomena such as electrostatics, gravitation, and heat conduction.
  • 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: Laplacian spectrum
Triple: [Can one hear the shape of a drum?, mainConcept, Laplacian spectrum]
Generated description
The Laplacian spectrum is the collection of eigenvalues of the Laplace operator on a domain or manifold, encoding how functions vibrate or diffuse over it and serving as a key tool in spectral geometry and mathematical physics.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Laplacian spectrum
Target entity description: The Laplacian spectrum is the collection of eigenvalues of the Laplace operator on a domain or manifold, encoding how functions vibrate or diffuse over it and serving as a key tool in spectral geometry and mathematical physics.
  • A. Weyl law
    The Weyl law is a fundamental result in spectral theory that describes the asymptotic distribution of eigenvalues of the Laplacian (or similar operators) in terms of the volume of the underlying domain or manifold.
  • B. Laplace operator
    The Laplace operator is a second-order differential operator widely used in mathematics and physics to describe phenomena such as diffusion, heat flow, and wave propagation.
  • C. Selberg trace formula
    The Selberg trace formula is a fundamental result in analytic number theory and spectral theory that relates lengths of closed geodesics on a Riemannian manifold to the spectrum of its Laplace operator, serving as a non-abelian analogue of the Poisson summation formula.
  • D. Can one hear the shape of a drum?
    "Can one hear the shape of a drum?" is a famous 1966 paper by mathematician Mark Kac that explores whether the geometric shape of a domain can be uniquely determined from the spectrum of its Laplacian, encapsulated in the question of whether one can infer a drum’s shape from the sound it makes.
  • E. Laplace equation
    The Laplace equation is a fundamental second-order partial differential equation widely used in physics and engineering to describe steady-state phenomena such as electrostatics, gravitation, and heat conduction.
  • 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_69aed9697de0819087c2559295ff3d12 completed March 9, 2026, 2:30 p.m.
NER Named-entity recognition batch_69aeec9029908190a7b36a3827734db1 completed March 9, 2026, 3:51 p.m.
NED1 Entity disambiguation (via context triple) batch_69b5125bee048190ba7553797e9fd254 completed March 14, 2026, 7:46 a.m.
NEDg Description generation batch_69b512e3721c8190accd26499191c153 completed March 14, 2026, 7:48 a.m.
NED2 Entity disambiguation (via description) batch_69b513618b888190acda94dcc91d24d2 completed March 14, 2026, 7:50 a.m.
Created at: March 9, 2026, 3:20 p.m.