Triple

T4927386
Position Surface form Disambiguated ID Type / Status
Subject Weierstrass substitution E110609 entity
Predicate relatedTo P37 FINISHED
Object Euler substitution
Euler substitution is a classical technique in integral calculus that simplifies integrals involving square roots of quadratic expressions by transforming them into rational functions through a specific change of variables.
E480876 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: Euler substitution | Statement: [Weierstrass substitution, relatedTo, Euler substitution]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Euler substitution
Context triple: [Weierstrass substitution, relatedTo, Euler substitution]
  • A. Weierstrass substitution
    The Weierstrass substitution is a trigonometric substitution technique that transforms integrals involving sine and cosine into rational functions, simplifying their evaluation.
  • B. Cauchy–Euler equation
    The Cauchy–Euler equation is a type of linear ordinary differential equation with variable coefficients that often appears in problems with power-law or scale-invariant behavior.
  • C. Euler–Maclaurin summation formula
    The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
  • D. Euler’s theorem
    Euler’s theorem is a fundamental result in number theory stating that for any integer a coprime to n, a raised to the power of φ(n) is congruent to 1 modulo n.
  • E. Euler’s method of rearranging absolutely convergent series
    Euler’s method of rearranging absolutely convergent series is a technique introduced by Leonhard Euler to systematically reorder and manipulate convergent infinite series in order to derive new identities and product expansions, such as those appearing in analytic number 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: Euler substitution
Triple: [Weierstrass substitution, relatedTo, Euler substitution]
Generated description
Euler substitution is a classical technique in integral calculus that simplifies integrals involving square roots of quadratic expressions by transforming them into rational functions through a specific change of variables.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Euler substitution
Target entity description: Euler substitution is a classical technique in integral calculus that simplifies integrals involving square roots of quadratic expressions by transforming them into rational functions through a specific change of variables.
  • A. Weierstrass substitution
    The Weierstrass substitution is a trigonometric substitution technique that transforms integrals involving sine and cosine into rational functions, simplifying their evaluation.
  • B. Cauchy–Euler equation
    The Cauchy–Euler equation is a type of linear ordinary differential equation with variable coefficients that often appears in problems with power-law or scale-invariant behavior.
  • C. Euler–Maclaurin summation formula
    The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
  • D. Euler’s theorem
    Euler’s theorem is a fundamental result in number theory stating that for any integer a coprime to n, a raised to the power of φ(n) is congruent to 1 modulo n.
  • E. Euler’s method of rearranging absolutely convergent series
    Euler’s method of rearranging absolutely convergent series is a technique introduced by Leonhard Euler to systematically reorder and manipulate convergent infinite series in order to derive new identities and product expansions, such as those appearing in analytic number theory.
  • 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_69bd4415190c8190817bee7ec9f9f944 completed March 20, 2026, 12:56 p.m.
NER Named-entity recognition batch_69bd7036d8e88190bc4be2975160da23 completed March 20, 2026, 4:05 p.m.
NED1 Entity disambiguation (via context triple) batch_69be77ac88148190a51fa2e9085d6897 completed March 21, 2026, 10:49 a.m.
NEDg Description generation batch_69be7847a378819081687ec783a8b862 completed March 21, 2026, 10:51 a.m.
NED2 Entity disambiguation (via description) batch_69be7915a26c81909b21a128daebf5b3 completed March 21, 2026, 10:55 a.m.
Created at: March 20, 2026, 1:30 p.m.