Triple
T4552232
| Position | Surface form | Disambiguated ID | Type / Status |
|---|---|---|---|
| Subject | Divergent Series |
E120391
|
entity |
| Predicate | topic |
P261
|
FINISHED |
| Object |
Abel summation
Abel summation is a method in mathematical analysis for assigning values to certain divergent series by considering the limit of their power series as the variable approaches 1 from below.
|
E451515
|
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: Abel summation | Statement: [Divergent Series, topic, Abel summation]
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Abel summation Context triple: [Divergent Series, topic, Abel summation]
-
A.
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.
-
B.
Ramanujan’s sum
Ramanujan’s sum is a number-theoretic function introduced by Srinivasa Ramanujan, expressing certain periodic arithmetic functions as finite trigonometric sums over primitive roots of unity.
-
C.
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.
-
D.
Bernoulli numbers
Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
-
E.
Poisson summation formula
The Poisson summation formula is a fundamental result in harmonic analysis that links sums of a function over the integers to sums of its Fourier transform, with deep applications in number theory, signal processing, and physics.
- 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: Abel summation Triple: [Divergent Series, topic, Abel summation]
Generated description
Abel summation is a method in mathematical analysis for assigning values to certain divergent series by considering the limit of their power series as the variable approaches 1 from below.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Abel summation Target entity description: Abel summation is a method in mathematical analysis for assigning values to certain divergent series by considering the limit of their power series as the variable approaches 1 from below.
-
A.
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.
-
B.
Ramanujan’s sum
Ramanujan’s sum is a number-theoretic function introduced by Srinivasa Ramanujan, expressing certain periodic arithmetic functions as finite trigonometric sums over primitive roots of unity.
-
C.
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.
-
D.
Bernoulli numbers
Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
-
E.
Poisson summation formula
The Poisson summation formula is a fundamental result in harmonic analysis that links sums of a function over the integers to sums of its Fourier transform, with deep applications in number theory, signal processing, and physics.
- 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_69bd4636f1648190a701445c2fcd9c17 |
completed | March 20, 2026, 1:05 p.m. |
| NER | Named-entity recognition | batch_69bd57f7b9748190af29d02fc77b02e0 |
completed | March 20, 2026, 2:21 p.m. |
| NED1 | Entity disambiguation (via context triple) | batch_69bdb95b01b0819094a600752e41aa09 |
completed | March 20, 2026, 9:17 p.m. |
| NEDg | Description generation | batch_69bdbdbf73508190b64a78ff9274ee6d |
completed | March 20, 2026, 9:35 p.m. |
| NED2 | Entity disambiguation (via description) | batch_69bdbe1bcd8c819094adea59c91c6f5b |
completed | March 20, 2026, 9:37 p.m. |
Created at: March 20, 2026, 1:09 p.m.