Hahn series
E159923
Hahn series are formal power series with exponents in an ordered abelian group and well-ordered supports, providing a general framework for constructing large ordered fields that include structures like the surreal numbers.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Hahn generalized power series | 1 |
| Hahn series canonical | 1 |
| Malcev–Neumann series | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1390514 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
Target entity: Hahn series Context triple: [Surreal numbers, relatedConcept, Hahn series]
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A.
Born–Huang expansion
The Born–Huang expansion is a quantum mechanical method that systematically improves upon the Born–Oppenheimer approximation by including couplings between electronic and nuclear motions in molecular systems.
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B.
Taylor series
A Taylor series is an infinite sum of terms calculated from the derivatives of a function at a single point, used to represent and approximate functions as power series.
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C.
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.
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D.
TrizecHahn
TrizecHahn was a major North American real estate investment and development company known for large-scale commercial and mixed-use projects.
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E.
Dyson series
The Dyson series is a perturbative expansion in quantum field theory that expresses time-ordered exponentials and scattering amplitudes as an infinite series of integrals, each term corresponding to a Feynman diagram.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
Target entity: Hahn series Target entity description: Hahn series are formal power series with exponents in an ordered abelian group and well-ordered supports, providing a general framework for constructing large ordered fields that include structures like the surreal numbers.
-
A.
Born–Huang expansion
The Born–Huang expansion is a quantum mechanical method that systematically improves upon the Born–Oppenheimer approximation by including couplings between electronic and nuclear motions in molecular systems.
-
B.
Taylor series
A Taylor series is an infinite sum of terms calculated from the derivatives of a function at a single point, used to represent and approximate functions as power series.
-
C.
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.
-
D.
TrizecHahn
TrizecHahn was a major North American real estate investment and development company known for large-scale commercial and mixed-use projects.
-
E.
Dyson series
The Dyson series is a perturbative expansion in quantum field theory that expresses time-ordered exponentials and scattering amplitudes as an infinite series of integrals, each term corresponding to a Feynman diagram.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic structure
ⓘ
formal power series ⓘ mathematical concept ⓘ ordered field construction ⓘ |
| appearsIn |
construction of exponential fields
ⓘ
construction of real-closed fields ⓘ valuation theory of ordered fields ⓘ |
| componentType |
coefficients from a base field or ring
ⓘ
exponents from an ordered abelian group ⓘ |
| definedOver | ordered abelian group ⓘ |
| ensuresWellDefinedProductBy | well-ordered support condition ⓘ |
| fieldOfStudy |
algebra
ⓘ
model theory ⓘ non-Archimedean analysis ⓘ ordered algebra ⓘ valuation theory ⓘ |
| generalizes |
Laurent series
ⓘ
Puiseux series ⓘ formal power series with integer exponents ⓘ |
| hasAlternativeName |
Hahn series
ⓘ
surface form:
Hahn generalized power series
|
| hasCanonicalValuation | support minimum exponent ⓘ |
| hasConstraintOnSupport | every nonempty subset of the support has a least element ⓘ |
| hasExponentGroup | ordered abelian group ⓘ |
| hasMathematicianNamesake | Hans Hahn ⓘ |
| hasProperty |
can be equipped with a natural valuation
ⓘ
can be made into an ordered field ⓘ supports transfinite exponents ⓘ supports well-ordered sets of exponents ⓘ |
| hasSupportCondition | well-ordered support ⓘ |
| isSpecialCaseOf | generalized power series ⓘ |
| namedAfter | Hans Hahn ⓘ |
| providesFrameworkFor |
constructing Hahn fields
ⓘ
embedding ordered fields ⓘ |
| relatedConcept |
Levi-Civita field
ⓘ
Hahn series self-linksurface differs ⓘ
surface form:
Malcev–Neumann series
transseries ⓘ |
| relatedTo | surreal numbers ⓘ |
| supportsOperation |
Cauchy-type product
ⓘ
termwise addition ⓘ |
| typicalNotation | K((G)) for coefficients in K and exponents in G ⓘ |
| usedIn |
asymptotic differential algebra
ⓘ
construction of saturated valued fields ⓘ model-theoretic examples of non-Archimedean fields ⓘ study of ordered exponential fields ⓘ |
| usedToConstruct |
large ordered fields
ⓘ
maximally valued fields ⓘ non-Archimedean ordered fields ⓘ |
How these facts were elicited
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You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Subject: Hahn series Description of subject: Hahn series are formal power series with exponents in an ordered abelian group and well-ordered supports, providing a general framework for constructing large ordered fields that include structures like the surreal numbers.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.