Bloch theorem
E898493
Bloch theorem is a fundamental result in complex analysis stating that any holomorphic function on the unit disk with bounded derivative maps some subdisk onto a disk of a universal minimum radius, known as Bloch's constant.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Bloch theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10991843 — 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.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Bloch theorem Context triple: [Schwarz lemma, relatedResult, Bloch theorem]
-
A.
Bloch waves
Bloch waves are quantum-mechanical wavefunctions that describe particles, such as electrons, propagating through a periodic crystal lattice and form the basis for understanding electronic band structure in solids.
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B.
Brillouin theorem
Brillouin theorem is a fundamental result in quantum chemistry and Hartree–Fock theory stating that single excitations from a Hartree–Fock ground state do not mix with the ground state and therefore do not lower its energy to first order.
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C.
Brillouin zone
The Brillouin zone is the fundamental region of reciprocal space in a crystal lattice, defining the allowed wavevectors for electrons and underpinning the structure of electronic band diagrams.
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D.
Bloch
Bloch is a surname most notably associated with Felix Bloch, the Swiss-American physicist and Nobel laureate known for his pioneering work in nuclear magnetic resonance.
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E.
Bloch oscillations
Bloch oscillations are periodic oscillations of electrons in a crystal lattice subjected to a constant electric field, arising from the band structure and leading to oscillatory motion in momentum space.
- 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: Bloch theorem Target entity description: Bloch theorem is a fundamental result in complex analysis stating that any holomorphic function on the unit disk with bounded derivative maps some subdisk onto a disk of a universal minimum radius, known as Bloch's constant.
-
A.
Bloch waves
Bloch waves are quantum-mechanical wavefunctions that describe particles, such as electrons, propagating through a periodic crystal lattice and form the basis for understanding electronic band structure in solids.
-
B.
Brillouin theorem
Brillouin theorem is a fundamental result in quantum chemistry and Hartree–Fock theory stating that single excitations from a Hartree–Fock ground state do not mix with the ground state and therefore do not lower its energy to first order.
-
C.
Brillouin zone
The Brillouin zone is the fundamental region of reciprocal space in a crystal lattice, defining the allowed wavevectors for electrons and underpinning the structure of electronic band diagrams.
-
D.
Bloch
Bloch is a surname most notably associated with Felix Bloch, the Swiss-American physicist and Nobel laureate known for his pioneering work in nuclear magnetic resonance.
-
E.
Bloch oscillations
Bloch oscillations are periodic oscillations of electrons in a crystal lattice subjected to a constant electric field, arising from the band structure and leading to oscillatory motion in momentum space.
- F. None of above. chosen
Statements (36)
| Predicate | Object |
|---|---|
| instanceOf | theorem in complex analysis ⓘ |
| appliesTo | holomorphic functions on simply connected domains via conformal maps ⓘ |
| assumption |
derivative of the function is bounded on the unit disk
ⓘ
function is holomorphic on the unit disk ⓘ |
| concerns | holomorphic functions ⓘ |
| conclusion | there exists a subdisk of the unit disk whose image contains a disk of radius at least Bloch constant ⓘ |
| domainCondition | unit disk in the complex plane ⓘ |
| field | complex analysis ⓘ |
| guarantees | existence of a universal positive lower bound for the radius of an image disk ⓘ |
| hasUnknownParameter | exact value of Bloch constant is not known ⓘ |
| historicalPeriod | early 20th century mathematics ⓘ |
| implies | normal families type estimates for holomorphic functions ⓘ |
| introduces | Bloch constant NERFINISHED ⓘ |
| involves |
derivative estimates for holomorphic functions
ⓘ
image disks under holomorphic maps ⓘ subdisks of the unit disk ⓘ |
| language | mathematical analysis ⓘ |
| lowerBound | Bloch constant has known positive lower bounds NERFINISHED ⓘ |
| mathematicalSubjectClassification |
30C80
ⓘ
30D45 ⓘ |
| namedAfter | André Bloch NERFINISHED ⓘ |
| objectOfStudy | local behavior of holomorphic functions ⓘ |
| quantifier |
existential subdisk of the unit disk
ⓘ
universal constant independent of the particular holomorphic function ⓘ |
| relatedTo |
Bloch constant
NERFINISHED
ⓘ
Koebe quarter theorem NERFINISHED ⓘ Landau theorem NERFINISHED ⓘ Schwarz lemma NERFINISHED ⓘ |
| statementForm | for every holomorphic function with bounded derivative on the unit disk there exists a point and radius such that the image contains a disk of universal radius ⓘ |
| typeOf |
distortion theorem
ⓘ
existence theorem ⓘ |
| upperBound | Bloch constant has known finite upper bounds NERFINISHED ⓘ |
| usedIn |
conformal mapping theory
ⓘ
geometric function theory ⓘ theory of normal families ⓘ value distribution theory ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
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.
Input
Subject: Bloch theorem Description of subject: Bloch theorem is a fundamental result in complex analysis stating that any holomorphic function on the unit disk with bounded derivative maps some subdisk onto a disk of a universal minimum radius, known as Bloch's constant.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.