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 | 3 |
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 ⓘ |
| 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
ⓘ
linked to:
Bloch constant
|
| mathematicalSubjectClassification |
30C80
ⓘ
30D45 ⓘ |
| namedAfter | André Bloch ⓘ |
| objectOfStudy | local behavior of holomorphic functions ⓘ |
| quantifier |
existential subdisk of the unit disk
ⓘ
universal constant independent of the particular holomorphic function ⓘ |
| relatedTo |
Bloch constant
ⓘ
Koebe quarter theorem ⓘ Landau theorem ⓘ Schwarz lemma ⓘ |
| 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
ⓘ
linked to:
Bloch constant
|
| 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
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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 (3)
Full triples — surface form annotated when it differs from this entity's canonical label.