Seite - 162 - in Differential Geometrical Theory of Statistics
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Entropy2016,18, 433
bysetting
âKV(w)(X)=âX ·w+w ·X if wâ J(W), (7a)
âKVf=â
[i<j] S[i,j](f) if q>0. (7b)
ByLemma2weobtain the followingstatement
Theorem3. Forevery twosidedKVmoduleWofaKValgebraA thepair (CâKV,âKV) is a cochaincomplex.
3.3.4. TheTotalCohomology
LetWbeatwo-sidedmoduleofaKValgebraA.Ourconcern is theZ-gradedvectorspace
CÏ =W+âq>0Cq(A,W).
Forourpresentpurpose themapsSij arenotsubject therequirementas inStep2.
WedeïŹnethecoboundaryoperatorâÏ bysetting
âÏw(X)=âX ·w+w ·X âwinW,
âÏ f(Ο)= â
1â€i<jâ€q+1 S[i,j](f)(Ο) âq>0.
Thequantity (â2Ï f(Ο)depends linearlyontheKVanomaly functionsof thepair (A,W). Thus the
pair (CâÏ,âÏ) isacochaincomplex. Its cohomologyiscalledtheW-valuedtotalKVcohomologyofA.
Wedenote itbyHâÏ(A,W).
3.3.5. TheResidualCohomology,SomeExactSequences,RelatedTopics,DTO-HEG-IGE-ENT
Inthenextsectionswewillseethatthelinksbetweentheinformationgeometryandthedifferential
topology involve therealvaluedtotalKVcohomologyofKValgebroids.Manyrelevant relationships
arebasedontheexact sequences
âHqâ1KVres(A,R)âHqe(A,R)âHqKV(A,R)âHqKVres(A,R)â
âHqâ1Ïres(A,R)âHqe(A,R)âHqÏ(A,R)âHqÏres(A,R)â
Nowweareprovidedwithcohomological toolswhichwillbeusedin thenextsections.
WeplantoperformKVcohomologicalmethodsforstudyingsomelinksbetweentheverticesof
thesquareâDTO, IGE,ENTHGEâas inFigure1.Werecallbasicnotions.
DTO IGE
ENTHGE
KVH
Figure1.Federation.
DTOstands forDifferentialTOpology.
Thepurposes: Riemannian foliationsandRiemannianwebs. Symplectic foliationsandsymplecticwebs.
Linearizationofwebs.
Our aims: We use cohomologicalmethods for constructingRiemannian foliations, Riemannianwebs,
linearizablewebs.
162
Differential Geometrical Theory of Statistics
- Titel
- Differential Geometrical Theory of Statistics
- Autoren
- Frédéric Barbaresco
- Frank Nielsen
- Herausgeber
- MDPI
- Ort
- Basel
- Datum
- 2017
- Sprache
- englisch
- Lizenz
- CC BY-NC-ND 4.0
- ISBN
- 978-3-03842-425-3
- Abmessungen
- 17.0 x 24.4 cm
- Seiten
- 476
- Schlagwörter
- Entropy, Coding Theory, Maximum entropy, Information geometry, Computational Information Geometry, Hessian Geometry, Divergence Geometry, Information topology, Cohomology, Shape Space, Statistical physics, Thermodynamics
- Kategorien
- Naturwissenschaften Physik