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Entropy2016,18, 433
bysetting
∂KV(w)(X)=−X ·w+w ·X if w∈ J(W), (7a)
∂KVf=∑
[i<j] S[i,j](f) if q>0. (7b)
ByLemma2weobtain the followingstatement
Theorem3. Forevery twosidedKVmoduleWofaKValgebraA thepair (C∗KV,∂KV) is a cochaincomplex.
3.3.4. TheTotalCohomology
LetWbeatwo-sidedmoduleofaKValgebraA.Ourconcern is theZ-gradedvectorspace
Cτ =W+⊕q>0Cq(A,W).
Forourpresentpurpose themapsSij arenotsubject therequirementas inStep2.
Wedefinethecoboundaryoperator∂τ bysetting
∂τw(X)=−X ·w+w ·X ∀winW,
∂τ f(ξ)= ∑
1≤i<j≤q+1 S[i,j](f)(ξ) ∀q>0.
Thequantity (∂2τ f(ξ)depends linearlyontheKVanomaly functionsof thepair (A,W). Thus the
pair (C∗τ,∂τ) isacochaincomplex. Its cohomologyiscalledtheW-valuedtotalKVcohomologyofA.
Wedenote itbyH∗τ(A,W).
3.3.5. TheResidualCohomology,SomeExactSequences,RelatedTopics,DTO-HEG-IGE-ENT
Inthenextsectionswewillseethatthelinksbetweentheinformationgeometryandthedifferential
topology involve therealvaluedtotalKVcohomologyofKValgebroids.Manyrelevant relationships
arebasedontheexact sequences
→Hq−1KVres(A,R)→Hqe(A,R)→HqKV(A,R)→HqKVres(A,R)→
→Hq−1τres(A,R)→Hqe(A,R)→Hqτ(A,R)→Hqτres(A,R)→
Nowweareprovidedwithcohomological toolswhichwillbeusedin thenextsections.
WeplantoperformKVcohomologicalmethodsforstudyingsomelinksbetweentheverticesof
thesquare“DTO, IGE,ENTHGE”as 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
- Title
- Differential Geometrical Theory of Statistics
- Authors
- Frédéric Barbaresco
- Frank Nielsen
- Editor
- MDPI
- Location
- Basel
- Date
- 2017
- Language
- English
- License
- CC BY-NC-ND 4.0
- ISBN
- 978-3-03842-425-3
- Size
- 17.0 x 24.4 cm
- Pages
- 476
- Keywords
- Entropy, Coding Theory, Maximum entropy, Information geometry, Computational Information Geometry, Hessian Geometry, Divergence Geometry, Information topology, Cohomology, Shape Space, Statistical physics, Thermodynamics
- Categories
- Naturwissenschaften Physik