A Cheeger-Type Inequality on Simplicial Complexes Scientific and - - PowerPoint PPT Presentation

A Cheeger-Type Inequality on Simplicial Complexes

Scientific and Statistical Computing Seminar – U Chicago Sayan Mukherjee

Departments of Statistical Science, Computer Science, Mathematics Institute for Genome Sciences & Policy, Duke University Joint work with:

• J. Steenbergen (Duke), Carly Klivans (Brown)

November 9, 2012

Motivation Definitions Results Open problems Acknowledgements

Dimension reduction algorithm

Examples include:

1

Isomap - 2000

2

Locally Linear Embedding (LLE) - 2000

3

Hessian LLE - 2003

4

Laplacian Eigenmaps - 2003

5

Diffusion Maps - 2004

Laplacian Eigenmaps is based directly on the graph Laplacian.

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Motivation Definitions Results Open problems Acknowledgements

Laplacian eigenmaps

Given data points x1, x2, . . . , xn ∈ I Rp which we wish to map into I Rk, k ≪ p,

1 Construct a graph (association matrix) A out of the data

points.

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Motivation Definitions Results Open problems Acknowledgements

Laplacian eigenmaps

Given data points x1, x2, . . . , xn ∈ I Rp which we wish to map into I Rk, k ≪ p,

1 Construct a graph (association matrix) A out of the data

points.

2 Compute L = D − A where Dii =

j Aij

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Motivation Definitions Results Open problems Acknowledgements

Laplacian eigenmaps

Given data points x1, x2, . . . , xn ∈ I Rp which we wish to map into I Rk, k ≪ p,

1 Construct a graph (association matrix) A out of the data

points.

2 Compute L = D − A where Dii =

j Aij

3 Compute eigenvalues and eigenvectors of L

0 = λ0 ≤ λ1 ≤ λ2 ≤ . . . ≤ λn, f0 = 1, ..., fn

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Motivation Definitions Results Open problems Acknowledgements

Laplacian eigenmaps

Given data points x1, x2, . . . , xn ∈ I Rp which we wish to map into I Rk, k ≪ p,

1 Construct a graph (association matrix) A out of the data

points.

2 Compute L = D − A where Dii =

j Aij

3 Compute eigenvalues and eigenvectors of L

0 = λ0 ≤ λ1 ≤ λ2 ≤ . . . ≤ λn, f0 = 1, ..., fn

4 Map the data points into I

Rk by the map xi → (f1(xi), f2(i), . . . , fk(xi))

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Motivation Definitions Results Open problems Acknowledgements

Laplacian eigenmaps example

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Motivation Definitions Results Open problems Acknowledgements

Laplacian eigenmaps example

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Motivation Definitions Results Open problems Acknowledgements

Near 0 homology

Near-Connected Components?

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Motivation Definitions Results Open problems Acknowledgements

Clustering

Clusters

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Motivation Definitions Results Open problems Acknowledgements

Near 0-homology for graphs

S S Cut

For a graph with vertex set V , the Cheeger number is defined to be h = min

∅SV

|δS| min

• |S|, |S|

.

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Motivation Definitions Results Open problems Acknowledgements

Near 0-homology for graphs

S S Cut

For a graph with vertex set V , the Cheeger number is defined to be h = min

∅SV

|δS| min

• |S|, |S|

. h = 0 ⇔ graph is disconnected.

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Motivation Definitions Results Open problems Acknowledgements

Near 0-homology for graphs

 ⇔

For a graph with vertex set V , the Fiedler number is defined to be λ = min

f :V →I R

f ⊥1

• u∼v(f (u) − f (v))2
• u(f (u))2

.

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Motivation Definitions Results Open problems Acknowledgements

Near 0-homology for graphs

 ⇔

For a graph with vertex set V , the Fiedler number is defined to be λ = min

f :V →I R

f ⊥1

• u∼v(f (u) − f (v))2
• u(f (u))2

. λ = 0 ⇔ graph is disconnected.

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Motivation Definitions Results Open problems Acknowledgements

Cheeger inequality for graphs

Theorem (Alon, Milman, Lawler & Sokal, Frieze & Kannan & Polson,...) For any graph with Cheeger number h and Fiedler number λ 2h ≥ λ1 > h2 2M , M = maxu ud, maximum vertex degree.

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Motivation Definitions Results Open problems Acknowledgements

Edge expansion

Expander graphs are families of graphs that are sparse and strongly connected.

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Motivation Definitions Results Open problems Acknowledgements

Edge expansion

Expander graphs are families of graphs that are sparse and strongly connected. A family of expander graphs G has the property h(G) > ǫ > 0 for all G ∈ G.

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Motivation Definitions Results Open problems Acknowledgements

Edge expansion

Expander graphs are families of graphs that are sparse and strongly connected. A family of expander graphs G has the property h(G) > ǫ > 0 for all G ∈ G. Cheeger inequality lets us use λ as criteria λ(G) > 0 for all G ∈ G.

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Motivation Definitions Results Open problems Acknowledgements

Higher-dimensional notions

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Motivation Definitions Results Open problems Acknowledgements

Homology

0-Homology 1-Homology 2-Homology

β0 = 1, β1 = 0, β2 = 1 β0 = 1, β1 = 1, β2 = 0 β0 = 2, β1 = 0, β2 = 0

Hole Void Connected Components

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Motivation Definitions Results Open problems Acknowledgements

Near one homology

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Motivation Definitions Results Open problems Acknowledgements

Near one homology

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Motivation Definitions Results Open problems Acknowledgements

Simplicial complexes

A k-simplex is the convex hull of k + 1 affinely independent points, σ = {u0, ..., uk}.

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Motivation Definitions Results Open problems Acknowledgements

Simplicial complexes

A k-simplex is the convex hull of k + 1 affinely independent points, σ = {u0, ..., uk}. A face τ of σ is the convex hull of a non-empty subset of the ui, τ ≤ σ.

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Motivation Definitions Results Open problems Acknowledgements

Simplicial complexes

A k-simplex is the convex hull of k + 1 affinely independent points, σ = {u0, ..., uk}. A face τ of σ is the convex hull of a non-empty subset of the ui, τ ≤ σ. A simplicial complex is a finite collection of simplices K such that σ ∈ K and τ ≤ σ implies τ ∈ K, and σ, σ0 ∈ K implies σ ∩ σ0 is either empty or a face of both.

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Motivation Definitions Results Open problems Acknowledgements

Chains and cochains

X = simplicial complex of dimension m, (X) = m. Ck(I F) = {I F-linear combinations of oriented k-simplices} C k(I F) = {I F-valued functions on oriented k-simplices}

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Motivation Definitions Results Open problems Acknowledgements

Chains and cochains

X = simplicial complex of dimension m, (X) = m. Ck(I F) = {I F-linear combinations of oriented k-simplices} C k(I F) = {I F-valued functions on oriented k-simplices} Chain Complex: 0 ← − C0

∂1(I F)

← − C1

∂2(I F)

← − · · ·

∂m(I F)

← − Cm ← − 0 Cochain Complex: 0 − → C 0 δ0(I

F)

− → C 1 δ1(I

F)

− → · · ·

δm−1(I F)

− → C m − → 0 I F = I R, Z2

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Motivation Definitions Results Open problems Acknowledgements

Chains and cochains

Given a simplicial complex σ = {v0, ..., vk} an orientation of [v0, ..., vk] is the equivalence class of even permutations of ordering.

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Motivation Definitions Results Open problems Acknowledgements

Chains and cochains

Given a simplicial complex σ = {v0, ..., vk} an orientation of [v0, ..., vk] is the equivalence class of even permutations of ordering. Boundary map ∂k(I F) : Ck(I F) → Ck−1(I F) ∂k[v0, ..., vk] =

k

• i=1

(−1)i[v0, ..., vi−1, vi+1, ..., vk].

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Motivation Definitions Results Open problems Acknowledgements

Chains and cochains

Given a simplicial complex σ = {v0, ..., vk} an orientation of [v0, ..., vk] is the equivalence class of even permutations of ordering. Boundary map ∂k(I F) : Ck(I F) → Ck−1(I F) ∂k[v0, ..., vk] =

k

• i=1

(−1)i[v0, ..., vi−1, vi+1, ..., vk]. Coboundary map δk−1(I F) : C k−1(I F) → C k(I F) is the transpose

• f the boundary map.

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Motivation Definitions Results Open problems Acknowledgements

∂(I R)

∂2(R) v1 v2 v4 v3 v1 v2 v4 v3 ∂2(R) [v1, v3, v2] + 2[v2, v4, v3] [v1, v3] + [v2, v1] + 3[v3, v2] +2[v2, v4] + 2[v4, v3] 1 1 3 2 2 1 2

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Motivation Definitions Results Open problems Acknowledgements

∂(Z2)

v1 v2 v4 v3 v1 v2 v4 v3 ∂2(Z2) [v1, v2, v3] + [v2, v3, v4] ∂2(Z2) [v1, v2] + [v1, v3] +[v2, v4] + [v3, v4] 1 1 1 1 1 1

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Motivation Definitions Results Open problems Acknowledgements

δ(Z2)

v1 v2 v4 v3 v1 v2 v4 v3 δ1(Z2) [v1, v2, v3] + [v2, v3, v4] δ1(Z2) [v1, v2] + [v1, v3] + [v2, v3] 1 1 1 1 1

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Motivation Definitions Results Open problems Acknowledgements

δ(I R)

δ1(R) v1 v2 v4 v3 v1 v2 v4 v3 δ1(R) 3[v2, v1] + 4[v3, v1] + 2[v3, v2] [v1, v3, v2] + 2[v2, v4, v3] 4 3 2 1 2

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Motivation Definitions Results Open problems Acknowledgements

Cheeger numbers

Z2 homology and cohomology Hk(Z2) = ker∂k im∂k+1 , Hk(Z2) = kerδk imδk+1 .

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Motivation Definitions Results Open problems Acknowledgements

Cheeger numbers

Z2 homology and cohomology Hk(Z2) = ker∂k im∂k+1 , Hk(Z2) = kerδk imδk+1 . Coboundary and Boundary Cheeger numbers: hk := min

φ∈C k(Z2)\imδ

• |δφ|

minδφ=δψ |ψ|

• ← distance from kerδk

← distance from imδk−1

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Motivation Definitions Results Open problems Acknowledgements

Cheeger numbers

Z2 homology and cohomology Hk(Z2) = ker∂k im∂k+1 , Hk(Z2) = kerδk imδk+1 . Coboundary and Boundary Cheeger numbers: hk := min

φ∈C k(Z2)\imδ

• |δφ|

minδφ=δψ |ψ|

• ← distance from kerδk

← distance from imδk−1 hk := min

φ∈Ck(Z2)\im∂

• |∂φ|

min∂φ=∂ψ |ψ|

• ← distance from ker∂k

← distance from im∂k+1 .

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Motivation Definitions Results Open problems Acknowledgements

Cheeger numbers

Z2 homology and cohomology Hk(Z2) = ker∂k im∂k+1 , Hk(Z2) = kerδk imδk+1 . Coboundary and Boundary Cheeger numbers: hk := min

φ∈C k(Z2)\imδ

• |δφ|

minδφ=δψ |ψ|

• ← distance from kerδk

← distance from imδk−1 hk := min

φ∈Ck(Z2)\im∂

• |∂φ|

min∂φ=∂ψ |ψ|

• ← distance from ker∂k

← distance from im∂k+1 . Hk(Z2) = 0 ⇔ hk = 0 and Hk(Z2) = 0 ⇔ hk = 0. h0 is the Cheeger number for graphs. hk defined by Dotterer and Kahle.

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Motivation Definitions Results Open problems Acknowledgements

Fiedler numbers

I R homology and cohomology Hk(I R) = ker∂k im∂k+1 , Hk(I R) = kerδk imδk+1 .

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Motivation Definitions Results Open problems Acknowledgements

Fiedler numbers

I R homology and cohomology Hk(I R) = ker∂k im∂k+1 , Hk(I R) = kerδk imδk+1 . Coboundary and Boundary Fiedler numbers: λk := min

f ∈C k(I R)\imδ

• δf 2

2

minδf =δg g2

2

• ← distance from kerδk

← distance from imδk−1

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Motivation Definitions Results Open problems Acknowledgements

Fiedler numbers

I R homology and cohomology Hk(I R) = ker∂k im∂k+1 , Hk(I R) = kerδk imδk+1 . Coboundary and Boundary Fiedler numbers: λk := min

f ∈C k(I R)\imδ

• δf 2

2

minδf =δg g2

2

• ← distance from kerδk

← distance from imδk−1 λk := min

fi∈Ck(I R)\im∂

• ∂f 2

2

min∂f =∂g g2

2

• ← distance from ker∂k

← distance from im∂k+1 .

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Motivation Definitions Results Open problems Acknowledgements

Fiedler numbers

I R homology and cohomology Hk(I R) = ker∂k im∂k+1 , Hk(I R) = kerδk imδk+1 . Coboundary and Boundary Fiedler numbers: λk := min

f ∈C k(I R)\imδ

• δf 2

2

minδf =δg g2

2

• ← distance from kerδk

← distance from imδk−1 λk := min

fi∈Ck(I R)\im∂

• ∂f 2

2

min∂f =∂g g2

2

• ← distance from ker∂k

← distance from im∂k+1 . Hk(I R) = 0 ⇔ λk = 0 and Hk(I R) = 0 ⇔ λk = 0. λ0 is the Fiedler number for graphs.

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Motivation Definitions Results Open problems Acknowledgements

Combinatorial Laplacian

The k-th Laplacian of X Lk := ∂k+1δk + δk−1∂k

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Motivation Definitions Results Open problems Acknowledgements

Combinatorial Laplacian

The k-th Laplacian of X Lk := ∂k+1δk + δk−1∂k Eigenvalues of Lk measure near (co)homology Theorem (Eckmann) C k = im(∂k+1)

• im(δk−1)
• ker(Lk),

and Hk(I R) ∼ = Hk(I R) ∼ = ker Lk.

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Motivation Definitions Results Open problems Acknowledgements

Combinatorial Laplacian

C 1 = im(δ0)

• ker(L1)
• im(∂1).

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Motivation Definitions Results Open problems Acknowledgements

Orientability

Two m-simplexes are lower adjacent if they share a common face

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Motivation Definitions Results Open problems Acknowledgements

Orientability

Two m-simplexes are lower adjacent if they share a common face Two oriented lower adjacent k-simplexes τ and σ are dissimilar on a face ν if ∂τ and ∂σ assign the same coefficient to ν.

a b c d

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Motivation Definitions Results Open problems Acknowledgements

Orientability

Two m-simplexes are lower adjacent if they share a common face Two oriented lower adjacent k-simplexes τ and σ are dissimilar on a face ν if ∂τ and ∂σ assign the same coefficient to ν.

a b c d

If X is a simplicial m-complex and all its m-simplices can be

• riented similarly, then X is called orientable.

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Motivation Definitions Results Open problems Acknowledgements

Pseudomanifold

A pseudomanifold is a combinatorial realization of a manifold with singularities.

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Motivation Definitions Results Open problems Acknowledgements

Pseudomanifold

A pseudomanifold is a combinatorial realization of a manifold with singularities. A topological space X endowed with a triangulation K is an m-dimensional pseudomanifold if the following conditions hold: (1) X = |K| is the union of all n-simplices. (2) Every (m1)-simplex is a face of exactly two m-simplices for m > 1. (3) X is a strongly connected, there is a path between any pair of m-simplices in K.

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Motivation Definitions Results Open problems Acknowledgements

Pseudomanifold

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Motivation Definitions Results Open problems Acknowledgements

Positive result: chain complex

Proposition (Steenbergen & Klivans & M) If X is an m-dimensional orientable pseudomanifold then hm ≥ λm ≥ h2

m

2(m + 1).

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Motivation Definitions Results Open problems Acknowledgements

Positive result: chain complex

Proposition (Steenbergen & Klivans & M) If X is an m-dimensional orientable pseudomanifold then hm ≥ λm ≥ h2

m

2(m + 1). Discrete analog of the Cheeger inequality for manifolds with Dirichlet boundary condition, every (m − 1)-simplex has at most two cofaces.

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Motivation Definitions Results Open problems Acknowledgements

Orientation hypothesis

X is a triangulation of the real projective plane. H2(Z2) = 0 ⇒ h2 = 0 and H2(I R) = 0 ⇒ λ2 = 0.

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Motivation Definitions Results Open problems Acknowledgements

Real projective plane

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Motivation Definitions Results Open problems Acknowledgements

Orientation hypothesis

Theorem (Gundert & Wagner) There is an infinite family of complexes that are not combinatorially expanding, h = 0, and whose spectral expansion is bounded away from zero, λ > 0.

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Motivation Definitions Results Open problems Acknowledgements

Boundary hypothesis

Consider the graph below h1 = 2

3 and λ1 = λ0 ≤ 2 k+1. k vertices k vertices Figure 4. The family of graphs Gk.

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Motivation Definitions Results Open problems Acknowledgements

Negative result: cochain complex

Proposition (Steenbergen & Klivans & M) For every m >1, there exist families of pseudomanifolds Xk and Yk

• f dimension m such that

(1) λm−1(Xk) ≥ (m−1)2

2(m+1) for all k but hm−1(Xk) → 0 as k → ∞

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Motivation Definitions Results Open problems Acknowledgements

Negative result: cochain complex

Proposition (Steenbergen & Klivans & M) For every m >1, there exist families of pseudomanifolds Xk and Yk

• f dimension m such that

(1) λm−1(Xk) ≥ (m−1)2

2(m+1) for all k but hm−1(Xk) → 0 as k → ∞

(2) λm−1(Yk) ≤

1 mk−1 for k > 1 but hm−1(Yk) ≥ 1 k for all k.

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Motivation Definitions Results Open problems Acknowledgements

Buser part

hm−1(Xk) ≤ λm−1(Xk)

X1 X2 X3 X4

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Motivation Definitions Results Open problems Acknowledgements

Cheeger part

λm−1(Yk) ≤ hm−1(Yk)

Y1 Y2 Y3

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Motivation Definitions Results Open problems Acknowledgements

Open problems

(1) Intermediate values of k – relation of hk and λk or hk and λk for 1 < k < m − 1.

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Motivation Definitions Results Open problems Acknowledgements

Open problems

(1) Intermediate values of k – relation of hk and λk or hk and λk for 1 < k < m − 1. (2) High-order eigenvalues – λk,j and hk,j where j > 1 indexes the

• rdering of Fiedler/Cheeger numbers.

(3) Manifolds – The k-dimensional coboundary/boundary Cheeger numbers of a manifold M might be hk = inf

S

Volm−k−1(∂S \ ∂M) inf∂T=∂S Volm−k(T), hk = inf

S

Volk−1(∂S) inf∂T=∂S Volk(T).

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Motivation Definitions Results Open problems Acknowledgements

Acknowledgements

Thanks: Matt Kahle, Shmuel Weinberger, Anna Gundert, Misha Belkin, Yusu Wang, Lek-Heng Lim, Yuan Yao, Bill Allard, Anil Hirani Funding: Center for Systems Biology at Duke NSF DMS and CCF AFOSR NIH

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